{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 }{CSTYLE "" -1 256 "" 1 24 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 24 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 24 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 24 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 24 0 0 0 0 1 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 1 24 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 24 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "Times" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 288 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 1 14 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 296 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 301 "" 1 18 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 302 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 303 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 304 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 305 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 306 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 307 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 308 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 309 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 310 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 311 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 312 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 313 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 314 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 316 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 317 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 318 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 319 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 320 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 321 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 322 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 323 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 324 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 326 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 327 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 328 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 329 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 330 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 331 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 332 "" 1 24 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 333 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 334 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 335 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 336 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 337 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 338 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 339 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 340 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 341 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 342 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 343 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 344 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 345 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 346 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 347 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 348 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 349 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 350 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 351 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 352 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 353 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 354 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 355 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 356 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 357 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 358 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 359 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 360 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 361 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " 18 362 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 363 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 364 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 365 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 366 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 367 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 368 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 369 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 370 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 371 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 372 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 373 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 374 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 375 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 376 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 377 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 378 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 379 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 380 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 381 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 382 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 383 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 384 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 385 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 386 "" 1 14 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 387 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 388 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 389 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 390 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 391 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 392 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 393 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 394 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 395 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 396 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 397 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 398 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 399 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 400 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 401 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 402 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 403 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 404 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 405 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 406 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 407 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 408 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 409 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 410 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 411 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 412 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 413 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 414 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 415 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 416 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 417 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 418 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 419 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 420 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 421 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 422 "" 0 18 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 423 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 424 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 425 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 18 426 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 427 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 428 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 429 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 