# LaTeX2HTML 2008 (1.71)
# Associate images original text with physical files.


$key = q/displaystyler-(rslashk)Q(t_r).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$\displaystyle r - (r/k) Q(t_r ) .$">|; 

$key = q/{figure}{epsfig{figure=fig1.ps,width=5in}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="383" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="\begin{figure}\epsfig{figure=fig1.ps,width=5in}\end{figure}">|; 

$key = q/displaystyle{df(t)overdt}>0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$\displaystyle {d f(t) \over dt}&gt; 0.$">|; 

$key = q/Q(-infty)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$Q(- \infty ) = 0$">|; 

$key = q/displaystyle{doverdt}{logf(t)-log[1-f(t)]};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$\displaystyle {d \over dt} \{ \log f(t) - \log [1-f(t)] \}$">|; 

$key = q/{P^i,t_r^i};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$\{P^i , t_r^i \}$">|; 

$key = q/displaystyle1-f(t)=1slash[1+exp(t)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\displaystyle 1- f(t) = 1/[ 1 + \exp(t)]$">|; 

$key = q/%latex2htmlidmarker180ref{eq:linear};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="% latex2html id marker 180
$\ref{eq:linear}$">|; 

$key = q/displaystyleQ(bar{t_r})=kslash2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$\displaystyle Q( \bar{t_r} ) = k / 2 .$">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$k$">|; 

$key = q/displaystylef(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$\displaystyle f(t)$">|; 

$key = q/textstylepropto;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$\textstyle \propto$">|; 

$key = q/displaystyler[1-Q(t_r-bar{t_r})slashk]Q(t_r-bar{t_r});;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$\displaystyle r [1-Q(t_r - \bar{t_r} )/k] Q(t_r - \bar{t_r} ) ;$">|; 

$key = q/displaystyleleft[{1overf(t)}+{1over1-f(t)}right]df(t)slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="220" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$\displaystyle \left [ {1 \over f(t)} + {1 \over 1-f(t)} \right ] df(t)/dt$">|; 

$key = q/displaystylef(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$\displaystyle f(t),$">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\textstyle =$">|; 

$key = q/displaystyle{df(0)overdt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$\displaystyle {d f (0) \over dt}$">|; 

$key = q/displaystyle-infty.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$\displaystyle - \infty .$">|; 

$key = q/displaystyle{dQ(t_r)overdt_r}slashQ(t_r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$\displaystyle {d Q(t_r ) \over dt_r } / Q(t_r )$">|; 

$key = q/displaystylef(0)=1slash2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$\displaystyle f(0) = 1/2.$">|; 

$key = q/rslashk;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$r/k$">|; 

$key = q/displaystylet,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$\displaystyle t ,$">|; 

$key = q/t_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$t_r$">|; 

$key = q/displaystyle{df(t)overdt};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$\displaystyle {d f(t) \over dt}$">|; 

$key = q/displaystyle{df(t)overdt}=f(t)[1-f(t)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$\displaystyle {d f(t) \over dt}= f(t) [1-f(t)] .$">|; 

$key = q/displaystyletequivr(t_r-bar{t_r}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$\displaystyle t \equiv r (t_r - \bar{t_r} ) ,$">|; 

$key = q/df(t)slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$df(t)/dt$">|; 

$key = q/{dQ(t_r)overdt_r}slashQ(t_r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="${d Q(t_r ) \over dt_r } / Q(t_r )$">|; 

$key = q/Q(t_r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$Q(t_r )$">|; 

$key = q/displaystylembox{as}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$\displaystyle \mbox{ as } t$">|; 

$key = q/displaystyle1.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$\displaystyle 1.$">|; 

$key = q/displaystylef(t)equivQ(t)slashk,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$\displaystyle f(t) \equiv Q(t)/k ,$">|; 

$key = q/P(t_r);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$P(t_r )$">|; 

$key = q/(k,r,bar{t_r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$(k, r, \bar{t_r} )$">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$i$">|; 

$key = q/displaystyle{df(t)overdt}proptof(t)[1-f(t)].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$\displaystyle {d f(t) \over dt}\propto f(t) [1-f(t)] .$">|; 

$key = q/displaystylelogf(t)-log[1-f(t)];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\displaystyle \log f(t) - \log [1-f(t)]$">|; 

