{VERSION 2018 2 "Windows 10" "2018.2" }
{USTYLETAB {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0
 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "War
ning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0
 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Maple Outp
ut" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 
}1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "
" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 
0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 
255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Head
ing 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0
 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1
 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 
0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "T
imes" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }
{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2
 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" 
-1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 
0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1
 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2
 2 -1 1 }{PSTYLE "Ordered List 1" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 
12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE 
"Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2
 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 2" 
-1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 
1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 202 1 {CSTYLE
 "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2
 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 203 1 {CSTYLE "" -1 -1 "Times
" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }
{PSTYLE "Ordered List 5" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 
1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Annota
tion Title" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2
 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Output12" -1 
206 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0
 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkError" -1 207 1 {CSTYLE "" 
-1 -1 "Courier New" 1 12 255 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 
2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkWarning" -1 208 1 {CSTYLE "" -1 -1
 "Courier New" 1 12 0 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0
 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12
 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "M
aple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1
 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3
 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times
" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }
{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2
 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 
1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 
12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Time
s" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }
{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2
 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{CSTYLE "Help Variable" -1
 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlin
ed Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Copy
right" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple \+
Comment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Po
pup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Atom
ic Variable" -1 200 "Times" 1 12 175 0 175 1 1 2 2 2 2 2 0 0 0 1 }
{CSTYLE "Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2
 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 
0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0
 0 0 1 }{CSTYLE "Code" -1 201 "Courier New" 1 12 255 0 0 1 2 2 2 2 2 2
 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2
 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2
 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1
 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2
 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 
2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 2
 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2
 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 202 "Times" 1 12 144 144 
144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 
255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 
12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Header and Footer" -1 203 "Ti
mes" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small20
4" -1 204 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math It
alic Small207208" -1 205 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }
{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 
}{CSTYLE "2D Math Italic Small206" -1 206 "Times" 1 1 0 0 0 1 1 2 2 2 
2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2
 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small205" -1 207 "Times" 1 1 0 0
 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 
0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small208209215" -1 
208 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined"
 -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Ital
ic Small" -1 209 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D \+
Math Italic Small208" -1 210 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }
{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 
0 1 }{CSTYLE "2D Math Italic Small207" -1 211 "Times" 1 1 0 0 0 1 1 2 
2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2
 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 
1 2 2 2 2 0 0 0 1 }{CSTYLE "Annotation Text" -1 212 "Times" 1 12 0 0 0
 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "Times" 1 12 
104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 213 "Times" 1 12 0 
0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0
 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Time
s" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Reference" -1 
214 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 
"Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" 
-1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input P
laceholder" -1 215 "Courier New" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 
}{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }
{CSTYLE "2D Math Italic Small208209" -1 216 "Times" 1 1 0 0 0 1 1 2 2 
2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 
2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 217 "Times" 1 12 0 0 0 
1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1
 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 1
 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1
 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1
 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2
 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Text" -1 218 "Times" 1 12 0 0 0 1
 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 219 "Times" 1 12 0 0 0 1 2 1 2 2 2
 2 0 0 0 1 }{CSTYLE "" -1 220 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1
 }{CSTYLE "" -1 221 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE 
"" -1 222 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 213 240 "In this Maple file, we \+
compute the evolution equations for the Painlev\351 3 equations using \+
the compatibility equation of the Lax system. We also obtain the expre
ssion of the Lax matrices in the geometric gauge without apparent sing
ularities." }{TEXT 219 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 223 56 "L
ax matrices in the oper gauge from previous Maple files" }{TEXT 223 0 
"" }}{EXCHG {PARA 0 "" 0 "" {TEXT 224 103 "Summary of previous files: \+
 We have the expression for some coefficients of the Lax matrix L and \+
of A.\n" }{TEXT 224 194 "The deformation operator is \\mathcal\{L\}=
\\hbar (alphainf11\\partial_\{t_\{\\infty^\{(1)\},1\} +alphainf21\\par
tial_\{t_\{\\infty^\{(2)\},1\}+alpha011\\partial_\{t_\{0^\{(1)\},1\} +
alpha021\\partial_\{t_\{0^\{(2)\},1\})) )" }}}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(LinearAlgebra):\n"
 }{MPLTEXT 1 0 19 "P042 := t011*t021;\n" }{MPLTEXT 1 0 29 "P032 := t01
0*t021+t011*t020;\n" }{MPLTEXT 1 0 91 "P012 := -1/2*(t010+t020)*(tinft
y11+tinfty21)-1/2*(tinfty10-tinfty20)*(-tinfty21+tinfty11);\n" }
{MPLTEXT 1 0 19 "P021 := t011+t021;\n" }{MPLTEXT 1 0 19 "P011 := t010+
t020;\n" }{MPLTEXT 1 0 32 "Pinfty01 := -tinfty11-tinfty21;\n" }
{MPLTEXT 1 0 31 "Pinfty02 := tinfty11*tinfty21;\n" }{MPLTEXT 1 0 9 "Co
herence" }{MPLTEXT 1 0 42 "Equation1 := tinfty10+tinfty20+t010+t020;\n
" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 49 "P1:=lambda-> P021/lambda^2+P0
11/lambda+Pinfty01;\n" }{MPLTEXT 1 0 77 "P2:=lambda-> P042/lambda^4+P0
32/lambda^3+P022/lambda^2+P012/lambda+Pinfty02;\n" }{MPLTEXT 1 0 53 "d
P1dlambda:=unapply(diff(P1(lambda),lambda),lambda):\n" }{MPLTEXT 1 0 
53 "dP2dlambda:=unapply(diff(P2(lambda),lambda),lambda):\n" }{MPLTEXT 
1 0 1 "\n" }{MPLTEXT 1 0 25 "tdP2:=unapply(P2(lambda)-" }{MPLTEXT 1 0 
35 "P022/lambda^2-P012/lambda,lambda);\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 18 "L:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 11 "L[1,1]:=0:\n" 
}{MPLTEXT 1 0 11 "L[1,2]:=1:\n" }{MPLTEXT 1 0 170 "L[2,1]:=-t011*t021/
lambda^4  -(t010*t021+t020*t011)/lambda^3-H/lambda^2- tinfty11*tinfty2
1 - (tinfty11*tinfty20+tinfty21*tinfty10+h*tinfty11-h*p)/lambda- p*h/(
lambda-q):\n" }{MPLTEXT 1 0 92 "L[2,2]:= (t011+t021)/lambda^2+ (t010+t
020-2*h)/lambda -(tinfty11+tinfty21)   +h/(lambda-q):\n" }{MPLTEXT 1 
0 1 "\n" }{MPLTEXT 1 0 5 "C01:=" }{MPLTEXT 1 0 26 "residue(L[2,1],lamb
da=0);\n" }{MPLTEXT 1 0 5 "C02:=" }{MPLTEXT 1 0 32 "residue(L[2,1]*lam
bda,lambda=0);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 
0 18 "A:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 29 "A[1,1]:=(alphainf11*tinft
y21-" }{MPLTEXT 1 0 10 "alphainf21" }{MPLTEXT 1 0 62 "*tinfty11)/(tinf
ty11-tinfty21)*lambda+ C+rho/(lambda-q)-(t011*" }{MPLTEXT 1 0 8 "alpha
021" }{MPLTEXT 1 0 6 "-t021*" }{MPLTEXT 1 0 8 "alpha011" }{MPLTEXT 1 
0 22 ")/(t011-t021)/lambda:\n" }{MPLTEXT 1 0 9 "A[1,2]:=(" }{MPLTEXT 
1 0 10 "alphainf11" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 10 "alphainf21" 
}{MPLTEXT 1 0 48 ")/(tinfty11-tinfty21)*lambda+nu+ mu/(lambda-q):\n" }
{MPLTEXT 1 0 22 "A[2,1]:=AA21(lambda):\n" }{MPLTEXT 1 0 22 "A[2,2]:=AA
22(lambda):\n" }{MPLTEXT 1 0 26 "dAdlambda:=Matrix(2,2,0):\n" }
{MPLTEXT 1 0 88 "for i from 1 to 2 do for j from 1 to 2 do dAdlambda[i
,j]:=diff(A[i,j],lambda):  od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 
1 0 3 "L;\n" }{MPLTEXT 1 0 3 "A;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 
0 61 "nuMinus1:=factor(-residue(A[1,2]/lambda^2,lambda=infinity));\n" 
}{MPLTEXT 1 0 53 "nu0:=factor(-residue(A[1,2]/lambda,lambda=infinity))
;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 60 "cinfty1:=factor(-residue(A[1
,1]/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 5 "c01:=" }{MPLTEXT 
1 0 33 "factor(residue(A[1,1],lambda=0));" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "Q2:=" }
{MPLTEXT 1 0 28 "unapply(-p*(q-0)^2,lambda):\n" }{MPLTEXT 1 0 18 "J:=M
atrix(2,2,0):\n" }{MPLTEXT 1 0 11 "J[1,1]:=1:\n" }{MPLTEXT 1 0 11 "J[1
,2]:=0:\n" }{MPLTEXT 1 0 31 "J[2,1]:=Q2(lambda)/(lambda-q):\n" }
{MPLTEXT 1 0 32 "J[2,2]:=(lambda-0)^2/(lambda-q):" }{MPLTEXT 1 0 1 "\n
" }{MPLTEXT 1 0 26 "dJdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 87 "for
 i from 1 to 2 do for j from 1 to 2 do dJdlambda[i,j]:=diff(J[i,j],lam
bda): od: od:\n" }{MPLTEXT 1 0 3 "J:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT
 1 0 19 "LJ:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 12 "LJ[1,1]:=0:\n" }
{MPLTEXT 1 0 12 "LJ[1,2]:=0:\n" }{MPLTEXT 1 0 63 "LJ[2,2]:=diff(J[2,2]
,q)*Lq+diff(J[2,2],p)*Lp+h*diff(J[2,2],t):\n" }{MPLTEXT 1 0 63 "LJ[2,1
]:=diff(J[2,1],q)*Lq+diff(J[2,1],p)*Lp+h*diff(J[2,1],t):\n" }{MPLTEXT 
1 0 4 "LJ:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 79 "checkL:=simplify(
Multiply(Multiply(J,L),J^(-1))+h*Multiply(dJdlambda,J^(-1))):\n" }
{MPLTEXT 1 0 70 "checkA:=simplify(Multiply(Multiply(J,A),J^(-1))+Multi
ply(LJ,J^(-1))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I%P042G6\"*&I%t011GF$\"\"\"I%t021GF$F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I%P032G6\",&*&I%t010GF$\"\"\"I%t021GF$F
(F(*&I%t011GF$F(I%t020GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P012
G6\",&*&,&I%t010GF$\"\"\"I%t020GF$F)F),&I)tinfty11GF$F)I)tinfty21GF$F)
F)#!\"\"\"\"#*&,&I)tinfty10GF$F)I)tinfty20GF$F/F),&F-F/F,F)F)F." }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I%P021G6\",&I%t011GF$\"\"\"I%t021GF$F'"
 }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P011G6\",&I%t010GF$\"\"\"I%t020GF$
F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty01G6\",&I)tinfty11GF$!\"
\"I)tinfty21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty02G6\"*&I)
tinfty11GF$\"\"\"I)tinfty21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I3C
oherenceEquation1G6\",*I)tinfty10GF$\"\"\"I)tinfty20GF$F'I%t010GF$F'I%
t020GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I'lambdaGF$F$6
$I)operatorGF$I&arrowGF$F$,(*&I%P021GF$\"\"\"9$!\"#F.*&I%P011GF$F.F/!
