diff --git a/main/2048isAcyclic.tex b/main/2048isAcyclic.tex
index 6bc127f..94458ce 100644
--- a/main/2048isAcyclic.tex
+++ b/main/2048isAcyclic.tex
@@ -1,10 +1,10 @@
-\section{Non-ergodicity of the 2048 game}
+\section{Aacyclicity of the 2048 game}
 
 
-The purpose of this section is to prove the non-ergodicity of the 2048 game
+The purpose of this section is to prove the acyclicity of the 2048 game
 and give some discussions.
 
-\subsection{Non-ergodicity of the 2048 game}
+\subsection{acyclicity of the 2048 game}
 
 The 2048 game consists of a 4$\times$4 grid board, totaling 16 squares.
  At the beginning of the game,  two squares are randomly filled
@@ -24,7 +24,7 @@ The game ends when all squares are filled, and no valid merge operations can be 
 
 
 \begin{theorem}
-2048 game is non-ergodic between non-absorbing states.
+2048 game is acyclic between non-absorbing states.
 \end{theorem}
 \begin{IEEEproof}
  To apply Theorem \ref{judgmentTheorem}, what we need 
@@ -99,26 +99,6 @@ the claim follows by applying Theorem \ref{judgmentTheorem}.
 
 \subsection{Discussions}
 
-行为策略采样
-$\langle s_t,a_t,r_{t+1},a_{t+1},s_{t+1} \rangle$，对应的特征
-$\langle \phi_t,r_{t+1},\phi_{t+1} \rangle$ 
-
-目标策略采样
-$\langle s_t,a_t,r_{t+1},a',s_{t+1} \rangle$，对应的特征
-$\langle \phi_t,r_{t+1},\phi' \rangle$ 
-
-\begin{equation}
-\theta_{t+1}=\theta_t+\alpha F_t (\rho_tR_t+\gamma \theta_t^{\top}\phi_t'-\theta_t^{\top}\phi_t-\mathbb{E}_{\pi}[\delta])\phi_t
-\end{equation}
-写的简单点是这样
-
-\begin{equation}
-\theta_{t+1}=\theta_t+\alpha F_t \rho_t(\delta_t-\mathbb{E}_{\mu}[\rho_t\delta_t])\phi_t,
-\end{equation}
-where
-$\delta_t=R_t+\gamma \theta_t^{\top}\phi_{t+1}-\theta_t^{\top}\phi_t$
-
-