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6.5 mm=0
<#2190#>;SPMnbsp;<#2190#>Springer-Verlag, Postfach 105280, D-6900 Heidelberg 1, FRG
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If you do not yet have an account, press RETURN
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The file name should be your name and country
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9 a.m. and 1 p.m. -- European local time):
Telephone (0049)(0)6221-487478
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!!Mailing address for your disk/magnetic tape and output:
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;SPMnbsp;Mailing address for your disk/magnetic tape and output:
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=N=<#2202#><#2202#><#2200#>=A
<#276#>=
Springer-Verlag New York, Inc.
175 Fifth Avenue,
New York, New York 10010 USA
<#276#>
The following formats are acceptable: 5.25#math279#′′ diskette
MS-DOS, 5.25#math280#′′ CP/M, 3.5#math281#′′ diskette
MS-DOS, 3.5#math282#′′ diskette Apple MacIntosh, 9-track 1600
bpi magnetic tape VAX/VMS, 9-track 1600 bpi magnetic tape ANSI with
label, 9-track 1600 bpi magnetic tape, SUN-Streamer Tape.
Once you have completed your work using this macro package, please
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Contributions coded with TEX<#284#><#284#> but not with the JNS style, cannot be
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N
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!!Table of Contents
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N!1!true pt
=N<#2209#>to
<#2210#>1<#2210#><#2211#>1=
1=1=Introduction<#2211#>!4!to 0.5em<#2212#>.<#2212#>to<#2213#>4<#2213#><#2209#>
N!2!true pt
=N<#2214#>to
<#2215#>2<#2215#><#2216#>1=
1=1=General Remarks<#2216#>!4!to 0.5em<#2217#>.<#2217#>to<#2218#>4<#2218#><#2214#>
truept
<#2219#>
to<#2220#>2.1<#2220#><#2221#>How to Proceed<#2221#>!4!to 0.5em<#2222#>.<#2222#>to<#2223#>4<#2223#><#2219#>
<#2224#>
to<#2225#>2.2<#2225#><#2226#>Contributions Coded with PlainTEX<#2229#><#2229#> without the<#2226#>!!to 0.5em<#2227#>.<#2227#>to<#2228#><#2228#><#2224#>
<#2230#>
to<#2231#><#2231#><#2232#>Springer-Verlag TEX<#2235#><#2235#> Macro Package<#2232#>!4!to 0.5em<#2233#>.<#2233#>to<#2234#>4<#2234#><#2230#>
<#2236#>
to<#2237#>2.3<#2237#><#2238#>Problems with PC TEX<#2238#>!5!to 0.5em<#2239#>.<#2239#>to<#2240#>5<#2240#><#2236#>
<#2241#>
to<#2242#>2.4<#2242#><#2243#>General Rules for Coding Mathematics<#2243#>!5!to 0.5em<#2244#>.<#2244#>to<#2245#>5<#2245#><#2241#>
<#2246#>
to<#2247#>2.4.1<#2247#><#2248#>Italic and Roman in Math Mode<#2248#>!6!to 0.5em<#2249#>.<#2249#>to<#2250#>6<#2250#><#2246#>
<#2251#>
to<#2252#>2.5<#2252#><#2253#>Capitalization and Non-capitalization in the<#2253#>!!to 0.5em<#2254#>.<#2254#>to<#2255#><#2255#><#2251#>
<#2256#>
to<#2257#><#2257#><#2258#>Input (Source) File<#2258#>!6!to 0.5em<#2259#>.<#2259#>to<#2260#>6<#2260#><#2256#>
<#2261#>
to<#2262#>2.6<#2262#><#2263#>Abbreviations of Words in the Input (Source) File<#2263#>!6!to 0.5em<#2264#>.<#2264#>to<#2265#>6<#2265#><#2261#>
N!3!true pt
=N<#2266#>to
<#2267#>3<#2267#><#2268#>1=
1=1=How to Handle Your Contribution<#2268#>!7!to 0.5em<#2269#>.<#2269#>to<#2270#>7<#2270#><#2266#>
N!4!true pt
=N<#2271#>to
<#2272#>4<#2272#><#2273#>1=
1=1=How to Code<#2273#>!7!to 0.5em<#2274#>.<#2274#>to<#2275#>7<#2275#><#2271#>
truept
<#2276#>
to<#2277#>4.1<#2277#><#2278#>Headings<#2278#>!7!to 0.5em<#2279#>.<#2279#>to<#2280#>7<#2280#><#2276#>
<#2281#>
to<#2282#>4.1.1<#2282#><#2283#>Defining Your Own Environments<#2283#>!9!to 0.5em<#2284#>.<#2284#>to<#2285#>9<#2285#><#2281#>
<#2286#>
to<#2287#>4.2<#2287#><#2288#>Text<#2288#>!13!to 0.5em<#2289#>.<#2289#>to<#2290#>13<#2290#><#2286#>
<#2291#>
to<#2292#>4.3<#2292#><#2293#>Special Typefaces<#2293#>!13!to 0.5em<#2294#>.<#2294#>to<#2295#>13<#2295#><#2291#>
<#2296#>
to<#2297#>4.4<#2297#><#2298#>Footnotes<#2298#>!14!to 0.5em<#2299#>.<#2299#>to<#2300#>14<#2300#><#2296#>
<#2301#>
to<#2302#>4.5<#2302#><#2303#>Lists<#2303#>!14!to 0.5em<#2304#>.<#2304#>to<#2305#>14<#2305#><#2301#>
<#2306#>
to<#2307#>4.6<#2307#><#2308#>Figures<#2308#>!15!to 0.5em<#2309#>.<#2309#>to<#2310#>15<#2310#><#2306#>
<#2311#>
to<#2312#>4.6.1<#2312#><#2313#>Two Figures Next to Each Other<#2313#>!16!to 0.5em<#2314#>.<#2314#>to<#2315#>16<#2315#><#2311#>
<#2316#>
to<#2317#>4.6.2<#2317#><#2318#>Modified Legend Arrangements<#2318#>!17!to 0.5em<#2319#>.<#2319#>to<#2320#>17<#2320#><#2316#>
<#2321#>
to<#2322#>4.7<#2322#><#2323#>Tables<#2323#>!18!to 0.5em<#2324#>.<#2324#>to<#2325#>18<#2325#><#2321#>
<#2326#>
to<#2327#>4.7.1<#2327#><#2328#>Tables Coded with TEX<#2331#><#2331#><#2328#>!18!to 0.5em<#2329#>.<#2329#>to<#2330#>18<#2330#><#2326#>
<#2332#>
to<#2333#>4.7.2<#2333#><#2334#>Tables Not Coded with TEX<#2337#><#2337#><#2334#>!19!to 0.5em<#2335#>.<#2335#>to<#2336#>19<#2336#><#2332#>
<#2338#>
to<#2339#>4.8<#2339#><#2340#>Signs and Special Characters<#2340#>!19!to 0.5em<#2341#>.<#2341#>to<#2342#>19<#2342#><#2338#>
<#2343#>
to<#2344#>4.8.1<#2344#><#2345#>Special Signs<#2345#>!19!to 0.5em<#2346#>.<#2346#>to<#2347#>19<#2347#><#2343#>
<#2348#>
to<#2349#>4.8.2<#2349#><#2350#>Gothic (Fraktur)<#2350#>!20!to 0.5em<#2351#>.<#2351#>to<#2352#>20<#2352#><#2348#>
<#2353#>
to<#2354#>4.8.3<#2354#><#2355#>Script<#2355#>!20!to 0.5em<#2356#>.<#2356#>to<#2357#>20<#2357#><#2353#>
<#2358#>
to<#2359#>4.8.4<#2359#><#2360#>Special Roman<#2360#>!20!to 0.5em<#2361#>.<#2361#>to<#2362#>20<#2362#><#2358#>
<#2363#>
to<#2364#>4.8.5<#2364#><#2365#>Sans Serif<#2365#>!20!to 0.5em<#2366#>.<#2366#>to<#2367#>20<#2367#><#2363#>
<#2368#>
to<#2369#>4.8.6<#2369#><#2370#>Invented Characters<#2370#>!20!to 0.5em<#2371#>.<#2371#>to<#2372#>20<#2372#><#2368#>
N!5!true pt
=N<#2373#>to
<#2374#>5<#2374#><#2375#>1=
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truept
<#2378#>
to<#2379#>5.1<#2379#><#2380#>Author--Year System<#2380#>!21!to 0.5em<#2381#>.<#2381#>to<#2382#>21<#2382#><#2378#>
<#2383#>
to<#2384#>5.2<#2384#><#2385#>References by Number Only and
by Letter--Number<#2385#>!22!to 0.5em<#2386#>.<#2386#>to<#2387#>22<#2387#><#2383#>
<#2388#>
to<#2389#>5.3<#2389#><#2390#>Examples<#2390#>!22!to 0.5em<#2391#>.<#2391#>to<#2392#>22<#2392#><#2388#>
N!6!true pt
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<#2394#>6<#2394#><#2395#>1=
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N
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1=1=1=
=10000
0pt plus 6em
0=<#2399#>1<#2402#><#2402#>
!1.!Introduction
1.;SPMnbsp;Introduction
<#2399#>0=0 by-
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0=00 by00 by
3
0 by
0;SPMgt;
!1.!Introduction
1.;SPMnbsp;Introduction
=<#2401#>
=N=<#2403#><#2403#><#2401#>=A
Authors wishing to code their contribution for the
Journal of Nonlinear Science,
with TEX<#401#><#401#>, as well as those who have already coded with TEX<#402#><#402#>, will
be provided with macros that will give the text the desired layout.
Using the macros will ease considerably your coding with TEX.<#403#><#403#> Authors
are requested to adhere strictly to these instructions; <#404#>the macros
must not be changed<#404#>.
The text output area is 12.2cm horizontal and 19.4cm vertical;
excluding running heads.
N
0=0=0=
1=1=1=
=10000
0pt plus 6em
0=<#2405#>1<#2408#><#2408#>
!2.!General Remarks
2.;SPMnbsp;General Remarks
<#2405#>0=0 by-
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0=00 by00 by
3
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!2.!General Remarks
2.;SPMnbsp;General Remarks
=<#2407#>
=N=<#2409#><#2409#><#2407#>=A
N
=10000
0pt plus 6em
0=<#2411#>1<#2414#><#2414#>
!2.1.!How to Proceed
2.1.;SPMnbsp;How to Proceed
<#2411#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
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!2.1.!How to Proceed
2.1.;SPMnbsp;How to Proceed
=<#2413#>
=N=<#2415#><#2415#><#2413#>=B
Please insert the enclosed diskette or tape into your computer. You will
find the following files:
xxxxxxxxxxx;SPMamp;xxxxxxxxxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
<#409#>jns.doc<#409#>;SPMamp;general instructions (this document)
<#410#>jns.amm<#410#>;SPMamp;the macro package with am-fonts (the old TEX<#411#><#411#> fonts)
<#412#>jns.cmm<#412#>;SPMamp;the macro package with cm-fonts (newer and preferable)
;SPMamp;(these are both macro files and
should not be changed)
<#413#>jns.dem<#413#>;SPMamp;an example showing how to code the
text
Please insert at the beginning of your text file (also called
input or source file) the macro file with:
2 or
3 (preferable).
Now some settings will be carried out automatically: for example, the
horizontal and vertical sizes, the page layout, the running heads and
other features. Some of TEX's internal variables are changed.
N
=10000
0pt plus 6em
0=<#2447#>1<#2454#><#2454#>
!2.2.!Contributions Already Coded with PlainTEX<#2455#><#2455#> without the
Springer-Verlag TEX<#2456#><#2456#> Macro Package
2.2.;SPMnbsp;Contributions Already Coded with PlainTEX<#2457#><#2457#> without the
Springer-Verlag TEX<#2458#><#2458#> Macro Package
<#2447#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!2.2.!Contributions Already Coded with PlainTEX<#2448#><#2448#> without the
Springer-Verlag TEX<#2449#><#2449#> Macro Package
2.2.;SPMnbsp;Contributions Already Coded with PlainTEX<#2450#><#2450#> without the
Springer-Verlag TEX<#2451#><#2451#> Macro Package
=<#2453#>
=N=<#2459#><#2459#><#2453#>=B
If your file already contains TEX coding,
then:
6.5 mm=0
<#2460#>--true mm;SPMnbsp;<#2460#>you will need to replace some of <#418#>your<#418#>
TEX<#419#><#419#> commands by <#420#>our<#420#> codes listed in section <#421#>4 How to
Code<#421#>;
6.5 mm=0
<#2461#>--true mm;SPMnbsp;<#2461#>you need only to insert 4
to get the desired page layout and fonts if your layout is close to the
one you can see in our demonstration file. It is most important to
change your macros for the headings (see 5 ...). Making
further improvements by using more of our macros is still better.
N-
Very important:
If your text or your own macros contain layout
codes such as 6, 7, 8 and
9, or special fonts, these should be taken out.
(There may nevertheless be exceptional occasions on which to use some of
them.)
If you have your <#424#>own macros<#424#> or definitions, insert them <#425#>before<#425#> the call 10, so that some of them may be
replaced or tailored according to Springer style. Please put in
sufficient comments with your macros to help us understand them.
N
=10000
0pt plus 6em
0=<#2489#>1<#2494#><#2494#>
!2.3.!Problems with PC TEX<#2495#><#2495#>
2.3.;SPMnbsp;Problems with PC TEX<#2496#><#2496#>
<#2489#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!2.3.!Problems with PC TEX<#2490#><#2490#>
2.3.;SPMnbsp;Problems with PC TEX<#2491#><#2491#>
=<#2493#>
=N=<#2497#><#2497#><#2493#>=B
In PC TEX<#428#><#428#> the default memory capacity is not sufficient to
accommodate our fonts. To cope with this problem, invoke TEX<#429#><#429#> with the
following command:
<#430#>tex myfile.tex /f=26000 /m=65000<#430#>
N
=10000
0pt plus 6em
0=<#2499#>1<#2502#><#2502#>
!2.4.!General Rules for Coding Mathematics
2.4.;SPMnbsp;General Rules for Coding Mathematics
<#2499#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!2.4.!General Rules for Coding Mathematics
2.4.;SPMnbsp;General Rules for Coding Mathematics
=<#2501#>
=N=<#2503#><#2503#><#2501#>=B
For mathematical expressions and tables please follow PlainTEX from
<#1294#>The TEX<#433#><#433#>book<#1294#> by Donald E. Knuth (1986), Addison-Wesley
Publishing Company (a comprehensive general reference).
In the case of long equations in the text (enclosed in single $)
that extend beyond the type area (12.2cm), insert an
11 where the equation can be divided. For displayed
equations (enclosed in 12) please refer to Section <#434#>3.
Long Formulas<#434#> in Chap.19, p. 195 of <#1295#>The TEX<#435#><#435#>book<#1295#>.
Equations should be numbered consecutively throughout your contribution
e.g. (1), (2) etc., on the right-hand side. Place e.g. 13
etc. as the last item in a displayed (14) equation.
If you include elements of ordinary text in math mode, please enclose
them in 15, e.g.
<#436#>Input <#436#>
=
`
=`{=
`}=`$=`=
`#=`=`=̃
`_=`=̂
<#2505#> <#2505#>` =
#math283#
#tex2html_wrap_indisplay15263##tex2html_wrap_indisplay15264##tex2html_wrap_indisplay15265# = 1;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;onlywhen;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;c≠0 .#tex2html_wrap_indisplay15266#(1)
<#439#>Output <#439#>
#math284#
#tex2html_wrap_indisplay15268##tex2html_wrap_indisplay15269##tex2html_wrap_indisplay15270# = 1;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;onlywhen;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;c≠0 .#tex2html_wrap_indisplay15271#(1)
Please note that the sizes of the parentheses or other so-called
delimiter symbols used in equations should ideally match the formulas
they enclose (see p.145ff. of <#1296#>The TEX<#442#><#442#>book<#1296#>).
After a displayed equation you have to insert a blank line or to give
the command 16 if you want a new paragraph with an indention.
If there is no new paragraph either do not insert a blank line or code
17 immediately before continuing the text.
Please punctuate a displayed equation the same way as any other written
statement but with an 18 before end punctuation.
N-
0=0=0=
1=1=1=
2.4.1.;SPMnbsp;Italic and Roman in Math Mode
6.5 mm=0
<#2508#>a);SPMnbsp;<#2508#>In math mode TEX<#445#><#445#> treats all letters as though
they were mathematical or physical variables, hence they are typeset in
italics. However, for certain components of formulas, like short texts,
this would be incorrect and therefore coding in roman is required.
Roman should also be used for
subscripts and superscripts <#446#>in formulas<#446#> where these are
merely labels and not in themselves variables,
e.g. #math285#Teff not Teff,
TK not TK (K = Kelvin),
me not me (e = electron).
However, do not code for roman
if the sub/superscripts represent variables,
e.g. #math286##tex2html_wrap_inline15279#ai.
6.5 mm=0
<#2509#>b);SPMnbsp;<#2509#>Please ensure that <#455#>physical units<#455#> (e.g. pc, erg
s-1 K, cm-3, W m-2 Hz-1, m kg s-2 A-2) and
<#462#>abbreviations<#462#> such as Ord, Var, GL, SL, Aut, Ker, sgn, const. are always set in roman type. To ensure this use the 19 command:
20. On p.162 of <#1297#>The TEX<#463#><#463#>book<#1297#> by Donald
E.~Knuth you will find the names of common mathematical functions,
such as log, sin, exp, max and sup. These should be coded as
21, 22, 23, 24, 25 and will
appear in roman.
6.5 mm=0
<#2510#>c);SPMnbsp;<#2510#>
Chemical symbols and formulas should be coded for roman,
e.g. Fe not Fe, H2O not <#465#>H2O<#465#>.
6.5 mm=0
<#2511#>d);SPMnbsp;<#2511#>Familiar foreign words and phrases, e.g. et al., a priori,
in situ, bremsstrahlung, eigenvalues should appear also in roman.
N
=10000
0pt plus 6em
0=<#2513#>1<#2516#><#2516#>
!2.5.!Capitalization and Non-capitalization in the Input
(Source) File
2.5.;SPMnbsp;Capitalization and Non-capitalization in the Input
(Source) File
<#2513#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!2.5.!Capitalization and Non-capitalization in the Input
(Source) File
2.5.;SPMnbsp;Capitalization and Non-capitalization in the Input
(Source) File
=<#2515#>
=N=<#2517#><#2517#><#2515#>=B
6.5 mm=0
<#2518#>a);SPMnbsp;<#2518#>The following should always be capitalized:
11.5 mm=0
<#2519#>--;SPMnbsp;<#2519#>Headings [see <#471#>4.1 Headings<#471#>]
11.5 mm=0
<#2520#>--;SPMnbsp;<#2520#>Abbreviations and expressions in the text such as
Fig(s)., Table(s), Sect(s)., Chap(s)., Theorem, Corollary, Definition
etc. when used with numbers, e.g. Fig.3, Table1, Theorem 2.
6.5 mm=0
<#2521#>;SPMnbsp;<#2521#>Please see below the special rules for referring to equations.
6.5 mm=0
<#2522#>b);SPMnbsp;<#2522#>The following should <#475#>not<#475#> be capitalized:
11.5 mm=0
<#2523#>--;SPMnbsp;<#2523#>The words figure(s), table(s), equation(s), theorem(s) in
the text when used without an accompanying number
11.5 mm=0
<#2524#>--;SPMnbsp;<#2524#>Figure legends and table captions except for names and
abbreviations.
