1\begin{patch}{ugxProblemSeriesBernoulliPagePatch1} 2\begin{paste}{ugxProblemSeriesBernoulliPageFull1}{ugxProblemSeriesBernoulliPageEmpty1} 3\pastebutton{ugxProblemSeriesBernoulliPageFull1}{\hidepaste} 4\tab{5}\spadcommand{reduce(+,[m^4 for m in 1..10])} 5\indentrel{3}\begin{verbatim} 6 (1) 25333 7 Type: PositiveInteger 8\end{verbatim} 9\indentrel{-3}\end{paste}\end{patch} 10 11\begin{patch}{ugxProblemSeriesBernoulliPageEmpty1} 12\begin{paste}{ugxProblemSeriesBernoulliPageEmpty1}{ugxProblemSeriesBernoulliPagePatch1} 13\pastebutton{ugxProblemSeriesBernoulliPageEmpty1}{\showpaste} 14\tab{5}\spadcommand{reduce(+,[m^4 for m in 1..10])} 15\end{paste}\end{patch} 16 17\begin{patch}{ugxProblemSeriesBernoulliPagePatch2} 18\begin{paste}{ugxProblemSeriesBernoulliPageFull2}{ugxProblemSeriesBernoulliPageEmpty2} 19\pastebutton{ugxProblemSeriesBernoulliPageFull2}{\hidepaste} 20\tab{5}\spadcommand{sum4 := sum(m^4, m = 1..k)\bound{sum4 }} 21\indentrel{3}\begin{verbatim} 22 5 4 3 23 6 k + 15 k + 10 k - k 24 (2) ------------------------ 25 30 26 Type: Fraction(Polynomial(Integer)) 27\end{verbatim} 28\indentrel{-3}\end{paste}\end{patch} 29 30\begin{patch}{ugxProblemSeriesBernoulliPageEmpty2} 31\begin{paste}{ugxProblemSeriesBernoulliPageEmpty2}{ugxProblemSeriesBernoulliPagePatch2} 32\pastebutton{ugxProblemSeriesBernoulliPageEmpty2}{\showpaste} 33\tab{5}\spadcommand{sum4 := sum(m^4, m = 1..k)\bound{sum4 }} 34\end{paste}\end{patch} 35 36\begin{patch}{ugxProblemSeriesBernoulliPagePatch3} 37\begin{paste}{ugxProblemSeriesBernoulliPageFull3}{ugxProblemSeriesBernoulliPageEmpty3} 38\pastebutton{ugxProblemSeriesBernoulliPageFull3}{\hidepaste} 39\tab{5}\spadcommand{eval(sum4, k = 10)\free{sum4 }} 40\indentrel{3}\begin{verbatim} 41 (3) 25333 42 Type: Fraction(Polynomial(Integer)) 43\end{verbatim} 44\indentrel{-3}\end{paste}\end{patch} 45 46\begin{patch}{ugxProblemSeriesBernoulliPageEmpty3} 47\begin{paste}{ugxProblemSeriesBernoulliPageEmpty3}{ugxProblemSeriesBernoulliPagePatch3} 48\pastebutton{ugxProblemSeriesBernoulliPageEmpty3}{\showpaste} 49\tab{5}\spadcommand{eval(sum4, k = 10)\free{sum4 }} 50\end{paste}\end{patch} 51 52\begin{patch}{ugxProblemSeriesBernoulliPagePatch4} 53\begin{paste}{ugxProblemSeriesBernoulliPageFull4}{ugxProblemSeriesBernoulliPageEmpty4} 54\pastebutton{ugxProblemSeriesBernoulliPageFull4}{\hidepaste} 55\tab{5}\spadcommand{f := t*exp(x*t) / (exp(t) - 1)\bound{f }} 56\indentrel{3}\begin{verbatim} 57 t x 58 t %e 59 (4) ------- 60 t 61 %e - 1 62 Type: Expression(Integer) 63\end{verbatim} 64\indentrel{-3}\end{paste}\end{patch} 65 66\begin{patch}{ugxProblemSeriesBernoulliPageEmpty4} 67\begin{paste}{ugxProblemSeriesBernoulliPageEmpty4}{ugxProblemSeriesBernoulliPagePatch4} 68\pastebutton{ugxProblemSeriesBernoulliPageEmpty4}{\showpaste} 69\tab{5}\spadcommand{f := t*exp(x*t) / (exp(t) - 1)\bound{f }} 70\end{paste}\end{patch} 71 72\begin{patch}{ugxProblemSeriesBernoulliPagePatch5} 73\begin{paste}{ugxProblemSeriesBernoulliPageFull5}{ugxProblemSeriesBernoulliPageEmpty5} 74\pastebutton{ugxProblemSeriesBernoulliPageFull5}{\hidepaste} 75\tab{5}\spadcommand{)set streams calculate 5\bound{set }} 76\indentrel{3}\begin{verbatim} 77\end{verbatim} 78\indentrel{-3}\end{paste}\end{patch} 79 80\begin{patch}{ugxProblemSeriesBernoulliPageEmpty5} 81\begin{paste}{ugxProblemSeriesBernoulliPageEmpty5}{ugxProblemSeriesBernoulliPagePatch5} 82\pastebutton{ugxProblemSeriesBernoulliPageEmpty5}{\showpaste} 83\tab{5}\spadcommand{)set streams calculate 5\bound{set }} 84\end{paste}\end{patch} 85 86\begin{patch}{ugxProblemSeriesBernoulliPagePatch6} 87\begin{paste}{ugxProblemSeriesBernoulliPageFull6}{ugxProblemSeriesBernoulliPageEmpty6} 88\pastebutton{ugxProblemSeriesBernoulliPageFull6}{\hidepaste} 89\tab{5}\spadcommand{ff := taylor(f,t = 0)\free{f set }\bound{ff }} 90\indentrel{3}\begin{verbatim} 91 (5) 92 2 93 2 x - 1 6 x - 6 x + 1 2 94 1 + ------- t + -------------- t 95 2 12 96 + 97 3 2 4 3 2 98 2 x - 3 x + x 3 30 x - 60 x + 30 x - 1 4 99 --------------- t + ------------------------- t 100 12 720 101 + 102 5 4 3 103 6 x - 15 x + 10 x - x 5 6 104 ------------------------ t + O(t ) 105 720 106 Type: UnivariateTaylorSeries(Expression(Integer),t,0) 107\end{verbatim} 108\indentrel{-3}\end{paste}\end{patch} 109 110\begin{patch}{ugxProblemSeriesBernoulliPageEmpty6} 111\begin{paste}{ugxProblemSeriesBernoulliPageEmpty6}{ugxProblemSeriesBernoulliPagePatch6} 112\pastebutton{ugxProblemSeriesBernoulliPageEmpty6}{\showpaste} 113\tab{5}\spadcommand{ff := taylor(f,t = 0)\free{f set }\bound{ff }} 114\end{paste}\end{patch} 115 116\begin{patch}{ugxProblemSeriesBernoulliPagePatch7} 117\begin{paste}{ugxProblemSeriesBernoulliPageFull7}{ugxProblemSeriesBernoulliPageEmpty7} 118\pastebutton{ugxProblemSeriesBernoulliPageFull7}{\hidepaste} 119\tab{5}\spadcommand{factorial(6) * coefficient(ff,6)\free{ff }} 120\indentrel{3}\begin{verbatim} 121 6 5 4 2 122 42 x - 126 x + 105 x - 21 x + 1 123 (6) ----------------------------------- 124 42 125 Type: Expression(Integer) 126\end{verbatim} 127\indentrel{-3}\end{paste}\end{patch} 128 129\begin{patch}{ugxProblemSeriesBernoulliPageEmpty7} 130\begin{paste}{ugxProblemSeriesBernoulliPageEmpty7}{ugxProblemSeriesBernoulliPagePatch7} 131\pastebutton{ugxProblemSeriesBernoulliPageEmpty7}{\showpaste} 132\tab{5}\spadcommand{factorial(6) * coefficient(ff,6)\free{ff }} 133\end{paste}\end{patch} 134 135\begin{patch}{ugxProblemSeriesBernoulliPagePatch8} 136\begin{paste}{ugxProblemSeriesBernoulliPageFull8}{ugxProblemSeriesBernoulliPageEmpty8} 137\pastebutton{ugxProblemSeriesBernoulliPageFull8}{\hidepaste} 138\tab{5}\spadcommand{g := eval(f, x = x + 1) - f\bound{g }\free{f }} 139\indentrel{3}\begin{verbatim} 140 t x + t t x 141 t %e - t %e 142 (7) --------------------- 143 t 144 %e - 1 145 Type: Expression(Integer) 146\end{verbatim} 147\indentrel{-3}\end{paste}\end{patch} 148 149\begin{patch}{ugxProblemSeriesBernoulliPageEmpty8} 150\begin{paste}{ugxProblemSeriesBernoulliPageEmpty8}{ugxProblemSeriesBernoulliPagePatch8} 151\pastebutton{ugxProblemSeriesBernoulliPageEmpty8}{\showpaste} 152\tab{5}\spadcommand{g := eval(f, x = x + 1) - f\bound{g }\free{f }} 153\end{paste}\end{patch} 154 155\begin{patch}{ugxProblemSeriesBernoulliPagePatch9} 156\begin{paste}{ugxProblemSeriesBernoulliPageFull9}{ugxProblemSeriesBernoulliPageEmpty9} 157\pastebutton{ugxProblemSeriesBernoulliPageFull9}{\hidepaste} 158\tab{5}\spadcommand{normalize(g)\free{g }} 159\indentrel{3}\begin{verbatim} 160 t x 161 (8) t %e 162 Type: Expression(Integer) 163\end{verbatim} 164\indentrel{-3}\end{paste}\end{patch} 165 166\begin{patch}{ugxProblemSeriesBernoulliPageEmpty9} 167\begin{paste}{ugxProblemSeriesBernoulliPageEmpty9}{ugxProblemSeriesBernoulliPagePatch9} 168\pastebutton{ugxProblemSeriesBernoulliPageEmpty9}{\showpaste} 169\tab{5}\spadcommand{normalize(g)\free{g }} 170\end{paste}\end{patch} 171 172\begin{patch}{ugxProblemSeriesBernoulliPagePatch10} 173\begin{paste}{ugxProblemSeriesBernoulliPageFull10}{ugxProblemSeriesBernoulliPageEmpty10} 174\pastebutton{ugxProblemSeriesBernoulliPageFull10}{\hidepaste} 175\tab{5}\spadcommand{taylor(g,t = 0)\free{g }} 176\indentrel{3}\begin{verbatim} 177 2 3 4 178 2 x 3 x 4 x 5 6 179 (9) t + x t + -- t + -- t + -- t + O(t ) 180 2 6 24 181 Type: UnivariateTaylorSeries(Expression(Integer),t,0) 182\end{verbatim} 183\indentrel{-3}\end{paste}\end{patch} 184 185\begin{patch}{ugxProblemSeriesBernoulliPageEmpty10} 186\begin{paste}{ugxProblemSeriesBernoulliPageEmpty10}{ugxProblemSeriesBernoulliPagePatch10} 187\pastebutton{ugxProblemSeriesBernoulliPageEmpty10}{\showpaste} 188\tab{5}\spadcommand{taylor(g,t = 0)\free{g }} 189\end{paste}\end{patch} 190 191\begin{patch}{ugxProblemSeriesBernoulliPagePatch11} 192\begin{paste}{ugxProblemSeriesBernoulliPageFull11}{ugxProblemSeriesBernoulliPageEmpty11} 193\pastebutton{ugxProblemSeriesBernoulliPageFull11}{\hidepaste} 194\tab{5}\spadcommand{B5 := factorial(5) * coefficient(ff,5)\free{ff }\bound{B5 }} 195\indentrel{3}\begin{verbatim} 196 5 4 3 197 6 x - 15 x + 10 x - x 198 (10) ------------------------ 199 6 200 Type: Expression(Integer) 201\end{verbatim} 202\indentrel{-3}\end{paste}\end{patch} 203 204\begin{patch}{ugxProblemSeriesBernoulliPageEmpty11} 205\begin{paste}{ugxProblemSeriesBernoulliPageEmpty11}{ugxProblemSeriesBernoulliPagePatch11} 206\pastebutton{ugxProblemSeriesBernoulliPageEmpty11}{\showpaste} 207\tab{5}\spadcommand{B5 := factorial(5) * coefficient(ff,5)\free{ff }\bound{B5 }} 208\end{paste}\end{patch} 209 210\begin{patch}{ugxProblemSeriesBernoulliPagePatch12} 211\begin{paste}{ugxProblemSeriesBernoulliPageFull12}{ugxProblemSeriesBernoulliPageEmpty12} 212\pastebutton{ugxProblemSeriesBernoulliPageFull12}{\hidepaste} 213\tab{5}\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1))\free{B5 }} 214\indentrel{3}\begin{verbatim} 215 5 4 3 216 6 k + 15 k + 10 k - k 217 (11) ------------------------ 218 30 219 Type: Expression(Integer) 220\end{verbatim} 221\indentrel{-3}\end{paste}\end{patch} 222 223\begin{patch}{ugxProblemSeriesBernoulliPageEmpty12} 224\begin{paste}{ugxProblemSeriesBernoulliPageEmpty12}{ugxProblemSeriesBernoulliPagePatch12} 225\pastebutton{ugxProblemSeriesBernoulliPageEmpty12}{\showpaste} 226\tab{5}\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1))\free{B5 }} 227\end{paste}\end{patch} 228 229\begin{patch}{ugxProblemSeriesBernoulliPagePatch13} 230\begin{paste}{ugxProblemSeriesBernoulliPageFull13}{ugxProblemSeriesBernoulliPageEmpty13} 231\pastebutton{ugxProblemSeriesBernoulliPageFull13}{\hidepaste} 232\tab{5}\spadcommand{sum4\free{sum4 }} 233\indentrel{3}\begin{verbatim} 234 5 4 3 235 6 k + 15 k + 10 k - k 236 (12) ------------------------ 237 30 238 Type: Fraction(Polynomial(Integer)) 239\end{verbatim} 240\indentrel{-3}\end{paste}\end{patch} 241 242\begin{patch}{ugxProblemSeriesBernoulliPageEmpty13} 243\begin{paste}{ugxProblemSeriesBernoulliPageEmpty13}{ugxProblemSeriesBernoulliPagePatch13} 244\pastebutton{ugxProblemSeriesBernoulliPageEmpty13}{\showpaste} 245\tab{5}\spadcommand{sum4\free{sum4 }} 246\end{paste}\end{patch} 247 248\begin{patch}{ugProblemLaplacePagePatch1} 249\begin{paste}{ugProblemLaplacePageFull1}{ugProblemLaplacePageEmpty1} 250\pastebutton{ugProblemLaplacePageFull1}{\hidepaste} 251\tab{5}\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)} 252\indentrel{3}\begin{verbatim} 253 3 254 4 a 255 (1) --------- 256 4 4 257 s + 4 a 258 Type: Expression(Integer) 259\end{verbatim} 260\indentrel{-3}\end{paste}\end{patch} 261 262\begin{patch}{ugProblemLaplacePageEmpty1} 263\begin{paste}{ugProblemLaplacePageEmpty1}{ugProblemLaplacePagePatch1} 264\pastebutton{ugProblemLaplacePageEmpty1}{\showpaste} 265\tab{5}\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)} 266\end{paste}\end{patch} 267 268\begin{patch}{ugProblemLaplacePagePatch2} 269\begin{paste}{ugProblemLaplacePageFull2}{ugProblemLaplacePageEmpty2} 270\pastebutton{ugProblemLaplacePageFull2}{\hidepaste} 271\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)} 272\indentrel{3}\begin{verbatim} 273 (2) - log(s - a) + log(s - b) 274 Type: Expression(Integer) 275\end{verbatim} 276\indentrel{-3}\end{paste}\end{patch} 277 278\begin{patch}{ugProblemLaplacePageEmpty2} 279\begin{paste}{ugProblemLaplacePageEmpty2}{ugProblemLaplacePagePatch2} 280\pastebutton{ugProblemLaplacePageEmpty2}{\showpaste} 281\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)} 282\end{paste}\end{patch} 283 284\begin{patch}{ugProblemLaplacePagePatch3} 