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Ian Jauslin
2018-01-11 22:48:14 +00:00
commit 01f47ace67
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70
docs/nstrophy_doc/Makefile Normal file
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PROJECTNAME=$(basename $(wildcard *.tex))
LIBS=$(notdir $(wildcard libs/*))
FIGS=$(notdir $(wildcard figs/*.fig))
PDFS=$(addsuffix .pdf, $(PROJECTNAME))
SYNCTEXS=$(addsuffix .synctex.gz, $(PROJECTNAME))
all: $(PROJECTNAME)
$(PROJECTNAME): $(LIBS) $(FIGS)
pdflatex -file-line-error $@.tex
pdflatex -file-line-error $@.tex
pdflatex -synctex=1 $@.tex
$(PROJECTNAME).aux: $(LIBS) $(FIGS)
pdflatex -file-line-error -draftmode $(PROJECTNAME).tex
$(SYNCTEXS): $(LIBS) $(FIGS)
pdflatex -synctex=1 $(patsubst %.synctex.gz, %.tex, $@)
install: $(PROJECTNAME)
cp $^.pdf $(INSTALLDIR)/
install-synctex: $(SYNCTEXS)
cp $^ $(INSTALLDIR)/
install-bibliography: bibliography/bibliography.tex
cp bibliography/bibliography.tex $(INSTALLDIR)/bibliography/
libs: $(LIBS)
$(LIBS):
ln -fs libs/$@ ./
bibliography/bibliography.tex: $(PROJECTNAME).aux
BBlog -c bibliography/conf.BBlog -d $(BIBLIOGRAPHY) -b bibliography/bibliography.tex
figs: $(FIGS)
$(FIGS):
make -C figs/$@
ln -fs figs/$@/*.pdf ./
clean-aux: clean-figs-aux
rm -f $(addsuffix .aux, $(PROJECTNAME))
rm -f $(addsuffix .log, $(PROJECTNAME))
rm -f $(addsuffix .out, $(PROJECTNAME))
rm -f $(addsuffix .toc, $(PROJECTNAME))
clean-libs:
rm -f $(LIBS)
clean-figs:
$(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean; )
rm -f $(notdir $(wildcard figs/*.fig/*.pdf))
clean-figs-aux:
$(foreach fig,$(addprefix figs/, $(FIGS)), make -C $(fig) clean-aux; )
clean-tex:
rm -f $(PDFS) $(SYNCTEXS)
clean-bibliography:
rm -f bibliography/bibliography.tex
clean: clean-aux clean-tex clean-libs clean-figs

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docs/nstrophy_doc/bibliography/conf.BBlog Normal file
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format: \bibitem[%token%]{%citeref%}%author:auth% - {\it %title%}, %journal%, %year%|doi?,\par\penalty10000%n%doi:{\tt\color{blue}\href{http://dx.doi.org/%doi%}{%doi%}}||arxiv?, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/%arxiv%}{%arxiv%}}|.\par\medskip%n% %n%
out_file: bibliography.tex
filter:auth: s/([A-Z])[^, ]* /\1. /g; s/ ([^ ,]*),/_\1,_/g; s/ ([^ ,]*)$/_\1/g; s/ //g; s/_/ /g;
filter:journal: s/([a-zA-Z]) ([0-9]+)/\1~\\-\2/g;
aux_cmd: \\citation{

659
docs/nstrophy_doc/libs/ian.cls Normal file
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%%
%% Ian's class file
%%
%% TeX format
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
%% class name
\ProvidesClass{ian}[2017/09/29]
%% boolean to signal that this class is being used
\newif\ifianclass
%% options
% no section numbering in equations
\DeclareOption{section_in_eq}{\sectionsineqtrue}
\DeclareOption{section_in_fig}{\sectionsinfigtrue}
\DeclareOption{section_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}}
\DeclareOption{section_in_all}{\sectionsineqtrue\sectionsinfigtrue\PassOptionsToPackage{section_in_theo}{iantheo}}
\DeclareOption{subsection_in_eq}{\subsectionsineqtrue}
\DeclareOption{subsection_in_fig}{\subsectionsinfigtrue}
\DeclareOption{subsection_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}}
\DeclareOption{subsection_in_all}{\subsectionsineqtrue\subsectionsinfigtrue\PassOptionsToPackage{subsection_in_theo}{iantheo}}
\DeclareOption{no_section_in_eq}{\sectionsineqfalse}
\DeclareOption{no_section_in_fig}{\sectionsinfigfalse}
