Changeset 113
- Timestamp:
- Dec 22, 2007, 3:31:50 AM (18 years ago)
- Author:
- neil.c.c.brown
- Message:
-
Added lots more messy maths to the slides
- File:
-
- 1 edited
- docs/trunk/omega-test-slides/omega-test.tex (modified) (3 diffs)
Legend:
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docs/trunk/omega-test-slides/omega-test.tex
r112 r113 214 214 215 215 %TODO probably best shown in two dimensions 216 (追記) (追記ここまで) 217 (追記) (追記ここまで) 218 (追記) %TODO show tightening inequalities during normalisation. Maybe show two normalising to an equality? (追記ここまで) 216 219 217 220 %TODO show a triangle of inequalities, then demonstrate projecting this onto one dimension: … … 276 279 \section{Answers to Code} 277 280 281 (追記) %Need an equality example with a step that has no unit coefficient: (追記ここまで) 282 (追記) (追記ここまで) 283 (追記) % Working backwards (solution: 5,7,-6): (追記ここまで) 284 (追記) % sigma = -3 (追記ここまで) 285 (追記) % -19sigma = 57 (追記ここまで) 286 (追記) % (Subst: z = -3sigma - 15) (追記ここまで) 287 (追記) % -12sigma - 4z = 60 (=> -3sigma - z = 15), z - 16sigma = 42 (追記ここまで) 288 (追記) % (subst: 4sigma = -x + z - 1, therefore x = z - 4sigma - 1) (追記ここまで) 289 (追記) % 3x - 7z = 57, 4x - 3z = 38 (追記ここまで) 290 (追記) % (subst: y = 5x + 3z) (追記ここまで) 291 (追記) % 5x - y + 3z = 0, 18x - 3y + 2z = 57, 14x - 2y + 3z = 38 (追記ここまで) 292 (追記) (追記ここまで) 293 (追記) \begin{frame}<1-5> [fragile] (追記ここまで) 294 (追記) \frametitle{A Worked Problem} (追記ここまで) 295 (追記) \begin{tabular}{ll} (追記ここまで) 296 (追記) Substitution & Equations \\ \hline (追記ここまで) 297 (追記) (追記ここまで) 298 (追記) ~ & $\begin{array}{rrrcr} (追記ここまで) 299 (追記) 5x &- y + & 3z & = & 0 \\ (追記ここまで) 300 (追記) 18x &- 3y + & 2z & = & 57 \\ (追記ここまで) 301 (追記) 14x &- 2y + &3z &=& 38 \end{array}$ (追記ここまで) 302 (追記) (追記ここまで) 303 (追記) \\ \hline (追記ここまで) 304 (追記) \only<2->{ (追記ここまで) 305 (追記) $y = 5x + 3z$ & (追記ここまで) 306 (追記) $\begin{array}{rrcr} (追記ここまで) 307 (追記) 3x &- 7z &=& 57 \\ (追記ここまで) 308 (追記) 4x &- 3z &=& 38 \end{array}$} (追記ここまで) 309 (追記) (追記ここまで) 310 (追記) \\ \hline (追記ここまで) 311 (追記) \only<3->{ (追記ここまで) 312 (追記) $x = -4\sigma + z - 1$ & (追記ここまで) 313 (追記) $\begin{array}{rrcr} (追記ここまで) 314 (追記) -3\sigma &- z &=& 15 \\ (追記ここまで) 315 (追記) -16\sigma &+ z &=& 42 \end{array}$} (追記ここまで) 316 (追記) (追記ここまで) 317 (追記) \\ \hline (追記ここまで) 318 (追記) \only<4->{ (追記ここまで) 319 (追記) $z = -3\sigma - 15$ & (追記ここまで) 320 (追記) $-19\sigma = 57$} (追記ここまで) 321 (追記) (追記ここまで) 322 (追記) \\ \hline (追記ここまで) 323 (追記) \only<5->{ (追記ここまで) 324 (追記) $\sigma = -3$ & ~ \\} (追記ここまで) 325 (追記) (追記ここまで) 326 (追記) \end{tabular} (追記ここまで) 327 (追記) \end{frame} (追記ここまで) 328 (追記) (追記ここまで) 329 (追記) \begin{frame}<1-4>[fragile] (追記ここまで) 330 (追記) \frametitle{A Worked Problem -- Solved} (追記ここまで) 331 (追記) \begin{tabular}{l} (追記ここまで) 332 (追記) Substitution/Values \\ \hline (追記ここまで) 333 (追記) (追記ここまで) 334 (追記) \only<1-3>{$y = 5x + 3z$}\only<4->{$y = 7$} \\ (追記ここまで) 335 (追記) (追記ここまで) 336 (追記) \only<1-2>{$x = -4\sigma + z - 1$}\only<3->{$x = 5$} \\ (追記ここまで) 337 (追記) (追記ここまで) 338 (追記) \only<1>{$z = -3\sigma - 15$}\only<2->{$z = -6$} \\ (追記ここまで) 339 (追記) (追記ここまで) 340 (追記) $\sigma = -3$ \\ (追記ここまで) 341 (追記) \end{tabular} (追記ここまで) 342 (追記) \end{frame} (追記ここまで) 343 (追記) (追記ここまで) 344 (追記) %Adapting previous problem: (追記ここまで) 345 (追記) (追記ここまで) 346 (追記) % 55 <= -19sigma <= 57 (追記ここまで) 347 (追記) % (Subst: z = -3sigma - 15) (追記ここまで) 348 (追記) % -12sigma - 4z = 60 (=> -3sigma - z = 15), 40 <= z - 16sigma <= 42 (追記ここまで) 349 (追記) % (subst: 4sigma = -x + z - 1, therefore x = z - 4sigma - 1) (追記ここまで) 350 (追記) % 3x - 7z = 57, 37 <= 4x - 3z <= 39 (追記ここまで) 351 (追記) % (subst: y = 5x + 3z) (追記ここまで) 352 (追記) % 5x - y + 3z = 0, 18x - 3y + 2z = 57, 37 <= 14x - 2y + 3z <= 39 (追記ここまで) 353 (追記) (追記ここまで) 354 (追記) %TODO change the bounds above (37,39) so that more than one value of sigma will lie between them. (追記ここまで) 355 (追記) %In the current problem, normalising sigma turns the problem into an equality (I think - check that too) (追記ここまで) 356 (追記) (追記ここまで) 278 357 \begin{frame}<1-4> [fragile] 279 358 \frametitle{A Worked Problem} … … 281 360 Substitution & Equations \\ \hline 282 361 283 ~ & $\begin{array}{rrr (削除) (削除ここまで)cr}284 (削除) 3x &- y + & z & = & 3 (削除ここまで)0 \\285 (削除) x &+ y + & 3z & = & 50 (削除ここまで)\\286 (削除) 4x &+ &z &=& 53 (削除ここまで)\end{array}$362 ~ & $\begin{array}{rrr(追記) r (追記ここまで)cr} 363 (追記) & 5x &- y + & 3z & = & (追記ここまで)0 \\ 364 (追記) & 18x &- 3y + & 2z & = & 57 (追記ここまで) \\ 365 (追記) 37 \leq & 14x &- 2y + &3z &\leq& 39 (追記ここまで) \end{array}$ 287 366 288 367 \\ \hline 289 368 \only<2->{ 290 $y = (削除) 3x + z - 30 (削除ここまで)$ &291 $\begin{array}{rr (削除) (削除ここまで)cr}292 (削除) x &+ z &=& 20 (削除ここまで)\\293 (削除) 4x &+ z &=& 53 (削除ここまで)\end{array}$}369 $y = (追記) 5x + 3z (追記ここまで)$ & 370 $\begin{array}{rr(追記) r (追記ここまで)cr} 371 (追記) & 3x &- 7z &=& 57 (追記ここまで) \\ 372 (追記) 37 \leq & 4x &- 3z &\leq& 39 (追記ここまで) \end{array}$} 294 373 295 374 \\ \hline 296 375 \only<3->{ 297 $x = 20 - z$ & 298 $\begin{array}{rcr} 299 -4z &=& -27 \end{array}$} 300 301 \\ \hline 302 \only<4->{ 303 $z = 9$ & ~ \\} 376 $x = -4\sigma + z - 1$ & 377 $\begin{array}{rrrcr} 378 &-3\sigma &- z &=& 15 \\ 379 40 \leq &-16\sigma &+ z &\leq& 42 \end{array}$} 380 381 \\ \hline 382 \only<4->{ 383 $z = -3\sigma - 15$ & 384 55ドル \leq -19\sigma \leq 57$} 304 385 305 386 \end{tabular} 306 \end{frame} 307 308 \begin{frame}<1-3>[fragile] 309 \frametitle{A Worked Problem -- Solved} 310 \begin{tabular}{l} 311 Substitution/Values \\ \hline 312 313 \only<1-2>{$y = 3x + z - 30$}\only<3->{$y = 12$} \\ 314 315 \only<1>{$x = 20 - z$}\only<2->{$x = 11$} \\ 316 317 $z = 9$ \\ 387 388 \only<4->{where $-19\sigma = \ldots$?} 389 390 \end{frame} 391 392 %TODO explain why we can't divide in these examples 393 394 \begin{frame}<1-4> [fragile] 395 \frametitle{A Worked Problem} 396 \begin{tabular}{ll} 397 Rev Substitution & Rev Equations \\ \hline 398 399 & \only<4->{$-120 \leq 19z \leq -114$} 400 401 402 \\ \hline 403 \only<1>{$z = -3\sigma - 15$} \only<2->{3ドル\sigma = -z - 15$} 404 & \only<1-2>{55ドル \leq -19\sigma \leq 57$} \only<3>{165ドル \leq -57\sigma \leq 171$} 405 318 406 \end{tabular} 319 \end{frame} 407 408 Error message: Unsafe when $-120 \leq 19z \leq -114,ドル 3ドルx + z = -63$ and $y = 5x + 3z$. Good luck! 409 410 %TODO need to understand the omega test a little better to know all possible ways to improve this 411 412 413 \end{frame} 414 320 415 321 416
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