%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%POSTSCRIPT-FILE gzCPT.ps %%%%%%%%%%%%%%%%% %!PS-Adobe-2.0 %%Creator: dvips 5.526 Copyright 1986, 1994 Radical Eye Software %%Title: gzCPT.dvi %%CreationDate: Wed Feb 25 12:00:56 1998 %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /sw/tex/bin/Dvips gzCPT %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1998.02.25:1200 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B 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Fu(g)j FB(gro)o(ws)e(faster)f(than)h(quadratically)l(.\))14 2465 y Fs(Scenario)e(A)p FB(.)d Fv(While)j(the)g(ferr)n(omagnetic)g (phase)g(tr)n(ansition)f(at)h Fu(z)g FB(=)f Fu(z)1364 2472 y FA(m)1397 2465 y FB(\()p Fu(\014)s FB(\))g Fv(is)g(only)h(of)f (se)n(c)n(ond)h(or)n(der)-59 2537 y(for)i(smal)r(l)h Fu(\014)s Fv(,)f(it)h(is)f(of)g(\014rst)h(or)n(der)e(when)i Fu(\014)i Fv(is)d(su\016ciently)i(lar)n(ge)p FB(.)14 2615 y(This)12 b(scenario)f(o)q(ccurs)h(if)f Fu(g)514 2597 y FF(00)547 2615 y FB(is)g(increasing)g(with)g(0)j Ft(\024)g FB(2)8 b Fu(g)1068 2597 y FF(00)1090 2615 y FB(\(0\))14 b Fu(<)g(J)5 b FB(;)12 b(cf.)f(Prop)q(osition)h(3.2)g(and)g (Theorem)-59 2688 y(3.3)17 b(b)q(elo)o(w.)k(The)16 b(simplest)e (example)g(is)i Fu(g)r FB(\()p Fu(\032)p FB(\))e(=)g Fu(c)8 b(\032)915 2670 y FC(3)952 2688 y FB(with)16 b Fu(c)d(>)h FB(0;)i(see)g(Figure)g(1.)933 2817 y(2)p eop %%Page: 3 4 3 3 bop -59 776 a @beginspecial -75 @llx 239 @lly 694 @urx 562 @ury 4818 @rwi @setspecial %%BeginDocument: CWfluid/scA2.ps %AI5_FileFormat 2.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 0 0 552 728 %AI3_DocumentPreview: Header %AI5_ArtSize: 842 595 %AI5_RulerUnits: 1 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: -222 756 1 1018 725 18 0 0 3 40 %AI5_OpenViewLayers: 7 userdict /Adobe_level2_AI5 23 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 17 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 14 dict def } if Adobe_ColorImage_AI6_Vars begin /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /plateindex -1 def /XIMask 0 def /XIBinary 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIBuffer null def /XIDataProc null def end /WalkRGBString null def /WalkCMYKString null def /StuffRGBIntoGrayString null def /RGBToGrayImageProc null def /StuffCMYKIntoGrayString null def /CMYKToGrayImageProc null def /ColorImageCompositeEmulator null def /SeparateCMYKImageProc null def /FourEqual null def /TestPlateIndex null def currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def } { /_colorimage null def } ifelse } if /_currenttransfer systemdict /currenttransfer get def /colorimage null def /XI null def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval { } forall 5 index exec 3 index } for 5 { pop } repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval { } forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat } { Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq { /RGBToGrayImageProc } { /CMYKToGrayImageProc } ifelse load end } if image } ifelse } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec } { sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /FourEqual { 4 index ne { pop pop pop false } { 4 index ne { pop pop false } { 4 index ne { pop false } { 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def } { 0 1 0 0 FourEqual { /plateindex 1 def } { 0 0 1 0 FourEqual { /plateindex 2 def } { 0 0 0 1 FourEqual { /plateindex 3 def } { 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index dup 8 eq exch 1 eq or not end { /_colorimage load null ne { _colorimage } { Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse } { dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator } { dup 1 eq { pop pop image } { Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse } { dup 1 eq { pop pop image } { pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIMask exch 0 ne def /XIBinary exch 0 ne def pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi } { XIImageWidth XIChannelCount mul } ifelse /XIBuffer exch string def XIBinary { /XIDataProc { currentfile XIBuffer readstring pop } def currentfile 128 string readline pop pop } { /XIDataProc { currentfile XIBuffer readhexstring pop } def } ifelse 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIMask { XIImageWidth XIImageHeight false [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load /_lp /null ddef _fc /_lp /imagemask ddef imagemask } { XIImageWidth XIImageHeight XIBitsPerPixel [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load XIChannelCount 1 eq { gsave 0 setgray image grestore } { false XIChannelCount colorimage } ifelse } ifelse grestore end } def end currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla 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216.25 297 L S 216.25 297 m 216 297 L S 216 297 m 215.75 297.25 L S 215.75 297.25 m 215.5 297.25 L S 215.5 297.25 m 215.25 297.25 L S 215.25 297.25 m 215 297.5 L S 215 297.5 m 214.75 297.5 L S 214.75 297.5 m 214.5 297.5 L S 214.5 297.5 m 214.25 297.5 L S 214.25 297.5 m 214 297.75 L S 214 297.75 m 213.75 297.75 L S 213.75 297.75 m 213.75 297.75 L S 213.75 297.75 m 213.5 297.75 L S 213.5 297.75 m 213.25 298 L S 213.25 298 m 213 298 L S 207 300 m 206.75 300 L S 206.75 300 m 206.5 300 L S 206.5 300 m 206.25 300.25 L S 206.25 300.25 m 206 300.25 L S 206 300.25 m 205.75 300.25 L S 205.75 300.25 m 205.5 300.5 L S 205.5 300.5 m 205.25 300.5 L S 205.25 300.5 m 205 300.5 L S 205 300.5 m 204.75 300.75 L S 204.75 300.75 m 204.5 300.75 L S 204.5 300.75 m 204.25 300.75 L S 204.25 300.75 m 204 300.75 L S 204 300.75 m 203.75 301 L S 203.75 301 m 203.5 301 L S 203.5 301 m 203.25 301 L S 203.25 301 m 203 301.25 L S 203 301.25 m 202.75 301.25 L S 202.75 301.25 m 202.5 301.25 L S 202.5 301.25 m 202.25 301.5 L S 202.25 301.5 m 202 301.5 L S 202 301.5 m 201.75 301.5 L S 201.75 301.5 m 201.5 301.75 L S 201.5 301.75 m 201.25 301.75 L S 201.25 301.75 m 201 301.75 L S 195 304 m 194.75 304.25 L S 194.75 304.25 m 194.5 304.25 L S 194.5 304.25 m 194.25 304.25 L S 194.25 304.25 m 194 304.5 L S 194 304.5 m 193.75 304.5 L S 193.75 304.5 m 193.5 304.5 L S 193.5 304.5 m 193.25 304.75 L S 193.25 304.75 m 193 304.75 L S 193 304.75 m 192.75 305 L S 192.75 305 m 192.5 305 L S 192.5 305 m 192.25 305 L S 192.25 305 m 192 305.25 L S 192 305.25 m 191.75 305.25 L S 191.75 305.25 m 191.5 305.25 L S 191.5 305.25 m 191.25 305.5 L S 191.25 305.5 m 191 305.5 L S 191 305.5 m 190.75 305.5 L S 190.75 305.5 m 190.5 305.75 L S 190.5 305.75 m 190.25 305.75 L S 190.25 305.75 m 190 306 L S 190 306 m 189.75 306 L S 189.75 306 m 189.5 306 L S 189.5 306 m 189.5 306 L S 189.5 306 m 189.25 306.25 L S 189.25 306.25 m 189 306.25 L S 183 308.75 m 182.75 309 L S 182.75 309 m 182.5 309 L S 182.5 309 m 182.25 309 L S 182.25 309 m 182 309.25 L S 182 309.25 m 181.75 309.25 L S 181.75 309.25 m 181.5 309.5 L S 181.5 309.5 m 181.25 309.5 L S 181.25 309.5 m 181 309.5 L S 181 309.5 m 180.75 309.75 L S 180.75 309.75 m 180.5 309.75 L S 180.5 309.75 m 180.25 310 L S 180.25 310 m 180 310 L S 180 310 m 179.75 310.25 L S 179.75 310.25 m 179.5 310.25 L S 179.5 310.25 m 179.5 310.25 L S 179.5 310.25 m 179.25 310.25 L S 179.25 310.25 m 179 310.5 L S 179 310.5 m 178.75 310.5 L S 178.75 310.5 m 178.5 310.75 L S 178.5 310.75 m 178.25 310.75 L S 178.25 310.75 m 178 311 L S 178 311 m 177.75 311 L S 177.75 311 m 177.5 311.25 L S 177.5 311.25 m 177.25 311.25 L S 177.25 311.25 m 177 311.5 L S 171 314.25 m 170.75 314.25 L S 170.75 314.25 m 170.5 314.5 L S 170.5 314.5 m 170.5 314.5 L S 170.5 314.5 m 170.25 314.5 L S 170.25 314.5 m 170 314.75 L S 170 314.75 m 169.75 314.75 L S 169.75 314.75 m 169.5 315 L S 169.5 315 m 169.25 315 L S 169.25 315 m 169 315.25 L S 169 315.25 m 168.75 315.25 L S 168.75 315.25 m 168.5 315.5 L S 168.5 315.5 m 168.25 315.5 L S 168.25 315.5 m 168 315.75 L S 168 315.75 m 167.75 315.75 L S 167.75 315.75 m 167.5 316 L S 167.5 316 m 167.25 316 L S 167.25 316 m 167 316.25 L S 167 316.25 m 166.75 316.25 L S 166.75 316.25 m 166.5 316.5 L S 166.5 316.5 m 166.25 316.75 L S 166.25 316.75 m 166 316.75 L S 166 316.75 m 165.75 317 L S 165.75 317 m 165.5 317 L S 165.5 317 m 165.25 317.25 L S 165.25 317.25 m 165 317.25 L S 159 320.5 m 158.75 320.75 L S 158.75 320.75 m 158.5 320.75 L S 158.5 320.75 m 158.25 321 L S 158.25 321 m 158 321 L S 158 321 m 157.75 321.25 L S 157.75 321.25 m 157.5 321.25 L S 157.5 321.25 m 157.25 321.5 L S 157.25 321.5 m 157 321.75 L S 157 321.75 m 156.75 321.75 L S 156.75 321.75 m 156.5 322 L S 156.5 322 m 156.25 322 L S 156.25 322 m 156 322.25 L S 156 322.25 m 155.75 322.25 L S 155.75 322.25 m 155.5 322.5 L S 155.5 322.5 m 155.25 322.5 L S 155.25 322.5 m 155 322.75 L S 155 322.75 m 155 322.75 L S 155 322.75 m 154.75 323 L S 154.75 323 m 154.5 323 L S 154.5 323 m 154.25 323.25 L S 154.25 323.25 m 154 323.25 L S 154 323.25 m 153.75 323.5 L S 153.75 323.5 m 153.5 323.75 L S 153.5 323.75 m 153.25 323.75 L S 153.25 323.75 m 153 324 L S 147 327.75 m 146.75 328 L S 146.75 328 m 146.5 328 L S 146.5 328 m 146.25 328.25 L S 146.25 328.25 m 146 328.25 L S 146 328.25 m 145.75 328.5 L S 145.75 328.5 m 145.5 328.75 L S 145.5 328.75 m 145.25 328.75 L S 145.25 328.75 m 145 329 L S 145 329 m 144.75 329.25 L S 144.75 329.25 m 144.5 329.25 L S 144.5 329.25 m 144.25 329.5 L S 144.25 329.5 m 144 329.75 L S 144 329.75 m 143.75 329.75 L S 143.75 329.75 m 143.5 330 L S 143.5 330 m 143.25 330 L S 143.25 330 m 143 330.25 L S 143 330.25 m 142.75 330.5 L S 142.75 330.5 m 142.5 330.5 L S 142.5 330.5 m 142.25 330.75 L S 142.25 330.75 m 142 331 L S 142 331 m 141.75 331 L S 141.75 331 m 141.75 331 L S 141.75 331 m 141.5 331.25 L S 141.5 331.25 m 141.25 331.5 L S 141.25 331.5 m 141 331.75 L S 135 336 m 134.75 336.25 L S 134.75 336.25 m 134.5 336.5 L S 134.5 336.5 m 134.25 336.5 L S 134.25 336.5 m 134 336.75 L S 134 336.75 m 133.75 337 L S 133.75 337 m 133.5 337.25 L S 133.5 337.25 m 133.25 337.25 L S 133.25 337.25 m 133 337.5 L S 133 337.5 m 132.75 337.75 L S 132.75 337.75 m 132.5 338 L S 132.5 338 m 132.25 338 L S 132.25 338 m 132 338.25 L S 132 338.25 m 131.75 338.5 L S 131.75 338.5 m 131.5 338.75 L S 131.5 338.75 m 131.25 338.75 L S 131.25 338.75 m 131 339 L S 131 339 m 130.75 339.25 L S 130.75 339.25 m 130.5 339.5 L S 130.5 339.5 m 130.5 339.5 L S 130.5 339.5 m 130.25 339.5 L S 130.25 339.5 m 130 339.75 L S 130 339.75 m 129.75 340 L S 129.75 340 m 129.5 340.25 L S 129.5 340.25 m 129.25 340.5 L S 129.25 340.5 m 129 340.5 L S 2.5 w 5 M 121 347.25 m 116.25 351.5 L S 116.25 351.5 m 111.75 355.5 L S 111.75 355.5 m 107.25 359.75 L S 107.25 359.75 m 103 364 L S 103 364 m 98.75 368 L S 98.75 368 m 94.75 372.25 L S 94.75 372.25 m 90.75 376.5 L S 90.75 376.5 m 87 380.5 L S 87 380.5 m 83.5 384.75 L S 83.5 384.75 m 79.5 389 L S 79.5 389 m 75.75 393 L S 75.75 393 m 72.25 397.25 L S 72.25 397.25 m 68.75 401.5 L S 68.75 401.5 m 65.25 405.5 L S 65.25 405.5 m 61.75 409.75 L S 61.75 409.75 m 58.25 414 L S 58.25 414 m 55 418 L S 55 418 m 51.5 422.25 L S 51.5 422.25 m 48.25 426.5 L S 48.25 426.5 m 45 430.5 L S 45 430.5 m 41.75 434.75 L S 41.75 434.75 m 38.5 439 L S 38.5 439 m 35.25 443 L S 35.25 443 m 32 447.25 L S 32 447.25 m 28.75 451.5 L S 28.75 451.5 m 25.75 455.5 L S 25.75 455.5 m 22.5 459.75 L S 22.5 459.75 m 19.25 464 L S 19.25 464 m 16.25 468 L S 16.25 468 m 13 472.25 L S 13 472.25 m 9.75 476.5 L S 9.75 476.5 m 6.75 480.5 L S 6.75 480.5 m 3.75 484.75 L S 3.75 484.75 m 0.5 489 L S 0.5 489 m -2.5 493 L S -2.5 493 m -5.75 497.25 L S -5.75 497.25 m -8.75 501.5 L S -8.75 501.5 m -11.75 505.5 L S -11.75 505.5 m -15 509.75 L S 0.75 w 1.5 M 342.875 410.375 m 662.875 410.375 L S 342.875 410.375 m 342.875 535.375 L S 342.875 260.375 m 662.875 260.375 L S 342.875 260.375 m 342.875 335.375 L S 339.875 421.625 m 345.875 421.625 L S 339.875 433.125 m 345.875 433.125 L S 339.875 444.375 m 345.875 444.375 L S 339.875 455.875 m 345.875 455.875 L S 339.875 467.125 m 345.875 467.125 L S 339.875 478.625 m 345.875 478.625 L S 339.875 489.875 m 345.875 489.875 L S 339.875 501.375 m 345.875 501.375 L S 339.875 512.625 m 345.875 512.625 L S 339.875 524.125 m 345.875 524.125 L S 339.875 535.375 m 345.875 535.375 L S 339.875 335.375 m 345.875 335.375 L S 0 To 1 0 0 1 322 530.75 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (11) Tx (\r) TX TO 0 To 1 0 0 1 330 330.75 0 Tp TP 0 Tr (1) Tx (\r) TX TO 0 To 1 0 0 1 338.25 547.75 0 Tp TP 0 Tr /_Symbol 14 Tf (r) Tx (\r) TX TO 0 To 1 0 0 1 667.5 406.25 0 Tp TP 0 Tr /_Helvetica 14 Tf (ln ) Tx (\r) TX TO 0 To 1 0 0 1 686.75 406.25 0 Tp TP 0 Tr (z) Tx (\r) TX TO 0 To 1 0 0 1 667.5 256.25 0 Tp TP 0 Tr (ln ) Tx (\r) TX TO 0 To 1 0 0 1 686.75 256.25 0 Tp TP 0 Tr (z) Tx (\r) TX TO 0 To 1 0 0 1 335.75 347.75 0 Tp TP 0 Tr (m) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 342.875 407.375 m 342.875 413.375 L S 342.875 257.375 m 342.875 263.375 L S 0 To 1 0 0 1 336.25 391.75 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M /_Helvetica 12 Tf (-1) Tx (\r) TX TO 0 To 1 0 0 1 336.25 241.75 0 Tp TP 0 Tr (-1) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 422.875 407.375 m 422.875 413.375 L S 422.875 257.375 m 422.875 263.375 L S 0 To 1 0 0 1 418.5 391.75 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (0) Tx (\r) TX TO 0 To 1 0 0 1 418.5 241.75 0 Tp TP 0 Tr (0) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 502.875 407.375 m 502.875 413.375 L S 502.875 257.375 m 502.875 263.375 L S 0 To 1 0 0 1 498.5 391.75 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (1) Tx (\r) TX TO 0 To 1 0 0 1 498.5 241.75 0 Tp TP 0 Tr (1) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 582.875 407.375 m 582.875 413.375 L S 582.875 257.375 m 582.875 263.375 L S 0 To 1 0 0 1 578.5 391.75 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (2) Tx (\r) TX TO 0 To 1 0 0 1 578.5 241.75 0 Tp TP 0 Tr (2) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 662.875 407.375 m 662.875 413.375 L S 662.875 257.375 m 662.875 263.375 L S 0 To 1 0 0 1 658.5 391.75 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (3) Tx (\r) TX TO 0 To 1 0 0 1 658.5 241.75 0 Tp TP 0 Tr (3) Tx (\r) TX TO 0 R 0 G 2 J 1.5 w 3 M 343.25 415.25 m 344 415.25 L S 344 415.25 m 344.75 415.25 L S 344.75 415.25 m 345.75 415.25 L S 345.75 415.25 m 346.5 415.25 L S 346.5 415.25 m 347.25 415.25 L S 347.25 415.25 m 348 415.5 L S 348 415.5 m 348.75 415.5 L S 348.75 415.5 m 349.75 415.5 L S 349.75 415.5 m 350.5 415.5 L S 350.5 415.5 m 351.25 415.5 L S 351.25 415.5 m 352 415.75 L S 352 415.75 m 352.75 415.75 L S 352.75 415.75 m 353.75 415.75 L S 353.75 415.75 m 354.5 415.75 L S 354.5 415.75 m 355.25 415.75 L S 355.25 415.75 m 356 415.75 L S 356 415.75 m 356.75 416 L S 356.75 416 m 357.75 416 L S 357.75 416 m 358.5 416 L S 358.5 416 m 359.25 416 L S 359.25 416 m 360 416 L S 360 416 m 360.75 416.25 L S 360.75 416.25 m 361.75 416.25 L S 361.75 416.25 m 362.5 416.25 L S 362.5 416.25 m 363.25 416.25 L S 363.25 416.25 m 364 416.25 L S 364 416.25 m 364.75 416.5 L S 364.75 416.5 m 365.75 416.5 L S 365.75 416.5 m 366.5 416.5 L S 366.5 416.5 m 367.25 416.5 L S 367.25 416.5 m 368 416.75 L S 368 416.75 m 368.75 416.75 L S 368.75 416.75 m 369.75 416.75 L S 369.75 416.75 m 370.5 416.75 L S 370.5 416.75 m 371.25 416.75 L S 371.25 416.75 m 372 417 L S 372 417 m 372.75 417 L S 372.75 417 m 373.75 417 L S 373.75 417 m 374.5 417 L S 374.5 417 m 375.25 417.25 L S 375.25 417.25 m 376 417.25 L S 376 417.25 m 376.75 417.25 L S 376.75 417.25 m 377.75 417.25 L S 377.75 417.25 m 378.5 417.5 L S 378.5 417.5 m 379.25 417.5 L S 379.25 417.5 m 380 417.5 L S 380 417.5 m 380.75 417.5 L S 380.75 417.5 m 381.75 417.75 L S 381.75 417.75 m 382.5 417.75 L S 382.5 417.75 m 383.25 417.75 L S 383.25 417.75 m 384 417.75 L S 384 417.75 m 384.75 418 L S 384.75 418 m 385.75 418 L S 385.75 418 m 386.5 418 L S 386.5 418 m 387.25 418 L S 387.25 418 m 388 418.25 L S 388 418.25 m 388.75 418.25 L S 388.75 418.25 m 389.75 418.25 L S 389.75 418.25 m 390.5 418.5 L S 390.5 418.5 m 391.25 418.5 L S 391.25 418.5 m 392 418.5 L S 392 418.5 m 392.75 418.5 L S 392.75 418.5 m 393.75 418.75 L S 393.75 418.75 m 394.5 418.75 L S 394.5 418.75 m 395.25 418.75 L S 395.25 418.75 m 396 419 L S 396 419 m 396.75 419 L S 396.75 419 m 397.75 419 L S 397.75 419 m 398.5 419 L S 398.5 419 m 399.25 419.25 L S 399.25 419.25 m 400 419.25 L S 400 419.25 m 400.75 419.25 L S 400.75 419.25 m 401.75 419.5 L S 401.75 419.5 m 402.5 419.5 L S 402.5 419.5 m 403.25 419.5 L S 403.25 419.5 m 404 419.75 L S 404 419.75 m 404.75 419.75 L S 404.75 419.75 m 405.75 419.75 L S 405.75 419.75 m 406.5 420 L S 406.5 420 m 407.25 420 L S 407.25 420 m 408 420 L S 408 420 m 408.75 420.25 L S 408.75 420.25 m 409.75 420.25 L S 409.75 420.25 m 410.5 420.25 L S 410.5 420.25 m 411.25 420.5 L S 411.25 420.5 m 412 420.5 L S 412 420.5 m 412.75 420.75 L S 412.75 420.75 m 413.75 420.75 L S 413.75 420.75 m 414.5 420.75 L S 414.5 420.75 m 415.25 421 L S 415.25 421 m 416 421 L S 416 421 m 416.75 421 L S 416.75 421 m 417.75 421.25 L S 417.75 421.25 m 418.5 421.25 L S 418.5 421.25 m 419.25 421.25 L S 419.25 421.25 m 420 421.5 L S 420 421.5 m 420.75 421.5 L S 420.75 421.5 m 421.75 421.75 L S 421.75 421.75 m 422.5 421.75 L S 422.5 421.75 m 423.25 421.75 L S 423.25 421.75 m 424 422 L S 424 422 m 424.75 422 L S 424.75 422 m 425.75 422.25 L S 425.75 422.25 m 426.5 422.25 L S 426.5 422.25 m 427.25 422.25 L S 427.25 422.25 m 428 422.5 L S 428 422.5 m 428.75 422.5 L S 428.75 422.5 m 429.75 422.75 L S 429.75 422.75 m 430.5 422.75 L S 430.5 422.75 m 431.25 423 L S 431.25 423 m 432 423 L S 432 423 m 432.75 423 L S 432.75 423 m 433.75 423.25 L S 433.75 423.25 m 434.5 423.25 L S 434.5 423.25 m 435.25 423.5 L S 435.25 423.5 m 436 423.5 L S 436 423.5 m 436.75 423.75 L S 436.75 423.75 m 437.75 423.75 L S 437.75 423.75 m 438.5 424 L S 438.5 424 m 439.25 424 L S 439.25 424 m 440 424 L S 440 424 m 440.75 424.25 L S 440.75 424.25 m 441.75 424.25 L S 441.75 424.25 m 442.5 424.5 L S 442.5 424.5 m 443.25 424.5 L S 443.25 424.5 m 444 424.75 L S 444 424.75 m 444.75 424.75 L S 444.75 424.75 m 445.75 425 L S 445.75 425 m 446.5 425 L S 446.5 425 m 447.25 425.25 L S 447.25 425.25 m 448 425.25 L S 448 425.25 m 448.75 425.5 L S 448.75 425.5 m 449.75 425.5 L S 449.75 425.5 m 450.5 425.75 L S 450.5 425.75 m 451.25 425.75 L S 451.25 425.75 m 452 426 L S 452 426 m 452.75 426 L S 452.75 426 m 453.75 426.25 L S 453.75 426.25 m 454.5 426.25 L S 454.5 426.25 m 455.25 426.5 L S 455.25 426.5 m 456 426.5 L S 456 426.5 m 456.75 426.75 L S 456.75 426.75 m 457.75 426.75 L S 457.75 426.75 m 458.5 427 L S 458.5 427 m 459.25 427 L S 459.25 427 m 460 427.25 L S 460 427.25 m 460.75 427.5 L S 460.75 427.5 m 461.75 427.5 L S 461.75 427.5 m 462.5 427.75 L S 462.5 427.75 m 463.25 427.75 L S 463.25 427.75 m 464 428 L S 464 428 m 464.75 428 L S 464.75 428 m 465.75 428.25 L S 465.75 428.25 m 466.5 428.25 L S 466.5 428.25 m 467.25 428.5 L S 467.25 428.5 m 468 428.75 L S 468 428.75 m 468.75 428.75 L S 468.75 428.75 m 469.75 429 L S 469.75 429 m 470.5 429 L S 470.5 429 m 471.25 429.25 L S 471.25 429.25 m 472 429.25 L S 472 429.25 m 472.75 429.5 L S 472.75 429.5 m 473.75 429.75 L S 473.75 429.75 m 474.5 429.75 L S 474.5 429.75 m 475.25 430 L S 475.25 430 m 476 430 L S 476 430 m 476.75 430.25 L S 476.75 430.25 m 477.75 430.5 L S 477.75 430.5 m 478.5 430.5 L S 478.5 430.5 m 479.25 430.75 L S 479.25 430.75 m 480 430.75 L S 480 430.75 m 480.75 431 L S 480.75 431 m 481.75 431.25 L S 481.75 431.25 m 482.5 431.25 L S 482.5 431.25 m 483.25 431.5 L S 483.25 431.5 m 484 431.5 L S 484 431.5 m 484.75 431.75 L S 484.75 431.75 m 485.75 432 L S 485.75 432 m 486.5 432 L S 486.5 432 m 487.25 432.25 L S 487.25 432.25 m 488 432.5 L S 488 432.5 m 488.75 432.5 L S 488.75 432.5 m 489.75 432.75 L S 489.75 432.75 m 490.5 432.75 L S 490.5 432.75 m 491.25 433 L S 491.25 433 m 492 433.25 L S 492 433.25 m 492.75 433.25 L S 492.75 433.25 m 493.75 433.5 L S 493.75 433.5 m 494.5 433.5 L S 494.5 448.25 m 495.25 450.25 L S 495.25 450.25 m 496 452 L S 496 452 m 496.75 453.5 L S 496.75 453.5 m 497.75 455 L S 497.75 455 m 498.5 456 L S 498.5 456 m 499.25 457.25 L S 499.25 457.25 m 500 458.25 L S 500 458.25 m 500.75 459.25 L S 500.75 459.25 m 501.75 460.25 L S 501.75 460.25 m 502.5 461 L S 502.5 461 m 503.25 462 L S 503.25 462 m 504 462.75 L S 504 462.75 m 504.75 463.5 L S 504.75 463.5 m 505.75 464.25 L S 505.75 464.25 m 506.5 465 L S 506.5 465 m 507.25 465.75 L S 507.25 465.75 m 508 466.5 L S 508 466.5 m 508.75 467 L S 508.75 467 m 509.75 467.75 L S 509.75 467.75 m 510.5 468.25 L S 510.5 468.25 m 511.25 469 L S 511.25 469 m 512 469.5 L S 512 469.5 m 512.75 470.25 L S 512.75 470.25 m 513.75 470.75 L S 513.75 470.75 m 514.5 471.25 L S 514.5 471.25 m 515.25 472 L S 515.25 472 m 516 472.5 L S 516 472.5 m 516.75 473 L S 516.75 473 m 517.75 473.5 L S 517.75 473.5 m 518.5 474 L S 518.5 474 m 519.25 474.5 L S 519.25 474.5 m 520 475 L S 520 475 m 520.75 475.5 L S 520.75 475.5 m 521.75 476 L S 521.75 476 m 522.5 476.5 L S 522.5 476.5 m 523.25 477 L S 523.25 477 m 524 477.5 L S 524 477.5 m 524.75 478 L S 524.75 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500.75 L S 577.75 500.75 m 578.5 501 L S 578.5 501 m 579.25 501.25 L S 579.25 501.25 m 580 501.5 L S 580 501.5 m 580.75 501.75 L S 580.75 501.75 m 581.75 502 L S 581.75 502 m 582.5 502.25 L S 582.5 502.25 m 583.25 502.5 L S 583.25 502.5 m 584 502.75 L S 584 502.75 m 584.75 503 L S 584.75 503 m 585.75 503.25 L S 585.75 503.25 m 586.5 503.75 L S 586.5 503.75 m 587.25 504 L S 587.25 504 m 588 504.25 L S 588 504.25 m 588.75 504.5 L S 588.75 504.5 m 589.75 504.75 L S 589.75 504.75 m 590.5 505 L S 590.5 505 m 591.25 505.25 L S 591.25 505.25 m 592 505.5 L S 592 505.5 m 592.75 505.75 L S 592.75 505.75 m 593.75 506 L S 593.75 506 m 594.5 506.25 L S 594.5 506.25 m 595.25 506.5 L S 595.25 506.5 m 596 506.75 L S 596 506.75 m 596.75 507 L S 596.75 507 