diff options
| -rw-r--r-- | iopams.sty | 88 | ||||
| -rw-r--r-- | iopart10.clo | 125 | ||||
| -rw-r--r-- | iopart12.clo | 126 | ||||
| -rw-r--r-- | setstack.sty | 140 | ||||
| -rw-r--r-- | stokes.tex | 44 |
5 files changed, 519 insertions, 4 deletions
@@ -1 +1,87 @@ -%%
%% 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 +%% +%% 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'. diff --git a/iopart10.clo b/iopart10.clo index 1805a0b..dec5747 100644 --- a/iopart10.clo +++ b/iopart10.clo @@ -1 +1,124 @@ -%%
%% This is file `iopart10.clo'
%%
%% 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.
%%
%% \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 \~}
\ProvidesFile{iopart10.clo}[1997/01/13 v1.0 IOP Book file (size option)]
\renewcommand\normalsize{%
\@setfontsize\normalsize\@xpt\@xiipt
\abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@
\abovedisplayshortskip \z@ \@plus3\p@
\belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@
\belowdisplayskip \abovedisplayskip
\let\@listi\@listI}
\normalsize
\newcommand\small{%
\@setfontsize\small\@ixpt{11}%
\abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@
\abovedisplayshortskip \z@ \@plus2\p@
\belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@
\def\@listi{\leftmargin\leftmargini
\topsep 4\p@ \@plus2\p@ \@minus2\p@
\parsep 2\p@ \@plus\p@ \@minus\p@
\itemsep \parsep}%
\belowdisplayskip \abovedisplayskip
}
\newcommand\footnotesize{%
\@setfontsize\footnotesize\@viiipt{9.5}%
\abovedisplayskip 6\p@ \@plus2\p@ \@minus4\p@
\abovedisplayshortskip \z@ \@plus\p@
\belowdisplayshortskip 3\p@ \@plus\p@ \@minus2\p@
\def\@listi{\leftmargin\leftmargini
\topsep 3\p@ \@plus\p@ \@minus\p@
\parsep 2\p@ \@plus\p@ \@minus\p@
\itemsep \parsep}%
\belowdisplayskip \abovedisplayskip
}
\newcommand\scriptsize{\@setfontsize\scriptsize\@viipt\@viiipt}
\newcommand\tiny{\@setfontsize\tiny\@vpt\@vipt}
\newcommand\large{\@setfontsize\large\@xiipt{14}}
\newcommand\Large{\@setfontsize\Large\@xivpt{18}}
\newcommand\LARGE{\@setfontsize\LARGE\@xviipt{22}}
\newcommand\huge{\@setfontsize\huge\@xxpt{25}}
\newcommand\Huge{\@setfontsize\Huge\@xxvpt{30}}
\if@twocolumn
\setlength\parindent{12\p@}
\else
\setlength\parindent{15\p@}
\fi
\setlength\headheight{12\p@}
\setlength\headsep {12\p@}
\setlength\topskip {10\p@}
\setlength\footskip{20\p@}
\setlength\maxdepth{.5\topskip}
\setlength\@maxdepth\maxdepth
\setlength\textwidth{31pc}
\setlength\textheight{49pc}
\setlength\oddsidemargin {24\p@}
\setlength\evensidemargin {24\p@}
\setlength\marginparwidth {72\p@}
\setlength\marginparsep {10\p@}
\setlength\marginparpush{5\p@}
\setlength\topmargin{\z@}
\setlength\footnotesep{6.65\p@}
\setlength{\skip\footins} {9\p@ \@plus 4\p@ \@minus 2\p@}
\setlength\floatsep {12\p@ \@plus 2\p@ \@minus 2\p@}
\setlength\textfloatsep {20\p@ \@plus 2\p@ \@minus 4\p@}
\setlength\intextsep {12\p@ \@plus 2\p@ \@minus 2\p@}
\setlength\dblfloatsep {12\p@ \@plus 2\p@ \@minus 2\p@}
\setlength\dbltextfloatsep{20\p@ \@plus 2\p@ \@minus 4\p@}
\setlength\@fptop{0\p@}
\setlength\@fpsep{8\p@ \@plus 2fil}
\setlength\@fpbot{0\p@}
\setlength\@dblfptop{0\p@}
\setlength\@dblfpsep{8\p@ \@plus 2fil}
\setlength\@dblfpbot{0\p@}
\setlength\partopsep{2\p@ \@plus 1\p@ \@minus 1\p@}
\def\@listI{\leftmargin\leftmargini
\parsep=\z@
\topsep=5\p@ \@plus3\p@ \@minus3\p@
\itemsep=3\p@ \@plus2\p@ \@minus\p@}
\let\@listi\@listI
\@listi
\def\@listii {\leftmargin\leftmarginii
\labelwidth\leftmarginii
\advance\labelwidth-\labelsep
\topsep=2\p@ \@plus2\p@ \@minus\p@
\parsep=\z@
