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\begin{document}

According equation 3 and 4, we have

\begin{equation}
P_{ij} = A_{ij} * c_j / b_i
\end{equation}

\begin{eqnarray}
\nonumber
b_1 & = & e^{6\beta} \\
\nonumber
b_2 & = & e^{4\beta} \\
\nonumber
b_3 & = & e^{2\beta} \\
\nonumber
b_4 & = & 1 \\
\nonumber
b_5 & = & e^{-2\beta} \\
\nonumber
b_6 & = & e^{-2\beta} \\
\nonumber
b_7 & = & e^{-4\beta} \\
\nonumber
b_8 & = & e^{-6\beta} 
\nonumber
\end{eqnarray}

% i = 1
\begin{eqnarray}
\nonumber
P_{11} & = & e^{-12\beta} \\
\nonumber
P_{12} & = & 6(e^{-10\beta} - e^{-12\beta}) \\
\nonumber
P_{13} & = & 3(e^{-8\beta} - 2e^{-10\beta}  + e^{-12\beta}) \\
\nonumber
P_{14} & = & 6(e^{-8\beta} - 2e^{-10\beta} + e^{-12\beta}) \\
\nonumber
P_{15} & = & 2(e^{-6\beta} - 3e^{-10\beta} + 2e^{-12\beta}) \\
\nonumber
P_{16} & = & 3(e^{-4\beta} - 2e^{-6\beta} + 2e^{-10\beta} - e^{-12\beta}) \\
\nonumber
P_{17} & = & 3(0.5e^{-2\beta} - e^{-4\beta} + e^{-6\beta} - 2e^{-8\beta} +
2.5e^{-10\beta} - e^{-12\beta}) \\
\nonumber
P_{18} & = & 1 - 3e^{-2\beta} + 3e^{-4\beta} - 2e^{-6\beta} +
3e^{-8\beta} - 2e^{-10\beta} + e^{-12\beta} 
\nonumber
\end{eqnarray}

% i = 2
\begin{eqnarray}
\nonumber
P_{21} & = & e^{-10\beta} \\
\nonumber
P_{22} & = & 5(e^{-8\beta} - e^{-10\beta}) \\
\nonumber
P_{23} & = & 2(e^{-6\beta} - 2e^{-8\beta}  + e^{-10\beta}) \\
\nonumber
P_{24} & = & 2(e^{-6\beta} - 2e^{-8\beta} + e^{-10\beta}) \\
\nonumber
P_{25} & = & 2(e^{-4\beta} - 3e^{-8\beta} + 2e^{-10\beta}) \\
\nonumber
P_{26} & = & 2(e^{-2\beta} - 2e^{-4\beta} + 2e^{-8\beta} - e^{-10\beta}) \\
\nonumber
P_{27} & = & 0.5 - e^{-2\beta} + e^{-4\beta} - 2e^{-6\beta} +
2.5e^{-8\beta} - e^{-10\beta} \\
\nonumber
P_{28} & = & 0
\nonumber
\end{eqnarray}

% i = 3
\begin{eqnarray}
\nonumber
P_{31} & = & e^{-8\beta} \\
\nonumber
P_{32} & = & 4(e^{-6\beta} - e^{-8\beta}) \\
\nonumber
P_{33} & = & 0 \\
\nonumber
P_{34} & = & 0 \\
\nonumber
P_{35} & = & 2(e^{-2\beta} - 3e^{-6\beta} + 2e^{-8\beta}) \\
\nonumber
P_{36} & = & 1 - 2e^{-2\beta} + 2e^{-6\beta} - e^{-8\beta} \\
\nonumber
P_{37} & = & 0 \\
\nonumber
P_{38} & = & 0
\nonumber
\end{eqnarray}

% i = 4
\begin{eqnarray}
\nonumber
P_{41} & = & e^{-6\beta} \\
\nonumber
P_{42} & = & 3(e^{-4\beta} - e^{-6\beta}) \\
\nonumber
P_{43} & = & 0 \\
\nonumber
P_{44} & = & 0 \\
\nonumber
P_{45} & = & 1 - 3e^{-4\beta} + 2e^{-6\beta} \\
\nonumber
P_{46} & = & 0 \\
\nonumber
P_{47} & = & 0 \\
\nonumber
P_{48} & = & 0
\nonumber
\end{eqnarray}

% i = 5
\begin{eqnarray}
\nonumber
P_{51} & = & e^{-4\beta} \\
\nonumber
P_{52} & = & 2(e^{-2\beta} - e^{-4\beta}) \\
\nonumber
P_{53} & = & 0 \\
\nonumber
P_{54} & = & 1 - 2e^{-2\beta} + e^{-4\beta} \\
\nonumber
P_{55} & = & 0 \\
\nonumber
P_{56} & = & 0 \\
\nonumber
P_{57} & = & 0 \\
\nonumber
P_{58} & = & 0
\nonumber
\end{eqnarray}


% i = 6
\begin{eqnarray}
\nonumber
P_{61} & = & e^{-4\beta} \\
\nonumber
P_{62} & = & 2(e^{-2\beta} - e^{-4\beta}) \\
\nonumber
P_{63} & = & 1 - 2e^{-2\beta}  + e^{-4\beta} \\
\nonumber
P_{64} & = & 0 \\
\nonumber
P_{65} & = & 0 \\
\nonumber
P_{66} & = & 0 \\
\nonumber
P_{67} & = & 0 \\
\nonumber
P_{68} & = & 0
\nonumber
\end{eqnarray}

% i = 7
\begin{eqnarray}
\nonumber
P_{71} & = & e^{-2\beta} \\
\nonumber
P_{72} & = & 1 - e^{-2\beta} \\
\nonumber
P_{73} & = & 0 \\
\nonumber
P_{74} & = & 0 \\
\nonumber
P_{75} & = & 0 \\
\nonumber
P_{76} & = & 0 \\
\nonumber
P_{77} & = & 0 \\
\nonumber
P_{78} & = & 0
\nonumber
\end{eqnarray}

% i = 8
\begin{eqnarray}
\nonumber
P_{81} & = & 1 \\
\nonumber
P_{82} & = & 0 \\
\nonumber
P_{83} & = & 0 \\
\nonumber
P_{84} & = & 0 \\
\nonumber
P_{85} & = & 0 \\
\nonumber
P_{86} & = & 0 \\
\nonumber
P_{87} & = & 0 \\
\nonumber
P_{88} & = & 0
\nonumber
\end{eqnarray}


\end{document}

