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+% lmatann.dem
+% LaTeX package LJour1 1.0: demo file for Mathematische Annalen
+% (c) Springer-Verlag HD
+%----------------------------------------------------------------------
+%
+% customization
+\documentstyle[bibay]{pljour1}
+\journalname{Mathematische Annalen } % State name of journal
+\newcommand{\DXDYCZ}[3]{\left( \frac{ \partial #1 }{ \partial #2 }
+ \right)_{#3}}
+% end of customization
+%
+\begin{document}
+%
+\title{ Optimality relationships for $p$-cyclic
+ SOR\thanks{Research supported in part by the US Air Force under
+ grant no. AFOSR-88-0285 and the National Science Foundation under
+ grant no. DMS-85-0285.}\fnmsep\thanks{In memory of J.L. Verdier}}
+\subtitle{A demonstration text}
+\author{Daniel J. Pierce\inst{1} \and Apostolos
+ Jadjidimos\inst{2}\fnmsep\thanks{{\it Present address:\/} Department
+ of Computer Science, Purdue University, West Lafayette, IN 47907, USA.}
+ \and Robert J. Plemmons\inst{3}}
+\mail{R. Plemmons}
+\titlerunning{Optimality relationships for $p$-cyclic SOR}
+\authorrunning{D. J. Pierce et al.}
+\institute{Boeing Computer Service, P.O. Box 24346, MS 7L-21,
+ Seattle, WA 98124-0346, USA \and
+ Department of Mathematics, University of Ioannina, GR-45 1210
+ Ionnanina, Greece \and
+ Department of Computer Science and Mathematics, North Carolina
+ State University, Raleigh, NC 27695-8205, USA}
+\date{Received: 20 January 1989 / Accepted: 3 August 1991}
+\maketitle
+
+\section{Introduction}
+This text was compiled to demonstrate the use of the Springer
+\LaTeX\ macropackages {\em LJour1\/} for one-column journals.
+Please refer to \cite{leslie} for general information on coding \LaTeX{}
+and to the \cite{springer} for information concerning the Springer
+layout.
+
+Parts of this ``article" were taken from different real articles, but
+may have been changed to show a special feature of a macro.
+
+\section{Notation}
+
+Here are a few examples of how to use special fonts. Vectors are denoted
+by boldface letters: $\vec V,\; \vec W$. Tensors are denoted by sans
+serif letters: $\tens{A, B}$. If no tensors are needed, sans serif
+letters may be reserved for other purposes. Vector spaces may be denoted
+by gothic letters: $\frak{G, H}$. Sets of functions are denoted by
+script letters: ${\cal W}_i,{\cal F}$. Sets of numbers are denoted by
+special roman letters ${\Bbb R}, {\Bbb C}$.
+
+You are of course (within limits) free to design your own notation but
+sticking to conventions makes your article easier for others to read.
+
+\section{Preliminaries}
+Let us state a few well known results and demonstrate how to typeset
+lists. The functions $f$ and $g$ of (1) and (2) fulfill the following
+assumptions:
+
+\begin{enumerate}
+\item $f: B_f \subset {\Bbb R}^n \times {\Bbb R}^n \times [a,b] \to
+{\Bbb R}^n$ \\
+$f^\prime _x$, $f^\prime_y$ exist and are continous
+\item ker$(f^\prime _y (y, x, t)) = N (t)\quad \forall (y, x, t)
+\in B_f$ \\
+${\rm rank} (f^\prime _y (y, x, t)) = r$ \\
+${\rm dim} (N (t)) = n - r$
+\item $Q(t)$ denotes a projection onto $N(t)$ \\
+$Q$ is smooth and $P(t) := I - Q (t)$
+\item The matrix $G (y, x, t) := f^\prime _y (y, x, t) + f^\prime
+_x (y, x, t) Q (t)$ is nonsingular \\
+$\forall (y, x, t) \in B_f$\quad (i.e. (1) is transferable)
+\item $g: B_g \subset {\Bbb R}^n \times {\Bbb R}^n \to M \subset
+{\Bbb R}^n$ \\
+$g^\prime _{x_a} , g^\prime _{x_b}$ exist and are continuous\\
+${\rm im} (g^\prime _{x_a} , g^\prime _{x_b}) =: M$
+\end{enumerate}
+
+Now we give another example of a list with changed indentation.
