Content-Type: multipart/mixed; boundary="-------------0405310510663" This is a multi-part message in MIME format. ---------------0405310510663 Content-Type: text/plain; name="04-171.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-171.keywords" Lippmann-Schwinger equation; Gamow vectors ---------------0405310510663 Content-Type: application/x-tex; name="03lisbon.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="03lisbon.tex" \documentstyle[proceedings,numreferences,epsfig]{tasy} \newcommand{\stt}{\small\tt} % This document needs the tasy style file to create a % document with font size 12pts. %\begin{document} \begin{opening} \title{Two analytic continuations of the Lippmann-Schwinger eigenfunctions} \author{Rafael de la Madrid} \institute{Departamento de F\'\i sica Te\'orica \\ Facultad de Ciencias \\ Universidad del Pa\'\i s Vasco \\ 48080 Bilbao, Spain \\ E-mail: wtbdemor@lg.ehu.es} \end{opening} \runningtitle{Two analytic continuations} \begin{document} \begin{abstract} We first present two possible analytic continuations of the Lippmann-Schwinger eigenfunctions to the second sheet of the Riemann surface, and then we compare the different Gamow vectors that are obtained through each analytic continuation. \end{abstract} \section{Introduction} Gamow vectors should be related with the analytic continuation of the Lippmann-Schwinger eigenfunctions to the resonance energy. However, the analytic continuation of the Lippmann-Schwinger eigenfunctions to the second sheet is not unique. In this paper, we address this non-uniqueness by constructing and comparing two possible analytic continuations. %\par \section{Preliminaries} Before proceeding with their analytic continuations, we recall some of the basic properties of the Lippmann-Schwinger eigenfunctions. Let us take a simple example, such as the spherical shell potential for zero angular momentum: \begin{equation} V(r)=\left\{ \begin{array}{ll} 0 &00 \label{wavenumber} \end{equation} is the wave number. In terms of $E$, the boundary conditions imposed by the $\pm i\epsilon$ are tantamount to obtaining boundary values from above ($+i\epsilon$) and from below ($-i\epsilon$) the spectrum of $H$. Thus, the Lippmann-Schwinger eigenfunctions are to be seen as boundary values from above and from below the cut. This is analogous to the calculation of the advanced and retarded Green functions $G^{\pm}(r,s;E)$. These functions are boundary values on the upper and lower rims of the cut of the Green function $G(r,s;z)$: \begin{equation} G^{\pm}(r,s;E) = \lim_{{\rm Im}(z) \to 0}G(r,s;z) \, , \qquad z=E\pm i {\rm Im}(z), \ {\rm Im}(z) >0 \, . \label{gpmE} \end{equation} As is well known, for the potential (\ref{potential}) all these functions (the Green function, the $S$ matrix, and the Lippmann-Schwinger eigenfunctions) depend explicitly not on $E$ but on $k$. This is why it is convenient to work with $k$ rather than with $E$. In terms of $k$, the Lippmann-Schwinger eigenfunctions (\ref{pmeigndu}) read as \begin{equation} \chi ^{\pm}(r;k)= \sqrt{\frac{1}{\pi} \, \frac{2m/\hbar ^2}{k}\,} \, \frac{\chi (r;k)}{{\cal J}_{\pm}(k)}\, . \label{kpmeigndu} \end{equation} These eigenfunctions are normalized to $\delta (E-E')$. In order to normalize them to $\delta (k-k')$, we define \begin{equation} \phi ^{\pm}(r;k) \equiv \sqrt{\frac{\hbar ^2}{2m} 2k\,} \chi ^{\pm}(r;k) =\sqrt{\frac{2}{\pi}} \, \frac{\chi (r;k)}{{\cal J}_{\pm}(k)}\, . \label{phpmeigndu} \end{equation} It is important to notice that in this equation, as well as in Eqs.~(\ref{bcoLS5}) and (\ref{bcoLS6}), $k$ is {\it positive}. We also can write Eq.~(\ref{gpmE}) in terms of $k$. If we denote the complex wave number by $q\equiv k+i{\rm Im}(q)$, we have that \begin{equation} G^{+}(r,s;k) = \lim_{{\rm Im}(q) \to 0}G(r,s;q) \, , \qquad q=k+ i {\rm Im}(q), \ {\rm Im}(q),k >0 \, . \label{gpk} \end{equation} Thus, $G^+$ is the boundary value of $G$ on the positive real line. In the same vein, $G^-$ is the boundary value of $G$ on the negative real line: \begin{equation} G^{-}(r,s;-k) = \lim_{{\rm Im}(q) \to 0}G(r,s;q) \, , \qquad q=-k+ i {\rm Im}(q), \ {\rm Im}(q),k >0 \, . \label{gmk} \end{equation} To finish this section, we recall that it is possible to obtain one of the Jost functions from the other one by way of the following relation: \begin{equation} {\cal J}_+(k)= {\cal J}_-(-k) \, , \quad k>0 \, . \label{jpminof} \end{equation} It also holds that \begin{equation} - \chi (r;k)= \chi (r;-k) \, , \quad k>0 \, . \label{siofchi} \end{equation} Equations~(\ref{phpmeigndu}), (\ref{jpminof}) and (\ref{siofchi}) yield \begin{equation} \phi ^+(r;k)= -\phi ^-(r;-k) \, , \quad k>0 \, ; \label{siofphi} \end{equation} that is, we can obtain one of the Lippmann-Schwinger eigenfunctions from the other one. \section{Two possible analytic continuations} After recalling some basic results in the previous section, we are in a position to analytically continue the Lippmann-Schwinger eigenfunctions into the second sheet of the Riemann surface. We shall examine two possible analytic continuations, which will be referred to as Continuation~1 and Continuation~2. \subsection{Continuation~1} %Let us start with the boundary conditions %(\ref{bcoLS5}) and (\ref{bcoLS6}). The asymptotic behavior of %Eq.~(\ref{bcoLS5}) means that far from the potential region, %$\langle r|E^+\rangle$ is a combination of an incoming spherical wave %$e^{-ikr}$ with an outgoing spherical wave $e^{ikr}$ multiplied %by $S(k)$. This interpretation assumes that $k$ is positive, because if $k$ %were negative, the interpretation would be the opposite. Analogously, the %asymptotic behavior of Eq.~(\ref{bcoLS6}) means that far from the potential %region, $\langle r|E^- \rangle$ is a linear combination of an outgoing wave %$e^{ikr}$ with an incoming wave $e^{-ikr}$ multiplied by $S^*(k)$. By means of Eq.~(\ref{siofphi}), we can obtain one of the Lippmann-Schwinger eigenfunctions in terms of the other one. What is more, in analogy with Eqs.~(\ref{gpk}) and (\ref{gmk}), the Lippmann-Schwinger eigenfunctions should be seen as limiting values on the positive and on the negative real $k$-axis of one and the same function, which we shall denote by $\phi _1$. In order to construct $\phi _1$, let us define the Jost function ${\cal J}_1(q)$ in such a way that ${\cal J}_1(q)$ coincides with ${\cal J}_+(k)$ on the positive $k$-axis and with ${\cal J}_-(k)$ on the negative $k$-axis.