430 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 431 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 432 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 433 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 434 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 435 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 436 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 437 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 438 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 439 "" 1 24 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 440 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 441 "" 1 14 0 0 0 0 1 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 442 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 443 "" 1 14 0 0 0 0 1 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 444 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 445 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 446 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 447 "" 1 24 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE " " -1 448 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 449 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" 18 450 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" 18 451 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 } {CSTYLE "" -1 452 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 453 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 454 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 455 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 456 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 457 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 458 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" 18 459 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 460 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" 18 461 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 462 "" 1 14 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 463 "" 1 24 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 464 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 465 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 466 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 467 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 468 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 469 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 470 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 471 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 472 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 473 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 474 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 475 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 476 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 477 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 478 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 479 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 480 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 481 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 482 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 483 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 484 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 485 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 486 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Norm al" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "Map Symbols" 1 18 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 " " 1 18 0 0 0 0 1 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 269 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 270 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 " " 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 280 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 283 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 3 284 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 285 1 {CSTYLE "" -1 -1 " " 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 286 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 287 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 288 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 290 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 291 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 292 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 294 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 295 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 296 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 297 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 298 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 299 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 300 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 301 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 302 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 303 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 0 304 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 305 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 306 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG }{EXCHG {PARA 18 "" 0 "" {TEXT 265 51 "Maple procedures for combined asymptotic expa nsions" }}}{SECT 0 {PARA 256 "" 0 "" {TEXT 260 49 "Preliminary remark \+ on a bug of Maple version V.4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "" 0 "" {TEXT 257 11 "The command" }{TEXT 418 2 " " }{TEXT 258 2 " " }{XPPEDIT 416 0 " int ( f , x= a .. infin ity) " "-%$intG6$%\"fG/%\"xG;%\"aG%)infinityG" }{TEXT 417 4 " " } {TEXT 419 55 "produces sometimes a bug then, it will be replaced by " }{TEXT 420 1 " " }{XPPEDIT 421 0 "limit(int(f,x=a..y),y=infinity)" " -%&limitG6$-%$intG6$%\"fG/%\"xG;%\"aG%\"yG/F-%)infinityG" }{TEXT 310 15 " for example :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := 1 / (1+ exp(x)) / ln(2); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "wrong_result:= int(f,x=0..infinity);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "exact_resu lt:=limit(int(f,x=0..y),y=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fG*&,&\"\"\"F'-%$expG6#%\"xGF'!\"\"-%#lnG6#\"\"#F," }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%-wrong_resultG-%#lnG6#\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%-exact_resultG\"\"\"" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 257 "" 0 "" {TEXT 261 13 "Introduction " }} {EXCHG {PARA 0 "" 0 "" {TEXT 266 2 "1)" }{TEXT 267 2 " " }{TEXT 284 56 "An asymptotic expansion (abbreviation : DA, from french " }{TEXT 293 26 "D\351veloppement Asymptotique" }{TEXT 294 17 ") is implemented " }{TEXT 269 2 " " }{TEXT 285 1 " " }}{PARA 258 "" 0 "" {TEXT 268 1 "[" }{XPPEDIT 270 0 "epsilon" "I(epsilonG6\"" }{TEXT 271 4 " , [" } {XPPEDIT 272 0 "f[0], f[1]" "6$&%\"fG6#\"\"!&F$6#\"\"\"" }{TEXT 273 7 ", ..., " }{XPPEDIT 274 0 "f[N-1]" "&%\"fG6#,&%\"NG\"\"\"\"\"\"!\"\"" }{TEXT 275 2 "]]" }}{PARA 261 "" 0 "" {TEXT -1 7 "where " }{TEXT 278 1 " " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT 276 3 " " } {TEXT -1 37 "is a name of an unevaluated variable," }}{PARA 259 "" 0 " " {TEXT 277 7 "where " }{TEXT -1 1 " " }{XPPEDIT 18 0 "f[i]" "&%\"fG6 #%\"iG" }{TEXT 280 35 " are expressions (independent of " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT 281 1 " " }{TEXT 282 1 ")" } {TEXT 279 1 "." }}{PARA 260 "" 0 "" {TEXT 283 93 "All the computations suppose that the studied object is a function with asymptotic expansi on " }{TEXT 422 1 ":" }}{PARA 262 "" 0 "" {XPPEDIT 18 0 "f[0] + f[ 1]*epsilon" ",&&%\"fG6#\"\"!\"\"\"*&&F$6#\"\"\"F'%(epsilonGF'F'" } {TEXT 286 14 " + ... + " }{XPPEDIT 18 0 " f[N-1]*epsilon^(N-1) \+ " "*&&%\"fG6#,&%\"NG\"\"\"\"\"\"!