+
+\begin{description}[Shoot.]
+
+\item[Shoot.]
+Collocation methods for this type of equations are considered in
+\cite{yser} and \cite{wendl}. Shooting and difference methods for
+linear, {\it solvable} DAE's in the sense of [9], also with higher
+index, are treated in [8] under the assumption that consistent initial
+values can be calculated and a stable integration method is available.
+
+\item[Diff.]
+This paper aims at constructing an algorithm for solving a BVP in
+transferable nonlinear DAE's with nonsingular Jacobian and the same
+dimension as in the ODE case.
+
+\begin{description}[Jacob.]
+
+\item[Jacob.] We also deal with Jacobians, which means that we
+explain the functions, advantages and inconveniences of calling them not
+Jacobians.....
+
+\item[Nonl.] Nonlinear functions play an important role in
+this connection. Please note that we always call them nonlinear whenever
+there is no............
+
+\end{description}
+
+\end{description}
+
+\section{The shooting method}
+The natural way to construct a shooting method for DAE's is described by
+\cite{yser}.
+
+The physical meaning of $ \sigma_0 $ and $K$ is clearly visible in
+the equations above. $\sigma_0$ represents a frequency of the order one
+per free-fall time. $K$ is
+proportional to the ratio of the free-fall time and the cooling time.
+Substituting into Baker's criteria, using thermodynamic identities
+and definitions of thermodynamic quantities,
+\begin{displaymath}
+ \Gamma_1 = \DXDYCZ{\ln P}{\ln \rho}{S} \, , \;
+ \chi^{}_\rho = \DXDYCZ{\ln P}{\ln \rho}{T} \, , \;
+ \kappa^{}_{P} = \DXDYCZ{\ln \kappa}{\ln P}{T}
+\end{displaymath}
+\begin{displaymath}
+ \nabla_{\rm ad} = \DXDYCZ{\ln T}{\ln P}{S} \, , \;
+ \chi^{}_T = \DXDYCZ{\ln P}{\ln T}{\rho} \, , \;
+ \kappa^{}_{T} = \DXDYCZ{\ln \kappa}{\ln T}{T}
+\end{displaymath}
+one obtains, after some pages of algebra, the conditions for
+{\em stability} given
+below:
+\begin{eqnarray}
+ \frac{\pi^2}{8} \frac{1}{\tau_{\rm ff}^2}
+ ( 3 \Gamma_1 - 4 )
+ & > & 0 \label{ZSDynSta} \\
+ \frac{\pi^2}{\tau_{\rm co}
+ \tau_{\rm ff}^2}
+ \Gamma_1 \nabla_{\rm ad}
+ \left[ \frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T }
+ ( \kappa^{}_T - 4 )
+ + \kappa^{}_P + 1
+ \right]
+ & > & 0 \label{ZSSecSta} \\
+ \frac{\pi^2}{4} \frac{3}{\tau_{ \rm co }
+ \tau_{ \rm ff }^2
+ }
+ \Gamma_1^2 \, \nabla_{\rm ad} \left[
+ 4 \nabla_{\rm ad}
+ - ( \nabla_{\rm ad} \kappa^{}_T
+ + \kappa^{}_P
+ )
+ - \frac{4}{3 \Gamma_1}
+ \right]
+ & > & 0 \label{ZSVibSta}
+\end{eqnarray}
+
+For a physical discussion of the stability criteria see \cite{tetz}
+or \cite{yser}.
