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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 210 168 "In this Maple file, we \+
conmpute the Lax pair in the oper gauge in the Painlev\351 1 case. We \+
also check that the formulas proposed in the article are correct in th
is case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }
{MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 12 "tinfty10:=0
;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "Pinfty11:=-(2*tinfty14)/2;\n
" }{MPLTEXT 1 0 26 "Pinfty01:=-(2*tinfty12)/2;" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 31 "Pinfty32 := -(1/4)*tinfty15^2;\n" }{MPLTEXT 1 0 55 "P
infty22 := -(1/2)*tinfty15*tinfty13+(1/4)*tinfty14^2;\n" }{MPLTEXT 1 
0 79 "Pinfty12 := -(1/2)*tinfty15*tinfty11+(1/2)*tinfty14*tinfty12-(1/
4)*tinfty13^2;\n" }{MPLTEXT 1 0 29 "P1:=x-> Pinfty01+Pinfty11*x;\n" }
{MPLTEXT 1 0 55 "P2:=x-> Pinfty02+Pinfty12*x+Pinfty22*x^2+Pinfty32*x^3
;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "Unknownn:=-" }{MPLTEXT 1 
0 9 "Unknown:\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 12 "Unknownn:=-L" }
{MPLTEXT 1 0 8 "Unknown:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "Unkno
wnn2:=" }{MPLTEXT 1 0 10 "Unknown2:\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 
1 0 22 "Unknownn2:=LUnknown2:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 
"L" }{MPLTEXT 1 0 22 "tinfty25:=-Ltinfty15;\n" }{MPLTEXT 1 0 1 "L" }
{MPLTEXT 1 0 22 "tinfty23:=-Ltinfty13;\n" }{MPLTEXT 1 0 1 "L" }
{MPLTEXT 1 0 22 "tinfty21:=-Ltinfty11;\n" }{MPLTEXT 1 0 1 "L" }
{MPLTEXT 1 0 22 "tinfty20:=-Ltinfty10;\n" }{MPLTEXT 1 0 1 "L" }
{MPLTEXT 1 0 21 "tinfty24:=Ltinfty14;\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT
 1 0 21 "tinfty22:=Ltinfty12;\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 12 
"tinfty10:=0:" }}{PARA 0 "" 0 "" {TEXT 227 0 "" }{MPLTEXT 1 0 1 "\n" }
}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tinfty10G6\"\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I)Pinfty11G6\",$I)tinfty14GF$!\"\"" }}{PARA 11 "" 1 ""
 {XPPMATH 20 ">I)Pinfty01G6\",$I)tinfty12GF$!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)Pinfty32G6\",$*$I)tinfty15GF$\"\"##!\"\"\"\"%" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6\",&*&I)tinfty15GF$\"\"\"I)
tinfty13GF$F(#!\"\"\"\"#*$I)tinfty14GF$F,#F(\"\"%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I)Pinfty12G6\",(*&I)tinfty15GF$\"\"\"I)tinfty11GF$F(#!\"
\"\"\"#*&I)tinfty14GF$F(I)tinfty12GF$F(#F(F,*$I)tinfty13GF$F,#F+\"\"%"
 }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F$6$I)operatorGF$
I&arrowGF$F$,&I)Pinfty01GF$\"\"\"*&I)Pinfty11GF$F-9$F-F-F$F$F$" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I\"xGF$F$6$I)operatorGF$I&a
rrowGF$F$,*I)Pinfty02GF$\"\"\"*&I)Pinfty12GF$F-9$F-F-*&I)Pinfty22GF$F-
F0\"\"#F-*&I)Pinfty32GF$F-F0\"\"$F-F$F$F$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I*Ltinfty25G6\",$I*Ltinfty15GF$!\"\"" }}{PARA 11 "" 1 ""
 {XPPMATH 20 ">I*Ltinfty23G6\",$I*Ltinfty13GF$!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 ">I*Ltinfty21G6\",$I*Ltinfty11GF$!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I*Ltinfty20G6\",$I*Ltinfty10GF$!\"\"" }}{PARA 11 "" 1
 "" {XPPMATH 20 ">I*Ltinfty24G6\"I*Ltinfty14GF$" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I*Ltinfty22G6\"I*Ltinfty12GF$" }}}{EXCHG {PARA 210 "" 0 
"" {TEXT 228 17 "Study at infinity" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "espilon:=1:\n" }{MPLTEXT 1 0 210 "logPsi1Infty:=-1/5*
tinfty15/h*lambda^(5/2)-1/4*tinfty14/h*lambda^2-1/3*tinfty13/h*lambda^
(3/2)-1/2*tinfty12/h*lambda^1-tinfty11/h*lambda^(1/2)+1/4*epsilon*ln(l
ambda)+A10+ Unknown/lambda^(1/2)+ Unknown2/lambda;\n" }{MPLTEXT 1 0 
145 "logPsi2Infty:=1/5*tinfty15/h*lambda^(5/2)-1/4*tinfty14/h*lambda^2
+1/3*tinfty13/h*lambda^(3/2)-1/2*tinfty12/h*lambda^1+tinfty11/h*lambda
^(1/2)+1/4" }{MPLTEXT 1 0 8 "*epsilon" }{MPLTEXT 1 0 56 "*ln(lambda)+A
20- Unknown/lambda^(1/2)+ Unknown2/lambda;\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 202 "GrosLlogpsi1Infty:=-1/5*Ltinfty
15/h*lambda^(5/2)-1/4*Ltinfty14/h*lambda^2-1/3*Ltinfty13/h*lambda^(3/2
)-1/2*Ltinfty12/h*lambda^1-Ltinfty11/h*lambda^(1/2)+LA10+ LUnknown/lam
bda^(1/2)+ LUnknown2/lambda  ;\n" }{MPLTEXT 1 0 199 "GrosLlogpsi2Infty
:=1/5*Ltinfty15/h*lambda^(5/2)-1/4*Ltinfty14/h*lambda^2+1/3*Ltinfty13/
h*lambda^(3/2)-1/2*Ltinfty12/h*lambda^1+Ltinfty11/h*lambda^(1/2)+LA20-
 LUnknown/lambda^(1/2)+ LUnknown2/lambda;\n" }{MPLTEXT 1 0 1 "\n" }
{MPLTEXT 1 0 155 "GrosLpsi1Infty := exp(-1/5*tinfty15/h*lambda^(5/2)-1
/4*tinfty14/h*lambda^2-1/3*tinfty13/h*lambda^(3/2)-1/2*tinfty12/h*lamb
da^1-tinfty11/h*lambda^(1/2)+1/4*" }{MPLTEXT 1 0 8 "epsilon*" }
{MPLTEXT 1 0 238 "ln(lambda)+A10+ Unknown/lambda^(1/2)+ Unknown2/lambd
a)*(-1/5*Ltinfty15/h*lambda^(5/2)-1/4*Ltinfty14/h*lambda^2-1/3*Ltinfty
13/h*lambda^(3/2)-1/2*Ltinfty12/h*lambda^1-Ltinfty11/h*lambda^(1/2)+LA
10+ LUnknown/lambda^(1/2)+ LUnknown2/lambda);\n" }{MPLTEXT 1 0 153 "Gr
