\batchmode
\documentstyle[11pt,leqno]{article}
\makeatletter

\textheight=20cm
\textwidth=16.5cm
\oddsidemargin=0cm
\evensidemargin=0cm
\renewcommand{\theequation}{\thesection.\arabic{equation}}


\renewcommand{\theequation}{\thesection.\arabic{equation}}\newcommand{\Oc}{{\Omega_c}}
\newcommand{\w}{{\bf x'}}
\newcommand{\y}{{t}}
\newcommand{\tp}{{t'}}
\newcommand{\n}{{\bf n}}
\newcommand{\x}{{\bf x}}
\newcommand{\ds}{\displaystyle}

\makeatother
\newenvironment{tex2html_wrap}{}{}
\setlength{\textheight}{10in}\begin{document}
\pagestyle{empty}
\newpage

{\samepage \clearpage ${\Omega _c}$
}


\newpage

{\samepage \clearpage $V = V(t)$
}


\newpage

{\samepage \clearpage ${\Omega _c}$
}


\newpage

{\samepage \clearpage $P({\string\pbf\space x},{t})$
}


\newpage

{\samepage \clearpage ${\string\pbf\space x}$
}


\newpage

{\samepage \clearpage ${t}$
}


\setcounter{section}{0}
\stepcounter{section}
\setcounter{equation}{0}
\newpage

{\samepage \clearpage $X_e$
}


\newpage

{\samepage \clearpage $t$
}


\newpage

{\samepage \clearpage $t + dt$
}


\newpage

{\samepage \clearpage $x$
}


\newpage

{\samepage \clearpage $0$
}


\stepcounter{section}
\setcounter{equation}{0}
\newpage

{\samepage \clearpage $1-P$
}


\newpage

{\samepage \clearpage $P$
}


\newpage

{\samepage \clearpage $\y$
}


\newpage

{\samepage \clearpage $\tau$
}


\newpage

{\samepage \clearpage $\w$
}


\newpage

{\samepage \clearpage $\Omega$
}


\newpage

{\samepage \clearpage $\omega $
}


\newpage

{\samepage \clearpage $\omega= (x,y,z;t) = ({\bf x};t).$
}


\newpage

{\samepage \clearpage $({\bf x}; {t})$
}


\newpage

{\samepage \clearpage $({\bf x'}; \tau)$
}


\newpage

{\samepage \clearpage $({\bf x}; {t})$
}


\newpage

{\samepage \clearpage $\Oc$
}


\newpage

{\samepage \clearpage $<N>_c$
}


\newpage

{\samepage \clearpage \begin{equation}\label{Poisson}
 P = e^{-<N>_c}.
\end{equation}
}


\stepcounter{subsection}
\newpage

{\samepage \clearpage \begin{equation}\label{radius}
R({t},\tau) = \int_{\tau}^{t}V({t'})d{t'}. 
\end{equation}
}


\newpage

{\samepage \clearpage $R$
}


\newpage

{\samepage \clearpage \begin{equation}\label{tdCone}
R({t},\tau)^2 - | {\bf x}-\w|^2 \ge 0,  
\end{equation}
}


\newpage

{\samepage \clearpage $\bf x$
}


\newpage

{\samepage \clearpage $({\bf x};t)$
}


\newpage

{\samepage \clearpage ${t}({\bf x})$
}


\newpage

{\samepage \clearpage ${t}({\bf x})$
}


\newpage

{\samepage \clearpage ${\bf n},$
}


\newpage

{\samepage \clearpage $1/V.$
}


\newpage

{\samepage \clearpage $ {\bf n} =
\nabla
{t} /|\nabla \y|$
}


\newpage

{\samepage \clearpage $|\nabla \y| = |1/V|$
}


\newpage

{\samepage \clearpage $V$
}


\newpage

{\samepage \clearpage ${\bf x},\y$
}


\newpage

{\samepage \clearpage $\n$
}


\newpage

{\samepage \clearpage \begin{equation}\label{Cone}
V^2({t}-\tau)^2 - | {\bf x}-\w|^2 \ge 0.
\end{equation}
}