285\begin{paste}{ugProblemLaplacePageFull3}{ugProblemLaplacePageEmpty3} 286\pastebutton{ugProblemLaplacePageFull3}{\hidepaste} 287\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)} 288\indentrel{3}\begin{verbatim} 289 2 2 290 (3) log(s + a ) - 2 log(s) 291 Type: Expression(Integer) 292\end{verbatim} 293\indentrel{-3}\end{paste}\end{patch} 294 295\begin{patch}{ugProblemLaplacePageEmpty3} 296\begin{paste}{ugProblemLaplacePageEmpty3}{ugProblemLaplacePagePatch3} 297\pastebutton{ugProblemLaplacePageEmpty3}{\showpaste} 298\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)} 299\end{paste}\end{patch} 300 301\begin{patch}{ugProblemLaplacePagePatch4} 302\begin{paste}{ugProblemLaplacePageFull4}{ugProblemLaplacePageEmpty4} 303\pastebutton{ugProblemLaplacePageFull4}{\hidepaste} 304\tab{5}\spadcommand{laplace(exp(-a*t) * sin(b*t) / b^2, t, s)} 305\indentrel{3}\begin{verbatim} 306 1 307 (4) ------------------------- 308 2 3 2 309 b s + 2 a b s + b + a b 310 Type: Expression(Integer) 311\end{verbatim} 312\indentrel{-3}\end{paste}\end{patch} 313 314\begin{patch}{ugProblemLaplacePageEmpty4} 315\begin{paste}{ugProblemLaplacePageEmpty4}{ugProblemLaplacePagePatch4} 316\pastebutton{ugProblemLaplacePageEmpty4}{\showpaste} 317\tab{5}\spadcommand{laplace(exp(-a*t) * sin(b*t) / b^2, t, s)} 318\end{paste}\end{patch} 319 320\begin{patch}{ugProblemLaplacePagePatch5} 321\begin{paste}{ugProblemLaplacePageFull5}{ugProblemLaplacePageEmpty5} 322\pastebutton{ugProblemLaplacePageFull5}{\hidepaste} 323\tab{5}\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)} 324\indentrel{3}\begin{verbatim} 325 2 2 2 2 326 log(s + b ) - log(s + a ) 327 (5) --------------------------- 328 2 329 Type: Expression(Integer) 330\end{verbatim} 331\indentrel{-3}\end{paste}\end{patch} 332 333\begin{patch}{ugProblemLaplacePageEmpty5} 334\begin{paste}{ugProblemLaplacePageEmpty5}{ugProblemLaplacePagePatch5} 335\pastebutton{ugProblemLaplacePageEmpty5}{\showpaste} 336\tab{5}\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)} 337\end{paste}\end{patch} 338 339\begin{patch}{ugProblemLaplacePagePatch6} 340\begin{paste}{ugProblemLaplacePageFull6}{ugProblemLaplacePageEmpty6} 341\pastebutton{ugProblemLaplacePageFull6}{\hidepaste} 342\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)} 343\indentrel{3}\begin{verbatim} 344 b s + c - a 345 %e log(---------) 346 c 347 (6) ----------------- 348 s - a 349 Type: Expression(Integer) 350\end{verbatim} 351\indentrel{-3}\end{paste}\end{patch} 352 353\begin{patch}{ugProblemLaplacePageEmpty6} 354\begin{paste}{ugProblemLaplacePageEmpty6}{ugProblemLaplacePagePatch6} 355\pastebutton{ugProblemLaplacePageEmpty6}{\showpaste} 356\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)} 357\end{paste}\end{patch} 358 359\begin{patch}{ugProblemLaplacePagePatch7} 360\begin{paste}{ugProblemLaplacePageFull7}{ugProblemLaplacePageEmpty7} 361\pastebutton{ugProblemLaplacePageFull7}{\hidepaste} 362\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)} 363\indentrel{3}\begin{verbatim} 364 2 2 365 s + b d 366 a log(-------) + 2 c atan(-) 367 2 s 368 b 369 (7) ---------------------------- 370 2 s 371 Type: Expression(Integer) 372\end{verbatim} 373\indentrel{-3}\end{paste}\end{patch} 374 375\begin{patch}{ugProblemLaplacePageEmpty7} 376\begin{paste}{ugProblemLaplacePageEmpty7}{ugProblemLaplacePagePatch7} 377\pastebutton{ugProblemLaplacePageEmpty7}{\showpaste} 378\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)} 379\end{paste}\end{patch} 380 381\begin{patch}{ugProblemLaplacePagePatch8} 382\begin{paste}{ugProblemLaplacePageFull8}{ugProblemLaplacePageEmpty8} 383\pastebutton{ugProblemLaplacePageFull8}{\hidepaste} 384\tab{5}\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t^2), t, s)} 385\indentrel{3}\begin{verbatim} 386 2 387 4 2 2 4 t 3 388 (s + 2 a s + a )laplace(%e ,t,s) + 2 a 389 (8) ------------------------------------------ 390 4 2 2 4 391 s + 2 a s + a 392 Type: Expression(Integer) 393\end{verbatim} 394\indentrel{-3}\end{paste}\end{patch} 395 396\begin{patch}{ugProblemLaplacePageEmpty8} 397\begin{paste}{ugProblemLaplacePageEmpty8}{ugProblemLaplacePagePatch8} 398\pastebutton{ugProblemLaplacePageEmpty8}{\showpaste} 399\tab{5}\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t^2), t, s)} 400\end{paste}\end{patch} 401 402\begin{patch}{ugxProblemSeriesFunctionsPagePatch1} 403\begin{paste}{ugxProblemSeriesFunctionsPageFull1}{ugxProblemSeriesFunctionsPageEmpty1} 404\pastebutton{ugxProblemSeriesFunctionsPageFull1}{\hidepaste} 405\tab{5}\spadcommand{x := series 'x\bound{x }} 406\indentrel{3}\begin{verbatim} 407 (1) x 408 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 409\end{verbatim} 410\indentrel{-3}\end{paste}\end{patch} 411 412\begin{patch}{ugxProblemSeriesFunctionsPageEmpty1} 413\begin{paste}{ugxProblemSeriesFunctionsPageEmpty1}{ugxProblemSeriesFunctionsPagePatch1} 414\pastebutton{ugxProblemSeriesFunctionsPageEmpty1}{\showpaste} 415\tab{5}\spadcommand{x := series 'x\bound{x }} 416\end{paste}\end{patch} 417 418\begin{patch}{ugxProblemSeriesFunctionsPagePatch2} 419\begin{paste}{ugxProblemSeriesFunctionsPageFull2}{ugxProblemSeriesFunctionsPageEmpty2} 420\pastebutton{ugxProblemSeriesFunctionsPageFull2}{\hidepaste} 421\tab{5}\spadcommand{rat := x^2 / (1 - 6*x + x^2)\free{x }\bound{rat }} 422\indentrel{3}\begin{verbatim} 423 (2) 424 2 3 4 5 6 7 425 x + 6 x + 35 x + 204 x + 1189 x + 6930 x 426 + 427 8 9 10 11 428 40391 x + 235416 x + 1372105 x + 7997214 x 429 + 430 12 13 431 46611179 x + O(x ) 432 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 433\end{verbatim} 434\indentrel{-3}\end{paste}\end{patch} 435 436\begin{patch}{ugxProblemSeriesFunctionsPageEmpty2} 