\"\"F.I)Pinfty01GF$F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"
f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$,,*&I%P042GF$\"\"\"9$!\"%F
.*&I%P032GF$F.F/!\"$F.*&I%P022GF$F.F/!\"#F.*&I%P012GF$F.F/!\"\"F.I)Pin
fty02GF$F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6#I'lam
bdaGF$F$6$I)operatorGF$I&arrowGF$F$,(*(I%t011GF$\"\"\"I%t021GF$F.9$!\"
%F.*&,&*&I%t010GF$F.F/F.F.*&F-F.I%t020GF$F.F.F.F0!\"$F.*&I)tinfty11GF$
F.I)tinfty21GF$F.F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$C01G6\",
$*&,**(I\"hGF$\"\"\"I\"pGF$F*I\"qGF$F*!\"\"*(F)F*F,F*I)tinfty11GF$F*F*
*(F,F*I)tinfty10GF$F*I)tinfty21GF$F*F**(F,F*F/F*I)tinfty20GF$F*F*F*F,F
-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$C02G6\",$I\"HGF$!\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typesett
ingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#m
nGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowal
ignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/%+columnspa
nGFJ-F56(-F86%FJF;F>FAFDFFFHFKFAFDFF-F26'-F56(-F,6.-I#moGF$6.Q*&uminus
0;F'F;F>/%&fenceGQ&falseF'/%*separatorGFgn/%)stretchyGFgn/%*symmetricG
Fgn/%(largeopGFgn/%.movablelimitsGFgn/%'accentGFgn/%'lspaceGQ,0.222222
2emF'/%'rspaceGFfo-I&mfracGF$6)-F,6%-I#miGF$6&Q%t011F'/%'italicGQ%true
F'F;/F?Q'italicF'-FX6-Q1⁢F'F>FenFhnFjnF\\oF^oF`oFbo/Feo
Q&0.0emF'/FhoF[q-F_p6&Q%t021F'FbpF;Fep-F,6#-I%msupGF$6%-F_p6&Q)&lambda
;F'/FcpFgnF;F>-F86%Q\"4F'F;F>/%1superscriptshiftGF:/%.linethicknessGFJ
/%+denomalignGQ'centerF'/%)numalignGFbr/%)bevelledGFgnF;-FX6.Q(−
F'F;F>FenFhnFjnF\\oF^oF`oFboFdoFgo-Fjo6)-F,6#-F,6%-F,6%-F_p6&Q%t010F'F
bpF;FepFgpF]q-FX6.Q\"+F'F;F>FenFhnFjnF\\oF^oF`oFboFdoFgo-F,6%F^pFgp-F_
p6&Q%t020F'FbpF;Fep-F,6#-Fcq6%Feq-F86%Q\"3F'F;F>F\\rF^rF`rFcrFerF;Fgr-
Fjo6)-F,6#-F_p6&Q\"HF'FbpF;Fep-F,6#-Fcq6%Feq-F86%Q\"2F'F;F>F\\rF^rF`rF
crFerF;Fgr-F,6%-F_p6&Q)tinfty11F'FbpF;FepFgp-F_p6&Q)tinfty21F'FbpF;Fep
Fgr-Fjo6)-F,6#-F,6*FW-F,6%-F_p6&Q\"hF'FbpF;FepFgp-F_p6&Q\"pF'FbpF;FepF
es-F,6%FbvFgpFduFes-F,6%-F_p6&Q)tinfty10F'FbpF;FepFgpFguFes-F,6%FduFgp
-F_p6&Q)tinfty20F'FbpF;Fep-F,6#FeqF^rF`rFcrFerF;Fgr-Fjo6)-F,6%FevFgpFb
v-F,6#-F,6%FeqFgr-F_p6&Q\"qF'FbpF;FepF^rF`rFcrFerF;FAFDFFFHFK-F56(-F,6
+-Fjo6)-F,6#-F,6%F^pFesF]qF[uF^rF`rFcrFerF;Fes-Fjo6)-F,6#-F,6'FbsFesFj
sFgr-F,6%F_uFgpFbvFdwF^rF`rFcrFerF;FgrFduFgrFguFes-Fjo6)-F,6#FbvFjwF^r
F`rFcrFerF;FAFDFFFHFKFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEFbr/FGQ'
|frleft|hrF'/%/alignmentscopeGFdp/%,columnwidthGQ%autoF'/%&widthGFcz/%
+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,
columnlinesGF^[l/%&frameGF^[l/%-framespacingGQ,0.4em~0.5exF'/%*equalro
wsGFgn/%-equalcolumnsGFgn/%-displaystyleGFgn/%%sideGQ&rightF'/%0minlab
elspacingGF[[lF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6
'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6)-I&mfracGF$6)-F,
6%-F#6%-F,6%-F,6%-I#miGF$6&Q+alphainf11F'/%'italicGQ%trueF'/%+foregrou
ndGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6-Q1⁢F'/
FOQ'normalF'/%&fenceGQ&falseF'/%*separatorGFY/%)stretchyGFY/%*symmetri
cGFY/%(largeopGFY/%.movablelimitsGFY/%'accentGFY/%'lspaceGQ&0.0emF'/%'
rspaceGFbo-FE6&Q)tinfty21F'FHFKFN-FR6.Q(−F'FKFUFWFZFfnFhnFjnF\\o
F^o/FaoQ,0.2222222emF'/FdoF\\p-F,6%-FE6&Q+alphainf21F'FHFKFNFQ-FE6&Q)t
infty11F'FHFKFNFKFUFQ-FE6&Q)λF'/FIFYFKFU-F,6#-F,6&-FR6.Q*&uminu
s0;F'FKFUFWFZFfnFhnFjnF\\oF^oF[pF]pFeo-FR6.Q\"+F'FKFUFWFZFfnFhnFjnF\\o
F^oF[pF]pFcp/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalign
GFiq/%)bevelledGFYFKFaq-FE6&Q\"CF'FHFKFNFaq-F:6)-F,6#-FE6&Q&ρF'Fip
FKFU-F,6#-F,6%FfpFho-FE6&Q\"qF'FHFKFNFdqFgqFjqF\\rFKFho-F:6)-F,6#-F,6%
-F,6%-FE6&Q%t011F'FHFKFNFQ-FE6&Q+α021F'FHFKFNFho-F,6%-FE6&Q%t021
F'FHFKFNFQ-FE6&Q+α011F'FHFKFN-F,6%-F#6%-F,6%FgsFhoF_tFKFUFQFfpFd
qFgqFjqF\\rFK/%)rowalignGQ!F'/%,columnalignGF]u/%+groupalignGF]u/%(row
spanGFfq/%+columnspanGFfq-F56(-F,6'-F:6)-F,6%-F#6%-F,6%FDFhoF`pFKFUFQF
fpFjpFdqFgqFjqF\\rFKFaq-FE6&Q%νF'FipFKFUFaq-F:6)-F,6#-FE6&Q%μF'F
ipFKFUFhrFdqFgqFjqF\\rFKF[uF^uF`uFbuFduF[uF^uF`u-F26'-F56(-F,6%-FE6&Q%
AA21F'FHFKFN-FR6-Q0⁡F'FUFWFZFfnFhnFjnF\\oF^oF`oFco-F#6%-
F,6#FfpFKFUF[uF^uF`uFbuFdu-F56(-F,6%-FE6&Q%AA22F'FHFKFNFewFhwF[uF^uF`u
FbuFduF[uF^uF`u/%&alignGQ%axisF'/F\\uQ)baselineF'/F_uFiq/FauQ'|frleft|
hrF'/%/alignmentscopeGFJ/%,columnwidthGQ%autoF'/%&widthGF_y/%+rowspaci
ngGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlin
esGFjy/%&frameGFjy/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFY/%-equ
alcolumnsGFY/%-displaystyleGFY/%%sideGQ&rightF'/%0minlabelspacingGFgyF
KFU/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)n
uMinus1G6\"*&,&I+alphainf11GF$\"\"\"I+alphainf21GF$!\"\"F(,&I)tinfty21
GF$F*I)tinfty11GF$F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu0G6\"I#nu
GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(cinfty1G6\"*&,&*&I+alphainf11G
F$\"\"\"I)tinfty21GF$F)F)*&I+alphainf21GF$F)I)tinfty11GF$F)!\"\"F),&F*
F.F-F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$c01G6\",$*&,&*&I%t011GF$
\"\"\"I)alpha021GF$F*F**&I%t021GF$F*I)alpha011GF$F*!\"\"F*,&F)F*F-F/F/
F/" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 223 73 "Solving the compatibilit
y equations to obtain the Hamiltonian evolutions." }{TEXT 223 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 224 69 "The compatibility equation is \\m
athcal\{L\}L=h\\partial_\\lambda A+[A,L]\n" }{TEXT 224 111 "Since the \+
first line of L is trivial, we may easily obtain A[2,1] et A[2,2] to o
btain the full expression for A" }{TEXT 224 0 "" }}}{EXCHG {PARA 0 "> 
" 0 "" {MPLTEXT 1 0 46 "LL:=h*dAdlambda+(Multiply(A,L)-Multiply(L,A)):
" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "Entry11:=LL[1,1]:\n" }
{MPLTEXT 1 0 18 "Entry12:=LL[1,2]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 
1 0 53 "AA21:=unapply(solve(Entry11=0,AA21(lambda)),lambda):\n" }
{MPLTEXT 1 0 41 "AA21bis:=h*dAdlambda[1,1]+A[1,2]*L[2,1]:\n" }{MPLTEXT
 1 0 1 "\n" }{MPLTEXT 1 0 32 "simplify(AA21(lambda)-AA21bis);\n" }
{MPLTEXT 1 0 53 "AA22:=unapply(solve(Entry12=0,AA22(lambda)),lambda):
\n" }{MPLTEXT 1 0 48 "AA22bis:=h*dAdlambda[1,2]+A[1,1]+A[1,2]*L[2,2]:
\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simplify(AA22(lambda)-AA22b
