7 files changed, 0 insertions, 1358 deletions
diff --git a/docs/nstrophy_doc/Makefile b/docs/nstrophy_doc/Makefile
deleted file mode 100644
index c63e53f..0000000
--- a/docs/nstrophy_doc/Makefile
+++ /dev/null
@@ -1,70 +0,0 @@
-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) $(SYNCTEXS)
-
-clean-bibliography:
-	rm -f bibliography/bibliography.tex
-
-clean: clean-aux clean-tex clean-libs clean-figs
diff --git a/docs/nstrophy_doc/bibliography/conf.BBlog b/docs/nstrophy_doc/bibliography/conf.BBlog
deleted file mode 100644
index abe5ad4..0000000
--- a/docs/nstrophy_doc/bibliography/conf.BBlog
+++ /dev/null
@@ -1,6 +0,0 @@
-format: \bibitem[%token%]{%citeref%}%author:auth% - {\it %title%}, %journal%, %year%|doi?,\par\penalty10000%n%doi:{\tt\color{blue}\href{http://dx.doi.org/%doi%}{%doi%}}||arxiv?, arxiv:{\tt\color{blue}\href{http://arxiv.org/abs/%arxiv%}{%arxiv%}}|.\par\medskip%n% %n%
-out_file: bibliography.tex
-filter:auth: s/([A-Z])[^, ]* /\1. /g; s/ ([^ ,]*),/_\1,_/g; s/ ([^ ,]*)$/_\1/g; s/ //g; s/_/ /g;
-filter:journal: s/([a-zA-Z]) ([0-9]+)/\1~\\-\2/g;
-aux_cmd: \\citation{
-extra: Gallavotti2022:Ga22:Ga22:\bibitem[%token%]{%citeref%}G.\-~Gallavotti - {\it Navier-Stokes and equivalence conjectures}, preprint, 2022\par\penalty10000%n%arxiv:{\tt\color{blue}\href{https://arxiv.org/abs/2211.02961}{2211.02961}}%n%%n%
diff --git a/docs/nstrophy_doc/libs/ian.cls b/docs/nstrophy_doc/libs/ian.cls
deleted file mode 100644
index 538a108..0000000
--- a/docs/nstrophy_doc/libs/ian.cls
+++ /dev/null
@@ -1,659 +0,0 @@
-%%
-%% 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
diff --git a/docs/nstrophy_doc/libs/iantheo.sty b/docs/nstrophy_doc/libs/iantheo.sty
deleted file mode 100644
index d33a93d..0000000
--- a/docs/nstrophy_doc/libs/iantheo.sty
+++ /dev/null
@@ -1,162 +0,0 @@
-%%
-%% 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
diff --git a/docs/nstrophy_doc/libs/largearray.sty b/docs/nstrophy_doc/libs/largearray.sty
deleted file mode 100644
index ad5753b..0000000
--- a/docs/nstrophy_doc/libs/largearray.sty
+++ /dev/null
@@ -1,19 +0,0 @@
-%%
-%% 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}}
diff --git a/docs/nstrophy_doc/libs/point.sty b/docs/nstrophy_doc/libs/point.sty
deleted file mode 100644
index 9f983dc..0000000
--- a/docs/nstrophy_doc/libs/point.sty
+++ /dev/null
@@ -1,106 +0,0 @@
-%%
-%% 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
diff --git a/docs/nstrophy_doc/nstrophy_doc.tex b/docs/nstrophy_doc/nstrophy_doc.tex
deleted file mode 100644
index 54e802e..0000000
--- a/docs/nstrophy_doc/nstrophy_doc.tex
+++ /dev/null
@@ -1,336 +0,0 @@
-\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 incompressible Navier-Stokes equation in 2 dimensions
-\begin{equation}
-  \partial_tU=\nu\Delta U+G-(U\cdot\nabla)U,\quad
-  \nabla\cdot U=0
-  \label{ins}
-\end{equation}
-in which $G$ is the forcing term.