437\begin{paste}{ugxProblemSeriesFunctionsPageEmpty2}{ugxProblemSeriesFunctionsPagePatch2} 438\pastebutton{ugxProblemSeriesFunctionsPageEmpty2}{\showpaste} 439\tab{5}\spadcommand{rat := x^2 / (1 - 6*x + x^2)\free{x }\bound{rat }} 440\end{paste}\end{patch} 441 442\begin{patch}{ugxProblemSeriesFunctionsPagePatch3} 443\begin{paste}{ugxProblemSeriesFunctionsPageFull3}{ugxProblemSeriesFunctionsPageEmpty3} 444\pastebutton{ugxProblemSeriesFunctionsPageFull3}{\hidepaste} 445\tab{5}\spadcommand{sin(rat)\free{rat }} 446\indentrel{3}\begin{verbatim} 447 (3) 448 2 3 4 5 7133 6 7 449 x + 6 x + 35 x + 204 x + ---- x + 6927 x 450 6 451 + 452 80711 8 9 164285281 10 31888513 11 453 ----- x + 235068 x + --------- x + -------- x 454 2 120 4 455 + 456 371324777 12 13 457 --------- x + O(x ) 458 8 459 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 460\end{verbatim} 461\indentrel{-3}\end{paste}\end{patch} 462 463\begin{patch}{ugxProblemSeriesFunctionsPageEmpty3} 464\begin{paste}{ugxProblemSeriesFunctionsPageEmpty3}{ugxProblemSeriesFunctionsPagePatch3} 465\pastebutton{ugxProblemSeriesFunctionsPageEmpty3}{\showpaste} 466\tab{5}\spadcommand{sin(rat)\free{rat }} 467\end{paste}\end{patch} 468 469\begin{patch}{ugxProblemSeriesFunctionsPagePatch4} 470\begin{paste}{ugxProblemSeriesFunctionsPageFull4}{ugxProblemSeriesFunctionsPageEmpty4} 471\pastebutton{ugxProblemSeriesFunctionsPageFull4}{\hidepaste} 472\tab{5}\spadcommand{y : UTS(FRAC INT,y,0) := y\bound{y }} 473\indentrel{3}\begin{verbatim} 474 (4) y 475 Type: UnivariateTaylorSeries(Fraction(Integer),y,0) 476\end{verbatim} 477\indentrel{-3}\end{paste}\end{patch} 478 479\begin{patch}{ugxProblemSeriesFunctionsPageEmpty4} 480\begin{paste}{ugxProblemSeriesFunctionsPageEmpty4}{ugxProblemSeriesFunctionsPagePatch4} 481\pastebutton{ugxProblemSeriesFunctionsPageEmpty4}{\showpaste} 482\tab{5}\spadcommand{y : UTS(FRAC INT,y,0) := y\bound{y }} 483\end{paste}\end{patch} 484 485\begin{patch}{ugxProblemSeriesFunctionsPagePatch5} 486\begin{paste}{ugxProblemSeriesFunctionsPageFull5}{ugxProblemSeriesFunctionsPageEmpty5} 487\pastebutton{ugxProblemSeriesFunctionsPageFull5}{\hidepaste} 488\tab{5}\spadcommand{exp(y)\free{y }} 489\indentrel{3}\begin{verbatim} 490 (5) 491 1 2 1 3 1 4 1 5 1 6 492 1 + y + - y + - y + -- y + --- y + --- y 493 2 6 24 120 720 494 + 495 1 7 1 8 1 9 1 10 11 496 ---- y + ----- y + ------ y + ------- y + O(y ) 497 5040 40320 362880 3628800 498 Type: UnivariateTaylorSeries(Fraction(Integer),y,0) 499\end{verbatim} 500\indentrel{-3}\end{paste}\end{patch} 501 502\begin{patch}{ugxProblemSeriesFunctionsPageEmpty5} 503\begin{paste}{ugxProblemSeriesFunctionsPageEmpty5}{ugxProblemSeriesFunctionsPagePatch5} 504\pastebutton{ugxProblemSeriesFunctionsPageEmpty5}{\showpaste} 505\tab{5}\spadcommand{exp(y)\free{y }} 506\end{paste}\end{patch} 507 508\begin{patch}{ugxProblemSeriesFunctionsPagePatch6} 509\begin{paste}{ugxProblemSeriesFunctionsPageFull6}{ugxProblemSeriesFunctionsPageEmpty6} 510\pastebutton{ugxProblemSeriesFunctionsPageFull6}{\hidepaste} 511\tab{5}\spadcommand{tan(y^2)\free{y }} 512\indentrel{3}\begin{verbatim} 513 2 1 6 2 10 11 514 (6) y + - y + -- y + O(y ) 515 3 15 516 Type: UnivariateTaylorSeries(Fraction(Integer),y,0) 517\end{verbatim} 518\indentrel{-3}\end{paste}\end{patch} 519 520\begin{patch}{ugxProblemSeriesFunctionsPageEmpty6} 521\begin{paste}{ugxProblemSeriesFunctionsPageEmpty6}{ugxProblemSeriesFunctionsPagePatch6} 522\pastebutton{ugxProblemSeriesFunctionsPageEmpty6}{\showpaste} 523\tab{5}\spadcommand{tan(y^2)\free{y }} 524\end{paste}\end{patch} 525 526\begin{patch}{ugxProblemSeriesFunctionsPagePatch7} 527\begin{paste}{ugxProblemSeriesFunctionsPageFull7}{ugxProblemSeriesFunctionsPageEmpty7} 528\pastebutton{ugxProblemSeriesFunctionsPageFull7}{\hidepaste} 529\tab{5}\spadcommand{cos(y + y^5)\free{y }} 530\indentrel{3}\begin{verbatim} 531 (7) 532 1 2 1 4 721 6 6721 8 1844641 10 533 1 - - y + -- y - --- y + ----- y - ------- y 534 2 24 720 40320 3628800 535 + 536 11 537 O(y ) 538 Type: UnivariateTaylorSeries(Fraction(Integer),y,0) 539\end{verbatim} 540\indentrel{-3}\end{paste}\end{patch} 541 542\begin{patch}{ugxProblemSeriesFunctionsPageEmpty7} 543\begin{paste}{ugxProblemSeriesFunctionsPageEmpty7}{ugxProblemSeriesFunctionsPagePatch7} 544\pastebutton{ugxProblemSeriesFunctionsPageEmpty7}{\showpaste} 545\tab{5}\spadcommand{cos(y + y^5)\free{y }} 546\end{paste}\end{patch} 547 548\begin{patch}{ugxProblemSeriesFunctionsPagePatch8} 549\begin{paste}{ugxProblemSeriesFunctionsPageFull8}{ugxProblemSeriesFunctionsPageEmpty8} 550\pastebutton{ugxProblemSeriesFunctionsPageFull8}{\hidepaste} 551\tab{5}\spadcommand{log(1 + sin(y))\free{y }} 552\indentrel{3}\begin{verbatim} 553 (8) 554 1 2 1 3 1 4 1 5 1 6 61 7 555 y - - y + - y - -- y + -- y - -- y + ---- y 556 2 6 12 24 45 5040 557 + 558 17 8 277 9 31 10 11 559 - ---- y + ----- y - ----- y + O(y ) 560 2520 72576 14175 561 Type: UnivariateTaylorSeries(Fraction(Integer),y,0) 562\end{verbatim} 563\indentrel{-3}\end{paste}\end{patch} 564 565\begin{patch}{ugxProblemSeriesFunctionsPageEmpty8} 566\begin{paste}{ugxProblemSeriesFunctionsPageEmpty8}{ugxProblemSeriesFunctionsPagePatch8} 567\pastebutton{ugxProblemSeriesFunctionsPageEmpty8}{\showpaste} 568\tab{5}\spadcommand{log(1 + sin(y))\free{y }} 569\end{paste}\end{patch} 570 571\begin{patch}{ugxProblemSeriesFunctionsPagePatch9} 572\begin{paste}{ugxProblemSeriesFunctionsPageFull9}{ugxProblemSeriesFunctionsPageEmpty9} 573\pastebutton{ugxProblemSeriesFunctionsPageFull9}{\hidepaste} 574\tab{5}\spadcommand{z : UTS(EXPR INT,z,0) := z\bound{z }} 575\indentrel{3}\begin{verbatim} 576 (9) z 577 Type: UnivariateTaylorSeries(Expression(Integer),z,0) 578\end{verbatim} 579\indentrel{-3}\end{paste}\end{patch} 580 581\begin{patch}{ugxProblemSeriesFunctionsPageEmpty9} 