diff --git a/theory/paper_spheric/IEEEtran.cls b/theory/paper_spheric/IEEEtran.cls
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@@ -0,0 +1,4702 @@
+%%
+%% IEEEtran.cls 2007/03/05 version V1.7a
+%%
+%%
+%% This is the official IEEE LaTeX class for authors of the Institute of
+%% Electrical and Electronics Engineers (IEEE) Transactions journals and
+%% conferences.
+%%
+%% Support sites:
+%% http://www.michaelshell.org/tex/ieeetran/
+%% http://www.ctan.org/tex-archive/macros/latex/contrib/IEEEtran/
+%% and
+%% http://www.ieee.org/
+%%
+%% Based on the original 1993 IEEEtran.cls, but with many bug fixes
+%% and enhancements (from both JVH and MDS) over the 1996/7 version.
+%%
+%%
+%% Contributors:
+%% Gerry Murray (1993), Silvano Balemi (1993),
+%% Jon Dixon (1996), Peter N"uchter (1996),
+%% Juergen von Hagen (2000), and Michael Shell (2001-2007)
+%%
+%%
+%% Copyright (c) 1993-2000 by Gerry Murray, Silvano Balemi,
+%% Jon Dixon, Peter N"uchter,
+%% Juergen von Hagen
+%% and
+%% Copyright (c) 2001-2007 by Michael Shell
+%%
+%% Current maintainer (V1.3 to V1.7): Michael Shell
+%% See:
+%% http://www.michaelshell.org/
+%% for current contact information.
+%%
+%% Special thanks to Peter Wilson (CUA) and Donald Arseneau
+%% for allowing the inclusion of the \@ifmtarg command
+%% from their ifmtarg LaTeX package.
+%%
+%%*************************************************************************
+%% Legal Notice:
+%% This code is offered as-is without any warranty either expressed or
+%% implied; without even the implied warranty of MERCHANTABILITY or
+%% FITNESS FOR A PARTICULAR PURPOSE!
+%% User assumes all risk.
+%% In no event shall IEEE or any contributor to this code be liable for
+%% any damages or losses, including, but not limited to, incidental,
+%% consequential, or any other damages, resulting from the use or misuse
+%% of any information contained here.
+%%
+%% All comments are the opinions of their respective authors and are not
+%% necessarily endorsed by the IEEE.
+%%
+%% This work is distributed under the LaTeX Project Public License (LPPL)
+%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used,
+%% distributed and modified. A copy of the LPPL, version 1.3, is included
+%% in the base LaTeX documentation of all distributions of LaTeX released
+%% 2003/12/01 or later.
+%% Retain all contribution notices and credits.
+%% ** Modified files should be clearly indicated as such, including **
+%% ** renaming them and changing author support contact information. **
+%%
+%% File list of work: IEEEtran.cls, IEEEtran_HOWTO.pdf, bare_adv.tex,
+%% bare_conf.tex, bare_jrnl.tex, bare_jrnl_compsoc.tex
+%%
+%% Major changes to the user interface should be indicated by an
+%% increase in the version numbers. If a version is a beta, it will
+%% be indicated with a BETA suffix, i.e., 1.4 BETA.
+%% Small changes can be indicated by appending letters to the version
+%% such as "IEEEtran_v14a.cls".
+%% In all cases, \Providesclass, any \typeout messages to the user,
+%% \IEEEtransversionmajor and \IEEEtransversionminor must reflect the
+%% correct version information.
+%% The changes should also be documented via source comments.
+%%*************************************************************************
+%%
+%
+% Available class options
+% e.g., \documentclass[10pt,conference]{IEEEtran}
+%
+% *** choose only one from each category ***
+%
+% 9pt, 10pt, 11pt, 12pt
+% Sets normal font size. The default is 10pt.
+%
+% conference, journal, technote, peerreview, peerreviewca
+% determines format mode - conference papers, journal papers,
+% correspondence papers (technotes), or peer review papers. The user
+% should also select 9pt when using technote. peerreview is like
+% journal mode, but provides for a single-column "cover" title page for
+% anonymous peer review. The paper title (without the author names) is
+% repeated at the top of the page after the cover page. For peer review
+% papers, the \IEEEpeerreviewmaketitle command must be executed (will
+% automatically be ignored for non-peerreview modes) at the place the
+% cover page is to end, usually just after the abstract (keywords are
+% not normally used with peer review papers). peerreviewca is like
+% peerreview, but allows the author names to be entered and formatted
+% as with conference mode so that author affiliation and contact
+% information can be easily seen on the cover page.
+% The default is journal.
+%
+% draft, draftcls, draftclsnofoot, final
+% determines if paper is formatted as a widely spaced draft (for
+% handwritten editor comments) or as a properly typeset final version.
+% draftcls restricts draft mode to the class file while all other LaTeX
+% packages (i.e., \usepackage{graphicx}) will behave as final - allows
+% for a draft paper with visible figures, etc. draftclsnofoot is like
+% draftcls, but does not display the date and the word "DRAFT" at the foot
+% of the pages. If using one of the draft modes, the user will probably
+% also want to select onecolumn.
+% The default is final.
+%
+% letterpaper, a4paper
+% determines paper size: 8.5in X 11in or 210mm X 297mm. CHANGING THE PAPER
+% SIZE WILL NOT ALTER THE TYPESETTING OF THE DOCUMENT - ONLY THE MARGINS
+% WILL BE AFFECTED. In particular, documents using the a4paper option will
+% have reduced side margins (A4 is narrower than US letter) and a longer
+% bottom margin (A4 is longer than US letter). For both cases, the top
+% margins will be the same and the text will be horizontally centered.
+% For final submission to IEEE, authors should use US letter (8.5 X 11in)
+% paper. Note that authors should ensure that all post-processing
+% (ps, pdf, etc.) uses the same paper specificiation as the .tex document.
+% Problems here are by far the number one reason for incorrect margins.
+% IEEEtran will automatically set the default paper size under pdflatex
+% (without requiring a change to pdftex.cfg), so this issue is more
+% important to dvips users. Fix config.ps, config.pdf, or ~/.dvipsrc for
+% dvips, or use the dvips -t papersize option instead as needed. See the
+% testflow documentation
+% http://www.ctan.org/tex-archive/macros/latex/contrib/IEEEtran/testflow
+% for more details on dvips paper size configuration.
+% The default is letterpaper.
+%
+% oneside, twoside
+% determines if layout follows single sided or two sided (duplex)
+% printing. The only notable change is with the headings at the top of
+% the pages.
+% The default is oneside.
+%
+% onecolumn, twocolumn
+% determines if text is organized into one or two columns per page. One
+% column mode is usually used only with draft papers.
+% The default is twocolumn.
+%
+% compsoc
+% Use the format of the IEEE Computer Society.
+%
+% romanappendices
+% Use the "Appendix I" convention when numbering appendices. IEEEtran.cls
+% now defaults to Alpha "Appendix A" convention - the opposite of what
+% v1.6b and earlier did.
+%
+% captionsoff
+% disables the display of the figure/table captions. Some IEEE journals
+% request that captions be removed and figures/tables be put on pages
+% of their own at the end of an initial paper submission. The endfloat
+% package can be used with this class option to achieve this format.
+%
+% nofonttune
+% turns off tuning of the font interword spacing. Maybe useful to those
+% not using the standard Times fonts or for those who have already "tuned"
+% their fonts.
+% The default is to enable IEEEtran to tune font parameters.
+%
+%
+%----------
+% Available CLASSINPUTs provided (all are macros unless otherwise noted):
+% \CLASSINPUTbaselinestretch
+% \CLASSINPUTinnersidemargin
+% \CLASSINPUToutersidemargin
+% \CLASSINPUTtoptextmargin
+% \CLASSINPUTbottomtextmargin
+%
+% Available CLASSINFOs provided:
+% \ifCLASSINFOpdf (TeX if conditional)
+% \CLASSINFOpaperwidth (macro)
+% \CLASSINFOpaperheight (macro)
+% \CLASSINFOnormalsizebaselineskip (length)
+% \CLASSINFOnormalsizeunitybaselineskip (length)
+%
+% Available CLASSOPTIONs provided:
+% all class option flags (TeX if conditionals) unless otherwise noted,
+% e.g., \ifCLASSOPTIONcaptionsoff
+% point size options provided as a single macro:
+% \CLASSOPTIONpt
+% which will be defined as 9, 10, 11, or 12 depending on the document's
+% normalsize point size.
+% also, class option peerreviewca implies the use of class option peerreview
+% and classoption draft implies the use of class option draftcls
+
+
+
+
+
+\ProvidesClass{IEEEtran}[2007/03/05 V1.7a by Michael Shell]
+\typeout{-- See the "IEEEtran_HOWTO" manual for usage information.}
+\typeout{-- http://www.michaelshell.org/tex/ieeetran/}
+\NeedsTeXFormat{LaTeX2e}
+
+% IEEEtran.cls version numbers, provided as of V1.3
+% These values serve as a way a .tex file can
+% determine if the new features are provided.
+% The version number of this IEEEtrans.cls can be obtained from
+% these values. i.e., V1.4
+% KEEP THESE AS INTEGERS! i.e., NO {4a} or anything like that-
+% (no need to enumerate "a" minor changes here)
+\def\IEEEtransversionmajor{1}
+\def\IEEEtransversionminor{7}
+
+% These do nothing, but provide them like in article.cls
+\newif\if@restonecol
+\newif\if@titlepage
+
+
+% class option conditionals
+\newif\ifCLASSOPTIONonecolumn \CLASSOPTIONonecolumnfalse
+\newif\ifCLASSOPTIONtwocolumn \CLASSOPTIONtwocolumntrue
+
+\newif\ifCLASSOPTIONoneside \CLASSOPTIONonesidetrue
+\newif\ifCLASSOPTIONtwoside \CLASSOPTIONtwosidefalse
+
+\newif\ifCLASSOPTIONfinal \CLASSOPTIONfinaltrue
+\newif\ifCLASSOPTIONdraft \CLASSOPTIONdraftfalse
+\newif\ifCLASSOPTIONdraftcls \CLASSOPTIONdraftclsfalse
+\newif\ifCLASSOPTIONdraftclsnofoot \CLASSOPTIONdraftclsnofootfalse
+
+\newif\ifCLASSOPTIONpeerreview \CLASSOPTIONpeerreviewfalse
+\newif\ifCLASSOPTIONpeerreviewca \CLASSOPTIONpeerreviewcafalse
+
+\newif\ifCLASSOPTIONjournal \CLASSOPTIONjournaltrue
+\newif\ifCLASSOPTIONconference \CLASSOPTIONconferencefalse
+\newif\ifCLASSOPTIONtechnote \CLASSOPTIONtechnotefalse
+
+\newif\ifCLASSOPTIONnofonttune \CLASSOPTIONnofonttunefalse
+
+\newif\ifCLASSOPTIONcaptionsoff \CLASSOPTIONcaptionsofffalse
+
+\newif\ifCLASSOPTIONcompsoc \CLASSOPTIONcompsocfalse
+
+\newif\ifCLASSOPTIONromanappendices \CLASSOPTIONromanappendicesfalse
+
+
+% class info conditionals
+
+% indicates if pdf (via pdflatex) output
+\newif\ifCLASSINFOpdf \CLASSINFOpdffalse
+
+
+% V1.6b internal flag to show if using a4paper
+\newif\if@IEEEusingAfourpaper \@IEEEusingAfourpaperfalse
+
+
+
+% IEEEtran class scratch pad registers
+% dimen
+\newdimen\@IEEEtrantmpdimenA
+\newdimen\@IEEEtrantmpdimenB
+% count
+\newcount\@IEEEtrantmpcountA
+\newcount\@IEEEtrantmpcountB
+% token list
+\newtoks\@IEEEtrantmptoksA
+
+% we use \CLASSOPTIONpt so that we can ID the point size (even for 9pt docs)
+% as well as LaTeX's \@ptsize to retain some compatability with some
+% external packages
+\def\@ptsize{0}
+% LaTeX does not support 9pt, so we set \@ptsize to 0 - same as that of 10pt
+\DeclareOption{9pt}{\def\CLASSOPTIONpt{9}\def\@ptsize{0}}
+\DeclareOption{10pt}{\def\CLASSOPTIONpt{10}\def\@ptsize{0}}
+\DeclareOption{11pt}{\def\CLASSOPTIONpt{11}\def\@ptsize{1}}
+\DeclareOption{12pt}{\def\CLASSOPTIONpt{12}\def\@ptsize{2}}
+
+
+
+\DeclareOption{letterpaper}{\setlength{\paperheight}{11in}%
+ \setlength{\paperwidth}{8.5in}%
+ \@IEEEusingAfourpaperfalse
+ \def\CLASSOPTIONpaper{letter}%
+ \def\CLASSINFOpaperwidth{8.5in}%
+ \def\CLASSINFOpaperheight{11in}}
+
+
+\DeclareOption{a4paper}{\setlength{\paperheight}{297mm}%
+ \setlength{\paperwidth}{210mm}%
+ \@IEEEusingAfourpapertrue
+ \def\CLASSOPTIONpaper{a4}%
+ \def\CLASSINFOpaperwidth{210mm}%
+ \def\CLASSINFOpaperheight{297mm}}
+
+\DeclareOption{oneside}{\@twosidefalse\@mparswitchfalse
+ \CLASSOPTIONonesidetrue\CLASSOPTIONtwosidefalse}
+\DeclareOption{twoside}{\@twosidetrue\@mparswitchtrue
+ \CLASSOPTIONtwosidetrue\CLASSOPTIONonesidefalse}
+
+\DeclareOption{onecolumn}{\CLASSOPTIONonecolumntrue\CLASSOPTIONtwocolumnfalse}
+\DeclareOption{twocolumn}{\CLASSOPTIONtwocolumntrue\CLASSOPTIONonecolumnfalse}
+
+% If the user selects draft, then this class AND any packages
+% will go into draft mode.
+\DeclareOption{draft}{\CLASSOPTIONdrafttrue\CLASSOPTIONdraftclstrue
+ \CLASSOPTIONdraftclsnofootfalse}
+% draftcls is for a draft mode which will not affect any packages
+% used by the document.
+\DeclareOption{draftcls}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclstrue
+ \CLASSOPTIONdraftclsnofootfalse}
+% draftclsnofoot is like draftcls, but without the footer.
+\DeclareOption{draftclsnofoot}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclstrue
+ \CLASSOPTIONdraftclsnofoottrue}
+\DeclareOption{final}{\CLASSOPTIONdraftfalse\CLASSOPTIONdraftclsfalse
+ \CLASSOPTIONdraftclsnofootfalse}
+
+\DeclareOption{journal}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
+ \CLASSOPTIONjournaltrue\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+
+\DeclareOption{conference}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
+ \CLASSOPTIONjournalfalse\CLASSOPTIONconferencetrue\CLASSOPTIONtechnotefalse}
+
+\DeclareOption{technote}{\CLASSOPTIONpeerreviewfalse\CLASSOPTIONpeerreviewcafalse
+ \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotetrue}
+
+\DeclareOption{peerreview}{\CLASSOPTIONpeerreviewtrue\CLASSOPTIONpeerreviewcafalse
+ \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+
+\DeclareOption{peerreviewca}{\CLASSOPTIONpeerreviewtrue\CLASSOPTIONpeerreviewcatrue
+ \CLASSOPTIONjournalfalse\CLASSOPTIONconferencefalse\CLASSOPTIONtechnotefalse}
+
+\DeclareOption{nofonttune}{\CLASSOPTIONnofonttunetrue}
+
+\DeclareOption{captionsoff}{\CLASSOPTIONcaptionsofftrue}
+
+\DeclareOption{compsoc}{\CLASSOPTIONcompsoctrue}
+
+\DeclareOption{romanappendices}{\CLASSOPTIONromanappendicestrue}
+
+
+% default to US letter paper, 10pt, twocolumn, one sided, final, journal
+\ExecuteOptions{letterpaper,10pt,twocolumn,oneside,final,journal}
+% overrride these defaults per user requests
+\ProcessOptions
+
+
+
+% Computer Society conditional execution command
+\long\def\@IEEEcompsoconly#1{\relax\ifCLASSOPTIONcompsoc\relax#1\relax\fi\relax}
+% inverse
+\long\def\@IEEEnotcompsoconly#1{\relax\ifCLASSOPTIONcompsoc\else\relax#1\relax\fi\relax}
+% compsoc conference
+\long\def\@IEEEcompsocconfonly#1{\relax\ifCLASSOPTIONcompsoc\ifCLASSOPTIONconference\relax#1\relax\fi\fi\relax}
+% compsoc not conference
+\long\def\@IEEEcompsocnotconfonly#1{\relax\ifCLASSOPTIONcompsoc\ifCLASSOPTIONconference\else\relax#1\relax\fi\fi\relax}
+
+
+% IEEE uses Times Roman font, so we'll default to Times.
+% These three commands make up the entire times.sty package.
+\renewcommand{\sfdefault}{phv}
+\renewcommand{\rmdefault}{ptm}
+\renewcommand{\ttdefault}{pcr}
+
+\@IEEEcompsoconly{\typeout{-- Using IEEE Computer Society mode.}}
+
+% V1.7 compsoc nonconference papers, use Palatino/Palladio as the main text font,
+% not Times Roman.
+\@IEEEcompsocnotconfonly{\renewcommand{\rmdefault}{ppl}}
+
+% enable Times/Palatino main text font
+\normalfont\selectfont
+
+
+
+
+
+% V1.7 conference notice message hook
+\def\@IEEEconsolenoticeconference{\typeout{}%
+\typeout{** Conference Paper **}%
+\typeout{Before submitting the final camera ready copy, remember to:}%
+\typeout{}%
+\typeout{ 1. Manually equalize the lengths of two columns on the last page}%
+\typeout{ of your paper;}%
+\typeout{}%
+\typeout{ 2. Ensure that any PostScript and/or PDF output post-processing}%
+\typeout{ uses only Type 1 fonts and that every step in the generation}%
+\typeout{ process uses the appropriate paper size.}%
+\typeout{}}
+
+
+% we can send console reminder messages to the user here
+\AtEndDocument{\ifCLASSOPTIONconference\@IEEEconsolenoticeconference\fi}
+
+
+% warn about the use of single column other than for draft mode
+\ifCLASSOPTIONtwocolumn\else%
+ \ifCLASSOPTIONdraftcls\else%
+ \typeout{** ATTENTION: Single column mode is not typically used with IEEE publications.}%
+ \fi%
+\fi
+
+
+% V1.7 improved paper size setting code.
+% Set pdfpage and dvips paper sizes. Conditional tests are similar to that
+% of ifpdf.sty. Retain within {} to ensure tested macros are never altered,
+% even if only effect is to set them to \relax.
+% if \pdfoutput is undefined or equal to relax, output a dvips special
+{\@ifundefined{pdfoutput}{\AtBeginDvi{\special{papersize=\CLASSINFOpaperwidth,\CLASSINFOpaperheight}}}{%
+% pdfoutput is defined and not equal to \relax
+% check for pdfpageheight existence just in case someone sets pdfoutput
+% under non-pdflatex. If exists, set them regardless of value of \pdfoutput.
+\@ifundefined{pdfpageheight}{\relax}{\global\pdfpagewidth\paperwidth
+\global\pdfpageheight\paperheight}%
+% if using \pdfoutput=0 under pdflatex, send dvips papersize special
+\ifcase\pdfoutput
+\AtBeginDvi{\special{papersize=\CLASSINFOpaperwidth,\CLASSINFOpaperheight}}%
+\else
+% we are using pdf output, set CLASSINFOpdf flag
+\global\CLASSINFOpdftrue
+\fi}}
+
+% let the user know the selected papersize
+\typeout{-- Using \CLASSINFOpaperwidth\space x \CLASSINFOpaperheight\space
+(\CLASSOPTIONpaper)\space paper.}
+
+\ifCLASSINFOpdf
+\typeout{-- Using PDF output.}
+\else
+\typeout{-- Using DVI output.}
+\fi
+
+
+% The idea hinted here is for LaTeX to generate markleft{} and markright{}
+% automatically for you after you enter \author{}, \journal{},
+% \journaldate{}, journalvol{}, \journalnum{}, etc.
+% However, there may be some backward compatibility issues here as
+% well as some special applications for IEEEtran.cls and special issues
+% that may require the flexible \markleft{}, \markright{} and/or \markboth{}.
+% We'll leave this as an open future suggestion.
+%\newcommand{\journal}[1]{\def\@journal{#1}}
+%\def\@journal{}
+
+
+
+% pointsize values
+% used with ifx to determine the document's normal size
+\def\@IEEEptsizenine{9}
+\def\@IEEEptsizeten{10}
+\def\@IEEEptsizeeleven{11}
+\def\@IEEEptsizetwelve{12}
+
+
+
+% FONT DEFINITIONS (No sizexx.clo file needed)
+% V1.6 revised font sizes, displayskip values and
+% revised normalsize baselineskip to reduce underfull vbox problems
+% on the 58pc = 696pt = 9.5in text height we want
+% normalsize #lines/column baselineskip (aka leading)
+% 9pt 63 11.0476pt (truncated down)
+% 10pt 58 12pt (exact)
+% 11pt 52 13.3846pt (truncated down)
+% 12pt 50 13.92pt (exact)
+%
+
+% we need to store the nominal baselineskip for the given font size
+% in case baselinestretch ever changes.
+% this is a dimen, so it will not hold stretch or shrink
+\newdimen\@IEEEnormalsizeunitybaselineskip
+\@IEEEnormalsizeunitybaselineskip\baselineskip
+
+\ifx\CLASSOPTIONpt\@IEEEptsizenine
+\typeout{-- This is a 9 point document.}
+\def\normalsize{\@setfontsize{\normalsize}{9}{11.0476pt}}%
+\setlength{\@IEEEnormalsizeunitybaselineskip}{11.0476pt}%
+\normalsize
+\abovedisplayskip 1.5ex plus3pt minus1pt%
+\belowdisplayskip \abovedisplayskip%
+\abovedisplayshortskip 0pt plus3pt%
+\belowdisplayshortskip 1.5ex plus3pt minus1pt
+\def\small{\@setfontsize{\small}{8.5}{10pt}}
+\def\footnotesize{\@setfontsize{\footnotesize}{8}{9pt}}
+\def\scriptsize{\@setfontsize{\scriptsize}{7}{8pt}}
+\def\tiny{\@setfontsize{\tiny}{5}{6pt}}
+% sublargesize is the same as large - 10pt
+\def\sublargesize{\@setfontsize{\sublargesize}{10}{12pt}}
+\def\large{\@setfontsize{\large}{10}{12pt}}
+\def\Large{\@setfontsize{\Large}{12}{14pt}}
+\def\LARGE{\@setfontsize{\LARGE}{14}{17pt}}
+\def\huge{\@setfontsize{\huge}{17}{20pt}}
+\def\Huge{\@setfontsize{\Huge}{20}{24pt}}
+\fi
+
+
+% Check if we have selected 10 points
+\ifx\CLASSOPTIONpt\@IEEEptsizeten
+\typeout{-- This is a 10 point document.}
+\def\normalsize{\@setfontsize{\normalsize}{10}{12.00pt}}%
+\setlength{\@IEEEnormalsizeunitybaselineskip}{12pt}%
+\normalsize
+\abovedisplayskip 1.5ex plus4pt minus2pt%
+\belowdisplayskip \abovedisplayskip%
+\abovedisplayshortskip 0pt plus4pt%
+\belowdisplayshortskip 1.5ex plus4pt minus2pt
+\def\small{\@setfontsize{\small}{9}{10pt}}
+\def\footnotesize{\@setfontsize{\footnotesize}{8}{9pt}}
+\def\scriptsize{\@setfontsize{\scriptsize}{7}{8pt}}
+\def\tiny{\@setfontsize{\tiny}{5}{6pt}}
+% sublargesize is a tad smaller than large - 11pt
+\def\sublargesize{\@setfontsize{\sublargesize}{11}{13.4pt}}
+\def\large{\@setfontsize{\large}{12}{14pt}}
+\def\Large{\@setfontsize{\Large}{14}{17pt}}
+\def\LARGE{\@setfontsize{\LARGE}{17}{20pt}}
+\def\huge{\@setfontsize{\huge}{20}{24pt}}
+\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+\fi
+
+
+% Check if we have selected 11 points
+\ifx\CLASSOPTIONpt\@IEEEptsizeeleven
+\typeout{-- This is an 11 point document.}
+\def\normalsize{\@setfontsize{\normalsize}{11}{13.3846pt}}%
+\setlength{\@IEEEnormalsizeunitybaselineskip}{13.3846pt}%
+\normalsize
+\abovedisplayskip 1.5ex plus5pt minus3pt%
+\belowdisplayskip \abovedisplayskip%
+\abovedisplayshortskip 0pt plus5pt%
+\belowdisplayshortskip 1.5ex plus5pt minus3pt
+\def\small{\@setfontsize{\small}{10}{12pt}}
+\def\footnotesize{\@setfontsize{\footnotesize}{9}{10.5pt}}
+\def\scriptsize{\@setfontsize{\scriptsize}{8}{9pt}}
+\def\tiny{\@setfontsize{\tiny}{6}{7pt}}
+% sublargesize is the same as large - 12pt
+\def\sublargesize{\@setfontsize{\sublargesize}{12}{14pt}}
+\def\large{\@setfontsize{\large}{12}{14pt}}
+\def\Large{\@setfontsize{\Large}{14}{17pt}}
+\def\LARGE{\@setfontsize{\LARGE}{17}{20pt}}
+\def\huge{\@setfontsize{\huge}{20}{24pt}}
+\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+\fi
+
+
+% Check if we have selected 12 points
+\ifx\CLASSOPTIONpt\@IEEEptsizetwelve
+\typeout{-- This is a 12 point document.}
+\def\normalsize{\@setfontsize{\normalsize}{12}{13.92pt}}%
+\setlength{\@IEEEnormalsizeunitybaselineskip}{13.92pt}%
+\normalsize
+\abovedisplayskip 1.5ex plus6pt minus4pt%
+\belowdisplayskip \abovedisplayskip%
+\abovedisplayshortskip 0pt plus6pt%
+\belowdisplayshortskip 1.5ex plus6pt minus4pt
+\def\small{\@setfontsize{\small}{10}{12pt}}
+\def\footnotesize{\@setfontsize{\footnotesize}{9}{10.5pt}}
+\def\scriptsize{\@setfontsize{\scriptsize}{8}{9pt}}
+\def\tiny{\@setfontsize{\tiny}{6}{7pt}}
+% sublargesize is the same as large - 14pt
+\def\sublargesize{\@setfontsize{\sublargesize}{14}{17pt}}
+\def\large{\@setfontsize{\large}{14}{17pt}}
+\def\Large{\@setfontsize{\Large}{17}{20pt}}
+\def\LARGE{\@setfontsize{\LARGE}{20}{24pt}}
+\def\huge{\@setfontsize{\huge}{22}{26pt}}
+\def\Huge{\@setfontsize{\Huge}{24}{28pt}}
+\fi
+
+
+% V1.6 The Computer Modern Fonts will issue a substitution warning for
+% 24pt titles (24.88pt is used instead) increase the substitution
+% tolerance to turn off this warning
+\def\fontsubfuzz{.9pt}
+% However, the default (and correct) Times font will scale exactly as needed.
+
+
+% warn the user in case they forget to use the 9pt option with
+% technote
+\ifCLASSOPTIONtechnote%
+ \ifx\CLASSOPTIONpt\@IEEEptsizenine\else%
+ \typeout{** ATTENTION: Technotes are normally 9pt documents.}%
+ \fi%
+\fi
+
+
+% V1.7
+% Improved \textunderscore to provide a much better fake _ when used with
+% OT1 encoding. Under OT1, detect use of pcr or cmtt \ttfamily and use
+% available true _ glyph for those two typewriter fonts.
+\def\@IEEEstringptm{ptm} % Times Roman family
+\def\@IEEEstringppl{ppl} % Palatino Roman family
+\def\@IEEEstringphv{phv} % Helvetica Sans Serif family
+\def\@IEEEstringpcr{pcr} % Courier typewriter family
+\def\@IEEEstringcmtt{cmtt} % Computer Modern typewriter family
+\DeclareTextCommandDefault{\textunderscore}{\leavevmode
+\ifx\f@family\@IEEEstringpcr\string_\else
+\ifx\f@family\@IEEEstringcmtt\string_\else
+\ifx\f@family\@IEEEstringptm\kern 0em\vbox{\hrule\@width 0.5em\@height 0.5pt\kern -0.3ex}\else
+\ifx\f@family\@IEEEstringppl\kern 0em\vbox{\hrule\@width 0.5em\@height 0.5pt\kern -0.3ex}\else
+\ifx\f@family\@IEEEstringphv\kern -0.03em\vbox{\hrule\@width 0.62em\@height 0.52pt\kern -0.33ex}\kern -0.03em\else
+\kern 0.09em\vbox{\hrule\@width 0.6em\@height 0.44pt\kern -0.63pt\kern -0.42ex}\kern 0.09em\fi\fi\fi\fi\fi\relax}
+
+
+
+
+% set the default \baselinestretch
+\def\baselinestretch{1}
+\ifCLASSOPTIONdraftcls
+ \def\baselinestretch{1.5}% default baselinestretch for draft modes
+\fi
+
+
+% process CLASSINPUT baselinestretch
+\ifx\CLASSINPUTbaselinestretch\@IEEEundefined
+\else
+ \edef\baselinestretch{\CLASSINPUTbaselinestretch} % user CLASSINPUT override
+ \typeout{** ATTENTION: Overriding \string\baselinestretch\space to
+ \baselinestretch\space via \string\CLASSINPUT.}
+\fi
+
+\normalsize % make \baselinestretch take affect
+
+
+
+
+% store the normalsize baselineskip
+\newdimen\CLASSINFOnormalsizebaselineskip
+\CLASSINFOnormalsizebaselineskip=\baselineskip\relax
+% and the normalsize unity (baselinestretch=1) baselineskip
+% we could save a register by giving the user access to
+% \@IEEEnormalsizeunitybaselineskip. However, let's protect
+% its read only internal status
+\newdimen\CLASSINFOnormalsizeunitybaselineskip
+\CLASSINFOnormalsizeunitybaselineskip=\@IEEEnormalsizeunitybaselineskip\relax
+% store the nominal value of jot
+\newdimen\IEEEnormaljot
+\IEEEnormaljot=0.25\baselineskip\relax
+
+% set \jot
+\jot=\IEEEnormaljot\relax
+
+
+
+
+% V1.6, we are now going to fine tune the interword spacing
+% The default interword glue for Times under TeX appears to use a
+% nominal interword spacing of 25% (relative to the font size, i.e., 1em)
+% a maximum of 40% and a minimum of 19%.
+% For example, 10pt text uses an interword glue of:
+%
+% 2.5pt plus 1.49998pt minus 0.59998pt
+%
+% However, IEEE allows for a more generous range which reduces the need
+% for hyphenation, especially for two column text. Furthermore, IEEE
+% tends to use a little bit more nominal space between the words.
+% IEEE's interword spacing percentages appear to be:
+% 35% nominal
+% 23% minimum
+% 50% maximum
+% (They may even be using a tad more for the largest fonts such as 24pt.)
+%
+% for bold text, IEEE increases the spacing a little more:
+% 37.5% nominal
+% 23% minimum
+% 55% maximum
+
+% here are the interword spacing ratios we'll use
+% for medium (normal weight)
+\def\@IEEEinterspaceratioM{0.35}
+\def\@IEEEinterspaceMINratioM{0.23}
+\def\@IEEEinterspaceMAXratioM{0.50}
+
+% for bold
+\def\@IEEEinterspaceratioB{0.375}
+\def\@IEEEinterspaceMINratioB{0.23}
+\def\@IEEEinterspaceMAXratioB{0.55}
+
+
+% command to revise the interword spacing for the current font under TeX:
+% \fontdimen2 = nominal interword space
+% \fontdimen3 = interword stretch
+% \fontdimen4 = interword shrink
+% since all changes to the \fontdimen are global, we can enclose these commands
+% in braces to confine any font attribute or length changes
+\def\@@@IEEEsetfontdimens#1#2#3{{%
+\setlength{\@IEEEtrantmpdimenB}{\f@size pt}% grab the font size in pt, could use 1em instead.
+\setlength{\@IEEEtrantmpdimenA}{#1\@IEEEtrantmpdimenB}%
+\fontdimen2\font=\@IEEEtrantmpdimenA\relax
+\addtolength{\@IEEEtrantmpdimenA}{-#2\@IEEEtrantmpdimenB}%
+\fontdimen3\font=-\@IEEEtrantmpdimenA\relax
+\setlength{\@IEEEtrantmpdimenA}{#1\@IEEEtrantmpdimenB}%
+\addtolength{\@IEEEtrantmpdimenA}{-#3\@IEEEtrantmpdimenB}%
+\fontdimen4\font=\@IEEEtrantmpdimenA\relax}}
+
+% revise the interword spacing for each font weight
+\def\@@IEEEsetfontdimens{{%
+\mdseries
+\@@@IEEEsetfontdimens{\@IEEEinterspaceratioM}{\@IEEEinterspaceMAXratioM}{\@IEEEinterspaceMINratioM}%
+\bfseries
+\@@@IEEEsetfontdimens{\@IEEEinterspaceratioB}{\@IEEEinterspaceMAXratioB}{\@IEEEinterspaceMINratioB}%
+}}
+
+% revise the interword spacing for each font shape
+% \slshape is not often used for IEEE work and is not altered here. The \scshape caps are
+% already a tad too large in the free LaTeX fonts (as compared to what IEEE uses) so we
+% won't alter these either.
+\def\@IEEEsetfontdimens{{%
+\normalfont
+\@@IEEEsetfontdimens
+\normalfont\itshape
+\@@IEEEsetfontdimens
+}}
+
+% command to revise the interword spacing for each font size (and shape
+% and weight). Only the \rmfamily is done here as \ttfamily uses a
+% fixed spacing and \sffamily is not used as the main text of IEEE papers.
+\def\@IEEEtunefonts{{\selectfont\rmfamily
+\tiny\@IEEEsetfontdimens
+\scriptsize\@IEEEsetfontdimens
+\footnotesize\@IEEEsetfontdimens
+\small\@IEEEsetfontdimens
+\normalsize\@IEEEsetfontdimens
+\sublargesize\@IEEEsetfontdimens
+\large\@IEEEsetfontdimens
+\LARGE\@IEEEsetfontdimens
+\huge\@IEEEsetfontdimens
+\Huge\@IEEEsetfontdimens}}
+
+% if the nofonttune class option is not given, revise the interword spacing
+% now - in case IEEEtran makes any default length measurements, and make
+% sure all the default fonts are loaded
+\ifCLASSOPTIONnofonttune\else
+\@IEEEtunefonts
+\fi
+
+% and again at the start of the document in case the user loaded different fonts
+\AtBeginDocument{\ifCLASSOPTIONnofonttune\else\@IEEEtunefonts\fi}
+
+
+
+% V1.6
+% LaTeX is a little to quick to use hyphenations
+% So, we increase the penalty for their use and raise
+% the badness level that triggers an underfull hbox
+% warning. The author may still have to tweak things,
+% but the appearance will be much better "right out
+% of the box" than that under V1.5 and prior.
+% TeX default is 50
+\hyphenpenalty=750
+% If we didn't adjust the interword spacing, 2200 might be better.
+% The TeX default is 1000
+\hbadness=1350
+% IEEE does not use extra spacing after punctuation
+\frenchspacing
+
+% V1.7 increase this a tad to discourage equation breaks
+\binoppenalty=1000 % default 700
+\relpenalty=800 % default 500
+
+
+% margin note stuff
+\marginparsep 10pt
+\marginparwidth 20pt
+\marginparpush 25pt
+
+
+% if things get too close, go ahead and let them touch
+\lineskip 0pt
+\normallineskip 0pt
+\lineskiplimit 0pt
+\normallineskiplimit 0pt
+
+% The distance from the lower edge of the text body to the
+% footline
+\footskip 0.4in
+
+% normally zero, should be relative to font height.
+% put in a little rubber to help stop some bad breaks (underfull vboxes)
+\parskip 0ex plus 0.2ex minus 0.1ex
+
+\parindent 1.0em
+
+\topmargin -49.0pt
+\headheight 12pt
+\headsep 0.25in
+
+% use the normal font baselineskip
+% so that \topskip is unaffected by changes in \baselinestretch
+\topskip=\@IEEEnormalsizeunitybaselineskip
+\textheight 58pc % 9.63in, 696pt
+% Tweak textheight to a perfect integer number of lines/page.
+% The normal baselineskip for each document point size is used
+% to determine these values.
+\ifx\CLASSOPTIONpt\@IEEEptsizenine\textheight=63\@IEEEnormalsizeunitybaselineskip\fi % 63 lines/page
+\ifx\CLASSOPTIONpt\@IEEEptsizeten\textheight=58\@IEEEnormalsizeunitybaselineskip\fi % 58 lines/page
+\ifx\CLASSOPTIONpt\@IEEEptsizeeleven\textheight=52\@IEEEnormalsizeunitybaselineskip\fi % 52 lines/page
+\ifx\CLASSOPTIONpt\@IEEEptsizetwelve\textheight=50\@IEEEnormalsizeunitybaselineskip\fi % 50 lines/page
+
+
+\columnsep 1pc
+\textwidth 43pc % 2 x 21pc + 1pc = 43pc
+
+
+% the default side margins are equal
+\if@IEEEusingAfourpaper
+\oddsidemargin 14.32mm
+\evensidemargin 14.32mm
+\else
+\oddsidemargin 0.680in
+\evensidemargin 0.680in
+\fi
+% compensate for LaTeX's 1in offset
+\addtolength{\oddsidemargin}{-1in}
+\addtolength{\evensidemargin}{-1in}
+
+
+
+% adjust margins for conference mode
+\ifCLASSOPTIONconference
+ \topmargin -0.25in
+ % we retain the reserved, but unused space for headers
+ \addtolength{\topmargin}{-\headheight}
+ \addtolength{\topmargin}{-\headsep}
+ \textheight 9.25in % The standard for conferences (668.4975pt)
+ % Tweak textheight to a perfect integer number of lines/page.
+ \ifx\CLASSOPTIONpt\@IEEEptsizenine\textheight=61\@IEEEnormalsizeunitybaselineskip\fi % 61 lines/page
+ \ifx\CLASSOPTIONpt\@IEEEptsizeten\textheight=56\@IEEEnormalsizeunitybaselineskip\fi % 56 lines/page
+ \ifx\CLASSOPTIONpt\@IEEEptsizeeleven\textheight=50\@IEEEnormalsizeunitybaselineskip\fi % 50 lines/page
+ \ifx\CLASSOPTIONpt\@IEEEptsizetwelve\textheight=48\@IEEEnormalsizeunitybaselineskip\fi % 48 lines/page
+\fi
+
+
+% compsoc conference
+\ifCLASSOPTIONcompsoc
+\ifCLASSOPTIONconference
+ % compsoc conference use a larger value for columnsep
+ \columnsep 0.375in
+ % compsoc conferences want 1in top margin, 1.125in bottom margin
+ \topmargin 0in
+ \addtolength{\topmargin}{-6pt}% we tweak this a tad to better comply with top of line stuff
+ % we retain the reserved, but unused space for headers
+ \addtolength{\topmargin}{-\headheight}
+ \addtolength{\topmargin}{-\headsep}
+ \textheight 8.875in % (641.39625pt)
+ % Tweak textheight to a perfect integer number of lines/page.
+ \ifx\CLASSOPTIONpt\@IEEEptsizenine\textheight=58\@IEEEnormalsizeunitybaselineskip\fi % 58 lines/page
+ \ifx\CLASSOPTIONpt\@IEEEptsizeten\textheight=53\@IEEEnormalsizeunitybaselineskip\fi % 53 lines/page
+ \ifx\CLASSOPTIONpt\@IEEEptsizeeleven\textheight=48\@IEEEnormalsizeunitybaselineskip\fi % 48 lines/page
+ \ifx\CLASSOPTIONpt\@IEEEptsizetwelve\textheight=46\@IEEEnormalsizeunitybaselineskip\fi % 46 lines/page
+ \textwidth 6.5in
+ % the default side margins are equal
+ \if@IEEEusingAfourpaper
+ \oddsidemargin 22.45mm
+ \evensidemargin 22.45mm
+ \else
+ \oddsidemargin 1in
+ \evensidemargin 1in
+ \fi
+ % compensate for LaTeX's 1in offset
+ \addtolength{\oddsidemargin}{-1in}
+ \addtolength{\evensidemargin}{-1in}
+\fi\fi
+
+
+
+% draft mode settings override that of all other modes
+% provides a nice 1in margin all around the paper and extra
+% space between the lines for editor's comments
+\ifCLASSOPTIONdraftcls
+ % want 1in from top of paper to text
+ \setlength{\topmargin}{-\headsep}%
+ \addtolength{\topmargin}{-\headheight}%
+ % we want 1in side margins regardless of paper type
+ \oddsidemargin 0in
+ \evensidemargin 0in
+ % set the text width
+ \setlength{\textwidth}{\paperwidth}%
+ \addtolength{\textwidth}{-2.0in}%
+ \setlength{\textheight}{\paperheight}%
+ \addtolength{\textheight}{-2.0in}%
+ % digitize textheight to be an integer number of lines.
+ % this may cause the bottom margin to be off a tad
+ \addtolength{\textheight}{-1\topskip}%
+ \divide\textheight by \baselineskip%
+ \multiply\textheight by \baselineskip%
+ \addtolength{\textheight}{\topskip}%
+\fi
+
+
+
+% process CLASSINPUT inner/outer margin
+% if inner margin defined, but outer margin not, set outer to inner.
+\ifx\CLASSINPUTinnersidemargin\@IEEEundefined
+\else
+ \ifx\CLASSINPUToutersidemargin\@IEEEundefined
+ \edef\CLASSINPUToutersidemargin{\CLASSINPUTinnersidemargin}
+ \fi
+\fi
+
+\ifx\CLASSINPUToutersidemargin\@IEEEundefined
+\else
+ % if outer margin defined, but inner margin not, set inner to outer.
+ \ifx\CLASSINPUTinnersidemargin\@IEEEundefined
+ \edef\CLASSINPUTinnersidemargin{\CLASSINPUToutersidemargin}
+ \fi
+ \setlength{\oddsidemargin}{\CLASSINPUTinnersidemargin}
+ \ifCLASSOPTIONtwoside
+ \setlength{\evensidemargin}{\CLASSINPUToutersidemargin}
+ \else
+ \setlength{\evensidemargin}{\CLASSINPUTinnersidemargin}
+ \fi
+ \addtolength{\oddsidemargin}{-1in}
+ \addtolength{\evensidemargin}{-1in}
+ \setlength{\textwidth}{\paperwidth}
+ \addtolength{\textwidth}{-\CLASSINPUTinnersidemargin}
+ \addtolength{\textwidth}{-\CLASSINPUToutersidemargin}
+ \typeout{** ATTENTION: Overriding inner side margin to \CLASSINPUTinnersidemargin\space and
+ outer side margin to \CLASSINPUToutersidemargin\space via \string\CLASSINPUT.}
+\fi
+
+
+
+% process CLASSINPUT top/bottom text margin
+% if toptext margin defined, but bottomtext margin not, set bottomtext to toptext margin
+\ifx\CLASSINPUTtoptextmargin\@IEEEundefined
+\else
+ \ifx\CLASSINPUTbottomtextmargin\@IEEEundefined
+ \edef\CLASSINPUTbottomtextmargin{\CLASSINPUTtoptextmargin}
+ \fi
+\fi
+
+\ifx\CLASSINPUTbottomtextmargin\@IEEEundefined
+\else
+ % if bottomtext margin defined, but toptext margin not, set toptext to bottomtext margin
+ \ifx\CLASSINPUTtoptextmargin\@IEEEundefined
+ \edef\CLASSINPUTtoptextmargin{\CLASSINPUTbottomtextmargin}
+ \fi
+ \setlength{\topmargin}{\CLASSINPUTtoptextmargin}
+ \addtolength{\topmargin}{-1in}
+ \addtolength{\topmargin}{-\headheight}
+ \addtolength{\topmargin}{-\headsep}
+ \setlength{\textheight}{\paperheight}
+ \addtolength{\textheight}{-\CLASSINPUTtoptextmargin}
+ \addtolength{\textheight}{-\CLASSINPUTbottomtextmargin}
+ % in the default format we use the normal baselineskip as topskip
+ % we only need 0.7 of this to clear typical top text and we need
+ % an extra 0.3 spacing at the bottom for descenders. This will
+ % correct for both.
+ \addtolength{\topmargin}{-0.3\@IEEEnormalsizeunitybaselineskip}
+ \typeout{** ATTENTION: Overriding top text margin to \CLASSINPUTtoptextmargin\space and
+ bottom text margin to \CLASSINPUTbottomtextmargin\space via \string\CLASSINPUT.}
+\fi
+
+
+
+
+
+
+
+% LIST SPACING CONTROLS
+
+% Controls the amount of EXTRA spacing
+% above and below \trivlist
+% Both \list and IED lists override this.
+% However, \trivlist will use this as will most
+% things built from \trivlist like the \center
+% environment.
+\topsep 0.5\baselineskip
+
+% Controls the additional spacing around lists preceded
+% or followed by blank lines. IEEE does not increase
+% spacing before or after paragraphs so it is set to zero.
+% \z@ is the same as zero, but faster.
+\partopsep \z@
+
+% Controls the spacing between paragraphs in lists.
+% IEEE does not increase spacing before or after paragraphs
+% so this is also zero.
+% With IEEEtran.cls, global changes to
+% this value DO affect lists (but not IED lists).
+\parsep \z@
+
+% Controls the extra spacing between list items.
+% IEEE does not put extra spacing between items.
+% With IEEEtran.cls, global changes to this value DO affect
+% lists (but not IED lists).
+\itemsep \z@
+
+% \itemindent is the amount to indent the FIRST line of a list
+% item. It is auto set to zero within the \list environment. To alter
+% it, you have to do so when you call the \list.
+% However, IEEE uses this for the theorem environment
+% There is an alternative value for this near \leftmargini below
+\itemindent -1em
+
+% \leftmargin, the spacing from the left margin of the main text to
+% the left of the main body of a list item is set by \list.
+% Hence this statement does nothing for lists.
+% But, quote and verse do use it for indention.
+\leftmargin 2em
+
+% we retain this stuff from the older IEEEtran.cls so that \list
+% will work the same way as before. However, itemize, enumerate and
+% description (IED) could care less about what these are as they
+% all are overridden.
+\leftmargini 2em
+%\itemindent 2em % Alternative values: sometimes used.
+%\leftmargini 0em
+\leftmarginii 1em
+\leftmarginiii 1.5em
+\leftmarginiv 1.5em
+\leftmarginv 1.0em
+\leftmarginvi 1.0em
+\labelsep 0.5em
+\labelwidth \z@
+
+
+% The old IEEEtran.cls behavior of \list is retained.
