%
% Commenting out blocks of text:
% ------------------------------
\newcommand{\ignore}[1]{}
%
% Theorem-like environments:
% --------------------------
\newtheorem{thm}{Theorem}[section]
\newtheorem{defn}[thm]{Definition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{fact}[thm]{Fact}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{rem}[thm]{Remark} 
\newtheorem{prob}{Problem}
%
% Figures, tables and pictures:
% ---------------------
\newcommand{\inputpictex}{\input{prepicte} \input{pictex} \input{postpict}}
                           % for making PiCTeX available, used in book.tex.
\newenvironment{pictex}{\vspace*{\intextsep} \hspace*{\fill}
\beginpicture}%
{\endpicture \hspace*{\fill}}  % PiCTeX environment, centered on page.
                               % to be used whenever needed.
\newenvironment{fig}{\begin{figure}[htb]\footnotesize}{\end{figure}}
\newenvironment{tbl}{\begin{table}[htb]\footnotesize}{\end{table}}
\newenvironment{figpage}{\begin{figure}[htbp] \footnotesize}{\end{figure}}
\renewcommand{\topfraction}{0.9}
\renewcommand{\bottomfraction}{0.9}
\renewcommand{\textfraction}{0.1}
%
% Added by Mahadev on December 15, 92, modified by Uri May 16, 1994
% -----------------------------------------------------------------
%
%The following command allows us to use texcad to draw lines and later
%modify the lines to dashed lines.  To do this, first enclose the texcad
%file between "\begin{pictex} \setunits{...}" and "\end{pictex}".  Comment
%out the \linethickness command and the \unitlength command.  Specify
%instead the units in \setunits.  (e.g. \setunits{1mm}.  To scale down the
%picture, use for e.g. \setunits{.5mm} instead of \setunits{1mm}.)  Then
%simply replace \emline by \dashline for each line that is to be changed.
\newcommand{\setunits}[1]{\setcoordinatesystem units <#1,#1>%
                       \setlength{\unitlength}{#1}}
%
\newcommand{\dashline}[6]{%
\setlinear%
\findlength {\plot #1 #2 #4 #5 /}%
\setdashesnear <5pt> for <\totalarclength>%
\plot #1 #2 #4 #5 /%
\setsolid % resume solid line and curve plotting
}
%
% Added by Uri on August 8, 1993, modified by Uri May 16, 1994
% ------------------------------------------------------------
%
% The following converts a Bezier command produced by TeXCAD (or otherwise)
% to a pictex dashed curve. TeXCAD produces the command
%         \bezier{n}(p,q)(u,v)(r,s)
% where (p,q) and (r,s) are the endpoints of the parabolic arc and (u,v) 
% is where the tangents at these points meet. Inside pictex, replace it with
%         \dashbezier{n}(p,q)(u,v)(r,s)
% It can be verified that the Bezier curve is given by
%      (1-t)^2*(p,q) + 2t(1-t)*(u,v) + t^2*(r,s),  0 <= t <= 1.
% \dashbezier computes the point for t = 1/2 on the curve between
% (p,q) and (r,s), as required by Pictex, to be (p,q)/4 + (u,v)/2 + (r,s)/4,
% and places its coordinates in \xmidpoint and \ymidpoint before plotting.
%
\newdimen\pdimen
\newdimen\qdimen
\newdimen\udimen
\newdimen\vdimen
\newdimen\rdimen
\newdimen\sdimen
\newdimen\xmidpoint
\newdimen\ymidpoint
%
\def\dashbezier#1(#2,#3)(#4,#5)(#6,#7){%
 \pdimen=\Xdistance{#2} % pictex command to transfer coordinates
 \qdimen=\Ydistance{#3} % to lengths. See Section 9.2 of the PicTeX manual.
 \udimen=\Xdistance{#4} %
 \vdimen=\Ydistance{#5} %
 \rdimen=\Xdistance{#6} %
 \sdimen=\Ydistance{#7} %
 \xmidpoint=0pt %
 \advance \xmidpoint by 0.25\pdimen % compute coordinates of midpoint
 \advance \xmidpoint by 0.50\udimen % using lengths to do real arithmetic
 \advance \xmidpoint by 0.25\rdimen %
 \ymidpoint=0pt %
 \advance \ymidpoint by 0.25\qdimen %
 \advance \ymidpoint by 0.50\vdimen %
 \advance \ymidpoint by 0.25\sdimen %
 \setdimensionmode % enter pictex dimension mode
 \setquadratic %     now do the actual plotting
 \findlength {\plot 
                   {\pdimen} {\qdimen} 
                   {\xmidpoint} {\ymidpoint} 
                   {\rdimen} {\sdimen} / %
              } %
 \setdashesnear <5pt> for <\totalarclength> %
 \plot {\pdimen} {\qdimen} {\xmidpoint} {\ymidpoint} {\rdimen} {\sdimen} / 
 \setsolid % resume solid line and curve plotting
 \setcoordinatemode % resume pictex coordinate mode
}
%
% Added by Uri on March 23, 1993
% ------------------------------
%
% The following environment is an improvement on \setunits (which is left
% intact for backwards compatibility). It gives a better placement of the
% figures. To do a TeXCAD picture with dashed
% lines, move the TeXCAD picture to the bottom left of the TeXCAD screen,
% save the file and comment out the \unitlength and \linethickness commands
% and replace \emline by \dashline where desired as before. Then enclose
% this file between the following lines
%     \begin{pictexcad}{0.5mm}{%
%        .....