N
=10000
0pt plus 6em
0=<#2526#>1<#2529#><#2529#>
!2.6.!Abbreviation of Words in the Input (Source) File
2.6.;SPMnbsp;Abbreviation of Words in the Input (Source) File
<#2526#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!2.6.!Abbreviation of Words in the Input (Source) File
2.6.;SPMnbsp;Abbreviation of Words in the Input (Source) File
=<#2528#>
=N=<#2530#><#2530#><#2528#>=B
6.5 mm=0
<#2531#>a);SPMnbsp;<#2531#>The following should be abbreviated in the text <#481#>unless<#481#>
they come at the beginning of a sentence: Chap., Sect., Fig.; e.g. The
results are shown in Fig.5. Figure 9 reveals that ....
<#482#>Please note<#482#>: Equations should be referred to solely by their
number in parentheses: e.g. (14). However, when the reference comes at
the beginning of a sentence, the unabbreviated word ``Equation'' should
be used: e.g. Equation (14) is very important. However, (15) makes it
clear that ....
6.5 mm=0
<#2532#>b);SPMnbsp;<#2532#>If abbreviations of names or concepts are used throughout the
text, they should be defined at first mention, e.g. Plurisubharmonic
(PSH) Functions, Strong Optimization (SOPT) Problem.
N
0=0=0=
1=1=1=
=10000
0pt plus 6em
0=<#2534#>1<#2537#><#2537#>
!3.!How to Handle Your Contribution
3.;SPMnbsp;How to Handle Your Contribution
<#2534#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!3.!How to Handle Your Contribution
3.;SPMnbsp;How to Handle Your Contribution
=<#2536#>
=N=<#2538#><#2538#><#2536#>=A
Once you have completed your work using this macro package, you should
send your printout <#486#>together<#486#> with the disk or magnetic tape
(concerning the acceptable formats see remark on p. 2) to the
editorial (see address on p.2). Please make sure
that the text of your printout and the disk or magnetic tape is <#487#>identical<#487#>.
Your contribution should begin with the following coding
(see <#488#>4 How to Code<#488#>), and please adhere strictly to this sequence:
xxxxxxxxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxxxxxxxx
26;SPMamp;your own macros if you have any
27;SPMamp;call for the macros and fonts
28;SPMamp;the title of your article
29;SPMamp;the subtitle of your article (it is
optional)
30;SPMamp;author(s) name(s)
31;SPMamp;address(es) of the author(s)
32;SPMamp;date of receipt of your article
33 ;SPMamp;text of the summary
34 ;SPMamp;appropriate keywords
35;SPMamp;see <#489#>4 How to Code<#489#>
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;&vellip#vdots;;SPMamp;<#490#>(here goes the body of your article)<#490#>
36;SPMamp;see <#491#>5 How to Code References<#491#>
N
0=0=0=
1=1=1=
=10000
0pt plus 6em
0=<#2540#>1<#2543#><#2543#>
!4.!How to Code
4.;SPMnbsp;How to Code
<#2540#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.!How to Code
4.;SPMnbsp;How to Code
=<#2542#>
=N=<#2544#><#2544#><#2542#>=A
N
=10000
0pt plus 6em
0=<#2546#>1<#2549#><#2549#>
!4.1.!Headings
4.1.;SPMnbsp;Headings
<#2546#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.1.!Headings
4.1.;SPMnbsp;Headings
=<#2548#>
=N=<#2550#><#2550#><#2548#>=B
No blank line should be left in the input between titles or headings and
the following text. Otherwise you will get an indentation which is not
allowed after a heading. If you want to structure the source text for
easier reading please use lines that begin with a percent sign
(38) at such places.
If a long title must be divided please use the code: 39.
<#2552#>0<#2552#>0=0=0=
1=1=1=
2=2=2=
<#2553#><#2553#>=
<#2554#><#2554#>==
=
<#2555#><#2555#>==
=
<#2556#><#2556#>=
=10pt
=<#2557#>height7pt depth2pt width0pt<#2557#>#1<#2558#><#2559#>1=1=
0=0=
<#2560#><#2564#>#math287###1<#2564#><#2560#><#2561#><#2565#>#math288###1<#2565#><#2561#>
<#2562#><#2566#>#math289###1<#2566#><#2562#><#2563#><#2567#>#math290###1<#2567#><#2563#><#2559#><#2558#>
=0pt plus 1fil
<#497#>1<#497#><#498#>Other initials are optional and may be inserted if this
is the usual way of writing your name, e.g. Alfred J.~Holmes, E.~Henry
Green.<#498#>
All words in titles should be capitalized except for conjunctions,
prepositions (e.g. on, of, by, and, or, but, from, with, without, under)
and definite and indefinite articles (the, a, an) unless they appear at
the beginning. Formula letters must be typeset as in the text.
<#499#>
40
41, it is optional
42143
44
45
46
47
48
49
50
51
52<#499#>
N-
In general:
If you need a heading without numbering, suppress the first argument
(the number) by coding an empty pair of braces:
53n54;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;n means <#501#>a<#501#> or <#502#>b<#502#> here.
The text elements of 55 and 56 have no end
punctuation, but their preceding number as any other numbering of a
section must be completed by a period. The text elements of
57;SPMnbsp;and 58;SPMnbsp;require end
punctuation.
N-
More than one author:
If there are more than one author, and the address of each is different,
the following coding may be used to indicate by a small superscript
number which author has which address (see also <#504#>jns.dem<#504#>):
=
`
=`{=
`}=`$=`=
`#=`=`=̃
`_=`=̂
<#2574#> <#2574#>` =
=10000
Ivar Ekeland@1 and Roger Temam@2
<#2577#>@#1<#2578#><#2578#> 0=<#2579#><#2586#>0<#2586#>0=0=0=
1=1=1=
2=2=2=
<#2587#><#2587#>=
<#2588#><#2588#>==
=
<#2589#><#2589#>==
=
<#2590#><#2590#>=
=10pt
=<#2591#>height7pt depth2pt width0pt<#2591#>1<#2592#><#2597#>1=1=
0=0=
<#2600#><#2604#>#math291##1<#2604#><#2600#><#2601#><#2605#>#math292##1<#2605#><#2601#>
<#2602#><#2606#>#math293##1<#2606#><#2602#><#2603#><#2607#>#math294##1<#2607#><#2603#><#2597#><#2592#>true ccIvar Ekeland@1 and Roger Temam@2<#2579#>0;SPMgt;
to2.5true cc<#2582#>;SPMlt;@
<#2593#>-<#2593#><#2582#>Missing name(s)
of the author(s)<#2583#>to2.5true cc<#2594#>;SPMlt;@
<#2598#>-<#2598#><#2594#>AUTHORS suppressed due to excessive
length<#2583#>
to2.5true cc<#2584#>;SPMlt;@
<#2595#>-<#2595#><#2584#>Missing name(s)
of the author(s)<#2585#>to2.5true
cc<#2596#>;SPMlt;@
<#2599#>-<#2599#><#2596#>Ivar Ekeland@1 and Roger Temam@2<#2585#>
<#2577#>=E
<#2609#>0<#2609#>0=0=0=
1=1=1=
2=2=2=
<#2610#><#2610#>=
<#2611#><#2611#>==
=
<#2612#><#2612#>==
=
<#2613#><#2613#>=
=10pt
=<#2614#>height7pt depth2pt width0pt<#2614#>1<#2615#><#2617#>1=1=
0=0=
<#2618#><#2622#>#math295##1<#2622#><#2618#><#2619#><#2623#>#math296##1<#2623#><#2619#>
<#2620#><#2624#>#math297##1<#2624#><#2620#><#2621#><#2625#>#math298##1<#2625#><#2621#><#2617#><#2615#>
@1Princeton University, Princeton, NJ 08544, USA
@2Université de Paris-Sud, Laboratoire d'Analyse Numérique,
Bâtiment 425, F-91405 Orsay Cedex, France
Note that 59 and 60 will create the page
header (running head), but it may happen that you need to shorten your
title for the page header, because only one line is allowed (our macro
will produce an error message whereupon you must provide a shortened
version of the title for the page header). In this case you should use
the following coding directly after the coding of 61
or 62.
63
64
The following bold run-in headings with italicized text are available
as built-in environments:
=
`
=`{=
`}=`$=`=
`#=`=`=̃
`_=`=̂
<#2627#> <#2627#>` =
<#507#>no.<#507#><#508#>Text<#508#>
<#509#>no.<#509#><#510#>Text<#510#>
<#511#>no.<#511#><#512#>Text<#512#>
<#513#>no.<#513#><#514#>Text<#514#>
|medskip
|bgroup|rm The following will generally appear as italic run-in heading:|egroup|smallskip
<#2628#>Proof!additional Text! additional Text. <#2628#>Text <#2629#><#2631#>#tex2html_wrap_inline15306#<#2631#>#tex2html_wrap_inline15308#<#2629#><#2630#>
<#2632#><#2633#>#tex2html_wrap_inline15310#<#2633#>#tex2html_wrap_inline15312#<#2632#>
=0pt=0
<#2630#>|vfill|eject
|bgroup|rm Further italic or bold run-in headings may also occur:|egroup|smallskip
<#516#>no.<#516#><#517#>Text<#517#>
<#518#>no.<#518#><#519#>Text<#519#>
<#2634#>Remark. <#2634#>
<#521#>no.<#521#><#522#>Text<#522#>
<#523#>no.<#523#><#524#>Text<#524#>
<#525#>no.<#525#><#526#>Text<#526#>
<#527#>no.<#527#><#528#>Text<#528#>
<#529#>no.<#529#><#530#>Text<#530#>
N-
0=0=0=
1=1=1=
4.1.1.;SPMnbsp;Defining Your Own Environments.
You can define additional environments like these using the command
65 which has four parameters. The first is the name
your environment should have (e.g. 66). Then the run-in
heading is to be given (e.g. 67). After this follows
the font family used for this heading (please use only 68
for bold or 69 for italic). Finally comes
the font family to use for the text of this new environment (e.g.
70 or 71).
Sample definition:
72
<#532#>Conjecture<#532#> <#533#><#533#> <#534#><#534#>
Use of that definition:
73 e.g. 74
It's output:
<#535#>17.<#535#><#536#>It is clear that...<#536#>
<#537#>Sample Input:<#537#>
=
`
=`{=
`}=`$=`=
`#=`=`=̃
`_=`=̂
<#2638#> <#2638#>` =
to<#2639#>
254=<#2649#> THE JOURNAL OF <#2649#><#2650#>
<#2665#><#2669#>254
to254<#2674#>NONLINEAR<#2674#>
to254<#2675#>SCIENCE<#2675#>
to254<#2676#>..<#2676#>
<#2669#><#2665#>
<#2650#><#2639#>=N
0=0=0=
1=1=1=
2=2=2=
=0pt
=10000
Hamiltonian Mechanics
=<#2642#>=N=<#2651#><#2651#><#2642#>=A 0=<#2643#><#2652#>0<#2652#>0=0=0=
1=1=1=
2=2=2=
<#2653#><#2653#>=
<#2654#><#2654#>==
=
<#2655#><#2655#>==
=
<#2656#><#2656#>=
=10pt
=<#2657#>height7pt depth2pt width0pt<#2657#>1<#2658#><#2666#>1=1=
0=0=
<#2670#><#2677#>#math299##1<#2677#><#2670#><#2671#><#2678#>#math300##1<#2678#><#2671#>
<#2672#><#2679#>#math301##1<#2679#><#2672#><#2673#><#2680#>#math302##1<#2680#><#2673#><#2666#><#2658#><#2659#> <#2659#>1<#2660#><#2660#>true
ccHamiltonian Mechanics<#2643#>0;SPMgt;
Missing MAINTITLEto2.5true
cc<#2645#>;SPMlt;@
<#2661#>-<#2661#><#2645#><#2646#>MAIN title
suppressed due to excessive lengthto2.5true cc<#2662#>;SPMlt;@
<#2667#>-<#2667#><#2662#><#2646#>
Missing MAINTITLEto2.5true
cc<#2647#>;SPMlt;@
<#2663#>-<#2663#><#2647#><#2648#>Hamiltonian Mechanicsto2.5true
cc<#2664#>;SPMlt;@
<#2668#>-<#2668#><#2664#><#2648#>
`=
=10000
Ivar Ekeland@<#2754#>1<#2754#> and Roger Temam@<#2755#>2<#2755#>
<#2757#>@#1<#2758#><#2758#> 0=<#2759#><#2766#>0<#2766#>0=0=0=
1=1=1=
2=2=2=
<#2767#><#2767#>=
<#2768#><#2768#>==
=
<#2769#><#2769#>==
=
<#2770#><#2770#>=
=10pt
=<#2771#>height7pt depth2pt width0pt<#2771#>1<#2772#><#2781#>1=1=
0=0=
<#2784#><#2788#>#math303##1<#2788#><#2784#><#2785#><#2789#>#math304##1<#2789#><#2785#>
<#2786#><#2790#>#math305##1<#2790#><#2786#><#2787#><#2791#>#math306##1<#2791#><#2787#><#2781#><#2772#>true ccIvar Ekeland@<#2773#>1<#2773#> and Roger Temam@<#2774#>2<#2774#><#2759#>0;SPMgt;
to2.5true cc<#2762#>;SPMlt;@
<#2775#>-<#2775#><#2762#>Missing name(s)
of the author(s)<#2763#>to2.5true cc<#2776#>;SPMlt;@
<#2782#>-<#2782#><#2776#>AUTHORS suppressed due to excessive
length<#2763#>
to2.5true cc<#2764#>;SPMlt;@
<#2777#>-<#2777#><#2764#>Missing name(s)
of the author(s)<#2765#>to2.5true
cc<#2778#>;SPMlt;@
<#2783#>-<#2783#><#2778#>Ivar Ekeland@<#2779#>1<#2779#> and Roger Temam@<#2780#>2<#2780#><#2765#>
<#2757#>=E
<#2793#>0<#2793#>0=0=0=
1=1=1=
2=2=2=
<#2794#><#2794#>=
<#2795#><#2795#>==
=
<#2796#><#2796#>==
=
<#2797#><#2797#>=
=10pt
=<#2798#>height7pt depth2pt width0pt<#2798#>1<#2799#><#2801#>1=1=
0=0=
<#2802#><#2806#>#math307##1<#2806#><#2802#><#2803#><#2807#>#math308##1<#2807#><#2803#>
<#2804#><#2808#>#math309##1<#2808#><#2804#><#2805#><#2809#>#math310##1<#2809#><#2805#><#2801#><#2799#>
@1Princeton University, Princeton NJ 08544, USA
@2Université de Paris-Sud,
Laboratoire d'Analyse Numérique, Bâtiment 425,
F-91405 Orsay Cedex, France
<#2811#>0<#2811#>0=0=0=
1=1=1=
2=2=2=
<#2812#><#2812#>=
<#2813#><#2813#>==
=
<#2814#><#2814#>==
=
<#2815#><#2815#>=
=10pt
=<#2816#>height7pt depth2pt width0pt<#2816#>1<#2817#><#2819#>1=1=
0=0=
<#2820#><#2824#>#math311##1<#2824#><#2820#><#2821#><#2825#>#math312##1<#2825#><#2821#>
<#2822#><#2826#>#math313##1<#2826#><#2822#><#2823#><#2827#>#math314##1<#2827#><#2823#><#2819#><#2817#>Received June 5, 1989
<#2828#>Summary. <#2828#>A new variant of the multi-grid algorithms is presented.
... to anisotropic problems is considered.
<#2829#>Key words. <#2829#>multi-grid method -- coarse-grid correction --
singular perturbation -- robustness.
N
0=0=0=
1=1=1=
=10000
0pt plus 6em
0=<#2831#>1<#2834#><#2834#>
!1.!Fixed-Period Problems: The Sublinear Case
1.;SPMnbsp;Fixed-Period Problems: The Sublinear Case
<#2831#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!1.!Fixed-Period Problems: The Sublinear Case
1.;SPMnbsp;Fixed-Period Problems: The Sublinear Case
=<#2833#>
=N=<#2835#><#2835#><#2833#>=A
With this chapter, the preliminaries are over, and we begin the
search for periodic solutions ...
N
=10000
0pt plus 6em
0=<#2837#>1<#2840#><#2840#>
!1.1.!Autonomous Systems
1.1.;SPMnbsp;Autonomous Systems
<#2837#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!1.1.!Autonomous Systems
1.1.;SPMnbsp;Autonomous Systems
=<#2839#>
=N=<#2841#><#2841#><#2839#>=B
In this section we will consider the case when the Hamiltonian
H(x) ...
N-
0=0=0=
1=1=1=
The General Case: Nontriviality.
We assume that H is #math315#(A∞, B∞)-subquadratic at
infinity, for some constant ...
N-
Notes and Comments.
The chronology is as follows. Palais and Smale introduced their
celebrated condition (PS) to extend ...