-We take periodic boundary conditions, so, at every given time, $U(t,\cdot)$ is a function on the torus $\mathbb T^2:=\mathbb R^2/(L\mathbb Z)^2$. We represent $U(t,\cdot)$ using its Fourier series
-\begin{equation}
-  \hat U_k(t):=\frac1{L^2}\int_{\mathbb T^2}dx\ e^{i\frac{2\pi}L kx}U(t,x)
-\end{equation}
-for $k\in\mathbb Z^2$, and rewrite~\-(\ref{ins}) as
-\begin{equation}
-  \partial_t\hat U_k=
-  -\frac{4\pi^2}{L^2}\nu k^2\hat U_k+\hat G_k
-  -i\frac{2\pi}L\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{i2\pi k^\perp}{L|k|}\hat u_k\equiv\frac{i2\pi}{L|k|}(-k_y\hat u_k,k_x\hat u_k)
-  \label{udef}
-\end{equation}
-in terms of which, multiplying both sides of the equation by $\frac L{i2\pi}\frac{k^\perp}{|k|}$,
-\begin{equation}
-  \partial_t\hat u_k=
-  -\frac{4\pi^2}{L^2}\nu k^2\hat u_k
-  +\hat g_k
-  +\frac{4\pi^2}{L^2|k|}\sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
-  \frac{(q\cdot p^\perp)(k^\perp\cdot q^\perp)}{|q||p|}\hat u_p\hat u_q
-  \label{ins_k}
-\end{equation}
-with
-\begin{equation}
-  \hat g_k:=\frac{Lk^\perp}{2i\pi|k|}\cdot\hat G_k
-  .
-  \label{gdef}
-\end{equation}
-Furthermore
-\begin{equation}
-  (q\cdot p^\perp)(k^\perp\cdot q^\perp)
-  =
-  (q\cdot p^\perp)(q^2+p\cdot q)
-\end{equation}
-and $q\cdot p^\perp$ is antisymmetric under exchange of $q$ and $p$. Therefore,
-\begin{equation}
-  \partial_t\hat u_k=
-  -\frac{4\pi^2}{L^2}\nu k^2\hat u_k+\hat g_k
-  +\frac{4\pi^2}{L^2|k|}T(\hat u,k)
-  \label{ins_k}
-\end{equation}
-with
-\begin{equation}
-  T(\hat u,k):=
-  \sum_{\displaystyle\mathop{\scriptstyle p,q\in\mathbb Z^2}_{p+q=k}}
-  \frac{(q\cdot p^\perp)|q|}{|p|}\hat u_p\hat u_q
-  .
-  \label{T}
-\end{equation}
-We truncate the Fourier modes and assume that $\hat u_k=0$ 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 Reality}.
-Since $U$ is real, $\hat U_{-k}=\hat U_k^*$, and so
-\begin{equation}
-  \hat u_{-k}=\hat u_k^*
-  .
-  \label{realu}
-\end{equation}
-Similarly,
-\begin{equation}
-  \hat g_{-k}=\hat g_k^*
-  .
-  \label{realg}
-\end{equation}
-Thus,
-\begin{equation}
-  T(\hat u,-k)
-  =
-  T(\hat u,k)^*
-  .
-  \label{realT}
-\end{equation}
-\bigskip
-
-\point{\bf FFT}. We compute T 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< N_1,\ 0\leqslant n_2< 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 u,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)}
-  (q\cdot p^\perp)\frac{|q|}{|p|}\hat u_q\hat u_p
-\end{equation}
-provided
-\begin{equation}
-  N_i>3K_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 $p,q,k\in\mathcal K$, then $|p_i+q_i-k_i|\leqslant3K_i$, so, as long as $N_i>3K_i$, then $(p_i+q_i-k_i)=0\%N_i$ implies $p_i+q_i=k_i$.
