{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 "2D Math Itali c Small209212" -1 200 "Times" 1 1 0 0 0 1 1 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 "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 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 "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 "Header and Footer" -1 203 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small204" -1 204 "Times " 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small207208 " -1 205 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Ita lic 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 Italic Small208" -1 209 "Ti mes" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small207 " -1 210 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Ita lic Small209" -1 211 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 212 "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 "Defaul t" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Itali c Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 213 "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 "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small211" -1 214 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Label" -1 215 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small210" -1 216 "Times " 1 1 0 0 0 1 1 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" -1 217 "Times " 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small220" -1 218 "Times" 1 1 0 0 0 1 1 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 "2D Mat h Italic Small226" -1 219 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic Small225" -1 220 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small228" -1 221 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small227" -1 222 "Time s" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small222" -1 223 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Itali c Small221" -1 224 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2 D Math Italic Small224" -1 225 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small223" -1 226 "Times" 1 1 0 0 0 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 "2D Math Italic Small229" -1 227 "Ti mes" 1 1 0 0 0 1 1 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 "2D Math Italic Small231" -1 228 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Ital ic Small230" -1 229 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE " 2D Math Italic Small237" -1 230 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small236" -1 231 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small239" -1 232 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small238" -1 233 "T imes" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small23 3" -1 234 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math It alic Small232" -1 235 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small235" -1 236 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small234" -1 237 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small210220" -1 238 "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 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 239 "Ti mes" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math Italic S mall240" -1 240 "Times" 1 1 0 0 0 1 1 2 2 2 2 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 "2D M ath Italic Small242" -1 241 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic Small241" -1 242 "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 Small244" -1 243 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small243" -1 244 "Tim es" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small245" -1 245 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Ital ic Small" -1 246 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D \+ Math Italic Small211223" -1 247 "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" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small211221" -1 248 "Times" 1 1 0 0 0 1 1 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 "2D Math Italic Small211222" -1 249 " Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small2 11220" -1 250 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Annota tion Text" -1 251 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "H elp Maple Name" -1 35 "Times" 1 12 104 64 92 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" -1 252 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small208209" -1 253 "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 "" -1 254 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 255 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 254 240 "In this Maple file, we \+ compute the evolution equations for the Painlev\351 4 equation using t he compatibility equation of the Lax system. \nWe also obtain the expr ession of the Lax matrices in the geometric gauge without apparent sin gularities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" } {MPLTEXT 1 0 20 "with(LinearAlgebra):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 17 "P011:=sX10+sX20;\n" }{MPLTEXT 1 0 17 "P022:=sX10*sX20;\n" } {MPLTEXT 1 0 20 "Pinfty11:=-s12-s22;\n" }{MPLTEXT 1 0 20 "Pinfty01:=-s 11-s21;\n" }{MPLTEXT 1 0 19 "Pinfty22:=s12*s22;\n" }{MPLTEXT 1 0 27 "P infty12:=s11*s22+s12*s21;\n" }{MPLTEXT 1 0 35 "Pinfty02:=s12*s20+s10*s 22+s11*s21;\n" }{MPLTEXT 1 0 38 "CoherenceEquation:=s10+s20+sX10+sX20; \n" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 52 "R1:=unapply( P011/(xi-X1)+ Pinfty01+Pinfty11*xi,xi);\n" }{MPLTEXT 1 0 4 "R2:=" }{MPLTEXT 1 0 8 "u napply(" }{MPLTEXT 1 0 7 " P022/(" }{MPLTEXT 1 0 15 "xi-X1)^2+P012/(" }{MPLTEXT 1 0 25 "xi-X1)+Pinfty02+Pinfty12*" }{MPLTEXT 1 0 12 "xi+Pinf ty22*" }{MPLTEXT 1 0 9 "xi^2,xi);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 21 "tdR2:=unapply(R2(xi)-" }{MPLTEXT 1 0 17 "P012/(xi-X1),xi);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 46 "c2bis:=(bet a12*s22-beta22*s12)/(2*(s12-s22));\n" }{MPLTEXT 1 0 95 "c1bis:=(1/2)*( s12*s21-s11*s22)/(s12-s22)^2*(beta12-beta22)+ (beta11*s22-beta21*s12)/ (s12-s22);\n" }{MPLTEXT 1 0 7 "mubis:=" }{MPLTEXT 1 0 113 "(1/2)*(X1*( s12-s22)-s11+s21)*(Q-X1)/(-s22+s12)^2*(beta12-beta22)+(beta11-beta21)* (Q-X1)/(s12-s22)+(Q-X1)*betaX1;\n" }{MPLTEXT 1 0 13 "nuMinus1bis:=" } {MPLTEXT 1 0 32 "(beta12-beta22)/(2*(-s22+s12));\n" }{MPLTEXT 1 0 8 "n u0bis:=" }{MPLTEXT 1 0 70 "(1/2)*(s21-s11)/(s12-s22)^2*(beta12-beta22) +(beta11-beta21)/(s12-s22);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 37 "dR1dxi:=unapply(diff(R1(xi),xi ),xi):\n" }{MPLTEXT 1 0 37 "dR2dxi:=unapply(diff(R2(xi),xi),xi):\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 60 "L[2,1]:=-R2(xi)+P012 /(xi-X1) +C/(xi-X1) -h*s12 -h*P/(xi-Q):\n" }{MPLTEXT 1 0 36 "L[2,2]:= \+ R1(xi)-h/(xi-X1)+h/(xi-Q);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 8 "C0 1:=C:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "A:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 40 "A[1,1]:= c2*xi^2+ c1*xi +c0+rho/(xi-Q):\n" }{MPLTEXT 1 0 35 "A[1,2]:=nuMinus1*xi+nu0+mu/(xi-Q):\n" }{MPLTEXT 1 0 19 "A[2,1] := AA21(xi):\n" }{MPLTEXT 1 0 19 "A[2,2]:= AA22(xi):\n" }{MPLTEXT 1 0 22 "dAdxi:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 79 "for i from 1 to 2 do fo r j from 1 to 2 do dAdxi[i,j]:=diff(A[i,j],xi): od: od:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 3 "L;\n" }{MPLTEXT 1 0 3 " A;\n" }{MPLTEXT 1 0 4 "Q2:=" }{MPLTEXT 1 0 22 "unapply(-P*(Q-X1),xi):" }{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 23 "J[2,1]:=Q2(xi)/(xi-Q):\n" }{MPLTEXT 1 0 26 "J[2,2]:=(xi-X1)^1/(xi- Q):\n" }{MPLTEXT 1 0 3 "J;\n" }{MPLTEXT 1 0 22 "dJdxi:=Matrix(2,2,0): \n" }{MPLTEXT 1 0 79 "for i from 1 to 2 do for j from 1 to 2 do dJdxi[ i,j]:=diff(J[i,j],xi): 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 71 "LJ[2,2] :=diff(J[2,2],Q)*LQ+diff(J[2,2],P)*LP+h*diff(J[2,2],X1)*betaX1:\n" } {MPLTEXT 1 0 71 "LJ[2,1]:=diff(J[2,1],Q)*LQ+diff(J[2,1],P)*LP+h*diff(J [2,1],X1)*betaX1:\n" }{MPLTEXT 1 0 4 "LJ:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 75 "checkL:=simplify(Multiply(Multiply(J,L),J^(-1))+h*Mul tiply(dJdxi,J^(-1))):\n" }{MPLTEXT 1 0 69 "checkA:=simplify(Multiply(M ultiply(J,A),J^(-1))+Multiply(LJ,J^(-1))):" }{MPLTEXT 1 0 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I%P011G6\",&I%sX10GF$\"\"\"I%sX20GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%P022G6\"*&I%sX10GF$\"\"\"I%sX20GF$ F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty11G6\",&I$s12GF$!\"\"I$s2 2GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty01G6\",&I$s11GF$!\"\" I$s21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty22G6\"*&I$s12GF$ \"\"\"I$s22GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty12G6\",&*&I $s11GF$\"\"\"I$s22GF$F(F(*&I$s12GF$F(I$s21GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)Pinfty02G6\",(*&I$s10GF$\"\"\"I$s22GF$F(F(*&I$s11GF$F( I$s21GF$F(F(*&I$s12GF$F(I$s20GF$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I2CoherenceEquationG6\",*I$s10GF$\"\"\"I$s20GF$F'I%sX10GF$F'I%sX20GF$ F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#R1G6\"f*6#I#xiGF$F$6$I)operator GF$I&arrowGF$F$,**&,&I%sX10GF$\"\"\"I%sX20GF$F/F/,&9$F/I#X1GF$!\"\"F4F /I$s11GF$F4I$s21GF$F4*&,&I$s12GF$F4I$s22GF$F4F/F2F/F/F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#R2G6\"f*6#I#xiGF$F$6$I)operatorGF$I&arrowGF $F$,0*(I%sX10GF$\"\"\"I%sX20GF$F.,&9$F.I#X1GF$!\"\"!\"#F.*&I%P012GF$F. F0F3F.*&I$s10GF$F.I$s22GF$F.F.*&I$s11GF$F.I$s21GF$F.F.*&I$s12GF$F.I$s2 0GF$F.F.*&,&*&F;F.F9F.F.*&F>F.FF.F9F.F1\"\"#F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdR2G6\"f*6#I#xiGF$F$6$I)operatorG F$I&arrowGF$F$,.*(I%sX10GF$\"\"\"I%sX20GF$F.,&9$F.I#X1GF$!\"\"!\"#F.*& I$s10GF$F.I$s22GF$F.F.*&I$s11GF$F.I$s21GF$F.F.*&I$s12GF$F.I$s20GF$F.F. *&,&*&F9F.F7F.F.*&FI&c2bisG6\"*&,&*&I'beta12GF$\"\"\"I$s22 GF$F)F)*&I'beta22GF$F)I$s12GF$F)!\"\"F),&F-\"\"#F*!\"#F." }}{PARA 11 " " 1 "" {XPPMATH 20 ">I&c1bisG6\",&*(,&*&I$s11GF$\"\"\"I$s22GF$F*!\"\"* &I$s12GF$F*I$s21GF$F*F*F*,&F.F*F+F,!\"#,&I'beta12GF$F*I'beta22GF$F,F*# F*\"\"#*&,&*&I'beta11GF$F*F+F*F**&I'beta21GF$F*F.F*F,F*F0F,F*" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I&mubisG6\",(**,(*&I#X1GF$\"\"\",&I$s12 GF$F*I$s22GF$!\"\"F*F*I$s11GF$F.I$s21GF$F*F*,&I\"QGF$F*F)F.F*F+!\"#,&I 'beta12GF$F*I'beta22GF$F.F*#F*\"\"#*(,&I'beta11GF$F*I'beta21GF$F.F*F1F *F+F.F**&F1F*I'betaX1GF$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,nuMin us1bisG6\"*&,&I'beta12GF$\"\"\"I'beta22GF$!\"\"F(,&I$s12GF$\"\"#I$s22G F$!\"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'nu0bisG6\",&*(,&I$s21GF$ \"\"\"I$s11GF$!\"\"F),&I$s12GF$F)I$s22GF$F+!\"#,&I'beta12GF$F)I'beta22 GF$F+F)#F)\"\"#*&,&I'beta11GF$F)I'beta21GF$F+F)F,F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I\"LG6\"6$\"\"#F',.*&,&I%sX10GF%\"\"\"I%sX20GF%F,F, ,&I#xiGF%F,I#X1GF%!\"\"F1F,I$s11GF%F1I$s21GF%F1*&,&I$s12GF%F1I$s22GF%F 1F,F/F,F,*&I\"hGF%F,F.F1F1*&F9F,,&F/F,I\"QGF%F1F1F," }}{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(-I#mnGF$6%Q\"0F'/%+ foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,colu mnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/%+columnspanGFJ-F56(-F86%F JF;F>FAFDFFFHFKFAFDFF-F26'-F56(-F,64-I#moGF$6.Q*&uminus0;F'F;F>/%&fenc eGQ&falseF'/%*separatorGFgn/%)stretchyGFgn/%*symmetricGFgn/%(largeopGF gn/%.movablelimitsGFgn/%'accentGFgn/%'lspaceGQ,0.2222222emF'/%'rspaceG Ffo-I&mfracGF$6)-F,6%-I#miGF$6&Q%sX10F'/%'italicGQ%trueF'F;/F?Q'italic F'-FX6-Q1⁢F'F>FenFhnFjnF\\oF^oF`oFbo/FeoQ&0.0emF'/FhoF[ q-F_p6&Q%sX20F'FbpF;Fep-F,6#-I%msupGF$6%-F#6%-F,6%-F_p6&Q%ξF'/FcpFg nF;F>-FX6.Q(−F'F;F>FenFhnFjnF\\oF^oF`oFboFdoFgo-F_p6&Q#X1F'FbpF; FepF;F>-F86%Q\"2F'F;F>/%1superscriptshiftGF:/%.linethicknessGFJ/%+deno malignGQ'centerF'/%)numalignGF\\s/%)bevelledGFgnF;F]r-F,6%-F_p6&Q$s10F 'FbpF;FepFgp-F_p6&Q$s22F'FbpF;FepF]r-F,6%-F_p6&Q$s11F'FbpF;FepFgp-F_p6 &Q$s21F'FbpF;FepF]r-F,6%-F_p6&Q$s12F'FbpF;FepFgp-F_p6&Q$s20F'FbpF;FepF ]r-F,6%-F#6%-F,6%-F,6%F[tFgpFfs-FX6.Q\"+F'F;F>FenFhnFjnF\\oF^oF`oFboFd oFgo-F,6%FctFgpF^tF;F>FgpFiqF]r-F,6'FctFgpFfsFgp-Fcq6%FiqFcrFfrFau-Fjo 6)-F,6#-F_p6&Q\"CF'FbpF;Fep-F,6#FgqFhrFjrF]sF_sF;F]r-F,6%-F_p6&Q\"hF'F bpF;FepFgpFctF]r-Fjo6)-F,6%FevFgp-F_p6&Q\"PF'FbpF;Fep-F,6#-F,6%FiqF]r- F_p6&Q\"QF'FbpF;FepFhrFjrF]sF_sF;FAFDFFFHFK-F56(-F,6--Fjo6)-F,6#-F,6%F ^pFauF]qFavFhrFjrF]sF_sF;F]rF[tF]rF^tFau-F,6%-F#6%-F,6&FWFctF]rFfsF;F> FgpFiqF]r-Fjo6)-F,6#FevFavFhrFjrF]sF_sF;Fau-Fjo6)FhxF_wFhrFjrF]sF_sF;F AFDFFFHFKFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEF\\s/FGQ'|frleft|hrF '/%/alignmentscopeGFdp/%,columnwidthGQ%autoF'/%&widthGFhy/%+rowspacing GQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlines GFcz/%&frameGFcz/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFgn/%-equa lcolumnsGFgn/%-displaystyleGFgn/%%sideGQ&rightF'/%0minlabelspacingGF`z F;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)-F,6%-I#miGF$6&Q#c2F'/%'italicGQ %trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6-Q1& InvisibleTimes;F'/FFQ'normalF'/%&fenceGQ&falseF'/%*separatorGFP/%)stre tchyGFP/%*symmetricGFP/%(largeopGFP/%.movablelimitsGFP/%'accentGFP/%'l spaceGQ&0.0emF'/%'rspaceGFin-I%msupGF$6%-F<6&Q%ξF'/F@FPFBFL-I#mnGF$ 6%Q\"2F'FBFL/%1superscriptshiftGQ\"0F'-FI6.Q\"+F'FBFLFNFQFSFUFWFYFen/F hnQ,0.2222222emF'/F[oF^p-F,6%-F<6&Q#c1F'F?FBFEFHF_oFjo-F<6&Q#c0F'F?FBF EFjo-I&mfracGF$6)-F,6#-F<6&Q&ρF'FboFBFL-F,6#-F,6%F_o-FI6.Q(− F'FBFLFNFQFSFUFWFYFenF]pF_p-F<6&Q\"QF'F?FBFE/%.linethicknessGQ\"1F'/%+ denomalignGQ'centerF'/%)numalignGF_r/%)bevelledGFPFB/%)rowalignGQ!F'/% ,columnalignGFfr/%+groupalignGFfr/%(rowspanGF\\r/%+columnspanGF\\r-F56 (-F,6'-F,6%-F<6&Q)nuMinus1F'F?FBFEFHF_oFjo-F<6&Q&ν0F'F?FBFEFjo-Fip6 )-F,6#-F<6&Q%μF'FboFBFLF`qFjqF]rF`rFbrFBFdrFgrFirF[sF]sFdrFgrFir-F2 6'-F56(-F,6%-F<6&Q%AA21F'F?FBFE-FI6-Q0⁡F'FLFNFQFSFUFWFYF enFgnFjn-F#6%-F,6#F_oFBFLFdrFgrFirF[sF]s-F56(-F,6%-F<6&Q%AA22F'F?FBFEF [uF^uFdrFgrFirF[sF]sFdrFgrFir/%&alignGQ%axisF'/FerQ)baselineF'/FhrF_r/ FjrQ'|frleft|hrF'/%/alignmentscopeGFA/%,columnwidthGQ%autoF'/%&widthGF ev/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF '/%,columnlinesGF`w/%&frameGF`w/%-framespacingGQ,0.4em~0.5exF'/%*equal rowsGFP/%-equalcolumnsGFP/%-displaystyleGFP/%%sideGQ&rightF'/%0minlabe lspacingGF]wFBFL/%%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(-I#mnGF$6%Q\"1F'/%+fo regroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,column alignGFC/%+groupalignGFC/%(rowspanGF:/%+columnspanGF:-F56(-F86%Q\"0F'F ;F>FAFDFFFHFJFAFDFF-F26'-F56(-F,6$-I#moGF$6.Q*&uminus0;F'F;F>/%&fenceG Q&falseF'/%*separatorGFgn/%)stretchyGFgn/%*symmetricGFgn/%(largeopGFgn /%.movablelimitsGFgn/%'accentGFgn/%'lspaceGQ,0.2222222emF'/%'rspaceGFf o-I&mfracGF$6)-F,6%-I#miGF$6&Q\"PF'/%'italicGQ%trueF'F;/F?Q'italicF'-F X6-Q1⁢F'F>FenFhnFjnF\\oF^oF`oFbo/FeoQ&0.0emF'/FhoF[q-F# 6%-F,6%-F_p6&Q\"QF'FbpF;Fep-FX6.Q(−F'F;F>FenFhnFjnF\\oF^oF`oFboF doFgo-F_p6&Q#X1F'FbpF;FepF;F>-F,6#-F,6%-F_p6&Q%ξF'/FcpFgnF;F>FdqFaq /%.linethicknessGF:/%+denomalignGQ'centerF'/%)numalignGFfr/%)bevelledG FgnF;FAFDFFFHFJ-F56(-Fjo6)-F,6#-F,6%F^rFdqFgqFjqFbrFdrFgrFirF;FAFDFFFH FJFAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEFfr/FGQ'|frleft|hrF'/%/alig nmentscopeGFdp/%,columnwidthGQ%autoF'/%&widthGF_t/%+rowspacingGQ&1.0ex F'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFjt/%&f rameGFjt/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFgn/%-equalcolumns GFgn/%-displaystyleGFgn/%%sideGQ&rightF'/%0minlabelspacingGFgtF;F>/%%o penGQ\"[F'/%&closeGQ\"]F'" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 256 73 "So lving the compatibility equations to obtain the Hamiltonian evolutions ." }{TEXT 256 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 257 65 "The compatib ility equation is \\mathcal\{L\}L=h\\partial_\\xi A+[A,L]\n" }{TEXT 257 111 "Since the first line of L is trivial, we may easily obtain A[ 2,1] et A[2,2] to obtain the full expression for A" }{TEXT 257 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "LL:=h*dAdxi+(Multiply(A,L)- Multiply(L,A)):\n" }{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 45 "AA21:=unapply(solve(Entry11=0,AA21(xi)),xi):\n" } {MPLTEXT 1 0 37 "AA21bis:=h*dAdxi[1,1]+A[1,2]*L[2,1]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "simplify(AA21(xi)-AA21bis);\n" }{MPLTEXT 1 0 45 "AA22:=unapply(solve(Entry12=0,AA22(xi)),xi):\n" }{MPLTEXT 1 0 43 "AA22bis:=h*dAdxi[1,2]+A[1,1]+A[1,2]*L[2,2]:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "simplify(AA22(xi)-AA22bis);\n" }{MPLTEXT 1 0 19 "simplify(Entry11);\n" }{MPLTEXT 1 0 18 "simplify(Ent ry12);" }}{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 257 95 "We now compute the action of \\mathcal\{L\} on L[2,2] et L[2,1] to obta in the evolution equations" }{TEXT 257 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 26 "Evolution of entry L_\{2,2\}" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "Entry22:=simplify(LL[2,2]);\n" }{MPLTEXT 1 0 65 "En try22TermxiMinusQCube:=factor(residue(Entry22*(xi-Q)^2,xi=Q));\n" } {MPLTEXT 1 0 65 "Entry22TermxiMinusQSquare:=factor(residue(Entry22*(xi -Q),xi=Q));\n" }{MPLTEXT 1 0 52 "Entry22TermxiMinusQ:=factor(residue(E ntry22,xi=Q));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 65 "Entry22Termxi Infty4:=factor(-residue(Entry22/xi^5,xi=infinity));\n" }{MPLTEXT 1 0 65 "Entry22TermxiInfty3:=factor(-residue(Entry22/xi^4,xi=infinity));\n " }{MPLTEXT 1 0 65 "Entry22TermxiInfty2:=factor(-residue(Entry22/xi^3, xi=infinity));\n" }{MPLTEXT 1 0 65 "Entry22TermxiInfty1:=factor(-resid ue(Entry22/xi^2,xi=infinity));\n" }{MPLTEXT 1 0 63 "Entry22TermxiInfty 0:=factor(-residue(Entry22/xi,xi=infinity));\n" }{MPLTEXT 1 0 69 "Entr y22TermxiInftyMinus1:=factor(-residue(Entry22/xi^2,xi=infinity));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 69 "Entry22TermxiInftyMinus2:=factor (-residue(Entry22/xi^3,xi=infinity));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "Entry22TermxiTMinus1:=factor(residue(Entry22,xi=X1));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 "Entry22TermxiTMinus2:=factor(res idue(Entry22*(xi-X1),xi=X1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 81 "simplify( Entry22-(Entry22TermxiMinusQSquare/(xi -Q)^2+Entry22TermxiMinusQ/(xi-Q)\n" }{MPLTEXT 1 0 119 "+Entry22TermxiI nfty0+Entry22TermxiInfty1*xi+Entry22TermxiInfty2*xi^2+Entry22TermxiInf ty3*xi^3+Entry22TermxiInfty4*xi^4+" }{MPLTEXT 1 0 29 "Entry22TermxiTMi nus1/(xi-X1)+" }{MPLTEXT 1 0 30 "Entry22TermxiTMinus2/(xi-X1)^2" } {MPLTEXT 1 0 5 ") );\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(Entry22G6 \",$**,2*&,&*&,&I$s12GF$\"\"#I$s22GF$F-\"\"\"I)nuMinus1GF$F/F/I#c2GF$! \"%F/I#xiGF$\"\"&F/*&,,*&,&*&,&F,F2F.F2F/F0F/F/F1\"\")F/I#X1GF$F/F/*&F 8F/I\"QGF$F/F/*&,&I$s21GF$F/I$s11GF$F/F/F0F/F/*&,&F,F/F.F/F/I$nu0GF$F/ F/I#c1GF$!\"#F/F3\"\"%F/*&,(*&F)F/FF/F/*&,&FBFGFAFGF/F0F/F/*&,&F,FGF.FGF/FEF/F/FFFHF/FF/, **&,&*&FDF/F0F/F/F1FGF/F>F/F/*&,&FB!\"\"FAFhnF/F0F/F/*&,&F,FhnF.FhnF/F EF/F/FFF-F/F-F/F3\"\"$F/*&,0*&,*F=F/F?F/FCF/FFFGF/FF- F/*&,(*&,&FBFHFAFHF/F0F/F/*&,&F,FHF.FHF/FEF/F/FF!\")F/F>F/F/*&F0F/,(I% sX10GF$F/I%sX20GF$F/I\"hGF$FhnF/F/F/FF-F/*&,& *&F0F/F^pF/F/*&I#muGF$F/FDF/FhnF/F>F/F/*&,&F\\pF/F]pF/F/FEF/F/*&FgnF/F epF/F/I$rhoGF$F-F/F3F-F/*&,(*(F>F/FXF/FF- F/*&,&*&,&F\\pFGF]pFGF/F0F/F/FdpF-F/F>F/F/*&F^pF/FEF/FG*&,&FBF-FAF-F/F epF/F/FipF2F/FF/FEF/F/FepFhnF/FGF/F3F/F/*&,,F_pF/Fap F/FeqF/FhpF/FipF-F/FF-F0F/F/FepFhnF/F[pF/FF/F[pF /FiqF/F/F/F^pF/,&F3FhnF>F/FG,&F3FhnFI8Entry22TermxiMinusQCubeG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:Entry22TermxiMinusQSquareG6\",$*(,D*(I\"QGF$\"\"#I\"hG F$\"\"\"I)nuMinus1GF$F,F,*(F)F*I#muGF$F,I$s12GF$F,!\"\"*(F)F*F/F,I$s22 GF$F,F1**F)F,I#X1GF$F,F+F,F-F,F1**F)F,F5F,F/F,F0F,F,**F)F,F5F,F/F,F3F, F,*(F)F,F+F,I$nu0GF$F,F,*(F)F,F/F,I$s11GF$F,F1*(F)F,F/F,I$s21GF$F,F1*( F5F,F+F,F9F,F1*(F5F,F/F,F;F,F,*(F5F,F/F,F=F,F,*&F)F,I$rhoGF$F,F**&F5F, FBF,!\"#*&F+F,F/F,F1*&F/F,I%sX10GF$F,F,*&F/F,I%sX20GF$F,F,F,F+F,,&F)F, F5F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry22TermxiMinusQG6\"\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry22TermxiInfty4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry22TermxiInfty3G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry22TermxiInfty2G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry22TermxiInfty1G6\",$*&,(*&I$s12G F$\"\"\"I)nuMinus1GF$F*!\"\"*&I$s22GF$F*F+F*F,I#c2GF$\"\"#F*I\"hGF$F*F 0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry22TermxiInfty0G6\"*&I\"hGF$ \"\"\",,*&I$nu0GF$F'I$s12GF$F'!\"\"*&F*F'I$s22GF$F'F,*&I$s11GF$F'I)nuM inus1GF$F'F,*&I$s21GF$F'F1F'F,I#c1GF$\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry22TermxiInftyMinus1G6\",$*&,(*&I$s12GF$\"\"\"I)nu Minus1GF$F*!\"\"*&I$s22GF$F*F+F*F,I#c2GF$\"\"#F*I\"hGF$F*F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9Entry22TermxiInftyMinus2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5Entry22TermxiTMinus1G6\"\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I5Entry22TermxiTMinus2G6\"**I\"hGF$\"\"\",(I%sX10 GF$!\"\"I%sX20GF$F*F&F'F',,*(I\"QGF$F'I#X1GF$F'I)nuMinus1GF$F'F'*&F/\" \"#F0F'F**&F.F'I$nu0GF$F'F'*&F/F'F4F'F*I#muGF$F*F',&F.F'F/F*F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", .*&,&I%sX10G6\"\"\"\"I%sX20GF&F'F',&I#xiGF&F'I#X1GF&!\"\"F,F'I$s11GF&F ,I$s21GF&F,*&,&I$s12GF&F,I$s22GF&F,F'F*F'F'*&I\"hGF&F'F)F,F,*&F4F',&F* F'I\"QGF&F,F,F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 35 "Since the def ormation operator is " }{TEXT 257 78 "\\hbar (beta12\\partial_\{t_\{ \\infty^\{(1)\},2\} +beta22\\partial_\{t_\{\\infty^\{(2)\},2\}+" } {TEXT 257 6 "beta11" }{TEXT 257 30 "\\partial_\{t_\{\\infty^\{(1)\},1 \} +" }{TEXT 257 6 "beta21" }{TEXT 257 28 "\\partial_\{t_\{\\infty^\{( 2)\},1\}" }{TEXT 257 22 ")) we can obtain L[Q]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "L22Orderxi2:=-residue(L[2,2]/xi^3,xi=infinity) ;\n" }{MPLTEXT 1 0 48 "L22Orderxi1:=-residue(L[2,2]/xi^2,xi=infinity); \n" }{MPLTEXT 1 0 48 "L22Orderxi0:=-residue(L[2,2]/xi^1,xi=infinity); \n" }{MPLTEXT 1 0 52 "L22OrderxiMinus1:=-residue(L[2,2]/xi^2,xi=infini ty);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "L22OrderS1:=residue(L[2,2 ],xi=X1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "factor(simplify(h*(b eta12*diff(L22Orderxi2,s12)+beta22*diff(L22Orderxi2,s22)\n" }{MPLTEXT 1 0 30 "+beta11*diff(L22Orderxi2,s11)+" }{MPLTEXT 1 0 6 "beta21" } {MPLTEXT 1 0 32 "*diff(L22Orderxi2,s21) +betaX1*" }{MPLTEXT 1 0 20 "d iff(L22Orderxi2,X1)" }{MPLTEXT 1 0 2 " )" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 24 "- Entry22TermxiInfty2));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "Equation0:=" }{MPLTEXT 1 0 44 "E ntry22TermxiTMinus2-h*(sX10+sX20-h)*betaX1;" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 11 "Equation1:=" }{MPLTEXT 1 0 77 "f actor(simplify(h*(beta12*diff(L22Orderxi1,s12)+beta22*diff(L22Orderxi1 ,s22)\n" }{MPLTEXT 1 0 30 "+beta11*diff(L22Orderxi1,s11)+" }{MPLTEXT 1 0 6 "beta21" }{MPLTEXT 1 0 22 "*diff(L22Orderxi1,s21)" }{MPLTEXT 1 0 8 "+betaX1*" }{MPLTEXT 1 0 20 "diff(L22Orderxi1,X1)" }{MPLTEXT 1 0 2 ")\n" }{MPLTEXT 1 0 24 "- Entry22TermxiInfty1));" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 81 "Equation2:=simplify(h*(beta 12*diff(L22Orderxi0,s12)+beta22*diff(L22Orderxi0,s22)\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 57 "beta11*diff(L22Orderxi0,s11)+beta21*diff(L22 Orderxi0,s21)" }{MPLTEXT 1 0 8 "+betaX1*" }{MPLTEXT 1 0 20 "diff(L22Or derxi0,X1)" }{MPLTEXT 1 0 1 ")" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "- Entry22TermxiInfty0);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "Equ ation3:=factor(simplify(h*(beta12*diff(L22OrderxiMinus1" }{MPLTEXT 1 0 34 ",s12)+beta22*diff(L22OrderxiMinus1" }{MPLTEXT 1 0 6 ",s22)\n" } {MPLTEXT 1 0 29 "+beta11*diff(L22OrderxiMinus1" }{MPLTEXT 1 0 34 ",s11 )+beta21*diff(L22OrderxiMinus1" }{MPLTEXT 1 0 5 ",s21)" }{MPLTEXT 1 0 8 "+betaX1*" }{MPLTEXT 1 0 25 "diff(L22OrderxiMinus1,X1)" }{MPLTEXT 1 0 1 ")" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 20 "- Entry22TermxiInfty" } {MPLTEXT 1 0 6 "Minus1" }{MPLTEXT 1 0 3 "));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 45 "Equation3bis:=factor(simplify(h* (beta12*diff(" }{MPLTEXT 1 0 28 "L22OrderS1,s12)+beta22*diff(" } {MPLTEXT 1 0 16 "L22OrderS1,s22)\n" }{MPLTEXT 1 0 13 "+beta11*diff(" } {MPLTEXT 1 0 28 "L22OrderS1,s11)+beta21*diff(" }{MPLTEXT 1 0 15 "L22Or derS1,s21)" }{MPLTEXT 1 0 8 "+betaX1*" }{MPLTEXT 1 0 5 "diff(" } {MPLTEXT 1 0 14 "L22OrderS1,X1)" }{MPLTEXT 1 0 2 ")\n" }{MPLTEXT 1 0 2 "- " }{MPLTEXT 1 0 20 "Entry22TermxiTMinus1" }{MPLTEXT 1 0 3 "));" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,L22Orderxi2G6\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,L22Orderxi1G6\",&I$s12GF$!\" \"I$s22GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,L22Orderxi0G6\",&I$s1 1GF$!\"\"I$s21GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1L22OrderxiMinu s1G6\",&I$s12GF$!\"\"I$s22GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation0G6\",&**I\"hGF$\"\"\",(I %sX10GF$!\"\"I%sX20GF$F+F'F(F(,,*(I\"QGF$F(I#X1GF$F(I)nuMinus1GF$F(F(* &F0\"\"#F1F(F+*&F/F(I$nu0GF$F(F(*&F0F(F5F(F+I#muGF$F+F(,&F/F(F0F+F+F(* (F'F(,(F*F(F,F(F'F+F(I'betaX1GF$F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I*Equation1G6\",$*&I\"hGF$\"\"\",,*&I$s12GF$F(I)nuMinus1GF$F(!\"#*&I$ s22GF$F(F,F(F-I'beta12GF$F(I'beta22GF$F(I#c2GF$\"\"%F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation2G6\"*&I\"hGF$\"\"\",,*&,&I$s21GF$F 'I$s11GF$F'F'I)nuMinus1GF$F'F'*&,&I$s12GF$F'I$s22GF$F'F'I$nu0GF$F'F'I# c1GF$!\"#I'beta11GF$!\"\"I'beta21GF$F6F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation3G6\",$*&I\"hGF$\"\"\",,*&I$s12GF$F(I)nuMinus1GF$F(!\"# *&I$s22GF$F(F,F(F-I'beta12GF$F(I'beta22GF$F(I#c2GF$\"\"%F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I-Equation3bisG6\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "LQ:=factor(Entry22TermxiMinusQSquar e/h):\n" }{MPLTEXT 1 0 26 "nu0:=solve(Equation0,nu0);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu0G6\",$*&,,*(I\"QGF$\"\"\"I#X1GF$F*I)nuMinus1GF$ F*F**&F+\"\"#F,F*!