$key = q/displaystyle{f(t)over1-f(t)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$\displaystyle {f(t) \over 1-f(t)}$">|; 

$key = q/Q(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$Q(t)$">|; 

$key = q/displaystyleexp(t).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$\displaystyle \exp(t) .$">|; 

$key = q/displaystyle0,mbox{and};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$\displaystyle 0 , \mbox{ and }$">|; 

$key = q/displaystylef(t)={exp(t)over1+exp(t)}={1over1+exp(-t)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$\displaystyle f(t) = {\exp(t) \over 1 + \exp(t)} = {1 \over 1 + \exp(-t)} .$">|; 

$key = q/Q(infty)=k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$Q(\infty ) = k$">|; 

$key = q/displaystylet,mbox{and};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\displaystyle t , \mbox{ and }$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$t$">|; 

$key = q/displaystyle1-f(t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$\displaystyle 1 - f(t),$">|; 

$key = q/displaystyleinfty.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$\displaystyle \infty .$">|; 

$key = q/f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$f(x)$">|; 

$key = q/displaystyle1slash4,mbox{and}{df(pminfty)overdt}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$\displaystyle 1/4 , \mbox { and }
{d f ( \pm \infty ) \over dt} = 0 .$">|; 

$key = q/displaystyleQ(t_r)={kover1+exp[-r(t_r-bar{t_r})]}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$\displaystyle Q(t_r ) = {k \over 1 + \exp[-r (t_r - \bar{t_r} )]} .$">|; 

$key = q/displaystyleP(t_r)equiv{dQ(t_r)overdt_r}={krover{exp[r(t_r-bar{t_r})slash2]+exp[-r(t_r-bar{t_r})slash2]}^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="494" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\displaystyle P(t_r ) \equiv
{d Q(t_r ) \over dt_r } =
{k r \over \{\exp[r (t_r - \bar{t_r} )/2] + \exp[-r (t_r - \bar{t_r} )/2]\}^2} .$">|; 

$key = q/displaystyle1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\displaystyle 1 ,$">|; 

$key = q/displaystyle{exp(-t)over[1+exp(-t)]^2}={1over[exp(tslash2)+exp(-tslash2)]^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$\displaystyle { \exp(-t) \over [ 1 + \exp(-t) ]^2 }
= {1 \over [ \exp(t/2) + \exp(-t/2) ]^2} ,$">|; 

$key = q/dQ(t_r)slashdt_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$d Q(t_r ) / dt_r$">|; 

$key = q/displaystyle{dQ(t_r-bar{t_r})overdt_r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$\displaystyle {d Q(t_r - \bar{t_r} ) \over dt_r }$">|; 

$key = q/bar{t_r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$ \bar{t_r} $">|; 

$key = q/displaystylef(-t);;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\displaystyle f(-t) ;$">|; 

$key = q/displaystylelogleft[{f(t)over1-f(t)}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\displaystyle \log \left [ {f(t) \over 1-f(t)} \right ]$">|; 

$key = q/{figure}{epsfig{figure=fig2.ps,width=5in}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="575" HEIGHT="383" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="\begin{figure}\epsfig{figure=fig2.ps,width=5in}\end{figure}">|; 

$key = q/f(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$f(t)$">|; 

$key = q/P(t_r)=dQ(bar{t_r})slashdt_r=krslash4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$P(t_r ) = d Q( \bar{t_r} )/dt_r = k r /4$">|; 

$key = q/displaystyle1,mbox{and};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$\displaystyle 1 , \mbox{ and }$">|; 

$key = q/displaystylef(t)+f(-t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\displaystyle f(t) + f(-t)$">|; 

$key = q/P(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$P(t)$">|; 

$key = q/r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$r$">|; 

$key = q/displaystylemin_{k,r,bar{t_r}}sum_ileft{P^i~log[P^i~slash~P(t_r^i)]right}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\displaystyle \min_{k, r, \bar{t_r} } \sum_i \left \{ P^i&nbsp; \log [ P^i&nbsp;/&nbsp; P( t_r^i ) ]\right \} .$">|; 

$key = q/textstylerightarrow;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\textstyle \rightarrow$">|; 

1;