osLpsi2Infty := exp(1/5*tinfty15/h*lambda^(5/2)-1/4*tinfty14/h*lambda^
2+1/3*tinfty13/h*lambda^(3/2)-1/2*tinfty12/h*lambda^1+tinfty11/h*lambd
a^(1/2)+1/4" }{MPLTEXT 1 0 8 "*epsilon" }{MPLTEXT 1 0 237 "*ln(lambda)
+A20- Unknown/lambda^(1/2)+ Unknown2/lambda)*(1/5*Ltinfty15/h*lambda^(
5/2)-1/4*Ltinfty14/h*lambda^2+1/3*Ltinfty13/h*lambda^(3/2)-1/2*Ltinfty
12/h*lambda^1+Ltinfty11/h*lambda^(1/2)+LA20- LUnknown/lambda^(1/2)+ LU
nknown2/lambda);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "psi1Infty:=ex
p(logPsi1Infty);\n" }{MPLTEXT 1 0 30 "psi2Infty:=exp(logPsi2Infty);\n"
 }{MPLTEXT 1 0 43 "dpsi1dlambdaInfty:=diff(psi1Infty,lambda):\n" }
{MPLTEXT 1 0 43 "dpsi2dlambdaInfty:=diff(psi2Infty,lambda):\n" }
{MPLTEXT 1 0 47 "d2psi1dlambda2Infty:=diff(psi1Infty,lambda$2):\n" }
{MPLTEXT 1 0 47 "d2psi2dlambda2Infty:=diff(psi2Infty,lambda$2):\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "WronskianLa
mbdaInfty:=h*factor(psi1Infty*dpsi2dlambdaInfty-psi2Infty*dpsi1dlambda
Infty):\n" }{MPLTEXT 1 0 132 "WronskianLambdabisInfty:=h*simplify(fact
or( (diff(logPsi2Infty,lambda)-diff(logPsi1Infty,lambda))*exp(logPsi1I
nfty+logPsi2Infty))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 116 "Wrons
kianTildeLambdaInfty:=h^3*factor(dpsi2dlambdaInfty*d2psi1dlambda2Infty
-dpsi1dlambdaInfty*d2psi2dlambda2Infty):\n" }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I-logPsi1InftyG6\",4*(I)tinfty15GF$\"\"\"I\"hGF$!\"\"I'l
ambdaGF$#\"\"&\"\"##F*F-*(I)tinfty14GF$F(F)F*F+F.#F*\"\"%*(I)tinfty13G
F$F(F)F*F+#\"\"$F.#F*F7*(I)tinfty12GF$F(F)F*F+F(#F*F.*(I)tinfty11GF$F(
F)F*F+#F(F.F**&I(epsilonGF$F(-I#lnG6$%*protectedGI(_syslibGF$6#F+F(#F(
F3I$A10GF$F(*&I(UnknownGF$F(F+F;F(*&I)Unknown2GF$F(F+F*F(" }}{PARA 11 
"" 1 "" {XPPMATH 20 ">I-logPsi2InftyG6\",4*(I)tinfty15GF$\"\"\"I\"hGF$
!\"\"I'lambdaGF$#\"\"&\"\"##F(F-*(I)tinfty14GF$F(F)F*F+F.#F*\"\"%*(I)t
infty13GF$F(F)F*F+#\"\"$F.#F(F7*(I)tinfty12GF$F(F)F*F+F(#F*F.*(I)tinft
y11GF$F(F)F*F+#F(F.F(*&I(epsilonGF$F(-I#lnG6$%*protectedGI(_syslibGF$6
#F+F(#F(F3I$A20GF$F(*&I(UnknownGF$F(F+F;F**&I)Unknown2GF$F(F+F*F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I2GrosLlogpsi1InftyG6\",2*(I*Ltinfty15G
F$\"\"\"I\"hGF$!\"\"I'lambdaGF$#\"\"&\"\"##F*F-*(I*Ltinfty14GF$F(F)F*F
+F.#F*\"\"%*(I*Ltinfty13GF$F(F)F*F+#\"\"$F.#F*F7*(I*Ltinfty12GF$F(F)F*
F+F(#F*F.*(I*Ltinfty11GF$F(F)F*F+#F(F.F*I%LA10GF$F(*&I)LUnknownGF$F(F+
F;F(*&I*LUnknown2GF$F(F+F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I2GrosL
logpsi2InftyG6\",2*(I*Ltinfty15GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$#\"\"&
\"\"##F(F-*(I*Ltinfty14GF$F(F)F*F+F.#F*\"\"%*(I*Ltinfty13GF$F(F)F*F+#
\"\"$F.#F(F7*(I*Ltinfty12GF$F(F)F*F+F(#F*F.*(I*Ltinfty11GF$F(F)F*F+#F(
F.F(I%LA20GF$F(*&I)LUnknownGF$F(F+F;F**&I*LUnknown2GF$F(F+F*F(" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I/GrosLpsi1InftyG6\"*&-I$expG6$%*protec
tedGI(_syslibGF$6#,4*(I)tinfty15GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$#\"\"&
\"\"##F1F4*(I)tinfty14GF$F/F0F1F2F5#F1\"\"%*(I)tinfty13GF$F/F0F1F2#\"
\"$F5#F1F>*(I)tinfty12GF$F/F0F1F2F/#F1F5*(I)tinfty11GF$F/F0F1F2#F/F5F1
*&I(epsilonGF$F/-I#lnGF(6#F2F/#F/F:I$A10GF$F/*&I(UnknownGF$F/F2FBF/*&I
)Unknown2GF$F/F2F1F/F/,2*(I*Ltinfty15GF$F/F0F1F2F3F6*(I*Ltinfty14GF$F/
F0F1F2F5F9*(I*Ltinfty13GF$F/F0F1F2F=F?*(I*Ltinfty12GF$F/F0F1F2F/FB*(I*
Ltinfty11GF$F/F0F1F2FEF1I%LA10GF$F/*&I)LUnknownGF$F/F2FBF/*&I*LUnknown
2GF$F/F2F1F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/GrosLpsi2InftyG6\"*
&-I$expG6$%*protectedGI(_syslibGF$6#,4*(I)tinfty15GF$\"\"\"I\"hGF$!\"
\"I'lambdaGF$#\"\"&\"\"##F/F4*(I)tinfty14GF$F/F0F1F2F5#F1\"\"%*(I)tinf
ty13GF$F/F0F1F2#\"\"$F5#F/F>*(I)tinfty12GF$F/F0F1F2F/#F1F5*(I)tinfty11
GF$F/F0F1F2#F/F5F/*&I(epsilonGF$F/-I#lnGF(6#F2F/#F/F:I$A20GF$F/*&I(Unk
nownGF$F/F2FBF1*&I)Unknown2GF$F/F2F1F/F/,2*(I*Ltinfty15GF$F/F0F1F2F3F6
*(I*Ltinfty14GF$F/F0F1F2F5F9*(I*Ltinfty13GF$F/F0F1F2F=F?*(I*Ltinfty12G
F$F/F0F1F2F/FB*(I*Ltinfty11GF$F/F0F1F2FEF/I%LA20GF$F/*&I)LUnknownGF$F/
F2FBF1*&I*LUnknown2GF$F/F2F1F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*p
si1InftyG6\"-I$expG6$%*protectedGI(_syslibGF$6#,4*(I)tinfty15GF$\"\"\"
I\"hGF$!\"\"I'lambdaGF$#\"\"&\"\"##F0F3*(I)tinfty14GF$F.F/F0F1F4#F0\"
\"%*(I)tinfty13GF$F.F/F0F1#\"\"$F4#F0F=*(I)tinfty12GF$F.F/F0F1F.#F0F4*
(I)tinfty11GF$F.F/F0F1#F.F4F0*&I(epsilonGF$F.-I#lnGF'6#F1F.#F.F9I$A10G
F$F.*&I(UnknownGF$F.F1FAF.*&I)Unknown2GF$F.F1F0F." }}{PARA 11 "" 1 "" 
{XPPMATH 20 ">I*psi2InftyG6\"-I$expG6$%*protectedGI(_syslibGF$6#,4*(I)
tinfty15GF$\"\"\"I\"hGF$!\"\"I'lambdaGF$#\"\"&\"\"##F.F3*(I)tinfty14GF
$F.F/F0F1F4#F0\"\"%*(I)tinfty13GF$F.F/F0F1#\"\"$F4#F.F=*(I)tinfty12GF$
F.F/F0F1F.#F0F4*(I)tinfty11GF$F.F/F0F1#F.F4F.*&I(epsilonGF$F.-I#lnGF'6
#F1F.#F.F9I$A20GF$F.*&I(UnknownGF$F.F1FAF0*&I)Unknown2GF$F.F1F0F." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "L21Infty:=simplify(Wronskian
TildeLambdaInfty/WronskianLambdabisInfty):\n" }{MPLTEXT 1 0 75 "L21Inf
tyOrdrelambda5:=factor(-residue(L21Infty/lambda^6,lambda=infinity));\n
" }{MPLTEXT 1 0 75 "L21InftyOrdrelambda4:=factor(-residue(L21Infty/lam