\DeclareOption{no_section_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}}
\DeclareOption{no_section_in_all}{\sectionsineqfalse\sectionsinfigfalse\PassOptionsToPackage{no_section_in_theo}{iantheo}}
\DeclareOption{no_subsection_in_eq}{\subsectionsineqfalse}
\DeclareOption{no_subsection_in_fig}{\subsectionsinfigfalse}
\DeclareOption{no_subsection_in_theo}{\PassOptionsToPackage{\CurrentOption}{iantheo}}
\DeclareOption{no_subsection_in_all}{\subsectionsineqfalse\subsectionsinfigfalse\PassOptionsToPackage{no_subsection_in_theo}{iantheo}}
% reset point
\DeclareOption{point_reset_at_section}{\PassOptionsToPackage{reset_at_section}{point}}
\DeclareOption{point_no_reset_at_section}{\PassOptionsToPackage{no_reset_at_section}{point}}
\DeclareOption{point_reset_at_theo}{\PassOptionsToPackage{reset_at_theo}{point}}
\DeclareOption{point_no_reset_at_theo}{\PassOptionsToPackage{no_reset_at_theo}{point}}
\def\ian@defaultoptions{
\ExecuteOptions{section_in_all, no_subsection_in_all}
\ProcessOptions
%% required packages
\RequirePackage{iantheo}
\RequirePackage{point}
\RequirePackage{color}
\RequirePackage{marginnote}
\RequirePackage{amssymb}
\PassOptionsToPackage{hidelinks}{hyperref}
\RequirePackage{hyperref}
}
%% paper dimensions
\setlength\paperheight{297mm}
\setlength\paperwidth{210mm}
%% fonts
\input{size11.clo}
\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
%% text dimensions
\hoffset=-50pt
\voffset=-72pt
\textwidth=460pt
\textheight=704pt
%% remove default indentation
\parindent=0pt
%% indent command
\def\indent{\hskip20pt}
%% something is wrong with \thepage, redefine it
\gdef\thepage{\the\c@page}
%% array lines (to use the array environment)
\setlength\arraycolsep{5\p@}
\setlength\arrayrulewidth{.4\p@}
%% correct vertical alignment at the end of a document
\AtEndDocument{
\vfill
\eject
}
%% hyperlinks
% hyperlinkcounter
\newcounter{lncount}
% hyperref anchor
\def\hrefanchor{%
\stepcounter{lncount}%
\hypertarget{ln.\thelncount}{}%
}
%% define a command and write it to aux file
\def\outdef#1#2{%
% define command%
\expandafter\xdef\csname #1\endcsname{#2}%
% hyperlink number%
\expandafter\xdef\csname #1@hl\endcsname{\thelncount}%
% write command to aux%
\immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1\endcsname{\csname #1\endcsname}}%
\immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1@hl\endcsname{\thelncount}}%
}
%% can call commands even when they are not defined
\def\safe#1{%
\ifdefined#1%
#1%
\else%
{\color{red}\bf?}%
\fi%
}
%% define a label for the latest tag
%% label defines a command containing the string stored in \tag
\def\deflabel{
\def\label##1{\expandafter\outdef{label@##1}{\safe\tag}}
\def\ref##1{%
% check whether the label is defined (hyperlink runs into errors if this check is omitted)
\ifcsname label@##1@hl\endcsname%
\hyperlink{ln.\csname label@##1@hl\endcsname}{{\color{blue}\safe\csname label@##1\endcsname}}%
\else%
\ifcsname label@##1\endcsname%
{\color{blue}\csname ##1\endcsname}%
\else%
{\bf ??}%
\fi%
\fi%
}
}
%% make a custom link at any given location in the document
\def\makelink#1#2{%
\hrefanchor%
\outdef{label@#1}{#2}%
}
%% section command
% counter
\newcounter{sectioncount}
% space before section
\newlength\secskip
\setlength\secskip{40pt}
% a prefix to put before the section number, e.g. A for appendices
\def\sectionprefix{}
% define some lengths
\newlength\secnumwidth
\newlength\sectitlewidth
\def\section#1{
% reset counters
\stepcounter{sectioncount}
\setcounter{subsectioncount}{0}
\ifsectionsineq
