qwqwqwq

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diff --git a/docs/math/basic/polar-coordinate.md b/docs/math/basic/polar-coordinate.md
new file mode 100644 (file)
index 0000000..5797d4a
--- /dev/null
+++ b/docs/math/basic/polar-coordinate.md
@@ -0,0 +1,39 @@
+## 极坐标与极坐标系
+
+（为人教版高中数学选修 4-4 内容）
+
+> 某同学：学平面直角坐标系都学烦了，有没有其他坐标系？
+
+我们考虑实际情况，比如航海，我们说「 $B$ 在 $A$ 的北偏东 $30^\circ$ 方向上，距离为 $100$ 米」，而不是「以 $A$ 为原点建立平面直角坐标系， $B(50,50\sqrt 3)$ 」。
+
+这样，我们在平面上选一定点 $O$ ，称为 **极点** ，自极点引出一条射线 $Ox$ ，称为 **极轴** ，再选择一个单位长度（在数学问题中通常为 $1$ ），一个角度单位（通常为弧度）及其正方向（通常为逆时针方向），这样就建立了 **极坐标系** 。
+
+在极坐标系下，我们怎么描述位置呢？
+
+设 $A$ 为平面上一点，极点 $O$ 与 $A$ 之间的距离 $|OA|$ 即为 **极径** ，记为 $\rho$ ；以极轴为始边， $OA$ 为终边的角 $\angle xOA$ 为 **极角** ，记为 $\theta$ ，那么有序数对 $(\rho,\theta)$ 即为 $A$ 的 **极坐标** 。
+
+由终边相同的角的定义可知， $(\rho,\theta)$ 与 $(\rho,\theta+2k\pi)\ (k\in \mathbb{Z})$ 其实表示的是一样的点，特别地，极点的极坐标为 $(0,\theta)\ (\theta\in \mathbb{R})$ ，于是平面内的点的极坐标表示有无数多种。
+
+如果规定 $\rho>0,0\le \theta<2\pi​$ ，那么除极点外，其他平面内的点可以用唯一有序数对 $(\rho,\theta)​$ 表示，而极坐标 $(\rho,\theta)​$ 表示的点是唯一确定的。
+
+当然，有时候研究极坐标系下的图形有些不方便，我们想要转到直角坐标系下研究，那么我们有互化公式。
+
+点 $A(\rho,\theta)$ 的直角坐标 $(x,y)$ 可以如下表示：
+
+$$
+\begin{cases}
+x=\rho \cos \theta\\
+y=\rho \sin \theta
+\end{cases}
+$$
+
+进而可知：
+
+$$
+\rho ^2=x^2+y^2\\
+\tan \theta=\frac{y}{x}\ \ \ \ (x\not =0)
+$$
+
+于是，极角 $\theta=\arctan \frac{y}{x}$ ，这样就可以求出极角了。
+
+在编程中，若要求反正切函数，尽量使用 `atan2(y, x)` ，这个函数用途比 `atan(x)` 广泛。

diff --git a/docs/math/misc.md b/docs/math/basic/vector.md
similarity index 84%
rename from docs/math/misc.md
rename to docs/math/basic/vector.md
index 25a2413..84fc8c2 100644 (file)
--- a/docs/math/misc.md
+++ b/docs/math/basic/vector.md
@@ -1,9 +1,3 @@
-本文主要介绍了在 OI 中可能用到的重要高中数学知识。
-
-如果您是高中 OIer，强烈建议您回班级听课，相比于自学，老师讲课可以使您理解得更透彻。
-
-下面按照从必修到选修的顺序介绍。所有内容均基于人教版高中数学 A 版教科书。
-
 ## 向量
 
 （为人教版高中数学必修四内容）
@@ -246,43 +240,3 @@ $$
 与角 $\alpha$ 终边位置相同的角的集合很容易得出，为 $\{\theta\mid \theta=\alpha+2k\pi,k\in \mathbb{Z}\}$ 。
 