\itemsep=\parsep}
\def\@listiii{\leftmargin\leftmarginiii
\labelwidth\leftmarginiii
\advance\labelwidth-\labelsep
\topsep=\z@
\parsep=\z@
\partopsep=\z@
\itemsep=\z@}
\def\@listiv {\leftmargin\leftmarginiv
\labelwidth\leftmarginiv
\advance\labelwidth-\labelsep}
\def\@listv {\leftmargin\leftmarginv
\labelwidth\leftmarginv
\advance\labelwidth-\labelsep}
\def\@listvi {\leftmargin\leftmarginvi
\labelwidth\leftmarginvi
\advance\labelwidth-\labelsep}
\endinput
%%
%% End of file `iopart.clo'.
\ No newline at end of file +%% +%% This is file `iopart10.clo' +%% +%% 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. +%% +%% \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 \~} +\ProvidesFile{iopart10.clo}[1997/01/13 v1.0 IOP Book file (size option)] +\renewcommand\normalsize{% + \@setfontsize\normalsize\@xpt\@xiipt + \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ + \belowdisplayskip \abovedisplayskip + \let\@listi\@listI} +\normalsize +\newcommand\small{% + \@setfontsize\small\@ixpt{11}% + \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus2\p@ + \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 4\p@ \@plus2\p@ \@minus2\p@ + \parsep 2\p@ \@plus\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\newcommand\footnotesize{% + \@setfontsize\footnotesize\@viiipt{9.5}% + \abovedisplayskip 6\p@ \@plus2\p@ \@minus4\p@ + \abovedisplayshortskip \z@ \@plus\p@ + \belowdisplayshortskip 3\p@ \@plus\p@ \@minus2\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 3\p@ \@plus\p@ \@minus\p@ + \parsep 2\p@ \@plus\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\newcommand\scriptsize{\@setfontsize\scriptsize\@viipt\@viiipt} +\newcommand\tiny{\@setfontsize\tiny\@vpt\@vipt} +\newcommand\large{\@setfontsize\large\@xiipt{14}} +\newcommand\Large{\@setfontsize\Large\@xivpt{18}} +\newcommand\LARGE{\@setfontsize\LARGE\@xviipt{22}} +\newcommand\huge{\@setfontsize\huge\@xxpt{25}} +\newcommand\Huge{\@setfontsize\Huge\@xxvpt{30}} +\if@twocolumn + \setlength\parindent{12\p@} +\else + \setlength\parindent{15\p@} +\fi +\setlength\headheight{12\p@} +\setlength\headsep {12\p@} +\setlength\topskip {10\p@} +\setlength\footskip{20\p@} +\setlength\maxdepth{.5\topskip} +\setlength\@maxdepth\maxdepth +\setlength\textwidth{31pc} +\setlength\textheight{49pc} +\setlength\oddsidemargin {24\p@} +\setlength\evensidemargin {24\p@} +\setlength\marginparwidth {72\p@} +\setlength\marginparsep {10\p@} +\setlength\marginparpush{5\p@} +\setlength\topmargin{\z@} +\setlength\footnotesep{6.65\p@} +\setlength{\skip\footins} {9\p@ \@plus 4\p@ \@minus 2\p@} +\setlength\floatsep {12\p@ \@plus 2\p@ \@minus 2\p@} +\setlength\textfloatsep {20\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\intextsep {12\p@ \@plus 2\p@ \@minus 2\p@} +\setlength\dblfloatsep {12\p@ \@plus 2\p@ \@minus 2\p@} +\setlength\dbltextfloatsep{20\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\@fptop{0\p@} +\setlength\@fpsep{8\p@ \@plus 2fil} +\setlength\@fpbot{0\p@} +\setlength\@dblfptop{0\p@} +\setlength\@dblfpsep{8\p@ \@plus 2fil} +\setlength\@dblfpbot{0\p@} +\setlength\partopsep{2\p@ \@plus 1\p@ \@minus 1\p@} +\def\@listI{\leftmargin\leftmargini + \parsep=\z@ + \topsep=5\p@ \@plus3\p@ \@minus3\p@ + \itemsep=3\p@ \@plus2\p@ \@minus\p@} +\let\@listi\@listI +\@listi +\def\@listii {\leftmargin\leftmarginii + \labelwidth\leftmarginii + \advance\labelwidth-\labelsep + \topsep=2\p@ \@plus2\p@ \@minus\p@ + \parsep=\z@ + \itemsep=\parsep} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii + \advance\labelwidth-\labelsep + \topsep=\z@ + \parsep=\z@ + \partopsep=\z@ + \itemsep=\z@} +\def\@listiv {\leftmargin\leftmarginiv + \labelwidth\leftmarginiv + \advance\labelwidth-\labelsep} +\def\@listv {\leftmargin\leftmarginv + \labelwidth\leftmarginv + \advance\labelwidth-\labelsep} +\def\@listvi {\leftmargin\leftmarginvi + \labelwidth\leftmarginvi + \advance\labelwidth-\labelsep} +\endinput +%% +%% End of file `iopart.clo'. diff --git a/iopart12.clo b/iopart12.clo index 48ec5c1..d392bf9 100644 --- a/iopart12.clo +++ b/iopart12.clo @@ -1 +1,125 @@ -%%
%% This is file `iopart12.clo'
%%
%% 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.
%%
%% \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 \~}
\ProvidesFile{iopart12.clo}[1997/01/15 v1.0 LaTeX2e file (size option)]
\renewcommand\normalsize{%
\@setfontsize\normalsize\@xiipt{16}%
\abovedisplayskip 12\p@ \@plus3\p@ \@minus7\p@
\abovedisplayshortskip \z@ \@plus3\p@
\belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@
\belowdisplayskip \abovedisplayskip
\let\@listi\@listI}
\normalsize
\newcommand\small{%
\@setfontsize\small\@xipt{14}%
\abovedisplayskip 11\p@ \@plus3\p@ \@minus6\p@
\abovedisplayshortskip \z@ \@plus3\p@
\belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@
\def\@listi{\leftmargin\leftmargini
\topsep 9\p@ \@plus3\p@ \@minus5\p@
\parsep 4.5\p@ \@plus2\p@ \@minus\p@
\itemsep \parsep}%
\belowdisplayskip \abovedisplayskip
}
\newcommand\footnotesize{%
% \@setfontsize\footnotesize\@xpt\@xiipt
\@setfontsize\footnotesize\@xpt{13}%
\abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@
\abovedisplayshortskip \z@ \@plus3\p@
\belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@
\def\@listi{\leftmargin\leftmargini
\topsep 6\p@ \@plus2\p@ \@minus2\p@
\parsep 3\p@ \@plus2\p@ \@minus\p@
\itemsep \parsep}%
\belowdisplayskip \abovedisplayskip
}
\newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9.5}}
\newcommand\tiny{\@setfontsize\tiny\@vipt\@viipt}
\newcommand\large{\@setfontsize\large\@xivpt{18}}
\newcommand\Large{\@setfontsize\Large\@xviipt{22}}
\newcommand\LARGE{\@setfontsize\LARGE\@xxpt{25}}
\newcommand\huge{\@setfontsize\huge\@xxvpt{30}}
\let\Huge=\huge
\if@twocolumn
\setlength\parindent{14\p@}
\else
\setlength\parindent{18\p@}
\fi
\setlength\headheight{14\p@}
\setlength\headsep{14\p@}
\setlength\topskip{12\p@}
\setlength\footskip{24\p@}
\setlength\maxdepth{.5\topskip}
\setlength\@maxdepth\maxdepth
\setlength\textwidth{37.2pc}
\setlength\textheight{56pc}
\setlength\oddsidemargin {\z@}
\setlength\evensidemargin {\z@}
\setlength\marginparwidth {72\p@}
\setlength\marginparsep{10\p@}
\setlength\marginparpush{5\p@}
\setlength\topmargin{-12pt}
\setlength\footnotesep{8.4\p@}
\setlength{\skip\footins} {10.8\p@ \@plus 4\p@ \@minus 2\p@}
\setlength\floatsep {14\p@ \@plus 2\p@ \@minus 4\p@}
\setlength\textfloatsep {24\p@ \@plus 2\p@ \@minus 4\p@}
\setlength\intextsep {16\p@ \@plus 4\p@ \@minus 4\p@}
\setlength\dblfloatsep {16\p@ \@plus 2\p@ \@minus 4\p@}
\setlength\dbltextfloatsep{24\p@ \@plus 2\p@ \@minus 4\p@}
\setlength\@fptop{0\p@}
\setlength\@fpsep{10\p@ \@plus 1fil}
\setlength\@fpbot{0\p@}
\setlength\@dblfptop{0\p@}
\setlength\@dblfpsep{10\p@ \@plus 1fil}
\setlength\@dblfpbot{0\p@}
\setlength\partopsep{3\p@ \@plus 2\p@ \@minus 2\p@}
\def\@listI{\leftmargin\leftmargini
\parsep=\z@
\topsep=6\p@ \@plus3\p@ \@minus3\p@
\itemsep=3\p@ \@plus2\p@ \@minus1\p@}
\let\@listi\@listI
\@listi
\def\@listii {\leftmargin\leftmarginii
\labelwidth\leftmarginii
\advance\labelwidth-\labelsep
\topsep=3\p@ \@plus2\p@ \@minus\p@