+% However, the new V1.3 IED list environments override all the
+% @list stuff (\@listX is called within \list for the
+% appropriate level just before the user's list_decl is called).
+% \topsep is now 2pt as IEEE puts a little extra space around
+% lists - used by those non-IED macros that depend on \list.
+% Note that \parsep and \itemsep are not redefined as in
+% the sizexx.clo \@listX (which article.cls uses) so global changes
+% of these values DO affect \list
+%
+\def\@listi{\leftmargin\leftmargini \topsep 2pt plus 1pt minus 1pt}
+\let\@listI\@listi
+\def\@listii{\leftmargin\leftmarginii\labelwidth\leftmarginii%
+ \advance\labelwidth-\labelsep \topsep 2pt}
+\def\@listiii{\leftmargin\leftmarginiii\labelwidth\leftmarginiii%
+ \advance\labelwidth-\labelsep \topsep 2pt}
+\def\@listiv{\leftmargin\leftmarginiv\labelwidth\leftmarginiv%
+ \advance\labelwidth-\labelsep \topsep 2pt}
+\def\@listv{\leftmargin\leftmarginv\labelwidth\leftmarginv%
+ \advance\labelwidth-\labelsep \topsep 2pt}
+\def\@listvi{\leftmargin\leftmarginvi\labelwidth\leftmarginvi%
+ \advance\labelwidth-\labelsep \topsep 2pt}
+
+
+% IEEE uses 5) not 5.
+\def\labelenumi{\theenumi)} \def\theenumi{\arabic{enumi}}
+
+% IEEE uses a) not (a)
+\def\labelenumii{\theenumii)} \def\theenumii{\alph{enumii}}
+
+% IEEE uses iii) not iii.
+\def\labelenumiii{\theenumiii)} \def\theenumiii{\roman{enumiii}}
+
+% IEEE uses A) not A.
+\def\labelenumiv{\theenumiv)} \def\theenumiv{\Alph{enumiv}}
+
+% exactly the same as in article.cls
+\def\p@enumii{\theenumi}
+\def\p@enumiii{\theenumi(\theenumii)}
+\def\p@enumiv{\p@enumiii\theenumiii}
+
+% itemized list label styles
+\def\labelitemi{$\scriptstyle\bullet$}
+\def\labelitemii{\textbf{--}}
+\def\labelitemiii{$\ast$}
+\def\labelitemiv{$\cdot$}
+
+
+
+% **** V1.3 ENHANCEMENTS ****
+% Itemize, Enumerate and Description (IED) List Controls
+% ***************************
+%
+%
+% IEEE seems to use at least two different values by
+% which ITEMIZED list labels are indented to the right
+% For The Journal of Lightwave Technology (JLT) and The Journal
+% on Selected Areas in Communications (JSAC), they tend to use
+% an indention equal to \parindent. For Transactions on Communications
+% they tend to indent ITEMIZED lists a little more--- 1.3\parindent.
+% We'll provide both values here for you so that you can choose
+% which one you like in your document using a command such as:
+% setlength{\IEEEilabelindent}{\IEEEilabelindentB}
+\newdimen\IEEEilabelindentA
+\IEEEilabelindentA \parindent
+
+\newdimen\IEEEilabelindentB
+\IEEEilabelindentB 1.3\parindent
+% However, we'll default to using \parindent
+% which makes more sense to me
+\newdimen\IEEEilabelindent
+\IEEEilabelindent \IEEEilabelindentA
+
+
+% This controls the default amount the enumerated list labels
+% are indented to the right.
+% Normally, this is the same as the paragraph indention
+\newdimen\IEEEelabelindent
+\IEEEelabelindent \parindent
+
+% This controls the default amount the description list labels
+% are indented to the right.
+% Normally, this is the same as the paragraph indention
+\newdimen\IEEEdlabelindent
+\IEEEdlabelindent \parindent
+
+% This is the value actually used within the IED lists.
+% The IED environments automatically set its value to
+% one of the three values above, so global changes do
+% not have any effect
+\newdimen\IEEElabelindent
+\IEEElabelindent \parindent
+
+% The actual amount labels will be indented is
+% \IEEElabelindent multiplied by the factor below
+% corresponding to the level of nesting depth
+% This provides a means by which the user can
+% alter the effective \IEEElabelindent for deeper
+% levels
+% There may not be such a thing as correct "standard IEEE"
+% values. What IEEE actually does may depend on the specific
+% circumstances.
+% The first list level almost always has full indention.
+% The second levels I've seen have only 75% of the normal indentation
+% Three level or greater nestings are very rare. I am guessing
+% that they don't use any indentation.
+\def\IEEElabelindentfactori{1.0} % almost always one
+\def\IEEElabelindentfactorii{0.75} % 0.0 or 1.0 may be used in some cases
+\def\IEEElabelindentfactoriii{0.0} % 0.75? 0.5? 0.0?
+\def\IEEElabelindentfactoriv{0.0}
+\def\IEEElabelindentfactorv{0.0}
+\def\IEEElabelindentfactorvi{0.0}
+
+% value actually used within IED lists, it is auto
+% set to one of the 6 values above
+% global changes here have no effect
+\def\IEEElabelindentfactor{1.0}
+
+% This controls the default spacing between the end of the IED
+% list labels and the list text, when normal text is used for
+% the labels.
+\newdimen\IEEEiednormlabelsep
+\IEEEiednormlabelsep 0.6em
+
+% This controls the default spacing between the end of the IED
+% list labels and the list text, when math symbols are used for
+% the labels (nomenclature lists). IEEE usually increases the
+% spacing in these cases
+\newdimen\IEEEiedmathlabelsep
+\IEEEiedmathlabelsep 1.2em
+
+% This controls the extra vertical separation put above and
+% below each IED list. IEEE usually puts a little extra spacing
+% around each list. However, this spacing is barely noticeable.
+\newskip\IEEEiedtopsep
+\IEEEiedtopsep 2pt plus 1pt minus 1pt
+
+
+% This command is executed within each IED list environment
+% at the beginning of the list. You can use this to set the
+% parameters for some/all your IED list(s) without disturbing
+% global parameters that affect things other than lists.
+% i.e., renewcommand{\IEEEiedlistdecl}{\setlength{\labelsep}{5em}}
+% will alter the \labelsep for the next list(s) until
+% \IEEEiedlistdecl is redefined.
+\def\IEEEiedlistdecl{\relax}
+
+% This command provides an easy way to set \leftmargin based
+% on the \labelwidth, \labelsep and the argument \IEEElabelindent
+% Usage: \IEEEcalcleftmargin{width-to-indent-the-label}
+% output is in the \leftmargin variable, i.e., effectively:
+% \leftmargin = argument + \labelwidth + \labelsep
+% Note controlled spacing here, shield end of lines with %
+\def\IEEEcalcleftmargin#1{\setlength{\leftmargin}{#1}%
+\addtolength{\leftmargin}{\labelwidth}%
+\addtolength{\leftmargin}{\labelsep}}
+
+% This command provides an easy way to set \labelwidth to the
+% width of the given text. It is the same as
+% \settowidth{\labelwidth}{label-text}
+% and useful as a shorter alternative.
+% Typically used to set \labelwidth to be the width
+% of the longest label in the list
+\def\IEEEsetlabelwidth#1{\settowidth{\labelwidth}{#1}}
+
+% When this command is executed, IED lists will use the
+% IEEEiedmathlabelsep label separation rather than the normal
+% spacing. To have an effect, this command must be executed via
+% the \IEEEiedlistdecl or within the option of the IED list
+% environments.
+\def\IEEEusemathlabelsep{\setlength{\labelsep}{\IEEEiedmathlabelsep}}
+
+% A flag which controls whether the IED lists automatically
+% calculate \leftmargin from \IEEElabelindent, \labelwidth and \labelsep
+% Useful if you want to specify your own \leftmargin
+% This flag must be set (\IEEEnocalcleftmargintrue or \IEEEnocalcleftmarginfalse)
+% via the \IEEEiedlistdecl or within the option of the IED list
+% environments to have an effect.
+\newif\ifIEEEnocalcleftmargin
+\IEEEnocalcleftmarginfalse
+
+% A flag which controls whether \IEEElabelindent is multiplied by
+% the \IEEElabelindentfactor for each list level.
+% This flag must be set via the \IEEEiedlistdecl or within the option
+% of the IED list environments to have an effect.
+\newif\ifIEEEnolabelindentfactor
+\IEEEnolabelindentfactorfalse
+
+
+% internal variable to indicate type of IED label
+% justification
+% 0 - left; 1 - center; 2 - right
+\def\@IEEEiedjustify{0}
+
+
+% commands to allow the user to control IED
+% label justifications. Use these commands within
+% the IED environment option or in the \IEEEiedlistdecl
+% Note that changing the normal list justifications
+% is nonstandard and IEEE may not like it if you do so!
+% I include these commands as they may be helpful to
+% those who are using these enhanced list controls for
+% other non-IEEE related LaTeX work.
+% itemize and enumerate automatically default to right
+% justification, description defaults to left.
+\def\IEEEiedlabeljustifyl{\def\@IEEEiedjustify{0}}%left
+\def\IEEEiedlabeljustifyc{\def\@IEEEiedjustify{1}}%center
+\def\IEEEiedlabeljustifyr{\def\@IEEEiedjustify{2}}%right
+
+
+
+
+% commands to save to and restore from the list parameter copies
+% this allows us to set all the list parameters within
+% the list_decl and prevent \list (and its \@list)
+% from overriding any of our parameters
+% V1.6 use \edefs instead of dimen's to conserve dimen registers
+% Note controlled spacing here, shield end of lines with %
+\def\@IEEEsavelistparams{\edef\@IEEEiedtopsep{\the\topsep}%
+\edef\@IEEEiedlabelwidth{\the\labelwidth}%
+\edef\@IEEEiedlabelsep{\the\labelsep}%
+\edef\@IEEEiedleftmargin{\the\leftmargin}%
+\edef\@IEEEiedpartopsep{\the\partopsep}%
+\edef\@IEEEiedparsep{\the\parsep}%
+\edef\@IEEEieditemsep{\the\itemsep}%
+\edef\@IEEEiedrightmargin{\the\rightmargin}%
+\edef\@IEEEiedlistparindent{\the\listparindent}%
+\edef\@IEEEieditemindent{\the\itemindent}}
+
+% Note controlled spacing here
+\def\@IEEErestorelistparams{\topsep\@IEEEiedtopsep\relax%
+\labelwidth\@IEEEiedlabelwidth\relax%
+\labelsep\@IEEEiedlabelsep\relax%
+\leftmargin\@IEEEiedleftmargin\relax%
+\partopsep\@IEEEiedpartopsep\relax%
+\parsep\@IEEEiedparsep\relax%
+\itemsep\@IEEEieditemsep\relax%
+\rightmargin\@IEEEiedrightmargin\relax%
+\listparindent\@IEEEiedlistparindent\relax%
+\itemindent\@IEEEieditemindent\relax}
+
+
+% v1.6b provide original LaTeX IED list environments
+% note that latex.ltx defines \itemize and \enumerate, but not \description
+% which must be created by the base classes
+% save original LaTeX itemize and enumerate
+\let\LaTeXitemize\itemize
+\let\endLaTeXitemize\enditemize
+\let\LaTeXenumerate\enumerate
+\let\endLaTeXenumerate\endenumerate
+
+% provide original LaTeX description environment from article.cls
+\newenvironment{LaTeXdescription}
+ {\list{}{\labelwidth\z@ \itemindent-\leftmargin
+ \let\makelabel\descriptionlabel}}
+ {\endlist}
+\newcommand*\descriptionlabel[1]{\hspace\labelsep
+ \normalfont\bfseries #1}
+
+
+% override LaTeX's default IED lists
+\def\itemize{\@IEEEitemize}
+\def\enditemize{\@endIEEEitemize}
+\def\enumerate{\@IEEEenumerate}
+\def\endenumerate{\@endIEEEenumerate}
+\def\description{\@IEEEdescription}
+\def\enddescription{\@endIEEEdescription}
+
+% provide the user with aliases - may help those using packages that
+% override itemize, enumerate, or description
+\def\IEEEitemize{\@IEEEitemize}
+\def\endIEEEitemize{\@endIEEEitemize}
+\def\IEEEenumerate{\@IEEEenumerate}
+\def\endIEEEenumerate{\@endIEEEenumerate}
+\def\IEEEdescription{\@IEEEdescription}
+\def\endIEEEdescription{\@endIEEEdescription}
+
+
+% V1.6 we want to keep the IEEEtran IED list definitions as our own internal
+% commands so they are protected against redefinition
+\def\@IEEEitemize{\@ifnextchar[{\@@IEEEitemize}{\@@IEEEitemize[\relax]}}
+\def\@IEEEenumerate{\@ifnextchar[{\@@IEEEenumerate}{\@@IEEEenumerate[\relax]}}
+\def\@IEEEdescription{\@ifnextchar[{\@@IEEEdescription}{\@@IEEEdescription[\relax]}}
+\def\@endIEEEitemize{\endlist}
+\def\@endIEEEenumerate{\endlist}
+\def\@endIEEEdescription{\endlist}
+
+
+% DO NOT ALLOW BLANK LINES TO BE IN THESE IED ENVIRONMENTS
+% AS THIS WILL FORCE NEW PARAGRAPHS AFTER THE IED LISTS
+% IEEEtran itemized list MDS 1/2001
+% Note controlled spacing here, shield end of lines with %
+\def\@@IEEEitemize[#1]{%
+ \ifnum\@itemdepth>3\relax\@toodeep\else%
+ \ifnum\@listdepth>5\relax\@toodeep\else%
+ \advance\@itemdepth\@ne%
+ \edef\@itemitem{labelitem\romannumeral\the\@itemdepth}%
+ % get the labelindentfactor for this level
+ \advance\@listdepth\@ne% we need to know what the level WILL be
+ \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+ \advance\@listdepth-\@ne% undo our increment
+ \def\@IEEEiedjustify{2}% right justified labels are default
+ % set other defaults
+ \IEEEnocalcleftmarginfalse%
+ \IEEEnolabelindentfactorfalse%
+ \topsep\IEEEiedtopsep%
+ \IEEElabelindent\IEEEilabelindent%
+ \labelsep\IEEEiednormlabelsep%
+ \partopsep 0ex%
+ \parsep 0ex%
+ \itemsep 0ex%
+ \rightmargin 0em%
+ \listparindent 0em%
+ \itemindent 0em%
+ % calculate the label width
+ % the user can override this later if
+ % they specified a \labelwidth
+ \settowidth{\labelwidth}{\csname labelitem\romannumeral\the\@itemdepth\endcsname}%
+ \@IEEEsavelistparams% save our list parameters
+ \list{\csname\@itemitem\endcsname}{%
+ \@IEEErestorelistparams% override any list{} changes
+ % to our globals
+ \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
+ \IEEEiedlistdecl% let user alter parameters
+ #1\relax%
+ % If the user has requested not to use the
+ % labelindent factor, don't revise \labelindent
+ \ifIEEEnolabelindentfactor\relax%
+ \else\IEEElabelindent=\IEEElabelindentfactor\labelindent%
+ \fi%
+ % Unless the user has requested otherwise,
+ % calculate our left margin based
+ % on \IEEElabelindent, \labelwidth and
+ % \labelsep
+ \ifIEEEnocalcleftmargin\relax%
+ \else\IEEEcalcleftmargin{\IEEElabelindent}%
+ \fi}\fi\fi}%
+
+
+% DO NOT ALLOW BLANK LINES TO BE IN THESE IED ENVIRONMENTS
+% AS THIS WILL FORCE NEW PARAGRAPHS AFTER THE IED LISTS
+% IEEEtran enumerate list MDS 1/2001
+% Note controlled spacing here, shield end of lines with %
+\def\@@IEEEenumerate[#1]{%
+ \ifnum\@enumdepth>3\relax\@toodeep\else%
+ \ifnum\@listdepth>5\relax\@toodeep\else%
+ \advance\@enumdepth\@ne%
+ \edef\@enumctr{enum\romannumeral\the\@enumdepth}%
+ % get the labelindentfactor for this level
+ \advance\@listdepth\@ne% we need to know what the level WILL be
+ \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+ \advance\@listdepth-\@ne% undo our increment
+ \def\@IEEEiedjustify{2}% right justified labels are default
+ % set other defaults
+ \IEEEnocalcleftmarginfalse%
+ \IEEEnolabelindentfactorfalse%
+ \topsep\IEEEiedtopsep%
+ \IEEElabelindent\IEEEelabelindent%
+ \labelsep\IEEEiednormlabelsep%
+ \partopsep 0ex%
+ \parsep 0ex%
+ \itemsep 0ex%
+ \rightmargin 0em%
+ \listparindent 0em%
+ \itemindent 0em%
+ % calculate the label width
+ % We'll set it to the width suitable for all labels using
+ % normalfont 1) to 9)
+ % The user can override this later
+ \settowidth{\labelwidth}{9)}%
+ \@IEEEsavelistparams% save our list parameters
+ \list{\csname label\@enumctr\endcsname}{\usecounter{\@enumctr}%
+ \@IEEErestorelistparams% override any list{} changes
+ % to our globals
+ \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
+ \IEEEiedlistdecl% let user alter parameters
+ #1\relax%
+ % If the user has requested not to use the
+ % IEEElabelindent factor, don't revise \IEEElabelindent
+ \ifIEEEnolabelindentfactor\relax%
+ \else\IEEElabelindent=\IEEElabelindentfactor\IEEElabelindent%
+ \fi%
+ % Unless the user has requested otherwise,
+ % calculate our left margin based
+ % on \IEEElabelindent, \labelwidth and
+ % \labelsep
+ \ifIEEEnocalcleftmargin\relax%
+ \else\IEEEcalcleftmargin{\IEEElabelindent}%
+ \fi}\fi\fi}%
+
+
+% DO NOT ALLOW BLANK LINES TO BE IN THESE IED ENVIRONMENTS
+% AS THIS WILL FORCE NEW PARAGRAPHS AFTER THE IED LISTS
+% IEEEtran description list MDS 1/2001
+% Note controlled spacing here, shield end of lines with %
+\def\@@IEEEdescription[#1]{%
+ \ifnum\@listdepth>5\relax\@toodeep\else%
+ % get the labelindentfactor for this level
+ \advance\@listdepth\@ne% we need to know what the level WILL be
+ \edef\IEEElabelindentfactor{\csname IEEElabelindentfactor\romannumeral\the\@listdepth\endcsname}%
+ \advance\@listdepth-\@ne% undo our increment
+ \def\@IEEEiedjustify{0}% left justified labels are default
+ % set other defaults
+ \IEEEnocalcleftmarginfalse%
+ \IEEEnolabelindentfactorfalse%
+ \topsep\IEEEiedtopsep%
+ \IEEElabelindent\IEEEdlabelindent%
+ % assume normal labelsep
+ \labelsep\IEEEiednormlabelsep%
+ \partopsep 0ex%
+ \parsep 0ex%
+ \itemsep 0ex%
+ \rightmargin 0em%
+ \listparindent 0em%
+ \itemindent 0em%
+ % Bogus label width in case the user forgets
+ % to set it.
+ % TIP: If you want to see what a variable's width is you
+ % can use the TeX command \showthe\width-variable to
+ % display it on the screen during compilation
+ % (This might be helpful to know when you need to find out
+ % which label is the widest)
+ \settowidth{\labelwidth}{Hello}%
+ \@IEEEsavelistparams% save our list parameters
+ \list{}{\@IEEErestorelistparams% override any list{} changes
+ % to our globals
+ \let\makelabel\@IEEEiedmakelabel% v1.6b setup \makelabel
+ \IEEEiedlistdecl% let user alter parameters
+ #1\relax%
+ % If the user has requested not to use the
+ % labelindent factor, don't revise \IEEElabelindent
+ \ifIEEEnolabelindentfactor\relax%
+ \else\IEEElabelindent=\IEEElabelindentfactor\IEEElabelindent%
+ \fi%
+ % Unless the user has requested otherwise,
+ % calculate our left margin based
+ % on \IEEElabelindent, \labelwidth and
+ % \labelsep
+ \ifIEEEnocalcleftmargin\relax%
+ \else\IEEEcalcleftmargin{\IEEElabelindent}\relax%
+ \fi}\fi}
+
+% v1.6b we use one makelabel that does justification as needed.
+\def\@IEEEiedmakelabel#1{\relax\if\@IEEEiedjustify 0\relax
+\makebox[\labelwidth][l]{\normalfont #1}\else
+\if\@IEEEiedjustify 1\relax
+\makebox[\labelwidth][c]{\normalfont #1}\else
+\makebox[\labelwidth][r]{\normalfont #1}\fi\fi}
+
+
+% VERSE and QUOTE
+% V1.7 define environments with newenvironment
+\newenvironment{verse}{\let\\=\@centercr
+ \list{}{\itemsep\z@ \itemindent -1.5em \listparindent \itemindent
+ \rightmargin\leftmargin\advance\leftmargin 1.5em}\item\relax}
+ {\endlist}
+\newenvironment{quotation}{\list{}{\listparindent 1.5em \itemindent\listparindent
+ \rightmargin\leftmargin \parsep 0pt plus 1pt}\item\relax}
+ {\endlist}
+\newenvironment{quote}{\list{}{\rightmargin\leftmargin}\item\relax}
+ {\endlist}
+
+
+% \titlepage
+% provided only for backward compatibility. \maketitle is the correct
+% way to create the title page.
+\newif\if@restonecol
+\def\titlepage{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn
+ \else \newpage \fi \thispagestyle{empty}\c@page\z@}
+\def\endtitlepage{\if@restonecol\twocolumn \else \newpage \fi}
+
+% standard values from article.cls
+\arraycolsep 5pt
+\arrayrulewidth .4pt
+\doublerulesep 2pt
+
+\tabcolsep 6pt
+\tabbingsep 0.5em
+
+
+%% FOOTNOTES
+%
+%\skip\footins 10pt plus 4pt minus 2pt
+% V1.6 respond to changes in font size
+% space added above the footnotes (if present)
+\skip\footins 0.9\baselineskip plus 0.4\baselineskip minus 0.2\baselineskip
+
+% V1.6, we need to make \footnotesep responsive to changes
+% in \baselineskip or strange spacings will result when in
+% draft mode. Here is a little LaTeX secret - \footnotesep
+% determines the height of an invisible strut that is placed
+% *above* the baseline of footnotes after the first. Since
+% LaTeX considers the space for characters to be 0.7/baselineskip
+% above the baseline and 0.3/baselineskip below it, we need to
+% use 0.7/baselineskip as a \footnotesep to maintain equal spacing
+% between all the lines of the footnotes. IEEE often uses a tad
+% more, so use 0.8\baselineskip. This slightly larger value also helps
+% the text to clear the footnote marks. Note that \thanks in IEEEtran
+% uses its own value of \footnotesep which is set in \maketitle.
+{\footnotesize
+\global\footnotesep 0.8\baselineskip}
+
+
+\skip\@mpfootins = \skip\footins
+\fboxsep = 3pt
+\fboxrule = .4pt
+% V1.6 use 1em, then use LaTeX2e's \@makefnmark
+% Note that IEEE normally *left* aligns the footnote marks, so we don't need
+% box resizing tricks here.
+\long\def\@makefntext#1{\parindent 1em\indent\hbox{\@makefnmark}#1}% V1.6 use 1em
+% V1.7 compsoc does not use superscipts for footnote marks
+\ifCLASSOPTIONcompsoc
+\def\@IEEEcompsocmakefnmark{\hbox{\normalfont\@thefnmark.\ }}
+\long\def\@makefntext#1{\parindent 1em\indent\hbox{\@IEEEcompsocmakefnmark}#1}
+\fi
+
+% IEEE does not use footnote rules
+\def\footnoterule{}
+
+% V1.7 for compsoc, IEEE uses a footnote rule only for \thanks. We devise a "one-shot"
+% system to implement this.
+\newif\if@IEEEenableoneshotfootnoterule
+\@IEEEenableoneshotfootnoterulefalse
+\ifCLASSOPTIONcompsoc
+\def\footnoterule{\relax\if@IEEEenableoneshotfootnoterule
+\kern-5pt
+\hbox to \columnwidth{\hfill\vrule width 0.5\columnwidth height 0.4pt\hfill}
+\kern4.6pt
+\global\@IEEEenableoneshotfootnoterulefalse
+\else
+\relax
+\fi}
+\fi
+
+% V1.6 do not allow LaTeX to break a footnote across multiple pages
+\interfootnotelinepenalty=10000
+
+% V1.6 discourage breaks within equations
+% Note that amsmath normally sets this to 10000,
+% but LaTeX2e normally uses 100.
+\interdisplaylinepenalty=2500
+
+% default allows section depth up to /paragraph
+\setcounter{secnumdepth}{4}
+
+% technotes do not allow /paragraph
+\ifCLASSOPTIONtechnote
+ \setcounter{secnumdepth}{3}
+\fi
+% neither do compsoc conferences
+\@IEEEcompsocconfonly{\setcounter{secnumdepth}{3}}
+
+
+\newcounter{section}
+\newcounter{subsection}[section]
+\newcounter{subsubsection}[subsection]
+\newcounter{paragraph}[subsubsection]
+
+% used only by IEEEtran's IEEEeqnarray as other packages may
+% have their own, different, implementations
+\newcounter{IEEEsubequation}[equation]
+
+% as shown when called by user from \ref, \label and in table of contents
+\def\theequation{\arabic{equation}} % 1
+\def\theIEEEsubequation{\theequation\alph{IEEEsubequation}} % 1a (used only by IEEEtran's IEEEeqnarray)
+\ifCLASSOPTIONcompsoc
+% compsoc is all arabic
+\def\thesection{\arabic{section}}
+\def\thesubsection{\thesection.\arabic{subsection}}
+\def\thesubsubsection{\thesubsection.\arabic{subsubsection}}
+\def\theparagraph{\thesubsubsection.\arabic{paragraph}}
+\else
+\def\thesection{\Roman{section}} % I
+% V1.7, \mbox prevents breaks around -
+\def\thesubsection{\mbox{\thesection-\Alph{subsection}}} % I-A
+% V1.7 use I-A1 format used by IEEE rather than I-A.1
+\def\thesubsubsection{\thesubsection\arabic{subsubsection}} % I-A1
+\def\theparagraph{\thesubsubsection\alph{paragraph}} % I-A1a
+\fi
+
+% From Heiko Oberdiek. Because of the \mbox in \thesubsection, we need to
+% tell hyperref to disable the \mbox command when making PDF bookmarks.
+% This done already with hyperref.sty version 6.74o and later, but
+% it will not hurt to do it here again for users of older versions.
+\@ifundefined{pdfstringdefPreHook}{\let\pdfstringdefPreHook\@empty}{}%
+\g@addto@macro\pdfstringdefPreHook{\let\mbox\relax}
+
+
+% Main text forms (how shown in main text headings)
+% V1.6, using \thesection in \thesectiondis allows changes
+% in the former to automatically appear in the latter
+\ifCLASSOPTIONcompsoc
+ \ifCLASSOPTIONconference% compsoc conference
+ \def\thesectiondis{\thesection.}
+ \def\thesubsectiondis{\thesectiondis\arabic{subsection}.}
+ \def\thesubsubsectiondis{\thesubsectiondis\arabic{subsubsection}.}
+ \def\theparagraphdis{\thesubsubsectiondis\arabic{paragraph}.}
+ \else% compsoc not conferencs
+ \def\thesectiondis{\thesection}
+ \def\thesubsectiondis{\thesectiondis.\arabic{subsection}}
+ \def\thesubsubsectiondis{\thesubsectiondis.\arabic{subsubsection}}
+ \def\theparagraphdis{\thesubsubsectiondis.\arabic{paragraph}}
+ \fi
+\else% not compsoc
+ \def\thesectiondis{\thesection.} % I.
+ \def\thesubsectiondis{\Alph{subsection}.} % B.
+ \def\thesubsubsectiondis{\arabic{subsubsection})} % 3)
+ \def\theparagraphdis{\alph{paragraph})} % d)
+\fi
+
+% just like LaTeX2e's \@eqnnum
+\def\theequationdis{{\normalfont \normalcolor (\theequation)}}% (1)
+% IEEEsubequation used only by IEEEtran's IEEEeqnarray
+\def\theIEEEsubequationdis{{\normalfont \normalcolor (\theIEEEsubequation)}}% (1a)
+% redirect LaTeX2e's equation number display and all that depend on
+% it, through IEEEtran's \theequationdis
+\def\@eqnnum{\theequationdis}
+
+
+
+% V1.7 provide string macros as article.cls does
+\def\contentsname{Contents}
+\def\listfigurename{List of Figures}
+\def\listtablename{List of Tables}
+\def\refname{References}
+\def\indexname{Index}
+\def\figurename{Fig.}
+\def\tablename{TABLE}
+\@IEEEcompsocconfonly{\def\figurename{Figure}\def\tablename{Table}}
+\def\partname{Part}
+\def\appendixname{Appendix}
+\def\abstractname{Abstract}
+% IEEE specific names
+\def\IEEEkeywordsname{Index Terms}
+\def\IEEEproofname{Proof}
+
+
+% LIST OF FIGURES AND TABLES AND TABLE OF CONTENTS
+%
+\def\@pnumwidth{1.55em}
+\def\@tocrmarg{2.55em}
+\def\@dotsep{4.5}
+\setcounter{tocdepth}{3}
+
+% adjusted some spacings here so that section numbers will not easily
+% collide with the section titles.
+% VIII; VIII-A; and VIII-A.1 are usually the worst offenders.
+% MDS 1/2001
+\def\tableofcontents{\section*{\contentsname}\@starttoc{toc}}
+\def\l@section#1#2{\addpenalty{\@secpenalty}\addvspace{1.0em plus 1pt}%
+ \@tempdima 2.75em \begingroup \parindent \z@ \rightskip \@pnumwidth%
+ \parfillskip-\@pnumwidth {\bfseries\leavevmode #1}\hfil\hbox to\@pnumwidth{\hss #2}\par%
+ \endgroup}
+% argument format #1:level, #2:labelindent,#3:labelsep
+\def\l@subsection{\@dottedtocline{2}{2.75em}{3.75em}}
+\def\l@subsubsection{\@dottedtocline{3}{6.5em}{4.5em}}
+% must provide \l@ defs for ALL sublevels EVEN if tocdepth
+% is such as they will not appear in the table of contents
+% these defs are how TOC knows what level these things are!
+\def\l@paragraph{\@dottedtocline{4}{6.5em}{5.5em}}
+\def\l@subparagraph{\@dottedtocline{5}{6.5em}{6.5em}}
+\def\listoffigures{\section*{\listfigurename}\@starttoc{lof}}
+\def\l@figure{\@dottedtocline{1}{0em}{2.75em}}
+\def\listoftables{\section*{\listtablename}\@starttoc{lot}}
+\let\l@table\l@figure
+
+
+%% Definitions for floats
+%%
+%% Normal Floats
+\floatsep 1\baselineskip plus 0.2\baselineskip minus 0.2\baselineskip
+\textfloatsep 1.7\baselineskip plus 0.2\baselineskip minus 0.4\baselineskip
+\@fptop 0pt plus 1fil
+\@fpsep 0.75\baselineskip plus 2fil
+\@fpbot 0pt plus 1fil
+\def\topfraction{0.9}
+\def\bottomfraction{0.4}
+\def\floatpagefraction{0.8}
+% V1.7, let top floats approach 90% of page
+\def\textfraction{0.1}
+
+%% Double Column Floats
+\dblfloatsep 1\baselineskip plus 0.2\baselineskip minus 0.2\baselineskip
+
+\dbltextfloatsep 1.7\baselineskip plus 0.2\baselineskip minus 0.4\baselineskip
+% Note that it would be nice if the rubber here actually worked in LaTeX2e.
+% There is a long standing limitation in LaTeX, first discovered (to the best
+% of my knowledge) by Alan Jeffrey in 1992. LaTeX ignores the stretchable
+% portion of \dbltextfloatsep, and as a result, double column figures can and
+% do result in an non-integer number of lines in the main text columns with
+% underfull vbox errors as a consequence. A post to comp.text.tex
+% by Donald Arseneau confirms that this had not yet been fixed in 1998.
+% IEEEtran V1.6 will fix this problem for you in the titles, but it doesn't
+% protect you from other double floats. Happy vspace'ing.
+
+\@dblfptop 0pt plus 1fil
+\@dblfpsep 0.75\baselineskip plus 2fil
+\@dblfpbot 0pt plus 1fil
+\def\dbltopfraction{0.8}
+\def\dblfloatpagefraction{0.8}
+\setcounter{dbltopnumber}{4}
+
+\intextsep 1\baselineskip plus 0.2\baselineskip minus 0.2\baselineskip
+\setcounter{topnumber}{2}
+\setcounter{bottomnumber}{2}
+\setcounter{totalnumber}{4}
+
+
+
+% article class provides these, we should too.
+\newlength\abovecaptionskip
+\newlength\belowcaptionskip
+% but only \abovecaptionskip is used above figure captions and *below* table
+% captions
+\setlength\abovecaptionskip{0.5\baselineskip}
+\setlength\belowcaptionskip{0pt}
+% V1.6 create hooks in case the caption spacing ever needs to be
+% overridden by a user
+\def\@IEEEfigurecaptionsepspace{\vskip\abovecaptionskip\relax}%
+\def\@IEEEtablecaptionsepspace{\vskip\abovecaptionskip\relax}%
+
+
+% 1.6b revise caption system so that \@makecaption uses two arguments
+% as with LaTeX2e. Otherwise, there will be problems when using hyperref.
+\def\@IEEEtablestring{table}
+
+\ifCLASSOPTIONcompsoc
+% V1.7 compsoc \@makecaption
+\ifCLASSOPTIONconference% compsoc conference
+\long\def\@makecaption#1#2{%
+% test if is a for a figure or table
+\ifx\@captype\@IEEEtablestring%
+% if a table, do table caption
+\normalsize\begin{center}{\normalfont\sffamily\normalsize {#1.}~ #2}\end{center}%
+\@IEEEtablecaptionsepspace
+% if not a table, format it as a figure
+\else
+\@IEEEfigurecaptionsepspace
+\setbox\@tempboxa\hbox{\normalfont\sffamily\normalsize {#1.}~ #2}%
+\ifdim \wd\@tempboxa >\hsize%
+% if caption is longer than a line, let it wrap around
+\setbox\@tempboxa\hbox{\normalfont\sffamily\normalsize {#1.}~ }%
+\parbox[t]{\hsize}{\normalfont\sffamily\normalsize \noindent\unhbox\@tempboxa#2}%
+% if caption is shorter than a line, center
+\else%
+\hbox to\hsize{\normalfont\sffamily\normalsize\hfil\box\@tempboxa\hfil}%
+\fi\fi}
+\else% nonconference compsoc
+\long\def\@makecaption#1#2{%
+% test if is a for a figure or table
+\ifx\@captype\@IEEEtablestring%
+% if a table, do table caption
+\normalsize\begin{center}{\normalfont\sffamily\normalsize #1}\\{\normalfont\sffamily\normalsize #2}\end{center}%
+\@IEEEtablecaptionsepspace
+% if not a table, format it as a figure
+\else
+\@IEEEfigurecaptionsepspace
+\setbox\@tempboxa\hbox{\normalfont\sffamily\normalsize {#1.}~ #2}%
+\ifdim \wd\@tempboxa >\hsize%
+% if caption is longer than a line, let it wrap around
+\setbox\@tempboxa\hbox{\normalfont\sffamily\normalsize {#1.}~ }%
+\parbox[t]{\hsize}{\normalfont\sffamily\normalsize \noindent\unhbox\@tempboxa#2}%
+% if caption is shorter than a line, left justify
+\else%
+\hbox to\hsize{\normalfont\sffamily\normalsize\box\@tempboxa\hfil}%
+\fi\fi}
+\fi
+
+\else% traditional noncompsoc \@makecaption
+\long\def\@makecaption#1#2{%
+% test if is a for a figure or table
+\ifx\@captype\@IEEEtablestring%
+% if a table, do table caption
+\footnotesize\begin{center}{\normalfont\footnotesize #1}\\{\normalfont\footnotesize\scshape #2}\end{center}%
+\@IEEEtablecaptionsepspace
+% if not a table, format it as a figure
+\else
+\@IEEEfigurecaptionsepspace
+% 3/2001 use footnotesize, not small; use two nonbreaking spaces, not one
+\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}~~ #2}%
+\ifdim \wd\@tempboxa >\hsize%
+% if caption is longer than a line, let it wrap around
+\setbox\@tempboxa\hbox{\normalfont\footnotesize {#1.}~~ }%
+\parbox[t]{\hsize}{\normalfont\footnotesize\noindent\unhbox\@tempboxa#2}%
+% if caption is shorter than a line, center if conference, left justify otherwise
+\else%
+\ifCLASSOPTIONconference \hbox to\hsize{\normalfont\footnotesize\hfil\box\@tempboxa\hfil}%
+\else \hbox to\hsize{\normalfont\footnotesize\box\@tempboxa\hfil}%
+\fi\fi\fi}
+\fi
+
+
+
+% V1.7 disable captions class option, do so in a way that retains operation of \label
+% within \caption
+\ifCLASSOPTIONcaptionsoff
+\long\def\@makecaption#1#2{\vspace*{2em}\footnotesize\begin{center}{\footnotesize #1}\end{center}%
+\let\@IEEEtemporiglabeldefsave\label
+\let\@IEEEtemplabelargsave\relax
+\def\label##1{\gdef\@IEEEtemplabelargsave{##1}}%
+\setbox\@tempboxa\hbox{#2}%
+\let\label\@IEEEtemporiglabeldefsave
+\ifx\@IEEEtemplabelargsave\relax\else\label{\@IEEEtemplabelargsave}\fi}
+\fi
+
+
+% V1.7 define end environments with \def not \let so as to work OK with
+% preview-latex
+\newcounter{figure}
+\def\thefigure{\@arabic\c@figure}
+\def\fps@figure{tbp}
+\def\ftype@figure{1}
+\def\ext@figure{lof}
+\def\fnum@figure{\figurename~\thefigure}
+\def\figure{\@float{figure}}
+\def\endfigure{\end@float}
+\@namedef{figure*}{\@dblfloat{figure}}
+\@namedef{endfigure*}{\end@dblfloat}
+\newcounter{table}
+\ifCLASSOPTIONcompsoc
+\def\thetable{\arabic{table}}
+\else
+\def\thetable{\@Roman\c@table}
+\fi
+\def\fps@table{tbp}
+\def\ftype@table{2}
+\def\ext@table{lot}
+\def\fnum@table{\tablename~\thetable}
+% V1.6 IEEE uses 8pt text for tables
+% to default to footnotesize, we hack into LaTeX2e's \@floatboxreset and pray
+\def\table{\def\@floatboxreset{\reset@font\footnotesize\@setminipage}\@float{table}}
+\def\endtable{\end@float}
+% v1.6b double column tables need to default to footnotesize as well.
+\@namedef{table*}{\def\@floatboxreset{\reset@font\footnotesize\@setminipage}\@dblfloat{table}}
+\@namedef{endtable*}{\end@dblfloat}
+
+
+
+
+%%
+%% START OF IEEEeqnarry DEFINITIONS
+%%
+%% Inspired by the concepts, examples, and previous works of LaTeX
+%% coders and developers such as Donald Arseneau, Fred Bartlett,
+%% David Carlisle, Tony Liu, Frank Mittelbach, Piet van Oostrum,
+%% Roland Winkler and Mark Wooding.
+%% I don't make the claim that my work here is even near their calibre. ;)
+
+
+% hook to allow easy changeover to IEEEtran.cls/tools.sty error reporting
+\def\@IEEEclspkgerror{\ClassError{IEEEtran}}
+
+\newif\if@IEEEeqnarraystarform% flag to indicate if the environment was called as the star form
+\@IEEEeqnarraystarformfalse
+
+\newif\if@advanceIEEEeqncolcnt% tracks if the environment should advance the col counter
+% allows a way to make an \IEEEeqnarraybox that can be used within an \IEEEeqnarray
+% used by IEEEeqnarraymulticol so that it can work properly in both
+\@advanceIEEEeqncolcnttrue
+
+\newcount\@IEEEeqnnumcols % tracks how many IEEEeqnarray cols are defined
+\newcount\@IEEEeqncolcnt % tracks how many IEEEeqnarray cols the user actually used
+
+
+% The default math style used by the columns
+\def\IEEEeqnarraymathstyle{\displaystyle}
+% The default text style used by the columns
+% default to using the current font
+\def\IEEEeqnarraytextstyle{\relax}
+
+% like the iedlistdecl but for \IEEEeqnarray
+\def\IEEEeqnarraydecl{\relax}
+\def\IEEEeqnarrayboxdecl{\relax}
+
+% \yesnumber is the opposite of \nonumber
+% a novel concept with the same def as the equationarray package
+% However, we give IEEE versions too since some LaTeX packages such as
+% the MDWtools mathenv.sty redefine \nonumber to something else.
+\providecommand{\yesnumber}{\global\@eqnswtrue}
+\def\IEEEyesnumber{\global\@eqnswtrue}
+\def\IEEEnonumber{\global\@eqnswfalse}
+
+
+\def\IEEEyessubnumber{\global\@IEEEissubequationtrue\global\@eqnswtrue%
+\if@IEEEeqnarrayISinner% only do something inside an IEEEeqnarray
+\if@IEEElastlinewassubequation\addtocounter{equation}{-1}\else\setcounter{IEEEsubequation}{1}\fi%
+\def\@currentlabel{\p@IEEEsubequation\theIEEEsubequation}\fi}
+
+% flag to indicate that an equation is a sub equation
+\newif\if@IEEEissubequation%
+\@IEEEissubequationfalse
+
+% allows users to "push away" equations that get too close to the equation numbers
+\def\IEEEeqnarraynumspace{\hphantom{\if@IEEEissubequation\theIEEEsubequationdis\else\theequationdis\fi}}
+
+% provides a way to span multiple columns within IEEEeqnarray environments
+% will consider \if@advanceIEEEeqncolcnt before globally advancing the
+% column counter - so as to work within \IEEEeqnarraybox
+% usage: \IEEEeqnarraymulticol{number cols. to span}{col type}{cell text}
+\long\def\IEEEeqnarraymulticol#1#2#3{\multispan{#1}%
+% check if column is defined
+\relax\expandafter\ifx\csname @IEEEeqnarraycolDEF#2\endcsname\@IEEEeqnarraycolisdefined%
+\csname @IEEEeqnarraycolPRE#2\endcsname#3\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST#2\endcsname%
+\else% if not, error and use default type
+\@IEEEclspkgerror{Invalid column type "#2" in \string\IEEEeqnarraymulticol.\MessageBreak
+Using a default centering column instead}%
+{You must define IEEEeqnarray column types before use.}%
+\csname @IEEEeqnarraycolPRE@IEEEdefault\endcsname#3\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST@IEEEdefault\endcsname%
+\fi%
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by #1\relax\fi}
+
+% like \omit, but maintains track of the column counter for \IEEEeqnarray
+\def\IEEEeqnarrayomit{\omit\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by 1\relax\fi}
+
+
+% provides a way to define a letter referenced column type
+% usage: \IEEEeqnarraydefcol{col. type letter/name}{pre insertion text}{post insertion text}
+\def\IEEEeqnarraydefcol#1#2#3{\expandafter\def\csname @IEEEeqnarraycolPRE#1\endcsname{#2}%
+\expandafter\def\csname @IEEEeqnarraycolPOST#1\endcsname{#3}%
+\expandafter\def\csname @IEEEeqnarraycolDEF#1\endcsname{1}}
+
+
+% provides a way to define a numerically referenced inter-column glue types
+% usage: \IEEEeqnarraydefcolsep{col. glue number}{glue definition}
+\def\IEEEeqnarraydefcolsep#1#2{\expandafter\def\csname @IEEEeqnarraycolSEP\romannumeral #1\endcsname{#2}%
+\expandafter\def\csname @IEEEeqnarraycolSEPDEF\romannumeral #1\endcsname{1}}
+
+
+\def\@IEEEeqnarraycolisdefined{1}% just a macro for 1, used for checking undefined column types
+
+
+% expands and appends the given argument to the \@IEEEtrantmptoksA token list
+% used to build up the \halign preamble
+\def\@IEEEappendtoksA#1{\edef\@@IEEEappendtoksA{\@IEEEtrantmptoksA={\the\@IEEEtrantmptoksA #1}}%
+\@@IEEEappendtoksA}
+
+% also appends to \@IEEEtrantmptoksA, but does not expand the argument
+% uses \toks8 as a scratchpad register
+\def\@IEEEappendNOEXPANDtoksA#1{\toks8={#1}%
+\edef\@@IEEEappendNOEXPANDtoksA{\@IEEEtrantmptoksA={\the\@IEEEtrantmptoksA\the\toks8}}%
+\@@IEEEappendNOEXPANDtoksA}
+
+% define some common column types for the user
+% math
+\IEEEeqnarraydefcol{l}{$\IEEEeqnarraymathstyle}{$\hfil}
+\IEEEeqnarraydefcol{c}{\hfil$\IEEEeqnarraymathstyle}{$\hfil}
+\IEEEeqnarraydefcol{r}{\hfil$\IEEEeqnarraymathstyle}{$}
+\IEEEeqnarraydefcol{L}{$\IEEEeqnarraymathstyle{}}{{}$\hfil}
+\IEEEeqnarraydefcol{C}{\hfil$\IEEEeqnarraymathstyle{}}{{}$\hfil}
+\IEEEeqnarraydefcol{R}{\hfil$\IEEEeqnarraymathstyle{}}{{}$}
+% text
+\IEEEeqnarraydefcol{s}{\IEEEeqnarraytextstyle}{\hfil}
+\IEEEeqnarraydefcol{t}{\hfil\IEEEeqnarraytextstyle}{\hfil}
+\IEEEeqnarraydefcol{u}{\hfil\IEEEeqnarraytextstyle}{}
+
+% vertical rules
+\IEEEeqnarraydefcol{v}{}{\vrule width\arrayrulewidth}
+\IEEEeqnarraydefcol{vv}{\vrule width\arrayrulewidth\hfil}{\hfil\vrule width\arrayrulewidth}
+\IEEEeqnarraydefcol{V}{}{\vrule width\arrayrulewidth\hskip\doublerulesep\vrule width\arrayrulewidth}
+\IEEEeqnarraydefcol{VV}{\vrule width\arrayrulewidth\hskip\doublerulesep\vrule width\arrayrulewidth\hfil}%
+{\hfil\vrule width\arrayrulewidth\hskip\doublerulesep\vrule width\arrayrulewidth}
+
+% horizontal rules
+\IEEEeqnarraydefcol{h}{}{\leaders\hrule height\arrayrulewidth\hfil}
+\IEEEeqnarraydefcol{H}{}{\leaders\vbox{\hrule width\arrayrulewidth\vskip\doublerulesep\hrule width\arrayrulewidth}\hfil}
+
+% plain
+\IEEEeqnarraydefcol{x}{}{}
+\IEEEeqnarraydefcol{X}{$}{$}
+
+% the default column type to use in the event a column type is not defined
+\IEEEeqnarraydefcol{@IEEEdefault}{\hfil$\IEEEeqnarraymathstyle}{$\hfil}
+
+
+% a zero tabskip (used for "-" col types)
+\def\@IEEEeqnarraycolSEPzero{0pt plus 0pt minus 0pt}
+% a centering tabskip (used for "+" col types)
+\def\@IEEEeqnarraycolSEPcenter{1000pt plus 0pt minus 1000pt}
+
+% top level default tabskip glues for the start, end, and inter-column
+% may be reset within environments not always at the top level, e.g., \IEEEeqnarraybox
+\edef\@IEEEeqnarraycolSEPdefaultstart{\@IEEEeqnarraycolSEPcenter}% default start glue
+\edef\@IEEEeqnarraycolSEPdefaultend{\@IEEEeqnarraycolSEPcenter}% default end glue
+\edef\@IEEEeqnarraycolSEPdefaultmid{\@IEEEeqnarraycolSEPzero}% default inter-column glue
+
+
+
+% creates a vertical rule that extends from the bottom to the top a a cell
+% Provided in case other packages redefine \vline some other way.
