{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 "2D M ath Italic Small209212" -1 200 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -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 "Popup" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Atomic Variable" -1 201 "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" -1 202 "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 "2D Math Italic Small210220" -1 203 "Times" 1 1 0 0 0 1 1 2 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 "T imes" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 204 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Help Nor mal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple In put" -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 "Heade r and Footer" -1 205 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small204" -1 206 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small207208" -1 207 "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" -1 208 "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" -1 209 "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 S mall208209215" -1 210 "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 Italic Small" -1 211 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small208" -1 212 "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" -1 213 "Ti mes" 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 "Cou rier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small2 09" -1 214 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Annotatio n Text" -1 215 "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" -1 216 "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 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Reference" -1 217 "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 Placeholder" -1 218 "Courier N ew" 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 Small20820 9" -1 219 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Empha sized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Mat h Italic Small211" -1 220 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "Equation Label" -1 221 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small210" -1 222 "Times" 1 1 0 0 0 1 1 2 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" -1 223 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 224 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 216 271 "In this Maple file, we \+ perform the gauge transformation to obtain the geometric Lax pairs for the second element of the Painlev\351 1 hierarchy. This Lax pair is t hen expressed in terms of the symmetric Darboux coordinates and formul as are checked with the theoretical ones." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 219 "We load the results on the Lax pair expressed in the op er gauge as well as some procedures to obtain the elementary symmetric polynomials. We define \\td\{L\} and \\td\{A\} using the gauge trans formation from the oper gauge." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 17 "with(ListTools):\n" }{MPLTEXT 1 0 16 "with(combinat):\n" } {MPLTEXT 1 0 23 "with(PolynomialTools):\n" }{MPLTEXT 1 0 16 "with(Groe bner):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "chk:=proc()\n" } {MPLTEXT 1 0 22 "local V,A,p,L,K,N,KK:\n" }{MPLTEXT 1 0 30 "V:=[seq(ar gs[i],i=2..nargs)];\n" }{MPLTEXT 1 0 33 "A:=[seq(sigma[i],i=1..nargs-1 )];\n" }{MPLTEXT 1 0 52 "p:=simplify(expand(mul(x_-args[i],i=2..nargs) ),x_);\n" }{MPLTEXT 1 0 60 "L := Reverse([seq((-1)^(r+nargs-1)*coeff(p , x_, r), r = 0..\n" }{MPLTEXT 1 0 14 "nargs-2)])-A;\n" }{MPLTEXT 1 0 23 "K:=Basis(L,tdeg(V[])):\n" }{MPLTEXT 1 0 36 "N:=NormalForm(args[1], K,tdeg(V[]));\n" }{MPLTEXT 1 0 24 "KK:=Basis(A,tdeg(A[]));\n" } {MPLTEXT 1 0 28 "NormalForm(N,KK,tdeg(A[]));\n" }{MPLTEXT 1 0 64 "if i s(NormalForm(N,KK,tdeg(A[]))=0) then print(\"symmetric\")else\n" } {MPLTEXT 1 0 26 "print(\"not symmetric\")fi;\n" }{MPLTEXT 1 0 10 "end \+ proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "es:=proc()\n" } {MPLTEXT 1 0 21 "local V, A, p, L, K;\n" }{MPLTEXT 1 0 62 "V:=[seq(arg s[i],i=2..nargs)];A:=[seq(sigma[i],i=1..nargs-1)];\n" }{MPLTEXT 1 0 52 "p:=simplify(expand(mul(x_-args[i],i=2..nargs)),x_);\n" }{MPLTEXT 1 0 60 "L := Reverse([seq((-1)^(r+nargs-1)*coeff(p, x_, r), r = 0..\n" }{MPLTEXT 1 0 14 "nargs-2)])-A;\n" }{MPLTEXT 1 0 23 "K:=Basis(L,tdeg( V[]));\n" }{MPLTEXT 1 0 33 "NormalForm(args[1],K,tdeg(V[]));\n" } {MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 38 "s s:=proc() local L, LL, t, LLL, H, K;\n" }{MPLTEXT 1 0 30 "L:=[seq(args [i],i=2..nargs)];\n" }{MPLTEXT 1 0 39 "LL:=[seq(map(x->x^r,L),r=1..nar gs-1)];\n" }{MPLTEXT 1 0 27 "t:=seq(s[i],i=1..nargs-1);\n" }{MPLTEXT 1 0 45 "LLL:=[seq(add(i,i in LL[u]),u=1..nops(LL))];\n" }{MPLTEXT 1 0 12 "H:=LLL-[t];\n" }{MPLTEXT 1 0 24 "K:=Basis(H,grlex(L[]));\n" } {MPLTEXT 1 0 34 "NormalForm(args[1],K,grlex(L[]));\n" }{MPLTEXT 1 0 9 "end proc:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "rinfty:=4:\n" }{MPLTEXT 1 0 13 "g:=rinfty-2:\n" }{MPLTEXT 1 0 21 " tinfty27:=-tinfty17:\n" }{MPLTEXT 1 0 21 "tinfty25:=-tinfty15:\n" } {MPLTEXT 1 0 21 "tinfty23:=-tinfty13:\n" }{MPLTEXT 1 0 21 "tinfty21:=- tinfty11:\n" }{MPLTEXT 1 0 21 "tinfty20:=-tinfty10:\n" }{MPLTEXT 1 0 20 "tinfty26:=tinfty16:\n" }{MPLTEXT 1 0 20 "tinfty24:=tinfty14:\n" } {MPLTEXT 1 0 20 "tinfty22:=tinfty12:\n" }{MPLTEXT 1 0 13 "tinfty10:=0: \n" }{MPLTEXT 1 0 23 "Pinfty01 := -tinfty12;\n" }{MPLTEXT 1 0 23 "Pinf ty11 := -tinfty14;\n" }{MPLTEXT 1 0 23 "Pinfty21 := -tinfty16;\n" } {MPLTEXT 1 0 103 "Pinfty22 := -(1/2)*tinfty11*tinfty17+(1/2)*tinfty12* tinfty16-(1/2)*tinfty13*tinfty15+(1/4)*tinfty14^2;\n" }{MPLTEXT 1 0 79 "Pinfty32 := -(1/2)*tinfty13*tinfty17+(1/2)*tinfty14*tinfty16-(1/4) *tinfty15^2;\n" }{MPLTEXT 1 0 55 "Pinfty42 := -(1/2)*tinfty15*tinfty17 +(1/4)*tinfty16^2;\n" }{MPLTEXT 1 0 31 "Pinfty52 := -(1/4)*tinfty17^2; \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "Pinfty2[1]:=" }{MPLTEXT 1 0 10 "Pinfty21:\n" }{MPLTEXT 1 0 22 "Pinfty2[2]:=Pinfty22:\n" } {MPLTEXT 1 0 22 "Pinfty2[3]:=Pinfty32:\n" }{MPLTEXT 1 0 22 "Pinfty2[4] :=Pinfty42:\n" }{MPLTEXT 1 0 21 "Pinfty2[5]:=Pinfty52:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 42 "P1:=x-> Pinfty01+Pinfty1 1*x+Pinfty21*x^2;\n" }{MPLTEXT 1 0 81 "P2:=x-> Pinfty02+Pinfty12*x+Pin fty22*x^2+Pinfty32*x^3+Pinfty42*x^4+Pinfty52*x^5;\n" }{MPLTEXT 1 0 69 "tdP2:=unapply(Pinfty22*x^2+Pinfty32*x^3+Pinfty42*x^4+Pinfty52*x^5,x); " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "dP1dlamb da:=unapply(diff(P1(lambda),lambda),lambda):\n" }{MPLTEXT 1 0 52 "dP2d lambda:=unapply(diff(P2(lambda),lambda),lambda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 56 "dtdP2dlambda:=unapply(diff(tdP2(lambda),lambda),lam bda):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "c3: =-(7*alpha6*tinfty17-6*alpha7*tinfty16)/(42*tinfty17):\n" }{MPLTEXT 1 0 2 "c2" }{MPLTEXT 1 0 128 ":=-(35*alpha4*tinfty17^2-28*alpha5*tinfty1 6*tinfty17-20*alpha7*tinfty14*tinfty17+20*alpha7*tinfty15*tinfty16)/(1 40*tinfty17^2):\n" }{MPLTEXT 1 0 2 "c1" }{MPLTEXT 1 0 276 ":=-(105*alp ha2*tinfty17^3-70*alpha3*tinfty16*tinfty17^2-42*alpha5*tinfty14*tinfty 17^2+42*alpha5*tinfty15*tinfty16*tinfty17-30*alpha7*tinfty12*tinfty17^ 2+30*alpha7*tinfty13*tinfty16*tinfty17+30*alpha7*tinfty14*tinfty15*tin fty17-30*alpha7*tinfty15^2*tinfty16)/(210*tinfty17^3):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 10 "c[0]:=c0:\n" }{MPLTEXT 1 0 10 "c[1]:=c1:\n" }{MPLTEXT 1 0 10 "c[2]:=c2:\n" }{MPLTEXT 1 0 10 "c[3]:=c3:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "alpha[1]:=alpha1:\n" }{MPLTEXT 1 0 18 "alpha[2]:=alpha2:\n" }{MPLTEXT 1 0 18 "alpha[3]:=alpha3:\n" } {MPLTEXT 1 0 18 "alpha[4]:=alpha4:\n" }{MPLTEXT 1 0 18 "alpha[5]:=alph a5:\n" }{MPLTEXT 1 0 18 "alpha[6]:=alpha6:\n" }{MPLTEXT 1 0 18 "alpha[ 7]:=alpha7:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "tinfty[1]:=" } {MPLTEXT 1 0 7 "tinfty1" }{MPLTEXT 1 0 3 "1:\n" }{MPLTEXT 1 0 6 "tinft y" }{MPLTEXT 1 0 5 "[2]:=" }{MPLTEXT 1 0 7 "tinfty1" }{MPLTEXT 1 0 3 " 2:\n" }{MPLTEXT 1 0 6 "tinfty" }{MPLTEXT 1 0 5 "[3]:=" }{MPLTEXT 1 0 7 "tinfty1" }{MPLTEXT 1 0 3 "3:\n" }{MPLTEXT 1 0 6 "tinfty" }{MPLTEXT 1 0 5 "[4]:=" }{MPLTEXT 1 0 7 "tinfty1" }{MPLTEXT 1 0 3 "4:\n" } {MPLTEXT 1 0 6 "tinfty" }{MPLTEXT 1 0 5 "[5]:=" }{MPLTEXT 1 0 7 "tinft y1" }{MPLTEXT 1 0 3 "5:\n" }{MPLTEXT 1 0 6 "tinfty" }{MPLTEXT 1 0 5 "[ 6]:=" }{MPLTEXT 1 0 7 "tinfty1" }{MPLTEXT 1 0 3 "6:\n" }{MPLTEXT 1 0 6 "tinfty" }{MPLTEXT 1 0 5 "[7]:=" }{MPLTEXT 1 0 7 "tinfty1" }{MPLTEXT 1 0 2 "7:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "nuMoins1:=2*alpha7/(7*tinfty17):\n" }{MPLTEXT 1 0 64 "nu0:=(2*(7*a lpha5*tinfty17-5*alpha7*tinfty15))/(35*tinfty17^2):\n" }{MPLTEXT 1 0 119 "nu1:=-(2*(tinfty13*tinfty17-tinfty15^2))*alpha7/(7*tinfty17^3)-2* tinfty15*alpha5/(5*tinfty17^2)+2*alpha3/(3*tinfty17):\n" }{MPLTEXT 1 0 203 "nu2:=-(2*(tinfty11*tinfty17^2-2*tinfty13*tinfty15*tinfty17+tinf