\"\"*&F)F*I'betaX1GF$F*F**&F+F*F1F*F/I#muGF$F/F*,&F) F*F+F/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 34 "We now look at \\m athcal\{L\}[L[2,1]]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Entr y21:=simplify(LL[2,1]):\n" }{MPLTEXT 1 0 65 "Entry21TermxiMinusQCube:= factor(residue(Entry21*(xi-Q)^2,xi=Q));\n" }{MPLTEXT 1 0 65 "Entry21Te rmxiMinusQSquare:=factor(residue(Entry21*(xi-Q),xi=Q));\n" }{MPLTEXT 1 0 51 "Entry21TermxiMinusQ:=factor(residue(Entry21,xi=Q));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 64 "Entry21TermxiInfty2:=factor(-residue(Ent ry21/xi^3,xi=infinity));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 65 "Entry 21TermxiInfty1:=factor(-residue(Entry21/xi^2,xi=infinity));\n" } {MPLTEXT 1 0 62 "Entry21TermxiInfty0:=factor(-residue(Entry21/xi,xi=in finity));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 48 "Entry21TermxiS1:=fac tor(residue(Entry21,xi=X1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 56 "E ntry21TermxiS2:=factor(residue(Entry21*(xi-X1),xi=X1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 114 "simplify( Entry21-(En try21TermxiMinusQCube/(xi-Q)^3+Entry21TermxiMinusQSquare/(xi-Q)^2+Entr y21TermxiMinusQ/(xi-Q)\n" }{MPLTEXT 1 0 44 "+Entry21TermxiInfty0+Entry 21TermxiInfty1*xi+" }{MPLTEXT 1 0 24 "Entry21TermxiInfty2*xi^2" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 24 "Entry21Termx iS1/(xi-X1)\n" }{MPLTEXT 1 0 5 ") );\n" }{MPLTEXT 1 0 7 "L[2,1];" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermxiMinusQCubeG6\",$*&,&*&I \"PGF$\"\"\"I#muGF$F*F*I$rhoGF$F*F*I\"hGF$\"\"#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I:Entry21TermxiMinusQSquareG6\",$*(,bq**I\"QGF$\"\"%I #muGF$\"\"\"I$s12GF$F,I$s22GF$F,!\"#*,F)\"\"$I#X1GF$F,F+F,F-F,F.F,F**, F)\"\"#F2F4F+F,F-F,F.F,F/**I\"PGF$F,F)F1I\"hGF$F,I)nuMinus1GF$F,!\"\"* ,F6F,F)F4F2F,F7F,F8F,F1*,F6F,F)F,F2F4F7F,F8F,!\"$**F6F,F2F1F7F,F8F,F,* *F)F1F+F,I$s11GF$F,F.F,F/**F)F1F+F,F-F,I$s21GF$F,F/*,F)F4F2F,F+F,F?F,F .F,F**,F)F4F2F,F+F,F-F,FAF,F**,F)F,F2F4F+F,F?F,F.F,F/*,F)F,F2F4F+F,F-F ,FAF,F/**F6F,F)F4I'betaX1GF$F,F7F,F,*,F6F,F)F,F2F,FGF,F7F,F/**F6F,F2F4 FGF,F7F,F,*(F)F1I$rhoGF$F,F-F,F,*(F)F1FKF,F.F,F,**F)F4F2F,FKF,F-F,F/** F)F4F2F,FKF,F.F,F/**F)F4F7F,F+F,F-F,F/**F)F4F+F,I$s10GF$F,F.F,F/**F)F4 F+F,F?F,FAF,F/**F)F4F+F,F-F,I$s20GF$F,F/**F)F,F2F4FKF,F-F,F,**F)F,F2F4 FKF,F.F,F,*,F)F,F2F,F7F,F+F,F-F,F**,F)F,F2F,F+F,FQF,F.F,F**,F)F,F2F,F+ F,F?F,FAF,F**,F)F,F2F,F+F,F-F,FTF,F***F2F4F7F,F+F,F-F,F/**F2F4F+F,FQF, F.F,F/**F2F4F+F,F?F,FAF,F/**F2F4F+F,F-F,FTF,F/**F6F,F)F,F7F,F+F,F9**F6 F,F2F,F7F,F+F,F,*(F)F4FKF,F?F,F,*(F)F4FKF,FAF,F,**F)F,F2F,FKF,F?F,F/** F)F,F2F,FKF,FAF,F/*(F2F4FKF,F?F,F,*(F2F4FKF,FAF,F,*(I\"CGF$F,F)F,F+F,F 4*(FboF,F2F,F+F,F/*(F)F,F7F,FKF,F,*(F)F,FKF,I%sX10GF$F,F9*(F)F,FKF,I%s X20GF$F,F9*(F2F,F7F,FKF,F9*(F2F,FKF,FfoF,F,*(F2F,FKF,FhoF,F,*(F+F,FfoF ,FhoF,F/F,F7F,,&F)F,F2F9F/F9" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry 21TermxiMinusQG6\"*(I\"hGF$\"\"\",^p**I\"QGF$\"\"%I#muGF$F'I$s12GF$F'I $s22GF$F'\"\"#*,F*\"\"$I#X1GF$F'F,F'F-F'F.F'!\"'*,F*F/F2F/F,F'F-F'F.F' \"\"'*,F*F'F2F1F,F'F-F'F.F'!\"#**I\"PGF$F'F*F1F&F'I)nuMinus1GF$F'!\"\" *,F9F'F*F/F2F'F&F'F:F'F1*,F9F'F*F'F2F/F&F'F:F'!\"$**F9F'F2F1F&F'F:F'F' *(F*F+I#c2GF$F'F&F'F7**F*F1F2F'FAF'F&F'F5**F*F1F,F'I$s11GF$F'F.F'F'**F *F1F,F'F-F'I$s21GF$F'F'**F*F/F2F/FAF'F&F'F3*,F*F/F2F'F,F'FDF'F.F'F>*,F *F/F2F'F,F'F-F'FFF'F>**F*F'F2F1FAF'F&F'F/*,F*F'F2F/F,F'FDF'F.F'F1*,F*F 'F2F/F,F'F-F'FFF'F1**F2F1F,F'FDF'F.F'F;**F2F1F,F'F-F'FFF'F;*(F*F1I#c1G F$F'F&F'F;*(F*F1I$rhoGF$F'F-F'F;*(F*F1FRF'F.F'F;**F*F/F2F'FPF'F&F'F1** F*F/F2F'FRF'F-F'F1**F*F/F2F'FRF'F.F'F1**F*F'F2F/FPF'F&F'F>**F*F'F2F/FR F'F-F'F>**F*F'F2F/FRF'F.F'F>*(F2F1FPF'F&F'F'*(F2F1FRF'F-F'F'*(F2F1FRF' F.F'F'*(I\"CGF$F'F*F'F,F'F'*(FhnF'F2F'F,F'F;*(F*F'F&F'FRF'F'*(F*F'FRF' I%sX10GF$F'F;*(F*F'FRF'I%sX20GF$F'F;*(F2F'F&F'FRF'F;*(F2F'FRF'F\\oF'F' *(F2F'FRF'F^oF'F'*(F,F'F\\oF'F^oF'F7F',&F*F'F2F;F>" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry21TermxiInfty2G6\",$*&,(*(I$s12GF$\"\"\"I$s22GF$ F*I)nuMinus1GF$F*!\"#*&I#c2GF$F*F)F*F**&F/F*F+F*F*F*I\"hGF$F*\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry21TermxiInfty1G6\"*(I\"hGF$\"\" \",D*,I\"QGF$F'I#X1GF$F'I$s12GF$F'I$s22GF$F'I)nuMinus1GF$F'\"\"#**F+F/ F,F'F-F'F.F'!\"#**F*F'I'betaX1GF$F'F,F'F-F'F/**F*F'I$s11GF$F'F-F'F.F'! \"$**F*F'F,F'I$s21GF$F'F.F'F6**F+F'F3F'F,F'F-F'F1**F+F'F5F'F-F'F.F'\" \"$**F+F'F,F'F8F'F.F'F;*(F*F'I#c1GF$F'F,F'F'*(F*F'F>F'F-F'F'*(F*F'I#c2 GF$F'F5F'F/*(F*F'FAF'F8F'F/*(F+F'F>F'F,F'!\"\"*(F+F'F>F'F-F'FD*(F+F'FA F'F5F'F1*(F+F'FAF'F8F'F1*(I#muGF$F'F,F'F-F'F1F',&F*F'F+FDFD" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I4Entry21TermxiInfty0G6\"*(I\"hGF$\"\"\",Z*,I \"QGF$F'I#X1GF$F'I$s11GF$F'I$s22GF$F'I)nuMinus1GF$F'F'*,F*F'F+F'I$s12G F$F'I$s21GF$F'F.F'F'**F+\"\"#F,F'F-F'F.F'!\"\"**F+F3F0F'F1F'F.F'F4**F* F'I'betaX1GF$F'F,F'F-F'F'**F*F'F7F'F0F'F1F'F'**F*F'F&F'F0F'F.F'!\"#**F *F'I$s10GF$F'F-F'F.F'F:**F*F'F,F'F1F'F.F'F:**F*F'F0F'I$s20GF$F'F.F'F:* *F+F'F7F'F,F'F-F'F4**F+F'F7F'F0F'F1F'F4**F+F'F&F'F0F'F.F'F3**F+F'FI0Entry21TermxiS1G6\"*(,fo**I\" QGF$\"\"$I#X1GF$\"\"\"I#c2GF$F+I\"hGF$F+\"\"#**F(F)F*F+F,F+I%sX10GF$F+ !\"#**F(F)F*F+F,F+I%sX20GF$F+F1**F(F.F*F.F,F+F-F+!\"'**F(F.F*F.F,F+F0F +\"\"'**F(F.F*F.F,F+F3F+F7**F(F+F*F)F,F+F-F+F7**F(F+F*F)F,F+F0F+F5**F( F+F*F)F,F+F3F+F5*(F*\"\"%F,F+F-F+F1*(F*F=F,F+F0F+F.*(F*F=F,F+F3F+F.*(I \"CGF$F+F(F)I)nuMinus1GF$F+F+**FAF+F(F.F*F+FBF+!\"$**FAF+F(F+F*F.FBF+F )*(FAF+F*F)FBF+!\"\"*(F(F)I#c1GF$F+F-F+F+*(F(F)FIF+F0F+FG*(F(F)FIF+F3F +FG**F(F.F*F+FIF+F-F+FD**F(F.F*F+FIF+F0F+F)**F(F.F*F+FIF+F3F+F)**F(F+F *F.FIF+F-F+F)**F(F+F*F.FIF+F0F+FD**F(F+F*F.FIF+F3F+FD*(F*F)FIF+F-F+FG* (F*F)FIF+F0F+F+*(F*F)FIF+F3F+F+*(FAF+F(F+I#muGF$F+FG*(FAF+F*F+FVF+F+*( F(F+F-F+I$rhoGF$F+FG*(F(F+FYF+F0F+F+*(F(F+FYF+F3F+F+*(F*F+F-F+FYF+F+*( F*F+FYF+F0F+FG*(F*F+FYF+F3F+FG*(FVF+F0F+F3F+F.F+F-F+,&F(F+F*FGFD" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I0Entry21TermxiS2G6\"*(I\"CGF$\"\"\"I\" hGF$F'I'betaX1GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "**I'betaX1G6\"\" \"\"I\"hGF$F%,&*&,&I#xiGF$!\"\"I#X1GF$F%F%I\"CGF$F%F%*&I%sX10GF$F%I%sX 20GF$F%\"\"#F%F)!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",4*(I%sX10G6\"\" \"\"I%sX20GF%F&,&I#xiGF%F&I#X1GF%!\"\"!\"#F+*&I$s10GF%F&I$s22GF%F&F+*& I$s11GF%F&I$s21GF%F&F+*&I$s12GF%F&I$s20GF%F&F+*&,&*&F1F&F/F&F&*&F4F&F2 F&F&F&F)F&F+*(F4F&F/F&F)\"\"#F+*&I\"CGF%F&F(F+F&*&I\"hGF%F&F4F&F+*(F?F &I\"PGF%F&,&F)F&I\"QGF%F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "rho:=factor(solve(Entry21TermxiMinusQCube,rho));\n" }{MPLTEXT 1 0 23 "simplify(rho-(-P*mu));\n" }{MPLTEXT 1 0 34 "simplify(Entry21Term xiMinusQCube);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$rhoG6\",$*&I\"PGF$ \"\"\"I#muGF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "L21Orderxi3:=-residue(L[2,1]/xi^4,xi=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 47 "L21Orderxi2:=-residue(L[2,1]/xi^3,xi=infinity );" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 48 "L21Orderxi1:=-residue(L[2,1 ]/xi^2,xi=infinity);\n" }{MPLTEXT 1 0 48 "L21Orderxi0:=-residue(L[2,1] /xi^1,xi=infinity);\n" }{MPLTEXT 1 0 52 "L21OrderxiMinus1:=-residue(L[ 2,1]/xi^2,xi=infinity);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 52 "L21Ord erxiMinus2:=-residue(L[2,1]/xi^3,xi=infinity);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 19 "L21TOrder2:=factor(" }{MPLTEXT 1 0 31 "residue(L[2,1] *(xi-X1),xi=X1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "L21TOrder1:= " }{MPLTEXT 1 0 22 "residue(L[2,1],xi=X1);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 11 "Equation4:=" }{MPLTEXT 1 0 70 "simplify(h*(beta12*dif f(L21Orderxi2,s12)+beta22*diff(L21Orderxi2,s22)+" }{MPLTEXT 1 0 57 "be ta11*diff(L21Orderxi2,s11)+beta21*diff(L21Orderxi2,s21)" }{MPLTEXT 1 0 28 "+betaX1*diff(L21Orderxi2,X1)" }{MPLTEXT 1 0 25 ")- Entry21Termxi Infty2);\n" }{MPLTEXT 1 0 11 "Equation5:=" }{MPLTEXT 1 0 12 "simplify( h*(" }{MPLTEXT 1 0 58 "beta12*diff(L21Orderxi1,s12)+beta22*diff(L21Ord erxi1,s22)+" }{MPLTEXT 1 0 57 "beta11*diff(L21Orderxi1,s11)+beta21*dif f(L21Orderxi1,s21)" }{MPLTEXT 1 0 28 "+betaX1*diff(L21Orderxi1,X1)" } {MPLTEXT 1 0 22 ")- Entry21TermxiInfty1" }{MPLTEXT 1 0 2 ");" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 35 "Equation6:=simplify(h*(beta12*di ff(" }{MPLTEXT 1 0 10 "L21TOrder2" }{MPLTEXT 1 0 18 ",s12)+beta22*diff (" }{MPLTEXT 1 0 10 "L21TOrder2" }{MPLTEXT 1 0 18 ",s22)+beta11*diff(" }{MPLTEXT 1 0 10 "L21TOrder2" }{MPLTEXT 1 0 18 ",s11)+beta21*diff(" } {MPLTEXT 1 0 10 "L21TOrder2" }{MPLTEXT 1 0 5 ",s21)" }{MPLTEXT 1 0 27 "+betaX1*diff(L21TOrder2,X1)" }{MPLTEXT 1 0 3 ")- " }{MPLTEXT 1 0 17 " Entry21TermxiS2);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,L21Orderxi3G6\" \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,L21Orderxi2G6\",$*&I$s12GF$ \"\"\"I$s22GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,L21Orderxi1G 6\",&*&I$s11GF$\"\"\"I$s22GF$F(!\"\"*&I$s12GF$F(I$s21GF$F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,L21Orderxi0G6\",**&I\"hGF$\"\"\"I$s12GF$F(! \"\"*&I$s10GF$F(I$s22GF$F(F**&I$s11GF$F(I$s21GF$F(F**&F)F(I$s20GF$F(F* " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1L21OrderxiMinus1G6\",&*&I$s11GF$ \"\"\"I$s22GF$F(!\"\"*&I$s12GF$F(I$s21GF$F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1L21OrderxiMinus2G6\",$*&I$s12GF$\"\"\"I$s22GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+L21TOrder2G6\",$*&I%sX10GF$\"\"\"I %sX20GF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+L21TOrder1G6\"*&,& *&I\"CGF$\"\"\"I\"QGF$F)F)*&F(F)I#X1GF$F)!\"\"F),&F*F)F,F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation4G6\",$*&,&*&,(*&I$s22GF$\"\"\"I)nu Minus1GF$F,!\"#I#c2GF$F,I'beta22GF$#F,\"\"#F,I$s12GF$F,F,*&F+F,,&F/F,I 'beta12GF$F1F,F,F,I\"hGF$F,F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equa tion5G6\",$*(,***I#X1GF$\"\"#I$s12GF$\"\"\"I$s22GF$F,I)nuMinus1GF$F,! \"#*&,,**I\"QGF$F,F+F,F-F,F.F,F**&,**&I'betaX1GF$F,F-F,F/*&I$s21GF$F,F .F,\"\"$I'beta21GF$!\"\"I#c1GF$FI*Equation6G6\",$*(I \"CGF$\"\"\"I\"hGF$F(I'betaX1GF$F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "c2:=factor(solve(Equation1,c2));\n" }{MPLTEXT 1 0 33 "mu:=factor(solve(Equation2,mu));\n" }{MPLTEXT 1 0 45 "nuMinus1:=facto r(solve(Equation4,nuMinus1));\n" }{MPLTEXT 1 0 33 "c1:=factor(solve(Eq uation5,c1));\n" }{MPLTEXT 1 0 18 "c2:=simplify(c2);\n" }{MPLTEXT 1 0 15 "mu:=factor(mu);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c2G6\",**&I$s1 2GF$\"\"\"I)nuMinus1GF$F(#F(\"\"#*&I$s22GF$F(F)F(F*I'beta12GF$#!\"\"\" \"%I'beta22GF$F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#muG6\"*(,4*(I#X1G F$\"\"\"I$s12GF$F)I)nuMinus1GF$F)F)*(F(F)I$s22GF$F)F+F)F)*&I'betaX1GF$ F)F*F)F)*&F/F)F-F)F)*&I$s11GF$F)F+F)!\"\"*&I$s21GF$F)F+F)F3I'beta11GF$ F)I'beta21GF$F)I#c1GF$\"\"#F),&F*F)F-F)F3,&I\"QGF$F)F(F3F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuMinus1G6\",$*&,&I'beta12GF$\"\"\"I'beta22GF $!\"\"F),&I$s12GF$F)I$s22GF$F+F+#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\",$*&,2*(I'beta11GF$\"\"\"I$s12GF$F*I$s22GF$F*\"\"#*&F)F*F ,F-!\"#*(I'beta12GF$F*I$s11GF$F*F,F*!\"\"*(F1F*F+F*I$s21GF$F*F**&I'bet a21GF$F*F+F-F/*(F7F*F+F*F,F*F-*(I'beta22GF$F*F2F*F,F*F**(F:F*F+F*F5F*F 3F*,&F+F*F,F3F/#F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c2G6\"*&,&*&I 'beta12GF$\"\"\"I$s22GF$F)F)*&I'beta22GF$F)I$s12GF$F)!\"\"F),&F-\"\"#F *!\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#muG6\",$*(,&I\"QGF$\"\"\"I #X1GF$!\"\"F),@*(F*F)I'beta12GF$F)I$s12GF$F)F)*(F*F)F.F)I$s22GF$F)F+*( F*F)I'beta22GF$F)F/F)F+*(F*F)F3F)F1F)F)*&I'betaX1GF$F)F/\"\"#F7*(F6F)F /F)F1F)!\"%*&F6F)F1F7F7*&I'beta11GF$F)F/F)F7*&F*&FDF)F1F)F7*&F3F)F@F)F)* &F3F)FBF)F+F),&F/F)F1F+F>#F)F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(Equation1);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 12 "Equation2);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 12 "E quation3);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 12 "Equation4) ;\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 12 "Equation5);\n" } {MPLTEXT 1 0 20 "simplify(c1-c1bis);\n" }{MPLTEXT 1 0 20 "simplify(c2- c2bis);\n" }{MPLTEXT 1 0 20 "simplify(mu-mubis);\n" }{MPLTEXT 1 0 22 " simplify(nu0-nu0bis);\n" }{MPLTEXT 1 0 31 "simplify(nuMinus1-nuMinus1b is);" }}{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 "\"\"!" }} {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 46 "simplify(Entry21TermxiMinusQSquare- (-h*P*LQ)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "LPfunction:=unapply(-Entry21TermxiMinusQ/h,C):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Equation7:=unapply( simplify(Entry2 1TermxiMinusQSquare-(-h*P*LQ)),C):\n" }{MPLTEXT 1 0 537 "Cter:=(Q^4*s1 2*s22-2*Q^3*X1*s12*s22+Q^2*X1^2*s12*s22+P*Q^3*s12+P*Q^3*s22-2*P*Q^2*X1 *s12-2*P*Q^2*X1*s22+P*Q*X1^2*s12+P*Q*X1^2*s22+Q^3*s11*s22+Q^3*s12*s21- 2*Q^2*X1*s11*s22-2*Q^2*X1*s12*s21+Q*X1^2*s11*s22+Q*X1^2*s12*s21+h*Q^2* s12-2*h*Q*X1*s12+h*X1^2*s12+P^2*Q^2-2*P^2*Q*X1+P^2*X1^2+P*Q^2*s11+P*Q^ 2*s21-2*P*Q*X1*s11-2*P*Q*X1*s21+P*X1^2*s11+P*X1^2*s21+Q^2*s10*s22+Q^2* s11*s21+Q^2*s12*s20-2*Q*X1*s10*s22-2*Q*X1*s11*s21-2*Q*X1*s12*s20+X1^2* s10*s22+X1^2*s11*s21+X1^2*s12*s20+h*P*Q-h*P*X1-P*Q*sX10-P*Q*sX20+P*sX1 0*X1+P*sX20*X1+sX10*sX20)/(Q-X1):\n" }{MPLTEXT 1 0 27 "simplify(Equati on7(Cter));\n" }{MPLTEXT 1 0 23 "solve(Equation7(C),C):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "Cbis:=(Q-X1)*P^2" }{MPLTEXT 1 0 4 "+h*P" }{MPLTEXT 1 0 47 "-(Q-X1)*R1(Q)*P+(Q-X1)*R2(Q)-P012+h*s12*(Q-X1);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "simplify(Equation7(Cbis));\n" } {MPLTEXT 1 0 31 "simplify(series(Cter-Cbis,P));\n" }{MPLTEXT 1 0 8 "C: =Cbis:" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%CbisG6\",.*&,&I\"QGF$\"\"\"I#X1GF$!\"\"F)I\"PGF$\"\"#F )*&I\"hGF$F)F,F)F)*(F'F),**&,&I%sX10GF$F)I%sX20GF$F)F)F'F+F)I$s11GF$F+ I$s21GF$F+*&,&I$s12GF$F+I$s22GF$F+F)F(F)F)F)F,F)F+*&F'F),0*(F4F)F5F)F' !\"#F)*&I%P012GF$F)F'F+F)*&I$s10GF$F)F;F)F)*&F6F)F7F)F)*&F:F)I$s20GF$F )F)*&,&*&F6F)F;F)F)*&F:F)F7F)F)F)F(F)F)*(F:F)F;F)F(F-F)F)F)FAF+*(F/F)F :F)F'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "LP:=f actor(simplify(LPfunction(" }{MPLTEXT 1 0 4 "Cbis" }{MPLTEXT 1 0 5 ")) ):\n" }{MPLTEXT 1 0 101 "LPbis:=mu*(P*diff(R1(Q),Q)+P*h*1/(Q-X1)^2-dif f(tdR2(Q),Q)- C01/(Q-X1)^2)+h*nuMinus1*P +h*c1+2*h*c2*Q:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "factor(series(LP-LPbis,P012=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "LQbis:=" }{MPLTEXT 1 0 31 "2*mu*(P-R1(Q)/2+1/2*h*1/(Q-X1))" } {MPLTEXT 1 0 20 "-h*nu0-h*nuMinus1*Q:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "simplify(LQ-LQbis);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "nuMinus1:=" }{MPLTEXT 1 0 8 "nuMinus1" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 10 "nu0:=nu0;\n" } {MPLTEXT 1 0 8 "c1:=c1;\n" }{MPLTEXT 1 0 7 "c2:=c2;" }{MPLTEXT 1 0 0 " " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)nuMinus1G6\",$*&,&I'beta12GF$\"\" \"I'beta22GF$!\"\"F),&I$s12GF$F)I$s22GF$F+F+#F)\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu0G6\",$*&,,**I\"QGF$\"\"\"I#X1GF$F*,&I'beta12GF$F *I'beta22GF$!\"\"F*,&I$s12GF$F*I$s22GF$F/F/#F*\"\"#*(F+F4F,F*F0F/#F/F4 *&F)F*I'betaX1GF$F*F**&F+F*F8F*F/*(,&F)F*F+F/F*,@*(F+F*F-F*F1F*F**(F+F *F-F*F2F*F/*(F+F*F.F*F1F*F/*(F+F*F.F*F2F*F**&F8F*F1F4F4*(F8F*F1F*F2F*! \"%*&F8F*F2F4F4*&I'beta11GF$F*F1F*F4*&FFF*F2F*!\"#*&F-F*I$s11GF$F*F/*& F-F*I$s21GF$F*F**&I'beta21GF$F*F1F*FH*&FNF*F2F*F4*&F.F*FJF*F**&F.F*FLF *F/F*F0FHF6F*F;F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c1G6\",$*&,2*( I'beta11GF$\"\"\"I$s12GF$F*I$s22GF$F*\"\"#*&F)F*F,F-!\"#*(I'beta12GF$F *I$s11GF$F*F,F*!\"\"*(F1F*F+F*I$s21GF$F*F**&I'beta21GF$F*F+F-F/*(F7F*F +F*F,F*F-*(I'beta22GF$F*F2F*F,F*F**(F:F*F+F*F5F*F3F*,&F+F*F,F3F/#F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#c2G6\"*&,&*&I'beta12GF$\"\"\"I$s22 GF$F)F)*&I'beta22GF$F)I$s12GF$F)!\"\"F),&F-\"\"#F*!\"#F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 17 "We thus get that\n" }{TEXT 212 5 "L[Q]=" }{TEXT 212 60 "2*mu*Q*(P-R1(Q)/2)-(1/2)*h*Q/(s12-s22)*(beta12-beta22) -h*mu" }}{PARA 0 "" 0 "" {TEXT 212 7 "L[P] = " }{TEXT 212 96 "-mu*P^2 +mu*diff(P*Q*R1(Q)-Q*R2(Q),Q)-h*s12*mu +1/2*h*(beta12-beta22)/(s12-s22 )*P+ h*c1+2*h*c2*Q\n" }{TEXT 212 6 "with \n" }{XPPEDIT 2 0 "Typesettin g:-mrow(Typesetting:-mi(\"nuMinus1\", bold = \"true\", italic = \"true \", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting :-mo(\"≔\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \"false\", stretchy = \"fa lse\", symmetric = \"false\", largeop = \"false\", movablelimits = \"f alse\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.277 7778em\"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mi(\"&alp ha;12\", bold = \"true\", italic = \"true\", mathvariant = \"bold-ital ic\", fontweight = \"bold\"), Typesetting:-mo(\"−\", bold = \"tr ue\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"& alpha;22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-i talic\", fontweight = \"bold\")), Typesetting:-mrow(Typesetting:-mn(\" 2\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), \+ Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence \+ = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced (Typesetting:-mrow(Typesetting:-mi(\"s12\", bold = \"true\", italic = \+ \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Types etting:-mo(\"−\", bold = \"true\", mathvariant = \"bold\", fontw eight = \"bold\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \" 0.2222222em\"), Typesetting:-mi(\"s22\", bold = \"true\", italic = \"t rue\", mathvariant = \"bold-italic\", fontweight = \"bold\")), bold = \+ \"true\", mathvariant = \"bold\", fontweight = \"bold\")), linethickne ss = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"false\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I#miGF$6'Q)nuMinus1F'/%%boldGQ%trueF'/%'italicGF1/%,mathvariantGQ,bo ld-italicF'/%+fontweightGQ%boldF'-I#moGF$6/Q)≔F'F//F5F9F7/%&fen ceGQ&falseF'/%*separatorGFA/%)stretchyGFA/%*symmetricGFA/%(largeopGFA/ %.movablelimitsGFA/%'accentGFA/%'lspaceGQ,0.2777778emF'/%'rspaceGFP-I& mfracGF$6(-F#6%-F,6'Q*α12F'F/F2F4F7-F;6/Q(−F'F/F>F7F?FBFDF FFHFJFL/FOQ,0.2222222emF'/FRFin-F,6'Q*α22F'F/F2F4F7-F#6%-I#mnGF$ 6&Q\"2F'F/F>F7-F;6-Q1⁢F'/F5Q'normalF'F?FBFDFFFHFJFL/FOQ &0.0emF'/FRFjo-I(mfencedGF$6&-F#6%-F,6'Q$s12F'F/F2F4F7Fen-F,6'Q$s22F'F /F2F4F7F/F>F7/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalig nGF\\q/%)bevelledGFA" }{TEXT 212 1 "\n" }{XPPEDIT 2 0 "Typesetting:-mr ow(Typesetting:-mi(\"ν0\", bold = \"true\", italic = \"true\", math variant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"& Assign;\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bol d\", fence = \"false\", separator = \"false\", stretchy = \"false\", s ymmetric = \"false\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\" ), Typesetting:-mrow(Typesetting:-mi(\"\", italic = \"true\", mathvari ant = \"italic\"), Typesetting:-mo(\"&uminus0;\", mathvariant = \"norm al\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em \"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-mfrac(Typesetti ng:-mrow(Typesetting:-mi(\"h\", bold = \"true\", italic = \"true\", ma thvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo( \"⁢\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"Q\", bold = \"true \", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"b old\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\" , fence = \"false\", separator = \"false\", stretchy = \"false\", symm etric = \"false\", largeop = \"false\", movablelimits = \"false\", acc ent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting: -mi(\"X1\", bold = \"true\", italic = \"true\", mathvariant = \"bold-i talic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\" α12\", bold = \"true\", italic = \"true\", mathvariant = \"bold- italic\", fontweight = \"bold\"), Typesetting:-mo(\"−\", bold = \+ \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi (\"α22\", bold = \"true\", italic = \"true\", mathvariant = \"bo ld-italic\", fontweight = \"bold\")), bold = \"true\", mathvariant = \+ \"bold\", fontweight = \"bold\")), Typesetting:-mrow(Typesetting:-mn( \"2\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\") , Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenc ed(Typesetting:-mrow(Typesetting:-mi(\"s12\", bold = \"true\", italic \+ = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typ esetting:-mo(\"−\", bold = \"true\", mathvariant = \"bold\", fon tweight = \"bold\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \+ \"0.2222222em\"), Typesetting:-mi(\"s22\", bold = \"true\", italic = \+ \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\")), linethic kness = \"1\", denomalign = \"center\", numalign = \"center\", bevelle d = \"false\"), Typesetting:-mo(\"−\", bold = \"true\", mathvari ant = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.222 2222em\", rspace = \"0.2222222em\"), Typesetting:-mfrac(Typesetting:-m row(Typesetting:-mi(\"h\", bold = \"true\", italic = \"true\", mathvar iant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"&Inv isibleTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting:-mi(\"X1\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fo ntweight = \"bold\"), Typesetting:-mn(\"2\", bold = \"true\", mathvari ant = \"bold\", fontweight = \"bold\"), superscriptshift = \"0\"), Typ esetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(T ypesetting:-mrow(Typesetting:-mi(\"α12\", bold = \"true\", itali c = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), T ypesetting:-mo(\"−\", bold = \"true\", mathvariant = \"bold\", f ontweight = \"bold\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace \+ = \"0.2222222em\"), Typesetting:-mi(\"α22\", bold = \"true\", it alic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\") ), bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\")), T ypesetting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), Typesetting:-mo(\"⁢ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi (\"s12\", bold = \"true\", italic = \"true\", mathvariant = \"bold-ita lic\", fontweight = \"bold\"), Typesetting:-mo(\"−\", bold = \"t rue\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi( \"s22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-ital ic\", fontweight = \"bold\")), bold = \"true\", mathvariant = \"bold\" , fontweight = \"bold\")), linethickness = \"1\", denomalign = \"cente r\", numalign = \"center\", bevelled = \"false\"), Typesetting:-mo(\"& plus;\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mrow(Typesetting:-mi(\"betaX1\", bold = \"true\", itali c = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), T ypesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"Q \", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"−\", bold = \"true\" , mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\", se parator = \"false\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Types etting:-mi(\"betaX1\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"&Invisib leTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mi(\"X1\", bold = \"true\", ital ic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"−\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspac e = \"0.2222222em\"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting :-mfenced(Typesetting:-mrow(Typesetting:-mi(\"Q\", bold = \"true\", it alic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\") , Typesetting:-mo(\"−\", bold = \"true\", mathvariant = \"bold\" , fontweight = \"bold\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspa ce = \"0.2222222em\"), Typesetting:-mi(\"X1\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), b old = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), Typese tting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Type setting:-mrow(Typesetting:-mi(\"\", italic = \"true\", mathvariant = \+ \"italic\"), Typesetting:-mrow(Typesetting:-mi(\"α12\", bold = \+ \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Types etting:-mi(\"h\", bold = \"true\", italic = \"true\", mathvariant = \" bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"&InvisibleTim es;\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.0em\"), Typesetting:-mi(\"X1\", bold = \"true\", italic = \+ \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Types etting:-mo(\"⁢\", 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:-mi(\"s12\", \+ bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fon tweight = \"bold\")), Typesetting:-mo(\"−\", bold = \"true\", ma thvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separa tor = \"false\", stretchy = \"false\", symmetric = \"false\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesett ing:-mi(\"α12\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"&Invisib leTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mi(\"h\", bold = \"true\", itali c = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), T ypesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"X1 \", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-mi(\"s22\", bold = \"true\", italic = \"true\", math variant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\" −\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bol d\", fence = \"false\", separator = \"false\", stretchy = \"false\", s ymmetric = \"false\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\" ), Typesetting:-mrow(Typesetting:-mi(\"α22\", bold = \"true\", i talic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\" ), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fen ce = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi( \"h\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic \", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", math variant = \"normal\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0. 