% defsNSF.tex

\newcommand{\Title}{Turbulence:
    \\
        A walk through a repertoire of unstable recurrent patterns}

%%              Predrag  29 oct 2003
%%              Predrag  22 sep 2003
%%                      Predrag  20 mar 1999
%% created              Predrag  26 oct 1998

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% define abreviations used throughout the text
%% to make it easier to change conventions
%% and also define Feynmann diagrams
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%% PAGE LAYOUT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epsfig}
\usepackage{wrapfig}
\usepackage[dvips]{color}     %% dvips allows for colors

%%%%%%%%%%%%%%%%% PUBLIC/EDITING VERSIONS CONTROL %%%%%%%%%%%%%%%%%
        \usepackage{ifthen} \newboolean{display_comments}
%\setboolean{display_comments}{true}     %% turn margin comments on
\setboolean{display_comments}{false}    %% turn margin comments off

%%%%%%%%%%%%%%%%% DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% regulation SNF margins from Kath
\newlength{\extramargin}
\setlength{\extramargin}{0.0in}
\setlength{\oddsidemargin}{\extramargin}
\setlength{\evensidemargin}{\extramargin}
\setlength{\textwidth}{6.5in}
\addtolength{\textwidth}{-2\extramargin}
%\setlength{\textheight}{9in}
\setlength{\textheight}{9.2in} % Predrag steretched it a bit
%\setlength{\topmargin}{\extramargin}
%\setlength{\topmargin}{1.6cm}
\setlength{\topmargin}{-0.2cm} % Needed to move it u, do not know why
\addtolength{\textheight}{-2\extramargin}
%% uncomment the next three lines to get more space
%%
\setlength{\headheight}{0pt}
\setlength{\headsep}{0pt}
\setlength{\footskip}{2\baselineskip}
%%
%% comment out the next three lines to get more space
%%
\addtolength{\textheight}{-\headheight}
%%\addtolength{\textheight}{-\headsep}
\addtolength{\textheight}{-\footskip}
%%\setlength{\headsep}{\baselineskip}

\parindent 6mm
% \parskip 3.5mm
\parskip 0.5mm

\renewcommand{\arraystretch}{0.9}
\renewcommand{\topfraction}{1}
\renewcommand{\bottomfraction}{1}
\renewcommand{\textfraction}{0}
\bibliographystyle{unsrt}

% to fit the NSF straghtjacket:
% \renewcommand{\thesubsection}{\Alph{subsection}}
%\renewcommand{\thesubsection}{\thesection\Alph{subsection}}
\newenvironment{NSFlist}[1]%%
{\begin{list}{$\bullet$}%%
{\topsep=1pt\parsep=0pt\itemsep=#1\leftmargin=3.0em}}{\end{list}}

%%%%%%%%%%%%%% SPECIFIC TO nsf03 proposal %%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\inertM}{{\mathcal M}}          % inertial manifold

\newcommand{\CNS}{Center for Nonlinear Science}
\newcommand{\NS}{Nonlinear Science}
\newcommand{\CCMST}{Center for Computational Molecular Science \& Technology}

\newcommand{\GT}{Georgia Institute of Technology}
\newcommand{\COS}{College of Sciences}
\newcommand{\COE}{College of Engineering}
\newcommand{\AsE}{School of Aerospace Engineering}
\newcommand{\Phys}{School of Physics}
\newcommand{\Math}{School of Mathematics}
\newcommand{\Bio}{School of Biology}
\newcommand{\Chem}{School of Chemistry and Biochemistry}
\newcommand{\ME}{School of Mechanical Engineering}
\newcommand{\BME}{Georgia Tech/Emory Biomedical Engineering Department}
\newcommand{\CE}{School of Civil \&\ Environmental Engineering}
\newcommand{\ECE}{School of  Electrical \&\ Computer Engineering}


    % while editing, keep these
% \newcommand{\fix}[1]          % purple text = needs to be rewritten
%              {{\color{magenta} #1 }}
    %