m 597.75 507.25 L S 597.75 507.25 m 598.5 507.5 L S 598.5 507.5 m 599.25 507.75 L S 599.25 507.75 m 600 508 L S 600 508 m 600.75 508.25 L S 600.75 508.25 m 601.75 508.5 L S 601.75 508.5 m 602.5 508.5 L S 602.5 508.5 m 603.25 508.75 L S 603.25 508.75 m 604 509 L S 604 509 m 604.75 509.25 L S 604.75 509.25 m 605.75 509.5 L S 605.75 509.5 m 606.5 509.75 L S 606.5 509.75 m 607.25 510 L S 607.25 510 m 608 510.25 L S 608 510.25 m 608.75 510.5 L S 608.75 510.5 m 609.75 510.75 L S 609.75 510.75 m 610.5 511 L S 610.5 511 m 611.25 511.25 L S 611.25 511.25 m 612 511.5 L S 612 511.5 m 612.75 511.75 L S 612.75 511.75 m 613.75 512 L S 613.75 512 m 614.5 512 L S 614.5 512 m 615.25 512.25 L S 615.25 512.25 m 616 512.5 L S 616 512.5 m 616.75 512.75 L S 616.75 512.75 m 617.75 513 L S 617.75 513 m 618.5 513.25 L S 618.5 513.25 m 619.25 513.5 L S 619.25 513.5 m 620 513.75 L S 620 513.75 m 620.75 514 L S 620.75 514 m 621.75 514.25 L S 621.75 514.25 m 622.5 514.25 L S 622.5 514.25 m 623.25 514.5 L S 623.25 514.5 m 624 514.75 L S 624 514.75 m 624.75 515 L S 624.75 515 m 625.75 515.25 L S 625.75 515.25 m 626.5 515.5 L S 626.5 515.5 m 627.25 515.75 L S 627.25 515.75 m 628 515.75 L S 628 515.75 m 628.75 516 L S 628.75 516 m 629.75 516.25 L S 629.75 516.25 m 630.5 516.5 L S 630.5 516.5 m 631.25 516.75 L S 631.25 516.75 m 632 517 L S 632 517 m 632.75 517.25 L S 632.75 517.25 m 633.75 517.25 L S 633.75 517.25 m 634.5 517.5 L S 634.5 517.5 m 635.25 517.75 L S 635.25 517.75 m 636 518 L S 636 518 m 636.75 518.25 L S 636.75 518.25 m 637.75 518.5 L S 637.75 518.5 m 638.5 518.5 L S 638.5 518.5 m 639.25 518.75 L S 639.25 518.75 m 640 519 L S 640 519 m 640.75 519.25 L S 640.75 519.25 m 641.75 519.5 L S 641.75 519.5 m 642.5 519.5 L S 642.5 519.5 m 643.25 519.75 L S 643.25 519.75 m 644 520 L S 644 520 m 644.75 520.25 L S 644.75 520.25 m 645.75 520.5 L S 645.75 520.5 m 646.5 520.5 L S 646.5 520.5 m 647.25 520.75 L S 647.25 520.75 m 648 521 L S 648 521 m 648.75 521.25 L S 648.75 521.25 m 649.75 521.5 L S 649.75 521.5 m 650.5 521.5 L S 650.5 521.5 m 651.25 521.75 L S 651.25 521.75 m 652 522 L S 652 522 m 652.75 522.25 L S 652.75 522.25 m 653.75 522.5 L S 653.75 522.5 m 654.5 522.5 L S 654.5 522.5 m 655.25 522.75 L S 655.25 522.75 m 656 523 L S 656 523 m 656.75 523.25 L S 656.75 523.25 m 657.75 523.25 L S 657.75 523.25 m 658.5 523.5 L S 658.5 523.5 m 659.25 523.75 L S 659.25 523.75 m 660 524 L S 660 524 m 660.75 524.25 L S 660.75 524.25 m 661.75 524.25 L S 661.75 524.25 m 662.5 524.5 L S 662.5 524.5 m 663.25 524.75 L S 343.25 261 m 344 261 L S 344 261 m 344.75 261 L S 344.75 261 m 345.75 261 L S 345.75 261 m 346.5 261 L S 346.5 261 m 347.25 261 L S 347.25 261 m 348 261 L S 348 261 m 348.75 261 L S 348.75 261 m 349.75 261 L S 349.75 261 m 350.5 261 L S 350.5 261 m 351.25 261 L S 351.25 261 m 352 261 L S 352 261 m 352.75 261 L S 352.75 261 m 353.75 261 L S 353.75 261 m 354.5 261 L S 354.5 261 m 355.25 261 L S 355.25 261 m 356 261 L S 356 261 m 356.75 261 L S 356.75 261 m 357.75 261 L S 357.75 261 m 358.5 261 L S 358.5 261 m 359.25 261 L S 359.25 261 m 360 261 L S 360 261 m 360.75 261 L S 360.75 261 m 361.75 261 L S 361.75 261 m 362.5 261 L S 362.5 261 m 363.25 261 L S 363.25 261 m 364 261 L S 364 261 m 364.75 261 L S 364.75 261 m 365.75 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m 426.5 261 L S 426.5 261 m 427.25 261 L S 427.25 261 m 428 261 L S 428 261 m 428.75 261 L S 428.75 261 m 429.75 261 L S 429.75 261 m 430.5 261 L S 430.5 261 m 431.25 261 L S 431.25 261 m 432 261 L S 432 261 m 432.75 261 L S 432.75 261 m 433.75 261 L S 433.75 261 m 434.5 261 L S 434.5 261 m 435.25 261 L S 435.25 261 m 436 261 L S 436 261 m 436.75 261 L S 436.75 261 m 437.75 261 L S 437.75 261 m 438.5 261 L S 438.5 261 m 439.25 261 L S 439.25 261 m 440 261 L S 440 261 m 440.75 261 L S 440.75 261 m 441.75 261 L S 441.75 261 m 442.5 261 L S 442.5 261 m 443.25 261 L S 443.25 261 m 444 261 L S 444 261 m 444.75 261 L S 444.75 261 m 445.75 261 L S 445.75 261 m 446.5 261 L S 446.5 261 m 447.25 261 L S 447.25 261 m 448 261 L S 448 261 m 448.75 261 L S 448.75 261 m 449.75 261 L S 449.75 261 m 450.5 261 L S 450.5 261 m 451.25 261 L S 451.25 261 m 452 261 L S 452 261 m 452.75 261 L S 452.75 261 m 453.75 261 L S 453.75 261 m 454.5 261 L S 454.5 261 m 455.25 261 L S 455.25 261 m 456 261 L S 456 261 m 456.75 261 L S 456.75 261 m 457.75 261 L S 457.75 261 m 458.5 261 L S 458.5 261 m 459.25 261 L S 459.25 261 m 460 261 L S 460 261 m 460.75 261 L S 460.75 261 m 461.75 261 L S 461.75 261 m 462.5 261 L S 462.5 261 m 463.25 261 L S 463.25 261 m 464 261 L S 464 261 m 464.75 261 L S 464.75 261 m 465.75 261 L S 465.75 261 m 466.5 261 L S 466.5 261 m 467.25 261 L S 467.25 261 m 468 261 L S 468 261 m 468.75 261 L S 468.75 261 m 469.75 261 L S 469.75 261 m 470.5 261 L S 470.5 261 m 471.25 261 L S 471.25 261 m 472 261 L S 472 261 m 472.75 261 L S 472.75 261 m 473.75 261 L S 473.75 261 m 474.5 261 L S 474.5 261 m 475.25 261 L S 475.25 261 m 476 261 L S 476 261 m 476.75 261 L S 476.75 261 m 477.75 261 L S 477.75 261 m 478.5 261 L S 478.5 261 m 479.25 261 L S 479.25 261 m 480 261 L S 480 261 m 480.75 261 L S 480.75 261 m 481.75 261 L S 481.75 261 m 482.5 261 L S 482.5 261 m 483.25 261 L S 483.25 261 m 484 261 L S 484 261 m 484.75 261 L S 484.75 261 m 485.75 261 L S 485.75 261 m 486.5 261 L S 486.5 261 m 487.25 261 L S 487.25 261 m 488 261 L S 488 261 m 488.75 261 L S 488.75 261 m 489.75 261 L S 489.75 261 m 490.5 261 L S 490.5 261 m 491.25 261 L S 491.25 261 m 492 261 L S 492 261 m 492.75 261 L S 492.75 261 m 493.75 261 L S 493.75 261 m 494.5 261 L S 494.5 328.5 m 495.25 330 L S 495.25 330 m 496 331 L S 496 331 m 496.75 331.75 L S 496.75 331.75 m 497.75 332.25 L S 497.75 332.25 m 498.5 332.75 L S 498.5 332.75 m 499.25 333 L S 499.25 333 m 500 333.25 L S 500 333.25 m 500.75 333.5 L S 500.75 333.5 m 501.75 333.75 L S 501.75 333.75 m 502.5 334 L S 502.5 334 m 503.25 334.25 L S 503.25 334.25 m 504 334.25 L S 504 334.25 m 504.75 334.5 L S 504.75 334.5 m 505.75 334.5 L S 505.75 334.5 m 506.5 334.5 L S 506.5 334.5 m 507.25 334.75 L S 507.25 334.75 m 508 334.75 L S 508 334.75 m 508.75 334.75 L S 508.75 334.75 m 509.75 335 L S 509.75 335 m 510.5 335 L S 510.5 335 m 511.25 335 L S 511.25 335 m 512 335 L S 512 335 m 512.75 335.25 L S 512.75 335.25 m 513.75 335.25 L S 513.75 335.25 m 514.5 335.25 L S 514.5 335.25 m 515.25 335.25 L S 515.25 335.25 m 516 335.25 L S 516 335.25 m 516.75 335.25 L S 516.75 335.25 m 517.75 335.25 L S 517.75 335.25 m 518.5 335.5 L S 518.5 335.5 m 519.25 335.5 L S 519.25 335.5 m 520 335.5 L S 520 335.5 m 520.75 335.5 L S 520.75 335.5 m 521.75 335.5 L S 521.75 335.5 m 522.5 335.5 L S 522.5 335.5 m 523.25 335.5 L S 523.25 335.5 m 524 335.5 L S 524 335.5 m 524.75 335.5 L S 524.75 335.5 m 525.75 335.5 L S 525.75 335.5 m 526.5 335.5 L S 526.5 335.5 m 527.25 335.5 L S 527.25 335.5 m 528 335.75 L S 528 335.75 m 528.75 335.75 L S 528.75 335.75 m 529.75 335.75 L S 529.75 335.75 m 530.5 335.75 L S 530.5 335.75 m 531.25 335.75 L S 531.25 335.75 m 532 335.75 L S 532 335.75 m 532.75 335.75 L S 532.75 335.75 m 533.75 335.75 L S 533.75 335.75 m 534.5 335.75 L S 534.5 335.75 m 535.25 335.75 L S 535.25 335.75 m 536 335.75 L S 536 335.75 m 536.75 335.75 L S 536.75 335.75 m 537.75 335.75 L S 537.75 335.75 m 538.5 335.75 L S 538.5 335.75 m 539.25 335.75 L S 539.25 335.75 m 540 335.75 L S 540 335.75 m 540.75 335.75 L S 540.75 335.75 m 541.75 335.75 L S 541.75 335.75 m 542.5 335.75 L S 542.5 335.75 m 543.25 335.75 L S 543.25 335.75 m 544 335.75 L S 544 335.75 m 544.75 335.75 L S 544.75 335.75 m 545.75 335.75 L S 545.75 335.75 m 546.5 335.75 L S 546.5 335.75 m 547.25 335.75 L S 547.25 335.75 m 548 335.75 L S 548 335.75 m 548.75 335.75 L S 548.75 335.75 m 549.75 335.75 L S 549.75 335.75 m 550.5 335.75 L S 550.5 335.75 m 551.25 335.75 L S 551.25 335.75 m 552 335.75 L S 552 335.75 m 552.75 335.75 L S 552.75 335.75 m 553.75 336 L S 553.75 336 m 554.5 336 L S 554.5 336 m 555.25 336 L S 555.25 336 m 556 336 L S 556 336 m 556.75 336 L S 556.75 336 m 557.75 336 L S 557.75 336 m 558.5 336 L S 558.5 336 m 559.25 336 L S 559.25 336 m 560 336 L S 560 336 m 560.75 336 L S 560.75 336 m 561.75 336 L S 561.75 336 m 562.5 336 L S 562.5 336 m 563.25 336 L S 563.25 336 m 564 336 L S 564 336 m 564.75 336 L S 564.75 336 m 565.75 336 L S 565.75 336 m 566.5 336 L S 566.5 336 m 567.25 336 L S 567.25 336 m 568 336 L S 568 336 m 568.75 336 L S 568.75 336 m 569.75 336 L S 569.75 336 m 570.5 336 L S 570.5 336 m 571.25 336 L S 571.25 336 m 572 336 L S 572 336 m 572.75 336 L S 572.75 336 m 573.75 336 L S 573.75 336 m 574.5 336 L S 574.5 336 m 575.25 336 L S 575.25 336 m 576 336 L S 576 336 m 576.75 336 L S 576.75 336 m 577.75 336 L S 577.75 336 m 578.5 336 L S 578.5 336 m 579.25 336 L S 579.25 336 m 580 336 L S 580 336 m 580.75 336 L S 580.75 336 m 581.75 336 L S 581.75 336 m 582.5 336 L S 582.5 336 m 583.25 336 L S 583.25 336 m 584 336 L S 584 336 m 584.75 336 L S 584.75 336 m 585.75 336 L S 585.75 336 m 586.5 336 L S 586.5 336 m 587.25 336 L S 587.25 336 m 588 336 L S 588 336 m 588.75 336 L S 588.75 336 m 589.75 336 L S 589.75 336 m 590.5 336 L S 590.5 336 m 591.25 336 L S 591.25 336 m 592 336 L S 592 336 m 592.75 336 L S 592.75 336 m 593.75 336 L S 593.75 336 m 594.5 336 L S 594.5 336 m 595.25 336 L S 595.25 336 m 596 336 L S 596 336 m 596.75 336 L S 596.75 336 m 597.75 336 L S 597.75 336 m 598.5 336 L S 598.5 336 m 599.25 336 L S 599.25 336 m 600 336 L S 600 336 m 600.75 336 L S 600.75 336 m 601.75 336 L S 601.75 336 m 602.5 336 L S 602.5 336 m 603.25 336 L S 603.25 336 m 604 336 L S 604 336 m 604.75 336 L S 604.75 336 m 605.75 336 L S 605.75 336 m 606.5 336 L S 606.5 336 m 607.25 336 L S 607.25 336 m 608 336 L S 608 336 m 608.75 336 L S 608.75 336 m 609.75 336 L S 609.75 336 m 610.5 336 L S 610.5 336 m 611.25 336 L S 611.25 336 m 612 336 L S 612 336 m 612.75 336 L S 612.75 336 m 613.75 336 L S 613.75 336 m 614.5 336 L S 614.5 336 m 615.25 336 L S 615.25 336 m 616 336 L S 616 336 m 616.75 336 L S 616.75 336 m 617.75 336 L S 617.75 336 m 618.5 336 L S 618.5 336 m 619.25 336 L S 619.25 336 m 620 336 L S 620 336 m 620.75 336 L S 620.75 336 m 621.75 336 L S 621.75 336 m 622.5 336 L S 622.5 336 m 623.25 336 L S 623.25 336 m 624 336 L S 624 336 m 624.75 336 L S 624.75 336 m 625.75 336 L S 625.75 336 m 626.5 336 L S 626.5 336 m 627.25 336 L S 627.25 336 m 628 336 L S 628 336 m 628.75 336 L S 628.75 336 m 629.75 336 L S 629.75 336 m 630.5 336 L S 630.5 336 m 631.25 336 L S 631.25 336 m 632 336 L S 632 336 m 632.75 336 L S 632.75 336 m 633.75 336 L S 633.75 336 m 634.5 336 L S 634.5 336 m 635.25 336 L S 635.25 336 m 636 336 L S 636 336 m 636.75 336 L S 636.75 336 m 637.75 336 L S 637.75 336 m 638.5 336 L S 638.5 336 m 639.25 336 L S 639.25 336 m 640 336 L S 640 336 m 640.75 336 L S 640.75 336 m 641.75 336 L S 641.75 336 m 642.5 336 L S 642.5 336 m 643.25 336 L S 643.25 336 m 644 336 L S 644 336 m 644.75 336 L S 644.75 336 m 645.75 336 L S 645.75 336 m 646.5 336 L S 646.5 336 m 647.25 336 L S 647.25 336 m 648 336 L S 648 336 m 648.75 336 L S 648.75 336 m 649.75 336 L S 649.75 336 m 650.5 336 L S 650.5 336 m 651.25 336 L S 651.25 336 m 652 336 L S 652 336 m 652.75 336 L S 652.75 336 m 653.75 336 L S 653.75 336 m 654.5 336 L S 654.5 336 m 655.25 336 L S 655.25 336 m 656 336 L S 656 336 m 656.75 336 L S 656.75 336 m 657.75 336 L S 657.75 336 m 658.5 336 L S 658.5 336 m 659.25 336 L S 659.25 336 m 660 336 L S 660 336 m 660.75 336 L S 660.75 336 m 661.75 336 L S 661.75 336 m 662.5 336 L S 662.5 336 m 663.25 336 L S 0 To 1 0 0 1 139 443 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M /_Helvetica 21 Tf 1 TA 36 0 Xb XB (FL) Tx (\r) TX TO 0 To 1 0 0 1 14 334 0 Tp TP 0 Tr (NV) Tx (\r) TX TO 0 To 1 0 0 1 125 350 0 Tp TP 0 Tr /_Helvetica 14 Tf (A) Tx (\r) TX TO LB %AI5_EndLayer-- gsave annotatepage grestore showpage Adobe_Illustrator_AI5 /terminate get exec Adobe_ColorImage_AI6 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EndDocument @endspecial 63 x Fr(Figur)n(e)19 b(1)p Fq(.)32 b(The)20 b(case)f Fp(J)24 b Fq(=)19 b(1,)h Fp(g)r Fq(\()p Fp(\032)p Fq(\))d(=)i Fp(\032)691 822 y FC(3)711 839 y Fp(=)p Fq(30.)30 b(\(a\))19 b(Phase)g(diagram.)31 b(The)19 b(ferromagnetic)g(transition) h(curv)o(e)-59 887 y Fp(z)15 b Fq(=)e Fp(z)46 894 y FA(m)79 887 y Fq(\()p Fp(\014)r Fq(\))g(splits)i(in)o(to)e(t)o(w)o(o)f(parts:) 18 b(a)c(b)q(old)g(part)f(indicating)j(a)d(\014rst-order)g(phase)h (transition)f(b)q(et)o(w)o(een)h(nonmag-)-59 935 y(netic)j(v)m(ap)q(or) g(\(NV\))f(and)h(ferromagnetic)f(liquid)i(\(FL\),)e(and)h(a)f(brok)o (en)g(part)g(corresp)q(onding)h(to)f(a)g(second-order)-59 983 y(transition)f(at)e(high)i(temp)q(eratures.)20 b(The)14 b(p)q(oin)o(t)h(A)g(is)g(the)f(liquid{v)m(ap)q(or)j(critical)f(p)q(oin) o(t;)e(the)h(asso)q(ciated)f(in)o(v)o(erse)-59 1031 y(temp)q(erature)i Fp(\014)227 1038 y FA(A)272 1031 y Fq(is)h(determined)h(b)o(y)e(the)h (equation)g(2)8 b Fp(g)944 1014 y FF(00)963 1031 y Fq(\(1)p Fp(=\014)1053 1038 y FA(A)1081 1031 y Fp(J)t Fq(\))15 b(=)g Fp(J)t Fq(.)24 b(\(b\))16 b(Densit)o(y)h(\()p Fp(\032)p Fq(\))e(and)i(magnetization)-59 1079 y(p)q(er)f(particle)g(\()p Fp(m)p Fq(\))e(for)h Fp(\014)g Fq(=)e(0)p Fp(:)p Fq(5.)14 1215 y Fs(Scenario)20 b(B)p FB(.)d Fv(The)i(ferr)n(omagnetic)g(phase)f (tr)n(ansition)g(at)h Fu(z)f FB(=)d Fu(z)1275 1222 y FA(m)1308 1215 y FB(\()p Fu(\014)s FB(\))j Fv(is)g(stil)r(l)i(of)f(se)n (c)n(ond)f(or)n(der)f(for)-59 1287 y(smal)r(l)d Fu(\014)h Fv(and)e(of)f(\014rst)h(or)n(der)e(for)i(interme)n(diate)g(values)h(of) e Fu(\014)s Fv(.)20 b(F)l(or)13 b(lar)n(ge)g Fu(\014)s Fv(,)f(however,)j(the)e(phase)g(tr)n(ansition)-59 1359 y(at)j(the)h(magnetization)h(thr)n(eshold)e Fu(z)619 1366 y FA(m)652 1359 y FB(\()p Fu(\014)s FB(\))g Fv(is)g(only)h(of)g (se)n(c)n(ond)f(or)n(der,)f(while)j(a)e(\014rst-or)n(der)g(tr)n (ansition)g(with)-59 1431 y(simultane)n(ous)f(jumps)f(of)g(density)h (and)g(magnetization)g(o)n(c)n(curs)f(at)g(some)h Fu(z)g(>)f(z)1424 1438 y FA(m)1457 1431 y FB(\()p Fu(\014)s FB(\))p Fv(,)g(i.e.,)h(in)g (the)g(interior)-59 1503 y(of)i(the)h(ferr)n(omagnetic)g(p)n(ar)n (ameter)e(r)n(e)n(gion.)14 1582 y FB(This)f(holds,)g(for)h(example,)c (if)j Fu(g)610 1564 y FF(00)646 1582 y FB(is)g(con)o(v)o(ex)f(with)h (minim)n(um)10 b(0)16 b(at)f(some)f Fu(\032)1423 1589 y FC(min)1498 1582 y Fu(>)g FB(0,)h(and)h(2)8 b Fu(g)1754 1564 y FF(00)1776 1582 y FB(\(0\))14 b Fu(>)g(J)5 b FB(;)-59 1654 y(see)18 b(Corollary)g(3.6)h(and)f(Theorem)f(3.4\(c\).)27 b(A)18 b(t)o(ypical)f(example)f(is)i Fu(g)r FB(\()p Fu(\032)p FB(\))g(=)f Fu(c)8 b FB(\()p Fu(\032)k Ft(\000)h FB(1\))1589 1636 y FC(4)1627 1654 y FB(with)18 b Fu(c)f(>)g(J)r(=)p FB(24;)-59 1727 y(cf.)k(Figure)16 b(2.)14 1818 y Fs(Scenario)j(C)p FB(.)d Fv(F)l(or)h(some)h(critic)n(al)g(inverse)g(temp)n(er)n(atur)n(e) f Fu(\014)1170 1825 y FA(c)1204 1818 y Fv(and)h Fu(z)1322 1825 y FA(c)1353 1818 y FB(=)c Fu(z)1428 1825 y FA(m)1461 1818 y FB(\()p Fu(\014)1508 1825 y FA(c)1525 1818 y FB(\))p Fv(,)k(ther)n(e)f(exist)i(phases)-59 1890 y(with)k(thr)n(e)n(e)e (di\013er)n(ent)i(densities)g Fu(\032)604 1897 y FF(\000)656 1890 y Fu(<)g(\032)742 1897 y FA(])780 1890 y Fu(<)g(\032)866 1897 y FC(+)896 1890 y Fv(.)36 b(The)23 b(phases)f(with)g(density)h Fu(\032)1518 1897 y FC(+)1570 1890 y Fv(ar)n(e)e(ferr)n(omagnetic)-59 1962 y(\(with)d(p)n(ositive)g(or)f(ne)n(gative)j(orientation\).)k(The) 18 b(triple)g(p)n(oint)g FB(\()p Fu(z)1183 1969 y FA(c)1200 1962 y Fu(;)8 b(\014)1250 1969 y FA(c)1267 1962 y FB(\))18 b Fv(is)f(the)i(endp)n(oint)f(of)g(a)f(\014rst-or)n(der)-59 2034 y(\(liquid-vap)n(or\))j(tr)n(ansition)f(line)i(in)f(the)g (\(nonmagnetic\))i(r)n(e)n(gion)d Ft(f)p Fu(z)h(<)e(z)1339 2041 y FA(m)1372 2034 y FB(\()p Fu(\014)s FB(\))p Fu(;)8 b(\014)19 b(<)f(\014)1594 2041 y FA(c)1611 2034 y Ft(g)p Fv(;)j(se)n(e)f(Figur)n(e)f(3.)-59 2107 y(F)l(ol)r(lowing)i(this)e (line)g(towar)n(ds)f FB(\()p Fu(z)574 2114 y FA(c)591 2107 y Fu(;)8 b(\014)641 2114 y FA(c)658 2107 y FB(\))18 b Fv(one)i(obtains)f(the)g(limiting)h(densities)g Fu(\032)1449 2114 y FF(\000)1497 2107 y Fv(and)f Fu(\032)1618 2114 y FA(])1634 2107 y Fv(.)25 b(On)20 b(the)f(other)-59 2179 y(hand,)h(in)f(a)g(neighb)n(orho)n(o)n(d)f(of)h(the)h(triple)g(p)n (oint)e(the)i(ferr)n(omagnetic)g(tr)n(ansition)f(at)g(the)h(line)g Fu(z)f FB(=)e Fu(z)1848 2186 y FA(m)1881 2179 y FB(\()p Fu(\014)s FB(\))-59 2251 y Fv(is)j(also)g(of)f(\014rst)h(or)n(der.)28 b(Appr)n(o)n(aching)20 b FB(\()p Fu(z)742 2258 y FA(c)759 2251 y Fu(;)8 b(\014)809 2258 y FA(c)825 2251 y FB(\))20 b Fv(along)h(this)f(line)h(fr)n(om)e(b)n(elow)h(\()p Fu(\014)h(<)d(\014)1591 2258 y FA(c)1608 2251 y Fv(\))i(one)g(arrives)g (at)-59 2323 y(the)h(limiting)i(densities)f Fu(\032)439 2330 y FA(])476 2323 y Fv(and)f Fu(\032)599 2330 y FC(+)628 2323 y Fv(,)h(and)g(c)n(oming)f(fr)n(om)f(ab)n(ove)h(\()p Fu(\014)i(>)d(\014)1345 2330 y FA(c)1362 2323 y Fv(\))h(one)h(ends)f (up)h(with)f Fu(\032)1821 2330 y FF(\000)1872 2323 y Fv(and)-59 2396 y Fu(\032)-34 2403 y FC(+)-4 2396 y Fv(.)31 b(In)21 b(other)f(wor)n(ds,)h(for)f Fu(\014)507 2385 y(<)507 2412 y Ft(\030)549 2396 y Fu(\014)577 2403 y FA(c)614 2396 y Fv(ther)n(e)h(ar)n(e)f(two)h(density)f(jumps)h(at)f (di\013er)n(ent)h(activities,)i(one)e(due)g(to)-59 2468 y(the)e(mole)n(cular)f(for)n(c)n(es)g(and)g(one)h(at)f(the)h(incipienc) n(e)g(of)f(ferr)n(omagnetic)h(or)n(der.)k(These)c(jumps)f(add)g(up)g (at)-59 2540 y Fu(\014)e FB(=)e Fu(\014)65 2547 y FA(c)82 2540 y Fv(,)j(and)h(an)g(enhanc)n(e)n(d)g(jump)f(p)n(ersists)g(for)g Fu(\014)894 2529 y(>)894 2556 y Ft(\030)936 2540 y Fu(\014)964 2547 y FA(c)981 2540 y Fv(.)14 2619 y FB(This)e(scenario)g(o)q(ccurs,)g (for)g(instance,)f(if)h Fu(g)817 2601 y FF(00)853 2619 y FB(is)g(con)o(v)o(ex)e(with)i(min)6 b Fu(g)1283 2601 y FF(00)1319 2619 y Fu(<)13 b FB(0,)i(and)h Fu(J)j FB(is)c(suitably)f (c)o(hosen;)-59 2691 y(see)i(Theorem)f(3.8)h(and)h(the)f(example)e(in)i (Figure)g(3.)933 2817 y(3)p eop %%Page: 4 5 4 4 bop -59 791 a @beginspecial -83 @llx 272 @lly 675 @urx 596 @ury 4818 @rwi @setspecial %%BeginDocument: CWfluid/scB2.ps %AI5_FileFormat 2.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 0 0 552 728 %AI3_DocumentPreview: Header %AI5_ArtSize: 842 595 %AI5_RulerUnits: 1 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: -174 756 1 1018 725 18 0 0 3 40 %AI5_OpenViewLayers: 7 userdict /Adobe_level2_AI5 23 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 17 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 14 dict def } if Adobe_ColorImage_AI6_Vars begin /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /plateindex -1 def /XIMask 0 def /XIBinary 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIBuffer null def /XIDataProc null def end /WalkRGBString null def /WalkCMYKString null def /StuffRGBIntoGrayString null def /RGBToGrayImageProc null def /StuffCMYKIntoGrayString null def /CMYKToGrayImageProc null def /ColorImageCompositeEmulator null def /SeparateCMYKImageProc null def /FourEqual null def /TestPlateIndex null def currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def } { /_colorimage null def } ifelse } if /_currenttransfer systemdict /currenttransfer get def /colorimage null def /XI null def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval { } forall 5 index exec 3 index } for 5 { pop } repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval { } forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat } { Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq { /RGBToGrayImageProc } { /CMYKToGrayImageProc } ifelse load end } if image } ifelse } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec } { sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /FourEqual { 4 index ne { pop pop pop false } { 4 index ne { pop pop false } { 4 index ne { pop false } { 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def } { 0 1 0 0 FourEqual { /plateindex 1 def } { 0 0 1 0 FourEqual { /plateindex 2 def } { 0 0 0 1 FourEqual { /plateindex 3 def } { 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index dup 8 eq exch 1 eq or not end { /_colorimage load null ne { _colorimage } { Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse } { dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator } { dup 1 eq { pop pop image } { Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse } { dup 1 eq { pop pop image } { pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIMask exch 0 ne def /XIBinary exch 0 ne def pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi } { XIImageWidth XIChannelCount mul } ifelse /XIBuffer exch string def XIBinary { /XIDataProc { currentfile XIBuffer readstring pop } def currentfile 128 string readline pop pop } { /XIDataProc { currentfile XIBuffer readhexstring pop } def } ifelse 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIMask { XIImageWidth XIImageHeight false [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load /_lp /null ddef _fc /_lp /imagemask ddef imagemask } { XIImageWidth XIImageHeight XIBitsPerPixel [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load XIChannelCount 1 eq { gsave 0 setgray image grestore } { false XIChannelCount colorimage } ifelse } ifelse grestore end } def end currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI3_BeginEncoding: _Helvetica Helvetica [/_Helvetica/Helvetica 0 0 1 TZ %AI3_EndEncoding AdobeType %AI3_BeginEncoding: _Symbol Symbol [/_Symbol/Symbol 0 0 0 TZ %AI3_EndEncoding AdobeType %AI5_Begin_NonPrinting 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0 1 317 275.25 0 Tp TP 0 Tr (-3) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 403.625 440.875 m 403.625 446.875 L S 403.625 290.875 m 403.625 296.875 L S 0 To 1 0 0 1 397 425.25 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (-2) Tx (\r) TX TO 0 To 1 0 0 1 397 275.25 0 Tp TP 0 Tr (-2) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 483.625 440.875 m 483.625 446.875 L S 483.625 290.875 m 483.625 296.875 L S 0 To 1 0 0 1 477 425.25 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (-1) Tx (\r) TX TO 0 To 1 0 0 1 477 275.25 0 Tp TP 0 Tr (-1) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 563.625 440.875 m 563.625 446.875 L S 563.625 290.875 m 563.625 296.875 L S 0 To 1 0 0 1 559.25 425.25 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (0) Tx (\r) TX TO 0 To 1 0 0 1 559.25 275.25 0 Tp TP 0 Tr (0) Tx (\r) TX TO 0 R 0 G 2 J 0.75 w 1.5 M 643.625 440.875 m 643.625 446.875 L S 643.625 290.875 m 643.625 296.875 L S 0 To 1 0 0 1 639.25 425.25 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M (1) Tx (\r) TX TO 0 To 1 0 0 1 639.25 275.25 0 Tp TP 0 Tr (1) Tx (\r) TX TO 0 R 0 G 2 J 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376.75 471.25 m 378.5 471.5 L S 378.5 471.5 m 380 471.5 L S 380 471.5 m 381.5 471.75 L S 381.5 471.75 m 383.25 471.75 L S 383.25 471.75 m 384.75 472 L S 384.75 472 m 386.5 472 L S 386.5 472 m 388 472.25 L S 388 472.25 m 389.5 472.25 L S 389.5 472.25 m 391.25 472.5 L S 391.25 472.5 m 392.75 472.5 L S 392.75 472.5 m 394.5 472.75 L S 394.5 472.75 m 396 472.75 L S 396 472.75 m 397.5 473 L S 397.5 473 m 399.25 473 L S 399.25 473 m 400.75 473.25 L S 400.75 473.25 m 402.5 473.25 L S 402.5 473.25 m 404 473.5 L S 404 473.5 m 405.5 473.75 L S 405.5 473.75 m 407.25 473.75 L S 407.25 473.75 m 408.75 474 L S 408.75 474 m 410.5 474 L S 410.5 474 m 412 474.25 L S 412 474.25 m 413.5 474.25 L S 413.5 474.25 m 415.25 474.5 L S 415.25 474.5 m 416.75 474.75 L S 416.75 474.75 m 418.5 474.75 L S 418.5 474.75 m 420 475 L S 420 475 m 421.5 475 L S 421.5 475 m 423.25 475.25 L S 423.25 475.25 m 424.75 475.5 L S 424.75 475.5 m 426.5 475.5 L S 426.5 475.5 m 428 475.75 L S 428 475.75 m 429.5 476 L S 429.5 476 m 431.25 476.25 L S 431.25 476.25 m 432.75 476.5 L S 432.75 476.5 m 434.5 476.75 L S 434.5 476.75 m 436 477 L S 436 477 m 437.5 477.25 L S 437.5 477.25 m 439.25 477.5 L S 439.25 477.5 m 440.75 477.75 L S 440.75 477.75 m 442.5 478.25 L S 442.5 478.25 m 444 478.5 L S 444 478.5 m 445.5 478.75 L S 445.5 478.75 m 447.25 479 L S 447.25 479 m 448.75 479.5 L S 448.75 479.5 m 450.5 479.75 L S 450.5 479.75 m 452 480.25 L S 452 480.25 m 453.5 480.5 L S 453.5 480.5 m 455.25 481 L S 455.25 481 m 456.75 481.25 L S 456.75 481.25 m 458.5 481.75 L S 458.5 481.75 m 460 482.25 L S 460 482.25 m 461.5 482.75 L S 461.5 482.75 m 463.25 483 L S 463.25 483 m 464.75 483 L S 464.75 532 m 466.5 532.5 L S 466.5 532.5 m 468 533 L S 468 533 m 469.5 533.5 L S 469.5 533.5 m 471.25 533.75 L S 471.25 533.75 m 472.75 534.25 L S 472.75 534.25 m 474.5 534.5 L S 474.5 534.5 m 476 535 L S 476 535 m 477.5 535.25 L S 477.5 535.25 m 479.25 535.75 L S 479.25 535.75 m 480.75 536 L S 480.75 536 m 482.5 536.25 L S 482.5 536.25 m 484 536.75 L S 484 536.75 m 485.5 537 L S 485.5 537 m 487.25 537.25 L S 487.25 537.25 m 488.75 537.5 L S 488.75 537.5 m 490.5 537.75 L S 490.5 537.75 m 492 538 L S 492 538 m 493.5 538.25 L S 493.5 538.25 m 495.25 538.75 L S 495.25 538.75 m 496.75 539 L S 496.75 539 m 498.5 539.25 L S 498.5 539.25 m 500 539.5 L S 500 539.5 m 501.5 539.75 L S 501.5 539.75 m 503.25 540 L S 503.25 540 m 504.75 540 L S 504.75 540 m 506.5 540.25 L S 506.5 540.25 m 508 540.5 L S 508 540.5 m 509.5 540.75 L S 509.5 540.75 m 511.25 541 L S 511.25 541 m 512.75 541.25 L S 512.75 541.25 m 514.5 541.5 L S 514.5 541.5 m 516 541.75 L S 516 541.75 m 517.5 541.75 L S 517.5 541.75 m 519.25 542 L S 519.25 542 m 520.75 542.25 L S 520.75 542.25 m 522.5 542.5 L S 522.5 542.5 m 524 542.75 L S 524 542.75 m 525.5 542.75 L S 525.5 542.75 m 527.25 543 L S 527.25 543 m 528.75 543.25 L S 528.75 543.25 m 530.5 543.5 L S 530.5 543.5 m 532 543.5 L S 532 543.5 m 533.5 543.75 L S 533.5 543.75 m 535.25 544 L S 535.25 544 m 536.75 544.25 L S 536.75 544.25 m 538.5 544.25 L S 538.5 544.25 m 540 544.5 L S 540 544.5 m 541.5 544.75 L S 541.5 544.75 m 543.25 544.75 L S 543.25 544.75 m 544.75 545 L S 544.75 545 m 546.5 545.25 L S 546.5 545.25 m 548 545.25 L S 548 545.25 m 549.5 545.5 L S 549.5 545.5 m 551.25 545.75 L S 551.25 545.75 m 552.75 545.75 L S 552.75 545.75 m 554.5 546 L S 554.5 546 m 556 546.25 L S 556 546.25 m 557.5 546.25 L S 557.5 546.25 m 559.25 546.5 L S 559.25 546.5 m 560.75 546.5 L S 560.75 546.5 m 562.5 546.75 L S 562.5 546.75 m 564 547 L S 564 547 m 565.5 547 L S 565.5 547 m 567.25 547.25 L S 567.25 547.25 m 568.75 547.25 L S 568.75 547.25 m 570.5 547.5 L S 570.5 547.5 m 572 547.75 L S 572 547.75 m 573.5 547.75 L S 573.5 547.75 m 575.25 548 L S 575.25 548 m 576.75 548 L S 576.75 548 m 578.5 548.25 L S 578.5 548.25 m 580 548.25 L S 580 548.25 m 581.5 548.5 L S 581.5 548.5 m 583.25 548.5 L S 583.25 548.5 m 584.75 548.75 L S 584.75 548.75 m 586.5 549 L S 586.5 549 m 588 549 L S 588 549 m 589.5 549.25 L S 589.5 549.25 m 591.25 549.25 L S 591.25 549.25 m 592.75 549.5 L S 592.75 549.5 m 594.5 549.5 L S 594.5 549.5 m 596 549.75 L S 596 549.75 m 597.5 549.75 L S 597.5 549.75 m 599.25 550 L S 599.25 550 m 600.75 550 L S 600.75 550 m 602.5 550.25 L S 602.5 550.25 m 604 550.25 L S 604 550.25 m 605.5 550.5 L S 605.5 550.5 m 607.25 550.5 L S 607.25 550.5 m 608.75 550.75 L S 608.75 550.75 m 610.5 550.75 L S 610.5 550.75 m 612 551 L S 612 551 m 613.5 551 L S 613.5 551 m 615.25 551.25 L S 615.25 551.25 m 616.75 551.25 L S 616.75 551.25 m 618.5 551.5 L S 618.5 551.5 m 620 551.5 L S 620 551.5 m 621.5 551.5 L S 621.5 551.5 m 623.25 551.75 L S 623.25 551.75 m 624.75 551.75 L S 624.75 551.75 m 626.5 552 L S 626.5 552 m 628 552 L S 628 552 m 629.5 552.25 L S 629.5 552.25 m 631.25 552.25 L S 631.25 552.25 m 632.75 552.5 L S 632.75 552.5 m 634.5 552.5 L S 634.5 552.5 m 636 552.75 L S 636 552.75 m 637.5 552.75 L S 637.5 552.75 m 639.25 552.75 L S 639.25 552.75 m 640.75 553 L S 640.75 553 m 642.5 553 L S 642.5 553 m 644 553.25 L S 324 294.5 m 325.5 294.5 L S 325.5 294.5 m 327.25 294.5 L S 327.25 294.5 m 328.75 294.5 L S 328.75 294.5 m 330.5 294.5 L S 330.5 294.5 m 332 294.5 L S 332 294.5 m 333.5 294.5 L S 333.5 294.5 m 335.25 294.5 L S 335.25 294.5 m 336.75 294.5 L S 336.75 294.5 m 338.5 294.5 L S 338.5 294.5 m 340 294.5 L S 340 294.5 m 341.5 294.5 L S 341.5 294.5 m 343.25 294.5 L S 343.25 294.5 m 344.75 294.5 L S 344.75 294.5 m 346.5 294.5 L S 346.5 294.5 m 348 294.5 L S 348 294.5 m 349.5 294.5 L S 349.5 294.5 m 351.25 294.5 L S 351.25 294.5 m 352.75 294.5 L S 352.75 294.5 m 354.5 294.5 L S 354.5 294.5 m 356 294.5 L S 356 294.5 m 357.5 294.5 L S 357.5 294.5 m 359.25 294.5 L S 359.25 294.5 m 360.75 294.5 L S 360.75 294.5 m 362.5 294.5 L S 362.5 294.5 m 364 294.5 L S 364 294.5 m 365.5 294.5 L S 365.5 294.5 m 367.25 294.5 L S 367.25 294.5 m 368.75 294.5 L S 368.75 294.5 m 370.5 294.5 L S 370.5 294.5 m 372 294.5 L S 372 294.5 m 373.5 294.5 L S 373.5 294.5 m 375.25 294.5 L S 375.25 294.5 m 376.75 294.5 L S 376.75 294.5 m 378.5 294.5 L S 378.5 294.5 m 380 294.5 L S 380 294.5 m 381.5 294.5 L S 381.5 294.5 m 383.25 294.5 L S 383.25 294.5 m 384.75 294.5 L S 384.75 294.5 m 386.5 294.5 L S 386.5 294.5 m 388 294.5 L S 388 294.5 m 389.5 294.5 L S 389.5 294.5 m 391.25 294.5 L S 391.25 294.5 m 392.75 294.5 L S 392.75 294.5 m 394.5 294.5 L S 394.5 294.5 m 396 294.5 L S 396 294.5 m 397.5 294.5 L S 397.5 294.5 m 399.25 294.5 L S 399.25 294.5 m 400.75 294.5 L S 400.75 294.5 m 402.5 294.5 L S 402.5 294.5 m 404 294.5 L S 404 294.5 m 405.5 294.5 L S 405.5 294.5 m 407.25 294.5 L S 407.25 294.5 m 408.75 294.5 L S 408.75 294.5 m 410.5 294.5 L S 410.5 294.5 m 412 294.5 L S 412 294.5 m 413.5 294.5 L S 413.5 294.5 m 415.25 294.5 L S 415.25 294.5 m 416.75 294.5 L S 416.75 294.5 m 418.5 294.5 L S 418.5 294.5 m 420 294.5 L S 420 294.5 m 421.5 294.5 L S 421.5 294.5 m 423.25 294.5 L S 423.25 294.5 m 424.75 294.5 L S 424.75 294.5 m 426.5 294.5 L S 426.5 294.5 m 428 294.5 L S 428 294.5 m 429.5 304.75 L S 429.5 304.75 m 431.25 309.75 L S 431.25 309.75 m 432.75 313.5 L S 432.75 313.5 m 434.5 316.75 L S 434.5 316.75 m 436 319.5 L S 436 319.5 m 437.5 321.75 L S 437.5 321.75 m 439.25 324 L S 439.25 324 m 440.75 326.25 L S 440.75 326.25 m 442.5 328 L S 442.5 328 m 444 330 L S 444 330 m 445.5 331.75 L S 445.5 331.75 m 447.25 333.25 L S 447.25 333.25 m 448.75 334.75 L S 448.75 334.75 m 450.5 336.25 L S 450.5 336.25 m 452 337.75 L S 452 337.75 m 453.5 339.25 L S 453.5 339.25 m 455.25 340.5 L S 455.25 340.5 m 456.75 341.75 L S 456.75 341.75 m 458.5 343 L S 458.5 343 m 460 344.25 L S 460 344.25 m 461.5 345.5 L S 461.5 345.5 m 463.25 346.75 L S 463.25 346.75 m 464.75 346.75 L S 464.75 369 m 466.5 369 L S 466.5 369 m 468 369 L S 468 369 m 469.5 369 L S 469.5 369 m 471.25 369 L S 471.25 369 m 472.75 369 L S 472.75 369 m 474.5 369 L S 474.5 369 m 476 369 L S 476 369 m 477.5 369 L S 477.5 369 m 479.25 369 L S 479.25 369 m 480.75 369 L S 480.75 369 m 482.5 369 L S 482.5 369 m 484 369 L S 484 369 m 485.5 369 L S 485.5 369 m 487.25 369 L S 487.25 369 m 488.75 369 L S 488.75 369 m 490.5 369 L S 490.5 369 m 492 369 L S 492 369 m 493.5 369 L S 493.5 369 m 495.25 369.25 L S 495.25 369.25 m 496.75 369.25 L S 496.75 369.25 m 498.5 369.25 L S 498.5 369.25 m 500 369.25 L S 500 369.25 m 501.5 369.25 L S 501.5 369.25 m 503.25 369.25 L S 503.25 369.25 m 504.75 369.25 L S 504.75 369.25 m 506.5 369.25 L S 506.5 369.25 m 508 369.25 L S 508 369.25 m 509.5 369.25 L S 509.5 369.25 m 511.25 369.25 L S 511.25 369.25 m 512.75 369.25 L S 512.75 369.25 m 514.5 369.25 L S 514.5 369.25 m 516 369.25 L S 516 369.25 m 517.5 369.25 L S 517.5 369.25 m 519.25 369.25 L S 519.25 369.25 m 520.75 369.25 L S 520.75 369.25 m 522.5 369.25 L S 522.5 369.25 m 524 369.25 L S 524 369.25 m 525.5 369.25 L S 525.5 369.25 m 527.25 369.25 L S 527.25 369.25 m 528.75 369.25 L S 528.75 369.25 m 530.5 369.25 L S 530.5 369.25 m 532 369.25 L S 532 369.25 m 533.5 369.25 L S 533.5 369.25 m 535.25 369.25 L S 535.25 369.25 m 536.75 369.25 L S 536.75 369.25 m 538.5 369.25 L S 538.5 369.25 m 540 369.25 L S 540 369.25 m 541.5 369.25 L S 541.5 369.25 m 543.25 369.25 L S 543.25 369.25 m 544.75 369.25 L S 544.75 369.25 m 546.5 369.25 L S 546.5 369.25 m 548 369.25 L S 548 369.25 m 549.5 369.25 L S 549.5 369.25 m 551.25 369.25 L S 551.25 369.25 m 552.75 369.25 L S 552.75 369.25 m 554.5 369.25 L S 554.5 369.25 m 556 369.25 L S 556 369.25 m 557.5 369.25 L S 557.5 369.25 m 559.25 369.25 L S 559.25 369.25 m 560.75 369.25 L S 560.75 369.25 m 562.5 369.25 L S 562.5 369.25 m 564 369.25 L S 564 369.25 m 565.5 369.25 L S 565.5 369.25 m 567.25 369.25 L S 567.25 369.25 m 568.75 369.25 L S 568.75 369.25 m 570.5 369.25 L S 570.5 369.25 m 572 369.25 L S 572 369.25 m 573.5 369.25 L S 573.5 369.25 m 575.25 369.25 L S 575.25 369.25 m 576.75 369.25 L S 576.75 369.25 m 578.5 369.25 L S 578.5 369.25 m 580 369.25 L S 580 369.25 m 581.5 369.25 L S 581.5 369.25 m 583.25 369.25 L S 583.25 369.25 m 584.75 369.25 L S 584.75 369.25 m 586.5 369.25 L S 586.5 369.25 m 588 369.25 L S 588 369.25 m 589.5 369.25 L S 589.5 369.25 m 591.25 369.25 L S 591.25 369.25 m 592.75 369.25 L S 592.75 369.25 m 594.5 369.25 L S 594.5 369.25 m 596 369.25 L S 596 369.25 m 597.5 369.25 L S 597.5 369.25 m 599.25 369.25 L S 599.25 369.25 m 600.75 369.25 L S 600.75 369.25 m 602.5 369.25 L S 602.5 369.25 m 604 369.25 L S 604 369.25 m 605.5 369.25 L S 605.5 369.25 m 607.25 369.25 L S 607.25 369.25 m 608.75 369.25 L S 608.75 369.25 m 610.5 369.25 L S 610.5 369.25 m 612 369.25 L S 612 369.25 m 613.5 369.25 L S 613.5 369.25 m 615.25 369.25 L S 615.25 369.25 m 616.75 369.25 L S 616.75 369.25 m 618.5 369.25 L S 618.5 369.25 m 620 369.25 L S 620 369.25 m 621.5 369.25 L S 621.5 369.25 m 623.25 369.25 L S 623.25 369.25 m 624.75 369.25 L S 624.75 369.25 m 626.5 369.25 L S 626.5 369.25 m 628 369.25 L S 628 369.25 m 629.5 369.25 L S 629.5 369.25 m 631.25 369.25 L S 631.25 369.25 m 632.75 369.25 L S 632.75 369.25 m 634.5 369.25 L S 634.5 369.25 m 636 369.25 L S 636 369.25 m 637.5 369.25 L S 637.5 369.25 m 639.25 369.25 L S 639.25 369.25 m 640.75 369.25 L S 640.75 369.25 m 642.5 369.25 L S 642.5 369.25 m 644 369.25 L S 0 To 1 0 0 1 195 365 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M /_Helvetica 14 Tf 1 TA 36 0 Xb XB (A) Tx (\r) TX TO 0 To 1 0 0 1 108 435 0 Tp TP 0 Tr (B) Tx (\r) TX TO 0 To 1 0 0 1 -33 499 0 Tp TP 0 Tr /_Helvetica 21 Tf (FV) Tx (\r) TX TO 0 To 1 0 0 1 166 474 0 Tp TP 0 Tr (FL) Tx (\r) TX TO 0 To 1 0 0 1 34 362 0 Tp TP 0 Tr (NV) Tx (\r) TX TO LB %AI5_EndLayer-- gsave annotatepage grestore showpage Adobe_Illustrator_AI5 /terminate get exec Adobe_ColorImage_AI6 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EndDocument @endspecial 63 x Fr(Figur)n(e)17 b(2)p Fq(.)22 b(The)16 b(case)g Fp(J)j Fq(=)14 b(1,)h Fp(g)r Fq(\()p Fp(\032)p Fq(\))d(=)j(\()p Fp(\032)9 b Fo(\000)i Fq(1\))762 837 y FC(4)781 854 y Fq(.)22 b(\(a\))15 b(Phase)h(diagram.)22 b(The)16 b(b)q(old)i(and)e(brok)o(en)g(curv)o(es)g(and)g(the)-59 902 y(critical)g(p)q(oin)o(t)g(A)e(ha)o(v)o(e)h(the)g(same)f(meaning)i (as)e(in)i(Fig.)f(1.)k(In)c(addition)h(to)f(the)g(t)o(w)o(o)e(phases)i (NV)g(and)g(FL,)g(there)-59 950 y(is)h(an)g(in)o(termediate)g(phase)g (FV)f(\(ferromagnetic)g(v)m(ap)q(or\);)g(the)h(solid)h(line)g (separating)e(FV)h(from)e(FL)i(indicates)h(a)-59 998 y(\014rst-order)e(phase)h(transition.)22 b(Since)17 b(the)f(transition) g(from)f(NV)g(to)h(FV)f(is)h(second)g(order,)g(the)f(critical)i(p)q (oin)o(t)g(B)-59 1046 y(is)f(not)e(a)h(triple)i(p)q(oin)o(t.)j(\(b\))15 b(Densit)o(y)g(and)g(magnetization)h(for)e Fp(\014)h Fq(=)e(2.)-59 1943 y @beginspecial -90 @llx 260 @lly 684 @urx 584 @ury 4818 @rwi @setspecial %%BeginDocument: CWfluid/scC2.ps %AI5_FileFormat 2.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 0 0 552 728 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 842 595 %AI5_RulerUnits: 1 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: -174 756 1 1018 725 18 0 0 3 40 %AI5_OpenViewLayers: 7 userdict /Adobe_level2_AI5 23 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 17 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 14 dict def } if Adobe_ColorImage_AI6_Vars begin /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /plateindex -1 def /XIMask 0 def /XIBinary 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIBuffer null def /XIDataProc null def end /WalkRGBString null def /WalkCMYKString null def /StuffRGBIntoGrayString null def /RGBToGrayImageProc null def /StuffCMYKIntoGrayString null def /CMYKToGrayImageProc null def /ColorImageCompositeEmulator null def /SeparateCMYKImageProc null def /FourEqual null def /TestPlateIndex null def currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def } { /_colorimage null def } ifelse } if /_currenttransfer systemdict /currenttransfer get def /colorimage null def /XI null def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval { } forall 5 index exec 3 index } for 5 { pop } repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval { } forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat } { Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq { /RGBToGrayImageProc } { /CMYKToGrayImageProc } ifelse load end } if image } ifelse } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec } { sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /FourEqual { 4 index ne { pop pop pop false } { 4 index ne { pop pop false } { 4 index ne { pop false } { 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def } { 0 1 0 0 FourEqual { /plateindex 1 def } { 0 0 1 0 FourEqual { /plateindex 2 def } { 0 0 0 1 FourEqual { /plateindex 3 def } { 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index dup 8 eq exch 1 eq or not end { /_colorimage load null ne { _colorimage } { Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse } { dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator } { dup 1 eq { pop pop image } { Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse } { dup 1 eq { pop pop image } { pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIMask exch 0 ne def /XIBinary exch 0 ne def pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi } { XIImageWidth XIChannelCount mul } ifelse /XIBuffer exch string def XIBinary { /XIDataProc { currentfile XIBuffer readstring pop } def currentfile 128 string readline pop pop } { /XIDataProc { currentfile XIBuffer readhexstring pop } def } ifelse 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIMask { XIImageWidth XIImageHeight false [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load /_lp /null ddef _fc /_lp /imagemask ddef imagemask } { XIImageWidth XIImageHeight XIBitsPerPixel [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load XIChannelCount 1 eq { gsave 0 setgray image grestore } { false XIChannelCount colorimage } ifelse } ifelse grestore end } def end currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI3_BeginEncoding: _Helvetica Helvetica [/_Helvetica/Helvetica 0 0 1 TZ %AI3_EndEncoding AdobeType %AI3_BeginEncoding: _Symbol Symbol [/_Symbol/Symbol 0 0 0 TZ %AI3_EndEncoding AdobeType %AI5_Begin_NonPrinting 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348.625 m 637.375 348.875 L S 637.375 348.875 m 638.875 348.875 L S 638.875 348.875 m 640.625 348.875 L S 640.625 348.875 m 642.125 348.875 L S 642.125 348.875 m 643.875 348.875 L S 643.875 348.875 m 645.375 349.125 L S 645.375 349.125 m 646.875 349.125 L S 646.875 349.125 m 648.625 349.125 L S 648.625 349.125 m 650.125 349.125 L S 650.125 349.125 m 651.875 349.125 L S 651.875 349.125 m 653.375 349.125 L S 0 To 1 0 0 1 94 386 0 Tp TP 0 Tr 0 O 0 g 0 J 1 w 4 M /_Helvetica 14 Tf 1 TA 36 0 Xb XB (A) Tx (\r) TX TO 0 To 1 0 0 1 64 326 0 Tp TP 0 Tr (B) Tx (\r) TX TO 0 To 1 0 0 1 13 423 0 Tp TP 0 Tr (C) Tx (\r) TX TO 0 To 1 0 0 1 85 343 0 Tp TP 0 Tr /_Helvetica 21 Tf (NL) Tx (\r) TX TO 0 To 1 0 0 1 -25 361 0 Tp TP 0 Tr (NV) Tx (\r) TX TO 0 To 1 0 0 1 102 474 0 Tp TP 0 Tr (FL') Tx (\r) TX TO LB %AI5_EndLayer-- gsave annotatepage grestore showpage Adobe_Illustrator_AI5 /terminate get exec Adobe_ColorImage_AI6 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EndDocument @endspecial 50 x Fr(Figur)n(e)i(3)p Fq(.)20 b(The)15 b(case)g Fp(J)i Fq(=)c(2)p Fp(:)p Fq(5,)h Fp(g)r Fq(\()p Fp(\032)p Fq(\))c(=)j Fp(\032)670 1977 y FC(4)699 1993 y Fo(\000)c Fq(4)p Fp(\032)790 1977 y FC(2)809 1993 y Fq(.)20 b(\(a\))14 b(Phase)g(diagram.)20 b(Bold)15 b(and)g(brok)o(en)g (lines)h(are)e(as)h(in)g(Fig.)-59 2041 y(1.)26 b(The)18 b(solid)g(line)h(indicates)g(a)e(\014rst-order)g(v)m(ap)q(or{liquid)j (transition)e(within)g(the)g(nonmagnetic)g(region.)27 b(The)-59 2089 y(p)q(oin)o(t)15 b(C)g(is)g(a)f(triple)i(p)q(oin)o(t)f (of)f(co)q(existence)j(of)d(three)h(phases:)k(nonmagnetic)d(v)m(ap)q (or)e(\(NV\),)g(nonmagnetic)h(liquid)-59 2137 y(\(NL\),)h(and)i(a)f (ferromagnetic)f(high-densit)o(y)j(phase)f(whic)o(h)g(should)g (probably)f(also)g(b)q(e)h(in)o(terpreted)g(as)f(a)f(liquid)-59 2185 y(\(FL'\).)f(B)i(is)g(the)g(V{L)g(critical)h(p)q(oin)o(t,)f(and)g (A)f(the)h(L{L')g(critical)h(p)q(oin)o(t.)25 b(\(b\))16 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y(pro)q(cesses)i(whic)o(h)e(allo)o(ws)i(us)f(to)h(express)f(the)g (in\014nite-v)o(olume)d(b)q(eha)o(vior)j(of)h(our)g(mo)q(del)e(in)h (terms)e(of)j(the)-59 235 y(minim)o(ize)o(rs)g(of)k(a)f(free)f(energy)h (functional)f([8].)32 b(The)20 b(necessary)f(details)h(will)f(b)q(e)h (pro)o(vided)f(in)g(Section)-59 307 y(4.)31 b(Section)18 b(5)i(con)o(tains)f(a)h(general)f(analysis)h(of)f(these)g(minimi)o(ze)o (rs.)28 b(The)19 b(pro)q(ofs)i(of)e(our)h(main)e(results)-59 379 y(then)e(follo)o(w)g(in)g(Section)f(6.)22 b(In)16 b(the)g(\014nal)g(Section)g(7)h(w)o(e)e(commen)o(t)e(on)k(the)f(ph)o (ysical)f(signi\014cance)h(of)g(our)-59 451 y(results.)14 567 y Fv(A)n(cknow)r(le)n(dgemen)q(t)p FB(.)54 b(H.O.G.)25 b(gratefully)g(ac)o(kno)o(wledges)h(w)o(arm)f(hospitalit)o(y)h(at)g (the)g(Cen)o(tre)g(de)-59 640 y(Ph)o(ysique)18 b(Th)o(\023)-23 b(eorique)18 b(in)h(Marseille-Lumin)o(y)d(and)j(V.Z.)f(at)h(the)g (Mathematical)e(Institute)h(of)i(the)e(Uni-)-59 712 y(v)o(ersit)o(y)c (of)j(Munic)o(h.)-59 903 y Fw(2)83 b(The)27 b(Mo)r(del)-59 1031 y FB(W)l(e)13 b(consider)g(a)g(system)f(of)h(particles)f(in)h (Euclidean)f(space)1039 1013 y FC(2)1072 1031 y FD(R)1114 1013 y FA(d)1134 1031 y FB(,)h Fu(d)i Ft(\025)e FB(1.)21 b(Eac)o(h)13 b(particle)f(is)h(equipp)q(ed)f(with)-59 1103 y(a)17 b(spin)f(taking)h(v)m(alues)f(in)g(the)g(unit)g(sphere)g Fu(E)h FB(=)d Ft(f)p Fu(s)g Ft(2)g FD(R)1033 1085 y FA(D)1079 1103 y FB(:)g Ft(j)p Fu(s)p Ft(j)f FB(=)h(1)p Ft(g)j FB(of)g FD(R)1387 1085 y FA(D)1418 1103 y FB(,)f Fu(D)g Ft(\025)e FB(1.)22 b(A)16 b(con\014guration)-59 1175 y(of)h(suc)o(h)g(particles)f(\(without)h(m)o(ultiple)d(o)q (ccupancies\))j(can)g(b)q(e)g(describ)q(ed)g(b)o(y)f(a)i(pair)f Fu(!)g FB(=)e(\()p Fu(X)q(;)8 b FB(\()p Fu(\033)1798 1182 y FA(x)1820 1175 y FB(\))1839 1182 y FA(x)p FF(2)p FA(X)1916 1175 y FB(\),)-59 1248 y(where)16 b Fu(X)j Ft(\032)14 b FD(R)236 1229 y FA(d)256 1248 y FB(,)j(the)f(set)g(of)h(o) q(ccupied)g(p)q(ositions,)g(is)f(lo)q(cally)g(\014nite,)g(in)g(that)h 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1544 y FC(\))p FF(2)p FA(B)r FF(g)1661 1536 y FB(is)f(measurable)-59 1609 y(for)g(an)o(y)h (Borel)e(subset)i Fu(B)h FB(of)f FD(R)540 1591 y FA(d)571 1609 y Ft(\002)11 b Fu(E)s FB(.)14 1687 y(The)23 b(a)h(priori)f (probabilit)o(y)g(measure)f(on)h(\012)h(is)f(the)g(P)o(oisson)h(p)q (oin)o(t)g(random)f(\014eld)g(describing)f(an)-59 1760 y(ideal)g(gas)h(of)g(particles)e(at)i(random)f(p)q(ositions)h(with)f (random)g(spins)g(in)g Fu(E)s FB(.)40 b(Sp)q(eci\014cally)l(,)22 b(let)f Fu(z)26 b(>)e FB(0)-59 1832 y(b)q(e)d(an)h(activit)o(y)d (parameter)h(and)i Fu(\026)f FB(an)h(arbitrary)f(Borel)f(probabilit)o (y)g(measure)g(on)h Fu(E)s FB(.)36 b(The)21 b(P)o(oisson)-59 1904 y(p)q(oin)o(t)d(random)f(\014eld)g Fu(Q)396 1886 y FA(z)q(\026)455 1904 y FB(with)h(in)o(tensit)o(y)e Fu(z)j FB(and)g(spin)e(distribution)h Fu(\026)g FB(is)f(then)h (de\014ned)g(as)g(the)g(unique)-59 1976 y(probabilit)o(y)13 b(measure)g(on)h(\012)g(suc)o(h)g(that,)g(for)h(an)o(y)e(measurable)g (function)h Fu(f)19 b FB(:)13 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Fu(z)r(\026)p FB(,)i(the)f(spin)g(in)o(tensit)o(y)f(measure.)24 b(So,)18 b(in)f(fact)h(w)o(e)f(ha)o(v)o(e)f(de\014ned)i Fu(Q)1852 0 y FA(\027)1891 18 y FB(for)-59 90 y(an)o(y)e(\014nite)g (measure)f Fu(\027)k FB(on)e Fu(E)s FB(.