0em\"), Typesetting:-mi(\"X1\", bold = \"true\", italic = \"true\", ma thvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo( \"⁢\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"s12\", bold = \"tr ue\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \+ \"bold\")), Typesetting:-mo(\"+\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \"fal se\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\" , movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222e m\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mi(\"&a lpha;22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-it alic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", \+ mathvariant = \"normal\", fence = \"false\", separator = \"false\", st retchy = \"false\", symmetric = \"false\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.0em\"), Typesetting:-mi(\"h\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-m o(\"⁢\", mathvariant = \"normal\", fence = \"false\", s eparator = \"false\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"X1\", bold = \"t rue\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \+ \"bold\"), Typesetting:-mo(\"⁢\", mathvariant = \"norma l\", fence = \"false\", separator = \"false\", stretchy = \"false\", s ymmetric = \"false\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetti ng:-mi(\"s22\", bold = \"true\", italic = \"true\", mathvariant = \"bo ld-italic\", fontweight = \"bold\")), Typesetting:-mo(\"+\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting: -mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", \+ fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-mi(\"betaX1\", bold = \"true\", italic = \"true\", m athvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo( \"⁢\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting:-mi(\" s12\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic \", fontweight = \"bold\"), Typesetting:-mn(\"2\", bold = \"true\", ma thvariant = \"bold\", fontweight = \"bold\"), superscriptshift = \"0\" )), Typesetting:-mo(\"−\", bold = \"true\", mathvariant = \"bold \", fontweight = \"bold\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rs pace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn(\"4\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), Typesetti ng:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"betaX1\", b old = \"true\", italic = \"true\", mathvariant = \"bold-italic\", font weight = \"bold\"), Typesetting:-mo(\"⁢\", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), \+ Typesetting:-mi(\"s12\", bold = \"true\", italic = \"true\", mathvaria nt = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"&Invis ibleTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0e m\", rspace = \"0.0em\"), Typesetting:-mi(\"s22\", bold = \"true\", it alic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\") ), Typesetting:-mo(\"+\", bold = \"true\", mathvariant = \"bold\" , fontweight = \"bold\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspa ce = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), Typesetting :-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting:- mi(\"s22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-i talic\", fontweight = \"bold\"), Typesetting:-mn(\"2\", bold = \"true \", mathvariant = \"bold\", fontweight = \"bold\"), superscriptshift = \"0\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal \", 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(\"betaX1\", bold = \"true\", italic = \"true\", mathvariant = \" bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"+\", bo ld = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesettin g:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\" , fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathva riant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0e m\"), Typesetting:-mi(\"α11\", bold = \"true\", italic = \"true \", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting :-mo(\"⁢\", mathvariant = \"normal\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", bold = \+ \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Types etting:-mi(\"s12\", bold = \"true\", italic = \"true\", mathvariant = \+ \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"−\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typese tting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bo ld\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \" 0.0em\"), Typesetting:-mi(\"α11\", bold = \"true\", italic = \"t rue\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesett ing:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontwei ght = \"bold\"), Typesetting:-mo(\"⁢\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Ty pesetting:-mi(\"s22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"− \", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fe nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Type setting:-mrow(Typesetting:-mi(\"α12\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Type setting:-mo(\"⁢\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", b old = \"true\", italic = \"true\", mathvariant = \"bold-italic\", font weight = \"bold\"), Typesetting:-mo(\"⁢\", mathvariant \+ = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"f alse\", symmetric = \"false\", largeop = \"false\", movablelimits = \" false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), \+ Typesetting:-mi(\"s11\", bold = \"true\", italic = \"true\", mathvaria nt = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"&plus ;\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typ esetting:-mrow(Typesetting:-mi(\"α12\", bold = \"true\", italic \+ = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typ esetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fo ntweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvarian t = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\" ), Typesetting:-mi(\"s21\", bold = \"true\", italic = \"true\", mathva riant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"&m inus;\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvarian t = \"bold\", fontweight = \"bold\"), Typesetting:-mo(\"&InvisibleTime s;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mi(\"α21\", bold = \"true\", ital ic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), \+ Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence \+ = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h \", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-mi(\"s12\", bold = \"true\", italic = \"true\", math variant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\" +\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvarian t = \"bold\", fontweight = \"bold\"), Typesetting:-mo(\"&InvisibleTime s;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mi(\"α21\", bold = \"true\", ital ic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), \+ Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fence \+ = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h \", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-mi(\"s22\", bold = \"true\", italic = \"true\", math variant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\" +\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mrow(Typesetting:-mi(\"α22\", bold = \"true\", it alic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\") , Typesetting:-mo(\"⁢\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\" h\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\" , fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathva riant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0e m\"), Typesetting:-mi(\"s11\", bold = \"true\", italic = \"true\", mat hvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo( \"−\", bold = \"true\", mathvariant = \"bold\", fontweight = \"b old\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em \"), Typesetting:-mrow(Typesetting:-mi(\"α22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold \"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi (\"h\", bold = \"true\", italic = \"true\", mathvariant = \"bold-itali c\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mat hvariant = \"normal\", fence = \"false\", separator = \"false\", stret chy = \"false\", symmetric = \"false\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0 .0em\"), Typesetting:-mi(\"s21\", bold = \"true\", italic = \"true\", \+ mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-m i(\"\", italic = \"true\", mathvariant = \"italic\")), bold = \"true\" , mathvariant = \"bold\", fontweight = \"bold\")), Typesetting:-mrow(T ypesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", fontwei ght = \"bold\"), Typesetting:-mo(\"⁢\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Ty pesetting:-msup(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi (\"s12\", bold = \"true\", italic = \"true\", mathvariant = \"bold-ita lic\", fontweight = \"bold\"), Typesetting:-mo(\"−\", bold = \"t rue\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi( \"s22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-ital ic\", fontweight = \"bold\")), bold = \"true\", mathvariant = \"bold\" , fontweight = \"bold\"), Typesetting:-mn(\"2\", bold = \"true\", math variant = \"bold\", fontweight = \"bold\"), superscriptshift = \"0\")) , linethickness = \"1\", denomalign = \"center\", numalign = \"center \", bevelled = \"false\")), Typesetting:-mrow(Typesetting:-mfenced(Typ esetting:-mrow(Typesetting:-mi(\"Q\", bold = \"true\", italic = \"true \", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting :-mo(\"−\", bold = \"true\", mathvariant = \"bold\", fontweight \+ = \"bold\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222 222em\"), Typesetting:-mi(\"X1\", bold = \"true\", italic = \"true\", \+ mathvariant = \"bold-italic\", fontweight = \"bold\")), bold = \"true \", mathvariant = \"bold\", fontweight = \"bold\"), Typesetting:-mo(\" ⁢\", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"h\", bold = \"true\" , italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bol d\")), linethickness = \"1\", denomalign = \"center\", numalign = \"ce nter\", bevelled = \"false\")), Typesetting:-mi(\"\", italic = \"true \", mathvariant = \"italic\"));" "-I%mrowG6#/I+modulenameG6\"I,Typeset tingGI(_syslibGF'6&-I#miGF$6'Q&ν0F'/%%boldGQ%trueF'/%'italicGF1/%,m athvariantGQ,bold-italicF'/%+fontweightGQ%boldF'-I#moGF$6/Q)≔F' F//F5F9F7/%&fenceGQ&falseF'/%*separatorGFA/%)stretchyGFA/%*symmetricGF A/%(largeopGFA/%.movablelimitsGFA/%'accentGFA/%'lspaceGQ,0.2777778emF' /%'rspaceGFP-F#6%-F,6%Q!F'F2/F5Q'italicF'-F;6-Q*&uminus0;F'/F5Q'normal F'F?FBFDFFFHFJFL/FOQ,0.2222222emF'/FRFjn-I&mfracGF$6(-F#6+-F]o6(-F#6)- F,6'Q\"hF'F/F2F4F7-F;6-Q1⁢F'FgnF?FBFDFFFHFJFL/FOQ&0.0em F'/FRF\\p-F,6'Q\"QF'F/F2F4F7Fho-F,6'Q#X1F'F/F2F4F7Fho-I(mfencedGF$6&-F #6%-F,6'Q*α12F'F/F2F4F7-F;6/Q(−F'F/F>F7F?FBFDFFFHFJFLFinF[ o-F,6'Q*α22F'F/F2F4F7F/F>F7-F#6%-I#mnGF$6&Q\"2F'F/F>F7Fho-Fep6&- F#6%-F,6'Q$s12F'F/F2F4F7F\\q-F,6'Q$s22F'F/F2F4F7F/F>F7/%.linethickness GQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFgr/%)bevelledGFAF\\q-F]o6( -F#6'FeoFho-I%msupGF$6%FapFdq/%1superscriptshiftGQ\"0F'FhoFdpFbqFbrFer FhrFjr-F;6/Q'+F'F/F>F7F?FBFDFFFHFJFLFinF[o-F#6%-F,6'Q'betaX1F'F/F 2F4F7FhoF^pF\\q-F#6%F[tFhoFapF\\q-F]o6(-F#6%-Fep6&-F#6%F^pF\\qFapF/F>F 7Fho-Fep6&-F#6AFU-F#6)FipFhoFeoFhoFapFhoF\\rF\\q-F#6)FipFhoFeoFhoFapFh oF_rF\\q-F#6)F_qFhoFeoFhoFapFhoF\\rFfs-F#6)F_qFhoFeoFhoFapFhoF_rFfs-F# 6'FdqFhoF[tFho-Fas6%F\\rFdqFcsF\\q-F#6)-Feq6&Q\"4F'F/F>F7FhoF[tFhoF\\r FhoF_rFfs-F#6'FdqFho-Fas6%F_rFdqFcsFhoF[tFfs-F#6)FdqFho-F,6'Q*α1 1F'F/F2F4F7FhoFeoFhoF\\rF\\q-F#6)FdqFhoFcvFhoFeoFhoF_rF\\q-F#6'FipFhoF eoFho-F,6'Q$s11F'F/F2F4F7Ffs-F#6'FipFhoFeoFho-F,6'Q$s21F'F/F2F4F7F\\q- F#6)FdqFho-F,6'Q*α21F'F/F2F4F7FhoFeoFhoF\\rFfs-F#6)FdqFhoFdwFhoF eoFhoF_rFfs-F#6'F_qFhoFeoFhoFjvF\\q-F#6'F_qFhoFeoFhoF_wFUF/F>F7-F#6%Fd qFho-Fas6%FhqFdqFcsFbrFerFhrFjr-F#6%FdtFhoFeoFbrFerFhrFjrFU" }}{PARA 0 "" 0 "" {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"c1\", bold \+ = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweig ht = \"bold\"), Typesetting:-mo(\"≔\", bold = \"true\", mathvar iant = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.27 77778em\", rspace = \"0.2777778em\"), Typesetting:-mfrac(Typesetting:- mrow(Typesetting:-mi(\"\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvaria nt = \"bold\", fontweight = \"bold\"), Typesetting:-mo(\"&InvisibleTim es;\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rs pace = \"0.0em\"), Typesetting:-mi(\"s12\", bold = \"true\", italic = \+ \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Types etting:-mo(\"⁢\", 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:-mi(\"α 11\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic \", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", math variant = \"normal\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0. 0em\"), Typesetting:-mi(\"s22\", bold = \"true\", italic = \"true\", m athvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo (\"−\", bold = \"true\", mathvariant = \"bold\", fontweight = \" bold\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222 em\"), Typesetting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathv ariant = \"bold\", fontweight = \"bold\"), Typesetting:-mo(\"&Invisibl eTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\" , rspace = \"0.0em\"), Typesetting:-msup(Typesetting:-mi(\"s22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontwei ght = \"bold\"), Typesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), superscriptshift = \"0\"), Typesett ing:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"α11 \", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"−\", bold = \"true\" , mathvariant = \"bold\", fontweight = \"bold\", fence = \"false\", se parator = \"false\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Types etting:-mi(\"s11\", bold = \"true\", italic = \"true\", mathvariant = \+ \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"&InvisibleT imes;\", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"α12\", bold = \"true\", \+ italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold \"), Typesetting:-mo(\"⁢\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi (\"s22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-ita lic\", fontweight = \"bold\")), Typesetting:-mo(\"+\", bold = \"t rue\", mathvariant = \"bold\", fontweight = \"bold\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow (Typesetting:-mi(\"s21\", bold = \"true\", italic = \"true\", mathvari ant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"&Invi sibleTimes;\", mathvariant = \"normal\", fence = \"false\", separator \+ = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0 em\", rspace = \"0.0em\"), Typesetting:-mi(\"α12\", bold = \"tru e\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \" bold\"), Typesetting:-mo(\"⁢\", mathvariant = \"normal \", 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(\"s12\", bold = \"true\", italic = \"true\", mathvariant = \"bol d-italic\", fontweight = \"bold\")), Typesetting:-mo(\"−\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting: -mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", \+ fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-msup(Typesetting:-mi(\"s12\", bold = \"true\", itali c = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), T ypesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", fontwei ght = \"bold\"), superscriptshift = \"0\"), Typesetting:-mo(\"&Invisib leTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em \", rspace = \"0.0em\"), Typesetting:-mi(\"α21\", bold = \"true \", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"b old\")), Typesetting:-mo(\"+\", bold = \"true\", mathvariant = \" bold\", fontweight = \"bold\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em \", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn(\"2\" , bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), Typ esetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"s12 \", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em \"), Typesetting:-mi(\"α21\", bold = \"true\", italic = \"true\" , mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:- mo(\"⁢\", mathvariant = \"normal\", fence = \"false\", \+ separator = \"false\", stretchy = \"false\", symmetric = \"false\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", lspa ce = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"s22\", bold = \+ \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mo(\"+\", bold = \"true\", mathvarian t = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.22222 22em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mi( \"s11\", bold = \"true\", italic = \"true\", mathvariant = \"bold-ital ic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \" 0.0em\"), Typesetting:-mi(\"α22\", bold = \"true\", italic = \"t rue\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesett ing:-mo(\"⁢\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"s22\", bo ld = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontw eight = \"bold\")), Typesetting:-mo(\"−\", bold = \"true\", math variant = \"bold\", fontweight = \"bold\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting :-mi(\"s21\", bold = \"true\", italic = \"true\", mathvariant = \"bold -italic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢ \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mi(\"α22\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Ty pesetting:-mo(\"⁢\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"s12 \", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mi(\"\", italic = \"true\", mat hvariant = \"italic\")), Typesetting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), Typesetti ng:-mo(\"⁢\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-msup(Typesetting :-mfenced(Typesetting:-mrow(Typesetting:-mi(\"s12\", bold = \"true\", \+ italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold \"), Typesetting:-mo(\"−\", bold = \"true\", mathvariant = \"bol d\", fontweight = \"bold\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", r space = \"0.2222222em\"), Typesetting:-mi(\"s22\", bold = \"true\", it alic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\") ), bold = \"true\", mathvariant = \"bold\", fontweight = \"bold\"), Ty pesetting:-mn(\"2\", bold = \"true\", mathvariant = \"bold\", fontweig ht = \"bold\"), superscriptshift = \"0\")), linethickness = \"1\", den omalign = \"center\", numalign = \"center\", bevelled = \"false\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6'Q#c1 F'/%%boldGQ%trueF'/%'italicGF1/%,mathvariantGQ,bold-italicF'/%+fontwei ghtGQ%boldF'-I#moGF$6/Q)≔F'F//F5F9F7/%&fenceGQ&falseF'/%*separa torGFA/%)stretchyGFA/%*symmetricGFA/%(largeopGFA/%.movablelimitsGFA/%' accentGFA/%'lspaceGQ,0.2777778emF'/%'rspaceGFP-I&mfracGF$6(-F#63-F,6%Q !F'F2/F5Q'italicF'-F#6)-I#mnGF$6&Q\"2F'F/F>F7-F;6-Q1⁢F' /F5Q'normalF'F?FBFDFFFHFJFL/FOQ&0.0emF'/FRFco-F,6'Q$s12F'F/F2F4F7F]o-F ,6'Q*α11F'F/F2F4F7F]o-F,6'Q$s22F'F/F2F4F7-F;6/Q(−F'F/F>F7F ?FBFDFFFHFJFL/FOQ,0.2222222emF'/FRFbp-F#6'FinF]o-I%msupGF$6%F[pFin/%1s uperscriptshiftGQ\"0F'F]oFhoF^p-F#6'-F,6'Q$s11F'F/F2F4F7F]o-F,6'Q*&alp ha;12F'F/F2F4F7F]oF[p-F;6/Q'+F'F/F>F7F?FBFDFFFHFJFLFapFcp-F#6'-F, 6'Q$s21F'F/F2F4F7F]oFaqF]oFeoF^p-F#6'FinF]o-Fgp6%FeoFinFipF]o-F,6'Q*&a lpha;21F'F/F2F4F7Fdq-F#6)FinF]oFeoF]oF`rF]oF[pFdq-F#6'F^qF]o-F,6'Q*&al pha;22F'F/F2F4F7F]oF[pF^p-F#6'FiqF]oFgrF]oFeoFX-F#6%FinF]o-Fgp6%-I(mfe ncedGF$6&-F#6%FeoF^pF[pF/F>F7FinFip/%.linethicknessGQ\"1F'/%+denomalig nGQ'centerF'/%)numalignGFjs/%)bevelledGFA" }}{PARA 0 "" 0 "" {XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"c2\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\"), Ty pesetting:-mo(\"≔\", bold = \"true\", mathvariant = \"bold\", f ontweight = \"bold\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace \+ = \"0.2777778em\"), Typesetting:-mfrac(Typesetting:-mrow(Typesetting:- mi(\"\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mr ow(Typesetting:-mi(\"α12\", bold = \"true\", italic = \"true\", \+ mathvariant = \"bold-italic\", fontweight = \"bold\"), Typesetting:-mo (\"⁢\", mathvariant = \"normal\", fence = \"false\", se parator = \"false\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"s22\", bold = \"t rue\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \+ \"bold\")), Typesetting:-mo(\"−\", bold = \"true\", mathvariant \+ = \"bold\", fontweight = \"bold\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.222222 2em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mi(\" α22\", bold = \"true\", italic = \"true\", mathvariant = \"bold- italic\", fontweight = \"bold\"), Typesetting:-mo(\"⁢\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace \+ = \"0.0em\"), Typesetting:-mi(\"s12\", bold = \"true\", italic = \"tru e\", mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetti ng:-mi(\"\", italic = \"true\", mathvariant = \"italic\")), Typesettin g:-mrow(Typesetting:-mi(\"\", italic = \"true\", mathvariant = \"itali c\"), Typesetting:-mrow(Typesetting:-mn(\"2\", bold = \"true\", mathva riant = \"bold\", fontweight = \"bold\"), Typesetting:-mo(\"&Invisible Times;\", mathvariant = \"normal\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"s12\", bold = \"true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), T ypesetting:-mo(\"−\", bold = \"true\", mathvariant = \"bold\", f ontweight = \"bold\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace \+ = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-mn(\"2\", bold = \" true\", mathvariant = \"bold\", fontweight = \"bold\"), Typesetting:-m o(\"⁢\", mathvariant = \"normal\", fence = \"false\", s eparator = \"false\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"s22\", bold = \" true\", italic = \"true\", mathvariant = \"bold-italic\", fontweight = \"bold\")), Typesetting:-mi(\"\", italic = \"true\", mathvariant = \" italic\")), linethickness = \"1\", denomalign = \"center\", numalign = \"center\", bevelled = \"false\"));" "-I%mrowG6#/I+modulenameG6\"I,Ty pesettingGI(_syslibGF'6%-I#miGF$6'Q#c2F'/%%boldGQ%trueF'/%'italicGF1/% ,mathvariantGQ,bold-italicF'/%+fontweightGQ%boldF'-I#moGF$6/Q)≔ F'F//F5F9F7/%&fenceGQ&falseF'/%*separatorGFA/%)stretchyGFA/%*symmetric GFA/%(largeopGFA/%.movablelimitsGFA/%'accentGFA/%'lspaceGQ,0.2777778em F'/%'rspaceGFP-I&mfracGF$6(-F#6'-F,6%Q!F'F2/F5Q'italicF'-F#6%-F,6'Q*&a lpha;12F'F/F2F4F7-F;6-Q1⁢F'/F5Q'normalF'F?FBFDFFFHFJFL/ FOQ&0.0emF'/FRFbo-F,6'Q$s22F'F/F2F4F7-F;6/Q(−F'F/F>F7F?FBFDFFFHF JFL/FOQ,0.2222222emF'/FRF[p-F#6%-F,6'Q*α22F'F/F2F4F7F\\o-F,6'Q$s 12F'F/F2F4F7FX-F#6'FX-F#6%-I#mnGF$6&Q\"2F'F/F>F7F\\oFbpFgo-F#6%FipF\\o FdoFX/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFdq/%) bevelledGFA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Hamiltonian bis:= mu*(P^2-R1(Q)*P+h*P/(Q-X1) +tdR2(Q)+h*s12 )-h*nu0*P-h*nuMinus1*Q *P-h*c1*Q-h*c2*Q^2 \n" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 48 "factor( simplify(LP-(-diff(Hamiltonianbis,Q))));\n" }{MPLTEXT 1 0 38 "simplify (LQ-(diff(Hamiltonianbis,P)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 139 "In order to match the notation with the article, we sha ll take Hamiltonianbis as the Hamiltonian including the purely time de pendent terms." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify( series(mu,betaX1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "+'I'betaX1G6\",$* (,&I\"QGF$\"\"\"I#X1GF$!\"\"F),(*&,(*&,&I'beta22GF$F)I'beta12GF$F+F)F* F)F)I'beta11GF$!\"#I'beta21GF$\"\"#F)I$s12GF$F)F)*&,(*&F*F),&F2F)F1F+F )F)F3F6F5F4F)I$s22GF$F)F)*&,&I$s21GF$F+I$s11GF$F)F)F0F)F+F),&F7F)F " 0 "" {MPLTEXT 1 0 83 "simplify (L[2,1]-(-h*P/(xi-Q)-tdR2(xi)-h*s12+((Q-X1)*(P^2-R1(Q)*P+h/(Q-X1)*P+td R2(Q)" }{MPLTEXT 1 0 22 "+h*s12) )/(xi-X1) ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 256 81 "Expressi on of the Lax matrix in the geometric gauge and normalisation at infin ity" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "simplify(checkL[1,1]) ;\n" }{MPLTEXT 1 0 23 "simplify(checkL[1,2]);\n" }{MPLTEXT 1 0 38 "che ckL22bis:=R1(xi)-P*(Q-X1)/(xi-X1);\n" }{MPLTEXT 1 0 35 "simplify(check L[2,2]-checkL22bis);\n" }{MPLTEXT 1 0 31 "checkL21:=factor(checkL[2,1] );\n" }{MPLTEXT 1 0 37 "simplify(series(checkL[2,1],xi=X1)):\n" } {MPLTEXT 1 0 201 "checkL21bis:=(P*(Q-X1)-sX10)*(P*(Q-X1)-sX20)/((Q-X1) *(xi-X1))-s12*s22*xi^2+(((-Q+X1)*s22-s21)*s12-s11*s22)*xi+((-Q^2+Q*X1) *s22+(-s21-P)*Q+(s21+P)*X1-h-s20)*s12+((-P-s11)*Q+(s11+P)*X1-s10)*s22- s11*s21:\n" }{MPLTEXT 1 0 31 "simplify(checkL21-checkL21bis);" }} {PARA 11 "" 1 "" {XPPMATH 20 ",$*(I\"PG6\"\"\"\",&I\"QGF%F&I#X1GF%!\" \"F&,&I#xiGF%F*F)F&F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I#xiG6\"! \"\"I#X1GF%\"\"\"F&,&F$F&I\"QGF%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I,checkL22bisG6\",,*&,&I%sX10GF$\"\"\"I%sX20GF$F)F),&I#xiGF$F)I#X1GF$ !\"\"F.F)I$s11GF$F.I$s21GF$F.*&,&I$s12GF$F.I$s22GF$F.F)F,F)F)*(I\"PGF$ F),&I\"QGF$F)F-F.F)F+F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I)checkL21G6\",$*(,\\s**I\"QGF$\"\"$I#X 1GF$\"\"\"I$s12GF$F,I$s22GF$F,F,**F)F*F-F,F.F,I#xiGF$F,!\"\"**F)\"\"#F +F3F-F,F.F,!\"#*,F)F3F+F,F-F,F.F,F0F,F***F)F3F-F,F.F,F0F3F1**F)F,F+F*F -F,F.F,F,*,F)F,F+F3F-F,F.F,F0F,!