diff options
| -rw-r--r-- | iopams.sty | 1 | ||||
| -rw-r--r-- | iopart.cls | 1107 | ||||
| -rw-r--r-- | iopart10.clo | 1 | ||||
| -rw-r--r-- | iopart12.clo | 1 | ||||
| -rw-r--r-- | setstack.sty | 1 | ||||
| -rw-r--r-- | stokes.tex | 145 |
6 files changed, 1183 insertions, 73 deletions
diff --git a/iopams.sty b/iopams.sty new file mode 100644 index 0000000..baf2b72 --- /dev/null +++ b/iopams.sty @@ -0,0 +1 @@ +%%
%% This is file `iopams.sty'
%% File to include AMS fonts and extra definitions for bold greek
%% characters for use with iopart.cls
%%
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{iopams}[1997/02/13 v1.0]
\RequirePackage{amsgen}[1995/01/01]
\RequirePackage{amsfonts}[1995/01/01]
\RequirePackage{amssymb}[1995/01/01]
\RequirePackage{amsbsy}[1995/01/01]
%
\iopamstrue % \newif\ifiopams in iopart.cls & iopbk2e.cls
% % allows optional text to be in author guidelines
%
% Bold lower case Greek letters
%
\newcommand{\balpha}{\boldsymbol{\alpha}}
\newcommand{\bbeta}{\boldsymbol{\beta}}
\newcommand{\bgamma}{\boldsymbol{\gamma}}
\newcommand{\bdelta}{\boldsymbol{\delta}}
\newcommand{\bepsilon}{\boldsymbol{\epsilon}}
\newcommand{\bzeta}{\boldsymbol{\zeta}}
\newcommand{\bfeta}{\boldsymbol{\eta}}
\newcommand{\btheta}{\boldsymbol{\theta}}
\newcommand{\biota}{\boldsymbol{\iota}}
\newcommand{\bkappa}{\boldsymbol{\kappa}}
\newcommand{\blambda}{\boldsymbol{\lambda}}
\newcommand{\bmu}{\boldsymbol{\mu}}
\newcommand{\bnu}{\boldsymbol{\nu}}
\newcommand{\bxi}{\boldsymbol{\xi}}
\newcommand{\bpi}{\boldsymbol{\pi}}
\newcommand{\brho}{\boldsymbol{\rho}}
\newcommand{\bsigma}{\boldsymbol{\sigma}}
\newcommand{\btau}{\boldsymbol{\tau}}
\newcommand{\bupsilon}{\boldsymbol{\upsilon}}
\newcommand{\bphi}{\boldsymbol{\phi}}
\newcommand{\bchi}{\boldsymbol{\chi}}
\newcommand{\bpsi}{\boldsymbol{\psi}}
\newcommand{\bomega}{\boldsymbol{\omega}}
\newcommand{\bvarepsilon}{\boldsymbol{\varepsilon}}
\newcommand{\bvartheta}{\boldsymbol{\vartheta}}
\newcommand{\bvaromega}{\boldsymbol{\varomega}}
\newcommand{\bvarrho}{\boldsymbol{\varrho}}
\newcommand{\bvarzeta}{\boldsymbol{\varsigma}} %NB really sigma
\newcommand{\bvarsigma}{\boldsymbol{\varsigma}}
\newcommand{\bvarphi}{\boldsymbol{\varphi}}
%
% Bold upright capital Greek letters
%
\newcommand{\bGamma}{\boldsymbol{\Gamma}}
\newcommand{\bDelta}{\boldsymbol{\Delta}}
\newcommand{\bTheta}{\boldsymbol{\Theta}}
\newcommand{\bLambda}{\boldsymbol{\Lambda}}
\newcommand{\bXi}{\boldsymbol{\Xi}}
\newcommand{\bPi}{\boldsymbol{\Pi}}
\newcommand{\bSigma}{\boldsymbol{\Sigma}}
\newcommand{\bUpsilon}{\boldsymbol{\Upsilon}}
\newcommand{\bPhi}{\boldsymbol{\Phi}}
\newcommand{\bPsi}{\boldsymbol{\Psi}}
\newcommand{\bOmega}{\boldsymbol{\Omega}}
%
% Bold versions of miscellaneous symbols
%
\newcommand{\bpartial}{\boldsymbol{\partial}}
\newcommand{\bell}{\boldsymbol{\ell}}
\newcommand{\bimath}{\boldsymbol{\imath}}
\newcommand{\bjmath}{\boldsymbol{\jmath}}
\newcommand{\binfty}{\boldsymbol{\infty}}
\newcommand{\bnabla}{\boldsymbol{\nabla}}
\newcommand{\bdot}{\boldsymbol{\cdot}}
%
% Symbols for caption
%
\renewcommand{\opensquare}{\mbox{$\square$}}
\renewcommand{\opentriangle}{\mbox{$\vartriangle$}}
\renewcommand{\opentriangledown}{\mbox{$\triangledown$}}
\renewcommand{\opendiamond}{\mbox{$\lozenge$}}
\renewcommand{\fullsquare}{\mbox{$\blacksquare$}}
\newcommand{\fulldiamond}{\mbox{$\blacklozenge$}}
\newcommand{\fullstar}{\mbox{$\bigstar$}}
\newcommand{\fulltriangle}{\mbox{$\blacktriangle$}}
\newcommand{\fulltriangledown}{\mbox{$\blacktriangledown$}}
\endinput
%%
%% End of file `iopams.sty'.
\ No newline at end of file diff --git a/iopart.cls b/iopart.cls new file mode 100644 index 0000000..99bfbe3 --- /dev/null +++ b/iopart.cls @@ -0,0 +1,1107 @@ +%% +%% This is file `iopart.cls' +%% +%% This file is distributed in the hope that it will be useful, +%% but WITHOUT ANY WARRANTY; without even the implied warranty of +%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. +%% +%% Licensed under the LPPL: http://www.latex-project.org/lppl.txt +%% Current Maintainer: IOP Publishing Ltd +%% +%% \CharacterTable +%% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z +%% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z +%% Digits \0\1\2\3\4\5\6\7\8\9 +%% Exclamation \! Double quote \" Hash (number) \# +%% Dollar \$ Percent \% Ampersand \& +%% Acute accent \' Left paren \( Right paren \) +%% Asterisk \* Plus \+ Comma \, +%% Minus \- Point \. Solidus \/ +%% Colon \: Semicolon \; Less than \< +%% Equals \= Greater than \> Question mark \? +%% Commercial at \@ Left bracket \[ Backslash \\ +%% Right bracket \] Circumflex \^ Underscore \_ +%% Grave accent \` Left brace \{ Vertical bar \| +%% Right brace \} Tilde \~} +% +\NeedsTeXFormat{LaTeX2e} +\ProvidesClass{iopart}[1996/06/10 v0.0 IOP Journals LaTeX article class] +\newcommand\@ptsize{0} +\newif\if@restonecol +\newif\if@titlepage +\newif\ifiopams +\@titlepagefalse +\DeclareOption{a4paper} + {\setlength\paperheight {297mm}% + \setlength\paperwidth {210mm}} +\DeclareOption{letterpaper} + {\setlength\paperheight {11in}% + \setlength\paperwidth {8.5in}} +\DeclareOption{landscape} + {\setlength\@tempdima {\paperheight}% + \setlength\paperheight {\paperwidth}% + \setlength\paperwidth {\@tempdima}} +\DeclareOption{10pt}{\renewcommand\@ptsize{0}} +\DeclareOption{11pt}{\renewcommand\@ptsize{2}} % No 11pt version +\DeclareOption{12pt}{\renewcommand\@ptsize{2}} +\DeclareOption{draft}{\setlength\overfullrule{5pt}} +\DeclareOption{final}{\setlength\overfullrule{0pt}} +\DeclareOption{titlepage}{\@titlepagetrue} +\DeclareOption{notitlepage}{\@titlepagefalse} +\ExecuteOptions{letterpaper,final} +\ProcessOptions +\DeclareMathAlphabet{\bi}{OML}{cmm}{b}{it} +\DeclareMathAlphabet{\bcal}{OMS}{cmsy}{b}{n} +\input{iopart1\@ptsize.clo} +\setlength\lineskip{1\p@} +\setlength\normallineskip{1\p@} +\renewcommand\baselinestretch{} +\setlength\parskip{0\p@ \@plus \p@} +\@lowpenalty 51 +\@medpenalty 151 +\@highpenalty 301 +\setlength\parindent{2em} +\setcounter{topnumber}{8} +\renewcommand\topfraction{1} +\setcounter{bottomnumber}{3} +\renewcommand\bottomfraction{.99} +\setcounter{totalnumber}{8} +\renewcommand\textfraction{0.01} +\renewcommand\floatpagefraction{.8} +\setcounter{dbltopnumber}{6} +\renewcommand\dbltopfraction{1} +\renewcommand\dblfloatpagefraction{.8} +% +\pretolerance=5000 +\tolerance=8000 +% +% Headings for all pages apart from first +% +\def\ps@headings{\let\@oddfoot\@empty + \let\@evenfoot\@empty + \def\@evenhead{\thepage\hfil\itshape\rightmark}% + \def\@oddhead{{\itshape\leftmark}\hfil\thepage}% + \let\@mkboth\markboth + \let\sectionmark\@gobble + \let\subsectionmark\@gobble} +% +% Headings for first page +% +\def\ps@myheadings{\let\@oddfoot\@empty\let\@evenfoot\@empty + \let\@oddhead\@empty\let\@evenhead\@empty + \let\@mkboth\@gobbletwo + \let\sectionmark\@gobble + \let\subsectionmark\@gobble} +% +% \maketitle just ends page +% +\newcommand\maketitle{\newpage} +% +% Article titles +% +% Usage: \title[Short