<#551#>1.<#551#><#552#>Assume H'(0) = 0 and H(0) = 0. Set ...<#552#>
<#2846#>Proof!of proposition.! of proposition.. <#2846#>Condition (8) means that, for every
#math316#δ' ;SPMgt; δ, there is some #math317##tex2html_wrap_inline15336# ;SPMgt; 0
such that ...<#2847#><#2849#>#tex2html_wrap_inline15338#<#2849#>#tex2html_wrap_inline15340#<#2847#><#2848#>
<#2850#><#2851#>#tex2html_wrap_inline15342#<#2851#>#tex2html_wrap_inline15344#<#2850#>
=0pt=0
<#2848#>
<#554#>1<#554#><#555#>(External forcing). Consider the system ...<#555#>
<#556#>2.<#556#><#557#>Assume H is C2 and #math318#(a∞, b∞)-subquadratic at infinity. Let ...<#557#>
<#558#>3.<#558#><#1300#>Assume that H is C2 on #math319#IR2n\{0}
and that H''(x) is non-degenerate ...<#1300#>
<#560#>4 (Ghoussoub-Preiss).<#560#><#561#>Let X be a Banach Space and
#math320#Φ : X→IR ...<#561#>
<#562#>5.<#562#><#563#>We shall say that a C1 function
#math321#Φ : X→IR satisfies ...<#563#>
<#564#>Sample Output<#564#> (follows on the next two pages together with
examples of the above run-in headings)
to<#2855#>
254=<#2865#> THE JOURNAL OF <#2865#><#2866#>
<#2881#><#2885#>254
to254<#2890#>NONLINEAR<#2890#>
to254<#2891#>SCIENCE<#2891#>
to254<#2892#>..<#2892#>
<#2885#><#2881#>
<#2866#><#2855#>=N
0=0=0=
1=1=1=
2=2=2=
=0pt
=10000
Hamiltonian Mechanics
=<#2858#>=N=<#2867#><#2867#><#2858#>=A 0=<#2859#><#2868#>0<#2868#>0=0=0=
1=1=1=
2=2=2=
<#2869#><#2869#>=
<#2870#><#2870#>==
=
<#2871#><#2871#>==
=
<#2872#><#2872#>=
=10pt
=<#2873#>height7pt depth2pt width0pt<#2873#>1<#2874#><#2882#>1=1=
0=0=
<#2886#><#2893#>#math322##1<#2893#><#2886#><#2887#><#2894#>#math323##1<#2894#><#2887#>
<#2888#><#2895#>#math324##1<#2895#><#2888#><#2889#><#2896#>#math325##1<#2896#><#2889#><#2882#><#2874#><#2875#> <#2875#>1<#2876#><#2876#>true
ccHamiltonian Mechanics<#2859#>0;SPMgt;
Missing MAINTITLEto2.5true
cc<#2861#>;SPMlt;@
<#2877#>-<#2877#><#2861#><#2862#>MAIN title
suppressed due to excessive lengthto2.5true cc<#2878#>;SPMlt;@
<#2883#>-<#2883#><#2878#><#2862#>
Missing MAINTITLEto2.5true
cc<#2863#>;SPMlt;@
<#2879#>-<#2879#><#2863#><#2864#>Hamiltonian Mechanicsto2.5true
cc<#2880#>;SPMlt;@
<#2884#>-<#2884#><#2880#><#2864#>
`=
=10000
Ivar Ekeland@<#2970#>1<#2970#> and Roger Temam@<#2971#>2<#2971#>
<#2973#>@#1<#2974#><#2974#> 0=<#2975#><#2982#>0<#2982#>0=0=0=
1=1=1=
2=2=2=
<#2983#><#2983#>=
<#2984#><#2984#>==
=
<#2985#><#2985#>==
=
<#2986#><#2986#>=
=10pt
=<#2987#>height7pt depth2pt width0pt<#2987#>1<#2988#><#2997#>1=1=
0=0=
<#3000#><#3004#>#math326##1<#3004#><#3000#><#3001#><#3005#>#math327##1<#3005#><#3001#>
<#3002#><#3006#>#math328##1<#3006#><#3002#><#3003#><#3007#>#math329##1<#3007#><#3003#><#2997#><#2988#>true ccIvar Ekeland@<#2989#>1<#2989#> and Roger Temam@<#2990#>2<#2990#><#2975#>0;SPMgt;
to2.5true cc<#2978#>;SPMlt;@
<#2991#>-<#2991#><#2978#>Missing name(s)
of the author(s)<#2979#>to2.5true cc<#2992#>;SPMlt;@
<#2998#>-<#2998#><#2992#>AUTHORS suppressed due to excessive
length<#2979#>
to2.5true cc<#2980#>;SPMlt;@
<#2993#>-<#2993#><#2980#>Missing name(s)
of the author(s)<#2981#>to2.5true
cc<#2994#>;SPMlt;@
<#2999#>-<#2999#><#2994#>Ivar Ekeland@<#2995#>1<#2995#> and Roger Temam@<#2996#>2<#2996#><#2981#>
<#2973#>=E
<#3009#>0<#3009#>0=0=0=
1=1=1=
2=2=2=
<#3010#><#3010#>=
<#3011#><#3011#>==
=
<#3012#><#3012#>==
=
<#3013#><#3013#>=
=10pt
=<#3014#>height7pt depth2pt width0pt<#3014#>1<#3015#><#3017#>1=1=
0=0=
<#3018#><#3022#>#math330##1<#3022#><#3018#><#3019#><#3023#>#math331##1<#3023#><#3019#>
<#3020#><#3024#>#math332##1<#3024#><#3020#><#3021#><#3025#>#math333##1<#3025#><#3021#><#3017#><#3015#>
@1Princeton University, Princeton NJ 08544, USA
@2Université de Paris-Sud,
Laboratoire d'Analyse Numérique, Bâtiment 425,
F-91405 Orsay Cedex, France
<#3027#>0<#3027#>0=0=0=
1=1=1=
2=2=2=
<#3028#><#3028#>=
<#3029#><#3029#>==
=
<#3030#><#3030#>==
=
<#3031#><#3031#>=
=10pt
=<#3032#>height7pt depth2pt width0pt<#3032#>1<#3033#><#3035#>1=1=
0=0=
<#3036#><#3040#>#math334##1<#3040#><#3036#><#3037#><#3041#>#math335##1<#3041#><#3037#>
<#3038#><#3042#>#math336##1<#3042#><#3038#><#3039#><#3043#>#math337##1<#3043#><#3039#><#3035#><#3033#>Received June 5, 1989
<#3044#>Summary. <#3044#>A new variant of the multi-grid algorithms is presented.
... to anisotropic problems is considered.
<#3045#>Key words. <#3045#>multi-grid method -- coarse-grid correction --
singular perturbation -- robustness.
N
0=0=0=
1=1=1=
=10000
0pt plus 6em
0=<#3047#>1<#3050#><#3050#>
!1.!Fixed-Period Problems: The Sublinear Case
1.;SPMnbsp;Fixed-Period Problems: The Sublinear Case
<#3047#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!1.!Fixed-Period Problems: The Sublinear Case
1.;SPMnbsp;Fixed-Period Problems: The Sublinear Case
=<#3049#>
=N=<#3051#><#3051#><#3049#>=A
With this chapter, the preliminaries are over, and we begin the
search for periodic solutions ...
N
=10000
0pt plus 6em
0=<#3053#>1<#3056#><#3056#>
!1.1.!Autonomous Systems
1.1.;SPMnbsp;Autonomous Systems
<#3053#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!1.1.!Autonomous Systems
1.1.;SPMnbsp;Autonomous Systems
=<#3055#>
=N=<#3057#><#3057#><#3055#>=B
In this section we will consider the case when the Hamiltonian
H(x) ...
N-
0=0=0=
1=1=1=
The General Case: Nontriviality.
We assume that H is #math338#(A∞, B∞)-subquadratic at
infinity, for some constant ...
N-
Notes and Comments.
The chronology is as follows. Palais and Smale introduced their
celebrated condition (PS) to extend ...
<#578#>1.<#578#><#579#>Assume H'(0) = 0 and H(0) = 0. Set ...<#579#>
<#3062#>Proof!of proposition.! of proposition.. <#3062#>Condition (8) means that, for every
#math339#δ' ;SPMgt; δ, there is some #math340##tex2html_wrap_inline15381# ;SPMgt; 0
such that ...<#3063#><#3065#>#tex2html_wrap_inline15383#<#3065#>#tex2html_wrap_inline15385#<#3063#><#3064#>
<#3066#><#3067#>#tex2html_wrap_inline15387#<#3067#>#tex2html_wrap_inline15389#<#3066#>
=0pt=0
<#3064#>
<#581#>1<#581#><#582#>(External forcing). Consider the system ...<#582#>
<#583#>2.<#583#><#584#>Assume H is C2 and #math341#(a∞, b∞)-subquadratic at infinity. Let ...<#584#>
<#585#>3.<#585#><#1302#>Assume that H is C2 on #math342#IR2n\{0}
and that H''(x) is non-degenerate ...<#1302#>
<#587#>4 (Ghoussoub-Preiss).<#587#><#588#>Let X be a Banach Space and
#math343#Φ : X→IR ...<#588#>
<#589#>5.<#589#><#590#>We shall say that a C1 function
#math344#Φ : X→IR satisfies ...<#590#>
<#591#>This completes the demonstration output. We go on with the
instructions.<#591#>
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0=<#3072#>1<#3075#><#3075#>
!4.2.!Text
4.2.;SPMnbsp;Text
<#3072#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.2.!Text
4.2.;SPMnbsp;Text
=<#3074#>
=N=<#3076#><#3076#><#3074#>=B
The following should be used to improve the readability of the
text:
xxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxx
75;SPMamp;a thin space, e.g. between numbers or
between
;SPMamp;units and numbers; furthermore, a line division will
;SPMamp;not be made following this space
<#594#>-<#594#>;SPMamp;hyphen: one stroke, no space at either end
<#595#>--<#595#> ;SPMamp; en-dash: two strokes (<#596#>without<#596#> a space at
either end)
;SPMamp;Please note: in TEX<#597#><#597#>, <#598#>---<#598#> gives an em-dash ``---'';
;SPMamp;Springer does not use this.
;SPMamp; Instead, please use the following:
<#599#> -- <#599#> ;SPMamp; en-dash: two strokes (<#600#>with<#600#>
a space at either end)
<#601#>$-$<#601#>;SPMamp; minus: in the text <#602#>only<#602#>
<#603#>Input<#603#>
;SPMamp; 76
;SPMamp; 77
;SPMamp; 78
;SPMamp; 79
;SPMamp; 80
;SPMamp; 81
<#604#>Output<#604#>
;SPMamp; 21oC etc., Dr h.c.Rockefellar-Smith ...
;SPMamp; 20,000km and Prof.Dr Mallory ...
;SPMamp; 1950--1985 ...
;SPMamp; this -- written on a computer -- is now printed
;SPMamp; -30K ...
N
=10000
0pt plus 6em
0=<#3078#>1<#3081#><#3081#>
!4.3.!Special Typefaces
4.3.;SPMnbsp;Special Typefaces
<#3078#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.3.!Special Typefaces
4.3.;SPMnbsp;Special Typefaces
=<#3080#>
=N=<#3082#><#3082#><#3080#>=B
Normal type (roman) need not be marked. Preferably <#608#>italic<#608#> (not
<#609#>slanted<#609#>) or, if necessary, <#610#>boldface<#610#> should be used to
emphasize words and expressions.
xxxxxxxxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxx
;SPMamp;xxxxxxxxxxxxxx
82;SPMamp;<#611#>Text<#611#> (preferable)
83;SPMamp;<#612#>Text<#612#>
84;SPMamp;
<#1504#>to.6<#1303#>=.6<#613#><#3083#>0<#3083#>0=0=0=
1=1=1=
2=2=2=
<#3084#><#3084#>=
<#3085#><#3085#>==
=
<#3086#><#3086#>==
=
<#3087#><#3087#>=
=10pt
=<#3088#>height7pt depth2pt width0pt<#3088#>#1<#3089#><#3090#>1=1=
0=0=
<#3091#><#3095#>#math345###1<#3095#><#3091#><#3092#><#3096#>#math346###1<#3096#><#3092#>
<#3093#><#3097#>#math347###1<#3097#><#3093#><#3094#><#3098#>#math348###1<#3098#><#3094#><#3090#><#3089#>Paragraph
in small print (petit) for passages in the text that the reader may skip
on first reading or for exercises or sections of similar
importance.<#613#><#1303#><#1504#>
85;SPMamp; Vectors may only appear
in math mode
;SPMamp;86 yields
#math349##tex2html_wrap_inline15410#1=#tex2html_wrap_inline15411##tex2html_wrap_inline15412#1=#tex2html_wrap_inline15413##tex2html_wrap_inline15414#0=#tex2html_wrap_inline15415##tex2html_wrap_inline15416#0=#tex2html_wrap_inline15417##tex2html_wrap_inline15418#$A×B⋅C$$A×B⋅C$$A×B⋅C$$A×B⋅C$
;SPMamp;or;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;87;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;yields
;SPMamp;#math350##tex2html_wrap_inline15420#1=#tex2html_wrap_inline15421##tex2html_wrap_inline15422#1=#tex2html_wrap_inline15423##tex2html_wrap_inline15424#0=#tex2html_wrap_inline15425##tex2html_wrap_inline15426#0=#tex2html_wrap_inline15427##tex2html_wrap_inline15428#$AaB#tex2html_wrap_indisplay15429##tex2html_wrap_indisplay15430#$$AaB#tex2html_wrap_inline15431##tex2html_wrap_inline15432#$$AaB#tex2html_wrap_inline15433##tex2html_wrap_inline15434#$$AaB#tex2html_wrap_inline15435##tex2html_wrap_inline15436#$
N
=10000
0pt plus 6em
0=<#3118#>1<#3121#><#3121#>
!4.4.!Footnotes
4.4.;SPMnbsp;Footnotes
<#3118#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.4.!Footnotes
4.4.;SPMnbsp;Footnotes
=<#3120#>
=N=<#3122#><#3122#><#3120#>=B
<#1304#>88Footnote within text
(<#618#>no<#618#> blank before 89)<#1304#>
<#619#>Input<#619#>
Text with a footnote90
and the text continues. You will find the footnote below.
<#620#>Output<#620#>
Text with a footnoteby1=#math351##tex2html_wrap_inline15438##tex2html_wrap_inline15439#<#3125#>0<#3125#>0=0=0=
1=1=1=
2=2=2=
<#3126#><#3126#>=
<#3127#><#3127#>==
=
<#3128#><#3128#>==
=
<#3129#><#3129#>=
=10pt
=<#3130#>height7pt depth2pt width0pt<#3130#>1<#3131#><#3136#>1=1=
0=0=
<#3139#><#3143#>#math352##1<#3143#><#3139#><#3140#><#3144#>#math353##1<#3144#><#3140#>
<#3141#><#3145#>#math354##1<#3145#><#3141#><#3142#><#3146#>#math355##1<#3146#><#3142#><#3136#><#3131#>
=0pt plus 1fil
1<#3132#>0.5
to0.5<#3137#>#1<#3137#><#3132#><#3133#>#math356##tex2html_wrap_inline15445##tex2html_wrap_inline15446#<#3133#><#3134#>The footnote is automatically numbered.<#3134#>
and the text continues. You will find the footnote below.
<#622#>Remark<#622#>: Please avoid using footnotes in headings.
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=10000
0pt plus 6em
0=<#3148#>1<#3151#><#3151#>
!4.5.!Lists
4.5.;SPMnbsp;Lists
<#3148#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.5.!Lists
4.5.;SPMnbsp;Lists
=<#3150#>
=N=<#3152#><#3152#><#3150#>=B
<#1305#><#625#>Input<#625#><#1305#>
=
`
=`{=
`}=`$=`=
`#=`=`=̃
`_=`=̂
<#3154#> <#3154#>` =
This is ordinary text extending over several lines and
interrupted by a list. The text continues ...
|bgroup|rm Do not forget to type |egroup
6.5 mm=0
<#3155#>1.;SPMnbsp;<#3155#>|bgroup|rm Start of list and first item|egroup
6.5 mm=0
<#3156#>2.;SPMnbsp;<#3156#>|bgroup|rm Second item in list|egroup
11.5 mm=0
<#3157#>a);SPMnbsp;<#3157#>|bgroup|rm Start of subdivision and its first item|egroup
11.5 mm=0
<#3158#>b);SPMnbsp;<#3158#>|bgroup|rm Second item in subdivision|egroup
6.5 mm=0
<#3159#>n.;SPMnbsp;<#3159#>|bgroup|rm Item |it n |rm in list|egroup
|bgroup|rm Do not forget to type |egroup
The text continues...
<#631#>Output<#631#>
This is ordinary text extending over several lines and
interrupted by a list. The text continues ...
6.5 mm=0
<#3160#>1.;SPMnbsp;<#3160#>Start of list and first item
6.5 mm=0
<#3161#>2.;SPMnbsp;<#3161#>Second item in list
11.5 mm=0
<#3162#>a);SPMnbsp;<#3162#>Start of subdivision and its first item
11.5 mm=0
<#3163#>b);SPMnbsp;<#3163#>Second item in subdivision
6.5 mm=0
<#3164#><#3165#>n.<#3165#>;SPMnbsp;<#3164#>Item <#637#>n<#637#> in list
The text continues ...
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=10000
0pt plus 6em
0=<#3167#>1<#3170#><#3170#>
!4.6.! Figures
4.6.;SPMnbsp;Figures
<#3167#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.6.! Figures
4.6.;SPMnbsp;Figures
=<#3169#>
=N=<#3171#><#3171#><#3169#>=B
Figure legends should be inserted at the end of (not in) the paragraph
in which the figure is first mentioned. They should be numbered (using
arabic numerals) sequentially throughout your contribution, as shown
below.
Figures should <#640#>never<#640#> be surrounded by text.
<#641#>The figures<#641#> (line drawings and those containing halftone inserts
as well as halftone figures) <#642#>should not be pasted into your
laserprinter output<#642#>. They should be enclosed separately in camera-ready
form (original artwork, glossy prints, photographs and/or slides). The
lettering should be suitable for reproduction, and after reduction the
capital letters should be at least 1.8mm and not more than 2.5mm
in height. Check that lines and other details are uniformly black and
that the lettering on figures is clearly legible.
To leave the desired amount of space for the height of
your figures,
please use the following coding.
As can be seen in the output, we will automatically
provide 1cm space above and below the figure,
so that you should only leave the space corresponding to the size of the
figure itself.
xxxxxxxxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxxx
91;SPMamp;Begin space of <#643#>x<#643#> cm (use only cm)
92;SPMamp;Figure legend (no
capitalization, see Sect. 2.5)
93;SPMamp;End space
<#644#>Sample Input<#644#>
|1.5 cm|;SPMamp;(This is the space
required for your figure)
94
95
<#645#>Output<#645#>
1.5 cm
Ytrue cmtrue mm
=N 0=<#3172#><#3174#>0<#3174#>0=0=0=
1=1=1=
2=2=2=
<#3175#><#3175#>=
<#3176#><#3176#>==
=
<#3177#><#3177#>==
=
<#3178#><#3178#>=
=10pt
=<#3179#>height7pt depth2pt width0pt<#3179#>1<#3180#><#3190#>1=1=
0=0=
<#3192#><#3200#>#math357##1<#3200#><#3192#><#3193#><#3201#>#math358##1<#3201#><#3193#>
<#3194#><#3202#>#math359##1<#3202#><#3194#><#3195#><#3203#>#math360##1<#3203#><#3195#><#3190#><#3180#><#3181#>Fig.1. <#3181#>This is a figure legend
255=0255by<#3172#>255;SPMgt;10
<#3182#>0<#3182#>0=0=0=
1=1=1=
2=2=2=
<#3183#><#3183#>=
<#3184#><#3184#>==
=
<#3185#><#3185#>==
=
<#3186#><#3186#>=
=10pt
=
1<#3188#><#3191#>1=1=
0=0=
<#3196#><#3204#>#math361##1<#3204#><#3196#><#3197#><#3205#>#math362##1<#3205#><#3197#>
<#3198#><#3206#>#math363##1<#3206#><#3198#><#3199#><#3207#>#math364##1<#3207#><#3199#><#3191#><#3188#><#3189#>Fig.1. <#3189#>This is a figure legend
The text continues ...
N-
0=0=0=
1=1=1=
4.6.1.;SPMnbsp;Two Figures Next to Each Other.
If you
have two narrow figures that you
want
to insert side by side in one
``paragraph;SPMquot;,
and if the total width is sufficient (type area 12.2cm),
then, giving the height of the larger figure, please code:
xxxxxxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxxx
<#649#>Sample Input<#649#>
|6.5 cm|;SPMamp;(This
is the space required for the larger figure)
|Ytrue cmtrue mm
=N 0=<#3210#><#3212#>0<#3212#>0=0=0=
1=1=1=
2=2=2=
<#3213#><#3213#>=
<#3214#><#3214#>==
=
<#3215#><#3215#>==
=
<#3216#><#3216#>=
=10pt
=<#3217#>height7pt depth2pt width0pt<#3217#>1<#3218#><#3228#>1=1=
0=0=
<#3230#><#3238#>#math365##1<#3238#><#3230#><#3231#><#3239#>#math366##1<#3239#><#3231#>
<#3232#><#3240#>#math367##1<#3240#><#3232#><#3233#><#3241#>#math368##1<#3241#><#3233#><#3228#><#3218#><#3219#>Fig.1. <#3219#>...text...
255=0255by<#3210#>255;SPMgt;10
<#3220#>0<#3220#>0=0=0=
1=1=1=
2=2=2=
<#3221#><#3221#>=
<#3222#><#3222#>==
=
<#3223#><#3223#>==
=
<#3224#><#3224#>=
=10pt
=
1<#3226#><#3229#>1=1=
0=0=
<#3234#><#3242#>#math369##1<#3242#><#3234#><#3235#><#3243#>#math370##1<#3243#><#3235#>
<#3236#><#3244#>#math371##1<#3244#><#3236#><#3237#><#3245#>#math372##1<#3245#><#3237#><#3229#><#3226#><#3227#>Fig.1. <#3227#>...text...
|
;SPMamp;First figure legend (no capitalization, see Sect. 2.5)
|Ytrue cmtrue mm
=N 0=<#3246#><#3248#>0<#3248#>0=0=0=
1=1=1=
2=2=2=
<#3249#><#3249#>=
<#3250#><#3250#>==
=
<#3251#><#3251#>==
=
<#3252#><#3252#>=
=10pt
=<#3253#>height7pt depth2pt width0pt<#3253#>1<#3254#><#3264#>1=1=
0=0=
<#3266#><#3274#>#math373##1<#3274#><#3266#><#3267#><#3275#>#math374##1<#3275#><#3267#>
<#3268#><#3276#>#math375##1<#3276#><#3268#><#3269#><#3277#>#math376##1<#3277#><#3269#><#3264#><#3254#><#3255#>Fig.2. <#3255#>...text...