\"$*,F)F,F+F,F-F,F.F,F0F3F***F)F,F-F,F .F,F0F*F1**F+F*F-F,F.F,F0F,F,**F+F3F-F,F.F,F0F3F4**F+F,F-F,F.F,F0F*F,* *I\"PGF$F,F)F3F+F,F-F,F,**F@F,F)F3F+F,F.F,F,**F@F,F)F3F-F,F0F,F1**F@F, F)F3F.F,F0F,F1**F@F,F)F,F+F3F-F,F4**F@F,F)F,F+F3F.F,F4*,F@F,F)F,F+F,F- F,F0F,F3*,F@F,F)F,F+F,F.F,F0F,F3*(F@F,F+F*F-F,F,*(F@F,F+F*F.F,F,**F@F, F+F3F-F,F0F,F1**F@F,F+F3F.F,F0F,F1**F)F3F+F,I$s11GF$F,F.F,F,**F)F3F+F, F-F,I$s21GF$F,F,**F)F3FMF,F.F,F0F,F1**F)F3F-F,FOF,F0F,F1**F)F,F+F3FMF, F.F,F4**F)F,F+F3F-F,FOF,F4*,F)F,F+F,FMF,F.F,F0F,F**,F)F,F+F,F-F,FOF,F0 F,F***F)F,FMF,F.F,F0F3F1**F)F,F-F,FOF,F0F3F1*(F+F*FMF,F.F,F,*(F+F*F-F, FOF,F,**F+F3FMF,F.F,F0F,F4**F+F3F-F,FOF,F0F,F4**F+F,FMF,F.F,F0F3F,**F+ F,F-F,FOF,F0F3F,*&F@F3F)F3F,*(F@F3F)F,F+F,F4*&F@F3F+F3F,**F)F,F+F,I\"h GF$F,F-F,F,**F)F,F+F,I$s10GF$F,F.F,F,**F)F,F+F,FMF,FOF,F,**F)F,F+F,F-F ,I$s20GF$F,F,**F)F,F\\oF,F-F,F0F,F1**F)F,F^oF,F.F,F0F,F1**F)F,FMF,FOF, F0F,F1**F)F,F-F,FaoF,F0F,F1*(F+F3F\\oF,F-F,F1*(F+F3F^oF,F.F,F1*(F+F3FM F,FOF,F1*(F+F3F-F,FaoF,F1**F+F,F\\oF,F-F,F0F,F,**F+F,F^oF,F.F,F0F,F,** F+F,FMF,FOF,F0F,F,**F+F,F-F,FaoF,F0F,F,*(F@F,F)F,I%sX10GF$F,F1*(F@F,F) F,I%sX20GF$F,F1*(F@F,F+F,F_pF,F,*(F@F,F+F,FapF,F,*&F_pF,FapF,F,F,,&F0F 1F+F,F1,&F)F,F+F1F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Verification:=simplify(LL-h* dAdxi-(Multiply(A,L)-Multiply(L,A)));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 25 "checkL:=simplify(checkL):" }}{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(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0, 0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+group alignGFC/%(rowspanGQ\"1F'/%+columnspanGFJF4FAFDFFF1/%&alignGQ%axisF'/F BQ)baselineF'/FEQ&rightF'/FGQ'|frleft|hrF'/%/alignmentscopeGQ%trueF'/% ,columnwidthGQ%autoF'/%&widthGFen/%+rowspacingGQ&1.0exF'/%.columnspaci ngGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF`o/%&frameGF`o/%-frame spacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFjo/%-dis playstyleGFjo/%%sideGFS/%0minlabelspacingGF]oF;F>/%%openGQ\"[F'/%&clos eGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "LcheckL:=Matrix (2,2,0):\n" }{MPLTEXT 1 0 273 "for i from 1 to 2 do for j from 1 to 2 \+ do LcheckL[i,j]:=diff(checkL[i,j],Q)*LQbis+diff(checkL[i,j],P)*LPbis+h *diff(checkL[i,j],s12)*beta12+ h*diff(checkL[i,j],s22)*beta22+h*diff(c heckL[i,j],s11)*beta11+h*diff(checkL[i,j],s21)*beta21+h*diff(checkL[i, j],X1)*betaX1: od: od: \n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 26 "chec kA:=simplify(checkA):\n" }{MPLTEXT 1 0 27 "dcheckAdxi:=Matrix(2,2,0): \n" }{MPLTEXT 1 0 89 "for i from 1 to 2 do for j from 1 to 2 do dcheck Adxi[i,j]:=diff(checkA[i,j],xi): od: od:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 95 "Verification:=simplify(LcheckL-h*dcheckAdxi-(Multiply (checkA,checkL)-Multiply(checkL,checkA)));" }{MPLTEXT 1 0 0 "" }} {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 nGFJF4FAFDFF-F26'-F56(-I&mfracGF$6)-F,6'-F#6%-F,6%-F,6%-I#miGF$6&Q)&be ta;12F'/%'italicGQ%trueF'F;/F?Q'italicF'-I#moGF$6-Q1⁢F' F>/%&fenceGQ&falseF'/%*separatorGFeo/%)stretchyGFeo/%*symmetricGFeo/%( largeopGFeo/%.movablelimitsGFeo/%'accentGFeo/%'lspaceGQ&0.0emF'/%'rspa ceGFdp-Fgn6&Q$s22F'FjnF;F]o-F`o6.Q(−F'F;F>FcoFfoFhoFjoF\\pF^pF`p /FcpQ,0.2222222emF'/FfpF^q-F,6%-Fgn6&Q)β22F'FjnF;F]oF_o-Fgn6&Q$s1 2F'FjnF;F]oF;F>F_o-F#6%-F,6)-Fgn6&Q$s10F'FjnF;F]o-F`o6.Q\"+F'F;F>FcoFf oFhoFjoF\\pF^pF`pF]qF_q-Fgn6&Q$s20F'FjnF;F]oF_r-Fgn6&Q%sX10F'FjnF;F]oF _r-Fgn6&Q%sX20F'FjnF;F]oF;F>F_o-Fgn6&Q\"hF'FjnF;F]o-F,6#-F,6%FeqFjpFgp /%.linethicknessGFJ/%+denomalignGQ'centerF'/%)numalignGFfs/%)bevelledG FeoF;FAFDFFFHFKF4FAFDFF/%&alignGQ%axisF'/FBQ)baselineF'/FEFfs/FGQ'|frl eft|hrF'/%/alignmentscopeGF\\o/%,columnwidthGQ%autoF'/%&widthGFgt/%+ro wspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,col umnlinesGFbu/%&frameGFbu/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFe o/%-equalcolumnsGFeo/%-displaystyleGFeo/%%sideGQ&rightF'/%0minlabelspa cingGF_uF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{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 22 "G1[2,1]:=s12*xi+eta0:\n" }{MPLTEXT 1 0 22 "eta0:=(Q- X1)*s12+s11;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "dG1dxi:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 81 "for i from 1 to 2 do for j from 1 to 2 do dG1dxi[i,j]:=diff(G1[i,j],xi): od: od:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 20 "LG1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 68 "for i from 1 to 2 do for j from 1 to 2 do LG1[i,j]:=diff(G1[i ,j],Q)*" }{MPLTEXT 1 0 167 "LQ+diff(G1[i,j],P)*LP+h*diff(G1[i,j],s12)* beta12+ h*diff(G1[i,j],s22)*beta22+h*diff(G1[i,j],s11)*beta11+h* diff( G1[i,j],s21)*beta21+h*diff(G1[i,j],X1)*betaX1: od: od: " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 81 "tdL:=simplify(Multiply (Multiply(G1,checkL),G1^(-1))+h*Multiply(dG1dxi,G1^(-1))):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 4 "LtdL" }{MPLTEXT 1 0 17 ":=Matrix( 2,2,0):\n" }{MPLTEXT 1 0 42 "for i from 1 to 2 do for j from 1 to 2 do " }{MPLTEXT 1 0 4 "LtdL" }{MPLTEXT 1 0 12 "[i,j]:=diff(" }{MPLTEXT 1 0 3 "tdL" }{MPLTEXT 1 0 9 "[i,j],Q)*" }{MPLTEXT 1 0 8 "LQ+diff(" } {MPLTEXT 1 0 3 "tdL" }{MPLTEXT 1 0 19 "[i,j],P)*LP+h*diff(" }{MPLTEXT 1 0 3 "tdL" }{MPLTEXT 1 0 26 "[i,j],s12)*beta12+ h*diff(" }{MPLTEXT 1 0 3 "tdL" }{MPLTEXT 1 0 25 "[i,j],s22)*beta22+h*diff(" }{MPLTEXT 1 0 3 "tdL" }{MPLTEXT 1 0 26 "[i,j],s11)*beta11+h* diff(" }{MPLTEXT 1 0 3 "tdL" }{MPLTEXT 1 0 26 "[i,j],s21)*beta21+ h*diff(" }{MPLTEXT 1 0 3 "t dL" }{MPLTEXT 1 0 26 "[i,j],X1)*betaX1: od: od: " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 76 "tdA:=simplify(Multiply(Multipl y(G1,checkA),G1^(-1))+Multiply(LG1,G1^(-1))):\n" }{MPLTEXT 1 0 24 "dtd Adxi:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 83 "for i from 1 to 2 do for j f rom 1 to 2 do dtdAdxi[i,j]:=diff(tdA[i,j],xi): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "Verification1:=simplify(" }{MPLTEXT 1 0 4 " LtdL" }{MPLTEXT 1 0 50 "-h*dtdAdxi-(Multiply(tdA,tdL)-Multiply(tdL,tdA )));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%eta0G6\",&*&,&I\"QGF$\"\"\"I# X1GF$!\"\"F)I$s12GF$F)F)I$s11GF$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(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0 ,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+grou palignGFC/%(rowspanGQ\"1F'/%+columnspanGFJF4FAFDFF-F26'-F56(-I&mfracGF $6)-F,6'-F#6%-F,6%-F,6%-I#miGF$6&Q)β12F'/%'italicGQ%trueF'F;/F?Q' italicF'-I#moGF$6-Q1⁢F'F>/%&fenceGQ&falseF'/%*separator GFeo/%)stretchyGFeo/%*symmetricGFeo/%(largeopGFeo/%.movablelimitsGFeo/ %'accentGFeo/%'lspaceGQ&0.0emF'/%'rspaceGFdp-Fgn6&Q$s22F'FjnF;F]o-F`o6 .Q(−F'F;F>FcoFfoFhoFjoF\\pF^pF`p/FcpQ,0.2222222emF'/FfpF^q-F,6%- Fgn6&Q)β22F'FjnF;F]oF_o-Fgn6&Q$s12F'FjnF;F]oF;F>F_o-F#6%-F,6)-Fgn 6&Q$s10F'FjnF;F]o-F`o6.Q\"+F'F;F>FcoFfoFhoFjoF\\pF^pF`pF]qF_q-Fgn6&Q$s 20F'FjnF;F]oF_r-Fgn6&Q%sX10F'FjnF;F]oF_r-Fgn6&Q%sX20F'FjnF;F]oF;F>F_o- Fgn6&Q\"hF'FjnF;F]o-F,6#-F,6%FeqFjpFgp/%.linethicknessGFJ/%+denomalign GQ'centerF'/%)numalignGFfs/%)bevelledGFeoF;FAFDFFFHFKF4FAFDFF/%&alignG Q%axisF'/FBQ)baselineF'/FEFfs/FGQ'|frleft|hrF'/%/alignmentscopeGF\\o/% ,columnwidthGQ%autoF'/%&widthGFgt/%+rowspacingGQ&1.0exF'/%.columnspaci ngGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFbu/%&frameGFbu/%-frame spacingGQ,0.4em~0.5exF'/%*equalrowsGFeo/%-equalcolumnsGFeo/%-displayst yleGFeo/%%sideGQ&rightF'/%0minlabelspacingGF_uF;F>/%%openGQ\"[F'/%&clo seGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(tdL); \n" }{MPLTEXT 1 0 32 "series(tdL[1,1],xi=infinity,1);\n" }{MPLTEXT 1 0 32 "series(tdL[1,2],xi=infinity,1);\n" }{MPLTEXT 1 0 31 "series(tdL[ 2,1],xi=infinity,1);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 53 "tdL11bis:=((Q-X1)*(Q*s12+P+s11))/(xi-X1)-s12*xi-s11; \n" }{MPLTEXT 1 0 28 "tdL12bis:=1+(X1-Q)/(xi-X1);\n" }{MPLTEXT 1 0 222 "tdL21bis:=(Q^2*s12+(-X1*s12+P+s11)*Q+(-P-s11)*X1-sX10)*(Q^2*s12+( -X1*s12+P+s11)*Q+(-P-s11)*X1-sX20)/(Q-X1)/(xi-X1)+(s12-s22)*(Q-X1)*P+( s12-s22)*s12*Q^2-(s12-s22)*(X1*s12-s11)*Q-(X1*s11+sX10+sX20+s20)*s12+s 22*(X1*s11-s10);\n" }{MPLTEXT 1 0 79 "tdL22bis:=(-Q^2*s12+(X1*s12-P-s1 1)*Q+(s11+P)*X1+sX10+sX20)/(xi-X1)-xi*s22-s21;\n" }{MPLTEXT 1 0 29 "si mplify(tdL[1,1]-tdL11bis);\n" }{MPLTEXT 1 0 29 "simplify(tdL[1,2]-tdL1 2bis);\n" }{MPLTEXT 1 0 29 "simplify(tdL[2,1]-tdL21bis);\n" }{MPLTEXT 1 0 29 "simplify(tdL[2,2]-tdL22bis);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 91 "L21bis:=-h*P/(xi-Q)-tdR2(xi)-h*s12+(Q-X1)/(xi-X1)*( P^2-R1(Q) *P+h/(Q-X1)*P+tdR2(Q)+h*s12):\n" }{MPLTEXT 1 0 26 "simplify( L[2,1]-L2 1bis);\n" }{MPLTEXT 1 0 25 "L22bis:=h/(xi-Q)+R1(xi):\n" }{MPLTEXT 1 0 40 "simplify(series(tdL[2,2]-tdL22bis,Q=0));" }}{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(-I&mfracGF$6)-F,6#-F, 6*-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'nor malF'/%&fenceGQ&falseF'/%*separatorGFJ/%)stretchyGFJ/%*symmetricGFJ/%( largeopGFJ/%.movablelimitsGFJ/%'accentGFJ/%'lspaceGQ,0.2222222emF'/%'r spaceGFY-F,6'-F#6%-F,6&F>-I#miGF$6&Q%ξF'/%'italicGFJFBFE-F?6.Q\"+F' FBFEFHFKFMFOFQFSFUFWFZ-F]o6&Q\"QF'/FaoQ%trueF'FB/FFQ'italicF'FBFE-F?6- Q1⁢F'FEFHFKFMFOFQFSFU/FXQ&0.0emF'/FenF`p-F#6%-F,6'F\\oF boFeo-F?6.Q(−F'FBFEFHFKFMFOFQFSFUFWFZ-F]o6&Q#X1F'FhoFBFjoFBFEF \\p-F]o6&Q$s12F'FhoFBFjoFbo-F,6%FipF\\p-F]o6&Q\"PF'FhoFBFjoFbo-F,6%-F] o6&Q$s11F'FhoFBFjoF\\pF\\oFfp-F,6%-F#6%-F,6%FaqFboFfqFBFEF\\pFeo-F,6#- F,6&F>F\\oFboFip/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numa lignGFhr/%)bevelledGFJFB/%)rowalignGQ!F'/%,columnalignGF_s/%+groupalig nGF_s/%(rowspanGFer/%+columnspanGFer-F56(-F86)-F,6#FjnF_rFcrFfrFirF[sF BF]sF`sFbsFdsFfsF]sF`sFbs-F26'-F56(-F86)-F,6#-F,61-F,6'-F#6%-F,6%F\\qF fp-F]o6&Q$s22F'FhoFBFjoFBFEF\\p-F#6%-F,6'-F,6%FeoF\\pF\\qFboFaqFboFfqF BFEF\\p-I%msupGF$6%Fip-I#mnGF$6%Q\"3F'FBFE/%1superscriptshiftGQ\"0F'Fb o-F,6%-F#6%-F,61-F,6%-F#6%-F,6&F>-F,6%FjuF\\p-Fhu6%F\\q-F[v6%Q\"2F'FBF EF^vFbo-F,6'FawF\\pF\\qF\\pF^uFBFEF\\p-Fhu6%FeoFawF^vFbo-F,6%-F#6%-F,6 (F>-F,6%F\\oF\\pF_wFbo-F,6%-F#6%-F,6'-F,6%F^uF\\pF\\oFfp-F,6%-F[v6%Q\" 4F'FBFEF\\pFaqFfp-F,6%FjxF\\pFfqFBFEF\\pF\\qFbo-F,6'FawF\\pF^uF\\pF[rF BFEF\\pFeoFbo-F,6%-F#6%-F,6)-F,6%-F#6%-F,6&F>FaqFfpFfqFBFEF\\pF\\oFbo- F]o6&Q%sX10F'FhoFBFjoFbo-F]o6&Q%sX20F'FhoFBFjoFbo-F]o6&Q$s20F'FhoFBFjo FBFEF\\pF\\qFbo-F,6'F^uF\\pF[rF\\pF\\oFfp-F,6#-Fhu6%FaqFawF^vFfp-F,6'F awF\\pFaqF\\pFfqFbo-F,6%-F]o6&Q$s10F'FhoFBFjoF\\pF^uFfp-F,6#-Fhu6%FfqF awF^vFBFEF\\p-Fhu6%FipFawF^vFbo-F,6%-F#6%-F,6--F,6%-F#6%-F,6%F]wFfp-F, 6%F\\qF\\pF^uFBFEF\\p-Fhu6%FeoFjuF^vFbo-F,6%-F#6%-F,6'-F,6'FawF\\pF\\o F\\pF_wFbo-F,6%-F#6%-F,6(F>-F,6'FawF\\pF^uF\\pF\\oFbo-F,6%-F[v6%Q\"5F' FBFEF\\pFaqFbo-F,6%F[^lF\\pFfqFBFEF\\pF\\qFfp-F,6%F^uF\\pF[rFBFEF\\pFf wFbo-F,6%-F#6%-F,6--F,6%-F#6%-F,6)-F,6%-F#6%-F,6%-F,6%FawF\\pFaqFbo-F, 6%FawF\\pFfqFBFEF\\pF\\oFfp-F,6%FawF\\pF]zFfp-F,6%FawF\\pF`zFfpFczFBFE F\\pF\\qFfp-F,6)FawF\\pF^uF\\pF[rF\\pF\\oFbo-F,6%FawF\\pFjzFbo-F,6'Fjx F\\pFaqF\\pFfqFfpF^[lFbo-F,6%FawF\\pFe[lFBFEF\\pFeoFfp-F,6'F\\oF\\p-F# 6%-F,6'F]zFboF`zFboFczFBFEF\\pF\\qFfp-F,6'F\\oF\\pF`[lF\\pF^uFfp-F,6%- F#6%-F,6%F]zFboF`zFBFEF\\pF[rFBFEF\\pFipFfp-F,6%-Fhu6%FeoFjxF^vF\\pF_w Ffp-F,6)FawF\\pF\\qF\\p-F#6%-F,6)-F,6%-F86)-F[v6%FerFBFEFawFcrFfrFirF[ sFBF\\p-F,6%F\\qF\\pF\\oFfp-F,6%F^blF\\pFfxFboFaqFboFfqFBFEF\\pFg\\lFb o-F,6%-F#6%-F,6%-F,6%-F#6%-F,6'FgyFboF]zFboF`zFBFEF\\pF\\qFfp-F,6%F[rF \\p-F#6%-F,6(F>FfxFboFaqFboFfqFBFEFBFEF\\pFfwFbo-F,6%-F#6%-F,6'Fd`lFbo Fj`lFboF\\alFBFEF\\pFeoFfp-F,6%F]zF\\pF`z-F,6%-F#6%FarFBFEF\\p-F#6%-F, 6%FeoFfpFipFBFEFcrFfrFirF[sFBF]sF`sFbsFdsFfs-F56(-F86)-F,6#-F,6/-F,6%- F#6%-F,6,F>FeuFfpFfxFfpFaqFfpFfqFfp-F]o6&Q$s21F'FhoFBFjoFBFEF\\pFipFbo -F,6%-Fhu6%F\\oFawF^vF\\pF^uFbo-F,6%FfelF\\pF\\oFbo-F,6%FfwF\\pF\\qFbo FiqFfpF`zFfpF]zF_rFcrFfrFirF[sFBF]sF`sFbsFdsFfsF]sF`sFbs/%&alignGQ%axi sF'/F^sQ)baselineF'/FasFhr/FcsQ'|frleft|hrF'/%/alignmentscopeGFio/%,co lumnwidthGQ%autoF'/%&widthGF]gl/%+rowspacingGQ&1.0exF'/%.columnspacing GQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFhgl/%&frameGFhgl/%-frame spacingGQ,0.4em~0.5exF'/%*equalrowsGFJ/%-equalcolumnsGFJ/%-displaystyl eGFJ/%%sideGQ&rightF'/%0minlabelspacingGFeglFBFE/%%openGQ\"[F'/%&close GQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I$s12G6\"\"\"\"I#xiGF%F&! \"\"I$s11GF%F(-I\"OG6$%*protectedGI(_syslibGF%6#*$F'F(F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "-I\"OG6$%*protectedGI(_syslibG6\"6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&,,*&,(*&,&*$I$s12G6\"\"\"#!\"\"*&F* \"\"\"I$s22GF+F/F/F/I\"QGF+F/F/*&,&I\"PGF+F-I$s11GF+F-F/F*F/F/*&F0F/,& F4F/F5F/F/F/F/I#X1GF+F,F-*&,**&,&F)F,F.!\"#F/F1F,F/*&,&*&,&F4F,F5F,F/F *F/F/F6F=F/F1F/F/*&,(I%sX10GF+F/I%sX20GF+F/I$s20GF+F/F/F*F/F-*&I$s10GF +F/F0F/F-F/F8F/F-*(F*F/,&F*#F/F,F0#F-F,F/F1\"\"$F,*&,&F2F/F6F/F/F1F,F- *&,&FBF/FGF/F/F1F/F-F/,&F1F/F8F-F-F/-I\"OG6$%*protectedGI(_syslibGF+6# *$I#xiGF+F-F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tdL11bisG6\",(*(,&I \"QGF$\"\"\"I#X1GF$!\"\"F),(*&F(F)I$s12GF$F)F)I\"PGF$F)I$s11GF$F)F),&I #xiGF$F)F*F+F+F)*&F.F)F2F)F+F0F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)t dL12bisG6\",&\"\"\"F&*&,&I\"QGF$!\"\"I#X1GF$F&F&,&I#xiGF$F&F+F*F*F&" } }{PARA 11 "" 1 "" {XPPMATH 20 ">I)tdL21bisG6\",.**,**&I\"QGF$\"\"#I$s1 2GF$\"\"\"F,*&,(*&I#X1GF$F,F+F,!\"\"I\"PGF$F,I$s11GF$F,F,F)F,F,*&,&F2F 1F3F1F,F0F,F,I%sX10GF$F1F,,*F(F,F-F,F4F,I%sX20GF$F1F,,&F)F,F0F1F1,&I#x iGF$F,F0F1F1F,*(,&F+F,I$s22GF$F1F,F9F,F2F,F,*(F=F,F+F,F)F*F,*(F=F,,&F/ F,F3F1F,F)F,F1*&,**&F0F,F3F,F,I$s20GF$F,F6F,F8F,F,F+F,F1*&F>F,,&FDF,I$ s10GF$F1F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tdL22bisG6\",(*&,,*&I \"QGF$\"\"#I$s12GF$\"\"\"!\"\"*&,(*&I#X1GF$F,F+F,F,I\"PGF$F-I$s11GF$F- F,F)F,F,*&,&F2F,F3F,F,F1F,F,I%sX10GF$F,I%sX20GF$F,F,,&I#xiGF$F,F1F-F-F ,*&I$s22GF$F,F9F,F-I$s21GF$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "GeneralSpectralCurvefunction:=unap ply(simplify(Determinant(y*IdentityMatrix(2)-tdL)),xi,Q,P,s12,s22,s11, s21,X1,s10,s20,sX10,sX20):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 97 "G eneralSpectralCurve:=GeneralSpectralCurvefunction(xi,Q,P,s12,s22,s11,s 21,X1,s10,s20,sX10,sX20);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "Ge neralSpectralCurvebis:=y^2 -R1(xi)*y+tdR2(xi) +(Q-X1)*(-P^2+R1(Q)*P-td R2(Q))/(xi-X1) ;\n" }{MPLTEXT 1 0 55 "simplify(GeneralSpectralCurve-Ge neralSpectralCurvebis);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5GeneralSpectralCurveG6\"*(,2*&,,*(I$s12GF$\"\"\"I$s22G F$F+I#xiGF$\"\"#!\"\"*&,&*&,&I$s21GF$F/I\"yGF$F/F+F*F+F+*&F,F+,&F5F+I$ s11GF$F+F+F/F+F-F+F+*(F*F+F,F+I\"QGF$F.F+*&,&*&,&F4F+I\"PGF$F+F+F*F+F+ *&F,F+,&F?F+F8F+F+F+F+F:F+F+*&,&F?F+F5F/F+,*F?F+F4F+F8F+F5F+F+F+F+I#X1 GF$\"\"$F+*&,0*(F,F+F-FFF*F+F.*&,(*(F:F+F*F+F,F+F+*&,&F4F.F5F.F+F*F+F+ F6F.F+F-F.F+*&,4F9F/*(,&F*F+F,F+F+FCF+F:F+F/*&F*F+I$s20GF$F+F+*&I$s10G F$F+F,F+F+*$F?F.F/*&,&F8F/F4F/F+F?F+F+*$F5F.F.*&,&F8F.F4F.F+F5F+F+*&F8 F+F4F+F+F+F-F+F+*(F*F+F,F+F:FF!\"#*&,&*&,&F?FinF4FinF+F*F+F+F@FinF+F:F .F+*&,0FSF/FUF/FWFin*&,&F8FinF4FinF+F?F+F+FZF+*&,&F4F+F8F+F+F5F+F+FgnF /F+F:F+F+*&,&I%sX10GF$F+I%sX20GF$F+F+FCF+F+F+FEF.F+*&,,*(F*F+F,F+F-\" \"%F/*&,(FLFinF2F+F6F/F+F-FFF+*&,**&,&*&,&F4FinF5FinF+F*F+F+F6FinF+F:F +F+FSF/FUF/*&F7F+,&F4F+F5F+F+F/F+F-F.F+*&,*FhnF.*&,&*&,&F4F.F?F.F+F*F+ F+F@F.F+F:F.F+*(FCF+FDF+F:F+F.FdoF/F+F-F+F+*&,*FhnF+*&FF+F+F+F:F+F+FdoF/F+F:F+F+F+FEF+F+**F:F+F*F+F,F+F-F[pF+*( F:F+,&*&FepF+F*F+F+F6F+F+F-FFF+*(,(FSF+FUF+FdpF+F+F:F+F-F.F+*&,,*(F:F[ pF*F+F,F+F/*&,&*&,&F4F/F?F/F+F*F+F+F@F/F+F:FFF+*&,(FSF/FUF/FbqF/F+F:F. F+*(FeoF+FCF+F:F+F+*&FfoF+FgoF+F/F+F-F+F+*(F:F+FfoF+FgoF+F+F+,&F-F/FEF +Fin,&F:F+FEF/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8GeneralSpectralCu rvebisG6\",4*$I\"yGF$\"\"#\"\"\"*&,**&,&I%sX10GF$F)I%sX20GF$F)F),&I#xi GF$F)I#X1GF$!\"\"F3F)I$s11GF$F3I$s21GF$F3*&,&I$s12GF$F3I$s22GF$F3F)F1F )F)F)F'F)F3*(F.F)F/F)F0!\"#F)*&I$s10GF$F)F9F)F)*&F4F)F5F)F)*&F8F)I$s20 GF$F)F)*&,&*&F4F)F9F)F)*&F8F)F5F)F)F)F1F)F)*(F8F)F9F)F1F(F)*(,&I\"QGF$ F)F2F3F),2*$I\"PGF$F(F3*&,**&F-F)FGF3F)F4F3F5F3*&F7F)FHF)F)F)FKF)F)*(F .F)F/F)FGF;F3FF3F?F3*&FBF)FHF)F3*(F8F)F9F)FHF(F3F)F0F3F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 57 "Computation of the Hamiltonian flows in various direction" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "LQfunction:=unapply(LQ,beta1 2,beta22,beta11,beta21,betaX1):\n" }{MPLTEXT 1 0 59 "LPfunction:=unapp ly(LP,beta12,beta22,beta11,beta21,betaX1):" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 100 "Hamiltonianfunction:=unapply(Hamiltonianbis,s12,s22, s11,s21,X1,beta12,beta22,beta11,beta21,betaX1):\n" }{MPLTEXT 1 0 20 "n u0:=simplify(nu0);\n" }{MPLTEXT 1 0 95 "c1bis:=-(1/2)*(s11*s22-s12*s21 )*(beta12-beta22)/(s12-s22)^2+(beta11*s22-beta21*s12)/(s12-s22);\n" } {MPLTEXT 1 0 43 "factor(series(simplify(c1-c1bis),beta12));\n" } {MPLTEXT 1 0 110 "mubis:=(Q-X1)*(betaX1 +(beta11-beta21)/(s12-s22)+ (X 1*(s12-s22)-s11+s21)*(beta12-beta22)/(2*(s12-s22)^2) ):\n" }{MPLTEXT 1 0 37 "simplify(series(mu-mubis,beta11=0)); " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$nu0G6\",$*&,(*&,&I'beta11GF$\"\"#I'beta21GF$!\"#\"\"\" I$s12GF$F.F.*&,&F*F-F,F+F.I$s22GF$F.F.*&,&I$s21GF$!\"\"I$s11GF$F.F.,&I 'beta12GF$F.I'beta22GF$F6F.F6F.,&F/F.F2F6F-#F.F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&c1bisG6\",&*(,&*&I$s11GF$\"\"\"I$s22GF$F*F**&I$s12GF$F *I$s21GF$F*!\"\"F*,&I'beta12GF$F*I'beta22GF$F/F*,&F-F*F+F/!\"##F/\"\"# *&,&*&I'beta11GF$F*F+F*F**&I'beta21GF$F*F-F*F/F*F3F/F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 212 18 "Trivial directions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "simplify(LQfunction(1,1,0,0,0));\n" }{MPLTEXT 1 0 33 "simplify(LPfunction(1,1,0,0,0));\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 20 "Hamiltonianfunction(" }{MPLTEXT 1 0 19 "s12,s22,s11,s21,X1," }{MPLTEXT 1 0 12 "1,1,0,0,0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "simplify(LQfunction(0,0,1,1,0));\n" } {MPLTEXT 1 0 33 "simplify(LPfunction(0,0,1,1,0));\n" }{MPLTEXT 1 0 9 " simplify(" }{MPLTEXT 1 0 51 "Hamiltonianfunction(s12,s22,s11,s21,X1,0, 0,1,1,0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 47 "simplify(LQfunction (2*s12,2*s22,s11,s21,-X1));\n" }{MPLTEXT 1 0 20 "simplify(LPfunction(" }{MPLTEXT 1 0 23 "2*s12,2*s22,s11,s21,-X1" }{MPLTEXT 1 0 4 "));\n" } {MPLTEXT 1 0 48 "simplify(Hamiltonianfunction(s12,s22,s11,s21,X1," } {MPLTEXT 1 0 23 "2*s12,2*s22,s11,s21,-X1" }{MPLTEXT 1 0 3 "));" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 38 "simplify(LQfunction(0,0,s12,s22, -1));\n" }{MPLTEXT 1 0 38 "simplify(LPfunction(0,0,s12,s22,-1));\n" } {MPLTEXT 1 0 48 "simplify(Hamiltonianfunction(s12,s22,s11,s21,X1," } {MPLTEXT 1 0 14 "0,0,s12,s22,-1" }{MPLTEXT 1 0 3 "));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&I\"QG6\"\" \"\"I\"hGF%F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&I\"QG6\"\"\"#I \"hGF%\"\"\"#F(F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 ",$I\"hG6\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "* &I\"QG6\"\"\"\"I\"hGF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&I\"QG6\" \"\"\"I\"hGF%F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&I\"hG6\"\"\"\" I\"PGF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(I\"hG6\"\"\"\"I\"PGF%F& I\"QGF%F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I\"hG6\"!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", $*&I\"hG6\"\"\"\"I\"PGF%F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 12 "Direction X1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "simplif y(LQfunction(0,0,0,0,1));\n" }{MPLTEXT 1 0 32 "simplify(LPfunction(0,0 ,0,0,1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "HamX1:=unapply( " } {MPLTEXT 1 0 52 "(Q-X1)*(P^2-P*R1(Q)+h*P/(Q-X1)+tdR2(Q)+h*s12),Q,P);\n " }{MPLTEXT 1 0 59 "simplify(Hamiltonianfunction(s12,s22,s11,s21,X1,0, 0,0,0,1)-" }{MPLTEXT 1 0 10 "HamX1(Q,P)" }{MPLTEXT 1 0 2 ");" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 43 "LQfu nction(0,0,0,0,1)-diff(HamX1(Q,P),P));\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 42 "LPfunction(0,0,0,0,1)+diff(HamX1(Q,P),Q));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&,&I$s12G6\"\"\"\"I$s22GF&F'F'I\"QGF&\"\"#F '*&,**&,&F%!\"\"F(F/F'I#X1GF&F'F'I\"PGF&F*I$s11GF&F'I$s21GF&F'F'F)F'F' *&,(F1!\"#F2F/F3F/F'F0F'F'I%sX20GF&F/I\"hGF&F'I%sX10GF&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,0*(I\"QG6\"\"\"%I$s12GF&\"\"\"I$s22GF&F)!\"$*& ,(*(I#X1GF&F)F(F)F*F)\"\")*&,&I\"PGF&!\"#I$s21GF&F4F)F(F)F)*&F*F),&F3F )I$s11GF&F)F)F4F)F%\"\"$F)*&,0*(F/\"\"#F(F)F*F)!\"(*&,&*&,&F3\"\"&F5FC F)F(F)F)F6FCF)F/F)F)*&,&I$s20GF&!\"\"I\"hGF&FGF)F(F)F)*$F3F=FG*&,&F8FG F5FGF)F3F)F)*&I$s10GF&F)F*F)FG*&F8F)F5F)FGF)F%F=F)*(,0FI&HamX1G6\"f*6$I\"QGF$I\"PGF$F$6$I)operatorGF$I&arrowGF$F$*&,&9$ \"\"\"I#X1GF$!\"\"F/,6*$9%\"\"#F/*&,**&,&I%sX10GF$F/I%sX20GF$F/F/F-F1F /I$s11GF$F1I$s21GF$F1*&,&I$s12GF$F1I$s22GF$F1F/F.F/F/F/F4F/F1*(I\"hGF$ F/F4F/F-F1F/*(F:F/F;F/F-!\"#F/*&I$s10GF$F/FAF/F/*&F " 0 "" {MPLTEXT 1 0 33 "simplify(LQfunction(0,0,1,0,0));\n" }{MPLTEXT 1 0 33 "simplify(LPfunction(0,0,1,0,0));\n" }{MPLTEXT 1 0 337 "Hams11bis:= unapply( (-Q^4*s12*s22+(2*X1*s12*s22-s11*s22-s12*s21)*Q^3+(-s12*s22*X1 ^2+(2*s11*s22+2*s12*s21)*X1+(-s20-h)*s12+(h-s10)*s22-s11*s21)*Q^2-((s1 1*s22+s12*s21)*X1+(-s20-h)*s12+(h-s10)*s22-s11*s21)*X1*Q-sX10*sX20)/(( Q-X1)*(s22-s12))+((s12+s22)*Q^2+((-s12-s22)*X1+s11+s21)*Q+(-s11-s21)*X 1-sX10-sX20)*P/(s12-s22)+(Q-X1)*P^2/(s12-s22)\n" }{MPLTEXT 1 0 44 "-(( s20+h)*s12+s10*s22+s11*s21)*X1/(s12-s22)\n" }{MPLTEXT 1 0 26 ",Q,P,s11 ,s12,s21,s22,X1);\n" }{MPLTEXT 1 0 50 "simplify(LQfunction(0,0,1,0,0)- diff(Hams11bis(Q,P," }{MPLTEXT 1 0 25 "s11,s12,s21,s22,X1),P));\n" } {MPLTEXT 1 0 50 "simplify(LPfunction(0,0,1,0,0)+diff(Hams11bis(Q,P," } {MPLTEXT 1 0 24 "s11,s12,s21,s22,X1),Q));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 72 "simplify(Hamiltonianfunction(s12,s22,s11,s21,X1,0,0,1 ,0,0)-Hams11bis(Q,P" }{MPLTEXT 1 0 19 ",s11,s12,s21,s22,X1" }{MPLTEXT 1 0 3 "));" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,,*&,&I$s12G6\"\"\"\"I$s 22GF'F(F(I\"QGF'\"\"#F(*&,**&,&F&!\"\"F)F0F(I#X1GF'F(F(I\"PGF'F+I$s11G F'F(I$s21GF'F(F(F*F(F(*&,(F2!\"#F3F0F4F0F(F1F(F(I%sX10GF'F0I%sX20GF'F0 F(,&F&F(F)F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(,0*(I\"QG6\"\"\"%I$s 12GF&\"\"\"I$s22GF&F)\"\"$*&,(*(I#X1GF&F)F(F)F*F)!\")*&,&I$s21GF&\"\"# I\"PGF&F4F)F(F)F)*&F*F),&F5F)I$s11GF&F)F)F4F)F%F+F)*&,,*(F/F4F(F)F*F) \"\"(*&,&*&,&F5!\"&F3FAF)F(F)F)F6FAF)F/F)F)*&,&I$s20GF&F)I\"hGF&F)F)F( F)F)*&,&FE!\"\"I$s10GF&F)F)F*F)F)*&F7F),&F3F)F5F)F)F)F)F%F4F)*(F/F),,F ;F)*&,&*&,&F5!\"#F3FRF)F(F)F)F6FRF)F/F)F)FBF)FFF)FJF)F)F%F)FR*&,&*&,&F 3FHF5FHF)F(F)F)F6FHF)F/F+F)*&,(FBF)FFF)FJF)F)F/F4F)*&I%sX10GF&F)I%sX20 GF&F)FHF),&F%F)F/FHFR,&F(FHF*F)FH" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I* Hams11bisG6\"f*6)I\"QGF$I\"PGF$I$s11GF$I$s12GF$I$s21GF$I$s22GF$I#X1GF$ F$6$I)operatorGF$I&arrowGF$F$,**(,,*(9$\"\"%9'\"\"\"9)F8!\"\"*&,(*(9*F 8F7F8F9F8\"\"#*&F9F89&F8F:*&F7F89(F8F:F8F5\"\"$F8*&,,*(F>F?F7F8F9F8F:* &,&F@F?FBF?F8F>F8F8*&,&I$s20GF$F:I\"hGF$F:F8F7F8F8*&,&FMF8I$s10GF$F:F8 F9F8F8*&FAF8FCF8F:F8F5F?F8*(,**&,&F@F8FBF8F8F>F8F8FJF8FNF8FQF:F8F>F8F5 F8F:*&I%sX10GF$F8I%sX20GF$F8F:F8,&F5F8F>F:F:,&F7F:F9F8F:F8*(,,*&,&F7F8 F9F8F8F5F?F8*&,(*&,&F7F:F9F:F8F>F8F8FAF8FCF8F8F5F8F8*&,&FAF:FCF:F8F>F8 F8FWF:FXF:F89%F8,&F7F8F9F:F:F8*(FYF8F_oF?F`oF:F8*(,(*&,&FLF8FMF8F8F7F8 F8*&FPF8F9F8F8FQF8F8F>F8F`oF:F:F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 21 "Non trivia l direction" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "solve(\{Sin fty1=s11+s21,Sinfty2=s12+s22, S2=sqrt(s12-s22)/sqrt(2), S1=(s11-s21)/s qrt(2)/sqrt(s12-s22),\n" }{MPLTEXT 1 0 45 "tdX1function=S2*X1+S1\},\{s 11,s12,s21,s22,X1\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'/I#X1G6\",$*& ,&I#S1GF%\"\"\"I-tdX1functionGF%!\"\"F*I#S2GF%F,F,/I$s11GF%,&I(Sinfty1 GF%#F*\"\"#*&F)F*F-F*F*/I$s12GF%,&I(Sinfty2GF%F2*$F-F3F*/I$s21GF%,&F4F ,F1F2/I$s22GF%,&F9F,F8F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "hpartialtdX1functionQ:=simplify(LQfunction(0,0,0,0,sqrt(2)/sqrt(s12-s 22) ));\n" }{MPLTEXT 1 0 23 "hpartialtdX1functionP:=" }{MPLTEXT 1 0 28 "simplify(LPfunction(0,0,0,0," }{MPLTEXT 1 0 24 "sqrt(2)/sqrt(s12-s 22) " }{MPLTEXT 1 0 3 "));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 272 " HamtdX1function:=unapply(2*P*((s12+s22)*Q^2+((-s12-s22)*X1+P+s11+s21)* Q+(-P-s11-s21)*X1+h-sX10-sX20)/sqrt(2*s12-2*s22)+(2*Q*(Q*(Q-X1)*s22+Q* s21-X1*s21+h+s20)*(Q-X1)*s12+2*Q*(Q-X1)*(Q*s11-X1*s11+s10)*s22+2*Q^2*s 11*s21-2*Q*X1*s11*s21+2*sX10*sX20)/(sqrt(2*s12-2*s22)*(Q-X1))\n" } {MPLTEXT 1 0 56 "-(2*((s20+h)*s12+s10*s22+s11*s21))*X1/sqrt(2*s12-2*s2 2)\n" }{MPLTEXT 1 0 7 ",Q,P);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 53 "hpartialtdX1functionQ-diff(HamtdX1function(Q,P),P));\n" } {MPLTEXT 1 0 62 "simplify(hpartialtdX1functionP+diff(HamtdX1function(Q ,P),Q));\n" }{MPLTEXT 1 0 56 "simplify(Hamiltonianfunction(s12,s22,s11 ,s21,X1,0,0,0,0," }{MPLTEXT 1 0 21 "sqrt(2)/sqrt(s12-s22)" }{MPLTEXT 1 0 1 ")" }{MPLTEXT 1 0 1 "-" }{MPLTEXT 1 0 15 "HamtdX1function" } {MPLTEXT 1 0 11 "(Q,P,s11));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6hpart ialtdX1functionQG6\",$*(,&I$s12GF$\"\"\"I$s22GF$!\"\"#F+\"\"#,.*&,&F(# F)F-F*F1F)I\"QGF$F-F)*&,**&,&F(F,F*F,F)I#X1GF$F)F)I\"PGF$F)I$s11GF$F1I $s21GF$F1F)F2F)F)*&,(F8F+F9F,F:F,F)F7F)F)I%sX20GF$F,I\"hGF$F1I%sX10GF$ F,F)F-F1F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I6hpartialtdX1functionPG6 \",$**,.*&,&I\"QGF$\"\"\"I#X1GF$!\"\"\"\"#I\"PGF$F.F+*(F)F.,**&,&F*F+F ,#F-F.F+I$s12GF$F+F+*&F3F+I$s22GF$F+F+I$s11GF$#F+F.I$s21GF$F9F+F/F+F.* (F)F.,**(F*F+,&F*F+F,#!\"#\"\"$F+F7F+F+*&,&F*#F.FAF,#F-FAF+F:F+F+I\"hG F$#F+FAI$s20GF$FGF+F5F+FA*(F)F.,&*&F3F+F8F+F+I$s10GF$F9F+F7F+F.*(F:F+F )F.F8F+F+*&I%sX10GF$F+I%sX20GF$F+F-F+F.F9,&F5F+F7F-F4F)F@F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0HamtdX1functionG6\"f*6$I\"QGF$I\"PGF$F$6$I) operatorGF$I&arrowGF$F$,(*(9%\"\"\",.*&,&I$s12GF$F/I$s22GF$F/F/9$\"\"# F/*&,**&,&F3!\"\"F4F;F/I#X1GF$F/F/F.F/I$s11GF$F/I$s21GF$F/F/F5F/F/*&,( F.F;F=F;F>F;F/FF/F/*&FF/F;FAF/I$s2 0GF$F/F/FLF/F3F/F6**F5F/FLF/,(*&F5F/F=F/F/*&FF/F6**F5F/FF/FE*&FBF/FCF/F6F/FDFFFLF;F/*(,(*&,& FOF/FAF/F/F3F/F/*&FTF/F4F/F/*&F=F/F>F/F/F/F " 0 "" {MPLTEXT 1 0 141 "KOldCoordinates := unapply( (1/2)*(sX10+s20)*(sX10+s 10)*ln((s12-s22)/2)+(1/2)*X1*((X1*s12+2*s11)*s10+s20*(X1*s22+2*s21)),s 11,s21,s12,s22,X1);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0KOldCoordinate sG6\"f*6'I$s11GF$I$s21GF$I$s12GF$I$s22GF$I#X1GF$F$6$I)operatorGF$I&arr owGF$F$,&*(,&I%sX10GF$\"\"\"I$s20GF$F3F3,&F2F3I$s10GF$F3F3-I#lnG6$%*pr otectedGI(_syslibGF$6#,&9&#F3\"\"#9'#!\"\"F@F3F?