is);\n" }{MPLTEXT 1 0 19 "simplify(Entry11);\n" }{MPLTEXT 1 0 19 "simp
lify(Entry12);\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 46 "L:=h*dAdlambda
+(Multiply(A,L)-Multiply(L,A)):\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 
1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 95 "We now compute the acti
on of  \\mathcal\{L\} on L[2,2] et L[2,1] to obtain the evolution equa
tions" }{TEXT 224 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 213 26 "Evoluti
on of entry L_\{2,2\}" }{TEXT 213 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "Entry22:=simplify(LL[2,2]):\n" }{MPLTEXT 1 0 77 "Entr
y22TermLambdaMinusqCube:=factor(residue(Entry22*(lambda-q)^2,lambda=q)
);\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaMinusqSquare:=factor(residue(
Entry22*(lambda-q),lambda=q));\n" }{MPLTEXT 1 0 60 "Entry22TermLambdaM
inusq:=factor(residue(Entry22,lambda=q));\n" }{MPLTEXT 1 0 73 "Entry22
TermLambdaZeroMinus4:=factor(residue(Entry22*lambda^3,lambda=0));\n" }
{MPLTEXT 1 0 73 "Entry22TermLambdaZeroMinus3:=factor(residue(Entry22*l
ambda^2,lambda=0));\n" }{MPLTEXT 1 0 71 "Entry22TermLambdaZeroMinus2:=
factor(residue(Entry22*lambda,lambda=0));\n" }{MPLTEXT 1 0 64 "Entry22
TermLambdaZeroMinus1:=factor(residue(Entry22,lambda=0));\n" }{MPLTEXT 
1 0 77 "Entry22TermLambdaInfty2:=factor(-residue(Entry22/lambda^3,lamb
da=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty1:=factor(-r
esidue(Entry22/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "Entry2
2TermLambdaInfty0:=factor(-residue(Entry22/lambda,lambda=infinity));\n
" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 97 "simplify( Entry22-(Entry22Ter
mLambdaMinusqSquare/(lambda-q)^2+Entry22TermLambdaMinusq/(lambda-q)\n"
 }{MPLTEXT 1 0 147 "+Entry22TermLambdaZeroMinus4/lambda^4+Entry22TermL
ambdaZeroMinus3/lambda^3+Entry22TermLambdaZeroMinus2/lambda^2+Entry22T
ermLambdaZeroMinus1/lambda\n" }{MPLTEXT 1 0 93 "+Entry22TermLambdaInft
y0+Entry22TermLambdaInfty1*lambda+Entry22TermLambdaInfty2*lambda^2) );
\n" }{MPLTEXT 1 0 8 "L[2,2];\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "
\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I<Entry22TermLambdaMinusqCubeG6\"
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I>Entry22TermLambdaMinusqSquar
eG6\"**,F**I\"hGF$\"\"\"I#nuGF$F)I\"qGF$\"\"#I)tinfty11GF$F)!\"\"**F(F
)F*F)F+F,I)tinfty21GF$F)F)*(F(F)F+\"\"$I+alphainf11GF$F)F.*(F(F)F+F2I+
alphainf21GF$F)F)*(I#muGF$F)F+F,F-F,F)*(F7F)F+F,F0F,F.**F(F)F7F)F+F)F-
F)F,**F(F)F7F)F+F)F0F)!\"#**F7F)F+F)I%t010GF$F)F-F)F.**F7F)F+F)F=F)F0F
)F)**F7F)F+F)I%t020GF$F)F-F)F.**F7F)F+F)F@F)F0F)F)*(F+F,I$rhoGF$F)F-F)
F;*(F+F,FCF)F0F)F,*(F7F)I%t011GF$F)F-F)F.*(F7F)FFF)F0F)F)*(F7F)I%t021G
F$F)F-F)F.*(F7F)FIF)F0F)F)F)F(F)F+F;,&F0F.F-F)F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I8Entry22TermLambdaMinusqG6\"\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I<Entry22TermLambdaZeroMinus4G6\"\"\"!" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I<Entry22TermLambdaZeroMinus3G6\",$**,&I%t011GF$\"\"
\"I%t021GF$F)F),&*&I#nuGF$F)I\"qGF$F)!\"\"I#muGF$F)F)I\"hGF$F)F.F/\"\"
#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I<Entry22TermLambdaZeroMinus2G6\",
$*,I\"hGF$\"\"\",do*,F'F(I#nuGF$F(I\"qGF$\"\"#I%t011GF$F(I)tinfty11GF$
F(!\"#*,F'F(F+F(F,F-F.F(I)tinfty21GF$F(F-*,F'F(F+F(F,F-I%t021GF$F(F/F(
F-*,F'F(F+F(F,F-F4F(F2F(F0*,F+F(F,F-I%t010GF$F(F.F(F/F(F(*,F+F(F,F-F7F
(F.F(F2F(!\"\"*,F+F(F,F-F7F(F4F(F/F(F9*,F+F(F,F-F7F(F4F(F2F(F(*,F+F(F,
F-F.F(I%t020GF$F(F/F(F(*,F+F(F,F-F.F(F=F(F2F(F9*,F+F(F,F-F=F(F4F(F/F(F
9*,F+F(F,F-F=F(F4F(F2F(F(*,F'F(I#muGF$F(F,F(F.F(F/F(F-*,F'F(FBF(F,F(F.
F(F2F(F0*,F'F(FBF(F,F(F4F(F/F(F0*,F'F(FBF(F,F(F4F(F2F(F-*,FBF(F,F(F7F(
F.F(F/F(F9*,FBF(F,F(F7F(F.F(F2F(F(*,FBF(F,F(F7F(F4F(F/F(F(*,FBF(F,F(F7
F(F4F(F2F(F9*,FBF(F,F(F.F(F=F(F/F(F9*,FBF(F,F(F.F(F=F(F2F(F(*,FBF(F,F(
F=F(F4F(F/F(F(*,FBF(F,F(F=F(F4F(F2F(F9*(F,F-F.F-I+alphainf11GF$F(F(*(F
,F-F.F-I+alphainf21GF$F(F9**F,F-F.F(I)alpha021GF$F(F/F(F0**F,F-F.F(FSF
(F2F(F-*(F,F-F4F-FOF(F9*(F,F-F4F-FQF(F(**F,F-F4F(I)alpha011GF$F(F/F(F-
**F,F-F4F(FXF(F2F(F0*(FBF(F.F-F/F(F9*(FBF(F.F-F2F(F(*(FBF(F4F-F/F(F(*(
FBF(F4F-F2F(F9F(F,F0,&F2F9F/F(F9,&F.F(F4F9F9F9" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I<Entry22TermLambdaZeroMinus1G6\"\"\"!" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I8Entry22TermLambdaInfty2G6\"\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I8Entry22TermLambdaInfty1G6\"\"\"!" }}{PARA 11 "" 1 ""
 {XPPMATH 20 ">I8Entry22TermLambdaInfty0G6\",$*&I\"hGF$\"\"\",&I+alpha
inf11GF$F(I+alphainf21GF$F(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&,&I%t011G6\"\"\"\"I%t021GF&F'
F'I'lambdaGF&!\"#F'*&,(I%t010GF&F'I%t020GF&F'I\"hGF&F*F'F)!\"\"F'I)tin
fty11GF&F0I)tinfty21GF&F0*&F/F',&F)F'I\"qGF&F0F0F'" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 224 34 "Since the deformation operator is " }{TEXT 224 
165 "mathcal\{L\}=\\hbar (alphainf11\\partial_\{t_\{\\infty^\{(1)\},1
\} +alphainf21\\partial_\{t_\{\\infty^\{(2)\},1\}+alpha011\\partial_\{
t_\{0^\{(1)\},1\} +alpha021\\partial_\{t_\{0^\{(2)\},1\})) )" }{TEXT 
224 37 ", the double pole at lambda=0  is h*(" }{TEXT 224 8 "alpha011"
 }{TEXT 224 1 "+" }{TEXT 224 8 "alpha021" }{TEXT 224 2 ") " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "solve(\{Entry22TermLambdaZeroMinus3
,Entry22TermLambdaZeroMinus2-h*(" }{MPLTEXT 1 0 8 "alpha011" }{MPLTEXT
 1 0 1 "+" }{MPLTEXT 1 0 8 "alpha021" }{MPLTEXT 1 0 13 ")\},\{nu,mu\})
;\n" }{MPLTEXT 1 0 207 "mu := q^2*(t011*alphainf11-t011*alphainf21-t02
1*alphainf11+t021*alphainf21+alpha011*tinfty11-alpha011*tinfty21-alpha
021*tinfty11+alpha021*tinfty21)/(t011*tinfty11-t011*tinfty21-t021*tinf
ty11+t021*tinfty21);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 204 "nu := \+
q*(t011*alphainf11-t011*alphainf21-t021*alphainf11+t021*alphainf21+alp
ha011*tinfty11-alpha011*tinfty21-alpha021*tinfty11+alpha021*tinfty21)/
(t011*tinfty11-t011*tinfty21-t021*tinfty11+t021*tinfty21);" }{MPLTEXT 
1 0 1 "\n" }{MPLTEXT 1 0 39 "simplify(Entry22TermLambdaZeroMinus3);\n"
 }{MPLTEXT 1 0 40 "simplify(Entry22TermLambdaZeroMinus2-h*(" }{MPLTEXT
 1 0 8 "alpha011" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 8 "alpha021" }