\footnote{Obviously, ${\cal J}_1(q)$ is just ${\cal J}_+(q)$, although it is better to use the subindexes $+/-$ only to refer to boundary values.} The eigenfunction $\phi _1$ is defined as follows:\footnote{The subindex 1 refers to Continuation~1.} \begin{equation} \phi _1 (r;q):= \sqrt{\frac{2}{\pi}} \, \frac{\chi (r;q)}{{\cal J}_{1}(q)} \, . \label{phpmeigndu1} \end{equation} Then, \begin{equation} \phi _1^+(r;k)=\lim_{{\rm Im}(q) \to 0} \phi _1 (r;q) \, , \quad q=k+i{\rm Im}(q) \, , \ k>0 \, , \end{equation} and \begin{equation} \phi _1^-(r;-k)=\lim_{{\rm Im}(q) \to 0} -\phi _1 (r;q) \, , \quad q=-k+i{\rm Im}(q) \, , \ k>0 \, . \end{equation} That is, we have obtained the Lippmann-Schwinger eigenfunctions as boundary values of a unique eigenfunction $\phi _1 (r;q)$ that is defined for all complex $q$. The function $\phi _1^+(r;k)$ is the boundary value on the positive $k$-axis, whereas $\phi _1^-(r;-k)$ is the boundary value on the negative $k$-axis. Now, to reach the resonance poles, we continue analytically $\phi _1^+(r;k)$ into the fourth quadrant and $\phi _1^-(r;k)$ into the third quadrant. Clearly, this looks like the natural analytic continuation. However, resonances seem to need a different analytic continuation, as explained in the next subsection. \subsection{Continuation~2} %Let's go back to the boundary conditions (\ref{bcoLS5})-(\ref{bcoLS6}). That %$e^{ikr}$ is an outgoing wave happens only when $k$ is positive. When %$k>0$, the outgoing wave can be written as %\begin{equation} % e^{-i(-k)r} \, , \quad k>0 \, . % \label{ouwkn} %\end{equation} %As long as $k$ is positive, the wave (\ref{ouwkn}) is outgoing. In the same %vein, the incoming wave $e^{-ikr}$ can be written as %\begin{equation} % e^{i(-k)r} \, , \quad k>0 \, . % \label{iuwkn} %\end{equation} %As long as $k$ is positive, the wave (\ref{iuwkn}) is incoming. By using this %somewhat awkward notation, the asymptotic behaviors of %Eqs.~(\ref{bcoLS5})-(\ref{bcoLS6}) can be written as %\begin{equation} % \langle r|E^+\rangle \sim % e^{i(-k)r}- S(-k)^*e^{-i(-k)r} \ {\rm as \ } r\to \infty \, , % \label{bcoLS5w} %\end{equation} %\begin{equation} % \langle r|E^-\rangle \sim % e^{ikr} - S^*(k)e^{-ikr} {\rm \ as \ } r\to \infty \, , % \label{bcoLS6w} %\end{equation} The second analytic continuation consists of continuing $\chi ^+(r;E)$ not to the lower half-plane of the second sheet as in Continuation~1, but to the upper half-plane of the second sheet. In terms of $k$, this means that instead of continuing $\phi ^+(r;k)$ into the fourth quadrant as in Continuation~1, we analytically continue $\phi ^+(r;k)$ into the third quadrant. For the out states, Continuation~2 means that $\chi ^-(r;E)$ is not continued into the upper but into the lower half-plane of the second sheet. In terms of $k$, $\phi ^-(r;k)$ is not continued into the third but into the fourth quadrant. If we denote the analytic continuation of ${\cal J}_-(k)$ by ${\cal J}_2(q)$,\footnote{Obviously, ${\cal J}_2(q)$ is just ${\cal J}_-(q)$.} we can define the eigenfunction\footnote{The subindex 2 refers to Continuation~2.} \begin{equation} \phi _2 (r;q):= \sqrt{\frac{2}{\pi}} \, \frac{\chi (r;q)}{{\cal J}_{2}(q)} \label{phpmeigndu2} \end{equation} so that \begin{equation} \phi _2^+(r;-k)=\lim_{{\rm Im}(q) \to 0} -\phi _2 (r;q) \, , \quad q=-k-i{\rm Im}(q) \, , \ k>0 \, , \end{equation} and \begin{equation} \phi _2^-(r;k)=\lim_{{\rm Im}(q) \to 0} \phi _2 (r;q) \, , \quad q=k-i{\rm Im}(q) \, , \ k>0 \, . \end{equation} %That is, we have obtained the Lippmann-Schwinger eigenfunctions as boundary values of a %unique eigenfunction $\phi _2 (r;q)$ that is defined for all complex $q$. In %contrast to Continuation~1, $\phi _2^+(r;-k)$ is the boundary value on the %negative real axis, and $\phi _2^-(r;k)$ is the boundary value on the %positive real axis. When we continue analytically the Lippmann-Schwinger eigenfunctions to %the second sheet using Continuation~2, $\phi _2^+(r;-k)$ gets continued to %the third quadrant, and $\phi _2^-(r;k)$ gets continued to the fourth %quadrant. %Similarly to the relations %\begin{equation} % e^{ikr}=e^{-i(-k)r} \, , \quad e^{-ikr}=e^{i(-k)r} \, , %\end{equation} %it holds that Notice that \begin{eqnarray} \phi _2^+(r;-k) = \phi _1^-(r;k) \, , \\ \phi _2^-(r;k) = \phi _1^-(r;-k) \, ; \end{eqnarray} that is, when the energy is real or lies on the first sheet, Continuations~1 and 2 yield the same Lippmann-Schwinger eigenfunctions. However, when the energy lies on the second sheet, Continuations~1 and 2 yield different Lippmann-Schwinger eigenfunctions. \section{Differences between Continuations~1 and~2} Although Continuations~1 and 2 yield the same results for real energies (thus, the standard results of scattering theory are unchanged), they do yield different Gamow vectors and different complex basis expansions, as we are going to outline in this section. \subsection{First difference: Gamow vectors} The analytic continuation of $\phi _1^+(r;k)$ to the fourth quadrant has poles. These poles are the same as those of the $S$ matrix. The Gamow vector corresponding to the resonance pole $k_n$ is given by \begin{equation} \langle r|k_n ^+ \rangle _1 = A_{n,1} \, {\rm res} \left[ \phi _1^+(r;q) \right]_{q=k_n} \, , \end{equation} where $k_n$ is a zero of ${\cal J}_+(q)\equiv {\cal J}_1(q)$ in the fourth quadrant, and $A_{n,1}$ is a normalization factor. The Gamow vector corresponding to a zero $-k_n^*$ of ${\cal J}_+(q)$ in the third quadrant is given by \begin{equation} \langle r|{\frac{\ \, }{\ \, }k_n^*} ^- \rangle = B_{n,1} \, {\rm res} \left[ \phi _1^-(r;q) \right]_{q=-k_n^*} \, , \end{equation} where $B_{n,1}$ is a normalization factor. By contrast, the continuation of $\phi _2^-(r;k)$ is analytic on the whole fourth quadrant. According to Continuation~2, the Gamow vector corresponding to the wave number $k_n$ is given by \begin{equation} \langle r|k_n ^- \rangle _2 = A_{n,2} \, \phi _2^-(r;q=k_n) \, . \end{equation} The Gamow vector corresponding to $-k_n^*$ is, according to Continuation~2, given by \begin{equation} \langle r|\frac{\ \, }{}{k_n^*}^+ \rangle _2 = B_{n,2} \, \phi _2^+(r;q=-k_n^*) \, . \end{equation} $A_{n,2}$ and $B_{n,2}$ are normalization factors. For the potential we are considering, $\langle r|k_n ^+ \rangle _1$ and $\langle r|k_n ^- \rangle _2$ are proportional to each other. However, their wave-number representations are quite different. In fact, \begin{equation} \langle ^+k|k_n ^+ \rangle _1 \equiv \mbox{a somewhat awkward distribution,} \end{equation} whereas~\cite{DIS} \begin{equation} \langle ^-k|k_n ^- \rangle _2 \equiv c_n \delta (k-k_n) \equiv d_n \frac{1}{k^2-k_n^2} \, ; \end{equation} that is, Continuation~2 yields Gamow vectors whose wave-number representations are given by the complex delta function and by the Breit-Wigner amplitude (up to normalization constants $c_n$ and $d_n$, and up to some technical details). \subsection{Second difference: Complex basis expansions} Continuation~2 enables the expansion of the wave functions in terms of Gamow vectors and a background. This expansion is possible because certain integral at infinity vanishes. However, Continuation~1 does not enable this expansion in a clear way. \section{Conclusion} The solutions of the Lippmann-Schwinger equation are to be interpreted as boundary values on the upper and lower rims of the cut. Since the upper (lower) rim of the cut corresponds to the positive (negative) $k$-axis, these boundary values have an easier interpretation in the $k$-plane. The Lippmann-Schwinger eigenfunctions can be analytically continued into the second sheet in two different ways: --According to Continuation~1, the eigenfunction $\langle r|k^+\rangle _1$ (respectively $\langle r|\frac{\ \, }{}k^-\rangle _1$) is analytically continued into the fourth (respectively third) quadrant of the complex $k$-plane. Continuation~1, although very natural, yields Gamow vectors whose wave-number representation is given by an awkward distribution. --According to Continuation~2, the eigenfunction $\langle r|k^-\rangle _2$ (respectively $\langle r|\frac{\ \, }{}{k^*}^+\rangle _2$) is analytically continued into the fourth (respectively third) quadrant of the complex $k$-plane. Continuation~2 yields Gamow vectors whose wave-number representation is given by the complex delta function and by the Breit-Wigner amplitude! Either continuation is mathematically meaningful and has both advantages and drawbacks. Continuation~1 has the advantage of providing a very natural analytical continuation. Continuation~2 implies an awkward analytic continuation, but it provides a link between the Gamow vectors and the Breit-Wigner amplitude. As of November 21, 2003, it is not known which of these solutions is the right one, and why the other should not be used. Work on this direction is under way. \acknowledgements This paper grew out of a question the author asked Prof.~Alfonso Mondrag\'on. The author wishes to thank Alfonso Mondrag\'on for enlightening, stimulating conversations on this and other topics. Thanks are also due to Prof.~Lidia Ferreira for her invitation to participate in the workshop ``Time Asymmetric Quantum Theory: The Theory of Resonances,'' Lisbon (2003). Research supported by the Basque Government through reintegration fellowship No.~BCI03.96. %\section{References} \begin{thebibliography}{99} \bibitem{NEWTON} R.~G.~Newton, {\it Scattering Theory of Waves and Particles}, McGraw-Hill, New York (1966); 2nd edition, Springer Verlag, New York (1982). \bibitem{TAYLOR} J.~R.~Taylor, {\it Scattering theory}, Jhon Wiley \& Sons, Inc., New York (1972). \bibitem{DIS} R.~de la Madrid, %{\it Quantum Mechanics in Rigged Hilbert %Space Language} Ph.D.~Thesis, Universidad de Valladolid (2001). Available at \texttt{http://www.ehu.es/$\sim$wtbdemor/}. \end{thebibliography} \end{document} ---------------0405310510663 Content-Type: application/x-tex; name="tasy.sty" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="tasy.sty" \def\author@uppercase{} % Author's name(s) uppercase \@ifundefined{abstractname}% {\def\abstractname{Abstract}}{\relax} \@ifundefined{keywordsname}% {\def\keywordsname{Key words}}{\relax} \@ifundefined{notesname}{\def\notesname{Notes}}{\relax} \@ifundefined{acknowledgementsname}% {\def\acknowledgementsname{Acknowledgements}}{\relax} \@ifundefined{refname}{\def\refname{References}}{\relax} \@ifundefined{appendixname}{\def\appendixname{Appendix}}{\relax} \def\@ptsize{2} \@namedef{ds@10pt}{\def\@ptsize{0}} \@namedef{ds@11pt}{\def\@ptsize{1}} \@namedef{ds@12pt}{\def\@ptsize{2}} \@twosidetrue % Defines twoside option. % \@mparswitchtrue % Marginpars go on outside of page. \def\ds@draft{\overfullrule 5pt} % Causes overfull hboxes to be marked. \newif\if@toto \newif\if@toto \newif\if@monograph \newif\if@editedvolume \newif\if@proceedings \def\ds@toto{\@tototrue\typeout{^^JKAP -- toto Series^^J}} \def\ds@toto{\@tototrue\typeout{^^JKAP -- toto Series^^J}} \def\ds@monograph{\@monographtrue\typeout{^^JKAP -- Monograph^^J}} \def\ds@editedvolume{\@editedvolumetrue\typeout{^^JKAP -- Edited Volume^^J}} \def\ds@proceedings{\@proceedingstrue\typeout{^^JKAP -- Proceedings^^J}} \newif\if@numreferences \def\ds@numreferences{\@numreferencestrue} \@ifundefined{default@numreferences}{\@numreferencesfalse}% {\@numreferencestrue} \newif\if@namedreferences \def\ds@namedreferences{\@namedreferencestrue} \@ifundefined{default@namedreferences}{\@namedreferencesfalse}% {\@namedreferencestrue} \@options \if@monograph \else \if@editedvolume \else \if@proceedings \else \if@toto \else \if@toto \else \typeout{^^JBook Type Not Specified! 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\def\@makessectionhead#1{ \vsecspace{16pt} { \parindent 0pt %\centering \normalsize\bf\baselineskip 12pt #1\par \nobreak \vskip 8pt} } \def\section{\@afterindentfalse \secdef\@section\@ssection} \def\@section[#1]#2{\ifnum \c@secnumdepth >\m@ne \refstepcounter{section} \addcontentsline{toc}{section}{\protect \numberline{\thesection}#1}\else \addcontentsline{toc}{section}{#1}\fi \sectionmark{#1} \if@twocolumn\@topnewpage[\@makesectionhead{#2}] \else \@makesectionhead{#2}\@afterheading \fi} \def\@ssection#1{\if@twocolumn \@topnewpage[\@makessectionhead{#1}] \else \@makessectionhead{#1}\@afterheading\fi} \def\@makesubsectionhead#1{ \vsecspace{16pt} { \parindent 0pt \raggedright \small\rm \thesubsection.