\parsep=\z@
\itemsep=\parsep}
\def\@listiii{\leftmargin\leftmarginiii
\labelwidth\leftmarginiii
\advance\labelwidth-\labelsep
\topsep=\z@
\parsep=\z@
\partopsep=\z@
\itemsep=\z@}
\def\@listiv {\leftmargin\leftmarginiv
\labelwidth\leftmarginiv
\advance\labelwidth-\labelsep}
\def\@listv{\leftmargin\leftmarginv
\labelwidth\leftmarginv
\advance\labelwidth-\labelsep}
\def\@listvi {\leftmargin\leftmarginvi
\labelwidth\leftmarginvi
\advance\labelwidth-\labelsep}
\endinput
%%
%% End of file `iopart12.clo'.
\ No newline at end of file +%% +%% This is file `iopart12.clo' +%% +%% 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. +%% +%% \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 \~} +\ProvidesFile{iopart12.clo}[1997/01/15 v1.0 LaTeX2e file (size option)] +\renewcommand\normalsize{% + \@setfontsize\normalsize\@xiipt{16}% + \abovedisplayskip 12\p@ \@plus3\p@ \@minus7\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ + \belowdisplayskip \abovedisplayskip + \let\@listi\@listI} +\normalsize +\newcommand\small{% + \@setfontsize\small\@xipt{14}% + \abovedisplayskip 11\p@ \@plus3\p@ \@minus6\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 9\p@ \@plus3\p@ \@minus5\p@ + \parsep 4.5\p@ \@plus2\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\newcommand\footnotesize{% +% \@setfontsize\footnotesize\@xpt\@xiipt + \@setfontsize\footnotesize\@xpt{13}% + \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ + \abovedisplayshortskip \z@ \@plus3\p@ + \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ + \def\@listi{\leftmargin\leftmargini + \topsep 6\p@ \@plus2\p@ \@minus2\p@ + \parsep 3\p@ \@plus2\p@ \@minus\p@ + \itemsep \parsep}% + \belowdisplayskip \abovedisplayskip +} +\newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9.5}} +\newcommand\tiny{\@setfontsize\tiny\@vipt\@viipt} +\newcommand\large{\@setfontsize\large\@xivpt{18}} +\newcommand\Large{\@setfontsize\Large\@xviipt{22}} +\newcommand\LARGE{\@setfontsize\LARGE\@xxpt{25}} +\newcommand\huge{\@setfontsize\huge\@xxvpt{30}} +\let\Huge=\huge +\if@twocolumn + \setlength\parindent{14\p@} +\else + \setlength\parindent{18\p@} +\fi +\setlength\headheight{14\p@} +\setlength\headsep{14\p@} +\setlength\topskip{12\p@} +\setlength\footskip{24\p@} +\setlength\maxdepth{.5\topskip} +\setlength\@maxdepth\maxdepth +\setlength\textwidth{37.2pc} +\setlength\textheight{56pc} +\setlength\oddsidemargin {\z@} +\setlength\evensidemargin {\z@} +\setlength\marginparwidth {72\p@} +\setlength\marginparsep{10\p@} +\setlength\marginparpush{5\p@} +\setlength\topmargin{-12pt} +\setlength\footnotesep{8.4\p@} +\setlength{\skip\footins} {10.8\p@ \@plus 4\p@ \@minus 2\p@} +\setlength\floatsep {14\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\textfloatsep {24\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\intextsep {16\p@ \@plus 4\p@ \@minus 4\p@} +\setlength\dblfloatsep {16\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\dbltextfloatsep{24\p@ \@plus 2\p@ \@minus 4\p@} +\setlength\@fptop{0\p@} +\setlength\@fpsep{10\p@ \@plus 1fil} +\setlength\@fpbot{0\p@} +\setlength\@dblfptop{0\p@} +\setlength\@dblfpsep{10\p@ \@plus 1fil} +\setlength\@dblfpbot{0\p@} +\setlength\partopsep{3\p@ \@plus 2\p@ \@minus 2\p@} +\def\@listI{\leftmargin\leftmargini + \parsep=\z@ + \topsep=6\p@ \@plus3\p@ \@minus3\p@ + \itemsep=3\p@ \@plus2\p@ \@minus1\p@} +\let\@listi\@listI +\@listi +\def\@listii {\leftmargin\leftmarginii + \labelwidth\leftmarginii + \advance\labelwidth-\labelsep + \topsep=3\p@ \@plus2\p@ \@minus\p@ + \parsep=\z@ + \itemsep=\parsep} +\def\@listiii{\leftmargin\leftmarginiii + \labelwidth\leftmarginiii + \advance\labelwidth-\labelsep + \topsep=\z@ + \parsep=\z@ + \partopsep=\z@ + \itemsep=\z@} +\def\@listiv {\leftmargin\leftmarginiv + \labelwidth\leftmarginiv + \advance\labelwidth-\labelsep} +\def\@listv{\leftmargin\leftmarginv + \labelwidth\leftmarginv + \advance\labelwidth-\labelsep} +\def\@listvi {\leftmargin\leftmarginvi + \labelwidth\leftmarginvi + \advance\labelwidth-\labelsep} +\endinput +%% +%% End of file `iopart12.clo'. diff --git a/setstack.sty b/setstack.sty index 1d0130f..da428a6 100644 --- a/setstack.sty +++ b/setstack.sty @@ -1 +1,139 @@ -%%
%% This is file `setstack.sty', created by VIK 15 Dec 1998
%% Reproduces useful macros from amsmath.sty, thus avoiding the need to load the entire
%% amsmath.sty and run into conflicts
%% Adds definitions for \overset, \underset, \sideset, \substack, \boxed, \leftroot,
%% \uproot, \dddot, \ddddot, \varrow, \harrow (see The LateX Companion, pp 225-227)
\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX)
[1998/12/15]% LaTeX date must December 1998 or later
\ProvidesPackage{setstack}
\DeclareRobustCommand{\text}{%
\ifmmode\expandafter\text@\else\expandafter\mbox\fi}
\let\nfss@text\text
\def\text@#1{\mathchoice
{\textdef@\displaystyle\f@size{#1}}%
{\textdef@\textstyle\tf@size{\firstchoice@false #1}}%
{\textdef@\textstyle\sf@size{\firstchoice@false #1}}%
{\textdef@\textstyle \ssf@size{\firstchoice@false #1}}%
\check@mathfonts
}
\def\textdef@#1#2#3{\hbox{{%
\everymath{#1}%
\let\f@size#2\selectfont #3}}}
% adds underset, overset, sideset and
% substack features from amsmath.sty (Companion p. 226)
\def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag}
\def\dft@tag{\invalid@tag{\string\tag\space not allowed here}}
\def\default@tag{\let\tag\dft@tag}
\def\Let@{\let\\\math@cr}
\def\restore@math@cr{\def\math@cr@@@{\cr}}
\def\overset#1#2{\binrel@{#2}%
\binrel@@{\mathop{\kern\z@#2}\limits^{#1}}}
\def\underset#1#2{\binrel@{#2}%
\binrel@@{\mathop{\kern\z@#2}\limits_{#1}}}
\def\sideset#1#2#3{%
\@mathmeasure\z@\displaystyle{#3}%
\global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@
\setbox\tw@\box\@ne
\@mathmeasure4\displaystyle{\copy\tw@#1}%
\@mathmeasure6\displaystyle{#3\nolimits#2}%
\dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@
\hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}%
}
\newenvironment{subarray}[1]{%
\vcenter\bgroup
\Let@ \restore@math@cr \default@tag
\baselineskip\fontdimen10 \scriptfont\tw@
\advance\baselineskip\fontdimen12 \scriptfont\tw@
\lineskip\thr@@\fontdimen8 \scriptfont\thr@@
\lineskiplimit\lineskip
\ialign\bgroup\ifx c#1\hfil\fi
$\m@th\scriptstyle##$\hfil\crcr
}{%
\crcr\egroup\egroup
}
\newcommand{\substack}[1]{\subarray{c}#1\endsubarray}
% definitions of dddot and ddddot(p. 225)
\def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
\hbox{\normalfont ...}\vss}}}}
\def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@
\hbox{\normalfont....}\vss}}}}
% definitions of leftroot, uproot (p.225)
\begingroup \catcode`\"=12
\gdef\@@sqrt#1{\radical"270370 {#1}}
\endgroup
\def\leftroot{\@amsmath@err{\Invalid@@\leftroot}\@eha}
\def\uproot{\@amsmath@err{\Invalid@@\uproot}\@eha}
\newcount\uproot@
\newcount\leftroot@
\def\root{\relaxnext@
\DN@{\ifx\@let@token\uproot\let\next@\nextii@\else
\ifx\@let@token\leftroot\let\next@\nextiii@\else
\let\next@\plainroot@\fi\fi\next@}%
\def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}%
\def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else