\))14 169 y(W)l(e)j(will)f (consider)h(the)g(follo)o(wing)g(c)o(hoices)g(of)g Fu(\026)p FB(.)34 b(A)20 b(priori,)g(w)o(e)g(will)f(set)i Fu(\026)g FB(=)f Fu(\025)p FB(,)i(the)e(normalized)-59 241 y(surface)14 b(measure)f(on)h Fu(E)s FB(.)21 b(A)14 b(p)q(osteriori,)g(it)g(will)f (b)q(ecome)f(necessary)i(to)g(consider,)g(for)h(eac)o(h)e(external)g (\014eld)-59 313 y FD(h)h Ft(2)g FD(R)75 295 y FA(D)107 313 y FB(,)i(the)g(tilted)f(probabilit)o(y)g(measure)594 428 y Fu(\025)622 435 y Fi(h)647 428 y FB(\()p Fu(d\033)r FB(\))e(=)h Fu(e)828 408 y Fi(h)o FF(\001)p FA(\033)884 428 y Fu(\025)p FB(\()p Fu(d\033)r FB(\))1005 380 y Fn(.)1047 370 y(Z)1097 428 y Fu(e)1120 408 y Fi(h)o FF(\001)p FA(\033)1175 428 y Fu(\025)p FB(\()p Fu(d\033)r FB(\))590 b(\(1\))-59 550 y(on)17 b 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Fu(\033)1215 702 y FA(x)1237 695 y FB(\))1256 702 y FA(x)p FF(2)p FA(X)1333 695 y FB(\))14 b Ft(!)f Fu(!)1459 702 y FA(n)1497 695 y FB(=)h(\()p Fu(X)5 b Ft(\\)q FB(\003)1681 702 y FA(n)1704 695 y Fu(;)j FB(\()p Fu(\033)1773 702 y FA(x)1794 695 y FB(\))1813 702 y FA(X)s FF(\\)p FC(\003)1893 706 y Fm(n)1916 695 y FB(\).)-59 767 y(Hence,)14 b Fu(Q)138 749 y FA(z)q(;)p Fi(h)138 779 y FA(n)207 767 y FB(is)i(a)g(probabilit)o (y)g(measure)f(on)h(\012)838 774 y FA(n)876 767 y FB(=)e Ft(f)p Fu(!)i FB(=)d(\()p Fu(X)q(;)8 b FB(\()p Fu(\033)1179 774 y FA(x)1201 767 y FB(\))1220 774 y FA(x)p FF(2)p FA(X)1297 767 y FB(\))14 b Ft(2)g FB(\012)g(:)g Fu(X)k Ft(\032)c FB(\003)1599 774 y FA(n)1622 767 y Ft(g)p FB(.)14 845 y(W)l(e)e(consider)g(a)h(particle)e(system)g(in)h(\003)734 852 y FA(n)770 845 y FB(with)g(an)h(in)o(teraction)e(of)i(Curie{W)l (eiss)f(t)o(yp)q(e.)19 b(More)12 b(precisely)l(,)-59 918 y(for)k Fu(!)g FB(=)e(\()p Fu(X)q(;)8 b FB(\()p Fu(\033)242 925 y FA(x)264 918 y FB(\))283 925 y FA(x)p FF(2)p FA(X)360 918 y FB(\))14 b Ft(2)g FB(\012)475 925 y FA(n)515 918 y FB(w)o(e)i(consider)g(the)g(Hamiltonian)511 1045 y Fu(H)551 1052 y FA(n)575 1045 y FB(\()p Fu(!)r FB(\))e(=)g Ft(\000)755 1012 y Fu(J)p 755 1034 32 2 v 759 1079 a FB(2)821 1004 y Fn(X)800 1096 y FA(x;y)q FF(2)p FA(X)916 1012 y Fu(\033)944 1019 y FA(x)977 1012 y Ft(\001)d Fu(\033)1030 1019 y FA(y)p 916 1034 135 2 v 957 1079 a Fu(V)985 1086 y FA(n)1066 1045 y FB(+)g Fu(V)1143 1052 y FA(n)1181 1045 y Fu(g)1206 997 y Fn(\020)1236 1012 y FB(#)p Fu(X)p 1236 1034 86 2 v 1253 1079 a(V)1281 1086 y FA(n)1326 997 y Fn(\021)1365 1045 y Fu(;)507 b FB(\(2\))-59 1191 y(where)11 b Fu(J)18 b(>)c FB(0)d(is)g(a)g(ferromagnetic)f(coupling)h (constan)o(t,)h Fu(g)k FB(:)d([0)p Fu(;)8 b Ft(1)p FB([)p Ft(!)13 b FD(R)e FB(is)g(a)g(suitable)g(function)f(describing)-59 1263 y(the)15 b(molecular)d(mean-\014eld)i(in)o(teraction,)f(and)j(#)p Fu(X)j FB(is)14 b(the)h(cardinalit)o(y)f(of)h Fu(X)t FB(.)21 b(The)15 b(essen)o(tial)f(features)h(of)-59 1335 y(this)h(Hamiltonian)e(are)j(the)f(follo)o(wing.)14 1414 y(The)k(spin)f(coupling)g(constan)o(t)h(is)f(prop)q(ortional)i(to)f(1)p Fu(=V)1097 1421 y FA(n)1141 1414 y FB(rather)f(than)h(1)p Fu(=)p FB(#)p Fu(X)t FB(.)31 b(Hence,)19 b(the)g(mean)-59 1486 y(\014eld)11 b(acting)i(on)f(a)h(single)e(spin)h(is)g(prop)q (ortional)i(to)e(the)g(particle)f(densit)o(y)g(#)p Fu(X=V)1441 1493 y FA(n)1477 1486 y FB(and)i(the)f(magnetization)-59 1558 y(p)q(er)h(particle.)19 b(This)13 b(fa)o(v)o(ors)g(high)g (particle)f(densities,)g(and)i(the)e(system)g(w)o(ould)h(b)q(ecome)e (unstable)j(without)-59 1630 y(an)k(additional)f(term)e(in)i(the)g (Hamiltonian.)23 b(This)17 b(is)g(a)h(formal)e(reason)i(for)f(in)o(tro) q(ducing)g(the)g(term)f(with)-59 1703 y Fu(g)r FB(.)22 b(In)16 b(addition,)g(w)o(e)f(w)o(an)o(t)i(to)f(ensure)h(that)f Fu(g)j FB(giv)o(es)c(rise)h(to)h(a)g(p)q(ositiv)o(e)e(feedbac)o(k)g(of) i(the)f(magnetization)-59 1775 y(to)h(the)f(particle)f(densit)o(y)l(.) 20 b(It)c(is)g(our)g(ob)s(jectiv)o(e)f(to)h(\014nd)h(the)f(relev)m(an)o (t)f(features)h(of)h Fu(g)h FB(for)f(this)f(to)h(o)q(ccur.)14 1854 y(Our)f(general)g(assumptions)h(on)f Fu(g)j FB(are)d(the)g(follo)o (wing.)-35 1968 y(\(A1\))g Fv(Smo)n(othness)p FB(.)22 b Fu(g)c FB(is)e(analytic)g(on)h(]0)p Fu(;)8 b Ft(1)p FB([,)15 b(and)i Fu(g)978 1950 y FF(00)999 1968 y FB(\(0\))d(=)g(lim) 1195 1975 y FA(\032)p FF(#)p FC(0)1258 1968 y Fu(g)1283 1950 y FF(00)1305 1968 y FB(\()p Fu(\032)p FB(\))i(exists.)-35 2079 y(\(A2\))g Fv(Str)n(ong)i(stability)p FB(.)k Fu(g)460 2061 y FF(00)481 2079 y FB(\()p Fu(\032)p FB(\))14 b Ft(!)g(1)i FB(as)h Fu(\032)d Ft(!)g(1)p FB(,)h(and)i(th)o(us)f Fu(g)r FB(\()p Fu(\032)p FB(\))p Fu(=\032)1269 2061 y FC(2)1304 2079 y Ft(!)d(1)j FB(as)h Fu(\032)d Ft(!)g(1)p FB(.)-59 2194 y(F)l(or)h(example,)e Fu(g)k FB(can)f(b)q(e)f(an)o(y)g(p) q(olynomial)f(of)i(degree)e(at)i(least)f(three)f(with)i(p)q(ositiv)o(e) 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b(W)l(e)18 b(migh)o(t)f(also)h (require)f(that)i(the)f Fu(x)928 2628 y FC(1)947 2621 y Fu(;)8 b(:)g(:)g(:)g(;)g(x)1085 2628 y FA(k)1124 2621 y FB(in)17 b(the)h(sum)g(ab)q(o)o(v)o(e)g(are)g(pairwise)g(distinct)-59 2693 y(b)q(ecause)e(this)h(condition)f(b)q(ecomes)f(irrelev)m(an)o(t)f (in)i(the)g(thermo)q(dynamic)e(limit.)933 2817 y(6)p eop %%Page: 7 8 7 7 bop 14 18 a FB(Let)523 90 y Fu(P)554 97 y FA(n;z)q(;\014)r(;J)669 90 y FB(\()p Fu(d!)r FB(\))14 b(=)g Fu(Z)867 70 y FF(\000)p FC(1)863 103 y FA(n;z)q(;\014)r(;J)992 90 y Fu(e)1015 70 y FF(\000)p FA(\014)7 b(H)1098 74 y Fm(n)1119 70 y FC(\()p FA(!)q FC(\))1186 90 y Fu(Q)1225 70 y FA(z)q(;)p FC(0)1225 102 y FA(n)1272 90 y FB(\()p Fu(d!)r FB(\))519 b(\(3\))-59 189 y(b)q(e)17 b(the)f(Gibbs)h(distribution)g(in)f(\003)592 196 y FA(n)632 189 y FB(with)h(Hamiltonian)e Fu(H)1065 196 y FA(n)1088 189 y FB(,)i(activit)o(y)e Fu(z)h(>)f FB(0)i(and)g(in)o(v)o(erse)e(temp)q(erature)-59 262 y Fu(\014)h(>)e FB(0.)21 b(As)c(usually)l(,)625 334 y Fu(Z)658 341 y FA(n;z)q(;\014)r(;J)787 334 y FB(=)839 275 y Fn(Z)889 334 y Fu(e)912 313 y FF(\000)p FA(\014)7 b(H)995 317 y Fm(n)1016 313 y FC(\()p FA(!)q FC(\))1083 334 y Fu(Q)1122 313 y FA(z)q(;)p FC(0)1122 346 y FA(n)1169 334 y FB(\()p Fu(d!)r FB(\))622 b(\(4\))-59 433 y(is)19 b(the)g(asso)q(ciated)i (partition)e(function.)31 b(\(Here)18 b(and)i(b)q(elo)o(w)f(w)o(e)g (consider)g Fu(g)j FB(and)e Fu(D)h FB(as)f(\014xed,)f(but)h(the)-59 505 y(other)c(parameters)g(ma)o(y)e(v)m(ary)j(and)f(are)h(th)o(us)f (included)f(in)o(to)h(our)h(notation.\))14 584 y(W)l(e)g(are)g(in)o (terested)f(in)h(the)g(set)g Ft(G)s FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))15 b(of)i(all)g(limiting)d(Gibbs)k(states,)f (i.e.,)f(of)h(all)g(accum)o(ulation)-59 656 y(p)q(oin)o(ts)d(of)f(the)h (sequence)e(\()p Fu(P)468 663 y FA(n;z)q(;\014)r(;J)583 656 y FB(\))602 663 y FA(n)p FF(\025)p FC(1)684 656 y FB(in)h(the)g(in\014nite{v)o(olume)e(limit)g Fu(n)j Ft(!)f(1)p FB(.)20 b(T)l(o)14 b(mak)o(e)e(this)h(de\014nition)-59 728 y(precise)e(w)o(e)g(need)h(to)g(sp)q(ecify)f(a)i(top)q(ology)g(for) f(probabilit)o(y)f(measures)g(on)i(\012.)20 b(Rather)12 b(than)g(the)g(usual)g(w)o(eak)-59 801 y(top)q(ology)l(,)i(w)o(e)e(can) g(use)h(here)e(a)i(\014ner)f(top)q(ology)i Fu(\034)845 808 y FF(L)884 801 y FB(whic)o(h)d(is)i(de\014ned)f(as)h(follo)o(ws.)19 b(Let)13 b Ft(L)g FB(b)q(e)f(the)g(class)h(of)f(all)-59 873 y(measurable)j(functions)i Fu(f)j FB(:)14 b(\012)h Ft(!)f FD(R)j FB(whic)o(h)f(are)h(lo)q(cal,)f(in)g(that)h Fu(f)5 b FB(\()p Fu(!)r FB(\))18 b(only)e(dep)q(ends)h(on)g(the)g (restriction)-59 945 y(of)g Fu(!)i FB(to)d(a)h(b)q(ounded)h(b)q(o)o(x)f (\003)d Ft(\032)g FD(R)583 927 y FA(d)603 945 y FB(,)i(and)h(suc)o(h)g (that,)f(for)h(some)e Fu(c)g(<)f Ft(1)p FB(,)i Ft(j)p Fu(f)5 b FB(\()p Fu(!)r FB(\))p Ft(j)15 b(\024)f Fu(c)p FB(\(1)d(+)g(#\()p Fu(X)16 b Ft(\\)11 b FB(\003\)\))17 b(for)-59 1017 y(all)i Fu(!)i FB(=)e(\()p Fu(X)q(;)8 b FB(\()p Fu(\033)249 1024 y FA(x)271 1017 y FB(\))290 1024 y FA(x)p FF(2)p 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Ft(G)s FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))16 b(is)i(a)g(singleton)g(is)-59 1385 y(equiv)m(alen)o(t)11 b(to)i(sa)o(ying)f(that)h Fu(P)503 1392 y FA(n;z)q(;\014)r(;J)630 1385 y FB(con)o(v)o(erges,)f (ev)o(en)f(in)h Fu(\034)1040 1392 y FF(L)1067 1385 y FB(,)h(to)o(w)o(ards)g(the)f(unique)f(elemen)o(t)f(of)i Ft(G)s FB(\()p Fu(z)r(;)c(\014)s(;)g(J)d FB(\).)-59 1576 y Fw(3)83 b(Results)-59 1704 y FB(Our)16 b(\014rst)h(theorem)d(describ) q(es)i(the)g(general)g(features)g(of)h(the)f(set)g(of)h(all)e(limiting) f(Gibbs)i(states.)-59 1820 y Fs(Theorem)d(3.1)59 b Fv(F)l(or)12 b(any)h Fu(z)r(;)8 b(\014)s(;)g(J)17 b(>)d FB(0)p Fv(,)g(ther)n(e)f(is) g(a)g(\014nite)h(set)g Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))12 b Ft(\032)p FB(]0)p Fu(;)c Ft(1)p FB([)j Fv(such)j(that)f Ft(G)s FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))-59 1892 y Fv(c)n(onsists)18 b(of)f(mixtur)n(es)g(of)g(the)h(me)n(asur)n (es)f Fu(P)754 1899 y FA(\032;\014)r(J)846 1892 y Fv(with)h Fu(\032)c Ft(2)g(M)p FB(\()p 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Fu(J)21 b FB(and)c(is)f(strictly)f (decreasing)h(in)f Fu(J)5 b FB(.)14 970 y(\(b\))24 b(F)l(or)f(an)o(y)h (parameter)e(triple)h(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\),)23 b(a)h(\014rst{order)g(phase)g(transition)g(o)q(ccurs)g(if)f (and)h(only)-59 1042 y(if)f Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))22 b(is)h(not)i(a)f(singleton.)43 b(W)l(e)23 b(then)h(ha)o(v)o(e)f(a)h(jump)e(of)i(the)g(particle)e (densit)o(y)h(and)h(of)g(the)-59 1114 y(magnetization)12 b(p)q(er)i(particle.)19 b(An)14 b(in)o(teresting)e(particular)h(case)h (o)q(ccurs)g(when)f Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))12 b(con)o(tains)i(t)o(w)o(o)-59 1186 y(elemen)o(ts)i Fu(\032)168 1193 y FF(\000)198 1186 y Fu(;)8 b(\032)245 1193 y FC(+)293 1186 y FB(with)18 b Fu(\032)431 1193 y FF(\000)479 1186 y Fu(<)g(D)q(=\014)s(J)23 b(<)18 b(\032)762 1193 y FC(+)792 1186 y FB(,)h(whic)o(h)f(is)h(only)f(p)q(ossible)h(for) g Fu(z)h FB(=)e Fu(z)1513 1193 y FA(m)1546 1186 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\).)28 b(Then)18 b(there)-59 1259 y(are)f(a)h(nonmagnetic)e(phase)i(of)f(densit)o(y)g Fu(\032)741 1266 y FF(\000)788 1259 y FB(and)g(magnetic)f(phases)i(of)g(strictly)e (larger)h(densit)o(y)f Fu(\032)1807 1266 y FC(+)1854 1259 y FB(with)-59 1331 y(arbitrary)e(orien)o(tations.)20 b(This)15 b(corresp)q(onds)g(to)f(a)h(discon)o(tin)o(uous)f(app)q (earance)g(of)h(magnetization)e(whic)o(h)-59 1403 y(is)j(coupled)g(to)g (a)h(v)m(ap)q(or{liquid)g(transition.)14 1482 y(\(c\))k(T)l(o)h(k)o (eep)f(things)g(as)i(simple)c(as)j(p)q(ossible)g(w)o(e)f(did)g(not)h (in)o(tro)q(duce)f(an)h(external)e(\014eld)h(in)o(to)g(our)-59 1554 y(Hamiltonian)15 b(\(2\).)24 b(This)17 b(is)g(the)g(ob)o(vious)g (reason)h(wh)o(y)f(the)g(limiting)d(Gibbs)j(measures)f Fu(P)1653 1561 y FA(\032;\014)r(J)1745 1554 y FB(ab)q(o)o(v)o(e)h(are) -59 1626 y Fv(uniform)h FB(mixtures)e(of)j(magnetic)d(phases)j(when)g Fu(\014)s(J)5 b(\032)16 b(>)h(D)q FB(.)28 b(In)o(tro)q(ducing)18 b(external)g(\014elds)f(whic)o(h)h(tend)-59 1699 y(to)g(0)g(su\016cien) o(tly)e(slo)o(wly)g(as)j Fu(n)d Ft(!)g(1)i FB(and)g(letting)f(also)h Fu(z)h FB(v)m(ary)f(with)g Fu(n)f FB(w)o(e)h(could)f(obtain)h(a)g (particular)-59 1771 y(magnetic)e(phase,)i(or)g(a)g(particular)f (mixture,)e(as)j(limiting)d(Gibbs)j(state,)f(cf.)25 b([1].)g(W)l(e)17 b(lea)o(v)o(e)f(this)h(to)h(the)-59 1843 y(in)o(terested)d(reader.)14 1951 y(W)l(e)20 b(no)o(w)h(turn)g(to)g(a)g(description)f(of)h(v)m (arious)g(scenarios)g(for)f(a)h(\014rst{order)h(transition.)34 b(T)l(o)21 b(b)q(egin,)-59 2024 y(w)o(e)15 b(note)h(that)g(suc)o(h)g(a) g(transition)g(neither)f(o)q(ccurs)h(at)g(high)g(temp)q(eratures)f(nor) h(for)g(w)o(eak)g(ferromagnetic)-59 2096 y(coupling,)e(pro)o(vided)e (suc)o(h)i(a)g(transition)g(is)f(not)i(already)e(induced)g(b)o(y)g(the) h(molecular)d(in)o(teraction)i Fu(g)j FB(alone)-59 2168 y(|)g(whic)o(h)g(can)g(b)q(e)g(excluded)f(b)o(y)h(a)h(con)o(v)o(exit)o (y)c(assumption)j(on)h Fu(g)r FB(.)-59 2276 y Fs(Pr)o(oposition)g(3.2) 64 b FB(\(Absence)16 b(of)g(densit)o(y)f(jumps\))h Fv(Supp)n(ose)h (that)h(either)14 2355 y FB(\(i\))e Fu(g)j Fv(and)f Fu(J)k Fv(ar)n(e)17 b(arbitr)n(arily)f(given)j(and)f Fu(\014)e(>)d FB(0)18 b Fv(is)g(su\016ciently)h(smal)r(l;)f(or)14 2434 y FB(\(ii\))d Fu(g)20 b Fv(is)d(c)n(onvex,)i Fu(\014)h Fv(is)d(\014xe)n(d,)h(and)g Fu(J)g Ft(\025)c FB(0)k Fv(is)f (su\016ciently)i(smal)r(l.)-59 2512 y(Then)f Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))16 b Fv(is)h(a)g(singleton)j(for)d (al)r(l)h Fu(z)e(>)e FB(0)p Fv(.)14 2621 y FB(Next)i(w)o(e)g(state)h(a) g(simple)d(su\016cien)o(t)h(condition)h(for)h(a)g(\014rst-order)g (transition)g(from)f(a)h(non-magnetic)-59 2693 y(v)m(ap)q(or)i(to)e(a)h (magnetic)e(liquid)g(at)i Fu(z)g FB(=)d Fu(z)706 2700 y FA(m)739 2693 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))16 b(according)i(to)g(Scenario)f(A)g(of)h(the)f(in)o(tro)q(duction.)25 b(This)933 2817 y(8)p eop %%Page: 9 10 9 9 bop -59 18 a FB(condition)13 b(holds)h(in)f(particular)g(when)h Fu(g)h FB(is)f(con)o(v)o(ex,)e(i.e.,)g(if)h(\(b)o(y)g(Prop)q(osition)h (3.2\))g(there)f(is)g(no)h(\014rst-order)-59 90 y(phase)j(transition)h (for)f(v)m(anishing)g(magnetic)e(coupling)i Fu(J)j FB(=)15 b(0.)24 b(In)16 b(this)h(case,)g(the)f(jump)g(of)h(densit)o(y)f(and)-59 162 y(magnetization)c(is)h(not)g(induced)g(b)o(y)g(the)f(molecular)g (forces)h(alone,)g(but)g(comes)f(from)g(their)g(in)o(terpla)o(y)g(with) -59 235 y(the)k(ferromagnetic)e(forces.)-59 341 y Fs(Theorem)h(3.3)62 b FB(\(First{order)14 b(transition)g(from)f(nonmagnetic)g(v)m(ap)q(or)i (to)f(ferromagnetic)e(liquid\))h Fv(Sup-)-59 414 y(p)n(ose)22 b Fu(\014)s(;)8 b(J)28 b(>)23 b FB(0)g Fv(ar)n(e)f(such)h(that)g Fu(J)28 b(>)c FB(2)8 b Fu(g)752 396 y FF(00)774 414 y FB(\()p Fu(D)q(=\014)s(J)d FB(\))p Fv(.)38 b(\(Evidently,)25 b(this)e(holds)g(when)g(either)h Fu(\014)h(>)f FB(0)f Fv(is)-59 486 y(arbitr)n(arily)17 b(\014xe)n(d)i(and)f Fu(J)23 b Fv(is)18 b(su\016ciently)i(lar)n(ge,)f(or)f Fu(J)i(>)15 b FB(2)8 b Fu(g)1086 468 y FF(00)1108 486 y FB(\(0\))19 b Fv(is)f(given)i(and)e Fu(\014)j Fv(is)d(su\016ciently)i (lar)n(ge.\))-59 558 y(Then,)e(for)f Fu(z)e FB(=)f Fu(z)274 565 y FA(m)307 558 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))p Fv(,)16 b Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))16 b Fv(c)n(ontains)i(two)g(densities)g Fu(\032)1216 565 y FF(\000)1246 558 y Fu(;)8 b(\032)1293 565 y FC(+)1340 558 y Fv(with)18 b Fu(\032)1471 565 y FF(\000)1514 558 y Fu(<)c(D)q(=\014)s(J)19 b(<)14 b(\032)1785 565 y FC(+)1814 558 y Fv(.)14 665 y FB(A)23 b(fairly)g(complete)e(picture)i(can)h(b)q (e)g(obtained)f(in)h(the)f(lo)o(w{temp)q(erature)f(and)j (strong{coupling)-59 737 y(limits.)g(The)18 b(lo)o(w{temp)q(erature)f (limit)e(dep)q(ends)k(on)g(pieces)e(of)h(non-con)o(v)o(exit)o(y)f(of)h (the)g(function)g Fu(g)r FB(\()p Fu(\032)p FB(\))13 b Ft(\000)-59 809 y Fu(J)5 b(\032)-2 791 y FC(2)17 809 y Fu(=)p FB(2,)17 b(as)g(is)f(sp)q(eci\014ed)g(in)g(the)g(next)f (de\014nition.)-59 916 y Fs(Definition.)48 b FB(Let)17 b Fu(f)k FB(:)15 b([0)p Fu(;)8 b Ft(1)p FB([)p Ft(!)15 b FD(R)i FB(b)q(e)g(con)o(tin)o(uous)g(and)h(b)q(ounded)g(from)f(b)q (elo)o(w,)g(and)g(let)g Fu(f)1753 898 y FF(\003\003)1808 916 y FB(denote)-59 988 y(its)f(con)o(v)o(ex)g(en)o(v)o(elop)q(e.)k(W)l (e)d(will)e(sa)o(y)i(a)g(nondegenerate)g(in)o(terv)m(al)e([)p Fu(r)1225 995 y FF(\000)1254 988 y Fu(;)8 b(r)1298 995 y FC(+)1327 988 y FB(])17 b(is)f Fv(critic)n(al)h FB(for)g Fu(f)22 b FB(with)16 b Fv(slop)n(e)h Fu(\015)-59 1060 y FB(if)14 1139 y(\(i\))f Fu(f)111 1121 y FF(\003\003)162 1139 y Fu(>)e(f)22 b FB(on)17 b(])p Fu(r)364 1146 y FF(\000)393 1139 y Fu(;)8 b(r)437 1146 y FC(+)466 1139 y FB([)13 b(;)14 1218 y(\(ii\))h(for)h(some)f Fu(")g(>)g FB(0,)h Fu(f)459 1200 y FF(\003\003)511 1218 y FB(and)h Fu(f)k FB(coincide)14 b(and)h(are)g(strictly)f(con)o(v)o(ex)f(on)j([)p Fu(r)1435 1225 y FC(+)1464 1218 y Fu(;)8 b(r)1508 1225 y FC(+)1545 1218 y FB(+)h Fu(")p FB(])14 b(and,)h(if)g Fu(r)1816 1225 y FF(\000)1859 1218 y Fu(>)f FB(0,)-59 1290 y(also)j(on)g([)p Fu(r)143 1297 y FF(\000)183 1290 y Ft(\000)10 b Fu(";)e(r)299 1297 y FF(\000)329 1290 y FB(];)15 b(and)14 1369 y(\(iii\))g Fu(\015)k FB(is)d(the)g(slop)q(e)h (of)f Fu(f)494 1351 y FF(\003\003)548 1369 y FB(on)h([)p Fu(r)652 1376 y FF(\000)681 1369 y Fu(;)8 b(r)725 1376 y FC(+)754 1369 y FB(].)-59 1475 y Fs(Theorem)25 b(3.4)71 b FB(\(Lo)o(w)23 b(temp)q(erature)e(b)q(eha)o(vior\))i Fv(L)n(et)g Fu(J)30 b Ft(\025)25 b FB(0)f Fv(b)n(e)f(\014xe)n(d.)41 b(Supp)n(ose)24 b FB([)p Fu(r)1675 1482 y FF(\000)1704 1475 y Fu(;)8 b(r)1748 1482 y FC(+)1777 1475 y FB(])24 b Fv(is)f(any)-59 1548 y(critic)n(al)18 b(interval)i(of)e(the)g (function)i Fu(g)640 1555 y FA(J)664 1548 y FB(\()p Fu(\032)p FB(\))15 b(=)g Fu(g)r FB(\()p Fu(\032)p FB(\))d Ft(\000)f Fu(J)i(\032)1010 1529 y FC(2)1030 1548 y Fu(=)p FB(2)p Fv(,)19 b(and)f Fu(\015)j Fv(is)d(the)h(asso)n(ciate)n(d)e(slop)n(e.)25 b(Then)18 b(the)-59 1620 y(fol)r(lowing)i(is)d(true.)14 1698 y FB(\(a\))12 b Fv(F)l(or)h(any)g Fu(")h(>)f FB(0)h Fv(and)f(su\016ciently)i(lar)n(ge)e Fu(\014)s Fv(,)h(ther)n(e)f(exists) h(a)f(unique)h(activity)g Fu(z)h Fv(with)f Ft(j)p Fu(\014)1685 1680 y FF(\000)p FC(1)1739 1698 y FB(ln)8 b Fu(z)t Ft(\000)r Fu(\015)s Ft(j)13 b Fu(<)-59 1771 y(")24 b Fv(such)h(that)f Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))22 b Fv(c)n(ontains)j(two)g(distinct)g(elements)h Fu(\032)1175 1778 y FF(\000)1205 1771 y Fu(;)8 b(\032)1252 1778 y FC(+)1306 1771 y Fv(satisfying)24 b Ft(j)p Fu(\032)1568 1778 y FF(\000)1614 1771 y Ft(\000)16 b Fu(r)1691 1778 y FF(\000)1720 1771 y Ft(j)26 b Fu(<)h(")d Fv(and)-59 1843 y Ft(j)p Fu(\032)-20 1850 y FC(+)21 1843 y Ft(\000)12 b Fu(r)94 1850 y FC(+)124 1843 y Ft(j)k Fu(<)g(")p Fv(.)26 b(\()18 b Fu(z)r Fv(,)h Fu(\032)394 1850 y FF(\000)443 1843 y Fv(and)f Fu(\032)563 1850 y FC(+)612 1843 y Fv(dep)n(end)h(on)g Fu(\014)s Fv(,)f Fu(")p Fv(,)h(the)g(critic)n(al)g(interval)h FB([)p Fu(r)1430 1850 y FF(\000)1459 1843 y Fu(;)8 b(r)1503 1850 y FC(+)1533 1843 y FB(])18 b Fv(and,)h(of)f(c)n(ourse,)h Fu(J)5 b Fv(,)-59 1915 y Fu(g)r Fv(,)17 b(and)h Fu(D)q Fv(.\))14 1994 y FB(\(b\))23 b Fv(Supp)n(ose)g(further)g(that)h Fu(g)593 1976 y FF(00)591 2006 y FA(J)616 1994 y FB(\()p Fu(r)657 2001 y FC(+)686 1994 y FB(\))h Fu(>)g FB(0)f Fv(and)f(either,)j(if)d Fu(r)1178 2001 y FF(\000)1232 1994 y Fu(>)i FB(0)p Fv(,)g Fu(g)1384 1976 y FF(00)1382 2006 y FA(J)1407 1994 y FB(\()p Fu(r)1448 2001 y FF(\000)1477 1994 y FB(\))g Fu(>)g FB(0)f Fv(or,)g(if)f Fu(r)1792 2001 y FF(\000)1847 1994 y FB(=)h(0)p Fv(,)-59 2066 y Fu(g)-34 2048 y FF(0)-36 2078 y FA(J)-11 2066 y FB(\(0\))14 b Fu(>)g(\015)s Fv(.)22 b(If)17 b Fu(\014)j Fv(is)d(lar)n(ge)h(enough)h (and)e Fu(z)j Fv(is)d(as)g(in)g FB(\(a\))f Fv(then)j Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))12 b(=)h Ft(f)p Fu(\032)1457 2073 y FF(\000)1487 2066 y Fu(;)8 b(\032)1534 2073 y FC(+)1564 2066 y Ft(g)p Fu(:)14 2145 y FB(\(c\))17 b Fv(Supp)n(ose,)i(in)g(addition,)g(that)g Fu(g)687 2127 y FF(00)685 2157 y FA(J)726 2145 y Fu(>)d FB(0)j Fv(on)g FB([)p Fu(r)930 2152 y FC(+)959 2145 y Fu(;)8 b Ft(1)p FB([)18 b Fv(and,)h(if)g Fu(r)1245 2152 y FF(\000)1290 2145 y Fu(>)d FB(0)p Fv(,)k(on)f FB([0)p Fu(;)8 b(r)1556 2152 y FF(\000)1585 2145 y FB(])p Fv(,)19 b(so)f(that)h FB([)p Fu(r)1833 2152 y FF(\000)1862 2145 y Fu(;)8 b(r)1906 2152 y FC(+)1935 2145 y FB(])-59 2217 y Fv(is)17 b(the)g(only)h(critic)n(al)f(interval)h(of)f Fu(g)600 2224 y FA(J)625 2217 y Fv(.)22 b(In)17 b(the)h(c)n(ase)f Fu(r)932 2224 y FF(\000)975 2217 y Fu(>)d FB(0)p Fv(,)j(we)h(ne)n(e)n (d)f(to)g(assume)h(further)e(that)i Fu(D)d FB(=)f(1)j Fv(or)-59 2289 y Fu(g)-34 2271 y FF(00)-36 2301 y FA(J)-11 2289 y FB(\(0\))d Fu(>)g(J)5 b FB(\()p Fu(b)p FB(\()p Fu(D)q FB(\))k Ft(\000)g FB(1\))17 b Fv(for)f(the)i(c)n(onstant)719 2271 y FC(3)756 2289 y Fu(b)p FB(\()p Fu(D)q FB(\))f Fv(de\014ne)n(d)h(by)e(\(19\))h(.)22 b(Then,)17 b(for)f(su\016ciently)i (lar)n(ge)f Fu(\014)s Fv(,)f(the)-59 2361 y(activity)i Fu(z)h Fv(in)e FB(\(a\))g Fv(is)g(the)h(unique)h(value)f(of)g Fu(z)h Fv(for)e(which)h Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))16 b Fv(is)h(not)h(a)f(singleton.)14 2468 y FB(The)g(b)q(eha)o (vior)f(for)h(large)f(magnetic)f(coupling)h(is)g(simpler)f(b)q(ecause)h (it)g(is)g(indep)q(enden)o(t)g(of)h(the)f(shap)q(e)-59 2540 y(of)g Fu(g)r FB(.)p -59 2587 804 2 v -3 2617 a Fx(3)16 2632 y Fy(Numerical)e(calculations)g(suggest)j(that)e Fl(b)p Fy(\()p Fl(D)q Fy(\))f(=)h(1)g(for)g Fl(D)g Fe(\025)f Fy(2,)h(so)h(that)f(the)h(additional)d(assumption)h(is)i(redundan)o(t.) -59 2692 y(But)e(w)o(e)h(could)e(not)h(pro)o(v)o(e)g(this.)933 2817 y FB(9)p eop %%Page: 10 11 10 10 bop -59 18 a Fs(Theorem)25 b(3.5)71 b FB(\(Strong)24 b(ferromagnetic)d(coupling\))i Fv(L)n(et)g Fu(\014)28 b(>)d FB(0)f Fv(b)n(e)g(given)i(and)d Fu(")j(>)f FB(0)p Fv(.)41 b(If)24 b Fu(J)k Fv(is)-59 90 y(su\016ciently)20 b(lar)n(ge,)g(ther)n(e)f(exists)g(some)g Fu(z)g(>)d FB(0)k Fv(with)f Fu(J)985 72 y FF(\000)p FC(1)1040 90 y FB(ln)8 b Fu(z)19 b(<)d Ft(\000)p Fu(")1247 72 y FF(\000)p FC(1)1313 90 y Fv(such)j(that)g Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))17 b Fv(c)n(ontains)-59 162 y(two)f(densities)g Fu(\032)253 169 y FF(\000)298 162 y Fv(and)f Fu(\032)415 169 y FC(+)460 162 y Fv(satisfying)h Fu(\032)700 169 y FF(\000)743 162 y Fu(<)e(")h Fv(and)h Fu(\032)951 169 y FC(+)994 162 y Fu(>)e(")1069 144 y FF(\000)p FC(1)1116 162 y Fv(.)22 b(F)l(or)14 b Fu(D)i FB(=)e(1)i Fv(and)f(any)g(other)g Fu(D)j Fv(for)c(which)-59 235 y(L)n(emma)i(5.2\(g\))i(holds)f(we)i (have)f Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))12 b(=)i Ft(f)p Fu(\032)911 242 y FF(\000)940 235 y Fu(;)8 b(\032)987 242 y FC(+)1017 235 y Ft(g)p Fv(,)17 b(and)h FB(#)p Ft(M)p FB(\()p Fu(z)1314 217 y FF(0)1325 235 y Fu(;)8 b(\014)s(;)g(J)d FB(\))12 b(=)i(1)k Fv(for)e(al)r(l)j Fu(z)1729 217 y FF(0)1754 235 y Ft(6)p FB(=)14 b Fu(z)r Fv(.)14 351 y FB(As)j(w)o(e)h(ha)o(v)o(e)e(claimed)g(in)h(Scenario)g(B) g(of)h(the)f(In)o(tro)q(duction)h(it)f(can)h(happ)q(en)g(that,)g(for)g (large)f Fu(\014)s FB(,)g(the)-59 423 y(ferromagnetic)e(transition)i (at)g Fu(z)555 430 y FA(m)604 423 y FB(is)f(of)h(second)g(order)g(but,) f(at)h(some)e Fu(z)i(>)d(z)1395 430 y FA(m)1428 423 y FB(,)i(a)h(jump)e(of)i(b)q(oth)h(densit)o(y)-59 495 y(and)i (magnetization)f(o)q(ccurs.)32 b(This)20 b(is)f(the)h(sub)s(ject)f(of)h (the)f(next)g(corollary)h(whic)o(h)f(applies)g(to)h(con)o(v)o(ex)-59 567 y(functions)c(lik)o(e)f Fu(g)r FB(\()p Fu(\032)p FB(\))f(=)f(\()p Fu(\032)f Ft(\000)e FB(1\))542 549 y FC(4)563 567 y FB(.)