\hskip 0.7em \uppercase{#1}\par \nobreak \vskip 8pt} } \def\@makessubsectionhead#1{ \vsecspace{16pt} { \parindent 0pt \raggedright \small\rm \uppercase{#1}\par \nobreak \vskip 8pt} } \def\subsection{\@afterindentfalse \secdef\@subsection\@ssubsection} \def\@subsection[#1]#2{\ifnum 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\addcontentsline{toc}{subsubsection}{\protect \numberline{\thesubsubsection}#1}\else \addcontentsline{toc}{subsubsection}{#1}\fi \if@twocolumn\@topnewpage[\@makesubsubsectionhead{#2}] \else \@makesubsubsectionhead{#2}\@afterheading \fi} \def\@ssubsubsection#1{\if@twocolumn \@topnewpage[\@makessubsubsectionhead{#1}] \else \@makessubsubsectionhead{#1}\@afterheading\fi} \fi \def\paragraph{\@startsection {paragraph}{4}{\z@}{3.25ex plus 1ex minus .2ex}{-12pt}{\normalsize\it}} \let\subparagraph=\paragraph %\def\subparagraph{\@startsection % {subparagraph}{4}{\parindent}{3.25ex plus 1ex minus % .2ex}{-1em}{\normalsize\bf}} \def\chaptermark#1{} \if@monograph \setcounter{secnumdepth}{2} \else \setcounter{secnumdepth}{3} \fi \def\acknowledgements{\section*{\acknowledgementsname}% \typeout{\acknowledgementsname}} \def\appendix{\par \setcounter{section}{0} \setcounter{subsection}{0}% \def\thesection{\Alph{section}}} \leftmargini 2em %17mm \leftmarginii 2.2em %22pt \leftmarginiii 1.87em %18.7pt \leftmarginiv 1.7em %17pt \leftmarginv 1em %10pt \leftmarginvi 1em %10pt \leftmargin\leftmargini \labelsep .4em %10mm \labelwidth\leftmargini \advance\labelwidth-\labelsep \def\@listI{\leftmargin\leftmargini \parsep 0pt \topsep 3pt plus 1pt minus 1pt \itemsep 0pt} \let\@listi\@listI \@listi \def\@listii{\leftmargin\leftmarginii \labelwidth\leftmarginii\advance\labelwidth-\labelsep \topsep 4pt plus 2pt minus 1pt \parsep 2pt plus 1pt minus 1pt \itemsep \parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii\advance\labelwidth-\labelsep \topsep 2pt plus 1pt minus 1pt \parsep \z@ \partopsep 1pt plus 0pt minus 1pt \itemsep \topsep} \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} \@ifundefined{refname}{\def\refname{References}}{\relax} % \@ifundefined{bibname}{\def\bibname{Bibliography}}{\relax} \@ifundefined{bibname}{\def\bibname{References}}{\relax} \@ifundefined{tablename}{\def\tablename{TABLE}}{\relax} \def\labelenumi{\theenumi.} \def\theenumi{\arabic{enumi}} \def\labelenumii{(\theenumii)} \def\theenumii{\alph{enumii}} \def\p@enumii{\theenumi} \def\labelenumiii{\theenumiii.} \def\theenumiii{\roman{enumiii}} \def\p@enumiii{\theenumi(\theenumii)} \def\labelenumiv{\theenumiv.} \def\theenumiv{\Alph{enumiv}} \def\p@enumiv{\p@enumiii\theenumiii} \def\labelitemi{$-$} \def\labelitemii{$\bullet$} \def\labelitemiii{$\ast$} \def\labelitemiv{{\small $+$}} \def\verse{\let\\=\@centercr \list{}{\itemsep\z@ \parskip 12pt \itemindent -1.5em \listparindent \itemindent \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]} \let\endverse\endlist \def\quotation{\par\normalsize \list{}{\listparindent 0pt % \leftmargin 0pt \leftmargin 1.5em \listparindent 1em \topsep .5ex plus 2pt minus 1pt \itemindent\listparindent \parsep 0pt plus 1pt}\item[]\hskip-1em} \def\endquotation{\endlist\par} \def\quote{\list{}{\leftmargin 1.5em \topsep.5ex plus 2pt minus 1pt}\item[]} \let\endquote=\endlist \def\descriptionlabel#1{\hspace\labelsep \bf #1} \def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin \let\makelabel\descriptionlabel}} \let\enddescription\endlist \newdimen\descriptionmargin \descriptionmargin=3em \def\titlepage{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn \else \newpage \fi \thispagestyle{empty}\c@page\z@} \def\endtitlepage{\if@restonecol\twocolumn \else \newpage \fi} \arraycolsep 5pt % Half the space between columns in an array environment. \tabcolsep 6pt % Half the space between columns in a tabular environment. \arrayrulewidth .4pt % Width of rules in array and tabular environment. \doublerulesep 2pt % Space between adjacent rules in array or tabular env. \tabbingsep \labelsep % Space used by the \' command. (See LaTeX manual.) \skip\@mpfootins = \skip\footins \fboxsep = 3pt % Space left between box and text by \fbox and \framebox. \fboxrule = .4pt % Width of rules in box made by \fbox and \framebox. \newcounter{part} \newcounter {chapter} \newcounter {section}[chapter] \newcounter {subsection}[section] \newcounter {subsubsection}[subsection] \newcounter {paragraph}[subsubsection] \newcounter {subparagraph}[paragraph] \def\thepart {\Roman{part}} \def\thechapter {\arabic{chapter}} \if@monograph \def\thesection {\thechapter.\arabic{section}} \else \def\thesection {\arabic{section}} \fi \def\thesubsection {\thesection.\arabic{subsection}} \def\thesubsubsection {\thesubsection .\arabic{subsubsection}} \def\theparagraph {\thesubsubsection.\arabic{paragraph}} \def\thesubparagraph {\theparagraph.\arabic{subparagraph}} \def\@chapapp{Chapter} \def\part{\cleardoublepage % Starts new page. \thispagestyle{plain} % Page style of part page is 'plain' \if@twocolumn % IF two-column style \onecolumn % THEN \onecolumn \@tempswatrue % @tempswa := true \else \@tempswafalse % ELSE @tempswa := false \fi % FI \hbox{}\vfil % Add fil glue to center title \secdef\@part\@spart} \def\@part[#1]#2{\ifnum \c@secnumdepth >-2\relax % IF secnumdepth > -2 \refstepcounter{part} % THEN step part counter \addcontentsline{toc}{part}{\thepart % add toc line \hspace{1em}#1}\else % ELSE add unnumbered line \addcontentsline{toc}{part}{#1}\fi % FI \markboth{}{} {\centering % %% added 19 Jan 88 \ifnum \c@secnumdepth >-2\relax % IF secnumdepth > -2 \huge\bf Part \thepart % THEN Print 'Part' and number \par % in \huge bold. \vskip 20pt \fi % Add space before title. \Huge \bf % FI #1\par}\@endpart} % Print Title in \Huge bold. \def\@endpart{\vfil\newpage % End page with 1fil glue. \if@twoside % IF twoside printing \hbox{} % THEN Produce totally blank page \thispagestyle{empty} \newpage \fi % FI \if@tempswa % IF @tempswa = true \twocolumn % THEN \twocolumn \fi} % FI \def\@spart#1{{\centering % %% added 19 Jan 88 \Huge \bf % Print title in \huge boldface #1\par}\@endpart} \def\@pnumwidth{1.55em} \def\@tocrmarg {2.55em} \def\@dotsep{4.5} \setcounter{tocdepth}{2} \def\tableofcontents{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn \fi\chapter*{{Table of Contents}\@mkboth{TABLE OF CONTENTS}{TABLE OF CONTENTS}} \@starttoc{toc}\if@restonecol\twocolumn\fi} \def\l@part#1#2{\addpenalty{-\@highpenalty} \addvspace{2.25em plus 1pt} % space above part line \begingroup \@tempdima 3em % width of box holding part number, used by \parindent \z@ \rightskip \@pnumwidth %% \numberline \parfillskip -\@pnumwidth {\large \bf % set line in \large boldface \leavevmode % TeX command to enter horizontal mode. #1\hfil \hbox to\@pnumwidth{\hss #2}}\par \nobreak % Never break after part entry \global\@nobreaktrue %% Added 24 May 89 as \everypar{\global\@nobreakfalse\everypar{}}%% suggested by Jerry Leichter \endgroup} \def\l@chapter#1#2{\addpenalty{-\@highpenalty} \vskip 1.0em plus 1pt % space above chapter line \@tempdima 1.5em % width of box holding chapter number \begingroup \parindent \z@ \rightskip \@pnumwidth \parfillskip -\@pnumwidth \bf % Boldface. \leavevmode % TeX command to