-Therefore,
-\begin{equation}
-  T(\hat u,k)
-  =
-  \textstyle
-  \frac1{N_1N_2}
-  \mathcal F^*\left(
-    \mathcal F\left(\frac{p_x\hat u_p}{|p|}\right)(n)
-    \mathcal F\left(q_y|q|\hat u_q\right)(n)
-    -
-    \mathcal F\left(\frac{p_y\hat u_p}{|p|}\right)(n)
-    \mathcal F\left(q_x|q|\hat u_q\right)(n)
-  \right)(k)
-\end{equation}
-\bigskip
-
-\point{\bf Energy}.
-We define the energy as
-\begin{equation}
-  E(t)=\frac12\int\frac{dx}{L^2}\ U^2(t,x)=\frac12\sum_{k\in\mathbb Z^2}|\hat U_k|^2
-  .
-\end{equation}
-We have
-\begin{equation}
-  \partial_t E=\int\frac{dx}{L^2}\ U\partial tU
-  =
-  \nu\int\frac{dx}{L^2}\ U\Delta U
-  +\int\frac{dx}{L^2}\ UG
-  -\int\frac{dx}{L^2}\ U(U\cdot\nabla)U
-  .
-\end{equation}
-Since we have periodic boundary conditions,
-\begin{equation}
-  \int dx\ U\Delta U=-\int dx\ |\nabla U|^2
-  .
-\end{equation}
-Furthermore,
-\begin{equation}
-  I:=\int dx\ U(U\cdot\nabla)U
-  =\sum_{i,j=1,2}\int dx\ U_iU_j\partial_jU_i
-  =
-  -\sum_{i,j=1,2}\int dx\ (\partial_jU_i)U_jU_i
-  -\sum_{i,j=1,2}\int dx\ U_i(\partial_jU_j)U_i
-\end{equation}
-and since $\nabla\cdot U=0$,
-\begin{equation}
-  I
-  =
-  -I
-\end{equation}
-and so $I=0$.
-Thus,
-\begin{equation}
-  \partial_t E=
-  \int\frac{dx}{L^2}\ \left(-\nu|\nabla U|^2+UG\right)
-  =
-  \sum_{k\in\mathbb Z^2}\left(-\frac{4\pi^2}{L^2}\nu k^2|\hat U_k|^2+\hat U_{-k}\hat G_k\right)
-  .
-\end{equation}
-Furthermore,
-\begin{equation}
-  \sum_{k\in\mathbb Z^2}k^2|\hat U_k|^2\geqslant
-  \sum_{k\in\mathbb Z^2}|\hat U_k|^2-|\hat U_0|^2
-  =2E-|\hat U_0|^2
-\end{equation}
-so
-\begin{equation}
-  \partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat U_0^2+\sum_{k\in\mathbb Z^2}\hat U_{-k}\hat G_k
-  \leqslant
-  -\frac{8\pi^2}{L^2}\nu E+\frac{4\pi^2}{L^2}\nu\hat U_0^2+
-  \|\hat G\|_2\sqrt{2E}
-  .
-\end{equation}
-In particular, if $\hat U_0=0$ (which corresponds to keeping the center of mass fixed),
-\begin{equation}
-  \partial_t E\leqslant -\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E}
-  .
-\end{equation}
-Now, if $\frac{8\pi^2}{L^2}\nu\sqrt E<\sqrt2\|\hat G\|_2$, then
-\begin{equation}
-  \frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E}}\leqslant1
-\end{equation}
-and so
-\begin{equation}
-  \frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(t)})}{-\frac{4\pi^2}{L^2}\nu}\leqslant t+
-  \frac{\log(1-\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(0)})}{-\frac{4\pi^2}{L^2}\nu}
-\end{equation}
-and
-\begin{equation}
-  E(t)
-  \leqslant
-  \left(
-    \frac{L^2\sqrt2\|\hat G\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t})
-    +e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)}
-  \right)^2
-  .