\setcounter{seqcount}0
\fi
\ifsectionsinfig
\setcounter{figcount}0
\fi
% space before section (if not first)
\ifnum\thesectioncount>1
\vskip\secskip
\penalty-1000
\fi
% hyperref anchor
\hrefanchor
% define tag (for \label)
\xdef\tag{\sectionprefix\thesectioncount}
% get widths
\def\@secnum{{\bf\Large\sectionprefix\thesectioncount.\hskip10pt}}
\settowidth\secnumwidth{\@secnum}
\setlength\sectitlewidth\textwidth
\addtolength\sectitlewidth{-\secnumwidth}
% print name
\parbox{\textwidth}{
\@secnum
\parbox[t]{\sectitlewidth}{\Large\bf #1}}
% write to table of contents
\iftoc
% save lncount in aux variable which is written to toc
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@sec.\thesectioncount\endcsname{\thelncount}}
\write\tocoutput{\noexpand\tocsection{#1}{\thepage}}
\fi
%space
\par\penalty10000
\bigskip\penalty10000
}
%% subsection
% counter
\newcounter{subsectioncount}
% space before subsection
\newlength\subsecskip
\setlength\subsecskip{30pt}
\def\subsection#1{
% counters
\stepcounter{subsectioncount}
\setcounter{subsubsectioncount}{0}
\ifsubsectionsineq
\setcounter{seqcount}0
\fi
\ifsubsectionsinfig
\setcounter{figcount}0
\fi
% space before subsection (if not first)
\ifnum\thesubsectioncount>1
\vskip\subsecskip
\penalty-500
\fi
% hyperref anchor
\hrefanchor
% define tag (for \label)
\xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount}
% get widths
\def\@secnum{{\bf\large\hskip.5cm\sectionprefix\thesectioncount.\thesubsectioncount.\hskip5pt}}
\settowidth\secnumwidth{\@secnum}
\setlength\sectitlewidth\textwidth
\addtolength\sectitlewidth{-\secnumwidth}
% print name
\parbox{\textwidth}{
\@secnum
\parbox[t]{\sectitlewidth}{\large\bf #1}}
% write to table of contents
\iftoc
% save lncount in aux variable which is written to toc
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsec.\thesectioncount.\thesubsectioncount\endcsname{\thelncount}}
\write\tocoutput{\noexpand\tocsubsection{#1}{\thepage}}
\fi
% space
\par\penalty10000
\medskip\penalty10000
}
%% subsubsection
% counter
\newcounter{subsubsectioncount}
% space before subsubsection
\newlength\subsubsecskip
\setlength\subsubsecskip{20pt}
\def\subsubsection#1{
% counters
\stepcounter{subsubsectioncount}
% space before subsubsection (if not first)
\ifnum\thesubsubsectioncount>1
\vskip\subsubsecskip
\penalty-500
\fi
% hyperref anchor
\hrefanchor
% define tag (for \label)
\xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount.\thesubsubsectioncount}
% get widths
\def\@secnum{{\bf\hskip1.cm\sectionprefix\thesectioncount.\thesubsectioncount.\thesubsubsectioncount.\hskip5pt}}
\settowidth\secnumwidth{\@secnum}
\setlength\sectitlewidth\textwidth
\addtolength\sectitlewidth{-\secnumwidth}
% print name
\parbox{\textwidth}{
\@secnum
\parbox[t]{\sectitlewidth}{\large\bf #1}}
% write to table of contents
\iftoc
% save lncount in aux variable which is written to toc
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsubsec.\thesectioncount.\thesubsectioncount.\thesubsubsectioncount\endcsname{\thelncount}}
\write\tocoutput{\noexpand\tocsubsubsection{#1}{\thepage}}
\fi
% space
\par\penalty10000
\medskip\penalty10000
}
%% itemize
\newlength\itemizeskip
% left margin for items
\setlength\itemizeskip{20pt}
\newlength\itemizeseparator
% space between the item symbol and the text
\setlength\itemizeseparator{5pt}
% penalty preceding an itemize
\newcount\itemizepenalty
\itemizepenalty=0
% counter counting the itemize level
\newcounter{itemizecount}
% item symbol
\def\itemizept#1{
\ifnum#1=1
\textbullet
