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whitepaper/automated-market-making-of-alex/README.md
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@@ -169,6 +169,6 @@ $$
Figure 1 plots $$L_{\text{Uniswap}}$$ against $$r$$ (which is proportional to $$p$$) regarding various levels of $$t$$. When $$0<t<1$$, $$L_{\text{Uniswap}}$$ is symmetric around 0% at which the maximum reaches . This is because Figure 1 plots $$L_{\text{Uniswap}}$$ against $$r$$ (which is proportional to $$p$$) regarding various levels of $$t$$. When $$0<t<1$$, $$L_{\text{Uniswap}}$$ is symmetric around 0% at which the maximum reaches . This is because
1. $$\cosh\left[(\frac{r(1-t)}{2})\right]$$ is symmetric around $$r$$= 0% with minimum at 0% and the minimum value 1; 1. $$\cosh\left[(\frac{r(1-t)}{2})\right]$$ is symmetric around $$r$$= 0% with minimum at 0% and the minimum value 1;
2. $$x^z$$ is a decreasing function of $$x$$ when $$x$$ is positive and power $$z$$ is negative. In our case, we have $$z=-2+t1-t<-1$$. Therefore, it is the maximum rather than minimum that $$L_{\text{Uniswap}}$$ achieves at 0. 2. $$x^z$$ is a decreasing function of $$x$$ when $$x$$ is positive and power $$z$$ is negative. In our case, we have $$z=\frac{-2+t}{1-t}<-1$$. Therefore, it is the maximum rather than minimum that $$L_{\text{Uniswap}}$$ achieves at 0.
Furthermore, the higher the $$t$$, the flatter the liquidity distribution is. When $$t$$ approaches 1, i.e. AMM converges to the constant product formula, the liquidity distribution is close to a flat line. When $$t$$ approaches 0, the distribution concentrates around 0%. Furthermore, the higher the $$t$$, the flatter the liquidity distribution is. When $$t$$ approaches 1, i.e. AMM converges to the constant product formula, the liquidity distribution is close to a flat line. When $$t$$ approaches 0, the distribution concentrates around 0%.