\"\"F()%(epsilonG,&F'F(\"\"\"F*F(" } {TEXT 287 5 "+ \243 " }{XPPEDIT 18 0 "epsilon^N" ")%(epsilonG%\"NG" } }{PARA 289 "" 0 "" {TEXT 328 1 "N" }{TEXT -1 37 " will be called the \+ order of the DA." }}{PARA 0 "" 0 "" {TEXT 288 2 "2)" }{TEXT 290 2 " \+ " }{TEXT 292 65 "A combined asymptotic expansion (abbreviation : DAC, \+ from french " }{TEXT 295 34 "D\351veloppement Asymptotique Combin\351 " }{TEXT 296 16 ") is implemented" }}{PARA 266 "" 0 "" {XPPEDIT 423 0 "[t=t[1]+epsilon*x,DAL,DAR] " "7%/%\"tG,&&F$6#\"\"\"\"\"\"*&%(epsilonG F)%\"xGF)F)%$DALG%$DARG" }}{PARA 264 "" 0 "" {TEXT -1 6 "where " } {TEXT 289 1 " " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "t , x" "6$%\"tG%\"xG" }{TEXT -1 54 " are names o f unevaluated variables with the relation" }{TEXT 291 4 " " } {XPPEDIT 424 0 "t=t[1]+epsilon*x" "/%\"tG,&&F#6#\"\"\"\"\"\"*&%(epsilo nGF(%\"xGF(F(" }}{PARA 265 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "DAL" "I$DALG6\"" }{TEXT -1 2 " (" }{TEXT 298 31 "D\351veloppement Asy mptotique Lent" }{TEXT -1 16 " : slow DA) and " }{XPPEDIT 18 0 "DAR" " I$DARG6\"" }{TEXT -1 2 " (" }{TEXT 299 33 "D\351veloppement Asymptotiq ue Rapide" }{TEXT -1 88 " : fast DA) are two asymptotic expansions imp lemented as before, with the same variable " }{TEXT 297 1 " " } {XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 95 ". Very often, th e order of the two asymptotic expansions will be the same. The coeffic ients of " }{XPPEDIT 18 0 "DAL" "I$DALG6\"" }{TEXT -1 11 " depend on \+ " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 26 ", and the coefficients o f " }{XPPEDIT 18 0 "DAR" "I$DARG6\"" }{TEXT -1 11 " depend on " } {XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 1 "\n" }{TEXT 300 103 "All the \+ computations suppose that the studied object is a function with combin ed asymptotic expansion :" }}{PARA 263 "" 0 "" {XPPEDIT 18 0 "f[0](t)+ f[1](t)*epsilon" ",&-&%\"fG6#\"\"!6#%\"tG\"\"\"*&-&F%6#\"\"\"6#F)F*%(e psilonGF*F*" }{TEXT -1 7 " +...+ " }{XPPEDIT 18 0 " f[N-1](t)*epsilon^ (N-1) " "*&-&%\"fG6#,&%\"NG\"\"\"\"\"\"!\"\"6#%\"tGF))%(epsilonG,&F(F )\"\"\"F+F)" }{MPLTEXT 0 21 5 " + " }{XPPEDIT 18 0 "g[0]((t-t[1])/ep silon) + g[1]((t-t[1])/epsilon)*epsilon " ",&-&%\"gG6#\"\"!6#*&,&% \"tG\"\"\"&F+6#\"\"\"!\"\"F,%(epsilonGF0F,*&-&F%6#\"\"\"6#*&,&F+F,&F+6 #\"\"\"F0F,F1F0F,F1F,F," }{TEXT -1 10 " + ... + " }{XPPEDIT 18 0 " g[ N-1]((t-t[1])/epsilon)*epsilon^(N-1) " "*&-&%\"gG6#,&%\"NG\"\"\"\"\" \"!\"\"6#*&,&%\"tGF)&F/6#\"\"\"F+F)%(epsilonGF+F))F3,&F(F)\"\"\"F+F)" }{MPLTEXT 0 21 6 " + \243 " }{XPPEDIT 18 0 "epsilon^N" ")%(epsilonG% \"NG" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 0 "" 0 " " {TEXT 301 52 "General bound of the order of asymptotic expansions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Nmax:=1000; " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%NmaxG\"%+5" }}}{PARA 267 "" 0 "" {TEXT -1 23 "Some useful procedures " }}{EXCHG {PARA 0 "" 0 "" {TEXT 311 17 "Simpl ification : " }{TEXT 344 59 "In some cases it is convenient to avoid t he Maple function " }{TEXT 302 8 "simplify" }{TEXT 345 50 ". For that \+ purpose, only a change of the function " }{TEXT 303 10 "simplifyEB" } {TEXT 346 16 " will be needed." }{TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "simplifyEB := proc(expr) simplify(expr); end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 312 18 "Taylor expansion :" }{TEXT 313 21 " The Maple function " }{TEXT 304 16 "taylor(f,var,N) " }{TEXT 347 42 " computes the Taylor series expansion of " }{TEXT 305 1 "f" } {TEXT 348 31 ", with respect to the variable " }{TEXT 306 3 "var" } {TEXT 349 15 ", up to order " }{TEXT 307 1 "N" }{TEXT 350 51 ". Unfo rtunately, the order is sometimes less than " }{TEXT 308 1 "N" }{TEXT 351 91 ". The following procedure makes computations to give the resu lt as a DA of order exactly " }{TEXT 309 1 "N" }{TEXT 352 1 "." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 578 "taylorEB := proc (f, var, N) # va r must be of type t=t0\n local tau,i,res,ordre_supplementaire; \n \+ res:=taylor(f,var,N);\n ordre_supplementaire := N-op(nops(res) ,res): # the last operand op(nops(\"),\") of a maple-series is the ord er ot the series.\n while (ordre_supplementaire>0 and op(nops(res) -1,res)=O(1) ) do \n res:=taylor(f,var, N+ordre_supplementair e):\n ordre_supplementaire := N-op(nops(res),res): od;\n \+ subs(lhs(var)=rhs(var)+tau,res); convert(\",polynom);\n [lhs(var) -rhs(var) , [seq ( simplifyEB(coeff(\",tau,i)) , i=0..N-1)]]; end:" }} {PARA 0 "" 0 "" {TEXT -1 9 "Example :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "f:=(1/(1+x**2)-1)/x**2;\nmaple_expansion_1 := taylor(f,x=0,6); \nmaple_expansion_2 := taylor(f,x=0,6);\nDA_expansion :=taylorEB(f,x=0 ,6);\nmaple_expansion_3 := taylor(f,x=0,6);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*&,&*$,&\"\"\"F)*$%\"xG\"\"#F)!\"\"F)F-F)F)F+!\"# " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2maple_expansion_1G+)%\"xG!\"\" \"\"!\"\"\"\"\"#-%\"OG6#F)\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2 maple_expansion_2G+)%\"xG!\"\"\"\"!\"\"\"\"\"#-%\"OG6#F)\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%-DA_expansionG7$%\"xG7(!\"\"\"\"!\" \"\"F)F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2maple_expansion_3G++% \"xG!\"\"\"\"!\"\"\"\"\"#F'\"\"%-%\"OG6#F)\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 314 37 "Change of unknowns for a vector fie ld" }{TEXT -1 5 "\n " }{TEXT 353 122 " eq is a list o f expressions: the components of the vector field, or the right hand \+ sides of the equations " }{XPPEDIT 354 0 "diff(x(t),t)=eq" "/-%%diffG6 $-%\"xG6#%\"tGF)%#eqG" }}{PARA 270 "" 0 "" {TEXT -1 45 " anc_var \+ is the list of old unknowns" }}{PARA 271 "" 0 "" {TEXT -1 44 " \+ nouv_var is the list of new unknowns" }}{PARA 272 "" 0 "" {TEXT -1 94 " formules is the list of formulae giving the old unknow ns in function of the new one.." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 396 "chgtvar:=proc(eq , anc_var , nouv_var , formules )\nlocal n, A, i, eq n:\nn:=nops(eq):\nif nops(anc_var)<>n or nops(nouv_var)<>n or nops(for mules)<>n then print (`error : wrong dimensions in chgtvar`);\nelse A :=linalg[jacobian](formules,nouv_var); \neq;\nfor i from 1 to n do \n subs( anc_var[i] = formules[i] , \" ) ; od;\neqn:=\";\nlinalg[mu ltiply](linalg[inverse](A),eqn);\nconvert(\",list);\nfi; end:" }} {PARA 0 "" 0 "" {TEXT -1 7 "Example" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "eq := [(u-(x**3/3-x))/epsilon,a-x] ;\nanc_var := [x,u];\nnouv_var := [X,Y];\nformules := [X,(X**3/3-X)+epsilon*Y];\nchgtvar(eq , anc_va r , nouv_var , formules );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG 7$*&,(%\"uG\"\"\"*$%\"xG\"\"$#!\"\"F,F+F)F)%(epsilonGF.,&%\"aGF)F+F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(anc_varG7$%\"xG%\"uG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)nouv_varG7$%\"XG%\"YG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)formulesG7$%\"XG,(*$F&\"\"$#\"\"\"F)F&!\"\"*&%(epsil onGF+%\"YGF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"YG,&*(,&*$%\"XG \"\"#\"\"\"!\"\"F+F+%(epsilonGF,F$F+F,*&F-F,,&%\"aGF+F)F,F+F+" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 268 "" 0 "" {TEXT 262 17 "Procedures on DA " }}{EXCHG {PARA 0 "" 0 "" {TEXT 315 18 "Nul DA \+ of order N" }{TEXT -1 2 ": " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "zero DA := proc (epsilon, N) local k; [epsilon, [seq(0,k=1..N)]]; end:" }} {PARA 0 "" 0 "" {TEXT 316 29 "Degree and Valuation of a DA " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "ordreDA := proc(f) nops(f[2]); end: #the n the remainder of the asymptotic expansion is \243*epsilon^ordreDA" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "valDA := proc(f) local d,res; d:= ordreDA(f): res:=0; \n while (res " 0 "" {MPLTEXT 1 0 74 "polyDA := proc(f)\n sum( f[2][i]*f[1]**(i-1),i=1..ordreDA(f));\n end:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 318 27 "Algebraic operations on DAs" }{TEXT -1 2 " " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "sommeDA := proc (f, g) local N, i;\n if (f[1]<>g[1]) then print (`error :`, f[1], `and` , g[1] , ` must be the same in sommeDA`); else\n N := min(ordreDA(f),ordreDA( g)) ;\n [f[1], [seq(simplifyEB(f[2][i]+g[2][i]),i=1..N)]]; fi:end :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 256 "soustractionDA := proc (f, g) local N,i;\n if (f[1]<>g[1]) then print (`error :`, f[1], `and ` , g[1] , `must be the same in soustractionDA`); else\n N := min( ordreDA(f),ordreDA(g)) ;\n [f[1], [seq(simplifyEB(f[2][i]-g[2][i]) ,i=1..N)]]; fi:end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "scalmulDA \+ := proc (a, f) local i;\n [f[1], [seq(simplifyEB(a*f[2][i]),i=1.. ordreDA(f))]]; end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 444 "multDA := proc (f, g)\n local N,val,j,k;\n if (f[1]<>g[1]) then print ( `error :`, f[1], `and` , g[1] , `must be the same in multDA`); else\n N:=min(ordreDA(f)+valDA(g) , ordreDA(g)+valDA(f) , Nmax); #order \+ of the product\n val:= min ( valDA(f)+valDA(g) , Nmax) ; #valuatio n of the product\n [f[1], [seq(0,k=1..val) , seq(simplifyEB\n \+ (sum(f[2][j+valDA(f)+1]*g[2][k-j-valDA(f)+1],j=0..k-val)),k=val.. N-1)]];\n fi:end: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 320 20 "Integr ation of a DA " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "primitiveDA := p roc(f,var) # analog of int(f,t) when f is a DA.