diff --git a/main/2048isAcyclic.tex b/main/2048isAcyclic.tex
index 36e3256..85fb81a 100644
--- a/main/2048isAcyclic.tex
+++ b/main/2048isAcyclic.tex
@@ -1,6 +1,6 @@
 
 
-\subsection{acyclicity 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
@@ -30,7 +30,7 @@ The game ends when all squares are filled, and no valid merge operations can be 
 
 
 In the 2048 game, each tile has 16 potential values,
- including empty and $2^k$, $k\in\{1,2,3,\ldots,15\}$.
+ including empty, and $2^k$, where $k\in\{1,2,3,\ldots,15\}$.
 Using 4 bits to represent a tile, the game board is a 4$\times$4 matrix 
 $B$. The corresponding tile is then computed as follows:
 \begin{equation}
diff --git a/main/acyclic.tex b/main/acyclic.tex
index 969d227..319617c 100644
--- a/main/acyclic.tex
+++ b/main/acyclic.tex
@@ -1,4 +1,4 @@
-\section{Aacyclicity of the 2048 game}
+\section{Acyclicity of the 2048 game}
 
 
 The purpose of this section is to prove the acyclicity of the 2048 game
@@ -17,7 +17,7 @@ It is easy to see that if a Markov chain is ergodic,
 then it is cyclic.
 
 
-\subsection{Boyan chain}
+%\subsection{Boyan chain}
 
 
 
@@ -25,7 +25,7 @@ then it is cyclic.
 
 \input{pic/boyanchain}
 
-Figure \ref{boyanchain} shows Boyan chain.
+Figure \ref{boyanchain} shows Boyan chain \cite{boyan2002technical}.
  The transition probabilities between
 non-absorbing states are as follows:
 \[
@@ -93,9 +93,11 @@ The $Q$ matrix (\ref{condition}) is an upper triangular matrix.
 The product of two upper triangular matrices is still an upper triangular matrix.
 Furthermore, the sum of two upper triangular matrices 
 is still an upper triangular matrix.
-Based on Definition \ref{definitionN}, 
+
+Based on Definition \ref{definitionN}, $N\dot{=} \sum_{i=0}^{\infty}Q^i$,
 the $N$ matrix is product and sum of upper triangular matrices.
 Then, the $N$ matrix is an upper triangular matrix.
+
 The claim now follows based on Definition \ref{definition3}.
 \end{IEEEproof}
 
diff --git a/references.bib b/references.bib
index 5ae999a..f7782a2 100644
--- a/references.bib
+++ b/references.bib
@@ -9,6 +9,16 @@
   year={2019},
   publisher={Nature Publishing Group}
 }
+@article{boyan2002technical,
+  title={Technical update: Least-squares temporal difference learning},
+  author={Boyan, Justin A},
+  journal={Machine learning},
+  volume={49},
+  number={2-3},
+  pages={233--246},
+  year={2002},
+  publisher={Springer}
+}
 @article{pakes1969some,
   title={Some conditions for ergodicity and recurrence of Markov chains},
   author={Pakes, Anthony G},