bda^5,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L21InftyOrdrelambda3:=fa
ctor(-residue(L21Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 
75 "L21InftyOrdrelambda2:=factor(-residue(L21Infty/lambda^3,lambda=inf
inity));\n" }{MPLTEXT 1 0 75 "L21InftyOrdrelambda1:=factor(-residue(L2
1Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 74 "L21InftyOrdrel
ambda0:=factor(-residue(L21Infty/lambda^1,lambda=infinity));" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda5G6\"\"\"!" }}{PARA 11 "
" 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda4G6\"\"\"!" }}{PARA 11 "" 1 
"" {XPPMATH 20 ">I5L21InftyOrdrelambda3G6\",$*$I)tinfty15GF$\"\"##\"\"
\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda2G6\",&
*&I)tinfty15GF$\"\"\"I)tinfty13GF$F(#F(\"\"#*$I)tinfty14GF$F+#!\"\"\"
\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L21InftyOrdrelambda1G6\",(*&I)
tinfty15GF$\"\"\"I)tinfty11GF$F(#F(\"\"#*&I)tinfty14GF$F(I)tinfty12GF$
F(#!\"\"F+*$I)tinfty13GF$F+#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">
I5L21InftyOrdrelambda0G6\",,*(I(UnknownGF$\"\"\"I\"hGF$F(I)tinfty15GF$
F(#F(\"\"#*(I(epsilonGF$F(F)F(I)tinfty14GF$F(#F(\"\"%*&F)F(F/F(F0*&I)t
infty11GF$F(I)tinfty13GF$F(F+*$I)tinfty12GF$F,#!\"\"F1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "factor(simplify(" }{MPLTEXT 1 0 20 
"L21InftyOrdrelambda4" }{MPLTEXT 1 0 10 "*lambda^4+" }{MPLTEXT 1 0 20 
"L21InftyOrdrelambda3" }{MPLTEXT 1 0 10 "*lambda^3+" }{MPLTEXT 1 0 20 
"L21InftyOrdrelambda2" }{MPLTEXT 1 0 10 "*lambda^2+" }{MPLTEXT 1 0 20 
"L21InftyOrdrelambda1" }{MPLTEXT 1 0 34 "*lambda- (-P2(lambda)+Pinfty0
2)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT 210 39 "We conclude that L_\{2,1\}is of the form " }{TEXT 210 
54 "-P2(lambda)+O(1) at infinity. Let us now study L_\{2,2\}" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "L22Infty:=factor(h*simplify(
diff(WronskianLambdabisInfty,lambda)/WronskianLambdabisInfty)):\n" }
{MPLTEXT 1 0 75 "L22InftyOrdrelambda5:=factor(-residue(L22Infty/lambda
^6,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrdrelambda4:=facto
r(-residue(L22Infty/lambda^5,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L
22InftyOrdrelambda3:=factor(-residue(L22Infty/lambda^4,lambda=infinity
));\n" }{MPLTEXT 1 0 75 "L22InftyOrdrelambda2:=factor(-residue(L22Inft
y/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 75 "L22InftyOrdrelambda
1:=factor(-residue(L22Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 
1 0 75 "L22InftyOrdrelambda0:=factor(-residue(L22Infty/lambda^1,lambda
=infinity));\n" }{MPLTEXT 1 0 80 "L22InftyOrdrelambdaMoins1:=factor(-r
esidue(L22Infty/lambda^0,lambda=infinity));\n" }{MPLTEXT 1 0 82 "L22In
ftyOrdrelambdaMoins2:=factor(-residue(L22Infty/lambda^(-1),lambda=infi
nity));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda5G6\"\"
\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda4G6\"\"\"!"
 }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda3G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda2G6\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda1G6\",$I)tinfty14G
F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5L22InftyOrdrelambda0G6\",$
I)tinfty12GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrdrela
mbdaMoins1G6\",$*&,&I(epsilonGF$\"\"\"\"\"$F)F)I\"hGF$F)#F)\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I:L22InftyOrdrelambdaMoins2G6\",$*(I\"h
GF$\"\"\",&*&I)Unknown2GF$F(I)tinfty15GF$F(\"\"#I)tinfty13GF$F(F(F,!\"
\"F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 2 140 "We conclude that L_\{2,2
\} behaves at infinity like -tinfty14*lambda -tinfty12+2*h/lambda +O(1
/lambda^2) =P1(lambda) +2*h/lambda+O(1/lambda^2)" }}}{EXCHG {PARA 0 ""
 0 "" {TEXT 215 58 "We end with the explicit formulas for L_\{2,2\} an
d L_\{2,1\}:" }{TEXT 216 39 "\nL_\{2,2\}=  P_1(lambda)  +2*h/(lambda-q
)" }{TEXT 2 6 "  and " }{TEXT 2 1 " " }{TEXT 217 7 "L_\{2,1\}" }{TEXT 
218 2 "= " }{TEXT 219 23 "-P_2(lambda)+Pinfty02+C" }{TEXT 220 17 "  -p
*h/(lambda-q)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "L21Form:=-
P2(lambda)+Pinfty02- p*h/(lambda-q);\n" }{MPLTEXT 1 0 38 "L22Form:=P1(
lambda)  +2*h/(lambda-q);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(L21For
mG6\",**&,(*&I)tinfty15GF$\"\"\"I)tinfty11GF$F*#!\"\"\"\"#*&I)tinfty14
GF$F*I)tinfty12GF$F*#F*F.*$I)tinfty13GF$F.#F-\"\"%F*I'lambdaGF$F*F-*&,
&*&F)F*F4F*F,*$F0F.#F*F6F*F7F.F-*&F)F.F7\"\"$F<*(I\"pGF$F*I\"hGF$F*,&F
7F*I\"qGF$F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(L22FormG6\",(*&I)
tinfty14GF$\"\"\"I'lambdaGF$F(!\"\"I)tinfty12GF$F**&I\"hGF$F(,&F)F(I\"
qGF$F*F*\"\"#" }}}{EXCHG {PARA 209 "" 0 "" {TEXT 221 56 "Conputation f
or the auxiliary matrix A in the oper gauge" }}}{EXCHG {PARA 0 "" 0 ""
 {TEXT 2 183 "The deformation operator is  \\mathcal\{L\}=\\hbar (alph
a15\\partial_\{t_\{\\infty^\{(1)\},5\} +alpha14\\partial_\{t_\{\\infty
^\{(1)\},4\}+e*\\partial_\{t_\{\\infty^\{(1)\},3\} +f*\\partial_\{t_\{