%%%%%%%%%%%% SHORTCUTS, loop.tex specific %%%%%%%%%%%%%

\newcommand{\Loop}{L}
\newcommand{\costF}{F^2}  % cost function, as in crete03.tex
% Lan's definition was \newcommand{\costF}{F}   % cost function
\newcommand{\lSpace}{\tilde{x}}         % a point on a loop
\newcommand{\lVeloc}{\tilde{v}}
\newcommand{\pSpaceDer}[1]{x^{(#1)}}
\newcommand{\lSpaceDer}[1]{\tilde{x}^{(#1)}}

%%%%%%%%%%%%  taken from defCrete.tex %%%%%%%%%%%%%%%%%%%%%%

\newcommand{\descent}{Newton descent}
\newcommand{\Descent}{Newton Descent}
\newcommand{\costFct}{cost function}    % functional to minimize
\newcommand{\pVeloc}{v}         % loop tangent
\newcommand{\unit}{{\bf 1}}
\newcommand{\Mvar}{{A}}            % matrix of variations
\def\Journal#1#2#3#4{{#1} {\bf #2}, #3 (#4)}

%%%%%%%%%%%%%% SPECIFIC TO hk.tex ARTICLE %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\Lnoise}[1]{{\mathcal L}_{#1}}   % noisy evolution operator
\newcommand{\Lmat}[1]{{{\bf L}_{#1}}}      % evolution matrix
\newcommand{\Bmat}[1]{{{\bf B}_{#1}}}      % perturbative evolution matrix
\newcommand{\Prpgtr}[1]{\Delta_{#1}}
\newcommand{\InvPrpgtr}[1]{\Delta^{-1}_{#1}}
\newcommand{\CrlMat}[1]{C_{#1}}
\newcommand{\InvCrlMat}[1]{C^{-1}_{#1}}
\newcommand{\field}{\phi}
\newcommand{\Df}[1]{f^{'}_{#1}}
\newcommand{\Shift}{S}
\newcommand{\mod}[1]{#1}
% \newcommand{\mod}[1]{\dot{#1}} % Dettmann's calculation
\newcommand{\const}[1]{\hat{#1}}
\newcommand{\nosum}{\not\!\!{\scriptstyle\sum}}
% \newcommand{\etal}{{\em et al.}}      % etcetera in italics


%%%%%%%%%%%%%%% REFERENCING EQUATIONS ETC. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\rf}     [1] {~\cite{#1}}
\newcommand{\refref} [1] {ref.~\cite{#1}}
\newcommand{\refrefs}[1] {refs.~\cite{#1}}
\newcommand{\Refrefs}[1] {Refs.~\cite{#1}}
\newcommand{\refeq}  [1] {(\ref{#1})}
\newcommand{\refeqs} [2]{(\ref{#1}--\ref{#2})}
\newcommand{\refpage}[1] {page~\pageref{#1}}
\newcommand{\reffig} [1] {figure~\ref{#1}}
\newcommand{\reffigs} [2] {figures~\ref{#1} and~\ref{#2}}
\newcommand{\refFig} [1] {Figure~\ref{#1}}
\newcommand{\reftab} [1] {table~\ref{#1}}
\newcommand{\refTab} [1] {Table~\ref{#1}}
\newcommand{\reftabs}[2] {tables~\ref{#1} and~\ref{#2}}
\newcommand{\refsect}[1] {sect.~\ref{#1}}
\newcommand{\refsects}[2] {sects.~\ref{#1} and \ref{#2}}
\newcommand{\refSect}[1] {Sect.~\ref{#1}}
\newcommand{\refappe}[1] {appendix~\ref{#1}}
\newcommand{\refappes}[2] {appendices~\ref{#1}--\ref{#2}}
\newcommand{\refAppe}[1] {Appendix~\ref{#1}}

%%%%%%%%%%%%%%% EQUATIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\beq}{\begin{equation}}
\newcommand{\continue}{\nonumber \\ }
\newcommand{\nnu}{\nonumber}
\newcommand{\eeq}{\end{equation}}
\newcommand{\ee}[1] {\label{#1} \end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\ceq}{\nonumber \\ & & }
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\barr}{\begin{array}}
\newcommand{\earr}{\end{array}}

%%%%%%%%%%%%%%% VECTORS, MATRICES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\MatrixII}[4]{
   \pmatrix{ {#1}  &  {#2} \cr
             {#3}  &  {#4} \cr} }