title]{Full title} +% [Short title] is optional; use where title is too long +% or contains footnotes, 50 characters maximum +% +\renewcommand{\title}{\@ifnextchar[{\@stitle}{\@ftitle}} +\def\@stitle[#1]#2{\markboth{#1}{#1}% + \thispagestyle{myheadings}% + \vspace*{3pc}{\exhyphenpenalty=10000\hyphenpenalty=10000 + \Large\raggedright\noindent + \bf#2\par}} +\def\@ftitle#1{\markboth{#1}{#1}% + \thispagestyle{myheadings}% + \vspace*{3pc}{\exhyphenpenalty=10000\hyphenpenalty=10000 + \Large\raggedright\noindent + \bf#1\par}} +% +% Can use \paper instead of \title +% +\let\paper=\title +% +% Generic title command for articles other than papers +% +% Usage: \article[Short title]{Article Type}{Full title} +% [Short title] is optional; use where title is too long +% or contains footnotes, 50 characters maximum +% +\newcommand{\article}{\@ifnextchar[{\@sarticle}{\@farticle}} +\def\@sarticle[#1]#2#3{\markboth{#1}{#1}% + \thispagestyle{myheadings}% + \vspace*{.5pc}% + {\parindent=\mathindent \bf #2\par}% + \vspace*{1.5pc}% + {\exhyphenpenalty=10000\hyphenpenalty=10000 + \Large\raggedright\noindent + \bf#3\par}}% +\def\@farticle#1#2{\markboth{#2}{#2}% + \thispagestyle{myheadings}% + \vspace*{.5pc}% + {\parindent=\mathindent \bf #1\par}% + \vspace*{1.5pc}% + {\exhyphenpenalty=10000\hyphenpenalty=10000 + \Large\raggedright\noindent + \bf#2\par}}% +% +% Letters to the Editor +% +% Usage \letter{Full title} +% No short title is required for Letters +% +\def\letter#1{\article[Letter to the Editor]{Letter to the Editor}{#1}} +% +% Fast Track Communications (added by sxb 9 March 2011) +% +% Usage \ftc{Full title} - there's no short title +\def\ftc#1{\article[Fast Track Communication]{Fast Track Communication}{#1}} +% +% +% Review articles +% +% Usage: \review[Short title]{Full title} +% [Short title] is optional; use where title is too long +% or contains footnotes, 50 characters maximum +% +\def\review{\@ifnextchar[{\@sreview}{\@freview}} +\def\@sreview[#1]#2{\@sarticle[#1]{Review Article}{#2}} +\def\@freview#1{\@farticle{Review Article}{#1}} +% +% Topical Review +% +% Usage: \topical[Short title]{Full title} +% [Short title] is optional; use where title is too long +% or contains footnotes, 50 characters maximum +% +\def\topical{\@ifnextchar[{\@stopical}{\@ftopical}} +\def\@stopical[#1]#2{\@sarticle[#1]{Topical Review}{#2}} +\def\@ftopical#1{\@farticle{Topical Review}{#1}} +% +% Comments +% +% Usage: \comment[Short title]{Full title} +% [Short title] is optional; use where title is too long +% or contains footnotes, 50 characters maximum +% +\def\comment{\@ifnextchar[{\@scomment}{\@fcomment}} +\def\@scomment[#1]#2{\@sarticle[#1]{Comment}{#2}} +\def\@fcomment#1{\@farticle{Comment}{#1}} +% +% Rapid Communications +% +% Usage: \rapid[Short title]{Full title} +% [Short title] is optional; use where title is too long +% or contains footnotes, 50 characters maximum +% +\def\rapid{\@ifnextchar[{\@srapid}{\@frapid}} +\def\@srapid[#1]#2{\@sarticle[#1]{Rapid Communication}{#2}} +\def\@frapid#1{\@farticle{Rapid Communication}{#1}} +% +% Notes +% +% Usage: \note[Short title]{Full title} +% [Short title] is optional; use where title is too long +% or contains footnotes, 50 characters maximum +% +\def\note{\@ifnextchar[{\@snote}{\@fnote}} +\def\@snote[#1]#2{\@sarticle[#1]{Note}{#2}} +\def\@fnote#1{\@farticle{Note}{#1}} +% +% Preliminary Communications +% +% Usage: \prelim[Short title]{Full title} +% [Short title] is optional; use where title is too long +% or contains footnotes, 50 characters maximum +% +\def\prelim{\@ifnextchar[{\@sprelim}{\@fprelim}} +\def\@sprelim[#1]#2{\@sarticle[#1]{Preliminary Communication}{#2}} +\def\@fprelim#1{\@farticle{Preliminary Communication}{#1}} +% +% List of authors +% +% Usage \author[Short form]{List of all authors} +% The short form excludes footnote symbols linking authors to addresses +% and is used for running heads in printed version (but not on preprints) +% +\renewcommand{\author}{\@ifnextchar[{\@sauthor}{\@fauthor}} +\def\@sauthor[#1]#2{\markright{#1} % for production only + \vspace*{1.5pc}% + \begin{indented}% + \item[]\normalsize\bf\raggedright#2 + \end{indented}% + \smallskip} +\def\@fauthor#1{%\markright{#1} for production only + \vspace*{1.5pc}% + \begin{indented}% + \item[]\normalsize\bf\raggedright#1 + \end{indented}% + \smallskip} +% +% Affiliation (authors address) +% +% Usage: \address{Address of first author} +% \address{Address of second author} +% Use once for each address, use symbols \dag \ddag \S \P $\|$ +% to connect authors with addresses +% +\newcommand{\address}[1]{\begin{indented} + \item[]\rm\raggedright #1 + \end{indented}} +% +% American Mathematical Society Classification Numbers +% Usage: \ams{57.XX, 58.XX} +% +\def\ams#1{\vspace{10pt} + \begin{indented} + \item[]\rm AMS classification scheme numbers: #1\par + \end{indented}} +% +% A single Physics & Astronomy Classification Number +% Usage \pacno{31.10} +% +\def\pacno#1{\vspace{10pt} + \begin{indented} + \item[]\rm PACS number: #1\par + \end{indented}} +% +% Physics & Astronomy Classification Numbers (more than one) +% Usage \pacs{31.10, 31.20T} +% +\def\pacs#1{\vspace{10pt} + \begin{indented} + \item[]\rm PACS numbers: #1\par + \end{indented}} +% +% Submission details. If \jl command used journals name printed +% otherwise Institute of Physics Publishing +% +\def\submitted{\vspace{28pt plus 10pt minus 18pt} + \noindent{\small\rm Submitted to: {\it \journal}\par}} +% +\def\submitto#1{\vspace{28pt plus 10pt minus 18pt} + \noindent{\small\rm Submitted to: {\it #1}\par}} +% +% For articles (other than Letters) not divided into sections +% Usage \nosections Start of text +% +\def\nosections{\vspace{30\p@ plus12\p@ minus12\p@} + \noindent\ignorespaces} +% +% Acknowledgments (no heading if letter) +% Usage \ack for Acknowledgments, \ackn for Acknowledgement +% +\def\ack{\ifletter\bigskip\noindent\ignorespaces\else + \section*{Acknowledgments}\fi} +\def\ackn{\ifletter\bigskip\noindent\ignorespaces\else + \section*{Acknowledgment}\fi} +% +% Footnotes: symbols selected in order \dag (1), \ddag (2), \S (3), +% $\|$ (4), $\P$ (5), $^+$ (6), $^*$ (7), \sharp (8), \dagger\dagger (9) +% unless optional argument of [<num>] use to specify required symbol, +% 1=\dag, 2=\ddag, etc +% Usage: \footnote{Text of footnote} +% \footnote[3]{Text of footnote} +% +\def\footnoterule{}% +\setcounter{footnote}{1} +\long\def\@makefntext#1{\parindent 1em\noindent + \makebox[1em][l]{\footnotesize\rm$\m@th{\fnsymbol{footnote}}$}% + \footnotesize\rm #1} +\def\@makefnmark{\hbox{${\fnsymbol{footnote}}\m@th$}} +\def\@thefnmark{\fnsymbol{footnote}} +\def\footnote{\@ifnextchar[{\@xfootnote}{\stepcounter{\@mpfn}% + \begingroup\let\protect\noexpand + \xdef\@thefnmark{\thempfn}\endgroup + \@footnotemark\@footnotetext}} +\def\@xfootnote[#1]{\setcounter{footnote}{#1}% + \addtocounter{footnote}{-1}\footnote} +\def\@fnsymbol#1{\ifcase#1\or \dagger\or \ddagger\or \S\or + \|\or \P\or ^{+}\or ^{\tsty *}\or \sharp + \or \dagger\dagger \else\@ctrerr\fi\relax} +% +% IOP Journals +% +\newcounter{jnl} +\newcommand{\jl}[1]{\setcounter{jnl}{#1}} +\def\journal{\ifnum\thejnl=0 Institute of Physics Publishing\fi + \ifnum\thejnl=1 J. Phys.