Content-Type: multipart/mixed; boundary="-------------0003290205499" This is a multi-part message in MIME format. ---------------0003290205499 Content-Type: text/plain; name="00-133.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-133.keywords" Stochasticity thresholds, FPU, energy equipartition, multitime, Nonlinear Schrodinger equation. ---------------0003290205499 Content-Type: application/postscript; name="ponno1.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ponno1.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: paper1.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips paper1.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.02.05:1936 %%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: special.pro %! 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y(mo)s(des)k(cen)m(tered)j(ab)s(out)d Fj(q)k Fp(with)d(a)f(width)g(of)g(size)1968 450 y Ff(p)p 2051 450 V 84 x Fj(R)q Fp(.)59 b(Since)37 b(w)m(e)i(are)f(taking)3094 450 y Ff(p)p 3177 450 V 84 x Fj(R)f Ff(\034)f Fp(1,)i(this)g(is)f(a)0 700 y(narro)m(w)c(\(or)f(quasi-mono)s(cromatic\))d(w)m(a)m(v)m(e)35 b(pac)m(k)m(et.)146 866 y(No)m(w,)f(substituting)f(the)h(ansatz)f (\(9\))g(in)m(to)f(the)i(equations)f(of)g(motion)e(\(7\))i(and)g(pro)s (ceeding)g(up)h(to)0 1032 y(order)39 b Fj(M)49 b Fp(=)38 b(2)g(as)h(sk)m(etc)m(hed)j(in)c(the)h(app)s(endix,)i(one)e(\014nds)g (that)g(the)g(zero-order)g(amplitude)e(of)h(the)0 1198 y(fundamen)m(tal)e(harmonic)g Fj( )k Ff(\021)d Fj( )1285 1147 y Fo(\(1\))1281 1220 y(0)1415 1198 y Fp(=)f Fj( )1594 1162 y Fo(\(1\))1688 1198 y Fp(\()p Fj(\020)d Ff(\000)1915 1150 y Fi(d!)1995 1158 y Fa(q)p 1915 1175 115 4 v 1937 1232 a Fi(dq)2039 1198 y Fj(\034)2081 1213 y Fo(1)2121 1198 y Fj(;)17 b(\034)2207 1213 y Fo(2)2246 1198 y Fp(;)g(0\))37 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b(reasons)g(wh)m(y)h(this)e(is)g(actually)f(exp)s(ected)j(are) e(explained)g(at)g(the)0 249 y(end)43 b(of)e(the)h(presen)m(t)i (section.)71 b(So)42 b(w)m(e)h(tak)m(e)f(the)h(lo)m(w)m(est)f(energy)h (violating)c(condition)h(\(13\))h(as)i(our)0 415 y(analytical)30 b(estimate)i(of)g(the)h(threshold.)146 581 y(W)-8 b(e)28 b(no)m(w)h(form)m(ulate)d(the)i(threshold)f(condition)g(in)g(terms)g (of)g(the)h(harmonic)f(energy)h(of)f(the)h(cen)m(tral)0 747 y(normal)40 b(mo)s(de)h(for)g(a)h(w)m(a)m(v)m(e)i(pac)m(k)m(et)f (giv)m(en)f(b)m(y)h(\(9\))e(at)h Fj(\034)55 b Fp(=)43 b(0.)71 b(T)-8 b(o)42 b(this)f(end,)k(w)m(e)e(remark)f(that)f(b)m(y)0 913 y(lo)s(oking)30 b(at)j(the)g(F)-8 b(ourier)31 b(transform)h(\(10\)) g(one)g(easily)g(\014nds)638 1170 y Fj(E)710 1185 y Fi(q)748 1170 y Fp(\()p Fj(\034)39 b Fp(=)28 b(0;)1064 1081 y Ff(p)p 1146 1081 76 4 v 1146 1170 a Fj(R)q Fp(\))g Ff(\021)1402 1102 y Fp(1)p 1402 1147 49 4 v 1402 1238 a(2)1477 995 y Fg(8)1477 1070 y(<)1477 1220 y(:)1551 1020 y(\014)1551 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b(sho)m(ws)i(that)e(solitons)f(con)m(tribute)0 5130 y(to)27 