\\infty^\{(1)\},2\}" }{TEXT 2 32 "+g*\\partial_\{t_\{\\infty^\{(1)\},1
\})" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "WronskianGrosLInfty:
=factor(psi1Infty*GrosLpsi2Infty-psi2Infty*GrosLpsi1Infty):\n" }
{MPLTEXT 1 0 70 "A12Infty:=factor(simplify(WronskianGrosLInfty/Wronski
anLambdaInfty)):\n" }{MPLTEXT 1 0 48 "Y1Infty:=h*factor(dpsi1dlambdaIn
fty/psi1Infty):\n" }{MPLTEXT 1 0 48 "Y2Infty:=h*factor(dpsi2dlambdaInf
ty/psi2Infty):\n" }{MPLTEXT 1 0 43 "Z1Infty:=factor(GrosLpsi1Infty/psi
1Infty):\n" }{MPLTEXT 1 0 43 "Z2Infty:=factor(GrosLpsi2Infty/psi2Infty
):\n" }{MPLTEXT 1 0 68 "A12bisInfty:=factor(simplify((Z2Infty-Z1Infty)
/(Y2Infty-Y1Infty))):\n" }{MPLTEXT 1 0 82 "A11Infty:=factor(simplify( \+
(Y2Infty*Z1Infty-Y1Infty*Z2Infty)/(Y2Infty-Y1Infty) )):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "factor(simplify(A12bisInfty-A12Inft
y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "Ltinfty15:=h*alpha15:\n" }{MPLTEXT 1 0 22 "Ltinfty
14:=h*alpha14:\n" }{MPLTEXT 1 0 22 "Ltinfty13:=h*alpha13:\n" }{MPLTEXT
 1 0 22 "Ltinfty12:=h*alpha12:\n" }{MPLTEXT 1 0 21 "Ltinfty11:=h*alpha
11:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 14 "Ltinfty10:=0:\n" }{MPLTEXT
 1 0 14 "Ltinfty20:=0:\n" }{MPLTEXT 1 0 12 "LA20:=LA10:\n" }{MPLTEXT 
1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A12InftyLambda
3:=factor(-residue(A12Infty/lambda^4,lambda=infinity));\n" }{MPLTEXT 
1 0 70 "A12InftyLambda2:=factor(-residue(A12Infty/lambda^3,lambda=infi
nity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda1:=factor(-residue(A12Infty
/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A12InftyLambda0:=fac
tor(-residue(A12Infty/lambda^1,lambda=infinity));\n" }{MPLTEXT 1 0 75 
"A12InftyLambdaMoins1:=factor(-residue(A12Infty/lambda^0,lambda=infini
ty));\n" }{MPLTEXT 1 0 78 "A12InftyLambdaMoins2:=factor(-residue(A12In
fty/lambda^(-1),lambda=infinity));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "
>I0A12InftyLambda3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12Inf
tyLambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A12InftyLambda1
G6\",$*&I(alpha15GF$\"\"\"I)tinfty15GF$!\"\"#\"\"#\"\"&" }}{PARA 11 ""
 1 "" {XPPMATH 20 ">I0A12InftyLambda0G6\",$*&,&*&I(alpha13GF$\"\"\"I)t
infty15GF$F*\"\"&*&I(alpha15GF$F*I)tinfty13GF$F*!\"$F*F+!\"##\"\"#\"#:
" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5A12InftyLambdaMoins1G6\",$*&,**&I
(alpha11GF$\"\"\"I)tinfty15GF$\"\"#\"#:*(I(alpha13GF$F*I)tinfty13GF$F*
F+F*!\"&*(I(alpha15GF$F*I)tinfty11GF$F*F+F*!\"$*&F3F*F0F,\"\"$F*F+F5#F
,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5A12InftyLambdaMoins2G6\",$*&,0
**I(UnknownGF$\"\"\"I(alpha15GF$F*I\"hGF$F*I)tinfty15GF$\"\"#\"\"$*(I(
alpha11GF$F*I)tinfty13GF$F*F-F.\"#:*(I(alpha13GF$F*I)tinfty11GF$F*F-F.
\"\"&*(F5F*F2F.F-F*!\"&**F+F*F6F*F2F*F-F*!\"'*&F+F*F2F/F/*&I)LUnknownG
F$F*F-F/F3F*F-!\"%#!\"#F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 2 12 "We ge
t that " }{TEXT 222 54 "A_\{1,2\}=2*alpha15/5/tinfty15*lambda+nu +mu/(
lambda-q)\n" }{TEXT 223 8 "with nu=" }{XPPEDIT 2 0 "Typesetting:-mrow(
Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal\", fence = \"fal
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ing:-mn(\"3\", mathvariant = \"normal\"), Typesetting:-mo(\"&Invisible
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pesetting:-mrow(Typesetting:-mn(\"5\", mathvariant = \"normal\"), Type
setting:-mo(\"&InvisibleTimes;\", mathvariant = \"normal\", fence = \"
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3\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"&
InvisibleTimes;\", mathvariant = \"normal\", fence = \"false\", separa
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\"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"tinfty15\", italic =
 \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), mathvar
iant = \"normal\")), Typesetting:-mrow(Typesetting:-mn(\"15\", mathvar
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Typesetting:-msup(Typesetting:-mi(\"tinfty15\", italic = \"true\", mat
hvariant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal
\"), superscriptshift = \"0\")), linethickness = \"1\", denomalign = \+
\"center\", numalign = \"center\", bevelled = \"false\"));" "-I%mrowG6
#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6$-I#moGF$6-Q*&uminus0;F'/
%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGF4/%)stretchyGF
4/%*symmetricGF4/%(largeopGF4/%.movablelimitsGF4/%'accentGF4/%'lspaceG
Q,0.2222222emF'/%'rspaceGFC-I&mfracGF$6(-F#6%-I#mnGF$6$Q\"2F'F/-F,6-Q1
&InvisibleTimes;F'F/F2F5F7F9F;F=F?/FBQ&0.0emF'/FEFS-I(mfencedGF$6$-F#6
'-I#miGF$6#Q!F'-F#6'-FL6$Q\"3F'F/FO-Fen6%Q(&alpha;F'/%'italicGF4F/FO-F
en6%Q)tinfty13F'/FaoQ%trueF'/F0Q'italicF'-F,6-Q(&minus;F'F/F2F5F7F9F;F
=F?FAFD-F#6'-FL6$Q\"5F'F/FO-Fen6%Q(alpha13F'FeoFgoFO-Fen6%Q)tinfty15F'
FeoFgoFZF/-F#6%-FL6$Q#15F'F/FO-I%msupGF$6%FdpFK/%1superscriptshiftGQ\"
0F'/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFgq/%)be
velledGF4" }{TEXT 224 5 " and\n" }{TEXT 225 3 "mu=" }{XPPEDIT 2 0 "Typ
esetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mo(\"&uminus0;\", m
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Typesetting:-mrow(Typesetting:-mn(\"5\", mathvariant = \"normal\"), Ty
pesetting:-mo(\"&InvisibleTimes;\", mathvariant = \"normal\", fence = \+
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\", fence = \"false\", separator = \"false\", stretchy = \"false\", sy
mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a
ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin
g:-mi(\"alpha11\", italic = \"true\", mathvariant = \"italic\"), Types
etting:-mo(\"&InvisibleTimes;\", mathvariant = \"normal\", fence = \"f
alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal
se\", largeop = \"false\", movablelimits = \"false\", accent = \"false
\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typeset
ting:-mi(\"tinfty15\", italic = \"true\", mathvariant = \"italic\"), T
ypesetting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"
0\")), Typesetting:-mi(\"\")), mathvariant = \"normal\")), Typesetting
:-mrow(Typesetting:-mn(\"15\", mathvariant = \"normal\"), Typesetting:
-mo(\"&InvisibleTimes;\", mathvariant = \"normal\", fence = \"false\",
 separator = \"false\", stretchy = \"false\", symmetric = \"false\", l
argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp
ace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting:-m
i(\"tinfty15\", italic = \"true\", mathvariant = \"italic\"), Typesett
ing:-mn(\"3\", mathvariant = \"normal\"), superscriptshift = \"0\")), \+
linethickness = \"1\", denomalign = \"center\", numalign = \"center\",
 bevelled = \"false\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(
_syslibGF'6%-I#miGF$6#Q!F'-I#moGF$6-Q*&uminus0;F'/%,mathvariantGQ'norm
alF'/%&fenceGQ&falseF'/%*separatorGF8/%)stretchyGF8/%*symmetricGF8/%(l
argeopGF8/%.movablelimitsGF8/%'accentGF8/%'lspaceGQ,0.2222222emF'/%'rs
paceGFG-I&mfracGF$6(-F#6%-I#mnGF$6$Q\"2F'F3-F06-Q1&InvisibleTimes;F'F3
F6F9F;F=F?FAFC/FFQ&0.0emF'/FIFW-I(mfencedGF$6$-F#6+F+-F#6)-FP6$Q\"3F'F
3FS-F,6%Q(&alpha;F'/%'italicGF8F3FS-F,6%Q)tinfty11F'/FaoQ%trueF'/F4Q'i
talicF'FS-F,6%Q)tinfty15F'FeoFgo-F06-Q(&minus;F'F3F6F9F;F=F?FAFCFEFH-F
#6'FjnFSF]oFS-I%msupGF$6%-F,6%Q)tinfty13F'FeoFgoFO/%1superscriptshiftG
Q\"0F'-F06-Q'&plus;F'F3F6F9F;F=F?FAFCFEFH-F#6)-FP6$Q\"5F'F3FS-F,6%Q(al
pha13F'FeoFgoFSFdpFSFioF\\p-F#6'-FP6$Q#15F'F3FS-F,6%Q(alpha11F'FeoFgoF
S-Fbp6%FioFOFgpF+F3-F#6%FgqFS-Fbp6%FioFjnFgp/%.linethicknessGQ\"1F'/%+
denomalignGQ'centerF'/%)numalignGFhr/%)bevelledGF8" }{TEXT 226 2 "  " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "A12Form:=" }{MPLTEXT 1 0 
27 "2*alpha15/5/tinfty15*lambda" }{MPLTEXT 1 0 20 "+nu+ mu/(lambda-q);
\n" }{MPLTEXT 1 0 71 "simplify(-residue(A12Form/lambda^2,lambda=infini
ty)-A12InftyLambda1); \n" }{MPLTEXT 1 0 154 "solve(\{factor(-residue(A
12Form/lambda,lambda=infinity))=A12InftyLambda0,factor(-residue(A12For
m,lambda=infinity))=factor(A12InftyLambdaMoins1)\},\{mu,nu\});\n" }
{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A12FormG6\",(*(
I(alpha15GF$\"\"\"I)tinfty15GF$!\"\"I'lambdaGF$F(#\"\"#\"\"&I#nuGF$F(*
&I#muGF$F(,&F+F(I\"qGF$F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "<$/I#muG6\",$*&,**&I(alpha11GF%\"\"\"I
)tinfty15GF%\"\"#\"#:*(I(alpha13GF%F+I)tinfty13GF%F+F,F+!\"&*(I(alpha1
5GF%F+I)tinfty11GF%F+F,F+!\"$*&F4F+F1F-\"\"$F+F,F6#F-F./I#nuGF%,$*&,&*
&F0F+F,F+\"\"&*&F4F+F1F+F6F+F,!\"#F9" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 129 "mu := -(2*(3*alpha15*tinfty11*tinfty15-3*alpha15*tin
fty13^2+5*alpha13*tinfty13*tinfty15-15*alpha11*tinfty15^2))/(15*tinfty
15^3);\n" }{MPLTEXT 1 0 127 "mubis := -2*(1/5*alpha15*tinfty11/tinfty1
5-1/5*alpha15*tinfty13^2/tinfty15^2 +1/3*alpha13*tinfty13/tinfty15-alp
ha11)/tinfty15;\n" }{MPLTEXT 1 0 28 "factor(simplify(mu-mubis));\n" }
{MPLTEXT 1 0 67 "nu := -(2*(3*alpha15*tinfty13-5*alpha13*tinfty15))/(1
5*tinfty15^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#muG6\",$*&,**&I(alp
ha11GF$\"\"\"I)tinfty15GF$\"\"#!#:*(I(alpha13GF$F*I)tinfty13GF$F*F+F*
\"\"&*(I(alpha15GF$F*I)tinfty11GF$F*F+F*\"\"$*&F3F*F0F,!\"$F*F+F7#!\"#
\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&mubisG6\",$*&,**(I(alpha15GF$
\"\"\"I)tinfty15GF$!\"\"I)tinfty11GF$F*#F*\"\"&*(F)F*I)tinfty13GF$\"\"
#F+!\"##F,F/*(I(alpha13GF$F*F1F*F+F,#F*\"\"$I(alpha11GF$F,F*F+F,F3" }}
{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">
I#nuG6\",$*&,&*&I(alpha13GF$\"\"\"I)tinfty15GF$F*!\"&*&I(alpha15GF$F*I
)tinfty13GF$F*\"\"$F*F+!\"##F1\"#:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
210 99 "These are theoretical results to compute the nu_i's, i.e. the \+
behavior of A_\{1,2\} at lambda->\\infty" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "Minfty:=Matrix(3,3,0):\n" }{MPLTEXT 1 0 23 "Minfty[1,
1]:=tinfty15:\n" }{MPLTEXT 1 0 23 "Minfty[2,2]:=tinfty15:\n" }{MPLTEXT
 1 0 23 "Minfty[3,3]:=tinfty15:\n" }{MPLTEXT 1 0 23 "Minfty[2,1]:=tinf
ty13:\n" }{MPLTEXT 1 0 23 "Minfty[3,2]:=tinfty13:\n" }{MPLTEXT 1 0 23 