\ A: Math.\ Gen.\ \fi + \ifnum\thejnl=2 J. Phys.\ B: At.\ Mol.\ Opt.\ Phys.\ \fi + \ifnum\thejnl=3 J. Phys.:\ Condens. Matter\ \fi + \ifnum\thejnl=4 J. Phys.\ G: Nucl.\ Part.\ Phys.\ \fi + \ifnum\thejnl=5 Inverse Problems\ \fi + \ifnum\thejnl=6 Class. Quantum Grav.\ \fi + \ifnum\thejnl=7 Network: Comput.\ Neural Syst.\ \fi + \ifnum\thejnl=8 Nonlinearity\ \fi + \ifnum\thejnl=9 J. Opt. B: Quantum Semiclass. Opt.\ \fi + \ifnum\thejnl=10 Waves Random Media\ \fi + \ifnum\thejnl=11 J. Opt. A: Pure Appl. Opt.\ \fi + \ifnum\thejnl=12 Phys. Med. Biol.\ \fi + \ifnum\thejnl=13 Modelling Simul.\ Mater.\ Sci.\ Eng.\ \fi + \ifnum\thejnl=14 Plasma Phys. Control. Fusion\ \fi + \ifnum\thejnl=15 Physiol. Meas.\ \fi + \ifnum\thejnl=16 Combust. Theory Modelling\ \fi + \ifnum\thejnl=17 High Perform.\ Polym.\ \fi + \ifnum\thejnl=18 Public Understand. Sci.\ \fi + \ifnum\thejnl=19 Rep.\ Prog.\ Phys.\ \fi + \ifnum\thejnl=20 J.\ Phys.\ D: Appl.\ Phys.\ \fi + \ifnum\thejnl=21 Supercond.\ Sci.\ Technol.\ \fi + \ifnum\thejnl=22 Semicond.\ Sci.\ Technol.\ \fi + \ifnum\thejnl=23 Nanotechnology\ \fi + \ifnum\thejnl=24 Measur.\ Sci.\ Technol.\ \fi + \ifnum\thejnl=25 Plasma.\ Sources\ Sci.\ Technol.\ \fi + \ifnum\thejnl=26 Smart\ Mater.\ Struct.\ \fi + \ifnum\thejnl=27 J.\ Micromech.\ Microeng.\ \fi + \ifnum\thejnl=28 Distrib.\ Syst.\ Engng\ \fi + \ifnum\thejnl=29 Bioimaging\ \fi + \ifnum\thejnl=30 J.\ Radiol. Prot.\ \fi + \ifnum\thejnl=31 Europ. J. Phys.\ \fi + \ifnum\thejnl=32 J. Opt. A: Pure Appl. Opt.\ \fi + \ifnum\thejnl=33 New. J. Phys.\ \fi} +% +% E-mail addresses (to provide links from headers) +% +\def\eads#1{\vspace*{5pt}\address{E-mail: #1}} +\def\ead#1{\vspace*{5pt}\address{E-mail: \mailto{#1}}} +\def\mailto#1{{\tt #1}} +% +% Switches +% +\newif\ifletter +% +\setcounter{secnumdepth}{3} +\newcounter {section} +\newcounter {subsection}[section] +\newcounter {subsubsection}[subsection] +\newcounter {paragraph}[subsubsection] +\newcounter {subparagraph}[paragraph] +\renewcommand\thesection {\arabic{section}} +\renewcommand\thesubsection {\thesection.\arabic{subsection}} +\renewcommand\thesubsubsection {\thesubsection.\arabic{subsubsection}} +\renewcommand\theparagraph {\thesubsubsection.\arabic{paragraph}} +\renewcommand\thesubparagraph {\theparagraph.\arabic{subparagraph}} +\def\@chapapp{Section} + +\newcommand\section{\@startsection {section}{1}{\z@}% + {-3.5ex \@plus -1ex \@minus -.2ex}% + {2.3ex \@plus.2ex}% + {\reset@font\normalsize\bfseries\raggedright}} +\newcommand\subsection{\@startsection{subsection}{2}{\z@}% + {-3.25ex\@plus -1ex \@minus -.2ex}% + {1.5ex \@plus .2ex}% + {\reset@font\normalsize\itshape\raggedright}} +\newcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% + {-3.25ex\@plus -1ex \@minus -.2ex}% + {-1em \@plus .2em}% + {\reset@font\normalsize\itshape}} +\newcommand\paragraph{\@startsection{paragraph}{4}{\z@}% + {3.25ex \@plus1ex \@minus.2ex}% + {-1em}% + {\reset@font\normalsize\itshape}} +\newcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}% + {3.25ex \@plus1ex \@minus .2ex}% + {-1em}% + {\reset@font\normalsize\itshape}} +\def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth + \let\@svsec\@empty\else + \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname. }\fi + \@tempskipa #5\relax + \ifdim \@tempskipa>\z@ + \begingroup #6\relax + \noindent{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par}% + \endgroup + \csname #1mark\endcsname{#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}\else + \def\@svsechd{#6\hskip #3\relax %% \relax added 2 May 90 + \@svsec #8\csname #1mark\endcsname + {#7}\addcontentsline + {toc}{#1}{\ifnum #2>\c@secnumdepth \else + \protect\numberline{\csname the#1\endcsname}\fi + #7}}\fi + \@xsect{#5}} +% +\def\@ssect#1#2#3#4#5{\@tempskipa #3\relax + \ifdim \@tempskipa>\z@ + \begingroup #4\noindent{\hskip #1}{\interlinepenalty \@M #5\par}\endgroup + \else \def\@svsechd{#4\hskip #1\relax #5}\fi + \@xsect{#3}} + +\setlength\leftmargini{2em} +\setlength\leftmarginii{2em} +\setlength\leftmarginiii{1.8em} +\setlength\leftmarginiv{1.6em} +\setlength\leftmarginv{1em} +\setlength\leftmarginvi{1em} +\setlength\leftmargin{\leftmargini} +\setlength\labelsep{0.5em} +\setlength\labelwidth{\leftmargini} +\addtolength\labelwidth{-\labelsep} +\@beginparpenalty -\@lowpenalty +\@endparpenalty -\@lowpenalty +\@itempenalty -\@lowpenalty +\renewcommand\theenumi{\roman{enumi}} +\renewcommand\theenumii{\alph{enumii}} +\renewcommand\theenumiii{\arabic{enumiii}} +\renewcommand\theenumiv{\Alph{enumiv}} +\newcommand\labelenumi{(\theenumi)} +\newcommand\labelenumii{(\theenumii)} +\newcommand\labelenumiii{\theenumiii.} +\newcommand\labelenumiv{(\theenumiv)} +\renewcommand\p@enumii{(\theenumi)} +\renewcommand\p@enumiii{(\theenumi.\theenumii)} +\renewcommand\p@enumiv{(\theenumi.\theenumii.\theenumiii)} +\newcommand\labelitemi{$\m@th\bullet$} +\newcommand\labelitemii{\normalfont\bfseries --} +\newcommand\labelitemiii{$\m@th\ast$} +\newcommand\labelitemiv{$\m@th\cdot$} +\newenvironment{description} + {\list{}{\labelwidth\z@ \itemindent-\leftmargin + \let\makelabel\descriptionlabel}} + {\endlist} +\newcommand\descriptionlabel[1]{\hspace\labelsep + \normalfont\bfseries #1} +\newenvironment{abstract}{% + \vspace{16pt plus3pt minus3pt} + \begin{indented} + \item[]{\bfseries \abstractname.}\quad\rm\ignorespaces} + {\end{indented}\if@titlepage\newpage\else\vspace{18\p@ plus18\p@}\fi} +\newenvironment{verse} + {\let\\=\@centercr + \list{}{\itemsep \z@ + \itemindent -1.5em% + \listparindent\itemindent + \rightmargin \leftmargin + \advance\leftmargin 1.5em}% + \item[]} + {\endlist} +\newenvironment{quotation} + {\list{}{\listparindent 1.5em% + \itemindent \listparindent + \rightmargin \leftmargin + \parsep \z@ \@plus\p@}% + \item[]} + {\endlist} +\newenvironment{quote} + {\list{}{\rightmargin\leftmargin}% + \item[]} + {\endlist} +\newenvironment{titlepage} + {% + \@restonecolfalse\newpage + \thispagestyle{empty}% + \if@compatibility + \setcounter{page}{0} + \else + \setcounter{page}{1}% + \fi}% + {\newpage\setcounter{page}{1}} +\def\appendix{\@ifnextchar*{\@appendixstar}{\@appendix}} +\def\@appendix{\eqnobysec\@appendixstar} +\def\@appendixstar{\@@par + \ifnumbysec % Added 30/4/94 to get Table A1, + \@addtoreset{table}{section} % Table B1 etc if numbering by + \@addtoreset{figure}{section}\fi % section + \setcounter{section}{0} + \setcounter{subsection}{0} + \setcounter{subsubsection}{0} + \setcounter{equation}{0} + \setcounter{figure}{0} + \setcounter{table}{0} + \def\thesection{Appendix \Alph{section}} + \def\theequation{\ifnumbysec + \Alph{section}.\arabic{equation}\else + \Alph{section}\arabic{equation}\fi} % Comment A\arabic{equation} maybe + \def\thetable{\ifnumbysec % better? 