\newpage

{\samepage \clearpage $({\bf x}; t)$
}


\newpage

{\samepage \clearpage $c^2(t-\tau)^2 -|{\bf x} -
\w|^2 = 0,$
}


\newpage

{\samepage \clearpage $c$
}


\newpage

{\samepage \clearpage $({\bf x},{t})$
}


\newpage

{\samepage \clearpage $2V({t}-\tau);$
}


\newpage

{\samepage \clearpage $V({t}-\tau);$
}


\newpage

{\samepage \clearpage $V({t}-\tau)$
}


\newpage

{\samepage \clearpage $({\bf x}; {t})$
}


\newpage

{\samepage \clearpage \begin{equation}\label{Coneset}
{\Omega_c}({\bf x}; t) = \{({\bf x'}; \tau): R({t},\tau)^2- |{\bf x}-\w|^2 \ge 0\}. 
\end{equation}
}


\newpage

{\samepage \clearpage $V(t)$
}


\newpage

{\samepage \clearpage $\alpha = 0$
}


\stepcounter{subsection}
\newpage

{\samepage \clearpage $\alpha({\bf x}; t)$
}


\newpage

{\samepage \clearpage $\alpha$
}


\newpage

{\samepage \clearpage \begin{equation}\label{nuclint} 
<N>_c =<N>_c({\bf x}; {t}) = \int_{{\Omega_c}} \alpha({\bf x'}; \tau)d\omega
\end{equation}
}


\newpage

{\samepage \clearpage $\alpha = \alpha (t)$
}


\newpage

{\samepage \clearpage $t \ge 0$
}


\newpage

{\samepage \clearpage \begin{equation}\label{fulltimedep}
<N>_c (t) = \frac{4\pi}{3} \int_0^t \alpha({t'}) R(t,{t'})^3 d{t'}.
\end{equation}
}


\newpage

{\samepage \clearpage $\|\Omega_N\|$
}


\newpage

{\samepage \clearpage \begin{equation}\label{nuclcnst}
<N>_c =  \alpha \|{\Omega_N}\|.
\end{equation}
}


\newpage

{\samepage \clearpage $\|{\Omega_N}\|$
}


\newpage

{\samepage \clearpage $B$
}


\newpage

{\samepage \clearpage $t=0$
}


\newpage

{\samepage \clearpage $(d+1).$
}


\newpage

{\samepage \clearpage \begin{equation}\label{classxe}
<N>_c =  \alpha B {t}/(d+1).
\end{equation}
}


\newpage

{\samepage \clearpage $\|{\Omega_N}\|$
}


\newpage

{\samepage \clearpage $2Vt$
}


\newpage

{\samepage \clearpage $Vt^2$
}


\newpage

{\samepage \clearpage $\pi V^2t^2$
}


\newpage

{\samepage \clearpage $\frac{\pi}{3}V^2t^3$
}


\newpage

{\samepage \clearpage $\frac{4\pi}{3}V^3t^3$
}


\newpage

{\samepage \clearpage $\frac{\pi}{3}V^3t^4$
}


\stepcounter{subsection}
\newpage

{\samepage \clearpage $P({\bf x},{t})$
}


\newpage

{\samepage \clearpage \begin{equation}\label{po}
P({\bf x}; {t}) = e^{-<N>_c}
\end{equation}
}


\newpage

{\samepage \clearpage $d\omega$
}


\newpage

{\samepage \clearpage \begin{equation}-dP  = P \cdot \alpha d\omega,
\end{equation}
}


\newpage

{\samepage \clearpage \begin{equation}log P = -\int_{{\Omega_c}} \alpha d\omega = -<N>_c.
\end{equation}
}