b(energy)i(tranfer)f(to)f(normal)f(mo)s(des)h(initially)d (not)j(excited.)43 b(When)28 b Fj(E)2720 5145 y Fi(q)2759 5130 y Fp(\(0\))f Ff(\035)g Fj(E)3116 5094 y Fi(c)3110 5155 y(q)3179 5130 y Fp(w)m(e)h(exp)s(ect)i(suc)m(h)f(a)0 5296 y(mec)m(hanism)h(to)g(b)s(ecome)h(v)m(ery)h(e\016cien)m(t)f(and)g (a)f(strong)g(in)m(teraction)g(b)s(et)m(w)m(een)j(normal)28 b(mo)s(des)i(to)h(tak)m(e)0 5462 y(place,)h(leading)f(to)i(an)f(in)m (tense)i(spreading)e(of)g(the)h(w)m(a)m(v)m(e)i(pac)m(k)m(et.)1918 5711 y(7)p eop %%Page: 8 8 8 7 bop 582 83 a Fk(I)s(I)s(I.)34 b(COMP)-9 b(ARISON)34 b(WITH)g(THE)g(NUMERICAL)h(RESUL)-9 b(TS)146 362 y Fp(W)h(e)37 b(concen)m(trate)g(our)f(atten)m(tion)g(on)f(the)i(n)m(umerical)e(w)m (orks)i([4])f(and)g([3].)54 b(In)36 b(the)h(latter)e(one)h(the)0 528 y Fj(\014)51 b Fp(mo)s(del)43 b(\(with)i Fj(\013)50 b Fp(=)e(0,)g(i.e.)81 b Fj(\026)49 b Fp(=)g(0\))44 b(is)h(studied,)k 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b(of)f(the)i(parameters)e(c)m(hosen)j(in)d([5])h(w)m(ere)h (suc)m(h)g(that)f Fj(\021)g Fp(=)d(4)p Fj(\017=m)3324 5260 y Fo(2)3391 5296 y Fp(=)f(40.)43 b(F)-8 b(rom)0 5462 y(\014gure)27 b(4)f(one)h(sees)h(that)e(the)g(agreemen)m(t)h(is)f (v)m(ery)i(go)s(o)s(d)d(for)h(the)h(high)e(frequencies,)30 b(while)25 b(the)i(situation)1918 5711 y(9)p eop %%Page: 10 10 10 9 bop 0 83 a Fp(concerning)47 b(the)h(lo)m(w)e(frequencies,)52 b(b)s(elo)m(w)47 b(the)h(cuto\013)f Fj(!)2242 98 y Fi(c)2276 83 y Fp(,)k(is)46 b(not)h(so)h(clear.)86 b(F)-8 b(urther)47 b(n)m(umerical)0 249 y(in)m(v)m(estigations)31 b(migh)m(t)f(clarify)g (this)h(p)s(oin)m(t.)42 b(In)32 b(an)m(y)g(case,)h(the)f(claim)d(that)i (some)h(relev)-5 b(an)m(t)31 b(role)g(migh)m(t)0 415 y(b)s(e)26 b(pla)m(y)m(ed)g(b)m(y)h(the)f(cuto\013)g(frequency)i Fj(!)1478 430 y Fi(c)1538 415 y Fp(for)d(the)h Fj(')1905 379 y Fo(4)1970 415 y Fp(mo)s(del)e(seems)j(to)e(b)s(e)h(supp)s(orted)h (b)m(y)g(the)f(results)g(of)0 581 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setfont 1700 820 m gsave 1700 820 translate 0 rotate 0 -30 m (m) show grestore newpath /Symbol findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 120 scalefont setfont /Symbol findfont 120 scalefont setfont 1169 195 m gsave 1169 195 translate 0 rotate 0 -40 m (w/W) show grestore newpath /Symbol findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 120 scalefont setfont /Times-Bold-ISOLatin1 findfont 120 scalefont setfont 216 915 m gsave 216 915 translate 90 rotate 0 -40 m (E) show /Times-Bold-ISOLatin1 findfont 71 scalefont setfont 0 71 rmoveto (c) show /Times-Bold-ISOLatin1 findfont 120 scalefont setfont 0 -71 rmoveto (/E) show /Times-Bold-ISOLatin1 findfont 71 scalefont setfont 0 71 rmoveto (c) show /Times-Bold-ISOLatin1 findfont 120 scalefont setfont 0 -71 rmoveto /Times-Bold-ISOLatin1 findfont 71 scalefont setfont 0 -35 rmoveto (max) show grestore newpath /Times-Bold-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 90 scalefont setfont /Times-Bold-ISOLatin1 findfont 90 scalefont setfont 493 1685 m gsave 493 1685 translate 0 rotate 0 -30 m (\(FPU\)) show grestore newpath /Times-Bold-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont end showpage %%Trailer %%EndDocument @endspecial 136 1921 a Fm(FIG.)43 b(1.)74 b(FPU)55 b(mo)s(del;)65 b(normalized)53 b(energy)h(threshold)f(vs.)112 b Fl(!)s(=)p Fm(\012)54 b(for)g(four)f(di\013eren)m(t)h(v)-5 b(alues)53 b(of)0 2120 y Fl(\026)25 b Fm(=)g(4)p Fl(\013)279 2087 y Fo(2)319 2120 y Fl(=)p Fm(\()p Fl(\014)5 b Fm(\012)521 2087 y Fo(2)561 2120 y Fm(\).)762 4004 y @beginspecial 24 @llx 40 @lly 511 @urx 434 @ury 2834 @rwi 1984 @rhi @setspecial %%BeginDocument: fpucomp.eps %!PS-Adobe-2.0 EPSF-1.2 %%BoundingBox: 24 40 511 434 %%Title: fpucomp.eps %%Creator: ponno@pcdotto6.pd.infn.it with xmgr %%CreationDate: Mon Sep 20 14:56:44 1999 %%EndComments 80 dict begin /languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix makepattern /Pat0 exch def <} image end } >> matrix makepattern /Pat1 exch def <} image end } >> matrix makepattern /Pat2 exch def <} image end } >> matrix makepattern /Pat3 exch def <} image end } >> matrix makepattern /Pat4 exch def <} image end } >> matrix makepattern /Pat5 exch def <} image end } >> matrix makepattern /Pat6 exch def <} image end } >> matrix makepattern /Pat7 exch def <} image end } >> matrix makepattern /Pat8 exch def <} image end } >> matrix makepattern /Pat9 exch def <} image end } >> matrix makepattern /Pat10 exch def <} image end } >> matrix makepattern /Pat11 exch def <} image end } >> matrix makepattern /Pat12 exch def <} image end } >> matrix makepattern /Pat13 exch def <} image end } >> matrix makepattern /Pat14 exch def <} image end } >> matrix 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closepath } bind def /dsq { sq s } bind def /fsq { sq f } bind def /osq { sq o } bind def % Triangle symbols /t1 { moveto dup 0 exch rmoveto dup neg dup 2 mul rlineto 2 mul 0 rlineto closepath } bind def /dt1 { t1 s } bind def /ft1 { t1 f } bind def /ot1 { t1 o } bind def /t2 { moveto dup neg 0 rmoveto dup dup 2 mul exch neg rlineto 2 mul 0 exch rlineto closepath } bind def /dt2 { t2 s } bind def /ft2 { t2 f } bind def /ot2 { t2 o } bind def /t3 { moveto dup neg 0 exch rmoveto dup dup 2 mul rlineto neg 2 mul 0 rlineto closepath } bind def /dt3 { t3 s } bind def /ft3 { t3 f } bind def /ot3 { t3 o } bind def /t4 { moveto dup 0 rmoveto dup dup -2 mul exch rlineto -2 mul 0 exch rlineto closepath } bind def /dt4 { t4 s } bind def /ft4 { t4 f } bind def /ot4 { t4 o } bind def % Diamond symbol /di { moveto dup 0 exch rmoveto dup neg dup rlineto dup dup neg rlineto dup dup rlineto closepath } bind def /ddi { di s } bind def /fdi { di f } bind def /odi { di o } bind def % Plus symbol /pl { dup 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%%Title: fi4sogliae.eps %%Creator: ponno@pcdotto6.pd.infn.it with xmgr %%CreationDate: Sun Sep 19 23:37:44 1999 %%EndComments 