255=0255by<#3246#>255;SPMgt;10
<#3256#>0<#3256#>0=0=0=
1=1=1=
2=2=2=
<#3257#><#3257#>=
<#3258#><#3258#>==
=
<#3259#><#3259#>==
=
<#3260#><#3260#>=
=10pt
=
1<#3262#><#3265#>1=1=
0=0=
<#3270#><#3278#>#math377##1<#3278#><#3270#><#3271#><#3279#>#math378##1<#3279#><#3271#>
<#3272#><#3280#>#math379##1<#3280#><#3272#><#3273#><#3281#>#math380##1<#3281#><#3273#><#3265#><#3262#><#3263#>Fig.2. <#3263#>...text...
|;SPMamp;Second figure legend
96
xxxxxxxxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxxx
<#654#>Sample Output<#654#>
6.5 cm
Ytrue cmtrue mm
=N 0=<#3282#><#3284#>0<#3284#>0=0=0=
1=1=1=
2=2=2=
<#3285#><#3285#>=
<#3286#><#3286#>==
=
<#3287#><#3287#>==
=
<#3288#><#3288#>=
=10pt
=<#3289#>height7pt depth2pt width0pt<#3289#>1<#3290#><#3300#>1=1=
0=0=
<#3302#><#3310#>#math381##1<#3310#><#3302#><#3303#><#3311#>#math382##1<#3311#><#3303#>
<#3304#><#3312#>#math383##1<#3312#><#3304#><#3305#><#3313#>#math384##1<#3313#><#3305#><#3300#><#3290#><#3291#>Fig.1. <#3291#>This is the first figure legend.
The width of this legend
is the same as for the second figure
255=0255by<#3282#>255;SPMgt;10
<#3292#>0<#3292#>0=0=0=
1=1=1=
2=2=2=
<#3293#><#3293#>=
<#3294#><#3294#>==
=
<#3295#><#3295#>==
=
<#3296#><#3296#>=
=10pt
=
1<#3298#><#3301#>1=1=
0=0=
<#3306#><#3314#>#math385##1<#3314#><#3306#><#3307#><#3315#>#math386##1<#3315#><#3307#>
<#3308#><#3316#>#math387##1<#3316#><#3308#><#3309#><#3317#>#math388##1<#3317#><#3309#><#3301#><#3298#><#3299#>Fig.1. <#3299#>This is the first figure legend.
The width of this legend
is the same as for the second figure
Ytrue cmtrue mm
=N 0=<#3318#><#3320#>0<#3320#>0=0=0=
1=1=1=
2=2=2=
<#3321#><#3321#>=
<#3322#><#3322#>==
=
<#3323#><#3323#>==
=
<#3324#><#3324#>=
=10pt
=<#3325#>height7pt depth2pt width0pt<#3325#>1<#3326#><#3336#>1=1=
0=0=
<#3338#><#3346#>#math389##1<#3346#><#3338#><#3339#><#3347#>#math390##1<#3347#><#3339#>
<#3340#><#3348#>#math391##1<#3348#><#3340#><#3341#><#3349#>#math392##1<#3349#><#3341#><#3336#><#3326#><#3327#>Fig.2. <#3327#>This is the legend of the second figure.
At the present stage of macro development it is not possible for
two figures side by side to have legends occupying different line
widths
255=0255by<#3318#>255;SPMgt;10
<#3328#>0<#3328#>0=0=0=
1=1=1=
2=2=2=
<#3329#><#3329#>=
<#3330#><#3330#>==
=
<#3331#><#3331#>==
=
<#3332#><#3332#>=
=10pt
=
1<#3334#><#3337#>1=1=
0=0=
<#3342#><#3350#>#math393##1<#3350#><#3342#><#3343#><#3351#>#math394##1<#3351#><#3343#>
<#3344#><#3352#>#math395##1<#3352#><#3344#><#3345#><#3353#>#math396##1<#3353#><#3345#><#3337#><#3334#><#3335#>Fig.2. <#3335#>This is the legend of the second figure.
At the present stage of macro development it is not possible for
two figures side by side to have legends occupying different line
widths
The text continues ...
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0=0=0=
1=1=1=
4.6.2.;SPMnbsp;Modified Legend Arrangements.
If the amount of text in the legends of two figures
(to be placed side by side)
is quite different,
please use the following coding:
<#660#>Sample Input<#660#>
|5.3 cm |;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;(This
is the space required for the larger figure)
97
98
99
<#661#>Sample Output<#661#>
5.3 cm
Ytrue cmtrue mm
=N 0=<#3356#><#3358#>0<#3358#>0=0=0=
1=1=1=
2=2=2=
<#3359#><#3359#>=
<#3360#><#3360#>==
=
<#3361#><#3361#>==
=
<#3362#><#3362#>=
=10pt
=<#3363#>height7pt depth2pt width0pt<#3363#>1<#3364#><#3374#>1=1=
0=0=
<#3376#><#3384#>#math397##1<#3384#><#3376#><#3377#><#3385#>#math398##1<#3385#><#3377#>
<#3378#><#3386#>#math399##1<#3386#><#3378#><#3379#><#3387#>#math400##1<#3387#><#3379#><#3374#><#3364#><#3365#>Fig.1. <#3365#>This is the first short figure legend
255=0255by<#3356#>255;SPMgt;10
<#3366#>0<#3366#>0=0=0=
1=1=1=
2=2=2=
<#3367#><#3367#>=
<#3368#><#3368#>==
=
<#3369#><#3369#>==
=
<#3370#><#3370#>=
=10pt
=
1<#3372#><#3375#>1=1=
0=0=
<#3380#><#3388#>#math401##1<#3388#><#3380#><#3381#><#3389#>#math402##1<#3389#><#3381#>
<#3382#><#3390#>#math403##1<#3390#><#3382#><#3383#><#3391#>#math404##1<#3391#><#3383#><#3375#><#3372#><#3373#>Fig.1. <#3373#>This is the first short figure legend
Ytrue cmtrue mm
=N 0=<#3392#><#3394#>0<#3394#>0=0=0=
1=1=1=
2=2=2=
<#3395#><#3395#>=
<#3396#><#3396#>==
=
<#3397#><#3397#>==
=
<#3398#><#3398#>=
=10pt
=<#3399#>height7pt depth2pt width0pt<#3399#>1<#3400#><#3410#>1=1=
0=0=
<#3412#><#3420#>#math405##1<#3420#><#3412#><#3413#><#3421#>#math406##1<#3421#><#3413#>
<#3414#><#3422#>#math407##1<#3422#><#3414#><#3415#><#3423#>#math408##1<#3423#><#3415#><#3410#><#3400#><#3401#>Fig.2. <#3401#>This is the extremely long legend of the second figure
and should therefore be typeset using the full width of the type area. For
typographical and aesthetic reasons it would be unacceptable if these
legends were
set next to one another in two columns, with a legend
of 2 lines for the first figure and a legend of 16 lines for the
second. Therefore we suggest that the legends for the
two figures should be placed one
below the other. It is not necessary to place
the figure numbers below or beside the two figures because it should
be clear
that the left one is the first figure and the right one the second
255=0255by<#3392#>255;SPMgt;10
<#3402#>0<#3402#>0=0=0=
1=1=1=
2=2=2=
<#3403#><#3403#>=
<#3404#><#3404#>==
=
<#3405#><#3405#>==
=
<#3406#><#3406#>=
=10pt
=
1<#3408#><#3411#>1=1=
0=0=
<#3416#><#3424#>#math409##1<#3424#><#3416#><#3417#><#3425#>#math410##1<#3425#><#3417#>
<#3418#><#3426#>#math411##1<#3426#><#3418#><#3419#><#3427#>#math412##1<#3427#><#3419#><#3411#><#3408#><#3409#>Fig.2. <#3409#>This is the extremely long legend of the second figure
and should therefore be typeset using the full width of the type area. For
typographical and aesthetic reasons it would be unacceptable if these
legends were
set next to one another in two columns, with a legend
of 2 lines for the first figure and a legend of 16 lines for the
second. Therefore we suggest that the legends for the
two figures should be placed one
below the other. It is not necessary to place
the figure numbers below or beside the two figures because it should
be clear
that the left one is the first figure and the right one the second
The text continues ...
xxxxxxxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxx
N
=10000
0pt plus 6em
0=<#3429#>1<#3432#><#3432#>
!4.7.!Tables
4.7.;SPMnbsp;Tables
<#3429#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.7.!Tables
4.7.;SPMnbsp;Tables
=<#3431#>
=N=<#3433#><#3433#><#3431#>=B
Table captions should be treated
in the same way as figure legends, except that
the table captions appear above the tables. The tables
should also be numbered (using arabic numerals) sequentially,
throughout your contribution.
N-
0=0=0=
1=1=1=
4.7.1.;SPMnbsp;Tables Coded with TEX.
The comand
100
will produce a table caption. Thereafter you should
code your table with TEX<#669#><#669#>. Leave 8 mm (not more) additional space
before the table caption and at the end of your table (101). Please make sure that all the material of your table will be
set in small print by using the command 102 inside a box or
a group.
<#670#>Sample Input<#670#>
=
`
=`{=
`}=`$=`=
`#=`=`=̃
`_=`=̂
<#3437#> <#3437#>` =
true mm
<#1593#>
<#3438#><#3439#>0<#3439#>0=0=0=
1=1=1=
2=2=2=
<#3440#><#3440#>=
<#3441#><#3441#>==
=
<#3442#><#3442#>==
=
<#3443#><#3443#>=
=10pt
=<#3444#>height7pt depth2pt width0pt<#3444#>1<#3445#><#3447#>1=1=
0=0=
<#3448#><#3452#>#math413##1<#3452#><#3448#><#3449#><#3453#>#math414##1<#3453#><#3449#>
<#3450#><#3454#>#math415##1<#3454#><#3450#><#3451#><#3455#>#math416##1<#3455#><#3451#><#3447#><#3445#><#3446#>Table1. <#3446#>Observational results from NGC 4827
<#3438#>
<#1505#><#3456#>0<#3456#>0=0=0=
1=1=1=
2=2=2=
<#3457#><#3457#>=
<#3458#><#3458#>==
=
<#3459#><#3459#>==
=
<#3460#><#3460#>=
=10pt
=<#3461#>height7pt depth2pt width0pt<#3461#>#1<#3462#><#3463#>1=1=
0=0=
<#3464#><#3468#>#math417###1<#3468#><#3464#><#3465#><#3469#>#math418###1<#3469#><#3465#>
<#3466#><#3470#>#math419###1<#3470#><#3466#><#3467#><#3471#>#math420###1<#3471#><#3467#><#3463#><#3462#>
<#1307#>;SPMnbsp;#;SPMnbsp;;SPMamp;;SPMamp;#;SPMnbsp;
;SPMamp;;SPMamp;3<#673#>RA (1950)<#673#>;SPMamp; ;SPMamp;3<#674#>Dec (1950)<#674#>
;SPMamp; S ;SPMamp; Pol ;SPMamp; ;SPMamp; log P
<#675#><#675#>
;SPMamp;;SPMamp;3;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMamp;;SPMamp;3;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMamp;;SPMamp;
<#676#><#676#>
;SPMamp; ;SPMamp;(h) ;SPMamp;(m) ;SPMamp; (s) ;SPMamp; ;SPMamp; (<#3472#>o<#3472#>) ;SPMamp; (<#3473#>′<#3473#>) ;SPMamp; (<#3474#>#math421#′′<#3474#>)
;SPMamp; (mJy) ;SPMamp; (mJy) ;SPMamp; ;SPMamp; (W Hz-1)
<#678#>
<#678#>
<#679#>
<#679#>
<#680#>
<#680#>
Core ;SPMamp; (5 GHz) ;SPMamp; 12 ;SPMamp; 54 ;SPMamp; 18.0 ;SPMamp; ;SPMamp; 27 ;SPMamp; 26 ;SPMamp; 56.2
;SPMamp; 8 ;SPMamp; ;SPMamp; ;SPMamp; 21.64
Total;SPMamp;(327 MHz);SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 210 ;SPMamp; ;SPMamp; ;SPMamp; 23.13
;SPMamp;(1.4 GHz);SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 57 ;SPMamp; 1.3 ;SPMamp; 2 ;SPMamp; 22.49
;SPMamp; (5 GHz) ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 26 ;SPMamp; 0.73 ;SPMamp; 3 ;SPMamp; 22.15 <#1307#>
<#1505#><#1593#>
true mm
<#681#>Sample Output<#681#>true mm
<#1594#>
<#3476#><#3477#>0<#3477#>0=0=0=
1=1=1=
2=2=2=
<#3478#><#3478#>=
<#3479#><#3479#>==
=
<#3480#><#3480#>==
=
<#3481#><#3481#>=
=10pt
=<#3482#>height7pt depth2pt width0pt<#3482#>1<#3483#><#3485#>1=1=
0=0=
<#3486#><#3490#>#math422##1<#3490#><#3486#><#3487#><#3491#>#math423##1<#3491#><#3487#>
<#3488#><#3492#>#math424##1<#3492#><#3488#><#3489#><#3493#>#math425##1<#3493#><#3489#><#3485#><#3483#><#3484#>Table1. <#3484#>Observational results from NGC 4827
<#3476#>
<#1506#><#3494#>0<#3494#>0=0=0=
1=1=1=
2=2=2=
<#3495#><#3495#>=
<#3496#><#3496#>==
=
<#3497#><#3497#>==
=
<#3498#><#3498#>=
=10pt
=<#3499#>height7pt depth2pt width0pt<#3499#>#1<#3500#><#3501#>1=1=
0=0=
<#3502#><#3506#>#math426###1<#3506#><#3502#><#3503#><#3507#>#math427###1<#3507#><#3503#>
<#3504#><#3508#>#math428###1<#3508#><#3504#><#3505#><#3509#>#math429###1<#3509#><#3505#><#3501#><#3500#>
<#1308#>;SPMnbsp;#;SPMnbsp;;SPMamp;;SPMamp;#;SPMnbsp;
;SPMamp;;SPMamp;3<#684#>RA (1950)<#684#>;SPMamp; ;SPMamp;3<#685#>Dec (1950)<#685#>
;SPMamp; S ;SPMamp; Pol ;SPMamp; ;SPMamp; log P
<#686#><#686#>
;SPMamp;;SPMamp;3;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMamp;;SPMamp;3;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMamp;;SPMamp;
<#687#><#687#>
;SPMamp; ;SPMamp;(h) ;SPMamp;(m) ;SPMamp; (s) ;SPMamp; ;SPMamp; (<#3510#>o<#3510#>) ;SPMamp; (<#3511#>′<#3511#>) ;SPMamp; (<#3512#>#math430#′′<#3512#>)
;SPMamp; (mJy) ;SPMamp; (mJy) ;SPMamp; ;SPMamp; (W Hz-1)
<#689#>
<#689#>
<#690#>
<#690#>
<#691#>
<#691#>
Core ;SPMamp; (5 GHz) ;SPMamp; 12 ;SPMamp; 54 ;SPMamp; 18.0 ;SPMamp; ;SPMamp; 27 ;SPMamp; 26 ;SPMamp; 56.2
;SPMamp; 8 ;SPMamp; ;SPMamp; ;SPMamp; 21.64
Total;SPMamp;(327 MHz);SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 210 ;SPMamp; ;SPMamp; ;SPMamp; 23.13
;SPMamp;(1.4 GHz);SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 57 ;SPMamp; 1.3 ;SPMamp; 2 ;SPMamp; 22.49
;SPMamp; (5 GHz) ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 26 ;SPMamp; 0.73 ;SPMamp; 3 ;SPMamp; 22.15 <#1308#>
<#1506#><#1594#>
true mm
Here your text continues ...
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1=1=1=
4.7.2.;SPMnbsp;Tables Not Coded with TEX.
If you do not wish to code your table using TEX<#693#><#693#>
but prefer to have it reproduced separately,
proceed as for figures and use the following coding:
xxxxxxxxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxxxxxx
103;SPMamp;<#1309#>Begin table of <#694#>x<#694#> cm (please use cm
only)<#1309#>
104
;SPMamp;Table caption (no capitalization, see Sect. 2.5)
105;SPMamp;End table
<#695#>Input<#695#>
|2.5 cm|;SPMamp;(This is the space required for your
table)
106
107
<#696#>Output<#696#>true cm
<#1507#>
<#3516#><#3517#>0<#3517#>0=0=0=
1=1=1=
2=2=2=
<#3518#><#3518#>=
<#3519#><#3519#>==
=
<#3520#><#3520#>==
=
<#3521#><#3521#>=
=10pt
=<#3522#>height7pt depth2pt width0pt<#3522#>1<#3523#><#3525#>1=1=
0=0=
<#3526#><#3530#>#math431##1<#3530#><#3526#><#3527#><#3531#>#math432##1<#3531#><#3527#>
<#3528#><#3532#>#math433##1<#3532#><#3528#><#3529#><#3533#>#math434##1<#3533#><#3529#><#3525#><#3523#><#3524#>Table2. <#3524#>This is another
table caption
<#3516#>to 2.5true cm<#1310#>
width2truecm<#699#>The distance between these two
lines indicates the height of your table.
In this case 2.5 true cm<#699#>135
width 2truecm<#1310#><#1507#>
N
=10000
0pt plus 6em
0=<#3535#>1<#3538#><#3538#>
!4.8.!Signs and Characters
4.8.;SPMnbsp;Signs and Characters
<#3535#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!4.8.!Signs and Characters
4.8.;SPMnbsp;Signs and Characters
=<#3537#>
=N=<#3539#><#3539#><#3537#>=B
N-
0=0=0=
1=1=1=
4.8.1.;SPMnbsp;Special Signs.
You may need to use special signs. The available ones are listed in
<#1311#>The TEX<#703#><#703#>book<#1311#>, by Donald E. Knuth (1986), Addison-Wesley
Publishing Company, pp. 434ff.