\else
$\scriptstyle\blacktriangleright$
\fi
}
\newlength\current@itemizeskip
\setlength\current@itemizeskip{0pt}
\def\itemize{%
\par\expandafter\penalty\the\itemizepenalty\medskip\expandafter\penalty\the\itemizepenalty%
\addtocounter{itemizecount}{1}%
\addtolength\current@itemizeskip{\itemizeskip}%
\leftskip\current@itemizeskip%
}
\def\enditemize{%
\addtocounter{itemizecount}{-1}%
\addtolength\current@itemizeskip{-\itemizeskip}%
\par\expandafter\penalty\the\itemizepenalty\leftskip\current@itemizeskip%
\medskip\expandafter\penalty\the\itemizepenalty%
}
\newlength\itempt@total
\def\item{
\settowidth\itempt@total{\itemizept\theitemizecount}
\addtolength\itempt@total{\itemizeseparator}
\par
\medskip
\hskip-\itempt@total\itemizept\theitemizecount\hskip\itemizeseparator
}
%% prevent page breaks after itemize
\newcount\previtemizepenalty
\def\nopagebreakafteritemize{
\previtemizepenalty=\itemizepenalty
\itemizepenalty=10000
}
%% back to previous value
\def\restorepagebreakafteritemize{
\itemizepenalty=\previtemizepenalty
}
%% enumerate
\newcounter{enumerate@count}
\def\enumerate{
\setcounter{enumerate@count}0
\let\olditem\item
\let\olditemizept\itemizept
\def\item{
% counter
\stepcounter{enumerate@count}
% set header
\def\itemizept{\theenumerate@count.}
% hyperref anchor
\hrefanchor
% define tag (for \label)
\xdef\tag{\theenumerate@count}
\olditem
}
\itemize
}
\def\endenumerate{
\enditemize
\let\item\olditem
\let\itemizept\olditemizept
}
%% equation numbering
% counter
\newcounter{seqcount}
% booleans (write section or subsection in equation number)
\newif\ifsectionsineq
\newif\ifsubsectionsineq
\def\seqcount{
\stepcounter{seqcount}
% the output
\edef\seqformat{\theseqcount}
% add subsection number
\ifsubsectionsineq
\let\tmp\seqformat
\edef\seqformat{\thesubsectioncount.\tmp}
\fi
% add section number
\ifsectionsineq
\let\tmp\seqformat
\edef\seqformat{\sectionprefix\thesectioncount.\tmp}
\fi
% define tag (for \label)
\xdef\tag{\seqformat}
% write number
\marginnote{\hfill(\seqformat)}
}
%% equation environment compatibility
\def\equation{\hrefanchor$$\seqcount}
\def\endequation{$$\@ignoretrue}
%% figures
% counter
\newcounter{figcount}
% booleans (write section or subsection in equation number)
\newif\ifsectionsinfig
\newif\ifsubsectionsinfig
% width of figures
\newlength\figwidth
\setlength\figwidth\textwidth
\addtolength\figwidth{-2.5cm}
% caption
\def\defcaption{
\long\def\caption##1{
\stepcounter{figcount}
% hyperref anchor
\hrefanchor
% the number of the figure
\edef\figformat{\thefigcount}
% add subsection number
\ifsubsectionsinfig
\let\tmp\figformat
\edef\figformat{\thesubsectioncount.\tmp}
\fi
% add section number
\ifsectionsinfig
\let\tmp\figformat
\edef\figformat{\sectionprefix\thesectioncount.\tmp}
\fi
% define tag (for \label)
\xdef\tag{\figformat}
% write
\hfil fig \figformat: \parbox[t]{\figwidth}{\leavevmode\small##1}
% space
\par\bigskip
}
}
%% short caption: centered
\def\captionshort#1{
\stepcounter{figcount}
% hyperref anchor
\hrefanchor
% the number of the figure
\edef\figformat{\thefigcount}
% add section number
\ifsectionsinfig
\let\tmp\figformat
\edef\figformat{\sectionprefix\thesectioncount.\tmp}
\fi
% define tag (for \label)
\xdef\tag{\figformat}
% write
\hfil fig \figformat: {\small#1}
%space
\par\bigskip
}
%% environment
\def\figure{
\par
\vfil\penalty100\vfilneg
\bigskip
}
\def\endfigure{
\par
\vfil\penalty100\vfilneg
\bigskip
}
%% start appendices
\def\appendix{
\vfill
\pagebreak
% counter
\setcounter{sectioncount}0
% prefix