15/4/95 + \Alph{section}\arabic{table}\else + A\arabic{table}\fi} + \def\thefigure{\ifnumbysec + \Alph{section}\arabic{figure}\else + A\arabic{figure}\fi}} +\def\noappendix{\setcounter{figure}{0} + \setcounter{table}{0} + \def\thetable{\arabic{table}} + \def\thefigure{\arabic{figure}}} +\setlength\arraycolsep{5\p@} +\setlength\tabcolsep{6\p@} +\setlength\arrayrulewidth{.4\p@} +\setlength\doublerulesep{2\p@} +\setlength\tabbingsep{\labelsep} +\skip\@mpfootins = \skip\footins +\setlength\fboxsep{3\p@} +\setlength\fboxrule{.4\p@} +\renewcommand\theequation{\arabic{equation}} +\newcounter{figure} +\renewcommand\thefigure{\@arabic\c@figure} +\def\fps@figure{tbp} +\def\ftype@figure{1} +\def\ext@figure{lof} +\def\fnum@figure{\figurename~\thefigure} +\newenvironment{figure}{\footnotesize\rm\@float{figure}}% + {\end@float\normalsize\rm} +\newenvironment{figure*}{\footnotesize\rm\@dblfloat{figure}}{\end@dblfloat} +\newcounter{table} +\renewcommand\thetable{\@arabic\c@table} +\def\fps@table{tbp} +\def\ftype@table{2} +\def\ext@table{lot} +\def\fnum@table{\tablename~\thetable} +\newenvironment{table}{\footnotesize\rm\@float{table}}% + {\end@float\normalsize\rm} +\newenvironment{table*}{\footnotesize\rm\@dblfloat{table}}% + {\end@dblfloat\normalsize\rm} +\newlength\abovecaptionskip +\newlength\belowcaptionskip +\setlength\abovecaptionskip{10\p@} +\setlength\belowcaptionskip{0\p@} +% +% Added redefinition of \@caption so captions are not written to +% aux file therefore less need to \protect fragile commands +% +\long\def\@caption#1[#2]#3{\par\begingroup + \@parboxrestore + \normalsize + \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par + \endgroup} +% +\long\def\@makecaption#1#2{\vskip \abovecaptionskip + \begin{indented} + \item[]{\bf #1.} #2 + \end{indented}\vskip\belowcaptionskip} +\let\@portraitcaption=\@makecaption + +\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} +\ifiopams +\renewcommand{\cal}{\protect\pcal} +\else +\newcommand{\cal}{\protect\pcal} +\fi +\newcommand{\pcal}{\@fontswitch{\relax}{\mathcal}} +\ifiopams +\renewcommand{\mit}{\protect\pmit} +\else +\newcommand{\mit}{\protect\pmit} +\fi +\newcommand{\pmit}{\@fontswitch{\relax}{\mathnormal}} +\newcommand\@pnumwidth{1.55em} +\newcommand\@tocrmarg {2.55em} +\newcommand\@dotsep{4.5} +\setcounter{tocdepth}{3} +\newcommand\tableofcontents{% + \section*{\contentsname + \@mkboth{\uppercase{\contentsname}}{\uppercase{\contentsname}}}% + \@starttoc{toc}% + } +\newcommand\l@part[2]{% + \ifnum \c@tocdepth >-2\relax + \addpenalty{\@secpenalty}% + \addvspace{2.25em \@plus\p@}% + \begingroup + \setlength\@tempdima{3em}% + \parindent \z@ \rightskip \@pnumwidth + \parfillskip -\@pnumwidth + {\leavevmode + \large \bfseries #1\hfil \hbox to\@pnumwidth{\hss #2}}\par + \nobreak + \if@compatibility + \global\@nobreaktrue + \everypar{\global\@nobreakfalse\everypar{}} + \fi + \endgroup + \fi} +\newcommand\l@section[2]{% + \ifnum \c@tocdepth >\z@ + \addpenalty{\@secpenalty}% + \addvspace{1.0em \@plus\p@}% + \setlength\@tempdima{1.5em}% + \begingroup + \parindent \z@ \rightskip \@pnumwidth + \parfillskip -\@pnumwidth + \leavevmode \bfseries + \advance\leftskip\@tempdima + \hskip -\leftskip + #1\nobreak\hfil \nobreak\hbox to\@pnumwidth{\hss #2}\par + \endgroup + \fi} +\newcommand\l@subsection {\@dottedtocline{2}{1.5em}{2.3em}} +\newcommand\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}} +\newcommand\l@paragraph {\@dottedtocline{4}{7.0em}{4.1em}} +\newcommand\l@subparagraph {\@dottedtocline{5}{10em}{5em}} +\newcommand\listoffigures{% + \section*{\listfigurename + \@mkboth{\uppercase{\listfigurename}}% + {\uppercase{\listfigurename}}}% + \@starttoc{lof}% + } +\newcommand\l@figure{\@dottedtocline{1}{1.5em}{2.3em}} +\newcommand\listoftables{% + \section*{\listtablename + \@mkboth{\uppercase{\listtablename}}{\uppercase{\listtablename}}}% + \@starttoc{lot}% + } +\let\l@table\l@figure +\newenvironment{theindex} + {\if@twocolumn + \@restonecolfalse + \else + \@restonecoltrue + \fi + \columnseprule \z@ + \columnsep 35\p@ + \twocolumn[\section*{\indexname}]% + \@mkboth{\uppercase{\indexname}}% + {\uppercase{\indexname}}% + \thispagestyle{plain}\parindent\z@ + \parskip\z@ \@plus .3\p@\relax + \let\item\@idxitem} + {\if@restonecol\onecolumn\else\clearpage\fi} +\newcommand\@idxitem {\par\hangindent 40\p@} +\newcommand\subitem {\par\hangindent 40\p@ \hspace*{20\p@}} +\newcommand\subsubitem{\par\hangindent 40\p@ \hspace*{30\p@}} +\newcommand\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax} +\newcommand\contentsname{Contents} +\newcommand\listfigurename{List of Figures} +\newcommand\listtablename{List of Tables} +\newcommand\refname{References} +\newcommand\indexname{Index} +\newcommand\figurename{Figure} +\newcommand\tablename{Table} +\newcommand\partname{Part} +\newcommand\appendixname{Appendix} +\newcommand\abstractname{Abstract} +\newcommand\today{\number\day\space\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\year} +\setlength\columnsep{10\p@} +\setlength\columnseprule{0\p@} + +\newcommand{\Tables}{\clearpage\section*{Tables and table captions} + \def\fps@table{hp}\noappendix} +\newcommand{\Figures}{\clearpage\section*{Figure captions} + \def\fps@figure{hp}\noappendix} +% +\newcommand{\Figure}[1]{\begin{figure} + \caption{#1} + \end{figure}} +% +\newcommand{\Table}[1]{\begin{table} + \caption{#1} + \begin{indented} + \lineup + \item[]\begin{tabular}{@{}l*{15}{l}}} + +\def\endTable{\end{tabular}\end{indented}\end{table}} +\let\endtab=\endTable +% +\newcommand{\fulltable}[1]{\begin{table} + \caption{#1} + \footnotesize + \lineup + \begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus 12pt}}l}}} +\def\endfulltable{\end{tabular*}\end{table}\normalsize} +% +% +\newcommand{\Bibliography}[1]{\section*{References}\par\numrefs{#1}} +\newcommand{\References}{\section*{References}\par\refs} + +\def\thebibliography#1{\list + {\hfil[\arabic{enumi}]}{\topsep=0\p@\parsep=0\p@ + \partopsep=0\p@\itemsep=0\p@ + \labelsep=5\p@\itemindent=-10\p@ + \settowidth\labelwidth{\footnotesize[#1]}% + \leftmargin\labelwidth + \advance\leftmargin\labelsep + \advance\leftmargin -\itemindent + \usecounter{enumi}}\footnotesize + \def\newblock{\ } + \sloppy\clubpenalty4000\widowpenalty4000 + \sfcode`\.=1000\relax} +\let\endthebibliography=\endlist +\def\numrefs#1{\begin{thebibliography}{#1}} +\def\endnumrefs{\end{thebibliography}} +\let\endbib=\endnumrefs +% +\def\thereferences{\list{}{\topsep=0\p@\parsep=0\p@ + \partopsep=0\p@\itemsep=0\p@\labelsep=0\p@\itemindent=-18\p@ +\labelwidth=0\p@\leftmargin=18\p@ +}\footnotesize\rm +\def\newblock{\ } +\sloppy\clubpenalty4000\widowpenalty4000 +\sfcode`\.=1000\relax +} +% +\let\endthereferences=\endlist +\newlength{\indentedwidth} +\newdimen\mathindent +\indentedwidth=\mathindent +% +% Macro to used for references in the Harvard system +% +\newenvironment{harvard}{\list{}{\topsep=0\p@\parsep=0\p@ +\partopsep=0\p@\itemsep=0\p@\labelsep=0\p@\itemindent=-18\p@ +\labelwidth=0\p@\leftmargin=18\p@ +}\footnotesize\rm +\def\newblock{\ } +\sloppy\clubpenalty4000\widowpenalty4000 +\sfcode`\.