+% usage: \IEEEeqnarrayvrule[rule thickness]
+% If no argument is provided, \arrayrulewidth will be used for the rule thickness.
+\newcommand\IEEEeqnarrayvrule[1][\arrayrulewidth]{\vrule\@width#1\relax}
+
+% creates a blank separator row
+% usage: \IEEEeqnarrayseprow[separation length][font size commands]
+% default is \IEEEeqnarrayseprow[0.25\normalbaselineskip][\relax]
+% blank arguments inherit the default values
+% uses \skip5 as a scratch register - calls \@IEEEeqnarraystrutsize which uses more scratch registers
+\def\IEEEeqnarrayseprow{\relax\@ifnextchar[{\@IEEEeqnarrayseprow}{\@IEEEeqnarrayseprow[0.25\normalbaselineskip]}}
+\def\@IEEEeqnarrayseprow[#1]{\relax\@ifnextchar[{\@@IEEEeqnarrayseprow[#1]}{\@@IEEEeqnarrayseprow[#1][\relax]}}
+\def\@@IEEEeqnarrayseprow[#1][#2]{\def\@IEEEeqnarrayseprowARGONE{#1}%
+\ifx\@IEEEeqnarrayseprowARGONE\@empty%
+% get the skip value, based on the font commands
+% use skip5 because \IEEEeqnarraystrutsize uses \skip0, \skip2, \skip3
+% assign within a bogus box to confine the font changes
+{\setbox0=\hbox{#2\relax\global\skip5=0.25\normalbaselineskip}}%
+\else%
+{\setbox0=\hbox{#2\relax\global\skip5=#1}}%
+\fi%
+\@IEEEeqnarrayhoptolastcolumn\IEEEeqnarraystrutsize{\skip5}{0pt}[\relax]\relax}
+
+% creates a blank separator row, but omits all the column templates
+% usage: \IEEEeqnarrayseprowcut[separation length][font size commands]
+% default is \IEEEeqnarrayseprowcut[0.25\normalbaselineskip][\relax]
+% blank arguments inherit the default values
+% uses \skip5 as a scratch register - calls \@IEEEeqnarraystrutsize which uses more scratch registers
+\def\IEEEeqnarrayseprowcut{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\@ifnextchar[{\@IEEEeqnarrayseprowcut}{\@IEEEeqnarrayseprowcut[0.25\normalbaselineskip]}}
+\def\@IEEEeqnarrayseprowcut[#1]{\relax\@ifnextchar[{\@@IEEEeqnarrayseprowcut[#1]}{\@@IEEEeqnarrayseprowcut[#1][\relax]}}
+\def\@@IEEEeqnarrayseprowcut[#1][#2]{\def\@IEEEeqnarrayseprowARGONE{#1}%
+\ifx\@IEEEeqnarrayseprowARGONE\@empty%
+% get the skip value, based on the font commands
+% use skip5 because \IEEEeqnarraystrutsize uses \skip0, \skip2, \skip3
+% assign within a bogus box to confine the font changes
+{\setbox0=\hbox{#2\relax\global\skip5=0.25\normalbaselineskip}}%
+\else%
+{\setbox0=\hbox{#2\relax\global\skip5=#1}}%
+\fi%
+\IEEEeqnarraystrutsize{\skip5}{0pt}[\relax]\relax}
+
+
+
+% draws a single rule across all the columns optional
+% argument determines the rule width, \arrayrulewidth is the default
+% updates column counter as needed and turns off struts
+% usage: \IEEEeqnarrayrulerow[rule line thickness]
+\def\IEEEeqnarrayrulerow{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\@ifnextchar[{\@IEEEeqnarrayrulerow}{\@IEEEeqnarrayrulerow[\arrayrulewidth]}}
+\def\@IEEEeqnarrayrulerow[#1]{\leaders\hrule height#1\hfil\relax% put in our rule
+% turn off any struts
+\IEEEeqnarraystrutsize{0pt}{0pt}[\relax]\relax}
+
+
+% draws a double rule by using a single rule row, a separator row, and then
+% another single rule row
+% first optional argument determines the rule thicknesses, \arrayrulewidth is the default
+% second optional argument determines the rule spacing, \doublerulesep is the default
+% usage: \IEEEeqnarraydblrulerow[rule line thickness][rule spacing]
+\def\IEEEeqnarraydblrulerow{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\@ifnextchar[{\@IEEEeqnarraydblrulerow}{\@IEEEeqnarraydblrulerow[\arrayrulewidth]}}
+\def\@IEEEeqnarraydblrulerow[#1]{\relax\@ifnextchar[{\@@IEEEeqnarraydblrulerow[#1]}%
+{\@@IEEEeqnarraydblrulerow[#1][\doublerulesep]}}
+\def\@@IEEEeqnarraydblrulerow[#1][#2]{\def\@IEEEeqnarraydblrulerowARG{#1}%
+% we allow the user to say \IEEEeqnarraydblrulerow[][]
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\@IEEEeqnarrayrulerow[\arrayrulewidth]%
+\else%
+\@IEEEeqnarrayrulerow[#1]\relax%
+\fi%
+\def\@IEEEeqnarraydblrulerowARG{#2}%
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\\\IEEEeqnarrayseprow[\doublerulesep][\relax]%
+\else%
+\\\IEEEeqnarrayseprow[#2][\relax]%
+\fi%
+\\\multispan{\@IEEEeqnnumcols}%
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\def\@IEEEeqnarraydblrulerowARG{#1}%
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\@IEEEeqnarrayrulerow[\arrayrulewidth]%
+\else%
+\@IEEEeqnarrayrulerow[#1]%
+\fi%
+}
+
+% draws a double rule by using a single rule row, a separator (cutting) row, and then
+% another single rule row
+% first optional argument determines the rule thicknesses, \arrayrulewidth is the default
+% second optional argument determines the rule spacing, \doublerulesep is the default
+% usage: \IEEEeqnarraydblrulerow[rule line thickness][rule spacing]
+\def\IEEEeqnarraydblrulerowcut{\multispan{\@IEEEeqnnumcols}\relax% span all the cols
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\@ifnextchar[{\@IEEEeqnarraydblrulerowcut}{\@IEEEeqnarraydblrulerowcut[\arrayrulewidth]}}
+\def\@IEEEeqnarraydblrulerowcut[#1]{\relax\@ifnextchar[{\@@IEEEeqnarraydblrulerowcut[#1]}%
+{\@@IEEEeqnarraydblrulerowcut[#1][\doublerulesep]}}
+\def\@@IEEEeqnarraydblrulerowcut[#1][#2]{\def\@IEEEeqnarraydblrulerowARG{#1}%
+% we allow the user to say \IEEEeqnarraydblrulerow[][]
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\@IEEEeqnarrayrulerow[\arrayrulewidth]%
+\else%
+\@IEEEeqnarrayrulerow[#1]%
+\fi%
+\def\@IEEEeqnarraydblrulerowARG{#2}%
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\\\IEEEeqnarrayseprowcut[\doublerulesep][\relax]%
+\else%
+\\\IEEEeqnarrayseprowcut[#2][\relax]%
+\fi%
+\\\multispan{\@IEEEeqnnumcols}%
+% advance column counter only if the IEEEeqnarray environment wants it
+\if@advanceIEEEeqncolcnt\global\advance\@IEEEeqncolcnt by \@IEEEeqnnumcols\relax\fi%
+\def\@IEEEeqnarraydblrulerowARG{#1}%
+\ifx\@IEEEeqnarraydblrulerowARG\@empty%
+\@IEEEeqnarrayrulerow[\arrayrulewidth]%
+\else%
+\@IEEEeqnarrayrulerow[#1]%
+\fi%
+}
+
+
+
+% inserts a full row's worth of &'s
+% relies on \@IEEEeqnnumcols to provide the correct number of columns
+% uses \@IEEEtrantmptoksA, \count0 as scratch registers
+\def\@IEEEeqnarrayhoptolastcolumn{\@IEEEtrantmptoksA={}\count0=1\relax%
+\loop% add cols if the user did not use them all
+\ifnum\count0<\@IEEEeqnnumcols\relax%
+\@IEEEappendtoksA{&}%
+\advance\count0 by 1\relax% update the col count
+\repeat%
+\the\@IEEEtrantmptoksA%execute the &'s
+}
+
+
+
+\newif\if@IEEEeqnarrayISinner % flag to indicate if we are within the lines
+\@IEEEeqnarrayISinnerfalse % of an IEEEeqnarray - after the IEEEeqnarraydecl
+
+\edef\@IEEEeqnarrayTHEstrutheight{0pt} % height and depth of IEEEeqnarray struts
+\edef\@IEEEeqnarrayTHEstrutdepth{0pt}
+
+\edef\@IEEEeqnarrayTHEmasterstrutheight{0pt} % default height and depth of
+\edef\@IEEEeqnarrayTHEmasterstrutdepth{0pt} % struts within an IEEEeqnarray
+
+\edef\@IEEEeqnarrayTHEmasterstrutHSAVE{0pt} % saved master strut height
+\edef\@IEEEeqnarrayTHEmasterstrutDSAVE{0pt} % and depth
+
+\newif\if@IEEEeqnarrayusemasterstrut % flag to indicate that the master strut value
+\@IEEEeqnarrayusemasterstruttrue % is to be used
+
+
+
+% saves the strut height and depth of the master strut
+\def\@IEEEeqnarraymasterstrutsave{\relax%
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+% remove stretchability
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% save values
+\edef\@IEEEeqnarrayTHEmasterstrutHSAVE{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEmasterstrutDSAVE{\the\dimen2}}
+
+% restores the strut height and depth of the master strut
+\def\@IEEEeqnarraymasterstrutrestore{\relax%
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutHSAVE\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutDSAVE\relax%
+% remove stretchability
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% restore values
+\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}}
+
+
+% globally restores the strut height and depth to the
+% master values and sets the master strut flag to true
+\def\@IEEEeqnarraystrutreset{\relax%
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+% remove stretchability
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% restore values
+\xdef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\xdef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\global\@IEEEeqnarrayusemasterstruttrue}
+
+
+% if the master strut is not to be used, make the current
+% values of \@IEEEeqnarrayTHEstrutheight, \@IEEEeqnarrayTHEstrutdepth
+% and the use master strut flag, global
+% this allows user strut commands issued in the last column to be carried
+% into the isolation/strut column
+\def\@IEEEeqnarrayglobalizestrutstatus{\relax%
+\if@IEEEeqnarrayusemasterstrut\else%
+\xdef\@IEEEeqnarrayTHEstrutheight{\@IEEEeqnarrayTHEstrutheight}%
+\xdef\@IEEEeqnarrayTHEstrutdepth{\@IEEEeqnarrayTHEstrutdepth}%
+\global\@IEEEeqnarrayusemasterstrutfalse%
+\fi}
+
+
+
+% usage: \IEEEeqnarraystrutsize{height}{depth}[font size commands]
+% If called outside the lines of an IEEEeqnarray, sets the height
+% and depth of both the master and local struts. If called inside
+% an IEEEeqnarray line, sets the height and depth of the local strut
+% only and sets the flag to indicate the use of the local strut
+% values. If the height or depth is left blank, 0.7\normalbaselineskip
+% and 0.3\normalbaselineskip will be used, respectively.
+% The optional argument can be used to evaluate the lengths under
+% a different font size and styles. If none is specified, the current
+% font is used.
+% uses scratch registers \skip0, \skip2, \skip3, \dimen0, \dimen2
+\def\IEEEeqnarraystrutsize#1#2{\relax\@ifnextchar[{\@IEEEeqnarraystrutsize{#1}{#2}}{\@IEEEeqnarraystrutsize{#1}{#2}[\relax]}}
+\def\@IEEEeqnarraystrutsize#1#2[#3]{\def\@IEEEeqnarraystrutsizeARG{#1}%
+\ifx\@IEEEeqnarraystrutsizeARG\@empty%
+{\setbox0=\hbox{#3\relax\global\skip3=0.7\normalbaselineskip}}%
+\skip0=\skip3\relax%
+\else% arg one present
+{\setbox0=\hbox{#3\relax\global\skip3=#1\relax}}%
+\skip0=\skip3\relax%
+\fi% if null arg
+\def\@IEEEeqnarraystrutsizeARG{#2}%
+\ifx\@IEEEeqnarraystrutsizeARG\@empty%
+{\setbox0=\hbox{#3\relax\global\skip3=0.3\normalbaselineskip}}%
+\skip2=\skip3\relax%
+\else% arg two present
+{\setbox0=\hbox{#3\relax\global\skip3=#2\relax}}%
+\skip2=\skip3\relax%
+\fi% if null arg
+% remove stretchability, just to be safe
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% dimen0 = height, dimen2 = depth
+\if@IEEEeqnarrayISinner% inner does not touch master strut size
+\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\@IEEEeqnarrayusemasterstrutfalse% do not use master
+\else% outer, have to set master strut too
+\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}%
+\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\@IEEEeqnarrayusemasterstruttrue% use master strut
+\fi}
+
+
+% usage: \IEEEeqnarraystrutsizeadd{added height}{added depth}[font size commands]
+% If called outside the lines of an IEEEeqnarray, adds the given height
+% and depth to both the master and local struts.
+% If called inside an IEEEeqnarray line, adds the given height and depth
+% to the local strut only and sets the flag to indicate the use
+% of the local strut values.
+% In both cases, if a height or depth is left blank, 0pt is used instead.
+% The optional argument can be used to evaluate the lengths under
+% a different font size and styles. If none is specified, the current
+% font is used.
+% uses scratch registers \skip0, \skip2, \skip3, \dimen0, \dimen2
+\def\IEEEeqnarraystrutsizeadd#1#2{\relax\@ifnextchar[{\@IEEEeqnarraystrutsizeadd{#1}{#2}}{\@IEEEeqnarraystrutsizeadd{#1}{#2}[\relax]}}
+\def\@IEEEeqnarraystrutsizeadd#1#2[#3]{\def\@IEEEeqnarraystrutsizearg{#1}%
+\ifx\@IEEEeqnarraystrutsizearg\@empty%
+\skip0=0pt\relax%
+\else% arg one present
+{\setbox0=\hbox{#3\relax\global\skip3=#1}}%
+\skip0=\skip3\relax%
+\fi% if null arg
+\def\@IEEEeqnarraystrutsizearg{#2}%
+\ifx\@IEEEeqnarraystrutsizearg\@empty%
+\skip2=0pt\relax%
+\else% arg two present
+{\setbox0=\hbox{#3\relax\global\skip3=#2}}%
+\skip2=\skip3\relax%
+\fi% if null arg
+% remove stretchability, just to be safe
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% dimen0 = height, dimen2 = depth
+\if@IEEEeqnarrayISinner% inner does not touch master strut size
+% get local strut size
+\expandafter\skip0=\@IEEEeqnarrayTHEstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEstrutdepth\relax%
+% add it to the user supplied values
+\advance\dimen0 by \skip0\relax%
+\advance\dimen2 by \skip2\relax%
+% update the local strut size
+\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\@IEEEeqnarrayusemasterstrutfalse% do not use master
+\else% outer, have to set master strut too
+% get master strut size
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+% add it to the user supplied values
+\advance\dimen0 by \skip0\relax%
+\advance\dimen2 by \skip2\relax%
+% update the local and master strut sizes
+\edef\@IEEEeqnarrayTHEmasterstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEmasterstrutdepth{\the\dimen2}%
+\edef\@IEEEeqnarrayTHEstrutheight{\the\dimen0}%
+\edef\@IEEEeqnarrayTHEstrutdepth{\the\dimen2}%
+\@IEEEeqnarrayusemasterstruttrue% use master strut
+\fi}
+
+
+% allow user a way to see the struts
+\newif\ifIEEEvisiblestruts
+\IEEEvisiblestrutsfalse
+
+% inserts an invisible strut using the master or local strut values
+% uses scratch registers \skip0, \skip2, \dimen0, \dimen2
+\def\@IEEEeqnarrayinsertstrut{\relax%
+\if@IEEEeqnarrayusemasterstrut
+% get master strut size
+\expandafter\skip0=\@IEEEeqnarrayTHEmasterstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEmasterstrutdepth\relax%
+\else%
+% get local strut size
+\expandafter\skip0=\@IEEEeqnarrayTHEstrutheight\relax%
+\expandafter\skip2=\@IEEEeqnarrayTHEstrutdepth\relax%
+\fi%
+% remove stretchability, probably not needed
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+% dimen0 = height, dimen2 = depth
+% allow user to see struts if desired
+\ifIEEEvisiblestruts%
+\vrule width0.2pt height\dimen0 depth\dimen2\relax%
+\else%
+\vrule width0pt height\dimen0 depth\dimen2\relax\fi}
+
+
+% creates an invisible strut, useable even outside \IEEEeqnarray
+% if \IEEEvisiblestrutstrue, the strut will be visible and 0.2pt wide.
+% usage: \IEEEstrut[height][depth][font size commands]
+% default is \IEEEstrut[0.7\normalbaselineskip][0.3\normalbaselineskip][\relax]
+% blank arguments inherit the default values
+% uses \dimen0, \dimen2, \skip0, \skip2
+\def\IEEEstrut{\relax\@ifnextchar[{\@IEEEstrut}{\@IEEEstrut[0.7\normalbaselineskip]}}
+\def\@IEEEstrut[#1]{\relax\@ifnextchar[{\@@IEEEstrut[#1]}{\@@IEEEstrut[#1][0.3\normalbaselineskip]}}
+\def\@@IEEEstrut[#1][#2]{\relax\@ifnextchar[{\@@@IEEEstrut[#1][#2]}{\@@@IEEEstrut[#1][#2][\relax]}}
+\def\@@@IEEEstrut[#1][#2][#3]{\mbox{#3\relax%
+\def\@IEEEstrutARG{#1}%
+\ifx\@IEEEstrutARG\@empty%
+\skip0=0.7\normalbaselineskip\relax%
+\else%
+\skip0=#1\relax%
+\fi%
+\def\@IEEEstrutARG{#2}%
+\ifx\@IEEEstrutARG\@empty%
+\skip2=0.3\normalbaselineskip\relax%
+\else%
+\skip2=#2\relax%
+\fi%
+% remove stretchability, probably not needed
+\dimen0\skip0\relax%
+\dimen2\skip2\relax%
+\ifIEEEvisiblestruts%
+\vrule width0.2pt height\dimen0 depth\dimen2\relax%
+\else%
+\vrule width0.0pt height\dimen0 depth\dimen2\relax\fi}}
+
+
+% enables strut mode by setting a default strut size and then zeroing the
+% \baselineskip, \lineskip, \lineskiplimit and \jot
+\def\IEEEeqnarraystrutmode{\IEEEeqnarraystrutsize{0.7\normalbaselineskip}{0.3\normalbaselineskip}[\relax]%
+\baselineskip=0pt\lineskip=0pt\lineskiplimit=0pt\jot=0pt}
+
+
+
+\def\IEEEeqnarray{\@IEEEeqnarraystarformfalse\@IEEEeqnarray}
+\def\endIEEEeqnarray{\end@IEEEeqnarray}
+
+\@namedef{IEEEeqnarray*}{\@IEEEeqnarraystarformtrue\@IEEEeqnarray}
+\@namedef{endIEEEeqnarray*}{\end@IEEEeqnarray}
+
+
+% \IEEEeqnarray is an enhanced \eqnarray.
+% The star form defaults to not putting equation numbers at the end of each row.
+% usage: \IEEEeqnarray[decl]{cols}
+\def\@IEEEeqnarray{\relax\@ifnextchar[{\@@IEEEeqnarray}{\@@IEEEeqnarray[\relax]}}
+\def\@@IEEEeqnarray[#1]#2{%
+ % default to showing the equation number or not based on whether or not
+ % the star form was involked
+ \if@IEEEeqnarraystarform\global\@eqnswfalse
+ \else% not the star form
+ \global\@eqnswtrue
+ \fi% if star form
+ \@IEEEissubequationfalse% default to no subequations
+ \@IEEElastlinewassubequationfalse% assume last line is not a sub equation
+ \@IEEEeqnarrayISinnerfalse% not yet within the lines of the halign
+ \@IEEEeqnarraystrutsize{0pt}{0pt}[\relax]% turn off struts by default
+ \@IEEEeqnarrayusemasterstruttrue% use master strut till user asks otherwise
+ \IEEEvisiblestrutsfalse% diagnostic mode defaults to off
+ % no extra space unless the user specifically requests it
+ \lineskip=0pt\relax
+ \lineskiplimit=0pt\relax
+ \baselineskip=\normalbaselineskip\relax%
+ \jot=\IEEEnormaljot\relax%
+ \mathsurround\z@\relax% no extra spacing around math
+ \@advanceIEEEeqncolcnttrue% advance the col counter for each col the user uses,
+ % used in \IEEEeqnarraymulticol and in the preamble build
+ \stepcounter{equation}% advance equation counter before first line
+ \setcounter{IEEEsubequation}{0}% no subequation yet
+ \def\@currentlabel{\p@equation\theequation}% redefine the ref label
+ \IEEEeqnarraydecl\relax% allow a way for the user to make global overrides
+ #1\relax% allow user to override defaults
+ \let\\\@IEEEeqnarraycr% replace newline with one that can put in eqn. numbers
+ \global\@IEEEeqncolcnt\z@% col. count = 0 for first line
+ \@IEEEbuildpreamble #2\end\relax% build the preamble and put it into \@IEEEtrantmptoksA
+ % put in the column for the equation number
+ \ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi% col separator for those after the first
+ \toks0={##}%
+ % advance the \@IEEEeqncolcnt for the isolation col, this helps with error checking
+ \@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}%
+ % add the isolation column
+ \@IEEEappendtoksA{\tabskip\z@skip\bgroup\the\toks0\egroup}%
+ % advance the \@IEEEeqncolcnt for the equation number col, this helps with error checking
+ \@IEEEappendtoksA{&\global\advance\@IEEEeqncolcnt by 1\relax}%
+ % add the equation number col to the preamble
+ \@IEEEappendtoksA{\tabskip\z@skip\hb@xt@\z@\bgroup\hss\the\toks0\egroup}%
+ % note \@IEEEeqnnumcols does not count the equation col or isolation col
+ % set the starting tabskip glue as determined by the preamble build
+ \tabskip=\@IEEEBPstartglue\relax
+ % begin the display alignment
+ \@IEEEeqnarrayISinnertrue% commands are now within the lines
+ $$\everycr{}\halign to\displaywidth\bgroup
+ % "exspand" the preamble
+ \span\the\@IEEEtrantmptoksA\cr}
+
+% enter isolation/strut column (or the next column if the user did not use
+% every column), record the strut status, complete the columns, do the strut if needed,
+% restore counters to correct values and exit
+\def\end@IEEEeqnarray{\@IEEEeqnarrayglobalizestrutstatus&\@@IEEEeqnarraycr\egroup%
+\if@IEEElastlinewassubequation\global\advance\c@IEEEsubequation\m@ne\fi%
+\global\advance\c@equation\m@ne%
+$$\@ignoretrue}
+
+% need a way to remember if last line is a subequation
+\newif\if@IEEElastlinewassubequation%
+\@IEEElastlinewassubequationfalse
+
+% IEEEeqnarray uses a modifed \\ instead of the plain \cr to
+% end rows. This allows for things like \\*[vskip amount]
+% This "cr" macros are modified versions those for LaTeX2e's eqnarray
+% the {\ifnum0=`} braces must be kept away from the last column to avoid
+% altering spacing of its math, so we use & to advance to the next column
+% as there is an isolation/strut column after the user's columns
+\def\@IEEEeqnarraycr{\@IEEEeqnarrayglobalizestrutstatus&% save strut status and advance to next column
+ {\ifnum0=`}\fi
+ \@ifstar{%
+ \global\@eqpen\@M\@IEEEeqnarrayYCR
+ }{%
+ \global\@eqpen\interdisplaylinepenalty \@IEEEeqnarrayYCR
+ }%
+}
+
+\def\@IEEEeqnarrayYCR{\@testopt\@IEEEeqnarrayXCR\z@skip}
+
+\def\@IEEEeqnarrayXCR[#1]{%
+ \ifnum0=`{\fi}%
+ \@@IEEEeqnarraycr
+ \noalign{\penalty\@eqpen\vskip\jot\vskip #1\relax}}%
+
+\def\@@IEEEeqnarraycr{\@IEEEtrantmptoksA={}% clear token register
+ \advance\@IEEEeqncolcnt by -1\relax% adjust col count because of the isolation column
+ \ifnum\@IEEEeqncolcnt>\@IEEEeqnnumcols\relax
+ \@IEEEclspkgerror{Too many columns within the IEEEeqnarray\MessageBreak
+ environment}%
+ {Use fewer \string &'s or put more columns in the IEEEeqnarry column\MessageBreak
+ specifications.}\relax%
+ \else
+ \loop% add cols if the user did not use them all
+ \ifnum\@IEEEeqncolcnt<\@IEEEeqnnumcols\relax
+ \@IEEEappendtoksA{&}%
+ \advance\@IEEEeqncolcnt by 1\relax% update the col count
+ \repeat
+ % this number of &'s will take us the the isolation column
+ \fi
+ % execute the &'s
+ \the\@IEEEtrantmptoksA%
+ % handle the strut/isolation column
+ \@IEEEeqnarrayinsertstrut% do the strut if needed
+ \@IEEEeqnarraystrutreset% reset the strut system for next line or IEEEeqnarray
+ &% and enter the equation number column
+ % is this line needs an equation number, display it and advance the
+ % (sub)equation counters, record what type this line was
+ \if@eqnsw%
+ \if@IEEEissubequation\theIEEEsubequationdis\addtocounter{equation}{1}\stepcounter{IEEEsubequation}%
+ \global\@IEEElastlinewassubequationtrue%
+ \else% display a standard equation number, initialize the IEEEsubequation counter
+ \theequationdis\stepcounter{equation}\setcounter{IEEEsubequation}{0}%
+ \global\@IEEElastlinewassubequationfalse\fi%
+ \fi%
+ % reset the eqnsw flag to indicate default preference of the display of equation numbers
+ \if@IEEEeqnarraystarform\global\@eqnswfalse\else\global\@eqnswtrue\fi
+ \global\@IEEEissubequationfalse% reset the subequation flag
+ % reset the number of columns the user actually used
+ \global\@IEEEeqncolcnt\z@\relax
+ % the real end of the line
+ \cr}
+
+
+
+
+
+% \IEEEeqnarraybox is like \IEEEeqnarray except the box form puts everything
+% inside a vtop, vbox, or vcenter box depending on the letter in the second
+% optional argument (t,b,c). Vbox is the default. Unlike \IEEEeqnarray,
+% equation numbers are not displayed and \IEEEeqnarraybox can be nested.
+% \IEEEeqnarrayboxm is for math mode (like \array) and does not put the vbox
+% within an hbox.
+% \IEEEeqnarrayboxt is for text mode (like \tabular) and puts the vbox within
+% a \hbox{$ $} construct.
+% \IEEEeqnarraybox will auto detect whether to use \IEEEeqnarrayboxm or
+% \IEEEeqnarrayboxt depending on the math mode.
+% The third optional argument specifies the width this box is to be set to -
+% natural width is the default.
+% The * forms do not add \jot line spacing
+% usage: \IEEEeqnarraybox[decl][pos][width]{cols}
+\def\IEEEeqnarrayboxm{\@IEEEeqnarraystarformfalse\@IEEEeqnarrayboxHBOXSWfalse\@IEEEeqnarraybox}
+\def\endIEEEeqnarrayboxm{\end@IEEEeqnarraybox}
+\@namedef{IEEEeqnarrayboxm*}{\@IEEEeqnarraystarformtrue\@IEEEeqnarrayboxHBOXSWfalse\@IEEEeqnarraybox}
+\@namedef{endIEEEeqnarrayboxm*}{\end@IEEEeqnarraybox}
+
+\def\IEEEeqnarrayboxt{\@IEEEeqnarraystarformfalse\@IEEEeqnarrayboxHBOXSWtrue\@IEEEeqnarraybox}
+\def\endIEEEeqnarrayboxt{\end@IEEEeqnarraybox}
+\@namedef{IEEEeqnarrayboxt*}{\@IEEEeqnarraystarformtrue\@IEEEeqnarrayboxHBOXSWtrue\@IEEEeqnarraybox}
+\@namedef{endIEEEeqnarrayboxt*}{\end@IEEEeqnarraybox}
+
+\def\IEEEeqnarraybox{\@IEEEeqnarraystarformfalse\ifmmode\@IEEEeqnarrayboxHBOXSWfalse\else\@IEEEeqnarrayboxHBOXSWtrue\fi%
+\@IEEEeqnarraybox}
+\def\endIEEEeqnarraybox{\end@IEEEeqnarraybox}
+
+\@namedef{IEEEeqnarraybox*}{\@IEEEeqnarraystarformtrue\ifmmode\@IEEEeqnarrayboxHBOXSWfalse\else\@IEEEeqnarrayboxHBOXSWtrue\fi%
+\@IEEEeqnarraybox}
+\@namedef{endIEEEeqnarraybox*}{\end@IEEEeqnarraybox}
+
+% flag to indicate if the \IEEEeqnarraybox needs to put things into an hbox{$ $}
+% for \vcenter in non-math mode
+\newif\if@IEEEeqnarrayboxHBOXSW%
+\@IEEEeqnarrayboxHBOXSWfalse
+
+\def\@IEEEeqnarraybox{\relax\@ifnextchar[{\@@IEEEeqnarraybox}{\@@IEEEeqnarraybox[\relax]}}
+\def\@@IEEEeqnarraybox[#1]{\relax\@ifnextchar[{\@@@IEEEeqnarraybox[#1]}{\@@@IEEEeqnarraybox[#1][b]}}
+\def\@@@IEEEeqnarraybox[#1][#2]{\relax\@ifnextchar[{\@@@@IEEEeqnarraybox[#1][#2]}{\@@@@IEEEeqnarraybox[#1][#2][\relax]}}
+
+% #1 = decl; #2 = t,b,c; #3 = width, #4 = col specs
+\def\@@@@IEEEeqnarraybox[#1][#2][#3]#4{\@IEEEeqnarrayISinnerfalse % not yet within the lines of the halign
+ \@IEEEeqnarraymasterstrutsave% save current master strut values
+ \@IEEEeqnarraystrutsize{0pt}{0pt}[\relax]% turn off struts by default
+ \@IEEEeqnarrayusemasterstruttrue% use master strut till user asks otherwise
+ \IEEEvisiblestrutsfalse% diagnostic mode defaults to off
+ % no extra space unless the user specifically requests it
+ \lineskip=0pt\relax%
+ \lineskiplimit=0pt\relax%
+ \baselineskip=\normalbaselineskip\relax%
+ \jot=\IEEEnormaljot\relax%
+ \mathsurround\z@\relax% no extra spacing around math
+ % the default end glues are zero for an \IEEEeqnarraybox
+ \edef\@IEEEeqnarraycolSEPdefaultstart{\@IEEEeqnarraycolSEPzero}% default start glue
+ \edef\@IEEEeqnarraycolSEPdefaultend{\@IEEEeqnarraycolSEPzero}% default end glue
+ \edef\@IEEEeqnarraycolSEPdefaultmid{\@IEEEeqnarraycolSEPzero}% default inter-column glue
+ \@advanceIEEEeqncolcntfalse% do not advance the col counter for each col the user uses,
+ % used in \IEEEeqnarraymulticol and in the preamble build
+ \IEEEeqnarrayboxdecl\relax% allow a way for the user to make global overrides
+ #1\relax% allow user to override defaults
+ \let\\\@IEEEeqnarrayboxcr% replace newline with one that allows optional spacing
+ \@IEEEbuildpreamble #4\end\relax% build the preamble and put it into \@IEEEtrantmptoksA
+ % add an isolation column to the preamble to stop \\'s {} from getting into the last col
+ \ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi% col separator for those after the first
+ \toks0={##}%
+ % add the isolation column to the preamble
+ \@IEEEappendtoksA{\tabskip\z@skip\bgroup\the\toks0\egroup}%
+ % set the starting tabskip glue as determined by the preamble build
+ \tabskip=\@IEEEBPstartglue\relax
+ % begin the alignment
+ \everycr{}%
+ % use only the very first token to determine the positioning
+ % this stops some problems when the user uses more than one letter,
+ % but is probably not worth the effort
+ % \noindent is used as a delimiter
+ \def\@IEEEgrabfirstoken##1##2\noindent{\let\@IEEEgrabbedfirstoken=##1}%
+ \@IEEEgrabfirstoken#2\relax\relax\noindent
+ % \@IEEEgrabbedfirstoken has the first token, the rest are discarded
+ % if we need to put things into and hbox and go into math mode, do so now
+ \if@IEEEeqnarrayboxHBOXSW \leavevmode \hbox \bgroup $\fi%
+ % use the appropriate vbox type
+ \if\@IEEEgrabbedfirstoken t\relax\vtop\else\if\@IEEEgrabbedfirstoken c\relax%
+ \vcenter\else\vbox\fi\fi\bgroup%
+ \@IEEEeqnarrayISinnertrue% commands are now within the lines
+ \ifx#3\relax\halign\else\halign to #3\relax\fi%
+ \bgroup
+ % "exspand" the preamble
+ \span\the\@IEEEtrantmptoksA\cr}
+
+% carry strut status and enter the isolation/strut column,
+% exit from math mode if needed, and exit
+\def\end@IEEEeqnarraybox{\@IEEEeqnarrayglobalizestrutstatus% carry strut status
+&% enter isolation/strut column
+\@IEEEeqnarrayinsertstrut% do strut if needed
+\@IEEEeqnarraymasterstrutrestore% restore the previous master strut values
+% reset the strut system for next IEEEeqnarray
+% (sets local strut values back to previous master strut values)
+\@IEEEeqnarraystrutreset%
+% ensure last line, exit from halign, close vbox
+\crcr\egroup\egroup%
+% exit from math mode and close hbox if needed
+\if@IEEEeqnarrayboxHBOXSW $\egroup\fi}
+
+
+
+% IEEEeqnarraybox uses a modifed \\ instead of the plain \cr to
+% end rows. This allows for things like \\[vskip amount]
+% This "cr" macros are modified versions those for LaTeX2e's eqnarray
+% For IEEEeqnarraybox, \\* is the same as \\
+% the {\ifnum0=`} braces must be kept away from the last column to avoid
+% altering spacing of its math, so we use & to advance to the isolation/strut column
+% carry strut status into isolation/strut column
+\def\@IEEEeqnarrayboxcr{\@IEEEeqnarrayglobalizestrutstatus% carry strut status
+&% enter isolation/strut column
+\@IEEEeqnarrayinsertstrut% do strut if needed
+% reset the strut system for next line or IEEEeqnarray
+\@IEEEeqnarraystrutreset%
+{\ifnum0=`}\fi%
+\@ifstar{\@IEEEeqnarrayboxYCR}{\@IEEEeqnarrayboxYCR}}
+
+% test and setup the optional argument to \\[]
+\def\@IEEEeqnarrayboxYCR{\@testopt\@IEEEeqnarrayboxXCR\z@skip}
+
+% IEEEeqnarraybox does not automatically increase line spacing by \jot
+\def\@IEEEeqnarrayboxXCR[#1]{\ifnum0=`{\fi}%
+\cr\noalign{\if@IEEEeqnarraystarform\else\vskip\jot\fi\vskip#1\relax}}
+
+
+
+% starts the halign preamble build
+\def\@IEEEbuildpreamble{\@IEEEtrantmptoksA={}% clear token register
+\let\@IEEEBPcurtype=u%current column type is not yet known
+\let\@IEEEBPprevtype=s%the previous column type was the start
+\let\@IEEEBPnexttype=u%next column type is not yet known
+% ensure these are valid
+\def\@IEEEBPcurglue={0pt plus 0pt minus 0pt}%
+\def\@IEEEBPcurcolname{@IEEEdefault}% name of current column definition
+% currently acquired numerically referenced glue
+% use a name that is easier to remember
+\let\@IEEEBPcurnum=\@IEEEtrantmpcountA%
+\@IEEEBPcurnum=0%
+% tracks number of columns in the preamble
+\@IEEEeqnnumcols=0%
+% record the default end glues
+\edef\@IEEEBPstartglue{\@IEEEeqnarraycolSEPdefaultstart}%
+\edef\@IEEEBPendglue{\@IEEEeqnarraycolSEPdefaultend}%
+% now parse the user's column specifications
+\@@IEEEbuildpreamble}
+
+
+% parses and builds the halign preamble
+\def\@@IEEEbuildpreamble#1#2{\let\@@nextIEEEbuildpreamble=\@@IEEEbuildpreamble%
+% use only the very first token to check the end
+% \noindent is used as a delimiter as \end can be present here
+\def\@IEEEgrabfirstoken##1##2\noindent{\let\@IEEEgrabbedfirstoken=##1}%
+\@IEEEgrabfirstoken#1\relax\relax\noindent
+\ifx\@IEEEgrabbedfirstoken\end\let\@@nextIEEEbuildpreamble=\@@IEEEfinishpreamble\else%
+% identify current and next token type
+\@IEEEgetcoltype{#1}{\@IEEEBPcurtype}{1}% current, error on invalid
+\@IEEEgetcoltype{#2}{\@IEEEBPnexttype}{0}% next, no error on invalid next
+% if curtype is a glue, get the glue def
+\if\@IEEEBPcurtype g\@IEEEgetcurglue{#1}{\@IEEEBPcurglue}\fi%
+% if curtype is a column, get the column def and set the current column name
+\if\@IEEEBPcurtype c\@IEEEgetcurcol{#1}\fi%
+% if curtype is a numeral, acquire the user defined glue
+\if\@IEEEBPcurtype n\@IEEEprocessNcol{#1}\fi%
+% process the acquired glue
+\if\@IEEEBPcurtype g\@IEEEprocessGcol\fi%
+% process the acquired col
+\if\@IEEEBPcurtype c\@IEEEprocessCcol\fi%
+% ready prevtype for next col spec.
+\let\@IEEEBPprevtype=\@IEEEBPcurtype%
+% be sure and put back the future token(s) as a group
+\fi\@@nextIEEEbuildpreamble{#2}}
+
+
+% executed just after preamble build is completed
+% warn about zero cols, and if prevtype type = u, put in end tabskip glue
+\def\@@IEEEfinishpreamble#1{\ifnum\@IEEEeqnnumcols<1\relax
+\@IEEEclspkgerror{No column specifiers declared for IEEEeqnarray}%
+{At least one column type must be declared for each IEEEeqnarray.}%
+\fi%num cols less than 1
+%if last type undefined, set default end tabskip glue
+\if\@IEEEBPprevtype u\@IEEEappendtoksA{\tabskip=\@IEEEBPendglue}\fi}
+
+
+% Identify and return the column specifier's type code
+\def\@IEEEgetcoltype#1#2#3{%
+% use only the very first token to determine the type
+% \noindent is used as a delimiter as \end can be present here
+\def\@IEEEgrabfirstoken##1##2\noindent{\let\@IEEEgrabbedfirstoken=##1}%
+\@IEEEgrabfirstoken#1\relax\relax\noindent
+% \@IEEEgrabfirstoken has the first token, the rest are discarded
+% n = number
+% g = glue (any other char in catagory 12)
+% c = letter
+% e = \end
+% u = undefined
+% third argument: 0 = no error message, 1 = error on invalid char
+\let#2=u\relax% assume invalid until know otherwise
+\ifx\@IEEEgrabbedfirstoken\end\let#2=e\else
+\ifcat\@IEEEgrabbedfirstoken\relax\else% screen out control sequences
+\if0\@IEEEgrabbedfirstoken\let#2=n\else
+\if1\@IEEEgrabbedfirstoken\let#2=n\else
+\if2\@IEEEgrabbedfirstoken\let#2=n\else
+\if3\@IEEEgrabbedfirstoken\let#2=n\else
+\if4\@IEEEgrabbedfirstoken\let#2=n\else
+\if5\@IEEEgrabbedfirstoken\let#2=n\else
+\if6\@IEEEgrabbedfirstoken\let#2=n\else
+\if7\@IEEEgrabbedfirstoken\let#2=n\else
+\if8\@IEEEgrabbedfirstoken\let#2=n\else
+\if9\@IEEEgrabbedfirstoken\let#2=n\else
+\ifcat,\@IEEEgrabbedfirstoken\let#2=g\relax
+\else\ifcat a\@IEEEgrabbedfirstoken\let#2=c\relax\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi
+\if#2u\relax
+\if0\noexpand#3\relax\else\@IEEEclspkgerror{Invalid character in column specifications}%
+{Only letters, numerals and certain other symbols are allowed \MessageBreak
+as IEEEeqnarray column specifiers.}\fi\fi}
+
+
+% identify the current letter referenced column
+% if invalid, use a default column
+\def\@IEEEgetcurcol#1{\expandafter\ifx\csname @IEEEeqnarraycolDEF#1\endcsname\@IEEEeqnarraycolisdefined%
+\def\@IEEEBPcurcolname{#1}\else% invalid column name
+\@IEEEclspkgerror{Invalid column type "#1" in column specifications.\MessageBreak
+Using a default centering column instead}%
+{You must define IEEEeqnarray column types before use.}%
+\def\@IEEEBPcurcolname{@IEEEdefault}\fi}
+
+
+% identify and return the predefined (punctuation) glue value
+\def\@IEEEgetcurglue#1#2{%
+% ! = \! (neg small) -0.16667em (-3/18 em)
+% , = \, (small) 0.16667em ( 3/18 em)
+% : = \: (med) 0.22222em ( 4/18 em)
+% ; = \; (large) 0.27778em ( 5/18 em)
+% ' = \quad 1em
+% " = \qquad 2em
+% . = 0.5\arraycolsep
+% / = \arraycolsep
+% ? = 2\arraycolsep
+% * = 1fil
+% + = \@IEEEeqnarraycolSEPcenter
+% - = \@IEEEeqnarraycolSEPzero
+% Note that all em values are referenced to the math font (textfont2) fontdimen6
+% value for 1em.
+%
+% use only the very first token to determine the type
+% this prevents errant tokens from getting in the main text
+% \noindent is used as a delimiter here
+\def\@IEEEgrabfirstoken##1##2\noindent{\let\@IEEEgrabbedfirstoken=##1}%
+\@IEEEgrabfirstoken#1\relax\relax\noindent
+% get the math font 1em value
+% LaTeX2e's NFSS2 does not preload the fonts, but \IEEEeqnarray needs
+% to gain access to the math (\textfont2) font's spacing parameters.
+% So we create a bogus box here that uses the math font to ensure
+% that \textfont2 is loaded and ready. If this is not done,
+% the \textfont2 stuff here may not work.
+% Thanks to Bernd Raichle for his 1997 post on this topic.
+{\setbox0=\hbox{$\displaystyle\relax$}}%
+% fontdimen6 has the width of 1em (a quad).