\DN@.{\FN@\nextv@}\fi\next@.}%
\def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else
\let\next@\plainroot@\fi\next@}%
\def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}%
\def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}%
\def\nextvii@{\ifx\@let@token\@sptoken
\DN@. {\FN@\nextviii@}\else
\DN@.{\FN@\nextviii@}\fi\next@.}%
\def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else
\let\next@\plainroot@\fi\next@}%
\def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}%
\bgroup\uproot@\z@\leftroot@\z@\FN@\next@}
\def\plainroot@#1\of#2{\setbox\rootbox\hbox{%
$\m@th\scriptscriptstyle{#1}$}%
\mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}}
{\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup}
\def\r@@t#1#2{\setboxz@h{$\m@th#1\@@sqrt{#2}$}%
\dimen@\ht\z@\advance\dimen@-\dp\z@
\setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}%
\advance\dimen@ by1.667\wd\@ne
\mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox
\mkern-10mu\mkern\leftroot@ mu\boxz@}
% definition of \boxed (math in frame, no dollars) p 225
\def\boxed#1{\fbox{\m@th$\displaystyle#1$}}
% definition of \smash for top and bottom parts of expression in braces (p.227)
\renewcommand{\smash}[2][tb]{%
\def\smash@{#1}%
\ifmmode\@xp\mathpalette\@xp\mathsm@sh\else
\@xp\makesm@sh\fi{#2}}
% additional difinitions for arrows
% zero mm wide #2 mm long vertical arrow shifted 1 mm to left, 1 mm up
\newcommand{\varrow}[2]{%
\unitlength=1mm
\begin{picture}(0,6) % (0,0)
% - for #1 to point arrow down,+ to point arrow up
\end{picture}%
\put(0,6){\vector(0,#1 3){#2}} % 6,- for down, 0, + for up
}
% 1 mm high #2 mm long horizontal arrow shifted 1 mm up (0,-1)
\newcommand{\harrow}[2]{%
\unitlength=1mm
\begin{picture}(8,1)(0,-1) %
% 1 mm distance between arrow and stackreled object over it (8,1)
\put(0,0){\vector(#1 2,0){#2}} % - to point arrow left, + right
\end{picture}%
}
\endinput
%% end of setstack.sty
%% Corrections history:
%%
\ No newline at end of file +%% +%% This is file `setstack.sty', created by VIK 15 Dec 1998 +%% Reproduces useful macros from amsmath.sty, thus avoiding the need to load the entire +%% amsmath.sty and run into conflicts +%% Adds definitions for \overset, \underset, \sideset, \substack, \boxed, \leftroot, +%% \uproot, \dddot, \ddddot, \varrow, \harrow (see The LateX Companion, pp 225-227) +\NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) +[1998/12/15]% LaTeX date must December 1998 or later +\ProvidesPackage{setstack} +\DeclareRobustCommand{\text}{% + \ifmmode\expandafter\text@\else\expandafter\mbox\fi} + \let\nfss@text\text + \def\text@#1{\mathchoice + {\textdef@\displaystyle\f@size{#1}}% + {\textdef@\textstyle\tf@size{\firstchoice@false #1}}% + {\textdef@\textstyle\sf@size{\firstchoice@false #1}}% + {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}% + \check@mathfonts +} +\def\textdef@#1#2#3{\hbox{{% + \everymath{#1}% + \let\f@size#2\selectfont #3}}} + +% adds underset, overset, sideset and +% substack features from amsmath.sty (Companion p. 226) + +\def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag} +\def\dft@tag{\invalid@tag{\string\tag\space not allowed here}} +\def\default@tag{\let\tag\dft@tag} + +\def\Let@{\let\\\math@cr} +\def\restore@math@cr{\def\math@cr@@@{\cr}} + +\def\overset#1#2{\binrel@{#2}% +\binrel@@{\mathop{\kern\z@#2}\limits^{#1}}} +\def\underset#1#2{\binrel@{#2}% +\binrel@@{\mathop{\kern\z@#2}\limits_{#1}}} +\def\sideset#1#2#3{% +\@mathmeasure\z@\displaystyle{#3}% +\global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@ + \setbox\tw@\box\@ne +\@mathmeasure4\displaystyle{\copy\tw@#1}% +\@mathmeasure6\displaystyle{#3\nolimits#2}% +\dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@ +\hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}% +} + +\newenvironment{subarray}[1]{% +\vcenter\bgroup +\Let@ \restore@math@cr \default@tag +\baselineskip\fontdimen10 \scriptfont\tw@ +\advance\baselineskip\fontdimen12 \scriptfont\tw@ +\lineskip\thr@@\fontdimen8 \scriptfont\thr@@ +\lineskiplimit\lineskip +\ialign\bgroup\ifx c#1\hfil\fi +$\m@th\scriptstyle##$\hfil\crcr +}{% +\crcr\egroup\egroup +} +\newcommand{\substack}[1]{\subarray{c}#1\endsubarray} + +% definitions of dddot and ddddot(p. 225) + +\def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ + \hbox{\normalfont ...}\vss}}}} +\def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ + \hbox{\normalfont....}\vss}}}} + +% definitions of leftroot, uproot (p.225) + +\begingroup \catcode`\"=12 +\gdef\@@sqrt#1{\radical"270370 {#1}} +\endgroup + +\def\leftroot{\@amsmath@err{\Invalid@@\leftroot}\@eha} +\def\uproot{\@amsmath@err{\Invalid@@\uproot}\@eha} +\newcount\uproot@ +\newcount\leftroot@ +\def\root{\relaxnext@ + \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else + \ifx\@let@token\leftroot\let\next@\nextiii@\else + \let\next@\plainroot@\fi\fi\next@}% + \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}% + \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else + \DN@.{\FN@\nextv@}\fi\next@.}% + \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else + \let\next@\plainroot@\fi\next@}% + \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}% + \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}% + \def\nextvii@{\ifx\@let@token\@sptoken + \DN@. {\FN@\nextviii@}\else + \DN@.{\FN@\nextviii@}\fi\next@.}% + \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else + \let\next@\plainroot@\fi\next@}% + \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}% + \bgroup\uproot@\z@\leftroot@\z@\FN@\next@} +\def\plainroot@#1\of#2{\setbox\rootbox\hbox{% + $\m@th\scriptscriptstyle{#1}$}% + \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}} + {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup} +\def\r@@t#1#2{\setboxz@h{$\m@th#1\@@sqrt{#2}$}% + \dimen@\ht\z@\advance\dimen@-\dp\z@ + \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}% + \advance\dimen@ by1.667\wd\@ne + \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox + \mkern-10mu\mkern\leftroot@ mu\boxz@} + +% definition of \boxed (math in frame, no dollars) p 225 + +\def\boxed#1{\fbox{\m@th$\displaystyle#1$}} + +% definition of \smash for top and bottom parts of expression in braces (p.227) +\renewcommand{\smash}[2][tb]{% + \def\smash@{#1}% + \ifmmode\@xp\mathpalette\@xp\mathsm@sh\else + \@xp\makesm@sh\fi{#2}} + +% additional difinitions for arrows +% zero mm wide #2 mm long vertical arrow shifted 1 mm to left, 1 mm up +\newcommand{\varrow}[2]{% +\unitlength=1mm + \begin{picture}(0,6) % (0,0) + % - for #1 to point arrow down,+ to point arrow up + \end{picture}% + \put(0,6){\vector(0,#1 3){#2}} % 6,- for down, 0, + for up +} +% 1 mm high #2 mm long horizontal arrow shifted 1 mm up (0,-1) +\newcommand{\harrow}[2]{% +\unitlength=1mm + \begin{picture}(8,1)(0,-1) % + % 1 mm distance between arrow and stackreled object over it (8,1) + \put(0,0){\vector(#1 2,0){#2}} % - to point arrow left, + right + \end{picture}% +} + +\endinput +%% end of setstack.sty +%% Corrections history: +%% @@ -40,6 +40,50 @@ \maketitle +\section{Preamble} + +Complex landscapes are basically functions of many variables having many minima and, inevitably, +many saddles of all indices, i.e. the number of unstable directions. Optimization problems +require us to find the deepest minima, often a difficult task. +For example, particles with a repulsive mutual potential enclosed in a box will have many stable +configurations, and we are asked to find the one with lowest energy. + +An aim of complexity theory is to be able to classify these landscapes in families having +common properties. Two simplifications make the task potentially tractable. The first is to consider +the limit of many variables. In the example of the particles, the limit of many particles (i.e. the thermodynamic limit) +may be expected to bring about simplifications. +The second simplification is of more technical nature: we consider functions that contain some random element to them, and we +study the average of an ensemble. The paradigm of this is the spin-glass, where the interactions are random, and we are asked to find the +ground state energy {\em on average over randomness}. + +Spin glass theory gave a surprise: random landscapes come in two kinds: those that +have a `threshold level' of energy, below which there are many minima but no saddles, +separated by high barriers, and those that have all sorts of saddles all the way down to the lowest +energy levels, and local minima are separated by relatively small barriers. +The latter are still complex, but good solutions are easier to find. +This classification is closely related to the structure of their Replica Trick solutions. +Armed with this solvable (random) example, it was easy to find non-random examples +that behave, at least approximately, in these two ways (e.g. sphere packings and the Travelling salesman Problem, +belong to first and second classes, respectively). + +What about systems whose variables are not real, but rather complex? +Recalling the Cauchy-Riemann conditions, we immediately see a difficulty: if our cost is, say, the real part of a function of +$N$ complex variables, in terms of the corresponding $2N$ real variables it has only saddles of index $N$! +Even worse: often not all saddles are equally interesting, so simply finding the lowest is not usually what we +need to do (more about this below). +As it turns out, there is a set of interesting questions to ask, as we describe below. For each saddle, there is a `thimble' +spanned by the +lines along which the cost function decreases. The way in which these thimbles fill the complex space is crucial for many +problems of analytic continuation, and is thus what we need to study. The central role played by saddles +in a real landscape, the `barriers', is now played by the Stokes lines, where thimbles exchange their properties. +Perhaps not surprisingly, the two classes of real landscapes described above retain their role in the complex case, but now +the distinction is that while in the first class the Stokes lines are rare, in the second class they proliferate. + +In this paper we shall start from a many-variable integral of a real function, and deform it in the many variable complex space. +The landscape one faces is the full one in this space, and we shall see that this is an example where the proliferation -- or lack of it -- +of Stokes lines is the interesting quantity in this context. + + \section{Introduction} Analytic continuation of physical theories is sometimes useful. Some theories |