COMMENT ⊗   VALID 00002 PAGES
C REC  PAGE   DESCRIPTION
C00001 00001
C00002 00002	\font\tenrm=cmr10 % roman text
C00014 ENDMK
C⊗;
\font\tenrm=cmr10 % roman text
\font\rmn=cmr9 %ninepoint
%\def\rmn{\font=amr9}
%\font\preloaded=amr9
\font\tte=cmtt8

\def\drawat#1#2#3{\vtop{\kern#2\rlap{\kern#1#3}}} %#3 goes in hbox

\newdimen\digitwidth
 \setbox0=\hbox{\rm0}
 \digitwidth=\wd0
%\qtoblank is set up.
\def\qtoblank%change ? to invisible digit
 {\catcode`?=\active
 \def?{\kern\digitwidth}}
%{\qtoblank ... 0.12579?58421?05.}
%\catcode'?=\other
%restores normal ?
%place { } around area that you wish to use \qtoblank
%or else say
%\catcode'?=\other %restores normal ?

\newdimen\unit
\def\point#1 #2 {\rlap{\kern#1\unit
  \raise#2\unit\hbox to 0\unit{\hss$\scriptstyle\bullet$\hss}}}
\def\ycoord#1 {\rlap{\kern -0.2\unit
 \raise#1\unit\hbox{#1}}}
\def\xcoord#1 {\rlap{\kern#1\unit
 \lower0.2\unit\hbox to 0\unit
 {\hss #1\hss}}}


\def\fnc#1{\mathop{\rm #1}\nolimits}
%usage: to get erf(x), write $\fnc{erf}(x)$

\mathsurround .5pt
\def\dminus{\mathrel{\vbox{\lineskip-1pt\baselineskip0pt
 \halign{\ctr{$##$}\cr
 .\cr-\cr}}}}
\def\ldt{\mathinner{\ldotp\ldotp}}
\def\relbuild#1\over#2{\mathrel{\mathop{#1}\limits_{#2}}}%to reverse buildrel
\def\tog{\relbuild \to\over G}%→ over G
\def\togp{\relbuild \to\over {G'}}%→ over G'
\def\Rag{\relbuild \Rightarrow\over G}%double → over G
\def\Ragp{\relbuild \Rightarrow\over{G'}}%double → over G'
\def\atog{\buildrel\ast\over\tog}%asterisk over → over G
\def\aRag{\buildrel\ast\over\Rag}%asterisk over double → over G
\def\aRagp{\buildrel\ast\over\Ragp}%asterisk over double → over G'
\def\aRa{\buildrel \ast\over\Rightarrow}%asterisk over double→
\def\mapstoa{\buildrel \ast\over\mapsto}%asterisk over \mapsto
\def\ato{\buildrel \ast\over\to}%asterisk over →
\def\indeq{\buildrel {\rm ind}\over =}%ind over =
\def\defeq{\buildrel {\rm def}\over =}%def over =