+\@IEEEtrantmpdimenA=\fontdimen6\textfont2\relax%
+% identify the glue value based on the first token
+% we discard anything after the first
+\if!\@IEEEgrabbedfirstoken\@IEEEtrantmpdimenA=-0.16667\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if,\@IEEEgrabbedfirstoken\@IEEEtrantmpdimenA=0.16667\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if:\@IEEEgrabbedfirstoken\@IEEEtrantmpdimenA=0.22222\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if;\@IEEEgrabbedfirstoken\@IEEEtrantmpdimenA=0.27778\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if'\@IEEEgrabbedfirstoken\@IEEEtrantmpdimenA=1\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if"\@IEEEgrabbedfirstoken\@IEEEtrantmpdimenA=2\@IEEEtrantmpdimenA\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if.\@IEEEgrabbedfirstoken\@IEEEtrantmpdimenA=0.5\arraycolsep\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if/\@IEEEgrabbedfirstoken\edef#2{\the\arraycolsep}\else
+\if?\@IEEEgrabbedfirstoken\@IEEEtrantmpdimenA=2\arraycolsep\edef#2{\the\@IEEEtrantmpdimenA}\else
+\if *\@IEEEgrabbedfirstoken\edef#2{0pt plus 1fil minus 0pt}\else
+\if+\@IEEEgrabbedfirstoken\edef#2{\@IEEEeqnarraycolSEPcenter}\else
+\if-\@IEEEgrabbedfirstoken\edef#2{\@IEEEeqnarraycolSEPzero}\else
+\edef#2{\@IEEEeqnarraycolSEPzero}%
+\@IEEEclspkgerror{Invalid predefined inter-column glue type "#1" in\MessageBreak
+column specifications. Using a default value of\MessageBreak
+0pt instead}%
+{Only !,:;'"./?*+ and - are valid predefined glue types in the\MessageBreak
+IEEEeqnarray column specifications.}\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi\fi}
+
+
+
+% process a numerical digit from the column specification
+% and look up the corresponding user defined glue value
+% can transform current type from n to g or a as the user defined glue is acquired
+\def\@IEEEprocessNcol#1{\if\@IEEEBPprevtype g%
+\@IEEEclspkgerror{Back-to-back inter-column glue specifiers in column\MessageBreak
+specifications. Ignoring consecutive glue specifiers\MessageBreak
+after the first}%
+{You cannot have two or more glue types next to each other\MessageBreak
+in the IEEEeqnarray column specifications.}%
+\let\@IEEEBPcurtype=a% abort this glue, future digits will be discarded
+\@IEEEBPcurnum=0\relax%
+\else% if we previously aborted a glue
+\if\@IEEEBPprevtype a\@IEEEBPcurnum=0\let\@IEEEBPcurtype=a%maintain digit abortion
+\else%acquire this number
+% save the previous type before the numerical digits started
+\if\@IEEEBPprevtype n\else\let\@IEEEBPprevsavedtype=\@IEEEBPprevtype\fi%
+\multiply\@IEEEBPcurnum by 10\relax%
+\advance\@IEEEBPcurnum by #1\relax% add in number, \relax is needed to stop TeX's number scan
+\if\@IEEEBPnexttype n\else%close acquisition
+\expandafter\ifx\csname @IEEEeqnarraycolSEPDEF\expandafter\romannumeral\number\@IEEEBPcurnum\endcsname\@IEEEeqnarraycolisdefined%
+\edef\@IEEEBPcurglue{\csname @IEEEeqnarraycolSEP\expandafter\romannumeral\number\@IEEEBPcurnum\endcsname}%
+\else%user glue not defined
+\@IEEEclspkgerror{Invalid user defined inter-column glue type "\number\@IEEEBPcurnum" in\MessageBreak
+column specifications. Using a default value of\MessageBreak
+0pt instead}%
+{You must define all IEEEeqnarray numerical inter-column glue types via\MessageBreak
+\string\IEEEeqnarraydefcolsep \space before they are used in column specifications.}%
+\edef\@IEEEBPcurglue{\@IEEEeqnarraycolSEPzero}%
+\fi% glue defined or not
+\let\@IEEEBPcurtype=g% change the type to reflect the acquired glue
+\let\@IEEEBPprevtype=\@IEEEBPprevsavedtype% restore the prev type before this number glue
+\@IEEEBPcurnum=0\relax%ready for next acquisition
+\fi%close acquisition, get glue
+\fi%discard or acquire number
+\fi%prevtype glue or not
+}
+
+
+% process an acquired glue
+% add any acquired column/glue pair to the preamble
+\def\@IEEEprocessGcol{\if\@IEEEBPprevtype a\let\@IEEEBPcurtype=a%maintain previous glue abortions
+\else
+% if this is the start glue, save it, but do nothing else
+% as this is not used in the preamble, but before
+\if\@IEEEBPprevtype s\edef\@IEEEBPstartglue{\@IEEEBPcurglue}%
+\else%not the start glue
+\if\@IEEEBPprevtype g%ignore if back to back glues
+\@IEEEclspkgerror{Back-to-back inter-column glue specifiers in column\MessageBreak
+specifications. Ignoring consecutive glue specifiers\MessageBreak
+after the first}%
+{You cannot have two or more glue types next to each other\MessageBreak
+in the IEEEeqnarray column specifications.}%
+\let\@IEEEBPcurtype=a% abort this glue
+\else% not a back to back glue
+\if\@IEEEBPprevtype c\relax% if the previoustype was a col, add column/glue pair to preamble
+\ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi
+\toks0={##}%
+% make preamble advance col counter if this environment needs this
+\if@advanceIEEEeqncolcnt\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}\fi
+% insert the column defintion into the preamble, being careful not to expand
+% the column definition
+\@IEEEappendtoksA{\tabskip=\@IEEEBPcurglue}%
+\@IEEEappendNOEXPANDtoksA{\begingroup\csname @IEEEeqnarraycolPRE}%
+\@IEEEappendtoksA{\@IEEEBPcurcolname}%
+\@IEEEappendNOEXPANDtoksA{\endcsname}%
+\@IEEEappendtoksA{\the\toks0}%
+\@IEEEappendNOEXPANDtoksA{\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST}%
+\@IEEEappendtoksA{\@IEEEBPcurcolname}%
+\@IEEEappendNOEXPANDtoksA{\endcsname\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\endgroup}%
+\advance\@IEEEeqnnumcols by 1\relax%one more column in the preamble
+\else% error: non-start glue with no pending column
+\@IEEEclspkgerror{Inter-column glue specifier without a prior column\MessageBreak
+type in the column specifications. Ignoring this glue\MessageBreak
+specifier}%
+{Except for the first and last positions, glue can be placed only\MessageBreak
+between column types.}%
+\let\@IEEEBPcurtype=a% abort this glue
+\fi% previous was a column
+\fi% back-to-back glues
+\fi% is start column glue
+\fi% prev type not a
+}
+
+
+% process an acquired letter referenced column and, if necessary, add it to the preamble
+\def\@IEEEprocessCcol{\if\@IEEEBPnexttype g\else
+\if\@IEEEBPnexttype n\else
+% we have a column followed by something other than a glue (or numeral glue)
+% so we must add this column to the preamble now
+\ifnum\@IEEEeqnnumcols>0\relax\@IEEEappendtoksA{&}\fi%col separator for those after the first
+\if\@IEEEBPnexttype e\@IEEEappendtoksA{\tabskip=\@IEEEBPendglue\relax}\else%put in end glue
+\@IEEEappendtoksA{\tabskip=\@IEEEeqnarraycolSEPdefaultmid\relax}\fi% or default mid glue
+\toks0={##}%
+% make preamble advance col counter if this environment needs this
+\if@advanceIEEEeqncolcnt\@IEEEappendtoksA{\global\advance\@IEEEeqncolcnt by 1\relax}\fi
+% insert the column definition into the preamble, being careful not to expand
+% the column definition
+\@IEEEappendNOEXPANDtoksA{\begingroup\csname @IEEEeqnarraycolPRE}%
+\@IEEEappendtoksA{\@IEEEBPcurcolname}%
+\@IEEEappendNOEXPANDtoksA{\endcsname}%
+\@IEEEappendtoksA{\the\toks0}%
+\@IEEEappendNOEXPANDtoksA{\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\csname @IEEEeqnarraycolPOST}%
+\@IEEEappendtoksA{\@IEEEBPcurcolname}%
+\@IEEEappendNOEXPANDtoksA{\endcsname\relax\relax\relax\relax\relax%
+\relax\relax\relax\relax\relax\endgroup}%
+\advance\@IEEEeqnnumcols by 1\relax%one more column in the preamble
+\fi%next type not numeral
+\fi%next type not glue
+}
+
+
+%%
+%% END OF IEEEeqnarry DEFINITIONS
+%%
+
+
+
+
+% set up the running headings, this complex because of all the different
+% modes IEEEtran supports
+\if@twoside
+ \ifCLASSOPTIONtechnote
+ \def\ps@headings{%
+ \def\@oddhead{\hbox{}\scriptsize\leftmark \hfil \thepage}
+ \def\@evenhead{\scriptsize\thepage \hfil \leftmark\hbox{}}
+ \ifCLASSOPTIONdraftcls
+ \ifCLASSOPTIONdraftclsnofoot
+ \def\@oddfoot{}\def\@evenfoot{}%
+ \else
+ \def\@oddfoot{\scriptsize\@date\hfil DRAFT}
+ \def\@evenfoot{\scriptsize DRAFT\hfil\@date}
+ \fi
+ \else
+ \def\@oddfoot{}\def\@evenfoot{}
+ \fi}
+ \else % not a technote
+ \def\ps@headings{%
+ \ifCLASSOPTIONconference
+ \def\@oddhead{}
+ \def\@evenhead{}
+ \else
+ \def\@oddhead{\hbox{}\scriptsize\rightmark \hfil \thepage}
+ \def\@evenhead{\scriptsize\thepage \hfil \leftmark\hbox{}}
+ \fi
+ \ifCLASSOPTIONdraftcls
+ \def\@oddhead{\hbox{}\scriptsize\rightmark \hfil \thepage}
+ \def\@evenhead{\scriptsize\thepage \hfil \leftmark\hbox{}}
+ \ifCLASSOPTIONdraftclsnofoot
+ \def\@oddfoot{}\def\@evenfoot{}%
+ \else
+ \def\@oddfoot{\scriptsize\@date\hfil DRAFT}
+ \def\@evenfoot{\scriptsize DRAFT\hfil\@date}
+ \fi
+ \else
+ \def\@oddfoot{}\def\@evenfoot{}%
+ \fi}
+ \fi
+\else % single side
+\def\ps@headings{%
+ \ifCLASSOPTIONconference
+ \def\@oddhead{}
+ \def\@evenhead{}
+ \else
+ \def\@oddhead{\hbox{}\scriptsize\leftmark \hfil \thepage}
+ \def\@evenhead{}
+ \fi
+ \ifCLASSOPTIONdraftcls
+ \def\@oddhead{\hbox{}\scriptsize\leftmark \hfil \thepage}
+ \def\@evenhead{}
+ \ifCLASSOPTIONdraftclsnofoot
+ \def\@oddfoot{}
+ \else
+ \def\@oddfoot{\scriptsize \@date \hfil DRAFT}
+ \fi
+ \else
+ \def\@oddfoot{}
+ \fi
+ \def\@evenfoot{}}
+\fi
+
+
+% title page style
+\def\ps@IEEEtitlepagestyle{\def\@oddfoot{}\def\@evenfoot{}%
+\ifCLASSOPTIONconference
+ \def\@oddhead{}%
+ \def\@evenhead{}%
+\else
+ \def\@oddhead{\hbox{}\scriptsize\leftmark \hfil \thepage}%
+ \def\@evenhead{\scriptsize\thepage \hfil \leftmark\hbox{}}%
+\fi
+\ifCLASSOPTIONdraftcls
+ \def\@oddhead{\hbox{}\scriptsize\leftmark \hfil \thepage}%
+ \def\@evenhead{\scriptsize\thepage \hfil \leftmark\hbox{}}%
+ \ifCLASSOPTIONdraftclsnofoot\else
+ \def\@oddfoot{\scriptsize \@date\hfil DRAFT}%
+ \def\@evenfoot{\scriptsize DRAFT\hfil \@date}%
+ \fi
+\else
+ % all non-draft mode footers
+ \if@IEEEusingpubid
+ % for title pages that are using a pubid
+ % do not repeat pubid if using peer review option
+ \ifCLASSOPTIONpeerreview
+ \else
+ \footskip 0pt%
+ \ifCLASSOPTIONcompsoc
+ \def\@oddfoot{\hss\normalfont\scriptsize\raisebox{-1.5\@IEEEnormalsizeunitybaselineskip}[0ex][0ex]{\@IEEEpubid}\hss}%
+ \def\@evenfoot{\hss\normalfont\scriptsize\raisebox{-1.5\@IEEEnormalsizeunitybaselineskip}[0ex][0ex]{\@IEEEpubid}\hss}%
+ \else
+ \def\@oddfoot{\hss\normalfont\footnotesize\raisebox{1.5ex}[1.5ex]{\@IEEEpubid}\hss}%
+ \def\@evenfoot{\hss\normalfont\footnotesize\raisebox{1.5ex}[1.5ex]{\@IEEEpubid}\hss}%
+ \fi
+ \fi
+ \fi
+\fi}
+
+
+% peer review cover page style
+\def\ps@IEEEpeerreviewcoverpagestyle{%
+\def\@oddhead{}\def\@evenhead{}%
+\def\@oddfoot{}\def\@evenfoot{}%
+\ifCLASSOPTIONdraftcls
+ \ifCLASSOPTIONdraftclsnofoot\else
+ \def\@oddfoot{\scriptsize \@date\hfil DRAFT}%
+ \def\@evenfoot{\scriptsize DRAFT\hfil \@date}%
+ \fi
+\else
+ % non-draft mode footers
+ \if@IEEEusingpubid
+ \footskip 0pt%
+ \ifCLASSOPTIONcompsoc
+ \def\@oddfoot{\hss\normalfont\scriptsize\raisebox{-1.5\@IEEEnormalsizeunitybaselineskip}[0ex][0ex]{\@IEEEpubid}\hss}%
+ \def\@evenfoot{\hss\normalfont\scriptsize\raisebox{-1.5\@IEEEnormalsizeunitybaselineskip}[0ex][0ex]{\@IEEEpubid}\hss}%
+ \else
+ \def\@oddfoot{\hss\normalfont\footnotesize\raisebox{1.5ex}[1.5ex]{\@IEEEpubid}\hss}%
+ \def\@evenfoot{\hss\normalfont\footnotesize\raisebox{1.5ex}[1.5ex]{\@IEEEpubid}\hss}%
+ \fi
+ \fi
+\fi}
+
+
+% start with empty headings
+\def\rightmark{}\def\leftmark{}
+
+
+%% Defines the command for putting the header. \footernote{TEXT} is the same
+%% as \markboth{TEXT}{TEXT}.
+%% Note that all the text is forced into uppercase, if you have some text
+%% that needs to be in lower case, for instance et. al., then either manually
+%% set \leftmark and \rightmark or use \MakeLowercase{et. al.} within the
+%% arguments to \markboth.
+\def\markboth#1#2{\def\leftmark{\@IEEEcompsoconly{\sffamily}\MakeUppercase{#1}}%
+\def\rightmark{\@IEEEcompsoconly{\sffamily}\MakeUppercase{#2}}}
+\def\footernote#1{\markboth{#1}{#1}}
+
+\def\today{\ifcase\month\or
+ January\or February\or March\or April\or May\or June\or
+ July\or August\or September\or October\or November\or December\fi
+ \space\number\day, \number\year}
+
+
+
+
+%% CITATION AND BIBLIOGRAPHY COMMANDS
+%%
+%% V1.6 no longer supports the older, nonstandard \shortcite and \citename setup stuff
+%
+%
+% Modify Latex2e \@citex to separate citations with "], ["
+\def\@citex[#1]#2{%
+ \let\@citea\@empty
+ \@cite{\@for\@citeb:=#2\do
+ {\@citea\def\@citea{], [}%
+ \edef\@citeb{\expandafter\@firstofone\@citeb\@empty}%
+ \if@filesw\immediate\write\@auxout{\string\citation{\@citeb}}\fi
+ \@ifundefined{b@\@citeb}{\mbox{\reset@font\bfseries ?}%
+ \G@refundefinedtrue
+ \@latex@warning
+ {Citation `\@citeb' on page \thepage \space undefined}}%
+ {\hbox{\csname b@\@citeb\endcsname}}}}{#1}}
+
+% V1.6 we create hooks for the optional use of Donald Arseneau's
+% cite.sty package. cite.sty is "smart" and will notice that the
+% following format controls are already defined and will not
+% redefine them. The result will be the proper sorting of the
+% citation numbers and auto detection of 3 or more entry "ranges" -
+% all in IEEE style: [1], [2], [5]--[7], [12]
+% This also allows for an optional note, i.e., \cite[mynote]{..}.
+% If the \cite with note has more than one reference, the note will
+% be applied to the last of the listed references. It is generally
+% desired that if a note is given, only one reference is listed in
+% that \cite.
+% Thanks to Mr. Arseneau for providing the required format arguments
+% to produce the IEEE style.
+\def\citepunct{], [}
+\def\citedash{]--[}
+
+% V1.7 default to using same font for urls made by url.sty
+\AtBeginDocument{\csname url@samestyle\endcsname}
+
+% V1.6 class files should always provide these
+\def\newblock{\hskip .11em\@plus.33em\@minus.07em}
+\let\@openbib@code\@empty
+
+
+% Provide support for the control entries of IEEEtran.bst V1.00 and later.
+% V1.7 optional argument allows for a different aux file to be specified in
+% order to handle multiple bibliographies. For example, with multibib.sty:
+% \newcites{sec}{Secondary Literature}
+% \bstctlcite[@auxoutsec]{BSTcontrolhak}
+\def\bstctlcite{\@ifnextchar[{\@bstctlcite}{\@bstctlcite[@auxout]}}
+\def\@bstctlcite[#1]#2{\@bsphack
+ \@for\@citeb:=#2\do{%
+ \edef\@citeb{\expandafter\@firstofone\@citeb}%
+ \if@filesw\immediate\write\csname #1\endcsname{\string\citation{\@citeb}}\fi}%
+ \@esphack}
+
+% V1.6 provide a way for a user to execute a command just before
+% a given reference number - used to insert a \newpage to balance
+% the columns on the last page
+\edef\@IEEEtriggerrefnum{0} % the default of zero means that
+ % the command is not executed
+\def\@IEEEtriggercmd{\newpage}
+
+% allow the user to alter the triggered command
+\long\def\IEEEtriggercmd#1{\long\def\@IEEEtriggercmd{#1}}
+
+% allow user a way to specify the reference number just before the
+% command is executed
+\def\IEEEtriggeratref#1{\@IEEEtrantmpcountA=#1%
+\edef\@IEEEtriggerrefnum{\the\@IEEEtrantmpcountA}}%
+
+% trigger command at the given reference
+\def\@IEEEbibitemprefix{\@IEEEtrantmpcountA=\@IEEEtriggerrefnum\relax%
+\advance\@IEEEtrantmpcountA by -1\relax%
+\ifnum\c@enumiv=\@IEEEtrantmpcountA\relax\@IEEEtriggercmd\relax\fi}
+
+
+\def\@biblabel#1{[#1]}
+
+% compsoc journals left align the reference numbers
+\@IEEEcompsocnotconfonly{\def\@biblabel#1{[#1]\hfill}}
+
+% controls bib item spacing
+\def\IEEEbibitemsep{0pt plus .5pt}
+
+\@IEEEcompsocconfonly{\def\IEEEbibitemsep{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}}
+
+
+\def\thebibliography#1{\section*{\refname}%
+ \addcontentsline{toc}{section}{\refname}%
+ % V1.6 add some rubber space here and provide a command trigger
+ \footnotesize\@IEEEcompsocconfonly{\small}\vskip 0.3\baselineskip plus 0.1\baselineskip minus 0.1\baselineskip%
+ \list{\@biblabel{\@arabic\c@enumiv}}%
+ {\settowidth\labelwidth{\@biblabel{#1}}%
+ \leftmargin\labelwidth
+ \advance\leftmargin\labelsep\relax
+ \itemsep \IEEEbibitemsep\relax
+ \usecounter{enumiv}%
+ \let\p@enumiv\@empty
+ \renewcommand\theenumiv{\@arabic\c@enumiv}}%
+ \let\@IEEElatexbibitem\bibitem%
+ \def\bibitem{\@IEEEbibitemprefix\@IEEElatexbibitem}%
+\def\newblock{\hskip .11em plus .33em minus .07em}%
+% originally:
+% \sloppy\clubpenalty4000\widowpenalty4000%
+% by adding the \interlinepenalty here, we make it more
+% difficult, but not impossible, for LaTeX to break within a reference.
+% IEEE almost never breaks a reference (but they do it more often with
+% technotes). You may get an underfull vbox warning around the bibliography,
+% but the final result will be much more like what IEEE will publish.
+% MDS 11/2000
+\ifCLASSOPTIONtechnote\sloppy\clubpenalty4000\widowpenalty4000\interlinepenalty100%
+\else\sloppy\clubpenalty4000\widowpenalty4000\interlinepenalty500\fi%
+ \sfcode`\.=1000\relax}
+\let\endthebibliography=\endlist
+
+
+
+
+% TITLE PAGE COMMANDS
+%
+%
+% \IEEEmembership is used to produce the sublargesize italic font used to indicate author
+% IEEE membership. compsoc uses a large size sans slant font
+\def\IEEEmembership#1{{\@IEEEnotcompsoconly{\sublargesize}\normalfont\@IEEEcompsoconly{\sffamily}\textit{#1}}}
+
+
+% \IEEEauthorrefmark{} produces a footnote type symbol to indicate author affiliation.
+% When given an argument of 1 to 9, \IEEEauthorrefmark{} follows the standard LaTeX footnote
+% symbol sequence convention. However, for arguments 10 and above, \IEEEauthorrefmark{}
+% reverts to using lower case roman numerals, so it cannot overflow. Do note that you
+% cannot use \footnotemark[] in place of \IEEEauthorrefmark{} within \author as the footnote
+% symbols will have been turned off to prevent \thanks from creating footnote marks.
+% \IEEEauthorrefmark{} produces a symbol that appears to LaTeX as having zero vertical
+% height - this allows for a more compact line packing, but the user must ensure that
+% the interline spacing is large enough to prevent \IEEEauthorrefmark{} from colliding
+% with the text above.
+% V1.7 make this a robust command
+\DeclareRobustCommand*{\IEEEauthorrefmark}[1]{\raisebox{0pt}[0pt][0pt]{\textsuperscript{\footnotesize\ensuremath{\ifcase#1\or *\or \dagger\or \ddagger\or%
+ \mathsection\or \mathparagraph\or \|\or **\or \dagger\dagger%
+ \or \ddagger\ddagger \else\textsuperscript{\expandafter\romannumeral#1}\fi}}}}
+
+
+% FONT CONTROLS AND SPACINGS FOR CONFERENCE MODE AUTHOR NAME AND AFFILIATION BLOCKS
+%
+% The default font styles for the author name and affiliation blocks (confmode)
+\def\@IEEEauthorblockNstyle{\normalfont\@IEEEcompsocnotconfonly{\sffamily}\sublargesize\@IEEEcompsocconfonly{\large}}
+\def\@IEEEauthorblockAstyle{\normalfont\@IEEEcompsocnotconfonly{\sffamily}\@IEEEcompsocconfonly{\itshape}\normalsize\@IEEEcompsocconfonly{\large}}
+% The default if the user does not use an author block
+\def\@IEEEauthordefaulttextstyle{\normalfont\@IEEEcompsocnotconfonly{\sffamily}\sublargesize}
+
+% spacing from title (or special paper notice) to author name blocks (confmode)
+% can be negative
+\def\@IEEEauthorblockconfadjspace{-0.25em}
+% compsoc conferences need more space here
+\@IEEEcompsocconfonly{\def\@IEEEauthorblockconfadjspace{0.75\@IEEEnormalsizeunitybaselineskip}}
+
+% spacing between name and affiliation blocks (confmode)
+% This can be negative.
+% IEEE doesn't want any added spacing here, but I will leave these
+% controls in place in case they ever change their mind.
+% Personally, I like 0.75ex.
+%\def\@IEEEauthorblockNtopspace{0.75ex}
+%\def\@IEEEauthorblockAtopspace{0.75ex}
+\def\@IEEEauthorblockNtopspace{0.0ex}
+\def\@IEEEauthorblockAtopspace{0.0ex}
+% baseline spacing within name and affiliation blocks (confmode)
+% must be positive, spacings below certain values will make
+% the position of line of text sensitive to the contents of the
+% line above it i.e., whether or not the prior line has descenders,
+% subscripts, etc. For this reason it is a good idea to keep
+% these above 2.6ex
+\def\@IEEEauthorblockNinterlinespace{2.6ex}
+\def\@IEEEauthorblockAinterlinespace{2.75ex}
+
+% This tracks the required strut size.
+% See the \@IEEEauthorhalign command for the actual default value used.
+\def\@IEEEauthorblockXinterlinespace{2.7ex}
+
+% variables to retain font size and style across groups
+% values given here have no effect as they will be overwritten later
+\gdef\@IEEESAVESTATEfontsize{10}
+\gdef\@IEEESAVESTATEfontbaselineskip{12}
+\gdef\@IEEESAVESTATEfontencoding{OT1}
+\gdef\@IEEESAVESTATEfontfamily{ptm}
+\gdef\@IEEESAVESTATEfontseries{m}
+\gdef\@IEEESAVESTATEfontshape{n}
+
+% saves the current font attributes
+\def\@IEEEcurfontSAVE{\global\let\@IEEESAVESTATEfontsize\f@size%
+\global\let\@IEEESAVESTATEfontbaselineskip\f@baselineskip%
+\global\let\@IEEESAVESTATEfontencoding\f@encoding%
+\global\let\@IEEESAVESTATEfontfamily\f@family%
+\global\let\@IEEESAVESTATEfontseries\f@series%
+\global\let\@IEEESAVESTATEfontshape\f@shape}
+
+% restores the saved font attributes
+\def\@IEEEcurfontRESTORE{\fontsize{\@IEEESAVESTATEfontsize}{\@IEEESAVESTATEfontbaselineskip}%
+\fontencoding{\@IEEESAVESTATEfontencoding}%
+\fontfamily{\@IEEESAVESTATEfontfamily}%
+\fontseries{\@IEEESAVESTATEfontseries}%
+\fontshape{\@IEEESAVESTATEfontshape}%
+\selectfont}
+
+
+% variable to indicate if the current block is the first block in the column
+\newif\if@IEEEprevauthorblockincol \@IEEEprevauthorblockincolfalse
+
+
+% the command places a strut with height and depth = \@IEEEauthorblockXinterlinespace
+% we use this technique to have complete manual control over the spacing of the lines
+% within the halign environment.
+% We set the below baseline portion at 30%, the above
+% baseline portion at 70% of the total length.
+% Responds to changes in the document's \baselinestretch
+\def\@IEEEauthorstrutrule{\@IEEEtrantmpdimenA\@IEEEauthorblockXinterlinespace%
+\@IEEEtrantmpdimenA=\baselinestretch\@IEEEtrantmpdimenA%
+\rule[-0.3\@IEEEtrantmpdimenA]{0pt}{\@IEEEtrantmpdimenA}}
+
+
+% blocks to hold the authors' names and affilations.
+% Makes formatting easy for conferences
+%
+% use real definitions in conference mode
+% name block
+\def\IEEEauthorblockN#1{\relax\@IEEEauthorblockNstyle% set the default text style
+\gdef\@IEEEauthorblockXinterlinespace{0pt}% disable strut for spacer row
+% the \expandafter hides the \cr in conditional tex, see the array.sty docs
+% for details, probably not needed here as the \cr is in a macro
+% do a spacer row if needed
+\if@IEEEprevauthorblockincol\expandafter\@IEEEauthorblockNtopspaceline\fi
+\global\@IEEEprevauthorblockincoltrue% we now have a block in this column
+%restore the correct strut value
+\gdef\@IEEEauthorblockXinterlinespace{\@IEEEauthorblockNinterlinespace}%
+% input the author names
+#1%
+% end the row if the user did not already
+\crcr}
+% spacer row for names
+\def\@IEEEauthorblockNtopspaceline{\cr\noalign{\vskip\@IEEEauthorblockNtopspace}}
+%
+% affiliation block
+\def\IEEEauthorblockA#1{\relax\@IEEEauthorblockAstyle% set the default text style
+\gdef\@IEEEauthorblockXinterlinespace{0pt}%disable strut for spacer row
+% the \expandafter hides the \cr in conditional tex, see the array.sty docs
+% for details, probably not needed here as the \cr is in a macro
+% do a spacer row if needed
+\if@IEEEprevauthorblockincol\expandafter\@IEEEauthorblockAtopspaceline\fi
+\global\@IEEEprevauthorblockincoltrue% we now have a block in this column
+%restore the correct strut value
+\gdef\@IEEEauthorblockXinterlinespace{\@IEEEauthorblockAinterlinespace}%
+% input the author affiliations
+#1%
+% end the row if the user did not already
+\crcr}
+% spacer row for affiliations
+\def\@IEEEauthorblockAtopspaceline{\cr\noalign{\vskip\@IEEEauthorblockAtopspace}}
+
+
+% allow papers to compile even if author blocks are used in modes other
+% than conference or peerreviewca. For such cases, we provide dummy blocks.
+\ifCLASSOPTIONconference
+\else
+ \ifCLASSOPTIONpeerreviewca\else
+ % not conference or peerreviewca mode
+ \def\IEEEauthorblockN#1{#1}%
+ \def\IEEEauthorblockA#1{#1}%
+ \fi
+\fi
+
+
+
+% we provide our own halign so as not to have to depend on tabular
+\def\@IEEEauthorhalign{\@IEEEauthordefaulttextstyle% default text style
+ \lineskip=0pt\relax% disable line spacing
+ \lineskiplimit=0pt\relax%
+ \baselineskip=0pt\relax%
+ \@IEEEcurfontSAVE% save the current font
+ \mathsurround\z@\relax% no extra spacing around math
+ \let\\\@IEEEauthorhaligncr% replace newline with halign friendly one
+ \tabskip=0pt\relax% no column spacing
+ \everycr{}% ensure no problems here
+ \@IEEEprevauthorblockincolfalse% no author blocks yet
+ \def\@IEEEauthorblockXinterlinespace{2.7ex}% default interline space
+ \vtop\bgroup%vtop box
+ \halign\bgroup&\relax\hfil\@IEEEcurfontRESTORE\relax ##\relax
+ \hfil\@IEEEcurfontSAVE\@IEEEauthorstrutrule\cr}
+
+% ensure last line, exit from halign, close vbox
+\def\end@IEEEauthorhalign{\crcr\egroup\egroup}
+
+% handle bogus star form
+\def\@IEEEauthorhaligncr{{\ifnum0=`}\fi\@ifstar{\@@IEEEauthorhaligncr}{\@@IEEEauthorhaligncr}}
+
+% test and setup the optional argument to \\[]
+\def\@@IEEEauthorhaligncr{\@testopt\@@@IEEEauthorhaligncr\z@skip}
+
+% end the line and do the optional spacer
+\def\@@@IEEEauthorhaligncr[#1]{\ifnum0=`{\fi}\cr\noalign{\vskip#1\relax}}
+
+
+
+% flag to prevent multiple \and warning messages
+\newif\if@IEEEWARNand
+\@IEEEWARNandtrue
+
+% if in conference or peerreviewca modes, we support the use of \and as \author is a
+% tabular environment, otherwise we warn the user that \and is invalid
+% outside of conference or peerreviewca modes.
+\def\and{\relax} % provide a bogus \and that we will then override
+
+\renewcommand{\and}[1][\relax]{\if@IEEEWARNand\typeout{** WARNING: \noexpand\and is valid only
+ when in conference or peerreviewca}\typeout{modes (line \the\inputlineno).}\fi\global\@IEEEWARNandfalse}
+
+\ifCLASSOPTIONconference%
+\renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
+\fi
+\ifCLASSOPTIONpeerreviewca
+\renewcommand{\and}[1][\hfill]{\end{@IEEEauthorhalign}#1\begin{@IEEEauthorhalign}}%
+\fi
+
+
+% page clearing command
+% based on LaTeX2e's \cleardoublepage, but allows different page styles
+% for the inserted blank pages
+\def\@IEEEcleardoublepage#1{\clearpage\if@twoside\ifodd\c@page\else
+\hbox{}\thispagestyle{#1}\newpage\if@twocolumn\hbox{}\thispagestyle{#1}\newpage\fi\fi\fi}
+
+
+% user command to invoke the title page
+\def\maketitle{\par%
+ \begingroup%
+ \normalfont%
+ \def\thefootnote{}% the \thanks{} mark type is empty
+ \def\footnotemark{}% and kill space from \thanks within author
+ \let\@makefnmark\relax% V1.7, must *really* kill footnotemark to remove all \textsuperscript spacing as well.
+ \footnotesize% equal spacing between thanks lines
+ \footnotesep 0.7\baselineskip%see global setting of \footnotesep for more info
+ % V1.7 disable \thanks note indention for compsoc
+ \@IEEEcompsoconly{\long\def\@makefntext##1{\parindent 1em\noindent\hbox{\@makefnmark}##1}}%
+ \normalsize%
+ \ifCLASSOPTIONpeerreview
+ \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+ \thispagestyle{IEEEpeerreviewcoverpagestyle}\@thanks%
+ \else
+ \if@twocolumn%
+ \ifCLASSOPTIONtechnote%
+ \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+ \else
+ \twocolumn[\@maketitle\@IEEEdynamictitlevspace\@IEEEaftertitletext]%
+ \fi
+ \else
+ \newpage\global\@topnum\z@ \@maketitle\@IEEEstatictitlevskip\@IEEEaftertitletext%
+ \fi
+ \thispagestyle{IEEEtitlepagestyle}\@thanks%
+ \fi
+ % pullup page for pubid if used.
+ \if@IEEEusingpubid
+ \enlargethispage{-\@IEEEpubidpullup}%
+ \fi
+ \endgroup
+ \setcounter{footnote}{0}\let\maketitle\relax\let\@maketitle\relax
+ \gdef\@thanks{}%
+ % v1.6b do not clear these as we will need the title again for peer review papers
+ % \gdef\@author{}\gdef\@title{}%
+ \let\thanks\relax}
+
+
+
+% V1.7 parbox to format \@IEEEcompsoctitleabstractindextext
+\long\def\@IEEEcompsoctitleabstractindextextbox#1{\parbox{0.915\textwidth}{#1}}
+
+% formats the Title, authors names, affiliations and special paper notice
+% THIS IS A CONTROLLED SPACING COMMAND! Do not allow blank lines or unintentional
+% spaces to enter the definition - use % at the end of each line
+\def\@maketitle{\newpage
+\begin{center}%
+\ifCLASSOPTIONtechnote% technotes
+ {\bfseries\large\@IEEEcompsoconly{\sffamily}\@title\par}\vskip 1.3em{\lineskip .5em\@IEEEcompsoconly{\sffamily}\@author
+ \@IEEEspecialpapernotice\par{\@IEEEcompsoconly{\vskip 1.5em\relax
+ \@IEEEcompsoctitleabstractindextextbox{\@IEEEcompsoctitleabstractindextext}\par
+ \hfill\@IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
+\else% not a technote
+ \vskip0.2em{\Huge\@IEEEcompsoconly{\sffamily}\@IEEEcompsocconfonly{\normalfont\normalsize\vskip 2\@IEEEnormalsizeunitybaselineskip
+ \bfseries\Large}\@title\par}\vskip1.0em\par%
+ % V1.6 handle \author differently if in conference mode
+ \ifCLASSOPTIONconference%
+ {\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
+ \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par}\relax
+ \else% peerreviewca, peerreview or journal
+ \ifCLASSOPTIONpeerreviewca
+ % peerreviewca handles author names just like conference mode
+ {\@IEEEcompsoconly{\sffamily}\@IEEEspecialpapernotice\mbox{}\vskip\@IEEEauthorblockconfadjspace%
+ \mbox{}\hfill\begin{@IEEEauthorhalign}\@author\end{@IEEEauthorhalign}\hfill\mbox{}\par
+ {\@IEEEcompsoconly{\vskip 1.5em\relax
+ \@IEEEcompsoctitleabstractindextextbox{\@IEEEcompsoctitleabstractindextext}\par\hfill
+ \@IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
+ \else% journal or peerreview
+ {\lineskip.5em\@IEEEcompsoconly{\sffamily}\sublargesize\@author\@IEEEspecialpapernotice\par
+ {\@IEEEcompsoconly{\vskip 1.5em\relax
+ \@IEEEcompsoctitleabstractindextextbox{\@IEEEcompsoctitleabstractindextext}\par\hfill
+ \@IEEEcompsocdiamondline\hfill\hbox{}\par}}}\relax
+ \fi
+ \fi
+\fi\end{center}}
+
+
+
+% V1.7 Computer Society "diamond line" which follows index terms for nonconference papers
+\def\@IEEEcompsocdiamondline{\vrule depth 0pt height 0.5pt width 4cm\hspace{7.5pt}%
+\raisebox{-3.5pt}{\fontfamily{pzd}\fontencoding{U}\fontseries{m}\fontshape{n}\fontsize{11}{12}\selectfont\char70}%
+\hspace{7.5pt}\vrule depth 0pt height 0.5pt width 4cm\relax}
+
+% V1.7 standard LateX2e \thanks, but with \itshape under compsoc. Also make it a \long\def
+% We also need to trigger the one-shot footnote rule
+\def\@IEEEtriggeroneshotfootnoterule{\global\@IEEEenableoneshotfootnoteruletrue}
+
+
+\long\def\thanks#1{\footnotemark
+ \protected@xdef\@thanks{\@thanks
+ \protect\footnotetext[\the\c@footnote]{\@IEEEcompsoconly{\itshape
+ \protect\@IEEEtriggeroneshotfootnoterule\relax}\ignorespaces#1}}}
+\let\@thanks\@empty
+
+% V1.7 allow \author to contain \par's. This is needed to allow \thanks to contain \par.
+\long\def\author#1{\gdef\@author{#1}}
+
+
+% in addition to setting up IEEEitemize, we need to remove a baselineskip space above and
+% below it because \list's \pars introduce blank lines because of the footnote struts.
+\def\@IEEEsetupcompsocitemizelist{\def\labelitemi{$\bullet$}%
+\setlength{\IEEElabelindent}{0pt}\setlength{\parskip}{0pt}%
+\setlength{\partopsep}{0pt}\setlength{\topsep}{0.5\baselineskip}\vspace{-1\baselineskip}\relax}
+
+
+% flag for fake non-compsoc \IEEEcompsocthanksitem - prevents line break on very first item
+\newif\if@IEEEbreakcompsocthanksitem \@IEEEbreakcompsocthanksitemfalse
+
+\ifCLASSOPTIONcompsoc
+% V1.7 compsoc bullet item \thanks
+% also, we need to redefine this to destroy the argument in \@IEEEdynamictitlevspace
+\long\def\IEEEcompsocitemizethanks#1{\relax\@IEEEbreakcompsocthanksitemfalse\footnotemark
+ \protected@xdef\@thanks{\@thanks
+ \protect\footnotetext[\the\c@footnote]{\itshape\protect\@IEEEtriggeroneshotfootnoterule
+ {\let\IEEEiedlistdecl\relax\protect\begin{IEEEitemize}[\protect\@IEEEsetupcompsocitemizelist]\ignorespaces#1\relax
+ \protect\end{IEEEitemize}}\protect\vspace{-1\baselineskip}}}}
+\DeclareRobustCommand*{\IEEEcompsocthanksitem}{\item}
+\else
+% non-compsoc, allow for dual compilation via rerouting to normal \thanks
+\long\def\IEEEcompsocitemizethanks#1{\thanks{#1}}
+% redirect to "pseudo-par" \hfil\break\indent after swallowing [] from \IEEEcompsocthanksitem[]
+\DeclareRobustCommand{\IEEEcompsocthanksitem}{\@ifnextchar [{\@IEEEthanksswallowoptionalarg}%
+{\@IEEEthanksswallowoptionalarg[\relax]}}
+% be sure and break only after first item, be sure and ignore spaces after optional argument
+\def\@IEEEthanksswallowoptionalarg[#1]{\relax\if@IEEEbreakcompsocthanksitem\hfil\break
+\indent\fi\@IEEEbreakcompsocthanksitemtrue\ignorespaces}
+\fi
+
+
+% V1.6b define the \IEEEpeerreviewmaketitle as needed
+\ifCLASSOPTIONpeerreview
+\def\IEEEpeerreviewmaketitle{\@IEEEcleardoublepage{empty}%
+\ifCLASSOPTIONtwocolumn
+\twocolumn[\@IEEEpeerreviewmaketitle\@IEEEdynamictitlevspace]
+\else
+\newpage\@IEEEpeerreviewmaketitle\@IEEEstatictitlevskip
+\fi
+\thispagestyle{IEEEtitlepagestyle}}
+\else
+% \IEEEpeerreviewmaketitle does nothing if peer review option has not been selected
+\def\IEEEpeerreviewmaketitle{\relax}
+\fi
+
+% peerreview formats the repeated title like the title in journal papers.
+\def\@IEEEpeerreviewmaketitle{\begin{center}\@IEEEcompsoconly{\sffamily}%
+\normalfont\normalsize\vskip0.2em{\Huge\@title\par}\vskip1.0em\par
+\end{center}}
+
+
+
+% V1.6
+% this is a static rubber spacer between the title/authors and the main text
+% used for single column text, or when the title appears in the first column
+% of two column text (technotes).
+\def\@IEEEstatictitlevskip{{\normalfont\normalsize
+% adjust spacing to next text
+% v1.6b handle peer review papers
+\ifCLASSOPTIONpeerreview
+% for peer review papers, the same value is used for both title pages
+% regardless of the other paper modes
+ \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip
+\else
+ \ifCLASSOPTIONconference% conference
+ \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip%
+ \else%
+ \ifCLASSOPTIONtechnote% technote
+ \vskip 1\baselineskip plus 0.375\baselineskip minus 0.1875\baselineskip%
+ \else% journal uses more space
+ \vskip 2.5\baselineskip plus 0.75\baselineskip minus 0.375\baselineskip%
+ \fi
+ \fi
+\fi}}
+
+
+% V1.6
+% This is a dynamically determined rigid spacer between the title/authors
+% and the main text. This is used only for single column titles over two
+% column text (most common)
+% This is bit tricky because we have to ensure that the textheight of the
+% main text is an integer multiple of \baselineskip
+% otherwise underfull vbox problems may develop in the second column of the
+% text on the titlepage
+% The possible use of \IEEEpubid must also be taken into account.
+\def\@IEEEdynamictitlevspace{{%
+ % we run within a group so that all the macros can be forgotten when we are done
+ \long\def\thanks##1{\relax}%don't allow \thanks to run when we evaluate the vbox height
+ \long\def\IEEEcompsocitemizethanks##1{\relax}%don't allow \IEEEcompsocitemizethanks to run when we evaluate the vbox height
+ \normalfont\normalsize% we declare more descriptive variable names
+ \let\@IEEEmaintextheight=\@IEEEtrantmpdimenA%height of the main text columns
+ \let\@IEEEINTmaintextheight=\@IEEEtrantmpdimenB%height of the main text columns with integer # lines
+ % set the nominal and minimum values for the title spacer
+ % the dynamic algorithm will not allow the spacer size to
+ % become less than \@IEEEMINtitlevspace - instead it will be
+ % lengthened
+ % default to journal values
+ \def\@IEEENORMtitlevspace{2.5\baselineskip}%
+ \def\@IEEEMINtitlevspace{2\baselineskip}%
+ % conferences and technotes need tighter spacing
+ \ifCLASSOPTIONconference%conference
+ \def\@IEEENORMtitlevspace{1\baselineskip}%
+ \def\@IEEEMINtitlevspace{0.75\baselineskip}%
+ \fi
+ \ifCLASSOPTIONtechnote%technote
+ \def\@IEEENORMtitlevspace{1\baselineskip}%
+ \def\@IEEEMINtitlevspace{0.75\baselineskip}%
+ \fi%
+ % get the height that the title will take up
+ \ifCLASSOPTIONpeerreview
+ \settoheight{\@IEEEmaintextheight}{\vbox{\hsize\textwidth \@IEEEpeerreviewmaketitle}}%
+ \else
+ \settoheight{\@IEEEmaintextheight}{\vbox{\hsize\textwidth \@maketitle}}%
+ \fi
+ \@IEEEmaintextheight=-\@IEEEmaintextheight% title takes away from maintext, so reverse sign
+ % add the height of the page textheight
+ \advance\@IEEEmaintextheight by \textheight%
+ % correct for title pages using pubid
+ \ifCLASSOPTIONpeerreview\else
+ % peerreview papers use the pubid on the cover page only.
+ % And the cover page uses a static spacer.
+ \if@IEEEusingpubid\advance\@IEEEmaintextheight by -\@IEEEpubidpullup\fi
+ \fi%
+ % subtract off the nominal value of the title bottom spacer
+ \advance\@IEEEmaintextheight by -\@IEEENORMtitlevspace%
+ % \topskip takes away some too
+ \advance\@IEEEmaintextheight by -\topskip%
+ % calculate the column height of the main text for lines
+ % now we calculate the main text height as if holding
+ % an integer number of \normalsize lines after the first
+ % and discard any excess fractional remainder
+ % we subtracted the first line, because the first line
+ % is placed \topskip into the maintext, not \baselineskip like the
+ % rest of the lines.
+ \@IEEEINTmaintextheight=\@IEEEmaintextheight%
+ \divide\@IEEEINTmaintextheight by \baselineskip%
+ \multiply\@IEEEINTmaintextheight by \baselineskip%
+ % now we calculate how much the title spacer height will
+ % have to be reduced from nominal (\@IEEEREDUCEmaintextheight is always
+ % a positive value) so that the maintext area will contain an integer
+ % number of normal size lines
+ % we change variable names here (to avoid confusion) as we no longer
+ % need \@IEEEINTmaintextheight and can reuse its dimen register
+ \let\@IEEEREDUCEmaintextheight=\@IEEEINTmaintextheight%
+ \advance\@IEEEREDUCEmaintextheight by -\@IEEEmaintextheight%
+ \advance\@IEEEREDUCEmaintextheight by \baselineskip%
+ % this is the calculated height of the spacer
+ % we change variable names here (to avoid confusion) as we no longer
+ % need \@IEEEmaintextheight and can reuse its dimen register
+ \let\@IEEECOMPENSATElen=\@IEEEmaintextheight%
+ \@IEEECOMPENSATElen=\@IEEENORMtitlevspace% set the nominal value
+ % we go with the reduced length if it is smaller than an increase
+ \ifdim\@IEEEREDUCEmaintextheight < 0.5\baselineskip\relax%
+ \advance\@IEEECOMPENSATElen by -\@IEEEREDUCEmaintextheight%
+ % if the resulting spacer is too small back out and go with an increase instead
+ \ifdim\@IEEECOMPENSATElen<\@IEEEMINtitlevspace\relax%
+ \advance\@IEEECOMPENSATElen by \baselineskip%
+ \fi%
+ \else%
+ % go with an increase because it is closer to the nominal than a decrease
+ \advance\@IEEECOMPENSATElen by -\@IEEEREDUCEmaintextheight%
+ \advance\@IEEECOMPENSATElen by \baselineskip%
+ \fi%
+ % set the calculated rigid spacer
+ \vspace{\@IEEECOMPENSATElen}}}
+
+
+
+% V1.6
+% we allow the user access to the last part of the title area
+% useful in emergencies such as when a different spacing is needed
+% This text is NOT compensated for in the dynamic sizer.
+\let\@IEEEaftertitletext=\relax
+\long\def\IEEEaftertitletext#1{\def\@IEEEaftertitletext{#1}}
+
+% V1.7 provide a way for users to enter abstract and keywords
+% into the onecolumn title are. This text is compensated for
+% in the dynamic sizer.
+\let\@IEEEcompsoctitleabstractindextext=\relax
+\long\def\IEEEcompsoctitleabstractindextext#1{\def\@IEEEcompsoctitleabstractindextext{#1}}
+% V1.7 provide a way for users to get the \@IEEEcompsoctitleabstractindextext if
+% not in compsoc journal mode - this way abstract and keywords can be placed
+% in their conventional position if not in compsoc mode.
+\def\IEEEdisplaynotcompsoctitleabstractindextext{%
+\ifCLASSOPTIONcompsoc% display if compsoc conf
+\ifCLASSOPTIONconference\@IEEEcompsoctitleabstractindextext\fi
+\else% or if not compsoc
+\@IEEEcompsoctitleabstractindextext\fi}
+
+
+% command to allow alteration of baselinestretch, but only if the current
+% baselineskip is unity. Used to tweak the compsoc abstract and keywords line spacing.
+\def\@IEEEtweakunitybaselinestretch#1{{\def\baselinestretch{1}\selectfont
+\global\@tempskipa\baselineskip}\ifnum\@tempskipa=\baselineskip%
+\def\baselinestretch{#1}\selectfont\fi\relax}
+
+
+% abstract and keywords are in \small, except
+% for 9pt docs in which they are in \footnotesize
+% Because 9pt docs use an 8pt footnotesize, \small
+% becomes a rather awkward 8.5pt
+\def\@IEEEabskeysecsize{\small}
+\ifx\CLASSOPTIONpt\@IEEEptsizenine
+ \def\@IEEEabskeysecsize{\footnotesize}
+\fi
+
+% compsoc journals use \footnotesize, compsoc conferences use normalsize
+\@IEEEcompsoconly{\def\@IEEEabskeysecsize{\footnotesize}}
+\@IEEEcompsocconfonly{\def\@IEEEabskeysecsize{\normalsize}}
+
+
+
+
+% V1.6 have abstract and keywords strip leading spaces, pars and newlines
+% so that spacing is more tightly controlled.