%     }% end of second argument of pictexcad environment
%     \end{pictexcad}
% where the first argument of the pictexcad environment (the 0.5mm) can
% be changed to scale the whole picture.
%
\newenvironment{pictexcad}[2]{
  \begin{pictex}
    \setcoordinatesystem units <#1,#1>
    \setlength{\unitlength}{#1}
    \put {#2} [Bl] at 0 0
                              }{\end{pictex}}
%
% The following is used as a gap between subfigures:
\newcommand{\goodgap}{\hspace{\subfigtopskip}\hspace{\subfigbottomskip}}
%
% If you use an empty caption [] for a subfigure to generate a subfigure
% number like (a), the subfigure number may not be aligned properly in
% the horizonbtal direction. To move the (a) a length len to the LEFT,
% where len may be negative, give the subfigure caption as [\hspace{len}].
%
% Running titles:
% ---------------
% Next 4 lines switch odd- and even-side margins:
% We don't need marginal notes, and we want plenty of margins for binding.
%\newlength{\templen}
%\setlength{\templen}{\oddsidemargin}
%\setlength{\oddsidemargin}{\evensidemargin}
%\setlength{\evensidemargin}{\templen}
%
% General math:
% -------------
\newcommand{\G}{$G=(V,E)$\ }
\newcommand{\pf}{\mbox{{\bf Proof.} }}
\newcommand{\qed}{\hspace*{\fill} $\scriptstyle \blacksquare$ \par}
\newcommand{\Iff}{if and only if\ }
\newcommand{\miff}{\quad \mbox{if and only if}\quad}
\renewcommand{\emptyset}{\varnothing}
\newcommand{\emp}[1]{{\em #1\/}}
\newcommand{\imply}[2]{{\bf #1) $\Rightarrow$ #2):}}
\newcommand{\hd}[1]{\noindent {\bf #1}}
           %Used for headline at beginning of paragraph
\newcommand{\bm}[1]{\mbox{\boldmath $#1$}}
           % After last example in LaTeX book, p.53
\newcommand{\BM}{\boldmath\everymath{}\everydisplay{}} 
       % \boldmath with subscripts in ordinary cmr unbold font
\newcommand{\BBM}{%
\font\twlbf=cmbx10 at 12pt
\font\egtbf=cmbx8
\font\sixbf=cmbx6
\textfont0\twlbf \scriptfont0\egtbf \scriptscriptfont0\sixbf
\boldmath
\everymath{}\everydisplay{}}
        % \boldmath with subscripts in ordinary bold font 
\newcommand{\twocond}[2]{\scriptstyle #1 \atop \scriptstyle #2}
%   for putting two conditions on top of each other under summations etc.
\newcommand{\udim}{\mbox{udim}}
\newcommand{\idim}{\mbox{idim}}
%
% Floors, ceilings and binomial:
% ------------------------------
\newcommand{\floor}[2]%
       {{\scriptstyle \left\lfloor \frac{#1}{#2} \right\rfloor}}
\newcommand{\ceil}[2]%
       {{\scriptstyle \left\lceil \frac{#1}{#2} \right\rceil}}
\newcommand{\dfmt}{D_{\floor{m}{2}}}
\newcommand{\dcmt}{D_{\ceil{m}{2}}}
\newcommand{\dmt}{D_{\frac{m}{2}}}
\newcommand{\binom}[2]{{{#1} \choose {#2}}}
%
% Listings:
% ---------
\newcommand{\clist}[3]{#1_{#2}, \ldots, #1_{#3}}
\newcommand{\rlist}[4]{#1_{#2} \; {#3} \; \cdots \; {#3} \; #1_{#4}}
\newcommand{\rdots}[1]{\; {#1} \; \cdots \; {#1} \;}
\newcommand{\lst}[3]{#1 = #2, \ldots, #3}
%
% Cases:
% ------
\newcommand{\twocases}[5]{#1 = \left\{ \begin{array}{ll}
                                         #2, & \mbox{#3} \\
                                         #4, & \mbox{#5}
                                        \end{array}
                               \right. }
\newcommand{\ntwocases}[5]{#1 = \left\{ \begin{array}{rl}
                                         #2, & \mbox{#3} \\
                                         #4, & \mbox{#5}
                                        \end{array}
                               \right. }
% \ntwocases is like \twocases, but the values are right-justified
% for mixed positives and negatives.
\newcommand{\rtwocases}[6]{#1 #2 \left\{ \begin{array}{ll}
                                         #3, & \mbox{#4} \\
                                         #5, & \mbox{#6}
                                        \end{array}
                                    \right. }
\newcommand{\threecases}[7]{#1 = \left\{ \begin{array}{ll}
                                         #2, & \mbox{#3} \\
                                         #4, & \mbox{#5} \\
                                         #6, & \mbox{#7}
                                        \end{array}
                               \right. }