We have created further symbols for math mode (enclosed in $):
<#1648#>=0pt1
<#1595#><#1508#><#1312#><#704#>#math435#\#<#704#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;
;SPMamp;yields #;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;
;SPMamp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;<#705#>#math436#\#<#705#>;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;
;SPMamp;yields #
grole;SPMamp;<#3542#><#3543#><#3547#>
<#3551#>
#math437###;SPMgt;<#3555#><#3555#>;SPMlt;<#3551#><#3547#><#3543#>
<#3544#><#3548#><#3552#>#math438###
;SPMgt;<#3556#><#3556#>;SPMlt;<#3552#><#3548#><#3544#>
<#3545#><#3549#><#3553#>#math439###
;SPMgt;<#3557#><#3557#>;SPMlt;<#3553#><#3549#><#3545#>
<#3546#><#3550#><#3554#>#math440###
;SPMgt;<#3558#><#3558#>;SPMlt;<#3554#><#3550#><#3546#><#3542#>;SPMamp;getsto;SPMamp;<#3559#><#3560#><#3564#>
<#3568#>
#math441###&larr#gets;&rarr#to;<#3568#><#3564#><#3560#>
<#3561#><#3565#><#3569#>#math442###&larr#gets;
&rarr#to;<#3569#><#3565#><#3561#>
<#3562#><#3566#><#3570#>#math443###&larr#gets;
&rarr#to;<#3570#><#3566#><#3562#>
<#3563#><#3567#><#3571#>#math444###
&larr#gets;&rarr#to;<#3571#><#3567#><#3563#><#3559#>
lid;SPMamp;<#3572#><#3573#><#3577#><#3581#>
#math445###;SPMlt;<#3585#><#3585#>=<#3581#><#3577#><#3573#>
<#3574#><#3578#><#3582#>#math446###;SPMlt;
<#3586#><#3586#>=<#3582#><#3578#><#3574#>
<#3575#><#3579#><#3583#>#math447###;SPMlt;
<#3587#><#3587#>=<#3583#><#3579#><#3575#>
<#3576#><#3580#><#3584#>#math448###
;SPMlt;
<#3588#><#3588#>=<#3584#><#3580#><#3576#><#3572#>;SPMamp;gid;SPMamp;<#3589#><#3590#><#3594#><#3598#>
#math449###;SPMgt;<#3602#><#3602#>=<#3598#><#3594#><#3590#>
<#3591#><#3595#><#3599#>#math450###;SPMgt;
<#3603#><#3603#>=<#3599#><#3595#><#3591#>
<#3592#><#3596#><#3600#>#math451###;SPMgt;
<#3604#><#3604#>=<#3600#><#3596#><#3592#>
<#3593#><#3597#><#3601#>#math452###
;SPMgt;
<#3605#><#3605#>=<#3601#><#3597#><#3593#><#3589#><#1312#><#1508#><#1595#>
<#1648#>
N-
0=0=0=
1=1=1=
4.8.2.;SPMnbsp;Gothic (Fraktur).
If gothic letters are <#707#>necessary<#707#>,
please use those of the relevant <#3608#>A<#3608#>
<#3609#>M<#3609#><#3610#>S<#3610#>-TEX<#708#><#708#> (American Mathematical
Society)
alphabet.
The
<#3611#>A<#3611#>
<#3612#>M<#3612#><#3613#>S<#3613#>-TEX<#709#><#709#> gothic alphabet is available from the American Mathematical
Society.
In PlainTEX<#710#><#710#> only the following gothic letters are
available: 108 yields ℜ and 109 yields ℑ.
These should <#711#>not<#711#> be used when you need gothic letters for your
contribution. Use <#3614#>A<#3614#>
<#3615#>M<#3615#><#3616#>S<#3616#>-TEX<#712#><#712#> gothic as explained above.
For the real and the imaginary parts of a complex number within math mode
you should use instead: 110 (which yields Re) or
111 (which yields Im).
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0=0=0=
1=1=1=
4.8.3.;SPMnbsp;Script.
For script capitals use the coding
<#714#>#math453#$\cal AB$,<#714#> which yields #tex2html_wrap_inline15560#B (see p. 164 of <#1313#>The TEX<#715#><#715#>book<#1313#>).
N-
0=0=0=
1=1=1=
4.8.4.;SPMnbsp;Special Roman.
If you need other or more than the symbols below please use throughout the
blackboard bold characters of <#3621#>A<#3621#>
<#3622#>M<#3622#><#3623#>S<#3623#>-TEX.
The following characters are built up by the use of combined symbols and signs:
=0pt
to<#1314#><#717#>#math454#\#<#717#>
;SPMamp;#;SPMamp; yields #=0pt plus1fil
;SPMamp;=0pt<#718#>#math455#\#<#718#>
;SPMamp;#;SPMamp; yields #
bbbc;SPMamp;(complex numbers);SPMamp;<#3624#><#3625#> 0=<#3629#>#math456#C<#3629#><#3630#>
to0pt<#3637#>0height0.90<#3637#>0<#3630#><#3625#>
<#3626#> 0=<#3631#>#math457#C<#3631#><#3632#>
to0pt<#3638#>0height0.90<#3638#>0<#3632#><#3626#>
<#3627#> 0=<#3633#>#math458#C<#3633#><#3634#>
to0pt<#3639#>0height0.90<#3639#>0<#3634#><#3627#>
<#3628#> 0=<#3635#>#math459#C<#3635#><#3636#>
to0pt<#3640#>0height0.90<#3640#>0<#3636#><#3628#><#3624#>;SPMamp;
bbbf;SPMamp;(blackboard bold F);SPMamp;<#3641#>IF<#3641#>
bbbh;SPMamp;(blackboard bold H);SPMamp;<#3642#>IH<#3642#>;SPMamp;
bbbk;SPMamp;(blackboard bold K);SPMamp;<#3643#>IK<#3643#>
bbbm;SPMamp;(blackboard bold M);SPMamp;<#3644#>IM<#3644#>;SPMamp;
bbbn;SPMamp;(natural numbers N);SPMamp;<#3645#>IN<#3645#>
bbbp;SPMamp;(blackboard bold P);SPMamp;<#3646#>IP<#3646#>;SPMamp;
bbbq;SPMamp;(rational numbers);SPMamp;<#3647#><#3648#> 0=<#3652#>#math460#Q<#3652#><#3653#>0to0pt<#3660#>0height0.80<#3660#>0<#3653#><#3648#>
<#3649#> 0=<#3654#>#math461#Q<#3654#><#3655#>0to0pt<#3661#>0height0.80<#3661#>0<#3655#><#3649#>
<#3650#> 0=<#3656#>#math462#Q<#3656#><#3657#>0to0pt<#3662#>0height0.70<#3662#>0<#3657#><#3650#>
<#3651#> 0=<#3658#>#math463#Q<#3658#><#3659#>0to0pt<#3663#>0height0.70<#3663#>0<#3659#><#3651#><#3647#>
bbbr;SPMamp;(real numbers);SPMamp;<#3664#>IR<#3664#>;SPMamp;
bbbs;SPMamp;(blackboard bold S);SPMamp;<#3665#>
<#3666#> 0=<#3670#>#math464#S<#3670#><#3671#>0
to0pt<#3678#>0height0.450<#3678#>
to0pt<#3679#>0height0.50<#3679#>0<#3671#><#3666#>
<#3667#> 0=<#3672#>#math465#S<#3672#><#3673#>0
to0pt<#3680#>0height0.450<#3680#>
to0pt<#3681#>0height0.50<#3681#>0<#3673#><#3667#>
<#3668#> 0=<#3674#>#math466#S<#3674#><#3675#>0
to0pt<#3682#>0height0.450<#3682#>0
to0pt<#3683#>0height0.450<#3683#>0<#3675#><#3668#>
<#3669#> 0=<#3676#>#math467#S<#3676#><#3677#>0
to0pt<#3684#>0height0.450<#3684#>0
to0pt<#3685#>0height0.450<#3685#>0<#3677#><#3669#><#3665#>
bbbt;SPMamp;(blackboard bold T);SPMamp;<#3686#><#3687#> 0=<#3691#>#math468#T<#3691#><#3692#>to0pt<#3699#>0height0.90<#3699#>0<#3692#><#3687#>
<#3688#> 0=<#3693#>#math469#T<#3693#><#3694#>
to0pt<#3700#>0height0.90<#3700#>0<#3694#><#3688#>
<#3689#> 0=<#3695#>#math470#T<#3695#><#3696#>
to0pt<#3701#>0height0.90<#3701#>0<#3696#><#3689#>
<#3690#> 0=<#3697#>#math471#T<#3697#><#3698#>
to0pt<#3702#>0height0.90<#3702#>0<#3698#><#3690#><#3686#>;SPMamp;
bbbz;SPMamp;(whole numbers);SPMamp;<#3703#><#3704#><#3708#>#math472##tex2html_wrap_inline15582##tex2html_wrap_inline15583##tex2html_wrap_inline15584#ZZ<#3708#><#3704#>
<#3705#><#3709#>#math473##tex2html_wrap_inline15586##tex2html_wrap_inline15587##tex2html_wrap_inline15588#ZZ<#3709#><#3705#>
<#3706#><#3710#>#math474##tex2html_wrap_inline15590##tex2html_wrap_inline15591##tex2html_wrap_inline15592#ZZ<#3710#><#3706#>
<#3707#><#3711#>#math475##tex2html_wrap_inline15594##tex2html_wrap_inline15595##tex2html_wrap_inline15596#ZZ<#3711#><#3707#><#3703#>
bbbone;SPMamp;(symbol one);SPMamp;<#3712#><#3713#>1-4mu l<#3713#> <#3714#>1-4mu l<#3714#>
<#3715#>1-4.5mu l<#3715#> <#3716#>1-5mu l<#3716#><#3712#>;SPMamp;
bbbe;SPMamp;(e symbol);SPMamp;<#3717#><#3718#> 0=<#3722#>#tex2html_accent_inline15597#<#3722#><#3723#>0to0pt<#3730#>0width0.3pt height0.70<#3730#>0<#3723#><#3718#>
<#3719#> 0=<#3724#>#tex2html_accent_inline15598#<#3724#><#3725#>0to0pt<#3731#>0width0.3pt height0.70<#3731#>0<#3725#><#3719#>
<#3720#> 0=<#3726#>#tex2html_accent_inline15599#<#3726#><#3727#>0to0pt<#3732#>0width0.2pt height0.70<#3732#>0<#3727#><#3720#>
<#3721#> 0=<#3728#>#tex2html_accent_inline15600#<#3728#><#3729#>0to0pt<#3733#>0width0.2pt height0.70<#3733#>0<#3729#><#3721#><#3717#><#1314#>
<#1315#><#719#>e.g.<#719#>#math476##tex2html_wrap_indisplay15604#0=$C$to0pt#tex2html_wrap_indisplay15606#0#tex2html_wrap_indisplay15607#0#tex2html_wrap_indisplay15608##tex2html_wrap_indisplay15609#00=$C$to0pt#tex2html_wrap_indisplay15611#0#tex2html_wrap_indisplay15612#0#tex2html_wrap_indisplay15613##tex2html_wrap_indisplay15614#00=$C$to0pt#tex2html_wrap_indisplay15616#0#tex2html_wrap_indisplay15617#0#tex2html_wrap_indisplay15618##tex2html_wrap_indisplay15619#00=$C$to0pt#tex2html_wrap_indisplay15625#0#tex2html_wrap_indisplay15626#0#tex2html_wrap_indisplay15627##tex2html_wrap_indisplay15628#0#tex2html_wrap_indisplay15630#0=$C$to0pt#tex2html_wrap_indisplay15632#0#tex2html_wrap_indisplay15633#0#tex2html_wrap_indisplay15634##tex2html_wrap_indisplay15635#00=$C$to0pt#tex2html_wrap_indisplay15637#0#tex2html_wrap_indisplay15638#0#tex2html_wrap_indisplay15639##tex2html_wrap_indisplay15640#00=$C$to0pt#tex2html_wrap_indisplay15642#0#tex2html_wrap_indisplay15643#0#tex2html_wrap_indisplay15644##tex2html_wrap_indisplay15645#00=$C$to0pt#tex2html_wrap_indisplay15649#0#tex2html_wrap_indisplay15650#0#tex2html_wrap_indisplay15651##tex2html_wrap_indisplay15652#0#tex2html_wrap_indisplay15653#0=$C$to0pt#tex2html_wrap_indisplay15655#0#tex2html_wrap_indisplay15656#0#tex2html_wrap_indisplay15657##tex2html_wrap_indisplay15658#00=$C$to0pt#tex2html_wrap_indisplay15660#0#tex2html_wrap_indisplay15661#0#tex2html_wrap_indisplay15662##tex2html_wrap_indisplay15663#00=$C$to0pt#tex2html_wrap_indisplay15665#0#tex2html_wrap_indisplay15666#0#tex2html_wrap_indisplay15667##tex2html_wrap_indisplay15668#00=$C$to0pt#tex2html_wrap_indisplay15670#0#tex2html_wrap_indisplay15671#0#tex2html_wrap_indisplay15672##tex2html_wrap_indisplay15673#0⊗IFIFIF⊗IHIHIH⊗IKIKIK⊗IMIMIM⊗INININ⊗IPIPIP<#1315#>
#math477#⊗#tex2html_wrap_indisplay15695#0=#tex2html_wrap_indisplay15696#0to0pt#tex2html_wrap_indisplay15697#0#tex2html_wrap_indisplay15698#0#tex2html_wrap_indisplay15699##tex2html_wrap_indisplay15700#00=#tex2html_wrap_indisplay15701#0to0pt#tex2html_wrap_indisplay15702#0#tex2html_wrap_indisplay15703#0#tex2html_wrap_indisplay15704##tex2html_wrap_indisplay15705#00=#tex2html_wrap_indisplay15706#0to0pt#tex2html_wrap_indisplay15707#0#tex2html_wrap_indisplay15708#0#tex2html_wrap_indisplay15709##tex2html_wrap_indisplay15710#00=#tex2html_wrap_indisplay15715#0to0pt#tex2html_wrap_indisplay15716#0#tex2html_wrap_indisplay15717#0#tex2html_wrap_indisplay15718##tex2html_wrap_indisplay15719#0#tex2html_wrap_indisplay15721#0=#tex2html_wrap_indisplay15722#0to0pt#tex2html_wrap_indisplay15723#0#tex2html_wrap_indisplay15724#0#tex2html_wrap_indisplay15725##tex2html_wrap_indisplay15726#00=#tex2html_wrap_indisplay15727#0to0pt#tex2html_wrap_indisplay15728#0#tex2html_wrap_indisplay15729#0#tex2html_wrap_indisplay15730##tex2html_wrap_indisplay15731#00=#tex2html_wrap_indisplay15732#0to0pt#tex2html_wrap_indisplay15733#0#tex2html_wrap_indisplay15734#0#tex2html_wrap_indisplay15735##tex2html_wrap_indisplay15736#00=#tex2html_wrap_indisplay15739#0to0pt#tex2html_wrap_indisplay15740#0#tex2html_wrap_indisplay15741#0#tex2html_wrap_indisplay15742##tex2html_wrap_indisplay15743#0#tex2html_wrap_indisplay15744#0=#tex2html_wrap_indisplay15745#0to0pt#tex2html_wrap_indisplay15746#0#tex2html_wrap_indisplay15747#0#tex2html_wrap_indisplay15748##tex2html_wrap_indisplay15749#00=#tex2html_wrap_indisplay15750#0to0pt#tex2html_wrap_indisplay15751#0#tex2html_wrap_indisplay15752#0#tex2html_wrap_indisplay15753##tex2html_wrap_indisplay15754#00=#tex2html_wrap_indisplay15755#0to0pt#tex2html_wrap_indisplay15756#0#tex2html_wrap_indisplay15757#0#tex2html_wrap_indisplay15758##tex2html_wrap_indisplay15759#00=#tex2html_wrap_indisplay15760#0to0pt#tex2html_wrap_indisplay15761#0#tex2html_wrap_indisplay15762#0#tex2html_wrap_indisplay15763##tex2html_wrap_indisplay15764#0⊗IRIRIR⊗#tex2html_wrap_indisplay15770#0=#tex2html_wrap_indisplay15771#0to0pt#tex2html_wrap_indisplay15772#0#tex2html_wrap_indisplay15773#0#tex2html_wrap_indisplay15774#to0pt#tex2html_wrap_indisplay15775#0#tex2html_wrap_indisplay15776#0#tex2html_wrap_indisplay15777##tex2html_wrap_indisplay15778#00=#tex2html_wrap_indisplay15779#0to0pt#tex2html_wrap_indisplay15780#0#tex2html_wrap_indisplay15781#0#tex2html_wrap_indisplay15782#to0pt#tex2html_wrap_indisplay15783#0#tex2html_wrap_indisplay15784#0#tex2html_wrap_indisplay15785##tex2html_wrap_indisplay15786#00=#tex2html_wrap_indisplay15787#0to0pt#tex2html_wrap_indisplay15788#0#tex2html_wrap_indisplay15789#0#tex2html_wrap_indisplay15790##tex2html_wrap_indisplay15791#0to0pt#tex2html_wrap_indisplay15792#0#tex2html_wrap_indisplay15793#0#tex2html_wrap_indisplay15794##tex2html_wrap_indisplay15795#00=#tex2html_wrap_indisplay15800#0to0pt#tex2html_wrap_indisplay15801#0#tex2html_wrap_indisplay15802#0#tex2html_wrap_indisplay15803##tex2html_wrap_indisplay15804#0to0pt#tex2html_wrap_indisplay15805#0#tex2html_wrap_indisplay15806#0#tex2html_wrap_indisplay15807##tex2html_wrap_indisplay15808#0#tex2html_wrap_indisplay15810#0=#tex2html_wrap_indisplay15811#0to0pt#tex2html_wrap_indisplay15812#0#tex2html_wrap_indisplay15813#0#tex2html_wrap_indisplay15814#to0pt#tex2html_wrap_indisplay15815#0#tex2html_wrap_indisplay15816#0#tex2html_wrap_indisplay15817##tex2html_wrap_indisplay15818#00=#tex2html_wrap_indisplay15819#0to0pt#tex2html_wrap_indisplay15820#0#tex2html_wrap_indisplay15821#0#tex2html_wrap_indisplay15822#to0pt#tex2html_wrap_indisplay15823#0#tex2html_wrap_indisplay15824#0#tex2html_wrap_indisplay15825##tex2html_wrap_indisplay15826#00=#tex2html_wrap_indisplay15827#0to0pt#tex2html_wrap_indisplay15828#0#tex2html_wrap_indisplay15829#0#tex2html_wrap_indisplay15830##tex2html_wrap_indisplay15831#0to0pt#tex2html_wrap_indisplay15832#0#tex2html_wrap_indisplay15833#0#tex2html_wrap_indisplay15834##tex2html_wrap_indisplay15835#00=#tex2html_wrap_indisplay15838#0to0pt#tex2html_wrap_indisplay15839#0#tex2html_wrap_indisplay15840#0#tex2html_wrap_indisplay15841##tex2html_wrap_indisplay15842#0to0pt#tex2html_wrap_indisplay15843#0#tex2html_wrap_indisplay15844#0#tex2html_wrap_indisplay15845##tex2html_wrap_indisplay15846#0#tex2html_wrap_indisplay15847#0=#tex2html_wrap_indisplay15848#0to0pt#tex2html_wrap_indisplay15849#0#tex2html_wrap_indisplay15850#0#tex2html_wrap_indisplay15851#to0pt#tex2html_wrap_indisplay15852#0#tex2html_wrap_indisplay15853#0#tex2html_wrap_indisplay15854##tex2html_wrap_indisplay15855#00=#tex2html_wrap_indisplay15856#0to0pt#tex2html_wrap_indisplay15857#0#tex2html_wrap_indisplay15858#0#tex2html_wrap_indisplay15859#to0pt#tex2html_wrap_indisplay15860#0#tex2html_wrap_indisplay15861#0#tex2html_wrap_indisplay15862##tex2html_wrap_indisplay15863#00=#tex2html_wrap_indisplay15864#0to0pt#tex2html_wrap_indisplay15865#0#tex2html_wrap_indisplay15866#0#tex2html_wrap_indisplay15867##tex2html_wrap_indisplay15868#0to0pt#tex2html_wrap_indisplay15869#0#tex2html_wrap_indisplay15870#0#tex2html_wrap_indisplay15871##tex2html_wrap_indisplay15872#00=#tex2html_wrap_indisplay15873#0to0pt#tex2html_wrap_indisplay15874#0#tex2html_wrap_indisplay15875#0#tex2html_wrap_indisplay15876##tex2html_wrap_indisplay15877#0to0pt#tex2html_wrap_indisplay15878#0#tex2html_wrap_indisplay15879#0#tex2html_wrap_indisplay15880##tex2html_wrap_indisplay15881#0⊗#tex2html_wrap_indisplay15884#0=to0pt#tex2html_wrap_indisplay15885#0#tex2html_wrap_indisplay15886#0#tex2html_wrap_indisplay15887##tex2html_wrap_indisplay15888#00=to0pt#tex2html_wrap_indisplay15889#0#tex2html_wrap_indisplay15890#0#tex2html_wrap_indisplay15891##tex2html_wrap_indisplay15892#00=to0pt#tex2html_wrap_indisplay15893#0#tex2html_wrap_indisplay15894#0#tex2html_wrap_indisplay15895##tex2html_wrap_indisplay15896#00=to0pt#tex2html_wrap_indisplay15901#0#tex2html_wrap_indisplay15902#0#tex2html_wrap_indisplay15903##tex2html_wrap_indisplay15904#0#tex2html_wrap_indisplay15906#0=to0pt#tex2html_wrap_indisplay15907#0#tex2html_wrap_indisplay15908#0#tex2html_wrap_indisplay15909##tex2html_wrap_indisplay15910#00=to0pt#tex2html_wrap_indisplay15911#0#tex2html_wrap_indisplay15912#0#tex2html_wrap_indisplay15913##tex2html_wrap_indisplay15914#00=to0pt#tex2html_wrap_indisplay15915#0#tex2html_wrap_indisplay15916#0#tex2html_wrap_indisplay15917##tex2html_wrap_indisplay15918#00=to0pt#tex2html_wrap_indisplay15921#0#tex2html_wrap_indisplay15922#0#tex2html_wrap_indisplay15923##tex2html_wrap_indisplay15924#0#tex2html_wrap_indisplay15925#0=to0pt#tex2html_wrap_indisplay15926#0#tex2html_wrap_indisplay15927#0#tex2html_wrap_indisplay15928##tex2html_wrap_indisplay15929#00=to0pt#tex2html_