\newcommand{\MatrixIII}[9]{
   \pmatrix{ {#1}  &  {#2} &  {#3} \cr
             {#4}  &  {#5} &  {#6} \cr
             {#7}  &  {#8} &  {#9} \cr} }

\newcommand{\transpVectorII}[2]{
   \pmatrix{ {#1}  &  {#2}  \cr} }

\newcommand{\VectorII}[2]{
   \pmatrix{ {#1} \cr
             {#2} \cr} }

\newcommand{\VectorIII}[3]{
   \pmatrix{ {#1} \cr
             {#2} \cr
             {#3} \cr} }

\newcommand{\combinatorial}[2]{ {#1 \choose #2}}

%% birdtracks:
\newcommand{\btrack}[1]{\raisebox{-2.0ex}[3.5ex][2.5ex]
        {\includegraphics[height=5ex]{figs/#1.eps}\negthinspace} }
\newcommand{\btrackA}[1]{\raisebox{-3.0ex}[4.5ex][3.5ex]
         {\includegraphics[height=7ex]{figs/#1.eps}\negthinspace} }
\newcommand{\btrackB}[1]{\raisebox{-4.0ex}[5.5ex][4.5ex]
          {\includegraphics[height=9ex]{figs/#1.eps}\negthinspace} }

\newlength{\Fsize}   % allow for easy resizing of diagrams
\setlength{\Fsize}{20pt}

%%%%%%%%%%%%%%  Abbreviations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\etc}{{\em etc}}        % etcetera in italics
\newcommand{\ie}{{that is}}     % use Latin or English?  Decide later.
\newcommand{\cf}{{\em cf.}}

%%%%%%%%%%%%%%% Evolution operators, zetas %%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\evOper}{evolution oper\-ator}
\newcommand{\EvOper}{Evolution oper\-ator}
\newcommand{\FPoper}{Perron-Frobenius oper\-ator} % Pesin's ordering
\newcommand{\FP}{Perron-Frobenius}
\newcommand{\jacobian}{Jacobian}        % determinant
\newcommand{\jacobianM}{Jacobian matrix}    % matrix
\newcommand{\jacobianMs}{Jacobian matrices} % matrices
\newcommand{\dzeta}{dyn\-am\-ic\-al zeta func\-tion}
\newcommand{\Dzeta}{Dyn\-am\-ic\-al zeta func\-tion}
\newcommand{\tzeta}{top\-o\-lo\-gi\-cal zeta func\-tion}
\newcommand{\Tzeta}{Top\-o\-lo\-gi\-cal zeta func\-tion}
\newcommand{\qS}{semi\-classical zeta func\-tion}
\newcommand{\Gt}{Gutz\-willer trace formula}
\newcommand{\Fd}{spec\-tral det\-er\-min\-ant}
\newcommand{\fd}{spec\-tral det\-er\-min\-ant}
\newcommand{\FD}{Spec\-tral det\-er\-min\-ant}
\newcommand{\cFd}{class\-ic\-al spec\-tral det\-er\-mi\-nant}
\newcommand{\cycForm}{cycle averaging formula}
\newcommand{\CycForm}{Cycle averaging formula}