-59 683 y Fs(Cor)o(ollar)m(y)20 b(3.6)67 b FB(\(First-order)20 b(transition)g(b)q(et)o(w)o(een)f(ferromagnetic)f (phases\))j Fv(Supp)n(ose)f Fu(g)1762 665 y FF(00)1804 683 y Fv(attains)-59 756 y(its)g(minimum)g(0)f(at)h(a)g(single)h(p)n (oint)f Fu(\032)675 763 y FC(min)754 756 y Fu(>)f FB(0)p Fv(.)30 b(If)19 b Fu(J)k(>)18 b FB(0)i Fv(is)g(su\016ciently)i(smal)r (l)e(and)g Fu(\014)j Fv(is)c(su\016ciently)-59 828 y(lar)n(ge,)f(ther)n (e)f(is)g(then)i(a)e(unique)i Fu(z)c(>)f(z)668 835 y FA(m)701 828 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))16 b Fv(such)i(that)g Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))12 b(=)h Ft(f)p Fu(\032)1394 835 y FF(\000)1424 828 y Fu(;)8 b(\032)1471 835 y FC(+)1501 828 y Ft(g)17 b Fv(with)h Fu(D)q(=\014)s(J)h(<)13 b(\032)1867 835 y FF(\000)1911 828 y Fu(<)-59 900 y(\032)-34 907 y FC(min)41 900 y Fu(<)h(\032)118 907 y FC(+)147 900 y Fv(.)14 1016 y FB(Next)d(w)o(e)h(consider)h(the)f(case)g(when)h Fu(g)h FB(is)f(not)f(con)o(v)o(ex,)g(whic)o(h)g(means)f(that)i(the)f (molecular)e(forces)j(alone)-59 1088 y(already)j(imply)e(a)i (\014rst{order)h(phase)g(transition)f(when)g Fu(\014)j FB(is)d(large)g(enough.)22 b(F)l(or)16 b(large)g Fu(J)5 b FB(,)16 b(Theorem)e(3.3)-59 1161 y(asserts)20 b(that)h(there)e(is)h (a)g(discon)o(tin)o(uous)f(transition)h(to)h(ferromagnetic)d(order)i (at)g Fu(z)h FB(=)f Fu(z)1664 1168 y FA(m)1697 1161 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\).)31 b(The)-59 1233 y(next)23 b(prop)q(osition)h(sho)o(ws)g(that,)h(for)e(small)f Fu(J)5 b FB(,)24 b(the)f(densit)o(y)f(jumps)h(induced)f(b)o(y)h Fu(g)i FB(do)f(not)g(imply)c(a)-59 1305 y(ferromagnetic)13 b(ordering,)i(so)h(that)f(the)g(magnetization)f(dep)q(ends)h(con)o(tin) o(uously)f(on)i Fu(z)r FB(.)k(\(This)c(statemen)o(t)-59 1377 y(ma)o(y)f(b)q(e)h(view)o(ed)f(as)i(a)f(coun)o(terpart)h(to)f (Prop)q(osition)i(3.2\(ii\))d(for)i(non{con)o(v)o(ex)f Fu(g)r FB(.\))-59 1493 y Fs(Pr)o(oposition)34 b(3.7)79 b FB(\(First{order)32 b(transition)f(b)q(et)o(w)o(een)g(non-magnetic)g (phases\))h Fv(Supp)n(ose)g(that)-59 1565 y FB(min)22 1572 y FA(\032>)p FC(0)96 1565 y Fu(\032)8 b(g)154 1547 y FF(00)176 1565 y FB(\()p Fu(\032)p FB(\))24 b Ft(\021)g(\000)p FB(1)p Fu(=\014)441 1572 y FC(min)526 1565 y Fu(<)g FB(0)p Fv(,)g Fu(\014)j(>)d(\014)796 1572 y FC(min)856 1565 y Fv(,)g(and)g Fu(J)j Fv(is)c(su\016ciently)i(smal)r(l.)40 b(Then)23 b(ther)n(e)g(exists)h(a)-59 1638 y(unique)i FB(0)i Fu(<)f(z)j(<)d(z)363 1645 y FA(m)396 1638 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))23 b Fv(such)i(that)g Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))26 b(=)h Ft(f)p Fu(\032)1138 1645 y FF(\000)1168 1638 y Fu(;)8 b(\032)1215 1645 y FC(+)1244 1638 y Ft(g)25 b Fv(with)g Fu(\032)1432 1645 y FF(\000)1489 1638 y Fu(<)j(\032)1580 1645 y FC(+)1637 1638 y Fu(<)f(D)q(=\014)s(J)5 b Fv(,)27 b(and)-59 1710 y FB(#)p Ft(M)p FB(\()p Fu(z)86 1692 y FF(0)97 1710 y Fu(;)8 b(\014)s(;)g(J)d FB(\))12 b(=)i(1)f Fv(whenever)i Fu(z)557 1692 y FF(0)583 1710 y Fu(>)e(z)r Fv(.)21 b(In)13 b(p)n(articular,)h(the)f(ferr)n(omagnetic)g(phase)g(tr)n(ansition)g(at) g Fu(z)1794 1717 y FA(m)1827 1710 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))-59 1782 y Fv(is)17 b(of)h(se)n(c)n(ond)f(or)n(der.)14 1898 y FB(Our)24 b(\014nal)h(result)f(pro)o(vides)g(conditions)g(for)g (the)h(existence)d(of)j(t)o(w)o(o)f(\014rst{order)h(transition)g(lines) -59 1970 y(whic)o(h)d(join)h(at)g(a)h(triple)d(p)q(oin)o(t)i(with)g (three)g(di\013eren)o(t)f(phases.)42 b(The)23 b(\014rst)g(of)g(these)g (transition)g(lines)-59 2043 y(comes)c(from)g(the)h(molecular)e(forces) i(alone,)h(whereas)g(the)f(second)g(is)g(again)h(the)f(result)g(of)g(a) h(feedbac)o(k)-59 2115 y(b)q(et)o(w)o(een)14 b(molecular)f(and)j (magnetic)e(forces.)20 b(After)14 b(the)h(joining)g(at)h(the)f(triple)f (p)q(oin)o(t,)h(one)g(has)h(the)f(same)-59 2187 y(picture)f(as)i(in)f (Theorem)f(3.3)i(|)f(a)h(discon)o(tin)o(uous)f(incipience)e(of)j (ferromagnetic)e(order)h(at)h Fu(z)f FB(=)f Fu(z)1794 2194 y FA(m)1827 2187 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))-59 2259 y(together)18 b(with)g(a)h(jump)e(of)h(densit)o(y)f(whic)o(h)h(is) g(partly)g(due)g(to)g(the)g(molecular)e(forces)i(but)g(enhanced)g(b)o (y)-59 2332 y(the)e(ferromagnetic)e(spin-in)o(teraction.)-59 2448 y Fs(Theorem)19 b(3.8)65 b FB(\(Existence)16 b(of)i(a)f(triple)f (p)q(oin)o(t)i(with)f(phases)h(of)f(three)g(di\013eren)o(t)f (densities\))h Fv(Supp)n(ose)-59 2520 y(ther)n(e)g(exists)i(some)e (unique)i Fu(\032)502 2527 y FC(min)577 2520 y Fu(>)14 b FB(0)k Fv(such)f(that)478 2642 y Fu(\032)503 2649 y FC(min)578 2642 y Fu(g)603 2621 y FF(00)624 2642 y FB(\()p Fu(\032)668 2649 y FC(min)729 2642 y FB(\))d(=)g(min)823 2670 y FA(\032>)p FC(0)903 2642 y Fu(\032)g(g)967 2621 y FF(00)989 2642 y FB(\()p Fu(\032)p FB(\))g Ft(\021)f(\000)p FB(1)p Fu(=\014)1233 2649 y FC(min)1308 2642 y Fu(<)h FB(0)g Fu(:)920 2817 y FB(10)p eop %%Page: 11 12 11 11 bop -59 18 a Fv(If)25 b Fu(\014)k(>)f(\014)151 25 y FC(min)236 18 y Fv(is)d(su\016ciently)h(close)g(to)f Fu(\014)770 25 y FC(min)831 18 y Fv(,)i(ther)n(e)d(exist)i(unique)h (numb)n(ers)e Fu(z)1511 25 y FA(c)1555 18 y FB(=)j Fu(z)1644 25 y FA(c)1661 18 y FB(\()p Fu(\014)s FB(\))f Fu(>)g FB(0)f Fv(and)-59 90 y Fu(J)-32 97 y FA(c)3 90 y FB(=)17 b Fu(J)85 97 y FA(c)102 90 y FB(\()p Fu(\014)s FB(\))g Fu(>)g FB(0)j Fv(such)f(that)h Ft(M)p FB(\()p Fu(z)603 97 y FA(c)620 90 y Fu(;)8 b(\014)s(;)g(J)722 97 y FA(c)738 90 y FB(\))17 b Ft(\033)h(f)p Fu(\032)881 97 y FF(\000)910 90 y Fu(;)8 b(\032)957 97 y FA(])973 90 y Fu(;)g(\032)1020 97 y FC(+)1049 90 y Ft(g)p Fv(,)20 b(wher)n(e)g FB(0)d Fu(<)h(\032)1371 97 y FF(\000)1418 90 y Fu(<)f(\032)1498 97 y FA(])1531 90 y Fu(<)g(D)q(=\014)s(J)1709 97 y FA(c)1745 90 y Fu(<)g(\032)1825 97 y FC(+)1855 90 y Fv(.)28 b(A)o(t)-59 162 y(le)n(ast)19 b(if)f Fu(D)g FB(=)d(1)k Fv(and)g Fu(g)376 144 y FF(00)416 162 y Fv(is)f(c)n(onvex,)i(we)f(have)g Ft(M)p FB(\()p Fu(z)930 169 y FA(c)947 162 y Fu(;)8 b(\014)s(;)g(J)1049 169 y FA(c)1066 162 y FB(\))15 b(=)h Ft(f)p Fu(\032)1204 169 y FF(\000)1233 162 y Fu(;)8 b(\032)1280 169 y FA(])1296 162 y Fu(;)g(\032)1343 169 y FC(+)1373 162 y Ft(g)p Fv(.)25 b(In)19 b(this)f(c)n(ase)h(we)g(c)n(an)g(also)-59 235 y(state,)f(writing)g Fu(z)262 242 y FA(J)300 235 y FB(=)c Fu(z)375 242 y FA(m)408 235 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))p Fv(:)14 313 y(\(a\))15 b(If)f Fu(J)19 b(>)13 b(J)266 320 y FA(c)299 313 y Fv(and)i Fu(J)k Fv(is)c(su\016ciently)h (close)g(to)f Fu(J)930 320 y FA(c)962 313 y Fv(then)h Fu(z)1091 320 y FA(J)1129 313 y Fu(<)e(z)1204 320 y FA(c)1221 313 y Fv(,)h Fu(z)1274 320 y FA(J)1313 313 y Fv(is)g(close)h(to)f Fu(z)1559 320 y FA(c)1576 313 y Fv(,)g(and)g Ft(M)p FB(\()p Fu(z)1800 320 y FA(J)1824 313 y Fu(;)8 b(\014)s(;)g(J)d FB(\))-59 386 y Fv(c)n(ontains)18 b(two)g(distinct)g(elements)i(arbitr) n(arily)c(close)i(to)g Fu(\032)1034 393 y FF(\000)1081 386 y Fv(r)n(esp.)f Fu(\032)1224 393 y FC(+)1253 386 y Fv(.)14 464 y(\(b\))h(If)e Fu(J)j(<)14 b(J)269 471 y FA(c)303 464 y Fv(and)k Fu(J)j Fv(is)c(su\016ciently)i(close)g(to)e Fu(J)949 471 y FA(c)983 464 y Fv(then)h Fu(\032)1116 471 y FF(\000)1146 464 y Fu(;)8 b(\032)1193 471 y FA(])1223 464 y Ft(2)14 b(M)p FB(\()p Fu(z)1372 471 y FA(c)1389 464 y Fu(;)8 b(\014)s(;)g(J)d FB(\))p Fv(,)15 b Fu(z)1568 471 y FA(J)1606 464 y Fu(>)f(z)1681 471 y FA(c)1698 464 y Fv(,)j Fu(z)1753 471 y FA(J)1795 464 y Fv(is)g(close)-59 536 y(to)g Fu(z)22 543 y FA(c)39 536 y Fv(,)h(and)g Ft(M)p FB(\()p Fu(z)269 543 y FA(J)292 536 y Fu(;)8 b(\014)s(;)g(J)d FB(\))16 b Fv(c)n(ontains)i(two)g(distinct)h(elements)g(arbitr)n(arily) d(close)j(to)e Fu(\032)1527 543 y FA(])1561 536 y Fv(r)n(esp.)f Fu(\032)1703 543 y FC(+)1733 536 y Fv(.)14 652 y FB(It)g(will)f(b)q(e)h (eviden)o(t)f(from)g(the)g(pro)q(of)j(that)e(if)g Fu(g)i FB(alone)e(already)g(admits)f(three)h(or)g(more)f(nonmagnetic)-59 725 y(phases)20 b(at)g(some)f Fu(\014)i FB(then,)f(for)g(suitable)f Fu(J)24 b FB(=)19 b Fu(J)864 732 y FA(c)882 725 y FB(\()p Fu(\014)s FB(\),)g(the)g(spin-in)o(teraction)g(leads)g(to)h(the)g (existence)e(of)-59 797 y(additional)e(phases)h(of)g(higher)f(densit)o (y)l(.)k(This)c(implies)e(the)i(existence)f(of)h(quadruple)g(p)q(oin)o (ts,)g(and)h(so)g(on.)14 876 y(The)12 b(pro)q(of)h(will)d(also)j(sho)o (w)f(that)g Fu(J)658 883 y FA(c)676 876 y FB(\()p Fu(\014)s FB(\))f(is)g(con)o(tin)o(uous)h(and)g(strictly)f(monotone)g(whenev)o (er)g(the)g(critical)-59 948 y(activit)o(y)j(for)j(the)e(densit)o(y)g (jump)g(induced)h(b)o(y)f Fu(g)j FB(alone)e(is)g(a)h(strictly)d (monotone)i(function)g(of)g Fu(\014)j FB(|)d(whic)o(h)-59 1020 y(can)g(b)q(e)h(c)o(hec)o(k)o(ed)d(in)i(sp)q(ecial)f(cases.)22 b(Then)16 b(it)g(is)g(p)q(ossible)h(to)f(replace)g(the)g(indep)q(enden) o(t)f(parameter)g Fu(J)21 b FB(in)-59 1092 y(Theorem)15 b(3.8)h(b)o(y)g Fu(\014)s FB(,)f(and)i(w)o(e)f(obtain)h(a)f(statemen)o (t)f(as)i(in)f(Scenario)g(C)g(of)h(the)f(In)o(tro)q(duction.)14 1171 y(Finally)h(w)o(e)g(note)h(that)g(the)f(\014rst-order)i (transition)f(line)e(describ)q(ed)i(in)f(statemen)o(t)f(\(a\))i(ma)o(y) e(split)h(at)-59 1243 y(some)g(critical)f(p)q(oin)o(t)h(in)o(to)h(t)o (w)o(o)f(branc)o(hes)h(corresp)q(onding)h(to)f(a)g(second-order)g (transition)g(to)g(ferromag-)-59 1315 y(netic)g(order)h(at)g Fu(z)h FB(=)e Fu(z)377 1322 y FA(m)429 1315 y FB(and,)h(on)h(the)e (other)h(hand,)h(to)f(jumps)f(of)h(densit)o(y)f(and)h(magnetization)f (in)g(the)-59 1388 y(regime)10 b(of)j(p)q(ositiv)o(e)e(magnetization.) 19 b(This)12 b(do)q(es)h(in)f(fact)g(o)q(ccur)g(in)g(situations)h (similar)d(to)j(those)f(describ)q(ed)-59 1460 y(in)k(Corollary)g(3.6.) -59 1651 y Fw(4)83 b(The)27 b(maxim)n(um)f(en)n(trop)n(y)g(principle) -59 1779 y FB(Theorem)13 b(3.1)i(will)e(b)q(e)h(deriv)o(ed)f(from)g(a)i (more)e(general)h(conditional)g(limit)d(theorem)i(whic)o(h)h(w)o(as)h (obtained)-59 1851 y(in)d([8].)20 b(In)12 b(this)h(section)f(w)o(e)g (will)g(pro)o(vide)g(the)g(necessary)h(details.)19 b(\(W)l(e)13 b(migh)o(t)e(replace)h Fu(E)j FB(b)o(y)e(an)g(arbitrary)-59 1923 y(P)o(olish)h(space)g(in)f(the)h(follo)o(wing,)f(but)i(for)f (simplicit)n(y)d(w)o(e)i(stic)o(k)g(to)h(the)g(setting)g(of)g(the)g (previous)f(sections.\))14 2002 y(Let)21 b Ft(M)166 2009 y FA(E)216 2002 y FB(denote)f(the)g(space)g(of)h(all)f(\014nite)g (Borel)f(measures)g(on)i Fu(E)s FB(.)34 b Ft(M)1425 2009 y FA(E)1475 2002 y FB(will)19 b(b)q(e)h(equipp)q(ed)g(with)-59 2074 y(the)g(smallest)e(top)q(ology)j(whic)o(h)e(is)h(suc)o(h)f(that,)i (for)f(an)o(y)g(b)q(ounded)h(measurable)d(function)i Fu(f)25 b FB(on)c Fu(E)s FB(,)f(the)-59 2146 y(ev)m(aluation)e(map)e Fu(\027)j Ft(!)395 2111 y Fn(R)431 2146 y Fu(f)14 b(d\027)21 b FB(is)c(con)o(tin)o(uous.)25 b(The)17 b Fv(r)n(elative)j(entr)n(opy)d FB(of)h(t)o(w)o(o)f(measures)f Fu(\026;)8 b(\027)20 b Ft(2)c(M)1869 2153 y FA(E)1916 2146 y FB(is)-59 2219 y(de\014ned)g(b)o(y)304 2320 y Fu(I)t FB(\()p Fu(\027)s FB(;)8 b Fu(\026)p FB(\))13 b(=)511 2233 y Fn(8)511 2270 y(<)511 2345 y(:)569 2250 y(R)596 2286 y FB(\(1)f Ft(\000)e Fu(f)17 b FB(+)11 b Fu(f)22 b FB(ln)8 b Fu(f)d FB(\))14 b Fu(d\026)42 b FB(if)16 b Fu(\027)h Ft(\034)d Fu(\026)i FB(with)g(densit)o(y)g Fu(f)5 b FB(,)760 2358 y Ft(1)233 b FB(otherwise)p Fu(:)1886 2320 y FB(\(5\))-59 2450 y(It)16 b(is)g(w)o(ell-kno)o(wn)f(and)i(easy)f(to)h(see)f(that)h Fu(I)t FB(\()p Fu(\027)s FB(;)8 b Fu(\026)p FB(\))13 b Ft(\025)h FB(0)j(with)f(equalit)o(y)e(if)i(and)h(only)f(if)g Fu(\027)h FB(=)d Fu(\026)p FB(.)21 b(Moreo)o(v)o(er,)-59 2523 y(for)c(an)o(y)g(\014xed)f Fu(\026)i FB(and)f Fu(c)e Ft(\025)g FB(0)i(the)f(sublev)o(el-set)g Ft(f)p Fu(\027)i Ft(2)d(M)1021 2530 y FA(E)1065 2523 y FB(:)g Fu(I)t FB(\()p Fu(\027)s FB(;)8 b Fu(\026)p FB(\))15 b Ft(\024)f Fu(c)p Ft(g)j FB(is)g(compact,)e(cf.)23 b([8].)f(W)l(e)17 b(will)-59 2595 y(need)i(the)g(relativ)o(e)e(en)o(trop)o(y)i(in)g(the)g(sp)q (ecial)g(case)g(when)g Fu(\026)h FB(is)f(a)h(m)o(ultiple)c(of)k(our)f (a)h(priori)f(measure)f Fu(\025)p FB(,)-59 2667 y(and)f(then)f(use)g (the)g(notation)h Fu(I)532 2674 y FA(z)552 2667 y FB(\()p Fu(\027)s FB(\))d(=)f Fu(I)t FB(\()p Fu(\027)s FB(;)8 b Fu(z)r(\025)p FB(\))16 b(for)h Fu(z)f(>)d FB(0.)920 2817 y(11)p eop %%Page: 12 13 12 12 bop 14 18 a FB(The)22 b(main)f(quan)o(tit)o(y)g(of)h(in)o(terest) f(is)h(the)f Fv(empiric)n(al)i(spin)g(intensity)h(me)n(asur)n(e)d FB(of)h(a)h(con\014guration)-59 90 y Fu(!)16 b Ft(2)e FB(\012)69 97 y FA(n)109 90 y FB(whic)o(h)i(is)g(de\014ned)g(b)o(y)633 162 y Fu(L)666 169 y FA(n;!)737 162 y FB(=)d Fu(V)828 142 y FF(\000)p FC(1)816 175 y FA(n)891 121 y Fn(X)883 213 y FA(x)p FF(2)p FA(X)967 162 y Fu(\016)989 169 y FA(\033)1009 173 y Fm(x)1092 162 y Ft(2)h(M)1199 169 y FA(E)1243 162 y Fu(;)629 b FB(\(6\))-59 285 y(where)12 b Fu(\016)100 292 y FA(a)133 285 y FB(stands)i(for)f(Dirac)f(measure)f (at)i Fu(a)p FB(.)20 b(Its)12 b(total)h(mass)f(is)h(nothing)g(other)f (than)i(the)e(particle)f(densit)o(y)l(,)-59 357 y Fu(L)-26 364 y FA(n;!)31 357 y FB(\()p Fu(E)s FB(\))i(=)h(#)p Fu(X=V)306 364 y FA(n)330 357 y FB(,)i(and)h(the)f(normalized)e(in)o (tegral)550 432 y Fn(Z)573 526 y FA(E)612 491 y Fu(\033)h(L)688 498 y FA(n;!)744 491 y FB(\()p Fu(d\033)r FB(\))837 443 y Fn(.)871 491 y Fu(L)904 498 y FA(n;!)961 491 y FB(\()p Fu(E)s FB(\))e(=)1138 457 y(1)p 1108 479 86 2 v 1108 525 a(#)p Fu(X)1214 449 y Fn(X)1206 541 y FA(x)p FF(2)p FA(X)1290 491 y Fu(\033)1318 498 y FA(x)-59 636 y FB(is)j(the)g(a)o(v)o (erage)g(magnetization.)14 714 y(The)f(follo)o(wing)g(theorem)e(is)i(a) h(sp)q(ecial)e(case)h(of)h(Corollary)f(3.6)g(of)h([8].)k(It)15 b(is)g(a)g(p)q(oin)o(t)g(pro)q(cess)h(coun)o(ter-)-59 787 y(part)g(of)f(Sano)o(v's)g(large)h(deviation)e(principle)g(and)h (Csiszar's)h(asso)q(ciated)g(conditional)f(limit)d(theorem,)i(cf.)-59 859 y([3].)14 938 y(Let)19 b Fu(H)j FB(:)17 b Ft(M)257 945 y FA(E)304 938 y Ft(!)h FD(R)g FB(b)q(e)h(a)g(con)o(tin)o(uous)g (functional)f(suc)o(h)g(that)h Fu(H)t FB(\()p Fu(\027)s FB(\))f Ft(\025)g(\000)p Fu(b)8 b(\027)s FB(\()p Fu(E)s FB(\))18 b(for)h(all)f Fu(\027)j Ft(2)d(M)1919 945 y FA(E)-59 1010 y FB(and)f(some)e Fu(b)f(<)f Ft(1)p FB(.)21 b(W)l(e)16 b(consider)g(the)g(lo)q(cal)g(Hamiltonians)608 1132 y Fu(H)648 1139 y FA(n)672 1132 y FB(\()p Fu(!)r FB(\))e(=)g Fu(V)836 1139 y FA(n)874 1132 y Fu(H)t FB(\()p Fu(L)970 1139 y FA(n;!)1027 1132 y FB(\))p Fu(;)56 b(!)16 b Ft(2)e FB(\012)1244 1139 y FA(n)1268 1132 y Fu(;)604 b FB(\(7\))-59 1254 y(and)12 b(the)g(asso)q(ciated)h(partition)f (functions)g Fu(Z)777 1261 y FA(n;z)q(;\014)r(;H)913 1254 y FB(and)g(Gibbs)g(distribution)g Fu(P)1433 1261 y FA(n;z)q(;\014)r(;H)1569 1254 y FB(de\014ned)f(in)h(analogy)-59 1326 y(to)17 b(\(4\))f(and)h(\(3\).)-59 1442 y Fs(Theorem)h(4.1)64 b Fv(F)l(or)17 b(al)r(l)i Fu(z)r(;)8 b(\014)15 b(>)f FB(0)p Fv(,)k(the)g(pr)n(essur)n(e)330 1564 y Fu(p)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(H)t FB(\))13 b(=)h(lim)6 b Fu(V)716 1544 y FF(\000)p FC(1)705 1577 y FA(n)780 1564 y FB(ln)i Fu(Z)862 1571 y FA(n;z)q(;\014)r(;H)1000 1564 y FB(=)14 b Ft(\000)23 b FB(min)1099 1594 y FA(\027)r FF(2M)1184 1600 y Fm(E)1218 1564 y FB([)p Fu(I)1254 1571 y FA(z)1273 1564 y FB(\()p Fu(\027)s FB(\))11 b(+)g Fu(\014)g(H)t FB(\()p Fu(\027)s FB(\)])-59 1695 y Fv(exists.)25 b(Mor)n(e)n(over,)18 b(the)g(se)n(quenc)n(e)i FB(\()p Fu(P)657 1702 y FA(n;z)q(;\014)r(;H) 781 1695 y FB(\))800 1702 y FA(n)p FF(\025)p FC(1)887 1695 y Fv(is)e(r)n(elatively)h(se)n(quential)r(ly)i(c)n(omp)n(act)d(in) g Fu(\034)1682 1702 y FF(L)1709 1695 y Fv(,)g(and)g(every)-59 1767 y(ac)n(cumulation)k(p)n(oint)f(has)g(the)h(form)667 1732 y Fn(R)703 1767 y Fu(Q)742 1749 y FA(\027)771 1767 y Fu(w)q FB(\()p Fu(d\027)s FB(\))g Fv(for)f(some)g(Bor)n(el)g(pr)n(ob) n(ability)g(me)n(asur)n(e)f(on)i(the)g(non-)-59 1839 y(empty)17 b(c)n(omp)n(act)g(set)h Ft(f)p Fu(\027)f Ft(2)d(M)522 1846 y FA(E)566 1839 y FB(:)f Fu(I)615 1846 y FA(z)635 1839 y FB(\()p Fu(\027)s FB(\))e(+)g Fu(\014)g(H)t FB(\()p Fu(\027)s FB(\))j(=)g Ft(\000)p Fu(p)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(H)t FB(\))p Ft(g)p Fv(.)14 1955 y FB(In)26 b(Example)e(4.2)j(of)f([8],)i(this)e(theorem)e(w)o(as)j(applied)f(to)g (the)g(particular)g(functional)g Fu(H)t FB(\()p Fu(\027)s FB(\))31 b(=)-59 2028 y Ft(\000j)2 1992 y Fn(R)38 2028 y Fu(\033)10 b(\027)s FB(\()p Fu(d\033)r FB(\))p Ft(j)210 2010 y FC(2)229 2028 y FB(/)q(2)p Fu(\027)s FB(\()p Fu(E)s FB(\))j(for)g(whic)o(h)f(the)g(Hamiltonian)f(\(7\))i(is)g(the)f(direct) g(con)o(tin)o(uum)f(analog)j(of)f(the)f(Curie-)-59 2100 y(W)l(eiss)k(lattice)f(mo)q(del.)20 b(It)c(w)o(as)h(sho)o(wn)f(that)h (in)f(this)g(case)g(a)h(second-order)g(phase)g(transition)f(o)q(ccurs.) 14 2178 y(The)g(mo)q(del)f(considered)h(in)g(this)g(pap)q(er)h(corresp) q(onds)g(to)g(the)f(c)o(hoice)548 2310 y Fu(H)t FB(\()p Fu(\027)s FB(\))e(=)g Ft(\000)767 2276 y Fu(J)p 767 2298 32 2 v 771 2344 a FB(2)812 2260 y Fn(\014)812 2285 y(\014)812 2310 y(\014)834 2251 y(Z)884 2310 y Fu(\033)c(\027)s FB(\()p Fu(d\033)r FB(\))1042 2260 y Fn(\014)1042 2285 y(\014)1042 2310 y(\014)1055 2273 y FC(2)1086 2310 y FB(+)h Fu(g)1160 2262 y Fn(\020)1185 2310 y Fu(\027)s FB(\()p Fu(E)s FB(\))1289 2262 y Fn(\021)1328 2310 y Fu(;)544 b FB(\(8\))-59 2432 y Fu(\027)17 b Ft(2)d(M)89 2439 y FA(E)119 2432 y FB(.)20 b(Indeed,)14 b(de\014ning)h Fu(H)547 2439 y FA(n)571 2432 y FB(\()p Fu(!)r FB(\))f(b)o(y)h(\(7\))g (in)f(terms)f(of)i(this)g Fu(H)k FB(w)o(e)14 b(arriv)o(e)f(at)i(\(2\).) 21 b(T)l(o)16 b(apply)e(Theorem)-59 2504 y(4.1)20 b(w)o(e)f(th)o(us)g (need)g(to)h(in)o(v)o(estigate)e(the)h(free)f(energy)h(functional)g Fu(I)1230 2511 y FA(z)1263 2504 y FB(+)13 b Fu(\014)e(H)t FB(.)30 b(W)l(e)19 b(note)h(\014rst)g(that)f Fu(H)24 b FB(is)-59 2576 y(ob)o(viously)13 b(con)o(tin)o(uous.)20 b(Also,)13 b Fu(I)559 2583 y FA(z)584 2576 y FB(+)5 b Fu(\014)10 b(H)18 b FB(attains)c(its)f(minim)n(um)c(b)q(ecause)14 b Fu(I)1363 2583 y FA(z)1395 2576 y FB(has)g(compact)f(sublev)o (el-sets.)-59 2649 y(W)l(e)j(will)f(sho)o(w)i(that)g(the)f(minim)o(iz)o (ers)d(are)k(of)f(the)g(form)f Fu(\032\025)1063 2656 y Fi(h)1105 2649 y FB(with)h(suitable)g Fu(\032)e Ft(\025)f FB(0)p Fu(;)8 b FD(h)15 b Ft(2)f FD(R)1670 2631 y FA(D)1702 2649 y FB(.)920 2817 y(12)p eop %%Page: 13 14 13 13 bop 14 18 a FB(T)l(o)17 b(this)f(end)g(w)o(e)g(\014rst)g(in)o (tro)q(duce)g(the)g(logarithmic)f(momen)o(t)e(generating)j(function)597 140 y Fu(')p FB(\()p Fu(h)p FB(\))e(=)g(ln)810 81 y Fn(Z)860 140 y Fu(e)883 119 y FA(h\033)923 124 y Fg(1)942 140 y Fu(\025)p FB(\()p Fu(d\033)r FB(\))49 b Fu(;)22 b(h)14 b Ft(2)g FD(R)p Fu(;)593 b FB(\(9\))-59 262 y(of)18 b(the)g(\014rst)g (co)q(ordinate)h Fu(\033)456 269 y FC(1)493 262 y FB(of)f Fu(\033)r FB(.)26 b(Ob)o(viously)l(,)17 b Fu(')h FB(is)g(analytic,)f (and)i(the)e(symmetry)e(of)j Fu(\025)h FB(implies)c(that,)-59 334 y(for)h(all)g FD(h)e Ft(2)g FD(R)217 316 y FA(D)265 334 y FB(and)j Fu(h)d FB(=)g Ft(j)p FD(h)p Ft(j)p FB(,)714 406 y Fu(')p FB(\()p Fu(h)p FB(\))g(=)f(ln)926 348 y Fn(Z)976 406 y Fu(e)999 386 y Fi(h)o FF(\001)p FA(\033)1055 406 y Fu(\025)p FB(\()p Fu(d\033)r FB(\))-59 506 y(and)709 578 y Fu(')741 557 y FF(0)752 578 y FB(\()p Fu(h)p FB(\))h(=)884 528 y Fn(\014)884 553 y(\014)884 578 y(\014)906 519 y(Z)956 578 y Fu(\033)c(\025)1022 585 y Fi(h)1047 578 y FB(\()p Fu(d\033)r FB(\))1140 528 y Fn(\014)1140 553 y(\014)1140 578 y(\014)j Fu(:)681 b FB(\(10\))-59 685 y(F)l(or)18 b Fu(D)g FB(=)f(1,)h Fu(')p FB(\()p Fu(h)p FB(\))e(=)g(ln)8 b(cosh)17 b Fu(h)h FB(and)h Fu(')700 667 y FF(0)711 685 y FB(\()p Fu(h)p FB(\))e(=)f(tanh)h Fu(h)p FB(,)h(whereas)g(for)g Fu(D)g FB(=)f(3)h(w)o(e)f(ha)o(v)o(e)g Fu(')p FB(\()p Fu(h)p FB(\))g(=)f(ln)1849 666 y FC(sinh)11 b FA(h)p 1849 674 96 2 v 1886 703 a(h)-59 758 y FB(and)17 b Fu(')68 740 y FF(0)79 758 y FB(\()p Fu(h)p FB(\))d(=)g(coth)j Fu(h)11 b Ft(\000)g Fu(h)437 740 y FF(\000)p FC(1)484 758 y FB(.)14 836 y(Next,)18 b(w)o(e)g(tak)o(e)g(an)h(arbitrary)f Fu(\027)j Ft(2)d(M)767 843 y FA(E)816 836 y FB(and)h(set)f Fu(\032)g FB(=)g Fu(\027)s FB(\()p Fu(E)s FB(\).)28 b(If)18 b Fu(I)1309 843 y FA(z)1329 836 y FB(\()p Fu(\027)s FB(\))f Fu(<)h Ft(1)p FB(,)h Fu(\027)i FB(is)e(not)g(supp)q(orted)-59 909 y(on)g(a)f(single)g(p)q(oin)o(t.)27 b(So)19 b(w)o(e)e(can)i(\014nd) f(some)f FD(h)h Ft(2)f FD(R)942 890 y FA(D)992 909 y FB(suc)o(h)h(that)1211 873 y Fn(R)1247 909 y Fu(\033)10 b(\027)s FB(\()p Fu(d\033)r FB(\))17 b(=)g Fu(\032)1510 873 y Fn(R)1547 909 y Fu(\033)10 b(\025)1613 916 y Fi(h)1638 909 y FB(\()p Fu(d\033)r FB(\).)26 b(Eq.)18 b(\(10\))-59 981 y(then)c(sho)o(ws)h(that)g Fu(H)t FB(\()p Fu(\027)s FB(\))f(=)g Fu(g)r FB(\()p Fu(\032)p FB(\))7 b Ft(\000)613 961 y FA(J)p 613 969 23 2 v 615 998 a FC(2)641 981 y Fu(\032)666 963 y FC(2)685 981 y Fu(')717 963 y FF(0)729 981 y FB(\()p Fu(h)p FB(\))795 963 y FC(2)815 981 y FB(,)14 b(where)g(again)h Fu(h)f FB(=)g Ft(j)p FD(h)p Ft(j)p FB(.)20 b(In)14 b(view)g(of)g(\(1\),)h Fu(\032)8 b(\025)1672 988 y Fi(h)1712 981 y FB(has)15 b(densit)o(y)-59 1053 y Fu(u)j FB(=)47 1032 y FA(\032)p 47 1041 19 2 v 47 1070 a(z)79 1053 y FB(exp)o([)p FD(h)12 b Ft(\001)h Fu(\033)h Ft(\000)f Fu(')p FB(\()p Fu(h)p FB(\)])18 b(with)g(resp)q(ect)g(to)h Fu(z)r(\025)p FB(.)29 b(Since)1031 1017 y Fn(R)1066 1053 y FB(ln)8 b Fu(u)g(d\027)22 b FB(=)1278 1017 y Fn(R)1314 1053 y FB(ln)8 b Fu(u)g(d)p FB(\()p Fu(\032\025)1496 1060 y Fi(h)1521 1053 y FB(\),)19 b(w)o(e)f(conclude)g(from)-59 1125 y(\(5\))f(that)f Fu(I)147 1132 y FA(z)167 1125 y FB(\()p Fu(\027)s FB(\))e(=)f Fu(I)t FB(\()p Fu(\027)s FB(;)8 b Fu(\032)g(\025)452 1132 y Fi(h)477 1125 y FB(\))k(+)f Fu(I)579 1132 y FA(z)598 1125 y FB(\()p Fu(\032)d(\025)678 1132 y Fi(h)703 1125 y FB(\).)22 b(On)16 b(the)g(other)g(hand,)h(using) f(\(10\))h(w)o(e)f(also)h(see)f(that)408 1247 y Fu(I)430 1254 y FA(z)449 1247 y FB(\()p Fu(\032)8 b(\025)529 1254 y Fi(h)555 1247 y FB(\))14 b(=)f Fu(z)664 1199 y Fn(h)684 1247 y FB(1)e Ft(\000)774 1214 y Fu(\032)p 774 1236 26 2 v 774 1281 a(z)815 1247 y FB(+)869 1214 y Fu(\032)p 869 1236 V 869 1281 a(z)916 1247 y FB(ln)970 1214 y Fu(\032)p 970 1236 V 970 1281 a(z)1000 1199 y Fn(i)1031 1247 y FB(+)g Fu(\032)1105 1199 y Fn(h)1125 1247 y Fu(h)i(')1198 1227 y FF(0)1210 1247 y FB(\()p Fu(h)p FB(\))e Ft(\000)g Fu(')p FB(\()p Fu(h)p FB(\))1435 1199 y Fn(i)1468 1247 y Fu(:)-59 1369 y FB(It)g(is)h(in)o(teresting)f(to)h(note)g(that)h(the) e(\014rst)i(term)d(on)i(the)g(righ)o(t-hand)h(side)e(is)h(simply)e(the) h(relativ)o(e)f(en)o(trop)o(y)i(of)-59 1442 y(the)g(P)o(oisson)i (distribution)e(with)h(parameter)f Fu(\032)h FB(relativ)o(e)e(to)i(the) f(P)o(oisson)i(distribution)e(with)h(parameter)e Fu(z)r FB(.)