wrap_indisplay15930#0#tex2html_wrap_indisplay15931#0#tex2html_wrap_indisplay15932##tex2html_wrap_indisplay15933#00=to0pt#tex2html_wrap_indisplay15934#0#tex2html_wrap_indisplay15935#0#tex2html_wrap_indisplay15936##tex2html_wrap_indisplay15937#00=to0pt#tex2html_wrap_indisplay15938#0#tex2html_wrap_indisplay15939#0#tex2html_wrap_indisplay15940##tex2html_wrap_indisplay15941#0⊗#tex2html_wrap_indisplay15944#$#tex2html_wrap_indisplay15945##tex2html_wrap_indisplay15946##tex2html_wrap_indisplay15947#ZZ$$#tex2html_wrap_indisplay15948##tex2html_wrap_indisplay15949##tex2html_wrap_indisplay15950#ZZ$$#tex2html_wrap_indisplay15951##tex2html_wrap_indisplay15952##tex2html_wrap_indisplay15953#ZZ$$#tex2html_wrap_indisplay15958##tex2html_wrap_indisplay15959##tex2html_wrap_indisplay15960#ZZ$#tex2html_wrap_indisplay15962#$#tex2html_wrap_indisplay15963##tex2html_wrap_indisplay15964##tex2html_wrap_indisplay15965#ZZ$$#tex2html_wrap_indisplay15966##tex2html_wrap_indisplay15967##tex2html_wrap_indisplay15968#ZZ$$#tex2html_wrap_indisplay15969##tex2html_wrap_indisplay15970##tex2html_wrap_indisplay15971#ZZ$$#tex2html_wrap_indisplay15974##tex2html_wrap_indisplay15975##tex2html_wrap_indisplay15976#ZZ$#tex2html_wrap_indisplay15977#$#tex2html_wrap_indisplay15978##tex2html_wrap_indisplay15979##tex2html_wrap_indisplay15980#ZZ$$#tex2html_wrap_indisplay15981##tex2html_wrap_indisplay15982##tex2html_wrap_indisplay15983#ZZ$$#tex2html_wrap_indisplay15984##tex2html_wrap_indisplay15985##tex2html_wrap_indisplay15986#ZZ$$#tex2html_wrap_indisplay15987##tex2html_wrap_indisplay15988##tex2html_wrap_indisplay15989#ZZ$⊗#tex2html_wrap_indisplay15992#1#tex2html_wrap_indisplay15993#-4mul1#tex2html_wrap_indisplay15994#-4mul1#tex2html_wrap_indisplay15995#-4.5mul1#tex2html_wrap_indisplay15998#-5mul#tex2html_wrap_indisplay16000#1#tex2html_wrap_indisplay16001#-4mul1#tex2html_wrap_indisplay16002#-4mul1#tex2html_wrap_indisplay16003#-4.5mul1#tex2html_wrap_indisplay16005#-5mul#tex2html_wrap_indisplay16006#1#tex2html_wrap_indisplay16007#-4mul1#tex2html_wrap_indisplay16008#-4mul1#tex2html_wrap_indisplay16009#-4.5mul1#tex2html_wrap_indisplay16010#-5mul⊗#tex2html_wrap_indisplay16013#0=#tex2html_wrap_indisplay16014#0to0pt#tex2html_wrap_indisplay16015#0#tex2html_wrap_indisplay16016#0#tex2html_wrap_indisplay16017##tex2html_wrap_indisplay16018#00=#tex2html_wrap_indisplay16019#0to0pt#tex2html_wrap_indisplay16020#0#tex2html_wrap_indisplay16021#0#tex2html_wrap_indisplay16022##tex2html_wrap_indisplay16023#00=#tex2html_wrap_indisplay16024#0to0pt#tex2html_wrap_indisplay16025#0#tex2html_wrap_indisplay16026#0#tex2html_wrap_indisplay16027##tex2html_wrap_indisplay16028#00=#tex2html_wrap_indisplay16033#0to0pt#tex2html_wrap_indisplay16034#0#tex2html_wrap_indisplay16035#0#tex2html_wrap_indisplay16036##tex2html_wrap_indisplay16037#0#tex2html_wrap_indisplay16039#0=#tex2html_wrap_indisplay16040#0to0pt#tex2html_wrap_indisplay16041#0#tex2html_wrap_indisplay16042#0#tex2html_wrap_indisplay16043##tex2html_wrap_indisplay16044#00=#tex2html_wrap_indisplay16045#0to0pt#tex2html_wrap_indisplay16046#0#tex2html_wrap_indisplay16047#0#tex2html_wrap_indisplay16048##tex2html_wrap_indisplay16049#00=#tex2html_wrap_indisplay16050#0to0pt#tex2html_wrap_indisplay16051#0#tex2html_wrap_indisplay16052#0#tex2html_wrap_indisplay16053##tex2html_wrap_indisplay16054#00=#tex2html_wrap_indisplay16057#0to0pt#tex2html_wrap_indisplay16058#0#tex2html_wrap_indisplay16059#0#tex2html_wrap_indisplay16060##tex2html_wrap_indisplay16061#0#tex2html_wrap_indisplay16062#0=#tex2html_wrap_indisplay16063#0to0pt#tex2html_wrap_indisplay16064#0#tex2html_wrap_indisplay16065#0#tex2html_wrap_indisplay16066##tex2html_wrap_indisplay16067#00=#tex2html_wrap_indisplay16068#0to0pt#tex2html_wrap_indisplay16069#0#tex2html_wrap_indisplay16070#0#tex2html_wrap_indisplay16071##tex2html_wrap_indisplay16072#00=#tex2html_wrap_indisplay16073#0to0pt#tex2html_wrap_indisplay16074#0#tex2html_wrap_indisplay16075#0#tex2html_wrap_indisplay16076##tex2html_wrap_indisplay16077#00=#tex2html_wrap_indisplay16078#0to0pt#tex2html_wrap_indisplay16079#0#tex2html_wrap_indisplay16080#0#tex2html_wrap_indisplay16081##tex2html_wrap_indisplay16082#0
N-
0=0=0=
1=1=1=
4.8.5.;SPMnbsp;Sans Serif.
Using our macros you will also be able to choose the font family of this
style; use the command 112 for <#735#>sans serif<#735#> (like 113 for <#736#>italic style<#736#>).
N-
0=0=0=
1=1=1=
4.8.6.;SPMnbsp;Invented Characters.
If you need to invent a special character not available in this list,
please start your coding as shown below.
Be sure your coding works without math mode; the characters invented
should be numbered using lower-case roman numerals.
<#1318#>#math478#\def#math479#\speciali{ #math480#\hbox{$=
#math481#\!#math482#\!#math483#\! ;SPMgt; $
}} <#739#>yields<#739#> <#4068#>#math484#= ;SPMgt; <#4068#>
#math485#\def#math486#\specialii{ Your definition of special ii
}
#math487#\def#math488#\specialiii{ Your definition of special iii
}
#math489#\def#math490#\specialiv{ Your definition of special
iv } <#740#>etc.<#740#> <#1318#>
N
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=10000
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0=<#4070#>1<#4073#><#4073#>
!5.!How to Code References
5.;SPMnbsp;How to Code References
<#4070#>0=0 by-
0;SPMlt;
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3
0 by
0;SPMgt;
!5.!How to Code References
5.;SPMnbsp;How to Code References
=<#4072#>
=N=<#4074#><#4074#><#4072#>=A
There are three reference systems available; only one, of course,
should be used for your contribution. With each system (by author--year,
by number only or by letter--number) a reference list,
preferably headed ``References;SPMquot; and containing all citations in the
text, should be included at the end of the your contribution.
xxxxxxxxxxxxxxxxxxxx;SPMamp;xxxxxxxxxxxxxxxxxxxx
114;SPMamp;Beginning of reference list: the
heading;
;SPMamp;the argument <#743#>name<#743#> stands for the chosen heading:
;SPMamp;References (preferable), Literature or Bibliography;
;SPMamp;the argument <#744#>mark<#744#> stands for the largest number or
;SPMamp;widest mark of your list, it is used for references
;SPMamp;by number only and by letter--number by the
;SPMamp;macros 115 and 116. In the
;SPMamp;author--year system it is not used but you have to
;SPMamp;code at least an empty pair of braces yet.
Use only one of the following three codings throughout
your reference list:
117;SPMamp;The coding in author--year system
118;SPMamp;The coding in number only system
119;SPMamp;The coding in letter--number
system
120;SPMamp;End of reference list
For detailed examples please see below, and also refer to the demo-file
(<#745#>jns.dem<#745#>).
<#746#>Very important<#746#>: For each entry in the reference list please
follow
<#747#>exactly<#747#> the order shown in the examples.
N
=10000
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0=<#4076#>1<#4079#><#4079#>
!5.1.!Author--Year System
5.1.;SPMnbsp;Author--Year System
<#4076#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!5.1.!Author--Year System
5.1.;SPMnbsp;Author--Year System
=<#4078#>
=N=<#4080#><#4080#><#4078#>=B
References are cited in the text by name and year in parentheses, e.g.
(Smith 1970, 1980), (Ekeland et al. 1985, Theorem 2), (Jones and Jaffe
1986; Farrow 1988, Chap.2) or only the year in parentheses if the
name is part of the sentence, e.g. Ekeland et al. (1985, Sect.2.1).
The reference list should contain all citations contained in the text,
ordered alphabetically by surname (with initials following). If there
are several works by the same author(s) the references should be listed
in the appropriate order indicated below:
6.5 mm=0
<#4081#>a);SPMnbsp;<#4081#>One author: list works chronologically;
6.5 mm=0
<#4082#>b);SPMnbsp;<#4082#>Author and same co-author(s): list works chronologically;
6.5 mm=0
<#4083#>c);SPMnbsp;<#4083#>Author and different co-authors: list works alphabetically
according to co-authors.
If there are several works by the same author(s) and in the same year,
but which are cited separately, they should be distinguished by
the use of ``a;SPMquot;, ``b;SPMquot; etc., e.g. (Smith 1982a), (Ekeland et al. 1982b).
xxxxxxxxxxxxxxxxxxx;SPMamp;xxxxxxxxxxxxxxxxxxxx
121;SPMamp;Do not forget to code at least an empty pair of
braces
;SPMamp;otherwise the first entry is not correctly indented
122;SPMamp;First entry in reference list
123;SPMamp;Second entry in reference list
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;#tex2html_wrap_inline16107#
124;SPMamp;<#753#>n<#753#>-th entry in reference list
125;SPMamp;End of reference list
N
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!5.2.!References by Number Only or by Letter--Number
5.2.;SPMnbsp;References by Number Only or by Letter--Number
<#4085#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!5.2.!References by Number Only or by Letter--Number
5.2.;SPMnbsp;References by Number Only or by Letter--Number
=<#4087#>
=N=<#4089#><#4089#><#4087#>=B
The author--year system is probably of more help to the reader.
However, referen<#756#><#756#>ces may instead be cited in the text by numbers in
square brackets, e.g. [1], [2] etc. used sequentially throughout your
contribution or by letter--number, e.g. [E1, S2], [P1] etc. or a
similar version.
For example, the first two references are given as [1] and [2] in the
text, and as 1. and 2. (i.e. the brackets are dropped) in the reference
list.
The coding is as follows:
xxxxxxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxxxx ;SPMamp;xxxxxxxxxxxxxx
126
127
128
;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;#tex2html_wrap_inline16109#
129
130
Instead of 131 in the number--only system use the coding
132 for the letter--number system (e.g.:
133, then you
should have coded the start of your references with
134, supposed ``[MB1]'' is your widest
mark).
<#757#>Important<#757#>:
You must ensure that the references cited in the text (name--year,
number or letter--number) correspond exactly with the entries
in the reference list.
N
=10000
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0=<#4091#>1<#4094#><#4094#>
!5.3.!Examples
5.3.;SPMnbsp;Examples
<#4091#>0=0 by-
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3
0 by
0;SPMgt;
!5.3.!Examples
5.3.;SPMnbsp;Examples
=<#4093#>
=N=<#4095#><#4095#><#4093#>=B
The following examples show the three ways of organizing a reference list.
Naturally, you will choose only one for your contribution.
<#760#>Examples of Typical Text Containing Reference Citations<#760#>
This is implicit in recent work of Arnold (1968) and Lerch et al. (1983)
...
Consider as an example the following theorem [1].
We refer now to the hypothesis as given in [S1].
<#761#>Input of Coding for Author--Year Reference List<#761#>
=
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=`{=
`}=`$=`=
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N
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1<#4137#>0.5
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!!References
;SPMnbsp;References
<#4099#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!!References
;SPMnbsp;References
=<#4101#>
=N=<#4118#><#4118#><#4101#>=A<#4103#>0<#4103#>0=0=0=
1=1=1=
2=2=2=
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<#4105#><#4105#>==
=
<#4106#><#4106#>==
=
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=10pt
=<#4108#>height7pt depth2pt width0pt<#4108#>1<#4109#><#4119#>1=1=
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<#4122#><#4126#>#math499##1<#4126#><#4122#><#4123#><#4127#>#math500##1<#4127#><#4123#><#4119#><#4109#>
0=<#4110#>here is no mark at all;SPMnbsp;<#4110#>=0
!here is no mark at all!
;SPMgt;0.5em
probably you missed the second argument of \if N
0=0=0=
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=10000
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0=<#4231#>1<#4234#><#4234#>
!!.
;SPMnbsp;.
<#4231#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!!.
;SPMnbsp;.
=<#4233#>
=N=<#4235#><#4235#><#4233#>=A<#4237#>0<#4237#>0=0=0=
1=1=1=
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=
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=
<#4241#><#4241#>=
=10pt
=<#4242#>height7pt depth2pt width0pt<#4242#>#1<#4243#><#4244#>1=1=
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<#4245#><#4249#>#math515###1<#4249#><#4245#><#4246#><#4250#>#math516###1<#4250#><#4246#>
<#4247#><#4251#>#math517###1<#4251#><#4247#><#4248#><#4252#>#math518###1<#4252#><#4248#><#4244#><#4243#>
0=<#4192#><;SPMnbsp;<#4192#>=0
!>
=1
Arnold, V.I. (1968): Singularities of smooth mappings. Usp.
Mat. Nauk <#771#>23<#771#>, 3--44 (Russian). [English transl.: Russ. Math.
Surv. <#772#>23<#772#> (1968) 1--43]
=1
Dirac, P.A.M. (1950): On generalized Hamiltonian dynamics.
Can. J. Math. <#773#>2<#773#>(2), 129--148
=1
Grötschel, M., Lovász, L., Schrijver, A. (1988):
Geometric algorithms and combinatorial optimization. (Algorithms
and Combinatorics, vol.2.) Springer, Berlin Heidelberg
=1
Lerch, F.J., Klosko, S.M., Patel, G.B. (1983): A refined
gravity model from LAGEOS (GEM-L2). NASA, Tech. Memo. TM 84986
=1
Rham, G. de (1931): Sur l'analysis situs de variétés
à n dimensions. J. Math. Pures Appl. <#774#>10<#774#>, 115--200
<#775#>Input of Coding for Number-Only Reference List<#775#>
=
`
=`{=
`}=`$=`=
`#=`=`=̃
`_=`=̂
<#4255#> <#4255#>` =
N
0=0=0=
1=1=1=
=10000
0pt plus 6em
0=<#4257#>by1=#math519##tex2html_wrap_inline16149##tex2html_wrap_inline16150#<#4288#>0<#4288#>0=0=0=
1=1=1=
2=2=2=
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=
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=
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=10pt
=<#4293#>height7pt depth2pt width0pt<#4293#>1<#4294#><#4299#>1=1=
0=0=
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<#4304#><#4308#>#math522##1<#4308#><#4304#><#4305#><#4309#>#math523##1<#4309#><#4305#><#4299#><#4294#>
=0pt plus 1fil
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!!References
;SPMnbsp;References
<#4257#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
0;SPMgt;
!!References
;SPMnbsp;References
=<#4259#>
=N=<#4276#><#4276#><#4259#>=A<#4261#>0<#4261#>0=0=0=
1=1=1=
2=2=2=
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=
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0=0=
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0=<#4268#>5.;SPMnbsp;<#4268#>=0
!5.!
;SPMgt;0.5em
probably you missed the second argument of \if N
0=0=0=
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!!.
;SPMnbsp;.
<#4349#>0=0 by-
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!!.
;SPMnbsp;.
=<#4351#>
=N=<#4381#><#4381#><#4351#>=A<#4353#>0<#4353#>0=0=0=
1=1=1=
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=
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=10pt
=<#4358#>height7pt depth2pt width0pt<#4358#>1<#4359#><#4382#>1=1=
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<#4392#><#4404#>#math559##1<#4404#><#4392#><#4393#><#4405#>#math560##1<#4405#><#4393#><#4382#><#4359#>
0=<#4360#><;SPMnbsp;<#4360#>=0
!><954>>
=1
<#4363#>1.;SPMnbsp;<#4363#>Dieck, T. tom: Bordism of G-manifolds and
integrality theorems. Topology <#779#>9<#779#> (1970) 345--358
0=<#4453#>2.;SPMnbsp;<#4453#>;SPMlt;0
|Your reference `3.' is wider than you pretended in using
\if N
0=0=0=
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0pt plus 6em
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0=<#4586#><;SPMnbsp;<#4586#>=0
! 0pt plus 6em
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;SPMnbsp;.
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;SPMnbsp;.
