Here are the useful blogs, packages and resources that I learned from to build this site.
And following the list of changes, I can reproduce this site quickly. This is why the blog is also helpful for me.
It's an elegant theme with a lot of effortless customizations.
The i18n package, which can generate directories for languages, but still not perfect.
Other solutions like moving the whole site to a sub-folder.
\[ f(t)=\frac{a_0}{2}+\sum_{n=1}^{\infty}\left[a_n\cos(n\omega t)+b_n\sin(n\omega t)\right] \]
其中
\[ \begin{align} a_n&=\frac{2}{T}\int_{t_0}^{t_0+T}f(t)\cos(n\omega t)\mathrm d t &,n\in \mathbb{Z}_+ \nonumber \\ b_n&=\frac{2}{T}\int_{t_0}^{t_0+T}f(t)\sin(n\omega t)\mathrm d t &,n\in \mathbb{Z}_+ \nonumber \end{align} \]
设有二次型矩阵 \(A\)，\(f(x)=\boldsymbol x^{\mathrm T}A\boldsymbol x\)
存在正交矩阵 \(P\) 使 \(\boldsymbol x=P\boldsymbol y\)，\(f(x)=\boldsymbol y^{\mathrm T}P^{\mathrm T}AP\boldsymbol x=d_1y_1^2+d_2y_2^2+\cdots+d_ny_n^2\)
步骤
DFA 是五元组 \((Q,\Sigma,\delta,q_0,F)\)，分别表示状态，符号集，迁移函数，开始状态，接收状态集
DFA 扩展转移函数 \(\hat\delta:Q\times\Sigma^*\to Q\)
NFA 是五元组 \((Q,\Sigma,\delta,q_0,F)\)，其中 \(\delta:Q\times\Sigma\to2^Q\)
NFA 扩展转移函数 \(\hat\delta:Q\times\Sigma^*\to 2^Q\)
\(\mathrm \varepsilon\)-NFA 是五元组 \((Q,\Sigma,\delta,q_0,F)\)，其中 \(\delta:Q\times(\Sigma\cup\{\varepsilon\})\to2^Q\)
泵引理证正则：
存在 \(n\) 使得对任意 \(w\in L\)，若 \(|w|\ge n\)，我们能写成 \(w=xyz\)，其中 \(|xy|\le n\)，\(|y|\ge 1\)，且 \(\forall k\geq 0, xy^kz\in L\)
rev0.3.1
本文档照例应用于考前记诵
但现在内容多得已经可以用来饱和式学习了
本文强调值的运算矢量性时利用箭头符号 \(\vec a\)，描绘一般原则时则使用加粗体 \(\mathbf a\)
所有常量均使用正体（如玻尔兹曼常量 \(\mathrm k\)）
电子电量 \(\mathrm e\) 与自然底数 \(\mathrm e\) 写法一致，但易于从上下文判断
rev 0.1，波动光学还没完善好！
rev 0.2，差不多该可以用了，但也快要考试了，加了一些没什么用的东西
rev 0.3，修正了一些已知错误
本文向量、矩阵符号均不以粗体标识，从上下文推断
求矩阵 \(A\) 的特征值 \(\lambda_i\) 以及特征向量 \(x\)：求解方程 \(|A-\lambda I|=0\)
设 \(A\) 是对称正定阵，\(\forall x\neq 0 \in R^n\) 二次型 \(x^{\mathrm T}Ax > 0\)
存在可逆矩阵 \(L\) 使 \(A = LL^T\)
\(A\) 特征值、顺序主子式均大于零
\[ P(n,r)=P_{n}^{r}=\frac{n!}{(n-r)!} \]
\[ \binom{n}{r} =C_n^r=C(n,r)=\frac{n!}{r!(n-r)!} \]
\(n\) 种不同元素取 \(r\) 个可重元素排列
\[ \overline{P}(n,r) = n^r \]
每种元素 \(x_i\) 选择 \(r_i\) 个按次序排列，\(n=\sum_i r_i\) \[ P(n;r_1,\cdots,r_k)=\binom{n}{r_1,r_2,\cdots,r_k}=\frac{n!}{r_1!r_2!\cdots r_k!} \]
由协方差性质推导 \[ E\left((XY)^2\right)\leq E(X^2)E(Y^2) \]
\(r\) 阶矩存在，则 \(\forall \varepsilon >0\) \[ P\{|X|\geq \varepsilon\}\leq \frac{(|X|^r)}{\varepsilon^r} \]
\[ \begin{align} P\{|X-E(X)|\geq \varepsilon\}&\leq\frac{D(X)}{\varepsilon^2} \nonumber\newline P\{|X-E(X)|< \varepsilon\}&\geq 1-\frac{D(X)}{\varepsilon^2}\nonumber \end{align} \]
随机现象的观察（随机试验） \(E\) 的样本空间为 \(\Omega=\{\omega\}\)，随机事件 \(A\subset \Omega\)。必然事件 \(\Omega\)，不可能事件 \(\varnothing\)
这个文档用来在考前清扫公式与部分定义，并不能有效地帮助理解，但是很有用
2021.1.7
请不要尝试记住 8 个公式，反之启发式地从以下二式推导 \[ \begin{align} \cos(\alpha + \beta) &+ \cos(\alpha - \beta) \nonumber \newline &= \cos(\alpha)\cos(\beta)-\sin(\alpha)\sin(\beta) + \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta) \nonumber \newline &= 2\cos(\alpha)\cos(\beta) \nonumber \newline \sin(\alpha+\beta) &+ \sin(\alpha - \beta) \nonumber \newline &= \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha) + \sin(\alpha)\cos(\beta) - \sin(\beta)\cos(\alpha) \nonumber \newline &= 2\sin(\alpha)\cos(\beta) \nonumber \end{align} \]
设周期量
\[ f(t) = A_m\cos\left(\frac{2\pi}{T}t+\phi\right) = A_m\cos\left(2\pi f t+\phi\right) = A_m\cos\left(\omega t+\phi\right) \]
其中 \(A_m\) 为最大值，\(\omega\) 为角频率，\(\phi\) 为初相角（\(|\phi|\leq \pi\)）
平均值 \(f_{avg} \triangleq \frac{1}{T}\int_{t_0}^{t_0+T}|f|\mathrm dt=\frac{2}{\pi}A_m\)
Hi, This is the first day that my new blog site and domain have been established. The new site is for sharing thoughts on specific problems, sharing works that I have done as well as introducing myself.
The previous site that I developed in my high school is no longer available, and blog posts are too insignificant to repost. Thus this site only contains posts from now on and some works not previously published before this date.
Now my blog is powered by Hexo, and operated by Github Pages (might change in the future). This is one of the static site frameworks that fit me the most. Several blogs indicating measures and pitfalls on establishing a Hexo site may be posted.
Hope it would operate well.