582\begin{paste}{ugxProblemSeriesFunctionsPageEmpty9}{ugxProblemSeriesFunctionsPagePatch9} 583\pastebutton{ugxProblemSeriesFunctionsPageEmpty9}{\showpaste} 584\tab{5}\spadcommand{z : UTS(EXPR INT,z,0) := z\bound{z }} 585\end{paste}\end{patch} 586 587\begin{patch}{ugxProblemSeriesFunctionsPagePatch10} 588\begin{paste}{ugxProblemSeriesFunctionsPageFull10}{ugxProblemSeriesFunctionsPageEmpty10} 589\pastebutton{ugxProblemSeriesFunctionsPageFull10}{\hidepaste} 590\tab{5}\spadcommand{exp(2 + tan(z))\free{z }} 591\indentrel{3}\begin{verbatim} 592 (10) 593 2 2 2 2 594 2 2 %e 2 %e 3 3 %e 4 37 %e 5 595 %e + %e z + --- z + --- z + ----- z + ------ z 596 2 2 8 120 597 + 598 2 2 2 2 599 59 %e 6 137 %e 7 871 %e 8 41641 %e 9 600 ------ z + ------- z + ------- z + --------- z 601 240 720 5760 362880 602 + 603 2 604 325249 %e 10 11 605 ---------- z + O(z ) 606 3628800 607 Type: UnivariateTaylorSeries(Expression(Integer),z,0) 608\end{verbatim} 609\indentrel{-3}\end{paste}\end{patch} 610 611\begin{patch}{ugxProblemSeriesFunctionsPageEmpty10} 612\begin{paste}{ugxProblemSeriesFunctionsPageEmpty10}{ugxProblemSeriesFunctionsPagePatch10} 613\pastebutton{ugxProblemSeriesFunctionsPageEmpty10}{\showpaste} 614\tab{5}\spadcommand{exp(2 + tan(z))\free{z }} 615\end{paste}\end{patch} 616 617\begin{patch}{ugxProblemSeriesFunctionsPagePatch11} 618\begin{paste}{ugxProblemSeriesFunctionsPageFull11}{ugxProblemSeriesFunctionsPageEmpty11} 619\pastebutton{ugxProblemSeriesFunctionsPageFull11}{\hidepaste} 620\tab{5}\spadcommand{w := taylor 'w\bound{w }} 621\indentrel{3}\begin{verbatim} 622 (11) w 623 Type: UnivariateTaylorSeries(Expression(Integer),w,0) 624\end{verbatim} 625\indentrel{-3}\end{paste}\end{patch} 626 627\begin{patch}{ugxProblemSeriesFunctionsPageEmpty11} 628\begin{paste}{ugxProblemSeriesFunctionsPageEmpty11}{ugxProblemSeriesFunctionsPagePatch11} 629\pastebutton{ugxProblemSeriesFunctionsPageEmpty11}{\showpaste} 630\tab{5}\spadcommand{w := taylor 'w\bound{w }} 631\end{paste}\end{patch} 632 633\begin{patch}{ugxProblemSeriesFunctionsPagePatch12} 634\begin{paste}{ugxProblemSeriesFunctionsPageFull12}{ugxProblemSeriesFunctionsPageEmpty12} 635\pastebutton{ugxProblemSeriesFunctionsPageFull12}{\hidepaste} 636\tab{5}\spadcommand{exp(2 + tan(w))\free{w }} 637\indentrel{3}\begin{verbatim} 638 (12) 639 2 2 2 2 640 2 2 %e 2 %e 3 3 %e 4 37 %e 5 641 %e + %e w + --- w + --- w + ----- w + ------ w 642 2 2 8 120 643 + 644 2 2 2 2 645 59 %e 6 137 %e 7 871 %e 8 41641 %e 9 646 ------ w + ------- w + ------- w + --------- w 647 240 720 5760 362880 648 + 649 2 650 325249 %e 10 11 651 ---------- w + O(w ) 652 3628800 653 Type: UnivariateTaylorSeries(Expression(Integer),w,0) 654\end{verbatim} 655\indentrel{-3}\end{paste}\end{patch} 656 657\begin{patch}{ugxProblemSeriesFunctionsPageEmpty12} 658\begin{paste}{ugxProblemSeriesFunctionsPageEmpty12}{ugxProblemSeriesFunctionsPagePatch12} 659\pastebutton{ugxProblemSeriesFunctionsPageEmpty12}{\showpaste} 660\tab{5}\spadcommand{exp(2 + tan(w))\free{w }} 661\end{paste}\end{patch} 662 663\begin{patch}{ugProblemGeneticPagePatch1} 664\begin{paste}{ugProblemGeneticPageFull1}{ugProblemGeneticPageEmpty1} 665\pastebutton{ugProblemGeneticPageFull1}{\hidepaste} 666\tab{5}\spadcommand{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ]\bound{segregationRates }} 667\indentrel{3}\begin{verbatim} 668 (1) 669 + 1 1 9+ + 1 1+ + 1 1+ 670 |1 - - --| |0 - 0 --| |0 0 - --| 671 | 2 2 20| | 2 20| | 2 20| 672 | | | | | | 673 |1 1 | |1 9 1 | | 9 | 674 |- 0 -- 0 | |- 1 -- - | |0 0 -- 0 | 675 |2 20 | |2 20 2 | | 20 | 676 [| |, | |, | |, 677 |1 1 | | 9 | |1 9 1 | 678 |- -- 0 0 | |0 -- 0 0 | |- -- 1 - | 679 |2 20 | | 20 | |2 20 2 | 680 | | | | | | 681 | 9 | | 1 1 | | 1 1 | 682 |-- 0 0 0 | |-- - 0 0 | |-- 0 - 0 | 683 +20 + +20 2 + +20 2 + 684 + 9+ 685 |0 0 0 --| 686 | 20| 687 | | 688 | 1 1 | 689 |0 0 -- - | 690 | 20 2 | 691 | |] 692 | 1 1 | 693 |0 -- 0 - | 694 | 20 2 | 695 | | 696 | 9 1 1 | 697 |-- - - 1 | 698 +20 2 2 + 699 Type: List(SquareMatrix(4,Fraction(Integer))) 700\end{verbatim} 701\indentrel{-3}\end{paste}\end{patch} 702 703\begin{patch}{ugProblemGeneticPageEmpty1} 704\begin{paste}{ugProblemGeneticPageEmpty1}{ugProblemGeneticPagePatch1} 705\pastebutton{ugProblemGeneticPageEmpty1}{\showpaste} 706\tab{5}\spadcommand{segregationRates : List SquareMatrix(4,FRAC INT) := [matrix [ [1, 1/2, 1/2, 9/20], [1/2, 0, 1/20, 0], [1/2, 1/20, 0, 0], [9/20, 0, 0, 0] ], matrix [ [0, 1/2, 0, 1/20], [1/2, 1, 9/20, 1/2], [0, 9/20, 0, 0], [1/20, 1/2, 0, 0] ], matrix [ [0, 0, 1/2, 1/20], [0, 0, 9/20, 0], [1/2, 9/20, 1, 1/2], [1/20, 0, 1/2, 0] ], matrix [ [0, 0, 0, 9/20], [0, 0, 1/20, 1/2], [0, 1/20, 0, 1/2], [9/20, 1/2, 1/2, 1] ] ]\bound{segregationRates }} 707\end{paste}\end{patch} 708 709\begin{patch}{ugProblemGeneticPagePatch2} 710\begin{paste}{ugProblemGeneticPageFull2}{ugProblemGeneticPageEmpty2} 711\pastebutton{ugProblemGeneticPageFull2}{\hidepaste} 712\tab{5}\spadcommand{gametes := ['AB,'Ab,'aB,'ab]\bound{gametes }} 713\indentrel{3}\begin{verbatim} 714 (2) [AB, Ab, aB, ab] 715 Type: List(OrderedVariableList([AB,Ab,aB,ab])) 716\end{verbatim} 717\indentrel{-3}\end{paste}\end{patch} 718 719\begin{patch}{ugProblemGeneticPageEmpty2} 720\begin{paste}{ugProblemGeneticPageEmpty2}{ugProblemGeneticPagePatch2} 721\pastebutton{ugProblemGeneticPageEmpty2}{\showpaste} 722\tab{5}\spadcommand{gametes := ['AB,'Ab,'aB,'ab]\bound{gametes }} 723\end{paste}\end{patch} 724 725\begin{patch}{ugProblemGeneticPagePatch3} 726\begin{paste}{ugProblemGeneticPageFull3}{ugProblemGeneticPageEmpty3} 727\pastebutton{ugProblemGeneticPageFull3}{\hidepaste} 728\tab{5}\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A }\free{gametes segregationRates }} 