\def\sectionprefix{A}
% write
{\bf \LARGE Appendices}\par\penalty10000\bigskip\penalty10000
% add a mention in the table of contents
\iftoc
\immediate\write\tocoutput{\noexpand\tocappendices}\penalty10000
\fi
%% uncomment for new page for each appendix
%\def\seqskip{\vfill\pagebreak}
}
%% bibliography
% size of header
\newlength\bibheader
\def\thebibliography#1{
\hrefanchor
% add a mention in the table of contents
\iftoc
% save lncount in aux variable which is written to toc
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@references\endcsname{\thelncount}}
\write\tocoutput{\noexpand\tocreferences{\thepage}}\penalty10000
\fi
% write
{\bf \LARGE References}\par\penalty10000\bigskip\penalty10000
% width of header
\settowidth\bibheader{[#1]}
\leftskip\bibheader
}
% end environment
\def\endthebibliography{
\par\leftskip0pt
}
%% bibitem command
\def\bibitem[#1]#2{%
\hrefanchor%
\outdef{label@cite#2}{#1}%
\hskip-\bibheader%
\makebox[\bibheader]{\cite{#2}\hfill}%
}
%% cite command (adapted from latex.ltx)
% @tempswa is set to true when there is an optional argument
\newif\@tempswa
\def\cite{%
% check whether there is an optional argument (if there is none, add on empty '[]')
\@ifnextchar [{\@tempswatrue\@citex}{\@tempswafalse\@citex[]}%
}
% command with optional argument
\def\@citex[#1]#2{\leavevmode%
% initialize loop
\let\@citea\@empty%
% format
\@cite{%
% loop over ',' separated list
\@for\@citeb:=#2\do{%
% text to add at each iteration of the loop (separator between citations)
\@citea\def\@citea{,\ }%
% add entry to citelist
\@writecitation{\@citeb}%
\ref{cite\@citeb}%
}%
}%
% add optional argument text (as an argument to '\@cite')
{#1}%
}
\def\@cite#1#2{%
[#1\if@tempswa , #2\fi]%
}
%% add entry to citelist after checking it has not already been added
\def\@writecitation#1{%
\ifcsname if#1cited\endcsname%
\else%
\expandafter\newif\csname if#1cited\endcsname%
\immediate\write\@auxout{\string\citation{#1}}%
\fi%
}
%% table of contents
% boolean
\newif\iftoc
\def\tableofcontents{
{\bf \large Table of contents:}\par\penalty10000\bigskip\penalty10000
% copy content from file
\IfFileExists{\jobname.toc}{\input{\jobname.toc}}{{\tt error: table of contents missing}}
% open new toc
\newwrite\tocoutput
\immediate\openout\tocoutput=\jobname.toc
\toctrue
}
%% close file
\AtEndDocument{
% close toc
\iftoc
\immediate\closeout\tocoutput
\fi
}
%% fill line with dots
\def\leaderfill{\leaders\hbox to 1em {\hss. \hss}\hfill}
%% same as sectionprefix
\def\tocsectionprefix{}
%% toc formats
\newcounter{tocsectioncount}
\def\tocsection #1#2{
\stepcounter{tocsectioncount}
\setcounter{tocsubsectioncount}{0}
\setcounter{tocsubsubsectioncount}{0}
% write
\smallskip\hyperlink{ln.\csname toc@sec.\thetocsectioncount\endcsname}{{\bf \tocsectionprefix\thetocsectioncount}.\hskip5pt {\color{blue}#1}\leaderfill#2}\par
}
\newcounter{tocsubsectioncount}
\def\tocsubsection #1#2{
\stepcounter{tocsubsectioncount}
\setcounter{tocsubsubsectioncount}{0}
% write
{\hskip10pt\hyperlink{ln.\csname toc@subsec.\thetocsectioncount.\thetocsubsectioncount\endcsname}{{\bf \thetocsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par
}
\newcounter{tocsubsubsectioncount}
\def\tocsubsubsection #1#2{
\stepcounter{tocsubsubsectioncount}
% write
{\hskip20pt\hyperlink{ln.\csname toc@subsubsec.\thetocsectioncount.\thetocsubsectioncount.\thetocsubsubsectioncount\endcsname}{{\bf \thetocsubsubsectioncount}.\hskip5pt {\color{blue}\small #1}\leaderfill#2}}\par
}
\def\tocappendices{
\medskip
\setcounter{tocsectioncount}0