 可以理解为：给这个角的边不停加转一圈，终边位置不变。
-
-### 极坐标与极坐标系
-
-（为人教版高中数学选修 4-4 内容）
-
-> 某同学：学平面直角坐标系都学烦了，有没有其他坐标系？
-
-我们考虑实际情况，比如航海，我们说「 $B$ 在 $A$ 的北偏东 $30^\circ$ 方向上，距离为 $100$ 米」，而不是「以 $A$ 为原点建立平面直角坐标系， $B(50,50\sqrt 3)$ 」。
-
-这样，我们在平面上选一定点 $O$ ，称为 **极点** ，自极点引出一条射线 $Ox$ ，称为 **极轴** ，再选择一个单位长度（在数学问题中通常为 $1$ ），一个角度单位（通常为弧度）及其正方向（通常为逆时针方向），这样就建立了 **极坐标系** 。
-
-在极坐标系下，我们怎么描述位置呢？
-
-设 $A$ 为平面上一点，极点 $O$ 与 $A$ 之间的距离 $|OA|$ 即为 **极径** ，记为 $\rho$ ；以极轴为始边， $OA$ 为终边的角 $\angle xOA$ 为 **极角** ，记为 $\theta$ ，那么有序数对 $(\rho,\theta)$ 即为 $A$ 的 **极坐标** 。
-
-由终边相同的角的定义可知， $(\rho,\theta)$ 与 $(\rho,\theta+2k\pi)\ (k\in \mathbb{Z})$ 其实表示的是一样的点，特别地，极点的极坐标为 $(0,\theta)\ (\theta\in \mathbb{R})$ ，于是平面内的点的极坐标表示有无数多种。
-
-如果规定 $\rho>0,0\le \theta<2\pi​$ ，那么除极点外，其他平面内的点可以用唯一有序数对 $(\rho,\theta)​$ 表示，而极坐标 $(\rho,\theta)​$ 表示的点是唯一确定的。
-
-当然，有时候研究极坐标系下的图形有些不方便，我们想要转到直角坐标系下研究，那么我们有互化公式。
-
-点 $A(\rho,\theta)$ 的直角坐标 $(x,y)$ 可以如下表示：
-
-$$
-\begin{cases}
-x=\rho \cos \theta\\
-y=\rho \sin \theta
-\end{cases}
-$$
-
-进而可知：
-
-$$
-\rho ^2=x^2+y^2\\
-\tan \theta=\frac{y}{x}\ \ \ \ (x\not =0)
-$$
-
-于是，极角 $\theta=\arctan \frac{y}{x}$ ，这样就可以求出极角了。
-
-在编程中，若要求反正切函数，尽量使用 `atan2(y, x)` ，这个函数用途比 `atan(x)` 广泛。

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diff --git a/docs/math/stirling.md b/docs/math/combinatorial/stirling.md
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diff --git a/docs/math/game-theory.md b/docs/math/game-theory/game-theory.md
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diff --git a/docs/math/index.md b/docs/math/index.md
index 99b27dc..0314f7c 100644 (file)
--- a/docs/math/index.md
+++ b/docs/math/index.md
@@ -91,4 +91,12 @@ NOIP 对数学的考察还处在一个比较简单的范围。
 2.  向下取整符号： $\lfloor x\rfloor$ ，表示小于等于 $x$ 的最大的整数。常用于分数，比如分数的向下取整 $\left\lfloor\frac{x}{y}\right\rfloor$ 。
 3.  向上取整符号： $\lceil x\rceil$ ，与向下取整符号相对，表示大于等于 $x$ 的最小的整数。
 
+## 温馨提示
+
+在「基础知识」一栏中介绍了在 OI 中可能用到的重要高中数学知识。
+
+如果您是高中 OIer，强烈建议您回班级听课，相比于自学，老师讲课可以使您理解得更透彻。
+
+下面按照从必修到选修的顺序介绍。所有内容均基于人教版高中数学 A 版教科书。
+
 另外，还请大家学好高一数学，这样学习数论的时侯会省很多功夫。

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index a3d64b9..0f2620d 100644 (file)
--- a/docs/math/primitive-root.md
+++ b/docs/math/number-theory/primitive-root.md
@@ -45,10 +45,12 @@ $$
 
 又有 $t<p-1$ ，故 $\gcd(t,p-1)\leqslant t<p-1$ 。
 
-又 $\gcd(t,p-1)\mid(p-1)$ ，故 $\gcd(t,p-1)$ 必至少整除 $a^{\frac{p-1}{d_{1}}},a^{\frac{p-1}{d_{2}}},\ldots,a^{\frac{p-1}{d_{m}}}$ 中的至少一个，设 $\gcd(t,p-1)\mid a^{\frac{p-1}{d_{i}}$ ，则 $a^{\frac{p-1}{d_{i}}}\equiv a^{\gcd(t,p-1)}\equiv 1\pmod{p}$ 。
+又 $\gcd(t,p-1)\mid(p-1)$ ，故 $\gcd(t,p-1)$ 必至少整除 $a^{\frac{p-1}{d_{1}}},a^{\frac{p-1}{d_{2}}},\ldots,a^{\frac{p-1}{d_{m}}}$ 中的至少一个，设 $\gcd(t,p-1)\mid a^{\frac{p-1}{d_{i}}}$ ，则 $a^{\frac{p-1}{d_{i}}}\equiv a^{\gcd(t,p-1)}\equiv 1\pmod{p}$ 。
 