729\indentrel{3}\begin{verbatim} 730 Type: Type 731\end{verbatim} 732\indentrel{-3}\end{paste}\end{patch} 733 734\begin{patch}{ugProblemGeneticPageEmpty3} 735\begin{paste}{ugProblemGeneticPageEmpty3}{ugProblemGeneticPagePatch3} 736\pastebutton{ugProblemGeneticPageEmpty3}{\showpaste} 737\tab{5}\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A }\free{gametes segregationRates }} 738\end{paste}\end{patch} 739 740\begin{patch}{ugProblemGeneticPagePatch4} 741\begin{paste}{ugProblemGeneticPageFull4}{ugProblemGeneticPageEmpty4} 742\pastebutton{ugProblemGeneticPageFull4}{\hidepaste} 743\tab{5}\spadcommand{a := basis()$A; a.1*a.4} 744\indentrel{3}\begin{verbatim} 745 9 1 1 9 746 (4) -- ab + -- aB + -- Ab + -- AB 747 20 20 20 20 748Type: AlgebraGivenByStructuralConstants(Fraction(Integer),4,[AB,Ab,aB,ab],[[[1,1/2,1/2,9/20],[1/2,0,1/20,0],[1/2,1/20,0,0],[9/20,0,0,0]],[[0,1/2,0,1/20],[1/2,1,9/20,1/2],[0,9/20,0,0],[1/20,1/2,0,0]],[[0,0,1/2,1/20],[0,0,9/20,0],[1/2,9/20,1,1/2],[1/20,0,1/2,0]],[[0,0,0,9/20],[0,0,1/20,1/2],[0,1/20,0,1/2],[9/20,1/2,1/2,1]]]) 749\end{verbatim} 750\indentrel{-3}\end{paste}\end{patch} 751 752\begin{patch}{ugProblemGeneticPageEmpty4} 753\begin{paste}{ugProblemGeneticPageEmpty4}{ugProblemGeneticPagePatch4} 754\pastebutton{ugProblemGeneticPageEmpty4}{\showpaste} 755\tab{5}\spadcommand{a := basis()$A; a.1*a.4} 756\end{paste}\end{patch} 757 758\begin{patch}{ugProblemGeneticPagePatch5} 759\begin{paste}{ugProblemGeneticPageFull5}{ugProblemGeneticPageEmpty5} 760\pastebutton{ugProblemGeneticPageFull5}{\hidepaste} 761\tab{5}\spadcommand{commutative?()$A\free{A }} 762\indentrel{3}\begin{verbatim} 763 algebra is commutative 764 (5) true 765 Type: Boolean 766\end{verbatim} 767\indentrel{-3}\end{paste}\end{patch} 768 769\begin{patch}{ugProblemGeneticPageEmpty5} 770\begin{paste}{ugProblemGeneticPageEmpty5}{ugProblemGeneticPagePatch5} 771\pastebutton{ugProblemGeneticPageEmpty5}{\showpaste} 772\tab{5}\spadcommand{commutative?()$A\free{A }} 773\end{paste}\end{patch} 774 775\begin{patch}{ugProblemGeneticPagePatch6} 776\begin{paste}{ugProblemGeneticPageFull6}{ugProblemGeneticPageEmpty6} 777\pastebutton{ugProblemGeneticPageFull6}{\hidepaste} 778\tab{5}\spadcommand{associative?()$A\free{A }} 779\indentrel{3}\begin{verbatim} 780 algebra is not associative 781 (6) false 782 Type: Boolean 783\end{verbatim} 784\indentrel{-3}\end{paste}\end{patch} 785 786\begin{patch}{ugProblemGeneticPageEmpty6} 787\begin{paste}{ugProblemGeneticPageEmpty6}{ugProblemGeneticPagePatch6} 788\pastebutton{ugProblemGeneticPageEmpty6}{\showpaste} 789\tab{5}\spadcommand{associative?()$A\free{A }} 790\end{paste}\end{patch} 791 792\begin{patch}{ugProblemGeneticPagePatch7} 793\begin{paste}{ugProblemGeneticPageFull7}{ugProblemGeneticPageEmpty7} 794\pastebutton{ugProblemGeneticPageFull7}{\hidepaste} 795\tab{5}\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]\bound{x }\free{gametes segregationRates }} 796\indentrel{3}\begin{verbatim} 797 (7) 0.4ab + 0.1aB + 0.2Ab + 0.3AB 798Type: AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab],[[[1,CONCAT(0,.,5),CONCAT(0,.,5),CONCAT(0,.,CONCAT(4,5))],[CONCAT(0,.,5),0,CONCAT(0,.,CONCAT(0,5)),0],[CONCAT(0,.,5),CONCAT(0,.,CONCAT(0,5)),0,0],[CONCAT(0,.,CONCAT(4,5)),0,0,0]],[[0,CONCAT(0,.,5),0,CONCAT(0,.,CONCAT(0,5))],[CONCAT(0,.,5),1,CONCAT(0,.,CONCAT(4,5)),CONCAT(0,.,5)],[0,CONCAT(0,.,CONCAT(4,5)),0,0],[CONCAT(0,.,CONCAT(0,5)),CONCAT(0,.,5),0,0]],[[0,0,CONCAT(0,.,5),CONCAT(0,.,CONCAT(0,5))],[0,0,CONCAT(0,.,CONCAT(4,5)),0],[CONCAT(0,.,5),CONCAT(0,.,CONCAT(4,5)),1,CONCAT(0,.,5)],[CONCAT(0,.,CONCAT(0,5)),0,CONCAT(0,.,5),0]],[[0,0,0,CONCAT(0,.,CONCAT(4,5))],[0,0,CONCAT(0,.,CONCAT(0,5)),CONCAT(0,.,5)],[0,CONCAT(0,.,CONCAT(0,5)),0,CONCAT(0,.,5)],[CONCAT(0,.,CONCAT(4,5)),CONCAT(0,.,5),CONCAT(0,.,5),1]]]) 799\end{verbatim} 800\indentrel{-3}\end{paste}\end{patch} 801 802\begin{patch}{ugProblemGeneticPageEmpty7} 803\begin{paste}{ugProblemGeneticPageEmpty7}{ugProblemGeneticPagePatch7} 804\pastebutton{ugProblemGeneticPageEmpty7}{\showpaste} 805\tab{5}\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]\bound{x }\free{gametes segregationRates }} 806\end{paste}\end{patch} 807 808\begin{patch}{ugProblemGeneticPagePatch8} 809\begin{paste}{ugProblemGeneticPageFull8}{ugProblemGeneticPageEmpty8} 810\pastebutton{ugProblemGeneticPageFull8}{\hidepaste} 811\tab{5}\spadcommand{plenaryPower(x,5)\free{x }} 812\indentrel{3}\begin{verbatim} 813 (8) 0.36561ab + 0.13439aB + 0.23439Ab + 0.26561AB 814Type: AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab],[[[1,CONCAT(0,.,5),CONCAT(0,.,5),CONCAT(0,.,CONCAT(4,5))],[CONCAT(0,.,5),0,CONCAT(0,.,CONCAT(0,5)),0],[CONCAT(0,.,5),CONCAT(0,.,CONCAT(0,5)),0,0],[CONCAT(0,.,CONCAT(4,5)),0,0,0]],[[0,CONCAT(0,.,5),0,CONCAT(0,.,CONCAT(0,5))],[CONCAT(0,.,5),1,CONCAT(0,.,CONCAT(4,5)),CONCAT(0,.,5)],[0,CONCAT(0,.,CONCAT(4,5)),0,0],[CONCAT(0,.,CONCAT(0,5)),CONCAT(0,.,5),0,0]],[[0,0,CONCAT(0,.,5),CONCAT(0,.,CONCAT(0,5))],[0,0,CONCAT(0,.,CONCAT(4,5)),0],[CONCAT(0,.,5),CONCAT(0,.,CONCAT(4,5)),1,CONCAT(0,.,5)],[CONCAT(0,.,CONCAT(0,5)),0,CONCAT(0,.,5),0]],[[0,0,0,CONCAT(0,.,CONCAT(4,5))],[0,0,CONCAT(0,.,CONCAT(0,5)),CONCAT(0,.,5)],[0,CONCAT(0,.,CONCAT(0,5)),0,CONCAT(0,.,5)],[CONCAT(0,.,CONCAT(4,5)),CONCAT(0,.,5),CONCAT(0,.,5),1]]]) 815\end{verbatim} 816\indentrel{-3}\end{paste}\end{patch} 817 818\begin{patch}{ugProblemGeneticPageEmpty8} 819\begin{paste}{ugProblemGeneticPageEmpty8}{ugProblemGeneticPagePatch8} 820\pastebutton{ugProblemGeneticPageEmpty8}{\showpaste} 821\tab{5}\spadcommand{plenaryPower(x,5)\free{x }} 822\end{paste}\end{patch} 823 824\begin{patch}{ugProblemGeneticPagePatch9} 825\begin{paste}{ugProblemGeneticPageFull9}{ugProblemGeneticPageEmpty9} 