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Bogdanov-Takens bifurcation, homoclinic orbits, 
 exponentially small phenomena
---------------0201150459879
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pop 1 add
} def
/ci6rgbtograyproc {
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec
XIBuffer3
dup 3 1 roll

/ci6putrgbtograystr load exch
ci6walkrgbstring
 end
} def
/ci6cmyktograyproc {
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec
XIBuffer3
dup 3 1 roll

/ci6putcmyktograystr load exch
ci6walkcmykstring
 end
} def
/ci6separatecmykproc {
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec

XIBuffer3

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
 end
} def

/ci6compositeimage {
dup 1 eq {
pop pop image
}{
/ci6colorimage load null ne {
ci6colorimage
}{
3 1 roll pop
sourcearray 0 3 -1 roll put
3 eq {/ci6rgbtograyproc}{/ci6cmyktograyproc} ifelse load
image
} ifelse
} ifelse
} def
/ci6knockoutimage {
gsave
0 ci6curtransfer exec 1 ci6curtransfer exec
eq {
0 ci6curtransfer exec 0.5 lt
}{
0 ci6curtransfer exec 1 ci6curtransfer exec gt
} ifelse
{{pop 0}}{{pop 1}} ifelse
systemdict /settransfer get exec
ci6compositeimage
grestore
} def
/ci6drawimage {
ci6testplate -1 eq {
pop ci6compositeimage
}{
dup type /arraytype eq {
dup length plateindex gt {plateindex get}{pop false} ifelse
}{
{
true
}{
dup 1 eq {plateindex 3 eq}{plateindex 3 le} ifelse
} ifelse
} ifelse
{
dup 1 eq {
pop pop ci6image
}{
dup 3 eq {
ci6compositeimage
}{
pop pop
sourcearray 0 3 -1 roll put
/ci6separatecmykproc load
ci6image
} ifelse
} ifelse
}{
ci6curoverprint {
7 {pop} repeat
}{
ci6knockoutimage
} ifelse
} ifelse
} ifelse
} def
/ci6proctintimage {
/_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store
/_pticomps _ptispace ci6numcomps store
/_ptiscale _pti1 _pti0 sub store
level2? {
_ptiname length 0 gt version cvr 2012 ge and {
[/Separation _ptiname _ptispace {_ptiproc}] setcolorspace
[_pti0 _pti1] ci6makeimagedict ci6image
}{
[/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace
[0 255] ci6makeimagedict ci6image
} ifelse
}{
_pticomps 1 eq {
{
dup
{
255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put
} ci6stringapply
} ci6concatprocs ci6image
}{
{
dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse
_ptibuf {
exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc
_pticomps 2 add -2 roll
_pticomps 1 sub -1 0 {
1 index add 2 index exch
5 -1 roll
255 mul cvi put
} for
pop pop
} ci6stringapply
} ci6concatprocs false _pticomps
/ci6colorimage load null eq {7 {pop} repeat}{ci6colorimage} ifelse
} ifelse
} ifelse
} def
/ci6graytintimage {
/_gtigray 5 -1 roll store
{1 _gtigray sub mul 1 exch sub} 4 1 roll
/DeviceGray ci6proctintimage
} def
/ci6cmyktintimage {
/_cticmyk 5 -1 roll store
{_cticmyk {1 index mul exch} forall pop} 4 1 roll
/DeviceCMYK ci6proctintimage
} def
/ci6rgbtintimage {
/_rtirgb 5 -1 roll store
{_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll
/DeviceRGB ci6proctintimage
} def
/ci6tintimage {
ci6testplate -1 eq {
ci6colorexpand
3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll
dup /DeviceGray eq {
pop ci6graytintimage
}{
dup /DeviceRGB eq {
pop ci6rgbtintimage
}{
pop ci6cmyktintimage
} ifelse
} ifelse
}{
dup ci6marksplate {
plateindex 5 lt {
ci6colortocmyk plateindex get
dup 0 eq ci6curoverprint and {
7 {pop} repeat
}{
1 exch sub
exch {1 0}{0 1} ifelse () ci6graytintimage
} ifelse
}{
pop exch {0}{0 exch} ifelse 0 3 1 roll () ci6graytintimage
} ifelse
}{
ci6curoverprint {
8 {pop} repeat
}{
pop pop pop
{pop 1} 0 1 () /DeviceGray ci6proctintimage
} ifelse
} ifelse
} ifelse
} def
/XINullImage {
} def
/XIImageMask {
XIImageWidth XIImageHeight false
[XIImageWidth 0 0 XIImageHeight neg 0 0]
/XIDataProc load
imagemask
} def
/XIImageTint {
XIImageWidth XIImageHeight XIBitsPerPixel
[XIImageWidth 0 0 XIImageHeight neg 0 0]
/XIDataProc load
XIType 3 eq XIColorValues XIColorSpace ci6tintimage
} def
/XIImage {
XIImageWidth XIImageHeight XIBitsPerPixel
[XIImageWidth 0 0 XIImageHeight neg 0 0]
/XIDataProc load
false XIChannelCount XIPlateList ci6drawimage
} def
/XG {
pop pop
} def
/XF {
13 {pop} repeat
} def
/Xh {
Adobe_ColorImage_AI6_Vars begin
gsave
/XIType exch def
/XIImageHeight exch def
/XIImageWidth exch def
/XIImageMatrix exch def
0 0 moveto
XIImageMatrix concat
XIImageWidth XIImageHeight scale

/_lp /null ddef
_fc
/_lp /imagemask ddef
 end
} def
/XH {
Adobe_ColorImage_AI6_Vars begin
grestore
 end
} def
/XIEnable {
Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put
} def
/XC {
Adobe_ColorImage_AI6_Vars begin
ci6colormake
/XIColorSpace exch def
/XIColorValues exch def
 end
} def
/XIPlates {
Adobe_ColorImage_AI6_Vars begin
/XIPlateList exch def
 end
} def
/XI
{
Adobe_ColorImage_AI6_Vars begin
gsave
/XIType exch def
cvi dup
256 idiv /XICompression exch store
256 mod /XIEncoding exch store
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
/XIRowBytes exch def
XIEnable {
/XIBuffer3 XIImageWidth string def
XICompression 0 eq {
/XIBuffer1 XIRowBytes string def
XIEncoding 0 eq {
{currentfile XIBuffer1 readhexstring pop}
}{
{currentfile XIBuffer1 readstring pop}
} ifelse
}{
/XIBuffer1 256 string def
/XIBuffer2 XIRowBytes string def
{currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if}
/ASCII85Decode filter /DCTDecode filter
/XIFile exch def
{XIFile XIBuffer2 readstring pop}
} ifelse
/XIDataProc exch def

XIType 1 ne {
0 setgray
} if
XIType 1 eq {
XIImageMask
}{
XIType 2 eq XIType 3 eq or {
XIImageTint
}{
XIImage
} ifelse
} ifelse
}{
XINullImage
} ifelse
/XIPlateList false def
grestore
 end
} def
end
%%EndProcSet
%%BeginResource: procset Adobe_Illustrator_AI5 1.3 0
%%Title: (Adobe Illustrator (R) Version 8.0 Full Prolog)
%%Version: 1.3 0
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1998 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI5_vars 112 dict dup begin
put
/_?cmyk false def
/_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
/_lineorientation 0 def
/_charorientation 0 def
/_yokoorientation 0 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
/_shift [0 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
/_fontSize 0 def
/_fontAscent 0 def
/_fontDescent 0 def
/_fontHeight 0 def
/_fontRotateAdjust 0 def
/Ss 256 string def
Ss 0 (fonts/) putinterval
/_cnt 0 def
/_scale [1 1] def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_hfname 100 string def
/_hffound false def
/Tx
{
} def
/Tj
{
} def
/CRender
{
} def
/_AI3_savepage
{
} def
/_gf null def
/_cf 4 array def
/_rgbf 3 array def
/_if null def
/_of false def
/_fc
{
} def
/_gs null def
/_cs 4 array def
/_rgbs 3 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
/_lobyte 0 def
/_hibyte 0 def
/_cproc null def
/_cscript 0 def
/_hvax 0 def
/_hvay 0 def
/_hvwb 0 def
/_hvcx 0 def
/_hvcy 0 def
/_bitfont null def
/_bitlobyte 0 def
/_bithibyte 0 def
/_bitkey null def
/_bitdata null def
/_bitindex 0 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 100 dict put
} if
userdict /Adobe_Illustrator_AI5 get begin
/initialize
{
Adobe_Illustrator_AI5 dup begin
Adobe_Illustrator_AI5_vars begin
/_aicmykps where {pop /_?cmyk _aicmykps def}if
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
/hswj
{
dup stringwidth 3 2 roll
{
_hvwb eq { exch _hvcx add exch _hvcy add } if
exch _hvax add exch _hvay add
} cforall
} def
/vswj
{
0 0 3 -1 roll
{
dup 255 le
_charorientation 1 eq
and
{
dup cstring stringwidth 5 2 roll
_hvwb eq { exch _hvcy sub exch _hvcx sub } if
exch _hvay sub exch _hvax sub
4 -1 roll sub exch
3 -1 roll sub exch
}
{
_hvwb eq { exch _hvcy sub exch _hvcx sub } if
exch _hvay sub exch _hvax sub
_fontHeight sub
} ifelse
} cforall
} def
/swj
{
6 1 roll
/_hvay exch ddef
/_hvax exch ddef
/_hvwb exch ddef
/_hvcy exch ddef
/_hvcx exch ddef
_lineorientation 0 eq { hswj } { vswj } ifelse
} def
/sw
{
0 0 0 6 3 roll swj
} def
/vjss
{
4 1 roll
{
dup cstring
dup length 1 eq
_charorientation 1 eq
and
{
-90 rotate
currentpoint
_fontRotateAdjust add
moveto
gsave
false charpath currentpoint
5 index setmatrix stroke
grestore
_fontRotateAdjust sub
moveto
_sp eq
{
5 index 5 index rmoveto
} if
2 copy rmoveto
90 rotate
}
{
currentpoint
_fontHeight sub
5 index sub
3 index _sp eq
{
9 index sub
} if

currentpoint
exch 4 index stringwidth pop 2 div sub
exch _fontAscent sub
moveto

gsave
2 index false charpath
6 index setmatrix stroke
grestore

moveto pop pop
} ifelse
} cforall
6 npop
} def
/hjss
{
4 1 roll
{
dup cstring
gsave
false charpath currentpoint
5 index setmatrix stroke
grestore
moveto
_sp eq
{
5 index 5 index rmoveto
} if
2 copy rmoveto
} cforall
6 npop
} def
/jss
{
_lineorientation 0 eq { hjss } { vjss } ifelse
} def
/ss
{
0 0 0 7 3 roll jss
} def
/vjsp
{
4 1 roll
{
dup cstring
dup length 1 eq
_charorientation 1 eq
and
{
-90 rotate
currentpoint
_fontRotateAdjust add
moveto
false charpath
            currentpoint
_fontRotateAdjust sub
moveto
_sp eq
{
5 index 5 index rmoveto
} if
2 copy rmoveto
90 rotate
}
{
currentpoint
_fontHeight sub
5 index sub
3 index _sp eq
{
9 index sub
} if