 故假设不成立。
 
 ## 用途
 
-我们发现原根 $g$ 拥有所有 FFT 所需的单位根 $\omega$ 的性质，于是我们用 $g^{\frac{p-1}{n}}\bmod{p}$ 来代替 $\omega_{n}$ ，就能把复数对应到一个整数，在模 $p$ 意义下进行快速变换了。
+我们发现原根 $g$ 拥有所有 DFT 所需的单位根 $\omega$ 的性质，于是我们用 $g^{\frac{p-1}{n}}\bmod{p}$ 来代替 $\omega_{n}$ ，理论上就能把复数对应到一个整数，在模 $p$ 意义下进行快速变换了。
+
+但实际上由于快速傅里叶变换（FFT）实现的多项式乘法的过程中要求序列长度是 $2$ 的幂次，因此这里模数 $p$ 还需要保证 $p-1$ 的标准分解式中素因子 $2$ 的幂次足够大，参见[快速数论变换](/math/poly/ntt)，

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similarity index 100%
rename from docs/math/images/linear-programming3.png
rename to docs/topic/optimization/lp/images/linear-programming3.png

diff --git a/docs/math/images/linear-programming4.png b/docs/topic/optimization/lp/images/linear-programming4.png
similarity index 100%
rename from docs/math/images/linear-programming4.png
rename to docs/topic/optimization/lp/images/linear-programming4.png

diff --git a/docs/math/linear-programming.md b/docs/topic/optimization/lp/linear-programming.md
similarity index 100%
rename from docs/math/linear-programming.md
rename to docs/topic/optimization/lp/linear-programming.md

diff --git a/docs/math/simplex.md b/docs/topic/optimization/lp/simplex.md
similarity index 100%
rename from docs/math/simplex.md
rename to docs/topic/optimization/lp/simplex.md