826\pastebutton{ugProblemGeneticPageFull9}{\hidepaste} 827\tab{5}\spadcommand{q := leftRankPolynomial()$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q }\free{gametes segregationRates }} 828\indentrel{3}\begin{verbatim} 829 (9) 830 3 29 29 29 29 2 831 Y + (- -- %x4 - -- %x3 - -- %x2 - -- %x1)Y 832 20 20 20 20 833 + 834 9 2 9 9 9 9 2 835 -- %x4 + (-- %x3 + -- %x2 + -- %x1)%x4 + -- %x3 836 20 10 10 10 20 837 + 838 9 9 9 2 9 839 (-- %x2 + -- %x1)%x3 + -- %x2 + -- %x1 %x2 840 10 10 20 10 841 + 842 9 2 843 -- %x1 844 20 845 * 846 Y 847Type: UnivariatePolynomial(Y,Polynomial(Fraction(Integer))) 848\end{verbatim} 849\indentrel{-3}\end{paste}\end{patch} 850 851\begin{patch}{ugProblemGeneticPageEmpty9} 852\begin{paste}{ugProblemGeneticPageEmpty9}{ugProblemGeneticPagePatch9} 853\pastebutton{ugProblemGeneticPageEmpty9}{\showpaste} 854\tab{5}\spadcommand{q := leftRankPolynomial()$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q }\free{gametes segregationRates }} 855\end{paste}\end{patch} 856 857\begin{patch}{ugProblemGeneticPagePatch10} 858\begin{paste}{ugProblemGeneticPageFull10}{ugProblemGeneticPageEmpty10} 859\pastebutton{ugProblemGeneticPageFull10}{\hidepaste} 860\tab{5}\spadcommand{factor(q :: POLY FRAC INT)\free{q }} 861\indentrel{3}\begin{verbatim} 862 (10) 863 (Y - %x4 - %x3 - %x2 - %x1) 864 * 865 9 9 9 9 866 (Y - -- %x4 - -- %x3 - -- %x2 - -- %x1)Y 867 20 20 20 20 868 Type: Factored(Polynomial(Fraction(Integer))) 869\end{verbatim} 870\indentrel{-3}\end{paste}\end{patch} 871 872\begin{patch}{ugProblemGeneticPageEmpty10} 873\begin{paste}{ugProblemGeneticPageEmpty10}{ugProblemGeneticPagePatch10} 874\pastebutton{ugProblemGeneticPageEmpty10}{\showpaste} 875\tab{5}\spadcommand{factor(q :: POLY FRAC INT)\free{q }} 876\end{paste}\end{patch} 877 878\begin{patch}{ugProblemGeneticPagePatch11} 879\begin{paste}{ugProblemGeneticPageFull11}{ugProblemGeneticPageEmpty11} 880\pastebutton{ugProblemGeneticPageFull11}{\hidepaste} 881\tab{5}\spadcommand{cI := conditionsForIdempotents()$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI }\free{gametes segregationRates }} 882\indentrel{3}\begin{verbatim} 883 (11) 884 9 1 2 885 [-- %x1 %x4 + (-- %x2 + %x1)%x3 + %x1 %x2 + %x1 - %x1, 886 10 10 887 1 9 2 888 (%x2 + -- %x1)%x4 + -- %x2 %x3 + %x2 + (%x1 - 1)%x2, 889 10 10 890 1 2 9 891 (%x3 + -- %x1)%x4 + %x3 + (-- %x2 + %x1 - 1)%x3, 892 10 10 893 2 9 1 894 %x4 + (%x3 + %x2 + -- %x1 - 1)%x4 + -- %x2 %x3] 895 10 10 896 Type: List(Polynomial(Fraction(Integer))) 897\end{verbatim} 898\indentrel{-3}\end{paste}\end{patch} 899 900\begin{patch}{ugProblemGeneticPageEmpty11} 901\begin{paste}{ugProblemGeneticPageEmpty11}{ugProblemGeneticPagePatch11} 902\pastebutton{ugProblemGeneticPageEmpty11}{\showpaste} 903\tab{5}\spadcommand{cI := conditionsForIdempotents()$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI }\free{gametes segregationRates }} 904\end{paste}\end{patch} 905 906\begin{patch}{ugProblemGeneticPagePatch12} 907\begin{paste}{ugProblemGeneticPageFull12}{ugProblemGeneticPageEmpty12} 908\pastebutton{ugProblemGeneticPageFull12}{\hidepaste} 909\tab{5}\spadcommand{gbs:= groebnerFactorize cI; gbs.1\free{cI }\bound{gbs }} 910\indentrel{3}\begin{verbatim} 911 (12) 912 [%x4 + %x3 + %x2 + %x1 - 1, 913 2 914 (%x2 + %x1)%x3 + %x1 %x2 + %x1 - %x1] 915 Type: List(Polynomial(Fraction(Integer))) 916\end{verbatim} 917\indentrel{-3}\end{paste}\end{patch} 918 919\begin{patch}{ugProblemGeneticPageEmpty12} 920\begin{paste}{ugProblemGeneticPageEmpty12}{ugProblemGeneticPagePatch12} 921\pastebutton{ugProblemGeneticPageEmpty12}{\showpaste} 922\tab{5}\spadcommand{gbs:= groebnerFactorize cI; gbs.1\free{cI }\bound{gbs }} 923\end{paste}\end{patch} 924 925\begin{patch}{ugProblemGeneticPagePatch13} 926\begin{paste}{ugProblemGeneticPageFull13}{ugProblemGeneticPageEmpty13} 927\pastebutton{ugProblemGeneticPageFull13}{\hidepaste} 928\tab{5}\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol }\free{gbs }} 929\indentrel{3}\begin{verbatim} 930 2 2 1 1 931 (13) [[%x4 = -, %x3 = -, %x2 = --, %x1 = --]] 932 5 5 10 10 933Type: List(List(Equation(Fraction(Polynomial(Integer))))) 934\end{verbatim} 935\indentrel{-3}\end{paste}\end{patch} 936 937\begin{patch}{ugProblemGeneticPageEmpty13} 938\begin{paste}{ugProblemGeneticPageEmpty13}{ugProblemGeneticPagePatch13} 939\pastebutton{ugProblemGeneticPageEmpty13}{\showpaste} 940\tab{5}\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol }\free{gbs }} 941\end{paste}\end{patch} 942 943\begin{patch}{ugProblemGeneticPagePatch14} 944\begin{paste}{ugProblemGeneticPageFull14}{ugProblemGeneticPageEmpty14} 945\pastebutton{ugProblemGeneticPageFull14}{\hidepaste} 946\tab{5}\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e }\free{A sol }} 947\indentrel{3}\begin{verbatim} 948 2 2 1 1 949 (14) - ab + - aB + -- Ab + -- AB 950 5 5 10 10 951Type: AlgebraGivenByStructuralConstants(Fraction(Integer),4,[AB,Ab,aB,ab],[[[1,1/2,1/2,9/20],[1/2,0,1/20,0],[1/2,1/20,0,0],[9/20,0,0,0]],[[0,1/2,0,1/20],[1/2,1,9/20,1/2],[0,9/20,0,0],[1/20,1/2,0,0]],[[0,0,1/2,1/20],[0,0,9/20,0],[1/2,9/20,1,1/2],[1/20,0,1/2,0]],[[0,0,0,9/20],[0,0,1/20,1/2],[0,1/20,0,1/2],[9/20,1/2,1/2,1]]]) 952\end{verbatim} 953\indentrel{-3}\end{paste}\end{patch} 954 955\begin{patch}{ugProblemGeneticPageEmpty14} 956\begin{paste}{ugProblemGeneticPageEmpty14}{ugProblemGeneticPagePatch14} 957\pastebutton{ugProblemGeneticPageEmpty14}{\showpaste} 958\tab{5}\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e }\free{A sol }} 959\end{paste}\end{patch} 960 961\begin{patch}{ugProblemGeneticPagePatch15} 962\begin{paste}{ugProblemGeneticPageFull15}{ugProblemGeneticPageEmpty15} 963\pastebutton{ugProblemGeneticPageFull15}{\hidepaste} 964\tab{5}\spadcommand{e*e-e\free{e }} 965\indentrel{3}\begin{verbatim} 