currentpoint
exch 4 index stringwidth pop 2 div sub
exch _fontAscent sub
moveto

2 index false charpath

moveto pop pop
} ifelse
} cforall
6 npop
} def
/hjsp
{
    4 1 roll
    {
        dup cstring
        false charpath
        _sp eq
        {
            5 index 5 index rmoveto
        } if
        2 copy rmoveto
    } cforall
    6 npop
} def
/jsp
{
matrix currentmatrix
    _lineorientation 0 eq {hjsp} {vjsp} ifelse
} def
/sp
{
    matrix currentmatrix
    0 0 0 7 3 roll
    _lineorientation 0 eq {hjsp} {vjsp} ifelse
} 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
/_rgbtocmyk
{
3
{
1 exch sub 3 1 roll
} repeat
3 copy 1 4 1 roll
3
{
3 index 2 copy gt
{
exch
} if
pop 4 1 roll
} repeat
pop pop pop
4 1 roll
3
{
3 index sub
3 1 roll
} repeat
4 -1 roll
} def
/setrgbfill
{
_rgbf astore pop
/_fc
{
_lp /fill ne
{
_of setoverprint
_rgbf aload pop setrgbcolor
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} ddef
/_lp /none ddef
} def
/setrgbstroke
{
_rgbs astore pop
/_sc
{
_lp /stroke ne
{
_os setoverprint
_rgbs aload pop setrgbcolor
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} 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
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} 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
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} 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
/Xa
{
_?cmyk {
3 npop k
}{
setrgbfill 4 npop
} ifelse
} def
/XA
{
_?cmyk {
3 npop K
}{
setrgbstroke 4 npop
} ifelse
} def
/Xs
{
/_gf exch ddef
5 npop
/_fc
{
_lp /fill ne
{
_of setoverprint
_gf setAIseparationgray
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} ddef
/_lp /none ddef
} def
/XS
{
/_gs exch ddef
5 npop
/_sc
{
_lp /stroke ne
{
_os setoverprint
_gs setAIseparationgray
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/Xx
{
exch
/_gf exch ddef
0 eq {
findcmykcustomcolor
}{
_?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse
{
4 1 roll 3 npop
findcmykcustomcolor
}{
8 -4 roll 4 npop
findrgbcustomcolor
} ifelse
} ifelse
/_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
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} ddef
/_lp /none ddef
} def
/XX
{
exch
/_gs exch ddef
0 eq {
findcmykcustomcolor
}{
_?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse
{
4 1 roll 3 npop
findcmykcustomcolor
}{
8 -4 roll 4 npop
findrgbcustomcolor
} ifelse
} ifelse
/_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
/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
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} 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
/XK
{
3 -1 roll pop
0 eq
{
1 exch sub
3 {dup 3 1 roll mul 5 1 roll} repeat
mul 4 1 roll
K
}
{
1 exch sub 4 1 roll
3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat
4 -1 roll pop
XA
} ifelse
} def
/Xk
{
3 -1 roll pop
0 eq
{
1 exch sub
3 {dup 3 1 roll mul 5 1 roll} repeat
mul 4 1 roll
k
}
{
1 exch sub 4 1 roll
3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat
4 -1 roll pop
Xa
} ifelse
} def
/A
{
pop
} def
/annotatepage
{
userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse
} def
/XT {
pop pop
} def
/Xt {
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} stopped {
% assume error was due to overfilling the buffer
}{
not
{
stop
} if
endString eq {
cleartomark stop
} if
}ifelse
} loop
} def
3
{
/beginString exch store
/layerCnt 1 store
mark
{
currentfile buffer {readline} stopped {
% assume error was due to overfilling the buffer
}{
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
}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
{
6 npop
7 2 roll
5 npop
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 4 dict def
nc begin
/setgray
{
pop
} bind def
/setcmykcolor
{
4 npop
} bind def
/setrgbcolor
{
3 npop
} bind def
/setcustomcolor
{
2 npop
} bind def
currentdict readonly pop
end
/XP
{
4 npop
} bind def
/XD
{
pop
} bind def
end
setpacking
%%EndResource
%%BeginResource: procset Adobe_cshow 2.0 8
%%Title: (Writing System Operators)
%%Version: 2.0 8
%%CreationDate: (1/23/89) ()
%%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_cshow 14 dict dup begin put
/initialize
{
Adobe_cshow begin
Adobe_cshow
{
dup xcheck
{
bind
} if
pop pop
} forall
 end
Adobe_cshow begin
} def
/terminate
{
currentdict Adobe_cshow eq
{
 end
} if
} def
/cforall
{
/_lobyte 0 ddef
/_hibyte 0 ddef
/_cproc exch ddef
/_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef
{
/_lobyte exch ddef
_hibyte 0 eq
_cscript 1 eq
_lobyte 129 ge _lobyte 159 le and
_lobyte 224 ge _lobyte 252 le and or and
_cscript 2 eq
_lobyte 161 ge _lobyte 254 le and and
_cscript 3 eq
_lobyte 161 ge _lobyte 254 le and and
     _cscript 25 eq
_lobyte 161 ge _lobyte 254 le and and
     _cscript -1 eq
or or or or and
{
/_hibyte _lobyte ddef
}
{
_hibyte 256 mul _lobyte add
_cproc
/_hibyte 0 ddef
} ifelse
} forall
} def
/cstring
{
dup 256 lt
{
(s) dup 0 4 3 roll put
}
{
dup 256 idiv exch 256 mod
(hl) dup dup 0 6 5 roll put 1 4 3 roll put
} ifelse
} def
/clength
{
0 exch
{ 256 lt { 1 } { 2 } ifelse add } cforall
} def
/hawidthshow
{
{
dup cstring
show
_hvax _hvay rmoveto
_hvwb eq { _hvcx _hvcy rmoveto } if
} cforall
} def
/vawidthshow
{
{
dup 255 le
_charorientation 1 eq
and
{
-90 rotate
0 _fontRotateAdjust rmoveto
cstring
_hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow
0 _fontRotateAdjust neg rmoveto
90 rotate
}
{
currentpoint
_fontHeight sub
exch _hvay sub exch _hvax sub
2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if
3 2 roll
cstring
dup stringwidth pop 2 div neg _fontAscent neg rmoveto
show
moveto
} ifelse
} cforall
} def
/hvawidthshow
{
6 1 roll
/_hvay exch ddef
/_hvax exch ddef
/_hvwb exch ddef
/_hvcy exch ddef
/_hvcx exch ddef
_lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse
} def
/hvwidthshow
{
0 0 3 -1 roll hvawidthshow
} def
/hvashow
{
0 0 0 6 -3 roll hvawidthshow
} def
/hvshow
{
0 0 0 0 0 6 -1 roll hvawidthshow
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_shading_AI8 1.0 0
%%Title: (Adobe Illustrator 8 Shading Procset)
%%Version: 1.0 0
%%CreationDate: (12/17/97) ()
%%Copyright: ((C) 1987-1997 Adobe Systems Incorporated All Rights Reserved)
userdict /defaultpacking currentpacking put true setpacking
userdict /Adobe_shading_AI8 10 dict dup begin put
/initialize {
Adobe_shading_AI8 begin
Adobe_shading_AI8 bdprocs
Mesh /initialize get exec
} def
/terminate {
currentdict Adobe_shading_AI8 eq {
 end
} if
} def
/bdprocs {
{
dup xcheck 1 index type /arraytype eq and {
bind
} if
pop pop
} forall
} def
/X! {pop} def
/X# {pop pop} def
/Mesh 40 dict def
Mesh begin
/initialize {
Mesh bdprocs
Mesh begin
/emulate? /AI8MeshEmulation where {
pop AI8MeshEmulation
}{
systemdict /shfill known not
} ifelse def
 end
} def
/bd {
shadingdict begin
} def
/paint {
emulate? {
 end
}{
/_lp /none ddef _fc /_lp /none ddef

/AIColorSpace AIColorSpace tocolorspace store
/ColorSpace AIColorSpace topsspace store

version_ge_3010.106 not systemdict /setsmoothness known and {
0.0001 setsmoothness
} if

composite? {
/DataSource getdatasrc def
Matrix concat
currentdict end
shfill
}{
AIColorSpace makesmarks AIPlateList markingplate and not isoverprint and {
 end
}{
/ColorSpace /DeviceGray store
/Decode [0 1 0 1 0 1] store
/DataSource getplatesrc def
Matrix concat
currentdict end
shfill
} ifelse
} ifelse
} ifelse
} def
/shadingdict 12 dict def
shadingdict begin
/ShadingType 6 def
/BitsPerCoordinate 16 def
/BitsPerComponent 8 def
/BitsPerFlag 8 def
end
/datafile null def
/databuf 256 string def
/dataptr 0 def
/srcspace null def
/srcchannels 0 def
/dstchannels 0 def
/dstplate 0 def
/srctodstcolor null def
/getplatesrc {
/srcspace AIColorSpace store
/srcchannels AIColorSpace getnchannels store
/dstchannels 1 store
/dstplate getplateindex store
/srctodstcolor srcspace makesmarks {
dstplate 4 eq {
{1 exch sub}
}{
{srcspace tocmyk 3 dstplate sub index 1 exch sub 5 1 roll 4 {pop} repeat}
} ifelse
}{
{srcchannels {pop} repeat 1}
} ifelse store
/datafile getdatasrc store
/rdpatch168 load DataLength () /SubFileDecode filter
} def
/getdatasrc {
/rdcmntline load /ASCII85Decode filter
} def
/rdpatch168 {
/dataptr 0 store
49 rdcount
4 {
dup {pop srcchannels getint8} if
dup {pop srctodstcolor dstchannels putint8 true} if
} repeat
{databuf 0 dataptr getinterval}{()} ifelse
} def
/rdpatch3216 {
/dataptr 0 store
97 rdcount
4 {
dup {pop srcchannels getint16} if
dup {pop srctodstcolor dstchannels putint16 true} if
} repeat
{databuf 0 dataptr getinterval}{()} ifelse
} def
/rdcount {
dup 0 gt {
datafile databuf dataptr 4 -1 roll getinterval readstring
exch length dataptr add /dataptr exch store
}{
true
} ifelse
} def
/getint8 {
mark true 3 -1 roll
{
dup {pop datafile read} if
dup {pop 255 div true} if
} repeat
{
counttomark 1 add -1 roll pop true
}{
cleartomark false
} ifelse
} def
/putint8 {
dup dataptr add /dataptr exch store
dataptr exch
{
1 sub exch
255 mul cvi
databuf 2 index
3 -1 roll put
} repeat
pop
} def
/getint16 {
mark true 3 -1 roll
{
dup {pop datafile read} if
dup {pop 256 mul datafile read} if
dup {pop add 65535 div true} if
} repeat
{
counttomark 1 add -1 roll pop true
}{
cleartomark false
} ifelse
} def
/putint16 {
dup 2 mul dataptr add /dataptr exch store
dataptr exch
{
2 sub exch
65535 mul cvi dup
256 idiv databuf 3 index 3 -1 roll put
256 mod databuf 2 index 1 add 3 -1 roll put
} repeat
pop
} def
/srcbuf 256 string def
/rdcmntline {
currentfile srcbuf readline pop
(%) anchorsearch {pop} if
} def
/getplateindex {
0 [cyan? magenta? yellow? black? customColor?] {{exit} if 1 add} forall
} def
/aicsarray 4 array def
/aicsaltvals 4 array def
/aicsaltcolr aicsaltvals def
/tocolorspace {
dup type /arraytype eq {
mark exch aload pop
aicsarray 0 3 -1 roll put
aicsarray 1 3 -1 roll put
dup aicsarray 2 3 -1 roll put
gettintxform aicsarray 3 3 -1 roll put
counttomark aicsaltvals 0 3 -1 roll getinterval /aicsaltcolr exch store
aicsaltcolr astore pop pop
aicsarray
} if
} def
/subtintxform {aicsaltcolr {1 index mul exch} forall pop} def
/addtintxform {aicsaltcolr {1 sub 1 index mul 1 add exch} forall pop} def
/gettintxform {
/DeviceRGB eq {/addtintxform}{/subtintxform} ifelse load
} def
/getnchannels {
dup type /arraytype eq {0 get} if
colorspacedict exch get begin Channels end
} def
/makesmarks {
composite? {
pop true
}{
dup dup type /arraytype eq {0 get} if
colorspacedict exch get begin MarksPlate end
} ifelse
} def
/markingplate {
composite? {
pop true
}{
dup type /arraytype eq {
dup length getplateindex gt {getplateindex get}{pop false} ifelse
} if
} ifelse
} def
/tocmyk {
dup dup type /arraytype eq {0 get} if
colorspacedict exch get begin ToCMYK end
} def
/topsspace {
dup dup type /arraytype eq {0 get} if
colorspacedict exch get begin ToPSSpace end
} def
/colorspacedict 5 dict dup begin
/DeviceGray 4 dict dup begin
/Channels 1 def
/MarksPlate {pop black?} def
/ToCMYK {pop 1 exch sub 0 0 0 4 -1 roll} def
/ToPSSpace {} def
 end def
/DeviceRGB 4 dict dup begin
/Channels 3 def
/MarksPlate {pop isCMYKSep?} def
/ToCMYK {pop _rgbtocmyk} def
/ToPSSpace {} def
 end def
/DeviceCMYK 4 dict dup begin
/Channels 4 def
/MarksPlate {pop isCMYKSep?} def
/ToCMYK {pop} def
/ToPSSpace {} def
 end def
/Separation 4 dict dup begin
/Channels 1 def
/MarksPlate {
/findcmykcustomcolor where {
pop dup 1 exch ToCMYK 5 -1 roll 1 get
findcmykcustomcolor 1 setcustomcolor
systemdict /currentgray get exec
1 ne
}{
pop false
} ifelse
} def
/ToCMYK {
dup 2 get mark exch 4 2 roll
3 get exec
counttomark -1 roll tocmyk
5 -1 roll pop
} def
/ToPSSpace {} def
 end def
/Process 4 dict dup begin
/Channels 1 def
/MarksPlate {
isCMYKSep? {
1 exch ToCMYK 4 array astore getplateindex get 0 ne
}{
pop false
} ifelse
} def
/ToCMYK {
dup 2 get mark exch 4 2 roll
3 get exec
counttomark -1 roll tocmyk
5 -1 roll pop
} def
/ToPSSpace {
4 array copy dup 0 /Separation put
} def
 end def
end def
/isoverprint {
/currentoverprint where {pop currentoverprint}{_of} ifelse
} def
/version_ge_3010.106 {
   version {cvr} stopped {
      pop
      false
   }{
      3010.106 ge
   } ifelse
} def
end
end
defaultpacking setpacking
%%EndResource
%%EndProlog
%%BeginSetup
userdict /_useSmoothShade false put
userdict /_aicmykps true put
userdict /_forceToCMYK true put
Adobe_level2_AI5 /initialize get exec
Adobe_cshow /initialize get exec
Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec
Adobe_ColorImage_AI6 /initialize get exec
Adobe_shading_AI8 /initialize get exec
Adobe_Illustrator_AI5 /initialize get exec
%AI3_BeginRider
currentpacking true setpacking
%%BeginFont: Symbol
%!PS-AdobeFont-1.0: Symbol 001.008
%%CreationDate: Fri Mar 28 22:03:48 1997
%%VMusage: 30820 39997
11 dict begin
/FontInfo 10 dict dup begin
/version (001.008) readonly def
/Notice (Copyright (c) 1985, 1987, 1989, 1990, 1997 Adobe Systems Incorporated. All rights reserved.) readonly def
/FullName (Symbol) readonly def
/FamilyName (Symbol) readonly def
/Weight (Medium) readonly def
/isFixedPitch false def
/ItalicAngle 0 def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
end readonly def
/FontName /Symbol def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 32 /space put
dup 33 /exclam put
dup 34 /universal put
dup 35 /numbersign put
dup 36 /existential put
dup 37 /percent put
dup 38 /ampersand put
dup 39 /suchthat put
dup 40 /parenleft put
dup 41 /parenright put
dup 42 /asteriskmath put
dup 43 /plus put
dup 44 /comma put
dup 45 /minus put
dup 46 /period put
dup 47 /slash put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 59 /semicolon put
dup 60 /less put
dup 61 /equal put
dup 62 /greater put
dup 63 /question put
dup 64 /congruent put
dup 65 /Alpha put
dup 66 /Beta put
dup 67 /Chi put
dup 68 /Delta put
dup 69 /Epsilon put
dup 70 /Phi put
dup 71 /Gamma put
dup 72 /Eta put
dup 73 /Iota put
dup 74 /theta1 put
dup 75 /Kappa put
dup 76 /Lambda put
dup 77 /Mu put
dup 78 /Nu put
dup 79 /Omicron put
dup 80 /Pi put
dup 81 /Theta put
dup 82 /Rho put
dup 83 /Sigma put
dup 84 /Tau put
dup 85 /Upsilon put
dup 86 /sigma1 put
dup 87 /Omega put
dup 88 /Xi put
dup 89 /Psi put
dup 90 /Zeta put
dup 91 /bracketleft put
dup 92 /therefore put
dup 93 /bracketright put
dup 94 /perpendicular put
dup 95 /underscore put
dup 96 /radicalex put
dup 97 /alpha put
dup 98 /beta put
dup 99 /chi put
dup 100 /delta put
dup 101 /epsilon put
dup 102 /phi put
dup 103 /gamma put
dup 104 /eta put
dup 105 /iota put
dup 106 /phi1 put
dup 107 /kappa put
dup 108 /lambda put
dup 109 /mu put
dup 110 /nu put
dup 111 /omicron put
dup 112 /pi put
dup 113 /theta put
dup 114 /rho put
dup 115 /sigma put
dup 116 /tau put
dup 117 /upsilon put
dup 118 /omega1 put
dup 119 /omega put
dup 120 /xi put
dup 121 /psi put
dup 122 /zeta put
dup 123 /braceleft put
dup 124 /bar put
dup 125 /braceright put
dup 126 /similar put
dup 160 /Euro put
dup 161 /Upsilon1 put
dup 162 /minute put
dup 163 /lessequal put
dup 164 /fraction put
dup 165 /infinity put
dup 166 /florin put
dup 167 /club put
dup 168 /diamond put
dup 169 /heart put
dup 170 /spade put
dup 171 /arrowboth put
dup 172 /arrowleft put
dup 173 /arrowup put
dup 174 /arrowright put
dup 175 /arrowdown put
dup 176 /degree put
dup 177 /plusminus put
dup 178 /second put
dup 179 /greaterequal put
dup 180 /multiply put
dup 181 /proportional put
dup 182 /partialdiff put
dup 183 /bullet put
dup 184 /divide put
dup 185 /notequal put
dup 186 /equivalence put
dup 187 /approxequal put
dup 188 /ellipsis put
dup 189 /arrowvertex put
dup 190 /arrowhorizex put
dup 191 /carriagereturn put
dup 192 /aleph put
dup 193 /Ifraktur put
dup 194 /Rfraktur put
dup 195 /weierstrass put
dup 196 /circlemultiply put
dup 197 /circleplus put
dup 198 /emptyset put
dup 199 /intersection put
dup 200 /union put
dup 201 /propersuperset put
dup 202 /reflexsuperset put
dup 203 /notsubset put
dup 204 /propersubset put
dup 205 /reflexsubset put
dup 206 /element put
dup 207 /notelement put
dup 208 /angle put
dup 209 /gradient put
dup 210 /registerserif put
dup 211 /copyrightserif put
dup 212 /trademarkserif put
dup 213 /product put
dup 214 /radical put
dup 215 /dotmath put
dup 216 /logicalnot put
dup 217 /logicaland put
dup 218 /logicalor put
dup 219 /arrowdblboth put
dup 220 /arrowdblleft put
dup 221 /arrowdblup put
dup 222 /arrowdblright put
dup 223 /arrowdbldown put
dup 224 /lozenge put
dup 225 /angleleft put
dup 226 /registersans put
dup 227 /copyrightsans put
dup 228 /trademarksans put
dup 229 /summation put
dup 230 /parenlefttp put
dup 231 /parenleftex put
dup 232 /parenleftbt put
dup 233 /bracketlefttp put
dup 234 /bracketleftex put
dup 235 /bracketleftbt put
dup 236 /bracelefttp put
dup 237 /braceleftmid put
dup 238 /braceleftbt put
dup 239 /braceex put
dup 241 /angleright put
dup 242 /integral put
dup 243 /integraltp put
dup 244 /integralex put
dup 245 /integralbt put
dup 246 /parenrighttp put
dup 247 /parenrightex put
dup 248 /parenrightbt put
dup 249 /bracketrighttp put
dup 250 /bracketrightex put
dup 251 /bracketrightbt put
dup 252 /bracerighttp put
dup 253 /bracerightmid put
dup 254 /bracerightbt put
readonly def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0]  def
/UniqueID 42996 def
/FontBBox{-180 -293 1090 1010}readonly def
currentdict end
currentfile eexec
8dda0385bbe2e28975521ce366a1efbcd19951c755b299f7de57699ffa32088176bbcfeb9d3d075a656fc2d3ffadac652e0956194030c65b6ac7dfb4cce0096cf30964c72ac7742ef0dfd71d6aeda00d45b6c86f2ebeadafcc658a64b076f436a780849440223f266d2dab36e19a62bf7550db533ad06fe9bb6bb7ebaf6f78
06f6852f23ff339a8c1fa991f1bf47f71268b6727020001c8171cb3acdb4eebea4d340dbfb5d95d8d24eb1c831a51121a967305aec34b69ab5a3a0c21ee4192997502d726972f6ccea2c58f4e7ccd90e299cdb76cdba9c42084edd3a07ecf08d5d8459c87f2d392b866eb9ca8d6b8e8cb4cf97599599df0d1c87704f3532f9
44b58dfe91e713ff665bc5a1d74808d77591d2989951c76bc0cb73e1b40597b250c9776948f8d60bead8cf0d4fb39c205662436ac4a6187b00b5645538a68c6ce44e7231c7492f6b4677329a9823d5510077ae7874e5cbd15e8a39797dd2b269d030912cbd063fcd5befc5065c16972fc994aef7110c16634aa476dd2beaf8
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cleartomark
%%EndFont
%%BeginFont: Times-Roman
%!PS-AdobeFont-1.0: Times-Roman 002.000
%%CreationDate: Thu May  1 14:12:30 1997
%%VMusage: 34396 41288
%% The digitally encoded machine readable software for producing the
%% Typefaces licensed to you is copyrighted (c) 1985, 1987, 1989, 1990, 1993,
%% 1997 Adobe Systems. All Rights Reserved. This software is the property of
%% Adobe Systems Incorporated and its licensors, and may not be reproduced,
%% used,  displayed, modified, disclosed or transferred without the express
%% written approval of Adobe.  The digitally encoded machine readable
%% outline data for producing the Typefaces licensed to you is copyrighted (c)
%% 1981, 1997 Linotype-Hell AG and/or its subsidiaries. All Rights Reserved.
%% This data is the property of Linotype-Hell AG and/or its subsidiaries and
%% may not be reproduced, used, displayed, modified, disclosed or  transferred
%% without the express written approval of Linotype-Hell AG  and/or its
%% subsidiaries.
%% Times is a trademark of Linotype-Hell AG and/or its subsidiaries.
11 dict begin
/FontInfo 10 dict dup begin
/version (002.000) readonly def
/Notice (Copyright (c) 1985, 1987, 1989, 1990, 1993, 1997 Adobe Systems Incorporated.  All Rights Reserved.Times is a trademark of Linotype-Hell AG and/or its subsidiaries.) readonly def
/Copyright ( The digitally encoded machine readable software for producing the
 Typefaces licensed to you is copyrighted (c) 1985, 1987, 1989, 1990,
 1993, 1997 Adobe Systems.
 All Rights Reserved. This software is the property of Adobe Systems
 Incorporated and its licensors, and may not be reproduced, used,
 displayed, modified, disclosed or transferred without the express
 written approval of Adobe.