enter horizontal mode. \advance\leftskip\@tempdima %% added 5 Feb 88 to conform to \hskip -\leftskip %% 25 Jan 88 change to \numberline #1\nobreak\hfil \nobreak\hbox to\@pnumwidth{\hss #2}\par \penalty\@highpenalty %% added 24 May 89 as suggested by J. Leichter \endgroup} \def\l@section{\@dottedtocline{1}{1.5em}{2.3em}} \def\l@subsection{\@dottedtocline{2}{3.8em}{3.2em}} \def\l@subsubsection{\@dottedtocline{3}{7.0em}{4.1em}} \def\l@paragraph{\@dottedtocline{4}{10em}{5em}} \def\l@subparagraph{\@dottedtocline{5}{12em}{6em}} \def\listoffigures{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn \fi\chapter*{List of Figures\@mkboth {LIST OF FIGURES}{LIST OF FIGURES}}\@starttoc{lof}\if@restonecol \twocolumn\fi} \def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}} \def\listoftables{\@restonecolfalse\if@twocolumn\@restonecoltrue\onecolumn \fi\chapter*{List of Tables\@mkboth {LIST OF TABLES}{LIST OF TABLES}}\@starttoc{lot}\if@restonecol \twocolumn\fi} \let\l@table\l@figure \def\thebibliography#1{{\section*{\refname}\@mkboth {\refname}{\refname}}\footnotesize \typeout{\refname}\def\bibwidthlabel{#1}\list {\kapbib@counter}{\kapbib@list} \let\makelabel\@biblabel \def\newblock{\hskip .11em plus .33em minus .07em} \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \let\endthebibliography=\endlist \if@numreferences \if@namedreferences \@numreferencesfalse \fi \fi \if@toto \@numreferencestrue \fi \if@toto \@numreferencestrue \fi \if@numreferences \typeout{KAP -- Numbered References} \def\kapbib@counter{\arabic{enumi}} \def\kapbib@list{\setlength{\labelsep}{1em}% \settowidth{\labelwidth}{\@biblabel{\bibwidthlabel}}% \setlength{\leftmargin}{\labelwidth}% \addtolength{\leftmargin}{\labelsep}% \setlength{\itemindent}{0pt}% \setlength{\itemsep}{0pt}% \setlength{\parsep}{0pt}% \usecounter{enumi}% \let\makelabel\kap@biblabel} \def\@biblabel#1{\hfill #1.} \def\@bibitem#1{\item\if@filesw\immediate\write\@auxout {\string\bibcite{#1}{\the\value{\@listctr}}}\fi\ignorespaces} \def\@lbibitem[#1]#2{\item\if@filesw {\def\protect##1{\string ##1\space}\immediate \write\@auxout{\string\bibcite{#2}{\the \value{\@listctr}}}}\fi\ignorespaces} \let\@internalcite\cite \def\cite{\@ifstar{\@internalcite}{\@internalcite}} \def\shortcite{\@ifstar{\@internalcite}{\@internalcite}} \def\citeauthor{\@ifstar{\@internalcite}{\@internalcite}} \def\citeyear{\@ifstar{\@internalcite}{\@internalcite}} \def\citeauthoryear{} \def\wer@cite#1#2{} \def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi \def\@citea{}\@cite{\@for\@citeb:=#2\do {\@citea\def\@citea{,\penalty\@m\ }\@ifundefined {b@\@citeb}{{\bf ?}\@warning {Citation `\@citeb' on page \thepage \space undefined}}% {\csname b@\@citeb\endcsname}}}{#1}} \else %%%% \if@namedreferences % IF AUTHOR USED DOCUMENTSTYLE [namedreferences] % NEW DEFINITIONS FROM NAMED.STY \typeout{KAP -- Documentstyle Named References} \def\kapbib@counter{\relax} \def\kapbib@list{\setlength{\labelsep}{0em}% \setlength{\labelwidth}{0pt}% \setlength{\itemindent}{-\bibhang}% \setlength{\itemsep}{0pt}% \setlength{\parsep}{0pt}% \setlength{\leftmargin}{\bibhang}} \newlength{\bibhang} \setlength{\bibhang}{1.4em} \def\@biblabel#1{} \def\@up#1{\raisebox{.2ex}{#1}}%[rmjw] \let\@internalcite\cite \def\cite{\def\@citeseppen{-1000}% \def\@cite##1##2{({##1\if@tempswa , ##2\fi})}% \def\citeauthoryear##1##2{\rm ##1, ##2}\@internalcite} \def\shortcite{\def\@citeseppen{1000}% \def\@cite##1##2{({##1\if@tempswa , ##2\fi})}% \def\citeauthoryear##1##2{\rm ##2}\@internalcite} \def\citeauthor#1{\def\@citeseppen{1000}% \def\@cite##1##2{{##1\if@tempswa , ##2\fi}}% \def\citeauthoryear##1##2{\rm ##1}\@citedata{#1}} \def\citeyear#1{\def\@citeseppen{1000}% \def\@cite##1##2{{##1\if@tempswa , ##2\fi}}% \def\citeauthoryear##1##2{\rm ##2}\@citedata{#1}} \def\@citedata#1{\@tempswafalse% \if@filesw\immediate\write\@auxout{\string\citation{#1}}\fi \def\@citea{}\@cite{\@for\@citeb:=#1\do {\@citea\def\@citea{,\penalty\@citeseppen\ }\@ifundefined {b@\@citeb}{{\bf ?}\@warning {Citation `\@citeb' on page \thepage \space undefined}}% {\csname b@\@citeb\endcsname}}}{}} \def\@citex[#1]#2{\if@filesw\immediate\write\@auxout{\string\citation{#2}}\fi \def\@citea{}\@cite{\@for\@citeb:=#2\do {\@citea\def\@citea{;\penalty\@citeseppen\ }\@ifundefined {b@\@citeb}{{\bf ?}\@warning {Citation `\@citeb' on page \thepage \space undefined}}% {\csname b@\@citeb\endcsname}}}{#1}} \def\wer@cite#1#2{} %%% \else %%% %%% %%% \typeout{KAP -- Named References} %%% %%% \def\kapbib@counter{\relax} %%% %%% \def\kapbib@list{\setlength{\labelsep}{0em}% %%% \setlength{\labelwidth}{0pt}% %%% \setlength{\itemindent}{-\bibhang}% %%% \setlength{\itemsep}{0pt}% %%% \setlength{\parsep}{0pt}% %%% \setlength{\leftmargin}{\bibhang}} %%% %%% \newlength{\bibhang} %%% \setlength{\bibhang}{1.4em} %%% %%% \def\@biblabel#1{} %%% %%% \def\@lbibitem[#1]#2{\item[\@biblabel{#1}]\if@filesw %%% {\def\protect##1{\string ##1\space}\immediate %%% \write\@auxout{\string\bibcite{#2}{\string %%% \bibnameyear #1}}}\fi\ignorespaces} %%% %%% \let\@internalcite\cite %%% %%% \def\cite{\@ifstar{\citeyear}{\citefull}} %%% \def\citefull{\def %%% \wer@cite##1##2{##1, ##2}\@internalcite} %%% \def\citeyear{\def %%% \wer@cite##1##2{##2}\@internalcite} %%% %%% \def\shortcite{} %%% \def\citeauthor{} %%% \def\citeyear{} %%% \def\citeauthoryear{} %%% %\let\shortcite=\cite %%% %\let\citeauthor=\cite %%% %\let\citeyear=\cite %%% % \def\citeauthoryear{} %%% %%% %%% \def\@citex[#1]#2{\if@filesw\immediate\write %%% \@auxout{\string\citation{#2}}\fi %%% \def\@citea{}\@cite{\@for\@citeb:=#2\do %%% {\@citea\def\@citea{;\penalty\@m\ }\@ifundefined %%% {b@\@citeb}{{\bf ?