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一些数学公式学习；
公式一：
$$
\begin{aligned}
V_j &= v_j &
X_i &= x_i - q_i x_j &
&= u_j + \sum_{i\ne j} q_i \\
V_i &= v_i - q_i v_j &
X_j &= x_j &
U_i &= u_i
\end{aligned}\tag{1.0.1}
$$
公式二：
$$
\begin{align}
A_1 &= N_0 (\lambda ; \Omega’)
- \phi ( \lambda ; \Omega’) \\
A_2 &= \phi (\lambda ; \Omega’)
\phi (\lambda ; \Omega) \\
\intertext{and finally}
A_3 &= \mathcal{N} (\lambda ; \omega)
\end{align}\tag{1.0.2}
$$
Fomula 3 :
$$
\begin{align}
x^2+y^2 &= z^2 \label{eq:A} \\
x^3+y^3 &= z^3 \notag \\
x^4+y^4 &= r^4 \tag{$*$} \\
x^5+y^5 &= r^5 \tag*{$*$} \\
x^6+y^6 &= r^6 \tag{\ref{eq:A}$’$} \\
A_1 &= N_0 (\lambda ; \Omega’)
- \phi ( \lambda ; \Omega’) \\
A_2 &= \phi (\lambda ; \Omega’)
\, \phi (\lambda ; \Omega)
\tag*{ALSO (\theequation)} \\
A_3 &= \mathcal{N} (\lambda ; \omega)
\end{align}
$$
fomula 4 :
$$
\begin{split}
\lvert I_2 \rvert &= \left\lvert \int_{0}^T \psi(t)
\left\{ u(a, t) - \int_{\gamma(t)}^a \frac{d\theta}{k}
(\theta, t) \int_{a}^\theta c (\xi) u_t (\xi, t) \,d\xi
\right\} dt \right\rvert \\
&\le C_6 \Biggl\lvert
\left\lvert f \int_\Omega \left\lvert
\widetilde{S}^{-1,0}_{a,-} W_2(\Omega, \Gamma_l)
\right\rvert \ \right\rvert
\left\lvert \lvert u \rvert
\overset{\circ}{\to} W_2^{\widetilde{A}} (\Omega; \Gamma_r,T)
\right\rvert \Biggr\rvert
\end{split}\tag{1.0.4}
$$
fomula 5:
$$
\begin{gather} \raisetag{-10pt}
\text{The sign function: \ }
\mathcal{S}(x) = \begin{cases}
-1 & x < 0 \\
0 & x = 0 \\
1 & x>0
\end{cases}
\end{gather}\tag{1.0.5}
$$
fomula 6:
$$
\begin{gather*}
\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \quad
\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \\
\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \quad
\begin{Bmatrix} 1 & 0 \\ 0 & -1 \end{Bmatrix} \\
\begin{vmatrix} a & b \\ c & d \end{vmatrix} \quad
\begin{Vmatrix} i & 0 \\ 0 & -i \end{Vmatrix}
\end{gather*}\tag{1.0.6}
$$
Equantion 7:
$$
\frac{1}{k} \log_2 c(f)
\quad \tfrac{1}{k} \log_2 c(f) \tag{1.0.7}$$
Text: $ \sqrt{ \frac{1}{k} \log_2 c(f) } \quad
\sqrt{ \dfrac{1}{k} \log_2 c(f) }\, $.
so,
Equantion 8:
$$
\biggl( \mathbf{E}_{y} \int_0^{t_\varepsilon}
L_{x, y^x(s)} \varphi(x)\, ds \biggr)\tag{1.0.8}
$$
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