{MPLTEXT 1 0 4 "));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$/I#muG6\"*(,2
*&I%t011GF%\"\"\"I+alphainf11GF%F*F**&F)F*I+alphainf21GF%F*!\"\"*&I%t0
21GF%F*F+F*F.*&F0F*F-F*F**&I)alpha011GF%F*I)tinfty11GF%F*F**&F3F*I)tin
fty21GF%F*F.*&I)alpha021GF%F*F4F*F.*&F8F*F6F*F*F*I\"qGF%\"\"#,**&F)F*F
4F*F**&F)F*F6F*F.*&F0F*F4F*F.*&F0F*F6F*F*F./I#nuGF%*(F'F*F:F*F<F." }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I#muG6\"*(,2*&I%t011GF$\"\"\"I+alphainf
11GF$F)F)*&F(F)I+alphainf21GF$F)!\"\"*&I%t021GF$F)F*F)F-*&F/F)F,F)F)*&
I)alpha011GF$F)I)tinfty11GF$F)F)*&F2F)I)tinfty21GF$F)F-*&I)alpha021GF$
F)F3F)F-*&F7F)F5F)F)F)I\"qGF$\"\"#,**&F(F)F3F)F)*&F(F)F5F)F-*&F/F)F3F)
F-*&F/F)F5F)F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#nuG6\"*(,2*&I%t01
1GF$\"\"\"I+alphainf11GF$F)F)*&F(F)I+alphainf21GF$F)!\"\"*&I%t021GF$F)
F*F)F-*&F/F)F,F)F)*&I)alpha011GF$F)I)tinfty11GF$F)F)*&F2F)I)tinfty21GF
$F)F-*&I)alpha021GF$F)F3F)F-*&F7F)F5F)F)F)I\"qGF$F),**&F(F)F3F)F)*&F(F
)F5F)F-*&F/F)F3F)F-*&F/F)F5F)F)F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "Lq:=factor(Entry22TermLambdaMinusqSquare/h):\n" }
{MPLTEXT 1 0 27 "Lqbis:=-2*rho-((alphainf11-" }{MPLTEXT 1 0 10 "alphai
nf21" }{MPLTEXT 1 0 102 ")/(tinfty11-tinfty21)+(alpha011-alpha021)/(t0
11-t021))*q^2*P1(q)+(alpha011-alpha021)*q/(t011-t021)*h;\n" }{MPLTEXT 
1 0 39 "factor(simplify(series(Lq-Lqbis,q=0)));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I&LqbisG6\",(I$rhoGF$!\"#*(,&*&,&I+alphainf11GF$\"\"\"I+
alphainf21GF$!\"\"F-,&I)tinfty21GF$F/I)tinfty11GF$F-F/F-*&,&I)alpha011
GF$F-I)alpha021GF$F/F-,&I%t011GF$F-I%t021GF$F/F/F-F-I\"qGF$\"\"#,**&,&
F8F-F9F-F-F:F'F-*&,&I%t010GF$F-I%t020GF$F-F-F:F/F-F2F/F1F/F-F/**F4F-F:
F-F7F/I\"hGF$F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 224 33 "Evolution of  \\mathcal\{L\}[L[2,1]]" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entry21:=simplify(LL[2,1]):
\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaMinusqCube:=factor(residue(Entr
y21*(lambda-q)^2,lambda=q));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaMin
usqSquare:=factor(residue(Entry21*(lambda-q),lambda=q));\n" }{MPLTEXT 
1 0 60 "Entry21TermLambdaMinusq:=factor(residue(Entry21,lambda=q));\n"
 }{MPLTEXT 1 0 73 "Entry21TermLambdaZeroMinus5:=factor(residue(Entry21
*lambda^4,lambda=0));\n" }{MPLTEXT 1 0 73 "Entry21TermLambdaZeroMinus4
:=factor(residue(Entry21*lambda^3,lambda=0));\n" }{MPLTEXT 1 0 73 "Ent
ry21TermLambdaZeroMinus3:=factor(residue(Entry21*lambda^2,lambda=0));
\n" }{MPLTEXT 1 0 71 "Entry21TermLambdaZeroMinus2:=factor(residue(Entr
y21*lambda,lambda=0));\n" }{MPLTEXT 1 0 64 "Entry21TermLambdaZeroMinus
1:=factor(residue(Entry21,lambda=0));\n" }{MPLTEXT 1 0 77 "Entry21Term
LambdaInfty3:=factor(-residue(Entry21/lambda^4,lambda=infinity));\n" }
{MPLTEXT 1 0 77 "Entry21TermLambdaInfty2:=factor(-residue(Entry21/lamb
da^3,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty1:=
factor(-residue(Entry21/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 
75 "Entry21TermLambdaInfty0:=factor(-residue(Entry21/lambda,lambda=inf
inity));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 138 "simplify( Entry21-
(Entry21TermLambdaMinusqCube/(lambda-q)^3+Entry21TermLambdaMinusqSquar
e/(lambda-q)^2+Entry21TermLambdaMinusq/(lambda-q)\n" }{MPLTEXT 1 0 
185 "+Entry21TermLambdaZeroMinus5/lambda^5+Entry21TermLambdaZeroMinus4
/lambda^4+Entry21TermLambdaZeroMinus3/lambda^3+ Entry21TermLambdaZeroM
inus2/lambda^2+Entry21TermLambdaZeroMinus1/lambda\n" }{MPLTEXT 1 0 
126 "+Entry21TermLambdaInfty0+Entry21TermLambdaInfty1*lambda+Entry21Te
rmLambdaInfty2*lambda^2+Entry21TermLambdaInfty3*lambda^3) );\n" }
{MPLTEXT 1 0 8 "L[2,1];\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I<Entry21TermLambdaMinusqCubeG6\",$**I\"hGF$\"\"#,:**I\"
pGF$\"\"\"I\"qGF$F(I%t011GF$F,I+alphainf11GF$F,F,**F+F,F-F(F.F,I+alpha
inf21GF$F,!\"\"**F+F,F-F(I%t021GF$F,F/F,F2**F+F,F-F(F4F,F1F,F,**F+F,F-
F(I)alpha011GF$F,I)tinfty11GF$F,F,**F+F,F-F(F7F,I)tinfty21GF$F,F2**F+F
,F-F(I)alpha021GF$F,F8F,F2**F+F,F-F(F<F,F:F,F,*(I$rhoGF$F,F.F,F8F,F,*(
F?F,F.F,F:F,F2*(F?F,F4F,F8F,F2*(F?F,F4F,F:F,F,F,,&F:F2F8F,F2,&F.F,F4F2
F2\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I>Entry21TermLambdaMinusqSqu
areG6\"*,,\\v*,I\"qGF$\"\"%I%t011GF$\"\"\"I+alphainf11GF$F+I)tinfty11G
F$F+I)tinfty21GF$F+\"\"#*,F(F)F*F+I+alphainf21GF$F+F-F+F.F+!\"#*,F(F)I
%t021GF$F+F,F+F-F+F.F+F2*,F(F)F4F+F1F+F-F+F.F+F/**F(F)I)alpha011GF$F+F
-F/F.F+F/**F(F)F7F+F-F+F.F/F2**F(F)I)alpha021GF$F+F-F/F.F+F2**F(F)F:F+
F-F+F.F/F/*,I\"hGF$F+I\"pGF$F+F(\"\"$F7F+F-F+!\"\"*,F=F+F>F+F(F?F7F+F.
F+F+*,F=F+F>F+F(F?F:F+F-F+F+*,F=F+F>F+F(F?F:F+F.F+F@*,F=F+F(F?F*F+F,F+
F-F+F/*,F=F+F(F?F*F+F1F+F-F+F2*,F=F+F(F?F4F+F,F+F-F+F2*,F=F+F(F?F4F+F1
F+F-F+F/**F=F+F(F?F7F+F-F/F/*,F=F+F(F?F7F+F-F+F.F+F2**F=F+F(F?F:F+F-F/
F2*,F=F+F(F?F:F+F-F+F.F+F/*,F(F?F*F+F,F+I)tinfty10GF$F+F.F+F/*,F(F?F*F
+F,F+F-F+I)tinfty20GF$F+F/*,F(F?F*F+F1F+FMF+F.F+F2*,F(F?F*F+F1F+F-F+FO
F+F2*,F(F?F4F+F,F+FMF+F.F+F2*,F(F?F4F+F,F+F-F+FOF+F2*,F(F?F4F+F1F+FMF+
F.F+F/*,F(F?F4F+F1F+F-F+FOF+F/*,F(F?F7F+FMF+F-F+F.F+F/**F(F?F7F+FMF+F.
F/F2**F(F?F7F+F-F/FOF+F/*,F(F?F7F+F-F+FOF+F.F+F2*,F(F?F:F+FMF+F-F+F.F+
F2**F(F?F:F+FMF+F.F/F/**F(F?F:F+F-F/FOF+F2*,F(F?F:F+F-F+FOF+F.F+F/**F(
F/I$rhoGF$F+F*F+F-F/F@**F(F/FinF+F*F+F.F/F+**F(F/FinF+F4F+F-F/F+**F(F/
FinF+F4F+F.F/F@**I\"HGF$F+F(F/F*F+F,F+F/**F^oF+F(F/F*F+F1F+F2**F^oF+F(
F/F4F+F,F+F2**F^oF+F(F/F4F+F1F+F/**F^oF+F(F/F7F+F-F+F/**F^oF+F(F/F7F+F
.F+F2**F^oF+F(F/F:F+F-F+F2**F^oF+F(F/F:F+F.F+F/*,F=F+F(F+FinF+F*F+F-F+
F2*,F=F+F(F+FinF+F*F+F.F+F/*,F=F+F(F+FinF+F4F+F-F+F/*,F=F+F(F+FinF+F4F
+F.F+F2*,F(F+FinF+I%t010GF$F+F*F+F-F+F+*,F(F+FinF+F[pF+F*F+F.F+F@*,F(F
+FinF+F[pF+F4F+F-F+F@*,F(F+FinF+F[pF+F4F+F.F+F+*,F(F+FinF+F*F+I%t020GF
$F+F-F+F+*,F(F+FinF+F*F+F`pF+F.F+F@*,F(F+FinF+F`pF+F4F+F-F+F@*,F(F+Fin
F+F`pF+F4F+F.F+F+*,F(F+F[pF+F*F+F4F+F,F+F/*,F(F+F[pF+F*F+F4F+F1F+F2**F
(F+F[pF+F4F/F,F+F2**F(F+F[pF+F4F/F1F+F/*,F(F+F[pF+F4F+F7F+F-F+F/*,F(F+
F[pF+F4F+F7F+F.F+F2*,F(F+F[pF+F4F+F:F+F-F+F2*,F(F+F[pF+F4F+F:F+F.F+F/*
*F(F+F*F/F`pF+F,F+F/**F(F+F*F/F`pF+F1F+F2*,F(F+F*F+F`pF+F4F+F,F+F2*,F(
F+F*F+F`pF+F4F+F1F+F/*,F(F+F*F+F`pF+F7F+F-F+F/*,F(F+F*F+F`pF+F7F+F.F+F