 The digitally encoded machine readable outline data for producing
 the Typefaces licensed to you is copyrighted (c) 1981, 1997 Linotype-Hell
 AG and/or its subsidiaries. All Rights Reserved.
 This data is the property of Linotype-Hell AG and/or its subsidiaries
 and may not be reproduced, used, displayed, modified, disclosed or
 transferred without the express written approval of Linotype-Hell AG
 and/or its subsidiaries. ) readonly def
/FullName (Times Roman) readonly def
/FamilyName (Times) readonly def
/Weight (Roman) readonly def
/isFixedPitch false def
/ItalicAngle 0 def
/UnderlinePosition -100 def
/UnderlineThickness 50 def
end readonly def
/FontName /Times-Roman def
/Encoding StandardEncoding def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0]  def
/UniqueID 43146 def
/FontBBox{-168 -218 1000 898}readonly def
currentdict end
currentfile eexec
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(Adobe Scatter Brush Tool)
(Marienk\344fer)
(1 /New Pattern 10/ 5 0.898039 0 0 / 0 1 1 0 0.803911 1.2 1 -1.55 1.55 ) -
(1 -180 180 1 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Scatter Brush Tool)
(Pin)
(1 /New Pattern 36/ 1 0.025 0.1 0.475 0 / 0 1 1 0 0.401676 1 1 -1.06145) -
( 1.06 1 -180 180 1 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Scatter Brush Tool)
(Stern)
(1 /New Pattern 2/ 0 1 / 1 0.5 1 1 0.25 1 1 -0.5 0.5 1 0 0 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe PatternOnPath Brush Tool)
(Doppelte Linie)
(1 / New Pattern 62/ New Pattern 63/ New Pattern 64/ / / 1 1 0.14 0.09 ) -
(0 /  1 0 1 0 1 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe PatternOnPath Brush Tool)
(Lorbeer)
(1 / New Pattern 65/ New Pattern 42/ New Pattern 67/ / New Pattern 69/ ) -
(1 0 0.55 1 0.3 /  1 0 1 0 1 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe PatternOnPath Brush Tool)
(Seil)
(1 / New Pattern 1/ / / New Pattern 3/ New Pattern 6/ 5 0 0 0 /  1 0 1 ) -
(0 1 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe ArtOnPath Brush Tool)
(Marker)
(1 / New Pattern 8/ / / / / 0 0 /  1 1 1 0 1 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe ArtOnPath Brush Tool)
(Pfeil)
(1 / New Pattern 45/ / / / / 5 0.898039 0 0 /  2 0 1 0 1 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe ArtOnPath Brush Tool)
(Pinsel)
(1 / New Pattern 5/ / / / / 1 0.5 0.85 1 0.45 /  0 0 1 0 1 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe ArtOnPath Brush Tool)
(Text)
(1 / New Pattern 50/ / / / / 1 0.952941 0.94902 0.196078 0.0745098 /  1) -
( 0 1 0 1 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe ArtOnPath Brush Tool)
(Zulaufende Linie)
(1 / New Pattern 83/ / / / / 1 0 0 0 1 /  1 1 1 0 1 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Calligraphic Brush Tool)
(6 pt Flach)
(1 4 8 10 10 90 90 2 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Calligraphic Brush Tool)
(10 pt Oval)
(1 1 19 15 15 130 130 2 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Calligraphic Brush Tool)
(12 pt Oval )
(1 7 17 45 45 0 0 2 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Calligraphic Brush Tool)
(20 pt Oval)
(1 20 20 20 100 40 80 0 2 1 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Calligraphic Brush Tool)
(25 pt Rund )
(1 10 40 100 100 0 0 2 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Calligraphic Brush Tool)
(50 pt Flach)
(1 40 60 0 0 44 44 0 0 0 0) .
%AI8_EndPluginObject
%AI8_BeginPluginObject
(Adobe Brush Manager Order)
(Adobe Brush Manager Order)
( Adobe Calligraphic Brush Tool/ 6 pt Flach/ Adobe Calligraphic Brush T) -
(ool/ 10 pt Oval/ Adobe Calligraphic Brush Tool/ 12 pt Oval / Adobe Cal) -
(ligraphic Brush Tool/ 20 pt Oval/ Adobe Calligraphic Brush Tool/ 25 pt) -
( Rund / Adobe Calligraphic Brush Tool/ 50 pt Flach/ Adobe Scatter Brus) -
(h Tool/ Hundespuren/ Adobe Scatter Brush Tool/ Blatt/ Adobe Scatter Br) -
(ush Tool/ Marienk\344fer/ Adobe Scatter Brush Tool/ Pin/ Adobe Scatter Br) -
(ush Tool/ Erdbeere/ Adobe Scatter Brush Tool/ Stern/ Adobe ArtOnPath B) -
(rush Tool/ Marker/ Adobe ArtOnPath Brush Tool/ Zulaufende Linie/ Adobe) -
( ArtOnPath Brush Tool/ Pfeil/ Adobe ArtOnPath Brush Tool/ Pinsel/ Adob) -
(e ArtOnPath Brush Tool/ Text/ Adobe PatternOnPath Brush Tool/ Doppelte) -
( Linie/ Adobe PatternOnPath Brush Tool/ Lorbeer/ Adobe PatternOnPath B) -
(rush Tool/ Seil/) .
%AI8_EndPluginObject
%AI5_End_NonPrinting--
%AI5_Begin_NonPrinting
Np
%AI8_PluginGroupInfo
(Adobe Path Blends) (Adobe Angleichungen-Zusatzmodul) (Angleichungen.aip)
%AI8_PluginGroupInfo
(Adobe PatternOnPath Brush Tool) (Adobe Musterpinsel-Zusatzmodul) (BildAufPfad.aip)
%AI8_PluginGroupInfo
(Adobe ArtOnPath Brush Tool) (Adobe Bildpinsel-Zusatzmodul) (BildAufPfad.aip)
%AI8_PluginGroupInfo
(Adobe Calligraphic Brush Tool) (Adobe Kalligraphiepinsel-Zusatzmodul) (Kalligraphiepinsel-Werkzeug.aip)
%AI8_PluginGroupInfo
(Adobe Scatter Brush Tool) (Adobe Spezialpinsel-Zusatzmodul) (Spezialpinsel-Werkzeug.aip)
%AI5_End_NonPrinting--
%%EndSetup
%AI5_BeginLayer
1 1 1 1 0 0 1 0 79 128 255 0 50 Lb
(Ebene 1) Ln
0 A
1 Ap
0 O
0.85 g
800 Ar
0 J 0 j 1 w 4 M []0 d
%AI3_Note:
0 D
0 XR
180.4302 686.1895 m
134.4741 686.1895 L
134.4741 786.1895 L
180.4302 786.1895 L
180.4302 686.1895 L
f
0 Ap
1 g
0 R
0 G
180.4863 686.4243 m
180.4863 788.395 l
B
133.8706 729.4785 m
245.5532 729.4785 l
B
0 To
1 0 0 1 247.4956 719.7671 0 Tp
0 Tv
TP
0 Tr
0 g
%_ 0 50 XQ
/_Symbol 10 10 -2.93 Tf
0 Ts
100 100 Tz
0 Tt
%_0 0 100 100 Xu
%AI55J_GlyphSubst: GlyphSubstNone
1 TA
%_ 0 XL
0 TY
0 TV
36 0 Xb
XB
0 0 5 TC
100 100 200 TW
25 TG
0 0 0 Ti
0 Ta
0 1 2 2 3 Th
0 Tq
240 Tg
0 0 Tl
0 Tc
0 Tw
(m) Tx 1 0 Tk
(\r) TX
TO
0 To
1 0 0 1 184.6943 788.0713 0 Tp
0 Tv
TP
0 Tr
(n) Tx 1 0 Tk
(\r) TX
TO
0 R
0 G
0.5 w
180.6465 729.9683 m
180.6465 737.3369 184.9863 747.8452 194.9917 757.0083 c
209.6313 770.4155 229.8936 776.0566 236.8677 778.6367 c
S
0 To
1 0 0 1 222.1641 765.3037 0 Tp
0 Tv
TP
0 Tr
0 O
0 g
1 w
(n \(m\)) Tx 1 0 Tk
(\r) TX
TO
0 To
1 0 0 1 226.1323 761.9922 0 Tp
0 Tv
TP
0 Tr
/_Symbol 8 8 -2.344 Tf
(-) Tx 1 0 Tk
(\r) TX
TO
0 To
1 0 0 1 220.0518 784.8389 0 Tp
0 Tv
TP
0 Tr
/_Symbol 10 10 -2.93 Tf
(n \(m\)) Tx 1 0 Tk
(\r) TX
TO
0 To
1 0 0 1 224.1919 782.2139 0 Tp
0 Tv
TP
0 Tr
/_Symbol 8 8 -2.344 Tf
(+) Tx 1 0 Tk
(\r) TX
TO
244.3906 729.5425 m
242.6533 730.9409 l
248.5854 729.6694 l
243.0767 727.8901 l
F
180.4375 788.2188 m
179.0625 786.7813 l
180.5 792.375 l
181.9766 786.9414 l
F
/BBAccumRotation (1.570796) XT
1 To
1 0 0 1 153.3491 794.8501 0 Tp
0 Tv
1 Ap
282.439 683.6128 m
282.439 794.8501 L
153.3491 794.8501 L
153.3491 683.6128 L
282.439 683.6128 L
n
TP
0 -9 Td
0 Tr
0 O
0 g
/_Symbol 10 10 -2.93 Tf
(\r) Tx 1 0 Tk
TO
0 To
1 0 0 1 181.7197 720.9702 0 Tp
0 Tv
TP
0 Tr
(0) Tx 1 0 Tk
(\r) TX
TO
0 To
1 0 0 1 138.6021 759.731 0 Tp
0 Tv
TP
0 Tr
/_Times-Roman 10 8.9799 -2.1799 Tf
(no f) Tx 1 20 Tk
(ix) Tx 1 15 Tk
(ed\r) Tx 1 0 Tk
0 -12 Td
(points) Tx 1 0 Tk
(\r) TX
TO
0 Ap
0 R
0 G
0.5 w
180.3442 728.0425 m
180.3442 736.0205 184.6841 747.3999 194.6899 757.3223 c
209.3296 771.8403 229.5913 777.9487 236.5659 780.7417 c
S
0.25 w [2 ]0 d
180.4541 729.5811 m
180.4541 739.4126 184.7939 753.4351 194.7998 765.6646 c
201.1519 773.4282 208.5625 779.2402 215.4868 783.6138 c
S
0 To
0.7071 0.7071 -0.7071 0.7071 188.9702 764.3657 0 Tp
0 Tv
TP
0 Tr
0 O
0 0 0 1 k
1 w []0 d
/_Times-Roman 9 8.082 -1.9619 Tf
(Hopf) Tx 1 0 Tk
(\r) TX
TO
/BBAccumRotation (0.785397) XT
0 To
0.7071 0.7071 -0.7071 0.7071 188.2671 735.4106 0 Tp
0 Tv
TP
0 Tr
(homoclinic) Tx 1 0 Tk
(\r) TX
TO
/BBAccumRotation (0.785397) XT
LB
%AI5_EndLayer--
%%PageTrailer
gsave annotatepage grestore showpage
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Adobe_Illustrator_AI5 /terminate get exec
Adobe_shading_AI8 /terminate get exec
Adobe_ColorImage_AI6 /terminate get exec
Adobe_typography_AI5 /terminate get exec
Adobe_cshow /terminate get exec
Adobe_level2_AI5 /terminate get exec
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%%Title: (Adobe Illustrator (R) Version 5.0 Level 2 Emulation)
%%Version: 1.2 0
%%CreationDate: (04/10/93) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
userdict /Adobe_level2_AI5 26 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
{
(AI8_CMYK_CustomColor)
6 packedarray
} bind def
/findrgbcustomcolor
{
(AI8_RGB_CustomColor)
5 packedarray
} bind def
/setcustomcolor
{
exch
aload pop dup
(AI8_CMYK_CustomColor) eq
{
pop pop
4
{
4 index mul
4 1 roll
} repeat
5 -1 roll pop
setcmykcolor
}
{
dup (AI8_RGB_CustomColor) eq
{
pop pop
3
{
1 exch sub
3 index mul
1 exch sub
3 1 roll
} repeat
4 -1 roll pop
setrgbcolor
}
{
pop
4
{
4 index mul 4 1 roll
} repeat
5 -1 roll pop
setcmykcolor
} ifelse
} ifelse
}
def
} if
/setAIseparationgray
{
false setoverprint
0 setgray
/setseparationgray where{
pop setseparationgray
}{
/setcolorspace where{
pop
[/Separation (All) /DeviceCMYK {dup dup dup}] setcolorspace
1 exch sub setcolor
}{
setgray
}ifelse
}ifelse
} def