{\bf Appendices}\par
\smallskip
\def\tocsectionprefix{A}
}
\def\tocreferences#1{
\medskip
{\hyperlink{ln.\csname toc@references\endcsname}{{\color{blue}\bf References}\leaderfill#1}}\par
\smallskip
}
%% definitions that must be loaded at begin document
\let\ian@olddocument\document
\def\document{
\ian@olddocument
\deflabel
\defcaption
}
%% end
\ian@defaultoptions
\endinput

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%%
%% iantheorem package:
%% Ian's customized theorem command
%%
%% boolean to signal that this package was loaded
\newif\ifiantheo
%% TeX format
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
%% package name
\ProvidesPackage{iantheo}[2016/11/10]
%% options
\newif\ifsectionintheo
\DeclareOption{section_in_theo}{\sectionintheotrue}
\DeclareOption{no_section_in_theo}{\sectionintheofalse}
\newif\ifsubsectionintheo
\DeclareOption{subsection_in_theo}{\subsectionintheotrue}
\DeclareOption{no_subsection_in_theo}{\subsectionintheofalse}
\def\iantheo@defaultoptions{
\ExecuteOptions{section_in_theo, no_subsection_in_theo}
\ProcessOptions
%%% reset at every new section
\ifsectionintheo
\let\iantheo@oldsection\section
\gdef\section{\setcounter{theocount}{0}\iantheo@oldsection}
\fi
%% reset at every new subsection
\ifsubsectionintheo
\let\iantheo@oldsubsection\subsection
\gdef\subsection{\setcounter{theocount}{0}\iantheo@oldsubsection}
\fi
}
%% delimiters
\def\delimtitle#1{
\par%
\leavevmode%
\raise.3em\hbox to\hsize{%
\lower0.3em\hbox{\vrule height0.3em}%
\hrulefill%
\ \lower.3em\hbox{#1}\ %
\hrulefill%
\lower0.3em\hbox{\vrule height0.3em}%
}%
\par\penalty10000%
}
%% callable by ref
\def\delimtitleref#1{
\par%
%
\ifdefined\ianclass%
% hyperref anchor%
\hrefanchor%
% define tag (for \label)%
\xdef\tag{#1}%
\fi%
%
\leavevmode%
\raise.3em\hbox to\hsize{%
\lower0.3em\hbox{\vrule height0.3em}%
\hrulefill%
\ \lower.3em\hbox{\bf #1}\ %
\hrulefill%
\lower0.3em\hbox{\vrule height0.3em}%
}%
\par\penalty10000%
}
%% no title
\def\delim{
\par%
\leavevmode\raise.3em\hbox to\hsize{%
\lower0.3em\hbox{\vrule height0.3em}%
\hrulefill%
\lower0.3em\hbox{\vrule height0.3em}%
}%
\par\penalty10000%
}
%% end delim
\def\enddelim{
\par\penalty10000%
\leavevmode%
\raise.3em\hbox to\hsize{%
\vrule height0.3em\hrulefill\vrule height0.3em%
}%
\par%
}
%% theorem
% counter
\newcounter{theocount}
% booleans (write section or subsection in equation number)
\def\theo#1{
\stepcounter{theocount}
\ifdefined\ianclass
% hyperref anchor
\hrefanchor
\fi
% the number
\def\formattheo{\thetheocount}
% add subsection number
\ifsubsectionintheo
\let\tmp\formattheo
\edef\formattheo{\thesubsectioncount.\tmp}
\fi
% add section number
\ifsectionintheo
\let\tmp\formattheo
\edef\formattheo{\sectionprefix\thesectioncount.\tmp}
\fi
% define tag (for \label)
\xdef\tag{\formattheo}
% write
\delimtitle{\bf #1 \formattheo}
}
\let\endtheo\enddelim
%% theorem headers with name
\def\theoname#1#2{
\theo{#1}\hfil({\it #2})\par\penalty10000\medskip%
}
%% qed symbol
\def\qedsymbol{$\square$}
\def\qed{\penalty10000\hfill\penalty10000\qedsymbol}
%% compatibility with article class
\ifdefined\ianclasstrue
\relax
\else
\def\thesectioncount{\thesection}
\def\thesubsectioncount{\thesubsection}
\def\sectionprefix{}
\fi
%% prevent page breaks after displayed equations
\newcount\prevpostdisplaypenalty
\def\nopagebreakaftereq{
\prevpostdisplaypenalty=\postdisplaypenalty
\postdisplaypenalty=10000
}
%% back to previous value
\def\restorepagebreakaftereq{
\postdisplaypenalty=\prevpostdisplaypenalty