-59 1514 y(The)i(term)f(in)h(the)h(second)f(square)h(brac)o(k)o (et)e(is)h(equal)g(to)h Fu(')1023 1496 y FF(\003)1043 1514 y FB(\()p Fu(m)p FB(\),)f(where)g Fu(m)h FB(=)f Fu(')1429 1496 y FF(0)1441 1514 y FB(\()p Fu(h)p FB(\))h(is)f(the)g (magnetization)-59 1586 y(corresp)q(onding)k(to)g(the)g(external)e (\014eld)h Fu(h)h FB(and)g Fu(')865 1568 y FF(\003)885 1586 y FB(\()p Fu(m)p FB(\))c(=)i(sup)1105 1598 y FA(h)p FF(2)p Fd(R)1184 1586 y FB([)p Fu(hm)c Ft(\000)g Fu(')p FB(\()p Fu(h)p FB(\)])16 b(,)g(the)g(Legendre{F)l(enc)o(hel)-59 1658 y(transform)i(of)i Fu(')p FB(,)f(is)g(the)f(so-called)h(Cram)o (\023)-23 b(er{transform)18 b(of)h Fu(\025)g FB(whic)o(h)g(go)o(v)o (erns)f(the)h(large)g(deviations)g(of)-59 1730 y(the)f(particle)f (spins.)27 b(\(F)l(rom)17 b(this)h(it)g(migh)o(t)e(seem)h(natural)h(to) h(replace)e(the)h(parameter)f Fu(h)h FB(b)o(y)g Fu(m)p FB(,)g(but)g(it)-59 1803 y(turns)f(out)f(that)h Fu(h)f FB(is)g(more)f(con)o(v)o(enien)o(t)f(to)j(w)o(ork)f(with.\))14 1881 y(Com)o(bining)f(the)h(preceding)g(calculations)f(w)o(e)h(\014nd)h (that)497 2003 y Fu(I)519 2010 y FA(z)539 2003 y FB(\()p Fu(\027)s FB(\))11 b(+)g Fu(\014)g(H)t FB(\()p Fu(\027)s FB(\))j(=)g Fu(I)t FB(\()p Fu(\027)s FB(;)8 b Fu(\032)g(\025)1033 2010 y Fi(h)1058 2003 y FB(\))j(+)g Fu(F)1169 2010 y FA(z)q(;\014)r(;J)1252 2003 y FB(\()p Fu(\032;)d(h)p FB(\))14 b Fu(;)469 b FB(\(11\))-59 2125 y(where)16 b Fu(h)e FB(=)g Ft(j)p FD(h)p Ft(j)i FB(and)g Fu(F)377 2132 y FA(z)q(;\014)r(;J)477 2125 y FB(is)g(giv)o(en)f(b)o(y)363 2247 y Fu(F)395 2254 y FA(z)q(;\014)r(;J)478 2247 y FB(\()p Fu(\032;)8 b(h)p FB(\))41 b(=)h Fu(I)734 2254 y FA(z)753 2247 y FB(\()p Fu(\032)8 b(\025)833 2254 y Fi(h)859 2247 y FB(\))j(+)g Fu(\014)f(H)t FB(\()p Fu(\032)e(\025)1100 2254 y Fi(h)1126 2247 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b(th)o(us)g(end)g(up)h(with)f(the)g(follo)o(wing)g (conclusion.)920 2817 y(13)p eop %%Page: 14 15 14 14 bop -59 18 a Fs(Pr)o(oposition)12 b(4.2)60 b Fv(F)l(or)13 b(the)h(functional)h Fu(H)i Fv(in)d(\(8\))f(and)h(any)f Fu(z)r(;)8 b(\014)16 b(>)e FB(0)p Fv(,)g Fu(I)1366 25 y FA(z)1388 18 y FB(+)r Fu(\014)c(H)18 b Fv(attains)13 b(its)h(minimum)-59 90 y(on)k(the)g(set)g(of)f(al)r(l)i(me)n(asur)n(es) d Fu(\032)8 b(\025)564 97 y Fi(h)607 90 y Fv(for)17 b(which)h FB(\()p Fu(\032;)8 b Ft(j)p FD(h)p Ft(j)p FB(\))17 b Fv(is)h(a)f(minimizer)g(of)g Fu(F)1399 97 y FA(z)q(;\014)r(;J)1482 90 y Fv(.)14 199 y FB(Since)c(the)h(measures)f Fu(P)459 206 y FA(n;z)q(;\014)r(;J)587 199 y FB(in)h(\(3\))g(are)g(in)o(v)m (arian)o(t)f(under)h(sim)o(ultaneous)f(rotations)h(of)h(all)e(spins,)h (the)-59 271 y(measure)k Fu(w)i FB(in)f(Theorem)f(4.1)i(m)o(ust)d(also) j(exhibit)e(this)h(rotational)h(in)o(v)m(ariance.)29 b(Hence,)19 b(Theorem)e(3.1)-59 343 y(will)f(follo)o(w)h(once)h(w)o(e)e 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FB(is)g(pro)o(v)o (ed)g(in)g(Theorem)f(I)q(I.13.5)h(of)g([20)q(].)k Fb(2)14 1426 y FB(In)i(view)g(of)g(Lemma)f(5.1)h(\(b\),)i(the)e(equation)g Fu(h)i FB(=)g Fu(a)8 b(')1104 1408 y FF(0)1116 1426 y FB(\()p Fu(h)p FB(\))22 b(with)g Fu(a)i(>)g FB(0)f(has)g(a)g(unique)f (p)q(ositiv)o(e)-59 1498 y(solution)c Fu(h)155 1505 y FF(\003)175 1498 y FB(\()p Fu(a)p FB(\))f Fu(>)f FB(0)j(if)e(and)i (only)f(if)f Fu(a)g(>)g(D)q FB(,)h(whereas)h Fu(h)1037 1505 y FF(\003)1057 1498 y FB(\()p Fu(a)p FB(\))d(=)h(0)h(is)g(the)g (only)g(solution)g(when)g Fu(a)f Ft(\024)f Fu(D)q FB(.)-59 1570 y(The)g(function)g Fu(h)260 1577 y FF(\003)280 1570 y FB(\()p Ft(\001)p FB(\))g(has)h(the)f(follo)o(wing)g(prop)q(erties.) -59 1679 y Fs(Lemma)j(5.2)64 b FB(\(a\))16 b Fv(On)i FB(])p Fu(D)q(;)8 b Ft(1)p FB([)p Fv(,)17 b Fu(h)624 1686 y FF(\003)644 1679 y FB(\()p Ft(\001)p FB(\))g Fv(is)h(strictly)f (incr)n(e)n(asing)h(and)g(analytic.)14 1758 y FB(\(b\))e Fu(h)123 1765 y FF(\003)143 1758 y FB(\()p Fu(a)p FB(\))e Ft(\024)f 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y FB(satis\014es)g(the)f(equation)808 862 y Fu(h)d FB(=)h Fu(h)929 869 y FF(\003)949 862 y FB(\()p Fu(\014)s(J)5 b(\032)p FB(\))787 b(\(14\))-59 984 y(where)16 b(\(as)h(ab)q(o)o(v)o (e\))f Fu(h)346 991 y FF(\003)366 984 y FB(\()p Fu(a)p FB(\))f(is)i(the)f(largest)g(solution)g(of)h(the)f(equation)g Fu(h)e FB(=)g Fu(a)8 b(')1419 966 y FF(0)1430 984 y FB(\()p Fu(h)p FB(\))17 b(.)14 1063 y(W)l(e)g(are)g(th)o(us)g(left)f(with)h(a)g (minimi)o(zation)d(o)o(v)o(er)i(the)h(single)f(parameter)g Fu(\032)p FB(,)h(and)g(w)o(e)g(need)f(to)i(\014nd)f(the)-59 1135 y(minim)o(ize)o(rs)d(of)i(the)g(function)g Fu(\032)e Ft(!)g Fu(F)653 1142 y FA(z)q(;\014)r(;J)736 1135 y FB(\()p Fu(\032;)8 b(h)830 1142 y FF(\003)850 1135 y FB(\()p Fu(\014)s(J)d(\032)p FB(\)\).)20 b(T)l(o)d(this)f(end)g(w)o(e)g(write) 503 1257 y Fu(F)535 1264 y FA(z)q(;\014)r(;J)618 1257 y FB(\()p Fu(\032;)8 b(h)712 1264 y FF(\003)731 1257 y FB(\()p Fu(\014)s(J)d(\032)p FB(\)\))13 b(=)h Fu(z)f FB(+)e Fu(G)1064 1264 y FA(\014)r(;J)1120 1257 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1672 y FB(\()p Fu(\032)p FB(\))f(=)g(ln)8 b Fu(\032)j FB(+)g Fu(\014)f(g)961 1652 y FF(0)973 1672 y FB(\()p Fu(\032)p FB(\))h Ft(\000)g Fu(')g Ft(\016)g Fu(h)1204 1679 y FF(\003)1224 1672 y FB(\()p Fu(\014)s(J)5 b(\032)p FB(\))512 b(\(17\))-59 1795 y(for)20 b(all)f Fu(\032)i(>)f FB(0)g(\(including)f(the)g(singular)h(v)m(alue)g Fu(\032)g FB(=)g Fu(D)q(=\014)s(J)5 b FB(\).)32 b(By)19 b(assumptions)h(\(A1\))g (and)g(\(A2\))g(and)-59 1867 y(Lemma)14 b(5.2\(d\),)j(w)o(e)f(ha)o(v)o (e)g Fu(G)496 1849 y FF(0)496 1879 y FA(\014)r(;J)552 1867 y FB(\()p Fu(\032)p FB(\))f Ft(!)f(\0001)j FB(as)g Fu(\032)d Ft(!)h FB(0)i(and)g Fu(G)1138 1849 y FF(0)1138 1879 y FA(\014)r(;J)1194 1867 y FB(\()p Fu(\032)p FB(\))e Ft(!)f(1)j FB(as)g Fu(\032)e Ft(!)f(1)p FB(.)22 b(Consequen)o(tly)l(,) -59 1939 y(for)16 b(eac)o(h)f Fu(c)f Ft(2)g FD(R)h FB(the)g(function)h Fu(c\032)9 b Ft(\000)h Fu(G)679 1946 y FA(\014)r(;J)735 1939 y FB(\()p Fu(\032)p FB(\))16 b(attains)g(its)f(maxim)o(um)c(o)o(v) o(er)j Fu(\032)i 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b(Consequen)o(tly)l(,)15 b Fu(G)576 2397 y FF(\003)576 2427 y FA(\014)r(;J)648 2415 y FB(coincides)g(with)h(the)g(Legendre{F)l(enc)o(hel)f(transform)h (of)g(the)g(con)o(v)o(ex)-59 2487 y(en)o(v)o(elop)q(e)21 b Fu(G)183 2469 y FF(\003\003)183 2499 y FA(\014)r(;J)263 2487 y FB(of)i Fu(G)363 2494 y FA(\014)r(;J)419 2487 y FB(,)h(whic)o(h)f(is)f(de\014ned)h(as)h(the)f(largest)g(con)o(v)o(ex) e(function)i(not)g(exceeding)f Fu(G)1879 2494 y FA(\014)r(;J)1935 2487 y FB(.)-59 2559 y(Theorem)15 b(12.2)j(of)f([18])g(therefore)f (implies)e(that)j(the)f(con)o(v)o(ex)g(en)o(v)o(elop)q(e)f(of)i Fu(G)1414 2566 y FA(\014)r(;J)1487 2559 y FB(is)g(indeed)f(equal)g(to)h (the)-59 2632 y(Legendre{F)l(enc)o(hel)e(transform)h(of)g Fu(G)651 2613 y FF(\003)651 2644 y FA(\014)r(;J)707 2632 y FB(;)g(this)g(justi\014es)g(our)h(notation.)920 2817 y(15)p eop %%Page: 16 17 16 16 bop 14 18 a FB(Let)23 b Fu(\032)133 25 y FA(\014)r(;J)o(;)p FF(\000)223 18 y FB(\()p Fu(z)r FB(\))f(and)i Fu(\032)435 25 y FA(\014)r(;J)o(;)p FC(+)524 18 y FB(\()p Fu(z)r FB(\))f(b)q(e)g(the)g(left)f(resp.)h(righ)o(t)f(deriv)m(ativ)o(e)g(of)h Fu(G)1445 0 y FF(\003)1445 30 y FA(\014)r(;J)1501 18 y FB(\()p Fu(c)p FB(\))g(at)h Fu(c)h FB(=)g(ln)8 b Fu(z)r FB(.)42 b(By)-59 90 y(con)o(v)o(exit)o(y)l(,)15 b(the)i(set)g Fu(D)371 97 y FA(\014)r(;J)443 90 y FB(=)e Ft(f)p Fu(z)j(>)d FB(0)h(:)f Fu(\032)709 97 y FA(\014)r(;J)o(;)p FF(\000)799 90 y FB(\()p Fu(z)r FB(\))g Fu(<)h(\032)956 97 y FA(\014)r(;J)o(;)p FC(+)1045 90 y FB(\()p Fu(z)r FB(\))p Ft(g)h FB(is)g(at)h(most)e(coun)o (table.)24 b(No)o(w,)17 b(it)g(is)g(w)o(ell-)-59 162 y(kno)o(wn)d(\(Theorem)f(23.5)i(of)g([18]\))f(and)g(easy)h(to)f(see)g (that,)g(for)h(eac)o(h)f Fu(z)h(>)f FB(0,)h(the)f(function)f Fu(G)1661 144 y FF(\003\003)1661 175 y FA(\014)r(;J)1718 162 y FB(\()p Fu(\032)p FB(\))7 b Ft(\000)g Fu(\032)16 b FB(ln)8 b Fu(z)-59 235 y FB(attains)19 b(its)f(global)h(minim)n(um)14 b(precisely)j(on)i(the)f(in)o(terv)m(al)f([)p Fu(\032)1117 242 y 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Fn(n)639 881 y Fu(\032)i(>)g FB(0)g(:)g Fu(G)834 888 y FA(\014)r(;J)890 881 y FB(\()p Fu(\032)p FB(\))g(=)g Fu(G)1057 860 y FF(\003\003)1057 893 y FA(\014)r(;J)1113 881 y FB(\()p Fu(\032)p FB(\))p Fu(;)8 b FB(\()p Fu(G)1255 860 y FF(\003\003)1255 893 y FA(\014)r(;J)1311 881 y FB(\))1330 860 y FF(0)1342 881 y FB(\()p Fu(\032)p FB(\))14 b(=)f(ln)8 b Fu(z)1544 833 y Fn(o)-59 985 y Fv(is)17 b(a)h(\014nite)h(set)f(with)g(c)n(onvex)h (hul)r(l)h FB([)p Fu(\032)637 992 y FA(\014)r(;J)o(;)p FF(\000)725 985 y FB(\()p Fu(z)r FB(\))p Fu(;)8 b(\032)835 992 y FA(\014)r(;J)o(;)p FC(+)924 985 y FB(\()p Fu(z)r FB(\)])p Fv(.)23 b(In)18 b(p)n(articular,)f FB(#)p Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))12 b Fu(>)i FB(1)k Fv(if)g(and)f(only)-59 1057 y(if)g Fu(G)26 1039 y FF(\003)26 1070 y FA(\014)r(;J)100 1057 y Fv(is)g(not)h(di\013er)n (entiable)h(at)f FB(ln)7 b Fu(z)r Fv(.)-59 1166 y(Pr)n(o)n(of.)47 b FB(It)16 b(only)g(remains)f(to)h(sho)o(w)h(that)g Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))14 b(is)i(\014nite.)21 b(Since)15 b Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))14 b(is)i(con)o(tained)g(in)g(the)-59 1238 y(set)k Ft(f)p Fu(\032)g(>)g FB(0)h(:)e Fu(G)265 1220 y FF(0)265 1250 y FA(\014)r(;J)322 1238 y FB(\()p Fu(\032)p FB(\))h(=)g(ln)8 b Fu(z)r Ft(g)p FB(,)20 b(this)g(follo)o(ws)g(from)f(the)h(analyticit)o (y)e(assumption)i(\(A1\))f(and)i(Lemma)-59 1310 y(5.2\(a\).)h Fb(2)14 1419 y FB(It)15 b(is)g(eviden)o(t)f(that)i(the)f(sets)h Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))14 b(exhibit)g(the)h (monotonicit)o(y)f(prop)q(ert)o(y)h(in)g Fu(z)j FB(stated)d(in)h(The-) -59 1491 y(orem)e(3.1\(ii\).)20 b(The)15 b(closed-graph)h(prop)q(ert)o (y)g(\(iii\))d(follo)o(ws)i(from)g(the)g(fact)g(that)g Fu(G)1490 1498 y FA(\014)r(;J)1547 1491 y FB(,)g Fu(G)1614 1473 y FF(\003\003)1614 1503 y FA(\014)r(;J)1685 1491 y FB(and,)h(b)o(y)e(con-)-59 1563 y(v)o(exit)o(y)h(and)i(di\013eren)o (tiabilit)o(y)l(,)d(also)j(\()p Fu(G)689 1545 y FF(\003\003)689 1575 y FA(\014)r(;J)745 1563 y FB(\))764 1545 y FF(0)793 1563 y FB(dep)q(end)g(con)o(tin)o(uously)f(on)h(\()p Fu(\014)s(;)8 b(J)d FB(\).)22 b(The)17 b(pro)q(of)h(of)f(Theorem)-59 1635 y(3.1)h(is)g(therefore)f(complete.)23 b(W)l(e)18 b(also)g(note)g(that)h Fu(z)933 1642 y FA(m)966 1635 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))16 b(is)i(ob)o(viously)f(equal)g (to)h(exp\()p Fu(G)1696 1617 y FF(\003\003)1696 1648 y FA(\014)r(;J)1752 1635 y FB(\))1771 1617 y FF(0)1783 1635 y FB(\()p Fu(D)q(=\014)s(J)5 b FB(\))-59 1708 y(and)19 b(therefore)e(con)o(tin)o(uous,)h(and)h(it)e(follo)o(ws)h(readily)f (from)g(\(17\))i(and)f(Lemma)e(5.2\(a\))j(that)f Fu(z)1743 1715 y FA(m)1776 1708 y FB(\()p Fu(\014)s(;)8 b(J)d FB(\))17 b(is)-59 1780 y(decreasing)f(in)g Fu(J)5 b FB(.)14 1858 y(Next,)14 b(b)o(y)g(Theorem)g(26.3)i(of)f([18)q(],)f(the)h(set)g Fu(D)876 1865 y FA(\014)r(;J)947 1858 y FB(is)g(empt)o(y)e(\(i.e.,)g Fu(G)1291 1840 y FF(\003)1291 1871 y FA(\014)r(;J)1363 1858 y FB(is)h(di\013eren)o(tiable\))g(if)g(and)i(only)-59 1931 y(if)c Fu(G)20 1913 y FF(\003\003)20 1943 y FA(\014)r(;J)90 1931 y FB(is)g(strictly)g(con)o(v)o(ex,)g(and)h(this)g(holds)g(if)g (and)g(only)g(if)f Fu(G)1093 1938 y FA(\014)r(;J)1163 1931 y FB(is)g(strictly)g(con)o(v)o(ex.)19 b(Moreo)o(v)o(er,)12 b(eq.)g(\(17\),)-59 2003 y(assumption)19 b(\(A1\))g(and)h(Lemma)d (5.2\(a\))j(imply)c(that)k(the)f(second)g(deriv)m(ativ)o(e)f Fu(G)1499 1985 y FF(00)1499 2015 y FA(\014)r(;J)1574 2003 y FB(exists)h(ev)o(erywhere)-59 2075 y(except)e(at)i(the)f (critical)e(p)q(oin)o(t)j Fu(\032)e FB(=)h Fu(D)q(=\014)s(J)5 b FB(.)27 b(A)o(t)17 b(this)i(p)q(oin)o(t,)f(w)o(e)g(will)f(alw)o(a)o (ys)h(consider)g(the)g(righ)o(t-sided)-59 2147 y(second)h(deriv)m(ativ) o(e)d Fu(G)364 2129 y FF(00)364 2160 y FA(\014)r(;J)420 2147 y FB(\()p Fu(D)q(=\014)s(J)5 b FB(\))19 b(whic)o(h)f(exists)f(b)q (ecause)i(of)g(Lemma)d(5.2\(e\).)27 b(Also,)19 b Fu(G)1628 2129 y FF(00)1628 2160 y FA(\014)r(;J)1702 2147 y FB(is)f(piecewise)-59 2220 y(analytic.)36 b(Its)21 b(zero)q(es)g(th)o(us)g(form)g(a)g(coun)o (table)g(set.)36 b(Hence)20 b Fu(G)1190 2227 y FA(\014)r(;J)1268 2220 y FB(is)h(strictly)f(con)o(v)o(ex)g(if)g(and)i(only)f(if)-59 2292 y Fu(G)-21 2274 y FF(00)-21 2304 y FA(\014)r(;J)35 2292 y FB(\()p Fu(\032)p FB(\))26 b Ft(\025)f FB(0)f(for)f(all)g Fu(\032)i(>)h FB(0.)42 b(This)23 b(leads)h(us)f(to)h(the)e(follo)o (wing)h(criterion)f(for)i(the)f(existence)e(of)i(a)-59 2364 y(\014rst{order)17 b(phase)g(transition.)-59 2473 y Fs(Pr)o(oposition)g(5.4)64 b Fv(F)l(or)17 b(any)h Fu(\014)e(>)e FB(0)j Fv(and)h Fu(J)g Ft(\025)c FB(0)k Fv(the)g(fol)r(lowing)i (assertions)d(ar)n(e)g(e)n(quivalent.)14 2551 y FB(\(a\))g Fv(F)l(or)g(at)g(le)n(ast)h(some)g Fu(\032)c(>)f FB(0)p Fv(,)436 2666 y Fu(G)474 2645 y FF(00)474 2678 y FA(\014)r(;J)531 2666 y FB(\()p Fu(\032)p FB(\))h Ft(\021)666 2632 y FB(1)p 665 2654 26 2 v 665 2700 a Fu(\032)706 2666 y FB(+)d Fu(\014)g(g)819 2645 y FF(00)840 2666 y FB(\()p Fu(\032)p FB(\))h Ft(\000)e Fu(\014)s(J)j FB(\()p Fu(')e Ft(\016)g Fu(h)1161 2673 y FF(\003)1180 2666 y FB(\))1199 2645 y FF(0)1211 2666 y FB(\()p Fu(\014)s(J)5 b(\032)p FB(\))13 b Fu(<)g FB(0)i Fu(:)408 b FB(\(18\))920 2817 y(16)p eop %%Page: 17 18 17 17 bop 14 18 a FB(\(b\))16 b Fv(Ther)n(e)h(exists)i(at)e(le)n(ast)h (one)g Fu(z)e(>)e FB(0)k Fv(such)f(that)h FB(#)p Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))12 b Fu(>)i FB(1)p Fv(.)-59 97 y(In)k(this)h(c)n(ase)f(one)h(c)n(an)g(cho)n(ose)f Fu(z)g FB(=)d(exp\()p Fu(G)770 79 y FF(\003\003)770 109 y FA(\014)r(;J)826 97 y FB(\))845 79 y FF(0)857 97 y FB(\()p Fu(\032)p FB(\))p Fv(,)j(and)h(this)f(is)h(the)g(unique)g Fu(z)i Fv(for)c(which)i Fu(\032)g Fv(b)n(elongs)h(to)-59 169 y(the)e(c)n(onvex)h(hul)r(l)g(of)e Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))16 b Fv(.)14 273 y FB(W)l(e)g(conclude)g(this)g(section)g(with)g(the)g(pro)q(of)h(of)g (Lemma)d(5.2.)-59 377 y Fv(Pr)n(o)n(of)i(of)h(L)n(emma)g(5.2.)71 b FB(\(a\))16 b(follo)o(ws)h(from)e(Lemma)f(5.1\(b\))j(and)g(the)f 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Fu(b)p FB(\(1\))15 b(=)h(\()p Fu(')11 b Ft(\016)h Fu(h)220 420 y FF(\003)240 413 y FB(\))259 395 y FF(0)270 413 y FB(\(1\))k(=)g(3)p Fu(=)p FB(2,)i(and)f(it)g(is)g(lik)o(ely)e(that)i Fu(b)p FB(\()p Fu(D)q FB(\))f(=)g(1)h(for)h(all)e Fu(D)i Ft(\025)d FB(2.)24 b(This)17 b(w)o(ould)h(imply)c(that)-59 485 y(Lemma)h(5.2\(g\))j(holds)g(also)f(for)h Fu(D)f(>)f FB(1,)h(so)h(that)g(the)f(pro)o(visos)g(on)h Fu(D)h FB(in)e(Theorems)f (3.4)i(and)f(3.5)h(could)-59 557 y(b)q(e)e(a)o(v)o(oided.)21 b(Unfortunately)l(,)15 b(w)o(e)h(could)g(not)h(pro)o(v)o(e)e(this.)14 636 y(W)l(e)h(b)q(egin)h(with)f(the)g(pro)q(of)h(of)g(Prop)q(osition)g (3.2)g(on)f(the)g(absence)g(of)h(\014rst{order)g(phase)g(transitions.) -59 752 y Fv(Pr)n(o)n(of)h(of)h(Pr)n(op)n(osition)f(3.2.)77 b FB(W)l(e)18 b(use)g(the)h(criterion)e(of)i(Prop)q(osition)g(5.4.)28 b(F)l(or)19 b(\014xed)f Fu(J)5 b FB(,)18 b(assumption)-59 824 y(\(A2\))e(implies)e(the)i(existence)e(of)j(some)e Fu(\032)716 831 y FC(0)750 824 y Fu(>)f 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FB(and)k Fu(J)i Ft(\024)13 b FB(1)k(so)g(small)d(that)j Fu(J)h Ft(\024)c Fu(D)q(=\014)s(\032)907 1048 y FC(0)927 1041 y FB(.)22 b(The)16 b(result)g(then)g(follo)o(ws)g(as)h(in)f(the)g (\014rst)g(case.)21 b Fb(2)14 1157 y FB(Next,)f(w)o(e)f(apply)h(Prop)q (osition)h(5.4)f(to)h(establish)e(the)h(existence)e(of)j(a)f (\014rst{order)h(transition)f(from)-59 1229 y(nonmagnetic)c(v)m(ap)q (or)j(to)f(ferromagnetic)d(liquid,)h(as)i(stated)g(in)f(Theorem)f(3.3.) 25 b(Since)17 b Fu(\032)f FB(=)f Fu(D)q(=\014)s(J)23 b FB(is)17 b(the)-59 1301 y(critical)i(densit)o(y)g(for)i(the)f (incipience)e(of)j(ferromagnetic)e(order,)i(w)o(e)f(only)g(need)g(to)h (insert)f(this)g(critical)-59 1373 y(v)m(alue)c(in)o(to)g(\(18\).)-59 1490 y Fv(Pr)n(o)n(of)e(of)h(The)n(or)n(em)f(3.3.)70 b FB(By)13 b(Lemma)f(5.2\(e\),)i(the)g(righ)o(t-sided)f(deriv)m(ativ)o (e)g(of)h Fu(')7 b Ft(\016)g Fu(h)1567 1497 y FF(\003)1586 1490 y FB(\()p Fu(a)p FB(\))13 b(at)i(the)e(critical)-59 1562 y(v)m(alue)20 b Fu(a)g FB(=)g Fu(D)i FB(exists)d(and)i(is)e(equal) h(to)g(\()p Fu(D)c FB(+)d(2\))p Fu(=)p FB(2)p Fu(D)q FB(.)35 b(Hence,)19 b Fu(G)1230 1544 y FF(00)1230 1574 y FA(\014)r(;J)1286 1562 y FB(\()p Fu(D)q(=\014)s(J)5 b FB(\))20 b Fu(<)g FB(0)h(if)e(and)i(only)f(if)f(the)-59 1634 y(h)o(yp)q(othesis)d(of)h(the)f(theorem)f(holds.)21 b(The)16 b(result)g(th)o(us)g(follo)o(ws)h(from)e(Prop)q(osition)i (5.4.)22 b Fb(2)14 1750 y FB(P)o(ostp)q(oning)e(the)e(pro)q(ofs)h(of)g (Theorems)e(3.4)h(and)h(3.5)g(un)o(til)e(the)h(end)g(of)g(this)h (section,)e(w)o(e)h(no)o(w)h(turn)-59 1822 y(to)e(the)f(pro)q(of)h(of)f (Corollary)h(3.6)f(on)h(the)f(\014rst{order)h(transition)g(b)q(et)o(w)o (een)e(magnetic)g(phases.)-59 1938 y Fv(Pr)n(o)n(of)i(of)h(Cor)n(ol)r (lary)g(3.6.)74 b FB(Let)17 b Fu(b)p FB(\()p Fu(D)q FB(\))h(b)q(e)g(as) g(in)f(\(19\).)26 b(W)l(e)17 b(\014x)g(some)g Fu(\032)1326 1945 y FC(0)1361 1938 y Ft(2)8 b FB(])g(0)p Fu(;)g(\032)1495 1945 y FC(min)1557 1938 y FB([)17 b(and)h(let)f Fu(J)j(>)c FB(0)i(b)q(e)-59 2010 y(so)f(small)d(that)14 2089 y(\(i\))i(min)163 2096 y FA(\032)p FF(\024)p FA(\032)226 2101 y Fg(0)254 2089 y Fu(g)279 2071 y FF(00)301 2089 y FB(\()p Fu(\032)p FB(\))e Ft(\025)f Fu(J)g(b)p FB(\()p Fu(D)q FB(\);)14 2168 y(\(ii\))i(the)g(set)g Ft(f)p Fu(g)303 2150 y FF(00)338 2168 y Fu(<)f(J)5 b Ft(g)15 b FB(is)g(an)h(in)o(terv)m(al)f(])p Fu(\032)792 2175 y FC(1)811 2168 y Fu(;)8 b(\032)858 2175 y FC(2)878 2168 y FB([)15 b(satisfying)h Fu(\032)1149 2175 y FC(0)1182 2168 y Fu(<)e(\032)1259 2175 y FC(1)1293 2168 y Fu(<)g(\032)1370 2175 y FC(min)1444 2168 y Fu(<)g(\032)1521 2175 y FC(2)1541 2168 y FB(,)h(so)h(that)g Fu(g)1759 2150 y FF(0)1771 2168 y FB(\()p Fu(\032)p FB(\))10 b Ft(\000)f Fu(J)c(\032)-59 2240 y FB(has)15 b(a)f(lo)q(cal)g(maxim)o(um) c(at)k Fu(\032)485 2247 y FC(1)519 2240 y FB(and)h(a)f(lo)q(cal)g (minim)n(um)c(at)k Fu(\032)1061 2247 y FC(2)1095 2240 y FB(and)h(is)f(strictly)f(monotone)g(on)i(the)f(in)o(terv)m(als)-59 2312 y(whic)o(h)h(are)i(separated)g(b)o(y)e(these)h(p)q(oin)o(ts;)g (and)14 2391 y(\(iii\))f Fu(g)134 2373 y FF(0)146 2391 y FB(\()p Fu(\032)190 2398 y FC(0)210 2391 y FB(\))c Ft(\000)f Fu(J)5 b(\032)346 2398 y FC(0)380 2391 y Fu(<)13 b(g)456 2373 y FF(0)468 2391 y FB(\()p Fu(\032)512 2398 y FC(2)532 2391 y FB(\))e Ft(\000)g Fu(J)5 b(\032)669 2398 y FC(2)688 2391 y FB(.)-59 2463 y(\(i\))13 b(is)g(p)q(ossible)g(b) q(ecause)h Fu(g)435 2445 y FF(00)470 2463 y FB(is)f(p)q(ositiv)o(e)f (on)i(the)f(left)g(of)g Fu(\032)999 2470 y FC(0)1019 2463 y FB(,)g(\(ii\))g(uses)g(the)g(T)l(a)o(ylor)h(expansion)f(of)h Fu(g)1758 2445 y FF(00)1793 2463 y FB(at)f Fu(\032)1874 2470 y FC(min)1935 2463 y FB(,)-59 2535 y(and)20 b(\(iii\))f(merely)e (sa)o(ys)j(that)h(the)e(in)o(terv)m(al)g(])p Fu(\032)823 2542 y FC(1)843 2535 y Fu(;)8 b(\032)890 2542 y FC(2)909 2535 y FB([)20 b(on)g(whic)o(h)g Fu(g)1183 2517 y FF(0)1194 2535 y FB(\()p Fu(\032)p FB(\))14 b Ft(\000)f Fu(J)5 b(\032)20 b FB(decreases)g(is)f(short)i(enough)-59 2608 y(compared)d(with)g(the)g(in)o(terv)m(al)g(])p Fu(\032)583 2615 y FC(0)602 2608 y Fu(;)8 b(\032)649 2615 y FC(1)669 2608 y FB([)18 b(of)h(increase.)28 b(Recalling)18 b(that)h(the)f (construction)h(of)g(the)f(con)o(v)o(ex)-59 2680 y(en)o(v)o(elop)q(e)c (corresp)q(onds)j(to)f(Maxw)o(ell's)e(equal{area)i(construction)g(for)g (the)f(deriv)m(ativ)o(e,)f(w)o(e)h(see)g(that)h(that)920 2817 y(18)p eop %%Page: 19 20 19 19 bop -59 18 a FB(our)17 b(conditions)f(imply)e(that)j(the)f(con)o (v)o(ex)e(en)o(v)o(elop)q(e)h(of)i Fu(g)r FB(\()p Fu(\032)p FB(\))11 b Ft(\000)g Fu(J)5 b(\032)1207 0 y FC(2)1226 18 y Fu(=)p FB(2)17 b(has)g(a)g(unique)f(a\016ne)g(piece)f(on)h(an)-59 90 y(in)o(terv)m(al)e([)p Fu(r)152 97 y FF(\000)181 90 y Fu(;)8 b(r)225 97 y FC(+)254 90 y FB(],)15 b(where)f Fu(r)458 97 y FF(\000)502 90 y Fu(>)f FB(0)j(due)f(to)g(\(iii\).)20 b(Hence,)13 b(if)i Fu(\014)i FB(is)e(large)g(enough)g(then)g Fu(D)q(=\014)s(J)k(<)14 b(\032)1763 97 y FF(\000)1808 90 y FB(for)h(the)-59 162 y Fu(\032)-34 169 y FF(\000)12 162 y FB(in)h(Theorem)f(3.4\(a\),)h(so)h(that)g(the)f(corollary)g (follo)o(ws)g(from)f(this)h(theorem.)k Fb(2)14 278 y FB(The)f(next)f(result)h(to)g(pro)o(v)o(e)f(is)h(Prop)q(osition)h(3.7)f (on)g(\014rst{order)h(transitions)f(in)g(the)g(non-magnetic)-59 351 y(regime.)