ty15^3))*alpha7/(7*tinfty17^4)-(2*(tinfty13*tinfty17-tinfty15^2))*alph a5/(5*tinfty17^3)-2*tinfty15*alpha3/(3*tinfty17^2)+2*alpha1/tinfty17: \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "nu[-1]:=nuMoins1:\n" } {MPLTEXT 1 0 12 "nu[0]:=nu0:\n" }{MPLTEXT 1 0 12 "nu[1]:=nu1:\n" } {MPLTEXT 1 0 12 "nu[2]:=nu2:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 359 "mu1:=2*(-35*alpha3*q2*tinfty17^3+21*alpha5*q2*tinfty15*tinfty17^2 +15*alpha7*q2*tinfty13*tinfty17^2-15*alpha7*q2*tinfty15^2*tinfty17+105 *alpha1*tinfty17^3-35*alpha3*tinfty15*tinfty17^2-21*alpha5*tinfty13*ti nfty17^2+21*alpha5*tinfty15^2*tinfty17-15*alpha7*tinfty11*tinfty17^2+3 0*alpha7*tinfty13*tinfty15*tinfty17-15*alpha7*tinfty15^3)/(105*tinfty1 7^4*(q1-q2)):\n" }{MPLTEXT 1 0 362 "mu2:=-1/(105*tinfty17^4*(q1-q2))* \+ 2*(-35*alpha3*q1*tinfty17^3+21*alpha5*q1*tinfty15*tinfty17^2+15*alpha7 *q1*tinfty13*tinfty17^2-15*alpha7*q1*tinfty15^2*tinfty17+105*alpha1*ti nfty17^3-35*alpha3*tinfty15*tinfty17^2-21*alpha5*tinfty13*tinfty17^2+2 1*alpha5*tinfty15^2*tinfty17-15*alpha7*tinfty11*tinfty17^2+30*alpha7*t infty13*tinfty15*tinfty17-15*alpha7*tinfty15^3):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 15 "rho1:=-mu1*p1:\n" }{MPLTEXT 1 0 14 "rho2:=-mu2*p2:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 661 "C0:=-(-q1^5*q2*tinfty17^2+q1*q2^5*tinfty17^2-2*q1^4* q2*tinfty15*tinfty17+q1^4*q2*tinfty16^2+2*q1*q2^4*tinfty15*tinfty17-q1 *q2^4*tinfty16^2-2*q1^3*q2*tinfty13*tinfty17+2*q1^3*q2*tinfty14*tinfty 16-q1^3*q2*tinfty15^2+2*q1*q2^3*tinfty13*tinfty17-2*q1*q2^3*tinfty14*t infty16+q1*q2^3*tinfty15^2+4*p1*q1^2*q2*tinfty16-4*p2*q1*q2^2*tinfty16 -2*q1^2*q2*tinfty11*tinfty17+2*q1^2*q2*tinfty12*tinfty16-2*q1^2*q2*tin fty13*tinfty15+q1^2*q2*tinfty14^2+2*q1*q2^2*tinfty11*tinfty17-2*q1*q2^ 2*tinfty12*tinfty16+2*q1*q2^2*tinfty13*tinfty15-q1*q2^2*tinfty14^2+4*p 1*q1*q2*tinfty14-4*p2*q1*q2*tinfty14+4*p1^2*q2+4*p1*q2*tinfty12-4*p2^2 *q1-4*p2*q1*tinfty12+4*h*p1-4*h*p2)/(4*(q1-q2)):\n" }{MPLTEXT 1 0 561 "C1:=1/4/(q1-q2)*(-q1^5*tinfty17^2+q2^5*tinfty17^2-2*q1^4*tinfty15*tin fty17+q1^4*tinfty16^2+2*q2^4*tinfty15*tinfty17-q2^4*tinfty16^2-2*q1^3* tinfty13*tinfty17+2*q1^3*tinfty14*tinfty16-q1^3*tinfty15^2+2*q2^3*tinf ty13*tinfty17-2*q2^3*tinfty14*tinfty16+q2^3*tinfty15^2+4*p1*q1^2*tinft y16-4*p2*q2^2*tinfty16-2*q1^2*tinfty11*tinfty17+2*q1^2*tinfty12*tinfty 16-2*q1^2*tinfty13*tinfty15+q1^2*tinfty14^2+2*q2^2*tinfty11*tinfty17-2 *q2^2*tinfty12*tinfty16+2*q2^2*tinfty13*tinfty15-q2^2*tinfty14^2+4*p1* q1*tinfty14-4*p2*q2*tinfty14+4*p1^2+4*p1*tinfty12-4*p2^2-4*p2*tinfty12 ):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 73 "CurlyL q1:=2*mu1*(p1-1/2*P1(q1))-h*nu0-h*nuMoins1*q1-h*(mu1+mu2)/(q1-q2):\n" }{MPLTEXT 1 0 72 "CurlyLq2:=2*mu2*(p2-1/2*P1(q2))-h*nu0-h*nuMoins1*q2- h*(mu2+mu1)/(q2-q1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 142 "CurlyLp1 :=h*(mu2+mu1)*(p2-p1)/(q1-q2)^2+mu1*(p1*dP1dlambda(q1)-dtdP2dlambda(q1 )+1*C1*q1^0) +h*nuMoins1*p1+ h*(1*c1*q1^0+2*c2*q1^1+3*c3*q1^2):\n" } {MPLTEXT 1 0 141 "CurlyLp2:=h*(mu1+mu2)*(p1-p2)/(q2-q1)^2+mu2*(p2*dP1d lambda(q2)-dtdP2dlambda(q2)+1*C1*q2^0) +h*nuMoins1*p2+ h*(1*c1*q2^0+2* c2*q2^1+3*c3*q2^2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 29 "CurlyLQ1:=CurlyLq1+CurlyLq2:\n" }{MPLTEXT 1 0 34 "Cur lyLQ2:=CurlyLq1*q2+q1*CurlyLq2:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "dP1dlambda:=unapply(diff(P1(lambda),lambda),lam bda):\n" }{MPLTEXT 1 0 53 "dP2dlambda:=unapply(diff(P2(lambda),lambda) ,lambda):\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 8 "L[2,1]: =" }{MPLTEXT 1 0 48 "-tdP2(lambda)+C1*lambda+C0 - h*p1/(lambda-q1)-h*" }{MPLTEXT 1 0 14 "p2/(lambda-q2)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 9 "L[2,2]:= " }{MPLTEXT 1 0 26 "P1(lambda) +h/(lambda-q1)" } {MPLTEXT 1 0 14 "+h/(lambda-q2)" }{MPLTEXT 1 0 3 " :\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "A:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 8 "A[1,1]: =" }{MPLTEXT 1 0 12 "c3*lambda^3+" }{MPLTEXT 1 0 60 "c2*lambda^2+c1*la mbda+c0+ rho1/(lambda-q1)+ rho2/(lambda-q2)" }{MPLTEXT 1 0 2 ":\n" } {MPLTEXT 1 0 8 "A[1,2]:=" }{MPLTEXT 1 0 53 "nuMoins1*lambda+nu0+ mu1/( lambda-q1)+ mu2/(lambda-q2)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 22 "A [2,1]:=AA21(lambda):\n" }{MPLTEXT 1 0 22 "A[2,2]:=AA22(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 73 "Q2Poly:=unapply( - p1*(lambda-q2)/(q1-q2)-p2*(lambda-q1)/(q2-q1),lambda);\n" }{MPLTEXT 1 0 11 "simplify(Q2" }{MPLTEXT 1 0 4 "Poly" }{MPLTEXT 1 0 7 "(q1));\n" } {MPLTEXT 1 0 11 "simplify(Q2" }{MPLTEXT 1 0 4 "Poly" }{MPLTEXT 1 0 7 " (q2));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "J:=Matrix(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 8 "J[2,1]:=" }{MPLTEXT 1 0 14 "Q2Poly(lambda)" }{MPLTEXT 1 0 12 "/(lambda-q1)" }{MPLTEXT 1 0 25 "/(lambda-q2)+1/2*tinfty16" } {MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 22 "J[2,2]:=1/(lambda-q1)/" } {MPLTEXT 1 0 11 "(lambda-q2)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 26 " dJdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 87 "for i from 1 to 2 do fo r j from 1 to 2 do dJdlambda[i,j]:=diff(J[i,j],lambda): od: od:\n" } {MPLTEXT 1 0 3 "J:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 78 "Ltilde:=s implify(Multiply(Multiply(J,L),J^(-1))+h*Multiply(dJdlambda,J^(-1))):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty01G6\",$I)tinfty12GF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty11G6\",$I)tinfty14GF$!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty21G6\",$I)tinfty16GF$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6\",**&I)tinfty11GF$\"\"\"I) tinfty17GF$F(#!\"\"\"\"#*&I)tinfty12GF$F(I)tinfty16GF$F(#F(F,*&I)tinft y13GF$F(I)tinfty15GF$F(F**$I)tinfty14GF$F,#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty32G6\",(*&I)tinfty13GF$\"\"\"I)tinfty17GF$F(#! \"\"\"\"#*&I)tinfty14GF$F(I)tinfty16GF$F(#F(F,*$I)tinfty15GF$F,#F+\"\" %" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty42G6\",&*&I)tinfty15GF$\" \"\"I)tinfty17GF$F(#!\"\"\"\"#*$I)tinfty16GF$F,#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty52G6\",$*$I)tinfty17GF$\"\"##!\"\"\"\"%" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I#P1G6\"f*6#I\"xGF$F$6$I)operatorGF$I& arrowGF$F$,(I)Pinfty01GF$\"\"\"*&I)Pinfty11GF$F-9$F-F-*&I)Pinfty21GF$F -F0\"\"#F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#P2G6\"f*6#I\"xGF$ F$6$I)operatorGF$I&arrowGF$F$,.I)Pinfty02GF$\"\"\"*&I)Pinfty12GF$F-9$F -F-*&I)Pinfty22GF$F-F0\"\"#F-*&I)Pinfty32GF$F-F0\"\"$F-*&I)Pinfty42GF$ F-F0\"\"%F-*&I)Pinfty52GF$F-F0\"\"&F-F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdP2G6\"f*6#I\"xGF$F$6$I)operatorGF$I&arrowGF$F$,**&9$ \"\"&I)tinfty17GF$\"\"##!\"\"\"\"%*&F-F3,&*&I)tinfty15GF$\"\"\"F/F8#F2 F0*$I)tinfty16GF$F0#F8F3F8F8*&F-\"\"$,(*&I)tinfty13GF$F8F/F8F9*&I)tinf ty14GF$F8F;F8#F8F0*$F7F0F1F8F8*&F-F0,**&I)tinfty11GF$F8F/F8F9*&I)tinft y12GF$F8F;F8FD*&FAF8F7F8F9*$FCF0FI'Q2PolyG6\"f*6#I'lambdaGF$F$6$I)operatorGF$I&arrowGF$F$ ,&*(I#p1GF$\"\"\",&9$F.I#q2GF$!\"\"F.,&I#q1GF$F.F1F2F2F2*(I#p2GF$F.,&F 0F.F4F2F.,&F1F.F4F2F2F2F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#p1G 6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#p2G6\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Elementaryh:= proc(k)\n" }{MPLTEXT 1 0 19 "local aux,i,Coeff:\n" }{MPLTEXT 1 0 55 "aux:=1: for i from 1 t o 2 do aux:=aux/(1-t*q[i]): od: \n" }{MPLTEXT 1 0 74 "Coeff:=unapply(e s(residue(aux/t^(k+1),t=0),q[1],q[2]),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 22 "return(Coeff(Q1,Q2)):\n" }{MPLTEXT 1 0 10 "end proc:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 7 "hh[0]:=" }{MPLTEXT 1 0 11 "Elemen taryh" }{MPLTEXT 1 0 5 "(0);\n" }{MPLTEXT 1 0 3 "hh[" }{MPLTEXT 1 0 4 "1]:=" }{MPLTEXT 1 0 11 "Elementaryh" }{MPLTEXT 1 0 5 "(1);\n" } {MPLTEXT 1 0 3 "hh[" }{MPLTEXT 1 0 4 "2]:=" }{MPLTEXT 1 0 11 "Elementa ryh" }{MPLTEXT 1 0 5 "(2);\n" }{MPLTEXT 1 0 18 "hh[3]:=Elementaryh" } {MPLTEXT 1 0 4 "(3);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "hh[4]:=El ementaryh(4);\n" }{MPLTEXT 1 0 22 "hh[5]:=Elementaryh(5);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "e[0]:=1:\n" }{MPLTEXT 1 0 17 "e[1]:=q[1]+q[2]:\n" }{MPLTEXT 1 0 17 "e[2]:=q[1]*q[2]:\n" } {MPLTEXT 1 0 45 "PolyP:=unapply(lambda^2-Q1*lambda+Q2,lambda);" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I#hhG6\"6#\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#hhG6\"6#\"\"\"I#Q1GF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#hhG6\"6#\"\"#,&*$I#Q1GF%F'\"\"\"I#Q2GF%!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I#hhG6\"6#\"\"$,&*$I#Q1GF%F'\"\"\"*&F* F+I#Q2GF%F+!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#hhG6\"6#\"\"%,(*$ I#Q1GF%F'\"\"\"*&F*\"\"#I#Q2GF%F+!\"$*$F.F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#hhG6\"6#\"\"&,(*$I#Q1GF%F'\"\"\"*&F*\"\"$I#Q2GF%F+!\" %*&F*F+F.\"\"#F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&PolyPG6\"f*6#I'la mbdaGF$F$6$I)operatorGF$I&arrowGF$F$,(*&I#Q1GF$\"\"\"9$F.!