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+% This is the sample paper for the AmSTeX SIAM style file, (amstex)siam.sty
+% for use with AmSTeX version 2.1 or later and amsppt.sty, version 2.1a.
+% RCS information: $Revision: 1.1 $, $Date: 93/01/25 15:33:19 $.
+\input amstex
+\documentstyle{amstexs1}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Macro definitions for running heads and first page %
+\accepted\SIMAF %
+\firstpageno{10} %
+\lastpageno{12} %
+\issuevolume{1} %
+\issuenumber{2} %
+\issuemonth{February} %
+\placenumber{002} % place of paper in this issue %
+\issueyear{1988} %
+\shortauthor{Bradley J. Lucier and Douglas N. Arnold} %
+\shorttitle{A Sample Paper} %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Macros specific to this paper %
+\define\loner{{L^1(\Bbb R)}} %
+\define\linfr{{L^\infty(\Bbb R)}} %
+\define\bvr{{\roman{BV}(\Bbb R)}} %
+\define\TV{{\roman {TV}}} %
+\define\sdot{\,\cdot\,} %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\topmatter
+\title
+A SAMPLE PAPER, WITH A RATHER LONG TITLE, TO ILLUSTRATE THE
+\AmSTeX\ SIAM STYLE\footnote[\boldkey*]{Unlikely to appear.}
+\endtitle
+\author
+BRADLEY J. LUCIER\footnote[\dag]{Department of Mathematics, Purdue University,
+West Lafayette, Indiana 47907. Present address, somewhere on the beach
+(lucier\@math.purdue.edu).
+The work of the first author was not supported by the
+Wolf Foundation.}\ and DOUGLAS N. ARNOLD\footnote[\ddag]{Department
+of Mathematics, Pennsylvania State University,
+University Park, Pennsylvania 16802.}
+\endauthor
+\abstract
+This sample paper illustrates many of the amstex
+macros as used with the \AmSTeX\ SIAM style file amstexsiam (version 2.0a).
+The \AmSTeX\ SIAM style file, which
+inputs and builds upon the amsppt style (version 2.1a or later)
+of Michael Spivak, gives authors easy
+access to most of the typographical constructions used in SIAM journals.
+It does not address the issues of the table of contents
+or tables, which must be set using more primitive \TeX\ macros.
+\endabstract
+\keywords
+porous medium, interface curves
+\endkeywords
+\subjclass
+65N60
+\endsubjclass
+\endtopmatter
+\document
+\subhead 1. Introduction\endsubhead
+We are concerned with numerical approximations to the so-called
+porous-medium equation \cite{6},
+$$
+\alignedat2
+ &u_t=\phi(u)_{xx},&&\qquad x\in\Bbb R,\quad t>0,\quad\phi(u)=u^m,\quad m>1,
+\\
+ &u(x,0)=u_0(x),&&\qquad x\in\Bbb R.
+\endalignedat
+\tag 1.1
+$$
+We assume that the initial data $u_0(x)$ has bounded support, that
+$0\leq u_0\leq M$, and that $\phi(u_0)_x\in\bvr$.
+It is well known that a unique solution $u(x,t)$ of (1.1) exists,
+and that $u$ satisfies
+$$
+ 0\leq u\leq M\text{ and }\TV\phi(u(\,\cdot\,,t))_x\leq\TV\phi(u_0)_x.
+\tag 1.2
+$$
+If the data has slightly more regularity, then this too is satisfied
+by the solution. Specifically, if $m$ is no greater than two and
+$u_0$ is Lipschitz continuous, then $u(\,\cdot\,,t)$ is also Lipschitz;
+if $m$ is greater than two and $(u_0^{m-1})_x\in\linfr$, then
+$(u(\,\cdot\,,t)^{m-1})_x\in\linfr$
+(see [3]). (This will follow from results presented here, also.)
+We also use the fact that the solution $u$ is H\"older continuous in $t$.
+
+\subhead 2. $\linfr$ error bounds\endsubhead
+After a simple definition, we state a theorem
+that expresses the error of approximations $u^h$ in
+terms of the weak truncation error $E$.
+\definition{Definition 2.1}\rm A {\it definition}
+is the same as a theorem set in roman
+type. In version 2 of the \AmSTeX\ style file for the SIAM journals,
+definitions are set with their own command.