\newpage

{\samepage \clearpage $N,$
}


\newpage

{\samepage \clearpage \begin{equation}p(n,N) = N^n e^{-N}/n!.
\end{equation}
}


\newpage

{\samepage \clearpage $n=0$
}


\newpage

{\samepage \clearpage $p(0,N) = e^{-N},$
}


\newpage

{\samepage \clearpage \begin{equation}\label{JMeq}
 P({\bf x}; t) = e^{-\frac{\pi}{3} \alpha V^3 t^4}.
\end{equation}
}


\stepcounter{section}
\setcounter{equation}{0}
\newpage

{\samepage \clearpage $({\bf x}; {t})$
}


\newpage

{\samepage \clearpage $({\bf x'}; \tau)$
}


\newpage

{\samepage \clearpage $({\bf x}; {t}),$
}


\newpage

{\samepage \clearpage $({\bf x'};\tau)$
}


\newpage

{\samepage \clearpage $1/V$
}


\newpage

{\samepage \clearpage $\|{\Omega_N}\|$
}


\newpage

{\samepage \clearpage $\|{\Omega_N}\|$
}


\newpage

{\samepage \clearpage $(\bf x ;t)$
}


\stepcounter{subsection}
\newpage

{\samepage \clearpage $z = 0$
}


\newpage

{\samepage \clearpage \begin{equation}\label{surf}
 P(0;t) = e^{-\frac{\pi}{6} \alpha V^3 t^4}.
\end{equation}
}


\newpage

{\samepage \clearpage $P(0;t)$
}


\newpage

{\samepage \clearpage $\sqrt {P({\bf x}; t)}$
}


\newpage

{\samepage \clearpage $(z,t)$
}


\newpage

{\samepage \clearpage $z \ge Vt$
}


\newpage

{\samepage \clearpage $z<V\y$
}


\newpage

{\samepage \clearpage $(\pi/6V)[(Vt)^4+2z(Vt)^3-2z^3Vt+z^4].$
}


\newpage

{\samepage \clearpage $Z = z/Vt$
}


\newpage

{\samepage \clearpage $0 \le Z \le 1$
}


\newpage

{\samepage \clearpage \begin{equation}\label{semi}
 P(Z;t) =  e^{-\frac{\pi}{6} \alpha V^3 t^4[1+2Z-2Z^3+Z^4]}, 
\end{equation}
}


\newpage

{\samepage \clearpage \begin{equation}P(Z;t) = f_1(Z;t) e^{-\frac{\pi}{3}\alpha V^3 t^4},
\end{equation}
}


\newpage

{\samepage \clearpage $0 \le Z <1$
}


\newpage

{\samepage \clearpage \begin{equation}f_1 = e^{\frac{\pi}{6}[1-2Z+2Z^3-Z^4] \alpha V^3 t^4};
\end{equation}
}


\newpage

{\samepage \clearpage $Z \ge 1, f_1 = 1.$
}


\newpage

{\samepage \clearpage $log\ log P(z)$
}


\newpage

{\samepage \clearpage $log t$
}


\newpage

{\samepage \clearpage $log P$
}


\newpage

{\samepage \clearpage $z$
}


\newpage

{\samepage \clearpage $\zeta$
}


\newpage

{\samepage \clearpage $0 \le z \le \zeta/2,$
}


\newpage

{\samepage \clearpage $\zeta - z$
}


\newpage

{\samepage \clearpage $Vt$
}


\newpage

{\samepage \clearpage $z<Vt$
}


\newpage

{\samepage \clearpage $(1-z) > Vt$
}


\newpage

{\samepage \clearpage $ (1-z) < Vt$
}


\newpage

{\samepage \clearpage $Vt = \zeta/2$
}


\newpage

{\samepage \clearpage $Vt = \zeta$
}


\newpage

{\samepage \clearpage \begin{equation}\label{filmomega}
\|\Omega_N\| = \frac{\pi}{3}V[\zeta(Vt)^3 - (z^3+(\zeta-z)^3)Vt + (z^4
+(\zeta-z)^4)/2]. 
\end{equation}
}