80 dict begin /languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix makepattern /Pat0 exch def <} image end } >> matrix makepattern /Pat1 exch def <} image end } >> matrix makepattern /Pat2 exch def <} image end } >> matrix makepattern /Pat3 exch def <} image end } >> matrix makepattern /Pat4 exch def <} image end } >> matrix makepattern /Pat5 exch def <} image end } >> matrix makepattern /Pat6 exch def <} image end } >> matrix makepattern /Pat7 exch def <} image end } >> matrix makepattern /Pat8 exch def <} image end } >> matrix makepattern /Pat9 exch def <} image end } >> matrix makepattern /Pat10 exch def <} image end } >> matrix makepattern /Pat11 exch def <} image end } >> matrix makepattern /Pat12 exch def <} image end } >> matrix makepattern /Pat13 exch def <} image 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rmoveto 2 mul dup neg 0 rlineto dup neg 0 exch rlineto 0 rlineto closepath } bind def /dsq { sq s } bind def /fsq { sq f } bind def /osq { sq o } bind def % Triangle symbols /t1 { moveto dup 0 exch rmoveto dup neg dup 2 mul rlineto 2 mul 0 rlineto closepath } bind def /dt1 { t1 s } bind def /ft1 { t1 f } bind def /ot1 { t1 o } bind def /t2 { moveto dup neg 0 rmoveto dup dup 2 mul exch neg rlineto 2 mul 0 exch rlineto closepath } bind def /dt2 { t2 s } bind def /ft2 { t2 f } bind def /ot2 { t2 o } bind def /t3 { moveto dup neg 0 exch rmoveto dup dup 2 mul rlineto neg 2 mul 0 rlineto closepath } bind def /dt3 { t3 s } bind def /ft3 { t3 f } bind def /ot3 { t3 o } bind def /t4 { moveto dup 0 rmoveto dup dup -2 mul exch rlineto -2 mul 0 exch rlineto closepath } bind def /dt4 { t4 s } bind def /ft4 { t4 f } bind def /ot4 { t4 o } bind def % Diamond symbol /di { moveto dup 0 exch rmoveto dup neg dup rlineto dup dup neg rlineto dup dup rlineto closepath } bind def /ddi { di s } bind def /fdi { di f } bind def /odi { di o } bind def % Plus symbol /pl { dup 0 rmoveto dup -2 mul 0 rlineto dup dup rmoveto -2 mul 0 exch rlineto } bind def /dpl { m pl s } bind def % x symbol /x { dup dup rmoveto dup -2 mul dup rlineto 2 mul dup 0 rmoveto dup neg exch rlineto } bind def /dx { m x s } bind def % Splat symbol /dsp { m dup pl dup 0 exch rmoveto 0.707 mul x s } bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.24 0.24 scale 1 setlinecap mark /ISOLatin1Encoding 8#000 1 8#054 {StandardEncoding exch get} for /minus 8#056 1 8#217 {StandardEncoding exch get} for /dotlessi 8#301 1 8#317 {StandardEncoding exch get} for /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis /ISOLatin1Encoding where not {256 array astore def} if cleartomark /makeISOEncoded { findfont /curfont exch def /newfont curfont maxlength dict def /ISOLatin1 (-ISOLatin1) def /curfontname curfont /FontName get dup length string cvs def /newfontname curfontname length ISOLatin1 length add string dup 0 curfontname putinterval dup curfontname length ISOLatin1 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