+
+\subsection{Disadvantages of the method}
+
+The disadvantage of Eq. (\ref{ZSVibSta}) is the singularity of the
+Jacobian. If we use the representation of
+$z_i = P_i z_i + Q_i z_i =: u_i + v_i$, we obtain the following system
+
+\begin{eqnarray}
+g (u_0 + v_0 , x (t_m, t_{m-1}, u_{m-1}))& = & 0 \label{dis}\\
+u_i - P_i x (t_i; t_{i-1}, u_{i-1}) & = & 0\;,
+ \quad i = 1, \ldots , m-1\;. \label{das}
+\end{eqnarray}
+
+\subsection{Specialization of $V$}
+
+Now we specialize $V := \hat S^\prime $ in. Let $P_D$ be a
+projector with ${\rm im} (P_D) = M$. If we demand Eq. (\ref{das}) and
+\begin{eqnarray*}
+VV^- &=& P_D \\
+V^-V &=& P\; ,
+\end{eqnarray*}
+%
+the generalized inverse $V^-$ in uniquely determined. Using Lemma 1 we
+construct a regular matrix $K$ so that ${\rm im} (P_D) \oplus {\rm im}
+(K^{-1} Q) = {\Bbb R}^n$. This provides the possibility to add without
+loss $(K^{-1} Q) = {\Bbb R}^n$. This provides the possibility to add,
+without loss of information, the Eqs.\ts (\ref{dis}) and (\ref{six})
+(after multiplying by $K^{-1})$. The following shooting operator is
+created
+\begin{equation}
+\quad S (\xi ) := \left\{
+\begin{array}{ll}
+ S_1 (\xi):= & \left\{
+ \begin{array} {ll}
+ g (u_0 + v_0, x (t_m; t_{m-1}, u_{m-1})) + K^{-1} Q_0 u_0
+ &\quad (a)\\
+ u_i - P_i x (t_i; t_{i-1} , u_{i-1})\; i = 1, \ldots , m-1
+ & \quad(b)
+ \end{array} \right. \\
+ S_2 (\xi) := & \left\{
+ \begin{array} {ll}
+ Q_0 y_0 + P_0 v_0 & \quad (c)\\
+ f(y_0, u_0 + v_0, t_0) & \quad (d) \quad ,
+ \end{array}
+ \right.
+\end{array} \right.\label{six}
+\end{equation}
+%
+with $\xi := (u_0 , u_1, \ldots , u_{m-1} , y_0, v_0)^{\rm T}$.
+
+\begin{lemma}
+Let $V$ be a singular matrix and $V^-$ a reflexive inverse of $V$ with
+Sect. (2.3) and $VV^- = P_D$, $V^-V = P$, where $P$ and $P_D$ satisfy
+the conditions of Lemma 2.1. Then the matrix $V + K^{-1} Q$ is
+nonsingular and
+%
+\[ (V + K^{-1} Q) ^{-1} = V^- + QK\; , \]
+%
+where $K$ is defined in Sect. (2.2).
+\end{lemma}
+
+\begin{proof}
+\begin{eqnarray*}
+(V + K^{-1}Q)(V^- + QK) & = & VV^- + VQK + K^{-1}QV^- + K^{-1} QK \\
+& = & P_D + 0 + 0 + Q_D = I\; . \quad\qed
+\end{eqnarray*}
+\end{proof}
+
+\begin{remark}
+The value $w := (P_s v_0 + Q_0 G^{-1} f (y_0, u_0 + v_0, t_0))$ at
+the right-hand side of Eq. (16) is the solution of the linear system
+\begin{equation}
+ J_4 \left(\begin{array}{c} \eta \\ w \end{array} \right)
+ = \left(\begin{array}{c} Q_0 y_0 + P_0 v_0 \\
+ f (y_0, u_0 + v_0, t_0) \end{array} \right)
+\end{equation}
+\end{remark}
+
+\begin{figure}\picplace {4 cm}
+\firstcaption{The doping profile $C (t)$ has the same structure as
+$N_-$}
+\secondcaption{The doping profile of $C (z)$}
+\end{figure}
+
+This leads to the following algorithm to compute the iteration $\xi^i$:
+\begin{description}[5 ---]
+\item[0 -- ] initial value $\xi^0 := (u_0^0 , \ldots , u^0_{m-1} , y_0^0
+, v_0^0)$
+\item[1 -- ] $i:= 0$
+\item[2 -- ] compute $u^{i+1}$ with (3.16)
+\item[3 -- ] compute $y^{i+1}_0, v_0^{i+1}$ with (3.17) using $\Delta
+u^{i+1} := u^{i+1} - u^i$
+\item[4 -- ]$i:= i + 1$
+\item[5 -- ]{\tt IF} accuracy not reached {\tt THEN GOTO 2 ELSE STOP}
+\end{description}
+
+\begin{theorem} Let the assumptions (A), (B) be fulfilled. Then the
+non-linear equation
+$$
+S (\xi) = 0
+$$
+has a nonsingular Jacobian in a neighbourhood of
+$$
+\xi = \xi_\star := (u_{\star 0}, \ldots , u_{\star m-1} , y_{\star 0},
+v_{\star 0})\; ,
+$$
+which corresponds with $x_\star$.