/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?
1 0 0 0 testCMYKColorThrough
0 1 0 0 testCMYKColorThrough
0 0 1 0 testCMYKColorThrough
0 0 0 1 testCMYKColorThrough
and and and
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
%%EndResource
%%BeginResource: procset Adobe_typography_AI5 1.0 1
%%Title: (Typography Operators)
%%Version: 1.0 1
%%CreationDate:(6/10/1996) ()
%%Copyright: ((C) 1987-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_typography_AI5 68 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
/havefont
{
systemdict /languagelevel known
{
/Font resourcestatus dup
{ exch pop exch pop }
if
}
{
systemdict /FontDirectory get 1 index known
{ pop true }
{
systemdict /fileposition known
{
dup length 6 add exch
Ss 6 250 getinterval
cvs pop
Ss exch 0 exch getinterval
status
{ pop pop pop pop true }
{ false }
ifelse
}
{
pop false
}
ifelse
}
ifelse
}
ifelse
} def
/TE
{
StandardEncoding 256 array copy modifyEncoding
/_nativeEncoding exch def
} def
/subststring {
exch 2 index exch search
{
exch pop
exch dup () eq
{
pop exch concatstring
}
{
3 -1 roll
exch concatstring
concatstring
} ifelse
exch pop true
}
{
pop pop false
} ifelse
} def
/concatstring {
1 index length 1 index length
1 index add
string
dup 0 5 index putinterval
dup 2 index 4 index putinterval
4 1 roll pop pop pop
} def
%
/TZ
{
dup type /arraytype eq
{
/_wv exch def
}
{
/_wv 0 def
} ifelse
/_useNativeEncoding exch def
2 index havefont
{
3 index
255 string
cvs

dup
(_Symbol_)
eq
{
pop
2 index
findfont

}
{
1 index 0 eq
{
dup length 1 sub
1 exch
getinterval

cvn
findfont
}
{
pop 2 index findfont
} ifelse
} ifelse
}
{
dup 1 eq
{
2 index 64 string cvs
dup (-90pv-RKSJ-) (-83pv-RKSJ-) subststring
{
exch pop dup havefont
{
findfont false
}
{
pop true
} ifelse
}
{
pop dup
(-90ms-RKSJ-) (-Ext-RKSJ-) subststring
{
exch pop dup havefont
{
findfont false
}
{
pop true
} ifelse
}
{
pop pop true
} ifelse
} ifelse
{
1 index 1 eq
{
/Ryumin-Light-Ext-RKSJ-V havefont
{/Ryumin-Light-Ext-RKSJ-V}
{/Courier}
ifelse
}
{
/Ryumin-Light-83pv-RKSJ-H havefont
{/Ryumin-Light-83pv-RKSJ-H}
{/Courier}
ifelse
} ifelse
findfont
[1 0 0.5 1 0 0] makefont
} if
}
{
/Courier findfont
} ifelse
} ifelse
_wv type /arraytype eq
{
_wv makeblendedfont
} if
dup length 10 add dict
 begin
mark exch
{
1 index /FID ne
{
def
} if
cleartomark mark
} forall
pop
/FontScript exch def
/FontDirection exch def
/FontRequest exch def
/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
{
W B
} def
/e0
{
/Tx
{
tr _psf
} ddef
/Tj
{
trj _pjsf
} ddef
} def
/e1
{
W F
} 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
{
W S
} 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
_shift aload pop _lineorientation 1 eq { exch } if translate
_scale aload pop _lineorientation 1 eq _yokoorientation 1 eq or { exch } if 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
{
1 index type /nametype eq
{
dup 0.75 mul 1 index 0.25 mul neg
} if
/_fontDescent exch ddef
/_fontAscent exch ddef
/_fontSize exch ddef
/_fontRotateAdjust _fontAscent _fontDescent add 2 div neg ddef
/_fontHeight _fontSize ddef
findfont _fontSize scalefont setfont
} def
/Tl
{
pop neg 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
{
0 exch
_shift astore pop
currentpoint
iTm
moveto
} def
/Ti
{
3 npop
} def
/Tz
{
count 1 eq { 100 } if
100 div exch 100 div exch
_scale astore pop
iTm
} def
/TA
{
pop
} def
/Tq
{
pop
} def
/Tg
{
pop
} def
/TG
{
pop
} def
/Tv
{
/_lineorientation exch ddef
} def
/TV
{
/_charorientation exch ddef
} def
/Ty
{
dup /_yokoorientation exch ddef 1 sub neg Tv
} def
/TY
{
pop
} def
/T~
{
Tx
} def
/Th
{
pop pop pop pop pop
} def
/TX
{
pop
} def
/Tk
{
_fontSize mul 1000 div
_lineorientation 0 eq { neg 0 } { 0 exch } ifelse
rmoveto
pop
} def
/TK
{
2 npop
} def
/T*
{
_leading aload pop
_lineorientation 0 ne { exch } if
Td
} def
/T*-
{
_leading aload pop
_lineorientation 0 ne { exch } if
exch neg exch neg
Td
} def
/T-
{
_ax neg 0 rmoveto
_lineorientation 1 eq _charorientation 0 eq and { 1 TV _hyphen Tx 0 TV } { _hyphen Tx } ifelse
} def
/T+
{
} def
/TR
{
_ctm currentmatrix pop
_tm astore pop
iTm 0 0 moveto
} def
/TS
{
currentfont 3 1 roll
/_Symbol_ findfont _fontSize scalefont setfont