"Minfty[3,1]:=tinfty11:\n" }{MPLTEXT 1 0 8 "Minfty;\n" }{MPLTEXT 1 0 
25 "NuVector:=Matrix(3,1,0):\n" }{MPLTEXT 1 0 25 "NuVector[1,1]:=nuMoi
ns1:\n" }{MPLTEXT 1 0 20 "NuVector[2,1]:=nu0:\n" }{MPLTEXT 1 0 20 "NuV
ector[3,1]:=nu1:\n" }{MPLTEXT 1 0 10 "NuVector;\n" }{MPLTEXT 1 0 26 "R
HSVector:=Matrix(3,1,0):\n" }{MPLTEXT 1 0 31 "RHSVector[1,1]:=2*alpha1
515/5:\n" }{MPLTEXT 1 0 31 "RHSVector[2,1]:=2*alpha1513/3:\n" }
{MPLTEXT 1 0 31 "RHSVector[3,1]:=2*alpha1511/1:\n" }{MPLTEXT 1 0 11 "R
HSVector;\n" }{MPLTEXT 1 0 46 "NuVectorSol:=Multiply(Minfty^(-1),RHSVe
ctor);\n" }{MPLTEXT 1 0 20 "alpha1515:=alpha15:\n" }{MPLTEXT 1 0 20 "a
lpha1514:=alpha14:\n" }{MPLTEXT 1 0 20 "alpha1513:=alpha13:\n" }
{MPLTEXT 1 0 20 "alpha1512:=alpha12:\n" }{MPLTEXT 1 0 19 "alpha1511:=a
lpha11:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "mu1:=mu:\n" }{MPLTEXT 
1 0 14 "NuVector[1,1]=" }{MPLTEXT 1 0 18 "NuVectorSol[1,1];\n" }
{MPLTEXT 1 0 32 "NuVector[2,1]=NuVectorSol[2,1];\n" }{MPLTEXT 1 0 32 "
NuVector[3,1]=NuVectorSol[3,1];\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 
12 "nuMoins1 := " }{MPLTEXT 1 0 17 "NuVectorSol[1,1];" }{MPLTEXT 1 0 
1 "\n" }{MPLTEXT 1 0 7 "nu0 := " }{MPLTEXT 1 0 17 "NuVectorSol[2,1];" 
}{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 "nu1:=" }{MPLTEXT 1 0 17 "NuVecto
rSol[3,1];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 
"" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibG
F'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6(-I$mtdGF$6(-I#miGF$6&Q)tinfty
15F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italic
F'/%)rowalignGQ!F'/%,columnalignGFF/%+groupalignGFF/%(rowspanGQ\"1F'/%
+columnspanGFM-F56(-I#mnGF$6%Q\"0F'F>/FBQ'normalF'FDFGFIFKFNFPFDFGFI-F
26(-F56(-F86&Q)tinfty13F'F;F>FAFDFGFIFKFNF4FPFDFGFI-F26(-F56(-F86&Q)ti
nfty11F'F;F>FAFDFGFIFKFNFZF4FDFGFI/%&alignGQ%axisF'/FEQ)baselineF'/FHQ
'centerF'/FJQ'|frleft|hrF'/%/alignmentscopeGF=/%,columnwidthGQ%autoF'/
%&widthGF]p/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowline
sGQ%noneF'/%,columnlinesGFhp/%&frameGFhp/%-framespacingGQ,0.4em~0.5exF
'/%*equalrowsGQ&falseF'/%-equalcolumnsGFbq/%-displaystyleGFbq/%%sideGQ
&rightF'/%0minlabelspacingGFepF>FV/%%openGQ\"[F'/%&closeGQ\"]F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typesett
ingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6&-I$mtdGF$6(-I#m
iGF$6&Q)nuMoins1F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathva
riantGQ'italicF'/%)rowalignGQ!F'/%,columnalignGFF/%+groupalignGFF/%(ro
wspanGQ\"1F'/%+columnspanGFMFDFGFI-F26&-F56(-F86&Q&&nu;0F'F;F>FAFDFGFI
FKFNFDFGFI-F26&-F56(-F86&Q&&nu;1F'F;F>FAFDFGFIFKFNFDFGFI/%&alignGQ%axi
sF'/FEQ)baselineF'/FHQ'centerF'/FJQ'|frleft|hrF'/%/alignmentscopeGF=/%
,columnwidthGQ%autoF'/%&widthGFeo/%+rowspacingGQ&1.0exF'/%.columnspaci
ngGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF`p/%&frameGF`p/%-frame
spacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFjp/%-dis
playstyleGFjp/%%sideGQ&rightF'/%0minlabelspacingGF]pF>/FBQ'normalF'/%%
openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfenced
G6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableG
F$67-I$mtrGF$6&-I$mtdGF$6(-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"2F'/%+foregro
undGQ([0,0,0]F'/%,mathvariantGQ'normalF'-F=6%Q\"5F'F@FC/%.linethicknes
sGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFN/%)bevelledGQ&falseF'F@-
I#moGF$6-Q1&InvisibleTimes;F'FC/%&fenceGFS/%*separatorGFS/%)stretchyGF
S/%*symmetricGFS/%(largeopGFS/%.movablelimitsGFS/%'accentGFS/%'lspaceG
Q&0.0emF'/%'rspaceGFbo-F,6#-I#miGF$6&Q,&alpha;1515F'/%'italicGQ%trueF'
F@/FDQ'italicF'/%)rowalignGQ!F'/%,columnalignGFbp/%+groupalignGFbp/%(r
owspanGFK/%+columnspanGFKF`pFcpFep-F26&-F56(-F,6%-F:6)F<-F=6%Q\"3F'F@F
CFIFLFOFQF@FT-F,6#-Fho6&Q,&alpha;1513F'F[pF@F^pF`pFcpFepFgpFipF`pFcpFe
p-F26&-F56(-F,6%F<FT-Fho6&Q,&alpha;1511F'F[pF@F^pF`pFcpFepFgpFipF`pFcp
Fep/%&alignGQ%axisF'/FapQ)baselineF'/FdpFN/FfpQ'|frleft|hrF'/%/alignme
ntscopeGF]p/%,columnwidthGQ%autoF'/%&widthGF`s/%+rowspacingGQ&1.0exF'/
%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF[t/%&fram
eGF[t/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFS/%-equalcolumnsGFS/
%-displaystyleGFS/%%sideGQ&rightF'/%0minlabelspacingGFhsF@FC/%%openGQ
\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+
modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67-I
$mtrGF$6&-I$mtdGF$6(-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"2F'/%+foregroundGQ(
[0,0,0]F'/%,mathvariantGQ'normalF'-F=6%Q\"5F'F@FC/%.linethicknessGQ\"1
F'/%+denomalignGQ'centerF'/%)numalignGFN/%)bevelledGQ&falseF'F@-I#moGF
$6-Q1&InvisibleTimes;F'FC/%&fenceGFS/%*separatorGFS/%)stretchyGFS/%*sy
mmetricGFS/%(largeopGFS/%.movablelimitsGFS/%'accentGFS/%'lspaceGQ&0.0e
mF'/%'rspaceGFbo-F:6)-F,6#-I#miGF$6&Q,&alpha;1515F'/%'italicGQ%trueF'F
@/FDQ'italicF'-F,6#-Fjo6&Q)tinfty15F'F]pF@F`pFIFLFOFQF@/%)rowalignGQ!F
'/%,columnalignGFip/%+groupalignGFip/%(rowspanGFK/%+columnspanGFKFgpFj
pF\\q-F26&-F56(-F,6&-FU6.Q*&uminus0;F'F@FCFXFZFfnFhnFjnF\\oF^o/FaoQ,0.