2*,F(F+F*F+F`pF+F:F+F-F+F2*,F(F+F*F+F`pF+F:F+F.F+F/*(FinF+F*F/F-F+F+*(
FinF+F*F/F.F+F@*(FinF+F4F/F-F+F@*(FinF+F4F/F.F+F+*(F*F/F4F+F,F+F/*(F*F
/F4F+F1F+F2*(F*F+F4F/F,F+F2*(F*F+F4F/F1F+F/**F*F+F4F+F7F+F-F+F/**F*F+F
4F+F7F+F.F+F2**F*F+F4F+F:F+F-F+F2**F*F+F4F+F:F+F.F+F/F+F=F+F(F2,&F.F@F
-F+F@,&F*F+F4F@F@" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambd
aMinusqG6\",$*,,`u*,I\"hGF$\"\"\"I\"pGF$F*I\"qGF$\"\"$I)alpha011GF$F*I
)tinfty11GF$F*!\"\"*,F)F*F+F*F,F-F.F*I)tinfty21GF$F*F**,F)F*F+F*F,F-I)
alpha021GF$F*F/F*F**,F)F*F+F*F,F-F4F*F2F*F0*,F)F*F,F-I%t011GF$F*I+alph
ainf11GF$F*F/F*F**,F)F*F,F-F7F*F8F*F2F*F**,F)F*F,F-F7F*I+alphainf21GF$
F*F/F*!\"#*,F)F*F,F-I%t021GF$F*F8F*F/F*F0*,F)F*F,F-F>F*F8F*F2F*F0*,F)F
*F,F-F>F*F;F*F/F*\"\"#**F)F*F,F-F.F*F/FAF**,F)F*F,F-F.F*F/F*F2F*F0**F)
F*F,F-F4F*F/FAF0*,F)F*F,F-F4F*F/F*F2F*F**,F,F-F7F*F8F*I)tinfty10GF$F*F
2F*F**,F,F-F7F*F8F*F/F*I)tinfty20GF$F*F**,F,F-F7F*F;F*FGF*F2F*F0*,F,F-
F7F*F;F*F/F*FIF*F0*,F,F-F>F*F8F*FGF*F2F*F0*,F,F-F>F*F8F*F/F*FIF*F0*,F,
F-F>F*F;F*FGF*F2F*F**,F,F-F>F*F;F*F/F*FIF*F**,F,F-F.F*FGF*F/F*F2F*F***
F,F-F.F*FGF*F2FAF0**F,F-F.F*F/FAFIF*F**,F,F-F.F*F/F*FIF*F2F*F0*,F,F-F4
F*FGF*F/F*F2F*F0**F,F-F4F*FGF*F2FAF***F,F-F4F*F/FAFIF*F0*,F,F-F4F*F/F*
FIF*F2F*F***I\"HGF$F*F,FAF7F*F8F*FA**FYF*F,FAF7F*F;F*F<**FYF*F,FAF>F*F
8F*F<**FYF*F,FAF>F*F;F*FA**FYF*F,FAF.F*F/F*FA**FYF*F,FAF.F*F2F*F<**FYF
*F,FAF4F*F/F*F<**FYF*F,FAF4F*F2F*FA*,F)F*F,F*I$rhoGF$F*F7F*F/F*F<*,F)F
*F,F*F\\oF*F7F*F2F*FA*,F)F*F,F*F\\oF*F>F*F/F*FA*,F)F*F,F*F\\oF*F>F*F2F
*F<*,F)F*F,F*F7F*F4F*F/F*F**,F)F*F,F*F7F*F4F*F2F*F0*,F)F*F,F*F>F*F.F*F
/F*F0*,F)F*F,F*F>F*F.F*F2F*F**,F,F*F\\oF*I%t010GF$F*F7F*F/F*F**,F,F*F
\\oF*FeoF*F7F*F2F*F0*,F,F*F\\oF*FeoF*F>F*F/F*F0*,F,F*F\\oF*FeoF*F>F*F2
F*F**,F,F*F\\oF*F7F*I%t020GF$F*F/F*F**,F,F*F\\oF*F7F*FjoF*F2F*F0*,F,F*
F\\oF*FjoF*F>F*F/F*F0*,F,F*F\\oF*FjoF*F>F*F2F*F**,F,F*FeoF*F7F*F>F*F8F
*F-*,F,F*FeoF*F7F*F>F*F;F*!\"$**F,F*FeoF*F>FAF8F*F`p**F,F*FeoF*F>FAF;F
*F-*,F,F*FeoF*F>F*F.F*F/F*F-*,F,F*FeoF*F>F*F.F*F2F*F`p*,F,F*FeoF*F>F*F
4F*F/F*F`p*,F,F*FeoF*F>F*F4F*F2F*F-**F,F*F7FAFjoF*F8F*F-**F,F*F7FAFjoF
*F;F*F`p*,F,F*F7F*FjoF*F>F*F8F*F`p*,F,F*F7F*FjoF*F>F*F;F*F-*,F,F*F7F*F
joF*F.F*F/F*F-*,F,F*F7F*FjoF*F.F*F2F*F`p*,F,F*F7F*FjoF*F4F*F/F*F`p*,F,
F*F7F*FjoF*F4F*F2F*F-*(F\\oF*F7FAF/F*FA*(F\\oF*F7FAF2F*F<*(F\\oF*F>FAF
/F*F<*(F\\oF*F>FAF2F*FA*(F7FAF>F*F8F*\"\"%*(F7FAF>F*F;F*!\"%*(F7F*F>FA
F8F*Ffq*(F7F*F>FAF;F*Fdq**F7F*F>F*F.F*F/F*Fdq**F7F*F>F*F.F*F2F*Ffq**F7
F*F>F*F4F*F/F*Ffq**F7F*F>F*F4F*F2F*FdqF*F)F*F,F`p,&F2F0F/F*F0,&F7F*F>F
0F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I<Entry21TermLambdaZeroMinus5G
6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I<Entry21TermLambdaZeroMinu
s4G6\",$*&,&*&I%t011GF$\"\"\"I)alpha021GF$F*F**&I%t021GF$F*I)alpha011G
F$F*F*F*I\"hGF$F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I<Entry21Term
LambdaZeroMinus3G6\",$*&,&*&I%t010GF$\"\"\"I)alpha021GF$F*F**&I%t020GF
$F*I)alpha011GF$F*F*F*I\"hGF$F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "
>I<Entry21TermLambdaZeroMinus2G6\"*,I\"hGF$\"\"\",\\p**I\"qGF$\"\"#I%t
011GF$F+I+alphainf11GF$F'I)tinfty21GF$F'!\"\"**F*F+F,F+I+alphainf21GF$
F'I)tinfty11GF$F'F'**F*F+F,F'I)alpha021GF$F'F2F+F'**F*F+F,F'F4F'F.F+F/
**F*F+I%t021GF$F+F-F'F.F'F'**F*F+F7F+F1F'F2F'F/**F*F+F7F'I)alpha011GF$
F'F2F+F/**F*F+F7F'F:F'F.F+F'*,F&F'F*F'F,F'F4F'F2F'F'*,F&F'F*F'F,F'F4F'
F.F'F/*,F&F'F*F'F7F'F:F'F2F'F/*,F&F'F*F'F7F'F:F'F.F'F'*,F*F'I%t010GF$F
'F,F'F7F'F-F'F'*,F*F'FAF'F,F'F7F'F1F'F/**F*F'FAF'F7F+F-F'F/**F*F'FAF'F
7F+F1F'F'*,F*F'FAF'F7F'F:F'F2F'F'*,F*F'FAF'F7F'F:F'F.F'F/*,F*F'FAF'F7F
'F4F'F2F'F/*,F*F'FAF'F7F'F4F'F.F'F'**F*F'F,F+I%t020GF$F'F-F'F'**F*F'F,
F+FJF'F1F'F/*,F*F'F,F'FJF'F7F'F-F'F/*,F*F'F,F'FJF'F7F'F1F'F'*,F*F'F,F'
FJF'F:F'F2F'F'*,F*F'F,F'FJF'F:F'F.F'F/*,F*F'F,F'FJF'F4F'F2F'F/*,F*F'F,
F'FJF'F4F'F.F'F'*(I$rhoGF$F'F,F+F2F'F'*(FSF'F,F+F.F'F/*(FSF'F7F+F2F'F/
*(FSF'F7F+F.F'F'*(F,F+F7F'F-F'F+*(F,F+F7F'F1F'!\"#*(F,F'F7F+F-F'FY*(F,
F'F7F+F1F'F+**F,F'F7F'F:F'F2F'F+**F,F'F7F'F:F'F.F'FY**F,F'F7F'F4F'F2F'
FY**F,F'F7F'F4F'F.F'F+F'F*FY,&F.F/F2F'F/,&F,F'F7F/F/" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I<Entry21TermLambdaZeroMinus1G6\"*,,\\u*,I\"hGF$\"\"
\"I\"pGF$F)I\"qGF$\"\"$I)alpha011GF$F)I)tinfty11GF$F)!\"\"*,F(F)F*F)F+
F,F-F)I)tinfty21GF$F)F)*,F(F)F*F)F+F,I)alpha021GF$F)F.F)F)*,F(F)F*F)F+
F,F3F)F1F)F/*,F(F)F+F,I%t011GF$F)I+alphainf11GF$F)F1F)\"\"#*,F(F)F+F,F
6F)I+alphainf21GF$F)F.F)!\"#*,F(F)F+F,I%t021GF$F)F7F)F1F)F;*,F(F)F+F,F
=F)F:F)F.F)F8**F(F)F+F,F-F)F.F8F)*,F(F)F+F,F-F)F.F)F1F)F/**F(F)F+F,F3F
)F.F8F/*,F(F)F+F,F3F)F.F)F1F)F)*,F+F,I%t010GF$F)F6F)F7F)F1F)F/*,F+F,FD
F)F6F)F:F)F.F)F)*,F+F,FDF)F=F)F7F)F1F)F)*,F+F,FDF)F=F)F:F)F.F)F/*,F+F,
F6F)I%t020GF$F)F7F)F1F)F/*,F+F,F6F)FIF)F:F)F.F)F)*,F+F,FIF)F=F)F7F)F1F
)F)*,F+F,FIF)F=F)F:F)F.F)F/*,F+F,F-F)I)tinfty10GF$F)F.F)F1F)F)**F+F,F-
F)FNF)F1F8F/**F+F,F-F)F.F8I)tinfty20GF$F)F)*,F+F,F-F)F.F)FQF)F1F)F/*,F
+F,F3F)FNF)F.F)F1F)F/**F+F,F3F)FNF)F1F8F)**F+F,F3F)F.F8FQF)F/*,F+F,F3F
)F.F)FQF)F1F)F)**I\"HGF$F)F+F8F6F)F7F)F8**FXF)F+F8F6F)F:F)F;**FXF)F+F8
F=F)F7F)F;**FXF)F+F8F=F)F:F)F8**FXF)F+F8F-F)F.F)F8**FXF)F+F8F-F)F1F)F;
**FXF)F+F8F3F)F.F)F;**FXF)F+F8F3F)F1F)F8*,F(F)F+F)I$rhoGF$F)F6F)F.F)F;
*,F(F)F+F)F[oF)F6F)F1F)F8*,F(F)F+F)F[oF)F=F)F.F)F8*,F(F)F+F)F[oF)F=F)F
1F)F;*,F(F)F+F)F6F)F3F)F.F)F)*,F(F)F+F)F6F)F3F)F1F)F/*,F(F)F+F)F=F)F-F
)F.F)F/*,F(F)F+F)F=F)F-F)F1F)F)*,F+F)F[oF)FDF)F6F)F.F)F)*,F+F)F[oF)FDF
)F6F)F1F)F/*,F+F)F[oF)FDF)F=F)F.F)F/*,F+F)F[oF)FDF)F=F)F1F)F)*,F+F)F[o
F)F6F)FIF)F.F)F)*,F+F)F[oF)F6F)FIF)F1F)F/*,F+F)F[oF)FIF)F=F)F.F)F/*,F+
F)F[oF)FIF)F=F)F1F)F)*,F+F)FDF)F6F)F=F)F7F)F,*,F+F)FDF)F6F)F=F)F:F)!\"
$**F+F)FDF)F=F8F7F)F]p**F+F)FDF)F=F8F:F)F,*,F+F)FDF)F=F)F-F)F.F)F,*,F+
F)FDF)F=F)F-F)F1F)F]p*,F+F)FDF)F=F)F3F)F.F)F]p*,F+F)FDF)F=F)F3F)F1F)F,
**F+F)F6F8FIF)F7F)F,**F+F)F6F8FIF)F:F)F]p*,F+F)F6F)FIF)F=F)F7F)F]p*,F+
F)F6F)FIF)F=F)F:F)F,*,F+F)F6F)FIF)F-F)F.F)F,*,F+F)F6F)FIF)F-F)F1F)F]p*
,F+F)F6F)FIF)F3F)F.F)F]p*,F+F)F6F)FIF)F3F)F1F)F,*(F[oF)F6F8F.F)F8*(F[o
F)F6F8F1F)F;*(F[oF)F=F8F.F)F;*(F[oF)F=F8F1F)F8*(F6F8F=F)F7F)\"\"%*(F6F
8F=F)F:F)!\"%*(F6F)F=F8F7F)Fcq*(F6F)F=F8F:F)Faq**F6F)F=F)F-F)F.F)Faq**
F6F)F=F)F-F)F1F)Fcq**F6F)F=F)F3F)F.F)Fcq**F6F)F=F)F3F)F1F)FaqF)F(F)F+F
]p,&F1F/F.F)F/,&F6F)F=F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21