=1000\relax}{\endlist} +% +\def\refs{\begin{harvard}} +\def\endrefs{\end{harvard}} +% +\newenvironment{indented}{\begin{indented}}{\end{indented}} +\newenvironment{varindent}[1]{\begin{varindent}{#1}}{\end{varindent}} +% +\def\indented{\list{}{\itemsep=0\p@\labelsep=0\p@\itemindent=0\p@ + \labelwidth=0\p@\leftmargin=\mathindent\topsep=0\p@\partopsep=0\p@ + \parsep=0\p@\listparindent=15\p@}\footnotesize\rm} + +\let\endindented=\endlist + +\def\varindent#1{\setlength{\varind}{#1}% + \list{}{\itemsep=0\p@\labelsep=0\p@\itemindent=0\p@ + \labelwidth=0\p@\leftmargin=\varind\topsep=0\p@\partopsep=0\p@ + \parsep=0\p@\listparindent=15\p@}\footnotesize\rm} + +\let\endvarindent=\endlist + +\def\[{\relax\ifmmode\@badmath\else + \begin{trivlist} + \@beginparpenalty\predisplaypenalty + \@endparpenalty\postdisplaypenalty + \item[]\leavevmode + \hbox to\linewidth\bgroup$ \displaystyle + \hskip\mathindent\bgroup\fi} +\def\]{\relax\ifmmode \egroup $\hfil \egroup \end{trivlist}\else \@badmath \fi} +\def\equation{\@beginparpenalty\predisplaypenalty + \@endparpenalty\postdisplaypenalty +\refstepcounter{equation}\trivlist \item[]\leavevmode + \hbox to\linewidth\bgroup $ \displaystyle +\hskip\mathindent} +\def\endequation{$\hfil \displaywidth\linewidth\@eqnnum\egroup \endtrivlist} +% +\@namedef{equation*}{\[} +\@namedef{endequation*}{\]} +% +\def\eqnarray{\stepcounter{equation}\let\@currentlabel=\theequation +\global\@eqnswtrue +\global\@eqcnt\z@\tabskip\mathindent\let\\=\@eqncr +\abovedisplayskip\topsep\ifvmode\advance\abovedisplayskip\partopsep\fi +\belowdisplayskip\abovedisplayskip +\belowdisplayshortskip\abovedisplayskip +\abovedisplayshortskip\abovedisplayskip +$$\halign to +\linewidth\bgroup\@eqnsel$\displaystyle\tabskip\z@ + {##{}}$&\global\@eqcnt\@ne $\displaystyle{{}##{}}$\hfil + &\global\@eqcnt\tw@ $\displaystyle{{}##}$\hfil + \tabskip\@centering&\llap{##}\tabskip\z@\cr} +\def\endeqnarray{\@@eqncr\egroup + \global\advance\c@equation\m@ne$$\global\@ignoretrue } +\mathindent = 6pc +% +\def\eqalign#1{\null\vcenter{\def\\{\cr}\openup\jot\m@th + \ialign{\strut$\displaystyle{##}$\hfil&$\displaystyle{{}##}$\hfil + \crcr#1\crcr}}\,} +% +\def\eqalignno#1{\displ@y \tabskip\z@skip + \halign to\displaywidth{\hspace{5pc}$\@lign\displaystyle{##}$% + \tabskip\z@skip + &$\@lign\displaystyle{{}##}$\hfill\tabskip\@centering + &\llap{$\@lign\hbox{\rm##}$}\tabskip\z@skip\crcr + #1\crcr}} +% +\newif\ifnumbysec +\def\theequation{\ifnumbysec + \arabic{section}.\arabic{equation}\else + \arabic{equation}\fi} +\def\eqnobysec{\numbysectrue\@addtoreset{equation}{section}} + +\newcounter{eqnval} + +% 2012: if you have an eqn numbered by parts (eg eqn 6a, 6b) this allows +% you to refer to it by just the number (eg eqn 6) if the label is between +% the \multiparteqn and the \numparts +\def\multiparteqn{\addtocounter{equation}{1} +\eqnarray\nonumber +\endeqnarray + +\vspace{-72pt} +} + +\def\numparts{\addtocounter{equation}{1}% + \setcounter{eqnval}{\value{equation}}% + \setcounter{equation}{0}% + \def\theequation{\ifnumbysec + \arabic{section}.\arabic{eqnval}{\it\alph{equation}}% + \else\arabic{eqnval}{\it\alph{equation}}\fi}} + +\def\endnumparts{\def\theequation{\ifnumbysec + \arabic{section}.\arabic{equation}\else + \arabic{equation}\fi}% + \setcounter{equation}{\value{eqnval}}} +% +\def\cases#1{% + \left\{\,\vcenter{\def\\{\cr}\normalbaselines\openup1\jot\m@th% + \ialign{\strut$\displaystyle{##}\hfil$&\tqs + \rm##\hfil\crcr#1\crcr}}\right.}% +% +\newcommand{\e}{\mathrm{e}} +\newcommand{\rme}{\mathrm{e}} +\newcommand{\rmi}{\mathrm{i}} +\newcommand{\rmd}{\mathrm{d}} +\renewcommand{\qquad}{\hspace*{25pt}} +\newcommand{\tdot}[1]{\stackrel{\dots}{#1}} % Added 1/9/94 +\newcommand{\tqs}{\hspace*{25pt}} +\newcommand{\fl}{\hspace*{-\mathindent}} +\newcommand{\Tr}{\mathop{\mathrm{Tr}}\nolimits} +\newcommand{\tr}{\mathop{\mathrm{tr}}\nolimits} +\newcommand{\Or}{\mathord{\mathrm{O}}} %changed from \mathop 20/1/95 +\newcommand{\lshad}{[\![} +\newcommand{\rshad}{]\!]} +\newcommand{\case}[2]{{\textstyle\frac{#1}{#2}}} +\def\pt(#1){({\it #1\/})} +\newcommand{\dsty}{\displaystyle} +\newcommand{\tsty}{\textstyle} +\newcommand{\ssty}{\scriptstyle} +\newcommand{\sssty}{\scriptscriptstyle} +\def\lo#1{\llap{${}#1{}$}} +\def\eql{\llap{${}={}$}} +\def\lsim{\llap{${}\sim{}$}} +\def\lsimeq{\llap{${}\simeq{}$}} +\def\lequiv{\llap{${}\equiv{}$}} +% +\newcommand{\eref}[1]{(\ref{#1})} +\newcommand{\sref}[1]{section~\ref{#1}} +\newcommand{\fref}[1]{figure~\ref{#1}} +\newcommand{\tref}[1]{table~\ref{#1}} +\newcommand{\Eref}[1]{Equation (\ref{#1})} +\newcommand{\Sref}[1]{Section~\ref{#1}} +\newcommand{\Fref}[1]{Figure~\ref{#1}} +\newcommand{\Tref}[1]{Table~\ref{#1}} + +\newcommand{\opencircle}{\mbox{\Large$\circ\,$}} % moved Large outside maths +\newcommand{\opensquare}{\mbox{$\rlap{$\sqcap$}\sqcup$}} +\newcommand{\opentriangle}{\mbox{$\triangle$}} +\newcommand{\opentriangledown}{\mbox{$\bigtriangledown$}} +\newcommand{\opendiamond}{\mbox{$\diamondsuit$}} +\newcommand{\fullcircle}{\mbox{{\Large$\bullet\,$}}} % moved Large outside maths +\newcommand{\fullsquare}{\,\vrule height5pt depth0pt width5pt} +\newcommand{\dotted}{\protect\mbox{${\mathinner{\cdotp\cdotp\cdotp\cdotp\cdotp\cdotp}}$}} +\newcommand{\dashed}{\protect\mbox{-\; -\; -\; -}} +\newcommand{\broken}{\protect\mbox{-- -- --}} +\newcommand{\longbroken}{\protect\mbox{--- --- ---}} +\newcommand{\chain}{\protect\mbox{--- $\cdot$ ---}} +\newcommand{\dashddot}{\protect\mbox{--- $\cdot$ $\cdot$ ---}} +\newcommand{\full}{\protect\mbox{------}} + +\def\;{\protect\psemicolon} +\def\psemicolon{\relax\ifmmode\mskip\thickmuskip\else\kern .3333em\fi} +\def\lineup{\def\0{\hbox{\phantom{\footnotesize\rm 0}}}% + \def\m{\hbox{$\phantom{-}$}}% + \def\-{\llap{$-$}}} +% +%%%%%%%%%%%%%%%%%%%%% +% Tables rules % +%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\boldarrayrulewidth}{1\p@} +% Width of bold rule in tabular environment. + +\def\bhline{\noalign{\ifnum0=`}\fi\hrule \@height +\boldarrayrulewidth \futurelet \@tempa\@xhline} + +\def\@xhline{\ifx\@tempa\hline\vskip \doublerulesep\fi + \ifnum0=`{\fi}} + +% +% Rules for tables with extra space around +% +\newcommand{\br}{\ms\bhline\ms} +\newcommand{\mr}{\ms\hline\ms} +% +\newcommand{\centre}[2]{\multispan{#1}{\hfill #2\hfill}} +\newcommand{\crule}[1]{\multispan{#1}{\hspace*{\tabcolsep}\hrulefill + \hspace*{\tabcolsep}}} +\newcommand{\fcrule}[1]{\ifnum\thetabtype=1\multispan{#1}{\hrulefill + \hspace*{\tabcolsep}}\else\multispan{#1}{\hrulefill}\fi} +% +% Extra spaces for tables and displayed equations +% +\newcommand{\ms}{\noalign{\vspace{3\p@ plus2\p@ minus1\p@}}} +\newcommand{\bs}{\noalign{\vspace{6\p@ plus2\p@ minus2\p@}}} +\newcommand{\ns}{\noalign{\vspace{-3\p@ plus-1\p@ minus-1\p@}}} +\newcommand{\es}{\noalign{\vspace{6\p@ plus2\p@ minus2\p@}}\displaystyle}% +% +\newcommand{\etal}{{\it et al\/}\ } +\newcommand{\dash}{------} +\newcommand{\nonum}{\par\item[]} %\par added 1/9/93 +\newcommand{\mat}[1]{\underline{\underline{#1}}} +% +% abbreviations for IOPP journals +% +\newcommand{\CQG}{{\it Class. Quantum Grav.