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
%
/X^
{
currentfont 5 1 roll
dup havefont
{
findfont _fontSize scalefont setfont
}
{
pop
exch
} ifelse
2 index 0 eq
{
Tx
}
{
Tj
} ifelse
pop pop
setfont
} def
/T^ /X^ load def
%%EndResource
%%BeginProcSet: Adobe_ColorImage_AI6 1.3 0
userdict /Adobe_ColorImage_AI6 known not
{
userdict /Adobe_ColorImage_AI6 53 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 41 dict def
} if
Adobe_ColorImage_AI6_Vars begin
/plateindex -1 def
/_newproc null def
/_proc1 null def
/_proc2 null def
/sourcearray 4 array def
/_ptispace null def
/_ptiname null def
/_pti0 0 def
/_pti1 0 def
/_ptiproc null def
/_ptiscale 0 def
/_pticomps 0 def
/_ptibuf 0 string def
/_gtigray 0 def
/_cticmyk null def
/_rtirgb null def
/XIEnable true def
/XIType 0 def
/XIEncoding 0 def
/XICompression 0 def
/XIChannelCount 0 def
/XIBitsPerPixel 0 def
/XIImageHeight 0 def
/XIImageWidth 0 def
/XIImageMatrix null def
/XIRowBytes 0 def
/XIFile null def
/XIBuffer1 null def
/XIBuffer2 null def
/XIBuffer3 null def
/XIDataProc null def
/XIColorSpace /DeviceGray def
/XIColorValues 0 def
/XIPlateList false def
end
/ci6colorimage /colorimage where {/colorimage get}{null} ifelse def
/ci6image systemdict /image get def
/ci6curtransfer systemdict /currenttransfer get def
/ci6curoverprint /currentoverprint where {/currentoverprint get}{{_of}} ifelse def
/ci6foureq {
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
/ci6testplate {
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 ci6foureq {
/plateindex 0 def
}{
0 1 0 0 ci6foureq {
/plateindex 1 def
}{
0 0 1 0 ci6foureq {
/plateindex 2 def
}{
0 0 0 1 ci6foureq {
/plateindex 3 def
}{
0 0 0 0 ci6foureq {
/plateindex 5 def
} if
} ifelse
} ifelse
} ifelse
} ifelse
pop pop pop pop
} if
plateindex
 end
} def
/ci6concatprocs {
/packedarray where {
pop dup type /packedarraytype eq 2 index type
/packedarraytype eq or
}{
false
} ifelse
{
/_proc2 exch cvlit def
/_proc1 exch cvlit def
_proc1 aload pop
_proc2 aload pop
_proc1 length
_proc2 length add
packedarray cvx
}{
/_proc2 exch cvlit def
/_proc1 exch cvlit def
/_newproc _proc1 length _proc2 length add array def
_newproc 0 _proc1 putinterval
_newproc _proc1 length _proc2 putinterval
_newproc cvx
} ifelse
} def
/ci6istint {
type /arraytype eq
} def
/ci6isspot {
dup type /arraytype eq {
dup length 1 sub get /Separation eq
}{
pop false
} ifelse
} def
/ci6spotname {
dup ci6isspot {dup length 2 sub get}{pop ()} ifelse
} def
/ci6altspace {
aload pop pop pop ci6colormake
} def
/ci6numcomps {
dup /DeviceGray eq {
pop 1
}{
dup /DeviceRGB eq {
pop 3
}{
/DeviceCMYK eq {
4
}{
1
} ifelse
} ifelse
} ifelse
} def
/ci6marksplate {
dup /DeviceGray eq {
pop plateindex 3 eq
}{
dup /DeviceRGB eq {
pop plateindex 5 ne
}{
dup /DeviceCMYK eq {
pop plateindex 5 ne
}{
dup ci6isspot {
/findcmykcustomcolor where {
pop
dup length 2 sub get
0.1 0.1 0.1 0.1 5 -1 roll
findcmykcustomcolor 1 setcustomcolor
systemdict /currentgray get exec
1 ne
}{
pop plateindex 5 ne
} ifelse
}{
pop plateindex 5 ne
} ifelse
} ifelse
} ifelse
} ifelse
} def
/ci6colormake {
dup ci6numcomps
exch 1 index 2 add 1 roll
dup 1 eq {pop}{array astore} ifelse
exch
} def
/ci6colorexpand {
dup ci6spotname exch
dup ci6istint {
ci6altspace
exch 4 1 roll
}{
1 3 1 roll
} ifelse
} def
/ci6colortint {
dup /DeviceGray eq {
3 1 roll 1 exch sub mul 1 exch sub exch
}{
dup /DeviceRGB eq {
3 1 roll {1 exch sub 1 index mul 1 exch sub exch} forall pop 3 array astore exch
}{
dup /DeviceCMYK eq {
3 1 roll {1 index mul exch} forall pop 4 array astore exch
}{
3 1 roll mul exch
} ifelse
} ifelse
} ifelse
} def
/ci6colortocmyk {
dup /DeviceGray eq {
pop 1 exch sub 0 0 0 4 -1 roll 4 array astore
}{
dup /DeviceRGB eq {
pop aload pop _rgbtocmyk 4 array astore
}{
dup /DeviceCMYK eq {
pop
}{
ci6altspace ci6colortint ci6colortocmyk
} ifelse
} ifelse
} ifelse
} def
/ci6makeimagedict {
7 dict begin
/ImageType 1 def
/Decode exch def
/DataSource exch def
/ImageMatrix exch def
/BitsPerComponent exch def
/Height exch def
/Width exch def
currentdict end
} def
/ci6stringinvert {
0 1 2 index length 1 sub {
dup 2 index exch get 255 exch sub 2 index 3 1 roll put
} for
} def
/ci6stringknockout {
0 1 2 index length 1 sub {
255 2 index 3 1 roll put
} for
} def
/ci6stringapply {
0 1 4 index length 1 sub {
dup
4 index exch get
3 index 3 1 roll
3 index exec
} for
pop exch pop
} def
/ci6walkrgbstring {
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
/ci6walkcmykstring
{
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
/ci6putrgbtograystr
{
.11 mul exch

.59 mul add exch

.3 mul add

cvi 3 copy put

pop 1 add
} def
/ci6putcmyktograystr
{
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
/ci6rgbtograyproc {
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec
XIBuffer3
dup 3 1 roll

/ci6putrgbtograystr load exch
ci6walkrgbstring
 end
} def
/ci6cmyktograyproc {
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec
XIBuffer3
dup 3 1 roll

/ci6putcmyktograystr load exch
ci6walkcmykstring
 end
} def
/ci6separatecmykproc {
Adobe_ColorImage_AI6_Vars begin
sourcearray 0 get exec

XIBuffer3

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
 end
} def

/ci6compositeimage {
dup 1 eq {
pop pop image
}{
/ci6colorimage load null ne {
ci6colorimage
}{
3 1 roll pop
sourcearray 0 3 -1 roll put
3 eq {/ci6rgbtograyproc}{/ci6cmyktograyproc} ifelse load
image
} ifelse
} ifelse
} def
/ci6knockoutimage {
gsave
0 ci6curtransfer exec 1 ci6curtransfer exec
eq {
0 ci6curtransfer exec 0.5 lt
}{
0 ci6curtransfer exec 1 ci6curtransfer exec gt
} ifelse
{{pop 0}}{{pop 1}} ifelse
systemdict /settransfer get exec
ci6compositeimage
grestore
} def
/ci6drawimage {
ci6testplate -1 eq {
pop ci6compositeimage
}{
dup type /arraytype eq {
dup length plateindex gt {plateindex get}{pop false} ifelse
}{
{
true
}{
dup 1 eq {plateindex 3 eq}{plateindex 3 le} ifelse
} ifelse
} ifelse
{
dup 1 eq {
pop pop ci6image
}{
dup 3 eq {
ci6compositeimage
}{
pop pop
sourcearray 0 3 -1 roll put
/ci6separatecmykproc load
ci6image
} ifelse
} ifelse
}{
ci6curoverprint {
7 {pop} repeat
}{
ci6knockoutimage
} ifelse
} ifelse
} ifelse
} def
/ci6proctintimage {
/_ptispace exch store /_ptiname exch store /_pti1 exch store /_pti0 exch store /_ptiproc exch store
/_pticomps _ptispace ci6numcomps store
/_ptiscale _pti1 _pti0 sub store
level2? {
_ptiname length 0 gt version cvr 2012 ge and {
[/Separation _ptiname _ptispace {_ptiproc}] setcolorspace
[_pti0 _pti1] ci6makeimagedict ci6image
}{
[/Indexed _ptispace 255 {255 div _ptiscale mul _pti0 add _ptiproc}] setcolorspace
[0 255] ci6makeimagedict ci6image
} ifelse
}{
_pticomps 1 eq {
{
dup
{
255 div _ptiscale mul _pti0 add _ptiproc 255 mul cvi put
} ci6stringapply
} ci6concatprocs ci6image
}{
{
dup length _pticomps mul dup _ptibuf length ne {/_ptibuf exch string store}{pop} ifelse
_ptibuf {
exch _pticomps mul exch 255 div _ptiscale mul _pti0 add _ptiproc
_pticomps 2 add -2 roll
_pticomps 1 sub -1 0 {
1 index add 2 index exch
5 -1 roll
255 mul cvi put
} for
pop pop
} ci6stringapply
} ci6concatprocs false _pticomps
/ci6colorimage load null eq {7 {pop} repeat}{ci6colorimage} ifelse
} ifelse
} ifelse
} def
/ci6graytintimage {
/_gtigray 5 -1 roll store
{1 _gtigray sub mul 1 exch sub} 4 1 roll
/DeviceGray ci6proctintimage
} def
/ci6cmyktintimage {
/_cticmyk 5 -1 roll store
{_cticmyk {1 index mul exch} forall pop} 4 1 roll
/DeviceCMYK ci6proctintimage
} def
/ci6rgbtintimage {
/_rtirgb 5 -1 roll store
{_rtirgb {1 exch sub 1 index mul 1 exch sub exch} forall pop} 4 1 roll
/DeviceRGB ci6proctintimage
} def
/ci6tintimage {
ci6testplate -1 eq {
ci6colorexpand
3 -1 roll 5 -1 roll {0}{0 exch} ifelse 4 2 roll
dup /DeviceGray eq {
pop ci6graytintimage
}{
dup /DeviceRGB eq {
pop ci6rgbtintimage
}{
pop ci6cmyktintimage
} ifelse
} ifelse
}{
dup ci6marksplate {
plateindex 5 lt {
ci6colortocmyk plateindex get
dup 0 eq ci6curoverprint and {
7 {pop} repeat
}{
1 exch sub
exch {1 0}{0 1} ifelse () ci6graytintimage
} ifelse
}{
pop exch {0}{0 exch} ifelse 0 3 1 roll () ci6graytintimage
} ifelse
}{
ci6curoverprint {
8 {pop} repeat
}{
pop pop pop
{pop 1} 0 1 () /DeviceGray ci6proctintimage
} ifelse
} ifelse
} ifelse
} def
/XINullImage {
} def
/XIImageMask {
XIImageWidth XIImageHeight false
[XIImageWidth 0 0 XIImageHeight neg 0 0]
/XIDataProc load
imagemask
} def
/XIImageTint {
XIImageWidth XIImageHeight XIBitsPerPixel
[XIImageWidth 0 0 XIImageHeight neg 0 0]
/XIDataProc load
XIType 3 eq XIColorValues XIColorSpace ci6tintimage
} def
/XIImage {
XIImageWidth XIImageHeight XIBitsPerPixel
[XIImageWidth 0 0 XIImageHeight neg 0 0]
/XIDataProc load
false XIChannelCount XIPlateList ci6drawimage
} def
/XG {
pop pop
} def
/XF {
13 {pop} repeat
} def
/Xh {
Adobe_ColorImage_AI6_Vars begin
gsave
/XIType exch def
/XIImageHeight exch def
/XIImageWidth exch def
/XIImageMatrix exch def
0 0 moveto
XIImageMatrix concat
XIImageWidth XIImageHeight scale

/_lp /null ddef
_fc
/_lp /imagemask ddef
 end
} def
/XH {
Adobe_ColorImage_AI6_Vars begin
grestore
 end
} def
/XIEnable {
Adobe_ColorImage_AI6_Vars /XIEnable 3 -1 roll put
} def
/XC {
Adobe_ColorImage_AI6_Vars begin
ci6colormake
/XIColorSpace exch def
/XIColorValues exch def
 end
} def
/XIPlates {
Adobe_ColorImage_AI6_Vars begin
/XIPlateList exch def
 end
} def
/XI
{
Adobe_ColorImage_AI6_Vars begin
gsave
/XIType exch def
cvi dup
256 idiv /XICompression exch store
256 mod /XIEncoding exch store
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
/XIRowBytes exch def
XIEnable {
/XIBuffer3 XIImageWidth string def
XICompression 0 eq {
/XIBuffer1 XIRowBytes string def
XIEncoding 0 eq {
{currentfile XIBuffer1 readhexstring pop}
}{
{currentfile XIBuffer1 readstring pop}
} ifelse
}{
/XIBuffer1 256 string def
/XIBuffer2 XIRowBytes string def
{currentfile XIBuffer1 readline pop (%) anchorsearch {pop} if}
/ASCII85Decode filter /DCTDecode filter
/XIFile exch def
{XIFile XIBuffer2 readstring pop}
} ifelse
/XIDataProc exch def

XIType 1 ne {
0 setgray
} if
XIType 1 eq {
XIImageMask
}{
XIType 2 eq XIType 3 eq or {
XIImageTint
}{
XIImage
} ifelse
} ifelse
}{
XINullImage
} ifelse
/XIPlateList false def
grestore
 end
} def
end
%%EndProcSet
%%BeginResource: procset Adobe_Illustrator_AI5 1.3 0
%%Title: (Adobe Illustrator (R) Version 8.0 Full Prolog)
%%Version: 1.3 0
%%CreationDate: (3/7/1994) ()
%%Copyright: ((C) 1987-1998 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_Illustrator_AI5_vars 112 dict dup begin
put
/_?cmyk false def
/_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
/_lineorientation 0 def
/_charorientation 0 def
/_yokoorientation 0 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
/_shift [0 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
/_fontSize 0 def
/_fontAscent 0 def
/_fontDescent 0 def
/_fontHeight 0 def
/_fontRotateAdjust 0 def
/Ss 256 string def
Ss 0 (fonts/) putinterval
/_cnt 0 def
/_scale [1 1] def
/_nativeEncoding 0 def
/_useNativeEncoding 0 def
/_tempEncode 0 def
/_pntr 0 def
/_tDict 2 dict def
/_hfname 100 string def
/_hffound false def
/Tx
{
} def
/Tj
{
} def
/CRender
{
} def
/_AI3_savepage
{
} def
/_gf null def
/_cf 4 array def
/_rgbf 3 array def
/_if null def
/_of false def
/_fc
{
} def
/_gs null def
/_cs 4 array def
/_rgbs 3 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
/_lobyte 0 def
/_hibyte 0 def
/_cproc null def
/_cscript 0 def
/_hvax 0 def
/_hvay 0 def
/_hvwb 0 def
/_hvcx 0 def
/_hvcy 0 def
/_bitfont null def
/_bitlobyte 0 def
/_bithibyte 0 def
/_bitkey null def
/_bitdata null def
/_bitindex 0 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 100 dict put
} if
userdict /Adobe_Illustrator_AI5 get begin
/initialize
{
Adobe_Illustrator_AI5 dup begin
Adobe_Illustrator_AI5_vars begin
/_aicmykps where {pop /_?cmyk _aicmykps def}if
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
/hswj
{
dup stringwidth 3 2 roll
{
_hvwb eq { exch _hvcx add exch _hvcy add } if
exch _hvax add exch _hvay add
} cforall
} def
/vswj
{
0 0 3 -1 roll
{
dup 255 le
_charorientation 1 eq
and
{
dup cstring stringwidth 5 2 roll
_hvwb eq { exch _hvcy sub exch _hvcx sub } if
exch _hvay sub exch _hvax sub
4 -1 roll sub exch
3 -1 roll sub exch
}
{
_hvwb eq { exch _hvcy sub exch _hvcx sub } if
exch _hvay sub exch _hvax sub
_fontHeight sub
} ifelse
} cforall
} def
/swj
{
6 1 roll
/_hvay exch ddef
/_hvax exch ddef
/_hvwb exch ddef
/_hvcy exch ddef
/_hvcx exch ddef
_lineorientation 0 eq { hswj } { vswj } ifelse
} def
/sw
{
0 0 0 6 3 roll swj
} def
/vjss
{
4 1 roll
{
dup cstring
dup length 1 eq
_charorientation 1 eq
and
{
-90 rotate
currentpoint
_fontRotateAdjust add
moveto
gsave
false charpath currentpoint
5 index setmatrix stroke
grestore
_fontRotateAdjust sub
moveto
_sp eq
{
5 index 5 index rmoveto
} if
2 copy rmoveto
90 rotate
}
{
currentpoint
_fontHeight sub
5 index sub
3 index _sp eq
{
9 index sub
} if