\n local i:\n i f (f[1]=var) then print(`error : it is forbidden to integrate a DA wit h respect to epsilon`): else\n [f[1],[seq ( int(f[2][i],var) , i= 1..ordreDA(f))]]:fi:end: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 647 "i ntDA := proc(f,var) # analog of int(f,t=a..b) when f is a DA. \n \+ # This procedure avoids the Maple's bug (cf preli minary remark above)\n # Take care of the functi on log : don't take negative argument, it will produce mistakes later. \n local F,t,a,b,i,t1,t2:\n t:=lhs(var): a:= op(1,rhs(var)): \+ b:= op(2,rhs(var)):\n F:= primitiveDA(f,t):\n for i from 1 to \+ ordreDA(F) do\n subs(t=t2,F[2][i]) - subs(t=t1,F[2][i]):\n \+ combine(\",ln,symbolic):\n limit(\",t1=a):\n combine( \",ln,symbolic):\n limit(\",t2=b):\n F[2][i]:= combine( \",ln,symbolic): \n od: F: end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 321 19 "Composition of DAs " }}{PARA 0 "" 0 "" {TEXT 364 13 "Compositi on " }{TEXT 322 9 "(g o f)" }{TEXT 365 9 " where " }{TEXT 323 5 " f " }{TEXT 366 24 " is an expression and " }{TEXT 324 1 "g" } {TEXT 367 12 " is a DA " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 501 "subs fDA := proc(substitution,g) \n # substitution is of type t=f wher e f can depend on the expanded variable of g, \n # g is a DA\n \+ local res, i, k, epsilon;\n epsilon := g[1]:\n # the substitutio n is made in each coefficient of g :\n res := zeroDA ( epsilon , o rdreDA(g) ): i:=0:\n while ( i " 0 "" {MPLTEXT 1 0 1309 "subsDAf := proc( substitution,g) \n local epsilon, f, variable, f0, g0, g1, val_f, val_g, ord_g, ord, k ; \n variable := lhs(substitution): f:=rhs (substitution) : epsilon:=f[1]:\n # V\351rification :\n if (dif f(g,epsilon)<>0) then print (`error : `, g, `depend on `, epsilon , `i n subsDAf`); else\n # Constant coefficients. Output in the case g=c onstant: \n f0:=f[2][1]; g0:=limit(g,variable=f0); \n if g= g0 then [epsilon,[g0,seq(0,k=1..Nmax)]]; else\n # Computation of t he valuation of g-g0 with respect to variable: \n val_g:=1; g1 : = simplifyEB((g-g0)/(variable-f0)); \n while (val_g " 0 "" {MPLTEXT 1 0 997 "subsDADA := proc(substitution,g) \n local epsilon, f, variabl e, res, i , k, Nmax_defaut; global Nmax;\n variable := lhs(substit ution): f:=rhs(substitution) : epsilon:=f[1]:\n # v\351rifications :\n if (f[1]<>g[1]) then print (`error :`, f[1], `and`, g[1], `mu st be the same in subsDADA`); \n elif (epsilon=variable) then pri nt (`error :`, epsilon, `and`, variable, `must be different in subsDAD A`); else\n # With subsDAf, all the compositions (g[i] o f) are com puted \n # The order of the result is bounded with an adaptative pro cedure to avoid unnecessary computations.\n Nmax_defaut := Nmax: \+ # to preserve the default value of Nmax. \n res := zeroDA ( epsilo n, ordreDA(g)): # this DA will accumulate the (g[i] o f)\n i:=0: \n while (i " 0 "" {MPLTEXT 1 0 186 "sommeDAC := proc(f,g)\n if (f[1]<>g[1] ) then print (`error :`, f[1], `and`, g[1], `must be the same in somme DAC`); else\n [f[1], sommeDA(f[2],g[2]) , sommeDA(f[3],g[3]) ]; f i:end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 215 "soustractionDAC := proc( f,g)\n if (f[1]<>g[1]) then print (`error :`, f[1], `and`, g[1], ` must be the same in sooustractionDAC`); else\n [f[1], soustraction DA(f[2],g[2]) , soustractionDA(f[3],g[3]) ]; fi:end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 428 "multDAC := proc(f,g)\n local DAL, DAR; # t he DAL and DAR of the result\n if (f[1]<>g[1]) then print (`error \+ :`, f[1], `and`, g[1], `must be the same in multDAC`); else\n DAL \+ := multDA(f[2],g[2]):\n DAR := multDA(f[3],g[3]): \n multDA ( \+ subsfDA(f[1],f[2]) , g[3]) :\n DAR := sommeDA ( \" , DAR ):\n \+ multDA ( subsfDA(g[1],g[2]) , f[3]) :\n DAR := sommeDA ( \" , DAR \+ ):\n [ f[1] , DAL, DAR ];\n fi:end:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 335 22 "Composition of DACs \n" }{TEXT 368 13 "Composition \+ " }{TEXT 336 9 "(g o f)" }{TEXT 369 9 " where " }{TEXT 337 3 " f " }{TEXT 370 12 " is a DAC" }{TEXT 338 2 " " }{TEXT 371 3 "and" } {TEXT 340 6 " g " }{TEXT 372 55 " is an expression which is indepen dent of the scale " }{XPPEDIT 373 0 "epsilon" "I(epsilonG6\"" } {TEXT 374 31 " and of the local variable " }{TEXT 339 1 "x" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 1264 "subsDACf := proc (substitution, g ) \n local phi, psi, g_x, g_phi_x, g_phi_t, g_phi_psi, epsilon, u, layer, t, x ;\n u := lhs(substitution): layer := rhs(substitut ion)[1]: phi := rhs(substitution)[2]: psi := rhs(substitution)[3]: \n epsilon:= phi[1]: t:= lhs(layer): x:= diff(rhs(layer),eps ilon):\n # Verifications :\n if (u=epsilon or u=t or u=x) then p rint (`error : incompatibility of variables`, u, `and`, layer, `in su bsDACf`); \n elif diff(g,epsilon)<>0 then print (`error : `, g, `d epend on `, epsilon , `in subsDACf`): \n elif diff(g,x)<>0 then pr int (`error : `, g, `depend on `, x , `in subsDAf`): else\n # comput ation of the slow asymptotic expansion :\n g_phi_t := subsDAf ( u= phi , g ) : # g(phi) with the variable t\n # comp utation of the fast asymptotic expansion :\n g_phi_x := subsfDA ( \+ layer , g_phi_t ) : # g(phi) with the variable x\n g_x := tay lorEB ( subs (layer ,g) , epsilon=0, ordreDA(g_phi_x)): \n sub sfDA ( layer, phi) : # phi with the variable x\n g_phi_psi := subsDADA ( u=sommeDA(\",psi) , g_x ) :# g(phi+psi)\n soustractionDA ( g_phi_psi , g_phi_x ) : # g(phi+psi)- g(phi)\n # result :\n [ layer, g_phi_t , \"]: fi: end:" }}{PARA 275 "" 0 "" {TEXT -1 13 "Composition " }{TEXT 341 9 "(g o f)" } {TEXT -1 9 " where " }{TEXT 342 3 " f" }{TEXT -1 8 " and " } {TEXT 343 1 "g" }{TEXT -1 11 " are DACs" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1837 "subsDACDAC := proc(substitution,g) \n local f, \+ variable, f_x, res, i , k, Nmax_defaut, NDAL, NDAR ; global Nmax;\n \+ variable := lhs(substitution): f:=rhs(substitution) :\n # Verific ations :\n if (f[1]<>g[1]) then print (`error :`, f[1], g[1], `mu st be the same in subsDACDAC`); \n elif (diff(f[1],variable)=0) t hen print (`error :`, f[1], `depend on`, variable, `in subsDACDAC`); e lse\n # With subsDACf, all the compositions (g[i] o f) are computed (with the slow and the fast components of g)\n # The order of the r esult is bounded with an adaptative procedure to avoid unnecessary com putations.\n Nmax_defaut:=Nmax: # to preserve the default value o f Nmax. \n res := [g[1] , zeroDA( g[2][1], ordreDA(g[2])), zeroDA ( g[3][1], ordreDA(g[3]))]: # this DA will accumulate the components \n i:=0: NDAL:= max(ordreDA(res[2]),ordreDA(res[3])):\n while (i " 0 "" {MPLTEXT 1 0 618 "intDAC := proc(f,signe) # signe is +1 if the slow c urve is attractive, -1 if it is repulsive.