*&9(F3,&*&,&*&F>F3FEF3 F39$F@F3F6F3F3*&F4F3,&*&FAF3FEF3F39%F@F3F3F3F?F$F$F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Hams11:=simplify(Hamiltonianfunction(" } {MPLTEXT 1 0 28 "s12,s22,s11,s21,X1,0,0,1,0,0" }{MPLTEXT 1 0 2 "))" } {MPLTEXT 1 0 2 "+\n" }{MPLTEXT 1 0 29 "U11(s11,s21,s12,s22,X1)+diff(" }{MPLTEXT 1 0 16 "KOldCoordinates(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22, X1)" }{MPLTEXT 1 0 5 ",s11)" }{MPLTEXT 1 0 1 ";" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 67 "Hams21:=simplify(Hamiltonianfunction(s12,s22,s11,s21 ,X1,0,0,0,1,0))" }{MPLTEXT 1 0 2 "+\n" }{MPLTEXT 1 0 23 "U21(s11,s21,s 12,s22,X1)" }{MPLTEXT 1 0 6 "+diff(" }{MPLTEXT 1 0 15 "KOldCoordinates " }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 5 ",s21)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 67 "Hams12:=simplif y(Hamiltonianfunction(s12,s22,s11,s21,X1,1,0,0,0,0))" }{MPLTEXT 1 0 2 "+\n" }{MPLTEXT 1 0 23 "U12(s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 6 "+dif f(" }{MPLTEXT 1 0 15 "KOldCoordinates" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 5 ",s12)" }{MPLTEXT 1 0 2 " ;\n" }{MPLTEXT 1 0 67 "Hams22:=simplify(Hamiltonianfunction(s12,s22,s1 1,s21,X1,0,1,0,0,0))" }{MPLTEXT 1 0 2 "+\n" }{MPLTEXT 1 0 23 "U22(s11, s21,s12,s22,X1)" }{MPLTEXT 1 0 6 "+diff(" }{MPLTEXT 1 0 15 "KOldCoordi nates" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" } {MPLTEXT 1 0 5 ",s22)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 66 "HamX1:= simplify(Hamiltonianfunction(s12,s22,s11,s21,X1,0,0,0,0,1))" }{MPLTEXT 1 0 2 "+\n" }{MPLTEXT 1 0 23 "UX1(s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 6 "+diff(" }{MPLTEXT 1 0 15 "KOldCoordinates" }{MPLTEXT 1 0 1 "(" } {MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 4 ",X1)" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 413 "Hams11formula := (-Q^4*s12*s22+(2*X1*s12*s22+(-s12-s22)*P-s11*s22 -s12*s21)*Q^3+(-X1^2*s12*s22+((2*s12+2*s22)*P+2*s11*s22+2*s12*s21)*X1- P^2+(-s11-s21)*P+(-s20-h)*s12+(h-s10)*s22-s11*s21)*Q^2+(((-s12-s22)*P- s12*s21-s11*s22)*X1^2+(2*P^2+(2*s11+2*s21)*P+(2*s20+2*h)*s12+(-h+2*s10 )*s22+2*s11*s21)*X1+P*(sX10+sX20))*Q+((-s20-h)*s12-P^2+(-s11-s21)*P-s1 0*s22-s11*s21)*X1^2-P*(sX10+sX20)*X1-sX10*sX20)/((Q-X1)*(-s12+s22))\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "U11(s11,s2 1,s12,s22,X1) " }{MPLTEXT 1 0 6 "+diff(" }{MPLTEXT 1 0 15 "KOldCoordin ates" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" } {MPLTEXT 1 0 5 ",s11)" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 6 "Hams21" }{MPLTEXT 1 0 7 "formula" }{MPLTEXT 1 0 382 " := (Q^4*s12*s22+(-2*X1*s12*s22+(s12+s22)*P+s11*s2 2+s12*s21)*Q^3+(X1^2*s12*s22+((-2*s12-2*s22)*P-2*s11*s22-2*s12*s21)*X1 +P^2+(s21+s11)*P+s10*s22+s11*s21+s12*s20)*Q^2+(((s12+s22)*P+s11*s22+s1 2*s21)*X1^2+(-2*P^2+(-2*s11-2*s21)*P+(-h-2*s20)*s12-2*s10*s22-2*s11*s2 1)*X1-P*(sX10+sX20))*Q+((s20+h)*s12+P^2+(s21+s11)*P+s10*s22+s11*s21)*X 1^2+P*(sX10+sX20)*X1+sX10*sX20)/((Q-X1)*(-s12+s22))\n" }{MPLTEXT 1 0 2 "+\n" }{MPLTEXT 1 0 23 "U21(s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 6 "+d iff(" }{MPLTEXT 1 0 15 "KOldCoordinates" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 5 ",s21)" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 6 "Hams12" }{MPLTEXT 1 0 7 "formula" }{MPLTEXT 1 0 1084 " := (-(s12*s22*Q^2+((s2 1+P)*s12+s22*(P+s11))*Q+(s20+h)*s12+s10*s22+(P+s11)*(s21+P))*(-s12+s22 )*X1^3+(2*s22*s12*(-s12+s22)*Q^3+((-2*P-2*s21)*s12^2-3*s22*(-s21+s11)* s12+2*s22^2*(P+s11))*Q^2+((-2*s20-2*h)*s12^2+((2*h-2*s10+2*s20)*s22-(2 *(P+3*s11*(1/2)-(1/2)*s21))*(s21+P))*s12+(2*(s10*s22+(P+s11)*(P-(1/2)* s11+3*s21*(1/2))))*s22)*Q+((sX10+sX20-h)*P-(-s21+s11)*(s20+h))*s12+((- sX10-sX20+h)*P-s10*(-s21+s11))*s22-(-s21+s11)*(s21+P)*(P+s11))*X1^2+(- s22*s12*(-s12+s22)*Q^4+((s21+P)*s12^2+3*s22*(-s21+s11)*s12-s22^2*(P+s1 1))*Q^3+((s20+h)*s12^2+((s10-s20)*s22+(s21+P)*(P+3*s11-2*s21))*s12-s22 *((h+s10)*s22+(P+s11)*(P-2*s11+3*s21)))*Q^2+(((-sX10-sX20+2*h)*P+(2*s2 0+2*h)*s11-s21*(h+2*s20))*s12+((sX10+sX20-2*h)*P+(-h+2*s10)*s11-2*s21* s10)*s22+(2*(-s21+s11))*(s21+P)*(P+s11))*Q+sX10*sX20*s12-sX10*sX20*s22 -P*(-s21+s11)*(sX10+sX20))*X1-s22*(-s21+s11)*s12*Q^4+((-s22*h-(-s21+s1 1)*(s21+P))*s12-(-s22*h+(-s21+s11)*(P+s11))*s22)*Q^3+((-h*P+(-s20-h)*s 11+s20*s21)*s12+(h*P+(h-s10)*s11+s21*s10)*s22-(-s21+s11)*(s21+P)*(P+s1 1))*Q^2+P*(-s21+s11)*(sX10+sX20)*Q-sX10*sX20*(-s21+s11))/(2*(-s12+s22) ^2*(Q-X1))\n" }{MPLTEXT 1 0 2 "+\n" }{MPLTEXT 1 0 23 "U12(s11,s21,s12, s22,X1)" }{MPLTEXT 1 0 6 "+diff(" }{MPLTEXT 1 0 15 "KOldCoordinates" } {MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 5 ",s12)" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 6 "Hams22" }{MPLTEXT 1 0 7 "formula" }{MPLTEXT 1 0 1085 " := ((s12*s22*Q^2+((s21+P)*s12+s22*(P+s11))*Q+(s20+h)*s12+s10*s2 2+(P+s11)*(s21+P))*(-s12+s22)*X1^3+(-2*s22*s12*(-s12+s22)*Q^3+((2*s21+ 2*P)*s12^2+3*s22*(-s21+s11)*s12-2*s22^2*(P+s11))*Q^2+((2*s20+2*h)*s12^ 2+((-2*h+2*s10-2*s20)*s22+(2*(P+3*s11*(1/2)-(1/2)*s21))*(s21+P))*s12-( 2*(s10*s22+(P+s11)*(P-(1/2)*s11+3*s21*(1/2))))*s22)*Q+((-sX10-sX20+h)* P+(-s21+s11)*(s20+h))*s12+((sX10+sX20-h)*P+s10*(-s21+s11))*s22+(-s21+s 11)*(s21+P)*(P+s11))*X1^2+(s22*s12*(-s12+s22)*Q^4+((-s21-P)*s12^2-3*s2 2*(-s21+s11)*s12+s22^2*(P+s11))*Q^3+((-2*h-s20)*s12^2+((2*h-s10+s20)*s 22-(s21+P)*(P+3*s11-2*s21))*s12+s22*(s10*s22+(P+s11)*(P-2*s11+3*s21))) *Q^2+(((sX10+sX20-2*h)*P+(-2*s20-2*h)*s11+s21*(h+2*s20))*s12+((-sX10-s X20+2*h)*P+(h-2*s10)*s11+2*s21*s10)*s22-(2*(-s21+s11))*(s21+P)*(P+s11) )*Q-sX10*sX20*s12+sX10*sX20*s22+P*(-s21+s11)*(sX10+sX20))*X1+s22*(-s21 +s11)*s12*Q^4+(s12^2*h+(-s22*h+(-s21+s11)*(s21+P))*s12+s22*(-s21+s11)* (P+s11))*Q^3+((h*P+(s20+h)*s11-s20*s21)*s12+(-h*P+(-h+s10)*s11-s21*s10 )*s22+(-s21+s11)*(s21+P)*(P+s11))*Q^2-P*(-s21+s11)*(sX10+sX20)*Q+sX10* sX20*(-s21+s11))/(2*(-s12+s22)^2*(Q-X1))\n" }{MPLTEXT 1 0 2 "+\n" } {MPLTEXT 1 0 23 "U22(s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 6 "+diff(" } {MPLTEXT 1 0 15 "KOldCoordinates" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" }{MPLTEXT 1 0 5 ",s22)" }{MPLTEXT 1 0 1 ":" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 5 "HamX1" } {MPLTEXT 1 0 7 "formula" }{MPLTEXT 1 0 379 " := (Q^4*s12*s22+(-2*X1*s1 2*s22+(s12+s22)*P+s11*s22+s12*s21)*Q^3+(X1^2*s12*s22+((-2*s12-2*s22)*P -2*s11*s22-2*s12*s21)*X1+(s20+h)*s12+P^2+(s21+s11)*P+s10*s22+s11*s21)* Q^2+(((s12+s22)*P+s11*s22+s12*s21)*X1^2+(-2*P^2+(-2*s11-2*s21)*P+(-2*s 20-2*h)*s12-2*s10*s22-2*s11*s21)*X1-(sX10+sX20-h)*P)*Q+((s20+h)*s12+P^ 2+(s21+s11)*P+s10*s22+s11*s21)*X1^2+(sX10+sX20-h)*P*X1+sX10*sX20)/(Q-X 1)\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "UX1(s 11,s21,s12,s22,X1)" }{MPLTEXT 1 0 6 "+diff(" }{MPLTEXT 1 0 15 "KOldCoo rdinates" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 19 "s11,s21,s12,s22,X1)" } {MPLTEXT 1 0 4 ",X1)" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 15 "simplify(HamX1-" }{MPLTEXT 1 0 14 "HamX1formula);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "simplify(Ha ms11-" }{MPLTEXT 1 0 15 "Hams11formula);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 32 "simplify(Hams21-Hams21formula);\n" }{MPLTEXT 1 0 32 " simplify(Hams12-Hams12formula);\n" }{MPLTEXT 1 0 31 "simplify(Hams22-H ams22formula);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Hams11G6\",(*(,0*(I \"QGF$\"\"%I$s12GF$\"\"\"I$s22GF$F,!\"\"*&,**(I#X1GF$F,F+F,F-F,\"\"#*& ,&F+F.F-F.F,I\"PGF$F,F,*&I$s11GF$F,F-F,F.*&F+F,I$s21GF$F,F.F,F)\"\"$F, *&,0*(F2F3F+F,F-F,F.*&,(*&,&F+F3F-F3F,F6F,F,F7F3F9F3F,F2F,F,*$F6F3F.*& ,&F8F.F:F.F,F6F,F,*&,&I$s20GF$F.I\"hGF$F.F,F+F,F,*&,&FIF,I$s10GF$F.F,F -F,F,*&F8F,F:F,F.F,F)F3F,*&,(*&,(F4F,F9F.F7F.F,F2F3F,*&,,FCF3*&,&F8F3F :F3F,F6F,F,*&,&FHF3FIF3F,F+F,F,*&,&FIF.FLF3F,F-F,F,FMF3F,F2F,F,*&F6F,, &I%sX10GF$F,I%sX20GF$F,F,F,F,F)F,F,*&,,FFF,FCF.FDF,*&FLF,F-F,F.FMF.F,F 2F3F,*(F6F,FenF,F2F,F.*&FfnF,FgnF,F.F,,&F)F,F2F.F.,&F+F.F-F,F.F,-I$U11 GF$6'F8F:F+F-F2F,*&F2F,FLF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Hams 21G6\",(*(,0*(I\"QGF$\"\"%I$s12GF$\"\"\"I$s22GF$F,F,*&,**(I#X1GF$F,F+F ,F-F,!\"#*&,&F+F,F-F,F,I\"PGF$F,F,*&I$s11GF$F,F-F,F,*&F+F,I$s21GF$F,F, F,F)\"\"$F,*&,0*(F1\"\"#F+F,F-F,F,*&,(*&,&F+F2F-F2F,F5F,F,F6F2F8F2F,F1 F,F,*$F5F>F,*&,&F9F,F7F,F,F5F,F,*&I$s10GF$F,F-F,F,*&F7F,F9F,F,*&F+F,I$ s20GF$F,F,F,F)F>F,*&,(*&,(F3F,F6F,F8F,F,F1F>F,*&,,FCF2*&,&F7F2F9F2F,F5 F,F,*&,&I\"hGF$!\"\"FJF2F,F+F,F,FFF2FHF2F,F1F,F,*&F5F,,&I%sX10GF$F,I%s X20GF$F,F,FVF,F)F,F,*&,,*&,&FJF,FUF,F,F+F,F,FCF,FDF,FFF,FHF,F,F1F>F,*( F5F,FXF,F1F,F,*&FYF,FZF,F,F,,&F)F,F1FVFV,&F+FVF-F,FVF,-I$U21GF$6'F7F9F +F-F1F,*&F1F,FJF,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Hams12G6\",**( ,2*(,&I$s12GF$!\"\"I$s22GF$\"\"\"F-,,*(F*F-F,F-I\"QGF$\"\"#F-*&,&*&,&I $s21GF$F-I\"PGF$F-F-F*F-F-*&F,F-,&F7F-I$s11GF$F-F-F-F-F0F-F-*&,&I$s20G F$F-I\"hGF$F-F-F*F-F-*&I$s10GF$F-F,F-F-*&F9F-F5F-F-F-I#X1GF$\"\"$F+*&, .**F,F-F*F-F)F-F0FCF1*&,(*&,&F7!\"#F6FKF-F*F1F-*(F,F-,&F6F+F:F-F-F*F-! \"$*&F,F1F9F-F1F-F0F1F-*&,(*&,&F=FKF>FKF-F*F1F-*&,&*&,(F>F1F@FKF=F1F-F ,F-F-*&,(F7F-F:#FCF1F6#F+F1F-F5F-FKF-F*F-F-*&F,F-,&F?F-*&F9F-,(F7F-F:F enF6FZF-F-F-F1F-F0F-F-*&,&*&,(I%sX10GF$F-I%sX20GF$F-F>F+F-F7F-F-*&FMF- FF-F-F7F-F-*&F@F-FMF-F+F-F,F-F-*(FMF- F5F-F9F-F+F-FBF1F-*&,0**F,F-F*F-F)F-F0\"\"%F+*&,(*&F5F-F*F1F-FLFCFOF+F -F0FCF-*&,(*&FF-F@F-F-F,F-F-*&F9F-,(F7F-F:FKF6FCF-F-F-F+F-F0F 1F-*&,(*&,(*&,(F^oF+F_oF+F>F1F-F7F-F-*&,&F=F1F>F1F-F:F-F-*&F6F-,&F>F-F =F1F-F+F-F*F-F-*&,(*&,(F^oF-F_oF-F>FKF-F7F-F-*&,&F>F+F@F1F-F:F-F-*&F6F -F@F-FKF-F,F-F-FfoF1F-F0F-F-*(F^oF-F_oF-F*F-F-*(F^oF-F_oF-F,F-F+*(F7F- FMF-,&F^oF-F_oF-F-F+F-FBF-F-**F,F-FMF-F*F-F0FjoF+*&,&*&,&*&F,F-F>F-F+* &FMF-F5F-F+F-F*F-F-*&F,F-,&FgrF+*&FMF-F9F-F-F-F+F-F0FCF-*&,(*&,(*&F>F- F7F-F+*&,&F=F+F>F+F-F:F-F-*&F=F-F6F-F-F-F*F-F-*&,(F`sF-*&,&F>F-F@F+F-F :F-F-F]rF-F-F,F-F-FfoF+F-F0F1F-**F7F-FMF-FarF-F0F-F-*(F^oF-F_oF-FMF-F+ F-F)FK,&F0F-FBF+F+#F-F1-I$U12GF$6'F:F6F*F,FBF-*(,&F^oF-F=F-F-,&F^oF-F@ F-F-,&F*F[tF,FenF+#F-Fjo*&FBF1F@F-F[t" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'Hams22G6\",**(,2*(,&I$s12GF$!\"\"I$s22GF$\"\"\"F-,,*(F*F-F,F-I\"QG F$\"\"#F-*&,&*&,&I$s21GF$F-I\"PGF$F-F-F*F-F-*&F,F-,&F7F-I$s11GF$F-F-F- F-F0F-F-*&,&I$s20GF$F-I\"hGF$F-F-F*F-F-*&I$s10GF$F-F,F-F-*&F9F-F5F-F-F -I#X1GF$\"\"$F-*&,.**F,F-F*F-F)F-F0FC!\"#*&,(*&,&F6F1F7F1F-F*F1F-*(F,F -,&F6F+F:F-F-F*F-FC*&F,F1F9F-FGF-F0F1F-*&,(*&,&F=F1F>F1F-F*F1F-*&,&*&, (F>FGF@F1F=FGF-F,F-F-*&,(F7F-F:#FCF1F6#F+F1F-F5F-F1F-F*F-F-*&F,F-,&F?F -*&F9F-,(F7F-F:FZF6FYF-F-F-FGF-F0F-F-*&,&*&,(I%sX10GF$F+I%sX20GF$F+F>F -F-F7F-F-*&FMF-FF+F-F7F-F-*&F@F-FMF-F -F-F,F-F-*(FMF-F5F-F9F-F-F-FBF1F-*&,0**F,F-F*F-F)F-F0\"\"%F-*&,(*&,&F6 F+F7F+F-F*F1F-FL!\"$FNF-F-F0FCF-*&,(*&,&F>FGF=F+F-F*F1F-*&,&*&,(F>F1F@ F+F=F-F-F,F-F-*&F5F-,(F7F-F:FCF6FGF-F+F-F*F-F-*&F,F-,&F?F-*&F9F-,(F7F- F:FGF6FCF-F-F-F-F-F0F1F-*&,(*&,(*&,(F]oF-F^oF-F>FGF-F7F-F-*&,&F=FGF>FG F-F:F-F-*&F6F-,&F>F-F=F1F-F-F-F*F-F-*&,(*&,(F]oF+F^oF+F>F1F-F7F-F-*&,& F>F-F@FGF-F:F-F-*&F6F-F@F-F1F-F,F-F-FeoFGF-F0F-F-*(F]oF-F^oF-F*F-F+*(F ]oF-F^oF-F,F-F-*(F7F-FMF-,&F]oF-F^oF-F-F-F-FBF-F-**F,F-FMF-F*F-F0FioF- *&,(*&F*F1F>F-F-*&,&*&F,F-F>F-F+*&FMF-F5F-F-F-F*F-F-*(F,F-FMF-F9F-F-F- F0FCF-*&,(*&,(*&F>F-F7F-F-*&FF+F@F-F-F:F-F-F]rF+F-F,F-F-FeoF-F-F0F1F-**F7F-FMF-FarF-F0F-F+*(F]oF -F^oF-FMF-F-F-F)FG,&F0F-FBF+F+#F-F1-I$U22GF$6'F:F6F*F,FBF-*(,&F]oF-F=F -F-,&F]oF-F@F-F-,&F*FisF,FZF+#F+Fio*&FBF1F=F-Fis" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&HamX1G6\",,*&,0*(I\"QGF$\"\"%I$s12GF$\"\"\"I$s22GF$F,F ,*&,**(I#X1GF$F,F+F,F-F,!\"#*&,&F+F,F-F,F,I\"PGF$F,F,*&I$s11GF$F,F-F,F ,*&F+F,I$s21GF$F,F,F,F)\"\"$F,*&,0*(F1\"\"#F+F,F-F,F,*&,(*&,&F+F2F-F2F ,F5F,F,F6F2F8F2F,F1F,F,*&,&I$s20GF$F,I\"hGF$F,F,F+F,F,*$F5F>F,*&,&F9F, F7F,F,F5F,F,*&I$s10GF$F,F-F,F,*&F7F,F9F,F,F,F)F>F,*&,(*&,(F3F,F6F,F8F, F,F1F>F,*&,,FGF2*&,&F7F2F9F2F,F5F,F,*&,&FEF2FFF2F,F+F,F,FJF2FLF2F,F1F, F,*&,(I%sX10GF$F,I%sX20GF$F,FF!\"\"F,F5F,FenF,F)F,F,*&,,FCF,FGF,FHF,FJ F,FLF,F,F1F>F,*(FXF,F5F,F1F,F,*&FYF,FZF,F,F,,&F)F,F1FenFenF,-I$UX1GF$6 'F7F9F+F-F1F,*&,&*&F1F,F+F,F,F7F>F,FKF,#F,F>*&FEF,,&*&F1F,F-F,F,F9F>F, Fao*&F1F,,&*&FKF,F+F,F,*&FEF,F-F,F,F,Fao" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "U11 := unapply( ((X1*s22+s21)*s10+(X1*s12+s11)*s20)/(s12-s22) +" }{MPLTEXT 1 0 18 "s22*X1/(s12-s22)*h" }{MPLTEXT 1 0 22 ",s11,s21,s12, s22,X1);\n" }{MPLTEXT 1 0 62 "U21 := unapply(-((X1*s22+s21)*s10+(X1*s1 2+s11)*s20)/(s12-s22) " }{MPLTEXT 1 0 19 "-s12*X1/(s12-s22)*h" } {MPLTEXT 1 0 22 ",s11,s21,s12,s22,X1);\n" }{MPLTEXT 1 0 94 "U12 := una pply( (X1*(s12-s22)-s11+s21)*((s10*s22+s12*s20)*X1+s21*s10+s11*s20)/(2 *(s12-s22)^2)\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 57 "(-s22^2*X1+(X1* s12-s11)*s22+s12*s21)*X1/(2*(s12-s22)^2)*h" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 22 ",s11,s21,s12,s22,X1);\n" }{MPLTEXT 1 0 94 "U22 := una pply(-(X1*(s12-s22)-s11+s21)*((s10*s22+s12*s20)*X1+s21*s10+s11*s20)/(2 *(s12-s22)^2)\n" }{MPLTEXT 1 0 58 "-h*(X1*s12^2+(-X1*s22+s21)*s12-s11* s22)*X1/(2*(s12-s22)^2)" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 ",s11,s 21,s12,s22,X1);\n" }{MPLTEXT 1 0 70 "UX1 := unapply( (X1*s22+s21)*s10+ (X1*s12+s11)*s20,s11,s21,s12,s22,X1);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$U11G6\"f*6'I$s11GF$I$s21GF$I$s12GF$I$s22GF$I#X1GF$F$6$I)operatorGF $I&arrowGF$F$,&*&,&*&,&*&9'\"\"\"9(F6F69%F6F6I$s10GF$F6F6*&,&*&9&F6F7F 6F69$F6F6I$s20GF$F6F6F6,&F=F6F5!\"\"FAF6**F5F6F7F6F@FAI\"hGF$F6F6F$F$F $" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$U21G6\"f*6'I$s11GF$I$s21GF$I$s12 GF$I$s22GF$I#X1GF$F$6$I)operatorGF$I&arrowGF$F$,&*&,&*&,&*&9'\"\"\"9(F 6F69%F6F6I$s10GF$F6F6*&,&*&9&F6F7F6F69$F6F6I$s20GF$F6F6F6,&F=F6F5!\"\" FAFA**F=F6F7F6F@FAI\"hGF$F6FAF$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I$U12G6\"f*6'I$s11GF$I$s21GF$I$s12GF$I$s22GF$I#X1GF$F$6$I)operatorGF$I &arrowGF$F$,&*(,(*&9(\"\"\",&9&F49'!\"\"F4F49$F89%F4F4,(*&,&*&I$s10GF$ F4F7F4F4*&F6F4I$s20GF$F4F4F4F3F4F4*&F?F4F:F4F4*&F9F4FAF4F4F4F5!\"##F4 \"\"#**,(*&F7FFF3F4F8*&,&*&F6F4F3F4F4F9F8F4F7F4F4*&F6F4F:F4F4F4F3F4F5F DI\"hGF$F4FEF$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$U22G6\"f*6'I$s1 1GF$I$s21GF$I$s12GF$I$s22GF$I#X1GF$F$6$I)operatorGF$I&arrowGF$F$,&*(,( *&9(\"\"\",&9&F49'!\"\"F4F49$F89%F4F4,(*&,&*&I$s10GF$F4F7F4F4*&F6F4I$s 20GF$F4F4F4F3F4F4*&F?F4F:F4F4*&F9F4FAF4F4F4F5!\"##F8\"\"#**I\"hGF$F4,( *&F3F4F6FFF4*&,&*&F7F4F3F4F8F:F4F4F6F4F4*&F9F4F7F4F8F4F3F4F5FDFEF$F$F$ " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$UX1G6\"f*6'I$s11GF$I$s21GF$I$s12G F$I$s22GF$I#X1GF$F$6$I)operatorGF$I&arrowGF$F$,&*&,&*&9'\"\"\"9(F4F49% F4F4I$s10GF$F4F4*&,&*&9&F4F5F4F49$F4F4I$s20GF$F4F4F$F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 16 "JMU differential" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 35 "First part: Contributions at xi=X_1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ExpMonodromiesX1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 46 "ExpMonodromiesX1[1,1]:=exp(sX10*ln(xi-X1)/h):\n" } {MPLTEXT 1 0 46 "ExpMonodromiesX1[2,2]:=exp(sX20*ln(xi-X1)/h):\n" } {MPLTEXT 1 0 18 "ExpMonodromiesX1;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "MonodromiesX1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 34 "Monodromies X1[1,1]:=sX10/(xi-X1):\n" }{MPLTEXT 1 0 34 "MonodromiesX1[2,2]:=sX20/( xi-X1):\n" }{MPLTEXT 1 0 15 "MonodromiesX1:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 25 "LambdaX1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 31 "LambdaX 1[1,1]:=sX10*ln(xi-X1):\n" }{MPLTEXT 1 0 31 "LambdaX1[2,2]:=sX20*ln(xi -X1):\n" }{MPLTEXT 1 0 10 "LambdaX1:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "N0:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 19 "N1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 19 "N2:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 19 "N3:=Matrix( 2,2,0):\n" }{MPLTEXT 1 0 19 "N4:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 43 "f or i from 1 to 2 do for j from 1 to 2 do \n" }{MPLTEXT 1 0 19 "N0[i,j] :=n0[i,j]: \n" }{MPLTEXT 1 0 19 "N1[i,j]:=n1[i,j]: \n" }{MPLTEXT 1 0 18 "N2[i,j]:=n2[i,j]:\n" }{MPLTEXT 1 0 18 "N3[i,j]:=n3[i,j]:\n" } {MPLTEXT 1 0 18 "N4[i,j]:=n4[i,j]:\n" }{MPLTEXT 1 0 8 "od: od:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 82 "HatPsiRegX1:=IdentityMatrix(2)+N 1*(xi-X1)+N2*(xi-X1)^2+N3*(xi-X1)^3+N4*(xi-X1)^4:\n" }{MPLTEXT 1 0 62 "tdPsiX1:=Multiply(N0,Multiply(HatPsiRegX1,ExpMonodromiesX1)):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "dHatPsiRegdxiX1:=Matrix(2,2,0): \n" }{MPLTEXT 1 0 99 "for i from 1 to 2 do for j from 1 to 2 do dHatPs iRegdxiX1[i,j]:=diff(HatPsiRegX1[i,j],xi): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "dtdPsidxiX1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 91 "for i from 1 to 2 do for j from 1 to 2 do dtdPsidxiX1[i,j]:=diff(t dPsiX1[i,j],xi): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "dLa mbdaX1dt:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 107 "for i from 1 to 2 do fo r j from 1 to 2 do dLambdaX1dt[i,j]:= simplify(diff(LambdaX1[i,j],X1)* dX1): od: od:\n" }{MPLTEXT 1 0 13 "dLambdaX1dt;\n" }{MPLTEXT 1 0 1 "\n " }{MPLTEXT 1 0 118 "omegaJMUPartX1:=simplify(-h*residue(Trace(Multipl y(Multiply(HatPsiRegX1^(-1),dHatPsiRegdxiX1), dLambdaX1dt)),xi=X1));\n " }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 58 "ToCancelX1:=simplify(h*dtdPsi dxiX1-Multiply(tdL,tdPsiX1)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfen cedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtab leGF$66-I$mtrGF$6'-I$mtdGF$6(-I%msupGF$6%-I#moGF$6.Q/ⅇF'/ %+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%* separatorGFF/%)stretchyGFF/%*symmetricGFF/%(largeopGFF/%.movablelimits GFF/%'accentGFF/%'lspaceGQ&0.0emF'/%'rspaceGQ,0.1111111emF'-I&mfracGF$ 6)-F,6%-I#miGF$6&Q%sX10F'/%'italicGQ%trueF'F>/FBQ'italicF'-F;6-Q1&Invi sibleTimes;F'FAFDFGFIFKFMFOFQFS/FWFU-F,6%-Fin6&Q#lnF'/F]oFFF>FA-F;6-Q0 ⁡F'FAFDFGFIFKFMFOFQFSFdo-F#6%-F,6#-F,6%-Fin6&Q%ξF'Fjo F>FA-F;6.Q(−F'F>FAFDFGFIFKFMFOFQ/FTQ,0.2222222emF'/FWF[q-Fin6&Q# X1F'F\\oF>F_oF>FA-F,6#-Fin6&Q\"hF'F\\oF>F_o/%.linethicknessGQ\"1F'/%+d enomalignGQ'centerF'/%)numalignGFjq/%)bevelledGFFF>/%1superscriptshift GQ\"0F'/%)rowalignGQ!F'/%,columnalignGFdr/%+groupalignGFdr/%(rowspanGF gq/%+columnspanGFgq-F56(-I#mnGF$6%FarF>FAFbrFerFgrFirF[sFbrFerFgr-F26' F]s-F56(-F86%F:-FZ6)-F,6%-Fin6&Q%sX20F'F\\oF>F_oFaoFeoF`qFeqFhqF[rF]rF >F_rFbrFerFgrFirF[sFbrFerFgr/%&alignGQ%axisF'/FcrQ)baselineF'/FfrFjq/F hrQ'|frleft|hrF'/%/alignmentscopeGF^o/%,columnwidthGQ%autoF'/%&widthGF [u/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF '/%,columnlinesGFfu/%&frameGFfu/%-framespacingGQ,0.4em~0.5exF'/%*equal rowsGFF/%-equalcolumnsGFF/%-displaystyleGFF/%%sideGQ&rightF'/%0minlabe lspacingGFcuF>FA/%%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(-I&mfracGF$6)-F,6%-I# miGF$6&Q%sX10F'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvaria ntGQ'italicF'-I#moGF$6-Q1⁢F'/FGQ'normalF'/%&fenceGQ&fal seF'/%*separatorGFQ/%)stretchyGFQ/%*symmetricGFQ/%(largeopGFQ/%.movabl elimitsGFQ/%'accentGFQ/%'lspaceGQ&0.0emF'/%'rspaceGFjn-F=6&Q$dX1F'F@FC FF-F,6#-F,6&-FJ6.Q*&uminus0;F'FCFMFOFRFTFVFXFZFfn/FinQ,0.2222222emF'/F \\oFho-F=6&Q%ξF'/FAFQFCFM-FJ6.Q\"+F'FCFMFOFRFTFVFXFZFfnFgoFio-F=6&Q #X1F'F@FCFF/%.linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignG Fip/%)bevelledGFQFC/%)rowalignGQ!F'/%,columnalignGF`q/%+groupalignGF`q /%(rowspanGFfp/%+columnspanGFfp-F56(-I#mnGF$6%Q\"0F'FCFMF^qFaqFcqFeqFg qF^qFaqFcq-F26'Fiq-F56(-F86)-F,6%-F=6&Q%sX20F'F@FCFFFIF]oF`oFdpFgpFjpF \\qFCF^qFaqFcqFeqFgqF^qFaqFcq/%&alignGQ%axisF'/F_qQ)baselineF'/FbqFip/ FdqQ'|frleft|hrF'/%/alignmentscopeGFB/%,columnwidthGQ%autoF'/%&widthGF fs/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF '/%,columnlinesGFat/%&frameGFat/%-framespacingGQ,0.4em~0.5exF'/%*equal rowsGFQ/%-equalcolumnsGFQ/%-displaystyleGFQ/%%sideGQ&rightF'/%0minlabe lspacingGF^tFCFM/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/omegaJMUPartX1G6\"*(I\"hGF$\"\"\"I$dX1GF$F',&*&I%sX10G F$F'&I#n1GF$6$F'F'F'F'*&I%sX20GF$F'&F-6$\"\"#F3F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "SingularParttdLxiX1:=Matrix(2,2,0): \n" }{MPLTEXT 1 0 111 "for i from 1 to 2 do for j from 1 to 2 do Singu larParttdLxiX1[i,j]:=simplify(residue(tdL[i,j],xi=X1)): od: od:\n" } {MPLTEXT 1 0 49 "MatrixEigenvectorsSingularPartX1:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 45 "MatrixEigenvectorsSingularPartX1[1,1]:=Q-X1:\n" } {MPLTEXT 1 0 45 "MatrixEigenvectorsSingularPartX1[1,2]:=Q-X1:\n" } {MPLTEXT 1 0 84 "MatrixEigenvectorsSingularPartX1[2,1]:=Q^2*s12-Q*X1*s 12+P*Q-P*X1+Q*s11-X1*s11-sX10:\n" }{MPLTEXT 1 0 84 "MatrixEigenvectors SingularPartX1[2,2]:=Q^2*s12-Q*X1*s12+P*Q-P*X1+Q*s11-X1*s11-sX20:\n" } {MPLTEXT 1 0 122 "simplify(Multiply(Multiply(MatrixEigenvectorsSingula rPartX1^(-1),SingularParttdLxiX1),MatrixEigenvectorsSingularPartX1)); \n" }{MPLTEXT 1 0 40 "GX10:=MatrixEigenvectorsSingularPartX1;\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "n" }{MPLTEXT 1 0 14 "0[1,1]:=Q-X1 ;\n" }{MPLTEXT 1 0 15 "n0[1,2]:=Q-X1;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowG F$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#miGF$6&Q%sX10F'/%'italicGQ% trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'/%)rowalignGQ! F'/%,columnalignGFF/%+groupalignGFF/%(rowspanGQ\"1F'/%+columnspanGFM-F 56(-I#mnGF$6%Q\"0F'F>/FBQ'normalF'FDFGFIFKFNFDFGFI-F26'FP-F56(-F86&Q%s X20F'F;F>FAFDFGFIFKFNFDFGFI/%&alignGQ%axisF'/FEQ)baselineF'/FHQ'center F'/FJQ'|frleft|hrF'/%/alignmentscopeGF=/%,columnwidthGQ%autoF'/%&width GFfo/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%non eF'/%,columnlinesGFap/%&frameGFap/%-framespacingGQ,0.4em~0.5exF'/%*equ alrowsGQ&falseF'/%-equalcolumnsGF[q/%-displaystyleGF[q/%%sideGQ&rightF '/%0minlabelspacingGF^pF>FV/%%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#miGF$6 &Q\"QF'/%'italicGQ%trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'ita licF'-I#moGF$6.Q(−F'F@/FDQ'normalF'/%&fenceGQ&falseF'/%*separato rGFN/%)stretchyGFN/%*symmetricGFN/%(largeopGFN/%.movablelimitsGFN/%'ac centGFN/%'lspaceGQ,0.2222222emF'/%'rspaceGFgn-F:6&Q#X1F'F=F@FC/%)rowal ignGQ!F'/%,columnalignGF_o/%+groupalignGF_o/%(rowspanGQ\"1F'/%+columns panGFfoF4F]oF`oFbo-F26'-F56(-F,6/-F,6%-I%msupGF$6%F9-I#mnGF$6%Q\"2F'F@ FJ/%1superscriptshiftGQ\"0F'-FG6-Q1⁢F'FJFLFOFQFSFUFWFY/ FfnQ&0.0emF'/FinF_q-F:6&Q$s12F'F=F@FCFF-F,6'F9F[qFjnF[qFaq-FG6.Q\"+F'F @FJFLFOFQFSFUFWFYFenFhn-F,6%-F:6&Q\"PF'F=F@FCF[qF9FF-F,6%F[rF[qFjnFfq- F,6%F9F[q-F:6&Q$s11F'F=F@FCFF-F,6%FjnF[qFbrFF-F:6&Q%sX10F'F=F@FCF]oF`o FboFdoFgo-F56(-F,6/F_pFFFdqFfqFiqFFF^rFfqF`rFFFerFF-F:6&Q%sX20F'F=F@FC F]oF`oFboFdoFgoF]oF`oFbo/%&alignGQ%axisF'/F^oQ)baselineF'/FaoQ'centerF '/FcoQ'|frleft|hrF'/%/alignmentscopeGF?/%,columnwidthGQ%autoF'/%&width GF^t/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%non eF'/%,columnlinesGFit/%&frameGFit/%-framespacingGQ,0.4em~0.5exF'/%*equ alrowsGFN/%-equalcolumnsGFN/%-displaystyleGFN/%%sideGQ&rightF'/%0minla belspacingGFftF@FJ/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#n0G6\"6$\"\"\"F',&I\"QGF%F'I#X1GF%!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 ">&I#n0G6\"6$\"\"\"\"\"#,&I\"QGF%F'I#X1GF%!\"\"" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "ToCancelX1Entry11:=simplify (series(simplify(ToCancelX1[1,1]*(xi-X1)^(-(sX10)/h)),xi=X1)):\n" } {MPLTEXT 1 0 89 "ToCancelX1Entry12:=simplify(series(simplify(ToCancelX 1[1,2]*(xi-X1)^(-(sX20)/h)),xi=X1)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "ToCancelX1Entry21:=simplify(series(simplify(ToCancelX1[2,1]*(x i-X1)^(-(sX10)/h)),xi=X1)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "To CancelX1Entry22:=simplify(series(simplify(ToCancelX1[2,2]*(xi-X1)^(-(s X20)/h)),xi=X1)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "n0[2,1 ]:=simplify(solve(residue( ToCancelX1Entry11" }{MPLTEXT 1 0 19 ",xi=X1 ),n0[2,1]));\n" }{MPLTEXT 1 0 50 "n0[2,2]:=simplify(solve(residue( ToC ancelX1Entry12" }{MPLTEXT 1 0 18 ",xi=X1),n0[2,2]));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "simplify(residue(ToCancelX1Entry21" }{MPLTEXT 1 0 9 ",xi=X1));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "simplify(resi due(ToCancelX1Entry22" }{MPLTEXT 1 0 10 ",xi=X1));\n" }{MPLTEXT 1 0 24 "n021:=P*(Q-X1)+s12*Q^2+(" }{MPLTEXT 1 0 3 "s11" }{MPLTEXT 1 0 24 " -X1*s12)*Q-X1*s11-sX10;\n" }{MPLTEXT 1 0 24 "n022:=P*(Q-X1)+s12*Q^2+(" }{MPLTEXT 1 0 3 "s11" }{MPLTEXT 1 0 23 "-X1*s12)*Q-X1*s11-sX20;" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "simplify(n0[2,1]-n021);\n" } {MPLTEXT 1 0 23 "simplify(n0[2,2]-n022);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#n0G6\"6$\"\"#\"\"\",**&I\"QGF%F'I$s12GF%F(F(*&,(*&I#X1GF%F(F,F (!\"\"I\"PGF%F(I$s11GF%F(F(F+F(F(*&,&F2F1F3F1F(F0F(F(I%sX10GF%F1" }} {PARA 11 "" 1 "" {XPPMATH 20 ">&I#n0G6\"6$\"\"#F',**&I\"QGF%F'I$s12GF% \"\"\"F,*&,(*&I#X1GF%F,F+F,!\"\"I\"PGF%F,I$s11GF%F,F,F*F,F,*&,&F2F1F3F 1F,F0F,F,I%sX20GF%F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%n021 G6\",,*&I\"PGF$\"\"\",&I\"QGF$F(I#X1GF$!\"\"F(F(*&F*\"\"#I$s12GF$F(F(* &,&*&F+F(F/F(F,I$s11GF$F(F(F*F(F(*&F+F(F3F(F,I%sX10GF$F," }}{PARA 11 " " 1 "" {XPPMATH 20 ">I%n022G6\",,*&I\"PGF$\"\"\",&I\"QGF$F(I#X1GF$!