} } +\newcommand{\CTM}{{\it Combust. Theory Modelling\/} } +\newcommand{\DSE}{{\it Distrib. Syst. Engng.\/} } +\newcommand{\EJP}{{\it Eur. J. Phys.} } +\newcommand{\JNE}{{\it J. Neural Eng.} } %added 30/11/2004 GMD +\newcommand{\PB}{{\it Phys. Biol.} } %added 30/11/2004 GMD +\newcommand{\SMS}{{\it Smart Mater. Struct.} } %added 30/11/2004 GMD +\newcommand{\HPP}{{\it High Perform. Polym.} } % added 4/5/93 +\newcommand{\IP}{{\it Inverse Problems\/} } +\newcommand{\JHM}{{\it J. Hard Mater.} } % added 4/5/93 +\newcommand{\JO}{{\it J. Opt.} } +\newcommand{\JOA}{{\it J. Opt. A: Pure Appl. Opt.} } +\newcommand{\JOB}{{\it J. Opt. B: Quantum Semiclass. Opt.} } +\newcommand{\JPA}{{\it J. Phys. A: Math. Gen.} } % superseded by \jpa below +\newcommand{\JPB}{{\it J. Phys. B: At. Mol. Phys.} } %1968-87 +\newcommand{\jpb}{{\it J. Phys. B: At. Mol. Opt. Phys.} } %1988 and onwards +\newcommand{\JPC}{{\it J. Phys. C: Solid State Phys.} } %1968--1988 +\newcommand{\JPCM}{{\it J. Phys.: Condens. Matter\/} } %1989 and onwards +\newcommand{\JPD}{{\it J. Phys. D: Appl. Phys.} } +\newcommand{\JPE}{{\it J. Phys. E: Sci. Instrum.} } +\newcommand{\JPF}{{\it J. Phys. F: Met. Phys.} } +\newcommand{\JPG}{{\it J. Phys. G: Nucl. Phys.} } %1975--1988 +\newcommand{\jpg}{{\it J. Phys. G: Nucl. Part. Phys.} } %1989 and onwards +\newcommand{\JMM}{{\it J. Micromech. Microeng.\/} } +\newcommand{\MSMSE}{{\it Modelling Simul. Mater. Sci. Eng.} } % sxb changed to 'Simul' 15 Mar 2011 +\newcommand{\MST}{{\it Meas. Sci. Technol.} } %1990 and onwards +\newcommand{\NET}{{\it Network: Comput. Neural Syst.} } +\newcommand{\NJP}{{\it New J. Phys.} } +\newcommand{\NL}{{\it Nonlinearity\/} } +\newcommand{\NT}{{\it Nanotechnology} } +\newcommand{\PAO}{{\it Pure Appl. Optics\/} } +\newcommand{\PM}{{\it Physiol. Meas.} } % added 4/5/93 +\newcommand{\PMB}{{\it Phys. Med. Biol.} } +\newcommand{\PPCF}{{\it Plasma Phys. Control. Fusion\/} } % added 4/5/93 +\newcommand{\PSST}{{\it Plasma Sources Sci. Technol.} } +\newcommand{\PUS}{{\it Public Understand. Sci.} } +\newcommand{\QO}{{\it Quantum Opt.} } +\newcommand{\QSO}{{\em Quantum Semiclass. Opt.} } +\newcommand{\RPP}{{\it Rep. Prog. Phys.} } +\newcommand{\SLC}{{\it Sov. Lightwave Commun.} } % added 4/5/93 +\newcommand{\SST}{{\it Semicond. Sci. Technol.} } +\newcommand{\SUST}{{\it Supercond. Sci. Technol.} } +\newcommand{\WRM}{{\it Waves Random Media\/} } +% +% Other commonly quoted journals +% +\newcommand{\AC}{{\it Acta Crystallogr.} } +\newcommand{\AM}{{\it Acta Metall.} } +\newcommand{\AP}{{\it Ann. Phys., Lpz.} } +\newcommand{\APNY}{{\it Ann. Phys., NY\/} } +\newcommand{\APP}{{\it Ann. Phys., Paris\/} } +\newcommand{\CJP}{{\it Can. J. Phys.} } +\newcommand{\JAP}{{\it J. Appl. Phys.} } +\newcommand{\JCP}{{\it J. Chem. Phys.} } +\newcommand{\JJAP}{{\it Jpn. J. Appl. Phys.} } +\newcommand{\JP}{{\it J. Physique\/} } +\newcommand{\JPhCh}{{\it J. Phys. Chem.} } +\newcommand{\JMMM}{{\it J. Magn. Magn. Mater.} } +\newcommand{\JMP}{{\it J. Math. Phys.} } +\newcommand{\JOSA}{{\it J. Opt. Soc. Am.} } +\newcommand{\JPSJ}{{\it J. Phys. Soc. Jpn.\/} } +\newcommand{\JQSRT}{{\it J. Quant. Spectrosc. Radiat. Transfer\/} } +\newcommand{\NC}{{\it Nuovo Cimento\/} } +\newcommand{\NIM}{{\it Nucl. Instrum. Methods\/} } +\newcommand{\NP}{{\it Nucl. Phys.} } +\newcommand{\PL}{{\it Phys. Lett.} } +\newcommand{\PR}{{\it Phys. Rev.} } +\newcommand{\PRL}{{\it Phys. Rev. Lett.} } +\newcommand{\PRS}{{\it Proc. R. Soc.} } +\newcommand{\PS}{{\it Phys. Scr.} } +\newcommand{\PSS}{{\it Phys. Status Solidi\/} } +\newcommand{\PTRS}{{\it Phil. Trans. R. Soc.} } +\newcommand{\RMP}{{\it Rev. Mod. Phys.} } +\newcommand{\RSI}{{\it Rev. Sci. Instrum.} } +\newcommand{\SSC}{{\it Solid State Commun.} } +\newcommand{\ZP}{{\it Z. Phys.} } +\newcommand{\GRG}{{\it Gen. Rel. Grav.} } +\newcommand{\PF}{{\it Phys. Fluids\/} } +\newcommand{\SPJ}{{\it Sov. Phys.--JETP\/} } +% +% More journals added 8 Mar 2011, below (sxb) +% +\newcommand{\jpa}{{\it J. Phys. A: Math. Theor.} } +\newcommand{\BF}{{\it Biofabrication\/} } +\newcommand{\BB}{{\it Bioinspir. Biomim.} } +\newcommand{\BMM}{{\it Biomed. Mater.} } +\newcommand{\CSD}{{\it Comput. Sci. Disc.} } +\newcommand{\ERL}{{\it Environ. Res. Lett.} } +\newcommand{\JBR}{{\it J. Breath Res.} } +\newcommand{\JGE}{{\it J. Geophys. Eng.} } +\newcommand{\JOPT}{{\it J. Opt.} } +\newcommand{\JRP}{{\it J. Radiol. Prot.} } +\newcommand{\MET}{{\it Metrologia\/} } +\newcommand{\NF}{{\it Nucl. Fusion\/} } +\newcommand{\PED}{{\it Phys. Educ.} } +% +% More journals added 6 Feb 2014, below (sxb) +% +\newcommand{\TDM}{{\it 2D Mater.} } +\newcommand{\MRE}{{\it Mater. Res. Express\/} } +\newcommand{\MAF}{{\it Methods Appl. Fluoresc.} } +\newcommand{\TMR}{{\it Transl. Mater. Res.} } +\newcommand{\STMP}{{\it Surf. Topogr.: Metrol. Prop.} } +% +% The following journals are externally-edited and the usual IOP Publishing +% submission guidelines do *not* apply if you're submitting to them. +% +\newcommand{\AJ}{{\it AJ\/} } +\newcommand{\APJ}{{\it ApJ\/} } +\newcommand{\APJL}{{\it ApJL\/} } +\newcommand{\APJS}{{\it ApJS\/} } +\newcommand{\ANSN}{{\it Adv. Nat. Sci: Nanosci. Nanotechnol.} } +\newcommand{\CJCP}{{\it Chin. J. Chem. Phys.} } +\newcommand{\CPB}{{\it Chinese Phys. B\/} } +\newcommand{\CPC}{{\it Chinese Phys. C\/} } +\newcommand{\CPL}{{\it Chinese Phys. Lett.} } +\newcommand{\CTP}{{\it Commun. Theor. Phys.} } +\newcommand{\EPL}{{\it EPL\/} } +\newcommand{\FDR}{{\it Fluid Dyn. Res.} } +\newcommand{\IZV}{{\it Izv. Math.} } +\newcommand{\JOS}{{\it J. Semicond.} } +\newcommand{\PHU}{{\it Phys.-Usp.} } +\newcommand{\PST}{{\it Plasma Sci. Technol.} } +\newcommand{\QEL}{{\it Quantum Electron.} } +\newcommand{\RAA}{{\it Res. Astron. Astrophys.} } +\newcommand{\RCR}{{\it Russ. Chem. Rev.} } +\newcommand{\RMS}{{\it Russ. Math. Surv.} } +\newcommand{\MSB}{{\it Sb. Math.} } +\newcommand{\SFC}{{\it Science Foundation in China\/} } +\newcommand{\STAM}{{\it Sci. Technol. Adv. Mater.} } +\newcommand{\LP}{{\it Laser Phys.} } +\newcommand{\LPL}{{\it Laser Phys. Lett.} } +\newcommand{\APEX}{{\it Appl. Phys. Express\/} } +% +% SISSA journals +% +\newcommand{\JCAP}{{\it JCAP\/} } +\newcommand{\JHEP}{{\it J. High Energy Phys.\/} } +\newcommand{\JSTAT}{{\it J. Stat. Mech.} } +\newcommand{\JINST}{{\it JINST\/} } +% +% These are the IOP Conference Series journals: again, if you're +% submitting to one of these journals you shouldn't be using iopart.cls +% +\newcommand{\JPCS}{{\it J. Phys.: Conf. Ser.} } +\newcommand{\EES}{{\it IOP Conf. Ser.: Earth Environ. Sci.} } +\newcommand{\MSE}{{\it IOP Conf. Ser.: Mater. Sci. Eng.} } + +% 2012 new option for twocolumn output +\def\ioptwocol{\setlength\hoffset{-0.5in}\setlength\voffset{-0.5in}\setlength\textwidth{6.75in} +\setlength\columnsep{0.2in}\setlength\textheight{9.25in}\mathindent=0in\twocolumn} +% +% +\pagestyle{headings} +\pagenumbering{arabic} % Arabic page numbers +\raggedbottom +\onecolumn +\endinput +%% +%% End of file `iopart.cls'. + diff --git a/iopart10.clo b/iopart10.clo new file mode 100644 index 0000000..1805a0b --- /dev/null +++ b/iopart10.clo @@ -0,0 +1 @@ +%%