+\enddefinition
+\proclaim{Theorem 2.1}
+Let $\{u^h\}$ be a family of approximate solutions satisfying
+the following conditions for $0\leq t\leq T${\rm:}
+\roster
+\item For all $x\in\Bbb R$ and positive $t$, $0\leq u^h(x,t)\leq M${\rm;}
+\item Both $u$ and $u^h$ are H\"older--$\alpha$ in $x$
+for some $\alpha\in(0,1\wedge 1/(m-1))${\rm;} $u^h$ is right
+continuous in $t${\rm;}
+and $u^h$ is H\"older continuous in $t$ on
+strips $\Bbb R\times(t^n,t^{n+1})$, with the set $\{t^n\}$ having no
+limit points\/{\rm;} and
+\item There exists a positive function $\omega(h,\epsilon)$ such that\/{\rm:}
+whenever $\{w^\epsilon\}_{0<\epsilon\leq\epsilon_0}$ is a family of functions
+in $\bold X$ for which
+{\roster
+\item"(a)" there is a sequence of positive numbers $\epsilon$ tending
+to zero, such that for these values of
+$\epsilon$, $\|w^\epsilon\|_\infty\leq 1/\epsilon$,
+\item"(b)" for all positive
+$\epsilon$, $\|w_x^\epsilon(\sdot,t)\|_\loner\leq 1/\epsilon^2$, and
+\item"(c)" for all $\epsilon>0$,
+$$
+\sup\Sb
+x\in\Bbb R\\0\leq t_1,t_2\leq T\endSb
+\dfrac{|w^\epsilon(x,t_2)-w^\epsilon(x,t_1)|}{|t_2-t_1|^p}\leq 1/\epsilon^2,
+$$
+where $p$ is some number not exceeding $1$,
+\endroster}%
+then\footnote{This is an obvious ploy, but we need a footnote.}
+ $|E (u^h,w^\epsilon,T)|\leq\omega(h,\epsilon).$
+\item
+This is the fourth item in the outer roster.
+\endroster
+Then, there is a constant $C=C(m,M,T)$ such that
+$$\multline
+\|u-u^h\|_{\infty,\Bbb R\times[0,T]}\leq C\biggl[
+\sup \biggl |\int_\Bbb R(u_0(x)-u^h(x,0)) w(x,0) \,dx\biggr|\\
++\omega(h,\epsilon)+\epsilon^\alpha\biggr],\endmultline
+\tag 2.1
+$$
+where the supremum is taken over all $w\in\bold X$.
+\endproclaim
+
+\demo{Proof}
+We assume first that $Q$ is decreasing and consider the following cases:
+\case{Case\/ {\rm1:}
+$b'\geq 1/2$} We have $P(1/8)\geq\delta>0$ where $\delta$
+depends only on $d$, for otherwise by (3.7) applied to $P$ and $p=\infty$,
+$P$ could not attain the value $1$ at $x=1$. Similarly, for
+$m=(a'+b')/2$, $Q(m)\geq\delta'>0$ for some $\delta'$ depending only on $d$
+since otherwise $Q$ cannot attain the value $1$ at $x=a'$. Hence, for
+$\delta''=\min(\delta,\delta')$,
+$|A(y)|\geq|m-1/8|\geq b'/4\geq\frac18\max(b',1)$ for
+$y\in[0,\delta'']$. On the other hand,
+$|A(y)|\leq \max(b',1)$ for all $y\in[0,1]$, so (4.2) follows for
+all $1\leq p\leq\infty$.
+\endcase
+\case{Case\/ {\rm2:}
+$b'\leq 1/2$} We have $P(3/4)\leq\delta<1$ with $\delta$
+depending only on $d$ for otherwise (3.7) applied to $1-P$ and $p=\infty$
+would show that $P$ could not attain the value $0$ at $x=0$. It follows
+that $|A(y)|\geq 3/4-b'\geq 1/4$, $y\in[\delta,1]$, while $|A(y)|\leq 1$
+for all $y\in[0,1]$. Hence (4.2) follows for
+all $1\leq p\leq\infty$.