\"\"*$F/\"\" #F.I#Q2GF$F.F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 225 66 "The compat ibility equation reads \\mathcal\{L\}L=h\\partial_x A+[A,L]\n" }{TEXT 225 107 "Because the first line of L is trivial (0,1) we may obtain A[ 2,1] and A[2,2] to complete the knowledge of A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "CurlyL:=h*dAdlambda+(Multiply(A,L)-Multiply(L,A )):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "Entree11:=CurlyL[1,1]:\n " }{MPLTEXT 1 0 23 "Entree12:=CurlyL[1,2]:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 54 "AA21:=unapply(solve(Entree11=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 54 "AA22:=unapply(solve(Entree12=0,AA22(lambda)),lam bda):\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)-A A22bis);\n" }{MPLTEXT 1 0 20 "simplify(Entree11);\n" }{MPLTEXT 1 0 20 "simplify(Entree12);\n" }{MPLTEXT 1 0 50 "CurlyL:=h*dAdlambda+(Multipl y(A,L)-Multiply(L,A)):" }}{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 11 "e1:=q1+q2:\n" }{MPLTEXT 1 0 11 "e2:=q1*q2:\n" } {MPLTEXT 1 0 35 "p1:=P1*diff(e1,q1)+P2*diff(e2,q1);\n" }{MPLTEXT 1 0 35 "p2:=P1*diff(e1,q2)+P2*diff(e2,q2);\n" }{MPLTEXT 1 0 111 "Q2Polyfun ction:=unapply(es(simplify(-p1*(lambda-q2)/(q1-q2)-p2*(lambda-q1)/(q2- q1)),q1,q2),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 17 "Q2Poly:=unapply( " }{MPLTEXT 1 0 25 "Q2Polyfunction(Q[1],Q[2])" }{MPLTEXT 1 0 9 ",lambd a);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "Q[0]:= 1:\n" }{MPLTEXT 1 0 10 "Q[1]:=Q1:\n" }{MPLTEXT 1 0 10 "Q[2]:=Q2:\n" } {MPLTEXT 1 0 10 "P[1]:=P1:\n" }{MPLTEXT 1 0 10 "P[2]:=P2:\n" }{MPLTEXT 1 0 10 "q[1]:=q1:\n" }{MPLTEXT 1 0 10 "q[2]:=q2:\n" }{MPLTEXT 1 0 10 "p[1]:=p1:\n" }{MPLTEXT 1 0 10 "p[2]:=p2:\n" }{MPLTEXT 1 0 20 "tinfty[ 1]:=tinfty11:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 21 "tinfty[2]:=tinft y12:\n" }{MPLTEXT 1 0 21 "tinfty[3]:=tinfty13:\n" }{MPLTEXT 1 0 21 "ti nfty[4]:=tinfty14:\n" }{MPLTEXT 1 0 21 "tinfty[5]:=tinfty15:\n" } {MPLTEXT 1 0 21 "tinfty[6]:=tinfty16:\n" }{MPLTEXT 1 0 21 "tinfty[7]:= tinfty17:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 20 "C0function:=unappl y(" }{MPLTEXT 1 0 43 "es(simplify(C0),q1,q2),sigma[1],sigma[2]):\n" } {MPLTEXT 1 0 63 "C1function:=unapply(es(simplify(C1),q1,q2),sigma[1],s igma[2]):\n" }{MPLTEXT 1 0 29 "C[0]:=C0function(Q[1],Q[2]):\n" } {MPLTEXT 1 0 28 "C[1]:=C1function(Q[1],Q[2]):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#p1G6\",&*&I#P2GF$\"\"\"I#q2GF$F(F(I#P1GF$F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#p2G6\",&*&I#P2GF$\"\"\"I#q1GF$F(F(I#P1GF$F( " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Q2PolyG6\"f*6#I'lambdaGF$F$6$I)op eratorGF$I&arrowGF$F$,(*&I#P2GF$\"\"\"9$F.F.*&F-F.&I\"QGF$6#F.F.!\"\"I #P1GF$F4F$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "solve(\{" }{MPLTEXT 1 0 36 "pp1=PP1*diff(e1,q1)+PP2*diff(e2,q1)," }{MPLTEXT 1 0 49 "pp2=PP1*diff(e1,q2)+PP2*diff(e2,q2)\},\{PP1,PP2\}):\n" }{MPLTEXT 1 0 32 "PP1 := (pp1*q1-pp2*q2)/(q1-q2):\n" }{MPLTEXT 1 0 27 "PP2 := -( pp1-pp2)/(q1-q2):\n" }{MPLTEXT 1 0 49 "CurlyLP1fonction:=unapply(diff( PP1,pp1)*CurlyLp1+" }{MPLTEXT 1 0 23 "diff(PP1,pp2)*CurlyLp2+" } {MPLTEXT 1 0 22 "diff(PP1,q1)*CurlyLq1+" }{MPLTEXT 1 0 32 "diff(PP1,q2 )*CurlyLq2,pp1,pp2):\n" }{MPLTEXT 1 0 49 "CurlyLP2fonction:=unapply(di ff(PP2,pp1)*CurlyLp1+" }{MPLTEXT 1 0 45 "diff(PP2,pp2)*CurlyLp2+diff(P P2,q1)*CurlyLq1+" }{MPLTEXT 1 0 31 "diff(PP2,q2)*CurlyLq2,pp1,pp2):" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "CurlyLP1:=simplify(" }{MPLTEXT 1 0 25 "CurlyLP1fonction(p1,p2)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 44 "CurlyLP2:=simplify(CurlyLP2fonction(p1,p2)):" }{MPLTEXT 1 0 2 " \n " }{MPLTEXT 1 0 53 "simplify(CurlyLp1-CurlyLP2*q2-P2*CurlyLq2-CurlyLP1 );\n" }{MPLTEXT 1 0 52 "simplify(CurlyLp2-CurlyLP2*q1-P2*CurlyLq1-Curl yLP1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 65 "Expression of \\td\{L\} in the Darboux coordinates (q_1,q_2,p_1,p_2)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Ltilde11:=simplify(Ltilde[1,1]);\n" }{MPLTEXT 1 0 10 "Ltilde12:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 14 "Ltilde[1,2]);\n" }{MPLTEXT 1 0 10 "Ltilde21:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 14 "Ltilde[2,1]):\n" }{MPLTEXT 1 0 10 "Lti lde22:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 13 "Ltilde[2,2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Ltilde11G6\",**(,&I'lambdaGF$!