\" \"F(F(*&F*\"\"#I$s12GF$F(F(*&,&*&F+F(F/F(F,I$s11GF$F(F(F*F(F(*&F+F(F3F (F,I%sX20GF$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 445 " n1[1,1]:=(Q^4*s12*s22+(-2*X1*s12*s22+(s12+s22)*P+s11*s22+s12*s21)*Q^3+ (X1^2*s12*s22+((-2*s12-2*s22)*P-2*s11*s22-2*s12*s21)*X1+P^2+(s21+s11)* P+s10*s22+s11*s21+s12*s20)*Q^2+(((s12+s22)*P+s11*s22+s12*s21)*X1^2+(-2 *P^2+(-2*s11-2*s21)*P+(-sX10-2*s20)*s12+(-sX10-2*s10)*s22-2*s11*s21)*X 1+(-sX10-sX20)*P-sX10*(s21+s11))*Q+(P^2+(s21+s11)*P+(sX10+s20)*s12+s22 *(sX10+s10)+s11*s21)*X1^2+(P*(sX10+sX20)+sX10*(s21+s11))*X1+sX10*sX20) /((Q-X1)*h*(sX10-sX20)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 464 "n1 [1,2]:=(-Q^4*s12*s22+(2*X1*s12*s22+(-s21-P)*s12-s22*(P+s11))*Q^3+(-X1^ 2*s12*s22+((2*s21+2*P)*s12+2*s22*(P+s11))*X1+(-sX10+sX20-s20)*s12-P^2+ (-s11-s21)*P-s10*s22-s11*s21)*Q^2+(((-s21-P)*s12-s22*(P+s11))*X1^2+((2 *sX10-sX20+2*s20)*s12+2*P^2+(2*s11+2*s21)*P+(sX20+2*s10)*s22+2*s11*s21 )*X1+2*sX20*(P+(1/2)*s11+(1/2)*s21))*Q+((-sX10-s20)*s12-P^2+(-s11-s21) *P+(-sX20-s10)*s22-s11*s21)*X1^2-2*sX20*(P+(1/2)*s11+(1/2)*s21)*X1-sX2 0^2)/((sX10-sX20)*(-h+sX10-sX20)*(Q-X1)):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 467 "n1[2,1]:=(-Q^4*s12*s22+(2*X1*s12*s22+(-s21-P)*s12-s2 2*(P+s11))*Q^3+(-X1^2*s12*s22+((2*s21+2*P)*s12+2*s22*(P+s11))*X1+(sX10 -sX20-s20)*s12-P^2+(-s11-s21)*P-s10*s22-s11*s21)*Q^2+(((-s21-P)*s12-s2 2*(P+s11))*X1^2+((-sX10+2*sX20+2*s20)*s12+2*P^2+(2*s11+2*s21)*P+(sX10+ 2*s10)*s22+2*s11*s21)*X1+(2*(P+(1/2)*s11+(1/2)*s21))*sX10)*Q+((-sX20-s 20)*s12-P^2+(-s11-s21)*P+(-sX10-s10)*s22-s11*s21)*X1^2-(2*(P+(1/2)*s11 +(1/2)*s21))*sX10*X1-sX10^2)/((h+sX10-sX20)*(sX10-sX20)*(Q-X1)):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 612 "n1[2,2]:=-(Q^4*s12*s22-2*Q^3*X1 *s12*s22+Q^2*X1^2*s12*s22+P*Q^3*s12+P*Q^3*s22-2*P*Q^2*X1*s12-2*P*Q^2*X 1*s22+P*Q*X1^2*s12+P*Q*X1^2*s22+Q^3*s11*s22+Q^3*s12*s21-2*Q^2*X1*s11*s 22-2*Q^2*X1*s12*s21+Q*X1^2*s11*s22+Q*X1^2*s12*s21+P^2*Q^2-2*P^2*Q*X1+P ^2*X1^2+P*Q^2*s11+P*Q^2*s21-2*P*Q*X1*s11-2*P*Q*X1*s21+P*X1^2*s11+P*X1^ 2*s21+Q^2*s10*s22+Q^2*s11*s21+Q^2*s12*s20-2*Q*X1*s10*s22-2*Q*X1*s11*s2 1-2*Q*X1*s12*s20-Q*X1*s12*sX20-Q*X1*s22*sX20+X1^2*s10*s22+X1^2*s11*s21 +X1^2*s12*s20+X1^2*s12*sX20+X1^2*s22*sX20-P*Q*sX10-P*Q*sX20+P*X1*sX10+ P*X1*sX20-Q*s11*sX20-Q*s21*sX20+X1*s11*sX20+X1*s21*sX20+sX10*sX20)/((Q -X1)*h*(sX10-sX20)):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "simplif y(residue( ToCancelX1Entry11/(xi-X1),xi=X1));\n" }{MPLTEXT 1 0 53 "sim plify(residue( ToCancelX1Entry12/(xi-X1),xi=X1));\n" }{MPLTEXT 1 0 53 "simplify(residue( ToCancelX1Entry21/(xi-X1),xi=X1));\n" }{MPLTEXT 1 0 52 "simplify(residue( ToCancelX1Entry22/(xi-X1),xi=X1));" }}{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 41 "omegaJMUPartX1:=si mplify(omegaJMUPartX1):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 38 "Secon d part: Contributions at infinity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ExpXiInfinity:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 49 "Ex pXiInfinity[1,1]:=exp(-(s11*xi+s12*xi^2/2)/h):\n" }{MPLTEXT 1 0 49 "Ex pXiInfinity[2,2]:=exp(-(s21*xi+s22*xi^2/2)/h):\n" }{MPLTEXT 1 0 15 "Ex pXiInfinity;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "XiInfinity:=Mat rix(2,2,0):\n" }{MPLTEXT 1 0 19 "XiInfinity[1,1]:=-(" }{MPLTEXT 1 0 11 "s10*ln(xi)+" }{MPLTEXT 1 0 20 "s11*xi+s12*xi^2/2):\n" }{MPLTEXT 1 0 19 "XiInfinity[2,2]:=-(" }{MPLTEXT 1 0 11 "s20*ln(xi)+" }{MPLTEXT 1 0 20 "s21*xi+s22*xi^2/2):\n" }{MPLTEXT 1 0 12 "XiInfinity:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "dXiInfinitydxi:=Matrix(2,2,0):\n " }{MPLTEXT 1 0 97 "for i from 1 to 2 do for j from 1 to 2 do dXiInfin itydxi[i,j]:=diff(XiInfinity[i,j],xi): od: od:\n" }{MPLTEXT 1 0 16 "dX iInfinitydxi:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 34 "ExpMonodromies Inf:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 44 "ExpMonodromiesInf[1,1]:=exp(- s10*ln(xi)/h):\n" }{MPLTEXT 1 0 44 "ExpMonodromiesInf[2,2]:=exp(-s20*l n(xi)/h):\n" }{MPLTEXT 1 0 19 "ExpMonodromiesInf;\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 31 "MonodromiesInf:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 30 "MonodromiesInf[1,1]:=-s10/xi:\n" }{MPLTEXT 1 0 30 "MonodromiesInf[ 2,2]:=-s20/xi:\n" }{MPLTEXT 1 0 16 "MonodromiesInf:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "J1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 19 "J2:=M atrix(2,2,0):\n" }{MPLTEXT 1 0 19 "J3:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 19 "J4:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 61 "for i from 1 to 2 do for j from 1 to 2 do J1[i,j]:=j1[i,j]: \n" }{MPLTEXT 1 0 18 "J2[i,j]:=j2[ i,j]:\n" }{MPLTEXT 1 0 18 "J3[i,j]:=j3[i,j]:\n" }{MPLTEXT 1 0 18 "J4[i ,j]:=j4[i,j]:\n" }{MPLTEXT 1 0 8 "od: od:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 63 "HatPsiRegInf:=IdentityMatrix(2)+ J1/xi+J2/xi^2+J3/xi^3+J4/xi^4:\n" }{MPLTEXT 1 0 76 "tdPsiInf:=Multiply (Multiply(HatPsiRegInf,ExpXiInfinity),ExpMonodromiesInf):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "dHatPsiRegdxiInf:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 101 "for i from 1 to 2 do for j from 1 to 2 do dHatPsiReg dxiInf[i,j]:=diff(HatPsiRegInf[i,j],xi): od: od:\n" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 29 "dtdPsidxiInf:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 93 "for i from 1 to 2 do for j from 1 to 2 do dtdPsidxiInf[i,j]:=diff( tdPsiInf[i,j],xi): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "d XiInfinitydt:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 205 "for i from 1 to 2 d o for j from 1 to 2 do dXiInfinitydt[i,j]:= simplify(diff(XiInfinity[i ,j],s11)*ds11+diff(XiInfinity[i,j],s21)*ds21+diff(XiInfinity[i,j],s12) *ds12+diff(XiInfinity[i,j],s22)*ds22) od: od:\n" }{MPLTEXT 1 0 15 "dXi Infinitydt:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 124 "omegaJMUPartInf inity:=-h*residue(Trace(Multiply(Multiply(HatPsiRegInf^(-1),dHatPsiReg dxiInf), dXiInfinitydt)),xi=infinity);\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 61 "ToCancelInf:=simplify(h*dtdPsidxiInf-Multiply(tdL,tdP siInf)):" }}{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(-I%msupGF$6%-I#moGF$6.Q/ⅇF'/%+foregroundGQ([0,0,0 ]F'/%,mathvariantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFF/%)stretc hyGFF/%*symmetricGFF/%(largeopGFF/%.movablelimitsGFF/%'accentGFF/%'lsp aceGQ&0.0emF'/%'rspaceGQ,0.1111111emF'-F,6$-F;6.Q*&uminus0;F'F>FAFDFGF IFKFMFOFQ/FTQ,0.2222222emF'/FWFin-I&mfracGF$6)-F,6#-F,6%-F,6%-I#miGF$6 &Q$s11F'/%'italicGQ%trueF'F>/FBQ'italicF'-F;6-Q1⁢F'FAFD FGFIFKFMFOFQFS/FWFU-Feo6&Q%ξF'/FioFFF>FA-F;6.Q\"+F'F>FAFDFGFIFKFMFO FQFhnFjn-F,6%-F\\o6)-I#mnGF$6%Q\"1F'F>FA-F]q6%Q\"2F'F>FA/%.linethickne ssGF_q/%+denomalignGQ'centerF'/%)numalignGFgq/%)bevelledGFFF>F]p-F,6%- F86%FapF`q/%1superscriptshiftGQ\"0F'F]p-Feo6&Q$s12F'FhoF>F[p-F,6#-Feo6 &Q\"hF'FhoF>F[pFcqFeqFhqFjqF>F`r/%)rowalignGQ!F'/%,columnalignGF]s/%+g roupalignGF]s/%(rowspanGF_q/%+columnspanGF_q-F56(-F]q6%FbrF>FAF[sF^sF` sFbsFdsF[sF^sF`s-F26'Ffs-F56(-F86%F:-F,6$Fen-F\\o6)-F,6#-F,6%-F,6%-Feo 6&Q$s21F'FhoF>F[pF]pFapFep-F,6%FjpF]p-F,6%F^rF]p-Feo6&Q$s22F'FhoF>F[pF frFcqFeqFhqFjqF>F`rF[sF^sF`sFbsFdsF[sF^sF`s/%&alignGQ%axisF'/F\\sQ)bas elineF'/F_sFgq/FasQ'|frleft|hrF'/%/alignmentscopeGFjo/%,columnwidthGQ% autoF'/%&widthGF`v/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%) rowlinesGQ%noneF'/%,columnlinesGF[w/%&frameGF[w/%-framespacingGQ,0.4em ~0.5exF'/%*equalrowsGFF/%-equalcolumnsGFF/%-displaystyleGFF/%%sideGQ&r ightF'/%0minlabelspacingGFhvF>FA/%%openGQ\"[F'/%&closeGQ\"]F'" }} {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 supGF$6%-I#moGF$6.Q/ⅇF'/%+foregroundGQ([0,0,0]F'/%,mathva riantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFF/%)stretchyGFF/%*symm etricGFF/%(largeopGFF/%.movablelimitsGFF/%'accentGFF/%'lspaceGQ&0.0emF '/%'rspaceGQ,0.1111111emF'-F,6$-F;6.Q*&uminus0;F'F>FAFDFGFIFKFMFOFQ/FT Q,0.2222222emF'/FWFin-I&mfracGF$6)-F,6%-I#miGF$6&Q$s10F'/%'italicGQ%tr ueF'F>/FBQ'italicF'-F;6-Q1⁢F'FAFDFGFIFKFMFOFQFS/FWFU-F, 6%-Fao6&Q#lnF'/FeoFFF>FA-F;6-Q0⁡F'FAFDFGFIFKFMFOFQFSF\\p -F#6%-F,6#-Fao6&Q%ξF'FbpF>FAF>FA-F,6#-Fao6&Q\"hF'FdoF>Fgo/%.linethi cknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFgq/%)bevelledGFFF>/% 1superscriptshiftGQ\"0F'/%)rowalignGQ!F'/%,columnalignGFar/%+groupalig nGFar/%(rowspanGFdq/%+columnspanGFdq-F56(-I#mnGF$6%F^rF>FAF_rFbrFdrFfr FhrF_rFbrFdr-F26'Fjr-F56(-F86%F:-F,6$Fen-F\\o6)-F,6%-Fao6&Q$s20F'FdoF> FgoFioF]pF]qFbqFeqFhqFjqF>F\\rF_rFbrFdrFfrFhrF_rFbrFdr/%&alignGQ%axisF '/F`rQ)baselineF'/FcrFgq/FerQ'|frleft|hrF'/%/alignmentscopeGFfo/%,colu mnwidthGQ%autoF'/%&widthGFjt/%+rowspacingGQ&1.0exF'/%.columnspacingGQ& 0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFeu/%&frameGFeu/%-framespaci ngGQ,0.4em~0.5exF'/%*equalrowsGFF/%-equalcolumnsGFF/%-displaystyleGFF/ %%sideGQ&rightF'/%0minlabelspacingGFbuF>FA/%%openGQ\"[F'/%&closeGQ\"]F '" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I5omegaJMUPartInfinityG6\",$*&I\"h GF$\"\"\",**&&I#j1GF$6$F(F(F(I%ds11GF$F(!\"\"*&,(*$F+\"\"#F(*&&F,6$F(F 3F(&F,6$F3F(F(F(&I#j2GF$F-!\"#F(I%ds12GF$F(#F(F3*&&F,6$F3F3F(I%ds21GF$ F(F/*&,(F4F(*$F?F3F(&F:F@F;F(I%ds22GF$F(F=F(F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "series(simplify(ToCancelInf[1,1]*exp((xi^2*s12+ 2*s10*ln(xi)+2*s11*xi)/(2*h))),xi=infinity):\n" }{MPLTEXT 1 0 1 "j" } {MPLTEXT 1 0 103 "1[2,1]:=simplify(solve(residue( simplify(ToCancelInf [1,1]*exp((xi^2*s12+2*s10*ln(xi)+2*s11*xi)/(2*h)))\n" }{MPLTEXT 1 0 25 ",xi=infinity),j1[2,1]));\n" }{MPLTEXT 1 0 107 "j2[2,1]:=simplify(s olve(residue( simplify(ToCancelInf[1,1]*exp((xi^2*s12+2*s10*ln(xi)+2*s 11*xi)/(2*h)))*xi\n" }{MPLTEXT 1 0 25 ",xi=infinity),j2[2,1]));\n" } {MPLTEXT 1 0 109 "j3[2,1]:=simplify(solve(residue( simplify(ToCancelIn f[1,1]*exp((xi^2*s12+2*s10*ln(xi)+2*s11*xi)/(2*h)))*xi^2\n" }{MPLTEXT 1 0 24 ",xi=infinity),j3[2,1])):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I# j1G6\"6$\"\"#\"\"\",**&I\"QGF%F'I$s12GF%F(!\"\"*&,(*&I#X1GF%F(F,F(F(I \"PGF%F-I$s11GF%F-F(F+F(F(*&,&F2F(F3F(F(F1F(F(I$s10GF%F-" }}{PARA 11 " " 1 "" {XPPMATH 20 ">&I#j2G6\"6$\"\"#\"\"\",,*&I$s12GF%F(I\"QGF%\"\"$! \"\"*&,**&I#X1GF%F(F+F(F(*&F+F(&I#j1GF%6$F(F(F(F.I\"PGF%F.I$s11GF%F.F( F,F'F(*&,(*&,(F1F(F7F.F8F.F(F4F(F(*&,&F7F(F8F(F(F2F(F(I$s10GF%F.F(F,F( F(*&,(F=F(I\"hGF%F.F?F.F(F4F(F(*&F2F(F?F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "series(simplify(ToCancelInf[1,2]*exp((xi^2*s22+2*s 20*ln(xi)+2*s21*xi)/(2*h))),xi=infinity):\n" }{MPLTEXT 1 0 1 "j" } {MPLTEXT 1 0 106 "1[1,2]:=simplify(solve(residue( simplify(ToCancelInf [1,2]*exp((xi^2*s22+2*s20*ln(xi)+2*s21*xi)/(2*h)))/xi\n" }{MPLTEXT 1 0 25 ",xi=infinity),j1[1,2]));\n" }{MPLTEXT 1 0 104 "j2[1,2]:=simplify (solve(residue( simplify(ToCancelInf[1,2]*exp((xi^2*s22+2*s20*ln(xi)+2 *s21*xi)/(2*h)))\n" }{MPLTEXT 1 0 25 ",xi=infinity),j2[1,2]));\n" } {MPLTEXT 1 0 107 "j3[1,2]:=simplify(solve(residue( simplify(ToCancelIn f[1,2]*exp((xi^2*s22+2*s20*ln(xi)+2*s21*xi)/(2*h)))*xi\n" }{MPLTEXT 1 0 24 ",xi=infinity),j3[1,2])):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#j1 G6\"6$\"\"\"\"\"#*$,&I$s12GF%F'I$s22GF%!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I#j2G6\"6$\"\"\"\"\"#*&,**&,(&I#j1GF%6$F(F(F'I\"QGF%!\" \"I#X1GF%F'F'I$s12GF%F'F'*&,(F0F'F2F1F-F1F'I$s22GF%F'F'I$s11GF%F1I$s21 GF%F'F',&F3F'F6F1!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sX 20:=-s10-s20-sX10:\n" }{MPLTEXT 1 0 117 "series(simplify(ToCancelInf[2 ,1]*exp((xi^2*s12+2*s10*ln(xi)+2*s11*xi)/(2*h))) -s12*(s10+s20+sX10+sX 20),xi=infinity):\n" }{MPLTEXT 1 0 1 "j" }{MPLTEXT 1 0 103 "1[1,1]:=si mplify(solve(residue( simplify(ToCancelInf[2,1]*exp((xi^2*s12+2*s10*ln (xi)+2*s11*xi)/(2*h)))\n" }{MPLTEXT 1 0 25 ",xi=infinity),j1[1,1]));\n " }{MPLTEXT 1 0 107 "j2[1,1]:=simplify(solve(residue( simplify(ToCance lInf[2,1]*exp((xi^2*s12+2*s10*ln(xi)+2*s11*xi)/(2*h)))*xi\n" }{MPLTEXT 1 0 24 ",xi=infinity),j2[1,1])):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">&I #j1G6\"6$\"\"\"F'**,0*(I\"QGF%\"\"%I$s12GF%F'I$s22GF%F'!\"\"*&,**(I#X1 GF%F'F-F'F.F'\"\"#*&,&F-F/F.F/F'I\"PGF%F'F'*&I$s11GF%F'F.F'F/*&F-F'I$s 21GF%F'F/F'F+\"\"$F'*&,0*(F3F4F-F'F.F'F/*&,(*&,&F-F4F.F4F'F7F'F'F8F4F: F4F'F3F'F'*$F7F4F/*&,&F9F/F;F/F'F7F'F'*&I$s10GF%F'F.F'F/*&F9F'F;F'F/*& F-F'I$s20GF%F'F/F'F+F4F'*&,,*&,(F5F'F:F/F8F/F'F3F4F'*&,,FDF4*&,&F9F4F; F4F'F7F'F'*&,&FHF/FKF'F'F-F'F'FGF4FIF4F'F3F'F'*&F7F',&FHF/FKF/F'F'*&F; F'FHF'F/*&F9F'FKF'F/F'F+F'F'*&,,*&FHF'F-F'F'FDF/FEF'FIF/FGF/F'F3F4F'*& ,(*&,&FHF'FKF'F'F7F'F'FXF'FYF'F'F3F'F'*&I%sX10GF%F',(F\\oF'FHF'FKF'F'F 'F',&F+F'F3F/F/,&F-F/F.F'F/I\"hGF%F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "series(simplify(ToCancelInf[2,2]*exp((xi^2*s22+2*s20* ln(xi)+2*s21*xi)/(2*h))),xi=infinity):\n" }{MPLTEXT 1 0 1 "j" } {MPLTEXT 1 0 106 "1[2,2]:=simplify(solve(residue( simplify(ToCancelInf [2,2]*exp((xi^2*s22+2*s20*ln(xi)+2*s21*xi)/(2*h)))*xi\n" }{MPLTEXT 1 0 25 ",xi=infinity),j1[2,2]));\n" }{MPLTEXT 1 0 109 "j2[2,2]:=simplify (solve(residue( simplify(ToCancelInf[2,2]*exp((xi^2*s22+2*s20*ln(xi)+2 *s21*xi)/(2*h)))*xi^2\n" }{MPLTEXT 1 0 24 ",xi=infinity),j2[2,2])):" } }{PARA 11 "" 1 "" {XPPMATH 20 ">&I#j1G6\"6$\"\"#F'**,0*(I\"QGF%\"\"%I$ s12GF%\"\"\"I$s22GF%F.F.*&,**(I#X1GF%F.F-F.F/F.!\"#*&,&F-F.F/F.F.I\"PG F%F.F.*&I$s11GF%F.F/F.F.*&F-F.I$s21GF%F.F.F.F+\"\"$F.*&,0*(F3F'F-F.F/F .F.*&,(*&,&F-F4F/F4F.F7F.F.F8F4F:F4F.F3F.F.*$F7F'F.*&,&F;F.F9F.F.F7F.F .*&I$s10GF%F.F/F.F.*&F9F.F;F.F.*&F-F.I$s20GF%F.F.F.F+F'F.*&,,*&,(F5F.F 8F.F:F.F.F3F'F.*&,,FDF4*&,&F9F4F;F4F.F7F.F.*&,&FH!\"\"FKF.F.F/F.F.FIF4 FJF4F.F3F.F.*&,&FHF.FKF.F.F7F.F.*&F;F.FHF.F.*&F9F.FKF.F.F.F+F.F.*&,,FD F.FEF.*&FKF.F/F.FVFIF.FJF.F.F3F'F.*&,(*&F7F.,&FHFVFKFVF.F.FYFVFZFVF.F3 F.F.*&I%sX10GF%F.,(F]oF.FHF.FKF.F.FVF.,&F-FVF/F.FV,&F+F.F3FVFVI\"hGF%F V" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "omegaJMUPartInfinity:= simplify(omegaJMUPartInfinity):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "omegaJMUP4:=omegaJMUPartX1+omegaJMUPartInfinity:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "omegaJMUP4ds11:=simplify(residue(om egaJMUP4/ds11^2,ds11=0)):\n" }{MPLTEXT 1 0 61 "omegaJMUP4ds21:=simplif y(residue(omegaJMUP4/ds21^2,ds21=0)):\n" }{MPLTEXT 1 0 61 "omegaJMUP4d s12:=simplify(residue(omegaJMUP4/ds12^2,ds12=0)):\n" }{MPLTEXT 1 0 61 "omegaJMUP4ds22:=simplify(residue(omegaJMUP4/ds22^2,ds22=0)):\n" } {MPLTEXT 1 0 57 "omegaJMUP4dX1:=simplify(residue(omegaJMUP4/dX1^2,dX1= 0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 10 "omegaJMUP4" }{MPLTEXT 1 0 5 "ds11-" }{MPLTEXT 1 0 8 "residue( " }{MPLTEXT 1 0 8 "Hams11/h" }{MPLTEXT 1 0 5 ",h=0)" }{MPLTEXT 1 0 3 " );\n" }{MPLTEXT 1 0 48 "simplify(omegaJMUP4ds21-residue(Hams21/h,h=0)) ;\n" }{MPLTEXT 1 0 48 "simplify(omegaJMUP4ds12-residue(Hams12/h,h=0)); \n" }{MPLTEXT 1 0 48 "simplify(omegaJMUP4ds22-residue(Hams22/h,h=0)); \n" }{MPLTEXT 1 0 46 "simplify(omegaJMUP4dX1 -residue(HamX1/h,h=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 21 "omegaJMUP4:=simplify(" }{MPLTEXT 1 0 12 "omegaJMUP4):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 3466 "omegaJMUP 4formula:= ((2*(-(1/2)*(ds12-ds22)*(-s12+s22)*X1+dX1*s12^2+(-2*dX1*s22 +ds11-ds21)*s12+dX1*s22^2+(-ds11+ds21)*s22-(1/2)*(-s21+s11)*(ds12-ds22 )))*s22*s12*Q^4+(2*(-(1/2)*(ds12-ds22)*(-s12+s22)*X1+dX1*s12^2+(-2*dX1 *s22+ds11-ds21)*s12+dX1*s22^2+(-ds11+ds21)*s22-(1/2)*(-s21+s11)*(ds12- ds22)))*(-2*X1*s12*s22+(s21+P)*s12+s22*(P+s11))*Q^3+(2*(-(1/2)*(ds12-d s22)*(-s12+s22)*X1+dX1*s12^2+(-2*dX1*s22+ds11-ds21)*s12+dX1*s22^2+(-ds 11+ds21)*s22-(1/2)*(-s21+s11)*(ds12-ds22)))*(X1^2*s12*s22+((-2*P-2*s21 )*s12-2*s22*(P+s11))*X1+s12*s20+s10*s22+(P+s11)*(s21+P))*Q^2+(-(-s12+s 22)*((s21+P)*s12+s22*(P+s11))*(ds12-ds22)*X1^3+(2*dX1*(s21+P)*s12^3+(- 2*dX1*(P-s11+2*s21)*s22+(2*ds11-2*ds21)*P+(2*ds11-2*ds21)*s21+(s10-s20 )*ds12+2*ds22*s20)*s12^2+(-2*dX1*(P+2*s11-s21)*s22^2+((2*ds11-2*ds21)* s11+(-2*ds11+2*ds21)*s21+(-3*s10+s20)*ds12+ds22*(s10-3*s20))*s22-(2*(P +3*s11*(1/2)-(1/2)*s21))*(ds12-ds22)*(s21+P))*s12+2*s22*(dX1*(P+s11)*s 22^2+((-ds11+ds21)*P+(-ds11+ds21)*s11+ds12*s10-(1/2)*ds22*(s10-s20))*s 22+(P-(1/2)*s11+3*s21*(1/2))*(ds12-ds22)*(P+s11)))*X1^2+(2*dX1*(s10-s2 0)*s12^3+(-6*dX1*(s10-s20)*s22-4*P^2*dX1-4*dX1*(s21+s11)*P-4*dX1*s11*s 21+2*ds11*s10-2*s20*(ds11-2*ds21))*s12^2+(6*dX1*(s10-s20)*s22^2+(8*P^2 *dX1+8*dX1*(s21+s11)*P+8*dX1*s11*s21+(-6*ds11+2*ds21)*s10+2*s20*(ds11- 3*ds21))*s22+(-4*ds11+4*ds21)*P^2+((-4*ds11+4*ds21)*s11+(-4*ds11+4*ds2 1)*s21+(s10+s20)*(ds12-ds22))*P+((-4*ds11+4*ds21)*s21+2*s20*(ds12-ds22 ))*s11+s21*(s10-s20)*(ds12-ds22))*s12-2*dX1*(s10-s20)*s22^3+(-4*P^2*dX 1-4*dX1*(s21+s11)*P-4*dX1*s11*s21+(4*ds11-2*ds21)*s10+2*ds21*s20)*s22^ 2+((4*ds11-4*ds21)*P^2+((4*ds11-4*ds21)*s11+(4*ds11-4*ds21)*s21-(s10+s 20)*(ds12-ds22))*P+((4*ds11-4*ds21)*s21+(s10-s20)*(ds12-ds22))*s11-2*s 21*s10*(ds12-ds22))*s22+(2*(-s21+s11))*(s21+P)*(P+s11)*(ds12-ds22))*X1 +2*dX1*(s10+s20)*(P+s11+s21)*s12^2+(-4*dX1*(s10+s20)*(P+s11+s21)*s22+( 2*(s10+s20))*(ds11-ds21)*P+2*s20*(ds11-ds21)*s11+2*s10*(ds11-ds21)*s21 +(sX10+s20)*(sX10+s10)*(ds12-ds22))*s12+2*dX1*(s10+s20)*(P+s11+s21)*s2 2^2+(-(2*(s10+s20))*(ds11-ds21)*P-2*s20*(ds11-ds21)*s11-2*s10*(ds11-ds 21)*s21-(sX10+s20)*(sX10+s10)*(ds12-ds22))*s22-(ds12-ds22)*((s10+s20)* P+s21*s10+s11*s20)*(-s21+s11))*Q-(-s12+s22)*((-ds12*s10-ds22*s20)*s12+ (ds12*s10+ds22*s20)*s22+(s21+P)*(P+s11)*(ds12-ds22))*X1^3+(-2*dX1*s10* s12^3+((4*(s10-(1/2)*s20))*dX1*s22+2*P^2*dX1+2*dX1*(s21+s11)*P+2*dX1*s 11*s21-2*ds11*s10-2*ds21*s20)*s12^2+(-2*dX1*(s10-2*s20)*s22^2+(-4*P^2* dX1-4*dX1*(s21+s11)*P-4*dX1*s11*s21+4*ds11*s10+4*ds21*s20)*s22+(2*ds11 -2*ds21)*P^2+((2*ds11-2*ds21)*s11+(2*ds11-2*ds21)*s21-(s10+s20)*(ds12- ds22))*P+((2*ds11-2*ds21)*s21-s20*(ds12-ds22))*s11-s21*s10*(ds12-ds22) )*s12-2*s22^3*dX1*s20+(2*P^2*dX1+2*dX1*(s21+s11)*P+2*dX1*s11*s21-2*ds1 1*s10-2*ds21*s20)*s22^2+((-2*ds11+2*ds21)*P^2+((-2*ds11+2*ds21)*s11+(- 2*ds11+2*ds21)*s21+(s10+s20)*(ds12-ds22))*P+((-2*ds11+2*ds21)*s21+s20* (ds12-ds22))*s11+s21*s10*(ds12-ds22))*s22-(-s21+s11)*(s21+P)*(P+s11)*( ds12-ds22))*X1^2+(-2*dX1*(s10+s20)*(P+s11+s21)*s12^2+(4*dX1*(s10+s20)* (P+s11+s21)*s22-(2*(s10+s20))*(ds11-ds21)*P-2*s20*(ds11-ds21)*s11-2*s1 0*(ds11-ds21)*s21-(2*((sX10+(1/2)*s20)*s10+sX10*(sX10+s20)))*(ds12-ds2 2))*s12-2*dX1*(s10+s20)*(P+s11+s21)*s22^2+((2*(s10+s20))*(ds11-ds21)*P +2*s20*(ds11-ds21)*s11+2*s10*(ds11-ds21)*s21+(2*((sX10+(1/2)*s20)*s10+ sX10*(sX10+s20)))*(ds12-ds22))*s22+(ds12-ds22)*((s10+s20)*P+s21*s10+s1 1*s20)*(-s21+s11))*X1-2*sX10*(sX10+s10+s20)*(dX1*s12^2+(-2*dX1*s22+ds1 1-ds21)*s12+dX1*s22^2+(-ds11+ds21)*s22-(1/2)*(-s21+s11)*(ds12-ds22)))/ ((2*(Q-X1))*(-s12+s22)^2):\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 11 "omegaJMUP4-" }{MPLTEXT 1 0 20 "omegaJMUP4formula);\n" }{MPLTEXT 1 0 12 "K0:=unapply(" }{MPLTEXT 1 0 36 "KOldCoordinates(s11,s21,s12,s2 2,X1)," }{MPLTEXT 1 0 20 "s11,s21,s12,s22,X1);" }{MPLTEXT 1 0 1 " " } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 223 "dK0:=unapply( diff(K0(s11,s21,s 12,s22,X1),s11)*ds11+ diff(K0(s11,s21,s12,s22,X1),s21)*ds21+diff(K0(s1 1,s21,s12,s22,X1),s12)*ds12+diff(K0(s11,s21,s12,s22,X1),s22)*ds22+diff (K0(s11,s21,s12,s22,X1),X1)*dX1,s11,s21,s12,s22,X1);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 21 "simplify(omegaJMUP4-(" }{MPLTEXT 1 0 27 "residu e(Hams11/h,h=0)*ds11+" }{MPLTEXT 1 0 26 "residue(Hams21/h,h=0)*ds21" } {MPLTEXT 1 0 2 " +" }{MPLTEXT 1 0 27 "residue(Hams12/h,h=0)*ds12+" } {MPLTEXT 1 0 27 "residue(Hams22/h,h=0)*ds22+" }{MPLTEXT 1 0 28 "residu e(HamX1/h,h=0)*dX1));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I#K0G6\"f*6'I$s11GF$I$s21GF$I$s12GF$I$s 22GF$I#X1GF$F$6$I)operatorGF$I&arrowGF$F$,&*(,&I%sX10GF$\"\"\"I$s20GF$ F3F3,&F2F3I$s10GF$F3F3-I#lnG6$%*protectedGI(_syslibGF$6#,&9&#F3\"\"#9' #!\"\"F@F3F?*&9(F3,&*&,&*&F>F3FEF3F39$F@F3F6F3F3*&F4F3,&*&FAF3FEF3F39% F@F3F3F3F?F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$dK0G6\"f*6'I$s11G F$I$s21GF$I$s12GF$I$s22GF$I#X1GF$F$6$I)operatorGF$I&arrowGF$F$,,*(9(\" \"\"I$s10GF$F2I%ds11GF$F2F2*(F1F2I$s20GF$F2I%ds21GF$F2F2*&,&*(,&I%sX10 GF$F2F6F2F2,&F " 0 "" {MPLTEXT 1 0 25 "checkQfunct ion:=S2*Q+S1;\n" }{MPLTEXT 1 0 6 "checkP" }{MPLTEXT 1 0 8 "function" } {MPLTEXT 1 0 21 "=1/S2*(P-1/2*R1(Q));\n" }{MPLTEXT 1 0 23 "SolQ:=-(S1- checkQ)/S2;\n" }{MPLTEXT 1 0 42 "SolP:=checkP*S2+(1/2)*R1(-(S1-checkQ) /S2);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 31 "simplify(checkQ-(S2*SolQ +S1));\n" }{MPLTEXT 1 0 17 "simplify(checkP-(" }{MPLTEXT 1 0 24 "1/S2* (SolP-1/2*R1(SolQ))" }{MPLTEXT 1 0 3 "));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/checkQfunctionG6\",&*&I\"QGF$\"\"\"I#S2GF$F(F(I#S1GF$F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "/I/checkPfunctionG6\"*&I#S2GF$!\"\" ,,I\"PGF$\"\"\"*&,&I$s10GF$F'I$s20GF$F'F*,&I\"QGF$F*I#X1GF$F'F'#F'\"\" #I$s11GF$#F*F3I$s21GF$F5*&,&I$s12GF$F'I$s22GF$F'F*F0F*F2F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%SolQG6\",$*&,&I#S1GF$\"\"\"I'checkQGF$!\"\"F) I#S2GF$F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%SolPG6\",,*&I'checkPGF $\"\"\"I#S2GF$F(F(*&,&I$s10GF$!\"\"I$s20GF$F-F(,&*&,&I#S1GF$F(I'checkQ GF$F-F(F)F-F-I#X1GF$F-F-#F(\"\"#I$s11GF$#F-F6I$s21GF$F8*(,&I$s12GF$F-I $s22GF$F-F(F1F(F)F-F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Sinfty2function:=s12+s22;\n" }{MPLTEXT 1 0 7 "Sinfty1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 11 ":=s11+s21;\n" }{MPLTEXT 1 0 2 "S2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 25 ":=sqrt(s12-s22)/sqrt(2); \n" }{MPLTEXT 1 0 2 "S1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 35 " :=(s11-s21)/sqrt(2)/sqrt(s12-s22);\n" }{MPLTEXT 1 0 19 "tdX1function:= X1*S2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 2 "S1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 8 "solve( \{" }{MPLTEXT 1 0 16 "s12+s22=Sinfty2," }{MPLTEXT 1 0 17 "s11+ s21=Sinfty1, " }{MPLTEXT 1 0 3 "S2=" }{MPLTEXT 1 0 21 "sqrt(s12-s22)/s qrt(2)" }{MPLTEXT 1 0 2 ",\n" }{MPLTEXT 1 0 3 "S1=" }{MPLTEXT 1 0 31 " (s11-s21)/sqrt(2)/sqrt(s12-s22)" }{MPLTEXT 1 0 2 ",\n" }{MPLTEXT 1 0 8 "tdX1=X1*" }{MPLTEXT 1 0 10 "S2function" }{MPLTEXT 1 0 3 "+S1" } {MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 24 "\},\{s12,s22,s11,s21,X1\}) ;" }{MPLTEXT 1 0 2 " \n" }{MPLTEXT 1 0 67 "X1function := unapply( -(S1 -tdX1)/S2 ,Sinfty1,Sinfty2,S1,S2,tdX1);\n" }{MPLTEXT 1 0 73 "s11functi on:= unapply( S2*S1+(1/2)*Sinfty1, Sinfty1,Sinfty2,S1,S2,tdX1);\n" } {MPLTEXT 1 0 72 "s12function:= unapply( S2^2+(1/2)*Sinfty2, Sinfty1,Si nfty2,S1,S2,tdX1);\n" }{MPLTEXT 1 0 74 "s21function:= unapply( -S2*S1+ (1/2)*Sinfty1, Sinfty1,Sinfty2,S1,S2,tdX1);\n" }{MPLTEXT 1 0 74 "s22fu nction:= unapply( -S2^2+(1/2)*Sinfty2, Sinfty1,Sinfty2,S1,S2,tdX1 );\n " }{MPLTEXT 1 0 27 "simplify(X1function(Sinfty1" }{MPLTEXT 1 0 8 "func tion" }{MPLTEXT 1 0 8 ",Sinfty2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 3 ",S1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 3 ",S2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 17 ",tdX1function));\n" }{MPLTEXT 1 0 28 "simplify(s11function(Sinfty1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 8 ",Sinfty2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 3 ",S1" } {MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 3 ",S2" }{MPLTEXT 1 0 8 "funct ion" }{MPLTEXT 1 0 17 ",tdX1function));\n" }{MPLTEXT 1 0 28 "simplify( s12function(Sinfty1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 8 ",Sinf ty2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 3 ",S1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 3 ",S2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 17 ",tdX1function));\n" }{MPLTEXT 1 0 28 "simplify(s21function(Sin fty1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 8 ",Sinfty2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 3 ",S1" }{MPLTEXT 1 0 8 "function" } {MPLTEXT 1 0 3 ",S2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 17 ",tdX 1function));\n" }{MPLTEXT 1 0 28 "simplify(s22function(Sinfty1" } {MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 8 ",Sinfty2" }{MPLTEXT 1 0 8 " function" }{MPLTEXT 1 0 3 ",S1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 3 ",S2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 16 ",tdX1function ));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 47 "parti altdX1function:=simplify(diff(s11function(" }{MPLTEXT 1 0 28 "Sinfty1, Sinfty2,S1,S2,tdX1)," }{MPLTEXT 1 0 18 "tdX1))*partials11+" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 54 "simplify(diff(s21function(Sinfty1,Sinfty2 ,S1,S2,tdX1)," }{MPLTEXT 1 0 7 "tdX1))*" }{MPLTEXT 1 0 11 "partials21 \n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 54 "simplify(diff(s12function(Si nfty1,Sinfty2,S1,S2,tdX1)," }{MPLTEXT 1 0 7 "tdX1))*" }{MPLTEXT 1 0 11 "partials12\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 54 "simplify(diff( s22function(Sinfty1,Sinfty2,S1,S2,tdX1)," }{MPLTEXT 1 0 7 "tdX1))*" } {MPLTEXT 1 0 11 "partials22\n" }{MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 53 "s implify(diff(X1function(Sinfty1,Sinfty2,S1,S2,tdX1)," }{MPLTEXT 1 0 17 "tdX1))*partialX1;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0Sinfty2funct ionG6\",&I$s12GF$\"\"\"I$s22GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0 Sinfty1functionG6\",&I$s21GF$\"\"\"I$s11GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+S2functionG6\",$*&,&I$s12GF$\"\"\"I$s22GF$!\"\"#F)\"\" #F-F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+S1functionG6\",$*(,&I$s21G F$!\"\"I$s11GF$\"\"\"F+\"\"##F+F,,&I$s12GF$F+I$s22GF$F)#F)F,F-" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I-tdX1functionG6\",&*(I#X1GF$\"\"\",&I$ s12GF$F(I$s22GF$!\"\"#F(\"\"#F.F-F-*(,&I$s21GF$F,I$s11GF$F(F(F.F-F)#F, F.F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'/I#X1G6\",$*&I#S2GF%!\"\",&I#S 1GF%\"\"\"I%tdX1GF%F)F,F)/I$s11GF%,&I(Sinfty1GF%#F,\"\"#*&F+F,F(F,F,/I $s12GF%,&I(Sinfty2GF%F2*$F(F3F,/I$s21GF%,&F4F)F1F2/I$s22GF%,&F9F)F8F2" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+X1functionG6\"f*6'I(Sinfty1GF$I(Si nfty2GF$I#S1GF$I#S2GF$I%tdX1GF$F$6$I)operatorGF$I&arrowGF$F$,$*&9'!\" \",&9&\"\"\"9(F2F5F2F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,s11func tionG6\"f*6'I(Sinfty1GF$I(Sinfty2GF$I#S1GF$I#S2GF$I%tdX1GF$F$6$I)opera torGF$I&arrowGF$F$,&9$#\"\"\"\"\"#*&9&F29'F2F2F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,s12functionG6\"f*6'I(Sinfty1GF$I(Sinfty2GF$I#S1GF$I #S2GF$I%tdX1GF$F$6$I)operatorGF$I&arrowGF$F$,&9%#\"\"\"\"\"#*$9'F3F2F$ F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,s21functionG6\"f*6'I(Sinfty1G F$I(Sinfty2GF$I#S1GF$I#S2GF$I%tdX1GF$F$6$I)operatorGF$I&arrowGF$F$,&*& 9&\"\"\"9'F2!\"\"9$#F2\"\"#F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I, s22functionG6\"f*6'I(Sinfty1GF$I(Sinfty2GF$I#S1GF$I#S2GF$I%tdX1GF$F$6$ I)operatorGF$I&arrowGF$F$,&*$9'\"\"#!\"\"9%#\"\"\"F2F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#X1G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$s11G 6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$s12G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$s21G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I$s22G6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I4partialtdX1functionG6\"*&I#S2GF$!\"\" I*partialX1GF$\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ds1 1function:=diff(s11function(" }{MPLTEXT 1 0 26 "Sinfty1,Sinfty2,S1,S2, tdX1" }{MPLTEXT 1 0 21 "),Sinfty1)*dSinfty1+ " }{MPLTEXT 1 0 17 "diff( s11function(" }{MPLTEXT 1 0 26 "Sinfty1,Sinfty2,S1,S2,tdX1" }{MPLTEXT 1 0 20 "),Sinfty2)*dSinfty2+" }{MPLTEXT 1 0 17 "diff(s11function(" } {MPLTEXT 1 0 26 "Sinfty1,Sinfty2,S1,S2,tdX1" }{MPLTEXT 1 0 11 "),S1)*d S1+ " }{MPLTEXT 1 0 17 "diff(s11function(" }{MPLTEXT 1 0 26 "Sinfty1,S infty2,S1,S2,tdX1" }{MPLTEXT 1 0 9 "),S2)*dS2" }{MPLTEXT 1 0 2 " +" } {MPLTEXT 1 0 17 "diff(s11function(" }{MPLTEXT 1 0 26 "Sinfty1,Sinfty2, S1,S2,tdX1" }{MPLTEXT 1 0 15 "),tdX1)*dtdX1;\n" }{MPLTEXT 1 0 297 "ds2 1function:=diff(s21function( Sinfty1,Sinfty2,S1,S2,tdX1),Sinfty1)*dSin fty1+ diff(s21function( Sinfty1,Sinfty2,S1,S2,tdX1),Sinfty2)*dSinfty2+ diff(s21function( Sinfty1,Sinfty2,S1,S2,tdX1),S1)*dS1+ diff(s21functio n( Sinfty1,Sinfty2,S1,S2,tdX1),S2)*dS2 +diff(s21function( Sinfty1,Sinf ty2,S1,S2,tdX1" }{MPLTEXT 1 0 14 "),tdX1)*dtdX1;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 297 "ds12function:=diff(s12function( Sinfty1,Sinfty2,S1 ,S2,tdX1),Sinfty1)*dSinfty1+ diff(s12function( Sinfty1,Sinfty2,S1,S2,t dX1),Sinfty2)*dSinfty2+diff(s12function( Sinfty1,Sinfty2,S1,S2,tdX1),S 1)*dS1+ diff(s12function( Sinfty1,Sinfty2,S1,S2,tdX1),S2)*dS2 +diff(s1 2function( Sinfty1,Sinfty2,S1,S2,tdX1" }{MPLTEXT 1 0 14 "),tdX1)*dtdX1 ;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 297 "ds22function:=diff(s22funct ion( Sinfty1,Sinfty2,S1,S2,tdX1),Sinfty1)*dSinfty1+ diff(s22function( \+ Sinfty1,Sinfty2,S1,S2,tdX1),Sinfty2)*dSinfty2+diff(s22function( Sinfty 1,Sinfty2,S1,S2,tdX1),S1)*dS1+ diff(s22function( Sinfty1,Sinfty2,S1,S2 ,tdX1),S2)*dS2 +diff(s22function( Sinfty1,Sinfty2,S1,S2,tdX1" } {MPLTEXT 1 0 14 "),tdX1)*dtdX1;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 82 "dX1function:=diff(X1function( Sinfty1,Sinfty2,S1,S2,tdX1),Sinfty1) *dSinfty1+ diff(" }{MPLTEXT 1 0 2 "X1" }{MPLTEXT 1 0 61 "function( Sin fty1,Sinfty2,S1,S2,tdX1),Sinfty2)*dSinfty2+diff(" }{MPLTEXT 1 0 2 "X1" }{MPLTEXT 1 0 52 "function( Sinfty1,Sinfty2,S1,S2,tdX1),S1)*dS1+ diff (" }{MPLTEXT 1 0 2 "X1" }{MPLTEXT 1 0 52 "function( Sinfty1,Sinfty2,S1 ,S2,tdX1),S2)*dS2 +diff(" }{MPLTEXT 1 0 2 "X1" }{MPLTEXT 1 0 36 "funct ion( Sinfty1,Sinfty2,S1,S2,tdX1" }{MPLTEXT 1 0 14 "),tdX1)*dtdX1;" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 25 "solve(\{ds1 1function=ds11," }{MPLTEXT 1 0 18 "ds21function=ds21," }{MPLTEXT 1 0 18 "ds12function=ds12," }{MPLTEXT 1 0 18 "ds22function=ds22," } {MPLTEXT 1 0 16 "dX1function=dX1," }{MPLTEXT 1 0 54 "ds11function=ds11 \},\{dS1,dS2,dSinfty1,dSinfty2,dtdX1\});" }{MPLTEXT 1 0 2 " \n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 "dS1function := ((2*ds11-2*ds21)* S2-S1*(ds12-ds22))/(4*S2^2);\n" }{MPLTEXT 1 0 35 "dS2function := (ds12 -ds22)/(4*S2);\n" }{MPLTEXT 1 0 90 "dtdX1function :=(4*S2^3*dX1+(2*ds1 1-2*ds21)*S2-(2*(S1-(1/2)*tdX1))*(ds12-ds22))/(4*S2^2);\n" }{MPLTEXT 1 0 31 "dSinfty1function := ds11+ds21;\n" }{MPLTEXT 1 0 30 "dSinfty2fu nction := ds12+ds22;" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-ds11functionG 6\",(I)dSinfty1GF$#\"\"\"\"\"#*&I#S2GF$F(I$dS1GF$F(F(*&I#S1GF$F(I$dS2G F$F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-ds21functionG6\",(I)dSinfty 1GF$#\"\"\"\"\"#*&I#S2GF$F(I$dS1GF$F(!