\n # the result is defined when (t-t1) and \"signe\" have the same sign. \+ \n local epsilon, t1, x, tau, xi, DAL, DAR, i :\n epsilon \+ := f[2][1] : t1 :=subs(epsilon=0, rhs(f[1])): tau:= lhs(f[1]): xi:=d iff(rhs(f[1]),epsilon): \n DAR := intDA ( f[3] , xi=signe*infinity ..x ): \n DAR[2] := [0,op(DAR[2])]: # multiplication by e psilon\n DAL:= soustractionDA ( intDA ( f[2] , tau=t1..lhs(f[1]) ) , subs(x=0,DAR) ):\n subs(x=diff(rhs(f[1]),epsilon),[f[1],DAL,D AR]):\n end:" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 276 "" 0 "" {TEXT 263 40 "Boundary layer in singular perturbation " }} {EXCHG {PARA 277 "" 0 "" {TEXT 376 14 "Slow solution " }{TEXT 386 9 "u _natural" }}{PARA 278 "" 0 "" {TEXT -1 43 "Computation of the DA of th e slow solution " }{TEXT 385 9 "u_natural" }{TEXT -1 45 " of the singu larly perturbed ODE : " }{XPPEDIT 425 0 "epsilon*diff(u,t)=f (t,u,epsilon)" "/*&%(epsilonG\"\"\"-%%diffG6$%\"uG%\"tGF%-%\"fG6%F*F)F $" }}{PARA 279 "" 0 "" {TEXT -1 8 "where " }{XPPEDIT 18 0 "f" "I\"fG 6\"" }{TEXT -1 86 " is an expression (for \"solution_lente\") \n \+ or a DA with respect to " }{XPPEDIT 18 0 "epsilon" "I(ep silonG6\"" }{TEXT -1 62 ", where each term is an expression (for \" solution_lenteDA\")" }}{PARA 280 "" 0 "" {TEXT -1 5 "and " }{XPPEDIT 18 0 "courbe_lente" "I-courbe_lenteG6\"" }{TEXT -1 39 " is an expressi on of independent of " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" } {TEXT -1 3 " ." }}{PARA 280 "> " 0 "" {MPLTEXT 1 0 202 "solution_lent e := proc(f,courbe_lente,N,epsilon,t,u) # epsilon, t and u are na mes of the variables\n local fDA;\n fDA := taylorEB(f,epsilon=0,N); \n solution_lenteDA(fDA,courbe_lente,t,u); end:" }}{PARA 0 "" 0 " " {TEXT 377 70 "In the procedure below, the main tool is identificatio n : we replace " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT 378 22 " by a \+ formal series " }{XPPEDIT 18 0 "sum(inconnue[i]*epsilon^i,i=0..N)" "- %$sumG6$*&&%)inconnueG6#%\"iG\"\"\")%(epsilonGF)F*/F);\"\"!%\"NG" } {MPLTEXT 0 21 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 749 "solution_len teDA := proc(f,courbe_lente,t,u) # t and u are names of the variabl es\n local inconnue,epsilon,N,df_sur_du, membre_droit, i;\n epsilon: =f[1]: N:=ordreDA(f):\n if (simplify(subs(u=courbe_lente,f[2][1]))<> 0) then print (`error :`, courbe_lente, `is not a slow curve`); els e \n df_sur_du := simplifyEB(subs(u=courbe_lente, diff( f[2][1],u))): \n inconnue[0]:=courbe_lente:\n membre_droit := subsDADA ( u = [epsi lon,[seq(inconnue[i],i=0..N)]] , f ): \n for i from 1 to N-1 do\n \+ simplify((-eval(membre_droit[2][i+1])+df_sur_du * inconnue[i] + d iff(inconnue[i-1],t) ) / df_sur_du ): \n # inconnue[i] must b e eliminated in this expression\n inconnue[i]:=\": od:\n \+ [epsilon,[seq(inconnue[i],i=0..N-1)]];\n fi: end:" }}{PARA 0 "" 0 "" {TEXT 379 88 "An other procedure doing exactly the same. We are doing the identification step by step" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 617 "solution_lenteDA_variante := proc(f,courbe_lente,t,u)\n local in connue, uDA, equ, epsilon, i, membre_droit;\n epsilon:=f[1]: \n if \+ (simplify(subs(u=courbe_lente,f[2][1]))<>0) then print (`error :`, co urbe_lente, `is not a slow curve`); else \n uDA:=[epsilon, [courbe_ lente]]: \n i:=1: \n while (true) do\n uDA := [epsilon, [op(uDA[ 2]),inconnue]]:\n membre_droit := subsDADA ( u=uDA, f):\n if ord reDA(membre_droit) <= i then break: else\n equ := diff(uDA[2][i],t) = membre_droit[2][i+1] :\n solve(equ,inconnue) : \n uDA[2][i+1 ] := \":\n i:=i+1: fi: od: \n uDA:=[uDA[1],[op(1..i,uDA[2])]]; f i: end:" }}}{EXCHG {PARA 281 "" 0 "" {TEXT 388 15 "Boundary layer " } {TEXT 387 7 "u_sharp" }{TEXT 389 2 " \n" }{TEXT -1 27 "Computation of \+ the DAC of " }{TEXT 384 7 "u_sharp" }{TEXT -1 20 ", the solution of \+ " }{XPPEDIT 426 0 "epsilon*diff(u,t)=f(t,u)" "/*&%(epsilonG\"\"\"-%%d iffG6$%\"uG%\"tGF%-%\"fG6$F*F)" }{TEXT 427 6 " , " }{XPPEDIT 428 0 "u(t[1])=c" "/-%\"uG6#&%\"tG6#\"\"\"%\"cG" }{TEXT 429 3 " ," }{TEXT -1 24 " \nwhere the arguments " }{TEXT 380 23 "f , courbe_lente , u \+ , " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT 383 4 " , N" } {TEXT -1 18 " are as before ;\n" }{TEXT 381 3 "var" }{TEXT -1 14 " is of type " }{XPPEDIT 18 0 "t=t[1]+epsilon*x" "/%\"tG,&&F#6#\"\"\"\" \"\"*&%(epsilonGF(%\"xGF(F(" }{TEXT -1 1 "." }}{PARA 282 "" 0 "" {TEXT 382 1 "c" }{TEXT -1 98 " is a constant (for \"couche_limite\") o r a DA with constant coefficients (for \"couche_limiteDA\").." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 211 "couche_limite := proc(f,courbe_len te,c,u,var,epsilon,N)\n local fDA, cDA;\n fDA := taylorEB(f,e psilon=0,N);\n cDA := taylorEB(c,epsilon=0,N);\n couche_limite DA(fDA,courbe_lente,cDA,u,var);\n end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 1106 "couche_limiteDA := proc(f,courbe_lente,c,u,var) \n \+ local epsilon, t, x, t1, DAL, DAR, y, i, eqy ; # y will be the d ifference between the result and the slow solution. \n epsilon:=f[ 1]: t:=lhs(var): x:=diff(rhs(var),epsilon): t1:=subs(epsilon=0,rhs( var)):\n if (f[1]<>c[1] ) then print (`the variables`, f[1], `and `, g[1], `are incompatible in couche_limiteDA`); else \n DAL:= sol ution_lenteDA ( f, courbe_lente, t, u);\n eqy := subsfDA( var , su bsDADA( u=DAL, soustractionDA(subsfDA(u=u+y,f),f ))); # ODE for y(x)\n # we will solve, step by step, this differential equation for each \+ coefficient of y:\n dsolve( \{diff(y(x),x)=subs(y=y(x),simplifyEB( eqy[2][1])) , y(0)=c[2][1]-subs(t=t1,DAL[2][1])\} , \{y(x)\} ):\n \+ DAR := [rhs(\")]: #will contain the coefficients of the expansion of y \n for i from 1 to ordreDA(f)-1 do \n eqy := subsfDA( y=DAR [i]+epsilon*y,eqy):\n dsolve( \{diff(y(x),x)=subs(y=y(x),simpli fyEB(eqy[2][i+1])) , y(0)=c[2][i+1]-subs(t=t1,DAL[2][i+1])\} , \{y(x) \} ):\n DAR := [op(DAR),rhs(\")]:\n od:\n [var, DAL, [epsilon, DAR]]:\n fi;end:" }}}{EXCHG {PARA 283 "" 0 "" {TEXT 390 16 "Computation of " }{TEXT 391 21 "(u_sharp - u_natural)" } {TEXT -1 129 "\nThis procedure computes the asymptotic expansion, coef ficient of the first exponential term in the transasymptotic expansion of " }{TEXT 392 8 "u_sharp " }{TEXT -1 1 "(" }{TEXT 393 7 "u_diese" } {TEXT -1 13 " in french ; " }{TEXT 408 9 "u_natural" }{TEXT 409 4 " is " }{TEXT 410 9 "u_becarre" }{TEXT 411 13 " in french ; " }{TEXT 412 6 "u_flat" }{TEXT 413 4 " is " }{TEXT 414 7 "u_bemol" }{TEXT 415 10 " \+ in french" }{TEXT -1 21 "), the solution of " }{XPPEDIT 18 0 "epsilo n*diff(u,t)=f(t,u)" "/*&%(epsilonG\"\"\"-%%diffG6$%\"uG%\"tGF%-%\"fG6$ F*F)" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "u(t[1])=c" "/-%\"uG6#&%\"tG 6#\"\"\"%\"cG" }{TEXT -1 61 " . The algorithm is explained in the art icle.\nThe arguments " }{TEXT 394 33 "f , courbe_lente , c , u , var , " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT 396 2 " " }{TEXT -1 3 "and" }{TEXT 395 2 " N" }{TEXT -1 101 " have the same meaning as before in the procedure \"couche_limite\". \n(the procedure \"ecartDA \", where " }{TEXT 406 1 "f" }{TEXT -1 5 " and " }{TEXT 407 1 "c" } {TEXT -1 95 " would be DA, could be written, but it is not needed for \+ the applications below)\nThe argument " }{TEXT 397 5 "signe" }{TEXT -1 140 " is +1 if the slow curve is attractive, -1 if it is repulsive. \nThe result is given as a list of two elements :\n \+ " }{TEXT 398 2 "[ " }{XPPEDIT 18 0 " A[0]" "&%\"AG6#\"\"!" } {TEXT 399 8 " , f ]" }{TEXT -1 8 " \nwhere " }{XPPEDIT 18 0 " A[0]" "&%\"AG6#\"\"!" }{TEXT -1 34 " is an expression independent of " } {XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 8 ", and " }{TEXT 400 1 "f" }{TEXT -1 10 " is a DA." }}{PARA 0 "" 0 "" {TEXT 401 31 "Th e meaning of this result is (" }{TEXT 402 9 "u_natural" }{TEXT 403 4 " is " }{TEXT 404 9 "u_becarre" }{TEXT 405 12 " in french):" }}{PARA 283 "" 0 "" {TEXT -1 27 " " }{XPPEDIT 430 0 " u[sharp] - u[natural] = exp(A[0]/epsilon) * f" "/,&&%\"uG6#%&sharp G\"\"\"&F%6#%(naturalG!\"\"*&-%$expG6#*&&%\"AG6#\"\"!F(%(epsilonGF,F(% \"fGF(" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 948 "ecart := proc( f, courbe _lente, signe, c, u, var, epsilon, N) \n local u_diese, u_becarre, delta, Delta2f, a, A, A0, A_sauf_0_sur_eps, u_becarre_0, u_diese_0, u _diese_transDA, i:\n u_diese := couche_limite (f, courbe_lente, c, u, var, epsilon, N+1):\n u_becarre:=[u_diese[1],u_diese[2],zeroDA (epsilon,N+1)]:\n # for technical reason, the order of the three sub stitutions below is important.