\"\"I #q2GF$\"\"\"F+,&F(F)I#q1GF$F+F+I)tinfty16GF$F+#F)\"\"#*&I#P2GF$F+F(F+F )*&,&F-F0F*F0F+F2F+#F+F0I#P1GF$F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I) Ltilde12G6\"*&,&I'lambdaGF$!\"\"I#q1GF$\"\"\"F*,&F'F(I#q2GF$F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)Ltilde22G6\",.*&I)tinfty16GF$\"\"\"I' lambdaGF$\"\"##!\"\"F**&,(*&,&I#q1GF$F,I#q2GF$F,F(F'F(F(I#P2GF$F*I)tin fty14GF$!\"#F(F)F(#F(F**(F1F(F2F(F'F(F6*&,&F1F5F2F5F(F3F(F6I#P1GF$F,I) tinfty12GF$F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 65 "Expression of \+ \\td\{L\} in the Darboux coordinates (Q_1,Q_2,P_1,P_2)" }{TEXT 216 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Ltilde11NewVar:=unapply ( es(simplify(" }{MPLTEXT 1 0 8 "Ltilde11" }{MPLTEXT 1 0 31 "), q1, q2 ),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 75 "Ltilde12NewVar:=unapply( es (simplify(Ltilde12), q1, q2),sigma[1],sigma[2]):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 76 "Ltilde22NewVar:=unapply( es(simplify(Ltilde22), q1, q2),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 22 "Ltilde11NewVar(Q1,Q2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Ltilde12NewVar(Q1,Q2);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Ltilde22NewVar(Q1,Q2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&I)tinfty16G6\"\"\"\"I'lambdaGF%\"\"##!\"\" F(*(F'F&F$F&I#Q1GF%F&#F&F(*&I#P2GF%F&F'F&F**&F/F&F,F&F&*&F$F&I#Q2GF%F& F)I#P1GF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I#Q1G6\"\"\"\"I'lambd aGF%F&!\"\"*$F'\"\"#F&I#Q2GF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*&I )tinfty16G6\"\"\"\"I'lambdaGF%\"\"##!\"\"F(*(F'F&F$F&I#Q1GF%F&F)*&I#P2 GF%F&F'F&F&*&F.F&F,F&F**&I)tinfty14GF%F&F'F&F**&F$F&I#Q2GF%F&#F&F(I#P1 GF%F*I)tinfty12GF%F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 78 "Let us n ow check that the matrix \\td\{L\} matches with the theoretical formul as." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ltilde11Theo:=proc() \n" }{MPLTEXT 1 0 19 "local res,j,aux,i:\n" }{MPLTEXT 1 0 8 "res:=0:\n " }{MPLTEXT 1 0 141 "for j from 0 to g-1 do aux:=0: if j+1<=g then fo r i from j+1 to g do aux:=aux+P[i]*Q[i-j-1]: od: fi: res:=res-(-1)^(j -1)*aux*lambda^j: od:\n" }{MPLTEXT 1 0 75 "for j from 0 to g do res:=r es-1/2*tinfty16*(-1)^(g-j)*Q[g-j]*lambda^j: od:\n" }{MPLTEXT 1 0 13 "r eturn(res):\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 15 "Ltilde11Theo();" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "simplify(Ltilde11NewVar(Q1,Q2)-" }{MPLTEXT 1 0 16 "Ltilde11Theo()) ;" }{MPLTEXT 1 0 1 " " }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&I)tinfty16G 6\"\"\"\"I'lambdaGF%\"\"##!\"\"F(*(F'F&F$F&I#Q1GF%F&#F&F(*&I#P2GF%F&F' F&F**&F/F&F,F&F&*&F$F&I#Q2GF%F&F)I#P1GF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ltild e12Theo:=proc()\n" }{MPLTEXT 1 0 13 "local res,m:\n" }{MPLTEXT 1 0 8 " res:=0:\n" }{MPLTEXT 1 0 63 "for m from 0 to g do res:=res+(-1)^(g-m) *Q[g-m]*lambda^m: od:\n" }{MPLTEXT 1 0 13 "return(res):\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "Ltilde12Th eo();\n" }{MPLTEXT 1 0 47 "simplify(Ltilde12NewVar(Q1,Q2)-Ltilde12Theo ());" }{MPLTEXT 1 0 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I#Q1G6\"\"\"\"I'lambdaGF%F&!\"\"*$F'\"\"#F&I#Q2GF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ltilde22Theo:=proc()\n" }{MPLTEXT 1 0 21 "local res,j ,aux,i,s:\n" }{MPLTEXT 1 0 8 "res:=0:\n" }{MPLTEXT 1 0 141 "for j from 0 to g-1 do aux:=0: if j+1<=g then for i from j+1 to g do aux:=aux+P [i]*Q[i-j-1]: od: fi: res:=res+(-1)^(j-1)*aux*lambda^j: od:\n" } {MPLTEXT 1 0 75 "for j from 0 to g do res:=res+1/2*tinfty16*(-1)^(g-j) *Q[g-j]*lambda^j: od:\n" }{MPLTEXT 1 0 66 "for s from 0 to rinfty-2 do res:=res- tinfty[2*s+2]*lambda^s: od:\n" }{MPLTEXT 1 0 13 "return(res ):\n" }{MPLTEXT 1 0 9 "end proc:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "Ltilde22Theo();\n" }{MPLTEXT 1 0 48 "simplify (Ltilde22NewVar(Q1,Q2)-Ltilde22Theo()); " }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*&I)tinfty16G6\"\"\"\"I'lambdaGF%\"\"##!