%%%%%%%%%%%%%%% Sundry symbols within math eviron.: %%%%%%%%%%%%
\newcommand{\DiffConst}{D}
\newcommand{\obser}{a}      % an observable from phase space to R^n
\newcommand{\Obser}{A}      % time integral of an observable
\newcommand{\onefun}{\iota} % the function that returns one no matter what
\newcommand{\defeq}{=}      % the different equal for a definition
\newcommand {\deff}{\stackrel{\rm def}{=}}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\integers}{\mathbb{Z}}
\newcommand{\rationals}{\mathbb{Q}}
\newcommand{\naturals}{\mathbb{N}}
\newcommand{\LieD}{{{\mathcal L}\!\!\llap{-}\,\,}}  % {{\pound}} % Lie Derivative
% \newcommand{\half}{{\scriptstyle{1\over2}}}
\newcommand{\pde}{\partial}
\renewcommand\Im{{\rm Im\,}}
\renewcommand\Re{{\rm Re\,}}
\renewcommand{\det}{\mbox{\rm det}}
\newcommand{\Det}{\mbox{\rm Det}\, }
\newcommand{\tr}{{\rm tr}\, }
\newcommand{\Tr}{\mbox{Tr}\,}
\newcommand{\sign}[1]{{\rm sign}(#1)}
\newcommand{\msr}{{\mu}}                % measure
\newcommand{\vol}{{V}}                  % volume of i-th tile
\newcommand{\prpgtr}[1]{\delta\negthinspace\left( {#1} \right)}
\newcommand{\Zqm}{Z_{qm}}           % Gutz-Voros zeta function
\newcommand{\Fqm}{F_{qm}}
\newcommand{\zfct}[1]{\zeta ^{-1}_{#1}}
\newcommand{\zetatop}{1/\zeta_{\mbox{\footnotesize top}}}
\newcommand{\expct}    [1]{\left\langle {#1} \right\rangle}
\newcommand{\spaceAver}[1]{\left\langle {#1} \right\rangle}
\newcommand{\timeAver} [1]{\overline{#1}}
\newcommand{\pS}{{\mathcal M}}          % symbol for phase space
\newcommand{\intM}[1]{{\int_\pS{\!d #1}\:}} %phasespace integral
\newcommand{\PoincS}{{\mathcal P}}          % symbol for Poincare section
\newcommand{\Lop}{{\mathcal L}}    % evolution operator
%\newcommand{\Lop}{{\bf L}}    % evolution operator
\newcommand{\Uop}{{\mathcal U}}    % Koopman operator
\newcommand{\Aop}{{\mathcal A}}    % evolution generator
\newcommand{\Top}{{\mathcal T}}    % transfer operator, like in statmech
\newcommand{\jMps}{{\bf J}}    %jacobiam matrix, full phase space
\newcommand{\oneMinJ}[1]{\left|\det\left({\bf 1}-{\bf J}_p^{#1}\right)\right|}
                   %Fredholm det jacobian weight
\newcommand{\monodromy}{{\bf J}}   %monodromy matrix, full Poincare cut
\newcommand{\maslov}[1]{{m_{#1}}}  %Maslov index
\newcommand{\ExpaEig}{\Lambda}
%\newcommand{\eigenvL}{{s}}
\newcommand{\eigenvL}{{\nu}}       %This is really eigenvalue, not log!
\newcommand{\eigCond}{F}           % eigenvalue cond. function
\newcommand{\inFix}[1]{{\in \mbox{\footnotesize Fix}f^{#1}}}
\newcommand{\inZero}[1]{{\in \mbox{\footnotesize Zero} \, f^{#1} }}
\newcommand{\xzero}[1]{{x_{#1}^{*}}}

\newcommand{\eps} {\varepsilon}
\newcommand{\twopi}{{2\pi}}
\newcommand{\avert}[1]{\overline{#1}}
\newcommand{\averx}{\int dx\,}
\newcommand{\derJ}{{d~\over dJ}}

%%%%%%%%%% flows: %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand\flow[2]{{f^{#1}(#2)}}
\newcommand\xInit{{x_0}}        %initial x
\newcommand\pSpace{x}       % phase space x=(q,p) coordinate
\newcommand{\para}{\parallel}

%%%%%%%%%%%%%%%% periods: %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand\period[1]{{T_{#1}}}         %continuous cycle period
\newcommand{\cl}[1]{{n_{#1}}}   % discrete length of a cycle, Predrag
\newcommand{\timeSegm}[1]{{\tau_{#1}}}      %billiard segment time period
\newcommand\Lyap{\lambda}           %Lyapunov exponent
\newcommand{\dz}{{\delta z}}

%%%%%%%%%%%%%%% symbolic dynamics %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\MarkGraph}{Markov graph}
\newcommand{\admissible}{admissible}
\newcommand{\inadmissible}{inadmissible}
\newcommand{\cycle}[1]{{$\overline{#1}$}}
\newcommand{\cycpt}{_{p,m}}
\newcommand\sumprime{\mathop{{\sum}'}}
\newcommand{\pseudos}{{p_1+p_2+\dots+p_k}}
\newcommand{\block}[1]{#1}
\newcommand{\Ksym}[1]{\sigma_{#1}}
\newcommand{\Ssym}[1]{{s_{#1}}}


%\newcommand{\F}{{\mathcal F}}
% \newcommand{\spao}[1]{\mbox{\hspace{#1}}}
% \newcommand{\spav}[1]{\parbox{1mm}{\vspace*{#1}}}
% \newcommand{\TTeject}{\vspace{1cm}}
%\newcommand{\sign}[1]{{\rm sign}[{#1}]}
    %sign redefined to have square brackets here...