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%% Corrections history:
%%
\ No newline at end of file @@ -1,4 +1,4 @@ -\documentclass[aps,reprint,longbibliography,floatfix]{revtex4-2} +\documentclass[]{iopart} \usepackage[utf8]{inputenc} % why not type "Stokes" with unicode? \usepackage[T1]{fontenc} % vector fonts @@ -9,24 +9,21 @@ filecolor=purple, linkcolor=purple ]{hyperref} % ref and cite links with pretty colors -\usepackage{amsmath, amssymb, graphicx, xcolor} % standard packages -\usepackage{newtxtext,newtxmath} % Times for PR +\usepackage{amsopn, amssymb, graphicx, xcolor} % standard packages \usepackage[subfolder]{gnuplottex} % need to compile separately for APS \begin{document} \title{Analytic continuation over complex landscapes} -\author{Jaron Kent-Dobias} -\author{Jorge Kurchan} +\author{Jaron Kent-Dobias and Jorge Kurchan} -\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} +\address{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} +\ead{jaron.dobias@phys.ens.fr} -\date\today - -\begin{abstract} \setcitestyle{authoryear,round} +\begin{abstract} In this paper we follow up the study of `complex complex landscapes' - [\cite{Kent-Dobias_2021_Complex}], rugged landscapes of many complex + \cite{Kent-Dobias_2021_Complex}, rugged landscapes of many complex variables. Unlike real landscapes, there is no useful classification of saddles by index. Instead, the spectrum at critical points determines their tendency to trade topological numbers under analytic continuation of the @@ -58,7 +55,7 @@ partition function. Recent developments have found that stationary points of the action are important for understanding another aspect of the partition function: its -analytic continuation. The integral \eqref{eq:partition.function} is first +analytic continuation. The integral \eref{eq:partition.function} is first interpreted as a contour in a larger complex phase space, then deformed into a linear combination of specially constructed contours each enumerated by a stationary point. Analytic continuation of parameters preserves this @@ -84,10 +81,10 @@ integrals of analytic functions which are not themselves analytic \section{Integration by Lefschetz thimble} -We return to the partition function \eqref{eq:partition.function}. If +We return to the partition function \eref{eq:partition.function}. If the action can be continued to a holomorphic function on the Kähler manifold $\tilde\Omega\supset\Omega$ and $\Omega$ is orientable in $\tilde\Omega$, -then \eqref{eq:partition.function} can be considered a contour integral. In +then \eref{eq:partition.function} can be considered a contour integral. In this case, the contour can be freely deformed without affecting the value of the integral. Two properties of this deformed contour would be ideal. First, that as $|s|\to\infty$ the real part of the action goes to $-\infty$, to ensure @@ -97,7 +94,7 @@ problems. Remarkably, there is an elegant recipe for accomplishing both these criteria at once, courtesy of Morse theory. For a more thorough review, see -\citet{Witten_2011_Analytic}. Consider a stationary point of the action. The +\cite{Witten_2011_Analytic}. Consider a stationary point of the action. The union of all gradient descent trajectories on the real part of the action that begin at the stationary point is known as a \emph{Lefschetz thimble}. Since each point on the Lefschetz thimble is found through descent from the @@ -119,7 +116,7 @@ exist for their value about each critical point. The integer weights $n_\sigma$ are fixed by comparison with the initial contour. For a real action, all maxima in $\Omega$ contribute in equal magnitude. -Under analytic continuation, the form of \eqref{eq:thimble.integral} +Under analytic continuation, the form of \eref{eq:thimble.integral} generically persists. When the relative homology of the thimbles is unchanged by the continuation, the integer weights are likewise unchanged, and one can therefore use the knowledge of these weights in one regime to compute the @@ -161,8 +158,8 @@ where $Dz=\partial z/\partial s$ is the Jacobian of the embedding. The embedding induces a metric on $\tilde\Omega$ by $g=(Dz)^\dagger Dz$. Writing $\partial=\partial/\partial z$, this gives \begin{equation} \label{eq:flow} - \dot z=-\tfrac12(\partial\mathcal S)^\dagger(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T - =-\tfrac12(\partial \mathcal S)^\dagger P + \dot z=-\frac12(\partial\mathcal S)^\dagger(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T + =-\frac12(\partial \mathcal S)^\dagger P \end{equation} which is nothing but the projection of $(\partial\mathcal S)^*$ into the tangent space of the manifold, with $P=(Dz)^*[(Dz)^\dagger(Dz)]^{-1}(Dz)^T$. @@ -172,22 +169,20 @@ Gradient descent on $\operatorname{Re}\mathcal S$ is equivalent to Hamiltonian dynamics with the Hamiltonian $\operatorname{Im}\mathcal S$. This is because $(\tilde\Omega, g)$ is Kähler and therefore admits a symplectic structure, but that the flow conserves $\operatorname{Im}\mathcal S$ can be shown using -\eqref{eq:flow} and the holomorphic property of $\mathcal S$: -\begin{equation} - \begin{aligned} - \frac d{dt}&\operatorname{Im}\mathcal S - =\dot z\partial\operatorname{Im}\mathcal S+\dot z^*\partial^*\operatorname{Im}\mathcal S \\ - &=\frac i4\left( - (\partial \mathcal S)^\dagger P\partial\mathcal S-(\partial\mathcal S)^TP^*(\partial\mathcal S)^* - \right) \\ - &=\frac i4\left( - (\partial\mathcal S)^\dagger P\partial\mathcal S-[(\partial\mathcal S)^\dagger P\partial\mathcal S]^* - \right) \\ - &=\frac i4\left( - \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2 - \right)=0. - \end{aligned} -\end{equation} +\eref{eq:flow} and the holomorphic property of $\mathcal S$: +\begin{eqnarray} + \frac d{dt}\operatorname{Im}\mathcal S + &=\dot z\partial\operatorname{Im}\mathcal S+\dot z^*\partial^*\operatorname{Im}\mathcal S \\ + &=\frac i4\left( + (\partial \mathcal S)^\dagger P\partial\mathcal S-(\partial\mathcal S)^TP^*(\partial\mathcal S)^* + \right) \\ + &=\frac i4\left( + (\partial\mathcal S)^\dagger P\partial\mathcal S-[(\partial\mathcal S)^\dagger P\partial\mathcal S]^* + \right) \\ + &=\frac i4\left( + \|\partial\mathcal S\|^2-(\|\partial\mathcal S\|^*)^2 + \right)=0. +\end{eqnarray} As a result of this conservation law, surfaces of constant imaginary action will be important when evaluting the possible endpoints of trajectories. A consequence of this conservation is that the flow in the action takes a simple @@ -209,28 +204,24 @@ single simply connected surface, locally diffeomorphic to $\mathbb