2222222emF'/FdoF\\r-F,6%F9FT-F:6)-F,6%-Fjo6&Q)tinfty13F'F]pF@F`pFTFio-
F,6#-I%msupGF$6%FdpF</%1superscriptshiftGQ\"0F'FIFLFOFQF@-FU6.Q\"+F'F@
FCFXFZFfnFhnFjnF\\oF^oF[rF]r-F,6%-F:6)F<-F=6%Q\"3F'F@FCFIFLFOFQF@FT-F:
6)-F,6#-Fjo6&Q,&alpha;1513F'F]pF@F`pFbpFIFLFOFQF@FgpFjpF\\qF^qF`qFgpFj
pF\\q-F26&-F56(-F,6(Fhq-F,6%F9FT-F:6)-F,6%-F#6%-F,6%-F,6%-Fjo6&Q)tinft
y11F'F]pF@F`pFTFdp-FU6.Q(&minus;F'F@FCFXFZFfnFhnFjnF\\oF^oF[rF]r-F,6#-
Fjr6%FdrF<F\\sF@FCFTFio-F,6#-Fjr6%FdpFfsF\\sFIFLFOFQF@Feu-F,6%FdsFT-F:
6)-F,6%FdrFTF]tFgrFIFLFOFQF@F_s-F:6)-F,6%F<FT-Fjo6&Q,&alpha;1511F'F]pF
@F`pFbpFIFLFOFQF@FgpFjpF\\qF^qF`qFgpFjpF\\q/%&alignGQ%axisF'/FhpQ)base
lineF'/F[qFN/F]qQ'|frleft|hrF'/%/alignmentscopeGF_p/%,columnwidthGQ%au
toF'/%&widthGFiw/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)ro
wlinesGQ%noneF'/%,columnlinesGFdx/%&frameGFdx/%-framespacingGQ,0.4em~0
.5exF'/%*equalrowsGFS/%-equalcolumnsGFS/%-displaystyleGFS/%%sideGQ&rig
htF'/%0minlabelspacingGFaxF@FC/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 
11 "" 1 "" {XPPMATH 20 "/I)nuMoins1G6\",$*&I(alpha15GF$\"\"\"I)tinfty1
5GF$!\"\"#\"\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I$nu0G6\",&*(I)
tinfty15GF$!\"#I)tinfty13GF$\"\"\"I(alpha15GF$F*#F(\"\"&*&F'!\"\"I(alp
ha13GF$F*#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I$nu1G6\",(*(,&
*&I)tinfty11GF$\"\"\"I)tinfty15GF$F*F**$I)tinfty13GF$\"\"#!\"\"F*F+!\"
$I(alpha15GF$F*#!\"#\"\"&*(F+F3F-F*I(alpha13GF$F*#F3\"\"$*&F+F/I(alpha
11GF$F*F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuMoins1G6\",$*&I(alpha1
5GF$\"\"\"I)tinfty15GF$!\"\"#\"\"#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I$nu0G6\",&*(I)tinfty15GF$!\"#I)tinfty13GF$\"\"\"I(alpha15GF$F*#F
(\"\"&*&F'!\"\"I(alpha13GF$F*#\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 
20 ">I$nu1G6\",(*(,&*&I)tinfty11GF$\"\"\"I)tinfty15GF$F*F**$I)tinfty13
GF$\"\"#!\"\"F*F+!\"$I(alpha15GF$F*#!\"#\"\"&*(F+F3F-F*I(alpha13GF$F*#
F3\"\"$*&F+F/I(alpha11GF$F*F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 58 
"We now check that the formula for the nu_\{i\}'s is correct." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "simplify(-residue(" }
{MPLTEXT 1 0 44 "A12Form/lambda^2,lambda=infinity)-nuMoins1);" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 56 "simplify(-residue(A12Form/lambda
,lambda=infinity)-nu0);\n" }{MPLTEXT 1 0 57 "simplify(-residue(A12Form
*lambda^0,lambda=infinity)-nu1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"
!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "NuMuVector:=Ma
trix(3,1,0):\n" }{MPLTEXT 1 0 27 "NuMuVector[1,1]:=nuMoins1:\n" }
{MPLTEXT 1 0 22 "NuMuVector[2,1]:=nu0:\n" }{MPLTEXT 1 0 21 "NuMuVector
[3,1]:=mu1:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "R:=Matrix(3,3,0):
\n" }{MPLTEXT 1 0 11 "R[1,1]:=1:\n" }{MPLTEXT 1 0 11 "R[2,2]:=1:\n" }
{MPLTEXT 1 0 11 "R[3,3]:=1:\n" }{MPLTEXT 1 0 54 "NuMuVectorTheo:=Multi
ply(Multiply(R^(-1),Minfty^(-1))," }{MPLTEXT 1 0 12 "RHSVector);\n" }
{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 11 "NuMuVector-" }{MPLTEXT 1 
0 16 "NuMuVectorTheo);" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/
I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$67
-I$mtrGF$6&-I$mtdGF$6(-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"2F'/%+foregroundG
Q([0,0,0]F'/%,mathvariantGQ'normalF'-F=6%Q\"5F'F@FC/%.linethicknessGQ
\"1F'/%+denomalignGQ'centerF'/%)numalignGFN/%)bevelledGQ&falseF'F@-I#m
oGF$6-Q1&InvisibleTimes;F'FC/%&fenceGFS/%*separatorGFS/%)stretchyGFS/%
*symmetricGFS/%(largeopGFS/%.movablelimitsGFS/%'accentGFS/%'lspaceGQ&0
.0emF'/%'rspaceGFbo-F:6)-F,6#-I#miGF$6&Q*&alpha;15F'/%'italicGQ%trueF'
F@/FDQ'italicF'-F,6#-Fjo6&Q)tinfty15F'F]pF@F`pFIFLFOFQF@/%)rowalignGQ!
F'/%,columnalignGFip/%+groupalignGFip/%(rowspanGFK/%+columnspanGFKFgpF
jpF\\q-F26&-F56(-F,6&-FU6.Q*&uminus0;F'F@FCFXFZFfnFhnFjnF\\oF^o/FaoQ,0
.2222222emF'/FdoF\\r-F,6%F9FT-F:6)-F,6%-Fjo6&Q)tinfty13F'F]pF@F`pFTFio
-F,6#-I%msupGF$6%FdpF</%1superscriptshiftGQ\"0F'FIFLFOFQF@-FU6.Q\"+F'F
@FCFXFZFfnFhnFjnF\\oF^oF[rF]r-F,6%-F:6)F<-F=6%Q\"3F'F@FCFIFLFOFQF@FT-F
:6)-F,6#-Fjo6&Q*&alpha;13F'F]pF@F`pFbpFIFLFOFQF@FgpFjpF\\qF^qF`qFgpFjp
F\\q-F26&-F56(-F,6(Fhq-F,6%F9FT-F:6)-F,6%-F#6%-F,6%-F,6%-Fjo6&Q)tinfty
11F'F]pF@F`pFTFdp-FU6.Q(&minus;F'F@FCFXFZFfnFhnFjnF\\oF^oF[rF]r-F,6#-F
jr6%FdrF<F\\sF@FCFTFio-F,6#-Fjr6%FdpFfsF\\sFIFLFOFQF@Feu-F,6%FdsFT-F:6
)-F,6%FdrFTF]tFgrFIFLFOFQF@F_s-F:6)-F,6%F<FT-Fjo6&Q*&alpha;11F'F]pF@F`
pFbpFIFLFOFQF@FgpFjpF\\qF^qF`qFgpFjpF\\q/%&alignGQ%axisF'/FhpQ)baselin
eF'/F[qFN/F]qQ'|frleft|hrF'/%/alignmentscopeGF_p/%,columnwidthGQ%autoF
'/%&widthGFiw/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowli
nesGQ%noneF'/%,columnlinesGFdx/%&frameGFdx/%-framespacingGQ,0.4em~0.5e
xF'/%*equalrowsGFS/%-equalcolumnsGFS/%-displaystyleGFS/%%sideGQ&rightF
'/%0minlabelspacingGFaxF@FC/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sys
libGF'6'-I%mrowGF$6#-I'mtableGF$67-I$mtrGF$6&-I$mtdGF$6(-I#mnGF$6%Q\"0
F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%
,columnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/%+columnspanGFJFAFDFF
F1F1/%&alignGQ%axisF'/FBQ)baselineF'/FEQ&rightF'/FGQ'|frleft|hrF'/%/al
ignmentscopeGQ%trueF'/%,columnwidthGQ%autoF'/%&widthGFen/%+rowspacingG
Q&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesG
F`o/%&frameGF`o/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%
-equalcolumnsGFjo/%-displaystyleGFjo/%%sideGFS/%0minlabelspacingGF]oF;
F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 
65 "We have checked that Proposition dealing with A_\{1,2\} is correct
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A11InftyLambda4:=facto
r(-residue(A11Infty/lambda^5,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A
11InftyLambda3:=factor(-residue(A11Infty/lambda^4,lambda=infinity));\n
" }{MPLTEXT 1 0 70 "A11InftyLambda2:=factor(-residue(A11Infty/lambda^3
,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11InftyLambda1:=factor(-resi
due(A11Infty/lambda^2,lambda=infinity));\n" }{MPLTEXT 1 0 70 "A11Infty
Lambda0:=factor(-residue(A11Infty/lambda^1,lambda=infinity));\n" }
{MPLTEXT 1 0 75 "A11InftyLambdaMoins1:=factor(-residue(A11Infty/lambda
^0,lambda=infinity)):\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLa
mbda4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda3G6\"
\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda2G6\",$*&,&*&I
(alpha14GF$\"\"\"I)tinfty15GF$F*\"\"&*&I(alpha15GF$F*I)tinfty14GF$F*!
