\setlength{\oddsidemargin}{0 in}
\setlength{\evensidemargin}{0 in}
\setlength{\topmargin}{-0.6 in}
\setlength{\textwidth}{6.5 in}
\setlength{\textheight}{9 in}
\setlength{\headsep}{0.5 in}
\setlength{\parindent}{0 in}
\setlength{\parskip}{0.1 in}

\newcommand{\lecture}[3]{
   \pagestyle{myheadings}
   \thispagestyle{plain}
   \newpage
   \setcounter{page}{1}
   \noindent
   \begin{center}
   \framebox{
      \vbox{\vspace{1mm}
    \hbox to 6.28in { #1 \hfill {\it #3} }
       \vspace{2mm}
       \hbox to 6.28in { {\Large \hfill #2 \hfill} }
      \vspace{1mm}}
   }
   \end{center}
   \markboth{#1}{#2}
}

\renewcommand{\i}{\item}
\newcommand{\bee}{\begin{enumerate}}
\newcommand{\ene}{\end{enumerate}}
\newcommand{\bei}{\begin{itemize}}
\newcommand{\eni}{\end{itemize}}
\newcommand{\tto}{~$\longrightarrow$~}
\newcommand{\tfr}{~$\longleftarrow$~}
\newcommand{\br}{\begin{tabular}{@{}p{6.5in}@{}}\\\hline\\\end{tabular}}

\newcommand{\leftright}[2]{\vspace{-2mm}\contentsline {section}{\large #1}{\textnormal{#2}}\vspace{-0mm}}


%%% ====================================================================
%%%  @LaTeX-style-file{
%%%     author          = "Mario Wolczko",
%%%     version         = "2",
%%%     date            = "21 May 1992",
%%%     time            = "20:55:01 BST",
%%%     filename        = "boxedminipage.sty",
%%%     email           = "mario@acm.org",
%%%     codetable       = "ISO/ASCII",
%%%     keywords        = "LaTeX, minipage, framebox",
%%%     supported       = "no",
%%%     docstring       = "LaTeX document-style option which defines
%%%     the boxedminipage environment -- just like minipage, but with
%%%     a box around it.",
%%%  }
%%% ====================================================================
%
% This file is in the public domain
%
% The thickness of the rules around the box is controlled by
% \fboxrule, and the distance between the rules and the edges of the
% inner box is governed by \fboxsep.
%
% This code is based on Lamport's minipage code.
%
% Fixed, 7 Jun 89 by Jerry Leichter
%	Leave \fboxsep worth of separation at top and bottom, not just at
%	the sides!
%
\def\boxedminipage{\@ifnextchar [{\@iboxedminipage}{\@iboxedminipage[c]}}

\def\@iboxedminipage[#1]#2{\leavevmode \@pboxswfalse
  \if #1b\vbox
    \else \if #1t\vtop
	     \else \ifmmode \vcenter
		       \else \@pboxswtrue $\vcenter
		    \fi
	  \fi
  \fi\bgroup % start of outermost vbox/vtop/vcenter
    \hsize #2
    \hrule\@height\fboxrule
    \hbox\bgroup % inner hbox
      \vrule\@width\fboxrule \hskip\fboxsep \vbox\bgroup % innermost vbox
	\vskip\fboxsep
	\advance\hsize -2\fboxrule \advance\hsize-2\fboxsep
	\textwidth\hsize \columnwidth\hsize
	\@parboxrestore
	\def\@mpfn{mpfootnote}\def\thempfn{\thempfootnote}\c@mpfootnote\z@
	\let\@footnotetext\@mpfootnotetext
	\let\@listdepth\@mplistdepth \@mplistdepth\z@
	\@minipagerestore\@minipagetrue
	\everypar{\global\@minipagefalse\everypar{}}}

\def\endboxedminipage{%
	\par\vskip-\lastskip
	\ifvoid\@mpfootins\else
	  \vskip\skip\@mpfootins\footnoterule\unvbox\@mpfootins\fi
	\vskip\fboxsep
      \egroup % ends the innermost \vbox
      \hskip\fboxsep \vrule\@width\fboxrule
    \egroup % ends the \hbox
    \hrule\@height\fboxrule
  \egroup% ends the vbox/vtop/vcenter
  \if@pboxsw $\fi}



\newenvironment{solution}
  {\linespread{1}\center%
   \begin{boxedminipage}{5.8in}%
     \vspace{+0.02in}%
     \begin{sffamily}%
     }
  {\end{sffamily} \vspace{+0.05in}%
   \end{boxedminipage}\vspace{+0.1in}\\}