=<#4711#>
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=10pt
=<#4718#>height7pt depth2pt width0pt<#4718#>1<#4719#><#4742#>1=1=
0=0=
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0=<#4720#><;SPMnbsp;<#4720#>=0
!><954>>
=1
<#4723#>4.;SPMnbsp;<#4723#>Peitgen, H.-O., Walther, H.-O. (eds.): Functional
differential equations and approximation of fixed points. (Lecture
Notes in Mathematics, vol.730.) Springer, New York Berlin
Heidelberg, 1979
0=<#4813#>5.;SPMnbsp;<#4813#>;SPMlt;0
|Your reference `1.' is wider than you pretended in using
\if N
0=0=0=
1=1=1=
=10000
0pt plus 6em
0=<#5013#>by1=#math687##tex2html_wrap_inline16366##tex2html_wrap_inline16367#<#5044#>0<#5044#>0=0=0=
1=1=1=
2=2=2=
<#5045#><#5045#>=
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=
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=
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=10pt
=<#5049#>height7pt depth2pt width0pt<#5049#><#5088#>1=1=
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0=<#5158#><;SPMnbsp;<#5158#>=0
!><954>>
=1
<#5161#>2.;SPMnbsp;<#5161#>Gantmacher, F.R., Krein, M.G.: Oscillation matrices and
kernels and small vibrations of mechanical systems. State Press
for Technical Literature, Moscow Leningrad, 1950 (Russian).
[German transl.: Oszillationsmatrizen, Oszillationskerne
und kleine Schwingungen mechanischer Systeme. Akademie-Verlag,
Berlin, 1960]
0=<#5251#>3.;SPMnbsp;<#5251#>;SPMlt;0
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<#5521#>5.;SPMnbsp;<#5521#>Redwood, R.: Personal communication, 1986
<#793#>Input of Coding for Letter--Number List<#793#>
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probably you missed the second argument of \if N
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to<#5721#>[B1]<#5721#>Brooks, M.: Automatic generation of test
data for recursive programs having simple errors. PhD thesis,
Stanford University, 1980
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to<#6081#>[T1]<#6081#>Thompson, d'A.: On growth and form. (Abriged
edition: J.T. Bonner, ed.) Cambridge University Press, 1961
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0=<#6516#><;SPMnbsp;<#6516#>=0
!><957>>
=1
to<#6519#>[GP1]<#6519#>Griswold, R.E., Poage, J.F., Polonsky, I.P.:
The SNOBOL4 programming language, 2nd edn. Prentice-Hall,
Englewood Cliffs, NJ, 1971
0=<#6609#>[R1];SPMnbsp;<#6609#>;SPMlt;0
|Your reference `[T1]' is wider than you pretended in using
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<#6963#><#6967#>#math990##1<#6967#><#6963#><#6964#><#6968#>#math991##1<#6968#><#6964#><#6958#><#6953#>
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<#6908#><#6920#>#math995##1<#6920#><#6908#><#6909#><#6921#>#math996##1<#6921#><#6909#><#6898#><#6875#>
0=<#6876#><;SPMnbsp;<#6876#>=0
!><957>>
=1
to<#6879#>[T2]<#6879#>Thompson, d'A.: Personal communication, 1963
<#1633#>
<#1596#><#1509#><#1319#>;SPMnbsp;To end your text you must use the TEX<#809#><#809#> command
<#810#>
else#math997#\byebye<#810#>;SPMnbsp;<#1319#><#1509#><#1596#>
<#1633#>
<#6970#> <#6970#>142
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=0
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=1.5em
150
151
='177 ='177
='60 ='60
157
&Gamma#Gamma;=;SPMquot;0100
&Delta#Delta;=;SPMquot;0101
&Theta#Theta;=;SPMquot;0102
&Lambda#Lambda;=;SPMquot;0103
&Xi#Xi;=;SPMquot;0104
&Pi#Pi;=;SPMquot;0105
&Sigma#Sigma;=;SPMquot;0106
&Upsi#Upsilon;=;SPMquot;0107
&Phi#Phi;=;SPMquot;0108
&Psi#Psi;=;SPMquot;0109
&Omega#Omega;=;SPMquot;010A
==
=
=Y
=`||
=Y
=N
=<#1400#>N=<#974#><#974#>
=Y
Y=N<#1400#>
=N
=<#1401#><#6971#>0<#6971#>0=0=0=
1=1=1=
2=2=2=
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=
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=10pt
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<#6979#><#6983#>#math998###1<#6983#><#6979#><#6980#><#6984#>#math999###1<#6984#><#6980#>
<#6981#><#6985#>#math1000###1<#6985#><#6981#><#6982#><#6986#>#math1001###1<#6986#><#6982#><#6978#><#6977#>
Missing MAINTITLEto2.5true
cc<#6987#>;SPMlt;@
<#6988#>-<#6988#><#6987#>to2.5true cc<#6989#>;SPMlt;@
<#6990#>-<#6990#><#6989#>Missing name(s)
of the author(s)<#1401#>
=Y
`=
=N
=
=0
<#1024#>Lemma<#1024#><#1025#><#1025#><#1026#><#1026#>
<#1027#>Proposition<#1027#><#1028#><#1028#><#1029#><#1029#>
<#1030#>Theorem<#1030#><#1031#><#1031#><#1032#><#1032#>
<#1033#>Corollary<#1033#><#1034#><#1034#><#1035#><#1035#>
<#1036#>Example<#1036#><#1037#><#1037#><#1038#><#1038#>
<#1039#>Exercise<#1039#><#1040#><#1040#><#1041#><#1041#>
<#1042#>Problem<#1042#><#1043#><#1043#><#1044#><#1044#>
<#1045#>Solution<#1045#><#1046#><#1046#><#1047#><#1047#>
<#1048#>Definition<#1048#><#1049#><#1049#><#1050#><#1050#>
<#1051#>Note<#1051#><#1052#><#1052#><#1053#><#1053#>
<#1054#>Question<#1054#><#1055#><#1055#><#1056#><#1056#>
=26260
=N 263
<#6991#>#math1002#= ;SPMgt; <#6991#>=264
to<#6992#>
254=<#7002#> THE JOURNAL OF <#7002#><#7003#>
<#7018#><#7022#>254
to254<#7027#>NONLINEAR<#7027#>
to254<#7028#>SCIENCE<#7028#>
to254<#7029#>..<#7029#>
<#7022#><#7018#>
<#7003#><#6992#>=N 0=0=0=
1=1=1=
2=2=2=
=0pt
=10000
Haupttitel 14pt halbfett/Title boldface -- 14/16
=<#6995#>=N=<#7004#><#7004#><#6995#>=A 0=<#6996#><#7005#>0<#7005#>0=0=0=
1=1=1=
2=2=2=
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<#7007#><#7007#>==
=
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=
<#7009#><#7009#>=
=10pt
=<#7010#>height7pt depth2pt width0pt<#7010#>1<#7011#><#7019#>1=1=
0=0=
<#7023#><#7030#>#math1003##1<#7030#><#7023#><#7024#><#7031#>#math1004##1<#7031#><#7024#>
<#7025#><#7032#>#math1005##1<#7032#><#7025#><#7026#><#7033#>#math1006##1<#7033#><#7026#><#7019#><#7011#><#7012#> <#7012#>1<#7013#><#7013#>true
ccHaupttitel 14pt halbfett/Title boldface -- 14/16<#6996#>0;SPMgt;
Missing MAINTITLEto2.5true
cc<#6998#>;SPMlt;@
<#7014#>-<#7014#><#6998#><#6999#>MAIN title
suppressed due to excessive lengthto2.5true cc<#7015#>;SPMlt;@
<#7020#>-<#7020#><#7015#><#6999#>
Missing MAINTITLEto2.5true
cc<#7000#>;SPMlt;@
<#7016#>-<#7016#><#7000#><#7001#>Haupttitel 14pt halbfett/Title boldface -- 14/16to2.5true
cc<#7017#>;SPMlt;@
<#7021#>-<#7021#><#7017#><#7001#>
`=
Missing MAINTITLEto2.5true
cc<#7035#>;SPMlt;@
<#7037#>-<#7037#><#7035#><#7036#>Expression of Cellular Oncogenes
to2.5true cc<#7038#>;SPMlt;@
<#7039#>-<#7039#><#7038#><#7036#>
0=0=0=
1=1=1=
=10000
0pt plus 6em
Untertitel 10pt halbfett/Subtitle boldface -- 10/11
=<#7042#>=N=<#7043#><#7043#><#7042#>=A
=10000
Ivar Ekeland@<#7117#>1<#7117#> and Roger Temam@<#7118#>2<#7118#>
<#7120#>@#1<#7121#><#7121#> 0=<#7122#><#7129#>0<#7129#>0=0=0=
1=1=1=
2=2=2=
<#7130#><#7130#>=
<#7131#><#7131#>==
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<#7132#><#7132#>==
=
<#7133#><#7133#>=
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to2.5true cc<#7125#>;SPMlt;@
<#7138#>-<#7138#><#7125#>Missing name(s)
of the author(s)<#7126#>to2.5true cc<#7139#>;SPMlt;@
<#7145#>-<#7145#><#7139#>AUTHORS suppressed due to excessive
length<#7126#>
to2.5true cc<#7127#>;SPMlt;@
<#7140#>-<#7140#><#7127#>Missing name(s)
of the author(s)<#7128#>to2.5true
cc<#7141#>;SPMlt;@
<#7146#>-<#7146#><#7141#>Ivar Ekeland@<#7142#>1<#7142#> and Roger Temam@<#7143#>2<#7143#><#7128#>
<#7120#>=E
to2.5true cc<#7156#>;SPMlt;@
<#7158#>-<#7158#><#7156#>Missing name(s)
of the author(s)<#7157#>to2.5true cc<#7159#>;SPMlt;@
<#7160#>-<#7160#>
<#7159#>R. Müller<#7157#>
<#7162#>0<#7162#>0=0=0=
1=1=1=
2=2=2=
<#7163#><#7163#>=
<#7164#><#7164#>==
=
<#7165#><#7165#>==
=
<#7166#><#7166#>=
=10pt
=<#7167#>height7pt depth2pt width0pt<#7167#>1<#7168#><#7170#>1=1=
0=0=
<#7171#><#7175#>#math1011##1<#7175#><#7171#><#7172#><#7176#>#math1012##1<#7176#><#7172#>
<#7173#><#7177#>#math1013##1<#7177#><#7173#><#7174#><#7178#>#math1014##1<#7178#><#7174#><#7170#><#7168#>
@1Princeton University, Princeton NJ 08544, USA
@2Université de Paris-Sud,
Laboratoire d'Analyse Numérique, Bâtiment 425,
F-91405 Orsay Cedex, France
<#7180#>0<#7180#>0=0=0=
1=1=1=
2=2=2=
<#7181#><#7181#>=
<#7182#><#7182#>==
=
<#7183#><#7183#>==
=
<#7184#><#7184#>=
=10pt
=<#7185#>height7pt depth2pt width0pt<#7185#>1<#7186#><#7188#>1=1=
0=0=
<#7189#><#7193#>#math1015##1<#7193#><#7189#><#7190#><#7194#>#math1016##1<#7194#><#7190#>
<#7191#><#7195#>#math1017##1<#7195#><#7191#><#7192#><#7196#>#math1018##1<#7196#><#7192#><#7188#><#7186#>Received June 5, 1989
<#7197#>Summary. <#7197#>A new variant of the multi-grid algorithms is presented. It
uses multiple coarse-grid corrections with particularly associated
prolongations and restrictions. In this paper the robustness with
respect to anisotropic problems is considered.
<#7198#>Key words. <#7198#>multi-grid method -- coarse--grid correction --
singular perturbation -- robustness.
N
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!1.!The Anisotropic Equation and Standard Multi-Grid Methods
1.;SPMnbsp;The Anisotropic Equation and Standard Multi-Grid Methods
<#7200#>0=0 by-
0;SPMlt;
0=00 by00 by
3
0 by
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!1.!The Anisotropic Equation and Standard Multi-Grid Methods
1.;SPMnbsp;The Anisotropic Equation and Standard Multi-Grid Methods
=<#7202#>
=N=<#7204#><#7204#><#7202#>=A
N
=10000
0pt plus 6em
0=<#7206#>1<#7209#><#7209#>
!1.1.!Introduction
1.1.;SPMnbsp;Introduction
<#7206#>0=0 by-
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0=00 by00 by
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!1.1.!Introduction
1.1.;SPMnbsp;Introduction
=<#7208#>
=N=<#7210#><#7210#><#7208#>=B
Multi-grid methods are known as very fast solvers of a large class of
discretised partial differential equations. However, the multi-grid
method cannot be understood as a fixed algorithm. Usually, the
components of the multi-grid iteration have to be adapted to the given
problem and sometimes the problems are modified in order to make them
acceptable for multi-grid methods. In particular, the smoothing
iteration is the most delicated part of the multi-grid process.
An iteration is called a <#1076#>robust<#1076#> one, if it works for a sufficient
large class of problems. Attempts have been made to construct robust
multi-grid iterations by means of sophisticated smoothing processes...
With this chapter, the preliminaries are over, and we begin the search
for periodic solutions to Hamiltonian systems. All this will be done in
the convex case; that is, we shall study the boundary-value problem
#math1019#
#tex2html_wrap_indisplay16967##tex2html_wrap_indisplay16968# ;SPMamp; =JH'(t, x)#tex2html_wrap_indisplay16969#x(0) ;SPMamp; =x(T)#tex2html_wrap_indisplay16970#
with #math1020#H(t,⋅) a convex function of x, going to + ∞ when
#math1021##tex2html_wrap_inline16975#x#tex2html_wrap_inline16976#→∞.
N
=10000
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!1.2.!Autonomous Systems
1.2.;SPMnbsp;Autonomous Systems
<#7212#>0=0 by-
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0=00 by00 by
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!1.2.!Autonomous Systems
1.2.;SPMnbsp;Autonomous Systems
=<#7214#>
=N=<#7216#><#7216#><#7214#>=B
In this section, we will consider the case when the Hamiltonian H(x)
is autonomous. For the sake of simplicity, we shall also assume that it
is C1.
We shall first consider the question of nontriviality, within the
general framework of #math1022##tex2html_wrap_inline16980#A∞, B∞#tex2html_wrap_inline16981#-subquadratic Hamiltonians. In
the second subsection, we shall look into the special case when H is
#math1023##tex2html_wrap_inline16984#0, b∞#tex2html_wrap_inline16985#-subquadratic, and we shall try to derive additional
information.
N-
0=0=0=
1=1=1=
The General Case: Nontriviality.