TermLambdaInfty3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21T
ermLambdaInfty2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21Te
rmLambdaInfty1G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21Ter
mLambdaInfty0G6\",$*&,&*&I+alphainf11GF$\"\"\"I)tinfty21GF$F*F**&I+alp
hainf21GF$F*I)tinfty11GF$F*F*F*I\"hGF$F*!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*(I%t011G6\"\"\"
\"I%t021GF%F&I'lambdaGF%!\"%!\"\"*&,&*&I%t010GF%F&F'F&F&*&F$F&I%t020GF
%F&F&F&F(!\"$F**&I\"HGF%F&F(!\"#F**&I)tinfty11GF%F&I)tinfty21GF%F&F**&
,**&I\"hGF%F&I\"pGF%F&F**&F;F&F6F&F&*&I)tinfty10GF%F&F7F&F&*&F6F&I)tin
fty20GF%F&F&F&F(F*F**(F<F&F;F&,&F(F&I\"qGF%F*F*F*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 53 "rho:=factor(solve(Entry21TermLambdaMinusqCub
e,rho));\n" }{MPLTEXT 1 0 22 "simplify(rho+p*q*nu);\n" }{MPLTEXT 1 0 
39 "simplify(Entry21TermLambdaMinusqCube);\n" }{MPLTEXT 1 0 44 "LH:=si
mplify(-Entry21TermLambdaZeroMinus1):\n" }{MPLTEXT 1 0 119 "EquationPo
leSimple:=simplify(-h*(alphainf11*tinfty20+alphainf21*tinfty10+h*alpha
inf11-Lp)-Entry21TermLambdaZeroMinus1):" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I$rhoG6\",$*,I\"pGF$\"\"\"I\"qGF$\"\"#,2*&I%t011GF$F(I+alphainf11
GF$F(F(*&F-F(I+alphainf21GF$F(!\"\"*&I%t021GF$F(F.F(F1*&F3F(F0F(F(*&I)
alpha011GF$F(I)tinfty11GF$F(F(*&F6F(I)tinfty21GF$F(F1*&I)alpha021GF$F(
F7F(F1*&F;F(F9F(F(F(,&F9F1F7F(F1,&F-F(F3F1F1F1" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "LpFunction:=unapply(-Entry21TermLam
bdaMinusq/h,H):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Equation
1:=simplify(Entry21TermLambdaMinusqSquare-(-p*h*Lq)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Hsol:=solve(Equation1,H):\n" }
{MPLTEXT 1 0 71 "Hsolbis:=-q^2*p^2+q^2*P1(q)*p-q^2*(P2(q)-P022/q^2)-p*
q*h-tinfty11*q*h;\n" }{MPLTEXT 1 0 49 "factor(series(Hsol-Hsolbis+(tin
fty11+tinfty21)/2*" }{MPLTEXT 1 0 9 "Coherence" }{MPLTEXT 1 0 19 "Equa
tion1*q,q=0));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(HsolbisG6\",,*&I
\"qGF$\"\"#I\"pGF$F(!\"\"*(F'F(,**&,&I%t011GF$\"\"\"I%t021GF$F0F0F'!\"
#F0*&,&I%t010GF$F0I%t020GF$F0F0F'F*F0I)tinfty11GF$F*I)tinfty21GF$F*F0F
)F0F0*&F'F(,**(F/F0F1F0F'!\"%F0*&,&*&F5F0F1F0F0*&F/F0F6F0F0F0F'!\"$F0*
&,&*&F4F0,&F7F0F8F0F0#F*F(*&,&I)tinfty10GF$F0I)tinfty20GF$F*F0,&F8F*F7
F0F0FFF0F'F*F0*&F7F0F8F0F0F0F**(I\"hGF$F0F)F0F'F0F**(F7F0F'F0FNF0F*" }
}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "Lp:=factor(simplify(LpFunction(Hsol))):\n" }{MPLTEXT 
1 0 8 "Lpbis:=\n" }{MPLTEXT 1 0 128 "((alphainf11-alphainf21)/(tinfty1
1-tinfty21)+(alpha011-alpha021)/(t011-t021) )*(-2*q*p^2+diff(q^2*P1(q)
,q)*p-diff(q^2*P2(q),q))\n" }{MPLTEXT 1 0 143 "-alphainf11*h- (alpha01
1-alpha021)*h*p/(t011-t021)-h*(alpha011-alpha021)*tinfty11/(t011-t021)
-(t021*alpha011-t011*alpha021)*h/q^2/(t011-t021):\n" }{MPLTEXT 1 0 1 "
\n" }{MPLTEXT 1 0 88 "Lpter:=((alphainf11-alphainf21)/(tinfty11-tinfty
21)+(alpha011-alpha021)/(t011-t021) )*(\n" }{MPLTEXT 1 0 193 "-2*q*p^2
+((t010+t020)-2*(tinfty11+tinfty21)*q)*p+2*t011*t021/q^3+(t010*t021+t0
11*t020)/q^2 +1/2*((t010+t020)*(tinfty11+tinfty21)+(tinfty10-tinfty20)
*(tinfty11-tinfty21))-2*tinfty11*tinfty21*q)\n" }{MPLTEXT 1 0 143 "-al
phainf11*h- (alpha011-alpha021)*h*p/(t011-t021)-h*(alpha011-alpha021)*
tinfty11/(t011-t021)-(t021*alpha011-t011*alpha021)*h/q^2/(t011-t021):
\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 193 "factor(series(Lp-Lpbis+1/2
*(tinfty11+tinfty21)*(-alphainf11+alphainf21)/(tinfty21-tinfty11)*Cohe
renceEquation1-1/2*(tinfty11+tinfty21)*(alpha011-alpha021)/(t021-t011)
*CoherenceEquation1,p=0));\n" }{MPLTEXT 1 0 159 "factor(Lp-Lpter-1/2*(
tinfty11+tinfty21)*(alphainf11-alphainf21)/(tinfty21-tinfty11)*Coheren
ceEquation1-1/2*(tinfty11+tinfty21)*(alpha011-alpha021)/(t021-t011)*" 
}{MPLTEXT 1 0 9 "Coherence" }{MPLTEXT 1 0 12 "Equation1);\n" }{MPLTEXT
 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 160 "fact
or(Lqbis-  (   ((alphainf11-alphainf21)/(tinfty11-tinfty21)+(alpha011-
alpha021)/(t011-t021) )*(2*q^2*p-q^2*P1(q))  +(alpha011-alpha021)*q*h/
(t011-t021))  );" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 224 12 "We get that\n" }{TEXT 220 19 "L[q]= ((al
phainf11-" }{TEXT 221 10 "alphainf21" }{TEXT 222 111 ")/(tinfty11-tinf
ty21)+(alpha011-alpha021)/(t011-t021))*(2*q^2*p-q^2*P1(q)) -(alpha011-
alpha021)*q*h/(t011-t021)" }}{PARA 0 "" 0 "" {TEXT 213 9 "L[p] = ((" }
{TEXT 213 10 "alphainf11" }{TEXT 213 1 "-" }{TEXT 213 10 "alphainf21" 
}{TEXT 213 247 ")/(tinfty11-tinfty21)+(alpha011-alpha021)/(t011-t021) \+
)*(-2*q*p^2+diff(q^2*P1(q),q)*p-diff(q^2*P2(q),q)) -alphainf11*h - (al
pha011-alpha021)*h*p/(t011-t021)-h*(alpha011-alpha021)*tinfty11/(t011-
t021)-(t021*alpha011-t011*alpha021)*h/q^2/(t011-t021)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Hamiltonian:=  ((alphainf11-" }
{MPLTEXT 1 0 10 "alphainf21" }{MPLTEXT 1 0 232 ")/(tinfty11-tinfty21)+
(alpha011-alpha021)/(t011-t021))*(q^2*p^2-q^2*P1(q)*p+q^2*P2(q))+alpha
inf11*q*h+(alpha011-alpha021)*p*q*h/(t011-t021)+h*(alpha011-alpha021)*
tinfty11/(t011-t021)*q-(t021*alpha011-t011*alpha021)*h/q/(t011-t021);
\n" }{MPLTEXT 1 0 42 "factor(simplify(diff(Hamiltonian,p)-Lq));\n" }
{MPLTEXT 1 0 106 "factor(simplify(diff(Hamiltonian,q)+Lp+(tinfty11+tin
fty21)*(-alphainf11+alphainf21)/(tinfty21-tinfty11)/2*" }{MPLTEXT 1 0 
9 "Coherence" }{MPLTEXT 1 0 11 "Equation1)\n" }{MPLTEXT 1 0 78 "-1/2*(
tinfty11+tinfty21)*(alpha011-alpha021)/(t021-t011)*CoherenceEquation1)
;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 100 "Hamiltonianbis:= mu*(p^2-
P1(q)*p+h*p*(2/q) +tdP2(q) )-h*nu0*p-h*nuMinus1*q*p -h*c01/q- h*cinfty
1*q \n" }{MPLTEXT 1 0 56 "+nu0*(tinfty11*tinfty20+tinfty21*tinfty10+h*
tinfty11) :\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 48 "factor(simplify(
Lp-(-diff(Hamiltonianbis,q))));\n" }{MPLTEXT 1 0 38 "simplify(Lq-(diff
(Hamiltonianbis,p)));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,HamiltonianG
6\",,*&,&*&,&I+alphainf11GF$\"\"\"I+alphainf21GF$!\"\"F+,&I)tinfty21GF
$F-I)tinfty11GF$F+F-F+*&,&I)alpha011GF$F+I)alpha021GF$F-F+,&I%t011GF$F
+I%t021GF$F-F-F+F+,(*&I\"qGF$\"\"#I\"pGF$F;F+*(F:F;,**&,&F6F+F7F+F+F:!