-59 467 y Fv(Pr)n(o)n(of)12 b(of)i(Pr)n(op)n(osition)f (3.7.)69 b FB(Our)13 b(assumptions)f(on)h Fu(g)i FB(and)e Fu(\014)i FB(imply)10 b(that)j(there)f(is)h(a)f(righ)o(tmost)g(in)o (terv)m(al)-59 539 y Fu(I)j FB(on)e(whic)o(h)f Fu(G)216 521 y FF(00)216 551 y FA(\014)r(;)p FC(0)279 539 y FB(is)g(negativ)o (e.)19 b(By)11 b(Prop)q(osition)j(4.3,)f Fu(I)i FB(is)d(con)o(tained)g (in)f(some)h(maxim)o(al)e(in)o(terv)m(al)h([)p Fu(\032)1830 546 y FF(\000)1859 539 y Fu(;)d(\032)1906 546 y FC(+)1935 539 y FB(])-59 611 y(on)17 b(whic)o(h)e Fu(G)186 593 y FF(\003\003)186 623 y FA(\014)r(;)p FC(0)254 611 y FB(is)h(a\016ne.)21 b(Let)16 b(ln)8 b Fu(z)19 b FB(b)q(e)d(the)g (corresp)q(onding)h(slop)q(e.)14 690 y(Next,)d(assumption)h(\(A2\))g (giv)o(es)g(us)g(some)f Fu(\032)843 697 y FC(1)877 690 y Fu(>)g(\032)954 697 y FC(+)999 690 y FB(suc)o(h)h(that)g Fu(g)1237 672 y FF(00)1273 690 y Ft(\025)e Fu(b)p FB(\()p Fu(D)q FB(\))j(on)g([)p Fu(\032)1547 697 y FC(1)1566 690 y Fu(;)8 b Ft(1)p FB([.)20 b(Let)15 b Fu(J)k Ft(2)p FB(]0)p Fu(;)8 b FB(1])-59 762 y(b)q(e)19 b(so)g(small)e(that)h Fu(D)q(=\014)s(J)23 b(>)18 b(\032)536 769 y FC(1)556 762 y FB(.)28 b(Lo)q(oking)19 b(at)g(\(18\))g(w)o(e)g(then)f(see)g (that)h Fu(G)1369 744 y FF(00)1369 774 y FA(\014)r(;J)1443 762 y Ft(\025)e FB(0)i(on)g([)p Fu(\032)1651 769 y FC(1)1670 762 y Fu(;)8 b Ft(1)p FB([.)28 b(On)18 b(the)-59 834 y(other)i(hand,)i(w)o(e)e(also)h(ha)o(v)o(e)f Fu(G)546 816 y FF(00)546 847 y FA(\014)r(;J)623 834 y FB(=)h Fu(G)720 816 y FF(00)720 847 y FA(\014)r(;)p FC(0)792 834 y Ft(\025)g FB(0)g(on)g([)p Fu(\032)1008 841 y FC(+)1037 834 y Fu(;)8 b(D)q(=\014)s(J)d FB(].)33 b(Consequen)o(tly)l(,)20 b Fu(G)1603 841 y FA(\014)r(;J)1680 834 y FB(is)g(con)o(v)o(ex)f(on)-59 906 y([)p Fu(\032)-20 913 y FC(+)9 906 y Fu(;)8 b Ft(1)p FB([)14 b(and)i(coincides)e(with)h Fu(G)556 913 y FA(\014)r(;)p FC(0)623 906 y FB(on)g(\(a)h(neigh)o(b)q(orho)q(o)q(d)h(of)s(\))f([0)p Fu(;)8 b(\032)1215 913 y FC(+)1244 906 y FB(].)20 b(This)c(sho)o(ws)f (that)h([)p Fu(\032)1685 913 y FF(\000)1714 906 y Fu(;)8 b(\032)1761 913 y FC(+)1791 906 y FB(])14 b(is)h(also)-59 979 y(the)h(righ)o(tmost)f(maximal)e(in)o(terv)m(al)i(on)i(whic)o(h)f Fu(G)867 961 y FF(\003\003)867 991 y FA(\014)r(;J)939 979 y FB(is)g(a\016ne,)g(and)h(the)f(prop)q(osition)h(follo)o(ws.)k Fb(2)14 1095 y FB(W)l(e)16 b(pro)q(ceed)g(with)g(the)g(pro)q(of)i(of)e (Theorem)f(3.8)i(on)g(the)f(existence)e(of)j(triple)e(p)q(oin)o(ts.)-59 1211 y Fv(Pr)n(o)n(of)20 b(of)i(The)n(or)n(em)f(3.8.)84 b(Step)23 b(1:)31 b(Existenc)n(e)23 b(of)f Fu(z)998 1218 y FA(c)1015 1211 y Fv(.)85 b FB(Let)21 b Fu(\014)j(>)e(\014)1347 1218 y FC(min)1428 1211 y FB(b)q(e)f(so)h(close)f(to)g Fu(\014)1778 1218 y FC(min)1859 1211 y FB(that)-59 1283 y Ft(f)p Fu(\032)h(>)h FB(0)f(:)g Fu(\032)8 b(g)214 1265 y FF(00)236 1283 y FB(\()p Fu(\032)p FB(\))23 b Fu(<)f Ft(\000)p FB(1)p Fu(=\014)s Ft(g)f FB(is)g(an)h(in)o(terv)m(al.)35 b(This)21 b(is)g(p)q(ossible)h(b)q(ecause,)g(b)o(y)f(analyticit)o(y)l (,)f Fu(\032)8 b(g)r FB(\()p Fu(\032)p FB(\))22 b(is)-59 1355 y(con)o(v)o(ex)14 b(in)h(a)h(neigh)o(b)q(orho)q(o)q(d)h(of)f Fu(\032)581 1362 y FC(min)642 1355 y FB(.)21 b(Then)16 b Ft(f)p Fu(G)867 1337 y FF(00)867 1368 y FA(\014)r(;)p FC(0)932 1355 y Fu(<)e FB(0)p Ft(g)h FB(is)h(an)g(in)o(terv)m(al)e(con) o(taining)h Fu(\032)1600 1362 y FC(min)1661 1355 y FB(.)21 b(Hence)14 b(there)-59 1427 y(is)h(a)h(unique)e(in)o(terv)m(al)g([)p Fu(\032)400 1434 y FF(\000)430 1427 y Fu(;)8 b(\032)477 1434 y FA(])492 1427 y FB(])14 b Ft(3)g Fu(\032)592 1434 y FC(min)668 1427 y FB(\(dep)q(ending)h(on)h Fu(\014)s FB(\))f(on)g(whic)o(h)g Fu(G)1295 1409 y FF(\003\003)1295 1440 y FA(\014)r(;)p FC(0)1362 1427 y FB(is)g(a\016ne.)21 b(Let)15 b Fu(z)1668 1434 y FA(c)1699 1427 y FB(=)f Fu(z)1774 1434 y FA(c)1791 1427 y FB(\()p Fu(\014)s FB(\))f Fu(>)g FB(0)-59 1500 y(b)q(e)j(suc)o(h)g(that)h(ln)8 b Fu(z)295 1507 y FA(c)328 1500 y FB(is)16 b(the)g(corresp)q(onding)i(slop)q(e.)14 1578 y(W)l(e)e(observ)o(e)g(that)h Fu(\032)403 1585 y FA(])434 1578 y FB(=)d Fu(\032)511 1585 y FA(])527 1578 y FB(\()p Fu(\014)s FB(\))g Ft(!)g Fu(\032)699 1585 y FC(min)776 1578 y FB(as)j Fu(\014)g Ft(!)d Fu(\014)973 1585 y FC(min)1034 1578 y FB(.)22 b(Indeed,)15 b(otherwise)i(there)f(w) o(ould)g(exist)g(some)-59 1651 y Fu(")22 b(>)g FB(0)g(and)g(a)f (sequence)g Fu(\014)473 1658 y FA(n)518 1651 y Ft(!)h Fu(\014)618 1658 y FC(min)700 1651 y FB(suc)o(h)f(that)g Fu(\032)950 1658 y FA(])966 1651 y FB(\()p Fu(\014)1013 1658 y FA(n)1036 1651 y FB(\))15 b Ft(\000)f Fu(\032)1148 1658 y FF(\000)1178 1651 y FB(\()p Fu(\014)1225 1658 y FA(n)1248 1651 y FB(\))22 b Ft(\025)g Fu(")f FB(for)h(all)e Fu(n)p FB(.)37 b(By)20 b(assumption)-59 1723 y(\(A2\),)15 b(the)g(sequence)g(\()p Fu(\032)398 1730 y FA(])414 1723 y FB(\()p Fu(\014)461 1730 y FA(n)484 1723 y FB(\)\))g(is)g(b)q (ounded.)22 b(Considering)16 b(suitable)f(limit)e(p)q(oin)o(ts)j(and)g (using)g(that)g Fu(G)1878 1705 y FF(0)1878 1735 y FA(\014)1898 1739 y Fm(n)1919 1735 y FA(;)p FC(0)-59 1795 y FB(con)o(v)o(erges)i(to) g Fu(G)261 1777 y FF(0)261 1807 y FA(\014)281 1818 y FC(min)342 1807 y FA(;)p FC(0)390 1795 y FB(lo)q(cally)g(uniformly)e(w) o(e)i(then)h(see)f(that)g Fu(G)1185 1777 y FF(0)1185 1807 y FA(\014)1205 1818 y FC(min)1267 1807 y FA(;)p FC(0)1315 1795 y FB(w)o(ould)g(tak)o(e)g(the)g(same)f(v)m(alue)h(at)-59 1867 y(t)o(w)o(o)c(p)q(oin)o(ts)g(of)g(distance)g Fu(")p FB(.)20 b(This)14 b(is)g(imp)q(ossible)e(b)q(ecause)i Fu(G)1080 1849 y FF(0)1080 1880 y FA(\014)1100 1890 y FC(min)1161 1880 y FA(;)p FC(0)1205 1867 y FB(is)g(strictly)e (increasing)i(b)o(y)f(assumption)-59 1939 y(on)k Fu(g)r FB(.)14 2018 y(It)d(follo)o(ws)g(from)f(the)h(preceding)g(observ)m (ation)h(that)g Fu(\014)10 b(\032)1060 2025 y FA(])1084 2018 y Fu(g)1109 2000 y FF(00)1131 2018 y FB(\()p Fu(\032)1175 2025 y FA(])1191 2018 y FB(\))j Ft(!)h(\000)p FB(1)h(as)f Fu(\014)j Ft(!)c Fu(\014)1558 2025 y FC(min)1619 2018 y FB(.)20 b(Consequen)o(tly)l(,)-59 2090 y(c)o(ho)q(osing)g Fu(\014)h FB(close)e(enough)g(to)h Fu(\014)572 2097 y FC(min)651 2090 y FB(w)o(e)e(can)i(ac)o(hiev)o(e)d(that)i Fu(\014)11 b(\032)1162 2097 y FA(])1186 2090 y Fu(g)1211 2072 y FF(00)1232 2090 y FB(\()p Fu(\032)1276 2097 y FA(])1292 2090 y FB(\))19 b Fu(<)f(D)q(=)p FB(2.)31 b(In)19 b(the)f(follo)o(wing,)h(w)o(e)-59 2163 y(will)d(consider)g(a)h(\014xed) g(suc)o(h)f Fu(\014)s FB(.)22 b(W)l(e)17 b(then)g(can)g(assume)f(for)h (simplicit)n(y)d(that)j(ln)8 b Fu(z)1493 2170 y FA(c)1525 2163 y FB(=)14 b(0)k(and)f Fu(G)1752 2170 y FA(\014)r(;)p FC(0)1804 2163 y FB(\()p Fu(\032)1848 2170 y FF(\000)1877 2163 y FB(\))e(=)-59 2235 y Fu(G)-21 2242 y FA(\014)r(;)p FC(0)30 2235 y FB(\()p Fu(\032)74 2242 y FA(])90 2235 y FB(\))f(=)g(0.)22 b(\(Otherwise)15 b(w)o(e)h(add)h(a)f(linear)g(term) e(to)j Fu(g)r FB(.\))14 2351 y Fv(Step)i(2:)24 b(Existenc)n(e)c(of)e Fu(J)489 2358 y FA(c)507 2351 y Fv(.)73 b FB(Let)17 b Fu(J)710 2358 y FA(])741 2351 y Fu(>)f FB(0)h(b)q(e)h(de\014ned)f(b)o (y)f(the)h(equation)g Fu(\032)1451 2358 y FA(])1483 2351 y FB(=)e Fu(D)q(=\014)s(J)1659 2358 y FA(])1675 2351 y FB(.)25 b(By)16 b(Lemma)-59 2423 y(5.2\(e\))g(and)h(\(18\),)f(it)g (then)g(follo)o(ws)h(from)e(Step)h(1)g(that)499 2557 y Fu(G)537 2537 y FF(00)537 2570 y FA(\014)r(;J)588 2576 y Fm(])605 2557 y FB(\()p Fu(\032)649 2564 y FA(])665 2557 y FB(\))e(=)763 2524 y(1)p 754 2546 41 2 v 754 2591 a Fu(\032)779 2598 y FA(])811 2557 y FB(+)d Fu(\014)g(g)924 2537 y FF(00)945 2557 y FB(\()p Fu(\032)989 2564 y FA(])1005 2557 y FB(\))g Ft(\000)1090 2524 y Fu(D)p 1090 2546 42 2 v 1090 2591 a(\032)1115 2598 y FA(])1142 2524 y Fu(D)i FB(+)e(2)p 1142 2546 127 2 v 1172 2591 a(2)p Fu(D)1287 2557 y(<)j FB(0)g Fu(:)920 2817 y FB(19)p eop %%Page: 20 21 20 20 bop -59 18 a FB(By)15 b(the)h(\014nal)h(assumptions)f(of)h(Step)f (1,)g(this)g(implies)e(that)i Fu(q)r FB(\()p Fu(J)1144 25 y FA(])1159 18 y FB(\))e Fu(<)g FB(0,)i(where)701 122 y Fu(q)r FB(\()p Fu(J)5 b FB(\))12 b(=)53 b(inf)859 154 y FA(\032>D)q(=\014)r(J)1004 122 y Fu(G)1042 129 y FA(\014)r(;J)1099 122 y FB(\()p Fu(\032)p FB(\))14 b Fu(:)-59 240 y FB(By)19 b(Lemma)e(5.2\(a\))j(and)g(\(17\),)h Fu(q)r FB(\()p Fu(J)5 b FB(\))18 b(is)h(strictly)g(decreasing)g(on)h ([0)p Fu(;)8 b(J)1293 247 y FA(])1308 240 y FB(].)31 b(Using)19 b(\(A2\))g(and)h(the)g(lo)q(cally)-59 312 y(uniform)11 b(con)o(v)o(ergence)g(of)i Fu(G)476 319 y FA(\014)r(;J)527 323 y Fm(n)563 312 y FB(to)g Fu(G)657 319 y FA(\014)r(;J)726 312 y FB(when)g Fu(J)877 319 y FA(n)914 312 y Ft(!)h Fu(J)5 b FB(,)13 b(w)o(e)f(also)h(see)f(that)i Fu(q)r FB(\()p Fu(J)5 b FB(\))11 b(is)i(con)o(tin)o(uous.)20 b(Finally)l(,)-59 385 y(w)o(e)d(kno)o(w)h(from)f(the)g(pro)q(of)i(of)f (Prop)q(osition)h(3.7)f(that)g Fu(q)r FB(\()p Fu(J)5 b FB(\))15 b Fu(>)h FB(0)i(if)f Fu(J)22 b FB(is)c(small)e(enough.)26 b(Consequen)o(tly)l(,)-59 457 y(there)18 b(is)g(a)h(unique)f Fu(J)350 464 y FA(c)385 457 y FB(=)g Fu(J)468 464 y FA(c)486 457 y FB(\()p Fu(\014)s FB(\))f Ft(2)8 b FB(])g(0)p Fu(;)g(J)708 464 y FA(])724 457 y FB([)18 b(suc)o(h)h(that)g Fu(q)r FB(\()p Fu(J)1047 464 y FA(c)1063 457 y FB(\))f(=)g(0.)29 b(On)18 b([)p Fu(\032)1345 464 y FA(])1361 457 y Fu(;)8 b(D)q(=\014)s(J)1506 464 y FA(c)1523 457 y FB([,)19 b Fu(G)1608 464 y FA(\014)r(;J)1659 468 y Fm(c)1695 457 y FB(equals)g Fu(G)1884 464 y FA(\014)r(;)p FC(0)1935 457 y FB(,)-59 529 y(whic)o(h)e(is)g(strictly)g(con)o(v)o(ex)f(and)i (increasing)f(on)h(this)g(in)o(terv)m(al.)24 b(So,)18 b(the)f(in\014m)o(um)e(de\014ning)i Fu(q)r FB(\()p Fu(J)1768 536 y FA(c)1785 529 y FB(\))f(=)g(0)i(is)-59 601 y(attained)e(for)h(at) f(least)h(one)f Fu(\032)496 608 y FC(+)539 601 y Fu(>)e(D)q(=\014)s(J) 714 608 y FA(c)732 601 y FB(.)21 b(This)c(pro)o(v)o(es)e(the)h(\014rst) h(assertion)g(of)f(the)g(theorem.)14 710 y Fv(Step)h(3:)k FB(#)p Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\))12 b(=)i(3)p Fv(.)70 b FB(Supp)q(ose)15 b(that)g Fu(D)h FB(=)e(1)h(and)g Fu(g)1184 692 y FF(00)1219 710 y FB(is)g(con)o(v)o (ex.)k(Then)14 b Fu(G)1608 692 y FF(00)1608 722 y FA(\014)r(;)p FC(0)1674 710 y FB(is)h(con)o(v)o(ex)e(\(cf.)-59 782 y(\(18\)\))18 b(and)h(thereb)o(y)e(strictly)f(increasing)i(on)g([)p Fu(\032)845 789 y FA(])860 782 y Fu(;)8 b Ft(1)p FB([.)26 b(Using)17 b(Lemma)f(5.2\(h\),)i(w)o(e)f(conclude)g(that)i Fu(G)1880 764 y FF(00)1880 794 y FA(\014)r(;J)1931 798 y Fm(c)-59 854 y FB(is)e(strictly)f(increasing)h(on)h([1)p Fu(=\014)s(J)577 861 y FA(c)594 854 y Fu(;)8 b Ft(1)p FB([,)17 b(and)h(therefore)f(strictly)f(p)q(ositiv)o(e)h(on)h([)p Fu(\032)1471 861 y FC(+)1500 854 y Fu(;)8 b Ft(1)p FB([,)16 b(where)i Fu(\032)1784 861 y FC(+)1831 854 y FB(is)f(the)-59 927 y(smallest)g(n)o(um)o(b)q(er)f Fu(>)h FB(1)p Fu(=\014)s(J)471 934 y FA(c)507 927 y FB(satisfying)h Fu(G)764 934 y FA(\014)r(;J)815 938 y Fm(c)834 927 y FB(\()p Fu(\032)878 934 y FC(+)907 927 y FB(\))g(=)f Fu(q)r FB(\()p Fu(J)1069 934 y FA(c)1085 927 y FB(\))h(=)f(0.)28 b(Hence)17 b Fu(G)1428 934 y FA(\014)r(;J)1479 938 y Fm(c)1515 927 y FB(is)h(strictly)f(con)o(v)o (ex)g(on)-59 999 y([)p Fu(\032)-20 1006 y FC(+)9 999 y Fu(;)8 b Ft(1)p FB([,)19 b(so)g(that)g(there)g(cannot)g(exist)f(an)o (y)h(other)g Fu(\032)g(>)f(\032)1056 1006 y FC(+)1104 999 y FB(with)h Fu(G)1256 1006 y FA(\014)r(;J)1307 1010 y Fm(c)1325 999 y FB(\()p Fu(\032)p FB(\))g(=)f Fu(q)r FB(\()p Fu(J)1533 1006 y FA(c)1550 999 y FB(\))g(=)g(0.)30 b(This)19 b(sho)o(ws)-59 1071 y(that)e Ft(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)253 1078 y FA(c)269 1071 y FB(\))13 b(=)h Ft(f)p Fu(\032)403 1078 y FF(\000)433 1071 y Fu(;)8 b(\032)480 1078 y FA(])496 1071 y Fu(;)g(\032)543 1078 y FC(+)572 1071 y Ft(g)p FB(.)21 b(In)16 b(the)g(next)g(t)o(w)o(o) g(steps,)g(w)o(e)g(assume)f(that)i(the)f(last)h(iden)o(tit)o(y)d (holds.)14 1179 y Fv(Step)23 b(4:)31 b(The)22 b(c)n(ase)g Fu(J)27 b(>)22 b(J)552 1186 y FA(c)569 1179 y Fv(.)85 b FB(Let)21 b Fu(J)27 b Ft(2)8 b FB(])p Fu(J)897 1186 y FA(c)914 1179 y Fu(;)g(J)963 1186 y FA(])979 1179 y FB(].)35 b(Then)21 b Fu(q)r FB(\()p Fu(J)5 b FB(\))21 b Fu(<)h FB(0,)g(but)f Fu(G)1541 1186 y FA(\014)r(;J)1620 1179 y FB(=)h Fu(G)1718 1186 y FA(\014)r(;)p FC(0)1791 1179 y Ft(\025)g FB(0)f(on)-59 1252 y(]0)p Fu(;)8 b(D)q(=\014)s(J)d FB(].)24 b(Hence)16 b Fu(G)365 1234 y FF(\003\003)365 1264 y FA(\014)r(;J)438 1252 y FB(is)h(a\016ne)g(at)h(least)f(on)h([)p Fu(\032)903 1259 y FF(\000)932 1252 y Fu(;)8 b(D)q(=\014)s(J)d FB(])17 b(with)g(negativ)o(e)f(slop)q(e)i(ln)8 b Fu(z)1615 1259 y FA(J)1654 1252 y Ft(\021)16 b FB(ln)8 b Fu(z)1781 1259 y FA(m)1814 1252 y FB(\()p Fu(\014)s(;)g(J)d FB(\).)-59 1324 y(Let)22 b([)p Fu(\032)73 1331 y FA(J)o(;)p FF(\000)130 1324 y Fu(;)8 b(\032)177 1331 y FA(J)o(;)p FC(+)235 1324 y FB(])23 b Ft(\033)g FB([)p Fu(\032)373 1331 y FF(\000)402 1324 y Fu(;)8 b(D)q(=\014)s(J)d FB(])21 b(b)q(e)h(the)f(maximal)e(in)o (terv)m(al)h(on)j(whic)o(h)e Fu(G)1391 1306 y FF(\003\003)1391 1336 y FA(\014)r(;J)1469 1324 y FB(has)h(slop)q(e)g(ln)8 b Fu(z)1762 1331 y FA(J)1786 1324 y FB(.)38 b(Then)-59 1396 y Fu(\032)-34 1403 y FA(J)o(;)p FF(\000)24 1396 y Fu(;)8 b(\032)71 1403 y FA(J)o(;)p FC(+)148 1396 y Ft(2)20 b(M)p FB(\()p Fu(z)303 1403 y FA(J)327 1396 y Fu(;)8 b(\014)s(;)g(J)d FB(\).)30 b(In)19 b(the)h(limit)d(as)j Fu(J)k Ft(#)c Fu(J)955 1403 y FA(c)972 1396 y FB(,)g(all)f(accum)o (ulation)f(p)q(oin)o(ts)i(of)g(\()p Fu(\032)1633 1403 y FA(J)o(;)p FF(\000)1691 1396 y FB(\))g(and)g(\()p Fu(\032)1872 1403 y FA(J)o(;)p FC(+)1930 1396 y FB(\))-59 1468 y(b)q(elong)d(to)g Ft(M)p FB(\()p Fu(z)259 1475 y FA(c)276 1468 y Fu(;)8 b(\014)s(;)g(J)378 1475 y FA(c)394 1468 y FB(\),)17 b(b)o(y)f(Theorem)f (3.1\(iii\).)22 b(Since)16 b Fu(\032)1049 1475 y FA(J)o(;)p FF(\000)1121 1468 y Fu(<)f(\032)1199 1475 y FF(\000)1245 1468 y FB(and)i Fu(\032)1365 1475 y FA(J)o(;)p FC(+)1438 1468 y Ft(\025)d Fu(D)q(=\014)s(J)5 b FB(,)17 b(it)f(follo)o(ws)g(that) -59 1541 y Fu(\032)-34 1548 y FA(J)o(;)p FF(\000)38 1541 y Ft(!)d Fu(\032)126 1548 y FF(\000)172 1541 y FB(and)k Fu(\032)292 1548 y FA(J)o(;)p FC(+)364 1541 y Ft(!)d Fu(\032)453 1548 y FC(+)482 1541 y FB(.)21 b(Assertion)16 b(\(a\))h(is)f(th)o(us)g(pro)o(v)o(ed.)14 1649 y Fv(Step)24 b(5:)33 b(The)23 b(c)n(ase)g Fu(J)28 b(<)c(J)560 1656 y FA(c)577 1649 y Fv(.)88 b FB(F)l(rom)21 b(Step)h(2)h(w)o(e)e(kno)o(w) h(that)h Fu(G)1338 1656 y FA(\014)r(;J)1389 1660 y Fm(c)1407 1649 y FB(\()p Fu(D)q(=\014)s(J)1549 1656 y FA(c)1567 1649 y FB(\))h Fu(>)f FB(0)i(=)e Fu(q)r FB(\()p Fu(J)1851 1656 y FA(c)1868 1649 y FB(\))h(=)-59 1721 y Fu(G)-21 1728 y FA(\014)r(;J)30 1732 y Fm(c)48 1721 y FB(\()p Fu(\032)92 1728 y FC(+)122 1721 y FB(\).)39 b(Let)22 b Fu(J)28 b(<)c(J)431 1728 y FA(c)471 1721 y FB(b)q(e)e(so)h(close)e (to)i Fu(J)824 1728 y FA(c)863 1721 y FB(that)g(still)e Fu(G)1114 1728 y FA(\014)r(;J)1170 1721 y FB(\()p Fu(D)q(=\014)s(J)5 b FB(\))24 b Fu(>)g(q)r FB(\()p Fu(J)5 b FB(\).)38 b(W)l(e)22 b(then)g(see)f(that)-59 1793 y Fu(G)-21 1800 y FA(\014)r(;J)35 1793 y FB(\()p Fu(D)q(=\014)s(J)5 b FB(\))14 b Fu(>)g(G)305 1775 y FF(\003\003)305 1806 y FA(\014)r(;J)361 1793 y FB(\()p Fu(D)q(=\014)s(J)5 b FB(\),)13 b(whence)f Fu(G)759 1775 y FF(\003\003)759 1806 y FA(\014)r(;J)828 1793 y FB(has)h(slop)q(e)g(ln)8 b Fu(z)1103 1800 y FA(J)1141 1793 y Ft(\021)14 b FB(ln)8 b Fu(z)1266 1800 y FA(m)1299 1793 y FB(\()p Fu(\014)s(;)g(J)d FB(\))11 b(on)i(an)g(in)o(terv)m(al)f ([)p Fu(\032)1773 1800 y FA(J)o(;)p FF(\000)1830 1793 y Fu(;)c(\032)1877 1800 y FA(J)o(;)p FC(+)1935 1793 y FB(])-59 1866 y(with)16 b Fu(\032)77 1873 y FA(])107 1866 y Fu(<)e(\032)184 1873 y FA(J)o(;)p FF(\000)255 1866 y Fu(<)g(D)q(=\014)s(J)19 b(<)14 b(\032)526 1873 y FA(J)o(;)p FC(+)584 1866 y FB(.)21 b(\(The)16 b(\014rst)h(inequalit)o (y)d(holds)j(b)q(ecause)f Fu(q)r FB(\()p Fu(J)5 b FB(\))13 b Fu(>)h FB(0\).)21 b(As)16 b Fu(J)j Ft(")14 b Fu(J)1794 1873 y FA(c)1811 1866 y FB(,)526 1970 y Fu(G)564 1950 y FF(0)564 1982 y FA(\014)r(;)p FC(0)616 1970 y FB(\()p Fu(\032)660 1977 y FA(J)o(;)p FF(\000)717 1970 y FB(\))g(=)g(ln)8 b Fu(z)874 1977 y FA(J)912 1970 y Ft(#)14 b FB(ln)8 b Fu(z)1023 1977 y FA(c)1054 1970 y FB(=)13 b(0)i(=)e Fu(G)1233 1950 y FF(0)1233 1982 y FA(\014)r(;)p FC(0)1285 1970 y FB(\()p Fu(\032)1329 1977 y FA(])1345 1970 y FB(\))-59 2074 y(and)i(therefore)f Fu(\032)262 2081 y FA(J)o(;)p FF(\000)334 2074 y Ft(#)f Fu(\032)397 2081 y FA(])413 2074 y FB(.)21 b(Arguing)15 b(as)g(in)f(Step)g(4)h(w)o(e)f(also)h(see)f (that)h Fu(\032)1268 2081 y FA(J)o(;)p FC(+)1340 2074 y Ft(!)f Fu(\032)1429 2081 y FC(+)1458 2074 y FB(.)21 b(The)14 b(rest)h(of)g(statemen)o(t)-59 2147 y(\(b\))h(is)g(eviden)o (t.)k Fb(2)14 2255 y FB(Finally)e(w)o(e)h(pro)o(vide)g(the)g(pro)q(of)h (of)g(Theorem)e(3.4)h(on)h(the)f(lo)o(w)g(temp)q(erature)f(limit.)28 b(The)19 b(pro)q(of)i(of)-59 2327 y(Theorem)15 b(3.5)h(is)h(similar)d (but)i(simpler)e(and)j(will)e(b)q(e)h(omitted.)-59 2436 y Fv(Pr)n(o)n(of)f(of)i(The)n(or)n(em)f(3.4.)70 b FB(Let)16 b Fu(J)i Ft(\025)c FB(0)i(b)q(e)g(giv)o(en)e(and)j([)p Fu(r)1028 2443 y FF(\000)1057 2436 y Fu(;)8 b(r)1101 2443 y FC(+)1130 2436 y FB(])15 b(b)q(e)h(an)o(y)g(critical)e(in)o (terv)m(al)g(of)i Fu(g)1732 2443 y FA(J)1757 2436 y FB(.)21 b(W)l(e)16 b(can)-59 2508 y(assume)e(that)h Fu(\015)i FB(=)d(0)h(and)h Fu(g)464 2490 y FF(\003\003)462 2520 y FA(J)515 2508 y FB(=)e(0)h(on)h([)p Fu(r)709 2515 y FF(\000)738 2508 y Fu(;)8 b(r)782 2515 y FC(+)811 2508 y FB(].)20 b(\(Otherwise)15 b(w)o(e)f(replace)g Fu(g)j FB(b)o(y)f(~)-25 b Fu(g)r FB(\()p Fu(\032)p FB(\))14 b(=)f Fu(g)r FB(\()p Fu(\032)p FB(\))c Ft(\000)f Fu(\015)s(\032)h Ft(\000)f Fu(c)15 b FB(for)-59 2580 y(suitable)h Fu(c)e Ft(2)g FD(R)p FB(.\))21 b(W)l(e)16 b(consider)g(the)g(scaled)f (functions)i Fu(g)1036 2587 y FA(\014)r(;J)1106 2580 y FB(=)c Fu(\014)1188 2562 y FF(\000)p FC(1)1235 2580 y Fu(G)1273 2587 y FA(\014)r(;J)1329 2580 y FB(.)21 b(F)l(or)c(an)o(y)f Fu(\032)e(>)g FB(0)i(w)o(e)g(ha)o(v)o(e)402 2685 y Fu(g)427 2664 y FF(0)425 2697 y FA(\014)r(;J)482 2685 y FB(\()p Fu(\032)p FB(\))e(=)f Fu(g)635 2664 y FF(0)633 2697 y FA(J)658 2685 y FB(\()p Fu(\032)p FB(\))e(+)g Fu(\014)812 2664 y FF(\000)p FC(1)867 2685 y FB(ln)d Fu(\032)j FB(+)1001 2637 y Fn(h)1021 2685 y Fu(J)5 b(\032)11 b Ft(\000)f Fu(\014)1169 2664 y FF(\000)p FC(1)1216 2685 y Fu(')h Ft(\016)g Fu(h)1323 2692 y FF(\003)1343 2685 y FB(\()p Fu(\014)s(J)5 b(\032)p FB(\))1469 2637 y Fn(i)1862 2685 y FB(\(20\))920 2817 y(20)p eop %%Page: 21 22 21 21 bop -59 18 a FB(and)450 90 y Fu(g)475 70 y FF(00)473 102 y FA(\014)r(;J)529 90 y FB(\()p Fu(\032)p FB(\))14 b(=)g Fu(g)683 70 y FF(00)681 102 y FA(J)706 90 y FB(\()p Fu(\032)p FB(\))d(+)850 56 y(1)p 834 79 56 2 v 834 124 a Fu(\014)s(\032)906 90 y FB(+)g Fu(J)987 42 y Fn(h)1006 90 y FB(1)g Ft(\000)g FB(\()p Fu(')g Ft(\016)g Fu(h)1217 97 y FF(\003)1237 90 y FB(\))1256 70 y FF(0)1267 90 y FB(\()p Fu(\014)s(J)5 b(\032)p FB(\))1393 42 y Fn(i)1426 90 y Fu(:)422 b FB(\(21\))14 305 y Fv(Step)21 b(1:)27 b(Choic)n(e)20 b(of)f Fu(\016)r Fv(.)78 b FB(Let)19 b Fu(")f(>)g FB(0)h(b)q(e)g(giv)o(en.)28 b(W)l(e)18 b(can)h(assume)f (that)i Fu(")e FB(is)h(so)g(small)e(that)j Fu(g)1860 287 y FF(\003\003)1858 318 y FA(J)1916 305 y FB(is)-59 378 y(strictly)e(con)o(v)o(ex)f(on)j([)p Fu(r)380 385 y FC(+)409 378 y Fu(;)8 b(r)453 385 y FC(+)495 378 y FB(+)13 b Fu(")p FB(])18 b(and,)i(if)e Fu(r)782 385 y FF(\000)830 378 y Fu(>)g FB(0,)i(also)f(on)h([)p Fu(r)1151 385 y FF(\000)1193 378 y Ft(\000)12 b Fu(";)c(r)1311 385 y FF(\000)1340 378 y FB(].)29 b(W)l(e)19 b(ma)o(y)e(also)j(require) d(that)-59 450 y Fu(L)h Ft(\021)f Fu(r)70 457 y FC(+)112 450 y Ft(\000)12 b Fu(r)185 457 y FF(\000)227 450 y Ft(\000)g FB(2)p Fu(")18 b(>)g FB(0)g(and,)h(if)f Fu(r)621 457 y FF(\000)668 450 y Fu(>)g FB(0,)h(also)g Fu(r)903 457 y FF(\000)945 450 y Ft(\000)12 b Fu(")17 b(>)h FB(0.)28 b(Since)17 b Fu(g)1310 457 y FA(J)1352 450 y Fu(>)h(g)1433 432 y FF(\003\003)1431 462 y FA(J)1488 450 y FB(=)f(0)i(on)g(])p Fu(r)1692 457 y FF(\000)1721 450 y Fu(;)8 b(r)1765 457 y FC(+)1794 450 y FB([,)18 b(there)-59 522 y(exists)d(some)g Fu(\016)g(>)f FB(0)i(suc)o(h)g(that)g Fu(g)564 529 y FA(J)603 522 y Fu(>)d FB(3)p Fu(\016)18 b FB(on)e([)p Fu(r)821 529 y FF(\000)861 522 y FB(+)10 b Fu(";)e(r)976 529 y FC(+)1015 522 y Ft(\000)i Fu(")p FB(].)20 b(W)l(e)c(also)g (require)f(that)h Fu(\016)h FB(is)f(so)g(small)e(that)-59 594 y Fu(g)-36 601 y FA(J)-11 594 y FB(\()p Fu(\032)p FB(\))k Ft(\025)g FB(2)p Fu(\016)c FB(+)f(\()p Fu(\032)g Ft(\000)f Fu(r)368 601 y FC(+)398 594 y FB(\)2)p Fu(\016)r(=L)19 b FB(for)g(all)f Fu(\032)h Ft(\025)e Fu(r)810 601 y FC(+)853 594 y FB(+)12 b Fu(")p FB(.)29 b(This)19 b(is)f(p)q(ossible)h(b)q (ecause)g Fu(g)1528 576 y FF(\003\003)1526 607 y FA(J)1584 594 y FB(is)g(strictly)e(con)o(v)o(ex)-59 667 y(on)i([)p Fu(r)47 674 y FC(+)76 667 y Fu(;)8 b(r)120 674 y FC(+)163 667 y FB(+)13 b Fu(")p FB(].)28 b(Similarly)l(,)16 b(if)j Fu(r)583 674 y FF(\000)631 667 y Fu(>)f FB(0)i(w)o(e)e(also)i (stipulate)e(that)i Fu(g)1241 674 y FA(J)1265 667 y FB(\()p Fu(\032)p FB(\))f Ft(\025)f FB(2)p Fu(\016)d Ft(\000)e FB(\()p Fu(\032)g Ft(\000)f Fu(r)1647 674 y FF(\000)1677 667 y FB(\)2)p Fu(\016)r(=L)19 b FB(for)g(all)-59 739 y Fu(\032)14 b Ft(\024)g Fu(r)55 746 y FF(\000)95 739 y Ft(\000)d Fu(")p FB(.)21 b(Finally)l(,)14 b(w)o(e)i(can)h(assume)e (that)i(2)p Fu(\016)r(=L)d(<)g(")p FB(.)14 855 y Fv(Step)k(2:)23 b(Condition)18 b(on)f Fu(\014)s Fv(.)70 b FB(Recall)15 b(that)191 977 y Fu(g)214 984 y FA(\014)r(;J)270 977 y FB(\()p Fu(\032)p FB(\))d Ft(\000)e Fu(g)417 984 y FA(J)442 977 y FB(\()p Fu(\032)p FB(\))42 b(=)f([)p Fu(\032)8 b FB(ln)g Fu(\032)j Ft(\000)g Fu(\032)p FB(])847 929 y Fn(.)881 977 y Fu(\014)626 1092 y FB(+)669 1058 y Fu(J)5 b(\032)726 1040 y FC(2)p 669 1080 77 2 v 695 1126 a FB(2)762 1092 y(+)11 b Fu(\032)855 1058 y(h)883 1065 y FF(\003)902 1058 y FB(\()p Fu(\014)s(J)5 b(\032)p FB(\))p 855 1080 173 2 v 914 1126 a(2)p Fu(\014)1046 1092 y(')1078 1071 y FF(0)1101 1092 y Ft(\016)11 b Fu(h)1165 1099 y FF(\003)1185 1092 y FB(\()p Fu(\014)s(J)5 b(\032)p FB(\))10 b Ft(\000)g Fu(\032)1414 1058 y(')h Ft(\016)g Fu(h)1521 1065 y FF(\003)1541 1058 y FB(\()p Fu(\014)s(J)5 b(\032)p FB(\))p 1414 1080 252 2 v 1525 1126 a Fu(\014)1685 1092 y(:)-59 1227 y FB(By)21 b(the)h(b)q(ounds)i(in)e(Lemma)e(5.2)i(\(b\)&\(d\),)i(there)d (exists)h(some)f Fu(\021)1245 1234 y FC(1)1289 1227 y Fu(>)j FB(0)e(suc)o(h)g(that,)i(for)e(all)g Fu(\014)k Ft(\025)e FB(1,)-59 1299 y Ft(j)p Fu(g)-22 1306 y FA(\014)r(;J)46 1299 y Ft(\000)12 b Fu(g)120 1306 y FA(J)145 1299 y Ft(j)k Fu(<)g(\016)j FB(on)f([0)p Fu(;)8 b(\021)423 1306 y FC(1)443 1299 y FB(].)25 b(By)17 b(assumption)h(\(A2\),)f(there)g(exists)g(some) g Fu(\021)1374 1306 y FC(2)1410 1299 y Fu(>)f FB(1)i(suc)o(h)g(that)g Fu(g)1750 1281 y FF(0)1748 1311 y FA(J)1789 1299 y Fu(>)e FB(2)p Fu(\016)r(=L)-59 1371 y FB(on)i([)p Fu(\021)48 1378 y FC(2)68 1371 y Fu(;)8 b Ft(1)p FB([.)25 b(\(Eq.)17 b(\(20\))i(and)f(again)h(the)e(b)q(ound)i(in)f(Lemma)d(5.2\(d\))j(then) g(sho)o(w)h(that,)f(for)g(all)f Fu(\014)s FB(,)g Fu(g)1840 1353 y FF(0)1838 1383 y FA(\014)r(;J)1911 1371 y Fu(>)-59 1443 y FB(2)p Fu(\016)r(=L)e(>)f FB(0)j(on)g([)p Fu(\021)260 1450 y FC(2)279 1443 y Fu(;)8 b Ft(1)p FB([.