~ubuntu-branches/debian/wheezy/texlive-extra/wheezy
» Revision
4
: texmf-dist/tex/amstex/siam/amsamp.tex
« back to all changes in this revision
Viewing changes to texmf-dist/tex/amstex/siam/amsamp.tex
- browse files at revision 4
- compare with another revision
- download diff
- download tarball
- view history from revision 4
- Committer: Bazaar Package Importer
- Author(s): Norbert Preining
- Date: 2007-01-12 19:08:37 UTC
- mfrom: (1.2.1 upstream) (3.1.2 feisty)
- Revision ID: james.westby@ubuntu.com-20070112190837-1cm7kyizrcdtk1ac
Tags: 2005.dfsg.3-1
blacklist siam.tpm and build new upstream, as the siam macros are not
DFSG free (no selling clause) (Closes: #406426)
DFSG free (no selling clause) (Closes: #406426)
- files added:
- debian/bug.control
- debian/texlive-fonts-extra.root/usr/share/texmf-texlive/fonts/map
- debian/texlive-fonts-extra.root/usr/share/texmf-texlive/fonts/map/dvips
- debian/texlive-fonts-extra.root/usr/share/texmf-texlive/fonts/map/dvips/doublestroke
- files removed:
- texmf-dist/doc/amstex/siam
- texmf-dist/doc/latex/siam
- texmf-dist/source/latex/siam
- texmf-dist/tex/amstex/siam
- texmf-dist/tex/latex/siam
- texmf-dist/tex/plain/siam
- files modified:
- debian/CHANGES.packaging
added
removed
texmf-dist/tex/amstex/siam/amsamp.tex
1
% This is the sample paper for the AmSTeX SIAM style file, (amstex)siam.sty
2
% for use with AmSTeX version 2.1 or later and amsppt.sty, version 2.1a.
3
% RCS information: $Revision: 1.1 $, $Date: 93/01/25 15:33:19 $.
4
\input amstex
5
\documentstyle{amstexs1}
6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
7
% Macro definitions for running heads and first page %
8
\accepted\SIMAF %
9
\firstpageno{10} %
10
\lastpageno{12} %
11
\issuevolume{1} %
12
\issuenumber{2} %
13
\issuemonth{February} %
14
\placenumber{002} % place of paper in this issue %
15
\issueyear{1988} %
16
\shortauthor{Bradley J. Lucier and Douglas N. Arnold} %
17
\shorttitle{A Sample Paper} %
18
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
19
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
20
% Macros specific to this paper %
21
\define\loner{{L^1(\Bbb R)}} %
22
\define\linfr{{L^\infty(\Bbb R)}} %
23
\define\bvr{{\roman{BV}(\Bbb R)}} %
24
\define\TV{{\roman {TV}}} %
25
\define\sdot{\,\cdot\,} %
26
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
27
\topmatter
28
\title
29
A SAMPLE PAPER, WITH A RATHER LONG TITLE, TO ILLUSTRATE THE
30
\AmSTeX\ SIAM STYLE\footnote[\boldkey*]{Unlikely to appear.}
31
\endtitle
32
\author
33
BRADLEY J. LUCIER\footnote[\dag]{Department of Mathematics, Purdue University,
34
West Lafayette, Indiana 47907. Present address, somewhere on the beach
35
(lucier\@math.purdue.edu).
36
The work of the first author was not supported by the
37
Wolf Foundation.}\ and DOUGLAS N. ARNOLD\footnote[\ddag]{Department
38
of Mathematics, Pennsylvania State University,
39
University Park, Pennsylvania 16802.}
40
\endauthor
41
\abstract
42
This sample paper illustrates many of the amstex
43
macros as used with the \AmSTeX\ SIAM style file amstexsiam (version 2.0a).
44
The \AmSTeX\ SIAM style file, which
45
inputs and builds upon the amsppt style (version 2.1a or later)
46
of Michael Spivak, gives authors easy
47
access to most of the typographical constructions used in SIAM journals.
