From 0c5d91ffefbba30aa18d704e06a4fbb362d67fb8 Mon Sep 17 00:00:00 2001
From: oliviasaa <oliviasaa@hotmail.com>
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 $m\leq \frac{-1}{\log(\text{Decay per Epoch})}$ and decreasing otherwise (i.e., it is a concave function). However, since the `Bootstrapping Duration` is set such that $\text{Bootstrapping Duration in Years}*\text{Decay per Year}=1$, we have that $\text{Bootstrapping Duration in Years} = \frac{1}{\text{Decay per Year}}<\frac{-1}{\log(\text{Decay per Year})}$, whenever $\text{Decay per year}>0.57$. Then, the maximum amount of Mana distributed as rewards in the bootstrapping Phase is achieved at `Bootstrapping Duration`, i.e., the maximum is
-
-$$
-\begin{align*}
-\frac{\text{Epochs per year}}{\text{Decay per Year}} \text{Max to Target Ratio}*\text{Final Target Rewards Rate}
-\end{align*}
-$$
--->
-
 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
 
-<!---
-$$
-\begin{align*}
-&\text{Token Supply}*\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\
-&*\left(\text{Reward To Generation Ratio}\right.\\
-&*\frac{\text{Epochs per year}}{\text{Decay per Year}} \text{Max to Target Ratio}\\
-+&\left.\frac{1}{1-\text{Decay per Epoch}}\right)\\
-&\leq \text{Token Supply}*\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\
-&\frac{1+\text{Reward To Generation Ratio}*\text{Max to Target Ratio}}{1-\text{Decay per Epoch}}
-\end{align*}
-$$
--->
-
 $$
 \begin{align*}
 &\text{Max Mana Supply Bootstrapping}\\
-&=\text{Token Supply}*\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\
-&\left(\text{Reward To Generation Ratio}*\frac{1}{-\log(\text{Decay per Epoch})}* \text{Max to Target Ratio}\right.\\
+&=\text{Token Supply}*\text{Generation Rate} * 2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\
+&\left(\text{Reward To Generation Ratio} * \frac{1}{-\log(\text{Decay per Epoch})} * \text{Max to Target Ratio}\right.\\
 +&\left.\frac{1}{1-\text{Decay per Epoch}}\right)\\
-&\leq \text{Token Supply}*\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\
-&\frac{1+\text{Reward To Generation Ratio}*\text{Max to Target Ratio}}{1-\text{Decay per Epoch}}
+&\leq \text{Token Supply} * \text{Generation Rate} * 2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\
+&\frac{1+\text{Reward To Generation Ratio} * \text{Max to Target Ratio}}{1-\text{Decay per Epoch}}
 \end{align*}
 $$
 
@@ -108,8 +85,8 @@ Then, we have that the total Mana in the system will never be larger than $\text
 $$
 \begin{align*}
 &\text{Max Mana Supply}\\
-&=\text{Token Supply}*\text{Generation Rate}*2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\
-&\frac{1+\text{Reward To Generation Ratio}*\text{Max to Target Ratio}}{1-\text{Decay per Epoch}}
+&=\text{Token Supply} * \text{Generation Rate} * 2^{\text{Slots per Epoch Exp}-\text{Generation Rate Exp}}\\
+&\frac{1+\text{Reward To Generation Ratio} * \text{Max to Target Ratio}}{1-\text{Decay per Epoch}}
 \end{align*}
 $$