\def\ld{\mathrel{\hbox{$<$\kern-4pt$\cdot$}}}
\def\gd{\mathrel{\hbox{$\cdot$\kern-4pt$>$}}}
\def\ej{\relbuild \equiv\over{j,k,x}}%≡ over j,k,x
\def\ez{\relbuild \equiv\over z}%≡ over z
\def\ext{\relbuild \equiv\over{x,t}}%≡ over x,t
\def\eqr{\relbuild =\over r}%= over r
\def\eqp{\relbuild =\over p}%= over p
\def\equU{\relbuild =\over U}%= over U
\def\eqq{\relbuild =\over q}%= over q
\def\eqs{\relbuild =\over s}%= over s
\def\eqj{\relbuild =\over j}%= over j
\def\appL{\relbuild \approx\over L}%approximate over L
\def\pg{\buildrel +\over <}
\def\tl{\buildrel \times\over <}
\def\peq{\buildrel +\over =}
\def\teq{\buildrel \times\over =}
\def\gd{\mathrel{\hbox{$\cdot$\kern-4pt$>$}}}
\def\pluss{\buildrel +\over\cup}%+ over ∪
\def\zap{\mathrel{\vbox{\lineskip1pt\baselineskip1pt
 \halign{\ctr{$##$}\cr
 \scriptscriptstyle L\cr \sim\cr}}}}
\def\zzap{\mathrel{\vbox{\lineskip1pt\baselineskip1pt
 \halign{\ctr{$##$}\cr
 \scriptscriptstyle L\cr \approx\cr}}}}
\def\zapp{\mathrel{\vbox{\lineskip1pt\baselineskip1pt
 \halign{\ctr{$##$}\cr
 \scriptscriptstyle L↓2\cr \sim\cr}}}}
\def\zappp{\mathrel{\vbox{\lineskip1pt\baselineskip1pt
 \halign{\ctr{$##$}\cr
 \scriptscriptstyle L↓3\cr \sim\cr}}}}
\def\alra{\buildrel a\over\longrightarrow}
\def\elra{\buildrel \epsilon\over\longrightarrow}
\def\Lap{\buildrel L\over\approx}
\def\Lapp{\buildrel L↓1\over\approx}
%\def\footnote#1#2{#1botinsert{\hrule width5pc \vskip3pt\baselineskip9pt
% \hbox par size{\eightpoint#1#2}}}
\def\qq{\qquad}
\def\q{\quad}
\def\narp{\buildrel n\over\Rightarrow}
\def\nnap{\buildrel n-1\over\Rightarrow}
\def\spa{\raise1pt\hbox{\tt\char'40 }}%use it as {\spa} or else changes everything after into \tt
\def\tmo{\hbox{$\vcenter{\hbox{\tt .}\vskip-8pt\hbox{\tt -}}$}} %. over -
 %use $\!$ on both sides of \tmo
\def\eol{\mskip1mu\raise2pt\vbox{\hbox{\tte E}}
 \raise1pt\vbox{\hbox{\tte O}}
 \hbox{\tte L}\mskip1mu}