currentpoint
exch 4 index stringwidth pop 2 div sub
exch _fontAscent sub
moveto

gsave
2 index false charpath
6 index setmatrix stroke
grestore

moveto pop pop
} ifelse
} cforall
6 npop
} def
/hjss
{
4 1 roll
{
dup cstring
gsave
false charpath currentpoint
5 index setmatrix stroke
grestore
moveto
_sp eq
{
5 index 5 index rmoveto
} if
2 copy rmoveto
} cforall
6 npop
} def
/jss
{
_lineorientation 0 eq { hjss } { vjss } ifelse
} def
/ss
{
0 0 0 7 3 roll jss
} def
/vjsp
{
4 1 roll
{
dup cstring
dup length 1 eq
_charorientation 1 eq
and
{
-90 rotate
currentpoint
_fontRotateAdjust add
moveto
false charpath
            currentpoint
_fontRotateAdjust sub
moveto
_sp eq
{
5 index 5 index rmoveto
} if
2 copy rmoveto
90 rotate
}
{
currentpoint
_fontHeight sub
5 index sub
3 index _sp eq
{
9 index sub
} if

currentpoint
exch 4 index stringwidth pop 2 div sub
exch _fontAscent sub
moveto

2 index false charpath

moveto pop pop
} ifelse
} cforall
6 npop
} def
/hjsp
{
    4 1 roll
    {
        dup cstring
        false charpath
        _sp eq
        {
            5 index 5 index rmoveto
        } if
        2 copy rmoveto
    } cforall
    6 npop
} def
/jsp
{
matrix currentmatrix
    _lineorientation 0 eq {hjsp} {vjsp} ifelse
} def
/sp
{
    matrix currentmatrix
    0 0 0 7 3 roll
    _lineorientation 0 eq {hjsp} {vjsp} ifelse
} 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
/_rgbtocmyk
{
3
{
1 exch sub 3 1 roll
} repeat
3 copy 1 4 1 roll
3
{
3 index 2 copy gt
{
exch
} if
pop 4 1 roll
} repeat
pop pop pop
4 1 roll
3
{
3 index sub
3 1 roll
} repeat
4 -1 roll
} def
/setrgbfill
{
_rgbf astore pop
/_fc
{
_lp /fill ne
{
_of setoverprint
_rgbf aload pop setrgbcolor
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} ddef
/_lp /none ddef
} def
/setrgbstroke
{
_rgbs astore pop
/_sc
{
_lp /stroke ne
{
_os setoverprint
_rgbs aload pop setrgbcolor
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} 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
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} 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
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} 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
/Xa
{
_?cmyk {
3 npop k
}{
setrgbfill 4 npop
} ifelse
} def
/XA
{
_?cmyk {
3 npop K
}{
setrgbstroke 4 npop
} ifelse
} def
/Xs
{
/_gf exch ddef
5 npop
/_fc
{
_lp /fill ne
{
_of setoverprint
_gf setAIseparationgray
/_lp /fill ddef
} if
} ddef
/_pf
{
_fc
_eo {eofill} {fill} ifelse
} ddef
/_psf
{
_fc
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} ddef
/_lp /none ddef
} def
/XS
{
/_gs exch ddef
5 npop
/_sc
{
_lp /stroke ne
{
_os setoverprint
_gs setAIseparationgray
/_lp /stroke ddef
} if
} ddef
/_ps
{
_sc
stroke
} ddef
/_pss
{
_sc
ss
} ddef
/_pjss
{
_sc
jss
} ddef
/_lp /none ddef
} def
/Xx
{
exch
/_gf exch ddef
0 eq {
findcmykcustomcolor
}{
_?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse
{
4 1 roll 3 npop
findcmykcustomcolor
}{
8 -4 roll 4 npop
findrgbcustomcolor
} ifelse
} ifelse
/_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
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} ddef
/_lp /none ddef
} def
/XX
{
exch
/_gs exch ddef
0 eq {
findcmykcustomcolor
}{
_?cmyk {true}{/findrgbcustomcolor where{pop false}{true}ifelse}ifelse
{
4 1 roll 3 npop
findcmykcustomcolor
}{
8 -4 roll 4 npop
findrgbcustomcolor
} ifelse
} ifelse
/_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
/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
hvashow
} ddef
/_pjsf
{
_fc
hvawidthshow
} 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
/XK
{
3 -1 roll pop
0 eq
{
1 exch sub
3 {dup 3 1 roll mul 5 1 roll} repeat
mul 4 1 roll
K
}
{
1 exch sub 4 1 roll
3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat
4 -1 roll pop
XA
} ifelse
} def
/Xk
{
3 -1 roll pop
0 eq
{
1 exch sub
3 {dup 3 1 roll mul 5 1 roll} repeat
mul 4 1 roll
k
}
{
1 exch sub 4 1 roll
3 {1 exch sub 3 index mul 1 exch sub 3 1 roll} repeat
4 -1 roll pop
Xa
} ifelse
} def
/A
{
pop
} def
/annotatepage
{
userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse
} def
/XT {
pop pop
} def
/Xt {
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} stopped {
% assume error was due to overfilling the buffer
}{
not
{
stop
} if
endString eq {
cleartomark stop
} if
}ifelse
} loop
} def
3
{
/beginString exch store
/layerCnt 1 store
mark
{
currentfile buffer {readline} stopped {
% assume error was due to overfilling the buffer
}{
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
}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
{
6 npop
7 2 roll
5 npop
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 4 dict def
nc begin
/setgray
{
pop
} bind def
/setcmykcolor
{
4 npop
} bind def
/setrgbcolor
{
3 npop
} bind def
/setcustomcolor
{
2 npop
} bind def
currentdict readonly pop
end
/XP
{
4 npop
} bind def
/XD
{
pop
} bind def
end
setpacking
%%EndResource
%%BeginResource: procset Adobe_cshow 2.0 8
%%Title: (Writing System Operators)
%%Version: 2.0 8
%%CreationDate: (1/23/89) ()
%%Copyright: ((C) 1992-1996 Adobe Systems Incorporated All Rights Reserved)
currentpacking true setpacking
userdict /Adobe_cshow 14 dict dup begin put
/initialize
{
Adobe_cshow begin
Adobe_cshow
{
dup xcheck
{
bind
} if
pop pop
} forall
 end
Adobe_cshow begin
} def
/terminate
{
currentdict Adobe_cshow eq
{
 end
} if
} def
/cforall
{
/_lobyte 0 ddef
/_hibyte 0 ddef
/_cproc exch ddef
/_cscript currentfont /FontScript known { currentfont /FontScript get } { -1 } ifelse ddef
{
/_lobyte exch ddef
_hibyte 0 eq
_cscript 1 eq
_lobyte 129 ge _lobyte 159 le and
_lobyte 224 ge _lobyte 252 le and or and
_cscript 2 eq
_lobyte 161 ge _lobyte 254 le and and
_cscript 3 eq
_lobyte 161 ge _lobyte 254 le and and
     _cscript 25 eq
_lobyte 161 ge _lobyte 254 le and and
     _cscript -1 eq
or or or or and
{
/_hibyte _lobyte ddef
}
{
_hibyte 256 mul _lobyte add
_cproc
/_hibyte 0 ddef
} ifelse
} forall
} def
/cstring
{
dup 256 lt
{
(s) dup 0 4 3 roll put
}
{
dup 256 idiv exch 256 mod
(hl) dup dup 0 6 5 roll put 1 4 3 roll put
} ifelse
} def
/clength
{
0 exch
{ 256 lt { 1 } { 2 } ifelse add } cforall
} def
/hawidthshow
{
{
dup cstring
show
_hvax _hvay rmoveto
_hvwb eq { _hvcx _hvcy rmoveto } if
} cforall
} def
/vawidthshow
{
{
dup 255 le
_charorientation 1 eq
and
{
-90 rotate
0 _fontRotateAdjust rmoveto
cstring
_hvcx _hvcy _hvwb _hvax _hvay 6 -1 roll awidthshow
0 _fontRotateAdjust neg rmoveto
90 rotate
}
{
currentpoint
_fontHeight sub
exch _hvay sub exch _hvax sub
2 index _hvwb eq { exch _hvcy sub exch _hvcx sub } if
3 2 roll
cstring
dup stringwidth pop 2 div neg _fontAscent neg rmoveto
show
moveto
} ifelse
} cforall
} def
/hvawidthshow
{
6 1 roll
/_hvay exch ddef
/_hvax exch ddef
/_hvwb exch ddef
/_hvcy exch ddef
/_hvcx exch ddef
_lineorientation 0 eq { hawidthshow } { vawidthshow } ifelse
} def
/hvwidthshow
{
0 0 3 -1 roll hvawidthshow
} def
/hvashow
{
0 0 0 6 -3 roll hvawidthshow
} def
/hvshow
{
0 0 0 0 0 6 -1 roll hvawidthshow
} def
currentdict readonly pop end
setpacking
%%EndResource
%%BeginResource: procset Adobe_shading_AI8 1.0 0
%%Title: (Adobe Illustrator 8 Shading Procset)
%%Version: 1.0 0
%%CreationDate: (12/17/97) ()
%%Copyright: ((C) 1987-1997 Adobe Systems Incorporated All Rights Reserved)
userdict /defaultpacking currentpacking put true setpacking
userdict /Adobe_shading_AI8 10 dict dup begin put
/initialize {
Adobe_shading_AI8 begin
Adobe_shading_AI8 bdprocs
Mesh /initialize get exec
} def
/terminate {
currentdict Adobe_shading_AI8 eq {
 end
} if
} def
/bdprocs {
{
dup xcheck 1 index type /arraytype eq and {
bind
} if
pop pop
} forall
} def
/X! {pop} def
/X# {pop pop} def
/Mesh 40 dict def
Mesh begin
/initialize {
Mesh bdprocs
Mesh begin
/emulate? /AI8MeshEmulation where {
pop AI8MeshEmulation
}{
systemdict /shfill known not
} ifelse def
 end
} def
/bd {
shadingdict begin
} def
/paint {
emulate? {
 end
}{
/_lp /none ddef _fc /_lp /none ddef

/AIColorSpace AIColorSpace tocolorspace store
/ColorSpace AIColorSpace topsspace store

version_ge_3010.106 not systemdict /setsmoothness known and {
0.0001 setsmoothness
} if