+\def\abstract{\normalfont
+ \if@twocolumn
+ \@IEEEabskeysecsize\bfseries\textit{\abstractname}---\relax
+ \else
+ \begin{center}\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\end{center}\quotation\@IEEEabskeysecsize
+ \fi\@IEEEgobbleleadPARNLSP}
+% V1.6 IEEE wants only 1 pica from end of abstract to introduction heading when in
+% conference mode (the heading already has this much above it)
+\def\endabstract{\relax\ifCLASSOPTIONconference\vspace{0ex}\else\vspace{1.34ex}\fi\par\if@twocolumn\else\endquotation\fi
+ \normalfont\normalsize}
+
+\def\IEEEkeywords{\normalfont
+ \if@twocolumn
+ \@IEEEabskeysecsize\bfseries\textit{\IEEEkeywordsname}---\relax
+ \else
+ \begin{center}\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\end{center}\quotation\@IEEEabskeysecsize
+ \fi\@IEEEgobbleleadPARNLSP}
+\def\endIEEEkeywords{\relax\ifCLASSOPTIONtechnote\vspace{1.34ex}\else\vspace{0.67ex}\fi
+ \par\if@twocolumn\else\endquotation\fi%
+ \normalfont\normalsize}
+
+% V1.7 compsoc keywords index terms
+\ifCLASSOPTIONcompsoc
+ \ifCLASSOPTIONconference% compsoc conference
+\def\abstract{\normalfont
+ \begin{center}\@IEEEabskeysecsize\textbf{\large\abstractname}\end{center}\vskip 0.5\baselineskip plus 0.1\baselineskip minus 0.1\baselineskip
+ \if@twocolumn\else\quotation\fi\itshape\@IEEEabskeysecsize%
+ \par\@IEEEgobbleleadPARNLSP}
+\def\IEEEkeywords{\normalfont\vskip 1.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip
+ \begin{center}\@IEEEabskeysecsize\textbf{\large\IEEEkeywordsname}\end{center}\vskip 0.5\baselineskip plus 0.1\baselineskip minus 0.1\baselineskip
+ \if@twocolumn\else\quotation\fi\itshape\@IEEEabskeysecsize%
+ \par\@IEEEgobbleleadPARNLSP}
+ \else% compsoc not conference
+\def\abstract{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\sffamily
+ \if@twocolumn
+ \@IEEEabskeysecsize\noindent\textbf{\abstractname}---\relax
+ \else
+ \begin{center}\vspace{-1.78ex}\@IEEEabskeysecsize\textbf{\abstractname}\end{center}\quotation\@IEEEabskeysecsize%
+ \fi\@IEEEgobbleleadPARNLSP}
+\def\IEEEkeywords{\normalfont\@IEEEtweakunitybaselinestretch{1.15}\sffamily
+ \if@twocolumn
+ \@IEEEabskeysecsize\vskip 0.5\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip\noindent
+ \textbf{\IEEEkeywordsname}---\relax
+ \else
+ \begin{center}\@IEEEabskeysecsize\textbf{\IEEEkeywordsname}\end{center}\quotation\@IEEEabskeysecsize%
+ \fi\@IEEEgobbleleadPARNLSP}
+ \fi
+\fi
+
+
+
+% gobbles all leading \, \\ and \par, upon finding first token that
+% is not a \ , \\ or a \par, it ceases and returns that token
+%
+% used to strip leading \, \\ and \par from the input
+% so that such things in the beginning of an environment will not
+% affect the formatting of the text
+\long\def\@IEEEgobbleleadPARNLSP#1{\let\@IEEEswallowthistoken=0%
+\let\@IEEEgobbleleadPARNLSPtoken#1%
+\let\@IEEEgobbleleadPARtoken=\par%
+\let\@IEEEgobbleleadNLtoken=\\%
+\let\@IEEEgobbleleadSPtoken=\ %
+\def\@IEEEgobbleleadSPMACRO{\ }%
+\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadPARtoken%
+\let\@IEEEswallowthistoken=1%
+\fi%
+\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadNLtoken%
+\let\@IEEEswallowthistoken=1%
+\fi%
+\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadSPtoken%
+\let\@IEEEswallowthistoken=1%
+\fi%
+% a control space will come in as a macro
+% when it is the last one on a line
+\ifx\@IEEEgobbleleadPARNLSPtoken\@IEEEgobbleleadSPMACRO%
+\let\@IEEEswallowthistoken=1%
+\fi%
+% if we have to swallow this token, do so and taste the next one
+% else spit it out and stop gobbling
+\ifx\@IEEEswallowthistoken 1\let\@IEEEnextgobbleleadPARNLSP=\@IEEEgobbleleadPARNLSP\else%
+\let\@IEEEnextgobbleleadPARNLSP=#1\fi%
+\@IEEEnextgobbleleadPARNLSP}%
+
+
+
+
+% TITLING OF SECTIONS
+\def\@IEEEsectpunct{:\ \,} % Punctuation after run-in section heading (headings which are
+ % part of the paragraphs), need little bit more than a single space
+ % spacing from section number to title
+% compsoc conferences use regular period/space punctuation
+\ifCLASSOPTIONcompsoc
+\ifCLASSOPTIONconference
+\def\@IEEEsectpunct{.\ }
+\fi\fi
+
+
+\def\@seccntformat#1{\csname the#1dis\endcsname\hskip 0.5em\relax}
+
+\ifCLASSOPTIONcompsoc
+% compsoc journals need extra spacing
+\ifCLASSOPTIONconference\else
+\def\@seccntformat#1{\csname the#1dis\endcsname\hskip 1em\relax}
+\fi\fi
+
+%v1.7 put {} after #6 to allow for some types of user font control
+%and use \@@par rather than \par
+\def\@sect#1#2#3#4#5#6[#7]#8{%
+ \ifnum #2>\c@secnumdepth
+ \let\@svsec\@empty
+ \else
+ \refstepcounter{#1}%
+ % load section label and spacer into \@svsec
+ \protected@edef\@svsec{\@seccntformat{#1}\relax}%
+ \fi%
+ \@tempskipa #5\relax
+ \ifdim \@tempskipa>\z@% tempskipa determines whether is treated as a high
+ \begingroup #6{\relax% or low level heading
+ \noindent % subsections are NOT indented
+ % print top level headings. \@svsec is label, #8 is heading title
+ % IEEE does not block indent the section title text, it flows like normal
+ {\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\@@par}}%
+ \endgroup
+ \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth\relax\else
+ \protect\numberline{\csname the#1\endcsname}\fi#7}%
+ \else % printout low level headings
+ % svsechd seems to swallow the trailing space, protect it with \mbox{}
+ % got rid of sectionmark stuff
+ \def\@svsechd{#6{\hskip #3\relax\@svsec #8\@IEEEsectpunct\mbox{}}%
+ \addcontentsline{toc}{#1}{\ifnum #2>\c@secnumdepth\relax\else
+ \protect\numberline{\csname the#1\endcsname}\fi#7}}%
+ \fi%skip down
+ \@xsect{#5}}
+
+
+% section* handler
+%v1.7 put {} after #4 to allow for some types of user font control
+%and use \@@par rather than \par
+\def\@ssect#1#2#3#4#5{\@tempskipa #3\relax
+ \ifdim \@tempskipa>\z@
+ %\begingroup #4\@hangfrom{\hskip #1}{\interlinepenalty \@M #5\par}\endgroup
+ % IEEE does not block indent the section title text, it flows like normal
+ \begingroup \noindent #4{\relax{\hskip #1}{\interlinepenalty \@M #5\@@par}}\endgroup
+ % svsechd swallows the trailing space, protect it with \mbox{}
+ \else \def\@svsechd{#4{\hskip #1\relax #5\@IEEEsectpunct\mbox{}}}\fi
+ \@xsect{#3}}
+
+
+%% SECTION heading spacing and font
+%%
+% arguments are: #1 - sectiontype name
+% (for \@sect) #2 - section level
+% #3 - section heading indent
+% #4 - top separation (absolute value used, neg indicates not to indent main text)
+% If negative, make stretch parts negative too!
+% #5 - (absolute value used) positive: bottom separation after heading,
+% negative: amount to indent main text after heading
+% Both #4 and #5 negative means to indent main text and use negative top separation
+% #6 - font control
+% You've got to have \normalfont\normalsize in the font specs below to prevent
+% trouble when you do something like:
+% \section{Note}{\ttfamily TT-TEXT} is known to ...
+% IEEE sometimes REALLY stretches the area before a section
+% heading by up to about 0.5in. However, it may not be a good
+% idea to let LaTeX have quite this much rubber.
+\ifCLASSOPTIONconference%
+% IEEE wants section heading spacing to decrease for conference mode
+\def\section{\@startsection{section}{1}{\z@}{1.5ex plus 1.5ex minus 0.5ex}%
+{0.7ex plus 1ex minus 0ex}{\normalfont\normalsize\centering\scshape}}%
+\def\subsection{\@startsection{subsection}{2}{\z@}{1.5ex plus 1.5ex minus 0.5ex}%
+{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
+\else % for journals
+\def\section{\@startsection{section}{1}{\z@}{3.0ex plus 1.5ex minus 1.5ex}% V1.6 3.0ex from 3.5ex
+{0.7ex plus 1ex minus 0ex}{\normalfont\normalsize\centering\scshape}}%
+\def\subsection{\@startsection{subsection}{2}{\z@}{3.5ex plus 1.5ex minus 1.5ex}%
+{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\itshape}}%
+\fi
+
+% for both journals and conferences
+% decided to put in a little rubber above the section, might help somebody
+\def\subsubsection{\@startsection{subsubsection}{3}{\parindent}{0ex plus 0.1ex minus 0.1ex}%
+{0ex}{\normalfont\normalsize\itshape}}%
+\def\paragraph{\@startsection{paragraph}{4}{2\parindent}{0ex plus 0.1ex minus 0.1ex}%
+{0ex}{\normalfont\normalsize\itshape}}%
+
+
+% compsoc
+\ifCLASSOPTIONcompsoc
+\ifCLASSOPTIONconference
+% compsoc conference
+\def\section{\@startsection{section}{1}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}{\normalfont\large\bfseries}}%
+\def\subsection{\@startsection{subsection}{2}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}{\normalfont\sublargesize\bfseries}}%
+\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{1\baselineskip plus 0.25\baselineskip minus 0.25\baselineskip}%
+{0ex}{\normalfont\normalsize\bfseries}}%
+\def\paragraph{\@startsection{paragraph}{4}{2\parindent}{0ex plus 0.1ex minus 0.1ex}%
+{0ex}{\normalfont\normalsize}}%
+\else% compsoc journals
+% use negative top separation as compsoc journals do not indent paragraphs after section titles
+\def\section{\@startsection{section}{1}{\z@}{-3ex plus -2ex minus -1.5ex}%
+{0.7ex plus 1ex minus 0ex}{\normalfont\large\sffamily\bfseries\scshape}}%
+% Note that subsection and smaller may not be correct for the Computer Society,
+% I have to look up an example.
+\def\subsection{\@startsection{subsection}{2}{\z@}{-3.5ex plus -1.5ex minus -1.5ex}%
+{0.7ex plus .5ex minus 0ex}{\normalfont\normalsize\sffamily\bfseries}}%
+\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-2.5ex plus -1ex minus -1ex}%
+{0.5ex plus 0.5ex minus 0ex}{\normalfont\normalsize\sffamily\itshape}}%
+\def\paragraph{\@startsection{paragraph}{4}{2\parindent}{-0ex plus -0.1ex minus -0.1ex}%
+{0ex}{\normalfont\normalsize}}%
+\fi\fi
+
+
+
+
+%% ENVIRONMENTS
+% "box" symbols at end of proofs
+\def\IEEEQEDclosed{\mbox{\rule[0pt]{1.3ex}{1.3ex}}} % for a filled box
+% V1.6 some journals use an open box instead that will just fit around a closed one
+\def\IEEEQEDopen{{\setlength{\fboxsep}{0pt}\setlength{\fboxrule}{0.2pt}\fbox{\rule[0pt]{0pt}{1.3ex}\rule[0pt]{1.3ex}{0pt}}}}
+\ifCLASSOPTIONcompsoc
+\def\IEEEQED{\IEEEQEDopen} % default to open for compsoc
+\else
+\def\IEEEQED{\IEEEQEDclosed} % otherwise default to closed
+\fi
+
+% v1.7 name change to avoid namespace collision with amsthm. Also add support
+% for an optional argument.
+\def\IEEEproof{\@ifnextchar[{\@IEEEproof}{\@IEEEproof[\IEEEproofname]}}
+\def\@IEEEproof[#1]{\par\noindent\hspace{2em}{\itshape #1: }}
+\def\endIEEEproof{\hspace*{\fill}~\IEEEQED\par}
+
+
+%\itemindent is set to \z@ by list, so define new temporary variable
+\newdimen\@IEEEtmpitemindent
+\def\@begintheorem#1#2{\@IEEEtmpitemindent\itemindent\topsep 0pt\rmfamily\trivlist%
+ \item[\hskip \labelsep{\indent\itshape #1\ #2:}]\itemindent\@IEEEtmpitemindent}
+\def\@opargbegintheorem#1#2#3{\@IEEEtmpitemindent\itemindent\topsep 0pt\rmfamily \trivlist%
+% V1.6 IEEE is back to using () around theorem names which are also in italics
+% Thanks to Christian Peel for reporting this.
+ \item[\hskip\labelsep{\indent\itshape #1\ #2\ (#3):}]\itemindent\@IEEEtmpitemindent}
+% V1.7 remove bogus \unskip that caused equations in theorems to collide with
+% lines below.
+\def\@endtheorem{\endtrivlist}
+
+% V1.6
+% display command for the section the theorem is in - so that \thesection
+% is not used as this will be in Roman numerals when we want arabic.
+% LaTeX2e uses \def\@thmcounter#1{\noexpand\arabic{#1}} for the theorem number
+% (second part) display and \def\@thmcountersep{.} as a separator.
+% V1.7 intercept calls to the section counter and reroute to \@IEEEthmcounterinsection
+% to allow \appendix(ices} to override as needed.
+%
+% special handler for sections, allows appendix(ices) to override
+\gdef\@IEEEthmcounterinsection#1{\arabic{#1}}
+% string macro
+\edef\@IEEEstringsection{section}
+
+% redefine the #1#2[#3] form of newtheorem to use a hook to \@IEEEthmcounterinsection
+% if section in_counter is used
+\def\@xnthm#1#2[#3]{%
+ \expandafter\@ifdefinable\csname #1\endcsname
+ {\@definecounter{#1}\@newctr{#1}[#3]%
+ \edef\@IEEEstringtmp{#3}
+ \ifx\@IEEEstringtmp\@IEEEstringsection
+ \expandafter\xdef\csname the#1\endcsname{%
+ \noexpand\@IEEEthmcounterinsection{#3}\@thmcountersep
+ \@thmcounter{#1}}%
+ \else
+ \expandafter\xdef\csname the#1\endcsname{%
+ \expandafter\noexpand\csname the#3\endcsname \@thmcountersep
+ \@thmcounter{#1}}%
+ \fi
+ \global\@namedef{#1}{\@thm{#1}{#2}}%
+ \global\@namedef{end#1}{\@endtheorem}}}
+
+
+
+%% SET UP THE DEFAULT PAGESTYLE
+\ps@headings
+\pagenumbering{arabic}
+
+% normally the page counter starts at 1
+\setcounter{page}{1}
+% however, for peerreview the cover sheet is page 0 or page -1
+% (for duplex printing)
+\ifCLASSOPTIONpeerreview
+ \if@twoside
+ \setcounter{page}{-1}
+ \else
+ \setcounter{page}{0}
+ \fi
+\fi
+
+% standard book class behavior - let bottom line float up and down as
+% needed when single sided
+\ifCLASSOPTIONtwoside\else\raggedbottom\fi
+% if two column - turn on twocolumn, allow word spacings to stretch more and
+% enforce a rigid position for the last lines
+\ifCLASSOPTIONtwocolumn
+% the peer review option delays invoking twocolumn
+ \ifCLASSOPTIONpeerreview\else
+ \twocolumn
+ \fi
+\sloppy
+\flushbottom
+\fi
+
+
+
+
+% \APPENDIX and \APPENDICES definitions
+
+% This is the \@ifmtarg command from the LaTeX ifmtarg package
+% by Peter Wilson (CUA) and Donald Arseneau
+% \@ifmtarg is used to determine if an argument to a command
+% is present or not.
+% For instance:
+% \@ifmtarg{#1}{\typeout{empty}}{\typeout{has something}}
+% \@ifmtarg is used with our redefined \section command if
+% \appendices is invoked.
+% The command \section will behave slightly differently depending
+% on whether the user specifies a title:
+% \section{My appendix title}
+% or not:
+% \section{}
+% This way, we can eliminate the blank lines where the title
+% would be, and the unneeded : after Appendix in the table of
+% contents
+\begingroup
+\catcode`\Q=3
+\long\gdef\@ifmtarg#1{\@xifmtarg#1QQ\@secondoftwo\@firstoftwo\@nil}
+\long\gdef\@xifmtarg#1#2Q#3#4#5\@nil{#4}
+\endgroup
+% end of \@ifmtarg defs
+
+
+% V1.7
+% command that allows the one time saving of the original definition
+% of section to \@IEEEappendixsavesection for \appendix or \appendices
+% we don't save \section here as it may be redefined later by other
+% packages (hyperref.sty, etc.)
+\def\@IEEEsaveoriginalsectiononce{\let\@IEEEappendixsavesection\section
+\let\@IEEEsaveoriginalsectiononce\relax}
+
+% neat trick to grab and process the argument from \section{argument}
+% we process differently if the user invoked \section{} with no
+% argument (title)
+% note we reroute the call to the old \section*
+\def\@IEEEprocessthesectionargument#1{%
+\@ifmtarg{#1}{%
+\@IEEEappendixsavesection*{\appendixname~\thesectiondis}%
+\addcontentsline{toc}{section}{\appendixname~\thesection}}{%
+\@IEEEappendixsavesection*{\appendixname~\thesectiondis \\* #1}%
+\addcontentsline{toc}{section}{\appendixname~\thesection: #1}}}
+
+% we use this if the user calls \section{} after
+% \appendix-- which has no meaning. So, we ignore the
+% command and its argument. Then, warn the user.
+\def\@IEEEdestroythesectionargument#1{\typeout{** WARNING: Ignoring useless
+\protect\section\space in Appendix (line \the\inputlineno).}}
+
+
+% remember \thesection forms will be displayed in \ref calls
+% and in the Table of Contents.
+% The \sectiondis form is used in the actual heading itself
+
+% appendix command for one single appendix
+% normally has no heading. However, if you want a
+% heading, you can do so via the optional argument:
+% \appendix[Optional Heading]
+\def\appendix{\relax}
+\renewcommand{\appendix}[1][]{\@IEEEsaveoriginalsectiononce\par
+ % v1.6 keep hyperref's identifiers unique
+ \gdef\theHsection{Appendix.A}%
+ % v1.6 adjust hyperref's string name for the section
+ \xdef\Hy@chapapp{appendix}%
+ \setcounter{section}{0}%
+ \setcounter{subsection}{0}%
+ \setcounter{subsubsection}{0}%
+ \setcounter{paragraph}{0}%
+ \gdef\thesection{A}%
+ \gdef\thesectiondis{}%
+ \gdef\thesubsection{\Alph{subsection}}%
+ \gdef\@IEEEthmcounterinsection##1{A}
+ \refstepcounter{section}% update the \ref counter
+ \@ifmtarg{#1}{\@IEEEappendixsavesection*{\appendixname}%
+ \addcontentsline{toc}{section}{\appendixname}}{%
+ \@IEEEappendixsavesection*{\appendixname~\\* #1}%
+ \addcontentsline{toc}{section}{\appendixname: #1}}%
+ % redefine \section command for appendix
+ % leave \section* as is
+ \def\section{\@ifstar{\@IEEEappendixsavesection*}{%
+ \@IEEEdestroythesectionargument}}% throw out the argument
+ % of the normal form
+}
+
+
+
+% appendices command for multiple appendices
+% user then calls \section with an argument (possibly empty) to
+% declare the individual appendices
+\def\appendices{\@IEEEsaveoriginalsectiononce\par
+ % v1.6 keep hyperref's identifiers unique
+ \gdef\theHsection{Appendix.\Alph{section}}%
+ % v1.6 adjust hyperref's string name for the section
+ \xdef\Hy@chapapp{appendix}%
+ \setcounter{section}{-1}% we want \refstepcounter to use section 0
+ \setcounter{subsection}{0}%
+ \setcounter{subsubsection}{0}%
+ \setcounter{paragraph}{0}%
+ \ifCLASSOPTIONromanappendices%
+ \gdef\thesection{\Roman{section}}%
+ \gdef\thesectiondis{\Roman{section}}%
+ \@IEEEcompsocconfonly{\gdef\thesectiondis{\Roman{section}.}}%
+ \gdef\@IEEEthmcounterinsection##1{A\arabic{##1}}
+ \else%
+ \gdef\thesection{\Alph{section}}%
+ \gdef\thesectiondis{\Alph{section}}%
+ \@IEEEcompsocconfonly{\gdef\thesectiondis{\Alph{section}.}}%
+ \gdef\@IEEEthmcounterinsection##1{\Alph{##1}}
+ \fi%
+ \refstepcounter{section}% update the \ref counter
+ \setcounter{section}{0}% NEXT \section will be the FIRST appendix
+ % redefine \section command for appendices
+ % leave \section* as is
+ \def\section{\@ifstar{\@IEEEappendixsavesection*}{% process the *-form
+ \refstepcounter{section}% or is a new section so,
+ \@IEEEprocessthesectionargument}}% process the argument
+ % of the normal form
+}
+
+
+
+% \IEEEPARstart
+% Definition for the big two line drop cap letter at the beginning of the
+% first paragraph of journal papers. The first argument is the first letter
+% of the first word, the second argument is the remaining letters of the
+% first word which will be rendered in upper case.
+% In V1.6 this has been completely rewritten to:
+%
+% 1. no longer have problems when the user begins an environment
+% within the paragraph that uses \IEEEPARstart.
+% 2. auto-detect and use the current font family
+% 3. revise handling of the space at the end of the first word so that
+% interword glue will now work as normal.
+% 4. produce correctly aligned edges for the (two) indented lines.
+%
+% We generalize things via control macros - playing with these is fun too.
+%
+% V1.7 added more control macros to make it easy for IEEEtrantools.sty users
+% to change the font style.
+%
+% the number of lines that are indented to clear it
+% may need to increase if using decenders
+\def\@IEEEPARstartDROPLINES{2}
+% minimum number of lines left on a page to allow a \@IEEEPARstart
+% Does not take into consideration rubber shrink, so it tends to
+% be overly cautious
+\def\@IEEEPARstartMINPAGELINES{2}
+% V1.7 the height of the drop cap is adjusted to match the height of this text
+% in the current font (when \IEEEPARstart is called).
+\def\@IEEEPARstartHEIGHTTEXT{T}
+% the depth the letter is lowered below the baseline
+% the height (and size) of the letter is determined by the sum
+% of this value and the height of the \@IEEEPARstartHEIGHTTEXT in the current
+% font. It is a good idea to set this value in terms of the baselineskip
+% so that it can respond to changes therein.
+\def\@IEEEPARstartDROPDEPTH{1.1\baselineskip}
+% V1.7 the font the drop cap will be rendered in,
+% can take zero or one argument.
+\def\@IEEEPARstartFONTSTYLE{\bfseries}
+% V1.7 any additional, non-font related commands needed to modify
+% the drop cap letter, can take zero or one argument.
+\def\@IEEEPARstartCAPSTYLE{\MakeUppercase}
+% V1.7 the font that will be used to render the rest of the word,
+% can take zero or one argument.
+\def\@IEEEPARstartWORDFONTSTYLE{\relax}
+% V1.7 any additional, non-font related commands needed to modify
+% the rest of the word, can take zero or one argument.
+\def\@IEEEPARstartWORDCAPSTYLE{\MakeUppercase}
+% This is the horizontal separation distance from the drop letter to the main text.
+% Lengths that depend on the font (e.g., ex, em, etc.) will be referenced
+% to the font that is active when \IEEEPARstart is called.
+\def\@IEEEPARstartSEP{0.15em}
+% V1.7 horizontal offset applied to the left of the drop cap.
+\def\@IEEEPARstartHOFFSET{0em}
+% V1.7 Italic correction command applied at the end of the drop cap.
+\def\@IEEEPARstartITLCORRECT{\/}
+
+% V1.7 compoc uses nonbold drop cap and small caps word style
+\ifCLASSOPTIONcompsoc
+\def\@IEEEPARstartFONTSTYLE{\mdseries}
+\def\@IEEEPARstartWORDFONTSTYLE{\scshape}
+\def\@IEEEPARstartWORDCAPSTYLE{\relax}
+\fi
+
+% definition of \IEEEPARstart
+% THIS IS A CONTROLLED SPACING AREA, DO NOT ALLOW SPACES WITHIN THESE LINES
+%
+% The token \@IEEEPARstartfont will be globally defined after the first use
+% of \IEEEPARstart and will be a font command which creates the big letter
+% The first argument is the first letter of the first word and the second
+% argument is the rest of the first word(s).
+\def\IEEEPARstart#1#2{\par{%
+% if this page does not have enough space, break it and lets start
+% on a new one
+\@IEEEtranneedspace{\@IEEEPARstartMINPAGELINES\baselineskip}{\relax}%
+% V1.7 move this up here in case user uses \textbf for \@IEEEPARstartFONTSTYLE
+% which uses command \leavevmode which causes an unwanted \indent to be issued
+\noindent
+% calculate the desired height of the big letter
+% it extends from the top of \@IEEEPARstartHEIGHTTEXT in the current font
+% down to \@IEEEPARstartDROPDEPTH below the current baseline
+\settoheight{\@IEEEtrantmpdimenA}{\@IEEEPARstartHEIGHTTEXT}%
+\addtolength{\@IEEEtrantmpdimenA}{\@IEEEPARstartDROPDEPTH}%
+% extract the name of the current font in bold
+% and place it in \@IEEEPARstartFONTNAME
+\def\@IEEEPARstartGETFIRSTWORD##1 ##2\relax{##1}%
+{\@IEEEPARstartFONTSTYLE{\selectfont\edef\@IEEEPARstartFONTNAMESPACE{\fontname\font\space}%
+\xdef\@IEEEPARstartFONTNAME{\expandafter\@IEEEPARstartGETFIRSTWORD\@IEEEPARstartFONTNAMESPACE\relax}}}%
+% define a font based on this name with a point size equal to the desired
+% height of the drop letter
+\font\@IEEEPARstartsubfont\@IEEEPARstartFONTNAME\space at \@IEEEtrantmpdimenA\relax%
+% save this value as a counter (integer) value (sp points)
+\@IEEEtrantmpcountA=\@IEEEtrantmpdimenA%
+% now get the height of the actual letter produced by this font size
+\settoheight{\@IEEEtrantmpdimenB}{\@IEEEPARstartsubfont\@IEEEPARstartCAPSTYLE{#1}}%
+% If something bogus happens like the first argument is empty or the
+% current font is strange, do not allow a zero height.
+\ifdim\@IEEEtrantmpdimenB=0pt\relax%
+\typeout{** WARNING: IEEEPARstart drop letter has zero height! (line \the\inputlineno)}%
+\typeout{ Forcing the drop letter font size to 10pt.}%
+\@IEEEtrantmpdimenB=10pt%
+\fi%
+% and store it as a counter
+\@IEEEtrantmpcountB=\@IEEEtrantmpdimenB%
+% Since a font size doesn't exactly correspond to the height of the capital
+% letters in that font, the actual height of the letter, \@IEEEtrantmpcountB,
+% will be less than that desired, \@IEEEtrantmpcountA
+% we need to raise the font size, \@IEEEtrantmpdimenA
+% by \@IEEEtrantmpcountA / \@IEEEtrantmpcountB
+% But, TeX doesn't have floating point division, so we have to use integer
+% division. Hence the use of the counters.
+% We need to reduce the denominator so that the loss of the remainder will
+% have minimal affect on the accuracy of the result
+\divide\@IEEEtrantmpcountB by 200%
+\divide\@IEEEtrantmpcountA by \@IEEEtrantmpcountB%
+% Then reequalize things when we use TeX's ability to multiply by
+% floating point values
+\@IEEEtrantmpdimenB=0.005\@IEEEtrantmpdimenA%
+\multiply\@IEEEtrantmpdimenB by \@IEEEtrantmpcountA%
+% \@IEEEPARstartfont is globaly set to the calculated font of the big letter
+% We need to carry this out of the local calculation area to to create the
+% big letter.
+\global\font\@IEEEPARstartfont\@IEEEPARstartFONTNAME\space at \@IEEEtrantmpdimenB%
+% Now set \@IEEEtrantmpdimenA to the width of the big letter
+% We need to carry this out of the local calculation area to set the
+% hanging indent
+\settowidth{\global\@IEEEtrantmpdimenA}{\@IEEEPARstartfont
+\@IEEEPARstartCAPSTYLE{#1\@IEEEPARstartITLCORRECT}}}%
+% end of the isolated calculation environment
+% add in the extra clearance we want
+\advance\@IEEEtrantmpdimenA by \@IEEEPARstartSEP\relax%
+% add in the optional offset
+\advance\@IEEEtrantmpdimenA by \@IEEEPARstartHOFFSET\relax%
+% V1.7 don't allow negative offsets to produce negative hanging indents
+\@IEEEtrantmpdimenB\@IEEEtrantmpdimenA
+\ifnum\@IEEEtrantmpdimenB < 0 \@IEEEtrantmpdimenB 0pt\fi
+% \@IEEEtrantmpdimenA has the width of the big letter plus the
+% separation space and \@IEEEPARstartfont is the font we need to use
+% Now, we make the letter and issue the hanging indent command
+% The letter is placed in a box of zero width and height so that other
+% text won't be displaced by it.
+\hangindent\@IEEEtrantmpdimenB\hangafter=-\@IEEEPARstartDROPLINES%
+\makebox[0pt][l]{\hspace{-\@IEEEtrantmpdimenA}%
+\raisebox{-\@IEEEPARstartDROPDEPTH}[0pt][0pt]{\hspace{\@IEEEPARstartHOFFSET}%
+\@IEEEPARstartfont\@IEEEPARstartCAPSTYLE{#1\@IEEEPARstartITLCORRECT}%
+\hspace{\@IEEEPARstartSEP}}}%
+{\@IEEEPARstartWORDFONTSTYLE{\@IEEEPARstartWORDCAPSTYLE{\selectfont#2}}}}
+
+
+
+
+
+
+% determines if the space remaining on a given page is equal to or greater
+% than the specified space of argument one
+% if not, execute argument two (only if the remaining space is greater than zero)
+% and issue a \newpage
+%
+% example: \@IEEEtranneedspace{2in}{\vfill}
+%
+% Does not take into consideration rubber shrinkage, so it tends to
+% be overly cautious
+% Based on an example posted by Donald Arseneau
+% Note this macro uses \@IEEEtrantmpdimenB internally for calculations,
+% so DO NOT PASS \@IEEEtrantmpdimenB to this routine
+% if you need a dimen register, import with \@IEEEtrantmpdimenA instead
+\def\@IEEEtranneedspace#1#2{\penalty-100\begingroup%shield temp variable
+\@IEEEtrantmpdimenB\pagegoal\advance\@IEEEtrantmpdimenB-\pagetotal% space left
+\ifdim #1>\@IEEEtrantmpdimenB\relax% not enough space left
+\ifdim\@IEEEtrantmpdimenB>\z@\relax #2\fi%
+\newpage%
+\fi\endgroup}
+
+
+
+% IEEEbiography ENVIRONMENT
+% Allows user to enter biography leaving place for picture (adapts to font size)
+% As of V1.5, a new optional argument allows you to have a real graphic!
+% V1.5 and later also fixes the "colliding biographies" which could happen when a
+% biography's text was shorter than the space for the photo.
+% MDS 7/2001
+% V1.6 prevent multiple biographies from making multiple TOC entries
+\newif\if@IEEEbiographyTOCentrynotmade
+\global\@IEEEbiographyTOCentrynotmadetrue
+
+% biography counter so hyperref can jump directly to the biographies
+% and not just the previous section
+\newcounter{IEEEbiography}
+\setcounter{IEEEbiography}{0}
+
+% photo area size
+\def\@IEEEBIOphotowidth{1.0in} % width of the biography photo area
+\def\@IEEEBIOphotodepth{1.25in} % depth (height) of the biography photo area
+% area cleared for photo
+\def\@IEEEBIOhangwidth{1.14in} % width cleared for the biography photo area
+\def\@IEEEBIOhangdepth{1.25in} % depth cleared for the biography photo area
+ % actual depth will be a multiple of
+ % \baselineskip, rounded up
+\def\@IEEEBIOskipN{4\baselineskip}% nominal value of the vskip above the biography
+
+\newenvironment{IEEEbiography}[2][]{\normalfont\@IEEEcompsoconly{\sffamily}\footnotesize%
+\unitlength 1in\parskip=0pt\par\parindent 1em\interlinepenalty500%
+% we need enough space to support the hanging indent
+% the nominal value of the spacer
+% and one extra line for good measure
+\@IEEEtrantmpdimenA=\@IEEEBIOhangdepth%
+\advance\@IEEEtrantmpdimenA by \@IEEEBIOskipN%
+\advance\@IEEEtrantmpdimenA by 1\baselineskip%
+% if this page does not have enough space, break it and lets start
+% with a new one
+\@IEEEtranneedspace{\@IEEEtrantmpdimenA}{\relax}%
+% nominal spacer can strech, not shrink use 1fil so user can out stretch with \vfill
+\vskip \@IEEEBIOskipN plus 1fil minus 0\baselineskip%
+% the default box for where the photo goes
+\def\@IEEEtempbiographybox{{\setlength{\fboxsep}{0pt}\framebox{%
+\begin{minipage}[b][\@IEEEBIOphotodepth][c]{\@IEEEBIOphotowidth}\centering PLACE\\ PHOTO\\ HERE \end{minipage}}}}%
+%
+% detect if the optional argument was supplied, this requires the
+% \@ifmtarg command as defined in the appendix section above
+% and if so, override the default box with what they want
+\@ifmtarg{#1}{\relax}{\def\@IEEEtempbiographybox{\mbox{\begin{minipage}[b][\@IEEEBIOphotodepth][c]{\@IEEEBIOphotowidth}%
+\centering%
+#1%
+\end{minipage}}}}% end if optional argument supplied
+% Make an entry into the table of contents only if we have not done so before
+\if@IEEEbiographyTOCentrynotmade%
+% link labels to the biography counter so hyperref will jump
+% to the biography, not the previous section
+\setcounter{IEEEbiography}{-1}%
+\refstepcounter{IEEEbiography}%
+\addcontentsline{toc}{section}{Biographies}%
+\global\@IEEEbiographyTOCentrynotmadefalse%
+\fi%
+% one more biography
+\refstepcounter{IEEEbiography}%
+% Make an entry for this name into the table of contents
+\addcontentsline{toc}{subsection}{#2}%
+% V1.6 properly handle if a new paragraph should occur while the
+% hanging indent is still active. Do this by redefining \par so
+% that it will not start a new paragraph. (But it will appear to the
+% user as if it did.) Also, strip any leading pars, newlines, or spaces.
+\let\@IEEEBIOORGparCMD=\par% save the original \par command
+\edef\par{\hfil\break\indent}% the new \par will not be a "real" \par
+\settoheight{\@IEEEtrantmpdimenA}{\@IEEEtempbiographybox}% get height of biography box
+\@IEEEtrantmpdimenB=\@IEEEBIOhangdepth%
+\@IEEEtrantmpcountA=\@IEEEtrantmpdimenB% countA has the hang depth
+\divide\@IEEEtrantmpcountA by \baselineskip% calculates lines needed to produce the hang depth
+\advance\@IEEEtrantmpcountA by 1% ensure we overestimate
+% set the hanging indent
+\hangindent\@IEEEBIOhangwidth%
+\hangafter-\@IEEEtrantmpcountA%
+% reference the top of the photo area to the top of a capital T
+\settoheight{\@IEEEtrantmpdimenB}{\mbox{T}}%
+% set the photo box, give it zero width and height so as not to disturb anything
+\noindent\makebox[0pt][l]{\hspace{-\@IEEEBIOhangwidth}\raisebox{\@IEEEtrantmpdimenB}[0pt][0pt]{%
+\raisebox{-\@IEEEBIOphotodepth}[0pt][0pt]{\@IEEEtempbiographybox}}}%
+% now place the author name and begin the bio text
+\noindent\textbf{#2\ }\@IEEEgobbleleadPARNLSP}{\relax\let\par=\@IEEEBIOORGparCMD\par%
+% 7/2001 V1.5 detect when the biography text is shorter than the photo area
+% and pad the unused area - preventing a collision from the next biography entry
+% MDS
+\ifnum \prevgraf <\@IEEEtrantmpcountA\relax% detect when the biography text is shorter than the photo
+ \advance\@IEEEtrantmpcountA by -\prevgraf% calculate how many lines we need to pad
+ \advance\@IEEEtrantmpcountA by -1\relax% we compensate for the fact that we indented an extra line
+ \@IEEEtrantmpdimenA=\baselineskip% calculate the length of the padding
+ \multiply\@IEEEtrantmpdimenA by \@IEEEtrantmpcountA%
+ \noindent\rule{0pt}{\@IEEEtrantmpdimenA}% insert an invisible support strut
+\fi%
+\par\normalfont}
+
+
+
+% V1.6
+% added biography without a photo environment
+\newenvironment{IEEEbiographynophoto}[1]{%
+% Make an entry into the table of contents only if we have not done so before
+\if@IEEEbiographyTOCentrynotmade%
+% link labels to the biography counter so hyperref will jump
+% to the biography, not the previous section
+\setcounter{IEEEbiography}{-1}%
+\refstepcounter{IEEEbiography}%
+\addcontentsline{toc}{section}{Biographies}%
+\global\@IEEEbiographyTOCentrynotmadefalse%
+\fi%
+% one more biography
+\refstepcounter{IEEEbiography}%
+% Make an entry for this name into the table of contents
+\addcontentsline{toc}{subsection}{#1}%
+\normalfont\@IEEEcompsoconly{\sffamily}\footnotesize\interlinepenalty500%
+\vskip 4\baselineskip plus 1fil minus 0\baselineskip%
+\parskip=0pt\par%
+\noindent\textbf{#1\ }\@IEEEgobbleleadPARNLSP}{\relax\par\normalfont}
+
+
+% provide the user with some old font commands
+% got this from article.cls
+\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}
+\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
+\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
+\DeclareRobustCommand*\cal{\@fontswitch\relax\mathcal}
+\DeclareRobustCommand*\mit{\@fontswitch\relax\mathnormal}
+
+
+% SPECIAL PAPER NOTICE COMMANDS
+%
+% holds the special notice text
+\def\@IEEEspecialpapernotice{\relax}
+
+% for special papers, like invited papers, the user can do:
+% \IEEEspecialpapernotice{(Invited Paper)} before \maketitle
+\def\IEEEspecialpapernotice#1{\ifCLASSOPTIONconference%
+\def\@IEEEspecialpapernotice{{\sublargesize\textit{#1}\vspace*{1em}}}%
+\else%
+\def\@IEEEspecialpapernotice{{\\*[1.5ex]\sublargesize\textit{#1}}\vspace*{-2ex}}%
+\fi}
+
+
+
+
+% PUBLISHER ID COMMANDS
+% to insert a publisher's ID footer
+% V1.6 \IEEEpubid has been changed so that the change in page size and style
+% occurs in \maketitle. \IEEEpubid must now be issued prior to \maketitle
+% use \IEEEpubidadjcol as before - in the second column of the title page
+% These changes allow \maketitle to take the reduced page height into
+% consideration when dynamically setting the space between the author
+% names and the maintext.
+%
+% the amount the main text is pulled up to make room for the
+% publisher's ID footer
+% IEEE uses about 1.3\baselineskip for journals,
+% dynamic title spacing will clean up the fraction
+\def\@IEEEpubidpullup{1.3\baselineskip}
+\ifCLASSOPTIONtechnote
+% for technotes it must be an integer of baselineskip as there can be no
+% dynamic title spacing for two column mode technotes (the title is in the
+% in first column) and we should maintain an integer number of lines in the
+% second column
+% There are some examples (such as older issues of "Transactions on
+% Information Theory") in which IEEE really pulls the text off the ID for
+% technotes - about 0.55in (or 4\baselineskip). We'll use 2\baselineskip
+% and call it even.
+\def\@IEEEpubidpullup{2\baselineskip}
+\fi
+
+% V1.7 compsoc does not use a pullup
+\ifCLASSOPTIONcompsoc
+\def\@IEEEpubidpullup{0pt}
+\fi
+
+% holds the ID text
+\def\@IEEEpubid{\relax}
+
+% flag so \maketitle can tell if \IEEEpubid was called
+\newif\if@IEEEusingpubid
+\global\@IEEEusingpubidfalse
+% issue this command in the page to have the ID at the bottom
+% V1.6 use before \maketitle
+\def\IEEEpubid#1{\def\@IEEEpubid{#1}\global\@IEEEusingpubidtrue}
+
+
+% command which will pull up (shorten) the column it is executed in
+% to make room for the publisher ID. Place in the second column of
+% the title page when using \IEEEpubid
+% Is smart enough not to do anything when in single column text or
+% if the user hasn't called \IEEEpubid
+% currently needed in for the second column of a page with the
+% publisher ID. If not needed in future releases, please provide this
+% command and define it as \relax for backward compatibility
+% v1.6b do not allow command to operate if the peer review option has been
+% selected because \IEEEpubidadjcol will not be on the cover page.
+% V1.7 do nothing if compsoc
+\def\IEEEpubidadjcol{\ifCLASSOPTIONcompsoc\else\ifCLASSOPTIONpeerreview\else
+\if@twocolumn\if@IEEEusingpubid\enlargethispage{-\@IEEEpubidpullup}\fi\fi\fi\fi}
+
+% Special thanks to Peter Wilson, Daniel Luecking, and the other
+% gurus at comp.text.tex, for helping me to understand how best to
+% implement the IEEEpubid command in LaTeX.
+
+
+
+%% Lockout some commands under various conditions
+
+% general purpose bit bucket
+\newsavebox{\@IEEEtranrubishbin}
+
+% flags to prevent multiple warning messages
+\newif\if@IEEEWARNthanks
+\newif\if@IEEEWARNIEEEPARstart
+\newif\if@IEEEWARNIEEEbiography
+\newif\if@IEEEWARNIEEEbiographynophoto
+\newif\if@IEEEWARNIEEEpubid
+\newif\if@IEEEWARNIEEEpubidadjcol
+\newif\if@IEEEWARNIEEEmembership
+\newif\if@IEEEWARNIEEEaftertitletext
+\@IEEEWARNthankstrue
+\@IEEEWARNIEEEPARstarttrue
+\@IEEEWARNIEEEbiographytrue
+\@IEEEWARNIEEEbiographynophototrue
+\@IEEEWARNIEEEpubidtrue
+\@IEEEWARNIEEEpubidadjcoltrue
+\@IEEEWARNIEEEmembershiptrue
+\@IEEEWARNIEEEaftertitletexttrue
+
+
+%% Lockout some commands when in various modes, but allow them to be restored if needed
+%%
+% save commands which might be locked out
+% so that the user can later restore them if needed
+\let\@IEEESAVECMDthanks\thanks
+\let\@IEEESAVECMDIEEEPARstart\IEEEPARstart
+\let\@IEEESAVECMDIEEEbiography\IEEEbiography
+\let\@IEEESAVECMDendIEEEbiography\endIEEEbiography
+\let\@IEEESAVECMDIEEEbiographynophoto\IEEEbiographynophoto
+\let\@IEEESAVECMDendIEEEbiographynophoto\endIEEEbiographynophoto
+\let\@IEEESAVECMDIEEEpubid\IEEEpubid
+\let\@IEEESAVECMDIEEEpubidadjcol\IEEEpubidadjcol
+\let\@IEEESAVECMDIEEEmembership\IEEEmembership
+\let\@IEEESAVECMDIEEEaftertitletext\IEEEaftertitletext
+
+
+% disable \IEEEPARstart when in draft mode
+% This may have originally been done because the pre-V1.6 drop letter
+% algorithm had problems with a non-unity baselinestretch
+% At any rate, it seems too formal to have a drop letter in a draft
+% paper.
+\ifCLASSOPTIONdraftcls
+\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** ATTENTION: \noexpand\IEEEPARstart
+ is disabled in draft mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+\fi
+% and for technotes
+\ifCLASSOPTIONtechnote
+\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** WARNING: \noexpand\IEEEPARstart
+ is locked out for technotes (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+\fi
+
+
+% lockout unneeded commands when in conference mode
+\ifCLASSOPTIONconference
+% when locked out, \thanks, \IEEEbiography, \IEEEbiographynophoto, \IEEEpubid,
+% \IEEEmembership and \IEEEaftertitletext will all swallow their given text.
+% \IEEEPARstart will output a normal character instead
+% warn the user about these commands only once to prevent the console screen
+% from filling up with redundant messages
+\def\thanks#1{\if@IEEEWARNthanks\typeout{** WARNING: \noexpand\thanks
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNthanksfalse}
+\def\IEEEPARstart#1#2{#1#2\if@IEEEWARNIEEEPARstart\typeout{** WARNING: \noexpand\IEEEPARstart
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEPARstartfalse}
+
+
+% LaTeX treats environments and commands with optional arguments differently.
+% the actual ("internal") command is stored as \\commandname
+% (accessed via \csname\string\commandname\endcsname )
+% the "external" command \commandname is a macro with code to determine
+% whether or not the optional argument is presented and to provide the
+% default if it is absent. So, in order to save and restore such a command
+% we would have to save and restore \\commandname as well. But, if LaTeX
+% ever changes the way it names the internal names, the trick would break.
+% Instead let us just define a new environment so that the internal
+% name can be left undisturbed.