+\endcase
+We consider now when $Q$ is increasing. We can assume that $Q$ is not
+a translate of $P$, i.e\., we do not have $P(x)=Q(x+\delta)$ for some $\delta$,
+for then (4.2) follows trivially. In what follows, $C$ and $\delta$
+depend on $d$, and $C$ may depend on $p$. We consider the following cases:
+\case{Case\/ {\rm3:} $a'\geq 1/4$ and $b'\leq 100$}
+From (3.7) for $P$
+and $p=\infty$, it follows that $P(1/8)\geq\delta$ since otherwise $P$ cannot
+attain the value $1$ at $x=1$. Hence $|A(y)|\geq a'-1/8\geq1/8$ on
+$[0,\delta]$. On the other hand $|A(y)|\leq b'$ for all $y\in[0,1]$ and hence
+(4.2) follows for all $1\leq p\leq\infty$.
+\endcase
+Let $z$ be in $\bold X$. Because $E(u,\sdot,\sdot)\equiv0$,
+Equation (1.5) implies that
+$$
+\int_\Bbb R\Delta uz|^T_0dx=\int_0^T\int_\Bbb R
+\Delta u(z_t+\phi[u,u^h]z_{xx})\,dx\,dt-
+E(u^h,z,t),
+\tag 2.2
+$$
+where $\Delta u=u-u^h$ and
+$$
+\phi[u,u^h]=\dfrac{\phi(u)-\phi(u^h)}{u-u^h}.
+$$
+Extend $\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,0)$ for negative $t$, and
+$\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,T)$
+for $t>T$.
+Fix a point $x_0$ and a number $\epsilon>0$. Let $j_\epsilon$
+be a smooth function of $x$ with integral $1$ and support in
+$[-\epsilon,\epsilon]$,
+and let $J_\delta$ be a smooth function of
+$x$ and $t$ with integral $1$ and support in
+$[-\delta,\delta]\times[-\delta,\delta]$; $\delta$ and $\epsilon$ are
+positive numbers to be specified later.
+We choose $z=z^{\epsilon\delta}$ to satisfy
+$$
+\aligned
+ &z_t+(\delta+J_\delta*\phi[u,u^h])z_{xx}=0,\qquad x\in\Bbb R,\;0
+\leq t\leq T,\\
+ &z(x,T)=j_\epsilon(x-x_0).
+\endaligned
+\tag 2.3
+$$
+The conclusion of the theorem now follows from (2.1) and the fact that
+$$
+|j_\epsilon*\Delta u(x_0,t)-\Delta u(x_0,t)|\leq C\epsilon^\alpha,
+$$
+which follows from Assumption 2.
+\qquad\qed
+\enddemo
+\example{Example\/ {\rm 1}} This is an example of an example.
+\endexample
+\remark{Remark\/ {\rm 1}} Examples are set the same as definitions in
+some styles,
+and the same as proofs in others. What convention does this style follow?
+\endremark
+Sometimes you want to include a figure, as in Fig.~1.
+\topinsert
+\def\Bif{{\bf if\/ }}\def\Bwhile{{\bf while\/ }}\def\Belse{{\bf else\/ }}
+\settabs\+\qquad&\qquad&\qquad&\qquad&\cr
+\+\smc Tree Partition Algorithm \{\cr
+\+&Let stack size denote the number of nodes in the\cr
+\+&&subtrees stored temporarily on the local stack\cr
+\+&pop I from global stack\cr
+\+&set stack size := 0\cr
+\+&\Bwhile (stack size $\leq$ max size and stack size +
+I$\rightarrow$tree size $>$ 3 (max size)) \{\cr
+\+&&process I as an interior node\cr
+\+&&let min tree be the smaller of the subtrees of the two children of I\cr
+\+&&let max tree be the larger of the subtrees of the two children of I\cr
+\+&&\Bif (min tree$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr
+\+&&&push min tree onto the global stack\cr
+\+&&\} \Belse \{\cr
+\+&&&push min tree onto the local stack\cr
+\+&&&set stack size := stack size + min tree$\rightarrow$tree size\cr
+\+&&\}\cr
+\+&&set I := max tree\cr
+\+&\}\cr
+\+&\Bif (I$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr
+\+&&push I onto the global stack\cr
+\+&\} \Belse \{\cr
+\+&&push I onto the local stack\cr
+\+&\}\cr
+\+&Process all subtrees on the local stack\cr
+\+\}\cr
+\botcaption{Fig.~1} Tree partition algorithm Tree partition algorithm
+Tree partition algorithm Tree partition algorithm Tree partition algorithm
+Tree partition algorithm Tree partition algorithm.\endcaption
+\endinsert
+
+We finish with a table of all SIAM journals.