\"#F+*&,&I%t010GF$F+I%t020GF$F+F+F:F-F+F0F-F/F-F+F<F+F-*&F:F;,,*(F6F+F
7F+F:!\"%F+*&,&*&FDF+F7F+F+*&F6F+FEF+F+F+F:!\"$F+*&I%P022GF$F+F:FAF+*&
,&*&FCF+,&F0F+F/F+F+#F-F;*&,&I)tinfty10GF$F+I)tinfty20GF$F-F+F.F+FUF+F
:F-F+*&F0F+F/F+F+F+F+F+F+*(F*F+F:F+I\"hGF$F+F+*,F2F+F<F+F:F+FfnF+F5F-F
+*,FfnF+F2F+F0F+F5F-F:F+F+**,&*&F6F+F4F+F-*&F7F+F3F+F+F+FfnF+F:F-F5F-F
-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 223 34 "Decompos
ition of the tangent space" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "tdp:=p-P1(q)/2:\n" }{MPLTEXT 1 0 161 "Ltdp:=Lp-1/2*diff(P1(q),q)*L
q-1/2*(h*alphainf11*diff(P1(q),tinfty11)+h*alphainf21*diff(P1(q),tinft
y21)+h*alpha011*diff(P1(q),t011)+h*alpha021*diff(P1(q),t021)):\n" }
{MPLTEXT 1 0 129 "Ltdpbis:=-((alphainf11-alphainf21)/(tinfty11-tinfty2
1)+(alpha011-alpha021)/(t011-t021))*(2*q*tdp^2+diff(q^2*(P2(q)-P1(q)^2
/4),q)\n" }{MPLTEXT 1 0 65 "+h/2*(tinfty11-tinfty21))-(alpha011-alpha0
21)*h*tdp/(t011-t021):\n" }{MPLTEXT 1 0 109 "factor(series(factor(simp
lify(Ltdp-Ltdpbis+(tinfty11+tinfty21)*(alphainf11-alphainf21)/2/(tinft
y11-tinfty21)*" }{MPLTEXT 1 0 9 "Coherence" }{MPLTEXT 1 0 11 "Equation
1)\n" }{MPLTEXT 1 0 81 "+(tinfty11+tinfty21)*(alpha011-alpha021)/2/(t0
11-t021)*CoherenceEquation1),h=0));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 
1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "
Lpfunction:=unapply(simplify(Lp),alphainf11," }{MPLTEXT 1 0 10 "alphai
nf21" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 9 "alpha011," }{MPLTEXT 1 0 8 
"alpha021" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 36 "Ltdpfunction:=unap
ply(simplify(Ltdp)" }{MPLTEXT 1 0 12 ",alphainf11," }{MPLTEXT 1 0 28 "
alphainf21,alpha011,alpha021" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 32 
"Lqfunction:=unapply(simplify(Lq)" }{MPLTEXT 1 0 12 ",alphainf11," }
{MPLTEXT 1 0 28 "alphainf21,alpha011,alpha021" }{MPLTEXT 1 0 3 "):\n" 
}{MPLTEXT 1 0 25 "cinfty1function:=unapply(" }{MPLTEXT 1 0 7 "cinfty1"
 }{MPLTEXT 1 0 12 ",alphainf11," }{MPLTEXT 1 0 28 "alphainf21,alpha011
,alpha021" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 21 "c01function:=unapp
ly(" }{MPLTEXT 1 0 3 "c01" }{MPLTEXT 1 0 12 ",alphainf11," }{MPLTEXT 
1 0 28 "alphainf21,alpha011,alpha021" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT
 1 0 34 "nuMinus1function:=unapply(nuMinus1" }{MPLTEXT 1 0 12 ",alphai
nf11," }{MPLTEXT 1 0 28 "alphainf21,alpha011,alpha021" }{MPLTEXT 1 0 
2 "):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 
"factor(Ltdpfunction(1,1,0,0));\n" }{MPLTEXT 1 0 29 "factor(Lqfunction
(1,1,0,0));\n" }{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 15 "cinfty1func
tion" }{MPLTEXT 1 0 12 "(1,1,0,0));\n" }{MPLTEXT 1 0 7 "factor(" }
{MPLTEXT 1 0 11 "c01function" }{MPLTEXT 1 0 12 "(1,1,0,0));\n" }
{MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 16 "nuMinus1function" }{MPLTEXT
 1 0 11 "(1,1,0,0));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 
20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 
"" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
31 "factor(Ltdpfunction(0,0,1,1));\n" }{MPLTEXT 1 0 18 "factor(Lqfunct
ion(" }{MPLTEXT 1 0 7 "0,0,1,1" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 
23 "factor(cinfty1function(" }{MPLTEXT 1 0 7 "0,0,1,1" }{MPLTEXT 1 0 
4 "));\n" }{MPLTEXT 1 0 19 "factor(c01function(" }{MPLTEXT 1 0 7 "0,0,
1,1" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 24 "factor(nuMinus1function
(" }{MPLTEXT 1 0 7 "0,0,1,1" }{MPLTEXT 1 0 3 "));" }{MPLTEXT 1 0 0 "" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 
"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0
 {PARA 3 "" 0 "" {TEXT 223 112 "Expression of the Lax matrices in the \+
geometric gauge after the symplectic reduction and the Painlev\351 3 e
quation" }{TEXT 223 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 224 109 "Expre
ssion of the geometric Lax matrix in the gauge without apparent singul
arities after symplectic reduction" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "tinfty21:=-tinfty11:\n" }{MPLTEXT 1 0 21 "tinfty20:=-
tinfty10:\n" }{MPLTEXT 1 0 13 "t021:=-t011:\n" }{MPLTEXT 1 0 13 "t020:
=-t010:\n" }{MPLTEXT 1 0 13 "tinfty11:=1:\n" }{MPLTEXT 1 0 11 "t011:=t
/2:\n" }{MPLTEXT 1 0 9 "H:=Hsol:\n" }{MPLTEXT 1 0 6 "C:=0:\n" }
{MPLTEXT 1 0 11 "q:=checkq:\n" }{MPLTEXT 1 0 11 "p:=checkp:\n" }
{MPLTEXT 1 0 11 "dcheckqdt:=" }{MPLTEXT 1 0 4 "Lq/h" }{MPLTEXT 1 0 2 "
:\n" }{MPLTEXT 1 0 11 "dcheckpdt:=" }{MPLTEXT 1 0 4 "Lp/h" }{MPLTEXT 
1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 15 "
alphainf11:=0:\n" }{MPLTEXT 1 0 15 "alphainf21:=0:\n" }{MPLTEXT 1 0 
15 "alpha011:=1/2:\n" }{MPLTEXT 1 0 8 "alpha021" }{MPLTEXT 1 0 7 ":=-1
/2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "G1:=Matrix(2,2,0):\n
" }{MPLTEXT 1 0 12 "G1[1,1]:=1:\n" }{MPLTEXT 1 0 12 "G1[2,2]:=1:\n" }
{MPLTEXT 1 0 12 "G1[1,2]:=0:\n" }{MPLTEXT 1 0 23 "G1[2,1]:=g1*lambda+g
0:\n" }{MPLTEXT 1 0 14 "g1:=tinfty11;\n" }{MPLTEXT 1 0 21 "g0:=checkq+
tinfty10;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "dG1dlambda:=Matrix
(2,2,0):\n" }{MPLTEXT 1 0 89 "for i from 1 to 2 do for j from 1 to 2 d
o dG1dlambda[i,j]:=diff(G1[i,j],lambda): od: od:\n" }{MPLTEXT 1 0 1 "
\n" }{MPLTEXT 1 0 24 "dG1dtau:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 93 "for
 i from 1 to 2 do for j from 1 to 2 do dG1dtau[i,j]:=diff(G1[i,j],t)+d
iff(G1[i,j],checkq)*" }{MPLTEXT 1 0 9 "dcheckqdt" }{MPLTEXT 1 0 13 "+d
iff(G1[i,j]" }{MPLTEXT 1 0 6 ",check" }{MPLTEXT 1 0 3 "p)*" }{MPLTEXT 
1 0 9 "dcheckpdt" }{MPLTEXT 1 0 11 " : od: od:\n" }{MPLTEXT 1 0 1 "\n"
 }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 85 "tdL:=simplify(Multiply(Multipl
y(G1,checkL),G1^(-1))+h*Multiply(dG1dlambda,G1^(-1))):\n" }{MPLTEXT 1 
0 82 "tdA:=simplify(Multiply(Multiply(G1,checkA),G1^(-1))+h*Multiply(d
G1dtau,G1^(-1))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 "checkL:=si
mplify(checkL);\n" }{MPLTEXT 1 0 25 "checkA:=simplify(checkA);" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 "tdL:=" }{MPLTEXT 1 0 15 "simplify
(tdL);\n" }{MPLTEXT 1 0 5 "tdA:=" }{MPLTEXT 1 0 14 "simplify(tdA);" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#g1G6\"\"\"\"" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I#g0G6\",&I'checkqGF$\"\"\"I)tinfty10G
F$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,
TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdG
F$6(-I&mfracGF$6)-F,6%-I#miGF$6&Q'checkpF'/%'italicGQ%trueF'/%+foregro
undGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6-Q1&InvisibleTimes;F'
/FGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFQ/%)stretchyGFQ/%*symmetr
icGFQ/%(largeopGFQ/%.movablelimitsGFQ/%'accentGFQ/%'lspaceGQ&0.0emF'/%
'rspaceGFjn-I%msupGF$6%-F=6&Q'checkqF'F@FCFF-I#mnGF$6%Q\"2F'FCFM/%1sup
erscriptshiftGQ\"0F'-F,6#-F^o6%-F=6&Q)&lambda;F'/FAFQFCFMFcoFgo/%.line
thicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFgp/%)bevelledGFQF
C/%)rowalignGQ!F'/%,columnalignGF^q/%+groupalignGF^q/%(rowspanGFdp/%+c
olumnspanGFdp-F56(-F86)-F,6#-F,6%F^p-FJ6.Q(&minus;F'FCFMFOFRFTFVFXFZFf
n/FinQ,0.2222222emF'/F\\oFcrF`oFjoFbpFepFhpFjpFCF\\qF_qFaqFcqFeqF\\qF_
qFaq-F26'-F56(-F,6%-F86)-Fdo6%FdpFCFM-Fdo6%Q\"4F'FCFMFbpFepFhpFjpFCFI-
F86)-F,6#-F,6--F,6'F_sFI-F^o6%F<FcoFgoFI-F^o6%F`o-Fdo6%Q\"5F'FCFMFgo-F
J6.Q\"+F'FCFMFOFRFTFVFXFZFfnFbrFdr-F,6)F_sFIFjsFI-F^o6%F`oF_sFgoFIF^pF
at-F,6'F_sFIF\\pFI-F^o6%F`o-Fdo6%Q\"3F'FCFMFgoF_r-F,6)F_sFIF\\pFI-F#6%
-F,6'-F=6&Q\"hF'F@FCFFF_rF^pF_r-F,6%FcoFI-F=6&Q)tinfty10F'F@FCFFFCFMFI
F]oFat-F,6%-F#6%-F,6&-FJ6.Q*&uminus0;F'FCFMFOFRFTFVFXFZFfnFbrFdr-F,6)F
_sFIF^pFI-F=6&Q\"tF'F@FCFFFI-F=6&Q%t010F'F@FCFFF_r-F,6#-F^o6%FhvFcoFgo
FCFMFIF`oF_r-F,6%F^pFIF`w-F,6%F\\pFIF]oFbpFepFhpFjpFCF\\qF_qFaqFcqFeq-
F56(-F,6$FcvF7F\\qF_qFaqFcqFeqF\\qF_qFaq/%&alignGQ%axisF'/F]qQ)baselin
eF'/F`qFgp/FbqQ'|frleft|hrF'/%/alignmentscopeGFB/%,columnwidthGQ%autoF
'/%&widthGFfx/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowli
nesGQ%noneF'/%,columnlinesGFay/%&frameGFay/%-framespacingGQ,0.4em~0.5e
xF'/%*equalrowsGFQ/%-equalcolumnsGFQ/%-displaystyleGFQ/%%sideGQ&rightF
'/%0minlabelspacingGF^yFCFM/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sys
libGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6$-I#moGF$6
.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fen
ceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*symmetricGFE/%(largeopGFE/
%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.2222222emF'/%'rspaceGFT-I&
mfracGF$6)-F,6%-I#miGF$6&Q'checkpF'/%'italicGQ%trueF'F=/FAQ'italicF'-F
:6-Q1&InvisibleTimes;F'F@FCFFFHFJFLFNFP/FSQ&0.0emF'/FVFco-I%msupGF$6%-
Fgn6&Q'checkqF'FjnF=F]o-I#mnGF$6%Q\"2F'F=F@/%1superscriptshiftGQ\"0F'-
F,6%-Fgn6&Q)&lambda;F'/F[oFEF=F@F_o-Fgn6&Q\"tF'FjnF=F]o/%.linethicknes
sGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGF`q/%)bevelledGFEF=/%)rowa
lignGQ!F'/%,columnalignGFgq/%+groupalignGFgq/%(rowspanGF]q/%+columnspa
nGF]q-F56(-FX6)-F,6#Fho-F,6%FhpF_oFdpF[qF^qFaqFcqF=FeqFhqFjqF\\rF^rFeq
FhqFjq-F26'-F56(-F,6%-FX6)-F\\p6%F]qF=F@-F\\p6%Q\"4F'F=F@F[qF^qFaqFcqF
=F_o-FX6)-F,6#-F,6*F9-F,6'FbsF_o-Ffo6%FfnF[pF_pF_o-Ffo6%FhoFbsF_p-F:6.