+\end{theorem}
+
+\section{Implementation}
+If listing of a program is desired, this is possible too \cite{darnell}
+
+\begin{verbatim}
+void get_two_kbd_chars()
+{
+ extern char KEYBOARD;
+ char c0, c1;
+
+ c0 = KEYBOARD;
+ c1 = KEYBOARD;
+}
+\end{verbatim}
+
+\section{Solutions}
+
+We solve this problem with the relative accuracy of integration $1d-4$.
+The physical meaning of $ \sigma_0 $ and $K$ is clearly visible in the
+equations above. $\sigma_0$ represents a frequency of the order one per
+free-fall time. $K$ is proportional to the ratio of the free-fall time
+and the cooling time. The experimental tests of the Standard Model and
+thereby of the unification of the weak and electromagnetic interactions
+have reached a new level of accuracy. The results are given in Table
+\ref{KapSou}.
+
+\begin{table}
+ \caption{Opacity sources}\label{KapSou}
+ \centering
+ \begin{tabular}{ll}
+ \hline\noalign{\smallskip}
+ Source & T/[K] \\
+ \noalign{\smallskip}
+ \hline
+ \noalign{\smallskip}
+ Yorke 1979, Yorke 1980a & $\leq 1700^{\rm a}$ \\
+ Kr\"ugel 1971 & $1700 \leq T \leq 5000$ \\
+ Cox and Stewart 1969 & $5000 \leq $ \\
+ \noalign{\smallskip}\hline\noalign{\smallskip}
+ $^{\rm a}$ This is a footnote.
+ \end{tabular}
+\end{table}
+
+\begin{acknowledgement}I wish to thank Prof. Dr. Roswitha M\"arz for
+many helpful discussions.\end{acknowledgement}
+
+\begin{thebibliography}[9]{References}
+% Note that space for square brackets is added to the width of the label
+% specified in the [] argument. If you don't use []s in your
+% bibliography, specify a narrower label or omit the specification
+% altogether. In this case \parindent is used.
+
+\bibitem{1.}{darnell}{[1]}
+Darnell, P.A., Margolis, P.E. (1988): C, A software engineering
+approach. Springer Verlag Berlin Heidelberg New York
+
+\bibitem{2.}{leslie}{[2]}
+Lamport, L. (1986): \LaTeX: A document preparation system.
+Addison-Wesley Publishing Company, Inc.
+
+\bibitem{3.}{seroul}{[3]}
+Seroul, R., Levy, S. (1989): A beginner's book of \TeX{}. Springer New
+York Berlin Heidelberg
+
+\bibitem{4.}{springer}{[4]}
+LJour1: Springer's \LaTeX{} style file for journals with one-column
+layout. Springer Heidelberg
+
+\bibitem{5.}{stroud}{[5]}
+Strout, A.H. (1971): approximate calculation of multiple integrals.
+Prentice Hall, Englewood Cliffs, N.J.
+
+\bibitem{6.}{tetz}{[6]}
+Tetzlaff, A. (1970): Stability in the Common Market. To appear.
+
+\bibitem{7.}{wendl}{[15]}
+Wendland, W.L., (1987): Strongly elliptic boundary integral equations.
+In: A. Iserles, M. Powell, eds., The state of the art in numerical
+analysis. Clarendon Press, Oxford, pp. 511--561
+
+\bibitem{8.}{yser}{[16]}
+Yserentant, H. (1983): A remark on the numerical computation of
+improper integrals. Computing {\bf 30}, 179--183
+
+\medskip\noindent
+\bibitem{Please}{}{}refer to a recent issue of the journal for further
+examples on how to format references.
+
+\end{thebibliography}
+\end{document}