diff --git a/mkdocs.yml b/mkdocs.yml
index e19039d..8d4fb21 100644 (file)
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -159,35 +159,31 @@ nav:
     - Lyndon 分解: string/lyndon.md
   - 数学:
     - 数学部分简介: math/index.md
-    - 进制: math/base.md
-    - 位运算: math/bit.md
-    - 高精度: math/bignum.md
-    - 快速幂: math/quick-pow.md
-    - 整除及其性质:
-      - 素数: math/prime.md
-      - 最大公约数: math/gcd.md
-      - 欧拉函数: math/euler.md
-      - 筛法: math/sieve.md
-      - 欧拉定理 & 费马小定理: math/fermat.md
-      - 类欧几里德算法: math/euclidean.md
-    - 同余方程相关:
-      - 裴蜀定理: math/bezouts.md
-      - 乘法逆元: math/inverse.md
-      - 线性同余方程: math/linear-equation.md
-      - 中国剩余定理: math/crt.md
-      - BSGS: math/bsgs.md
-      - 原根: math/primitive-root.md
-    - 线性代数:
-      - 矩阵: math/matrix.md
-      - 高斯消元: math/gauss.md
-      - 线性基: math/basis.md
-    - 复数: math/complex.md
-    - 分段打表: math/dictionary.md
-    - 牛顿迭代法: math/newton.md
-    - 数论函数相关:
-      - 莫比乌斯反演: math/mobius.md
-      - 杜教筛: math/du.md
-      - Min_25 筛: math/min-25.md
+    - 基础知识:
+      - 向量: math/basic/vector.md
+      - 复数: math/basic/complex.md
+      - 极坐标系: math/basic/polar-coordinate.md
+    - 数论与数论函数:
+      - 整除及其性质:
+        - 素数: math/number-theory/prime.md
+        - 最大公约数: math/number-theory/gcd.md
+        - 欧拉函数: math/number-theory/euler.md
+        - 筛法: math/number-theory/sieve.md
+        - 欧拉定理 & 费马小定理: math/number-theory/fermat.md
+        - 类欧几里德算法: math/number-theory/euclidean.md
+      - 同余与同余方程:
+        - 裴蜀定理: math/number-theory/bezouts.md
+        - 乘法逆元: math/number-theory/inverse.md
+        - 线性同余方程: math/number-theory/linear-equation.md
+        - 中国剩余定理: math/number-theory/crt.md
+        - BSGS: math/number-theory/bsgs.md
+        - 原根: math/number-theory/primitive-root.md
+        - 卢卡斯定理: math/number-theory/lucas.md
+      - 进位制: math/number-theory/base.md
+      - 数论函数:
+        - 莫比乌斯反演: math/number-theory/mobius.md
+        - 杜教筛: math/number-theory/du.md
+        - Min_25 筛: math/number-theory/min-25.md
     - 多项式:
       - 多项式部分简介: math/poly/intro.md
       - 拉格朗日插值: math/poly/lagrange.md
@@ -202,22 +198,30 @@ nav:
       - 多项式多点求值|快速插值: math/poly/multipoint-eval-interpolation.md
       - 多项式三角函数: math/poly/tri-func.md
       - 多项式反三角函数: math/poly/inv-tri-func.md
+    - 线性代数:
+      - 矩阵: math/linear-algebra/matrix.md
+      - 高斯消元: math/linear-algebra/gauss.md
+      - 线性基: math/linear-algebra/basis.md
+    - 抽象代数:
+      - 置换群: math/abstract-algebra/permutation-group.md
     - 组合数学:
-      - 排列组合: math/combination.md
-      - 卡特兰数: math/catalan.md
-      - 斯特林数: math/stirling.md
-      - 康托展开: math/cantor.md
-      - 容斥原理: math/inclusion-exclusion-principle.md
-      - 抽屉原理: math/drawer-principle.md
-    - 概率 & 期望: math/expectation.md
+      - 排列组合: math/combinatorial/combination.md
+      - 卡特兰数: math/combinatorial/catalan.md
+      - 斯特林数: math/combinatorial/stirling.md
+      - 康托展开: math/combinatorial/cantor.md
+      - 容斥原理: math/combinatorial/inclusion-exclusion-principle.md
+      - 抽屉原理: math/combinatorial/drawer-principle.md
+      - 概率 & 期望: math/combinatorial/expectation.md
+    - 博弈论:
+      - 博弈论简介: math/game-theory/game-theory.md
+    - 应试技巧与常用算法:
+      - 位运算: math/bit.md
+      - 快速幂: math/quick-pow.md
+      - 分段打表: math/dictionary.md
+      - 高精度: math/bignum.md
+      - 牛顿迭代法: math/newton.md
+      - 数值积分: math/integral.md
     - 斐波那契数列: math/fibonacci.md
-    - 置换群: math/permutation-group.md
-    - 数值积分: math/integral.md
-    - 线性规划: math/linear-programming.md
-    - 单纯形: math/simplex.md
-    - 博弈论: math/game-theory.md
-    - 卢卡斯定理: math/lucas.md
-    - 数学杂项: math/misc.md
   - 数据结构:
     - 数据结构部分简介: ds/index.md
     - 栈: ds/stack.md
@@ -304,13 +308,6 @@ nav:
     - 最小环: graph/min-circle.md
     - 平面图: graph/planar.md
     - 图的着色: graph/color.md
-    - 网络流:
-      - 网络流简介: graph/flow.md
-      - 拆点: graph/flow/node.md
-      - 最大流: graph/flow/max-flow.md
-      - 最小割: graph/flow/min-cut.md
-      - 费用流: graph/flow/min-cost.md
-      - 上下界网络流: graph/flow/bound.md
     - Prufer 序列: graph/prufer.md
     - 图论杂项: graph/misc.md
   - 计算几何:
@@ -351,6 +348,17 @@ nav:
     - 表达式求值: misc/expression.md
   - 专题:
     - RMQ: topic/rmq.md
+    - 组合优化问题:
+      - 线性规划:
+        - 线性规划简介: topic/optimization/lp/linear-programming.md
+        - 单纯形算法: topic/optimization/lp/simplex.md
+      - 网络流:
+        - 网络流简介: topic/optimization/flow/flow.md
+        - 拆点: topic/optimization/flow/node.md
+        - 最大流: topic/optimization/flow/max-flow.md
+        - 最小割: topic/optimization/flow/min-cut.md
+        - 费用流: topic/optimization/flow/min-cost.md
+        - 上下界网络流: topic/optimization/flow/bound.md
 
 # Theme
 theme: