From 0c5d91ffefbba30aa18d704e06a4fbb362d67fb8 Mon Sep 17 00:00:00 2001 From: oliviasaa Date: Thu, 28 Mar 2024 15:20:03 +0000 Subject: [PATCH] fix LaTeX --- tips/TIP-0040/constraints.md | 43 +++++++++--------------------------- 1 file changed, 10 insertions(+), 33 deletions(-) diff --git a/tips/TIP-0040/constraints.md b/tips/TIP-0040/constraints.md index 1c72916e4..e0f2be1f4 100644 --- a/tips/TIP-0040/constraints.md +++ b/tips/TIP-0040/constraints.md @@ -18,7 +18,7 @@ The maximum Mana in the system generated by holding tokens $\text{Max Mana Holde $$ \begin{align*} -\text{Max Mana Holders} = \frac{\text{Token Supply}*\text{Generation Rate}}{1-\text{Decay per Epoch}}2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}} +\text{Max Mana Holders} = \frac{\text{Token Supply} * \text{Generation Rate}}{1-\text{Decay per Epoch}}2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}} \end{align*} $$ @@ -45,22 +45,12 @@ $$ \end{align*} $$ - - the function above is increasing if $n\leq \frac{-1}{\log(\text{Decay per Epoch})}$ and decreasing otherwise (i.e., it is a concave function). However, since the $\text{Bootstrapping Duration}$ is set such that $\text{Bootstrapping Duration in Years}*\text{Beta per Year}=1$, we have that $\text{Bootstrapping Duration in Years} = \frac{1}{\text{Beta per Year}}=\frac{-1}{\log(\text{Decay per Year})}$. Then, the maximum amount of Mana distributed as rewards in the bootstrapping Phase is achieved at $\text{Bootstrapping Duration}$, i.e., the maximum is $$ \begin{align*} &\text{Max Mana Supply Bootstrapping}\\ -&=\frac{1}{-\log(\text{Decay per Epoch})}* \text{Max to Target Ratio}*\text{Final Target Rewards Rate} +&=\frac{1}{-\log(\text{Decay per Epoch})} * \text{Max to Target Ratio} * \text{Final Target Rewards Rate} \end{align*} $$ @@ -68,34 +58,21 @@ By definition, we have that $\text{Final Target Rewards Rate}$ is $$ \begin{align*} -&\text{Final Target Rewards Rate}=\text{Token Supply}*\text{Reward To Generation Ratio}\\ -*&\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}} +&\text{Final Target Rewards Rate}=\text{Token Supply} * \text{Reward To Generation Ratio}\\ +* &\text{Generation Rate} * 2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}} \end{align*} $$ Then, the Maximum Mana supply in the bootstrapping phase is - - $$ \begin{align*} &\text{Max Mana Supply Bootstrapping}\\ -&=\text{Token Supply}*\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\ -&\left(\text{Reward To Generation Ratio}*\frac{1}{-\log(\text{Decay per Epoch})}* \text{Max to Target Ratio}\right.\\ +&=\text{Token Supply}*\text{Generation Rate} * 2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\ +&\left(\text{Reward To Generation Ratio} * \frac{1}{-\log(\text{Decay per Epoch})} * \text{Max to Target Ratio}\right.\\ +&\left.\frac{1}{1-\text{Decay per Epoch}}\right)\\ -&\leq \text{Token Supply}*\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\ -&\frac{1+\text{Reward To Generation Ratio}*\text{Max to Target Ratio}}{1-\text{Decay per Epoch}} +&\leq \text{Token Supply} * \text{Generation Rate} * 2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\ +&\frac{1+\text{Reward To Generation Ratio} * \text{Max to Target Ratio}}{1-\text{Decay per Epoch}} \end{align*} $$ @@ -108,8 +85,8 @@ Then, we have that the total Mana in the system will never be larger than $\text $$ \begin{align*} &\text{Max Mana Supply}\\ -&=\text{Token Supply}*\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\ -&\frac{1+\text{Reward To Generation Ratio}*\text{Max to Target Ratio}}{1-\text{Decay per Epoch}} +&=\text{Token Supply} * \text{Generation Rate} * 2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\ +&\frac{1+\text{Reward To Generation Ratio} * \text{Max to Target Ratio}}{1-\text{Decay per Epoch}} \end{align*} $$