\n Delta2f:= simplify((subs(u=u+delt a,f)-f)/delta) : \n subsDACf ( delta=soustractionDAC(u_diese,u_bec arre) , Delta2f ): \n a:=subsDACDAC ( u=u_becarre , \"):\n \+ A:=simplifyEB(intDAC(a,signe))[2]:\n A0:=A[2][1]; A_sauf_0_sur_ eps:=[A[1],[seq(A[2][i],i=2..ordreDA(A))]]:\n u_becarre_0:=subs(t= t1,u_becarre[2]):\n u_diese_0:=taylorEB(c,epsilon=0,ordreDA(u_beca rre_0));\n multDA(soustractionDA(u_diese_0,u_becarre_0),subsDAf(s= A_sauf_0_sur_eps,exp(s)));\n u_diese_transDA:=map(expand,\");\n \+ [A0 , u_diese_transDA]: end:" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 284 "" 0 "" {TEXT 259 14 "Turning points" }}{PARA 285 " " 0 "" {TEXT -1 117 "This procedure computes the asymptotic expansion \+ of the canrad-values and of the canards solutions of the equation:\n \+ " }{TEXT 431 15 " " }{XPPEDIT 432 0 "epsilon*diff(u,t)=f (t,u,a)" "/*&%(epsilonG\"\"\"-%%diffG6$%\"uG%\"tGF%-%\"fG6%F*F)%\"aG" }}{PARA 286 "" 0 "" {TEXT -1 7 "where " }{TEXT 433 1 "f" }{TEXT -1 33 " is an expression depending on " }{XPPEDIT 18 0 "t " "I\"tG6\"" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "epsi lon" "I(epsilonG6\"" }{TEXT -1 64 " (for the procedure \"canard \") \n or a DA in " }{XPPEDIT 18 0 "epsilon" "I(epsilon G6\"" }{TEXT -1 56 ", where each coefficient is an expression dependin g on " }{XPPEDIT 18 0 "t " "I\"tG6\"" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "u" "I\"uG6\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "a " "I\"aG6 \"" }{TEXT -1 18 " (for \"canardDA\")" }}{PARA 287 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "courbe_lente" "I-courbe_lenteG6\"" }{TEXT -1 31 " is an expression depending on " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 13 ", but not on " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 1 "." }}{PARA 288 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a[0]" "&%\"aG6#\"\"!" }{TEXT -1 56 " is the standard part of the c anards-values, \nwhere " }{XPPEDIT 18 0 "t[0]" "&%\"tG6#\"\"!" } {TEXT -1 19 " is the value of " }{TEXT 434 1 "t" }{TEXT -1 67 " for the turning point.\nThe result is given as a list of two DAs.." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "canard := proc(f,courbe_len te, a0, t0, N, epsilon, t, u, a) # epsilon, t, u et a are names o f unevaluated variables\n local fDA;\n fDA := taylorEB(f,epsilon=0, N);\n canardDA(fDA, courbe_lente, a0, t0, t, u, a); end:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 1199 "canardDA := proc(f, courbe_lente, a0, t0, t, u, a)\n local inconnue_a, inconnue_u, aDA, uDA, equ, epsi lon, df_sur_du, i, droite;\n epsilon:=f[1]:\n df_sur_du := simplifyE B(subs(a=a0,u=courbe_lente, diff( f[2][1],u))):\n# V\351rifications : \n if simplifyEB(subs(a=a0,u=courbe_lente,f[2][1]))<>0 then print (` error :`, courbe_lente, `is not a slow curve`);\n elif eval(subs(t=t0 ,df_sur_du))<>0 then print (`error :`, t0, `is not a turning point`); \n elif eval(subs(t=t0,diff(df_sur_du,t)))<=0 then print (`error : th ere is no canard at point`, t0);\n else\n# Computations : \n aDA:=[ epsilon,[a0]]: uDA:=[epsilon, [courbe_lente]]: \n i:=1: \n while ( true) do\n inconnue_a:='inconnue_a': inconnue_u:='inconnue_u':\n \+ aDA := [epsilon, [op(aDA[2]),inconnue_a]]: uDA := [epsilon, [op(uDA [2]),inconnue_u]]:\n droite := subsDADA ( a=aDA , subsDADA ( u=uDA, f)):\n if ordreDA(\") <= i then break: else\n equ := simplifyEB (diff(uDA[2][i],t)-droite[2][i+1] ) :\n solve(subs(t=t0,equ),inconn ue_a):\n aDA[2][i+1]:=\": inconnue_a:=\":\n solve(eval(equ),in connue_u) : \n uDA[2][i+1]:=\":\n i:=i+1: fi: od: \n aDA:=[aDA [1],[op(1..i,aDA[2])]]; uDA:=[uDA[1],[op(1..i,uDA[2])]]; [aDA,uDA]: \+ fi: end:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 291 "" 0 " " {TEXT 264 32 "Steepest descent (proposition 2)" }}{EXCHG {PARA 292 " " 0 "" {TEXT -1 5 "Let " }{TEXT 435 1 "f" }{TEXT -1 7 " and " } {TEXT 436 1 "g" }{TEXT -1 28 " be two DA's in powers of " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 55 " . The procedure returns the DA of the real number " }{XPPEDIT 18 0 "Int(exp(-f(t)/epsilon) *g(t),t=t1..t2):" "-%$IntG6$*&-%$expG6#,$*&-%\"fG6#%\"tG\"\"\"%(epsilo nG!\"\"F1F/-%\"gG6#F.F//F.;%#t1G%#t2G" }{TEXT -1 20 " in powers of \+ " }{XPPEDIT 18 0 "eta=epsilon**(1/k)" "/%$etaG)%(epsilonG*&\"\"\"\" \"\"%\"kG!\"\"" }{TEXT -1 10 ", where " }{XPPEDIT 18 0 "k" "I\"kG6\" " }{TEXT -1 58 " is the (even) order of the first nonzero derivative o f " }{XPPEDIT 18 0 "f[0]^(\{k\})*\{t[0]\}" "*&)&%\"fG6#\"\"!<#%\"kG \"\"\"<#&%\"tG6#F'F*" }{TEXT -1 46 ". \nIt is assumed (without verific ation) that " }{XPPEDIT 18 0 "f[0]" "&%\"fG6#\"\"!" }{TEXT -1 34 " h as a unique global minimum at " }{XPPEDIT 18 0 "t[0]" "&%\"tG6#\"\"! " }{TEXT -1 18 " in the interval ]" }{XPPEDIT 18 0 "t[1],t[2]" "6$&%\" tG6#\"\"\"&F$6#\"\"#" }{TEXT -1 44 "[. \nIt is assumed (with verificat ion) that " }{XPPEDIT 18 0 "f[0](t[0])=0" "/-&%\"fG6#\"\"!6#&%\"tG6#F 'F'" }{TEXT -1 18 ". \nIn the result, " }{XPPEDIT 18 0 "eta " "I$etaG6 \"" }{TEXT -1 39 " is the name of an unassigned variable." }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 1610 "methode_du_col := proc(f,g,t,t0,eta) # t e t eta sont des noms de variables non assign\351es. \n local k, derivee_f0, i, j, tau, f_eta, f_sur_epsilon, g_eta, integrand, u ;\n \+ if (f[1]<>g[1] or f[1]=t) then print (`error : the epsilon variabl e is not compatible in methode_du_col for`, f, g); \n elif (f[2][ 1]=0) then print(`error : the first approximation of`,f,`is zero in me thode_du_col`); \n elif (eval(subs(t=t0,f[2][1]))<>0) then print(` error : `, f[2][1], `must be zero at point`, t0);\n else \n# k is the valuation of f0(t) at point t0: \n k:=0; derivee_f0:=f[2] [1]; \n while ( eval(subs(t=t0,derivee_f0))=0) do derivee_f0:=d iff(derivee_f0,t); k:=k+1; od:\n if ( (k mod 2)=1 or simplifyEB (subs(t=t0,derivee_f0))<0 ) \n then print (`error : the \+ local behaviour of`,f,`is not compatible with the procedure methode_du _col`); else\n# Rewriting of f_sur_epsilon and g_eta as DAs in eta: \+ \n [f[2][1],seq(0,j=1..k-1)];\n for i from 2 to ordreDA(f) d o \n [op(\"),f[2][i],seq(0,j=1..k-1)]; od;\n f_eta:=[eta, \"]; \n subsfDA ( t=t0+eta*tau , f_eta):\n f_sur_epsilon : = [eta,[seq(\"[2][i],i=k+1..k*(ordreDA(f)))]];\n [g[2][1],seq(0,j= 1..k-1)];\n for i from 2 to ordreDA(g) do \n [op(\"),g[2] [i],seq(0,j=1..k-1)]; od;\n g_eta:=[eta,\"]; \n subsfDA ( t=t0+eta*tau , g_eta):\n g_eta := [eta,[0,op(\"[2])]]; # g_eta(ta u) dtau = g(t) dt with dt = eta dtau \n integrand := multDA( subsD Af(u=f_sur_epsilon, exp(-u)) , g_eta ):\n# Int\351gration : \n int DA ( integrand, tau=-infinity..infinity): \n fi: fi: end: " } }}}{EXCHG {PARA 293 "" 0 "" {TEXT 256 54 "Applications of this procedu res to classical problems " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 438 4 "1) " }{TEXT 439 13 "Airy equation" }{TEXT 444 13 " " } {XPPEDIT 18 0 "epsilon*diff(u,t)=u^2-t" "/*&%(epsilonG\"\"\"-%%diffG6$ %\"uG%\"tGF%,&*$F)\"\"#F%F*!\"\"" }{TEXT 440 1 "\n" }{TEXT 445 60 "We \+ will compute the asymptotic expansion of a slow solution " }{TEXT 441 9 "u_natural" }{TEXT 442 52 " and the combined asymptotic equation of \+ a solution " }{TEXT 443 7 "u_sharp" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "f:=u**2-t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG, &*$%\"uG\"\"#\"\"\"%\"tG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "courbe_lente:=+sqrt(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-co urbe_lenteG*$%\"tG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "c:=0;t1:=1;var:=t=t1+epsilon*x; # the initial condition of u_s harp is u_sharp(t1)=c" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t1G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$varG/%\"tG,&\"\"\"F(*&%(epsilonGF(%\"xGF(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "solution_lente(f,courbe_lent e,5,epsilon,t,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%(epsilonG7'*$% \"tG#\"\"\"\"\"#,$*$F'!\"\"#F)\"\"%,$*$F'#!\"&F*#F3\"#K,$*$F'!\"%#\"#: \"#k,$*$F'#!#6F*#!%06\"%[?" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 485 31 "Th e meaning of this result is :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "u[ natural] = polyDA(\")+limited*epsilon^ordreDA(\");" }}{PARA 0 "" 0 "" {TEXT -1 43 "and it is valid when the standard part of " }{TEXT 486 1 "t" }{TEXT -1 10 " is in ]" }{XPPEDIT 18 0 "0,+infinity" "6$\"\"!% )infinityG" }{TEXT -1 2 " [" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"uG 6#%(naturalG,.