Content-Type: multipart/mixed; boundary="-------------0107191013608" This is a multi-part message in MIME format. ---------------0107191013608 Content-Type: text/plain; name="01-281.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-281.keywords" numerical analysis, parabolic PDE, dynamical systems, epsilon-entropy ---------------0107191013608 Content-Type: application/postscript; name="paper.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="paper.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: paper.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman Times-Italic Courier XYATIP10 %%+ XYBTIP10 Times-BoldItalic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o paper.ps paper %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.07.12:1555 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYBTIP10) readonly def /FamilyName (XYBTIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -276 def /UnderlineThickness 138 def end readonly def /FontName /XYBTIP10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 15 /d15 put dup 47 /d47 put readonly def /FontBBox{-542 -542 542 542}readonly def /UniqueXX 5092839 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041987a409a2d06b6b3057738213cee08cd789eeaa 097caceb2738a78b2f437638f0d63dd9e45ce613ae94486e726c4ed202501d61 51965c5c865a24933f21e0b1c67ff0d74bea0b8003496a2b1c9e3cde218dfa02 7343f1561243c5419412a440b6d4682c4dd92bf310718d73d28f47559a653346 c8fa6a8e3ec0a68d6661b293a71328a0bd0521249f1263070e67d0c20ca4a48d 221bcd864852e33289496155416b7cc05e73dd2b7f9ba0977ab328be862ac7e0 139c8eef1237e57525cbc853d7cbe3c9a8b54c378e8af02257a8daa736c3d9ae fb18fd198a33681c334984d81e2d783d32adf54549f5bea0bf351b1016032908 81685bde8d44703654d97063c8ebfb896e029b2383f5754d467163ec07f3398a d88196c720fd98b9a2260de8d7d3aa6453f831ce18233cfbf6cb098bc3ca2cd1 495386a279ced386537228ec08f3b3e400cc040ab2e763b0cd93c9a2c5ee0436 f0a2f033ba5d3e4231aacc9b0aa820f7ad72a3cec593a1153ee5527693ad3bc8 eaef55ac2f52fdf27146c04dcc825181a275e632e75a94cb9b3d3f7d17c1c08b 83bbf5c681f864e234d10b0f7c64839aa1671931f39a001e4134030b91d9a473 6c7d5e101e04feb20a04907ab46ab24902c1844b018beefd9014c8b629674e57 f1f0d63ad79dfa8ce4d1fffabeb4315386d494a3ab66cc9f291a714ef0ee4f9f 1687f0ecbcd2acea0e98dd5f94dfd700e546599e58d1f25bc54ef6ec0f12b91e 6690287b7c527a51724cea71da655f2b2974633ba5484cc6c2300ba28dff89e2 0c37542986ec1e4613cc8a16521e5c2720d88fa18111a1083dcee82a4b65f66f 22aaea93d794cdef4a30d0256760a484140899bd59081ea69d780c90cee5e94f afc2585b622df9a698e5b25ea1aea20eb885a85b21e94ded0ec01d65d64bccac 9693b853178f4872ba4450e36fe6caac2034993f15c2f8d5d599a32d5dc0b91e eb8a009d36b894114cf6d4eca2c3db208c80cb8b675d8bfccb8264d88d4a16b6 b80559737c56783e2175d7c469ba1b481e72c60483fc6c7b0c637f7e27e194b5 6616dac64797e0f717d9fe59e11003a1b3e5cab2a5ce809d019a4ac5a0478125 8f79868b6cad543270e5a1f761cd1aee2aeef7874301eb64d9af9dee85c55a2c 5c2d5cd124f599fb53d4d2b623919f5032e90ff04519b8fba53dd541ded5923d e43313657754bd56409bc9b011338213c00dd6006ba254768e8319bca858b354 0d7a80ee5d1d6e4b0efcd53fcca99094a062fff19fe061eb1880c471d291eb1a 47e8faac1d2594a2558c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: XYATIP10 %!PS-AdobeFont-1.1: XYATIP10 001.104 %%CreationDate: 1997 Jul 20 21:19:17 %%RevisionDate: 1997 Sep 14 19:58:47 % % XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style". % % Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose. % PostScript adaptation Copyright (C) 1994-1997 Ross Moore. % Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc. % % This file is part of the Xy-pic macro package. % Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose % % The Xy-pic macro package is free software; you can redistribute it % and/or modify it under the terms of the GNU General Public License % as published by the Free Software Foundation; either version 2 % of the License, or (at your option) any later version. % % The Xy-pic macro package is distributed in the hope that it will % be useful, but WITHOUT ANY WARRANTY; without even the implied % warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. % See the GNU General Public License for more details. % % You should have received a copy of the GNU General Public License % along with this macro package; if not, write to the % Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. 11 dict begin /FontInfo 9 dict dup begin /version (001.104) readonly def /Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly def /FullName (XYATIP10) readonly def /FamilyName (XYATIP) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch 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b(1993\),)e(Kluwer)h(Acad.)g(Publ.,)f(Dordrecht,)h (1994,)f(pp.)h(97\226104.)p eop %%Page: 22 23 22 22 bop 0 100 a Fv(Lar)o(ge)25 b(V)-10 b(olume)25 b(Limits)g(of)g (Discr)n(etised)g(P)o(arabolic)g(Equations)1332 b(22)50 407 y Ft([2])p 215 394 191 4 v 240 w(,)45 b Fh(Thermodynamic)40 b(limit)g(of)h(the)g(Ginzb)n(ur)l(g-Landau)e(equations)p Ft(,)k(Nonlinearity)-6 b(,)44 b(7)c(\(1994\),)215 527 y(pp.)24 b(1175\2261190.)50 728 y([3])p 215 715 V 240 w(,)f Fh(Extended)h(dynamical)e(systems)p Ft(,)h(in)g(Proceedings)h(of) f(the)h(International)e(Congress)i(of)f(Mathe-)215 848 y(maticians,)g(V)-13 b(ol.)25 b(III)g(\(Berlin,)g(1998\),)g(Doc.)f (Math.,)g(1998,)g(pp.)h(123\226132.)50 1049 y([4])51 b(P)-7 b(.)35 b(C)t Fb(O)t(L)t(L)t(E)t(T)i(A)t(N)t(D)d Ft(J)t(.)t(-)t(P)-7 b(.)38 b(E)t Fb(C)t(K)t(M)t(A)t(N)t(N)r Ft(,)d Fh(The)29 b(de\002nition)e(and)i(measur)l(ement)f(of)g(the)h (topolo)o(gical)e(en-)215 1169 y(tr)l(opy)c(per)i(unit)f(volume)h(in)f (par)o(abolic)f(PDEs)p Ft(,)h(Nonlinearity)-6 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Fb(I)t(B)t(B)t(O)t(N)t Ft(,)j(D)t(.)d(D)t(.) f(H)t Fb(O)t(L)t(M)t Ft(,)j Fb(A)t(N)t(D)d Ft(B)t(.)h(N)t Fb(I)t(C)t(O)t(L)t(A)t(E)t(N)t(K)r(O)r Ft(,)f Fh(Low-dimensional)215 2333 y(behaviour)22 b(in)h(the)g(comple)n(x)g(Ginzb)n(ur)l(g-Landau)f (equation)p Ft(,)g(Nonlinearity)-6 b(,)22 b(1)h(\(1988\),)g(pp.)g (279\226309.)50 2534 y([9])51 b(J)t(.)t(-)t(M)t(.)63 b(G)t Fb(H)t(I)t(D)q(A)q(G)t(L)t(I)t(A)g(A)t(N)t(D)d Ft(B)t(.)h(H)1543 2526 y(\264)1535 2534 y Fb(E)t(R)q(O)t(N)r Ft(,)66 b Fh(Dimension)53 b(of)i(the)f(attr)o(actor)o(s)f(associated)g (to)h(the)215 2654 y(Ginzb)n(ur)l(g-Landau)23 b(partial)g(dif)n(fer)l (ential)g(equation)p Ft(,)h(Phys.)g(D,)h(28)f(\(1987\),)h(pp.)f (282\226304.)0 2855 y([10])51 b(Z)t(.)34 b(G)t Fb(R)q(U)t(J)t(I)609 2847 y Ft(\264)599 2855 y Fb(C)j(A)t(N)t(D)c Ft(I)t(.)h(K)t Fb(U)t(K)t(A)-7 b(V)t(I)t(C)t(A)r Ft(,)34 b Fh(Space)29 