We assume that H is #math1024##tex2html_wrap_inline16988#A∞, B∞#tex2html_wrap_inline16989#-subquadratic at infinity,
for some constant symmetric matrices A∞ and B∞, with #math1025#B∞ - A∞ positive definite. Set:
#math1026#
#tex2html_wrap_indisplay16994##tex2html_wrap_indisplay16995#
Theorem 21 tells us that if #math1027#λ + γ ;SPMlt; 0, the boundary-value
problem:
#math1028#
#tex2html_wrap_indisplay16998##tex2html_wrap_indisplay16999# ;SPMamp; =JH'(x)#tex2html_wrap_indisplay17000#x(0) ;SPMamp; =x(T)#tex2html_wrap_indisplay17001##tex2html_wrap_indisplay17002#(3)
has at least one solution #math1029##tex2html_wrap_inline17004#, which is found by minimizing the dual
action functional:
#math1030#
ψ(u) = #tex2html_wrap_indisplay17006##tex2html_wrap_indisplay17007##tex2html_wrap_indisplay17008#2#tex2html_wrap_indisplay17009#Λo-1u, u#tex2html_wrap_indisplay17010# + N * (- u)#tex2html_wrap_indisplay17011#dt#tex2html_wrap_indisplay17012#(4)
on the range of Λ, which is a subspace #math1031#R(Λ)#tex2html_wrap_inline17015#L2 with
finite codimension. Here
#math1032#
N(x) : = H(x) - #tex2html_wrap_indisplay17017#2#tex2html_wrap_indisplay17018#A∞x, x#tex2html_wrap_indisplay17019##tex2html_wrap_indisplay17020#(5)
is a convex function, and
#math1033#
N(x)≤#tex2html_wrap_indisplay17022#2#tex2html_wrap_indisplay17023##tex2html_wrap_indisplay17024#B∞ - A∞#tex2html_wrap_indisplay17025#x, x#tex2html_wrap_indisplay17026# + c ∀x˙#tex2html_wrap_indisplay17027#(6)
<#1086#> 1.<#1086#> <#1433#> Assume H'(0) = 0 and H(0) = 0. Set:
#math1034#
δ : = #tex2html_wrap_indisplay17031#2N(x)#tex2html_wrap_indisplay17032#x#tex2html_wrap_indisplay17033# .#tex2html_wrap_indisplay17034#(7)
If #math1035#γ ;SPMlt; - λ ;SPMlt; δ, the solution #math1036##tex2html_wrap_inline17037# is non-zero:
#math1037#
#tex2html_wrap_indisplay17039#(t)≠0 ∀t .#tex2html_wrap_indisplay17040#(8)
<#1433#>
<#7223#>Proof!! . <#7223#>Condition (7) means that, for every #math1038#δ' ;SPMgt; δ, there is
some #math1039#ε ;SPMgt; 0 such that
#math1040#
#tex2html_wrap_indisplay17044#x#tex2html_wrap_indisplay17045#≤ε⇒N(x)≤#tex2html_wrap_indisplay17046##tex2html_wrap_indisplay17047#x#tex2html_wrap_indisplay17048#˙#tex2html_wrap_indisplay17049#(9)
It is an exercise in convex analysis, into which we shall not go, to
show that this implies that there is an η ;SPMgt; 0 such that
#math1041#
f#tex2html_wrap_indisplay17052#x#tex2html_wrap_indisplay17053#≤η⇒N * (y)≤#tex2html_wrap_indisplay17054##tex2html_wrap_indisplay17055#y#tex2html_wrap_indisplay17056# .#tex2html_wrap_indisplay17057#(10)
1.5cm
Ytrue cmtrue mm
=N 0=<#7224#><#7226#>0<#7226#>0=0=0=
1=1=1=
2=2=2=
<#7227#><#7227#>=
<#7228#><#7228#>==
=
<#7229#><#7229#>==
=
<#7230#><#7230#>=
=10pt
=<#7231#>height7pt depth2pt width0pt<#7231#>1<#7232#><#7242#>1=1=
0=0=
<#7244#><#7252#>#math1042##1<#7252#><#7244#><#7245#><#7253#>#math1043##1<#7253#><#7245#>
<#7246#><#7254#>#math1044##1<#7254#><#7246#><#7247#><#7255#>#math1045##1<#7255#><#7247#><#7242#><#7232#><#7233#>Fig.1. <#7233#>This is the caption of the figure displaying a white eagle
and a white horse on a snow field
255=0255by<#7224#>255;SPMgt;10
<#7234#>0<#7234#>0=0=0=
1=1=1=
2=2=2=
<#7235#><#7235#>=
<#7236#><#7236#>==
=
<#7237#><#7237#>==
=
<#7238#><#7238#>=
=10pt
=
1<#7240#><#7243#>1=1=
0=0=
<#7248#><#7256#>#math1046##1<#7256#><#7248#><#7249#><#7257#>#math1047##1<#7257#><#7249#>
<#7250#><#7258#>#math1048##1<#7258#><#7250#><#7251#><#7259#>#math1049##1<#7259#><#7251#><#7243#><#7240#><#7241#>Fig.1. <#7241#>This is the caption of the figure displaying a white eagle
and a white horse on a snow field
Since u1 is a smooth function, we will have #math1050##tex2html_wrap_inline17068#hu1#tex2html_wrap_inline17069#≤η for h small enough, and inequality (10) will hold, yielding
thereby:
#math1051#
ψ(hu1)≤#tex2html_wrap_indisplay17072##tex2html_wrap_indisplay17073##tex2html_wrap_indisplay17074#u1#tex2html_wrap_indisplay17075# + #tex2html_wrap_indisplay17076##tex2html_wrap_indisplay17077##tex2html_wrap_indisplay17078#u1#tex2html_wrap_indisplay17079# .#tex2html_wrap_indisplay17080#(11)
If we choose δ' close enough to δ, the quantity #math1052##tex2html_wrap_inline17084##tex2html_wrap_inline17085# + #tex2html_wrap_inline17086##tex2html_wrap_inline17087# will be negative, and we end up with
#math1053#
ψ(hu1) ;SPMlt; 0 for h≠0 small .#tex2html_wrap_indisplay17089#(12)
On the other hand, we check directly that #math1054#ψ(0) = 0. This shows
that 0 cannot be a minimizer of ψ, not even a local one. So #math1055##tex2html_wrap_inline17093#≠ 0 and #math1056##tex2html_wrap_inline17095#≠Λo-1(0) = 0. <#7261#><#7263#>#tex2html_wrap_inline17097#<#7263#>#tex2html_wrap_inline17099#<#7261#><#7262#>
<#7264#><#7265#>#tex2html_wrap_inline17101#<#7265#>#tex2html_wrap_inline17103#<#7264#>
=0pt=0
<#7262#>
<#1102#> 2.<#1102#> <#1434#> Assume H is C2 and #math1057##tex2html_wrap_inline17107#a∞, b∞#tex2html_wrap_inline17108#-subquadratic at infinity. Let
#math1058#ξ1,..., ξN be the
equilibria, that is, the solutions of #math1059#H'(ξ) = 0. Denote by ωk
the smallest eigenvalue of #math1060#H''#tex2html_wrap_inline17113#ξk#tex2html_wrap_inline17114#, and set:
#math1061#
ω : = Min #tex2html_wrap_indisplay17116#ω1,..., ωk#tex2html_wrap_indisplay17117# .#tex2html_wrap_indisplay17118#(13)
If:
#math1062#
#tex2html_wrap_indisplay17120#b∞ ;SPMlt; - E#tex2html_wrap_indisplay17121# - #tex2html_wrap_indisplay17122#a∞#tex2html_wrap_indisplay17123# ;SPMlt; #tex2html_wrap_indisplay17124#ω#tex2html_wrap_indisplay17125#(14)
then minimization of ψ yields a non-constant T-periodic solution
#math1063##tex2html_wrap_inline17129#.<#1434#>
We recall once more that by the integer part #math1064#E[α] of #math1065#α∈IR, we mean the #math1066#a∈#tex2html_wrap_inline17133#$#tex2html_wrap_inline17134##tex2html_wrap_inline17135##tex2html_wrap_inline17136#ZZ$$#tex2html_wrap_inline17137##tex2html_wrap_inline17138##tex2html_wrap_inline17139#ZZ$$#tex2html_wrap_inline17140##tex2html_wrap_inline17141##tex2html_wrap_inline17142#ZZ$$#tex2html_wrap_inline17143##tex2html_wrap_inline17144##tex2html_wrap_inline17145#ZZ$ such that #math1067#a ;SPMlt; α≤a + 1. For instance,
if we take #math1068#a∞ = 0, Corollary 2 tells us that #math1069##tex2html_wrap_inline17149# exists and is
non-constant provided that:
#math1070#
#tex2html_wrap_indisplay17151#b∞ ;SPMlt; 1 ;SPMlt; #tex2html_wrap_indisplay17152##tex2html_wrap_indisplay17153#(15)
or
#math1071#
T∈#tex2html_wrap_indisplay17155##tex2html_wrap_indisplay17156#,#tex2html_wrap_indisplay17157##tex2html_wrap_indisplay17158# .#tex2html_wrap_indisplay17159#(16)
<#7277#>Proof!! . <#7277#>The spectrum of Λ is #math1072##tex2html_wrap_inline17162##tex2html_wrap_inline17163#$#tex2html_wrap_inline17164##tex2html_wrap_inline17165##tex2html_wrap_inline17166#ZZ$$#tex2html_wrap_inline17167##tex2html_wrap_inline17168##tex2html_wrap_inline17169#ZZ$$#tex2html_wrap_inline17170##tex2html_wrap_inline17171##tex2html_wrap_inline17172#ZZ$$#tex2html_wrap_inline17173##tex2html_wrap_inline17174##tex2html_wrap_inline17175#ZZ$ + a∞. The
largest negative eigenvalue λ is given by #math1073##tex2html_wrap_inline17178#ko + a∞,
where
#math1074#
#tex2html_wrap_indisplay17180#ko + a∞ ;SPMlt; 0≤#tex2html_wrap_indisplay17181#(ko +1) + a∞˙#tex2html_wrap_indisplay17182#(17)
Hence:
#math1075#
ko = E#tex2html_wrap_indisplay17184# - #tex2html_wrap_indisplay17185#a∞#tex2html_wrap_indisplay17186# .#tex2html_wrap_indisplay17187#(18)
The condition #math1076#γ ;SPMlt; - λ ;SPMlt; δ now becomes:
#math1077#
b∞ - a∞ ;SPMlt; - #tex2html_wrap_indisplay17190#ko - a∞ ;SPMlt; ω - a∞#tex2html_wrap_indisplay17191#(19)
which is precisely condition (14).<#7287#><#7289#>#tex2html_wrap_inline17193#<#7289#>#tex2html_wrap_inline17195#<#7287#><#7288#>
<#7290#><#7291#>#tex2html_wrap_inline17197#<#7291#>#tex2html_wrap_inline17199#<#7290#>
=0pt=0
<#7288#>
<#1117#>3.<#1117#> <#1118#> Assume that H is C2 on #math1078#IR2n #tex2html_wrap_inline17205# {0} and
that H''(x) is non-degenerate for any x≠ 0. Then any local
minimizer #math1079##tex2html_wrap_inline17209# of ψ has minimal period T.<#1118#>
<#7294#>Proof!! . <#7294#>We know that #math1080##tex2html_wrap_inline17213#, or #math1081##tex2html_wrap_inline17215# + ξ for some constant #math1082#ξ∈IR2n, is a T-periodic solution of the Hamiltonian system:
#math1083#
#tex2html_wrap_indisplay17221# = JH'(x) .#tex2html_wrap_indisplay17222#(20)
There is no loss of generality in taking ξ = 0. So #math1084#ψ(x)≥ψ(#tex2html_wrap_inline17225#) for all #math1085##tex2html_wrap_inline17227# in some neighbourhood of x in #math1086#W1, 2#tex2html_wrap_inline17230#IR/T#tex2html_wrap_inline17231#$#tex2html_wrap_inline17232##tex2html_wrap_inline17233##tex2html_wrap_inline17234#ZZ$$#tex2html_wrap_inline17235##tex2html_wrap_inline17236##tex2html_wrap_inline17237#ZZ$$#tex2html_wrap_inline17238##tex2html_wrap_inline17239##tex2html_wrap_inline17240#ZZ$$#tex2html_wrap_inline17241##tex2html_wrap_inline17242##tex2html_wrap_inline17243#ZZ$;IR2n#tex2html_wrap_inline17246#.
But this index is precisely the index #math1087#iT(#tex2html_wrap_inline17248#) of the T-periodic
solution #math1088##tex2html_wrap_inline17251# over the interval (0, T), as defined in Sect.~2.6. So
#math1089#
iT(#tex2html_wrap_indisplay17254#) = 0 .#tex2html_wrap_indisplay17255#(21)
Now if #math1090##tex2html_wrap_inline17257# has a lower period, T/k say, we would have, by Corollary
31:
#math1091#
iT(#tex2html_wrap_indisplay17260#) = ikT/k(#tex2html_wrap_indisplay17261#)≥kiT/k(#tex2html_wrap_indisplay17262#) + k - 1≥k - 1≥1 .#tex2html_wrap_indisplay17263#(22)
This would contradict (21), and thus cannot happen.<#7309#><#7311#>#tex2html_wrap_inline17265#<#7311#>#tex2html_wrap_inline17267#<#7309#><#7310#>
<#7312#><#7313#>#tex2html_wrap_inline17269#<#7313#>#tex2html_wrap_inline17271#<#7312#>
=0pt=0
<#7310#>
N-
Notes and Comments.
The results in this section are a
refined
version of [CE1]; the minimality result of Proposition 14 was the first
of its kind.
To understand the nontriviality conditions, such as the one in formula
(16), one may think of a one-parameter family xT, #math1092#T∈#tex2html_wrap_inline17274#2πω-1, 2πb∞-1#tex2html_wrap_inline17275# of periodic solutions, #math1093#xT(0) = xT(T), with xT going away to infinity when #math1094#T→2πω-1, which
is the period of the linearized system at 0.
true mm
<#7316#><#7317#>0<#7317#>0=0=0=
1=1=1=
2=2=2=
<#7318#><#7318#>=
<#7319#><#7319#>==
=
<#7320#><#7320#>==
=
<#7321#><#7321#>=
=10pt
=<#7322#>height7pt depth2pt width0pt<#7322#>1<#7323#><#7325#>1=1=
0=0=
<#7326#><#7330#>#math1095##1<#7330#><#7326#><#7327#><#7331#>#math1096##1<#7331#><#7327#>
<#7328#><#7332#>#math1097##1<#7332#><#7328#><#7329#><#7333#>#math1098##1<#7333#><#7329#><#7325#><#7323#><#7324#>Table1. <#7324#>Observational results from NGC 4827
<#7316#>
<#1620#><#7334#>0<#7334#>0=0=0=
1=1=1=
2=2=2=
<#7335#><#7335#>=
<#7336#><#7336#>==
=
<#7337#><#7337#>==
=
<#7338#><#7338#>=
=10pt
=<#7339#>height7pt depth2pt width0pt<#7339#>#1<#7340#><#7341#>1=1=
0=0=
<#7342#><#7346#>#math1099###1<#7346#><#7342#><#7343#><#7347#>#math1100###1<#7347#><#7343#>
<#7344#><#7348#>#math1101###1<#7348#><#7344#><#7345#><#7349#>#math1102###1<#7349#><#7345#><#7341#><#7340#>
<#1568#><#1435#>;SPMnbsp;#;SPMnbsp;;SPMamp;;SPMamp;#;SPMnbsp;
;SPMamp;;SPMamp;3<#1129#>RA (1950)<#1129#>;SPMamp; ;SPMamp;3<#1130#>Dec (1950)<#1130#>
;SPMamp; S ;SPMamp; Pol ;SPMamp; ;SPMamp; log P
<#1131#><#1131#>
;SPMamp;;SPMamp;3;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMamp;;SPMamp;3;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMnbsp;;SPMamp;;SPMamp;
<#1132#><#1132#>
;SPMamp; ;SPMamp;(h) ;SPMamp;(m) ;SPMamp; (s) ;SPMamp; ;SPMamp; (<#7350#>o<#7350#>) ;SPMamp; (<#7351#>′<#7351#>) ;SPMamp; (<#7352#>#math1103#′′<#7352#>)
;SPMamp; (mJy) ;SPMamp; (mJy) ;SPMamp; ;SPMamp; (W Hz-1)
<#1134#>
<#1134#>
<#1135#>
<#1135#>
<#1136#>
<#1136#>
Core ;SPMamp; (5 GHz) ;SPMamp; 12 ;SPMamp; 54 ;SPMamp; 18.0 ;SPMamp; ;SPMamp; 27 ;SPMamp; 26 ;SPMamp; 56.2
;SPMamp; 8 ;SPMamp; ;SPMamp; ;SPMamp; 21.64
Total;SPMamp;(327 MHz);SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 210 ;SPMamp; ;SPMamp; ;SPMamp; 23.13
;SPMamp;(1.4 GHz);SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 57 ;SPMamp; 1.3 ;SPMamp; 2 ;SPMamp; 22.49
;SPMamp; (5 GHz) ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp; ;SPMamp;
;SPMamp; 26 ;SPMamp; 0.73 ;SPMamp; 3 ;SPMamp; 22.15 <#1435#>
<#1568#><#1620#>
true mm
<#1137#>4 (Ghoussoub-Preiss).<#1137#> <#1436#> Assume H(t, x) is
#math1104#(0, ε)-subquadratic at
infinity for all #math1105#ε ;SPMgt; 0, and T-periodic in t
#math1106#
H(t,⋅) is convex ∀t#tex2html_wrap_indisplay17299#(23)
#math1107#
H(⋅, x) is T-periodic ∀x#tex2html_wrap_indisplay17301#(24)
#math1108#
H(t, x)≥n#tex2html_wrap_indisplay17303##tex2html_wrap_indisplay17304#x#tex2html_wrap_indisplay17305##tex2html_wrap_indisplay17306# with n(s)s-1→∞ as s→∞#tex2html_wrap_indisplay17307#(25)
#math1109#
∀ε ;SPMgt; 0 , ∃c : H(t, x)≤#tex2html_wrap_indisplay17309##tex2html_wrap_indisplay17310#x#tex2html_wrap_indisplay17311# + c .#tex2html_wrap_indisplay17312#(26)
Assume also that H is C2, and H''(t, x) is positive definite
everywhere. Then there is a sequence xk, #math1110#k∈IN, of kT-periodic
solutions of the system
#math1111#
#tex2html_wrap_indisplay17320# = JH'(t, x)#tex2html_wrap_indisplay17321#(27)
such that, for every #math1112#k∈IN, there is some #math1113#po∈IN with:
#math1114#
p≥po⇒xpk≠xk .#tex2html_wrap_indisplay17325#(28)
<#7357#><#7359#>#tex2html_wrap_inline17327#<#7359#>#tex2html_wrap_inline17329#<#7357#><#7358#>
<#7360#><#7361#>#tex2html_wrap_inline17331#<#7361#>#tex2html_wrap_inline17333#<#7360#>
=0pt=0
<#7358#><#1436#>
<#1437#>1 <#1146#>(External forcing).<#1146#><#1437#><#1438#> Consider the system:
#math1115#
#tex2html_wrap_indisplay17335# = JH'(x) + f (t)#tex2html_wrap_indisplay17336#(29)
where the Hamiltonian H is #math1116##tex2html_wrap_inline17339#0, b∞#tex2html_wrap_inline17340#-subquadratic, and the
forcing term is a distribution on the circle:
#math1117#
f = #tex2html_wrap_indisplay17342#F + fo with F∈L2#tex2html_wrap_indisplay17343#IR/T#tex2html_wrap_indisplay17344#$#tex2html_wrap_indisplay17345##tex2html_wrap_indisplay17346##tex2html_wrap_indisplay17347#ZZ$$#tex2html_wrap_indisplay17348##tex2html_wrap_indisplay17349##tex2html_wrap_indisplay17350#ZZ$$#tex2html_wrap_indisplay17351##tex2html_wrap_indisplay17352##tex2html_wrap_indisplay17353#ZZ$$#tex2html_wrap_indisplay17354##tex2html_wrap_indisplay17355##tex2html_wrap_indisplay17356#ZZ$;IR2n#tex2html_wrap_indisplay17359# ,#tex2html_wrap_indisplay17360#(30)
where #math1118#fo : = T-1#tex2html_wrap_inline17362#f (t)dt. For instance,
#math1119#
f (t) = #tex2html_wrap_indisplay17364#δkξ ,#tex2html_wrap_indisplay17365#(31)
where δk is the Dirac mass at t = k and #math1120#ξ∈IR2n is a
constant, fits the prescription. This means that the system #math1121##tex2html_wrap_inline17372# = JH'(x) is being excited by a series of identical shocks at interval T.<#1438#>
<#1151#>5.<#1151#><#1439#>Let #math1122#A∞(t) and #math1123#B∞(t) be symmetric
operators in #math1124#IR2n, depending continuously on #math1125#t∈[0, T], such that
#math1126#A∞(t)≤B∞(t) for all t.
A Borelian function #math1127#H : [0, T]×IR2n→IR is called #math1128##tex2html_wrap_inline17386#A∞, B∞#tex2html_wrap_inline17387#-<#1152#>subquadratic at infinity<#1152#> if there exists a function
N(t, x) such that:
#math1129#
H(t, x) = #tex2html_wrap_indisplay17390#2#tex2html_wrap_indisplay17391#A∞(t)x, x#tex2html_wrap_indisplay17392# + N(t, x)#tex2html_wrap_indisplay17393#(32)
#math1130#
∀t , N(t, x) is convex with respect tonbsp;x#tex2html_wrap_indisplay17395#(33)
#math1131#
N(t, x)≥n#tex2html_wrap_indisplay17397##tex2html_wrap_indisplay17398#x#tex2html_wrap_indisplay17399##tex2html_wrap_indisplay17400# with n(s)s-1→ + ∞ as s→ + ∞#tex2html_wrap_indisplay17401#(34)
#math1132#
∃c∈IR : H(t, x)≤#tex2html_wrap_indisplay17403#2#tex2html_wrap_indisplay17404#B∞(t)x, x#tex2html_wrap_indisplay17405# + c nbsp;∀x .#tex2html_wrap_indisplay17406#(35)
<#1439#>
If #math1133#A∞(t) = a∞I and #math1134#B∞(t) = b∞I, with #math1135#a∞≤b∞∈IR, we shall say that H is #math1136##tex2html_wrap_inline17412#a∞, b∞#tex2html_wrap_inline17413#-subquadratic at infinity. As an example, the function #math1137##tex2html_wrap_inline17415#x#tex2html_wrap_inline17416#, with #math1138#1≤α ;SPMlt; 2, is #math1139#(0, ε)-subquadratic at infinity
for every #math1140#ε ;SPMgt; 0. Similarly, the Hamiltonian
#math1141#
H(t, x) = #tex2html_wrap_indisplay17421#2k#tex2html_wrap_indisplay17422#k#tex2html_wrap_indisplay17423# + #tex2html_wrap_indisplay17424#x#tex2html_wrap_indisplay17425##tex2html_wrap_indisplay17426#(36)
is #math1142#(k, k + ε)-subquadratic for every #math1143#ε ;SPMgt; 0. Note that, if k ;SPMlt; 0,
it is not convex.
N-
Notes and Comments.
The first results on subharmonics were
obtained by Rabinowitz in [Ra1], who showed the existence of infinitely
many subharmonics both in the subquadratic and superquadratic case, with
suitable growth conditions on H'. Again the duality approach enabled
Clarke and Ekeland in [CE2] to treat the same problem in the
convex-subquadratic case, with growth conditions on H only.
Recently, Michalek and Tarantello (see [MT1] and [Ta1]) have obtained
lower bound on the number of subharmonics of period kT, based on
symmetry considerations and on pinching estimates, as in Sect.~5.2 of
this article.
N
0=0=0=
1=1=1=
=10000
0pt plus 6em
0=<#7387#>by1=#math1144##tex2html_wrap_inline17434##tex2html_wrap_inline17435#<#7418#>0<#7418#>0=0=0=
1=1=1=
2=2=2=
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to<#7493#>[CE1]<#7493#>Clarke, F., Ekeland, I.: Nonlinear oscillations and
boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal.
<#1161#>78<#1161#> (1982) 315--333
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to<#7982#>[Ra1]<#7982#>Rabinowitz, P.: On subharmonic solutions of a Hamiltonian
system. Comm. Pure Appl. Math. <#1168#>33<#1168#> (1980) 609--633
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