*$%\"tG#\"\"\"\"\"#F,*&F*!\"\"%(epsilonGF,#F,\"\"%*&F*#! \"&F-F0F-#F5\"#K*&F*!\"%F0\"\"$#\"#:\"#k*&F*#!#6F-F0F2#!%06\"%[?*&%(li mitedGF,F0\"\"&F," }}}{EXCHG {PARA 306 "> " 1 "" {MPLTEXT 1 0 118 "res :=ecart ( f, courbe_lente, -1, c, u, var, epsilon, 4); # this DAC is v alid when the standard part of t is in ]0,t1]" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$resG7$,&*$%\"tG#\"\"$\"\"##\"\"%F*#!\"%F*\"\"\"7$%(e psilonG7&,$*$F(#F0F+!\"#,&F5#!\"(\"#7*$F(!\"\"#!\"&F;,(*$F(#F?F+#\"$b \"\"$w&F5#!#\\FEF<#!#N\"$)G,**$F(F/#!&:t\"\"&s9%F5#!&xm#FOFA#\"%&3\"\" &CQ\"F<#!$X#FT" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "u[sharp]- u[natural] = e^(res[1]/epsilon)*(polyDA(res[2])+limited*epsilon^ordreD A(res[2]));" }}{PARA 294 "" 0 "" {TEXT -1 43 "and it is valid when the standard part of " }{TEXT 437 1 "t" }{TEXT -1 10 " is in ]" } {XPPEDIT 18 0 "0,t[1]" "6$\"\"!&%\"tG6#\"\"\"" }{TEXT -1 2 "] " }} {PARA 12 "" 1 "" {XPPMATH 20 "6#/,&&%\"uG6#%&sharpG\"\"\"&F&6#%(natura lG!\"\"*&)%\"eG*&,&*$%\"tG#\"\"$\"\"##\"\"%F6#!\"%F6F)F)%(epsilonGF-F) ,,*$F4#F)F7!\"#*&,&F>#!\"(\"#7*$F4F-#!\"&FEF)F#!#\\FOFF#!#N\"$)GF)F#!&xm#FZFK #\"%&3\"\"&CQ\"FF#!$X#FinF)F " 0 "" {MPLTEXT 1 0 39 "eqvdP:=[(u-(X**3 /3-X))/epsilon , -X+a];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&eqvdPG7$ *&,(%\"uG\"\"\"*$%\"XG\"\"$#!\"\"F,F+F)F)%(epsilonGF.,&F+F.%\"aGF)" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "chgtvar(eqvdP,[X,u],[x,v], [x,x**3/3-x+epsilon*v]); # the old variables [X,u] and the new one [x, v] must have different names" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$%\"v G,&*(,&*$%\"xG\"\"#\"\"\"!\"\"F+F+%(epsilonGF,F$F+F,*&F-F,,&%\"aGF+F)F ,F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "eq:=(\"[2]/\"[1]* epsilon); # transformation of a planar vector field into a one dimensi onal differential equation for v(x)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#eqG*(,&*(,&*$%\"xG\"\"#\"\"\"!\"\"F,F,%(epsilonGF-%\"vGF,F-*&F.F-, &%\"aGF,F*F-F,F,F,F/F-F.F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "a0:=1; x0:=1; courbe_lente:=solve(subs(a=a0,eq),v);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#a0G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#x0G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-courbe_lenteG,$*$,& %\"xG\"\"\"F)F)!\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "r es := canard(eq,courbe_lente,a0,x0,10,epsilon,x,v,a):" }}{PARA 0 "" 0 "" {TEXT 453 50 "The asymptotic expansion of the canard-values is :" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "a[becarre] = polyDA(res[1]) + limi ted*epsilon^ordreDA(res[1]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"a G6#%(becarreG,8\"\"\"F)%(epsilonG#!\"\"\"\")*$F*\"\"##!\"$\"#K*$F*\"\" $#!$t\"\"%C5*$F*\"\"%#!%$f(\"&%Q;*$F*\"\"&#!'xnV\"'W@E*$F*\"\"'#!)f,M: \"(_r4#*$F*\"\"(#!*.A5K'\");sx;*$F*F-#!-V>'\\NQ#\"+C=ut5*$F*\"\"*#!/`E R&[f_#\",%=p)zr\"*&%(limitedGF)F*\"#5F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 454 53 "The asymptotic expansion of the canard-solutions is :" } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "u[becarre] = polyDA(res[2]) + limi ted*epsilon^ordreDA(res[2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/&%\"u G6#%(becarreG,8*$,&%\"xG\"\"\"F,F,!\"\"F-**,(*$F+\"\"#F,F+\"\"%\"\"(F, F,,(F0F,F+F1F,F,F-F*!\"#%(epsilonGF,#F-\"\")**,.*$F+\"\"&\"\"$*$F+F2\" #@*$F+F=\"#mF0\"$E\"F+\"$f\"\"$@\"F,F,F4F5F*!\"$F6F1#F-\"#K**,4*$F+F8 \"$t\"*$F+F3\"%I<*$F+\"\"'\"%ezF;\"&!\\AF>\"&+S%F@\"&eN'F0\"&I*pF+\"&a q&\"&.%GF,F,F4FEF*!\"%F6F=#F-\"%C5**,:*$F+\"#6\"%$f(*$F+\"#5\"&4()**$F +\"\"*\"'Z)*fFJ\"(2.F#FL\"(Q0-'FN\")][%>\"F;\")'o3&=F>\")yU2BF@\")d\"= N#F0\")l1V>F+\").Y=7\"(^Dd%F,F,F4FEF*!\"(F6F2#F-\"&%Q;**,@*$F+\"#9\"'x nV*$F+\"#8\"(K%))p*$F+\"#7\")<+&G&Ffn\"*_Ne^#Fin\"*$p>u%)F\\o\"+kK0a@F J\"+\\l_2VFL\"+Ces$)pFN\"+v)=wR*F;\",G\"ywo5F>\",n/J$Q5F@\"+s)R7f)F0\" +v*oe(eF+\"+#>%HWI\"*:'GG#*F,F,F4FEF*!#5F6F<#F-\"'W@E**,F*$F+\"#<\")f, M:*$F+\"#;\"*@IY\"H*$F+\"#:\"+[t]QEF]p\",#4xg::F`p\",%)4['3iFcp\"-!oLc M$>Ffn\"-;H+AnZFin\"-s(y#3y&*F\\o\".q%R_@.;FJ\".%f7$3pF#FL\".7#pF'py#F N\"._mR>y(HF;\".KgpY(*z#F>\".SowbWJ#F@\".#p()oAb;F0\"-3pkD2)*F+\"-f'H= -O%\"-*Gf@>6\"F,F,F4FEF*!#8F6FO#F-\"(_r4#**,L*$F+\"#?\"*.A5K'*$F+\"#> \",m%[i!R\"*$F+\"#=\"-'46xkY\"Fgq\"-'3TOM()*Fjq\".DA;2,x%F]r\"/?d['\\D w\"F]p\"/'pXDGd=&F`p\"0Grx`M%\\7Fcp\"0Au'*)*ew^#Ffn\"0CT\"Q)y_J%Fin\"0 ON;66JQ'F\\o\"0)e<=(pWD)FJ\"0i$*o*ybT%*FL\"0/dl[rfk*FN\"0!o\"\\ioz&))F ;\"0!3:+o_7tF>\"02#Hj$*)QP&F@\"09xRnuzU$F0\"0c))=OHM!=F+\"/5,tN$H/(\"/ <:-*HAb\"F,F,F4FWF*!#9F6F3#F-\");sx;**,R*$F+\"#B\"-V>'\\NQ#*$F+\"#A\". v&[S()ef*$F+F?\"/V[aV[urFds\"0vu^'GvTbFgs\"1<3][P\"o3$Fjs\"2t'*)y?(4:K \"Fgq\"2(fyT(fem_%Fjq\"3(RW?F81bF\"F]r\"39!fL:52y,$F]p\"3qOIi+rN#4'F`p \"4%pVXV>E`j5Fcp\"4q1iJBcYUi\"Ffn\"4Ix@rEiiM>#Fin\"4UEe&oExtXEF\\o\"4E [f`%41'p(GFJ\"4A,M*yv:^UGFL\"4J<\"GJHd!\\c#FN\"4Zr6ua5;Z6#F;\"4*zf4b` \"\\Ge\"F>\"4Nr\"3:,[;e5F@\"3Xg7]A.bEhF0\"3X-3@6XF1HF+\"3X05tbNH85\"2 \\Mv[3;y'>F,F,F4FWF*!#$[F\"F+\"5[9&4425WKFL\"7_TRZz%4zR$>;;[(3#F>\"6aZ!fPykh!*[b\"4> ;v1!piVvpF,Fgq\"6gk%*y%z#Rd(f?Fjs\"5*fk#oqw\"Q\\$))Fdu\"27bKP(RtX$)*$F +\"#C\"1([:Q'\\Mt&**$F+\"#D\"0%G%*4flsqFN\"7Qy/T@MC/5/8F;\"6)[lL$*el^M 1!*Ffn\"7#RHbNMa0t!GBFcp\"7'zuEMcn,Wi.#F`p\"7_Z6NQ]F8s;;Fgu\"3)R7-Mn) \\n_Fju\"4!3K5HsSllDFds\"51.Heuah!R+\"F,F4FWF*!#?F6F]o#F-\",%=p)zr\"*& %(limitedGF,F6FjnF," }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 455 4 "3) " } {TEXT 463 37 "Energy levels of Schr\366dinger equation" }{TEXT -1 3 " \+ " }{XPPEDIT 18 0 "epsilon^2*diff(psi,t,t) = (U(t)-E)*psi" "/*&%(epsi lonG\"\"#-%%diffG6%%$psiG%\"tGF*\"\"\"*&,&-%\"UG6#F*F+%\"EG!\"\"F+F)F+ " }{TEXT -1 1 "\n" }{TEXT 458 58 "We will compute the splitting of the first energy level : " }{XPPEDIT 459 0 "E[bemol]" "&%\"EG6#%&bemolG" }{TEXT 460 5 " and " }{XPPEDIT 461 0 "E[sharp]" "&%\"EG6#%&sharpG" } {TEXT 462 2 " " }}{EXCHG {PARA 295 "" 0 "" {TEXT 456 6 "Data :" } {TEXT -1 17 " The potential " }{XPPEDIT 18 0 "U(t)" "-%\"UG6#%\"tG" }{TEXT -1 57 " must be even, posistive. He has two minima at points \+ " }{XPPEDIT 18 0 "t[0]" "&%\"tG6#\"\"!" }{TEXT -1 7 " and " } {XPPEDIT 18 0 "-t[0]" ",$&%\"tG6#\"\"!!\"\"" }{TEXT -1 17 " , with va lue 0." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "U := (t**2-1)**2; t0:=1; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG*$,&*$%\"tG\"\"#\"\"\"!\"\"F *F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#t0G\"\"\"" }}}{EXCHG {PARA 297 "" 0 "" {TEXT -1 48 "The Riccati form of the Schr\366dinger equat ion : " }{XPPEDIT 18 0 " epsilon diff(v,t) = U(t)-E-v^2" "/*&%(epsilon G\"\"\"-%%diffG6$%\"vG%\"tGF%,(-%\"UG6#F*F%%\"EG!\"\"*$F)\"\"#F0" }} {PARA 296 "> " 0 "" {MPLTEXT 1 0 20 "eqv := U - E - v**2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqvG,(*$,&*$%\"tG\"\"#\"\"\"!\"\"F+F*F+%\"EG F,*$%\"vGF*F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT 464 18 "Initial conditi ons" }{TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "v0_bemol:=0 \+ ; v0_diese := +infinity;" }}{PARA 11 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)v0_bemolG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)v0_dieseG%)infinityG" }}}{EXCHG {PARA 298 "" 0 "" {TEXT -1 56 "Change of unknown : rotation of the cylinder with angle " } {MPLTEXT 0 21 0 "" }{TEXT -1 1 " " }{XPPEDIT 18 0 "atan(alpha)" "-%%at anG6#%&alphaG" }{TEXT -1 41 " . We have to choose a positive number \+ " }{XPPEDIT 18 0 "alpha" "I&alphaG6\"" }{TEXT -1 62 " not too big.\nL ater, all the computations will be made with " }{TEXT 465 1 "u" } {TEXT -1 46 " to avoid the infinite initial condition in " }{TEXT 466 1 "v" }{TEXT -1 57 " . \nBut, for printing we will translate the r esults on " }{TEXT 467 1 "v" }{TEXT -1 3 " ." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "alpha:=1: \nchgt := (v-alpha)/(1+alpha*v): \nchgt_in v := solve(chgt=u,v):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 468 26 "Order o f the expected DA :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N := 5;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"\"&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 457 15 "Computations :\n" }{TEXT 469 50 "slow curve : watch on the sign of the expression !" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 " \+ courbe_lente_v := sqrt(U,symbolic):\n if evalf(subs(t=t0,diff(courbe _lente_v,t))) >=0 then courbe_lente_v:=-courbe_lente_v; fi ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/courbe_lente_vG,&*$%\"tG\"\"#!\"\"\"\"\"F *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 470 31 "Transformation of the data \+ on " }{TEXT 471 1 "v" }{TEXT 472 16 " into data on " }{TEXT 473 1 " u" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 236 " equ := chgtvar ( [eqv/epsil on] , [v] , [u] , [chgt_inv] )[1]*epsilon: # the equation for the unkn own u \n courbe_lente_u := subs(v=courbe_lente_v, chgt):\n u0_bemol := limit(chgt,v=v0_bemol): \n u0_diese := limit(chgt,v=v0_diese): \n" }}}{EXCHG {PARA 299 "" 0 "" {TEXT 474 16 "Computation of " } {XPPEDIT 18 0 "E[natural]" "&%\"EG6#%(naturalG" }{TEXT 475 8 " and \+ " }{XPPEDIT 18 0 "u[natural]" "&%\"uG6#%(naturalG" }{TEXT 476 2 " " } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 153 " canard (equ, courbe_lente_u, 0, t0, N, epsilon, t, u, E): \n E_becarre := \"[1] ; u_becarre := \"\"[ 2]:\n v_becarre := subsDAf ( u=u_becarre, chgt_inv) ; " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%*E_becarreG7$%(epsilonG7'\"\"!\"\"##!\"\"F)#! \"*\"#K#!#*)\"$c#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*v_becarreG7$%( epsilonG7',&*$%\"tG\"\"#!\"\"\"\"\"F-,$*$,&F*F-F-F-F,F,,$*&,&F*F-\"\"$ F-F-F0!\"$#F,\"\"%,$*&,**$F*F4\"\"*F)\"#XF*\"#**\"$.\"F-F-F0!\"&#F,\"# k,$*&,.*$F*\"\"&\"#*)*$F*F7\"$B'F;\"%e>F)\"%%f$F*\"%@T\"%VDF-F-F0!\"(# F,\"$7&" }}}{EXCHG {PARA 301 "" 0 "" {TEXT 477 16 "Computation of " } {XPPEDIT 18 0 "u[flat]" "&%\"uG6#%%flatG" }{TEXT 478 7 " and " } {XPPEDIT 18 0 "u[sharp]" "&%\"uG6#%&sharpG" }{TEXT 479 1 " " }}{PARA 300 "> " 1 "" {MPLTEXT 1 0 461 " equ_becarre := subsDAf ( E=E_becarre , equ):\n u0_bemol_DA := taylorEB(u0_bemol,epsilon=0,N):\n u0_diese_ DA := taylorEB(u0_diese,epsilon=0,N):\n u0_becarre_DA := subs(t=0,u_b ecarre):\n u_bemol := couche_limiteDA (equ_becarre, courbe_lente_u, u 0_bemol_DA, u, t=0+epsilon*x):\n u_diese := couche_limiteDA (equ_beca rre, courbe_lente_u, u0_diese_DA, u, t=0+epsilon*x): \n v_bemol := su bsDACf ( u=u_bemol, chgt_inv) ;\n v_diese := subsDACf ( u=u_diese, ch gt_inv) ;\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(v_bemolG7%/%\"tG*&%(epsilonG\"\"\"%\"xGF*7$F)7',&*$F '\"\"#!\"\"F*F*,$*$,&F'F*F*F*F1F1,$*&,&F'F*\"\"$F*F*F4!\"$#F1\"\"%,$*& ,**$F'F8\"\"*F/\"#XF'\"#**\"$.\"F*F*F4!\"&#F1\"#k,$*&,.*$F'\"\"&\"#*)* $F'F;\"$B'F?\"%e>F/\"%%f$F'\"%@T\"%VDF*F*F4!\"(#F1\"$7&7$F)7',$*&-%$ex pG6#,$F+!\"#F*,&F*F*FZF*F1Fhn,$*(FZF*,(F+FhnFZF*F*F*F*FinFhnF0,$*(FZF* ,6!\"*F*F+!\"'*$F+F0\"#7-Fen6#,$F+!\"%F`oFZ!#=*$F+F8\"\")*&F+F0FdoF*!# 7*&F+F8FZF*Fjo*&F+F0FZF*!#[*&FZF*F+F*FaoF*FinF9#F1\"\"',$*(FZF*,@\"$4$ F*F+!#UFdo\"$F*-Fen6#,$F+FaoFfpFZFhpFio\"$G\"F[p\"$o(*&F+F8FdoF*F\\q*& F+F;FdoF*\"$c#*&FdoF*F+F*Fgp*$F+F;!$c#F]p\"%#z\"F^p\"$w&F`p!#%)*&F+F0F ipF*\"$#>F*FinFgo#F*\"#'*,$*(FZF*,hnFio!%/B!&()G#F*Fip!&[:*FZF`rFaq\"& yF$*&FZF*F+Fbp\"%C5Fbo!%oJF+\"&E4\"*&F+FKFdoF*!&[w#F`pFar*&F+F;FipF*\" %O:Fbq!%O:*&F+FKFZF*Fgr*&F+F;FZF*!&+%Q*&F+FKFipF*\"%sIF^p!&3]&*$F+FbpF cr*&F+FbpFdoF*!%C5*&FipF*F+F*Fer-Fen6#,$F+!\")F_r*$F+FKF_sF_q\"&+%Q*&F ipF*F+FbpFcsFdo!'At8F]p!&#*H&Fgq!&KK)F^qF]tF[p!'#z<\"*&F+F0FesF*!&!G<* &F+F8FipF*F^rF*FinFD#F1\"%/B" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(v_d ieseG7%/%\"tG*&%(epsilonG\"\"\"%\"xGF*7$F)7',&*$F'\"\"#!\"\"F*F*,$*$,& F'F*F*F*F1F1,$*&,&F'F*\"\"$F*F*F4!\"$#F1\"\"%,$*&,**$F'F8\"\"*F/\"#XF' \"#**\"$.\"F*F*F4!\"&#F1\"#k,$*&,.*$F'\"\"&\"#*)*$F'F;\"$B'F?\"%e>F/\" %%f$F'\"%@T\"%VDF*F*F4!\"(#F1\"$7&7$F)7',$*&-%$expG6#,$F+!\"#F*,&F1F*F ZF*F1Fhn,$*(FZF*,(FZF*F+F0F1F*F*FinFhnF0,$*(FZF*,6F@F*F+\"\"'*$F+F0!#7 -Fen6#,$F+!\"%F@FZ!#=*&FZF*F+F*!\"'*$F+F8!\")*&F+F0FcoF*\"#7*&F+F8FZF* \"\")*&F+F0FZF*!#[F*FinF9#F*F`o,$*(FZF*,@!$4$F*F+\"#U-Fen6#,$F+Fio\"$4 $Fco!$F*FZ\"$F*Fho!#%)Fjo!$G\"F\\p!$o(*&F+F8FcoF*F_q*&F+F0FhpF*\"$#>*& F+F;FcoF*!$c#*&FcoF*F+F*FgpF^p\"%#z\"F`p\"$w&*$F+F;\"$c#F*FinFfo#F*\"# '*,$*(FZF*,hnFjo\"%/BFhp!&[:*FZFar\"&()G#F*Ffq!&yF$*&FZF*F+F`o\"%C5Fao \"%oJF+!&E4\"*&F+FKFcoF*\"&[w#Fho\"&yF$*&F+F;FhpF*\"%O:FiqF\\s*&F+FKFZ F*!&[w#*&F+F;FZF*!&+%Q*&F+FKFhpF*\"%sIF`p!&3]&*$F+F`o!%C5*&F+F`oFcoF*F er*&FhpF*F+F*\"&E4\"-Fen6#,$F+F[pFbr*$F+FK!%sIFdqF`s*&FhpF*F+F`oFesFco \"'At8F^p!&#*H&Fbq!&KK)Faq\"&#*H&F\\p\"'#z<\"*&F+F0FisF*\"&!G<*&F+F8Fh pF*!%/BF*FinFD#F*F`r" }}}{EXCHG {PARA 302 "" 0 "" {TEXT -1 32 "Computa tion of the expression " }{TEXT 480 1 "A" }{TEXT -1 7 " and " } {TEXT 481 1 "B" }{TEXT -1 47 " needed for the linear equation satisfi ed by " }{XPPEDIT 18 0 "u[sharp] - u[flat]" ",&&%\"uG6#%&sharpG\"\"\" &F$6#%%flatG!\"\"" }{TEXT -1 19 " (see the article)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 487 " Delta_u_equ := normal((subs(u=u+du,equ)-equ)/ du):\n Delta_u_equ_becarre := subsDAf ( E=E_becarre , Delta_u_equ):\n subsDACDAC ( du=soustractionDAC(u_diese,u_bemol), [u_diese[1],Delta_ u_equ_becarre,zeroDA(epsilon,N)]) :\n A := subsDACDAC ( u=u_bemol , \+ \"):\n Delta_E_equ := normal((subs(E=E+dE,equ)-equ)/dE): \n # \+ \"Miracle\" : this expression doesn't depend on dE nor on E \n # \+ (due to the linearity of the equation with respect to E)\n B := subsD ACf ( u=u_diese , Delta_E_equ ):" }}{PARA 303 "" 0 "" {TEXT -1 16 "Com putation of " }{XPPEDIT 18 0 " E[sharp] - E[flat]" ",&&%\"EG6#%&sharp G\"\"\"&F$6#%%flatG!\"\"" }{TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 609 " f:=intDAC ( A , +1)[2]:\n f00:=subs(t=t0,f[2][1]) :\n f_sans_f00 := simplifyEB(subs(f[2][1]=f[2][1]-f00,f)): # The con stant f00 will be restore before the end of the procedure\n methode_d u_col ( f_sans_f00, B[2], t, t0, eta ): # the even terms in eta va nish\n [epsilon,[seq(\"[2][2*i],i=1..ordreDA(\")/2)]]: # the factor \+ 1/sqrt(epsilon) will be restore before the end of the procedure \n su bsDAf(s=\",1/s):\n scalmulDA ( -(u0_diese-u0_bemol) ,\"):\n E_diese_ moins_E_bemol := exp(f00/epsilon) * sqrt(epsilon) * \"[2][1] *(polyDA( scalmulDA(1/\"[2][1],\"))+limited*epsilon^(ordreDA(\"))) ; #formattin g the result" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%6E_diese_moins_E_bem olG,$*,-%$expG6#,$*$%(epsilonG!\"\"#!\"%\"\"$\"\"\"F,#F1\"\"#F3F2%#PiG #F-F3,,F1F1F,#!#r\"#'**$F,F3#!%*H'\"&K%=*$F,F0#!(26p#\"(;%3`*&%(limite dGF1F,\"\"%F1F1\"#;" }}}{EXCHG {PARA 304 "" 0 "" {TEXT -1 32 "Computat ion of the expression " }{TEXT 482 1 "A" }{TEXT -1 7 " and " } {TEXT 483 1 "B" }{TEXT -1 47 " needed for the linear equation satisfi ed by " }{XPPEDIT 18 0 "u[sharp] - u[natural]" ",&&%\"uG6#%&sharpG\" \"\"&F$6#%(naturalG!\"\"" }{TEXT -1 19 " (see the article)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 216 " u_becarre_DAC := [u_diese[1], u_becarr e, zeroDA(epsilon,N)]:\n subsDACDAC ( du=soustractionDAC(u_diese,u_be carre_DAC), [u_diese[1],Delta_u_equ_becarre,zeroDA(epsilon,N)]) :\n A := subsDACDAC ( u=u_becarre_DAC , \"):" }}{PARA 305 "" 0 "" {TEXT 484 16 "Computation of " }{XPPEDIT 18 0 " E[sharp] - E[natural]" ",&& %\"EG6#%&sharpG\"\"\"&F$6#%(naturalG!\"\"" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 686 " u0_diese_DA := taylorEB(u0_diese,epsilon=0,N):\n \+ f:=intDAC ( A , +1)[2]:\n f00:=subs(t=t0,f[2][1]):\n f_sans_f00 := s implifyEB(subs(f[2][1]=f[2][1]-f00,f)): # The constant f00 will be re store before the end of the procedure\n methode_du_col ( f_sans_f00, \+ B[2], t, t0, eta ): # the even terms in eta vanish\n [epsilon,[s eq(\"[2][2*i],i=1..ordreDA(\")/2)]]: # the factor 1/sqrt(epsil on) will be restore before the end of the procedure \n subsDAf(s=\",1 /s):\n multDA ( soustractionDA(u0_becarre_DA,u0_diese_DA) ,\"):\n E_ diese_moins_E_becarre := exp(f00/epsilon) * sqrt(epsilon) * \"[2][1] * (polyDA( scalmulDA(1/\"[2][1],\"))+limited*epsilon^(ordreDA(\"))) ; #f ormatting the result" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%8E_diese_moi ns_E_becarreG,$*,-%$expG6#,$*$%(epsilonG!\"\"#!\"%\"\"$\"\"\"F,#F1\"\" #F3F2%#PiG#F-F3,,F1F1F,#!#r\"#'**$F,F3#!%*H'\"&K%=*$F,F0#!(26p#\"(;%3` *&%(limitedGF1F,\"\"%F1F1\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "21 12 0 0" 0 }{VIEWOPTS 1 1 0 3 4 1802 }