b(analyticity)d(for)i(the)g(Navier)n(-Stok)o(es)g(and)f(r)l(elated)h (equa-)215 2975 y(tions)23 b(with)i(initial)e(data)h(in)g Fs(L)1263 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b(and)i(upper)n(-)215 5142 y(semicontinuity)20 b(for)i(a)g(\002nite)g(dif)n(fer)l(ence)g(appr)l (oximation)e(to)i(the)g(comple)n(x)g(Ginzb)n(ur)l(g-Landau)f(equa-)215 5262 y(tion)p Ft(,)j(Num.)g(Func.)h(Anal.)f(Opt.,)g(16)h(\(1995\),)f (pp.)h(1003\2261049.)0 5463 y([17])51 b(Y)-9 b(.)31 b(M)t Fb(E)t(Y)t(E)t(R)r Ft(,)d Fh(W)-9 b(avelets)24 b(and)g(oper)o(ator)o(s) p Ft(,)f(Cambridge)i(Uni)n(v)o(ersity)d(Press,)k(Cambridge,)e(1992.)p eop %%Page: 23 24 23 23 bop 0 100 a Fv(Lar)o(ge)25 b(V)-10 b(olume)25 b(Limits)g(of)g (Discr)n(etised)g(P)o(arabolic)g(Equations)1332 b(23)0 407 y Ft([18])51 b(A)t(.)38 b(M)t Fb(I)t(E)t(L)t(K)t(E)r Ft(,)f Fh(The)c(comple)n(x)f(Ginzb)n(ur)l(g-Landau)f(equation)g(on)h (lar)l(g)o(e)g(and)g(unbounded)f(domains:)215 527 y(sharper)24 b(bounds)f(and)i(attr)o(actor)o(s)p Ft(,)d(Nonlinearity)-6 b(,)23 b(10)h(\(1997\),)h(pp.)f(199\226222.)0 731 y([19])51 b(A)t(.)d(M)t Fb(I)t(E)t(L)t(K)t(E)i(A)t(N)t(D)e Ft(G)t(.)h(S)t Fb(C)t(H)t(N)t(E)t(I)t(D)t(E)t(R)s Ft(,)k 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Ft(,)h(SIAM)g(J.)h(Math.)f(Anal.,)h(27)f(\(1996\),)h(pp.)f(424\226448.) 0 2877 y([26])51 b(P)-7 b(.)49 b(T)l Fb(A)t(K)532 2869 y Ft(\264)520 2877 y Fb(A)592 2869 y Ft(\020)582 2877 y Fb(C)h(A)t(N)t(D)e Ft(A)t(.)h(J)1120 2869 y(\250)1108 2877 y Fb(U)t(N)t(G)t(E)t(L)r Ft(,)h Fh(A)43 b(nonstif)n(f)e(Euler)i (discr)l(etization)d(of)j(the)f(comple)n(x)h(Ginzb)n(ur)l(g-)215 2997 y(Landau)24 b(equation)g(in)g(one)h(space)g(dimension)p Ft(,)e(SIAM)i(J.)g(Numer)-5 b(.)24 b(Anal.,)g(38)g(\(2000\),)h(pp.)f (292\226328.)0 3200 y([27])51 b(R)t(.)38 b(T)t Fb(E)t(M)t(A)t(M)r Ft(,)f Fh(In\002nite-dimensional)29 b(dynamical)i(systems)g(in)h(mec)o (hanics)f(and)g(physics)p Ft(,)i(Springer)n(-)215 3321 y(V)-11 b(erlag,)25 b(Ne)n(w)f(Y)-11 b(ork,)25 b(second)f(ed.,)h(1997.) 0 3524 y([28])51 b(V)-9 b(.)33 b(M)t(.)g(T)t Fb(I)t(K)t(H)t(O)t(M)t(I)t (R)q(OV)r Ft(,)h Fh(On)28 b(the)f Fs(")p Fh(-entr)l(opy)f(of)h(some)h (classes)e(of)i(analytic)e(functions)p Ft(,)h(Dokl.)f(Akad.)215 3645 y(Nauk)e(SSSR)j(\(N.S.\),)e(117)g(\(1957\),)f(pp.)g(191\226194.)0 3848 y([29])51 b(Y)-9 b(.)39 b(Y)-7 b Fb(A)t(N)r Ft(,)36 b Fh(Dimensions)c(of)g(attr)o(actor)o(s)e(for)i(discr)l(etizations)f (for)h(Navier)n(-Stok)o(es)g(equations)p Ft(,)i(J.)e(Dy-)215 3968 y(nam.)24 b(Dif)n(ferential)g(Equations,)g(4)g(\(1992\),)h(pp.)f (275\226340.)0 4172 y([30])51 b(S)t(.)33 b(V)-9 b(.)33 b(Z)t Fb(E)t(L)t(I)t(K)r Ft(,)d Fh(An)c(attr)o(actor)f(of)h(a)h (nonlinear)e(system)i(of)f(r)l(eaction-dif)n(fusion)f(equations)g(in)h Fa(R)3602 4132 y Fr(n)3676 4172 y Fh(and)215 4292 y(estimates)d(for)i (its)f Fs(")p Fh(-entr)l(opy)p Ft(,)f(Mat.)i(Zametki,)e(65)i(\(1999\),) f(pp.)h(941\226944.)0 4496 y([31])p 215 4483 191 4 v 240 w(,)f Fh(The)g(attr)o(actor)e(of)h(a)h(quasilinear)e(hyperbolic)h (equation)f(with)i(dissipation)d(in)i Fa(R)3305 4456 y Fr(n)3352 4496 y Fh(:)30 b(dimension)215 4616 y(and)24 b Fs(")p Fh(-entr)l(opy)p Ft(,)g(Mat.)g(Zametki,)g(67)g(\(2000\),)h (pp.)f(304\226308.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0107191013608--