\"\" F(*(F'F&F$F&I#Q1GF%F&F)*&I#P2GF%F&F'F&F&*&F.F&F,F&F**&I)tinfty14GF%F&F 'F&F**&F$F&I#Q2GF%F&#F&F(I#P1GF%F*I)tinfty12GF%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Ltild e21Theo:=proc()\n" }{MPLTEXT 1 0 33 "local res,j,i,j1,j2,i1,i2,r,s,m: \n" }{MPLTEXT 1 0 8 "res:=0:\n" }{MPLTEXT 1 0 69 "for i from 0 to rinf ty-1 do for j from g+i to 2*rinfty-3 do res:=res-" }{MPLTEXT 1 0 20 "P infty2[j]*hh[j-g-i]" }{MPLTEXT 1 0 19 "*lambda^i: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 93 "for i from 0 to g-1 do for j from i to g do for s from g+i-j to g do for r from j+1 to g do \n" }{MPLTEXT 1 0 9 "res:=res+" }{MPLTEXT 1 0 1 " " }{MPLTEXT 1 0 79 "(-1)^(j-1)*tinfty[ 2*s+2]*P[r]*Q[r-j-1]*hh[s+j-i-g]*lambda^(i): od: od: od: od:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 168 "for i from 0 to g-2 do for j1 f rom i+1 to g-1 do for j2 from g+i-j1 to g-1 do for i1 from j1+1 to g d o for i2 from j2+1 to g do res:=res- (-1)^(j1+j2)*P[i1]*Q[i1-j1-1]*" } {MPLTEXT 1 0 41 "P[i2]*Q[i2-j2-1]*hh[j1+j2-g-i]*lambda^i:\n" }{MPLTEXT 1 0 20 "od: od: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 87 "for m from 0 to g do res:=res-1/4*tinfty[2*rinfty-2]^2*(-1)^(g-m)*Q[g -m]*lambda^m: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 81 "for s from 0 to g do res:=res+1/2*tinfty[2*rinfty-2]*tinfty[2*s+2]*lambda^s: od: \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 118 "for j from 0 to g-1 do for i from j+1 to g do res:=res-tinfty[2*rinfty-2]*(-1)^(j-1)*P[i]*Q[i-j- 1]*lambda^j: od: od: \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 13 "return (res):\n" }{MPLTEXT 1 0 9 "end proc:" }{MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Ltilde21NewVar:=unapply( es(simplif y(Ltilde21), q1, q2),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 22 "Ltilde21 NewVar(Q1,Q2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 72 "factor(series(s implify(Ltilde21NewVar(Q1,Q2)-Ltilde21Theo()),lambda=0));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ",N*&I)tinfty17G6\"\"\"#I'lamb daGF%\"\"$#\"\"\"\"\"%*(F'F&F$F&I#Q1GF%F*F)*(F'F*F$F&F-F&F)*&F$F&F-F(F )*(F'F&I)tinfty15GF%F*F$F*#F*F&**F'F*F1F*F$F*F-F*F2*(F'F*F$F&I#Q2GF%F* #!\"\"F+*(F1F*F$F*F-F&F2*&I)tinfty16GF%F&F-F&F6*(F$F&F-F*F5F*#F7F&*(I# P2GF%F*F:F*F-F*F**(F'F*I)tinfty13GF%F*F$F*F2*&F'F*F1F&F)*(F@F*F$F*F-F* F2*(I)tinfty14GF%F*F:F*F-F*F<*&F1F&F-F*F)*(F1F*F$F*F5F*F<*$F>F&F7*&FDF *F>F*F**&I)tinfty11GF%F*F$F*F2*&F@F*F1F*F2*$FDF&F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 105 "Let us n ow compute the auxiliary matrix \\td\{A\} and check that it correspond s to the theoretical formulas." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "J:=simplify(J):\n" }{MPLTEXT 1 0 24 "CurlyLJ:=Matrix(2,2,0):\n " }{MPLTEXT 1 0 57 "for i from 1 to 2 do for j from 1 to 2 do CurlyLJ[ i,j]:=\n" }{MPLTEXT 1 0 207 "h*(alpha7*diff(J[i,j],tinfty17)+alpha6*di ff(J[i,j],tinfty16)+alpha5*diff(J[i,j],tinfty15)+alpha4*diff(J[i,j],ti nfty14)+alpha3*diff(J[i,j],tinfty13)+alpha2*diff(J[i,j],tinfty12)+alph a1*diff(J[i,j],tinfty11))\n" }{MPLTEXT 1 0 26 "+CurlyLq1*diff(J[i,j],q 1)+" }{MPLTEXT 1 0 24 "CurlyLq2*diff(J[i,j],q2)" }{MPLTEXT 1 0 26 "+Cu rlyLP1*diff(J[i,j],P1)+" }{MPLTEXT 1 0 24 "CurlyLP2*diff(J[i,j],P2)" } {MPLTEXT 1 0 9 ": od: od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 74 "Atil de:=simplify(Multiply(Multiply(J,A),J^(-1))+Multiply(CurlyLJ,J^(-1))): " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 107 "Let us n ow check that the formula for \\td\{A\} matches with the theoretical f ormulas for each of its entries." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Atilde12Theo:=proc()\n" }{MPLTEXT 1 0 15 "local res,j,m:\n" } {MPLTEXT 1 0 8 "res:=0:\n" }{MPLTEXT 1 0 110 "for j from 0 to g+1 do f or m from max(-1,-j) to g-j do res:=res+(-1)^(g-j-m)*nu[m]*Q[g-j-m]*la mbda^j: od: od:\n" }{MPLTEXT 1 0 13 "return(res):\n" }{MPLTEXT 1 0 9 " end proc:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Atilde12fonction:=unapply(es(Atilde[1,2],q[1],q[2]),sigma[1],sig ma[2]):\n" }{MPLTEXT 1 0 48 "Atilde12NewCoordinates:=Atilde12fonction( Q1,Q2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 48 "simplify(Atilde12NewCo ordinates-Atilde12Theo());" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Atilde11Theo:=proc()\n" } {MPLTEXT 1 0 19 "local res,i,r,j,m:\n" }{MPLTEXT 1 0 8 "res:=0:\n" } {MPLTEXT 1 0 57 "for i from 0 to rinfty-1 do res:=res+c[i]*lambda^i: \+ od:\n" }{MPLTEXT 1 0 32 "for i from 0 to g do for m from " }{MPLTEXT 1 0 12 "(max(-1,-i))" }{MPLTEXT 1 0 102 " to g-1-i do for r from i+m+1 to g do res:=res+(-1)^(i+m)*nu[m]*P[r]*Q[r-i-m-1]*lambda^i: od: od: o d:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 135 "for i from 0 to g+1 do f or m from (max(-1,-i)) to g-i do res:=res-1/2*tinfty[2*rinfty-2]*(-1)^ (g-i-m)*Q[g-i-m]*nu[m]*lambda^i: od: od:\n" }{MPLTEXT 1 0 13 "return(r es):\n" }{MPLTEXT 1 0 9 "end proc:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Atilde11fonction:=unapply(es(simplify(Atilde[1,1]),q[ 