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\title{\textbf{A SCIENTIFIC WRITING EXAMPLE}}
\author{XIAOWEI WU,\footnote{Postal address: Department of Statistics, Rice University, 6100 Main Street, Houston, TX 77006, USA. Email address: xwwu@rice.edu} \textit{Rice University}}

\maketitle

\begin{abstract}
We investigate some extinction problems in Markov branching processes.

\textit{Keywords:} Branching process; Markov process
\end{abstract}

\section{Introduction}
\section{Markov branching processes}

A MBP $Z_t$ is completely determined by its offspring distribution and expected life span. Suppose the life length of each particle follows exponential distribution with parameter $a$, and the probability generating function (pgf) of the offspring distribution is given by $f(s)=\sum_{k=0}^{\infty} p_ks^k$. Denote the pgf of the process $Z_t$ by $F(s,t)=E_1\left[s^{Z_t}\right]$, then $F(s,t)$ satisfies the forward Kolmogorov equation
\begin{equation*}
\frac{\partial F(s,t)}{\partial t}=\phi(s)\frac{\partial F(s,t)}{\partial s},
\end{equation*}
where $\phi(s)=a(f(s)-s)$,
and the backward Kolmogorov equation
\begin{equation}
\frac{\partial F(s,t)}{\partial t}=\phi(F(s,t)).\label{Eq_BKE}
\end{equation}

\noindent
\textbf{Theorem.} \textit{Suppose $Z_t$ is a Markov branching process, define the survival probability $Q(t)=P_1(Z_t>0)$. In the subcritical case,}
\begin{equation}
Q(t)\sim b^{-1}e^{-rt}, \textrm{ as }t\rightarrow \infty.
\end{equation}

\textit{Proof.} By Equation (\ref{Eq_BKE}).

This result is confirmed by Figure \ref{Fig_QT}.

\begin{figure}[htb]
\begin{center}
\epsfig{file=fig1.eps,width=14cm}
\end{center}
\caption{\small Survival frequency $\hat{Q}(t)$ in a MBP, based on $500$ simulations and its asymptotic equivalent function.} \label{Fig_QT}
\end{figure}

\section{Acknowledgements}

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