48
It does not address the issues of the table of contents
49
or tables, which must be set using more primitive \TeX\ macros.
50
\endabstract
51
\keywords
52
porous medium, interface curves
53
\endkeywords
54
\subjclass
55
65N60
56
\endsubjclass
57
\endtopmatter
58
\document
59
\subhead 1. Introduction\endsubhead
60
We are concerned with numerical approximations to the so-called
61
porous-medium equation \cite{6},
62
$$
63
\alignedat2
64
&u_t=\phi(u)_{xx},&&\qquad x\in\Bbb R,\quad t>0,\quad\phi(u)=u^m,\quad m>1,
65
\\
66
&u(x,0)=u_0(x),&&\qquad x\in\Bbb R.
67
\endalignedat
68
\tag 1.1
69
$$
70
We assume that the initial data $u_0(x)$ has bounded support, that
71
$0\leq u_0\leq M$, and that $\phi(u_0)_x\in\bvr$.
72
It is well known that a unique solution $u(x,t)$ of (1.1) exists,
73
and that $u$ satisfies
74
$$
75
0\leq u\leq M\text{ and }\TV\phi(u(\,\cdot\,,t))_x\leq\TV\phi(u_0)_x.
76
\tag 1.2
77
$$
78
If the data has slightly more regularity, then this too is satisfied
79
by the solution. Specifically, if $m$ is no greater than two and
80
$u_0$ is Lipschitz continuous, then $u(\,\cdot\,,t)$ is also Lipschitz;
81
if $m$ is greater than two and $(u_0^{m-1})_x\in\linfr$, then
82
$(u(\,\cdot\,,t)^{m-1})_x\in\linfr$
83
(see [3]). (This will follow from results presented here, also.)
84
We also use the fact that the solution $u$ is H\"older continuous in $t$.
85
86
\subhead 2. $\linfr$ error bounds\endsubhead
87
After a simple definition, we state a theorem
88
that expresses the error of approximations $u^h$ in
89
terms of the weak truncation error $E$.
90
\definition{Definition 2.1}\rm A {\it definition}
91
is the same as a theorem set in roman
92
type. In version 2 of the \AmSTeX\ style file for the SIAM journals,
93
definitions are set with their own command.
94
\enddefinition
95
\proclaim{Theorem 2.1}
96
Let $\{u^h\}$ be a family of approximate solutions satisfying
97
the following conditions for $0\leq t\leq T${\rm:}
98
\roster
99
\item For all $x\in\Bbb R$ and positive $t$, $0\leq u^h(x,t)\leq M${\rm;}
100
\item Both $u$ and $u^h$ are H\"older--$\alpha$ in $x$
101
for some $\alpha\in(0,1\wedge 1/(m-1))${\rm;} $u^h$ is right
102
continuous in $t${\rm;}
103
and $u^h$ is H\"older continuous in $t$ on
104
strips $\Bbb R\times(t^n,t^{n+1})$, with the set $\{t^n\}$ having no
105
limit points\/{\rm;} and
106
\item There exists a positive function $\omega(h,\epsilon)$ such that\/{\rm:}
107
whenever $\{w^\epsilon\}_{0<\epsilon\leq\epsilon_0}$ is a family of functions
108
in $\bold X$ for which
109
{\roster
110
\item"(a)" there is a sequence of positive numbers $\epsilon$ tending
111
to zero, such that for these values of
112
$\epsilon$, $\|w^\epsilon\|_\infty\leq 1/\epsilon$,
113
\item"(b)" for all positive
114
$\epsilon$, $\|w_x^\epsilon(\sdot,t)\|_\loner\leq 1/\epsilon^2$, and
115
\item"(c)" for all $\epsilon>0$,
116
$$
117
\sup\Sb
118
x\in\Bbb R\\0\leq t_1,t_2\leq T\endSb
119
\dfrac{|w^\epsilon(x,t_2)-w^\epsilon(x,t_1)|}{|t_2-t_1|^p}\leq 1/\epsilon^2,
120
$$
121
where $p$ is some number not exceeding $1$,
122
\endroster}%
123
then\footnote{This is an obvious ploy, but we need a footnote.}
124
$|E (u^h,w^\epsilon,T)|\leq\omega(h,\epsilon).$
125
\item
126
This is the fourth item in the outer roster.