+\newenvironment{@IEEEbogusbiography}[2][]{\if@IEEEWARNIEEEbiography\typeout{** WARNING: \noexpand\IEEEbiography
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEbiographyfalse%
+\setbox\@IEEEtranrubishbin\vbox\bgroup}{\egroup\relax}
+% and make biography point to our bogus biography
+\let\IEEEbiography=\@IEEEbogusbiography
+\let\endIEEEbiography=\end@IEEEbogusbiography
+
+\renewenvironment{IEEEbiographynophoto}[1]{\if@IEEEWARNIEEEbiographynophoto\typeout{** WARNING: \noexpand\IEEEbiographynophoto
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEbiographynophotofalse%
+\setbox\@IEEEtranrubishbin\vbox\bgroup}{\egroup\relax}
+
+\def\IEEEpubid#1{\if@IEEEWARNIEEEpubid\typeout{** WARNING: \noexpand\IEEEpubid
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEpubidfalse}
+\def\IEEEpubidadjcol{\if@IEEEWARNIEEEpubidadjcol\typeout{** WARNING: \noexpand\IEEEpubidadjcol
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEpubidadjcolfalse}
+\def\IEEEmembership#1{\if@IEEEWARNIEEEmembership\typeout{** WARNING: \noexpand\IEEEmembership
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEmembershipfalse}
+\def\IEEEaftertitletext#1{\if@IEEEWARNIEEEaftertitletext\typeout{** WARNING: \noexpand\IEEEaftertitletext
+ is locked out when in conference mode (line \the\inputlineno).}\fi\global\@IEEEWARNIEEEaftertitletextfalse}
+\fi
+
+
+% provide a way to restore the commands that are locked out
+\def\IEEEoverridecommandlockouts{%
+\typeout{** ATTENTION: Overriding command lockouts (line \the\inputlineno).}%
+\let\thanks\@IEEESAVECMDthanks%
+\let\IEEEPARstart\@IEEESAVECMDIEEEPARstart%
+\let\IEEEbiography\@IEEESAVECMDIEEEbiography%
+\let\endIEEEbiography\@IEEESAVECMDendIEEEbiography%
+\let\IEEEbiographynophoto\@IEEESAVECMDIEEEbiographynophoto%
+\let\endIEEEbiographynophoto\@IEEESAVECMDendIEEEbiographynophoto%
+\let\IEEEpubid\@IEEESAVECMDIEEEpubid%
+\let\IEEEpubidadjcol\@IEEESAVECMDIEEEpubidadjcol%
+\let\IEEEmembership\@IEEESAVECMDIEEEmembership%
+\let\IEEEaftertitletext\@IEEESAVECMDIEEEaftertitletext}
+
+
+
+% need a backslash character for typeout output
+{\catcode`\|=0 \catcode`\\=12
+|xdef|@IEEEbackslash{\}}
+
+
+% hook to allow easy disabling of all legacy warnings
+\def\@IEEElegacywarn#1#2{\typeout{** ATTENTION: \@IEEEbackslash #1 is deprecated (line \the\inputlineno).
+Use \@IEEEbackslash #2 instead.}}
+
+
+% provide for legacy commands
+\def\authorblockA{\@IEEElegacywarn{authorblockA}{IEEEauthorblockA}\IEEEauthorblockA}
+\def\authorblockN{\@IEEElegacywarn{authorblockN}{IEEEauthorblockN}\IEEEauthorblockN}
+\def\authorrefmark{\@IEEElegacywarn{authorrefmark}{IEEEauthorrefmark}\IEEEauthorrefmark}
+\def\PARstart{\@IEEElegacywarn{PARstart}{IEEEPARstart}\IEEEPARstart}
+\def\pubid{\@IEEElegacywarn{pubid}{IEEEpubid}\IEEEpubid}
+\def\pubidadjcol{\@IEEElegacywarn{pubidadjcol}{IEEEpubidadjcol}\IEEEpubidadjcol}
+\def\QED{\@IEEElegacywarn{QED}{IEEEQED}\IEEEQED}
+\def\QEDclosed{\@IEEElegacywarn{QEDclosed}{IEEEQEDclosed}\IEEEQEDclosed}
+\def\QEDopen{\@IEEElegacywarn{QEDopen}{IEEEQEDopen}\IEEEQEDopen}
+\def\specialpapernotice{\@IEEElegacywarn{specialpapernotice}{IEEEspecialpapernotice}\IEEEspecialpapernotice}
+
+
+
+% provide for legacy environments
+\def\biography{\@IEEElegacywarn{biography}{IEEEbiography}\IEEEbiography}
+\def\biographynophoto{\@IEEElegacywarn{biographynophoto}{IEEEbiographynophoto}\IEEEbiographynophoto}
+\def\keywords{\@IEEElegacywarn{keywords}{IEEEkeywords}\IEEEkeywords}
+\def\endbiography{\endIEEEbiography}
+\def\endbiographynophoto{\endIEEEbiographynophoto}
+\def\endkeywords{\endIEEEkeywords}
+
+
+% provide for legacy IED commands/lengths when possible
+\let\labelindent\IEEElabelindent
+\def\calcleftmargin{\@IEEElegacywarn{calcleftmargin}{IEEEcalcleftmargin}\IEEEcalcleftmargin}
+\def\setlabelwidth{\@IEEElegacywarn{setlabelwidth}{IEEEsetlabelwidth}\IEEEsetlabelwidth}
+\def\usemathlabelsep{\@IEEElegacywarn{usemathlabelsep}{IEEEusemathlabelsep}\IEEEusemathlabelsep}
+\def\iedlabeljustifyc{\@IEEElegacywarn{iedlabeljustifyc}{IEEEiedlabeljustifyc}\IEEEiedlabeljustifyc}
+\def\iedlabeljustifyl{\@IEEElegacywarn{iedlabeljustifyl}{IEEEiedlabeljustifyl}\IEEEiedlabeljustifyl}
+\def\iedlabeljustifyr{\@IEEElegacywarn{iedlabeljustifyr}{IEEEiedlabeljustifyr}\IEEEiedlabeljustifyr}
+
+
+
+% let \proof use the IEEEtran version even after amsthm is loaded
+% \proof is now deprecated in favor of \IEEEproof
+\AtBeginDocument{\def\proof{\@IEEElegacywarn{proof}{IEEEproof}\IEEEproof}\def\endproof{\endIEEEproof}}
+
+% V1.7 \overrideIEEEmargins is no longer supported.
+\def\overrideIEEEmargins{%
+\typeout{** WARNING: \string\overrideIEEEmargins \space no longer supported (line \the\inputlineno).}%
+\typeout{** Use the \string\CLASSINPUTinnersidemargin, \string\CLASSINPUToutersidemargin \space controls instead.}}
+
+
+\endinput
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%% End of IEEEtran.cls %%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% That's all folks!
+
diff --git a/theory/paper_spheric/data/CosmoVolume_fixed.totals b/theory/paper_spheric/data/CosmoVolume_fixed.totals
new file mode 100644
index 0000000000000000000000000000000000000000..abac1e116ea934b9f94731c251ff3cd1f4d3ded0
--- /dev/null
+++ b/theory/paper_spheric/data/CosmoVolume_fixed.totals
@@ -0,0 +1,32 @@
+7.270 119.497 155.347 6.122 3317.845 2360.465 4400.177 4395.134 14972.749 12886.245 0.012 136.311 35.580 0.000 0.000 14865.624 14993.010 0.000
+12.035 90.284 169.023 6.599 3337.034 2375.781 4529.404 4553.796 15220.888 13184.254 0.014 141.518 55.892 3.945 7.886 7620.544 7763.003 0.000
+15.354 72.609 167.747 6.817 3444.881 2449.035 4672.190 4662.636 15727.882 13533.309 0.013 153.451 50.851 1.598 2.758 5227.670 5316.372 0.000
+15.323 52.278 178.587 6.927 3483.830 2485.267 4761.675 4776.669 15972.550 13803.354 0.015 202.509 71.143 12.870 23.100 4011.785 4080.204 0.000
+11.534 44.731 348.040 7.281 3589.043 2556.458 5041.711 5040.412 16686.249 14413.378 0.019 215.657 90.239 22.030 40.538 3396.116 3453.081 0.000
+12.545 38.532 278.191 7.511 3601.699 2560.577 4951.670 4990.886 16540.336 14345.367 0.021 178.791 101.765 26.809 47.899 2794.993 2846.659 0.000
+10.175 36.651 394.136 7.633 3732.559 2655.667 5139.090 5210.672 17163.345 14931.578 0.022 181.657 71.569 7.932 13.062 2498.693 2546.011 0.000
+11.343 32.602 397.427 7.534 3726.899 2654.810 5172.310 5234.477 17214.610 14963.049 0.015 165.689 65.790 4.523 7.130 2190.421 2234.864 0.000
+12.702 30.825 281.684 7.999 3682.353 2626.846 5198.771 5242.748 17189.326 14920.823 0.016 164.790 99.592 22.239 38.149 1936.858 1980.907 0.000
+12.009 33.486 192.219 8.184 3659.554 2596.816 5045.135 5036.763 16920.684 14531.245 0.013 129.621 129.834 42.818 77.601 1693.883 1739.896 0.000
+12.437 32.934 308.973 8.548 3662.234 2612.139 5192.838 5241.599 17171.056 14931.297 0.016 164.652 202.649 84.510 155.837 1598.624 1644.661 0.000
+12.111 30.994 123.376 8.673 3636.930 2581.050 4936.979 4893.490 16698.033 14273.933 0.012 106.307 248.821 120.746 225.816 1392.471 1435.981 0.000
+13.096 30.803 364.729 9.392 3680.226 2630.680 5257.635 5348.369 17323.970 15139.252 0.017 172.226 341.390 168.405 328.803 1389.611 1434.105 0.000
+12.545 30.195 464.309 9.845 3685.860 2634.623 5325.628 5476.462 17425.001 15355.808 0.021 200.112 117.813 22.582 37.744 1293.928 1337.200 0.000
+12.302 28.103 406.452 10.297 3749.431 2676.724 5365.000 5468.364 17684.591 15468.683 0.022 185.327 267.080 116.869 233.295 1226.343 1267.322 0.000
+12.765 27.296 485.241 10.645 3750.883 2679.194 5429.781 5586.855 17782.309 15657.020 0.019 199.635 414.375 217.416 435.402 1180.415 1221.126 0.000
+13.119 27.737 450.328 11.783 3727.484 2663.799 5417.082 5580.733 17721.502 15610.151 0.019 201.141 211.001 81.972 151.487 1089.924 1131.252 0.000
+11.767 26.399 297.131 13.217 3708.497 2629.163 5232.915 5281.770 17408.604 15050.150 0.016 155.340 234.858 106.303 193.836 988.916 1027.487 0.000
+13.225 24.763 424.335 13.835 3712.101 2649.464 5354.551 5495.852 17580.160 15449.964 0.017 188.358 210.412 80.966 154.771 965.175 1003.792 0.000
+13.313 26.549 507.648 16.069 3707.530 2645.606 5374.041 5525.296 17632.712 15496.198 0.019 193.405 249.005 106.764 190.364 927.033 967.458 0.000
+10.209 24.131 191.382 16.359 3706.373 2644.646 5167.641 5209.469 17279.440 14981.982 0.014 133.814 310.797 160.685 307.746 841.957 876.783 0.000
+13.169 28.065 535.502 19.406 3726.445 2664.014 5410.972 5618.777 17710.644 15690.068 0.019 202.476 317.783 149.837 299.033 857.896 899.683 0.000
+12.130 25.664 539.390 21.629 3770.959 2692.093 5463.640 5637.247 17937.280 15803.382 0.021 195.905 317.808 148.239 284.611 827.037 865.523 0.000
+11.819 23.754 239.064 23.389 3746.091 2674.205 5273.958 5326.309 17580.837 15251.621 0.016 146.457 471.662 282.021 525.850 762.302 798.457 0.000
+12.339 26.242 205.295 27.320 3740.444 2660.520 5214.464 5245.851 17469.391 15078.002 0.015 134.168 520.273 330.989 599.690 726.008 765.148 0.000
+11.176 23.398 197.461 30.245 3722.195 2654.508 5186.031 5252.136 17371.802 15084.906 0.015 134.320 750.292 472.145 934.072 709.681 744.824 0.000
+12.775 27.499 210.800 35.997 3724.768 2657.764 5252.088 5309.686 17528.550 15193.504 0.017 140.457 571.645 332.836 666.731 680.414 721.302 0.000
+12.419 26.140 266.836 44.264 3716.139 2655.713 5271.434 5381.740 17534.329 15297.050 0.017 151.400 706.651 433.070 954.618 672.571 716.344 0.000
+10.224 23.257 216.329 43.789 3729.618 2667.172 5243.710 5314.650 17529.001 15226.652 0.015 136.691 503.661 300.650 598.282 632.232 666.200 0.000
+10.915 23.869 220.144 49.482 3727.576 2671.798 5278.794 5342.227 17601.025 15282.138 0.018 140.935 676.734 410.277 873.626 623.523 658.906 0.000
+12.779 23.073 248.924 55.402 3768.740 2693.987 5314.346 5393.554 17765.960 15430.990 0.015 144.559 640.871 401.646 793.763 608.064 644.357 0.000
+10.917 23.751 271.559 56.550 3757.127 2691.782 5321.232 5417.237 17751.072 15455.746 0.018 146.392 473.453 284.687 559.326 582.791 617.943 0.000
diff --git a/theory/paper_spheric/data/CosmoVolume_fixed_Gadget2.totals b/theory/paper_spheric/data/CosmoVolume_fixed_Gadget2.totals
new file mode 100644
index 0000000000000000000000000000000000000000..e54a728f0fbc84dfaa97812637bed37640e0b9f2
--- /dev/null
+++ b/theory/paper_spheric/data/CosmoVolume_fixed_Gadget2.totals
@@ -0,0 +1,13 @@
+1 1.290e+02 1.280e-06 9.961e-09 1.841e+06 2.438e+11
+2 1.290e+02 1.280e-06 9.961e-09 9.523e+05 1.336e+11
+3 1.290e+02 1.280e-06 9.961e-09 6.302e+05 9.192e+10
+4 1.290e+02 1.280e-06 9.961e-09 4.658e+05 7.151e+10
+6 1.290e+02 1.280e-06 9.961e-09 3.079e+05 4.918e+10
+8 1.290e+02 1.280e-06 9.961e-09 2.368e+05 3.888e+10
+10 1.290e+02 1.280e-06 9.961e-09 1.930e+05 3.323e+10
+12 1.290e+02 1.280e-06 9.961e-09 1.634e+05 2.907e+10
+16 1.290e+02 1.280e-06 9.961e-09 1.229e+05 2.113e+10
+20 1.290e+02 1.280e-06 9.961e-09 9.841e+04 1.773e+10
+24 1.290e+02 1.280e-06 9.961e-09 8.158e+04 1.444e+10
+28 1.290e+02 1.280e-06 9.961e-09 6.934e+04 1.370e+10
+32 1.290e+02 1.280e-06 9.961e-09 6.032e+04 1.244e+10
diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_1.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_1.dat
new file mode 100644
index 0000000000000000000000000000000000000000..cc14b5ac6ce1b0c1d617b34e5bb532032b96d47a
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_1.dat
@@ -0,0 +1,259 @@
+# Number of cores: 1
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 1841127 2.49541e+11
+2 1e-08 1e-08 1841127 2.41972e+11
+3 2e-08 1e-08 1841127 2.46989e+11
+4 3e-08 1e-08 1841127 2.47354e+11
+5 4e-08 1e-08 1841127 2.48764e+11
+6 5e-08 1e-08 1841127 2.39867e+11
+7 6e-08 1e-08 1841127 2.41151e+11
+8 7e-08 1e-08 1841127 2.47176e+11
+9 8e-08 1e-08 1841127 2.40415e+11
+10 9e-08 1e-08 1841127 2.44373e+11
+11 1e-07 1e-08 1841127 2.43127e+11
+12 1.1e-07 1e-08 1841127 2.44901e+11
+13 1.2e-07 1e-08 1841127 2.41489e+11
+14 1.3e-07 1e-08 1841127 2.43391e+11
+15 1.4e-07 1e-08 1841127 2.4912e+11
+16 1.5e-07 1e-08 1841127 2.48539e+11
+17 1.6e-07 1e-08 1841127 2.45593e+11
+18 1.7e-07 1e-08 1841127 2.46147e+11
+19 1.8e-07 1e-08 1841127 2.38684e+11
+20 1.9e-07 1e-08 1841127 2.4488e+11
+21 2e-07 1e-08 1841127 2.37401e+11
+22 2.1e-07 1e-08 1841127 2.4023e+11
+23 2.2e-07 1e-08 1841127 2.38668e+11
+24 2.3e-07 1e-08 1841127 2.47569e+11
+25 2.4e-07 1e-08 1841127 2.38949e+11
+26 2.5e-07 1e-08 1841127 2.42014e+11
+27 2.6e-07 1e-08 1841127 2.38534e+11
+28 2.7e-07 1e-08 1841127 2.38274e+11
+29 2.8e-07 1e-08 1841127 2.49901e+11
+30 2.9e-07 1e-08 1841127 2.39624e+11
+31 3e-07 1e-08 1841127 2.43427e+11
+32 3.1e-07 1e-08 1841127 2.4817e+11
+33 3.2e-07 1e-08 1841127 2.44629e+11
+34 3.3e-07 1e-08 1841127 2.40832e+11
+35 3.4e-07 1e-08 1841127 2.45043e+11
+36 3.5e-07 1e-08 1841127 2.43631e+11
+37 3.6e-07 1e-08 1841127 2.43124e+11
+38 3.7e-07 1e-08 1841127 2.49266e+11
+39 3.8e-07 1e-08 1841127 2.40842e+11
+40 3.9e-07 1e-08 1841127 2.46852e+11
+41 4e-07 1e-08 1841127 2.44451e+11
+42 4.1e-07 1e-08 1841127 2.4711e+11
+43 4.2e-07 1e-08 1841127 2.42068e+11
+44 4.3e-07 1e-08 1841127 2.48401e+11
+45 4.4e-07 1e-08 1841127 2.40454e+11
+46 4.5e-07 1e-08 1841127 2.41779e+11
+47 4.6e-07 1e-08 1841127 2.47104e+11
+48 4.7e-07 1e-08 1841127 2.48655e+11
+49 4.8e-07 1e-08 1841127 2.37705e+11
+50 4.9e-07 1e-08 1841127 2.50405e+11
+51 5e-07 1e-08 1841127 2.43756e+11
+52 5.1e-07 1e-08 1841127 2.38041e+11
+53 5.2e-07 1e-08 1841127 2.39564e+11
+54 5.3e-07 1e-08 1841127 2.45563e+11
+55 5.4e-07 1e-08 1841127 2.48208e+11
+56 5.5e-07 1e-08 1841127 2.41248e+11
+57 5.6e-07 1e-08 1841127 2.37633e+11
+58 5.7e-07 1e-08 1841127 2.37176e+11
+59 5.8e-07 1e-08 1841127 2.42828e+11
+60 5.9e-07 1e-08 1841127 2.37799e+11
+61 6e-07 1e-08 1841127 2.40112e+11
+62 6.1e-07 1e-08 1841127 2.49319e+11
+63 6.2e-07 1e-08 1841127 2.49081e+11
+64 6.3e-07 1e-08 1841127 2.477e+11
+65 6.4e-07 1e-08 1841127 2.40199e+11
+66 6.5e-07 1e-08 1841127 2.44252e+11
+67 6.6e-07 1e-08 1841127 2.41898e+11
+68 6.7e-07 1e-08 1841127 2.46553e+11
+69 6.8e-07 1e-08 1841127 2.43482e+11
+70 6.9e-07 1e-08 1841127 2.45474e+11
+71 7e-07 1e-08 1841127 2.43577e+11
+72 7.1e-07 1e-08 1841127 2.37344e+11
+73 7.2e-07 1e-08 1841127 2.43476e+11
+74 7.3e-07 1e-08 1841127 2.48881e+11
+75 7.4e-07 1e-08 1841127 2.48735e+11
+76 7.5e-07 1e-08 1841127 2.46232e+11
+77 7.6e-07 1e-08 1841127 2.40849e+11
+78 7.7e-07 1e-08 1841127 2.46472e+11
+79 7.8e-07 1e-08 1841127 2.45576e+11
+80 7.9e-07 1e-08 1841127 2.41463e+11
+81 8e-07 1e-08 1841127 2.45939e+11
+82 8.1e-07 1e-08 1841127 2.39002e+11
+83 8.2e-07 1e-08 1841127 2.43038e+11
+84 8.3e-07 1e-08 1841127 2.48318e+11
+85 8.4e-07 1e-08 1841127 2.47645e+11
+86 8.5e-07 1e-08 1841127 2.41054e+11
+87 8.6e-07 1e-08 1841127 2.39986e+11
+88 8.7e-07 1e-08 1841127 2.48552e+11
+89 8.8e-07 1e-08 1841127 2.4166e+11
+90 8.9e-07 1e-08 1841127 2.45632e+11
+91 9e-07 1e-08 1841127 2.49184e+11
+92 9.1e-07 1e-08 1841127 2.44541e+11
+93 9.2e-07 1e-08 1841127 2.45425e+11
+94 9.3e-07 1e-08 1841127 2.48154e+11
+95 9.4e-07 1e-08 1841127 2.42583e+11
+96 9.5e-07 1e-08 1841127 2.48707e+11
+97 9.6e-07 1e-08 1841127 2.42242e+11
+98 9.7e-07 1e-08 1841127 2.47416e+11
+99 9.8e-07 1e-08 1841127 2.45948e+11
+100 9.9e-07 1e-08 1841127 2.4869e+11
+101 1e-06 1e-08 1841127 2.43131e+11
+102 1.01e-06 1e-08 1841127 2.40918e+11
+103 1.02e-06 1e-08 1841127 2.49165e+11
+104 1.03e-06 1e-08 1841127 2.48708e+11
+105 1.04e-06 1e-08 1841127 2.39101e+11
+106 1.05e-06 1e-08 1841127 2.48381e+11
+107 1.06e-06 1e-08 1841127 2.45242e+11
+108 1.07e-06 1e-08 1841127 2.42491e+11
+109 1.08e-06 1e-08 1841127 2.45098e+11
+110 1.09e-06 1e-08 1841127 2.40874e+11
+111 1.1e-06 1e-08 1841127 2.47036e+11
+112 1.11e-06 1e-08 1841127 2.40896e+11
+113 1.12e-06 1e-08 1841127 2.42822e+11
+114 1.13e-06 1e-08 1841127 2.40389e+11
+115 1.14e-06 1e-08 1841127 2.39214e+11
+116 1.15e-06 1e-08 1841127 2.40481e+11
+117 1.16e-06 1e-08 1841127 2.44091e+11
+118 1.17e-06 1e-08 1841127 2.42227e+11
+119 1.18e-06 1e-08 1841127 2.39134e+11
+120 1.19e-06 1e-08 1841127 2.48641e+11
+121 1.2e-06 1e-08 1841127 2.38446e+11
+122 1.21e-06 1e-08 1841127 2.38798e+11
+123 1.22e-06 1e-08 1841127 2.43307e+11
+124 1.23e-06 1e-08 1841127 2.46562e+11
+125 1.24e-06 1e-08 1841127 2.49537e+11
+126 1.25e-06 1e-08 1841127 2.49218e+11
+127 1.26e-06 1e-08 1841127 2.4558e+11
+128 1.27e-06 1e-08 1841127 2.41956e+11
+129 1.28e-06 1e-08 1841127 2.46577e+11
+130 1.29e-06 1e-08 1841127 2.41805e+11
+131 1.3e-06 1e-08 1841127 2.4071e+11
+132 1.31e-06 1e-08 1841127 2.4009e+11
+133 1.32e-06 1e-08 1841127 2.44535e+11
+134 1.33e-06 1e-08 1841127 2.41269e+11
+135 1.34e-06 1e-08 1841127 2.38852e+11
+136 1.35e-06 1e-08 1841127 2.46536e+11
+137 1.36e-06 1e-08 1841127 2.38541e+11
+138 1.37e-06 1e-08 1841127 2.47088e+11
+139 1.38e-06 1e-08 1841127 2.3928e+11
+140 1.39e-06 1e-08 1841127 2.46866e+11
+141 1.4e-06 1e-08 1841127 2.37898e+11
+142 1.41e-06 1e-08 1841127 2.49495e+11
+143 1.42e-06 1e-08 1841127 2.378e+11
+144 1.43e-06 1e-08 1841127 2.43855e+11
+145 1.44e-06 1e-08 1841127 2.39209e+11
+146 1.45e-06 1e-08 1841127 2.40331e+11
+147 1.46e-06 1e-08 1841127 2.47196e+11
+148 1.47e-06 1e-08 1841127 2.46447e+11
+149 1.48e-06 1e-08 1841127 2.45558e+11
+150 1.49e-06 1e-08 1841127 2.49346e+11
+151 1.5e-06 1e-08 1841127 2.45179e+11
+152 1.51e-06 1e-08 1841127 2.46992e+11
+153 1.52e-06 1e-08 1841127 2.38439e+11
+154 1.53e-06 1e-08 1841127 2.38665e+11
+155 1.54e-06 1e-08 1841127 2.44009e+11
+156 1.55e-06 1e-08 1841127 2.37937e+11
+157 1.56e-06 1e-08 1841127 2.37968e+11
+158 1.57e-06 1e-08 1841127 2.39537e+11
+159 1.58e-06 1e-08 1841127 2.43e+11
+160 1.59e-06 1e-08 1841127 2.48704e+11
+161 1.6e-06 1e-08 1841127 2.44625e+11
+162 1.61e-06 1e-08 1841127 2.46884e+11
+163 1.62e-06 1e-08 1841127 2.37603e+11
+164 1.63e-06 1e-08 1841127 2.39332e+11
+165 1.64e-06 1e-08 1841127 2.49885e+11
+166 1.65e-06 1e-08 1841127 2.40072e+11
+167 1.66e-06 1e-08 1841127 2.48453e+11
+168 1.67e-06 1e-08 1841127 2.38501e+11
+169 1.68e-06 1e-08 1841127 2.4996e+11
+170 1.69e-06 1e-08 1841127 2.37827e+11
+171 1.7e-06 1e-08 1841127 2.48082e+11
+172 1.71e-06 1e-08 1841127 2.38273e+11
+173 1.72e-06 1e-08 1841127 2.36946e+11
+174 1.73e-06 1e-08 1841127 2.48783e+11
+175 1.74e-06 1e-08 1841127 2.44543e+11
+176 1.75e-06 1e-08 1841127 2.39674e+11
+177 1.76e-06 1e-08 1841127 2.39833e+11
+178 1.77e-06 1e-08 1841127 2.41799e+11
+179 1.78e-06 1e-08 1841127 2.48682e+11
+180 1.79e-06 1e-08 1841127 2.4746e+11
+181 1.8e-06 1e-08 1841127 2.41678e+11
+182 1.81e-06 1e-08 1841127 2.44008e+11
+183 1.82e-06 1e-08 1841127 2.4564e+11
+184 1.83e-06 1e-08 1841127 2.42708e+11
+185 1.84e-06 1e-08 1841127 2.45847e+11
+186 1.85e-06 1e-08 1841127 2.38661e+11
+187 1.86e-06 1e-08 1841127 2.43879e+11
+188 1.87e-06 1e-08 1841127 2.4662e+11
+189 1.88e-06 1e-08 1841127 2.40857e+11
+190 1.89e-06 1e-08 1841127 2.49643e+11
+191 1.9e-06 1e-08 1841127 2.44895e+11
+192 1.91e-06 1e-08 1841127 2.48156e+11
+193 1.92e-06 1e-08 1841127 2.4673e+11
+194 1.93e-06 1e-08 1841127 2.45526e+11
+195 1.94e-06 1e-08 1841127 2.37373e+11
+196 1.95e-06 1e-08 1841127 2.46674e+11
+197 1.96e-06 1e-08 1841127 2.47827e+11
+198 1.97e-06 1e-08 1841127 2.48903e+11
+199 1.98e-06 1e-08 1841127 2.48118e+11
+200 1.99e-06 1e-08 1841127 2.47462e+11
+201 2e-06 1e-08 1841127 2.49679e+11
+202 2.01e-06 1e-08 1841127 2.4648e+11
+203 2.02e-06 1e-08 1841127 2.48482e+11
+204 2.03e-06 1e-08 1841127 2.50963e+11
+205 2.04e-06 1e-08 1841127 2.45634e+11
+206 2.05e-06 1e-08 1841127 2.4314e+11
+207 2.06e-06 1e-08 1841127 2.39173e+11
+208 2.07e-06 1e-08 1841127 2.47692e+11
+209 2.08e-06 1e-08 1841127 2.48322e+11
+210 2.09e-06 1e-08 1841127 2.38007e+11
+211 2.1e-06 1e-08 1841127 2.45285e+11
+212 2.11e-06 1e-08 1841127 2.40321e+11
+213 2.12e-06 1e-08 1841127 2.4532e+11
+214 2.13e-06 1e-08 1841127 2.41105e+11
+215 2.14e-06 1e-08 1841127 2.45989e+11
+216 2.15e-06 1e-08 1841127 2.41174e+11
+217 2.16e-06 1e-08 1841127 2.41287e+11
+218 2.17e-06 1e-08 1841127 2.40051e+11
+219 2.18e-06 1e-08 1841127 2.37963e+11
+220 2.19e-06 1e-08 1841127 2.45039e+11
+221 2.2e-06 1e-08 1841127 2.40084e+11
+222 2.21e-06 1e-08 1841127 2.45689e+11
+223 2.22e-06 1e-08 1841127 2.4369e+11
+224 2.23e-06 1e-08 1841127 2.49472e+11
+225 2.24e-06 1e-08 1841127 2.40499e+11
+226 2.25e-06 1e-08 1841127 2.4389e+11
+227 2.26e-06 1e-08 1841127 2.46203e+11
+228 2.27e-06 1e-08 1841127 2.38354e+11
+229 2.28e-06 1e-08 1841127 2.43015e+11
+230 2.29e-06 1e-08 1841127 2.44692e+11
+231 2.3e-06 1e-08 1841127 2.49186e+11
+232 2.31e-06 1e-08 1841127 2.43954e+11
+233 2.32e-06 1e-08 1841127 2.41146e+11
+234 2.33e-06 1e-08 1841127 2.47836e+11
+235 2.34e-06 1e-08 1841127 2.42775e+11
+236 2.35e-06 1e-08 1841127 2.37132e+11
+237 2.36e-06 1e-08 1841127 2.41883e+11
+238 2.37e-06 1e-08 1841127 2.44703e+11
+239 2.38e-06 1e-08 1841127 2.47572e+11
+240 2.39e-06 1e-08 1841127 2.39929e+11
+241 2.4e-06 1e-08 1841127 2.46093e+11
+242 2.41e-06 1e-08 1841127 2.43104e+11
+243 2.42e-06 1e-08 1841127 2.43483e+11
+244 2.43e-06 1e-08 1841127 2.40992e+11
+245 2.44e-06 1e-08 1841127 2.46368e+11
+246 2.45e-06 1e-08 1841127 2.39375e+11
+247 2.46e-06 1e-08 1841127 2.45623e+11
+248 2.47e-06 1e-08 1841127 2.37358e+11
+249 2.48e-06 1e-08 1841127 2.42598e+11
+250 2.49e-06 1e-08 1841127 2.47466e+11
+251 2.5e-06 1e-08 1841127 2.45522e+11
+252 2.51e-06 1e-08 1841127 2.45115e+11
+253 2.52e-06 1e-08 1841127 2.41498e+11
+254 2.53e-06 1e-08 1841127 2.39537e+11
+255 2.54e-06 1e-08 1841127 2.44713e+11
+256 2.55e-06 1e-08 1841127 2.44787e+11
+257 2.56e-06 1e-08 1841127 2.46263e+11
diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_10.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_10.dat
new file mode 100644
index 0000000000000000000000000000000000000000..daea219a89c21f58688ea6854f027ee209d6452b
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_10.dat
@@ -0,0 +1,259 @@
+# Number of cores: 10
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 183963 3.45886e+10
+2 1e-08 1e-08 183963 3.19859e+10
+3 2e-08 1e-08 183963 3.27121e+10
+4 3e-08 1e-08 183963 3.2731e+10
+5 4e-08 1e-08 183963 3.29447e+10
+6 5e-08 1e-08 183963 3.17616e+10
+7 6e-08 1e-08 183963 3.23126e+10
+8 7e-08 1e-08 183964 3.31991e+10
+9 8e-08 1e-08 183963 3.23147e+10
+10 9e-08 1e-08 183963 3.28677e+10
+11 1e-07 1e-08 183964 3.26094e+10
+12 1.1e-07 1e-08 183963 3.28295e+10
+13 1.2e-07 1e-08 183963 3.23607e+10
+14 1.3e-07 1e-08 183963 3.279e+10
+15 1.4e-07 1e-08 186256 3.35803e+10
+16 1.5e-07 1e-08 186256 3.31371e+10
+17 1.6e-07 1e-08 186255 3.2841e+10
+18 1.7e-07 1e-08 186255 3.27458e+10
+19 1.8e-07 1e-08 186255 3.1756e+10
+20 1.9e-07 1e-08 186255 3.26209e+10
+21 2e-07 1e-08 186255 3.15661e+10
+22 2.1e-07 1e-08 186255 3.22889e+10
+23 2.2e-07 1e-08 186255 3.18118e+10
+24 2.3e-07 1e-08 186255 3.29793e+10
+25 2.4e-07 1e-08 186255 3.18468e+10
+26 2.5e-07 1e-08 186255 3.21964e+10
+27 2.6e-07 1e-08 186255 3.17404e+10
+28 2.7e-07 1e-08 186254 3.17026e+10
+29 2.8e-07 1e-08 186254 3.3258e+10
+30 2.9e-07 1e-08 186254 3.19298e+10
+31 3e-07 1e-08 186254 3.24714e+10
+32 3.1e-07 1e-08 186254 3.30908e+10
+33 3.2e-07 1e-08 186254 3.2542e+10
+34 3.3e-07 1e-08 186254 3.20705e+10
+35 3.4e-07 1e-08 186254 3.25962e+10
+36 3.5e-07 1e-08 186254 3.26849e+10
+37 3.6e-07 1e-08 186255 3.25801e+10
+38 3.7e-07 1e-08 186255 3.3074e+10
+39 3.8e-07 1e-08 186255 3.20638e+10
+40 3.9e-07 1e-08 188306 3.31353e+10
+41 4e-07 1e-08 188305 3.30021e+10
+42 4.1e-07 1e-08 188306 3.3234e+10
+43 4.2e-07 1e-08 188305 3.22633e+10
+44 4.3e-07 1e-08 188306 3.32161e+10
+45 4.4e-07 1e-08 188306 3.2068e+10
+46 4.5e-07 1e-08 188305 3.23431e+10
+47 4.6e-07 1e-08 188305 3.29472e+10
+48 4.7e-07 1e-08 188305 3.31947e+10
+49 4.8e-07 1e-08 188305 3.1646e+10
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diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_12.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_12.dat
new file mode 100644
index 0000000000000000000000000000000000000000..bc07a64d5933f12a3ff075db99cbdce767cea48a
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_12.dat
@@ -0,0 +1,259 @@
+# Number of cores: 12
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 154096 3.07545e+10
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diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_16.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_16.dat
new file mode 100644
index 0000000000000000000000000000000000000000..1f5a6f68d5046a94b246242dcd4e85664d1cb296
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_16.dat
@@ -0,0 +1,259 @@
+# Number of cores: 16
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 114429 2.24214e+10
+2 1e-08 1e-08 114429 2.00007e+10
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+232 2.31e-06 1e-08 129335 2.30626e+10
+233 2.32e-06 1e-08 129335 2.21077e+10
+234 2.33e-06 1e-08 129335 2.29848e+10
+235 2.34e-06 1e-08 129334 2.24433e+10
+236 2.35e-06 1e-08 130478 2.19376e+10
+237 2.36e-06 1e-08 130477 2.23001e+10
+238 2.37e-06 1e-08 130478 2.26289e+10
+239 2.38e-06 1e-08 130478 2.28204e+10
+240 2.39e-06 1e-08 130478 2.2154e+10
+241 2.4e-06 1e-08 130478 2.27857e+10
+242 2.41e-06 1e-08 130478 2.2471e+10
+243 2.42e-06 1e-08 130478 2.25567e+10
+244 2.43e-06 1e-08 130478 2.22932e+10
+245 2.44e-06 1e-08 130478 2.27705e+10
+246 2.45e-06 1e-08 130478 2.21415e+10
+247 2.46e-06 1e-08 130478 2.28544e+10
+248 2.47e-06 1e-08 130478 2.23037e+10
+249 2.48e-06 1e-08 130478 2.274e+10
+250 2.49e-06 1e-08 130477 2.38549e+10
+251 2.5e-06 1e-08 130477 2.29446e+10
+252 2.51e-06 1e-08 130477 2.28871e+10
+253 2.52e-06 1e-08 130477 2.25942e+10
+254 2.53e-06 1e-08 130477 2.22376e+10
+255 2.54e-06 1e-08 130477 2.2916e+10
+256 2.55e-06 1e-08 130477 2.28703e+10
+257 2.56e-06 1e-08 130477 2.30775e+10
diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_2.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_2.dat
new file mode 100644
index 0000000000000000000000000000000000000000..a9b9a4d393745213e4a44c4a3a97d285dea8368a
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_2.dat
@@ -0,0 +1,259 @@
+# Number of cores: 2
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 911542 1.34916e+11
+2 1e-08 1e-08 911542 1.28584e+11
+3 2e-08 1e-08 911542 1.31289e+11
+4 3e-08 1e-08 911542 1.31676e+11
+5 4e-08 1e-08 932084 1.33758e+11
+6 5e-08 1e-08 932083 1.27086e+11
+7 6e-08 1e-08 932083 1.27818e+11
+8 7e-08 1e-08 932082 1.31405e+11
+9 8e-08 1e-08 932082 1.27953e+11
+10 9e-08 1e-08 932084 1.3046e+11
+11 1e-07 1e-08 932083 1.29911e+11
+12 1.1e-07 1e-08 932083 1.30009e+11
+13 1.2e-07 1e-08 932083 1.28319e+11
+14 1.3e-07 1e-08 932083 1.29305e+11
+15 1.4e-07 1e-08 932084 1.32868e+11
+16 1.5e-07 1e-08 932084 1.31629e+11
+17 1.6e-07 1e-08 932083 1.30066e+11
+18 1.7e-07 1e-08 932083 1.30509e+11
+19 1.8e-07 1e-08 932082 1.27009e+11
+20 1.9e-07 1e-08 932082 1.30107e+11
+21 2e-07 1e-08 932083 1.2577e+11
+22 2.1e-07 1e-08 932084 1.27452e+11
+23 2.2e-07 1e-08 932084 1.26453e+11
+24 2.3e-07 1e-08 932084 1.31589e+11
+25 2.4e-07 1e-08 932083 1.27116e+11
+26 2.5e-07 1e-08 932083 1.28329e+11
+27 2.6e-07 1e-08 932083 1.26248e+11
+28 2.7e-07 1e-08 932083 1.26168e+11
+29 2.8e-07 1e-08 932082 1.32595e+11
+30 2.9e-07 1e-08 932082 1.26879e+11
+31 3e-07 1e-08 932082 1.28978e+11
+32 3.1e-07 1e-08 932081 1.31641e+11
+33 3.2e-07 1e-08 932081 1.29628e+11
+34 3.3e-07 1e-08 932082 1.28051e+11
+35 3.4e-07 1e-08 932082 1.29736e+11
+36 3.5e-07 1e-08 932082 1.29163e+11
+37 3.6e-07 1e-08 932082 1.28768e+11
+38 3.7e-07 1e-08 932082 1.31997e+11
+39 3.8e-07 1e-08 932083 1.27837e+11
+40 3.9e-07 1e-08 932083 1.30919e+11
+41 4e-07 1e-08 932082 1.31156e+11
+42 4.1e-07 1e-08 932083 1.30917e+11
+43 4.2e-07 1e-08 932083 1.28205e+11
+44 4.3e-07 1e-08 932083 1.31739e+11
+45 4.4e-07 1e-08 932083 1.27371e+11
+46 4.5e-07 1e-08 932083 1.28178e+11
+47 4.6e-07 1e-08 932083 1.30836e+11
+48 4.7e-07 1e-08 932083 1.31663e+11
+49 4.8e-07 1e-08 932083 1.25789e+11
+50 4.9e-07 1e-08 932083 1.34892e+11
+51 5e-07 1e-08 932082 1.29761e+11
+52 5.1e-07 1e-08 932082 1.25947e+11
+53 5.2e-07 1e-08 932083 1.27165e+11
+54 5.3e-07 1e-08 932084 1.30431e+11
+55 5.4e-07 1e-08 932084 1.31507e+11
+56 5.5e-07 1e-08 932084 1.27885e+11
+57 5.6e-07 1e-08 932084 1.25972e+11
+58 5.7e-07 1e-08 932084 1.25947e+11
+59 5.8e-07 1e-08 932084 1.28519e+11
+60 5.9e-07 1e-08 932085 1.26553e+11
+61 6e-07 1e-08 932083 1.279e+11
+62 6.1e-07 1e-08 932082 1.32833e+11
+63 6.2e-07 1e-08 932082 1.33784e+11
+64 6.3e-07 1e-08 932082 1.31486e+11
+65 6.4e-07 1e-08 932082 1.27294e+11
+66 6.5e-07 1e-08 932081 1.30687e+11
+67 6.6e-07 1e-08 932080 1.28594e+11
+68 6.7e-07 1e-08 932080 1.3057e+11
+69 6.8e-07 1e-08 932080 1.2897e+11
+70 6.9e-07 1e-08 932080 1.30086e+11
+71 7e-07 1e-08 932080 1.29051e+11
+72 7.1e-07 1e-08 932080 1.25884e+11
+73 7.2e-07 1e-08 932081 1.30658e+11
+74 7.3e-07 1e-08 932081 1.31871e+11
+75 7.4e-07 1e-08 932081 1.3165e+11
+76 7.5e-07 1e-08 932081 1.30604e+11
+77 7.6e-07 1e-08 932080 1.27965e+11
+78 7.7e-07 1e-08 932081 1.26871e+11
+79 7.8e-07 1e-08 932081 1.27085e+11
+80 7.9e-07 1e-08 932081 1.29338e+11
+81 8e-07 1e-08 932081 1.31962e+11
+82 8.1e-07 1e-08 932081 1.27764e+11
+83 8.2e-07 1e-08 932082 1.30643e+11
+84 8.3e-07 1e-08 932082 1.32703e+11
+85 8.4e-07 1e-08 932082 1.32532e+11
+86 8.5e-07 1e-08 932082 1.28777e+11
+87 8.6e-07 1e-08 932082 1.28244e+11
+88 8.7e-07 1e-08 932082 1.33321e+11
+89 8.8e-07 1e-08 932082 1.29311e+11
+90 8.9e-07 1e-08 932082 1.3115e+11
+91 9e-07 1e-08 932082 1.33329e+11
+92 9.1e-07 1e-08 932081 1.31141e+11
+93 9.2e-07 1e-08 932080 1.31293e+11
+94 9.3e-07 1e-08 932081 1.32827e+11
+95 9.4e-07 1e-08 956728 1.33383e+11
+96 9.5e-07 1e-08 956728 1.36494e+11
+97 9.6e-07 1e-08 956728 1.33268e+11
+98 9.7e-07 1e-08 956728 1.36061e+11
+99 9.8e-07 1e-08 956727 1.35457e+11
+100 9.9e-07 1e-08 956727 1.3689e+11
+101 1e-06 1e-08 956727 1.33503e+11
+102 1.01e-06 1e-08 956727 1.34783e+11
+103 1.02e-06 1e-08 956726 1.3714e+11
+104 1.03e-06 1e-08 956726 1.3678e+11
+105 1.04e-06 1e-08 956727 1.31548e+11
+106 1.05e-06 1e-08 956727 1.36773e+11
+107 1.06e-06 1e-08 956727 1.34946e+11
+108 1.07e-06 1e-08 956727 1.33351e+11
+109 1.08e-06 1e-08 956727 1.35053e+11
+110 1.09e-06 1e-08 956727 1.32849e+11
+111 1.1e-06 1e-08 956728 1.35776e+11
+112 1.11e-06 1e-08 956728 1.32476e+11
+113 1.12e-06 1e-08 956727 1.3352e+11
+114 1.13e-06 1e-08 956728 1.32584e+11
+115 1.14e-06 1e-08 956728 1.31304e+11
+116 1.15e-06 1e-08 956728 1.32242e+11
+117 1.16e-06 1e-08 956728 1.34295e+11
+118 1.17e-06 1e-08 956728 1.3298e+11
+119 1.18e-06 1e-08 956728 1.31501e+11
+120 1.19e-06 1e-08 956728 1.36688e+11
+121 1.2e-06 1e-08 956728 1.31291e+11
+122 1.21e-06 1e-08 956729 1.31538e+11
+123 1.22e-06 1e-08 956729 1.33825e+11
+124 1.23e-06 1e-08 956729 1.35305e+11
+125 1.24e-06 1e-08 956729 1.37236e+11
+126 1.25e-06 1e-08 956729 1.37412e+11
+127 1.26e-06 1e-08 956729 1.34788e+11
+128 1.27e-06 1e-08 956729 1.33239e+11
+129 1.28e-06 1e-08 956729 1.35593e+11
+130 1.29e-06 1e-08 956729 1.33185e+11
+131 1.3e-06 1e-08 956729 1.32326e+11
+132 1.31e-06 1e-08 956729 1.32278e+11
+133 1.32e-06 1e-08 956730 1.34637e+11
+134 1.33e-06 1e-08 956730 1.34965e+11
+135 1.34e-06 1e-08 956730 1.31371e+11
+136 1.35e-06 1e-08 956730 1.35725e+11
+137 1.36e-06 1e-08 956731 1.31216e+11
+138 1.37e-06 1e-08 956731 1.35923e+11
+139 1.38e-06 1e-08 956731 1.31948e+11
+140 1.39e-06 1e-08 956732 1.36298e+11
+141 1.4e-06 1e-08 956732 1.30856e+11
+142 1.41e-06 1e-08 956731 1.3756e+11
+143 1.42e-06 1e-08 956731 1.30876e+11
+144 1.43e-06 1e-08 956731 1.3402e+11
+145 1.44e-06 1e-08 956731 1.31568e+11
+146 1.45e-06 1e-08 956730 1.32501e+11
+147 1.46e-06 1e-08 956730 1.36034e+11
+148 1.47e-06 1e-08 956730 1.35648e+11
+149 1.48e-06 1e-08 956730 1.35135e+11
+150 1.49e-06 1e-08 956730 1.37269e+11
+151 1.5e-06 1e-08 956730 1.35013e+11
+152 1.51e-06 1e-08 956729 1.36674e+11
+153 1.52e-06 1e-08 956729 1.31349e+11
+154 1.53e-06 1e-08 956729 1.31536e+11
+155 1.54e-06 1e-08 956728 1.34724e+11
+156 1.55e-06 1e-08 956728 1.30947e+11
+157 1.56e-06 1e-08 956728 1.30924e+11
+158 1.57e-06 1e-08 956728 1.31852e+11
+159 1.58e-06 1e-08 956728 1.34011e+11
+160 1.59e-06 1e-08 956728 1.39224e+11
+161 1.6e-06 1e-08 956727 1.34964e+11
+162 1.61e-06 1e-08 956726 1.36342e+11
+163 1.62e-06 1e-08 956726 1.3077e+11
+164 1.63e-06 1e-08 956727 1.32306e+11
+165 1.64e-06 1e-08 956727 1.37689e+11
+166 1.65e-06 1e-08 956725 1.32779e+11
+167 1.66e-06 1e-08 956725 1.36881e+11
+168 1.67e-06 1e-08 956725 1.31266e+11
+169 1.68e-06 1e-08 956725 1.37871e+11
+170 1.69e-06 1e-08 956725 1.31081e+11
+171 1.7e-06 1e-08 956725 1.36484e+11
+172 1.71e-06 1e-08 956725 1.3148e+11
+173 1.72e-06 1e-08 956725 1.30419e+11
+174 1.73e-06 1e-08 956725 1.36945e+11
+175 1.74e-06 1e-08 956725 1.34623e+11
+176 1.75e-06 1e-08 956726 1.32351e+11
+177 1.76e-06 1e-08 956726 1.32772e+11
+178 1.77e-06 1e-08 956726 1.33037e+11
+179 1.78e-06 1e-08 956726 1.36965e+11
+180 1.79e-06 1e-08 956726 1.36224e+11
+181 1.8e-06 1e-08 956726 1.3265e+11
+182 1.81e-06 1e-08 956726 1.34221e+11
+183 1.82e-06 1e-08 956726 1.37484e+11
+184 1.83e-06 1e-08 956726 1.33511e+11
+185 1.84e-06 1e-08 956726 1.35164e+11
+186 1.85e-06 1e-08 956725 1.31497e+11
+187 1.86e-06 1e-08 956725 1.34405e+11
+188 1.87e-06 1e-08 956725 1.35673e+11
+189 1.88e-06 1e-08 956725 1.32458e+11
+190 1.89e-06 1e-08 956725 1.37316e+11
+191 1.9e-06 1e-08 956724 1.34962e+11
+192 1.91e-06 1e-08 956724 1.36521e+11
+193 1.92e-06 1e-08 956724 1.3567e+11
+194 1.93e-06 1e-08 956725 1.35587e+11
+195 1.94e-06 1e-08 956725 1.30579e+11
+196 1.95e-06 1e-08 956725 1.36047e+11
+197 1.96e-06 1e-08 956724 1.36744e+11
+198 1.97e-06 1e-08 956724 1.36961e+11
+199 1.98e-06 1e-08 956724 1.36478e+11
+200 1.99e-06 1e-08 956724 1.35887e+11
+201 2e-06 1e-08 956724 1.37332e+11
+202 2.01e-06 1e-08 956724 1.35562e+11
+203 2.02e-06 1e-08 979628 1.41671e+11
+204 2.03e-06 1e-08 979628 1.45766e+11
+205 2.04e-06 1e-08 979628 1.40248e+11