-\end{equation}
-If $\frac{8\pi^2}{L^2}\nu\sqrt E>\sqrt2\|\hat G\|_2$,
-\begin{equation}
-  \frac{\partial_t E}{-\frac{8\pi^2}{L^2}\nu E+\|\hat G\|_2\sqrt{2E}}\geqslant1
-\end{equation}
-and so
-\begin{equation}
-  \frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(t)}-1)}{-\frac{4\pi^2}{L^2}\nu}\geqslant t+
-  \frac{\log(\frac{8\pi^2\nu}{L^2\sqrt2\|\hat G\|_2}\sqrt{E(0)})-1}{-\frac{4\pi^2}{L^2}\nu}
-\end{equation}
-and
-\begin{equation}
-  E(t)
-  \leqslant
-  \left(
-    \frac{L^2\sqrt2\|\hat G\|_2}{8\pi^2\nu}(1-e^{-\frac{4\pi^2}{L^2}\nu t})
-    +e^{-\frac{4\pi^2}{L^2}\nu t}\sqrt{E(0)}
-  \right)^2
-  .
-\end{equation}
-\bigskip
-
-\point{\bf Enstrophy}.
-The enstrophy is defined as
-\begin{equation}
-  \mathcal En(t)=\int\frac{dx}{L^2}\ |\nabla U|^2
-  =\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}k^2|\hat U_k|^2
-  .
-\end{equation}
-\bigskip
-
-\point{\bf Numerical instability}.
-In order to prevent the algorithm from blowing up, it is necessary to impose the reality of $u(x)$ by hand, otherwise, truncation errors build up, and lead to divergences.
-It is sufficient to ensure that the convolution term $T(\hat u,k)$ satisfies $T(\hat u,-k)=T(\hat u,k)^*$.
-After imposing this condition, the algorithm no longer blows up, but it is still unstable (for instance, increasing $K_1$ or $K_2$ leads to very different results).
-
-\subsection{Reversible equation}
-\indent The reversible equation is similar to\-~(\ref{ins}) but instead of fixing the viscosity, we fix the enstrophy\-~\cite{Ga22}.
-It is defined directly in Fourier space:
-\begin{equation}
-  \partial_t\hat U_k=
-  -\frac{4\pi^2}{L^2}\alpha(\hat U) k^2\hat U_k+\hat G_k
-  -i\frac{2\pi}L\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
-\end{equation}
-where $\alpha$ is chosen such that the enstrophy is constant.
-In terms of $\hat u$\-~(\ref{udef}), (\ref{gdef}), (\ref{T}):
-\begin{equation}
-  \partial_t\hat u_k=
-  -\frac{4\pi^2}{L^2}\alpha(\hat u) k^2\hat u_k
-  +\hat g_k
-  +\frac{4\pi^2}{L^2|k|}T(\hat u,k)
-  .
-  \label{rns_k}
-\end{equation}
-To compute $\alpha$, we use the constancy of the enstrophy:
-\begin{equation}
-  \sum_{k\in\mathbb Z^2}k^2\hat U_k\cdot\partial_t\hat U_k
-  =0
-\end{equation}
-which, in terms of $\hat u$ is
-\begin{equation}
-  \sum_{k\in\mathbb Z^2}k^2\hat u_k^*\partial_t\hat u_k
-  =0
-\end{equation}
-that is
-\begin{equation}
-  \frac{4\pi^2}{L^2}\alpha(\hat u)\sum_{k\in\mathbb Z^2}k^4|\hat u_k|^2
-  =
-  \sum_{k\in\mathbb Z^2}k^2\hat u_k^*\hat g_k
-  +\frac{4\pi^2}{L^2}\sum_{k\in\mathbb Z^2}|k|\hat u_k^*T(\hat u,k)
-\end{equation}
-and so
-\begin{equation}
-  \alpha(\hat u)
-  =\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2}
-  .
-\end{equation}
-Note that, by\-~(\ref{realu})-(\ref{realT}),
-\begin{equation}
-  \alpha(\hat u)\in\mathbb R
-  .
-\end{equation}
-
-
-
-\vfill
-\eject
-
-\begin{thebibliography}{WWW99}
-\small
-\IfFileExists{bibliography/bibliography.tex}{\input bibliography/bibliography.tex}{}
-\end{thebibliography}
-
-\end{document}