\"\"*&I#S1GF$F(I$dS2GF$F(F-" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I-ds12functionG6\",&I)dSinfty2GF$#\"\" \"\"\"#*&I#S2GF$F(I$dS2GF$F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-ds2 2functionG6\",&I)dSinfty2GF$#\"\"\"\"\"#*&I#S2GF$F(I$dS2GF$F(!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I,dX1functionG6\",(*&I#S2GF$!\"\"I$dS1G F$\"\"\"F(*(F'!\"#,&I#S1GF$F*I%tdX1GF$F(F*I$dS2GF$F*F**&F'F(I&dtdX1GF$ F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "<'/I$dS1G6\",$*&,**&I#S1GF%\"\" \"I%ds12GF%F+F+*&F*F+I%ds22GF%F+!\"\"*&I#S2GF%F+I%ds11GF%F+!\"#*&F1F+I %ds21GF%F+\"\"#F+F1F3#F/\"\"%/I$dS2GF%,$*&,&F,F+F.F/F+F1F/#F+F8/I&dtdX 1GF%,$*&,0*&F1\"\"$I$dX1GF%F+!\"%F)F6F-F3F0F3F4F6*&F,F+I%tdX1GF%F+F/*& F.F+FIF+F+F+F1F3F7/I)dSinfty1GF%,&F2F+F5F+/I)dSinfty2GF%,&F,F+F.F+" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I,dS1functionG6\",$*&,&*&,&I%ds11GF$\" \"#I%ds21GF$!\"#\"\"\"I#S2GF$F.F.*&I#S1GF$F.,&I%ds12GF$F.I%ds22GF$!\" \"F.F5F.F/F-#F.\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I,dS2functionG6 \",$*&,&I%ds12GF$\"\"\"I%ds22GF$!\"\"F)I#S2GF$F+#F)\"\"%" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I.dtdX1functionG6\",$*&,(*&I#S2GF$\"\"$I$dX1GF$\" \"\"\"\"%*&,&I%ds11GF$\"\"#I%ds21GF$!\"#F,F)F,F,*&,&I#S1GF$F,I%tdX1GF$ #!\"\"F1F,,&I%ds12GF$F,I%ds22GF$F9F,F3F,F)F3#F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1dSinfty1functionG6\",&I%ds11GF$\"\"\"I%ds21GF$F'" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I1dSinfty2functionG6\",&I%ds12GF$\"\"\" I%ds22GF$F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 107 "Rewriting the JM U differential in terms of (\\check\{Q\},\\check\{P\},S_1,S_2,S_\{\\in fty,1\},S_\{\\infty,2\},\\td\{X\}_1)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "omegaJMUP4function1:=unapply(" }{MPLTEXT 1 0 10 "omeg aJMUP4" }{MPLTEXT 1 0 7 ",Q,P):\n" }{MPLTEXT 1 0 38 "omegaJMUP4functio n2:=unapply(simplify(" }{MPLTEXT 1 0 19 "omegaJMUP4function1" } {MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 4 "SolQ" }{MPLTEXT 1 0 1 "," } {MPLTEXT 1 0 4 "SolP" }{MPLTEXT 1 0 48 ")),s11,s21,s12,s22,X1,ds11,ds2 1,ds12,ds22,dX1):\n" }{MPLTEXT 1 0 31 "omegaJMUP4function3:=simplify( " }{MPLTEXT 1 0 20 "omegaJMUP4function2(" }{MPLTEXT 1 0 12 "s11functio n(" }{MPLTEXT 1 0 26 "Sinfty1,Sinfty2,S1,S2,tdX1" }{MPLTEXT 1 0 2 ")," }{MPLTEXT 1 0 38 "s21function(Sinfty1,Sinfty2,S1,S2,tdX1" }{MPLTEXT 1 0 2 ")," }{MPLTEXT 1 0 38 "s12function(Sinfty1,Sinfty2,S1,S2,tdX1" } {MPLTEXT 1 0 2 ")," }{MPLTEXT 1 0 38 "s22function(Sinfty1,Sinfty2,S1,S 2,tdX1" }{MPLTEXT 1 0 39 "),X1function(Sinfty1,Sinfty2,S1,S2,tdX1" } {MPLTEXT 1 0 15 "),ds11function," }{MPLTEXT 1 0 13 "ds21function," } {MPLTEXT 1 0 13 "ds12function," }{MPLTEXT 1 0 13 "ds22function," } {MPLTEXT 1 0 15 "dX1function) ):" }{MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "omegaJMUP4NewCoordinates:=" }{MPLTEXT 1 0 31 "simplify(omegaJMUP4function3):\n" }{MPLTEXT 1 0 27 "CoeffdS1:=si mplify(residue(" }{MPLTEXT 1 0 24 "omegaJMUP4NewCoordinates" }{MPLTEXT 1 0 16 "/dS1^2,dS1=0));\n" }{MPLTEXT 1 0 27 "CoeffdS2:=simplify(resid ue(" }{MPLTEXT 1 0 24 "omegaJMUP4NewCoordinates" }{MPLTEXT 1 0 16 "/dS 2^2,dS2=0));\n" }{MPLTEXT 1 0 32 "CoeffdSinfty1:=simplify(residue(" } {MPLTEXT 1 0 24 "omegaJMUP4NewCoordinates" }{MPLTEXT 1 0 26 "/dSinfty1 ^2,dSinfty1=0));\n" }{MPLTEXT 1 0 32 "CoeffdSinfty2:=simplify(residue( " }{MPLTEXT 1 0 24 "omegaJMUP4NewCoordinates" }{MPLTEXT 1 0 26 "/dSinf ty2^2,dSinfty2=0));\n" }{MPLTEXT 1 0 29 "CoeffdtdX1:=simplify(residue( " }{MPLTEXT 1 0 24 "omegaJMUP4NewCoordinates" }{MPLTEXT 1 0 19 "/dtdX1 ^2,dtdX1=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)CoeffdS1G6\",$*&,(*( I#S1GF$\"\"\",&I$s10GF$F*I$s20GF$!\"\"F*I#S2GF$\"\"#!\"#*(I(Sinfty1GF$ F*,&F,F*F-F*F*F/F*F.*(I(Sinfty2GF$F*F4F*,&F)F*I%tdX1GF$F.F*F*F*F/F1#F* F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)CoeffdS2G6\",$*&,(*(,&I%sX10GF$ \"\"\"I$s20GF$F+F+,&F*F+I$s10GF$F+F+I#S2GF$\"\"#F0**I(Sinfty1GF$F+,&F. F+F,F+F+,&I#S1GF$F+I%tdX1GF$!\"\"F+F/F+F+*(I(Sinfty2GF$F+F4F0F3F+F7F+F /!\"$#F+F0" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.CoeffdSinfty1G6\",$*(,& I$s10GF$\"\"\"I$s20GF$F)F),&I#S1GF$F)I%tdX1GF$!\"\"F)I#S2GF$F.#F.\"\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I.CoeffdSinfty2G6\",$*(,&I#S1GF$\" \"\"I%tdX1GF$!\"\"\"\"#,&I$s10GF$F)I$s20GF$F)F)I#S2GF$!\"##F)\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I+CoeffdtdX1G6\",$*(,(*&,,*&,**$I'check PGF$\"\"#!\"%*$I'checkQGF$F.\"\"%I$s10GF$F2I$s20GF$F/\"\"\"I%tdX1GF$F. F5*(F1F5,*F,!\"\"F0F5F3F5F4F9F5F6F5!\")*$F1F2F2*&,(F,F/F3F2F4F/F5F1F.F 5*$,(F3F5F4F5I%sX10GF$F.F.F5F5I#S2GF$F.F5**I(Sinfty1GF$F5,&F6F5F1F9F5, &F3F5F4F5F5FAF5F.**I(Sinfty2GF$F5FDF5FEF5,&I#S1GF$F5F6F9F5!\"#F5FAFJFD F9#F5F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "CoeffdtdX1bis:=" }{MPLTEXT 1 0 47 "(checkQ-tdX1)*checkP^2-checkQ^3+tdX1*checkQ^2-(" } {MPLTEXT 1 0 16 "s10-s20)*checkQ\n" }{MPLTEXT 1 0 35 "+(-(sX10-sX20)^2 )/4/(checkQ-tdX1):\n" }{MPLTEXT 1 0 35 "PartConstantCoefftdX1bis:=simp lify(" }{MPLTEXT 1 0 11 "CoeffdtdX1-" }{MPLTEXT 1 0 15 "CoeffdtdX1bis) ;" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I9PartConstantC oefftdX1bisG6\",$*&,(*(I%tdX1GF$\"\"\",&I$s10GF$F*I$s20GF$!\"\"F*I#S2G F$\"\"#F0*(I(Sinfty1GF$F*,&F,F*F-F*F*F/F*F**(I(Sinfty2GF$F*F3F*,&I#S1G F$F*F)F.F*F.F*F/!\"##F*F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "pdsolve(\{CoeffdS1=diff(MM(Sinfty1,Sinfty2,S1,S2,tdX1),S1),\n" } {MPLTEXT 1 0 50 "CoeffdS2=diff(MM(Sinfty1,Sinfty2,S1,S2,tdX1),S2),\n" }{MPLTEXT 1 0 60 "CoeffdSinfty1=diff(MM(Sinfty1,Sinfty2,S1,S2,tdX1),Si nfty1),\n" }{MPLTEXT 1 0 60 "CoeffdSinfty2=diff(MM(Sinfty1,Sinfty2,S1, S2,tdX1),Sinfty2),\n" }{MPLTEXT 1 0 67 "PartConstantCoefftdX1bis=diff( MM(Sinfty1,Sinfty2,S1,S2,tdX1),tdX1)\n" }{MPLTEXT 1 0 36 "\},\{MM(Sinf ty1,Sinfty2,S1,S2,tdX1)\});" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 172 "M:=unapply( (4*S2^2*(sX10+s20)*(sX10+s10)*ln(S2)+( (-2*S1^2+2*tdX1^2)*s10+(2*S1^2-2*tdX1^2)*s20)*S2^2-2*Sinfty1*(s10+s20) *(S1-tdX1)*S2+Sinfty2*(S1-tdX1)^2*(s10+s20))/4/S2^2," }{MPLTEXT 1 0 29 "Sinfty1,Sinfty2,S1,S2,tdX1);\n" }{MPLTEXT 1 0 10 "simplify( " } {MPLTEXT 1 0 50 "CoeffdS1-diff(M(Sinfty1,Sinfty2,S1,S2,tdX1),S1));\n" }{MPLTEXT 1 0 10 "simplify( " }{MPLTEXT 1 0 43 "CoeffdS2-diff(M(Sinfty 1,Sinfty2,S1,S2,tdX1)" }{MPLTEXT 1 0 4 ",S2)" }{MPLTEXT 1 0 3 ");\n" } {MPLTEXT 1 0 10 "simplify( " }{MPLTEXT 1 0 48 "CoeffdSinfty1-diff(M(Si nfty1,Sinfty2,S1,S2,tdX1)" }{MPLTEXT 1 0 9 ",Sinfty1)" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 10 "simplify( " }{MPLTEXT 1 0 48 "CoeffdSinfty2 -diff(M(Sinfty1,Sinfty2,S1,S2,tdX1)" }{MPLTEXT 1 0 9 ",Sinfty2)" } {MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 10 "simplify( " }{MPLTEXT 1 0 24 " PartConstantCoefftdX1bis" }{MPLTEXT 1 0 35 "-diff(M(Sinfty1,Sinfty2,S1 ,S2,tdX1)" }{MPLTEXT 1 0 6 ",tdX1)" }{MPLTEXT 1 0 2 ");" }}{PARA 11 "" 1 "" {XPPMATH 20 "<#/-I#MMG6\"6'I(Sinfty1G6\"I(Sinfty2G6\"I#S1G6\"I#S 2G6\"I%tdX1G6\",$*&,***I#S2G6\"\"\"#,&I%sX10G6\"\"\"\"I$s20G6\"\"\"\" \"\"\",&I%sX10G6\"\"\"\"I$s10G6\"\"\"\"\"\"\"-I#lnG6$%*protectedGI(_sy slibG6\"6#I#S2G6\"\"\"\"\"\"%*&,(*&,&*$I#S1G6\"\"\"#!\"#*$I%tdX1G6\"\" \"#\"\"#\"\"\"I$s10G6\"\"\"\"\"\"\"*&,&*$I#S1G6\"\"\"#\"\"#*$I%tdX1G6 \"\"\"#!\"#\"\"\"I$s20G6\"\"\"\"\"\"\"I$_C1G6\"\"\"%\"\"\"I#S2G6\"\"\" #\"\"\"**I(Sinfty1G6\"\"\"\",&I$s10G6\"\"\"\"I$s20G6\"\"\"\"\"\"\",&I# S1G6\"\"\"\"I%tdX1G6\"!\"\"\"\"\"I#S2G6\"\"\"\"!\"#*(I(Sinfty2G6\"\"\" \",&I#S1G6\"\"\"\"I%tdX1G6\"!\"\"\"\"#,&I$s10G6\"\"\"\"I$s20G6\"\"\"\" \"\"\"\"\"\"\"\"\"I#S2G6\"!\"##\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I\"MG6\"f*6'I(Sinfty1GF$I(Sinfty2GF$I#S1GF$I#S2GF$I%tdX1 GF$F$6$I)operatorGF$I&arrowGF$F$,$*&,***9'\"\"#,&I%sX10GF$\"\"\"I$s20G F$F7F7,&F6F7I$s10GF$F7F7-I#lnG6$%*protectedGI(_syslibGF$6#F3F7\"\"%*&, &*&,&*$9&F4!\"#*$9(F4F4F7F:F7F7*&,&FFF4FIFHF7F8F7F7F7F3F4F7**9$F7,&F:F 7F8F7F7,&FGF7FJ!\"\"F7F3F7FH*(9%F7FPF4FOF7F7F7F3FH#F7FAF$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "KNewCoordinates:=unapply( simplify( KOldCoordinates(s11function(" }{MPLTEXT 1 0 40 "Sinfty1,Sinfty2,S1,S2, tdX1),s21function(" }{MPLTEXT 1 0 40 "Sinfty1,Sinfty2,S1,S2,tdX1),s12f unction(" }{MPLTEXT 1 0 40 "Sinfty1,Sinfty2,S1,S2,tdX1),s22function(" }{MPLTEXT 1 0 39 "Sinfty1,Sinfty2,S1,S2,tdX1),X1function(" }{MPLTEXT 1 0 28 "Sinfty1,Sinfty2,S1,S2,tdX1))" }{MPLTEXT 1 0 9 ",symbolic" } {MPLTEXT 1 0 3 ") ," }{MPLTEXT 1 0 27 "Sinfty1,Sinfty2,S1,S2,tdX1)" } {MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 16 "KNe wCoordinates(" }{MPLTEXT 1 0 30 "Sinfty1,Sinfty2,S1,S2,tdX1)-M(" } {MPLTEXT 1 0 30 "Sinfty1,Sinfty2,S1,S2,tdX1));\n" }{MPLTEXT 1 0 92 "KN ewCoordinates2:=unapply((sX10+s20)*(sX10+s10)*ln(S2)-(1/2)*(S1-tdX1)*( s10+s20)*Sinfty1/S2\n" }{MPLTEXT 1 0 78 "+(1/4)*(S1-tdX1)^2*(s10+s20)* Sinfty2/S2^2-(1/2)*(S1-tdX1)*(s10-s20)*(S1+tdX1)\n" }{MPLTEXT 1 0 1 ", " }{MPLTEXT 1 0 26 "Sinfty1,Sinfty2,S1,S2,tdX1" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 16 "KNewCoordinates(" } {MPLTEXT 1 0 28 "Sinfty1,Sinfty2,S1,S2,tdX1)-" }{MPLTEXT 1 0 17 "KNewC oordinates2(" }{MPLTEXT 1 0 38 "Sinfty1,Sinfty2,S1,S2,tdX1),symbolic); " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0KNewCoordinatesG6\"f*6'I(Sinfty1G F$I(Sinfty2GF$I#S1GF$I#S2GF$I%tdX1GF$F$6$I)operatorGF$I&arrowGF$F$,$*& ,&**9'\"\"#,&I%sX10GF$\"\"\"I$s20GF$F7F7,&F6F7I$s10GF$F7F7-I#lnG6$%*pr otectedGI(_syslibGF$6#F3F7\"\"%*&,&9&F79(!\"\"F7,(*(,&F:F7F8FFF7,&FDF7 FEF7F7F3F4F7*(9$F7,&F:F7F8F7F7F3F7F7*(9%F7FMF7FCF7#FFF4F7!\"#F7F3FQ#F7 FAF$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I1KNewCoordinates2G6\"f*6'I(Sinfty1GF$I(Sinfty2GF$I#S1GF $I#S2GF$I%tdX1GF$F$6$I)operatorGF$I&arrowGF$F$,**(,&I%sX10GF$\"\"\"I$s 20GF$F3F3,&F2F3I$s10GF$F3F3-I#lnG6$%*protectedGI(_syslibGF$6#9'F3F3**, &F6F3F4F3F3,&9&F39(!\"\"F3F=FC9$F3#FC\"\"#**F@FFF?F3F=!\"#9%F3#F3\"\"% *(F@F3,&F6F3F4FCF3,&FAF3FBF3F3FEF$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 255 63 "Obtaining the only no n-trivial Hamiltonians for (checkQ,checkP)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "assume(S2>0):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 45 "hpartialtdX1functioncheckQfunction:= unapply(" }{MPLTEXT 1 0 54 "d iff(S2*Q+S1,Q)*hpartialtdX1functionQ+diff(S2*Q+S1,P)*" }{MPLTEXT 1 0 22 "hpartialtdX1functionP\n" }{MPLTEXT 1 0 27 "+h/S2*(diff(S2function* Q+S1" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 5 ",X1))" }{MPLTEXT 1 0 7 ",Q,P):\n" }{MPLTEXT 1 0 55 "hpartialtdX1functioncheckQfunction2:= unapply( simplify(" }{MPLTEXT 1 0 68 "hpartialtdX1functioncheckQfuncti on(SolQ,SolP)),s12,s22,s11,s21,X1):\n" }{MPLTEXT 1 0 37 "hpartialtdX1f unctioncheckQ:=simplify(" }{MPLTEXT 1 0 48 "hpartialtdX1functioncheckQ function2(s12function(" }{MPLTEXT 1 0 28 "Sinfty1,Sinfty2,S1,S2,tdX1), " }{MPLTEXT 1 0 40 "s22function(Sinfty1,Sinfty2,S1,S2,tdX1)," } {MPLTEXT 1 0 40 "s11function(Sinfty1,Sinfty2,S1,S2,tdX1)," }{MPLTEXT 1 0 40 "s21function(Sinfty1,Sinfty2,S1,S2,tdX1)," }{MPLTEXT 1 0 42 "X1 function(Sinfty1,Sinfty2,S1,S2,tdX1))):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 45 "hpartialtdX1functioncheckPfunction:= unapply(" } {MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 18 "1/S2*(P-1/2*R1(Q))" }{MPLTEXT 1 0 31 ",Q)*hpartialtdX1functionQ+diff(" }{MPLTEXT 1 0 18 "1/S2*(P-1/ 2*R1(Q))" }{MPLTEXT 1 0 26 ",P)*hpartialtdX1functionP\n" }{MPLTEXT 1 0 12 "+h/S2*(diff(" }{MPLTEXT 1 0 4 "1/S2" }{MPLTEXT 1 0 8 "function" }{MPLTEXT 1 0 14 "*(P-1/2*R1(Q))" }{MPLTEXT 1 0 11 ",X1)),Q,P):" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 123 "hpartialtdX1functioncheckPfunct ion2:=unapply( simplify(hpartialtdX1functioncheckPfunction(SolQ,SolP)) ,s12,s22,s11,s21,X1):\n" }{MPLTEXT 1 0 113 "hpartialtdX1functioncheckP :=simplify(hpartialtdX1functioncheckPfunction2(s12function(Sinfty1,Sin fty2,S1,S2,tdX1)," }{MPLTEXT 1 0 160 "s22function(Sinfty1,Sinfty2,S1,S 2,tdX1),s11function(Sinfty1,Sinfty2,S1,S2,tdX1),s21function(Sinfty1,Si nfty2,S1,S2,tdX1),X1function(Sinfty1,Sinfty2,S1,S2,tdX1))\n" }{MPLTEXT 1 0 74 "+1/(4*S2^2*(tdX1-checkQ)^2)*2*Sinfty2*(tdX1-checkQ)^2*(s10+s2 0+sX10+sX20)\n" }{MPLTEXT 1 0 2 "):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 61 "HamtdX1checkQcheckP:=unapply(((checkQ-tdX1)*checkP+h)*checkP\n" }{MPLTEXT 1 0 151 "+(4*checkQ^4-8*tdX1*checkQ^3+(4*tdX1^2-4*h+4*s10-4* s20)*checkQ^2+4*tdX1*(h+s20-s10)*checkQ+sX10^2+(-2*sX20)*sX10-2*sX20*( -(1/2)*sX20))/4/(tdX1-checkQ)\n" }{MPLTEXT 1 0 17 ",checkQ,checkP):\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 32 "hpartialtdX1functionche ckQ-diff(" }{MPLTEXT 1 0 3 "Ham" }{MPLTEXT 1 0 4 "tdX1" }{MPLTEXT 1 0 13 "checkQcheckP(" }{MPLTEXT 1 0 14 "checkQ,checkP)" }{MPLTEXT 1 0 11 ",checkP));\n" }{MPLTEXT 1 0 16 "simplify(series(" }{MPLTEXT 1 0 9 "si mplify(" }{MPLTEXT 1 0 35 "hpartialtdX1functioncheckP+diff(Ham" } {MPLTEXT 1 0 4 "tdX1" }{MPLTEXT 1 0 43 "checkQcheckP(checkQ,checkP),ch eckQ)),h=0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 3 "Ham" }{MPLTEXT 1 0 4 "tdX1" }{MPLTEXT 1 0 81 "checkQcheckPbis:=un apply((checkQ-tdX1)*checkP^2+h*checkP-checkQ^3+tdX1*checkQ^2-(" } {MPLTEXT 1 0 8 "s10-s20-" }{MPLTEXT 1 0 10 "h)*checkQ\n" }{MPLTEXT 1 0 52 "+(-(sX10-sX20)^2)/4/(checkQ-tdX1) ,checkQ,checkP);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 3 "Ham" }{MPLTEXT 1 0 4 "tdX1" } {MPLTEXT 1 0 28 "checkQcheckP(checkQ,checkP)-" }{MPLTEXT 1 0 3 "Ham" } {MPLTEXT 1 0 4 "tdX1" }{MPLTEXT 1 0 32 "checkQcheckPbis(checkQ,checkP) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I7HamtdX1checkQch eckPbisG6\"f*6$I'checkQGF$I'checkPGF$F$6$I)operatorGF$I&arrowGF$F$,.*& ,&I%tdX1GF$!\"\"9$\"\"\"F29%\"\"#F2*&I\"hGF$F2F3F2F2*$F1\"\"$F0*&F/F2F 1F4F2*&,(F6F0I$s10GF$F2I$s20GF$F0F2F1F2F0*&,(F " 0 "" {MPLTEXT 1 0 26 "K0NewCoordinates:=unapply(" } {MPLTEXT 1 0 81 "KNewCoordinates(Sinfty1, Sinfty2, S1, S2, tdX1),Sinft y1, Sinfty2, S1, S2, tdX1): " }{MPLTEXT 1 0 1 "d" }{MPLTEXT 1 0 16 "K0 NewCoordinates" }{MPLTEXT 1 0 16 ":=unapply( diff(" }{MPLTEXT 1 0 16 " K0NewCoordinates" }{MPLTEXT 1 0 52 "(Sinfty1, Sinfty2, S1, S2, tdX1),S infty1)*dSinfty1+ " }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 16 "K0NewCoo rdinates" }{MPLTEXT 1 0 51 "(Sinfty1, Sinfty2, S1, S2, tdX1),Sinfty2)* dSinfty2+" }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 16 "K0NewCoordinates" }{MPLTEXT 1 0 41 "(Sinfty1, Sinfty2, S1, S2, tdX1),S1)*dS1\n" } {MPLTEXT 1 0 1 "+" }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 16 "K0NewCoor dinates" }{MPLTEXT 1 0 41 "(Sinfty1, Sinfty2, S1, S2, tdX1),S2)*dS2+" }{MPLTEXT 1 0 5 "diff(" }{MPLTEXT 1 0 16 "K0NewCoordinates" }{MPLTEXT 1 0 44 "(Sinfty1, Sinfty2, S1, S2, tdX1),tdX1)*dtdX1" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 30 "Sinfty1, Sinfty2, S1, S2, t dX1" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 24 "omegaJMUP4NewCoordinates" }{MPLTEXT 1 0 62 "-(residue(HamtdX1check QcheckPbis(checkQ,checkP)/h,h=0)*dtdX1 +" }{MPLTEXT 1 0 17 "dK0NewCoor dinates" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 30 "Sinfty1, Sinfty2, S1, S 2, tdX1" }{MPLTEXT 1 0 4 ")));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "tdL:= tdL:\n" }{MPLTEXT 1 0 19 "tdA:=simplify(tdA):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 256 33 "Gener al expressions for reduction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "c0:=0:\n" }{MPLTEXT 1 0 53 "tdL11function:=unapply(tdL[1,1],s12,s22 ,s11,s21,X1):\n" }{MPLTEXT 1 0 53 "tdL12function:=unapply(tdL[1,2],s12 ,s22,s11,s21,X1):\n" }{MPLTEXT 1 0 53 "tdL21function:=unapply(tdL[2,1] ,s12,s22,s11,s21,X1):\n" }{MPLTEXT 1 0 53 "tdL22function:=unapply(tdL[ 2,2],s12,s22,s11,s21,X1):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 88 "td A11function:=unapply(tdA[1,1],s12,s22,s11,s21,X1,beta12,beta22,beta11, beta21,betaX1):\n" }{MPLTEXT 1 0 50 "tdA12function:=unapply(tdA[1,2],s 12,s22,s11,s21,X1" }{MPLTEXT 1 0 36 ",beta12,beta22,beta11,beta21,beta X1)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 50 "tdA21function:=unapply(td A[2,1],s12,s22,s11,s21,X1" }{MPLTEXT 1 0 36 ",beta12,beta22,beta11,bet a21,betaX1)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 50 "tdA22function:=un apply(tdA[2,2],s12,s22,s11,s21,X1" }{MPLTEXT 1 0 36 ",beta12,beta22,be ta11,beta21,betaX1)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 38 "SolQfunct ion:=unapply(simplify(SolQ), " }{MPLTEXT 1 0 21 "s12,s22,s11,s21,X1); \n" }{MPLTEXT 1 0 59 "SolPfunction:=unapply(simplify(SolP), s12,s22,s1 1,s21,X1);\n" }{MPLTEXT 1 0 65 "tdX1functionfunction:=unapply(tdX1func tion,s12,s22,s11,s21,X1); \n" }{MPLTEXT 1 0 44 "S1function:=unapply(S1 ,s12,s22,s11,s21,X1);\n" }{MPLTEXT 1 0 44 "S2function:=unapply(S2,s12, s22,s11,s21,X1);\n" }{MPLTEXT 1 0 51 "R1function:=unapply(R1(xi),xi,s1 2,s22,s11,s21,X1);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "LQfunctio n:=unapply(LQ" }{MPLTEXT 1 0 57 ",s12,s22,s11,s21,X1,beta12,beta22,bet a11,beta21,betaX1):\n" }{MPLTEXT 1 0 79 "LPfunction:=unapply(LP,s12,s2 2,s11,s21,X1,beta12,beta22,beta11,beta21,betaX1):\n" }{MPLTEXT 1 0 99 "Hamiltonianfunction:=unapply(Hamiltonianbis,s12,s22,s11,s21,X1,beta12 ,beta22,beta11,beta21,betaX1):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-Sol QfunctionG6\"f*6'I$s12GF$I$s22GF$I$s11GF$I$s21GF$I#X1GF$F$6$I)operator GF$I&arrowGF$F$*&,&I#S1GF$!\"\"I'checkQGF$\"\"\"F4I$S2|irG6#/I'_localG %*protectedGI0_n2177231096352GF$F2F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I-SolPfunctionG6\"f*6'I$s12GF$I$s22GF$I$s11GF$I$s21GF$I#X1GF$F$6$ I)operatorGF$I&arrowGF$F$,$*(,**(I$S2|irG6#/I'_localG%*protectedGI0_n2 177231096352GF$\"\"$9(\"\"\"I'checkPGF$F;\"\"#*&,,*&FI5tdX1functionfunctionG6\"f*6'I$s12GF$I$s22GF$I$s11GF$I$s21GF$I#X1GF$ F$6$I)operatorGF$I&arrowGF$F$,&*(9(\"\"\",&9$F29%!\"\"#F2\"\"#F8F7F7*( ,&9&F29'F6F2F8F7F3#F6F8F7F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+S1 functionG6\"f*6'I$s12GF$I$s22GF$I$s11GF$I$s21GF$I#X1GF$F$6$I)operatorG F$I&arrowGF$F$I#S1GF$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+S2func tionG6\"f*6'I$s12GF$I$s22GF$I$s11GF$I$s21GF$I#X1GF$F$6$I)operatorGF$I& arrowGF$F$I$S2|irG6#/I'_localG%*protectedGI0_n2177231096352GF$F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+R1functionG6\"f*6(I#xiGF$I$s12GF$I$ s22GF$I$s11GF$I$s21GF$I#X1GF$F$6$I)operatorGF$I&arrowGF$F$,**&,&I$s10G F$!\"\"I$s20GF$F4\"\"\",&9$F69)F4F4F69'F49(F4*&,&9%F49&F4F6F8F6F6F$F$F $" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 256 18 "Jimbo-Miwa case: " } {TEXT 256 112 "We take s12=1 and s22=-1 and X_1=0 and s21=-s11. The ti me is sigma=s11. The direction is beta21=-1 and beta11=1." }{TEXT 256 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "simplify(SolQfunction (1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 11 "sigma,0));\n" } {MPLTEXT 1 0 29 "simplify(series(SolPfunction(" }{MPLTEXT 1 0 12 "1,-1 ,sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 12 "),checkP));\n" } {MPLTEXT 1 0 21 "tdX1functionfunction(" }{MPLTEXT 1 0 5 "1,-1," } {MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 3 "); \n" }{MPLTEXT 1 0 11 "S1function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 3 ");\n" } {MPLTEXT 1 0 11 "S2function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 14 "R1function(xi," }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma ,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 20 "solve(\{SolQfunction(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma ,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 18 ")=QQ,SolPfunction(" } {MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigm a,0" }{MPLTEXT 1 0 24 ")=PP\},\{checkQ,checkP\});\n" }{MPLTEXT 1 0 27 "tdLReduced:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 39 "tdL11function0:=unapp ly( tdL11function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 8 "),Q,P):\n" }{MPLTEXT 1 0 39 "tdL12function0:=unapply( tdL12function(" }{MPLTEXT 1 0 5 "1,-1," } {MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 8 "), Q,P):\n" }{MPLTEXT 1 0 39 "tdL21function0:=unapply( tdL21function(" } {MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigm a,0" }{MPLTEXT 1 0 8 "),Q,P):\n" }{MPLTEXT 1 0 39 "tdL22function0:=una pply( tdL22function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,- " }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 8 "),Q,P):\n" }{MPLTEXT 1 0 40 "tdLReduced[1,1]:=simplify(tdL11function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 40 "tdLReduced[1,2]:=simplify(tdL12function(" } {MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigm a,0" }{MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 40 "tdLReduced[2,1]:=simpl ify(tdL21function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 40 "t dLReduced[2,2]:=simplify(tdL22function(" }{MPLTEXT 1 0 5 "1,-1," } {MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 5 ") \+ ):\n" }{MPLTEXT 1 0 12 "tdLReduced:\n" }{MPLTEXT 1 0 28 "tdLReduced2:= Matrix(2,2,0):\n" }{MPLTEXT 1 0 55 "tdLReduced2[1,1]:=simplify(tdL11fu nction0(SolQfunction(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma, -" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 15 "),SolPfunction(" } {MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigm a,0" }{MPLTEXT 1 0 6 ")) ):\n" }{MPLTEXT 1 0 55 "tdLReduced2[1,2]:=sim plify(tdL12function0(SolQfunction(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 15 "),SolPfunc tion(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 6 ")) ):\n" }{MPLTEXT 1 0 55 "tdLReduced2[ 2,1]:=simplify(tdL21function0(SolQfunction(" }{MPLTEXT 1 0 5 "1,-1," } {MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 15 ") ,SolPfunction(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" } {MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 6 ")) ):\n" }{MPLTEXT 1 0 55 "t dLReduced2[2,2]:=simplify(tdL22function0(SolQfunction(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" } {MPLTEXT 1 0 15 "),SolPfunction(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 6 ")) ):\n" } {MPLTEXT 1 0 13 "tdLReduced2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I #S1G6\"!\"\"I'checkQGF%\"\"\"F(I$S2|irG6#/I'_localG%*protectedGI0_n217 7231096352GF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "+'I'checkPG6\"*(,&I$s 10GF$\"\"\"I$s20GF$F(F(I$S2|irG6#/I'_localG%*protectedGI0_n21772310963 52GF$F(,&I#S1GF$\"\"#I'checkQGF$!\"#!\"\"\"\"!F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&sigmaG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#S1G6\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "-I#miG6#/I+modulenameG6\"I,TypesettingG I(_syslibGF'6%Q$S2|irF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "*&,&I$s10G6\"!\"\"I$s20GF%F&\"\"\"I#xiGF %F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$/I'checkPG6\",$*(,(*&I#PPGF%\" \"\"I#QQGF%F+\"\"#I$s10GF%F+I$s20GF%F+F+F,!\"\"I$S2|irG6#/I'_localG%*p rotectedGI0_n2177231096352GF%F0#F+F-/I'checkQGF%,&*&F,F+F1F+F+I#S1GF%F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6 (-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0,0]F'/%,mathvar iantGQ'normalF'-F=6%Q\"2F'F@FC/%.linethicknessGF?/%+denomalignGQ'cente rF'/%)numalignGFM/%)bevelledGQ&falseF'F@-I#moGF$6-Q1⁢F' FC/%&fenceGFR/%*separatorGFR/%)stretchyGFR/%*symmetricGFR/%(largeopGFR /%.movablelimitsGFR/%'accentGFR/%'lspaceGQ&0.0emF'/%'rspaceGFao-F:6)-F ,6#-F,6'-F,6%-F#6%-F,6.-FT6.Q*&uminus0;F'F@FCFWFYFenFgnFinF[oF]o/F`oQ, 0.2222222emF'/FcoFdp-F,6'FFFS-I#miGF$6&Q#S1F'/%'italicGQ%trueF'F@/FDQ' italicF'FS-Fip6&Q'checkPF'F\\qF@F_q-FT6.Q\"+F'F@FCFWFYFenFgnFinF[oF]oF cpFep-F,6'FFFSFaqFS-Fip6&Q'checkQF'F\\qF@F_q-FT6.Q(−F'F@FCFWFYFe nFgnFinF[oF]oFcpFep-F,6'FFFS-Fip6&Q(σF'/F]qFRF@FCFS-Fip6&Q%ξF 'FdrF@FCF\\r-F,6%FFFS-I%msupGF$6%FerFF/%1superscriptshiftGQ\"0F'F\\r-F ip6&Q$s10F'F\\qF@F_qF\\r-Fip6&Q$s20F'F\\qF@F_qF@FCFS-F[s6%-Fip6&Q$S2|i rF'F\\qF@F_qFFF]sF\\r-F,6)FFFSFarFS-F#6%-F,6%FhpF\\rFiqF@FCFSFhsFdq-F, 6%FFFS-F[s6%F]tFFF]s-F,6%FerFSFfsFIFKFNFPF@/%)rowalignGQ!F'/%,columnal ignGFit/%+groupalignGFit/%(rowspanGF?/%+columnspanGF?-F56(-F:6)-F,6#-F ,6'-F,6%FhsFSFerFdqFhpF\\rFiq-F,6%FerFSFhsFIFKFNFPF@FgtFjtF\\uF^uF`uFg tFjtF\\u-F26'-F56(-F,6%-F:6)F<-F=6%Q\"4F'F@FCFIFKFNFPF@FS-F:6)-F,6#-F, 6,F`p-F,6)FfvFS-F#6%-F,6+-F,6%FhpFSFaqF\\r-F,6%FaqFSFiqFdq-Fip6&Q%sX10 F'F\\qF@F_qFdq-F,6%F9FS-F,6#F`sFdq-F,6%F9FS-F,6#FcsF@FCFS-F#6%-F,6+Few F\\rFgwF\\rFiwF\\rF\\xF\\rF`xF@FCFS-F[s6%FhsFfvF]sF\\r-F,6)-F=6%Q\"8F' F@FCFSF]tFS-F#6%-F,6(F`p-F,6'FaqFS-F#6%-F,6%FerFdqFarF@FCFSFiqFdq-F,6' FaqFSFeyFSFhpF\\r-F,6%F9FS-F,6%FerFS-F#6%-F,6%F`sF\\rFcsF@FCF@FCFS-F[s 6%Fhs-F=6%Q\"3F'F@FCF]sFdq-F,6)F\\yFSFctFS-F#6%-F,6(F`pFgwFdqFewF\\r-F ,6%-F#6%-F,6%FerFdq-F,6%F9FS-F,6#FarF@FCFSFarF@FCFSFfsFdq-F,6)F\\yFS-F [s6%F]tFezF]sFSFeyFSFhsF\\r-F,6%FfvFS-F[s6%F]tFfvF]s-F,6'FczFSFerFSF]t FIFKFNFPF@FgtFjtF\\uF^uF`u-F56(-F,6%F9FS-F:6)-F,6#-F,6'-F,6%-F#6%-F,6- FfpF\\rFgqFdqF_rFdqFhrF\\rF`sF\\rFcsF@FCFSFfsFdqF[tF\\rFatFetFIFKFNFPF @FgtFjtF\\uF^uF`uFgtFjtF\\u/%&alignGQ%axisF'/FhtQ)baselineF'/F[uFM/F]u Q'|frleft|hrF'/%/alignmentscopeGF^q/%,columnwidthGQ%autoF'/%&widthGF^^ l/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF' /%,columnlinesGFi^l/%&frameGFi^l/%-framespacingGQ,0.4em~0.5exF'/%*equa lrowsGFR/%-equalcolumnsGFR/%-displaystyleGFR/%%sideGQ&rightF'/%0minlab elspacingGFf^lF@FC/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "series(tdLReduced2[1,1],xi=infinity);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 38 "series(tdLReduced2[1,2],xi=infin ity);\n" }{MPLTEXT 1 0 38 "series(tdLReduced2[2,2],xi=infinity);\n" } {MPLTEXT 1 0 31 "series(tdLReduced2[2,1],xi=0):\n" }{MPLTEXT 1 0 68 "t dLReduced221bis:=((checkP+checkQ)^2*(checkQ-sigma)-(sX10-sX20)^2/4/" } {MPLTEXT 1 0 14 "(checkQ-sigma)" }{MPLTEXT 1 0 5 ")/xi\n" }{MPLTEXT 1 0 44 "+2*(checkQ-sigma)*(checkP+checkQ)+s10-s20 ;\n" }{MPLTEXT 1 0 16 "simplify(series(" }{MPLTEXT 1 0 17 "tdLReduced2[2,1]-" }{MPLTEXT 1 0 24 "tdLReduced221bis,xi=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(I#xiG6 \"!