composite? {
/DataSource getdatasrc def
Matrix concat
currentdict end
shfill
}{
AIColorSpace makesmarks AIPlateList markingplate and not isoverprint and {
 end
}{
/ColorSpace /DeviceGray store
/Decode [0 1 0 1 0 1] store
/DataSource getplatesrc def
Matrix concat
currentdict end
shfill
} ifelse
} ifelse
} ifelse
} def
/shadingdict 12 dict def
shadingdict begin
/ShadingType 6 def
/BitsPerCoordinate 16 def
/BitsPerComponent 8 def
/BitsPerFlag 8 def
end
/datafile null def
/databuf 256 string def
/dataptr 0 def
/srcspace null def
/srcchannels 0 def
/dstchannels 0 def
/dstplate 0 def
/srctodstcolor null def
/getplatesrc {
/srcspace AIColorSpace store
/srcchannels AIColorSpace getnchannels store
/dstchannels 1 store
/dstplate getplateindex store
/srctodstcolor srcspace makesmarks {
dstplate 4 eq {
{1 exch sub}
}{
{srcspace tocmyk 3 dstplate sub index 1 exch sub 5 1 roll 4 {pop} repeat}
} ifelse
}{
{srcchannels {pop} repeat 1}
} ifelse store
/datafile getdatasrc store
/rdpatch168 load DataLength () /SubFileDecode filter
} def
/getdatasrc {
/rdcmntline load /ASCII85Decode filter
} def
/rdpatch168 {
/dataptr 0 store
49 rdcount
4 {
dup {pop srcchannels getint8} if
dup {pop srctodstcolor dstchannels putint8 true} if
} repeat
{databuf 0 dataptr getinterval}{()} ifelse
} def
/rdpatch3216 {
/dataptr 0 store
97 rdcount
4 {
dup {pop srcchannels getint16} if
dup {pop srctodstcolor dstchannels putint16 true} if
} repeat
{databuf 0 dataptr getinterval}{()} ifelse
} def
/rdcount {
dup 0 gt {
datafile databuf dataptr 4 -1 roll getinterval readstring
exch length dataptr add /dataptr exch store
}{
true
} ifelse
} def
/getint8 {
mark true 3 -1 roll
{
dup {pop datafile read} if
dup {pop 255 div true} if
} repeat
{
counttomark 1 add -1 roll pop true
}{
cleartomark false
} ifelse
} def
/putint8 {
dup dataptr add /dataptr exch store
dataptr exch
{
1 sub exch
255 mul cvi
databuf 2 index
3 -1 roll put
} repeat
pop
} def
/getint16 {
mark true 3 -1 roll
{
dup {pop datafile read} if
dup {pop 256 mul datafile read} if
dup {pop add 65535 div true} if
} repeat
{
counttomark 1 add -1 roll pop true
}{
cleartomark false
} ifelse
} def
/putint16 {
dup 2 mul dataptr add /dataptr exch store
dataptr exch
{
2 sub exch
65535 mul cvi dup
256 idiv databuf 3 index 3 -1 roll put
256 mod databuf 2 index 1 add 3 -1 roll put
} repeat
pop
} def
/srcbuf 256 string def
/rdcmntline {
currentfile srcbuf readline pop
(%) anchorsearch {pop} if
} def
/getplateindex {
0 [cyan? magenta? yellow? black? customColor?] {{exit} if 1 add} forall
} def
/aicsarray 4 array def
/aicsaltvals 4 array def
/aicsaltcolr aicsaltvals def
/tocolorspace {
dup type /arraytype eq {
mark exch aload pop
aicsarray 0 3 -1 roll put
aicsarray 1 3 -1 roll put
dup aicsarray 2 3 -1 roll put
gettintxform aicsarray 3 3 -1 roll put
counttomark aicsaltvals 0 3 -1 roll getinterval /aicsaltcolr exch store
aicsaltcolr astore pop pop
aicsarray
} if
} def
/subtintxform {aicsaltcolr {1 index mul exch} forall pop} def
/addtintxform {aicsaltcolr {1 sub 1 index mul 1 add exch} forall pop} def
/gettintxform {
/DeviceRGB eq {/addtintxform}{/subtintxform} ifelse load
} def
/getnchannels {
dup type /arraytype eq {0 get} if
colorspacedict exch get begin Channels end
} def
/makesmarks {
composite? {
pop true
}{
dup dup type /arraytype eq {0 get} if
colorspacedict exch get begin MarksPlate end
} ifelse
} def
/markingplate {
composite? {
pop true
}{
dup type /arraytype eq {
dup length getplateindex gt {getplateindex get}{pop false} ifelse
} if
} ifelse
} def
/tocmyk {
dup dup type /arraytype eq {0 get} if
colorspacedict exch get begin ToCMYK end
} def
/topsspace {
dup dup type /arraytype eq {0 get} if
colorspacedict exch get begin ToPSSpace end
} def
/colorspacedict 5 dict dup begin
/DeviceGray 4 dict dup begin
/Channels 1 def
/MarksPlate {pop black?} def
/ToCMYK {pop 1 exch sub 0 0 0 4 -1 roll} def
/ToPSSpace {} def
 end def
/DeviceRGB 4 dict dup begin
/Channels 3 def
/MarksPlate {pop isCMYKSep?} def
/ToCMYK {pop _rgbtocmyk} def
/ToPSSpace {} def
 end def
/DeviceCMYK 4 dict dup begin
/Channels 4 def
/MarksPlate {pop isCMYKSep?} def
/ToCMYK {pop} def
/ToPSSpace {} def
 end def
/Separation 4 dict dup begin
/Channels 1 def
/MarksPlate {
/findcmykcustomcolor where {
pop dup 1 exch ToCMYK 5 -1 roll 1 get
findcmykcustomcolor 1 setcustomcolor
systemdict /currentgray get exec
1 ne
}{
pop false
} ifelse
} def
/ToCMYK {
dup 2 get mark exch 4 2 roll
3 get exec
counttomark -1 roll tocmyk
5 -1 roll pop
} def
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% Finally, here is the rest of gelfreich_mp_arc_2002.tex:
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%\documentclass[12pt]{article} 
% V. Gelfreich
% Jan. 2002
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\title {Chaotic zone in the Bogdanov-Takens bifurcation for diffeomorphisms
\protect\footnote{Mathematical Physics Preprint Archive, January 2002,
\newline
{\tt http://www.ma.utexas.edu/mp\_arc}} }
\author{Vassili Gelfreich\\
\normalsize
Institut f\"ur Mathematik I, Freie Universit\"at Berlin,\\
\normalsize
Arnimallee 2--6, 14195 Berlin, Germany\\
\normalsize
gelf@math.fu-berlin.de}
\date{January 14, 2002}
\begin{document}
\maketitle

\begin{abstract}
We consider a two-parametric analytic family of diffeomorphisms
near the Bogdanov-Takens bifurcation. It is known that if the
parameters belong to a homoclinic zone, the map has homoclinic points.
The width of the homoclinic zone is exponentially small.
We derive an asymptotic formula for the width of the homoclinic zone.
An analytic invariant associated with a parabolic fixed point
is an important ingredient of this formula. The proof of the asymptotic
formula is not complete. Additionally we provide results of
computations of the invariant for model families.
\end{abstract}
\begin{center} Keywords: 
Bogdanov-Takens bifurcation, homoclinic orbits,
 exponentially small phenomena
\end{center}

\section{Introduction}
There is a remarkable similarity between bifurcations of equilibria
for planar vector fields and bifurcations of periodic orbits for 
3 dimensional vector fields. This similarity is due to the fact that
in many cases the Poincar\'e map near a bifurcating periodic orbit
can be formally written as a time-$1$ map of a planar vector field.
Although all Taylor coefficients of the interpolating vector
field can be determined, there is no reason to expect convergence
of the corresponding series since the dynamics in the 3 dimensional
space can be much richer than the dynamics on the plane. 
On the other hand the interpolating planar vector field
can be used to construct normal forms for bifurcations of 
the periodic orbits. Thus the difference between these
two different types of bifurcations is moved beyond all
algebraic orders. In real-analytic theory this difference
is typically exponentially small
\cite{BroerR2001,FontichS1990,Gelfreich2000}.


The Bogdanov-Takens bifurcation \cite{Bogdanov1975,Takens1974,ChowLW1994}
has codimension 2. In the case of a planar vector field it corresponds
to an equilibrium with two vanishing eigenvalues. In the case of a
periodic orbit in the 3 dimensional space 
both multipliers of the periodic orbit equal $1$.
It is more convenient to study two dimensional Poincar\'e
maps instead of 3 dimensional vector fields.

The Bogdanov-Takens bifurcation for planar diffeomorphisms has
been intensively studied in the recent time \cite{BroerRS1996,BroerR2001}.
It was shown that, similar to planar vector fields,
there are both Hopf and homoclinic bifurcations in a neighborhood
of the Bogdanov-Takens bifurcation. 
In the vector field case the values of parameters, which correspond 
to the homoclinic bifurcations, form a line on the plane
of parameters. In the case of diffeomorphisms transversal homoclinic
orbits are possible. The latter are stable with respect to
small changes of parameters, so one can expect that there is
a sector (instead of the line)
on the parameter plane, where the map has homoclinic points. 
We call this sector a {\em homoclinic zone}. 
An exponentially small upper bound for its width
was established in \cite{BroerR2001}. The aim of this paper is
to present a lower bound.
Of course this lower bound is exponentially small, and we
conjecture that it is positive for a generic map.
In order to prove this conjecture it is necessary to show that
a certain constant $\Theta$ (see Section~\ref{Se:theta}) does
not vanish. We checked this conjecture numerically 
for several examples.

Let us formulate our main result more precisely.
We consider a two-parametric analytic family
$F_{\mu\nu}:(x,y)\mapsto(x_1,y_1)$:  
\begin{equation}\label{Eq:nfl}
\begin{array}{rcl}
x_1&=&x+y,\\
y_1&=&y+f(x,y,\mu,\nu)
\end{array}
\end{equation}
with analytic $f$. We assume $f$ to be of the form
\begin{equation}\label{Eq:f}
f(x,y,\mu,\nu)=-\mu+x^2+\nu y+\gamma xy+\dots
\end{equation}
It is convenient to order terms of Taylor series according to
values of the calibration function, which assigns orders to monomials:
$$
\delta(\mu^j\nu^kx^ly^m)=4 j+2k+2l+3m.
$$
The coefficients $\mu$, $\nu$ and $\gamma$ are invariant under 
close-to-identity transformations,
which preserve the form of the map $F_{\mu\nu}$.

\begin{figure}
\begin{center}
\includegraphics{gelfreich_fig_1.eps}
\end{center}
\caption{Bifurcation diagram}
\label{Fig:bifdiag}
\end{figure}

This form of the map is not exceptional: a generic two-parametric
family of maps near its Bogdanov-Takens bifurcation
can be transformed to the form (\ref{Eq:nfl}), (\ref{Eq:f}) 
by a change of coordinates and parameters~\cite{BroerRS1996}.

\smallskip

If $\mu=\nu=0$ the map $F_{\mu\nu}$ has a parabolic fixed point
at the origin.  Following \cite{BroerRS1996,BroerR2001} we study
study the bifurcation of this point on the plane of $(\mu,\nu)$.
The corresponding bifurcation diagram is shown on Fig.~\ref{Fig:bifdiag}.

If a point $(\mu,\nu)$ is situated between the lines $\nu=\nu_-(\mu)$
and $\nu=\nu_+(\mu)$, $0<\mu<\mu_0$, then the
map $F_{\mu\nu}$ has a homoclinic trajectory.
On the lower and upper boundary the homoclinic trajectory disappears.
The first and the last homoclinic points are shown on Fig.~\ref{Fig:tangent}.
The pictures correspond to $\nu=\nu_-(\mu)$ and $\nu=\nu_+(\mu)$ respectively.

It is not very difficult to show that 
$\nu_\pm(\mu)=\frac57(\gamma-2)\mu^{1/2}+O(\mu^{3/4})$.
A much more difficult problem is to prove the following estimate:
\begin{equation}\label{Eq:mainas}
\nu_+(\mu)- \nu_-(\mu)=
\frac{A_\gamma}
{\mu^{5/4}}
\cdot
\e^{-\sqrt2\pi/\root 4 \of {\mu}},
\end{equation}
\begin{equation}
A_\gamma=
\frac{ 5|\Theta|}{6\sqrt2}\,\e^{-6\pi^2(\gamma-2)/7}+O(\mu^{1/4}\log\mu).
\end{equation}
Here $\Theta$ is a constant defined later in Section~\ref{Se:theta}.
It depends on $\gamma$ and on all other Taylor coefficients
of the function $f(x,y,0,0)$. If the map $F_{00}$ is 
exactly the time-$1$ map of a planar flow, the corresponding
constant $\Theta$ vanishes. Each $F_{00}$ can be interpolated by
a flow up to an error, which is smaller than any power of $x$ and $y$.
Thus all Taylor coefficients of $f(x,y,0,0)$ are relevant for $\Theta$.

\begin{figure}
\begin{center}
\includegraphics{gelfreich_fig_2.eps}
\end{center}
\caption{The first and last homoclinic points of the map $F_{\mu\nu}$}
\label{Fig:tangent}
\end{figure}

The formula (\ref{Eq:mainas}) implies a lower bound
for the width of the zone provided $\Theta$ does not vanish.
It was proved analytically that $\Theta$ does not vanish
in the cases of the H\'enon family \cite{GelfreichS2001};
numerically $|\Theta(\mbox{H\'enon})|=2.47\ldots\cdot 10^6$. 
 One should expect that $\Theta$ does not
vanish generically, since it depends analytically on $F_{00}$.
We recall that the {H\'enon map}
\begin{eqnarray*}
u_1&=&v,\\
v_1&=&a v^2-b u+1
\end{eqnarray*}
can be transformed to the form (\ref{Eq:nfl}) with
$$
f(x,y,\mu,\nu) = (x+y)^2-\mu +\nu y.
$$
The coefficients $\mu$ and $\nu$ are in one-to-one correspondence
with the original parameters: 
$$
a=(1+\nu/2)^2-\mu,
\qquad
b=1+\nu.
$$
The H\'enon family has the special property: it is area-preserving
for $\nu=0$.
The separatrices of the area-preserving H\'enon map split transversally
\cite{Gelfreich2000,GelfreichS2001,Chernov2001}. 


\medskip

The rest of the paper contains an informal derivation of the asymptotic 
formula (\ref{Eq:mainas}). It can not be considered as a complete 
proof. The approach is based on Lazutkin's method
originally proposed for a study of the standard map
\cite{Lazutkin1984,Gelfreich1999}.
 Section~\ref{Se:numerics} provides a summary
of numerical experiments used to evaluate the splitting constant $\Theta$
for some model families and to check the correctness of the asymptotic
formula (\ref{Eq:mainas}). 

\section{Formal interpolation by a flow\label{Se:interpol}}
In this section we discuss the formal interpolation
of the family $F_{\mu\nu}$ by an autonomous vector-field
$$
X=X^1\partial_x+X^2\partial_y,
$$
where $\partial_x$ and $\partial_y$ stand for derivatives in $x$ and
$y$ respectively; $X^1$ and $X^2$ are power series in $x,y,\mu$ and $\nu$.
The vector field is a differential operator and
can be expanded into a sum of homogeneous operators:
$$
X=\sum_{k\ge1}X_k,
$$
where $X_k$ increases the order of a $\delta$-homogeneous polynomial
by $k$.