Q\"+F'F=F@FCFFFHFJFLFNFPFRFU-F,6'-F\\p6%Q\"8F'F=F@F_oFdpF_o-Ffo6%Fho-F
\\p6%Q\"3F'F=F@F_p-F:6.Q(&minus;F'F=F@FCFFFHFJFLFNFPFRFU-F,6)FbsF_oFdp
F_o-F#6%-F,6'-Fgn6&Q\"hF'FjnF=F]oF^uFdpF^u-F,6%F[pF_o-Fgn6&Q)tinfty10F
'FjnF=F]oF=F@F_oFeoFat-F,6#-Ffo6%FhpF[pF_p-F,6'FdpF_oFhoF_oFhpF[qF^qFa
qFcqF=FeqFhqFjqF\\rF^r-F56(FWFeqFhqFjqF\\rF^rFeqFhqFjq/%&alignGQ%axisF
'/FfqQ)baselineF'/FiqF`q/F[rQ'|frleft|hrF'/%/alignmentscopeGF\\o/%,col
umnwidthGQ%autoF'/%&widthGFcw/%+rowspacingGQ&1.0exF'/%.columnspacingGQ
&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF^x/%&frameGF^x/%-framespac
ingGQ,0.4em~0.5exF'/%*equalrowsGFE/%-equalcolumnsGFE/%-displaystyleGFE
/%%sideGQ&rightF'/%0minlabelspacingGF[xF=F@/%%openGQ\"[F'/%&closeGQ\"]
F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Ty
pesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$
6(-I&mfracGF$6)-F,6#-F,6'-F,6%-F#6%-F,6%-I#miGF$6&Q'checkpF'/%'italicG
Q%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6.Q
\"+F'FK/FOQ'normalF'/%&fenceGQ&falseF'/%*separatorGFY/%)stretchyGFY/%*
symmetricGFY/%(largeopGFY/%.movablelimitsGFY/%'accentGFY/%'lspaceGQ,0.
2222222emF'/%'rspaceGFbo-I#mnGF$6%Q\"1F'FKFUFKFU-FR6-Q1&InvisibleTimes
;F'FUFWFZFfnFhnFjnF\\oF^o/FaoQ&0.0emF'/FdoF]p-I%msupGF$6%-FE6&Q'checkq
F'FHFKFN-Ffo6%Q\"2F'FKFU/%1superscriptshiftGQ\"0F'FQ-F,6%-FE6&Q)tinfty
10F'FHFKFNFioFbp-FR6.Q(&minus;F'FKFUFWFZFfnFhnFjnF\\oF^oF`oFco-F,6%-FE
6&Q)&lambda;F'/FIFYFKFUFio-F#6%-F,6%FeqFQF]qFKFU-F,6#-F`p6%FeqFepFhp/%
.linethicknessGFho/%+denomalignGQ'centerF'/%)numalignGFer/%)bevelledGF
YFK/%)rowalignGQ!F'/%,columnalignGF\\s/%+groupalignGF\\s/%(rowspanGFho
/%+columnspanGFho-F56(-F86)-F,6#-F,6%FeqF`qFbpF]rFarFcrFfrFhrFKFjrF]sF
_sFasFcsFjrF]sF_s-F26'-F56(-F,6%-F86)Feo-Ffo6%Q\"4F'FKFUFarFcrFfrFhrFK
Fio-F86)-F,6#-F,6--F,6'FetFio-F`p6%F@FepFhpFio-F`p6%Fbp-Ffo6%Q\"5F'FKF
UFhpFQ-F,6)FetFio-F#6%-F,6%-F,6%F@FioFeqFQ-F,6%FepFioF]qFKFUFioF@Fio-F
`p6%FbpFetFhpFQ-F,6'FetFio-F`p6%F]qFepFhpFio-F`p6%Fbp-Ffo6%Q\"3F'FKFUF
hpF`q-F,6)FetFioFeqFioFevFioF_pFQ-F,6%-F#6%-F,6&-FR6.Q*&uminus0;F'FKFU
FWFZFfnFhnFjnF\\oF^oF`oFco-F,6)FetFioFeqFio-FE6&Q\"tF'FHFKFNFio-FE6&Q%
t010F'FHFKFNF`q-F,6#-F`p6%FiwFepFhpFKFUFioFbpF`q-F,6%FeqFioFax-F,6%F_r
FioF_pFarFcrFfrFhrFKFjrF]sF_sFasFcs-F56(-F86)-F,6#-F,6'-F,6%-F#6%-F,6&
FdwFDF`qFeoFKFUFioF_pF`qF[qFQFcqF]rFarFcrFfrFhrFKFjrF]sF_sFasFcsFjrF]s
F_s/%&alignGQ%axisF'/F[sQ)baselineF'/F^sFer/F`sQ'|frleft|hrF'/%/alignm
entscopeGFJ/%,columnwidthGQ%autoF'/%&widthGFaz/%+rowspacingGQ&1.0exF'/
%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF\\[l/%&fr
ameGF\\[l/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFY/%-equalcolumns
GFY/%-displaystyleGFY/%%sideGQ&rightF'/%0minlabelspacingGFizFKFU/%%ope
nGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#
/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$6
6-I$mtrGF$6'-I$mtdGF$6(-F,6$-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,
0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFE/%)str
etchyGFE/%*symmetricGFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE/%'
lspaceGQ,0.2222222emF'/%'rspaceGFT-I&mfracGF$6)-F,6%-F#6%-F,6)-F,6%-I#
miGF$6&Q'checkpF'/%'italicGQ%trueF'F=/FAQ'italicF'-F:6-Q1&InvisibleTim
es;F'F@FCFFFHFJFLFNFP/FSQ&0.0emF'/FVFio-F]o6&Q'checkqF'F`oF=Fco-F:6.Q
\"+F'F=F@FCFFFHFJFLFNFPFRFUF[pF^p-F]o6&Q)&lambda;F'/FaoFEF=F@F^p-F]o6&
Q)tinfty10F'F`oF=FcoF=F@FeoF[p-F,6%FapFeo-F]o6&Q\"tF'F`oF=Fco/%.lineth
icknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFbq/%)bevelledGFEF=/
%)rowalignGQ!F'/%,columnalignGFiq/%+groupalignGFiq/%(rowspanGF_q/%+col
umnspanGF_q-F56(-FX6)-F,6#F[p-F,6%FjpFeoFapF]qF`qFcqFeqF=FgqFjqF\\rF^r
F`rFgqFjqF\\r-F26'-F56(-F,6%-FX6)-I#mnGF$6%F_qF=F@-Fcs6%Q\"4F'F=F@F]qF
`qFcqFeqF=Feo-FX6)-F,6%-F#6%-F,6'-F,6%-F#6%-F,6%-F,6%-Fcs6%Q\"2F'F=F@F
eoF\\oF^pFhtF=F@Feo-I%msupGF$6%F[pFht/%1superscriptshiftGQ\"0F'F^p-F,6
'FhtFeoFepFeoF[pF^pFjpF=F@Feo-F#6%-F,6'-F,6%-F#6%-F,6&F9Fft-F:6.Q(&min
us;F'F=F@FCFFFHFJFLFNFPFRFUFhtF=F@FeoF[uF]vFauF^pFjpF=F@-F,6'FapFeoF[p
FeoFjpF]qF`qFcqFeqF=FgqFjqF\\rF^rF`r-F56(FWFgqFjqF\\rF^rF`rFgqFjqF\\r/
%&alignGQ%axisF'/FhqQ)baselineF'/F[rFbq/F]rQ'|frleft|hrF'/%/alignments
copeGFbo/%,columnwidthGQ%autoF'/%&widthGF`w/%+rowspacingGQ&1.0exF'/%.c
olumnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF[x/%&frameGF
[x/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFE/%-equalcolumnsGFE/%-d
isplaystyleGFE/%%sideGQ&rightF'/%0minlabelspacingGFhwF=F@/%%openGQ\"[F
'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "series(
tdL[1,1],lambda=infinity);\n" }{MPLTEXT 1 0 34 "series(tdL[1,2],lambda
=infinity);\n" }{MPLTEXT 1 0 7 "residue" }{MPLTEXT 1 0 34 "(tdL[2,1]/l
ambda,lambda=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(!\"\"\"\"\"
*&I)tinfty10G6\"F$I'lambdaGF'F#F#*&,&*&,&I'checkpGF'F$F$F$F$I'checkqGF
'\"\"#F$*&F&F$F.F$F$F$F(!\"#F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$I'
lambdaG6\"!\"\"\"\"\"*&I'checkqGF%F'F$!\"#F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 213 30 "Reduced Ha
miltonian evolutions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sim
plify(series(" }{MPLTEXT 1 0 9 "dcheckqdt" }{MPLTEXT 1 0 12 ",checkp=0
));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "simplify(series(" }
{MPLTEXT 1 0 9 "dcheckpdt" }{MPLTEXT 1 0 12 ",checkp=0));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "+'I'checkpG6\"*&I'checkqGF$\"\"\"I\"tGF$!\"\"\"\"
!,$*(F&\"\"#F(F)I\"hGF$F)F-F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "+)I'che
ckpG6\",$**,**$I'checkqGF$\"\"%F**&,&I\"hGF$!\"#I)tinfty10GF$F*\"\"\"F
)\"\"$F0*(F)F0I%t010GF$F0I\"tGF$F0F.*$F4\"\"#!\"\"F0F)!\"$F4F7F-F7#F0F
6\"\"!,$*$F4F7F7F0,$*(F)F0F4F7F-F7F.F6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2
 33 1 1 }