+\midinsert
+\topcaption{Table 1}{SIAM journal acronyms and titles}\endcaption
+\settabs\+\indent&Acronym\indent&Title&\cr
+\hbox to \hsize{\hrulefill}
+\+&Acronym&Title&\cr
+\hbox to \hsize{\hrulefill}
+\+&SINUM&SIAM Journal on Numerical Analysis&\cr
+\+&SIREV&SIAM Review&\cr
+\+&SIMA&SIAM Journal on Mathematical Analysis&\cr
+\+&SIMAX&SIAM Journal on Matrix Analysis and Applications&\cr
+\+&SICOMP&SIAM Journal on Computing&\cr
+\+&SISC&SIAM Journal on Scientific Computing&\cr
+\+&SIOPT&SIAM Journal on Optimization&\cr
+\+&SIAP&SIAM Journal on Applied Mathematics&\cr
+\+&SICON&SIAM Journal on Control and Optimization&\cr
+\+&SIDMA&SIAM Journal on Discrete Mathematics&\cr
+\+&TVP&Theory of Probability and Its Applications&\cr
+\hbox to \hsize{\hrulefill}
+\endinsert
+
+\Refs
+\ref
+ \no 1
+ \by L. A. Caffarelli and A. Friedman
+ \paper Regularity of the free boundary of a gas flow in an
+ $n$-dimensional porous medium
+ \jour Indiana Math. J.
+ \vol 29
+ \yr 1980
+ \pages 361--391
+\endref
+\ref\no 2
+ \by R. DeVore and B. Lucier
+ \paper High order regularity for solutions of the inviscid Burgers equation
+ \inbook Nonlinear Hyperbolic Problems
+\procinfo Proceedings of an Advanced Research Workshop, Bordeaux,
+France, June 1988
+ \bookinfo Lecture Notes in Mathematics
+ \vol 1402
+ \eds C. Carasso, P. Charrier, B. Hanouzet, and J.-L. Joly
+ \yr 1989
+ \publ Springer-Verlag
+ \publaddr New York
+ \pages 147--154
+\endref
+\ref \no 3
+ \bysame
+ \paper Wavelets
+ \jour Acta Numerica
+ \yr 1992
+ \ed A. Iserles
+ \publ Cambridge University Press
+ \publaddr New York
+ \pages 1--56
+\endref
+\ref \no 4
+ \by R. A. DeVore and V. A. Popov
+ \paper Interpolation spaces and non-linear approximation
+ \inbook Function Spaces and Applications
+ \bookinfo Lecture Notes in Mathematics
+ \procinfo Proceedings of the US--Swedish Seminar held in Lund,
+Sweden, June 15--21, 1986
+ \vol 1302
+ \eds M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin
+ \publ Springer-Verlag
+ \publaddr New York
+ \yr 1988
+ \pages 191--205
+ \endref
+\ref \no 5
+ \by R. A. DeVore and X. M. Yu
+ \paper Nonlinear $n$-widths in Besov spaces
+ \inbook Approximation Theory VI: Vol. 1
+ \eds C. K. Chui, L. L. Schumaker, and J. D. Ward
+ \publ Academic Press
+ \publaddr New York
+ \yr 1989
+ \pages 203--206
+ \lang In Russian
+ \endref
+\ref
+ \no 6
+ \by K. Hollig and M. Pilant
+ \paper Regularity of the free boundary for the porous medium equation
+ \paperinfo MRC Tech. Rep. 2742
+\endref
+\ref
+ \no 7
+ \by J. Jerome
+ \book Approximation of Nonlinear Evolution Systems
+ \publ Academic Press
+ \publaddr New York
+ \yr 1983
+\endref
+\ref
+ \no 8
+ \manyby R. J. LeVeque
+ \paper Convergence of a large time step generalization of Godunov's method
+ for conservation laws
+ \jour Comm. Pure Appl. Math.
+ \vol 37
+ \yr 1984
+ \pages 463--478
+\endref
+\ref\no 9
+ \by O. Rioul and M. Vetterli
+ \paper Wavelets and signal processing
+ \jour IEEE Signal Processing Magazine
+ \vol 8
+ \issue 4
+ \yr 1991
+ \toappear
+\endref
+\endRefs
+\enddocument
+