R^{2D-1}$. No (nondegenerate) critical point in the function at $z_0$. In the vicinity of the critical point, the flow is locally \begin{equation} - \begin{aligned} - \dot z - &\simeq-\tfrac12(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* - \end{aligned} + \dot z + \simeq-\frac12(\partial\partial\mathcal S|_{z=z_0})^\dagger P(z-z_0)^* \end{equation} The matrix $(\partial\partial\mathcal S)^\dagger P$ has a spectrum identical to that of $(\partial\partial\mathcal S)^\dagger$ save marginal directions corresponding to the normals to manifold. Assuming we are working in a diagonal basis, this becomes \begin{equation} - \dot z_i=-\tfrac12\lambda_i\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2) + \dot z_i=-\frac12\lambda_i\Delta z_i^*+O(\Delta z^2,(\Delta z^*)^2) \end{equation} Breaking into real and imaginary parts gives -\begin{equation} - \begin{aligned} - \frac{d\Delta x_i}{dt}&=-\frac12\left( - \operatorname{Re}\lambda_i\Delta x_i+\operatorname{Im}\beta\lambda_i\Delta y_i - \right) \\ - \frac{d\Delta y_i}{dt}&=-\frac12\left( - \operatorname{Im}\lambda_i\Delta x_i-\operatorname{Re}\beta\lambda_i\Delta y_i - \right) - \end{aligned} -\end{equation} +\begin{eqnarray} + \frac{d\Delta x_i}{dt}&=-\frac12\left( + \operatorname{Re}\lambda_i\Delta x_i+\operatorname{Im}\beta\lambda_i\Delta y_i + \right) \\ + \frac{d\Delta y_i}{dt}&=-\frac12\left( + \operatorname{Im}\lambda_i\Delta x_i-\operatorname{Re}\beta\lambda_i\Delta y_i + \right) +\end{eqnarray} Therefore, in the complex plane defined by each eigenvector of $(\partial\partial\mathcal S)^\dagger P$ there is a separatrix flow of the form in Figure \ref{fig:local_flow}. The effect of these separatrices in each complex @@ -243,7 +234,7 @@ lying in each of two quadrants corresponding with its value relative to that at the critical point. \begin{figure} - \includegraphics[width=\columnwidth]{figs/local_flow.pdf} + \includegraphics{figs/local_flow.pdf} \caption{ Gradient descent in the vicinity of a critical point, in the $z$--$z^*$ plane for an eigenvector $z$ of $(\partial\partial\mathcal S)^\dagger P$. The flow @@ -288,14 +279,17 @@ energies tend to be nearby, Stokes lines will proliferate, while in regions where critical points with the same energies tend to be distant compared to those with different energies, Stokes lines will be rare. -\textcolor{teal}{ - Here we make a generic argument that, for high-dimensional landscapes with - exponentially many critical points, the existence of exponentially many - Stokes points depends on the spectrum of the Hessian $\partial\partial H$ of - critical points. -} +\section{Analytic continuation} -\section{p-spin spherical models} +\begin{eqnarray} + Z(\beta) + &=\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)} \\ + &=\sum_{\sigma\in\Sigma_0}(-1)^{k_\sigma}\int_{\mathcal J_\sigma}ds\,e^{-\beta\mathcal S(s)} \\ + &=\sum_{\sigma\in\Sigma_0}(-1)^{k_\sigma}\left(\frac{2\pi}{\beta}\right)^{N/2}(\det\partial\partial\mathcal S(s_\sigma))^{-N/2}e^{-\beta\mathcal S_(s_\sigma)} \\ + &=\sum_k(-1)^k\int d\epsilon\,\mathcal N_k(\epsilon)\left(\frac{2\pi}{\beta}\right)^{N/2}\exp\left\{-\beta N\epsilon-\frac N2\int_0^\infty d\lambda\,\rho(\lambda\mid\epsilon)\log\lambda\right\} +\end{eqnarray} + +\section{The \textit{p}-spin spherical models} The $p$-spin spherical models are statistical mechanics models defined by the action $\mathcal S=-\beta H$ for the Hamiltonian @@ -310,7 +304,7 @@ $a_i=\delta_{ip}$ and $\kappa=1$. The phase space manifold $\Omega=\{x\mid x^2=N, x\in\mathbb R^N\}$ has a complex extension $\tilde\Omega=\{z\mid z^2=N, z\in\mathbb C^N\}$. The natural -extension of the hamiltonian \eqref{eq:p-spin.hamiltonian} to this complex manifold is +extension of the hamiltonian \eref{eq:p-spin.hamiltonian} to this complex manifold is holomorphic. The normal to this manifold at any point $z\in\tilde\Omega$ is always in the direction $z$. The projection operator onto the tangent space of this manifold is given by @@ -319,7 +313,7 @@ this manifold is given by \end{equation} where indeed $Pz=z-z|z|^2/|z|^2=0$ and $Pz'=z'$ for any $z'$ orthogonal to $z$. -To find critical points, we use the method of Lagrange multipliers. Introducing $\mu\in\mathbb C$, +To find critical points, we use the method of Lagrange multipliers. Introducing $\mu\in\mathbb C$, \subsection{2-spin} @@ -347,21 +341,17 @@ of generality, assume these are associated with the first and second cardinal directions. Since the gradient is proportional to $z$, any components that are zero at some time will be zero at all times. The dynamics for the components of interest assuming all others are zero are -\begin{equation} - \begin{aligned} - \dot z_1 - &=-z_1^*\left(d_1^*-\frac{d_1^*z_1^*z_1+d_2^*z_2^*z_2}{|z_1|^2+|z_2|^2}\right) \\ - &=-(d_1-d_2)^*z_1^*\frac{|z_2|^2}{|z_1|^2+|z_2|^2} - \end{aligned} -\end{equation} +\begin{eqnarray} + \dot z_1 + &=-z_1^*\left(d_1^*-\frac{d_1^*z_1^*z_1+d_2^*z_2^*z_2}{|z_1|^2+|z_2|^2}\right) \\ + &=-(d_1-d_2)^*z_1^*\frac{|z_2|^2}{|z_1|^2+|z_2|^2} +\end{eqnarray} and the same for $z_2$ with all indices swapped. Since $\Delta=d_1-d_2$ is real, if $z_1$ begins real it remains real, with the same for $z_2$. Since the critical points are at real $z$, we make this restriction, and find \begin{equation} - \begin{aligned} - \frac d{dt}(z_1^2+z_2^2)=0\\ - \frac d{dt}\frac{z_2}{z_1}=\Delta\frac{z_2}{z_1} - \end{aligned} + \frac d{dt}(z_1^2+z_2^2)=0 \qquad + \frac d{dt}\frac{z_2}{z_1}=\Delta\frac{z_2}{z_1} \end{equation} Therefore $z_2/z_1=e^{\Delta t}$, with $z_1^2+z_2^2=N$ as necessary. Depending on the sign of $\Delta$, $z$ flows from one critical point to the other over @@ -392,6 +382,18 @@ separatrix of a third. This means that when the imaginary energies of two critical points are brought to the same value, their surfaces of constant imaginary energy join. +\begin{eqnarray} + Z(\beta) + &=\int_{S^{N-1}}dx\,e^{-\beta H(x)} + =\int_{\mathbb R^N}dx\,\delta(x^2-N)e^{-\beta H(x)} \\ + &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12\beta x^TJx-\lambda(x^Tx-N)} \\ + &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12x^T(\beta J+\lambda I)x+\lambda N} \\ + &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\det(\beta J+\lambda I)}}e^{\lambda N} \\ + &=\frac1{2\pi}\int d\lambda\,\sqrt{\frac{(2\pi)^N}{\prod_i(\beta\lambda_i+\lambda)}}e^{\lambda N} \\ + &=(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac12\sum_i\log(\beta\lambda_i+\lambda)} \\ + &\simeq(2\pi)^{N/2-1}\int d\lambda\,e^{\lambda N-\frac N2\int d\lambda'\,\rho(\lambda')\log(\beta\lambda'+\lambda)} \\ +\end{eqnarray} + \subsection{Pure \textit{p}-spin} \begin{equation} @@ -444,10 +446,7 @@ imaginary energy join. To confirm the presence of Stokes lines under certain processes in the $p$-spin, we studied the problem numerically. -\begin{acknowledgments} - MIT mathematicians have been no help -\end{acknowledgments} - +\bibliographystyle{unsrt} \bibliography{stokes} \appendix |