%%%%%%%%%%%%%%%%%%%%%% COMMENTS IN THE TEXT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifthenelse{\boolean{display_comments}}
{       % if display_comments {true}, display comments in text
  \newcommand{\Preliminary}[1]
         {\color{magenta}
          \marginpar{$\Downarrow${\color{magenta}\footnotesize PRELIMINARY}}
           {#1}
              \marginpar{$\Uparrow${\color{magenta}\footnotesize PRELIMINARY}}
          \color{black}
         }
\newcommand{\PC}[1]{$\footnotemark\footnotetext{PC: #1}$}
\newcommand{\GrantDescr}[1]{$\footnotemark\footnotetext{McDonnell requires: #1}$}
\newcommand{\file}[1]{$\footnotemark\footnotetext{{\bf file} #1}$}
\newcommand{\Marsha}[1]{\marginpar{\footnotesize Marsha: #1}}
\newcommand{\YL}[1]{$\footnotemark\footnotetext{YL: #1}$}
}{      % if display_comments {false}, drop comments
\newcommand{\Preliminary}[1]{}
\newcommand{\PublicPrivate}[2]{#1}
\newcommand{\PC}[1]{}
\newcommand{\Marsha}[1]{}
\newcommand{\YL}[1]{}
\newcommand{\GrantDescr}[1]{}
\newcommand{\file}[1]{}
}

\renewcommand{\topfraction}{1}
\renewcommand{\bottomfraction}{1}
\renewcommand{\textfraction}{0}

\newcommand{\course}[4]
     {\item[#1] {{\bf #3} \hfill #2\\}{#4}}

%%%%%%%%%%%%%%%%% FIGURE MACROS     %%%%%%%%%%%%%%%%%%%%%%%%%%%%
% from DasBuch def.tex
% last edit             Predrag          2dec2001

%   \FIG{#1}    % \includegraphics[width=0.40\textwidth]{figs/f_name.ps}
%   {#2}    % short caption text
%   {#3}    % full caption text
%   {#4}    % f-figure-label
%       defined here:
\newcommand{\FIG}[4]{\begin{figure}
              %\hspace*{0.10\textwidth}%
              \begin{minipage}[b]{1.00\textwidth}
              \noindent{#1}
              %\centering{#1}
                      \caption[#2]{#3}
                      \label{#4}
              \end{minipage}
              \end{figure} }


%   \BFIG{#1}   % width=#1\textwidth
%        {#2}   % f_name.ps
%    {#3}   % short caption text
%    {#4}   % full caption text
%    {#5}   % f-figure-label
%       defined here:
\newcommand{\BFIG}[5]{\begin{figure}
              %\hspace*{0.10\textwidth}%
              \begin{minipage}[b]{1.00\textwidth}
              \centering{
                      \includegraphics[width=#1\textwidth]{figs/#2}
                     }
                      \caption[#3]{#4}
                      \label{#5}
              \end{minipage}
              \end{figure} }

%  \SFIG{#1}    % f_name.eps
%       {#2}    % short caption text
%       {#3}    % full caption text
%       {#4}    % f-figure-label
\newcommand{\SFIG}[4]{\begin{figure}[!]
              %\hspace*{-0.10\textwidth}
              %\hspace*{0.10\textwidth}
              \begin{minipage}[b]{0.45\textwidth}
                      \caption[#2]{#3}
                      \label{#4}
              \end{minipage}~~~~~%
              \begin{minipage}[b]{0.40\textwidth}
                      \includegraphics[width=1.00\textwidth]{figs/#1}
              \end{minipage}
              %\hfill
              \end{figure} }