+206 2.05e-06 1e-08 979628 1.38385e+11
+207 2.06e-06 1e-08 979628 1.36647e+11
+208 2.07e-06 1e-08 979627 1.41688e+11
+209 2.08e-06 1e-08 979627 1.41568e+11
+210 2.09e-06 1e-08 979627 1.3564e+11
+211 2.1e-06 1e-08 979627 1.39814e+11
+212 2.11e-06 1e-08 979627 1.37222e+11
+213 2.12e-06 1e-08 979628 1.40251e+11
+214 2.13e-06 1e-08 979628 1.39761e+11
+215 2.14e-06 1e-08 979628 1.40354e+11
+216 2.15e-06 1e-08 979628 1.37652e+11
+217 2.16e-06 1e-08 979628 1.37752e+11
+218 2.17e-06 1e-08 979627 1.37174e+11
+219 2.18e-06 1e-08 979627 1.35707e+11
+220 2.19e-06 1e-08 979627 1.39714e+11
+221 2.2e-06 1e-08 979628 1.37199e+11
+222 2.21e-06 1e-08 979627 1.40414e+11
+223 2.22e-06 1e-08 979627 1.39221e+11
+224 2.23e-06 1e-08 979626 1.42605e+11
+225 2.24e-06 1e-08 979626 1.37133e+11
+226 2.25e-06 1e-08 979626 1.39119e+11
+227 2.26e-06 1e-08 979625 1.40811e+11
+228 2.27e-06 1e-08 979625 1.36746e+11
+229 2.28e-06 1e-08 979624 1.39167e+11
+230 2.29e-06 1e-08 979623 1.40807e+11
+231 2.3e-06 1e-08 979624 1.43003e+11
+232 2.31e-06 1e-08 979624 1.33665e+11
+233 2.32e-06 1e-08 979624 1.33116e+11
+234 2.33e-06 1e-08 979624 1.41347e+11
+235 2.34e-06 1e-08 979623 1.39126e+11
+236 2.35e-06 1e-08 979623 1.36025e+11
+237 2.36e-06 1e-08 979622 1.38406e+11
+238 2.37e-06 1e-08 979622 1.39714e+11
+239 2.38e-06 1e-08 979622 1.40979e+11
+240 2.39e-06 1e-08 979622 1.36796e+11
+241 2.4e-06 1e-08 979621 1.40941e+11
+242 2.41e-06 1e-08 979621 1.38638e+11
+243 2.42e-06 1e-08 979622 1.39217e+11
+244 2.43e-06 1e-08 979622 1.37608e+11
+245 2.44e-06 1e-08 979623 1.40854e+11
+246 2.45e-06 1e-08 979623 1.36665e+11
+247 2.46e-06 1e-08 979622 1.41165e+11
+248 2.47e-06 1e-08 979622 1.35123e+11
+249 2.48e-06 1e-08 979621 1.38714e+11
+250 2.49e-06 1e-08 979620 1.43942e+11
+251 2.5e-06 1e-08 979620 1.40262e+11
+252 2.51e-06 1e-08 979620 1.39817e+11
+253 2.52e-06 1e-08 979619 1.38107e+11
+254 2.53e-06 1e-08 979621 1.3667e+11
+255 2.54e-06 1e-08 979621 1.39551e+11
+256 2.55e-06 1e-08 979621 1.39821e+11
+257 2.56e-06 1e-08 979621 1.4098e+11
diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_20.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_20.dat
new file mode 100644
index 0000000000000000000000000000000000000000..608025d166b2e611c86aa63833e09c8c4eaf1aaa
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_20.dat
@@ -0,0 +1,259 @@
+# Number of cores: 20
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 93289 2.06894e+10
+2 1e-08 1e-08 93289 1.67912e+10
+3 2e-08 1e-08 93289 1.70677e+10
+4 3e-08 1e-08 93289 1.69947e+10
+5 4e-08 1e-08 93289 1.71858e+10
+6 5e-08 1e-08 93289 1.64812e+10
+7 6e-08 1e-08 93289 1.6604e+10
+8 7e-08 1e-08 93289 1.70321e+10
+9 8e-08 1e-08 93289 1.66028e+10
+10 9e-08 1e-08 93289 1.67655e+10
+11 1e-07 1e-08 93289 1.66797e+10
+12 1.1e-07 1e-08 93289 1.68001e+10
+13 1.2e-07 1e-08 93289 1.66857e+10
+14 1.3e-07 1e-08 93289 1.66713e+10
+15 1.4e-07 1e-08 93289 1.70856e+10
+16 1.5e-07 1e-08 93289 1.70158e+10
+17 1.6e-07 1e-08 93289 1.68251e+10
+18 1.7e-07 1e-08 93289 1.68769e+10
+19 1.8e-07 1e-08 93289 1.64159e+10
+20 1.9e-07 1e-08 93289 1.68667e+10
+21 2e-07 1e-08 93289 1.62652e+10
+22 2.1e-07 1e-08 93289 1.63109e+10
+23 2.2e-07 1e-08 93289 1.63327e+10
+24 2.3e-07 1e-08 93289 1.68737e+10
+25 2.4e-07 1e-08 93289 1.62421e+10
+26 2.5e-07 1e-08 93289 1.65039e+10
+27 2.6e-07 1e-08 93289 1.6515e+10
+28 2.7e-07 1e-08 93289 1.62839e+10
+29 2.8e-07 1e-08 93289 1.70674e+10
+30 2.9e-07 1e-08 93289 1.63732e+10
+31 3e-07 1e-08 93289 1.66523e+10
+32 3.1e-07 1e-08 93289 1.72052e+10
+33 3.2e-07 1e-08 93289 1.68264e+10
+34 3.3e-07 1e-08 93289 1.65167e+10
+35 3.4e-07 1e-08 93289 1.68443e+10
+36 3.5e-07 1e-08 93289 1.67293e+10
+37 3.6e-07 1e-08 93289 1.6671e+10
+38 3.7e-07 1e-08 93289 1.7163e+10
+39 3.8e-07 1e-08 93289 1.66669e+10
+40 3.9e-07 1e-08 93289 1.71767e+10
+41 4e-07 1e-08 93289 1.74901e+10
+42 4.1e-07 1e-08 93289 1.70611e+10
+43 4.2e-07 1e-08 93289 1.67744e+10
+44 4.3e-07 1e-08 93289 1.71916e+10
+45 4.4e-07 1e-08 93289 1.65589e+10
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diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_24.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_24.dat
new file mode 100644
index 0000000000000000000000000000000000000000..16ece0cc0415e3d4362b7a892b32b1ba9fc7d645
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_24.dat
@@ -0,0 +1,259 @@
+# Number of cores: 24
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 75128 1.77483e+10
+2 1e-08 1e-08 75128 1.40325e+10
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diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_28.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_28.dat
new file mode 100644
index 0000000000000000000000000000000000000000..d40a8bdbc981d6376e1967525c7ec01f86a97b93
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_28.dat
@@ -0,0 +1,259 @@
+# Number of cores: 28
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 65506 1.58176e+10
+2 1e-08 1e-08 65506 1.23058e+10
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+256 2.55e-06 1e-08 72412 1.50235e+10
+257 2.56e-06 1e-08 72412 1.52097e+10
diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_3.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_3.dat
new file mode 100644
index 0000000000000000000000000000000000000000..a005c092e7330d381ea14e8fc1cc895d7820a1f6
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_3.dat
@@ -0,0 +1,259 @@
+# Number of cores: 3
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 620067 9.21148e+10
+2 1e-08 1e-08 620067 8.90722e+10
+3 2e-08 1e-08 620067 9.08199e+10
+4 3e-08 1e-08 620068 9.11088e+10
+5 4e-08 1e-08 620068 9.14273e+10
+6 5e-08 1e-08 620067 8.81768e+10
+7 6e-08 1e-08 620067 8.86408e+10
+8 7e-08 1e-08 620067 9.10254e+10
+9 8e-08 1e-08 620067 8.85437e+10
+10 9e-08 1e-08 620069 9.00623e+10
+11 1e-07 1e-08 620070 8.95361e+10
+12 1.1e-07 1e-08 620070 9.01124e+10
+13 1.2e-07 1e-08 620070 8.88226e+10
+14 1.3e-07 1e-08 620070 8.95589e+10
+15 1.4e-07 1e-08 620070 9.16851e+10
+16 1.5e-07 1e-08 620070 9.14193e+10
+17 1.6e-07 1e-08 620069 9.03277e+10
+18 1.7e-07 1e-08 620069 9.06566e+10
+19 1.8e-07 1e-08 620069 8.78154e+10
+20 1.9e-07 1e-08 620069 9.01345e+10
+21 2e-07 1e-08 620070 8.73703e+10
+22 2.1e-07 1e-08 620071 8.84787e+10
+23 2.2e-07 1e-08 620071 8.79001e+10
+24 2.3e-07 1e-08 620069 9.13356e+10
+25 2.4e-07 1e-08 620069 8.79989e+10
+26 2.5e-07 1e-08 620069 8.92299e+10
+27 2.6e-07 1e-08 620069 8.78817e+10
+28 2.7e-07 1e-08 620069 8.77727e+10
+29 2.8e-07 1e-08 620069 9.21461e+10
+30 2.9e-07 1e-08 620069 8.82028e+10
+31 3e-07 1e-08 620069 8.96016e+10
+32 3.1e-07 1e-08 620068 9.13525e+10
+33 3.2e-07 1e-08 620068 9.00609e+10
+34 3.3e-07 1e-08 620067 8.87275e+10
+35 3.4e-07 1e-08 620067 9.02337e+10
+36 3.5e-07 1e-08 620067 8.9693e+10
+37 3.6e-07 1e-08 620067 8.94807e+10
+38 3.7e-07 1e-08 620067 9.17468e+10
+39 3.8e-07 1e-08 620069 8.87734e+10
+40 3.9e-07 1e-08 620069 9.09994e+10
+41 4e-07 1e-08 622674 8.9635e+10
+42 4.1e-07 1e-08 622675 9.02803e+10
+43 4.2e-07 1e-08 622675 8.84681e+10
+44 4.3e-07 1e-08 622675 9.07539e+10
+45 4.4e-07 1e-08 622675 8.78511e+10
+46 4.5e-07 1e-08 622674 8.83898e+10
+47 4.6e-07 1e-08 622674 9.02884e+10
+48 4.7e-07 1e-08 622674 9.08574e+10
+49 4.8e-07 1e-08 622674 8.6827e+10
+50 4.9e-07 1e-08 622674 9.14228e+10
+51 5e-07 1e-08 622675 8.92378e+10
+52 5.1e-07 1e-08 622674 8.72839e+10
+53 5.2e-07 1e-08 622675 8.76453e+10
+54 5.3e-07 1e-08 622676 8.98898e+10
+55 5.4e-07 1e-08 622676 9.07765e+10
+56 5.5e-07 1e-08 622676 8.81583e+10
+57 5.6e-07 1e-08 622676 8.68001e+10
+58 5.7e-07 1e-08 622676 8.66365e+10
+59 5.8e-07 1e-08 622675 8.88746e+10
+60 5.9e-07 1e-08 622676 8.71848e+10
+61 6e-07 1e-08 622676 8.82078e+10
+62 6.1e-07 1e-08 622676 9.11607e+10
+63 6.2e-07 1e-08 622676 9.10512e+10
+64 6.3e-07 1e-08 622676 9.05616e+10
+65 6.4e-07 1e-08 622676 8.80654e+10
+66 6.5e-07 1e-08 622676 8.96531e+10
+67 6.6e-07 1e-08 622674 8.94852e+10
+68 6.7e-07 1e-08 622674 9.07883e+10
+69 6.8e-07 1e-08 622674 8.98464e+10
+70 6.9e-07 1e-08 622673 9.06877e+10
+71 7e-07 1e-08 622673 8.98142e+10
+72 7.1e-07 1e-08 622673 8.74642e+10
+73 7.2e-07 1e-08 622674 8.96056e+10
+74 7.3e-07 1e-08 622675 9.18292e+10
+75 7.4e-07 1e-08 622675 9.1734e+10
+76 7.5e-07 1e-08 622675 9.08336e+10
+77 7.6e-07 1e-08 622675 8.89604e+10
+78 7.7e-07 1e-08 622675 9.11103e+10
+79 7.8e-07 1e-08 622675 9.01511e+10
+80 7.9e-07 1e-08 629739 9.12621e+10
+81 8e-07 1e-08 629740 9.247e+10
+82 8.1e-07 1e-08 629740 9.01123e+10
+83 8.2e-07 1e-08 629741 9.15352e+10
+84 8.3e-07 1e-08 629741 9.41497e+10
+85 8.4e-07 1e-08 629741 9.37232e+10
+86 8.5e-07 1e-08 629741 9.11557e+10
+87 8.6e-07 1e-08 629741 9.05401e+10
+88 8.7e-07 1e-08 622677 9.21935e+10
+89 8.8e-07 1e-08 622677 8.89391e+10
+90 8.9e-07 1e-08 622677 9.04036e+10
+91 9e-07 1e-08 622677 9.17439e+10
+92 9.1e-07 1e-08 622677 9.00682e+10
+93 9.2e-07 1e-08 622677 9.04219e+10
+94 9.3e-07 1e-08 629740 9.40044e+10
+95 9.4e-07 1e-08 629740 9.1363e+10
+96 9.5e-07 1e-08 629740 9.36024e+10
+97 9.6e-07 1e-08 629740 9.12063e+10
+98 9.7e-07 1e-08 629741 9.32694e+10
+99 9.8e-07 1e-08 629740 9.27141e+10
+100 9.9e-07 1e-08 629741 9.38093e+10
+101 1e-06 1e-08 629741 9.15285e+10
+102 1.01e-06 1e-08 629741 9.06647e+10
+103 1.02e-06 1e-08 629741 9.38903e+10
+104 1.03e-06 1e-08 629741 9.3651e+10
+105 1.04e-06 1e-08 629741 9.0037e+10
+106 1.05e-06 1e-08 629741 9.35537e+10
+107 1.06e-06 1e-08 629742 9.2457e+10
+108 1.07e-06 1e-08 629742 9.13339e+10
+109 1.08e-06 1e-08 629742 9.22917e+10
+110 1.09e-06 1e-08 629743 9.07597e+10
+111 1.1e-06 1e-08 632361 9.3829e+10
+112 1.11e-06 1e-08 632361 9.11085e+10
+113 1.12e-06 1e-08 632361 9.20634e+10
+114 1.13e-06 1e-08 632361 9.10268e+10
+115 1.14e-06 1e-08 632361 9.04895e+10
+116 1.15e-06 1e-08 632361 9.0982e+10
+117 1.16e-06 1e-08 632361 9.23436e+10
+118 1.17e-06 1e-08 632361 9.1584e+10
+119 1.18e-06 1e-08 632361 9.0476e+10
+120 1.19e-06 1e-08 632361 9.40346e+10
+121 1.2e-06 1e-08 632361 9.03261e+10
+122 1.21e-06 1e-08 632362 9.03677e+10
+123 1.22e-06 1e-08 632362 9.20867e+10
+124 1.23e-06 1e-08 632362 9.30438e+10
+125 1.24e-06 1e-08 632362 9.42037e+10
+126 1.25e-06 1e-08 632363 9.43857e+10
+127 1.26e-06 1e-08 632363 9.29961e+10
+128 1.27e-06 1e-08 632363 9.16577e+10
+129 1.28e-06 1e-08 632363 9.33193e+10
+130 1.29e-06 1e-08 632363 9.15709e+10
+131 1.3e-06 1e-08 632364 9.12575e+10
+132 1.31e-06 1e-08 632363 9.09696e+10
+133 1.32e-06 1e-08 632363 9.25831e+10
+134 1.33e-06 1e-08 632363 9.13483e+10
+135 1.34e-06 1e-08 635935 9.12308e+10
+136 1.35e-06 1e-08 635935 9.35735e+10
+137 1.36e-06 1e-08 635935 9.07085e+10
+138 1.37e-06 1e-08 635935 9.39329e+10
+139 1.38e-06 1e-08 635935 9.09921e+10
+140 1.39e-06 1e-08 635936 9.39485e+10
+141 1.4e-06 1e-08 635936 9.04063e+10
+142 1.41e-06 1e-08 635935 9.45888e+10
+143 1.42e-06 1e-08 635935 9.04093e+10
+144 1.43e-06 1e-08 635935 9.2711e+10
+145 1.44e-06 1e-08 635935 9.08685e+10
+146 1.45e-06 1e-08 635934 9.143e+10
+147 1.46e-06 1e-08 635934 9.40227e+10
+148 1.47e-06 1e-08 635934 9.36338e+10
+149 1.48e-06 1e-08 635934 9.33984e+10
+150 1.49e-06 1e-08 635934 9.53957e+10
+151 1.5e-06 1e-08 635934 9.31968e+10
+152 1.51e-06 1e-08 635936 9.40491e+10
+153 1.52e-06 1e-08 635935 9.06751e+10
+154 1.53e-06 1e-08 635935 9.0702e+10
+155 1.54e-06 1e-08 635935 9.28578e+10
+156 1.55e-06 1e-08 635935 9.04298e+10
+157 1.56e-06 1e-08 635935 9.03826e+10
+158 1.57e-06 1e-08 635935 9.10523e+10
+159 1.58e-06 1e-08 635936 9.24843e+10
+160 1.59e-06 1e-08 635936 9.44973e+10
+161 1.6e-06 1e-08 635935 9.29647e+10
+162 1.61e-06 1e-08 635936 9.42233e+10
+163 1.62e-06 1e-08 635936 9.0508e+10
+164 1.63e-06 1e-08 635938 9.12429e+10
+165 1.64e-06 1e-08 635937 9.53743e+10
+166 1.65e-06 1e-08 635936 9.15678e+10
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+169 1.68e-06 1e-08 635937 9.5311e+10
+170 1.69e-06 1e-08 635936 9.07849e+10
+171 1.7e-06 1e-08 635936 9.45152e+10
+172 1.71e-06 1e-08 635936 9.08811e+10
+173 1.72e-06 1e-08 635936 9.02786e+10
+174 1.73e-06 1e-08 635936 9.47844e+10
+175 1.74e-06 1e-08 635936 9.31547e+10
+176 1.75e-06 1e-08 635937 9.1391e+10
+177 1.76e-06 1e-08 635937 9.15756e+10
+178 1.77e-06 1e-08 635937 9.23586e+10
+179 1.78e-06 1e-08 635937 9.49454e+10
+180 1.79e-06 1e-08 635937 9.44922e+10
+181 1.8e-06 1e-08 635937 9.23368e+10
+182 1.81e-06 1e-08 635937 9.31474e+10
+183 1.82e-06 1e-08 635937 9.37298e+10
+184 1.83e-06 1e-08 635937 9.26994e+10
+185 1.84e-06 1e-08 635937 9.39192e+10
+186 1.85e-06 1e-08 635936 9.12307e+10
+187 1.86e-06 1e-08 635936 9.30873e+10
+188 1.87e-06 1e-08 635936 9.41464e+10
+189 1.88e-06 1e-08 635936 9.22974e+10
+190 1.89e-06 1e-08 635936 9.56267e+10
+191 1.9e-06 1e-08 635935 9.36791e+10
+192 1.91e-06 1e-08 635936 9.51747e+10
+193 1.92e-06 1e-08 635936 9.45343e+10
+194 1.93e-06 1e-08 635937 9.3947e+10
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+199 1.98e-06 1e-08 635936 9.50155e+10
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+201 2e-06 1e-08 635936 9.58063e+10
+202 2.01e-06 1e-08 635936 9.46754e+10
+203 2.02e-06 1e-08 635936 9.54947e+10
+204 2.03e-06 1e-08 635937 9.65659e+10
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+206 2.05e-06 1e-08 635938 9.34386e+10
+207 2.06e-06 1e-08 635939 9.19939e+10
+208 2.07e-06 1e-08 635940 9.53309e+10
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+222 2.21e-06 1e-08 635940 9.45427e+10
+223 2.22e-06 1e-08 635940 9.38455e+10
+224 2.23e-06 1e-08 635940 9.61528e+10
+225 2.24e-06 1e-08 635940 9.26638e+10
+226 2.25e-06 1e-08 635940 9.39773e+10
+227 2.26e-06 1e-08 635940 9.50097e+10
+228 2.27e-06 1e-08 635940 9.18267e+10
+229 2.28e-06 1e-08 635940 9.37487e+10
+230 2.29e-06 1e-08 635940 9.43436e+10
+231 2.3e-06 1e-08 635941 9.61412e+10
+232 2.31e-06 1e-08 635941 9.39551e+10
+233 2.32e-06 1e-08 635941 9.29203e+10
+234 2.33e-06 1e-08 635941 9.54917e+10
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+238 2.37e-06 1e-08 635940 9.42072e+10
+239 2.38e-06 1e-08 635940 9.54928e+10
+240 2.39e-06 1e-08 635940 9.23819e+10
+241 2.4e-06 1e-08 635939 9.471e+10
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+244 2.43e-06 1e-08 635940 9.30259e+10
+245 2.44e-06 1e-08 635941 9.5465e+10
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+248 2.47e-06 1e-08 635940 9.19748e+10
+249 2.48e-06 1e-08 635939 9.39665e+10
+250 2.49e-06 1e-08 635940 9.59843e+10
+251 2.5e-06 1e-08 635941 9.50218e+10
+252 2.51e-06 1e-08 635941 9.46638e+10
+253 2.52e-06 1e-08 635940 9.34086e+10
+254 2.53e-06 1e-08 635941 9.25127e+10
+255 2.54e-06 1e-08 635941 9.46144e+10
+256 2.55e-06 1e-08 635941 9.46838e+10
+257 2.56e-06 1e-08 635941 9.52526e+10
diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_32.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_32.dat
new file mode 100644
index 0000000000000000000000000000000000000000..57e3593ee6f41cebce82e6208cbcee465cec34c2
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_32.dat
@@ -0,0 +1,259 @@
+# Number of cores: 32
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 56993 1.70321e+10
+2 1e-08 1e-08 56993 1.54719e+10
+3 2e-08 1e-08 56993 1.52835e+10
+4 3e-08 1e-08 56993 1.55199e+10
+5 4e-08 1e-08 56993 1.55924e+10
+6 5e-08 1e-08 56993 1.451e+10
+7 6e-08 1e-08 56993 1.3493e+10
+8 7e-08 1e-08 56993 1.51209e+10
+9 8e-08 1e-08 56993 1.47319e+10
+10 9e-08 1e-08 56993 1.49832e+10
+11 1e-07 1e-08 56993 1.48851e+10
+12 1.1e-07 1e-08 56993 1.50034e+10
+13 1.2e-07 1e-08 56993 1.49255e+10
+14 1.3e-07 1e-08 56993 1.50402e+10
+15 1.4e-07 1e-08 56993 1.53878e+10
+16 1.5e-07 1e-08 56993 1.53222e+10
+17 1.6e-07 1e-08 56993 1.48736e+10
+18 1.7e-07 1e-08 56993 1.53236e+10
+19 1.8e-07 1e-08 56993 1.48875e+10
+20 1.9e-07 1e-08 56993 1.52932e+10
+21 2e-07 1e-08 56993 1.49154e+10
+22 2.1e-07 1e-08 56993 1.49663e+10
+23 2.2e-07 1e-08 56993 1.50898e+10
+24 2.3e-07 1e-08 56993 1.5659e+10
+25 2.4e-07 1e-08 56993 1.51425e+10
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diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_4.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_4.dat
new file mode 100644
index 0000000000000000000000000000000000000000..b97b698bdbc532bebc7c5d95be9e648427ebc2f6
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_4.dat
@@ -0,0 +1,259 @@
+# Number of cores: 4
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 454244 7.04031e+10
+2 1e-08 1e-08 454244 6.79738e+10
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diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_48.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_48.dat
new file mode 100644
index 0000000000000000000000000000000000000000..886c9ee52a2008c170f0da7793996fb73bb6a6f0
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_48.dat
@@ -0,0 +1,259 @@
+# Number of cores: 48
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 38287 1.57667e+10
+2 1e-08 1e-08 38287 1.15414e+10
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diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_6.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_6.dat
new file mode 100644
index 0000000000000000000000000000000000000000..3d045a2d2aee9d0384fead0e30ee512687dd5316
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_6.dat
@@ -0,0 +1,259 @@
+# Number of cores: 6
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 302137 4.96219e+10
+2 1e-08 1e-08 302137 4.56983e+10
+3 2e-08 1e-08 302136 4.67071e+10
+4 3e-08 1e-08 302137 4.75594e+10
+5 4e-08 1e-08 302136 4.76418e+10
+6 5e-08 1e-08 302135 4.53619e+10
+7 6e-08 1e-08 302135 4.58541e+10
+8 7e-08 1e-08 302134 4.70621e+10
+9 8e-08 1e-08 302134 4.65289e+10
+10 9e-08 1e-08 302135 4.68613e+10
+11 1e-07 1e-08 302136 4.68624e+10
+12 1.1e-07 1e-08 302136 4.7033e+10
+13 1.2e-07 1e-08 302136 4.59732e+10
+14 1.3e-07 1e-08 302136 4.65647e+10
+15 1.4e-07 1e-08 302136 4.80509e+10
+16 1.5e-07 1e-08 302136 4.77631e+10
+17 1.6e-07 1e-08 302135 4.71936e+10
+18 1.7e-07 1e-08 302135 4.72235e+10
+19 1.8e-07 1e-08 302135 4.60729e+10
+20 1.9e-07 1e-08 302135 4.77964e+10
+21 2e-07 1e-08 302136 4.60601e+10
+22 2.1e-07 1e-08 302137 4.60312e+10
+23 2.2e-07 1e-08 302137 4.56379e+10
+24 2.3e-07 1e-08 302137 4.80227e+10
+25 2.4e-07 1e-08 302137 4.60907e+10
+26 2.5e-07 1e-08 302137 4.64192e+10
+27 2.6e-07 1e-08 302135 4.586e+10
+28 2.7e-07 1e-08 302134 4.56443e+10
+29 2.8e-07 1e-08 302135 4.79384e+10
+30 2.9e-07 1e-08 302135 4.58044e+10
+31 3e-07 1e-08 302135 4.72698e+10
+32 3.1e-07 1e-08 302134 4.76552e+10
+33 3.2e-07 1e-08 302134 4.73179e+10
+34 3.3e-07 1e-08 302132 4.63257e+10
+35 3.4e-07 1e-08 302132 4.69786e+10
+36 3.5e-07 1e-08 302132 4.67443e+10
+37 3.6e-07 1e-08 302132 4.65631e+10
+38 3.7e-07 1e-08 302132 4.77397e+10
+39 3.8e-07 1e-08 302134 4.6182e+10
+40 3.9e-07 1e-08 302133 4.75298e+10
+41 4e-07 1e-08 302132 4.72261e+10
+42 4.1e-07 1e-08 302133 4.74072e+10
+43 4.2e-07 1e-08 302132 4.67617e+10
+44 4.3e-07 1e-08 302133 4.75916e+10
+45 4.4e-07 1e-08 302133 4.59777e+10
+46 4.5e-07 1e-08 302131 4.67546e+10
+47 4.6e-07 1e-08 302131 4.73067e+10
+48 4.7e-07 1e-08 302131 4.76557e+10
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+51 5e-07 1e-08 302132 4.70695e+10
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+55 5.4e-07 1e-08 302132 4.76997e+10
+56 5.5e-07 1e-08 302132 4.70338e+10
+57 5.6e-07 1e-08 302132 4.62136e+10
+58 5.7e-07 1e-08 302132 4.60725e+10
+59 5.8e-07 1e-08 302131 4.7175e+10
+60 5.9e-07 1e-08 302132 4.60245e+10
+61 6e-07 1e-08 302132 4.68975e+10
+62 6.1e-07 1e-08 302132 4.86505e+10
+63 6.2e-07 1e-08 302132 4.88463e+10
+64 6.3e-07 1e-08 302132 4.9214e+10
+65 6.4e-07 1e-08 302132 4.78197e+10
+66 6.5e-07 1e-08 302131 4.88679e+10
+67 6.6e-07 1e-08 302131 4.77433e+10
+68 6.7e-07 1e-08 302131 4.84437e+10
+69 6.8e-07 1e-08 302130 4.8861e+10
+70 6.9e-07 1e-08 302130 4.85343e+10
+71 7e-07 1e-08 302130 4.75982e+10
+72 7.1e-07 1e-08 302130 4.76899e+10
+73 7.2e-07 1e-08 302130 4.89383e+10
+74 7.3e-07 1e-08 302130 4.98649e+10
+75 7.4e-07 1e-08 302130 4.94736e+10
+76 7.5e-07 1e-08 302130 4.91155e+10
+77 7.6e-07 1e-08 302131 4.78495e+10
+78 7.7e-07 1e-08 302131 5.05074e+10
+79 7.8e-07 1e-08 302131 5.02391e+10
+80 7.9e-07 1e-08 302132 4.91054e+10
+81 8e-07 1e-08 302133 4.99962e+10
+82 8.1e-07 1e-08 302132 4.83428e+10
+83 8.2e-07 1e-08 302130 4.87282e+10
+84 8.3e-07 1e-08 302129 5.0398e+10
+85 8.4e-07 1e-08 302129 4.99319e+10
+86 8.5e-07 1e-08 302129 4.86149e+10
+87 8.6e-07 1e-08 302130 4.88858e+10
+88 8.7e-07 1e-08 302130 5.02568e+10
+89 8.8e-07 1e-08 302129 4.89147e+10
+90 8.9e-07 1e-08 302129 4.95304e+10
+91 9e-07 1e-08 302129 5.03665e+10
+92 9.1e-07 1e-08 302129 4.95171e+10
+93 9.2e-07 1e-08 302129 5.00109e+10
+94 9.3e-07 1e-08 302128 5.05803e+10
+95 9.4e-07 1e-08 302128 4.93158e+10
+96 9.5e-07 1e-08 302128 5.07171e+10
+97 9.6e-07 1e-08 302128 4.959e+10
+98 9.7e-07 1e-08 302129 5.05094e+10
+99 9.8e-07 1e-08 302128 4.96888e+10
+100 9.9e-07 1e-08 302128 5.02847e+10
+101 1e-06 1e-08 302129 4.91618e+10
+102 1.01e-06 1e-08 302129 4.9187e+10
+103 1.02e-06 1e-08 302129 5.02806e+10
+104 1.03e-06 1e-08 302129 5.02056e+10
+105 1.04e-06 1e-08 302128 4.83619e+10
+106 1.05e-06 1e-08 302127 5.06234e+10
+107 1.06e-06 1e-08 302128 4.958e+10
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+109 1.08e-06 1e-08 302128 4.9815e+10
+110 1.09e-06 1e-08 302129 4.89653e+10
+111 1.1e-06 1e-08 309892 4.9766e+10
+112 1.11e-06 1e-08 309892 4.79804e+10
+113 1.12e-06 1e-08 309892 4.8465e+10
+114 1.13e-06 1e-08 309890 4.8182e+10
+115 1.14e-06 1e-08 309890 4.77479e+10
+116 1.15e-06 1e-08 309890 4.78739e+10
+117 1.16e-06 1e-08 309891 4.87382e+10
+118 1.17e-06 1e-08 309892 4.8453e+10
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+122 1.21e-06 1e-08 309892 4.76577e+10
+123 1.22e-06 1e-08 302128 5.06254e+10
+124 1.23e-06 1e-08 302128 5.04824e+10
+125 1.24e-06 1e-08 302128 5.10987e+10
+126 1.25e-06 1e-08 302129 5.10771e+10
+127 1.26e-06 1e-08 302129 5.01143e+10
+128 1.27e-06 1e-08 309892 4.86607e+10
+129 1.28e-06 1e-08 302128 5.22919e+10
+130 1.29e-06 1e-08 302127 5.09562e+10
+131 1.3e-06 1e-08 309891 4.8468e+10
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+133 1.32e-06 1e-08 309892 4.8902e+10
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+137 1.36e-06 1e-08 313690 4.78001e+10
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+148 1.47e-06 1e-08 313690 4.97996e+10
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+157 1.56e-06 1e-08 309892 4.73758e+10
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+159 1.58e-06 1e-08 309893 4.88587e+10
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+162 1.61e-06 1e-08 309894 4.93286e+10
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+164 1.63e-06 1e-08 309896 4.77544e+10
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+169 1.68e-06 1e-08 316826 5.09635e+10
+170 1.69e-06 1e-08 316827 4.8363e+10
+171 1.7e-06 1e-08 316827 5.03288e+10
+172 1.71e-06 1e-08 313694 4.93718e+10
+173 1.72e-06 1e-08 313694 4.82749e+10
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+175 1.74e-06 1e-08 302132 5.27984e+10
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+177 1.76e-06 1e-08 302133 5.20319e+10
+178 1.77e-06 1e-08 302133 5.22089e+10
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+184 1.83e-06 1e-08 313695 5.00078e+10
+185 1.84e-06 1e-08 313695 5.0078e+10
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+226 2.25e-06 1e-08 313700 5.07631e+10
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+229 2.28e-06 1e-08 313700 5.0716e+10
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+233 2.32e-06 1e-08 313701 5.01326e+10
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+238 2.37e-06 1e-08 313699 5.10723e+10
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+241 2.4e-06 1e-08 313699 5.10268e+10
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+247 2.46e-06 1e-08 320400 5.24377e+10
+248 2.47e-06 1e-08 320400 5.00069e+10
+249 2.48e-06 1e-08 320399 5.17427e+10
+250 2.49e-06 1e-08 320398 5.32681e+10
+251 2.5e-06 1e-08 320398 5.23023e+10
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+253 2.52e-06 1e-08 320398 5.14788e+10
+254 2.53e-06 1e-08 320399 5.11103e+10
+255 2.54e-06 1e-08 320399 5.21107e+10
+256 2.55e-06 1e-08 320399 5.20618e+10
+257 2.56e-06 1e-08 320399 5.24441e+10
diff --git a/theory/paper_spheric/data/Cosmo_fixed/performance_8.dat b/theory/paper_spheric/data/Cosmo_fixed/performance_8.dat
new file mode 100644
index 0000000000000000000000000000000000000000..01dc53a4f2e210eb5ec8f2861d9428e47ae23a24
--- /dev/null
+++ b/theory/paper_spheric/data/Cosmo_fixed/performance_8.dat
@@ -0,0 +1,259 @@
+# Number of cores: 8
+# Iteration | Time | TimeStep | NumPartUpdates | ElapsedTicks (cycle.h)
+1 0 0 229103 4.01597e+10
+2 1e-08 1e-08 229103 3.65139e+10
+3 2e-08 1e-08 229103 3.7268e+10
+4 3e-08 1e-08 229103 3.72267e+10
+5 4e-08 1e-08 229103 3.73918e+10
+6 5e-08 1e-08 229103 3.59081e+10
+7 6e-08 1e-08 229103 3.61374e+10
+8 7e-08 1e-08 229103 3.7056e+10
+9 8e-08 1e-08 229103 3.60692e+10
+10 9e-08 1e-08 229104 3.6609e+10
+11 1e-07 1e-08 229105 3.65127e+10
+12 1.1e-07 1e-08 229105 3.66936e+10
+13 1.2e-07 1e-08 229105 3.64289e+10
+14 1.3e-07 1e-08 229105 3.69561e+10
+15 1.4e-07 1e-08 229105 3.75631e+10
+16 1.5e-07 1e-08 229105 3.76187e+10
+17 1.6e-07 1e-08 229104 3.74079e+10
+18 1.7e-07 1e-08 229104 3.71372e+10
+19 1.8e-07 1e-08 229104 3.60809e+10
+20 1.9e-07 1e-08 229104 3.71056e+10
+21 2e-07 1e-08 229104 3.61313e+10
+22 2.1e-07 1e-08 229104 3.63133e+10
+23 2.2e-07 1e-08 233845 3.61672e+10
+24 2.3e-07 1e-08 233845 3.71315e+10
+25 2.4e-07 1e-08 233845 3.57881e+10
+26 2.5e-07 1e-08 233845 3.64072e+10
+27 2.6e-07 1e-08 233845 3.59621e+10
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diff --git a/theory/paper_spheric/data/SedovBlast_exact_10.dat b/theory/paper_spheric/data/SedovBlast_exact_10.dat
new file mode 100644
index 0000000000000000000000000000000000000000..826acebc86b9d5e4e9eaa1b0906eb29d5dd3a15c
--- /dev/null
+++ b/theory/paper_spheric/data/SedovBlast_exact_10.dat
@@ -0,0 +1,1000 @@
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diff --git a/theory/paper_spheric/data/SedovBlast_exact_15.dat b/theory/paper_spheric/data/SedovBlast_exact_15.dat
new file mode 100644
index 0000000000000000000000000000000000000000..f974ed7e661bffefb9a0d5f3295109a0e1977338
--- /dev/null
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diff --git a/theory/paper_spheric/data/SedovBlast_exact_20.dat b/theory/paper_spheric/data/SedovBlast_exact_20.dat
new file mode 100644
index 0000000000000000000000000000000000000000..27c8c7b53aae621972e47f70cb604bad290abc82
--- /dev/null
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diff --git a/theory/paper_spheric/data/SedovBlast_fixed.totals b/theory/paper_spheric/data/SedovBlast_fixed.totals
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index 0000000000000000000000000000000000000000..8bc3bb604580e958b27a88ea6eceaf43eb79cdd7
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diff --git a/theory/paper_spheric/data/SedovBlast_rdf_10.dat b/theory/paper_spheric/data/SedovBlast_rdf_10.dat
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diff --git a/theory/paper_spheric/data/SedovBlast_rdf_15.dat b/theory/paper_spheric/data/SedovBlast_rdf_15.dat
new file mode 100644
index 0000000000000000000000000000000000000000..7c90b12fbe85615869c2854b51ead9b2a4c84531
--- /dev/null
+++ b/theory/paper_spheric/data/SedovBlast_rdf_15.dat
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diff --git a/theory/paper_spheric/data/SedovBlast_rdf_20.dat b/theory/paper_spheric/data/SedovBlast_rdf_20.dat
new file mode 100644
index 0000000000000000000000000000000000000000..85849c36c21c16dea2efc5e0d0f822f96233e25c
--- /dev/null
+++ b/theory/paper_spheric/data/SedovBlast_rdf_20.dat
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diff --git a/theory/paper_spheric/data/SodShock_exact_P.dat b/theory/paper_spheric/data/SodShock_exact_P.dat
new file mode 100644
index 0000000000000000000000000000000000000000..27d9b5b6e7ebfeb01447d7e0ea1aa2ae478a600a
--- /dev/null
+++ b/theory/paper_spheric/data/SodShock_exact_P.dat
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diff --git a/theory/paper_spheric/data/SodShock_exact_rho.dat b/theory/paper_spheric/data/SodShock_exact_rho.dat
new file mode 100644
index 0000000000000000000000000000000000000000..0ff5268c696c20809860602dffcb28da8a233932
--- /dev/null
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diff --git a/theory/paper_spheric/data/SodShock_exact_v.dat b/theory/paper_spheric/data/SodShock_exact_v.dat
new file mode 100644
index 0000000000000000000000000000000000000000..9ea27cbaeca7c2842efba172fc1b245fe9586bdc
--- /dev/null
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diff --git a/theory/paper_spheric/data/SodShock_fixed.totals b/theory/paper_spheric/data/SodShock_fixed.totals
new file mode 100644
index 0000000000000000000000000000000000000000..7d01231831c308aecee843ea29334788ad9052af
--- /dev/null
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diff --git a/theory/paper_spheric/data/SodShock_rhox.dat b/theory/paper_spheric/data/SodShock_rhox.dat
new file mode 100644
index 0000000000000000000000000000000000000000..b0e592d904acdbb7ae96d87cc483e40f175ea94f
--- /dev/null
+++ b/theory/paper_spheric/data/SodShock_rhox.dat
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new file mode 100644
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+(0) s
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+3103 2669 mt 3074 2669 L
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+(5) s
+ 282 2238 mt 310 2238 L
+3103 2238 mt 3074 2238 L
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+(10) s
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+3103 1808 mt 3074 1808 L
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+ 282 3100 mt 282 344 L
+3103 3100 mt 3103 344 L
+gs 282 344 2822 2757 rc
+DO
+16.8 w
+2821 -2669 282 3013 2 MP stroke
+gr
+
+16.8 w
+DO
+1452 249 mt
+(Speedup) s
+SO
+6 w
+1 sg
+0 2757 5642 0 0 -2757 3573 3100 4 MP
+PP
+-5642 0 0 2757 5642 0 0 -2757 3573 3100 5 MP stroke
+4 w
+DO
+SO
+6 w
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+(0.8) s
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+3573 3100 mt 3573 343 L
+9215 3100 mt 9215 343 L
+gs 3573 344 5643 2757 rc
+DO
+16.8 w
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+gr
+
+16.8 w
+DO
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+(Parallel Efficiency) s
+SO
+6 w
+
+end %%Color Dict
+
+eplot
+%%EndObject
+
+epage
+end
+
+showpage
+
+%%Trailer
+%%EOF
diff --git a/theory/paper_spheric/data/count.sh b/theory/paper_spheric/data/count.sh
new file mode 100755
index 0000000000000000000000000000000000000000..79db4fe3c64dd6d713f45ed3db8042a5f90dcf8a
--- /dev/null
+++ b/theory/paper_spheric/data/count.sh
@@ -0,0 +1,17 @@
+#!/bin/bash
+
+# Collect the average data in each file
+for d in Cosmo Sedov SodShock
+do
+ rm -f ${d}.tot
+ for f in `ls $d/perf*.dat | sort -V`
+ do
+ awk -v c=${f//[^0-9]/} '!/^#/{for(k=1;k<=NF;k++)sum[k]+=$k;}END{printf("%i ",c);for(k=1;k<=NF;k++)printf("%.3e ",sum[k]);printf("\n");}' $f >> ${d}.tot
+ done
+ rm -f ${d}_fixed.tot
+ for f in `ls ${d}_fixed/perf*.dat | sort -V`
+ do
+ awk -v c=${f//[^0-9]/} '!/^#/{tot+=1;for(k=1;k<=NF;k++)sum[k]+=$k;}END{printf("%i ",c);for(k=1;k<=NF;k++)printf("%.3e ",sum[k]/tot);printf("\n");}' $f >> ${d}_fixed.tot
+ done
+done
+
diff --git a/theory/paper_spheric/figures b/theory/paper_spheric/figures
new file mode 120000
index 0000000000000000000000000000000000000000..17a43913aa8518c13602f475cada503854d7aa90
--- /dev/null
+++ b/theory/paper_spheric/figures
@@ -0,0 +1 @@
+../paper_algs/figures/
\ No newline at end of file
diff --git a/theory/paper_spheric/mkplots.m b/theory/paper_spheric/mkplots.m
new file mode 100644
index 0000000000000000000000000000000000000000..78147d4a4a8e272b6d628e5ca282a133a4f7d0a7
--- /dev/null
+++ b/theory/paper_spheric/mkplots.m
@@ -0,0 +1,195 @@
+
+%% Make a combined plot for the CosmoVolume
+gadget = importdata( 'data/CosmoVolume_fixed_Gadget2.totals' );
+gadget = [ gadget(:,1) , gadget(:,6)/2.4e9*1000 ];
+swift = importdata( 'data/CosmoVolume_fixed.totals');
+swift = [ (1:size(swift,1))' , swift(:,end-1) ];
+ncores = max( [ swift(:,1) ] );
+nparts = 1841127;
+
+clf
+subplot('position',[ 0.03 , 0.1 , 0.3 , 0.8 ]);
+plot( swift(:,1) , swift(1,2) ./ swift(:,2) , '-k' , 'LineWidth' , 2 ); hold on;
+plot( gadget(:,1) , gadget(1,2) ./ gadget(:,2) , '-.k' , 'LineWidth' , 2 );
+% plot( gadget(:,1) , swift(1,2) ./ gadget(:,2) , '--k' , 'LineWidth' , 1 ); hold on;
+text(ncores-0.2,gadget(1,2)/gadget(end,2)+0.2,sprintf('%.0f',min(gadget(:,2))),'HorizontalAlignment','Right','VerticalAlignment','Bottom','FontSize',12);
+text(ncores-0.2,swift(1,2)/swift(end,2)+0.2,sprintf('%.0f',min(swift(:,2))),'HorizontalAlignment','Right','VerticalAlignment','Bottom','FontSize',12);
+xlabel('nr. cores');
+plot( [1,ncores] , [1,ncores] , ':k' , 'LineWidth' , 1.4 );
+hold off;
+title('Speedup Cosmological Volume');
+axis([ 1 , ncores , 0 , ncores ]);
+
+subplot('position',[ 0.38 0.1 , 0.6 , 0.8 ]);
+plot( swift(:,1) , swift(1,2) ./ (swift(:,2).*swift(:,1)) , '-k' , 'LineWidth' , 2 ); hold on;
+plot( gadget(:,1) , gadget(1,2) ./ (gadget(:,2).*gadget(:,1)) , '-.k' , 'LineWidth' , 2 );
+xlabel('nr. cores');
+legend('SWIFT','Gadget2','Location','SouthWest');
+plot( [1,ncores] , [1,1] , ':k' , 'LineWidth' , 1.4 );
+hold off;
+title('Parallel Efficiency Cosmological Volume');
+axis([ 1 , ncores , 0 , 1.1 ]);
+
+% Print this plot
+set( gcf , 'PaperSize' , 2.3*[ 12 4.5 ] );
+set( gcf , 'PaperPosition' , 2.3*[ 0.25 0.25 12 4.4 ] );
+print -depsc2 figures/CosmoVolume_scaling.eps
+!epstopdf figures/CosmoVolume_scaling.eps
+
+
+%% Plot the timings for each part of the code.