}\@warning %%% {Citation `\@citeb' on page \thepage %%% \space undefined}}% %%% {\csname b@\@citeb\endcsname}}}{#1}} %%% %%% %%% \fi % \if@namedreferences \fi % \if@numreferences \def\astroncite{\wer@cite} \def\bibnameyear{\wer@cite} \newif\if@restonecol \def\theindex{\@restonecoltrue\if@twocolumn\@restonecolfalse\fi \columnseprule \z@ \columnsep 35pt\twocolumn[\@makeschapterhead{INDEX}] \@mkboth{INDEX}{INDEX}\thispagestyle{plain}\parindent\z@ \parskip\z@ plus .3pt\relax\let\item\@idxitem} \def\@idxitem{\par\hangindent 40pt} \def\subitem{\par\hangindent 40pt \hspace*{20pt}} \def\subsubitem{\par\hangindent 40pt \hspace*{30pt}} \def\endtheindex{\if@restonecol\onecolumn\else\clearpage\fi} \def\indexspace{\par \vskip 10pt plus 5pt minus 3pt\relax} \def\footnoterule{} \@addtoreset{footnote}{chapter} % Numbers footnotes within chapters \long\def\@makefntext#1{\parindent 1em\noindent \hbox to 1.8em{\hss$^{\@thefnmark}$}#1} \setcounter{topnumber}{2} \def\topfraction{.99} \setcounter{bottomnumber}{1} \def\bottomfraction{.3} \setcounter{totalnumber}{3} \def\textfraction{.01} \def\floatpagefraction{.9} \setcounter{dbltopnumber}{2} \def\dbltopfraction{.7} \def\dblfloatpagefraction{.5} % FROM wer.STY \newif\if@adjustcaption \newif\if@captionabovetabular \newif\if@tabularabovecaption \newdimen\@tabularwidth \def\@tablecapfont{\footnotesize} \def\@figcapfont{\footnotesize} \long\def\@makecaption#1#2{\vskip 10pt \ifnum\cap@type=1 \typeout{Figure \thefigure}% \setbox\@tempboxa\hbox{\@figcapfont #1 #2}% \ifdim \wd\@tempboxa >\hsize \unhbox\@tempboxa\@figcapfont\par \else \hbox to\hsize{\hbox{}\hfil\box\@tempboxa\hfil} \fi \fi \ifnum\cap@type=2 \typeout{\tablename\space\thetable}% \if@adjustcaption \if@tabularabovecaption\@maketablecaption{#1}{#2}\else \global\@captionabovetabulartrue\gdef\@captab{{#1}{#2}}\fi\else \@maketablecaption{#1}{#2}\fi\fi} \long\def\@maketablecaption#1#2{\if@adjustcaption \setbox\@tempboxa\hbox{\@tablecapfont #1#2}% \ifdim\wd\@tempboxa>\@tabularwidth \@tablecapfont \centerline{\parbox{\@tabularwidth}{\unhbox\@tempboxa}}\par \else \hbox to\hsize{\hfil\box\@tempboxa\hfil} \fi \else \setbox\@tempboxa\hbox{\@tablecapfont #2}% \ifdim\wd\@tempboxa>\hsize #1 \unhbox\@tempboxa\@tablecapfont \else \hbox to\hsize{\hfil\box\@tempboxa\hfil} \fi \fi \vspace{3pt}} \let\kap@tabular\tabular \let\kap@endtabular\endtabular % \hline redefined within tabular environment: 3pt extra space above and below \def\tabular{% \def\hline{\noalign{\ifnum0=`}\fi\vskip 3pt\hrule \@height \arrayrulewidth \vskip 3pt \futurelet \@tempa\@xhline} \begingroup\centering\if@captionabovetabular \else\global\@tabularabovecaptiontrue\fi \setbox\strutbox\hbox{\vrule height.8\baselineskip depth.4\baselineskip width\z@}\setbox0=\hbox\bgroup\kap@tabular} \def\endtabular{\kap@endtabular\egroup\global\@tabularwidth\wd0 \if@adjustcaption\if@captionabovetabular \expandafter\@maketablecaption\@captab\fi\fi\unhbox0\endgroup} \newcounter{figure}[chapter] \if@monograph \def\thefigure{\thechapter.\@arabic\c@figure} \else \def\thefigure{\@arabic\c@figure} \fi \def\fps@figure{tbp} \def\ftype@figure{1} \def\ext@figure{lof} \def\fnum@figure{\footnotesize{\it Figure~\thefigure.~~}} % \def\figure{\@float{figure}} \def\figure{\def\cap@type{1} \@float{figure}} % FROM wer.STY \let\endfigure\end@float \@namedef{figure*}{\def\cap@type{1} \@dblfloat{figure}} \@namedef{endfigure*}{\end@dblfloat} \newcounter{table}[chapter] \if@monograph \def\thetable{\thechapter.\@arabic\c@table} \else \def\thetable{\@arabic\c@table} \fi \def\fps@table{tbp} \def\ftype@table{2} \def\ext@table{lot} \def\fnum@table{\footnotesize{TABLE~\thetable.~~}} % \def\table{\@float{table}} \def\table{\def\cap@type{2}\@adjustcaptiontrue\@captionabovetabularfalse \@tabularabovecaptionfalse\footnotesize \@float{table}} \let\endtable\end@float \@namedef{table*}{\def\cap@type{2}\@adjustcaptiontrue\@captionabovetabularfalse \@tabularabovecaptionfalse\footnotesize \@dblfloat{table}} \@namedef{endtable*}{\end@dblfloat} \def\maketitle{\kap@warning \typeout{ -- '\string\maketitle' is not needed, title is made by document style^^J}} \def\subtitle@#1{\gdef\@subtitle{#1}% \global\@subtitledefinedtrue} \def\maketitle@@{% \begingroup \setcounter{footnote}{0}% \def\thefootnote{\fnsymbol{footnote}}% \gdef\thanks##1{\footnotemark\begingroup \def\protect{\noexpand\protect\noexpand}\xdef\@thanks{\@thanks \protect\footnotetext[\the\c@footnote]{##1}}\endgroup}% \@maketitle \@thanks \endgroup \setcounter{footnote}{0} \let\maketitle\relax \let\@maketitle\relax \gdef\@thanks{}% \gdef\@title{}% \gdef\@authors{}% \let\thanks\relax} \newif\if@keywords \@keywordsfalse \def\@keywords{} \long\def\keywords#1\end@keywords{\gdef\@keywords{#1}% \global\@keywordstrue} \newif\if@abstract \@abstractfalse \long\def\abstract#1\end#2{\gdef\@abstract{#1\end@keywords }\@abstracttrue\end{#2}} \def\endabstract{\make@abstract\vskip24pt\relax} \def\@abstract{} \def\end@keywords{} % \def\make@abstract{{\normalsize\typeout{\abstractname} % {\noindent{\bf \abstractname.}\@abstract}\gdef\@abstract{} % \if@keywords{\vskip 1\baselineskip\typeout{\keywordsname} % \noindent{\bf \keywordsname:\/} \@keywords\gdef % \@keywords{}}\fi % \par}} \def\make@abstract{{\normalsize\typeout{\abstractname} {\noindent{\bf \abstractname.}\@abstract}\gdef\@abstract{} \if@keywords{% \vskip 1\baselineskip\typeout{\keywordsname} \noindent{\bf \keywordsname:\/} \@keywords\gdef \@keywords{}% }\fi \par}} \if@monograph \def\@maketitle{\begingroup \chapter{\@title} \thispagestyle{myheadings}% THISPAGESTYLE SHOULD COME AFTER TITLE \if@subtitledefined \begin{center} {\large\it \@subtitle}\par \end{center} \vskip 33pt \fi \endgroup } \else % proceedings, editedvolume or toto \newskip\afterauthinsskip \def\@maketitle{\begingroup \chapter{\@title} % \thispagestyle{myheadings}% THISPAGESTYLE SHOULD COME AFTER TITLE \thispagestyle{empty}% \if@subtitledefined {\normalsize\it \@subtitle}\par \vskip 2\baselineskip%3pc \fi {\advance\leftskip by 5pc % For indentation of \author & \insitute \setbox\@tempboxa\vbox{\@formatauthorsandinstitutes}% \ifdim\ht\@tempboxa<1.5pc \afterauthinsskip 4pc \else\ifdim\ht\@tempboxa<2.5pc \afterauthinsskip 3pc \else\afterauthinsskip 2pc \fi \fi \normalsize\@formatauthorsandinstitutes \vskip \afterauthinsskip\relax} \endgroup } \fi \newif\ifkern@mathmode \kern@mathmodefalse \def\kern@charspace{\kern 0.5ex} \def\kern@wordspace{\unskip\hskip 0.5em} \def\@kap@tfor#1:=#2\do#3{\def\@fortmp{#2}\ifx\@fortmp\@empty \else\@tforloop#2\@nil\@nil\@@#1{#3}\fi} \def\charkern#1{\begingroup \def\@@kc{#1} \def\c@@k{@s@e@v@y}% \def\ck@@##1 @s@e@v@y\@@ck{\def\@@kc{##1}}% \def\ck@##1 ##2\@@ck{\def\@kc{##1}\def\@@kc{##2}}% \def\iterate{\expandafter\expandafter\expandafter\ck@ \expandafter\@@kc\space @s@e@v@y\@@ck \expandafter\@kernword\expandafter{\@kc}% \ifx\@@kc\c@@k \let\next\relax \else \expandafter\ck@@\@@kc\@@ck \let\next\iterate\kern@wordspace \fi\next}\iterate \endgroup} \def\@kernword#1{\def\d@ll@rm@th{$}\@kap@tfor\@nextchar:=#1\do{% \ifx\@nextchar\d@ll@rm@th \ifkern@mathmode \kern@mathmodefalse $\kern@math$% \else \kern@mathmodetrue \xdef\kern@math{}% \fi \else \ifkern@mathmode \begingroup \def\protect{\noexpand}% \xdef\kern@math{\kern@math\@nextchar}% \endgroup \else \uppercase\expandafter{\@nextchar}\kern@charspace \fi \fi}} \def\numtoword#1{\ifcase#1\or one\or two\or three\or four\or