127
\endroster
128
Then, there is a constant $C=C(m,M,T)$ such that
129
$$\multline
130
\|u-u^h\|_{\infty,\Bbb R\times[0,T]}\leq C\biggl[
131
\sup \biggl |\int_\Bbb R(u_0(x)-u^h(x,0)) w(x,0) \,dx\biggr|\\
132
+\omega(h,\epsilon)+\epsilon^\alpha\biggr],\endmultline
133
\tag 2.1
134
$$
135
where the supremum is taken over all $w\in\bold X$.
136
\endproclaim
137
138
\demo{Proof}
139
We assume first that $Q$ is decreasing and consider the following cases:
140
\case{Case\/ {\rm1:}
141
$b'\geq 1/2$} We have $P(1/8)\geq\delta>0$ where $\delta$
142
depends only on $d$, for otherwise by (3.7) applied to $P$ and $p=\infty$,
143
$P$ could not attain the value $1$ at $x=1$. Similarly, for
144
$m=(a'+b')/2$, $Q(m)\geq\delta'>0$ for some $\delta'$ depending only on $d$
145
since otherwise $Q$ cannot attain the value $1$ at $x=a'$. Hence, for
146
$\delta''=\min(\delta,\delta')$,
147
$|A(y)|\geq|m-1/8|\geq b'/4\geq\frac18\max(b',1)$ for
148
$y\in[0,\delta'']$. On the other hand,
149
$|A(y)|\leq \max(b',1)$ for all $y\in[0,1]$, so (4.2) follows for
150
all $1\leq p\leq\infty$.
151
\endcase
152
\case{Case\/ {\rm2:}
153
$b'\leq 1/2$} We have $P(3/4)\leq\delta<1$ with $\delta$
154
depending only on $d$ for otherwise (3.7) applied to $1-P$ and $p=\infty$
155
would show that $P$ could not attain the value $0$ at $x=0$. It follows
156
that $|A(y)|\geq 3/4-b'\geq 1/4$, $y\in[\delta,1]$, while $|A(y)|\leq 1$
157
for all $y\in[0,1]$. Hence (4.2) follows for
158
all $1\leq p\leq\infty$.
159
\endcase
160
We consider now when $Q$ is increasing. We can assume that $Q$ is not
161
a translate of $P$, i.e\., we do not have $P(x)=Q(x+\delta)$ for some $\delta$,
162
for then (4.2) follows trivially. In what follows, $C$ and $\delta$
163
depend on $d$, and $C$ may depend on $p$. We consider the following cases:
164
\case{Case\/ {\rm3:} $a'\geq 1/4$ and $b'\leq 100$}
165
From (3.7) for $P$
166
and $p=\infty$, it follows that $P(1/8)\geq\delta$ since otherwise $P$ cannot
167
attain the value $1$ at $x=1$. Hence $|A(y)|\geq a'-1/8\geq1/8$ on
168
$[0,\delta]$. On the other hand $|A(y)|\leq b'$ for all $y\in[0,1]$ and hence
169
(4.2) follows for all $1\leq p\leq\infty$.
170
\endcase
171
Let $z$ be in $\bold X$. Because $E(u,\sdot,\sdot)\equiv0$,
172
Equation (1.5) implies that
173
$$
174
\int_\Bbb R\Delta uz|^T_0dx=\int_0^T\int_\Bbb R
175
\Delta u(z_t+\phi[u,u^h]z_{xx})\,dx\,dt-
176
E(u^h,z,t),
177
\tag 2.2
178
$$
179
where $\Delta u=u-u^h$ and
180
$$
181
\phi[u,u^h]=\dfrac{\phi(u)-\phi(u^h)}{u-u^h}.