\newpage

{\samepage \clearpage $4$
}


\newpage

{\samepage \clearpage \begin{equation}\label{filmp}
 P(z;t,\zeta) = f_2(z;t) e^{-\frac{\pi}{3} \alpha \zeta V^2 t^3}.
\end{equation}
}


\newpage

{\samepage \clearpage $(\alpha\zeta)$
}


\newpage

{\samepage \clearpage $f_2$
}


\newpage

{\samepage \clearpage $X_1
= z/\zeta$
}


\newpage

{\samepage \clearpage $X_1 + X_2 = 1,$
}


\newpage

{\samepage \clearpage % latex2html id marker 895
$\theta = Vt/\zeta >1$
}


\newpage

{\samepage \clearpage \begin{equation}% latex2html id marker 170
f_2 = e^{\frac{\pi}{3}[\alpha\zeta^4/V][(X_1^3 +X_2^3)\theta +( X_1^3
+X_2^3)/2]}.
\end{equation}
}


\newpage

{\samepage \clearpage \begin{equation}<P> = \frac{1}{\zeta}\int_0^\zeta P(z)dz.
\end{equation}
}


\newpage

{\samepage \clearpage $P(z)$
}


\newpage

{\samepage \clearpage $z = 0, \zeta/2, \zeta$
}


\newpage

{\samepage \clearpage \begin{equation}\label{apx}
<P> = \frac{1}{3}(P(0) + 2P(\zeta/2))
\end{equation}
}


\newpage

{\samepage \clearpage $t = \zeta/2V$
}


\newpage

{\samepage \clearpage $$<P> = \frac{1}{3}((1 +
e^{-\frac{\pi}{6} \alpha V^3 t^4})^2 - 1).$$
}


\newpage

{\samepage \clearpage $ z_s = V_s \tau$
}


\newpage

{\samepage \clearpage $V_s>V = \beta V_s$
}


\newpage

{\samepage \clearpage $z \le
V_s\tau.$
}


\newpage

{\samepage \clearpage $ z = V_s \tau.$
}


\newpage

{\samepage \clearpage $V_s$
}


\newpage

{\samepage \clearpage $\tau = const = 0,$
}


\newpage

{\samepage \clearpage $V \ge
V_s$
}


\newpage

{\samepage \clearpage $V_s = 0$
}


\newpage

{\samepage \clearpage $(\frac{\pi}{3}) V^3 (t-
z/V_s)^4/(1-\beta^2)^2.$
}


\newpage

{\samepage \clearpage $t \ge 0,$
}


\newpage

{\samepage \clearpage \begin{equation}\label{Lorenz}
 P(0;t) = e^{-\frac{\pi}{3} \alpha V^3
(\frac{t}{\sqrt{1-\beta^2}})^4}.
\end{equation}
}


\newpage

{\samepage \clearpage $\sqrt{1-(V/V_s)^2}$
}


\newpage

{\samepage \clearpage $(1-(V/V_s)^2)^{3/4}.$
}


\newpage

{\samepage \clearpage $ z \ge V_st$
}


\stepcounter{section}
\setcounter{equation}{0}
\newpage

{\samepage \clearpage $X_e = <N_c>$
}


\newpage

{\samepage \clearpage $log\ log P$
}


\newpage

{\samepage \clearpage $log \y$
}


\newpage

{\samepage \clearpage $t^{-1/2}$
}


\newpage

{\samepage \clearpage $3/2$
}


\newpage

{\samepage \clearpage $5/2$
}


\newpage

{\samepage \clearpage $t^{-1/2}$
}


\newpage

{\samepage \clearpage $1$
}


\newpage

{\samepage \clearpage $<N_c>$
}


\newpage

{\samepage \clearpage $({\bf x},{t})$
}


\stepcounter{section}

\end{document}