five\or six\or seven\or eight\or nine\or ten\or eleven\or twelve\or thirteen\or fourteen\or fifteen\or sixteen\or seventeen\or eighteen\or nineteen\fi} \mark{{}{}} % Initializes TeX's marks \if@monograph \def\ps@headings{\let\@mkboth\@gobbletwo \def\@oddfoot{}\def\@evenfoot{}% No feet. \def\@evenhead{\rm\normalsize\hbox to 0pt{\thepage\hss}\hfil \small\leftmark\hfil}% \def\@oddhead{\hbox{}\rm\small\hfil\rightmark\hfil \rm\normalsize\hbox to 0pt{\hss\thepage}}% \def\chaptermark##1{% \markboth{\expandafter\protect \expandafter{CHAPTER \thechapter}}% {\expandafter\protect\expandafter{\@runningtitle}}}% \def\sectionmark##1{}} \def\ps@myheadings{\let\@mkboth\@gobbletwo \def\@oddhead{\hbox{}\rm\small\hfil\leftmark\hfil}% \def\@oddfoot{\hbox{}\hfil\rm\normalsize\thepage\hfil}% \let\@evenhead\@oddhead% \let\@evenfoot\@oddfoot% \def\chaptermark##1{}% \def\sectionmark##1{}% \def\subsectionmark##1{}} \else\if@proceedings \def\ps@headings{\let\@mkboth\@gobbletwo \def\@oddfoot{}\def\@evenfoot{}% No feet. \def\@evenhead{\rm\normalsize\hbox to 0pt{\thepage\hss}\hfil \small\@runningauthor\hfil}% \def\@oddhead{\hbox{}\rm\small\hfil\rightmark\hfil \rm\normalsize\hbox to 0pt{\hss\thepage}}% \def\chaptermark##1{% \markboth{\expandafter\protect\expandafter{\@runningauthor}}% {\expandafter\protect\expandafter{\@runningtitle}}}% \def\sectionmark##1{}} \def\ps@myheadings{\let\@mkboth\@gobbletwo \def\@oddhead{\hbox{}\rm\small\hfil \expandafter{\@runningauthor}\hfil}% \def\@oddfoot{{\parindent 0pt \parbox{\textwidth}{\hbox{}\vskip 3pt \hbox{}\hfil\rm\normalsize\thepage\hfil\\ \vskip 0pt\small\@copyright}}} \let\@evenhead\@oddhead% \let\@evenfoot\@oddfoot% \def\chaptermark##1{}% \def\sectionmark##1{}% \def\subsectionmark##1{}} \else\if@editedvolume \def\ps@headings{\let\@mkboth\@gobbletwo \def\@oddfoot{}\def\@evenfoot{}% No feet. \def\@evenhead{\rm\normalsize\hbox to 0pt{\thepage\hss}\hfil \small\@runningauthor\hfil}% \def\@oddhead{\hbox{}\rm\small\hfil\rightmark\hfil \rm\normalsize\hbox to 0pt{\hss\thepage}}% \def\chaptermark##1{% \markboth{\expandafter\protect\expandafter{\@runningauthor}}% {\expandafter\protect\expandafter{\@runningtitle}}}% \def\sectionmark##1{}} \def\ps@myheadings{\let\@mkboth\@gobbletwo \def\@oddhead{\hbox{}\rm\small\hfil \expandafter{\@runningauthor}\hfil}% \def\@oddfoot{{\parindent 0pt \parbox{\textwidth}{\hbox{}\vskip 3pt \hbox{}\hfil\rm\normalsize\thepage\hfil\\ \vskip 0pt\small\@copyright}}} \let\@evenhead\@oddhead% \let\@evenfoot\@oddfoot% \def\chaptermark##1{}% \def\sectionmark##1{}% \def\subsectionmark##1{}} \else % toto \def\ps@headings{\let\@mkboth\@gobbletwo \def\@oddfoot{}\def\@evenfoot{}% No feet. \def\@evenhead{\rm\normalsize\hbox to 0pt{\thepage\hss}\hfil}% \def\@oddhead{\hfil\rm\normalsize\hbox to 0pt{\hss\thepage}}% \def\chaptermark##1{}% \def\sectionmark##1{}} \def\ps@myheadings{\let\@mkboth\@gobbletwo \def\@oddfoot{}\def\@evenfoot{}% No feet. \def\@evenhead{}\def\@oddhead{}% \def\chaptermark##1{}% \def\sectionmark##1{}} \fi \fi \fi \newif\if@runningtitleset \newif\if@runningauthorset \def\runningtitle#1{\if@runningtitleset\else \xdef\@runningtitle{{\uppercase{#1}}}\fi \global\@runningtitlesettrue} \gdef\@runningtitle{} % \def\runningauthor#1{\if@runningauthorset\else % \xdef\@runningauthor{#1}\global\@runningauthorsettrue\fi} % \gdef\@runningauthor{} \def\runningauthor#1{\if@runningauthorset\else \xdef\@runningauthor{{\uppercase{#1}}}\fi \global\@runningauthorsettrue} \gdef\@runningauthor{} \newif\ifopening@opened \opening@openedfalse \newif\ifopening@closed \opening@closedfalse \def\opening{\global\opening@openedtrue \let\title=\title@ \let\subtitle=\subtitle@ \let\author=\author@ \let\institute=\institute@ \let\date=\date@ \let\thanks=\footnote} \def\endopening{\global\opening@closedtrue \let\title=\opening@only \let\subtitle=\opening@only \let\author=\opening@only \let\institute=\opening@only \let\date=\opening@only} \def\opening@only#1{\kap@warning\typeout{ -- \string\title, \string\subtitle, \string\author, \string\institute, or \string\date { }effective only inside}\typeout{ -- opening environment. Please refer to Author's Instructions.^^J}} \let\title=\opening@only \let\subtitle=\opening@only \let\author=\opening@only \let\institute=\opening@only \let\date=\opening@only \newtoks\t@a \newtoks\t@b \long\def\append@item#1\to#2{\t@a={\@k@p{#1}}\t@b=\expandafter{#2}% \xdef#2{\the\t@b\the\t@a}} \def\get@left#1\to#2{\expandafter\g@l#1\g@l#1#2} \long\def\g@l\@k@p#1#2\g@l#3#4{\def#4{#1}\def#3{#2}} \def\noauthors{} \def\@authors{} \def\@allauthors{} \def\@institutes{} \def\authors@institute{} \def\title@#1{\gdef\@title{\uppercase{#1}}\def\\{ }\xdef\@runningtitle{#1} \typeout{KAP -- Title Uppercase}} \def\subtitle@#1{\gdef\@subtitle{#1}\global\@subtitledefinedtrue} \def\author@#1{\append@item#1\to\@authors \append@item#1\to\@allauthors \xdef\@runningauthor{#1}} \def\institute@#1{\append@item#1\to\@institutes \expandafter\append@item\expandafter{\@authors}\to\authors@institute \gdef\@authors{}} % \def\date@#1{\gdef\@date{#1}} \def\date@#1{} \newif\ifthanks \newbox\@kapbox \newcount\@kapcount \@ifundefined{author@uppercase}{ \typeout{KAP -- Author(s) Normal} \def\@formatname#1#2{\@kapcount\c@footnote\begingroup \def\thanks##1{\global\thankstrue}\setbox\@kapbox\vbox{#1}\endgroup \c@footnote\@kapcount \begingroup \ifthanks \def\fn##1\thanks##2{{##1}\hbox to 0pt{#2\hss}\thanks{##2}}\expandafter\fn#1 \else \def\fn##1{{##1}#2}\expandafter\fn\expandafter{#1}\fi \endgroup\global\thanksfalse} }{ \typeout{KAP -- Author(s) Uppercase} \def\@formatname#1#2{\@kapcount\c@footnote\begingroup \def\thanks##1{\global\thankstrue}\setbox\@kapbox\vbox{#1}\endgroup \c@footnote\@kapcount \begingroup \ifthanks \def\fn##1\thanks##2{\uppercase{##1}\hbox to 0pt{#2\hss}\thanks{##2}}\expandafter\fn#1 \else \def\fn##1{\uppercase{##1}#2}\expandafter\fn\expandafter{#1}\fi \endgroup\global\thanksfalse} } \def\@formatauthors{{\small\begingroup \newcount\cnt@authors \cnt@authors=0\def\@k@p##1{\advance\cnt@authors by 1}\@curauths \def\@k@p##1{\advance\cnt@authors by -1 \ifnum\cnt@authors>1 \hbox{\@formatname{##1}{,}\penalty-1000}\space\fi \ifnum\cnt@authors=1 \hbox{\@formatname{##1}{} AND\penalty-1000}\space\fi \ifnum\cnt@authors<1 \hbox{\@formatname{##1}{}}\par\fi}\@curauths \endgroup}} \def\@formatinstitute{{\normalsize\it \@curinst\par}} \def\@formatauthorsandinstitutes{\begingroup \newcount\cnt@institutes \newcount\cnt@authors \newbox\@authbox \cnt@authors=0\def\@k@p##1{\advance\cnt@authors by 1}\@allauthors \xdef\@authnames{}% \begingroup \def\thanks##1{}% \def\@k@p##1{\advance\cnt@authors by -1 \ifnum\cnt@authors=0 \xdef\@authnames{\@authnames ##1}\fi \ifnum\cnt@authors=1 \xdef\@authnames{##1\ AND }\fi \ifnum\cnt@authors>1 \xdef\@authnames{##1\ ET AL.