182
$$
183
Extend $\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,0)$ for negative $t$, and
184
$\phi[u,u^h](\cdot,t)=\phi[u,u^h](\cdot,T)$
185
for $t>T$.
186
Fix a point $x_0$ and a number $\epsilon>0$. Let $j_\epsilon$
187
be a smooth function of $x$ with integral $1$ and support in
188
$[-\epsilon,\epsilon]$,
189
and let $J_\delta$ be a smooth function of
190
$x$ and $t$ with integral $1$ and support in
191
$[-\delta,\delta]\times[-\delta,\delta]$; $\delta$ and $\epsilon$ are
192
positive numbers to be specified later.
193
We choose $z=z^{\epsilon\delta}$ to satisfy
194
$$
195
\aligned
196
&z_t+(\delta+J_\delta*\phi[u,u^h])z_{xx}=0,\qquad x\in\Bbb R,\;0
197
\leq t\leq T,\\
198
&z(x,T)=j_\epsilon(x-x_0).
199
\endaligned
200
\tag 2.3
201
$$
202
The conclusion of the theorem now follows from (2.1) and the fact that
203
$$
204
|j_\epsilon*\Delta u(x_0,t)-\Delta u(x_0,t)|\leq C\epsilon^\alpha,
205
$$
206
which follows from Assumption 2.
207
\qquad\qed
208
\enddemo
209
\example{Example\/ {\rm 1}} This is an example of an example.
210
\endexample
211
\remark{Remark\/ {\rm 1}} Examples are set the same as definitions in
212
some styles,
213
and the same as proofs in others. What convention does this style follow?
214
\endremark
215
Sometimes you want to include a figure, as in Fig.~1.
216
\topinsert
217
\def\Bif{{\bf if\/ }}\def\Bwhile{{\bf while\/ }}\def\Belse{{\bf else\/ }}
218
\settabs\+\qquad&\qquad&\qquad&\qquad&\cr
219
\+\smc Tree Partition Algorithm \{\cr
220
\+&Let stack size denote the number of nodes in the\cr
221
\+&&subtrees stored temporarily on the local stack\cr
222
\+&pop I from global stack\cr
223
\+&set stack size := 0\cr
224
\+&\Bwhile (stack size $\leq$ max size and stack size +
225
I$\rightarrow$tree size $>$ 3 (max size)) \{\cr
226
\+&&process I as an interior node\cr
227
\+&&let min tree be the smaller of the subtrees of the two children of I\cr
228
\+&&let max tree be the larger of the subtrees of the two children of I\cr
229
\+&&\Bif (min tree$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr
230
\+&&&push min tree onto the global stack\cr
231
\+&&\} \Belse \{\cr
232
\+&&&push min tree onto the local stack\cr
233
\+&&&set stack size := stack size + min tree$\rightarrow$tree size\cr
234
\+&&\}\cr
235
\+&&set I := max tree\cr
236
\+&\}\cr
237
\+&\Bif (I$\rightarrow$tree size + stack size $>$ 3 (max size)) \{\cr
238
\+&&push I onto the global stack\cr
239
\+&\} \Belse \{\cr
240
\+&&push I onto the local stack\cr
241
\+&\}\cr
242
\+&Process all subtrees on the local stack\cr
243
\+\}\cr
244
\botcaption{Fig.~1} Tree partition algorithm Tree partition algorithm
245
Tree partition algorithm Tree partition algorithm Tree partition algorithm
246
Tree partition algorithm Tree partition algorithm.\endcaption
247
\endinsert
248
249
We finish with a table of all SIAM journals.