It is not too difficult to
check that there is a unique formal vector-field, such that
$$
F_{\mu\nu}(\x)=\mathrm e^X\x,\qquad
\x=(x,y)^T.
$$
In general one expects that the series for $X$ diverges.
Comparing the first two orders of the series on both sides we obtain
\begin{eqnarray*}
X_1&=&y\partial_x+(x^2-\mu)\partial_y,\\
X_2&=&-\tfrac12(x^2-\mu)\partial_x+(\nu+(\gamma-1)x)y\partial_y.
\end{eqnarray*}
The first oder vector-field $X_1$ is Hamiltonian with
the Hamiltonian function
$$
h_0=\frac{y^2}2+\mu x-\frac{x^3}3.
$$
If $\mu>0$ it has a homoclinic loop defined by  
$h_0(x,y)=\frac23\mu^{3/2}$.  The perturbation by $X_2$
can destroy this loop.
Let us study the homoclinic bifurcation of the vector field
$X_1+X_2$ using the Melnikov method \cite{GuckenheimerH1983}.
The unperturbed homoclinic orbit can be find explicitly:
\begin{eqnarray*}
x _0(t,\mu)&=&\sqrt\mu\left(
1-\frac3{\cosh^2(t\root4\of\mu/\sqrt2)}\right),\\
y_0(t,\mu)&=&\dot x_0=
3\sqrt2\mu^{3/4}\frac{\sinh(t\root4\of\mu/\sqrt2)}
{\cosh^3(t\root4\of\mu/\sqrt2)}\;.
\end{eqnarray*}
The Melnikov function describes the distance between the
stable and unstable separatrices of the flow generated by
$X_1+X_2$ measured in units of $h_0$:
\begin{eqnarray*}
M(\mu,\nu)&=&
\int_{-\infty}^{+\infty} \left.(X_2h_0)\right|_{x_0(t),y_0(t)}\,\d t
\\
&=&(\gamma-2)\int_{-\infty}^{+\infty} x_0(t)y_0^2(t)\,\d t+
\nu \int_{-\infty}^{+\infty} y_0^2(t)\,\d t.
\end{eqnarray*}
It is not too difficult to calculate this integral explicitly:
$$
M(\mu,\nu)=\frac{24\sqrt2}{35}\, \mu^{5/4}
\left(-5(\gamma-2)\sqrt\mu+7\nu \right).
$$
The Melnikov function vanishes along the line $\nu=5(\gamma-2)\sqrt\mu/7$,
which gives an asymptotic of the actual line of homoclinic
bifurcations.  In the following we will need the following 
derivative:
\begin{equation}
\frac{\partial M }{\partial \nu} (\mu,\nu) =\frac{24\sqrt2}{5} \mu^{5/4},
\end{equation}
which describes how fast the distance between separatrices 
changes as a function of $\nu$ for a fixed $\mu$.

\smallskip

Of course,  Melnikov-type computations do not give the width of the 
homoclinic zone for the map $F_{\mu\nu}$, because for a flow
it reduces to a line.

\smallskip

Let $\nu=5(\gamma-2)\sqrt\mu/7$ and find the first correction to
the unperturbed homoclinic: $x=x_0(t)+x_1(t)+\ldots$, $y=y_0(t)+y_1(t)+\ldots$
We have
\begin{eqnarray*}
\dot x_1&=&y_1-\tfrac12 (x_0^2-\mu),\\
\dot y_1&=&2 x_0 x_1+(\gamma-1)x_0 y_0+\nu y_0.
\end{eqnarray*}
For the purposes of the the present paper it is sufficient to compute
$x_1$ only:
$$
x_1(t)=\tfrac37(\gamma-2)y_0(t)\log\left(\sqrt\mu-x_0(t)\right).
$$
This calculation can be continued and approximations for the separatrix
loop for all $\delta$-orders can be constructed. All these approximations
give a closed homoclinic loop. In order to detect the exponentially
small splitting of separatrices we analytically continue in $t$
and study the approximations of the separatrices near a complex singularity.

It is clear that $x_0$ and $x_1$ have a common
singularity at $t=\pm\i\rho$ with 
\begin{equation}\label{Eq:rho}
\rho= \frac{\pi}{\sqrt2\mu^{1/4}}\;.
\end{equation}
Both $x_0$ and $x_1$ are analytic in the complex
strip $|\Im t|<\rho$.
The main parts of $x_0$ and $x_1$ at the singularity are:
\begin{equation}\label{Eq:mainsing}
\begin{array}{rcl}
x_0(t)&=&\displaystyle\frac6{(t-\i\rho)^2}+O(1)\;,\\
x_1(t)&=&\displaystyle\tfrac{36}{7}(\gamma-2)
\frac{\i\pi+\log 6-\log(t-\i\rho)^2}{(t-\i\rho)^3}
+O\left((t-\i\rho)^{-1}\right)\;.
\end{array}
\end{equation}
We will use this expansions for the complex matching method described later
in Section~\ref{Se:flowbox}.

\section{Splitting constant $\Theta$\label{Se:theta}}
Let us consider the map $F_{0}\equiv F_{00}$, which has a parabolic fixed 
point at the origin. The fixed point has stable and unstable invariant
manifold, which we parameterize by solutions of the finite-difference
equation:
$$
\x_0^\pm(\tau+1)=F_0(\x_0^\pm(\tau)).
$$
We supply this equation with asymptotic boundary conditions
for the first components of $\x_0^+$ and $\x_0^-$:
\begin{equation}\label{Eq:asimp0}
x_0^\pm(\tau)=\frac{6}{\tau^2}-\tfrac{36}{7}(\gamma-2)\frac{\log\tau^2}{\tau^3}
+O\left(\tau^{-4}\log^2\tau\right).
\end{equation}
The unstable solution $\x_0^-$ is analytic in a sectorial domain 
$$
D_-=\{\,\tau\in\mathbb C: \delta<\arg(\tau+c)<2\pi-\delta\,\}
$$
with $\delta\in(0,\frac{\pi}2)$ and sufficiently large $c>0$.
The stable solution $\x_0^+$ is analytic in the domain $D_+$,
which is a reflection of $D_-$ with respect to the imaginary axis.
These conditions define $\x_0^+$ and $\x_0^-$ uniquely.

Consider the auxiliary scalar equation 
$$
w_0(\tau+1)=J_0(\tau)w_0(\tau),
\qquad
\mbox{where $J_0(\tau)=\det F_0'(\x_0^-(\tau))$.}
$$
In $D_-$ this linear finite-difference equation has a unique analytic
solution, which satisfies the asymptotic boundary condition
$w_0(\tau)=1+o(\tau^{-1})$.  Then 
$$
\Theta=
\lim_{\Im\tau\to-\infty}\e^{2\pi\i\tau}\frac{\tilde\theta(\tau)}{w_0(\tau)}
$$
gives the same value of $\Theta$; but the expression under the
limit sign converges exponentially as $|\Im\tau|\to-\infty$.
Let us define the splitting function
\begin{equation}\label{Eq:tildetheta}
\tilde\theta(\tau)=
\frac1{w_0(\tau)}
\det\left(\frac{\d \x_0^-}{\d\tau}(\tau);\ \x_0^+(\tau)-\x_0^-(\tau)\right)
\end{equation}
for $\tau\in\Omega_-$, the lower component of $D_-\cap D_+$.
Then the splitting constant $\Theta$ is defined by
$$
\Theta=\lim_{\Im\tau\to-\infty}\e^{2\pi\i\tau}\tilde\theta(\tau).
$$
This definition generalizes the definition from \cite{Gelfreich2000}.
The existence of the limit can be proved by methods of 
\cite{Gelfreich1999,GelfreichS2001}. 
Given a map $F_0$, the
constant $\Theta$ can be numerically computed directly from the definition.

Finally we rewrite the definition of $\Theta$ in the form
\begin{equation}\label{Eq:approxtheta}
\tilde\theta(\tau)\approx w_0(\tau)\Theta\,\e^{-2\pi\i\tau}
\end{equation}
in oder to use it later in Section~\ref{Se:flowbox}.

\section{Flow box theorem\label{Se:flowbox}}

Now let us come back to the study of separatrices of the 
original map $F_{\mu\nu}$. We assume that parameters $\mu$
and $\nu$ are both small and close to the unperturbed 
homoclinic line $\nu=\frac57(\gamma-2)\mu^{1/2}$. So the
stable and unstable separatrices of the map are close to
each other.


We parameterize the stable and unstable separatrices of its 
hyperbolic fixed point by solutions of finite-difference equation
$$
\x^{u,s}(t+1)=F_{\mu\nu}(\x^{u,s}(t)),
$$
which satisfy appropriate boundary conditions
\cite{Gelfreich1999,Gelfreich2000}.
These functions are close to the separatrix solution
of the differential equations of Section~\ref{Se:interpol}.

In a neighborhood of the first intersection of the curve $\x^u(t)$
with the $y$-axes we introduce the time-energy coordinates 
$(E,T)$ \cite{Gelfreich1996}. In these coordinates the map
$F_{\mu\nu}$ takes the form of the translation
$$
(E,T)\mapsto(E,T+1).
$$
The coordinates can be normalized in such a way that
$$
E(\x^u(t))=0,\qquad
T(\x^u(t))= t.
$$
Moreover the energy $E$ is close to the Hamiltonian function $h_0$.
In the $(E,T)$ coordinates a segment of the stable manifold is a graph
of a periodic function $E=A(T,\mu,\nu)$, $A(T+1,\mu,\nu)=A(T,\mu,\nu)$.
We can expand this function in Fourier series
$$
A(T,\mu,\nu)=A_0(\mu,\nu)+A_1(\mu,\nu)\sin 2\pi (T-T_0(\mu,\nu))+h.o.t.
$$
In \cite{Gelfreich1996} it was shown the h.o.t. is an exponentially
small quantity of higher order.
The zeroes of $A(T,\mu,\nu)$ correspond to homoclinic points.
Moreover, if this function have nor zeroes, it can be shown
that the separatrices do not intersect at all.

The first term $A_0$ is approximated by the Melnikov function
described in Section~\ref{Se:interpol}:
$$
A_0(\mu,\nu)=M(\mu,\nu)+O(\mu^{2}).
$$
The amplitude $A_1$ of the second term is exponentially small.
The product $A_1\sin2\pi T$ can be large for non-real $T$.
In order to approximate it we use the following complex matching
method. We expand $E(\x^s(t))-E(\x^u(t))$ in powers of $\x^s(t)-\x^u(t)$
and use
$$
\partial_x E(\x^u(t))=\frac{\dot{y}^u(t)}{w(t)},\qquad
\partial_y E(\x^u(t))=-\frac{\dot{x}^u(t)}{w(t)},
$$
where $1/w(t)$ is the Jacobian of the map $(x,y)\mapsto(E,T)$
computed at the  point $\x^u(t)$.
We obtain that
\begin{equation}\label{Eq:sgraph}
A(t,\mu,\nu)=\frac1{w(t)}
\det(\dot {\x}^u(t);\x^s(t)-\x^u(t))+O(|\x^s(t)-\x^u(t)|^2)\;.
\end{equation}
In oder to approximate this function we consider its analytic
continuation in the variable $t$.  The functions $\x^u(t)$ and $\x^s(t)$
are approximated by $\x_0^-(\tau)$ and  $\x_0^+(\tau)$ respectively,
where
$$
\tau=t-\i\rho-\rho_1,
$$
$\rho$ is defined in (\ref{Eq:rho}); and
$\rho_1=\tfrac{3}{7}(\gamma-2)(\i\pi+\log6) $ 
takes into account the matching of the expansions
(\ref{Eq:asimp0}) and (\ref{Eq:mainsing}).

Comparing (\ref{Eq:approxtheta}), (\ref{Eq:tildetheta}) and (\ref{Eq:sgraph})
we obtain
$$
A_1\approx 2|\Theta|\, |\e^{-2\pi\i\rho_1}| \, \e^{-2\pi\rho}
=2|\Theta|\, \e^{-6\pi^2(\gamma-2)/7}
\,
\e^{-\sqrt2\pi^2/\root 4\of\mu}. 
$$

Now we calculate the width of the homoclinic zone:
$$
\nu_+(\mu)-\nu_-(\mu)
=\left.\frac{2 A_1(\mu,\nu)}
{ \frac{\partial A_0 }{\partial \nu} (\mu,\nu)
}\right|_{\nu:A_0(\mu,\nu)=0}=
\frac{ 5|\Theta|\e^{-6\pi^2(\gamma-2)/7}
}{6 \mu^{5/4} }
\,
\e^{-\sqrt2\pi^2/\root 4\of\mu},
$$
where we used the approximations for $A_0$ and $A_1$.
Thus we obtained the desired asymptotic (\ref{Eq:mainas}) for
the width of the homoclinic zone on the parameter plane of
the Bogdanov-Takens bifurcation for diffeomorphisms.
The error term comes from the consideration of 
next order corrections to this asymptotic.

\section{Numerical experiments\label{Se:numerics}}

In the most cases there is not known ways to find the
splitting constant $\Theta$ analyticly. Given a map $F_0$,
this constant is not too difficult to compute with 
any reasonable precision. For the area-preserving H\'enon
map this constant was computed with several thousands digits
\cite{Simo2000}. We did high-precision computations of $\Theta$
for the family (\ref{Eq:nfl}) with
$
f=x^2+\gamma xy
$
for several $\gamma$. The method is
essentially the same as in \cite{Gelfreich2000}.
Numerically the splitting constant $\Theta$ does 
not vanish on the interval $[-0.5,2]$.
Several values are given in the following table:

\begin{center}
\begin{tabular}{|l|l|}
\hline
$\gamma$&\hfil$\Theta$\hfil\\
\hline
0 & $5.202169191369226\cdot 10^{-5}$\\
1 & $6.9898737i51682470943...$\\
2 & $1.203229677075980622\cdot 10 ^{8}$\\
\hline
\end{tabular}
\end{center}

The estimates for the width of the homoclinic zone
were derived non-rigorously. So it is interesting to compare
them with results of direct numerical computations.
We choose the family (\ref{Eq:nfl}) with
$$
f(x,y,\mu,\nu)=-\mu+x^2+\nu y+\gamma xy,
$$
and the H\'enon map.
There is a 
reasonably good agreement between our theory and
experiments. For $\mu^{1/4}=0.08$ the relative error
of the asymptotic formula is typically around 20\%. 

It is difficult to perform computations for essentially smaller
values of $\mu$. So we took the data for several $\mu$ and then
extrapolated towards $\mu=0$. The values of $\Theta$ obtained
in this way coincide in several digits with the values from the table.
This  provides numerical evidence for the correctness of the asymptotic
formula (\ref{Eq:mainas}).

%% optional
%\section{Summary}

\section*{Acknowledgments}
The author thanks the International Society for Analysis,
Applications and Computation for presenting him
the "ISAAC Award for Young Scientists".


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