}\cnt@authors=0\fi}% \global\setbox\@authbox=\hbox{\@allauthors}% \expandafter\runningauthor\expandafter{\@authnames} \endgroup \cnt@institutes=0 \def\@k@p##1{\advance\cnt@institutes by 1}\@institutes \loop\ifnum\cnt@institutes>0 \get@left\authors@institute\to\@curauths \noindent\@formatauthors \get@left\@institutes\to\@curinst \noindent\@formatinstitute \ifnum\cnt@institutes=2 \vskip 0.4\baselineskip \noindent {\small AND} % and \fi \ifnum\cnt@institutes>1 \vskip 0.4\baselineskip \fi \advance\cnt@institutes by -1 \repeat\endgroup} \def\kap@warning{\typeout{^^J !! KAP Style File Warning}} \newread\@editorfile \newread\@kapjrnls \let\kapdocument\document \def\document{\kapdocument \ifopening@opened \ifopening@closed \else \@latexerr{`opening' environment not closed}{Close `opening' environment before beginning of document} \fi \else \@latexerr{No `opening' environment present}{Use `opening' environment before beginning of document} \fi \let\kap@chapter\chapter \def\chapter##1{\setcounter{chapter}{##1}\addtocounter{chapter}{-1}} \openin\@editorfile=\jobname.kap \ifeof\@editorfile \closein\@editorfile \else \closein\@editorfile \input{\jobname.kap} \setlastpage \typeout{^^J -- File `\jobname.tex' processed with wer Editor file for:} \typeout{ -- \@journal^^J} \fi\par \let\chapter\kap@chapter \maketitle@@} \let\kapenddocument\enddocument \def\enddocument{\label{lastpage} \kapenddocument} \def\editor#1{\gdef\@kapident{\hbox{}\hfill\elvtt #1\\ \hbox{}\hfill\jobname.tex - Date: \today~Time: \timenow\par}} \let\typesetter\editor \gdef\@kapident{} \newif\if@editorsset \def\editors#1{\gdef\@editors{#1}\global\@editorssettrue} \def\@editors{} \def\@copyright{\footnotesize\vskip 2\baselineskip\vskip1em \normalsize\@kapident} \newcount\@hour \newcount\@minute \def\timenow{\@hour=\time \divide\@hour by 60 \number\@hour:% \multiply\@hour by 60 \@minute=\time \global\advance\@minute by -\@hour% \ifnum\@minute<10 0\number\@minute\else\number\@minute\fi} \def\journal#1{\@ifundefined{@#1}{\kap@warning\typeout{ -- Journal code `#1' does not exist.^^J}}{\csname @#1\endcsname} \gdef\@copyright{\footnotesize \it \if@editorsset \@editors~(eds), \fi \@journal, \@firstpage--\@lastpage. \hfill\hbox{}\\ \copyright~\number\@pubyear~{\it\@printline} \hfill\hbox{}\par \vskip 1em \normalsize\@kapident}} \gdef\@journal{NAME OF JOURNAL} \def\book#1{\@ifundefined{@#1}{\kap@warning\typeout{ -- Book code `#1' does not exist.^^J}}{\csname @#1\endcsname} \gdef\@copyright{\footnotesize \it \if@editorsset \@editors~(eds), \fi \@book, \@firstpage--\@lastpage. \hfill\hbox{}\\ \copyright~\number\@pubyear~{\it\@printline} \hfill\hbox{}\par \vskip 1em \normalsize\@kapident}} \gdef\@book{NAME OF BOOK PROJECT} \def\@printline{\@kap. \@prin\ \@country.} \def\@kap{wer} \def\@prin{Printed in} \def\country#1{\gdef\@country{#1}} \gdef\@country{\@defaultcountry} \def\defaultcountry#1{\gdef\@defaultcountry{#1}} \defaultcountry{the Netherlands} \def\volume#1{\gdef\@volume{#1}} \gdef\@volume{0} \def\pubyear#1{\gdef\@pubyear{#1}} \gdef\@pubyear{\year} \def\firstpage#1{\c@page=0 \global\advance\c@page by #1 \xdef\@firstpage{\thepage}} \gdef\@firstpage{1} \def\lastpage#1{\gdef\@lastpage{#1}} \def\setlastpage{\@ifundefined{r@lastpage}{}{\expandafter\@setlastp\r@lastpage}} \def\@setlastp#1#2{\gdef\@lastpage{#2}} \gdef\@lastpage{1} \def\received#1{\gdef\@pubdate{(#1)}} \gdef\@pubdate{\@date} \def\today{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \@addtoreset{equation}{chapter} % Makes \chapter reset 'equation' counter. \if@monograph \def\theequation{\thechapter.\arabic{equation}} \else \def\theequation{\arabic{equation}} \fi \clubpenalty=4000 \widowpenalty=4000 \ps@headings % `headings' page style \pagenumbering{arabic} % Arabic page numbers \onecolumn % Single-column. \frenchspacing \openin\@kapjrnls=kapjrnls.tex \ifeof\@kapjrnls \closein\@kapjrnls \else \closein\@kapjrnls \input{kapjrnls.tex} \fi\par \@definecounter{endnote} \def\theendnote{\arabic{endnote}} \def\@makeenmark{\hbox{$^{\@theenmark}$}} \newdimen\endnotesep \def\endnote{\@ifnextchar[{\@xendnote}{\stepcounter {endnote}\xdef\@theenmark{\theendnote}\@endnotemark\@endnotetext}} \def\@xendnote[#1]{\begingroup \c@endnote=#1\relax \xdef\@theenmark{\theendnote}\endgroup \@endnotemark\@endnotetext} \let\@doanenote=0 \let\@endanenote=0 \newwrite\@enotes \newif\if@enotesopen \global\@enotesopenfalse \def\@openenotes{\immediate\openout\@enotes=\jobname.ent\relax \global\@enotesopentrue} \long\def\@endnotetext#1{% \if@enotesopen \else \@openenotes \fi \immediate\write\@enotes{\@doanenote{\@theenmark}}% \begingroup \def\next{#1}% \newlinechar='40 \immediate\write\@enotes{\meaning\next}% \endgroup \immediate\write\@enotes{\@endanenote}% \edef\@currentlabel{\csname p@endnote\endcsname\@theenmark}} \long\def\addtoendnotes#1{% \if@enotesopen \else \@openenotes \fi \begingroup \newlinechar='40 \let\protect\string \immediate\write\@enotes{#1}% \endgroup} \def\endnotemark{\@ifnextchar[{\@xendnotemark }{\stepcounter{endnote}\xdef\@theenmark{\theendnote}\@endnotemark}} \def\@xendnotemark[#1]{\begingroup \c@endnote #1\relax \xdef\@theenmark{\theendnote}\endgroup \@endnotemark} \def\@endnotemark{\leavevmode\ifhmode \edef\@x@sf{\the\spacefactor}\fi \@makeenmark \ifhmode\spacefactor\@x@sf\fi\relax} \def\endnotetext{\@ifnextchar [{\@xendnotenext}{\xdef\@theenmark{\theendnote}\@endnotetext}} \def\@xendnotenext[#1]{\begingroup \c@endnote=#1\relax \xdef\@theenmark{\theendnote}\endgroup \@endnotetext} \def\enoteheading{\section*{\notesname \@mkboth{\uppercase{\notesname}}{\uppercase{\notesname}}}% \leavevmode\par} % \def\enoteformat{\rightskip\z@ \leftskip\z@ \parindent=1.8em % \leavevmode\llap{\hbox{$^{\@theenmark}$}}} \def\enoteformat{\rightskip\z@ \leftskip\z@ \parindent=1.8em \leavevmode\llap{\hbox{$^{\@theenmark}$ }}} \def\enotesize{\footnotesize} \def\theendnotes{\immediate\closeout\@enotes \global\@enotesopenfalse \begingroup \makeatletter \def\@doanenote##1##2>{\def\@theenmark{##1}\par\begingroup \enoteformat} \def\@endanenote{\par\endgroup}% \def\ETC.{\errmessage{Some long endnotes will be truncated; % use BIGLATEX to avoid this}% \def\ETC.{\relax}} \enoteheading \enotesize \@input{\jobname.ent}% \endgroup} ---------------0405310510663--