User:Zyang666/sandbox

===Math=== $$g_i(x)\le 0,\frac{x}{y}, l(x)=\min\{j\mid f_j(x)\le x_j>0\}.$$



Math 1
$$\int_a^b f(x)dx=X.$$

Math 2
$$\begin{align} && \max(a_{01}x_1+a_{02}x_2+\cdots+a_{0n}x_n)\\ &&{\operatorname{s.t.}}\quad a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ &&\quad a_{21}x_1+a_{22}x_2+\cdots+a_{1n}x_n\\ &&\quad \vdots\quad \vdots\quad \vdots\quad\\ &&\quad a_{m1}x_1+a_{m2}x_2+\cdots+a_{mn}x_n \end{align} $$

where $$x_1,x_2,\cdots, x_n$$ are integers.

Math 3
$$\mathbf{A}=\begin{bmatrix} 1 & 0& \cdots & 0 &a(1,n+1)& \cdots & a(1,m)\\ 0 & 1& \cdots & 0 &a(2,n+2)& \cdots & a(2,m)\\ & & \cdots &   &        & \cdots &        \\ 0 & 0& \cdots & 1 &a(n,n+1)& \cdots &a(n,m)\end{bmatrix} $$

and $$\mathbf{C}=\begin{bmatrix} c(1,1) & \cdots & c(1,n) &c(1,n+1)& \cdots & c(1,m)\\ c(2,1) & \cdots & c(2,n) &c(2,n+2)& \cdots & c(2,m)\\ & \cdots &       &        & \cdots &        \\ c(n,1) & \cdots & c(n,m) &c(n,n+1)& \cdots &c(n,m)\end{bmatrix} $$

$$f:S^n\to S^n$$

$$S^n=\{x\in \mathbb{R}^n_{+} \mid \sum_{i=1}^nx_i=1\}$$

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