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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
-