\def\ctrline{\centerline}
\def\lft#1{{#1}\hfill}
\def\ctr#1{\hfill{#1}\hfill}
\def\rt#1{\hfill{#1}}
\def\naive{na\"\i ve} %\naive\

\def\xeq{\buildrel \times\over =}%x over = 
\def\xleq{\buildrel \times\over ≤}%x over ≤ 
\def\xgeq{\buildrel \times\over ≥}%x over ≥ 

\def\twodots{\mathinner{\ldotp\ldotp}} %same as \ldt
\def\derives{\buildrel\ast\over\rightarrow} %same as \ato
\def\transitiveclosure#1{\buildrel\ast\over 1} 

\def\eqaligntwo #1{\,\vcenter{\openup1\jot 
 \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil&\quad
 \hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil\crcr#1\crcr}}\,}


\def\nopagenumber{\output{\shipout\box255}}
\def\upsidea{\mathop{\forall}} % ∀ as math operator in formula
\def\backe{\mathop{\exists}} % ∃ as math operator in displayed formula
\def\therefore{\mathinner{\mskip2mu\raise1pt\vbox{\kern7pt\hbox{.}}
 \mskip2mu\raise7pt\hbox{.}
 \mskip2mu\raise1pt\hbox{.}\mskip6mu}}
\def\per{\mathop{\rm per}\nolimits}
\def\Det{\mathop{\rm Det}\nolimits}
\def\ihat{{\hat\imath}}
\def\jhat{{\hat\jmath}}
\def\xskip{\hskip .7em plus .3em minus .4em}
\def\adx#1:#2\par{\par\halign{\hskip #1##\hfill\cr #2}\par}
\def\disleft#1:#2:#3\par{\par\hangindent#1\noindent
			 \hbox to #1{#2 \hfill \hskip .1em}\ignorespaces#3\par}
\def\display#1:#2:#3\par{\par\hangindent #1 \noindent
			\hbox to #1{\hfill #2 \hskip .1em}\ignorespaces#3 \par}
\def\blackslug{\hbox{\hskip 1pt \vrule width 4pt height 6pt depth 1.5pt \hskip 1pt}}
\def\bblackslug{%
\hbox{\hskip 1pt \vrule width 2pt height 2pt depth 1pt \hskip 1pt}}
\def\bib{\noindent\hangindent 20pt}
\def\biba{\par\parindent 20pt\hangindent 40pt}
\def\figbox#1:{\vbox{\null\vskip#1\null}}%cut and paste space
\def\undertext#1{$\underline{\hbox{#1}}$}
\def\table#1{$$\hbox{[Table #1]}$$}
\def\fig#1{$$\hbox{[Figure #1]}$$}
\def\pfbox
 {\hbox{\vbox{
 \def\|{\vrule height 2pt depth 2pt}
 \hrule
 \hbox to 4pt{\|\hfill\|}
 \hrule
 }}\hskip 3pt}
\def\outlineone#1:{\par\hangindent 19pt\noindent
  \hbox to 19 pt{#1\hfill}\ignorespaces}
\def\outlinetwo#1:{\par\hangindent 38pt\noindent
  \hbox to 38 pt{\hbox to 19pt{\hfill}#1\hfill}\ignorespaces}
\def\outlinethree#1:{\par\hangindent 57pt\noindent
  \hbox to 57 pt{\hbox to 38pt{\hfill}#1\hfill}\ignorespaces}
\uchyph 1 % hyphenate Capital letter words

\def\boxbinary#1%
{\hbox to 1em{\hbox to 1em{\hfil#1\hfil}\hskip-1.05em minus1em
\vbox to 9.8222pt{\hrule height .5pt
\hbox to 1.2em{\vrule height 1.2em width .5pt\hfil\vrule height 
1.2em width .5pt}
\hrule height .5pt\vss}\hskip-.1emminus1em}}

\def\boxp{\mathbin{\boxbinary{$+$}}}
\def\boxm{\mathbin{\boxbinary{$-$}}}
\def\boxd{\mathbin{\boxbinary{$/$}}}
\def\boxa{\mathbin{\boxbinary{$\ast$}}}

\def\boxbinarya#1%
{\hbox to .6em{\hbox to .6em{\hfil#1\hfil}\hskip-.5em minus.45em
\vbox to 4.9111pt{\hrule height .25pt
\hbox to .6em{\vrule height .6em width .25pt\hfil\vrule height 
.6em width .25pt}
\hrule height .25pt\vss}\hskip-.05emminus.5em}}

%\def\box2{\boxbinarya{$\scriptstyle 2$}}
\def\boxtwo{\boxbinarya{$\scriptstyle 2$}}
\def\boxaa{\boxbinarya{$\scriptstyle a$}}

\def\aot{\buildrel\rightarrow\over\otimes}%→  over \otimes 

\def\dddots{\mathinner{\mskip1mu\raise1pt\vbox{\kern7pt\hbox{.}}\mskip2mu
    \raise4pt\hbox{.}\mskip2mu\raise7pt\hbox{.}\mskip1mu}}
\def\]{\leavevmode\hbox{\tt\char`\ }} % visible space

\def\return{\mskip1mu\raise2pt\vbox{\hbox{$\scriptstyle\rm c$}}
 \hbox{$\scriptstyle\rm r$}}
\def\esc{\mskip1mu\raise2pt\vbox{\hbox{$\scriptstyle\rm e$}}
 \raise1pt\vbox{\hbox{$\scriptstyle\rm s$}}
 \hbox{$\scriptstyle\rm c$}\mskip1mu}

\newbox\bgcrc \setbox\bgcrc=\hbox{$\bigcirc$}
\def\orightarrow{\mathop{\hbox to 0pt{\kern-.5pt\raise1pt\hbox to 1\wd\bgcrc{\hfil
	$\scriptstyle\rightarrow$\hfil}\hss}\bigcirc}}