250
\midinsert
251
\topcaption{Table 1}{SIAM journal acronyms and titles}\endcaption
252
\settabs\+\indent&Acronym\indent&Title&\cr
253
\hbox to \hsize{\hrulefill}
254
\+&Acronym&Title&\cr
255
\hbox to \hsize{\hrulefill}
256
\+&SINUM&SIAM Journal on Numerical Analysis&\cr
257
\+&SIREV&SIAM Review&\cr
258
\+&SIMA&SIAM Journal on Mathematical Analysis&\cr
259
\+&SIMAX&SIAM Journal on Matrix Analysis and Applications&\cr
260
\+&SICOMP&SIAM Journal on Computing&\cr
261
\+&SISC&SIAM Journal on Scientific Computing&\cr
262
\+&SIOPT&SIAM Journal on Optimization&\cr
263
\+&SIAP&SIAM Journal on Applied Mathematics&\cr
264
\+&SICON&SIAM Journal on Control and Optimization&\cr
265
\+&SIDMA&SIAM Journal on Discrete Mathematics&\cr
266
\+&TVP&Theory of Probability and Its Applications&\cr
267
\hbox to \hsize{\hrulefill}
268
\endinsert
269
270
\Refs
271
\ref
272
\no 1
273
\by L. A. Caffarelli and A. Friedman
274
\paper Regularity of the free boundary of a gas flow in an
275
$n$-dimensional porous medium
276
\jour Indiana Math. J.
277
\vol 29
278
\yr 1980
279
\pages 361--391
280
\endref
281
\ref\no 2
282
\by R. DeVore and B. Lucier
283
\paper High order regularity for solutions of the inviscid Burgers equation
284
\inbook Nonlinear Hyperbolic Problems
285
\procinfo Proceedings of an Advanced Research Workshop, Bordeaux,
286
France, June 1988
287
\bookinfo Lecture Notes in Mathematics
288
\vol 1402
289
\eds C. Carasso, P. Charrier, B. Hanouzet, and J.-L. Joly
290
\yr 1989
291
\publ Springer-Verlag
292
\publaddr New York
293
\pages 147--154
294
\endref
295
\ref \no 3
296
\bysame
297
\paper Wavelets
298
\jour Acta Numerica
299
\yr 1992
300
\ed A. Iserles
301
\publ Cambridge University Press
302
\publaddr New York
303
\pages 1--56
304
\endref
305
\ref \no 4
306
\by R. A. DeVore and V. A. Popov
307
\paper Interpolation spaces and non-linear approximation
308
\inbook Function Spaces and Applications
309
\bookinfo Lecture Notes in Mathematics
310
\procinfo Proceedings of the US--Swedish Seminar held in Lund,
311
Sweden, June 15--21, 1986
312
\vol 1302
313
\eds M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin
314
\publ Springer-Verlag
315
\publaddr New York
316
\yr 1988
317
\pages 191--205
318
\endref
319
\ref \no 5
320
\by R. A. DeVore and X. M. Yu
321
\paper Nonlinear $n$-widths in Besov spaces
322
\inbook Approximation Theory VI: Vol. 1
323
\eds C. K. Chui, L. L. Schumaker, and J. D. Ward
324
\publ Academic Press
325
\publaddr New York
326
\yr 1989
327
\pages 203--206
328
\lang In Russian
329
\endref
330
\ref
331
\no 6
332
\by K. Hollig and M. Pilant
333
\paper Regularity of the free boundary for the porous medium equation
334
\paperinfo MRC Tech. Rep. 2742
335
\endref
336
\ref
337
\no 7
338
\by J. Jerome
339
\book Approximation of Nonlinear Evolution Systems
340
\publ Academic Press
341
\publaddr New York
342
\yr 1983
343
\endref
344
\ref
345
\no 8
346
\manyby R. J. LeVeque
347
\paper Convergence of a large time step generalization of Godunov's method
348
for conservation laws
349
\jour Comm. Pure Appl. Math.
350
\vol 37
351
\yr 1984
352
\pages 463--478
353
\endref
354
\ref\no 9
355
\by O. Rioul and M. Vetterli
356
\paper Wavelets and signal processing
357
\jour IEEE Signal Processing Magazine
358
\vol 8
359
\issue 4
360
\yr 1991
361
\toappear
362
\endref
363
\endRefs
364
\enddocument
365
Loggerhead is a web-based interface for Breezy
Version: 2.0.1