}
%% end
\iantheo@defaultoptions
\endinput

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%%
%% largearray package:
%% Array spanning the entire line
%%
%% TeX format
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
%% package name
\ProvidesPackage{largearray}[2016/11/10]
\RequirePackage{array}
%% array spanning the entire line
\newlength\largearray@width
\setlength\largearray@width\textwidth
\addtolength\largearray@width{-10pt}
\def\largearray{\begin{array}{@{}>{\displaystyle}l@{}}\hphantom{\hspace{\largearray@width}}\\[-.5cm]}
\def\endlargearray{\end{array}}

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%%
%% Points package:
%% \point commands
%%
%% TeX format
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
%% package name
\ProvidesPackage{point}[2017/06/13]
%% options
\newif\ifresetatsection
\DeclareOption{reset_at_section}{\resetatsectiontrue}
\DeclareOption{no_reset_at_section}{\resetatsectionfalse}
\newif\ifresetatsubsection
\DeclareOption{reset_at_subsection}{\resetatsubsectiontrue}
\DeclareOption{no_reset_at_subsection}{\resetatsubsectionfalse}
\newif\ifresetattheo
\DeclareOption{reset_at_theo}{\resetattheotrue}
\DeclareOption{no_reset_at_theo}{\resetattheofalse}
\def\point@defaultoptions{
\ExecuteOptions{reset_at_section, reset_at_subsection, no_reset_at_theo}
\ProcessOptions
%% reset at every new section
\ifresetatsection
\let\point@oldsection\section
\gdef\section{\resetpointcounter\point@oldsection}
\fi
%% reset at every new subsection
\ifresetatsubsection
\let\point@oldsubsection\subsection
\gdef\subsection{\resetpointcounter\point@oldsubsection}
\fi
%% reset at every new theorem
\ifresetattheo
\ifdefined\iantheotrue
\let\point@oldtheo\theo
\gdef\theo{\resetpointcounter\point@oldtheo}
\fi
\fi
}
%% point
% counter
\newcounter{pointcount}
\def\point{
\stepcounter{pointcount}
\setcounter{subpointcount}{0}
% hyperref anchor (only if the class is 'ian')
\ifdefined\ifianclass
\hrefanchor
% define tag (for \label)
\xdef\tag{\thepointcount}
\fi
% header
\indent{\bf \thepointcount\ - }
}
%% subpoint
% counter
\newcounter{subpointcount}
\def\subpoint{
\stepcounter{subpointcount}
\setcounter{subsubpointcount}0
% hyperref anchor (only if the class is 'ian')
\ifdefined\ifianclass
\hrefanchor
% define tag (for \label)
\xdef\tag{\thepointcount-\thesubpointcount}
\fi
% header
\indent\hskip.5cm{\bf \thepointcount-\thesubpointcount\ - }
}
%% subsubpoint
% counter
\newcounter{subsubpointcount}
\def\subsubpoint{
\stepcounter{subsubpointcount}
% hyperref anchor (only if the class is 'ian')
\ifdefined\ifianclass
\hrefanchor
% define tag (for \label)
\xdef\tag{\thepointcount-\thesubpointcount-\thesubsubpointcount}
\fi
\indent\hskip1cm{\bf \thepointcount-\thesubpointcount-\thesubsubpointcount\ - }
}
%% reset point counters
\def\resetpointcounter{
\setcounter{pointcount}{0}
\setcounter{subpointcount}{0}
\setcounter{subsubpointcount}{0}
}
%% end
\point@defaultoptions
\endinput

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\documentclass{ian}
\usepackage{largearray}
\begin{document}
\hbox{}
\hfil{\bf\LARGE
{\tt nstrophy}
}
\vfill
\tableofcontents
\vfill
\eject
\setcounter{page}1
\pagestyle{plain}
\section{Description of the computation}
\subsection{Irreversible equation}
\indent Consider the {\it irreversible} Navier-Stokes equation in 2 dimensions
\begin{equation}
\partial_tu=\nu\Delta u+g-\nabla w-(u\cdot\nabla)u,\quad
\nabla\cdot u=0
\label{ins}
\end{equation}
in which $g$ is the forcing term and $w$ is the pressure.