\))21 b(W)l(e)c(can)f (assume)g(that)h Fu(\021)892 1450 y FC(2)926 1443 y Fu(>)d(r)1000 1450 y FC(+)1041 1443 y FB(+)d Fu(")p FB(.)22 b(In)17 b(view)e(of)i(the)f(asymptotics)g(in)g(Lemma)-59 1516 y(5.2\(b\)&\(d\),)g(w)o(e)g(no)o(w)g(can)h(c)o(ho)q(ose)g Fu(\014)h FB(so)f(large)f(that)h Ft(j)p Fu(g)987 1523 y FA(\014)r(;J)1054 1516 y Ft(\000)11 b Fu(g)1127 1523 y FA(J)1152 1516 y Ft(j)i Fu(<)h(\016)k FB(on)f([)p Fu(\021)1377 1523 y FC(1)1396 1516 y Fu(;)8 b(\021)1442 1523 y FC(2)1462 1516 y FB(].)14 1632 y Fv(Step)22 b(3:)28 b(Pr)n(o)n(of)20 b(of)g(assertion)h(\(a\).)81 b FB(Let)20 b Fu(\014)i FB(satisfy)d(the)h(requiremen)o(t)d(of)j(Step)f(2.)32 b(Then,)21 b(on)f(the)-59 1704 y(in)o(terv)m(al)15 b([)p Fu(r)153 1711 y FF(\000)192 1704 y FB(+)c Fu(";)d(r)308 1711 y FC(+)347 1704 y Ft(\000)i Fu(")p FB(],)15 b(the)h(inequalit)o(y) e Fu(g)795 1711 y FA(\014)r(;J)865 1704 y Fu(>)g(g)940 1711 y FA(J)975 1704 y Ft(\000)c Fu(\016)15 b(>)f FB(2)p Fu(\016)k FB(holds.)k(On)16 b(the)f(other)h(hand,)g(since)g Fu(g)1895 1686 y FF(\003\003)1893 1716 y FA(\014)r(;J)-59 1776 y FB(is)g(con)o(v)o(ex)g(and)h Fu(g)271 1758 y FF(\003\003)269 1788 y FA(\014)r(;J)325 1776 y FB(\()p Fu(\032)369 1784 y FC(+)p FA(=)p FF(\000)444 1776 y FB(\))d Ft(\024)g Fu(g)553 1783 y FA(\014)r(;J)609 1776 y FB(\()p Fu(\032)653 1784 y FC(+)p FA(=)p FF(\000)728 1776 y FB(\))g Fu(<)h(\016)r FB(,)g(w)o(e)i(ha)o(v)o(e)e Fu(g)1077 1758 y FF(\003\003)1075 1788 y FA(\014)r(;J)1146 1776 y Fu(<)f(\016)k FB(on)f([)p Fu(r)1342 1783 y FF(\000)1371 1776 y Fu(;)8 b(r)1415 1783 y FC(+)1445 1776 y FB(].)22 b(Hence)15 b Fu(g)1663 1783 y FA(\014)r(;J)1733 1776 y Fu(>)g(g)1811 1758 y FF(\003\003)1809 1788 y FA(\014)r(;J)1876 1776 y FB(+)c Fu(\016)-59 1848 y FB(on)17 b([)p Fu(r)45 1855 y FF(\000)86 1848 y FB(+)12 b Fu(";)c(r)203 1855 y FC(+)243 1848 y Ft(\000)k Fu(")p FB(].)22 b(This)17 b(sho)o(ws)h(that)g Fu(g)752 1830 y FF(\003\003)750 1861 y FA(\014)r(;J)823 1848 y FB(is)f(a\016ne)f(at)i(least)e(on)i(this)f(in)o(terv)m(al.)22 b(Let)17 b Fu(q)g FB(=)e Fu(q)1742 1855 y FA(\014)r(;J)1814 1848 y FB(b)q(e)i(the)-59 1921 y(asso)q(ciated)h(slop)q(e)e(and)i([)p Fu(\032)432 1928 y FF(\000)461 1921 y Fu(;)8 b(\032)508 1928 y FC(+)537 1921 y FB(])14 b Ft(\033)h FB([)p Fu(r)655 1928 y FF(\000)695 1921 y FB(+)c Fu(";)d(r)811 1928 y FC(+)852 1921 y Ft(\000)j Fu(")p FB(])16 b(b)q(e)g(the)h(maximal)c(in)o (terv)m(al)j(on)h(whic)o(h)f Fu(g)1715 1902 y FF(\003\003)1713 1933 y FA(\014)r(;J)1786 1921 y FB(is)g(a\016ne)-59 1993 y(with)f(this)g(slop)q(e.)22 b(\(By)14 b(assumption)i(\(A2\),)f Fu(\032)793 2000 y FC(+)836 1993 y Fu(<)f Ft(1)p FB(.\))21 b(It)15 b(is)g(then)g(eviden)o(t)f(that,)i(for)f Fu(z)h FB(=)e Fu(z)1678 2000 y FA(\014)r(;J)1747 1993 y FB(=)g(exp)8 b Fu(\014)s(q)r FB(,)-59 2065 y Fu(\032)-34 2072 y FF(\000)-4 2065 y Fu(;)g(\032)43 2072 y FC(+)86 2065 y Ft(2)14 b(M)p FB(\()p Fu(z)r(;)8 b(\014)s(;)g(J)d FB(\).)14 2144 y(T)l(o)19 b(estimate)e Fu(q)j FB(and)f Fu(\032)450 2151 y FF(\000)p FA(=)p FC(+)543 2144 y FB(w)o(e)f(note)g(that)h Fu(g)858 2151 y FA(\014)r(;J)932 2144 y Fu(>)f Ft(\000)p Fu(\016)h FB(on)g([0)p Fu(;)8 b(\021)1222 2151 y FC(2)1242 2144 y FB(])18 b(and)h Fu(g)1396 2126 y FF(0)1394 2156 y FA(\014)r(;J)1467 2144 y Fu(>)f FB(0)h(on)g([)p Fu(\021)1674 2151 y FC(2)1693 2144 y Fu(;)8 b Ft(1)p FB([.)27 b(Hence)-59 2216 y Fu(g)-36 2223 y FA(\014)r(;J)34 2216 y Fu(>)14 b Ft(\000)p Fu(\016)j FB(ev)o(erywhere)e(and)h(therefore)g Fu(g)742 2198 y FF(\003\003)740 2228 y FA(\014)r(;J)810 2216 y Fu(>)e Ft(\000)p Fu(\016)r FB(.)20 b(This)c(giv)o(es)g(us)h(the)f(estimate)418 2338 y Ft(j)p Fu(q)r Ft(j)d FB(=)h Ft(j)p Fu(g)574 2317 y FF(\003\003)572 2350 y FA(\014)r(;J)628 2338 y FB(\()p Fu(r)669 2345 y FC(+)709 2338 y Ft(\000)d Fu(")p FB(\))g Ft(\000)g Fu(g)887 2317 y FF(\003\003)885 2350 y FA(\014)r(;J)941 2338 y FB(\()p Fu(r)982 2345 y FF(\000)1023 2338 y FB(+)g Fu(")p FB(\))p Ft(j)p Fu(=L)j(<)f FB(2)p Fu(\016)r(=L)i(<)e(")h(:)-59 2460 y FB(This)i(in)g(turn)h(implies)c(that)k(for)f(all)g Fu(\032)e Ft(2)g FB([)p Fu(r)753 2467 y FC(+)793 2460 y FB(+)d Fu(";)d(\021)911 2467 y FC(2)931 2460 y FB(])487 2582 y Fu(g)510 2589 y FA(\014)r(;J)566 2582 y FB(\()p Fu(\032)p FB(\))42 b Fu(>)f(g)773 2589 y FA(J)798 2582 y FB(\()p Fu(\032)p FB(\))11 b Ft(\000)g Fu(\016)29 b(>)f(\016)12 b FB(+)f(\()p Fu(\032)g Ft(\000)g Fu(r)1249 2589 y FC(+)1279 2582 y FB(\)2)p Fu(\016)r(=L)671 2667 y(>)41 b(g)775 2646 y FF(\003\003)773 2679 y FA(\014)r(;J)829 2667 y FB(\()p Fu(r)870 2674 y FC(+)900 2667 y FB(\))11 b(+)g(\()p Fu(\032)g Ft(\000)g Fu(r)1106 2674 y FC(+)1135 2667 y FB(\))p Fu(q)k(:)920 2817 y FB(21)p eop %%Page: 22 23 22 22 bop -59 18 a FB(Hence)16 b Fu(\032)112 25 y FC(+)163 18 y Fu(=)-30 b Ft(2)15 b FB([)p Fu(r)241 25 y FC(+)282 18 y FB(+)d Fu(";)c(\021)401 25 y FC(2)420 18 y FB(].)24 b(By)16 b(our)i(assumption)f(on)g Fu(\021)988 25 y FC(2)1008 18 y FB(,)g(the)g(case)g Fu(\032)1253 25 y FC(+)1298 18 y Fu(>)e(\021)1375 25 y FC(2)1412 18 y FB(is)i(also)h(imp)q (ossible.)k(Hence)-59 90 y Fu(\032)-34 97 y FC(+)11 90 y Fu(<)14 b(r)85 97 y FC(+)126 90 y FB(+)e Fu(")p FB(.)23 b(In)16 b(the)h(case)g Fu(r)508 97 y FF(\000)552 90 y Fu(>)e FB(0,)i(the)g(same)f(reasoning)h(sho)o(ws)h(that)f Fu(\032)1357 97 y FF(\000)1402 90 y Fu(>)e(r)1477 97 y FF(\000)1518 90 y Ft(\000)c Fu(")p FB(.)23 b(This)17 b(completes)-59 162 y(the)f(pro)q(of)h(of)g(\(a\).)14 278 y Fv(Step)h(4:)k(Pr)n(o)n(of)16 b(of)h(\(b\).)70 b FB(W)l(e)16 b(supp)q(ose)h(\014rst)g(that)f Fu(r)1014 285 y FF(\000)1057 278 y Fu(>)e FB(0.)22 b(In)15 b(addition)h(to)h(the) e(conditions)h(in)g(Step)-59 351 y(1,)e(w)o(e)g(assume)g(that)g Fu(")g FB(is)g(so)h(small)d(that)j Fu(g)729 333 y FF(00)727 363 y FA(J)766 351 y Fu(>)e FB(0)i(on)g([)p Fu(r)958 358 y FF(\000)993 351 y Ft(\000)7 b Fu(";)h(r)1106 358 y FF(\000)1142 351 y FB(+)f Fu(")p FB(])g Ft([)g FB([)p Fu(r)1307 358 y FC(+)1342 351 y Ft(\000)g Fu(";)h(r)1455 358 y FC(+)1490 351 y FB(+)f Fu(")p FB(],)13 b(and)i(then)f(require)-59 423 y(that)20 b Fu(\016)i FB(is)d(also)i(suc)o(h)f(that)g Fu(g)496 405 y FF(00)494 435 y FA(J)539 423 y Fu(>)g(\016)h FB(on)g(the)f(same)e(set.)33 b(Lemma)17 b(5.2\(f)s(\))k(implies)c(that) k Ft(j)p Fu(g)1666 405 y FF(00)1664 435 y FA(\014)r(;J)1733 423 y Ft(\000)14 b Fu(g)1811 405 y FF(00)1809 435 y FA(J)1833 423 y Ft(j)20 b Fu(<)g(\016)-59 495 y FB(on)h([)p Fu(r)49 502 y FF(\000)92 495 y Ft(\000)14 b Fu(";)8 b Ft(1)p FB([)20 b(when)g Fu(\014)j FB(is)e(large)f(enough.)36 b(F)l(or)20 b(these)h Fu(\014)s FB(,)f Fu(g)1143 502 y FA(\014)r(;J)1220 495 y FB(is)h(strictly)e(con)o(v)o(ex)g(on)i(the)g (in)o(terv)m(als)-59 567 y([)p Fu(r)-23 574 y FF(\000)15 567 y Ft(\000)8 b Fu(";)g(r)129 574 y FF(\000)166 567 y FB(+)h Fu(")p FB(])14 b(and)i([)p Fu(r)394 574 y FC(+)431 567 y Ft(\000)9 b Fu(";)f(r)546 574 y FC(+)583 567 y FB(+)h Fu(")p FB(])14 b(con)o(taining)h Fu(\032)942 574 y FF(\000)987 567 y FB(resp.)f Fu(\032)1127 574 y FC(+)1157 567 y FB(,)h(whence)f Fu(g)1378 574 y FA(\014)r(;J)1448 567 y Fu(>)g(g)1525 549 y FF(\003\003)1523 580 y FA(\014)r(;J)1594 567 y FB(on)i(])p Fu(\032)1700 574 y FF(\000)1729 567 y Fu(;)8 b(\032)1776 574 y FC(+)1805 567 y FB([.)21 b(This)-59 640 y(pro)o(v)o(es)16 b(assertion)g(\(b\))h(in)f(the)g(case)g Fu(r)643 647 y FF(\000)686 640 y Fu(>)e FB(0.)14 718 y(If)19 b Fu(r)88 725 y FF(\000)136 718 y FB(=)f(0,)i(the)f(conditions) g(on)h Fu(")f FB(and)g Fu(\016)i FB(in)o(v)o(olving)c Fu(g)1062 700 y FF(00)1060 731 y FA(J)1104 718 y FB(in)i(the)g(neigh)o (b)q(orho)q(o)q(d)i(of)e Fu(r)1640 725 y FF(\000)1689 718 y FB(are)g(replaced)-59 790 y(b)o(y)h(the)h(condition)g(that)g Fu(g)456 772 y FF(0)454 803 y FA(J)500 790 y Fu(>)h(\016)16 b FB(+)e(2)p Fu(\016)r(=L)21 b FB(on)h([0)p Fu(;)8 b(")p FB(].)34 b(Let)21 b Fu(b)p FB(\()p Fu(D)q FB(\))g(b)q(e)g(as)h(in)e (\(19\))i(and)f Fu(b)h(<)f Ft(1)g FB(b)q(e)g(suc)o(h)-59 863 y(that)d Fu(g)73 845 y FF(00)107 863 y Ft(\000)11 b Fu(J)i(b)p FB(\()p Fu(D)q FB(\))k Fu(>)g Ft(\000)p Fu(b)g FB(on)h([0)p Fu(;)8 b(")p FB(].)25 b(By)17 b(\(21\),)h(it)g (then)f(follo)o(ws)h(that)g(,)g(for)g(an)o(y)f Fu(\014)i(>)d FB(1)p Fu(="b)p FB(,)i Fu(g)1730 845 y FF(00)1728 875 y FA(\014)r(;J)1801 863 y Fu(>)e FB(0)i(on)-59 935 y([0)p Fu(;)8 b FB(1)p Fu(=\014)s(b)p FB(].)21 b(Let)c Fu(\014)i FB(b)q(e)e(so)g(large)g(that,)f(in)h(addition)g(to)g(our)g(previous)f (assumptions,)g Fu(\014)1565 917 y FF(\000)p FC(1)1620 935 y FB(ln\()p Fu(\014)s(b)p FB(\))d Fu(<)i(\016)r FB(.)21 b(Eq.)-59 1007 y(\(20\))f(and)f(Lemma)e(5.2\(d\))i(then)g(sho)o(w)h (that)f Fu(g)838 989 y FF(0)836 1019 y FA(\014)r(;J)911 1007 y Fu(>)f(g)992 989 y FF(0)990 1019 y FA(J)1028 1007 y Ft(\000)12 b Fu(\016)20 b(>)f FB(2)p Fu(\016)r(=L)f(>)h(q)h FB(on)g([1)p Fu(=\014)s(b;)8 b(")p FB(].)27 b(This)20 b(implies)-59 1079 y(that)f Fu(\032)74 1086 y FF(\000)127 1079 y Fu(=)-29 b Ft(2)18 b FB([1)p Fu(=\014)s(b;)8 b(")p FB(].)27 b(It)18 b(follo)o(ws)g(that)i Fu(g)736 1086 y FA(\014)r(;J)810 1079 y FB(is)f(strictly)e(con)o(v)o(ex)g(on)i(a)h (neigh)o(b)q(orho)q(o)q(d)g(of)f Fu(\032)1698 1086 y FF(\000)1728 1079 y FB(,)g(and)g(th)o(us)-59 1152 y Fu(g)-36 1159 y FA(\014)r(;J)34 1152 y Fu(>)14 b(g)111 1134 y FF(\003\003)109 1164 y FA(\014)r(;J)181 1152 y FB(on)j(])p Fu(\032)288 1159 y FF(\000)317 1152 y Fu(;)8 b(\032)364 1159 y FC(+)394 1152 y FB([)15 b(also)i(in)f(the)g(case)g Fu(r)787 1159 y FF(\000)831 1152 y FB(=)d(0.)14 1268 y Fv(Step)22 b(5:)28 b(Pr)n(o)n(of)20 b(of)g(\(c\).)81 b FB(By)19 b(Prop)q(osition)i(5.4,)g(the)e(uniqueness)h(of)g Fu(z)h FB(is)f(equiv)m(alen)o(t)e(to)i(the)g(fact)-59 1340 y(that)f Fu(g)72 1347 y FA(\014)r(;J)147 1340 y FB(is)g(strictly)e(con)o(v)o(ex)g(on)j(the)e(in)o(terv)m(als)g([0)p Fu(;)8 b(\032)970 1347 y FF(\000)1000 1340 y FB(])18 b(and)h([)p Fu(\032)1168 1347 y FC(+)1198 1340 y Fu(;)8 b Ft(1)p FB([.)28 b(If)18 b Fu(D)i FB(=)e(1,)h(Lemma)e(5.2\(g\))i(and) -59 1412 y(\(21\))d(sho)o(w)f(that)g Fu(g)291 1394 y FF(00)289 1424 y FA(\014)r(;J)359 1412 y Fu(>)f(g)436 1394 y FF(00)434 1424 y FA(J)474 1412 y FB(ev)o(erywhere.)19 b(So,)c(in)f(case)h Fu(D)h FB(=)e(1)h(the)f(assertion)i(follo)o(ws)f (imm)o(ediatel)o(y)d(from)-59 1484 y(our)17 b(assumptions.)23 b(In)16 b(the)h(alternate)f(case)h(w)o(e)f(argue)h(as)h(follo)o(ws.)k (F)l(or)17 b Fu(")g FB(as)g(in)g(Step)f(4,)h(it)f(follo)o(ws)h(from)-59 1557 y(\(A2\))e(that)h Fu(g)185 1538 y FF(00)183 1569 y FA(J)222 1557 y Fu(>)e(\016)j FB(on)e([)p Fu(r)415 1564 y FC(+)454 1557 y Ft(\000)9 b Fu(";)f Ft(1)p FB([)15 b(for)g(some)g Fu(\016)g(>)f FB(0.)21 b(F)l(rom)14 b(Lemma)g(5.2\(f)s (\))i(and)g(\(21\))g(w)o(e)f(then)g(see)g(that)-59 1629 y Fu(g)-34 1611 y FF(00)-36 1641 y FA(\014)r(;J)34 1629 y Fu(>)f FB(0)e(on)g(this)g(in)o(terv)m(al)f(when)h Fu(\014)i FB(is)e(large)g(enough.)20 b(If)12 b Fu(r)1025 1636 y FF(\000)1068 1629 y Fu(>)i FB(0,)f(w)o(e)e(can)h(further)g(assume)f (that)h Fu(g)1773 1611 y FF(00)1771 1641 y FA(J)1810 1629 y Fu(>)i(\016)f FB(on)-59 1701 y([0)p Fu(;)8 b(r)23 1708 y FF(\000)57 1701 y FB(+)d Fu(")p FB(].)19 b(Again)13 b(b)o(y)g(Lemma)e(5.2\(f)s(\),)j(w)o(e)f(can)g(\014nd)h(some)e Fu(\013)i(>)g FB(0)f(suc)o(h)g(that)h(1)5 b Ft(\000)g FB(\()p Fu(')g Ft(\016)g Fu(h)1591 1708 y FF(\003)1609 1701 y FB(\))1628 1683 y FF(0)1640 1701 y FB(\()p Fu(\014)s(J)g(\032)p FB(\))13 b Fu(>)h Ft(\000)p Fu(\016)r(=J)-59 1773 y FB(for)j(all)f Fu(\032)f Ft(\025)f Fu(\013=\014)s FB(.)22 b(In)17 b(view)f(of)h (\(21\),)g(this)f(sho)o(ws)i(that)f Fu(g)1014 1755 y FF(00)1012 1786 y FA(\014)r(;J)1082 1773 y Fu(>)e FB(0)i(on)g([)p 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y(is)h(the)g(main)f(ph)o(ysical)g(justi\014cation)h(of)h(the)f (presen)o(t)g(w)o(ork.)14 2532 y(As)21 b(w)o(e)g(ha)o(v)o(e)f(stated)i (in)e(the)h(in)o(tro)q(duction,)h(a)g(main)d(feature)i(of)h(`soft')e (magnetic)g(materials)f(is)i(an)-59 2604 y(in)o(terpla)o(y)c(of)h (magnetic)f(and)h(molecular)f(forces.)27 b(F)l(or)18 b(magnets)g(on)g(soft)h(lattices,)e(this)i(phenomenon)e(is)-59 2677 y(kno)o(wn)e(as)h(magnetostriction)d(and)j(leads)f(to)g(a)g (\014rst-order)h(phase)f(transition)h(with)e(sim)o(ultaneous)g(jumps) 920 2817 y(22)p eop %%Page: 23 24 23 23 bop -59 18 a FB(of)16 b(magnetization)g(and)g(densit)o(y;)f(cf.,) g(e.g.,)g(Ref.)g([16)q(])h(and)g(the)g(references)f(therein.)14 97 y(The)k(ric)o(h)e(v)m(ariet)o(y)h(of)h(phase)g(transitions)g(in)f (our)h(mo)q(del)e(with)i(Hamiltonian)e(\(2\))h(is)h(also)g(similar)d (to)-59 169 y(the)g(b)q(eha)o(vior)h(of)f(another)h(kind)f(of)h(`soft') f(material,)e(namely)h(liquid)g(crystals.)22 b(In)16 b(fact)g(it)g(is)h(kno)o(wn)f(\(see)-59 241 y(e.g.)24 b([17)q(]\))17 b(that)g(a)h(transition)g(b)q(et)o(w)o(een)e(the)i (so-called)f(C-A)g(smectic)e(phases)j(can)g(b)q(e)f(con)o(tin)o(uous)h (while)-59 313 y(the)g(next)g(transition)h(from)f(the)g(smectic)e(A)i (to)h(the)f(nematic)f(phase)i(N)f(is)g(usually)h(discon)o(tin)o(uous)f (with)-59 386 y(a)k(jump)d(of)j(densit)o(y)l(,)f(cf.)f(our)i(scenario)f (B.)f(The)h(standard)i(explanation)e(of)g(this)h(phenomenon)e([17])h (is)-59 458 y(based)f(on)f(an)h(in)o(terpla)o(y)d(of)j(smectic)c (densit)o(y)i(order)h(and)h(the)f(magnitude)f(of)h(the)g(nematic)e (alignmen)o(t;)-59 530 y(this)i(corresp)q(onds)h(to)f(p)q(ositional)h (resp.)f(orien)o(tational)f(order)h(in)g(our)h(mo)q(del.)28 b(Our)19 b(scenario)g(A)f(can)i(b)q(e)-59 602 y(realized)d(in)h(the)g (case)g(of)g(the)h(order-disorder)f(transition)h(in)e(nematics)g(when)h (the)g(third)g(order)h(term)d(in)-59 674 y(the)d(Landau)i(expansion)e (of)h(the)f(free)g(energy)f(is)i(absen)o(t.)20 b(In)13 b(this)g(case)g(it)g(is)g(kno)o(wn)h(that)g(there)e(is)h(a)h(`w)o(eak') -59 747 y(\014rst-order)h(phase)g(transition)g(to)g(an)h(ordered)e (phase)h(together)g(with)g(a)g(jump)e(of)i(densit)o(y)l(,)f(whic)o(h)g (is)g(again)-59 819 y(due)i(to)h(the)f(in)o(terpla)o(y)e(of)j(p)q (ositional)g(and)f(orien)o(tational)g(order;)g(cf.)g(Refs.)21 b([17])16 b(or)g([5].)14 898 y(The)23 b(in)o(terpla)o(y)f(of)i (magnetic)d(and)j(molecular)e(forces)h(also)h(explains)e(transitions)i (in)f(a)h(new)f(kind)-59 970 y(of)d(magnetic)e(\015uids,)i(the)f(ferro) q(colloids,)h(whic)o(h)e(are)i(in)o(tensiv)o(ely)d(studied.)31 b(These)19 b(systems)f(are)i(stable)-59 1042 y(susp)q(ensions)e(of)e (disp)q(ersed)h(ferroparticles)e(in)h(liquids)f(\(ferro\015uid)h(em)o (ulsions\).)k(An)c(e\013ectiv)o(e)e(attraction)-59 1114 y(b)q(et)o(w)o(een)f(particles)h(due)g(to)g(magnetic)f(momen)o(ts)e (leads)k(to)f(a)h(v)m(ariet)o(y)e(of)h(phenomena)g(suc)o(h)g(as)h(n)o (ucleation)-59 1186 y(in)o(to)20 b(magnetic)f(dropletlik)o(e)f (aggregates,)23 b(phase)e(separation)g(induced)f(b)o(y)g(magnetic)f (\014elds,)i(and)g(ev)o(en)-59 1259 y(solidi\014cation)13 b(in)o(to)g(c)o(hains)g(|)g(whic)o(h)g(is)g(a)h(\014rst-order)g(phase)g (transition;)g(cf.)f([23,)g(24)q(])g(and)h(the)f(references)-59 1331 y(there.)28 b(T)l(o)19 b(study)g(these)g(phenomena)f(w)o(e)g(w)o (ould)h(need)f(to)h(mo)q(dify)f(our)h(mo)q(del)f(in)g(suc)o(h)h(a)g(w)o (a)o(y)f(that)h(it)-59 1403 y(pro)o(vides)11 b(a)h(lo)q(cal)g (description)f(of)h(the)f(in)o(terpla)o(y)f(of)i(forces.)20 b(This)11 b(means)g(that)h(the)g(mean-\014eld)e(in)o(teraction)-59 1475 y(of)16 b(our)h(mo)q(del)e(should)i(b)q(e)f(replaced)g(b)o(y)f(a)i (suitable)f(short-range)i(in)o(teraction)d(as)i(in)f(Refs.)f([7,)h(9].) -59 1642 y Fw(References)-36 1750 y Fq([1])23 b(Angelescu,)13 b(N.)e(and)h(Zagrebno)o(v,)e(V.A.)h(\(1985\))e(A)j(generalized)h (quasia)o(v)o(erage)d(approac)o(h)h(to)g(the)g(description)36 1806 y(of)k(the)g(limit)i(states)d(of)h(the)g(n-v)o(ector)g(Curie-W)l (eiss)i(ferromagnet.)c Fr(J.)j(Stat.)g(Phys.)f Fa(41)p Fq(,)g(323{334)f(.)-36 1900 y([2])23 b(Burk)o(e,)f(T.,)f(Leb)q(o)o (witz,)i(J.L.)e(and)g(Lieb,)i(E.)e(\(1966\))e(Phase)i(transition)g(in)h (a)e(mo)q(del)i(quan)o(tum)f(system:)36 1956 y(quan)o(tum)15 b(corrections)g(to)g(the)g(lo)q(cation)h(of)f(the)g(critical)i(p)q(oin) o(t.)e Fr(Phys.)h(R)n(ev.)f Fa(149)p Fq(,)g(118{122)e(.)-36 2050 y([3])23 b(Csisz\023)-23 b(ar,)14 b(I.)g(\(1984\))f(Sano)o(v)h (prop)q(ert)o(y)l(,)g(generalized)i(I-pro)s(jection)f(and)f(a)g (conditional)i(limit)g(theorem.)d Fr(A)o(nn.)36 2107 y(Pr)n(ob.)i Fa(12)p Fq(,)g(768-798)f(.)-36 2200 y([4])23 b(Gates,)12 b(D.J.)f(and)i(P)o(enrose,)f(O.)h(\(1969\))d(The)j(v)m(an)g (der)f(W)l(aals)h(limit)g(for)f(classical)i(systems.)e(I)g(:)g(A)h(v)m (ariational)36 2257 y(principle.)18 b Fr(Commun.)e(Math.)g(Phys.)f Fa(15)p Fq(,)g(255{276)f(.)-36 2351 y([5])23 b(de)16 b(Gennes)f(P)l(.G.)g(\(1974\))e Fr(The)j(physics)g(of)g(liquid)g (crystals)p Fq(.)e(Oxford:)20 b(Oxford)15 b(Univ)o(ersit)o(y)h(Press.) -36 2444 y([6])23 b(Georgii,)14 b(H.O.)f(\(1988\))e Fr(Gibbs)j(Me)n (asur)n(es)g(and)g(Phase)h(T)m(r)n(ansitions.)c Fq(De)i(Gruyter)g (Studies)h(in)g(Mathematics)36 2501 y(V)l(ol.)h(9.)g(Berlin:)22 b(de)15 b(Gruyter)g(1988)f(.)-36 2595 y([7])23 b(Georgii,)15 b(H.O.)f(and)h(H\177)-23 b(aggstr\177)g(om,)13 b(O.)i(\(1996\))e(Phase) i(transition)g(in)h(con)o(tin)o(uum)f(P)o(otts)f(mo)q(dels.)h Fr(Commun.)36 2651 y(Math.)i(Phys.)d Fa(181)p Fq(,)i(507{528)d(.)920 2817 y FB(23)p eop %%Page: 24 25 24 24 bop -36 18 a Fq([8])23 b(Georgii,)e(H.O.)e(and)h(Zessin,)h(H.)f (\(1993\))d(Large)j(deviations)h(and)e(the)h(maxim)o(um)g(en)o(trop)o (y)f(principle)j(for)36 74 y(mark)o(ed)15 b(p)q(oin)o(t)g(random)g (\014elds.)h Fr(Pr)n(ob)n(ab.)g(The)n(ory)g(R)n(elat.)g(Fields)e Fa(96)p Fq(,)h(177{204)f(.)-36 168 y([9])23 b(Grub)q(er,)14 b(Ch.)g(and)h(Gri\016ths,)e(R.B.)i(\(1986\))d(Phase)i(transition)h(in)g (a)f(ferromagnetic)g(\015uid.)h Fr(Physic)n(a)g(A)f Fa(138)p Fq(,)36 225 y(220{230)f(.)-59 318 y([10])23 b(Hemmer,)15 b(P)l(.C.)f(and)i(Im)o(bro,)e(D.)h(\(1977\))e(F)l(erromagnetic)i (\015uids.)h Fr(Phys.)g(R)n(ev.)f Fa(A)i(16)p Fq(,)e(380{386)e(.)-59 412 y([11])23 b(Johansson,)17 b(K.)f(\(1995\))f(On)j(separation)e(of)g (phases)h(in)h(one-dimensional)h(gases.)d Fr(Commun.)h(Math.)h(Phys.)36 469 y Fa(169)p Fq(,)d(521{561)e(.)-59 563 y([12])23 b(Leb)q(o)o(witz,) 15 b(J.L.)g(and)g(P)o(enrose,)f(O.)h(\(1966\))d(Rigorous)j(treatmen)o (t)f(of)g(the)h(V)l(an)g(der)g(W)l(aals)f(Maxw)o(ell)h(theory)36 619 y(of)g(the)g(liquid)j(v)m(ap)q(our)d(transition.)g Fr(J.)h(Math.)h(Phys.)e Fa(7)p Fq(,)g(98{113)e(.)-59 713 y([13])23 b(Leb)q(o)o(witz,)16 b(J.L.,)f(Mazel,)g(A.E.)f(and)i (Presutti,)e(E.)h(\(1997\))f(in)i(preparation.)-59 807 y([14])23 b(Lieb,)16 b(E.)e(\(1966\))e(Quan)o(tum-mec)o(hanical)17 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b(Ruelle,)g(D.)18 b(\(1971\))g(Existence)j(of)e(a)g(phase)h(transition) f(in)i(a)e(con)o(tin)o(uous)h(classical)h(system.)d Fr(Phys.)i(R)n(ev.) 36 1502 y(L)n(ett.)14 b Fa(27)p Fq(,)h(1040{1041)e(.)-59 1595 y([20])23 b(Simon,)17 b(B.)g(\(1993\))e Fr(The)i(Statistic)n(al)g (Me)n(chanics)f(of)i(L)n(attic)n(e)f(Gases)p Fq(,)f(V)l(ol.)h(I,)f (Princeton,)i(N.J.:)k(Princeton)36 1652 y(Univ)o(ersit)o(y)16 b(Press.)-59 1746 y([21])23 b(Widom,)14 b(B.)g(and)g(Ro)o(wlinson,)g (J.S.)g(\(1970\))e(New)i(mo)q(del)h(for)e(the)h(study)g(of)f(liquid-v)m (ap)q(or)k(phase)d(transition.)36 1802 y Fr(J.)i(Chem.)g(Phys.)f Fa(52)p Fq(,)g(1670{1684.)-59 1896 y([22])23 b(Zhang,)e(H.)f(and)g (Widom,)h(M.)f(\(1994\))e(Global)j(phase)f(diagrams)g(for)g(dip)q(olar) h(\015uids.)g Fr(Phys.R)n(ev.)e Fa(E)24 b(49)p Fq(,)36 1952 y(R3591{R3593.)-59 2046 y([23])f(Zh)o(u)d(Y)l(un,)h(Haddadian,)g (E.,)f(Mou,)g(T.,)f(Gross,)h(M.)e(and)i(Jing)h(Liu)40 b(\(1996\))18 b(R^)-23 b(ole)20 b(of)f(n)o(ucleation)i(in)g(the)36 2103 y(structure)15 b(ev)o(olution)h(of)f(a)g(magnetorheological)g (\015uid.)h Fr(Phys.R)n(ev.)f Fa(E)j(53)p Fq(,)d(1753{1759)d(.)-59 2196 y([24])23 b(Zubarev,)17 b(A.Y)l(u.)h(and)f(Iv)m(ano)o(v,)h(A.O.)f (\(1997\))e(Kinetics)k(of)e(a)g(magnetic)g(\015uid)i(phase)f (separation)f(induced)36 2253 y(b)o(y)e(an)g(external)h(magnetic)f (\014eld.)h Fr(Phys.R)n(ev.)f Fa(E)j(55)p Fq(,)d(7192{7202)d(.)920 2817 y FB(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF