User:Zyang666/sandbox

===Math===

$g_{i}(x)\leq 0,{\frac {x}{y}},l(x)=\min\{j\mid f_{j}(x)\leq x_{j}>0\}.$

An illustration of Scarf's method

Figure 1: The Illustration of Scarf's Algorithm

Math 1

$\int _{a}^{b}f(x)dx=X.$

Math 2

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

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

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

$S^{n}=\{x\in \mathbb {R} _{+}^{n}\mid \sum _{i=1}^{n}x_{i}=1\}$

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