\"\"I&sigmaGF$F%*(,(*&,**&I#S1GF$\"\"\"I'checkPGF$F-!\"#*&F.F-I'che ckQGF$F-\"\"#I$s10GF$F%I$s20GF$F%F-I$S2|irG6#/I'_localG%*protectedGI0_ n2177231096352GF$F2F-*(F&F-,&F,F-F1F%F-F5F-F/*$FI1tdLReduced221bisG6\",**&,&*&,&I'checkPGF$\"\"\"I'checkQGF$F+\" \"#,&F,F+I&sigmaGF$!\"\"F+F+*&,(I$s10GF$F+I$s20GF$F+I%sX10GF$F-F-F.F0# F0\"\"%F+I#xiGF$F0F+*&F.F+F)F+F-F3F+F4F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "+'I#xiG6\",$**,,**,&I'checkQGF$!\"\"I&sigmaGF$\"\"\"F-,,*&I#S1GF$F -I'checkPGF$F-F-*&F1F-F*F-F+I%sX10GF$F+I$s10GF$#F+\"\"#I$s20GF$F5F-,,F /F-F2F+F3F-F4#F-F6F7F9F-I$S2|irG6#/I'_localG%*protectedGI0_n2177231096 352GF$\"\"%!\"%*(,.*$F*F@F5*&,&F,F-F1F+F-F*\"\"$F-*&,(*$F,F6F5*&F1F-F, F-FG*$F1F6F5F-F*F6F-**F,F-F1F-,(F0F-F,F6F1F+F-F*F-F+*(F1F-,&F0F-F1F5F- F,F6F-*$,(F4F-F7F-F3F6F6#F-\"\")F-,&F0F-F*F+F-F:FG!\")**F)F-FUF6,(F2F+ F/F-FJF5F-F:F6FT**F,F-FUFGF)F-F:F-FT*&FUF@F)F-FAF-FUF+F:!\"$F)F+#F-F@F +*&,(*&,(*$F*F6!\"#*&F,F-F*F-F6*&F1F-,&F0F-F,F+F-F\\oF-F:F6F-*(F,F-FUF -F:F-F\\o*$FUF6F6F-F:F\\o\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "tdAReduced:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 39 "tdA11function0 :=unapply( tdA11function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "si gma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 19 ",0,0,1,-1,0),Q,P): \n" }{MPLTEXT 1 0 39 "tdA12function0:=unapply( tdA12function(" } {MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigm a,0" }{MPLTEXT 1 0 11 ",0,0,1,-1,0" }{MPLTEXT 1 0 8 "),Q,P):\n" } {MPLTEXT 1 0 39 "tdA21function0:=unapply( tdA21function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" } {MPLTEXT 1 0 11 ",0,0,1,-1,0" }{MPLTEXT 1 0 8 "),Q,P):\n" }{MPLTEXT 1 0 39 "tdA22function0:=unapply( tdA22function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 11 ",0,0,1,-1,0" }{MPLTEXT 1 0 8 "),Q,P):\n" }{MPLTEXT 1 0 40 "tdAReduced [1,1]:=simplify(tdA11function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 11 ",0,0,1,-1,0" } {MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 40 "tdAReduced[1,2]:=simplify(td A12function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" } {MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 11 ",0,0,1,-1,0" }{MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 40 "tdAReduced[2,1]:=simplify(tdA21function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "si gma,0" }{MPLTEXT 1 0 11 ",0,0,1,-1,0" }{MPLTEXT 1 0 5 ") ):\n" } {MPLTEXT 1 0 40 "tdAReduced[2,2]:=simplify(tdA22function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" } {MPLTEXT 1 0 11 ",0,0,1,-1,0" }{MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 12 "tdAReduced:\n" }{MPLTEXT 1 0 28 "tdAReduced2:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 62 "tdAReduced2[1,1]:=factor(simplify(tdA11function0(SolQ function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 15 "),SolPfunction(" }{MPLTEXT 1 0 5 "1 ,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 7 ")) )):\n" }{MPLTEXT 1 0 62 "tdAReduced2[1,2]:=factor(simplify(tdA 12function0(SolQfunction(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "si gma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 15 "),SolPfunction(" } {MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigm a,0" }{MPLTEXT 1 0 7 ")) )):\n" }{MPLTEXT 1 0 62 "tdAReduced2[2,1]:=fa ctor(simplify(tdA21function0(SolQfunction(" }{MPLTEXT 1 0 5 "1,-1," } {MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 15 ") ,SolPfunction(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" } {MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 7 ")) )):\n" }{MPLTEXT 1 0 62 " tdAReduced2[2,2]:=factor(simplify(tdA22function0(SolQfunction(" } {MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigm a,0" }{MPLTEXT 1 0 15 "),SolPfunction(" }{MPLTEXT 1 0 5 "1,-1," } {MPLTEXT 1 0 7 "sigma,-" }{MPLTEXT 1 0 7 "sigma,0" }{MPLTEXT 1 0 7 ")) )):\n" }{MPLTEXT 1 0 13 "tdAReduced2;\n" }{MPLTEXT 1 0 66 "factor(2*c heckP*checkQ-2*checkP*sigma+2*checkQ^2-2*checkQ*sigma);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_sy slibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I&mfracGF$6) -F,6#-F,6*-I#moGF$6.Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvaria ntGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFJ/%)stretchyGFJ/%*symmetr icGFJ/%(largeopGFJ/%.movablelimitsGFJ/%'accentGFJ/%'lspaceGQ,0.2222222 emF'/%'rspaceGFY-F,6%-I#miGF$6&Q$S2|irF'/%'italicGQ%trueF'FB/FFQ'itali cF'-F?6-Q1⁢F'FEFHFKFMFOFQFSFU/FXQ&0.0emF'/FenFeo-Fin6&Q (σF'/F]oFJFBFE-F?6.Q(−F'FBFEFHFKFMFOFQFSFUFWFZ-F,6%FhnFao- Fin6&Q%ξF'FjoFBFE-F?6.Q\"+F'FBFEFHFKFMFOFQFSFUFWFZ-Fin6&Q#S1F'F\\oF BF_oF[p-Fin6&Q'checkQF'F\\oFBF_o-F,6#Fhn/%.linethicknessGQ\"1F'/%+deno malignGQ'centerF'/%)numalignGFcq/%)bevelledGFJFB/%)rowalignGQ!F'/%,col umnalignGFjq/%+groupalignGFjq/%(rowspanGF`q/%+columnspanGF`q-F56(-I#mn GF$6%F`qFBFEFhqF[rF]rF_rFarFhqF[rF]r-F26'-F56(-F86)-F,6#-F,64F>-F,6)-F fr6%Q\"2F'FBFEFaoFfpFao-I%msupGF$6%FhnFds/%1superscriptshiftGQ\"0F'Fao -Fin6&Q'checkPF'F\\oFBF_oFcp-F,6)FdsFaoFgsFaoF]tFaoFipF[p-F,6)FdsFaoFf pFaoFhnFaoFgoFcp-F,6%FgsFao-Fin6&Q$s10F'F\\oFBF_oF[p-F,6%FgsFao-Fin6&Q $s20F'F\\oFBF_oFcp-F,6)FdsFaoFhnFaoFipFaoFgoFcp-F,6%FdsFao-Fhs6%FfpFds FjsF[p-F,6'-Ffr6%Q\"4F'FBFEFaoFfpFaoFipFcp-F,6%FdsFao-Fhs6%FipFdsFjs-F ,6#FgsF^qFaqFdqFfqFBFhqF[rF]rF_rFar-F56(-F,6$F>F7FhqF[rF]rF_rFarFhqF[r F]r/%&alignGQ%axisF'/FiqQ)baselineF'/F\\rFcq/F^rQ'|frleft|hrF'/%/align mentscopeGF^o/%,columnwidthGQ%autoF'/%&widthGF_w/%+rowspacingGQ&1.0exF '/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFjw/%&fr ameGFjw/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFJ/%-equalcolumnsGF J/%-displaystyleGFJ/%%sideGQ&rightF'/%0minlabelspacingGFgwFBFE/%%openG Q\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&,&I'checkQG 6\"\"\"\"I&sigmaGF&!\"\"F',&I'checkPGF&F'F%F'F'\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "tdR2function:=unapply(tdR2(xi)," } {MPLTEXT 1 0 8 "s12,s22," }{MPLTEXT 1 0 13 "s11,s21,X1):\n" }{MPLTEXT 1 0 13 "tdR2function(" }{MPLTEXT 1 0 5 "1,-1," }{MPLTEXT 1 0 17 "sigma ,-sigma,0);\n" }{MPLTEXT 1 0 35 "R1function:=unapply(R1(xi),s12,s22," }{MPLTEXT 1 0 13 "s11,s21,X1):\n" }{MPLTEXT 1 0 11 "R1function(" } {MPLTEXT 1 0 22 "1,-1,sigma,-sigma,0);\n" }{MPLTEXT 1 0 37 "L22functio n:=unapply(simplify(L[2,2])" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 8 "s12, s22," }{MPLTEXT 1 0 10 "s11,s21,X1" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 21 "simplify(L22function(" }{MPLTEXT 1 0 32 "1,-1,sigma,-sigma,0)- h/(xi-Q));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 96 "A11function:=unap ply( simplify(A[1,1]), s12,s22,s11,s21,X1,beta12,beta22,beta11,beta21, betaX1):\n" }{MPLTEXT 1 0 96 "A12function:=unapply( simplify(A[1,2]), \+ s12,s22,s11,s21,X1,beta12,beta22,beta11,beta21,betaX1):\n" }{MPLTEXT 1 0 96 "A21function:=unapply( simplify(A[2,1]), s12,s22,s11,s21,X1,bet a12,beta22,beta11,beta21,betaX1):\n" }{MPLTEXT 1 0 95 "A22function:=un apply( simplify(A[2,2]), s12,s22,s11,s21,X1,beta12,beta22,beta11,beta2 1,betaX1):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "A11function(" } {MPLTEXT 1 0 30 "1,-1,sigma,-sigma,0,0,0,1,-1,0" }{MPLTEXT 1 0 2 ");" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "A12function(" }{MPLTEXT 1 0 33 "1,-1,sigma,-sigma,0,0,0,1,-1,0);\n" }{MPLTEXT 1 0 9 "simplify(" } {MPLTEXT 1 0 12 "A21function(" }{MPLTEXT 1 0 34 "1,-1,sigma,-sigma,0,0 ,0,1,-1,0)):\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 45 "A22funct ion(1,-1,sigma,-sigma,0,0,0,1,-1,0)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 15 "factor(residue(" }{MPLTEXT 1 0 21 "simplify(A21function(" } {MPLTEXT 1 0 40 "1,-1,sigma,-sigma,0,0,0,1,-1,0)),xi=0));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 102 "A21bis:=P*(P*Q-sX10-sX20)/(xi-Q)+sX10*sX 20/Q/xi+xi^2+xi*Q+2*xi*sigma+Q^2+2*Q*sigma+sigma^2-h+s10-s20;\n" } {MPLTEXT 1 0 7 "factor(" }{MPLTEXT 1 0 44 "A21function(1,-1,sigma,-sig ma,0,0,0,1,-1,0)-" }{MPLTEXT 1 0 9 "A21bis);\n" }{MPLTEXT 1 0 1 "\n" } }{PARA 11 "" 1 "" {XPPMATH 20 ",.*(I%sX10G6\"\"\"\",(I$s10GF%!\"\"I$s2 0GF%F)F$F)F&I#xiGF%!\"#F&F(F)*$I&sigmaGF%\"\"#F)F*F&*&F.F&F+F&F,*$F+F/ F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I$s10G6\"!\"\"I$s20GF%F&\"\"\" I#xiGF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,(I$s10G6\"!\"\"I$s20GF%F &I\"hGF%F&\"\"\"I#xiGF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "*(I\"PG6\" \"\"\"I\"QGF$F%,&I#xiGF$!\"\"F&F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ", $*&I#xiG6\"\"\"\",&F$!\"\"I\"QGF%F&F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*(I%sX10G6\"\"\"\",(F$F&I$s10GF%F&I$s20GF%F&F&I\"QGF%!\"\"F+" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I'A21bisG6\",8*(I\"PGF$\"\"\",(*&F'F(I \"QGF$F(F(I$s10GF$F(I$s20GF$F(F(,&I#xiGF$F(F+!\"\"F0F(**I%sX10GF$F(,(F ,F0F-F0F2F0F(F+F0F/F0F(*$F/\"\"#F(*&F/F(F+F(F(*&I&sigmaGF$F(F/F(F5*$F+ F5F(*&F+F(F8F(F5*$F8F5F(I\"hGF$F0F,F(F-F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "simpl ify(LQfunction(" }{MPLTEXT 1 0 13 "1,-1,X1,-X1,0" }{MPLTEXT 1 0 11 ",0 ,0,1,-1,0" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 20 "simplify(LPfuncti on(" }{MPLTEXT 1 0 13 "1,-1,X1,-X1,0" }{MPLTEXT 1 0 11 ",0,0,1,-1,0" } {MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 29 "simplify(Hamiltonianfunction( " }{MPLTEXT 1 0 13 "1,-1,X1,-X1,0" }{MPLTEXT 1 0 11 ",0,0,1,-1,0" } {MPLTEXT 1 0 3 "));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I\"PG6\"\"\" \"I\"QGF%F&\"\"#I$s10GF%F&I$s20GF%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 " *&,**$I\"QG6\"\"\"%\"\"$*&F%F(I#X1GF&\"\"\"F'*&,,*$I\"PGF&\"\"#!\"\"*$ F*F0F+I\"hGF&F1I$s10GF&F+I$s20GF&F1F+F%F0F+*&I%sX10GF&F+,(F7F+F4F+F5F+ F+F1F+F%!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,,*$I\"QG6\"\"\"%!\"\" *&F%\"\"$I#X1GF&\"\"\"!\"#*&,,*$I\"PGF&\"\"#F,*$F+F2F(I\"hGF&F,I$s10GF &F(I$s20GF&F,F,F%F2F,*(,&F5F,F6F,F,F1F,F%F,F,*&I%sX10GF&F,,(F:F,F5F,F6 F,F,F(F,F%F(" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 256 89 "Case of t_\{1, 2\}=1, t_\{2,2\}=-1 , t_\{1,1\}=0, t_\{2,1\}=0 and sigma=X_1 position of the pole." }}{EXCHG {PARA 0 "" 0 "" {TEXT 212 44 "We take s12=1 an d s22=-1 and s11=0 and s21=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "SolQfunction(1,-1,0,0,sigma);\n" }{MPLTEXT 1 0 38 "simplify(seri es(SolPfunction(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 12 ") ,checkP));\n" }{MPLTEXT 1 0 30 "tdX1functionfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 20 "S1func tion(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 3 ");\n" } {MPLTEXT 1 0 20 "S2function(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" } {MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 23 "R1function(xi,1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 29 "solve( \{SolQfunction(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 27 ")= QQ,SolPfunction(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 ") =PP\},\{checkQ,checkP\});\n" }{MPLTEXT 1 0 27 "tdLReduced:=Matrix(2,2, 0):\n" }{MPLTEXT 1 0 48 "tdL11function0:=unapply( tdL11function(1,-1,0 ,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 8 "),Q,P):\n" }{MPLTEXT 1 0 48 "tdL12function0:=unapply( tdL12function(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 8 "),Q,P):\n" }{MPLTEXT 1 0 48 "tdL21function0 :=unapply( tdL21function(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 8 "),Q,P):\n" }{MPLTEXT 1 0 48 "tdL22function0:=unapply( tdL22func tion(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 8 "),Q,P):\n" } {MPLTEXT 1 0 49 "tdLReduced[1,1]:=simplify(tdL11function(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 49 "tdLR educed[1,2]:=simplify(tdL12function(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 49 "tdLReduced[2,1]:=simplify( tdL21function(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 5 ") ): \n" }{MPLTEXT 1 0 49 "tdLReduced[2,2]:=simplify(tdL22function(1,-1,0,0 ," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 5 ") ):\n" }{MPLTEXT 1 0 12 " tdLReduced;\n" }{MPLTEXT 1 0 28 "tdLReduced2:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 64 "tdLReduced2[1,1]:=simplify(tdL11function0(SolQfunctio n(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 "),SolPfunction( 1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 6 ")) ):\n" }{MPLTEXT 1 0 64 "tdLReduced2[1,2]:=simplify(tdL12function0(SolQfunction(1,-1,0 ,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 "),SolPfunction(1,-1,0,0 ," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 6 ")) ):\n" }{MPLTEXT 1 0 64 "tdLReduced2[2,1]:=simplify(tdL21function0(SolQfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 "),SolPfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 6 ")) ):\n" }{MPLTEXT 1 0 64 "tdL Reduced2[2,2]:=simplify(tdL22function0(SolQfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 "),SolPfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 6 ")) ):\n" }{MPLTEXT 1 0 13 "tdL Reduced2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I'checkQG6\"\"\"\"I#S 1GF%!\"\"F&I$S2|irG6#/I'_localG%*protectedGI0_n1277212501088GF%F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "+'I'checkPG6\"*(,&I$s10GF$\"\"\"I$s20GF$ F(F(I$S2|irG6#/I'_localG%*protectedGI0_n1277212501088GF$F(,(*(\"\"#F(F *F(I&sigmaGF$F(F(*&F2F(I#S1GF$F(F(*&F2F(I'checkQGF$F(!\"\"F8\"\"!F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "I&sigmaG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I#S1G6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I#miG6#/I+modu lenameG6\"I,TypesettingGI(_syslibGF'6%Q$S2|irF'/%'italicGQ%trueF'/%,ma thvariantGQ'italicF'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&,&I$s10G6\"! \"\"I$s20GF&F'\"\"\"I#xiGF&F'F)F)F)*&,&F*F'F)F'F)F*F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "<$/I'checkPG6\",$**#\"\"\"\"\"#F),**(F*F)I#PPGF%F)I #QQGF%F)F)*(F*F)F-F)I&sigmaGF%F)!\"\"I$s10GF%F)I$s20GF%F)F)I$S2|irG6#/ I'_localG%*protectedGI0_n1277212501088GF%F1,&F.F)F0F1F1F)/I'checkQGF%, &*&F.F)F4F)F)I#S1GF%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(-I&mfracGF$6)-F,6#-F,6'-F,6%-F#6%-F,6(-I#moGF$6. Q*&uminus0;F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%&fenc eGQ&falseF'/%*separatorGFP/%)stretchyGFP/%*symmetricGFP/%(largeopGFP/% .movablelimitsGFP/%'accentGFP/%'lspaceGQ,0.2222222emF'/%'rspaceGFin-I# miGF$6&Q\"QF'/%'italicGQ%trueF'FH/FLQ'italicF'-FE6.Q\"+F'FHFKFNFQFSFUF WFYFenFgnFjn-F]o6&Q%ξF'/FaoFPFHFK-FE6.Q(−F'FHFKFNFQFSFUFWFYFe nFgnFjn-F]o6&Q\"PF'F`oFHFcoFHFK-FE6-Q1⁢F'FKFNFQFSFUFWFY Fen/FhnQ&0.0emF'/F[oFfp-F]o6&Q(σF'F[pFHFKF\\p-F,6#-I%msupGF$6%Fh o-I#mnGF$6%Q\"2F'FHFK/%1superscriptshiftGQ\"0F'Feo-F,6%F\\oFbp-F#6%-F, 6%F\\oFeoF_pFHFK-F,6#-F,6%FhoF\\pFhp/%.linethicknessGQ\"1F'/%+denomali gnGQ'centerF'/%)numalignGFfr/%)bevelledGFPFH/%)rowalignGQ!F'/%,columna lignGF]s/%+groupalignGF]s/%(rowspanGFcr/%+columnspanGFcr-F56(-F86)-F,6 #-F,6&FDFhoFeoF\\o-F,6#-F,6&FDFhoFeoFhpFarFdrFgrFirFHF[sF^sF`sFbsFdsF[ sF^sF`s-F26'-F56(-F86)-F,6#-F,61-F,6#-F^q6%F\\o-Faq6%Q\"4F'FHFKFdqFeo- F,6%-F#6%-F,6'-F,6%F`qFbpF_pF\\p-F,6%F`uFbpFhpFeo-F,6%F`qFbpFhoFHFKFbp -F^q6%F\\o-Faq6%Q\"3F'FHFKFdqFeo-F,6%-F#6%-F,6--F,6%-Faq6%Q\"5F'FHFKFb p-F^q6%FhpF`qFdqFeo-F,6%-F#6%-F,6&FD-F,6%-Faq6%Q\"6F'FHFKFbpF_pF\\p-F, 6%F`uFbpFhoFHFKFbpFhpFeo-F,6#-F^q6%F_pF`qFdqFeo-F,6'F`qFbpFhoFbpF_pFeo -F]o6&Q$s10F'F`oFHFcoFeo-F]o6&Q$s20F'F`oFHFcoFHFKFbp-F^q6%F\\oF`qFdqFe o-F,6%-F#6%-F,6,FD-F,6%F`qFbp-F^q6%FhpFavFdqFeo-F,6%-F#6%-F,6%FgwFeoF] vFHFKFbpF_wFeo-F,6%-F#6%-F,6*FD-F,6%F`qFbpF`xF\\p-F,6'F`uFbpF_pFbpFhoF \\p-F,6%FavFbpFdxF\\pFgxFHFKFbpFhpFeo-F,6%-F#6%-F,6%FdxFeoFgxFHFKFbpF_ pFeo-F,6'F`qFbpFdxFbpFhoFHFKFbpF\\oF\\p-F,6'F`qFbpFdyFbpF_pFeo-F,6%-F# 6%-F,6'F^xFeo-F,6'F`qFbpF_pFbpFhoFeo-F,6%F`qFbpFdxFHFKFbpF_wFeo-F,6%-F #6%-F,6&FDF^[lFeo-F,6%F_pFbp-F#6%-F,6&FDFdxF\\pFgxFHFKFHFKFbpFhpF\\p-F ,6%-F]o6&Q%sX10F'F`oFHFcoFbp-F#6%-F,6'Fj\\lFeoFdxFeoFgxFHFK-F,6%-F#6%F _rFHFKFbp-F#6%-F,6%F\\oF\\pFhpFHFKFarFdrFgrFirFHF[sF^sF`sFbsFds-F56(-F 86)-F,6#-F,6--F,6%-F#6%-F,6'F\\oF\\pFhoFeoF_pFHFKFbpFhpF\\p-F,6%F_pFbp F\\oF\\p-F,6#FjxFeoF[qF\\pFdxF\\pFgxF]rFarFdrFgrFirFHF[sF^sF`sFbsFdsF[ sF^sF`s/%&alignGQ%axisF'/F\\sQ)baselineF'/F_sFfr/FasQ'|frleft|hrF'/%/a lignmentscopeGFbo/%,columnwidthGQ%autoF'/%&widthGFg_l/%+rowspacingGQ&1 .0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGFb` l/%&frameGFb`l/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFP/%-equalco lumnsGFP/%-displaystyleGFP/%%sideGQ&rightF'/%0minlabelspacingGF_`lFHFK /%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfen cedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtab leGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"1F'/%+fore groundGQ([0,0,0]F'/%,mathvariantGQ'normalF'-F=6%Q\"2F'F@FC/%.linethick nessGF?/%+denomalignGQ'centerF'/%)numalignGFM/%)bevelledGQ&falseF'F@-I #moGF$6-Q1⁢F'FC/%&fenceGFR/%*separatorGFR/%)stretchyGFR /%*symmetricGFR/%(largeopGFR/%.movablelimitsGFR/%'accentGFR/%'lspaceGQ &0.0emF'/%'rspaceGFao-F:6)-F,6#-F,6*-FT6.Q*&uminus0;F'F@FCFWFYFenFgnFi nF[oF]o/F`oQ,0.2222222emF'/FcoF^p-F,6)FFFS-I%msupGF$6%-I#miGF$6&Q$S2|i rF'/%'italicGQ%trueF'F@/FDQ'italicF'-F=6%Q\"3F'F@FC/%1superscriptshift GQ\"0F'FS-Ffp6&Q(σF'/FjpFRF@FCFS-Ffp6&Q'checkPF'FipF@F\\q-FT6.Q \"+F'F@FCFWFYFenFgnFinF[oF]oF]pF_p-F,6%-F#6%-F,6.Fjo-F,6'FFFS-Ffp6&Q#S 1F'FipF@F\\qFSFhqF[r-F,6'FFFSFhqFS-Ffp6&Q'checkQF'FipF@F\\qF[r-F,6'FFF SFdqFS-Ffp6&Q%ξF'FgqF@FC-FT6.Q(−F'F@FCFWFYFenFgnFinF[oF]oF]pF _p-F,6%FFFS-Fcp6%F`sFFFaqFcs-Ffp6&Q$s10F'FipF@F\\qFcs-Ffp6&Q$s20F'FipF @F\\qF@FCFS-Fcp6%FepFFFaqF[r-F,6)FFFSFdqFS-F#6%-F,6%FfrFcsF[sF@FCFSFep F[r-F,6%FFFS-Fcp6%FdtFFFaq-F,6%-F#6%-F,6%F`sFcsFdqF@FCFSF`tFIFKFNFPF@/ %)rowalignGQ!F'/%,columnalignGFdu/%+groupalignGFdu/%(rowspanGF?/%+colu mnspanGF?-F56(-F:6)-F,6#-F,6'-F,6%FepFSF`sF[rFfrFcsF[s-F,6%F^uFSFepFIF KFNFPF@FbuFeuFguFiuF[vFbuFeuFgu-F26'-F56(-F,6%-F:6)F<-F=6%Q\"4F'F@FCFI FKFNFPF@FS-F:6)-F,6#-F,60Fjo-F,6)FawFS-Fcp6%Fep-F=6%Q\"6F'F@FCFaqFS-Fc p6%FhqFFFaqFS-Fcp6%FdqFFFaqFcs-F,6+-F=6%Q\"8F'F@FCFSFhqFSFdqFS-F#6%-F, 6(Fjo-F,6#FcxF[r-F,6%FdqFSF`sF[r-F,6%FhqFSFdtF@FCFS-Fcp6%Fep-F=6%Q\"5F 'F@FCFaqF[r-F,6%-F#6%-F,6'-F,6%-F#6%-F,6)-F,6'-F=6%Q#24F'F@FCFSFfrFSFh qFcs-F,6'FgzFSFhqFSF[sFcs-F,6%FawFSFjsF[r-F,6%FawFSF]tF@FCFSFcxFcs-F,6 )-F=6%Q#16F'F@FCFS-F#6%-F,6)-F,6%FfrFSFhqFcs-F,6%FhqFSF[sFcs-F,6%F_wFS -F,6#FjsF[r-F,6%F_wFS-F,6#F]tF@FCFSF`sFSFdqFcs-F,6'FawFS-F#6%-F,6+Fi[l FcsF[\\lF[r-Ffp6&Q%sX10F'FipF@F\\qF[r-F,6%F9FSF_\\lF[r-F,6%F9FSFc\\lF@ FCFS-F#6%-F,6+Fi[lFcsF[\\lFcsF[]lFcsF^]lFcsF`]lF@FCF@FCFS-Fcp6%FepFawF aqFcs-F,6)FgxFSFdtFS-F#6%-F,6)-F,6#-Fcp6%FdqF^qFaqFcs-F,6%FcxFSF`sF[r- F,6%-F#6%-F,6*Fjo-F,6'F^qFSFfrFSFhqF[r-F,6'F^qFSFhqFSF[sF[rF^]lFcsF`]l F@FCFSFdqF[r-F,6%F`sFS-F#6%-F,6)Fi[lFcsF[\\lFcsF^]lF[rF`]lF@FCF@FCFSFb pF[r-F,6)FgxFSFjtFS-F#6%-F,6(Fjo-F,6%-F:6)FfyFFFIFKFNFPF@FSF^yF[rF^sF[ rFbyF@FCFSF`tF[r-F,6)FgxFS-Fcp6%FdtF^qFaqFS-F#6%-F,6%F`sFcs-F,6%FFFSFd qF@FCFSFepFcs-F,6%FawFS-Fcp6%FdtFawFaq-F,6'FbpFSF^uFS-F#6%-F,6'-F,6%Fe pFSFdqF[rFfrFcsF[sF@FCFIFKFNFPF@FbuFeuFguFiuF[v-F56(-F,6%F9FS-F:6)-F,6 #-F,6)F`pF[r-F,6%-F#6%-F,6-FdrFcsFirFcsF^sF[rFfsFcsFjsFcsF]tF@FCFSF`tF csFbtFcsFhtF\\uFIFKFNFPF@FbuFeuFguFiuF[vFbuFeuFgu/%&alignGQ%axisF'/Fcu Q)baselineF'/FfuFM/FhuQ'|frleft|hrF'/%/alignmentscopeGF[q/%,columnwidt hGQ%autoF'/%&widthGF`cl/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8em F'/%)rowlinesGQ%noneF'/%,columnlinesGF[dl/%&frameGF[dl/%-framespacingG Q,0.4em~0.5exF'/%*equalrowsGFR/%-equalcolumnsGFR/%-displaystyleGFR/%%s ideGQ&rightF'/%0minlabelspacingGFhclF@FC/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "tdAReduced:=Matrix(2,2,0 ):\n" }{MPLTEXT 1 0 48 "tdA11function0:=unapply( tdA11function(1,-1,0, 0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 18 ",0,0,0,0,1),Q,P):\n" } {MPLTEXT 1 0 48 "tdA12function0:=unapply( tdA12function(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 18 ",0,0,0,0,1),Q,P):\n" } {MPLTEXT 1 0 48 "tdA21function0:=unapply( tdA21function(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 18 ",0,0,0,0,1),Q,P):\n" } {MPLTEXT 1 0 48 "tdA22function0:=unapply( tdA22function(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 18 ",0,0,0,0,1),Q,P):\n" } {MPLTEXT 1 0 49 "tdAReduced[1,1]:=simplify(tdA11function(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 15 ",0,0,0,0,1) ):\n" }{MPLTEXT 1 0 49 "tdAReduced[1,2]:=simplify(tdA12function(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 15 ",0,0,0,0,1) ):\n" }{MPLTEXT 1 0 49 "td AReduced[2,1]:=simplify(tdA21function(1,-1,0,0," }{MPLTEXT 1 0 5 "sigm a" }{MPLTEXT 1 0 15 ",0,0,0,0,1) ):\n" }{MPLTEXT 1 0 49 "tdAReduced[2, 2]:=simplify(tdA22function(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" } {MPLTEXT 1 0 15 ",0,0,0,0,1) ):\n" }{MPLTEXT 1 0 12 "tdAReduced;\n" } {MPLTEXT 1 0 28 "tdAReduced2:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 71 "tdAR educed2[1,1]:=factor(simplify(tdA11function0(SolQfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 "),SolPfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 7 ")) )):\n" }{MPLTEXT 1 0 71 "td AReduced2[1,2]:=factor(simplify(tdA12function0(SolQfunction(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 "),SolPfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 7 ")) )):\n" }{MPLTEXT 1 0 71 "td AReduced2[2,1]:=factor(simplify(tdA21function0(SolQfunction(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 "),SolPfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 7 ")) )):\n" }{MPLTEXT 1 0 71 "td AReduced2[2,2]:=factor(simplify(tdA22function0(SolQfunction(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 24 "),SolPfunction(1,-1,0,0," } {MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 7 ")) )):\n" }{MPLTEXT 1 0 12 "td AReduced2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'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'/%,math variantGQ'normalF'/%&fenceGQ&falseF'/%*separatorGFE/%)stretchyGFE/%*sy mmetricGFE/%(largeopGFE/%.movablelimitsGFE/%'accentGFE/%'lspaceGQ,0.22 22222emF'/%'rspaceGFT-I&mfracGF$6)-F,6%-F#6%-F,6%-I#miGF$6&Q\"QF'/%'it alicGQ%trueF'F=/FAQ'italicF'-F:6.Q(−F'F=F@FCFFFHFJFLFNFPFRFU-F[o 6&Q(σF'/F_oFEF=F@F=F@-F:6-Q1⁢F'F@FCFFFHFJFLFNFP/F SQ&0.0emF'/FVF^p-F#6%-F,6)Fjn-F:6.Q\"+F'F=F@FCFFFHFJFLFNFPFRFU-F[o6&Q \"PF'F^oF=FaoFdp-F[o6&Q%ξF'FioF=F@FcoFfoF=F@-F,6#-F,6&F9FfoFdpFjp/% .linethicknessGQ\"1F'/%+denomalignGQ'centerF'/%)numalignGFfq/%)bevelle dGFEF=/%)rowalignGQ!F'/%,columnalignGF]r/%+groupalignGF]r/%(rowspanGFc q/%+columnspanGFcq-F56(-FX6)-F,6#FhnF]qFaqFdqFgqFiqF=F[rF^rF`rFbrFdrF[ rF^rF`r-F26'-F56(-F,6$F9-FX6)-F,6%-F#6%-F,6)-F,6#-I%msupGF$6%Fjn-I#mnG F$6%Q\"2F'F=F@/%1superscriptshiftGQ\"0F'Fdp-F,6%-F#6%-F,6&F9FfoFdpFgpF =F@FjoFjnFco-F,6%FgpFjoFfoFco-F[o6&Q%sX20F'F^oF=FaoF=F@Fjo-F#6%-F,6)Fj sFdpFftFcoF\\uFco-F[o6&Q%sX10F'F^oF=FaoF=F@-F,6%-F#6%F_qF=F@FjoFfnFaqF dqFgqFiqF=F[rF^rF`rFbrFdr-F56(-FX6)-F,6#-F,6--F,6#-F]t6%FfoF_tFctFdp-F ,6%-F#6%-F,6(F9FgpFco-F,6%F_tFjoFjnFcoFjpF=F@FjoFfoFdpFjsFdp-F,6%-F#6% -F,6%FjpFdpFgpF=F@FjoFjnFcoFeuFcoF^uF]qFaqFdqFgqFiqF=F[rF^rF`rFbrFdrF[ rF^rF`r/%&alignGQ%axisF'/F\\rQ)baselineF'/F_rFfq/FarQ'|frleft|hrF'/%/a lignmentscopeGF`o/%,columnwidthGQ%autoF'/%&widthGFbx/%+rowspacingGQ&1. 0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF]y/ %&frameGF]y/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFE/%-equalcolum nsGFE/%-displaystyleGFE/%%sideGQ&rightF'/%0minlabelspacingGFjxF=F@/%%o penGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG 6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF $66-I$mtrGF$6'-I$mtdGF$6(-F,6%-I&mfracGF$6)-I#mnGF$6%Q\"1F'/%+foregrou ndGQ([0,0,0]F'/%,mathvariantGQ'normalF'-F=6%Q\"2F'F@FC/%.linethickness GF?/%+denomalignGQ'centerF'/%)numalignGFM/%)bevelledGQ&falseF'F@-I#moG F$6-Q1⁢F'FC/%&fenceGFR/%*separatorGFR/%)stretchyGFR/%*s ymmetricGFR/%(largeopGFR/%.movablelimitsGFR/%'accentGFR/%'lspaceGQ&0.0 emF'/%'rspaceGFao-F:6)-F,6#-F,63-F,6'FFFS-I#miGF$6&Q'checkPF'/%'italic GQ%trueF'F@/FDQ'italicF'FS-F]p6&Q'checkQF'F`pF@Fcp-FT6.Q(−F'F@FC FWFYFenFgnFinF[oF]o/F`oQ,0.2222222emF'/FcoF\\q-F,6'FFFSF\\pFS-F]p6&Q(& sigma;F'/FapFRF@FC-FT6.Q\"+F'F@FCFWFYFenFgnFinF[oF]oF[qF]q-F,6%FFFS-I% msupGF$6%FepFF/%1superscriptshiftGQ\"0F'Fhp-F,6'-F=6%Q\"4F'F@FCFSFepFS F`qFdq-F,6'FFFSFepFS-F]p6&Q%ξF'FcqF@FCFdq-F,6%FFFS-Fjq6%F`qFFF\\rFh p-F,6'FFFSF`qFSFfrFdq-F]p6&Q%sX10F'F`pF@FcpFdq-F]p6&Q%sX20F'F`pF@Fcp-F ,6#-F,6&-FT6.Q*&uminus0;F'F@FCFWFYFenFgnFinF[oF]oF[qF]qFfrFdqF`qFIFKFN FPF@/%)rowalignGQ!F'/%,columnalignGF^t/%+groupalignGF^t/%(rowspanGF?/% +columnspanGF?-F56(-F,6$Fis-F:6)-F,6#-F,6%FepFhpF`qFesFIFKFNFPF@F\\tF_ tFatFctFetF\\tF_tFat-F26'-F56(-F,6%-F:6)F " 0 "" {MPLTEXT 1 0 29 "simplify(LQfunction(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 14 ",0,0,0,0,1));\n" }{MPLTEXT 1 0 29 "simplify(LPfunction(1,-1,0, 0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 14 ",0,0,0,0,1));\n" } {MPLTEXT 1 0 38 "simplify(Hamiltonianfunction(1,-1,0,0," }{MPLTEXT 1 0 5 "sigma" }{MPLTEXT 1 0 13 ",0,0,0,0,1));" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&,&*&\"\"#\"\"\"I\"QG6\"F'F'*&F&F'I&sigmaGF)F'!\"\"F'I \"PGF)F'F'I\"hGF)F'I%sX10GF)F,I%sX20GF)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,.*&\"\"$\"\"\")I\"QG6\"\"\"%F&F&*(\"\")F&)F(F%F&I&sigm aGF)F&!\"\"*&,,*$)I\"PGF)\"\"#F&F/*&\"\"(F&)F.F5F&F&I\"hGF)F/I$s10GF)F &I$s20GF)F/F&)F(F5F&F&**F5F&F.F&,,F2F&*$F8F&F/F9F&F:F/F;F&F&F(F&F&*&,* F2F/F9F/F:F&F;F/F&F8F&F&*&I%sX10GF)F&I%sX20GF)F&F&F&),&F(F&F.F/F5F/" } }{PARA 11 "" 1 "" {XPPMATH 20 "*&,0*$)I\"QG6\"\"\"%\"\"\"!\"\"*(\"\"#F ))F&\"\"$F)I&sigmaGF'F)F)*&,,*$)I\"PGF'F,F)F)*$)F/F,F)F*I\"hGF'F)I$s10 GF'F*I$s20GF'F)F))F&F,F)F)*&,&*&,**&F,F)F3F)F**&F,F)F7F)F**&F,F)F8F)F) *&F,F)F9F)F*F)F/F)F)*&,(I%sX10GF'F*I%sX20GF'F*F7F)F)F4F)F)F)F&F)F)*&,* F2F)F7F)F8F*F9F)F)F6F)F)*(F4F)FDF)F/F)F**&FEF)FFF)F)F),&F&F)F/F*F*" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{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 }