\"%F*F+!\"\"#F1\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0A11InftyLambda
1G6\",$*&,**&I(alpha12GF$\"\"\"I)tinfty15GF$\"\"#\"#:*(I(alpha13GF$F*I
)tinfty14GF$F*F+F*!#5*(I(alpha15GF$F*I)tinfty12GF$F*F+F*!\"'*(F3F*I)ti
nfty13GF$F*F0F*\"\"'F*F+!\"##!\"\"\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20
 ">I0A11InftyLambda0G6\",$*&,2**I(alpha15GF$\"\"\"I(epsilonGF$F*I\"hGF
$F*I)tinfty15GF$\"\"#!\"$*&I%LA10GF$F*F-\"\"$\"#I*(I(alpha11GF$F*I)tin
fty14GF$F*F-F.F3*(I(alpha13GF$F*I)tinfty12GF$F*F-F.\"#5**F8F*I)tinfty1
3GF$F*F6F*F-F*!#5**F)F*I)tinfty11GF$F*F6F*F-F*!\"'**F)F*F9F*F<F*F-F*F@
*(F)F*F<F.F6F*\"\"'F*F-F/#F*F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 210 
25 "We conclude that A_\{1,1\}=" }{TEXT 210 53 "(4*alpha15*tinfty14-5*
alpha14*tinfty15)/(20*tinfty15)" }{TEXT 210 38 "*lambda^2+c1*lambda+c0
+ rho/(lambda-q)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "A11Form:
=" }{MPLTEXT 1 0 53 "(4*alpha15*tinfty14-5*alpha14*tinfty15)/(20*tinft
y15)" }{MPLTEXT 1 0 40 "*lambda^2+c1*lambda+c0+ rho/(lambda-q);\n" }
{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 71 "simplify(-residue(A11Form/lambda
^4,lambda=infinity)-A11InftyLambda3); \n" }{MPLTEXT 1 0 148 "solve(\{f
actor(-residue(A11Form/lambda^3,lambda=infinity))=A11InftyLambda2,fact
or(-residue(A11Form/lambda^2,lambda=infinity))=A11InftyLambda1\},\{c1
\});" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(A11FormG6\",**(,&*&I(alpha14G
F$\"\"\"I)tinfty15GF$F*!\"&*&I(alpha15GF$F*I)tinfty14GF$F*\"\"%F*F+!\"
\"I'lambdaGF$\"\"##F*\"#?*&I#c1GF$F*F2F*F*I#c0GF$F**&I$rhoGF$F*,&F2F*I
\"qGF$F1F1F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "
" {XPPMATH 20 "<#/I#c1G6\",$*&,**&I(alpha12G6\"\"\"\"I)tinfty15G6\"\"
\"#\"#:*(I(alpha13G6\"\"\"\"I)tinfty14G6\"\"\"\"I)tinfty15G6\"\"\"\"!#
5*(I(alpha15G6\"\"\"\"I)tinfty12G6\"\"\"\"I)tinfty15G6\"\"\"\"!\"'*(I(
alpha15G6\"\"\"\"I)tinfty13G6\"\"\"\"I)tinfty14G6\"\"\"\"\"\"'\"\"\"I)
tinfty15G6\"!\"##!\"\"\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
131 "c1 := (6*alpha15*tinfty12*tinfty15-6*alpha15*tinfty13*tinfty14+10
*alpha13*tinfty14*tinfty15-15*alpha12*tinfty15^2)/(30*tinfty15^2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\",$*&,**&I(alpha12GF$\"\"\"I)tin
fty15GF$\"\"#!#:*(I(alpha13GF$F*I)tinfty14GF$F*F+F*\"#5*(I(alpha15GF$F
*I)tinfty12GF$F*F+F*\"\"'*(F3F*I)tinfty13GF$F*F0F*!\"'F*F+!\"##F*\"#I"
 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "c2:=" }{MPLTEXT 1 0 54 "(
4*alpha15*tinfty14-5*alpha14*tinfty15)/(20*tinfty15);" }}{PARA 11 "" 1
 "" {XPPMATH 20 ">I#c2G6\",$*&,&*&I(alpha14GF$\"\"\"I)tinfty15GF$F*!\"
&*&I(alpha15GF$F*I)tinfty14GF$F*\"\"%F*F+!\"\"#F*\"#?" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "c2theo:=1/tinfty15*(alpha1515/5*tin
fty14-alpha1514/4*tinfty15);\n" }{MPLTEXT 1 0 20 "c1theo:=1/tinfty15*(
" }{MPLTEXT 1 0 18 "-tinfty13*c2theo  " }{MPLTEXT 1 0 22 "+ alpha1515/
5*tinfty12" }{MPLTEXT 1 0 21 "-alpha1514/4*tinfty13" }{MPLTEXT 1 0 45 
"+alpha1513/3*tinfty14-alpha1512/2*tinfty15);\n" }{MPLTEXT 1 0 21 "sim
plify(c2-c2theo);\n" }{MPLTEXT 1 0 20 "simplify(c1-c1theo);" }}{PARA 
11 "" 1 "" {XPPMATH 20 ">I'c2theoG6\"*&I)tinfty15GF$!\"\",&*&I(alpha15
GF$\"\"\"I)tinfty14GF$F+#F+\"\"&*&I(alpha14GF$F+F&F+#F'\"\"%F+" }}
{PARA 11 "" 1 "" {XPPMATH 20 ">I'c1theoG6\"*&I)tinfty15GF$!\"\",,*(I)t
infty13GF$\"\"\"F&F',&*&I(alpha15GF$F+I)tinfty14GF$F+#F+\"\"&*&I(alpha
14GF$F+F&F+#F'\"\"%F+F'*&F.F+I)tinfty12GF$F+F0*&F3F+F*F+F4*&I(alpha13G
F$F+F/F+#F+\"\"$*&I(alpha12GF$F+F&F+#F'\"\"#F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 210 61 "We have checked that the formula for A_
\{1,1\} is also correct." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}
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 33 1 1 }