%% preamble.tex %% this should be included with a command like %% \input{preamble.tex} %% \lecture{1}{September 4, 1996 }{Daniel A. Spielman}{name %% of poor scribe} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 18.405J/6.841J Advanced Complexity Theory} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\lecture}[4]{\handout{#1}{#2}{Lecturer: #3}{Scribe: #4}{Lecture #1}} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{observation}[theorem]{Observation} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{claim}[theorem]{Claim} \newtheorem{fact}[theorem]{Fact} \newtheorem{assumption}[theorem]{Assumption} \newcommand{\qed}{\rule{7pt}{7pt}} \newcommand{\dis}{\mathop{\mbox{\rm d}}\nolimits} \newcommand{\per}{\mathop{\mbox{\rm per}}\nolimits} \newcommand{\area}{\mathop{\mbox{\rm area}}\nolimits} \newcommand{\cw}{\mathop{\rm cw}\nolimits} \newcommand{\ccw}{\mathop{\rm ccw}\nolimits} \newcommand{\DIST}{\mathop{\mbox{\rm DIST}}\nolimits} \newcommand{\OP}{\mathop{\mbox{\it OP}}\nolimits} \newcommand{\OPprime}{\mathop{\mbox{\it OP}^{\,\prime}}\nolimits} \newcommand{\ihat}{\hat{\imath}} \newcommand{\jhat}{\hat{\jmath}} \newcommand{\abs}[1]{\mathify{\left| #1 \right|}} \newenvironment{proof}{\noindent{\bf Proof}\hspace*{1em}}{\qed\bigskip} \newenvironment{proof-sketch}{\noindent{\bf Sketch of Proof}\hspace*{1em}}{\qed\bigskip} \newenvironment{proof-idea}{\noindent{\bf Proof Idea}\hspace*{1em}}{\qed\bigskip} \newenvironment{proof-of-lemma}[1]{\noindent{\bf Proof of Lemma #1}\hspace*{1em}}{\qed\bigskip} \newenvironment{proof-attempt}{\noindent{\bf Proof Attempt}\hspace*{1em}}{\qed\bigskip} \newenvironment{proofof}[1]{\noindent{\bf Proof} of #1:\hspace*{1em}}{\qed\bigskip} \newenvironment{remark}{\noindent{\bf Remark}\hspace*{1em}}{\bigskip} % \makeatletter % \@addtoreset{figure}{section} % \@addtoreset{table}{section} % \@addtoreset{equation}{section} % \makeatother \newcommand{\FOR}{{\bf for}} \newcommand{\TO}{{\bf to}} \newcommand{\DO}{{\bf do}} \newcommand{\WHILE}{{\bf while}} \newcommand{\AND}{{\bf and}} \newcommand{\IF}{{\bf if}} \newcommand{\THEN}{{\bf then}} \newcommand{\ELSE}{{\bf else}} % \renewcommand{\thefigure}{\thesection.\arabic{figure}} % \renewcommand{\thetable}{\thesection.\arabic{table}} % \renewcommand{\theequation}{\thesection.\arabic{equation}} \makeatletter \def\fnum@figure{{\bf Figure \thefigure}} \def\fnum@table{{\bf Table \thetable}} \long\def\@mycaption#1[#2]#3{\addcontentsline{\csname ext@#1\endcsname}{#1}{\protect\numberline{\csname the#1\endcsname}{\ignorespaces #2}}\par \begingroup \@parboxrestore \small \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par \endgroup} \def\mycaption{\refstepcounter\@captype \@dblarg{\@mycaption\@captype}} \makeatother \newcommand{\figcaption}[1]{\mycaption[]{#1}} \newcommand{\tabcaption}[1]{\mycaption[]{#1}} \newcommand{\head}[1]{\chapter[Lecture \##1]{}} \newcommand{\mathify}[1]{\ifmmode{#1}\else\mbox{$#1$}\fi} \renewcommand{\Pr}[1]{\mathify{\mbox{Pr}\left[#1\right]}} \newcommand{\Exp}[1]{\mathify{\mbox{Exp}\left[#1\right]}} \newcommand{\bigO}O \newcommand{\set}[1]{\mathify{\left\{ #1 \right\}}} \def\half{\frac{1}{2}} \def\implies{\Rightarrow} \def\prob#1#2{{\mathop{{\rm Prob}}_{#1}}\left[#2 \right]} \def\var#1#2{{\mathop{{\rm Var}}_{#1}}[#2]} \def\expec#1#2{{\mathop{{\rm E}}_{#1}}[#2]} \def\sizeof#1{\left| #1\right|} \def\setof#1{\left\{ #1\right\} }