We take periodic boundary conditions, so, at every given time, $u(t,\cdot)$ is a function on the unit torus $\mathbb T^2:=\mathbb R^2/\mathbb Z^2$. We represent $u(t,\cdot)$ using its Fourier series
\begin{equation}
\hat u_k(t):=\int_{\mathbb T^2}dx\ e^{2i\pi kx}u(t,x)
\end{equation}
for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
\begin{equation}
\partial_t\hat u_k=
-4\pi^2\nu k^2\hat u_k+\hat g_k-2i\pi k\hat w_k
-2i\pi\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
(q\cdot\hat u_p)\hat u_q
,\quad
k\cdot\hat u_k=0
\label{ins_k}
\end{equation}
We then reduce the equation to a scalar one, by writing
\begin{equation}
\hat u_k=\frac{2i\pi k^\perp}{|k|}\hat\varphi_k\equiv\frac{2i\pi}{|k|}(-k_y\hat\varphi_k,k_x\hat\varphi_k)
\end{equation}
in terms of which, multiplying both sides of the equation by $\frac{k^\perp}{|k|}$,
\begin{equation}
\partial_t\hat \varphi_k=
-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
-\frac{2i\pi}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
(q\cdot p^\perp)(k^\perp\cdot q^\perp)\hat\varphi_p\hat\varphi_q
\label{ins_k}
\end{equation}
which, since $q\cdot p^\perp=q\cdot(k^\perp-q^\perp)=q\cdot k^\perp$, is
\begin{equation}
\partial_t\hat \varphi_k=
-4\pi^2\nu k^2\hat \varphi_k+\frac{k^\perp}{2i\pi|k|}\cdot\hat g_k
+\frac{4\pi^2}{|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_p\hat\varphi_q}{|p||q|}
.
\label{ins_k}
\end{equation}
We truncate the Fourier modes and assume that $\hat\varphi_k=0$ if $|k_1|>K_1$ or $|k_2|>K_2$. Let
\begin{equation}
\mathcal K:=\{(k_1,k_2),\ |k_1|\leqslant K_1,\ |k_2|\leqslant K_2\}
.
\end{equation}
\bigskip
\point{\bf FFT}. We compute the last term in~\-(\ref{ins_k})
\begin{equation}
T(\hat\varphi,k):=
\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
\frac{\hat\varphi_p}{|p|}
(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|}
\end{equation}
using a fast Fourier transform, defined as
\begin{equation}
\mathcal F(f)(n):=\sum_{m\in\mathcal N}e^{-\frac{2i\pi}{N_1}m_1n_1-\frac{2i\pi}{N_2}m_2n_2}f(m_1,m_2)
\end{equation}
where
\begin{equation}
\mathcal N:=\{(n_1,n_2),\ 0\leqslant n_1\leqslant N_1,\ 0\leqslant n_2\leqslant N_2\}
\end{equation}
for some fixed $N_1,N_2$. The transform is inverted by
\begin{equation}
\frac1{N_1N_2}\mathcal F^*(\mathcal F(f))(n)=f(n)
\end{equation}
in which $\mathcal F^*$ is defined like $\mathcal F$ but with the opposite phase.
\bigskip
\indent The condition $p+q=k$ can be rewritten as
\begin{equation}
T(\hat\varphi,k)
=
\sum_{p,q\in\mathcal K}
\frac1{N_1N_2}
\sum_{n\in\mathcal N}e^{-\frac{2i\pi}{N_1}n_1(p_1+q_1-k_1)-\frac{2i\pi}{N_2}n_2(p_2+q_2-k_2)}
\frac{\hat\varphi_p}{|p|}
(q\cdot k^\perp)(k\cdot q)\frac{\hat\varphi_q}{|q|}
\end{equation}
provided
\begin{equation}
N_i>4K_i.
\end{equation}
Indeed, $\sum_{n_i=0}^{N_i}e^{-\frac{2i\pi}{N_i}n_im_i}$ vanishes unless $m_i=0\%N_i$ (in which $\%N_i$ means `modulo $N_i$'), and, if $p,q\in\mathcal K$, then $|p_i+q_i|\leqslant2K_i$. Therefore,
\begin{equation}
T(\hat\varphi,k)
=
\frac1{N_1N_2}
\mathcal F^*\left(
\mathcal F(|p|^{-1}\hat\varphi_p)(n)
\mathcal F((|q|^{-1}(q\cdot k^\perp)(k\cdot q)\hat\varphi_q)_q)(n)
\right)(k)
\end{equation}
which we expand
\begin{equation}
T(\hat\varphi,k)
=
\textstyle
\frac{k_x^2-k_y^2}{N_1N_2}
\mathcal F^*\left(
\mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n)
\mathcal F\left(\frac{q_xq_y}{|q|}\hat\varphi_q\right)(n)
\right)(k)
-
\frac{k_xk_y}{N_1N_2}
\mathcal F^*\left(
\mathcal F\left(\frac{\hat\varphi_p}{|p|}\right)(n)
\mathcal F\left(\frac{q_x^2-q_y^2}{|q|}\hat\varphi_q\right)(n)
\right)(k)
\end{equation}
\vfill
\eject
\begin{thebibliography}{WWW99}
\small
\IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{}
\end{thebibliography}
\end{document}