1],q[2]),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 48 "Atilde11NewCoordinat es:=Atilde11fonction(Q1,Q2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 48 "s implify(Atilde11NewCoordinates-Atilde11Theo());" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Atild e22Theo:=proc()\n" }{MPLTEXT 1 0 15 "local res,s,j:\n" }{MPLTEXT 1 0 6 "res:=-" }{MPLTEXT 1 0 14 "Atilde11Theo()" }{MPLTEXT 1 0 2 ":\n" } {MPLTEXT 1 0 30 "res:=res+2*c0+h*(g+1)*nu[-1]:\n" }{MPLTEXT 1 0 66 "fo r s from 1 to rinfty-1 do res:=res-1/s*alpha[2*s]*lambda^s: od:\n" } {MPLTEXT 1 0 62 "for j from 0 to rinfty-2 do res:=res-tinfty[2*j+2]*nu [j]: od:\n" }{MPLTEXT 1 0 13 "return(res):\n" }{MPLTEXT 1 0 9 "end pro c:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Atilde22fonction:=unapply( es(simplify(Atilde[2,2]),q[1],q[2]),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 49 "Atilde22NewCoordinates:=Atilde22fonction(Q1,Q2):\n" }{MPLTEXT 1 0 48 "simplify(Atilde22NewCoordinates-Atilde22Theo());" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Atilde21Theo:=proc()\n" }{MPLTEXT 1 0 33 "local r es,i,j,s,m,r,j1,j2,r1,r2:\n" }{MPLTEXT 1 0 111 "res:=-1/2*h*(g+1)*nu[- 1]*tinfty[2*rinfty-2]+h/2*alpha[2*rinfty-2]+h*(rinfty-1)*c[rinfty-1]+n u[-1]*C[rinfty-3]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "for i fro m 0 to g do for j from " }{MPLTEXT 1 0 10 "max(0,i-1)" }{MPLTEXT 1 0 89 " to g-1 do for s from g+i-j-1 to g do for r from j+1 to g do for m from -1 to s+j-g-i do\n" }{MPLTEXT 1 0 75 "res:=res-(-1)^j*tinfty[2*s +2]*nu[m]*hh[s+j-g-m-i]*P[r]*Q[r-j-1]*lambda^i: \n" }{MPLTEXT 1 0 19 " od: od: od: od: od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 96 "for i from 0 to rinfty do for j from (max(g,g+i-1)) t o 2*rinfty-3 do for m from -1 to j-g-i do \n" }{MPLTEXT 1 0 9 "res:=re s-" }{MPLTEXT 1 0 51 "nu[m]*hh[j-g-m-i]*Pinfty2[j]*lambda^i: od: od: o d:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 "for i from 0 to g do for \+ j1 from 0 to g-1 do for j2 from 0 to g-1 do\n" }{MPLTEXT 1 0 22 "if j1 +j2-g-i>=-1 then " }{MPLTEXT 1 0 29 "for m from -1 to j1+j2-g-i do" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 52 "for r1 from j1+1 to g do for r2 \+ from j2+1 to g do \n" }{MPLTEXT 1 0 9 "res:=res-" }{MPLTEXT 1 0 80 "( -1)^(j1+j2)*nu[m]*hh[j1+j2-g-i-m]*P[r1]*P[r2]*Q[r1-j1-1]*Q[r2-j2-1]*la mbda^i: \n" }{MPLTEXT 1 0 14 " od: od: od: " }{MPLTEXT 1 0 4 "fi: " } {MPLTEXT 1 0 12 "od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 137 "for i from 0 to rinfty-1 do for s from max(0,i-1) to rinfty-2 do \+ res:=res+1/2*tinfty[2*rinfty-2]*tinfty[2*s+2]*nu[s-i]*lambda^i: od: od :\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "for i from 0 to g do for j from " }{MPLTEXT 1 0 10 "max(0,i-1)" }{MPLTEXT 1 0 43 " to g-1 do for r from j+1 to g do res:=res-" }{MPLTEXT 1 0 74 "tinfty[2*rinfty-2]*(- 1)^(j-1)*nu[j-i]*P[r]*Q[r-j-1]*lambda^i: od: od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "for i from 0 to g+1 do for j from " }{MPLTEXT 1 0 10 "max(0,i-1)" }{MPLTEXT 1 0 88 " to g do res:=res-1/4*(tinfty[2 *rinfty-2])^2*(-1)^(g-j)*Q[g-j]*nu[j-i]*lambda^i: od: od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 13 "return(res):\n" } {MPLTEXT 1 0 9 "end proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Atilde21fonction:=unapply(es(simplify(Atilde[2,1]),q[1],q[2]),sigm a[1],sigma[2]):\n" }{MPLTEXT 1 0 48 "Atilde21NewCoordinates:=Atilde21f onction(Q1,Q2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "Atilde21NewCoo rdinates:\nsimplify(" }{MPLTEXT 1 0 49 "factor(Atilde21NewCoordinates- Atilde21Theo())):\n\n" }{MPLTEXT 1 0 79 "nu3fonction:=unapply(es(simpl ify(q1^g*mu1+q2^g*mu2),q1,q2),sigma[1],sigma[2]):\n" }{MPLTEXT 1 0 20 "nu3fonction(Q1,Q2):\n" }{MPLTEXT 1 0 27 "nu[3]:=nu3fonction(Q1,Q2):\n " }{MPLTEXT 1 0 46 "factor(Atilde21NewCoordinates-Atilde21Theo());" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "VerifSomme:=proc()\n" }{MPLTEXT 1 0 13 "local res,k: \n" }{MPLTEXT 1 0 8 "res:=0:\n" }{MPLTEXT 1 0 55 "for k from 0 to g-1 \+ do res:=res+(g-k)*Q[k]*P[k+1]: od:\n" }{MPLTEXT 1 0 13 "return(res):\n " }{MPLTEXT 1 0 11 "end proc:\n\n" }{MPLTEXT 1 0 14 "VerifSomme();\n" }{MPLTEXT 1 0 26 "es(simplify(p1+p2),q1,q2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I#P2G6\"\"\"\"I#Q1GF%F&F&I#P1GF%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I#P2G6\"\"\"\"&I&sigmaGF%6#F&F&F&I#P1GF%\"\"#" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{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 }