+swift = importdata( 'data/CosmoVolume_fixed.totals');
+swift = [ (1:size(swift,1))' , swift(:,end-1).*(1:size(swift,1))' , sum(swift(:,[7,8]),2) , sum(swift(:,[5,6]),2) , swift(:,4) , swift(:,13) , sum(swift(:,[1:3]),2) ];
+ncores = max( swift(:,1) );
+
+clf
+subplot( 'position' , [ 0.1 , 0.1 , 0.8 , 0.8 ] );
+plot( swift(:,1) , swift(:,2:end)/1000 , 'LineWidth' , 1.4 );
+legend( 'total' , 'pair' , 'self' , 'sort' , 'task' , 'int' , 'Location' , 'NorthWest' );
+title('Total times');
+axis([ 1 , ncores , 0 , 1.1*max(swift(:,2))/1000 ]);
+xlabel('nr. cores');
+ylabel('s');
+
+% Print this plot
+set( gcf , 'PaperSize' , 2.3*[ 8 4.5 ] );
+set( gcf , 'PaperPosition' , 2.3*[ 0.25 0.25 8 4.4 ] );
+print -depsc2 figures/CosmoVolume_times.eps
+!epstopdf figures/CosmoVolume_times.eps
+
+
+
+%% Make a combined plot for the SodShock
+swift = importdata( 'data/SodShock_fixed.totals');
+swift = [ (1:size(swift,1))' , swift(:,end-1) ];
+ncores = max( [ swift(:,1) ] );
+nparts = 1841127;
+
+clf
+subplot('position',[ 0.03 , 0.1 , 0.3 , 0.8 ]);
+plot( swift(:,1) , swift(1,2) ./ swift(:,2) , '-k' , 'LineWidth' , 2 ); hold on;
+text(ncores-0.2,swift(1,2)/swift(end,2)+0.2,sprintf('%.0f',min(swift(:,2))),'HorizontalAlignment','Right','VerticalAlignment','Bottom','FontSize',12);
+xlabel('nr. cores');
+plot( [1,ncores] , [1,ncores] , ':k' , 'LineWidth' , 1.4 );
+hold off;
+title('Speedup Sod-Shock');
+axis([ 1 , ncores , 0 , ncores ]);
+
+subplot('position',[ 0.38 0.1 , 0.6 , 0.8 ]);
+plot( swift(:,1) , swift(1,2) ./ (swift(:,2).*swift(:,1)) , '-k' , 'LineWidth' , 2 ); hold on;
+xlabel('nr. cores');
+% legend('Sod-Shock','Location','SouthWest');
+plot( [1,ncores] , [1,1] , ':k' , 'LineWidth' , 1.4 );
+hold off;
+title('Parallel Efficiency Sod-Shock');
+axis([ 1 , ncores , 0 , 1.1 ]);
+
+% Print this plot
+set( gcf , 'PaperSize' , 2.3*[ 12 4.5 ] );
+set( gcf , 'PaperPosition' , 2.3*[ 0.25 0.25 12 4.4 ] );
+print -depsc2 figures/SodShock_scaling.eps
+!epstopdf figures/SodShock_scaling.eps
+
+
+
+%% Make a combined plot for the SedovBlast
+swift = importdata( 'data/SedovBlast_fixed.totals');
+swift = [ (1:size(swift,1))' , swift(:,end-1) ];
+ncores = max( [ swift(:,1) ] );
+nparts = 1841127;
+
+clf
+subplot('position',[ 0.03 , 0.1 , 0.3 , 0.8 ]);
+plot( swift(:,1) , swift(1,2) ./ swift(:,2) , '-k' , 'LineWidth' , 2 ); hold on;
+text(ncores-0.2,swift(1,2)/swift(end,2)+0.2,sprintf('%.0f',min(swift(:,2))),'HorizontalAlignment','Right','VerticalAlignment','Bottom','FontSize',12);
+xlabel('nr. cores');
+plot( [1,ncores] , [1,ncores] , ':k' , 'LineWidth' , 1.4 );
+hold off;
+title('Speedup Sedov Blast');
+axis([ 1 , ncores , 0 , ncores ]);
+
+subplot('position',[ 0.38 0.1 , 0.6 , 0.8 ]);
+plot( swift(:,1) , swift(1,2) ./ (swift(:,2).*swift(:,1)) , '-k' , 'LineWidth' , 2 ); hold on;
+xlabel('nr. cores');
+% legend('Sedov Blast','Location','SouthWest');
+plot( [1,ncores] , [1,1] , ':k' , 'LineWidth' , 1.4 );
+hold off;
+title('Parallel Efficiency Sedov Blast');
+axis([ 1 , ncores , 0 , 1.1 ]);
+
+% Print this plot
+set( gcf , 'PaperSize' , 2.3*[ 12 4.5 ] );
+set( gcf , 'PaperPosition' , 2.3*[ 0.25 0.25 12 4.4 ] );
+print -depsc2 figures/SedovBlast_scaling.eps
+!epstopdf figures/SedovBlast_scaling.eps
+
+
+
+
+%% Plot the Sod-shock density, pressure and velocity profiles
+swift = importdata( 'data/SodShock_rhox.dat');
+exact_rho = importdata( 'data/SodShock_exact_rho.dat' );
+exact_P = importdata( 'data/SodShock_exact_P.dat' );
+exact_v = importdata( 'data/SodShock_exact_v.dat' );
+
+clf
+subplot('position',[ 0.05 , 0.1 , 0.28 , 0.8 ]);
+plot( 0.5*(swift(:,2)+swift(:,3)) - 0.5 , swift(:,9) , '-k' , 'LineWidth' , 2 ); hold on;
+plot( exact_rho(:,1) , exact_rho(:,2) , ':k' , 'LineWidth' , 2 ); hold on;
+xlabel('x'); ylabel('\rho');
+hold off;
+title('Sod-shock Density');
+axis([ -0.25 0.25 0.2 1 ]);
+
+subplot('position',[ 1/3+0.05 , 0.1 , 0.28 , 0.8 ]);
+plot( 0.5*(swift(:,2)+swift(:,3)) - 0.5 , swift(:,10) , '-k' , 'LineWidth' , 2 ); hold on;
+plot( exact_P(:,1) , exact_P(:,2) , ':k' , 'LineWidth' , 2 ); hold on;
+xlabel('x'); ylabel('P');
+hold off;
+title('Sod-shock Pressure');
+axis([ -0.25 0.25 0.1 1 ]);
+
+subplot('position',[ 2/3+0.05 , 0.1 , 0.28 , 0.8 ]);
+plot( 0.5*(swift(:,2)+swift(:,3)) - 0.5 , swift(:,5) , '-k' , 'LineWidth' , 2 ); hold on;
+plot( exact_v(:,1) , exact_v(:,2) , ':k' , 'LineWidth' , 2 ); hold on;
+xlabel('x'); ylabel('v_x');
+hold off;
+title('Sod-shock Velocity');
+axis([ -0.25 0.25 -0.8 0.8 ]);
+
+% Print this plot
+set( gcf , 'PaperSize' , 2.3*[ 12 4.2 ] );
+set( gcf , 'PaperPosition' , 2.3*[ 0.25 0.25 12 4.2 ] );
+print -depsc2 figures/SodShock_profile.eps
+!epstopdf figures/SodShock_profile.eps
+
+
+
+%% Plot the Sedov blast density profile
+swift_10 = importdata( 'data/SedovBlast_rdf_10.dat');
+swift_15 = importdata( 'data/SedovBlast_rdf_15.dat');
+swift_20 = importdata( 'data/SedovBlast_rdf_20.dat');
+exact_10 = importdata( 'data/SedovBlast_exact_10.dat');
+exact_15 = importdata( 'data/SedovBlast_exact_15.dat');
+exact_20 = importdata( 'data/SedovBlast_exact_20.dat');
+
+clf
+subplot('position',[ 0.03 , 0.1 , 0.8 , 0.8 ]);
+plot( exact_15(:,1) , exact_15(:,2) , ':k' , 'LineWidth' , 2 ); hold on;
+plot( 0.5*(swift_15(:,2)+swift_15(:,3)) , swift_15(:,9) , 'ok' , 'LineWidth' , 2 );
+plot( exact_10(:,1) , exact_10(:,2) , ':k' , 'LineWidth' , 2 ); hold on;
+plot( 0.5*(swift_10(:,2)+swift_10(:,3)) , swift_10(:,9) , 'xk' , 'LineWidth' , 2 );
+plot( exact_20(:,1) , exact_20(:,2) , ':k' , 'LineWidth' , 2 ); hold on;
+plot( 0.5*(swift_20(:,2)+swift_20(:,3)) , swift_20(:,9) , '^k' , 'LineWidth' , 2 );
+xlabel('r');
+ylabel('\rho');
+hold off;
+title('Density Sedov blast');
+axis([ 0 , 2.5 , 0 , max(exact_20(:,2)) ]);
+
+% Print this plot
+set( gcf , 'PaperSize' , 2.3*[ 8 4.5 ] );
+set( gcf , 'PaperPosition' , 2.3*[ 0.25 0.25 8 4.4 ] );
+print -depsc2 figures/SedovBlast_density.eps
+!epstopdf figures/SedovBlast_density.eps
+
diff --git a/theory/paper_spheric/paper.tex b/theory/paper_spheric/paper.tex
new file mode 100644
index 0000000000000000000000000000000000000000..060eaf1320a708d868b85234574b63a99ff425bf
--- /dev/null
+++ b/theory/paper_spheric/paper.tex
@@ -0,0 +1,937 @@
+%% 7th_SPHERIC_paper_template.tex
+%% V1.0
+%% updated 08/03/2013
+%% courtesy of nicolas.grenier@ec-nantes
+%% courtesy of pierre.maruzewski@epfl.ch
+%%
+%% NOTE: This text file uses MS Windows line feed conventions. When (human)
+%% reading this file on other platforms, you may have to use a text
+%% editor that can handle lines terminated by the MS Windows line feed
+%% characters (0x0D 0x0A).
+
+% Note that the a4paper option is mainly intended so that authors in
+% countries using A4 can easily print to A4 and see how their papers will
+% look in print. Authors are encouraged to use U.S.
+%
+% Also note that the "draftcls" or "draftclsnofoot", not "draft", option
+% should be used if it is desired that the figures are to be displayed in
+% draft mode.
+%
+% This paper can be formatted using the peerreviewca
+% (instead of conference) mode.
+\documentclass[a4paper,conference]{IEEEtran}
+
+% Dimensions of margins (DO NOT MODIFY THEM)
+\setlength{\textheight} {23.4cm}%
+\setlength{\topmargin} {-0.8cm}%
+\setlength{\headheight} {0.6cm}%
+\setlength{\headsep} {0.9cm}%
+
+% some very useful LaTeX packages include:
+
+\usepackage{cite} % Written by Donald Arseneau
+
+\usepackage{graphicx} % Written by David Carlisle and Sebastian Rahtz
+
+\usepackage{subfigure} % Written by Steven Douglas Cochran
+
+
+% *** Do not adjust lengths that control margins, column widths, etc. ***
+% *** Do not use packages that alter fonts (such as pslatex). ***
+% There should be no need to do such things with IEEEtran.cls V1.6 and later.
+
+
+% correct bad hyphenation here
+\hyphenation{op-tical net-works semi-conduc-tor IEEEtran}
+
+
+\usepackage{fancyheadings}
+\pagestyle{fancy}
+
+\lhead{$8^{th}$ international SPHERIC workshop}
+\rhead{Trondheim, Norway, June 4-6, 2013}
+\cfoot{} % to avoid page numbering
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Set options for the listings package
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{listings}
+\lstset{%
+ language=C,
+ basicstyle=\scriptsize\tt,
+ numbers=left,
+ numberstyle=\tiny
+ }
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Some useful commands
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\usepackage{amsfonts}
+\usepackage{amsmath}
+\usepackage{epsfig}
+\usepackage{xspace}
+\usepackage{flushend}
+\newcommand{\oh}[1]
+ {\mbox{$ {\mathcal O}( #1 ) $}}
+\newcommand{\eqn}[1]
+ {(\ref{eqn:#1})}
+\newcommand{\fig}[1]
+ {Figure~\ref{fig:#1}}
+\newcommand{\tabl}[1]
+ {Table~\ref{tab:#1}}
+\newcommand{\tab}[1]
+ {Table~\ref{tab:#1}}
+\newcommand{\figs}[1]
+ {Figures~\ref{fig:#1}}
+\newcommand{\cuda}
+ {{\sc cuda}\xspace}
+\newcommand{\namd}
+ {{\sc namd}\xspace}
+\newcommand{\acemd}
+ {{\sc acemd}\xspace}
+\newcommand{\fenzi}
+ {{\sc fenzi}\xspace}
+\newcommand{\mdcore}
+ {{\tt mdcore}\xspace}
+\newcommand{\swift}
+ {{\sc swift}\xspace}
+\newcommand{\bsf}[1]
+ {\textbf{\textsf{#1}}}
+
+
+\begin{document}
+
+% paper title
+\title{{\sc swift}: Fast algorithms for multi-resolution SPH on
+multi-core architectures}
+
+
+% author names and affiliations
+\author{
+\IEEEauthorblockN{Pedro Gonnet\IEEEauthorrefmark{1}, Matthieu Schaller\IEEEauthorrefmark{2}, Tom Theuns\IEEEauthorrefmark{2}\IEEEauthorrefmark{3}, Aidan B. G. Chalk\IEEEauthorrefmark{1}}
+\IEEEauthorblockA{\IEEEauthorrefmark{1}School of Engineering and Computing Sciences,\\
+Durham University,\\
+DH1 3LE Durham, UK\\
+\{pedro.gonnet,aidan.chalk\}@durham.ac.uk \\
+~\\}
+\IEEEauthorblockA{\IEEEauthorrefmark{2}Institute for Computational Cosmology,\\
+Durham University, \\
+DH1 3LE Durham, UK\\
+\{matthieu.schaller,tom.theuns\}@durham.ac.uk \\
+~\\}
+\IEEEauthorblockA{\IEEEauthorrefmark{3}Department of Physics, \\
+ University of Antwerp, Groenenborgerlaan 171, \\
+B-2020 Antwerp, Belgium}
+}
+
+
+% use only for invited papers
+%\specialpapernotice{(Invited Paper)}
+
+% make the title area
+\maketitle
+
+\begin{abstract}
+This paper describes a novel approach to neighbour-finding in
+Smoothed Particle Hydrodynamics (SPH) simulations with large dynamic
+range in smoothing length.
+This approach is based on hierarchical cell decompositions, sorted interactions,
+and a task-based formulation.
+It is shown to be faster than traditional tree-based
+codes, and to scale better than domain decomposition-based approaches on
+shared-memory parallel architectures such as multi-cores.
+\end{abstract}
+
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Introduction
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Introduction}
+
+Since the past few years, due to the physical limitations
+on the speed of individual processor cores, instead of
+getting {\em faster}, computers are getting {\em more parallel}.
+This increase in parallelism comes mainly in the form of
+{\em multi-core} computers in which the number cores can be
+expected to continue growing exponentially, e.g.~following
+Moore's law, much in the same way processor speeds were up until
+a few years ago.
+
+The predominant paradigm for parallel
+computing is currently distributed-memory parallelism using MPI
+(Message Passing Interface) \cite{ref:Snir1998},
+in which large simulations are generally
+parallelized by means of data decompositions, i.e.~by assigning
+each node or core a portion of the data on which to work.
+The cores execute the same code
+in parallel, each on its own part on the data, intermittently exchanging data.
+The amount of {\em computation} local to the node is then proportional
+to the amount of data it contains, e.g.~its volume, while
+the amount of {\em communication} is proportional to the
+amount of computation spanning two or more nodes, e.g.~its
+surface.
+If the number of nodes increases, or
+smaller problems are considered, the surface-to-volume ratio,
+i.e. the ratio of communication to computation,
+grows, and the time spent on communication will increasingly
+dominate the entire simulation, reducing scaling and parallel
+efficiency.
+
+Assuming the individual cores do not get any faster,
+this means that small simulations, for which the
+maximum degree of parallelism has already been reached, will never
+become any faster.
+In order to speed up small simulations, or to continue
+scaling for large simulations, new approaches on how
+computations are parallelized need to be considered.
+
+With the above in mind, we will, in the following,
+describe a reformulation of the
+underlying algorithms for Smoothed Particle Hydrodynamics (SPH)
+simulations which uses asynchronous
+task-based shared-memory parallelism to achieve better parallel
+scaling and efficiency on multi-core architectures.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Algorithms, i.e. decomposition, interactions, and parallelization
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Algorithms}
+
+The interactions in compressible gas dynamics using SPH
+are computed in two distinct stages that are
+evaluated separately:
+\begin{enumerate}
+ \item {\em Density} computation: For each particle $p_i$,
+ loop over all particles $p_j$ within $h_i$ of $p_i$ and
+ compute the particle densities.
+ \item {\em Force} computation: For each particle $p_i$,
+ loop over all particles $p_j$
+ within $\max\{h_i,h_j\}$ and compute the particle forces.
+\end{enumerate}
+The identification of these interacting particle pairs
+incurs the main computational cost,
+as will be shown in the following sections,
+and therefore also presents the main challenge in implementing efficient
+SPH simulations.
+
+
+\subsection{Tree-based approach}
+
+In its simplest formulation, all particles in an SPH simulation have
+a constant smoothing length $h$.
+In such a setup, finding the particles in range of any other particle
+is similar to Molecular Dynamics simulations, in which all particles
+interact within a constant cutoff radius, and approaches which are used
+in the latter, e.g. cell-linked lists
+\cite{ref:Allen1989} or Verlet lists \cite{ref:Verlet1967} can be used
+\cite{ref:Dominguez2011,ref:Viccione2008}.
+
+The neighbour-finding problem becomes more interesting, or difficult,
+in SPH simulations with variable smoothing lengths, i.e.~in which
+each particle has its own smoothing length $h_i$, with ranges spawning
+up to several orders of magnitude.
+In such cases, e.g. in Astrophysics simulations \cite{ref:Gingold1977},
+the above-mentioned approaches cease to work efficiently.
+Such codes therefore usually rely on {\em trees}
+for neighbour finding \cite{ref:Hernquist1989,ref:Springel2005,ref:Wadsley2004},
+i.e.~$k$-d trees \cite{ref:Bentley1975} or octrees \cite{ref:Meagher1982}
+are used to decompose the simulation space.
+The particle interactions are then computed by traversing the list of
+particles and searching for their neighbours in the tree.
+
+Using such trees, it is in principle trivial to parallelize
+the neighbour finding and the actual computation on shared-memory
+computers,
+e.g.~each thread walks the tree for a different particle,
+identifies its neighbours and computes the densities and/or
+the second derivatives of the physical quantities for that particle.
+
+Despite their simple and elegant formulation, the tree-based
+approach to neighbour-finding has three main problems:
+\begin{itemize}
+ \item Computational efficiency: Finding all neighbours
+ of a particle in the tree is, on average, in \oh{\log N},
+ and worst-case behavior in \oh{N^{2/3}} \cite{ref:Lee1977},
+ i.e.~the cost per particle grows with the
+ total number of particles $N$.
+ \item Cache efficiency: When searching for the neighbours of a
+ given particle, the data of all potential neighbours, which may
+ not be contiguous in memory, is traversed separately.
+ \item Symmetry: The parallel tree search can not exploit symmetry,
+ i.e.~a pair $p_i$ and $p_j$ will always be found twice,
+ once when walking the tree for each particle.
+\end{itemize}
+
+These problems are all inherently linked to the use of
+spatial trees, i.e. more specifically their traversal,
+for neighbour-finding.
+As of here we will proceed differently, using a hierarchical cell
+decomposition, and computing the interactions by cell pairs,
+avoiding the above-mentioned problems.
+
+
+\subsection{Spatial decomposition}
+
+If $h_\mathsf{max} := \max_i h_i $ is the maximum smoothing
+length of any particle in the simulation, we start by splitting
+the simulation domain into rectangular cells of edge length
+larger or equal to $h_\mathsf{max}$.
+
+Given such an initial decomposition, we then generate
+a list of cell {\em self-interactions}, which contains all
+non-empty cells in the grid.
+This list of interactions is then extended by the cell
+{\em pair-interactions}, i.e. a list of all non-empty cell pairs
+sharing either a face, edge, or corner.
+For periodic domains, cell pair-interactions need also be
+specified for cell neighbouring each other across
+periodic boundaries.
+These self- and pair-interactions encode, conceptually at least,
+the evaluation of the interactions between all particles
+in the same cell, or all interactions between particle pairs
+spanning a pair of cells, respectively.
+
+In this first coarse decomposition, if a particle $p_j$
+is within range of a particle $p_i$, both will be either
+in the same cell, or in neighbouring cells for which a
+cell self-interaction or cell pair-interaction has been
+specified respectively (see \fig{InitialDecomp}).
+
+In the best of cases, i.e.~if each cell contains
+only particles with smoothing length equal to the cell
+edge length, if for any particle $p_i$ we inspect each
+particle $p_j$ in the same
+and neighbouring cells, only roughly 16\% of the $p_j$
+will actually be within range of $p_i$ \cite{ref:Gonnet2007}.
+and is even worse if the cells contain particles who's smoothing length
+is less than the cell edge length.
+We therefore recursively refine the cell decomposition,
+bisecting each cell along
+all spatial dimension if: (a) The cell contains more than
+some minimal number of particles, and (b) the smoothing
+length of a reasonable fraction of the particles within
+the cell is less than half the cell edge length.
+
+After the cells have been split, the cell self-interactions
+of each cell can be split up into the self-interaction
+of its sub-cells and the pair-interactions between
+them.
+Likewise, the cell pair-interactions between two cells
+that have been split can themselves be split up into
+the pair-interactions of the sub-cells spanning the
+original pair boundary if, and only if,
+all particles in both cells have a smoothing length of
+less than half the cell edge length (see \fig{SplitCell}).
+
+If the cells, self-interactions, and pair-interactions are split
+in such a way, if two particles are within range of each other,
+they will (a) either share a cell for which a cell self-interaction
+is defined, or (b) they will be located in two cells which share
+a cell pair-interaction.
+In order to identify all the particles within range of each other,
+it is therefore sufficient to traverse the list of
+self-interactions and pair-interactions and inspect the particle
+pairs therein.
+
+
+\begin{figure}
+ \centerline{\epsfig{file=figures/InitialDecomp.pdf,width=0.5\columnwidth}}
+
+ \caption{Initial spatial decomposition: The simulation volume is divided into cells of
+ edge length greater or equal to the largest smoothing length in the
+ system. All neighbours of any given particle (small red circle) within
+ that particle's smoothing length (large red circle) are guaranteed to lie
+ either within that particle's own cell (green) or the directly
+ adjacent cells (orange).}
+ \label{fig:InitialDecomp}
+\end{figure}
+
+
+\begin{figure}
+ \centerline{{\scriptsize\bsf A} \raisebox{-0.85\height}{\epsfig{file=figures/SplitCell.pdf,width=0.44\columnwidth}}}
+
+ \vspace{2ex}
+
+ \centerline{{\scriptsize\bsf B} \raisebox{-0.85\height}{\epsfig{file=figures/SplitPair.pdf,width=0.8\columnwidth}}}
+
+ \caption{({\bsf A}) Large cells can be split, and their
+ self-interaction replaced by the self- and pair-interactions
+ of their sub-cells.
+ ({\bsf B}) If all particles in a pair of interacting cells have a smoothing
+ length less or equal to half of the cell edge length, both cells can be
+ split, and their pair-interaction replaced by the pair-interactions
+ of the neighbouring sub-cells across the interface.}
+ \label{fig:SplitCell}
+ \label{fig:SplitPair}
+\end{figure}
+
+
+\subsection{Particle interactions}
+
+The interactions between all particle pairs within a given cell,
+i.e. the cell's self-interaction, can be computed by means of
+a double {\tt for}-loop over the cell's particle array.
+The algorithm, in C-like pseudo code, can be written as follows:
+
+\begin{center}\begin{minipage}{0.9\columnwidth}
+ \begin{lstlisting}
+for ( i = 0 ; i < count-1 ; i++ ) {
+ for ( j = i+1 ; j < count ; j++ ) {
+ rij = ||parts[i] - parts[j]||.
+ if ( rij < h[i] || rij < h[j] ) {
+ compute interaction.
+ }
+ }
+ }
+ \end{lstlisting}
+\end{minipage}\end{center}
+
+\noindent where {\tt count} is the number of particles in the
+cell and {\tt parts} and {\tt h} refers to an array of those
+particles and their smoothing lengths respectively.
+
+The interactions between all particles in a pair of cells
+can be computed similarly, e.g.:
+
+\begin{center}\begin{minipage}{0.9\columnwidth}
+ \begin{lstlisting}
+for ( i = 0 ; i < count_i ; i++ ) {
+ for ( j = 0 ; j < count_j ; j++ ) {
+ rij = ||parts_i[i] - parts_j[j]||.
+ if ( rij < h_i[i] || rij < h_j[j] ) {
+ compute interaction.
+ }
+ }
+ }
+ \end{lstlisting}
+\end{minipage}\end{center}
+
+\noindent where {\tt count\_i} and {\tt count\_j} refer to
+the number of particles in each cell and {\tt parts\_i} and
+{\tt parts\_j}, and {\tt h\_i} and {\tt h\_j}, refer to the
+particles of each cell and their smoothing lengths respectively.
+
+As described in \cite{ref:Gonnet2007}, using this
+naive double {\tt for}-loop, for uniform particle
+distributions only $33.5\%$, $16.2\%$,
+and $3.6\%$ of all particle
+pairs between cells sharing a common face, edge, or corner, respectively,
+will be within range of each other, leading
+to an excessive number of spurious pairwise distance evaluations (line~3).
+We will therefore use the sorted cell
+interactions described in \cite{ref:Gonnet2007} and \cite{ref:Gonnet2013},
+yet with some minor modifications, as
+the original algorithm is designed for systems in which the
+smoothing lengths of all particles are equal:
+We first sort the particles in both cells along the vector joining
+the centers of the two cells and then loop over the
+parts $p_i$ on the left and interact them with the sorted parts $p_j$
+on the right which are within $h_i$ {\em along the cell pair axis}.
+The same procedure is repeated for each particle $p_j$ on the
+right, interacting with each other particle $p_i$ on the
+left, which is within $h_j$, {\em but not within} $h_i$, along
+the cell pair axis (see \fig{SortedInteractions}).
+The resulting algorithm, in C-like pseudo-code, can be written as follows:
+
+\begin{center}\begin{minipage}{0.9\columnwidth}
+ \begin{lstlisting}
+for ( i = 0 ; i < count_i ; i++ ) {
+ for ( jj = 0 ; jj < count_j ; jj++ ) {
+ j = ind_j[jj];
+ if ( r_i[i] + h_i[i] < r_j[j] )
+ break;
+ rij = ||parts_i[i] - parts_j[j]||.
+ if ( rij < h_i[i] ) {
+ compute interaction.
+ }
+ }
+ }
+for ( j = 0 ; j < count_j ; j++ ) {
+ for ( ii = count_i-1 ; ii >= 0 ; ii-- ) {
+ i = ind_i[i];
+ if ( r_i[i] < r_j[j] - h_j[j] )
+ break;
+ rij = ||parts_i[i] - parts_j[j]||.
+ if ( rij < h_j[j] && rij > h_i[i] ) {
+ compute interaction.
+ }
+ }
+ }
+ \end{lstlisting}
+\end{minipage}\end{center}
+
+\noindent where {\tt r\_i} and {\tt r\_j} contains the position
+the particles of both cells along the cell axis, and
+{\tt ind\_i} and {\tt ind\_j} contain the particle indices
+sorted with respect to these positions respectively.
+
+This may seem like quite a bit of sorting, but
+if a cell has been split and its sub-cells have been sorted,
+the sorted indices of the higher-level cell can be constructed
+by shifting and merging the indices of its eight sub-cells.
+This reduces the \oh{n\log{n}} sorting to \oh{n} for merging.
+Furthermore, instead of sorting the particles every time we compute the
+pairwise interactions between two cells, we can pre-compute
+the sorted indices along the 13 possible cell-pair axes and
+store them for each cell, i.e. as is done in \cite{ref:Gonnet2013}.
+The sorted indices can also be re-used over several time steps
+until the hierarchical cell decomposition needs to be
+re-computed.
+
+
+\begin{figure}
+ \centerline{\epsfig{file=figures/SortedInteractions.pdf,width=0.8\columnwidth}}
+
+ \caption{Sorted cell pair-interactions. ({\bsf A}) Starting from a pair of
+ neighbouring cells, ({\bsf B}) the particles from both cells
+ are projected onto the axis joining the centers of the two cells.
+ The particles on the left (blue) and right (orange) are
+ then sorted in descending and ascending order respectively.
+ Each particle on the left is then only interacted with
+ the particles on the right within the cutoff radius along the cell axis.
+ }
+ \label{fig:SortedInteractions}
+\end{figure}
+
+
+\subsection{Parallel implementation}
+
+OpenMP \cite{ref:Dagum1998} is arguably the most well-known
+paradigm for shared-memory, or multi-threaded parallelism.
+In OpenMP, compiler annotations are used to describe if and when
+specific loops or portions of the code can be executed
+in parallel.
+When such a parallel section, e.g.~a parallel loop, is
+encountered, the sections of the loop are split statically
+or dynamically over the available threads, each executing
+on a single core, and merge at the end of the parallel section.
+Unfortunately, this can lead to a lot of inefficient
+branch-and-bound type operations, which generally lead to
+low performance and bad scaling on even moderate numbers
+of cores (see \fig{OMPScaling}).
+
+Furthermore, this form of shared-memory parallelism provides
+no implicit mechanism to avoid or handle concurrency problems,
+e.g.~two threads attempting to modify the same data at the same time,
+or data dependencies between them.
+These must be implemented explicitly using either redundancy, barriers,
+critical sections, or atomic memory operations, which can further degrade
+parallel performance.
+
+\begin{figure}
+ \centerline{\epsfig{file=figures/OMPScaling.pdf,width=0.8\columnwidth}}
+
+ \caption{Branch-and-bound parallelism as is commonly used in OpenMP.
+ The horizontal arrows indicate the program flow over time, and
+ branching arrows indicate a parallel section. The dotted vertical
+ bars are the synchronization points at the end of each such section.
+ Parallel efficiency is lost due to two factors: The grey shaded areas
+ along the main horizontal area indicate parts of the program that
+ do not execute in parallel and restrict the maximum parallel
+ efficiency, e.g.~as described by Amdahl's law, and the red
+ areas indicate the difference between the fastest and slowest
+ threads in each parallel block, i.e.~the time lost to
+ thread synchronization.
+ }
+ \label{fig:OMPScaling}
+\end{figure}
+
+
+In order to better exploit shared-memory parallelism,
+we have to change the underlying paradigm, i.e. instead
+of annotating an essentially serial computation with parallel
+bits, it is preferable to describe the whole computation in a way that
+is inherently parallelizable.
+One such approach is {\em task-based parallelism}, in which the
+computation is divided into a number of inter-dependent
+computational tasks, which are then dynamically allocated
+to a number of processors.
+In order to ensure that the tasks are executed in the right
+order, e.g.~that data needed by one task is only used once it
+has been produced by another task, and that no two tasks
+update the same data at the same time, constraints are specified
+and enforced by the task scheduler.
+
+Several middle-wares providing such task-based
+parallelism exist, e.g.~Cilk \cite{ref:Blumofe1995}, QUARK \cite{ref:QUARK},
+StarPU \cite{ref:Augonnet2011}, SMP~Superscalar \cite{ref:SMPSuperscalar},
+OpenMP~3.0 \cite{ref:Duran2009}, and Intel's TBB \cite{ref:Reinders2007}.
+These implementations allow for the specification of individual tasks
+along with their {\em dependencies}, i.e.~hierarchical relationships
+specifying which tasks must have completed before another task
+can be executed.
+
+We will, however, differ from these approaches in that we introduce the
+concept of {\em conflicts} between tasks.
+Conflicts occur when two tasks operate on the same data,
+but the order in which these operations must occur is not defined.
+In previous task-based models, conflicts can be modeled as dependencies,
+yet this introduces an artificial ordering between the tasks
+and imposes unnecessary constraints on the task scheduler
+(e.g.~mutual and non-mutual interactions in \cite{ref:Ltaief2012}).
+These conflicts can be modelled using exclusive {\em locks} on shared
+resources, i.e.~a task will only be scheduled if it can obtain an
+exclusive lock on, and thus exclusive access to, potentially shared data.
+
+The particle interactions described in the previous subsection
+lead to three basic task types:
+%
+\begin{itemize}
+ \item Cell {\em sorting}, in which the particles in a given
+ cell are sorted with respect to their position along a
+ one-dimensional axis,
+ \item Cell {\em self-interaction}, in which all the particles
+ of a given cell are interacted with all the other particles
+ within the same cell,
+ \item Cell {\em pair-interaction}, in which the interactions for
+ all particle pairs spanning a pair of cells are computed.
+\end{itemize}
+%
+The self-interaction and pair-interaction tasks exist in
+two flavors, one for the density computation
+and one for the force computation.
+Each pair-interaction task requires the sorted indices of
+the particles in each cell provided by the sorting tasks.
+Since the tasks are restricted to operating on the data of a
+single cell, or pair of cells, two tasks conflict if they
+operate on overlapping sets of cells.
+Due to the hierarchical nature of the spatial decomposition,
+two tasks also conflict if the cells of one task are sub-cells
+of the other.
+
+Since the SPH computations have two phases, i.e.~density and force
+computation, we introduce a {\em ghost} task in between for each cell.
+This ghost task depends on all the density computations
+for a given cell, and, in turn, all force computations involving
+that cell depend on its ghost task.
+Using this mechanism, we can enforce that all density computations
+for a set of particles have completed before we use this
+density in the force computations.
+
+The dependencies and conflicts between tasks can thus be
+formulated as follows:
+
+\begin{itemize}
+
+ \item A cell sorting task on a cell with sub-cells depends
+ on the sorting tasks of all its sub-cells.
+
+ \item A cell pair-interaction depends on the cell sorting
+ tasks of both its cells.
+
+ \item Cell pair-interaction and cell self-interaction tasks
+ operating on overlapping sets of cells or sub-cells
+ conflict with each other.
+
+ \item The ghost task of each cell depends on all the density cell pair
+ interactions and self-interactions which involve the particles
+ in that cell.
+
+ \item The ghost task of each cell depends on the ghost tasks of
+ its sub-cells.
+
+ \item The force cell pair-interaction and self-interaction tasks
+ depend on the ghost tasks of the cells on which they operate.
+
+\end{itemize}
+
+\noindent These task dependencies are illustrated in \fig{Hierarchy2}.
+
+This task decomposition has significant advantages over the use of
+spatial trees:
+First of all, the cost of identifying all particles in range of
+a given particle does not depend on the total number of
+particles.
+Furthermore, the particle interactions in each task are computed
+symmetrically, i.e.~each particle pair is identified only
+once for each interaction type.
+The sorted particle indices can be re-used for both the
+density and force computation, and over several time-steps,
+thus reducing the computational cost even further.
+Finally, if the particles are stored grouped by cell, each task
+then only involves accessing and modifying a contained
+and contiguous regions of memory, thus potentially improving
+cache re-use \cite{ref:Fomin2011}.
+
+\begin{figure}
+ \centerline{\epsfig{file=figures/Hierarchy2.pdf,width=0.4\columnwidth}}
+
+ \caption{Task dependencies and conflicts:
+ Arrows indicate the dependencies
+ between different task types, i.e.~and arrow from task A to task
+ B indicates that A depends on B.
+ Horizontal dashed lines between tasks indicate conflicts,
+ i.e.~the two tasks can not be executed concurrently.
+ Each sort task (circles) depends
+ only on the sort tasks of its sub-cells.
+ The pair-interactions (rectangles) for the particle
+ density computation depend on the sort tasks of the respective cells,
+ whereas self-interaction tasks (squares) for the density computation
+ do not, as they do not require sorting.
+ Self- and pair-Interactions on overlapping cells (same colour)
+ conflict with each other.
+ The ghost task of each cell (triangles) depends on the self-
+ and pair-interaction density tasks.
+ The self- and pair-interaction tasks for the force computation,
+ finally, depend on the ghost tasks of the respective cells.
+ }
+ \label{fig:Hierarchy2}
+\end{figure}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Validation of the algorithms
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Validation}
+
+In the following, we describe how the algorithms shown in the previous section
+are implemented and tested against exiting codes on specific test problems.
+
+\subsection{Implementation details}
+
+The algorithms described above are all implemented as part
+of \swift (\underline{S}PH \underline{W}ith
+\underline{I}nter-dependent \underline{F}ine-grained
+\underline{T}asking),
+an Open-Source platform for hybrid shared/distributed-memory
+SPH simulations\footnote{See http://swiftsim.sourceforge.net/}.
+
+\swift is implemented in C, and can be compiled with the
+{\tt gcc} compiler.
+Although SIMD-vectorized code, using the {\tt gcc} vector types
+and SSE/AVX intrinsics, has been implemented, it was switched
+off in the following to allow for a fair comparison against
+unvectorized codes.
+
+The code for the task-based parallelism is implemented using
+standard {\tt pthread}s \cite{ref:pthreads}, and in some places,
+e.g.~in the time-stepper
+or task list creation, OpenMP is used.
+Each thread is assigned its own task queue, over which the tasks
+are distributed evenly in topological order of the dependencies.
+If a thread runs out of tasks, it can steal from another task's
+queue \cite{ref:Blumofe1999}.
+The threads are initialized once at the start of the simulation
+and synchronize via a barrier between time steps.
+
+
+\subsection{Simulation setup}
+
+In order to test their accuracy, efficiency, and scaling,
+the algorithms described in the previous section
+were tested in \swift using the following three
+simulation setups:
+%
+\begin{itemize}
+ \item {\em Sod-shock} \cite{ref:Sod1978}: A cubic periodic domain containing a
+ high-density region of
+ 800\,000 particles with $P_i=1$ and $\rho_i=1$ on one half,
+ and a low-density region of 200\,000
+ particles with $P_i=0.1795$ and $\rho_i=1/4$ on the other.
+ \item {\em Sedov blast}: A cubic grid of $101\times 101\times 101$
+ particles at rest with $P_i=10^{-5}$ and $\rho_i=1$, yet with the
+ energy of the central 33 particles set to
+ $u_i=100/33/m_i$.
+ \item {\em Cosmological volume}: Realistic distribution of matter
+ in a periodic volume of universe. The simulation consists
+ of 1\,841\,127 particles with
+ a mix of smoothing lengths spanning three orders of magnitude,
+ providing a test-case for neighbour finding and parallel
+ scaling in a real-world scenario.
+\end{itemize}
+%
+The first two cases are used both for benchmarking and for
+validating the correctness of the code.
+In all simulations, the constants $N_{ngb}=48$, $\gamma=5/3$,
+$C_{CFL}=0.3$, and $\alpha = 2.0$ (viscosity) were used.
+In \swift, cells were split if they contained more than 400
+particles and more than 87.5\% of the particles had a smoothing
+length less than half the cell edge length.
+
+In all three test cases, results were
+computed using a fixed time step and updating the forces on all
+particles in each time-step.
+
+The simulation results were compared with Gadget-2 \cite{ref:Springel2005}
+in terms of speed and parallel scaling.
+Gadget-2 was compiled with the Intel C Compiler version 2013.0.028
+using the options \texttt{\small -DUSE\_IRECV -O3 -ip -fp-model fast -ftz
+-no-prec-div -mcmodel=medium}.
+For the parallel runs, OpenMPI version 1.6.3 was used \cite{ref:Gabriel2004}.
+
+\swift v.~0.1.0 was compiled with the GNU C Compiler version 4.7.2
+using the options \texttt{\small -O3 -ffast-math -fstrict-aliasing
+-ftree-vectorize -funroll-loops -mmmx -msse -msse2 -msse3 -mssse3
+-msse4.1 -msse4.2 -mavx -fopenmp -march=native -pthread}.
+Note that although the compiler switches for the SSE and AVX
+vector instruction sets were activated, explicit
+SIMD-vectorization was not used.
+
+All simulations were run on a $4\times$Intel Xeon E5-4640
+32-core machine running at 2.4\,GHz with
+CentOS release 6.2 Linux for x86\_64.
+
+
+\subsection{Results}
+
+The scaling and efficiency of \swift for the simulations described
+in the previous subsection are shown in \fig{Results}.
+In all three cases, \swift obtains a parallel efficiency of over
+75\% on 32 cores.
+For the Cosmological volume, it is also more than eight times
+faster than Gadget-2.
+
+\fig{Times} shows the total contribution of each task type, as well
+as the overheads for task acquisition and time integration
+for the Cosmological volume simulation.
+Note that efficiency is lost principally in the self and pair
+interactions, which is most probably due to the reduction of
+the effective memory bandwidth with the increasing number of cores
+used.
+The overheads for task acquisition contribute less than 3.5\% to
+the total costs.
+
+In order to assess the correctness of the simulation, average radial
+density, pressure, and velocity profiles were computed for
+the Sedov blast (see \fig{Sedov_density}) and Sod-shock
+(see \fig{SodShock_profiles}) simulations and compared with
+the analytical results given in \cite{ref:Sedov1959} and
+\cite{ref:Sod1978} respectively.
+These resutls are comparable to the output produced by
+Gadget-2 for the same initial conditions, and sharper
+results could be achieved at higher resolutions and/or
+different parameterizations.
+
+
+\begin{figure*}
+ \centerline{\epsfig{file=figures/SodShock_scaling.pdf,width=0.8\textwidth}}
+ \centerline{\epsfig{file=figures/SedovBlast_scaling.pdf,width=0.8\textwidth}}
+ \centerline{\epsfig{file=figures/CosmoVolume_scaling.pdf,width=0.8\textwidth}}
+
+ \caption{Parallel scaling and efficiency for the Sod-shock, Sedov blast
+ Cosmological volume simulations.
+ The numbers in the scaling plots indicate the average milliseconds
+ per time step when running on all 32 cores.}
+ \label{fig:Results}
+\end{figure*}
+
+
+\begin{figure}
+ \centerline{\epsfig{file=figures/CosmoVolume_times.pdf,width=0.9\columnwidth}}
+
+ \caption{Total time for each task type, plus overheads for task
+ management and time integration.
+ The values, in seconds, are summed over all processors.}
+ \label{fig:Times}
+\end{figure}
+
+
+\begin{figure}
+ \centerline{\epsfig{file=figures/SedovBlast_density.pdf,width=0.9\columnwidth}}
+
+ \caption{Average radial density profiles and analytical solutions
+ (dotted lines)
+ for the Sedov blast simulation at
+ times $t=\{0.075, 0.15, 0.275\}$.}
+ \label{fig:Sedov_density}
+\end{figure}
+
+\begin{figure*}
+ \centerline{\epsfig{file=figures/SodShock_profile.pdf,width=0.8\textwidth}}
+
+ \caption{Average density, pressure, and velocity in $x$-direction,
+ along with the corresponding analytical solutions (dotted lines)
+ for the Sod-shock simulations at $t=0.12$.}
+ \label{fig:SodShock_profiles}
+\end{figure*}
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Conclusions
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Conclusions}
+
+The results show that \swift is significantly more efficient than
+current state-of-the-art tree-based methods.
+The improved performance is the result of several features:
+%
+\begin{itemize}
+ \item Cell-based neighbour searching which requires only
+ constant time per particle,
+ \item Sorted particle interactions which significantly
+ reduce the number of spurious pairwise distance
+ computations in the neighbour search,
+ \item Use of symmetry in computing the particle densities
+ and forces, thus requiring each particle pair to be
+ found only once,
+ \item Better cache efficiency of the cell-based interactions.
+\end{itemize}
+%
+Since no hardware-specific tricks or explicit
+SIMD-vectorization were used, although the latter is available
+in the code, they can not be made responsible for the
+improved efficiency.
+
+In addition to the increased speed, \swift
+also shows good scaling, in excess
+of 75\% parallel efficiency on 32 cores, for all three simulations.
+This is due to the fine-grained load balancing and low number
+of synchronization points as a result of the task-based parallelism scheme.
+
+We are currently working to extend \swift for hybrid
+shared/distributed-memory parallelism, and to GPUs
+\cite{ref:Chalk2013}.
+Task-based parallelism will play an important role for
+both these developments: In hybrid schemes, asynchronous communication
+between distributed-memory nodes can be modelled as simply another
+set of tasks with their corresponding dependencies, thus avoiding
+excessive synchronization points or latencies.
+As has already been shown for molecular dynamics simulations,
+the task-based parallel scheme can also easily be extended
+for computations on GPUs, using the same set of tasks and
+dependencies.
+Work is also ongoing in adding different and better implementations
+of particle-based hydrodynamics, i.e.~for cosmological
+simulations \cite{ref:Schaller2013}.
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Acknowledgments
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{Acknowledgments}
+
+The authors would like to thank Lydia Heck of the Institute for Computational
+Cosmology, Durham University, for having initiated this interdisciplinary
+collaboration, and for providing exclusive access to the hardware on which
+all benchmarks were run.
+
+
+% trigger a \newpage just before the given reference
+% number - used to balance the columns on the last page
+% adjust value as needed - may need to be readjusted if
+% the document is modified later
+%\IEEEtriggeratref{8}
+% The "triggered" command can be changed if desired:
+%\IEEEtriggercmd{\enlargethispage{-5in}}
+
+% references section
+% NOTE: BibTeX documentation can be easily obtained at:
+% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
+
+% can use a bibliography generated by BibTeX as a .bbl file
+% standard IEEE bibliography style from:
+% http://www.ctan.org/tex-archive/macros/latex/contrib/supported/IEEEtran/bibtex
+\bibliographystyle{IEEEtran.bst}
+% argument is your BibTeX string definitions and bibliography database(s)
+\bibliography{sph.bib}
+%
+
+
+% that's all folks
+\end{document}
+
+
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