966 (15) 0 967Type: AlgebraGivenByStructuralConstants(Fraction(Integer),4,[AB,Ab,aB,ab],[[[1,1/2,1/2,9/20],[1/2,0,1/20,0],[1/2,1/20,0,0],[9/20,0,0,0]],[[0,1/2,0,1/20],[1/2,1,9/20,1/2],[0,9/20,0,0],[1/20,1/2,0,0]],[[0,0,1/2,1/20],[0,0,9/20,0],[1/2,9/20,1,1/2],[1/20,0,1/2,0]],[[0,0,0,9/20],[0,0,1/20,1/2],[0,1/20,0,1/2],[9/20,1/2,1/2,1]]]) 968\end{verbatim} 969\indentrel{-3}\end{paste}\end{patch} 970 971\begin{patch}{ugProblemGeneticPageEmpty15} 972\begin{paste}{ugProblemGeneticPageEmpty15}{ugProblemGeneticPagePatch15} 973\pastebutton{ugProblemGeneticPageEmpty15}{\showpaste} 974\tab{5}\spadcommand{e*e-e\free{e }} 975\end{paste}\end{patch} 976 977\begin{patch}{ugxProblemOnePolPagePatch1} 978\begin{paste}{ugxProblemOnePolPageFull1}{ugxProblemOnePolPageEmpty1} 979\pastebutton{ugxProblemOnePolPageFull1}{\hidepaste} 980\tab{5}\spadcommand{solve(x^3 = 8,x)} 981\indentrel{3}\begin{verbatim} 982 2 983 (1) [x = 2, x + 2 x + 4 = 0] 984 Type: List(Equation(Fraction(Polynomial(Integer)))) 985\end{verbatim} 986\indentrel{-3}\end{paste}\end{patch} 987 988\begin{patch}{ugxProblemOnePolPageEmpty1} 989\begin{paste}{ugxProblemOnePolPageEmpty1}{ugxProblemOnePolPagePatch1} 990\pastebutton{ugxProblemOnePolPageEmpty1}{\showpaste} 991\tab{5}\spadcommand{solve(x^3 = 8,x)} 992\end{paste}\end{patch} 993 994\begin{patch}{ugxProblemOnePolPagePatch2} 995\begin{paste}{ugxProblemOnePolPageFull2}{ugxProblemOnePolPageEmpty2} 996\pastebutton{ugxProblemOnePolPageFull2}{\hidepaste} 997\tab{5}\spadcommand{radicalSolve(x^3 = 8,x)} 998\indentrel{3}\begin{verbatim} 999 +---+ +---+ 1000 (2) [x = - \|- 3 - 1, x = \|- 3 - 1, x = 2] 1001 Type: List(Equation(Expression(Integer))) 1002\end{verbatim} 1003\indentrel{-3}\end{paste}\end{patch} 1004 1005\begin{patch}{ugxProblemOnePolPageEmpty2} 1006\begin{paste}{ugxProblemOnePolPageEmpty2}{ugxProblemOnePolPagePatch2} 1007\pastebutton{ugxProblemOnePolPageEmpty2}{\showpaste} 1008\tab{5}\spadcommand{radicalSolve(x^3 = 8,x)} 1009\end{paste}\end{patch} 1010 1011\begin{patch}{ugxProblemOnePolPagePatch3} 1012\begin{paste}{ugxProblemOnePolPageFull3}{ugxProblemOnePolPageEmpty3} 1013\pastebutton{ugxProblemOnePolPageFull3}{\hidepaste} 1014\tab{5}\spadcommand{solve(x^4 - 10*x^3 + 35*x^2 - 50*x + 25,.0001)} 1015\indentrel{3}\begin{verbatim} 1016 (3) [x = 3.6180114746_09375, x = 1.3819885253_90625] 1017 Type: List(Equation(Polynomial(Float))) 1018\end{verbatim} 1019\indentrel{-3}\end{paste}\end{patch} 1020 1021\begin{patch}{ugxProblemOnePolPageEmpty3} 1022\begin{paste}{ugxProblemOnePolPageEmpty3}{ugxProblemOnePolPagePatch3} 1023\pastebutton{ugxProblemOnePolPageEmpty3}{\showpaste} 1024\tab{5}\spadcommand{solve(x^4 - 10*x^3 + 35*x^2 - 50*x + 25,.0001)} 1025\end{paste}\end{patch} 1026 1027\begin{patch}{ugxProblemOnePolPagePatch4} 1028\begin{paste}{ugxProblemOnePolPageFull4}{ugxProblemOnePolPageEmpty4} 1029\pastebutton{ugxProblemOnePolPageFull4}{\hidepaste} 1030\tab{5}\spadcommand{solve(x^3-2,1/1000)} 1031\indentrel{3}\begin{verbatim} 1032 2581 1033 (4) [x = ----] 1034 2048 1035 Type: List(Equation(Polynomial(Fraction(Integer)))) 1036\end{verbatim} 1037\indentrel{-3}\end{paste}\end{patch} 1038 1039\begin{patch}{ugxProblemOnePolPageEmpty4} 1040\begin{paste}{ugxProblemOnePolPageEmpty4}{ugxProblemOnePolPagePatch4} 1041\pastebutton{ugxProblemOnePolPageEmpty4}{\showpaste} 1042\tab{5}\spadcommand{solve(x^3-2,1/1000)} 1043\end{paste}\end{patch} 1044 1045\begin{patch}{ugxProblemOnePolPagePatch5} 1046\begin{paste}{ugxProblemOnePolPageFull5}{ugxProblemOnePolPageEmpty5} 1047\pastebutton{ugxProblemOnePolPageFull5}{\hidepaste} 1048\tab{5}\spadcommand{complexSolve(x^3-2,.0001)} 1049\indentrel{3}\begin{verbatim} 1050 (5) 1051 [x = 1.2599210498_1560260057_4493408203_125, 1052 1053 x 1054 = 1055 - 0.6299605247_3711410282 1056 + 1057 - 1.0911236358_806490898 %i 1058 , 1059 1060 x 1061 = 1062 - 0.6299605247_3711410282 1063 + 1064 1.0911236358_8064908981_3232421875 %i 1065 ] 1066 Type: List(Equation(Polynomial(Complex(Float)))) 1067\end{verbatim} 1068\indentrel{-3}\end{paste}\end{patch} 1069 1070\begin{patch}{ugxProblemOnePolPageEmpty5} 1071\begin{paste}{ugxProblemOnePolPageEmpty5}{ugxProblemOnePolPagePatch5} 1072\pastebutton{ugxProblemOnePolPageEmpty5}{\showpaste} 1073\tab{5}\spadcommand{complexSolve(x^3-2,.0001)} 1074\end{paste}\end{patch} 1075 1076\begin{patch}{ugxProblemOnePolPagePatch6} 1077\begin{paste}{ugxProblemOnePolPageFull6}{ugxProblemOnePolPageEmpty6} 1078\pastebutton{ugxProblemOnePolPageFull6}{\hidepaste} 1079\tab{5}\spadcommand{complexSolve(x^2-2*\%i+1,1/100)} 1080\indentrel{3}\begin{verbatim} 1081 (6) 1082 [ 1083 x 1084 = 1085 - 1086 2941342867319750366655494440259707227933201_ 1087 09632199 1088 / 1089 3741444191567111470601433171753684530319187_ 1090 31001856 1091 + 1092 10670475 1093 - -------- %i 1094 8388608 1095 , 1096 1097 x 1098 = 1099 2941342867319750366655494440259707227933201096_ 1100 32199 1101 / 1102 3741444191567111470601433171753684530319187310_ 1103 01856 1104 + 1105 10670475 1106 -------- %i 1107 8388608 1108 ] 1109Type: List(Equation(Polynomial(Complex(Fraction(Integer))))) 1110\end{verbatim} 1111\indentrel{-3}\end{paste}\end{patch} 1112 1113\begin{patch}{ugxProblemOnePolPageEmpty6} 1114\begin{paste}{ugxProblemOnePolPageEmpty6}{ugxProblemOnePolPagePatch6} 1115\pastebutton{ugxProblemOnePolPageEmpty6}{\showpaste} 1116\tab{5}\spadcommand{complexSolve(x^2-2*\%i+1,1/100)} 1117\end{paste}\end{patch} 1118 1119\begin{patch}{ugxProblemOnePolPagePatch7} 1120\begin{paste}{ugxProblemOnePolPageFull7}{ugxProblemOnePolPageEmpty7} 1121\pastebutton{ugxProblemOnePolPageFull7}{\hidepaste} 1122\tab{5}\spadcommand{radicalSolve(1/x^3 + 1/x^2 + 1/x = 0,x)} 1123\indentrel{3}\begin{verbatim} 1124 +---+ +---+ 1125 - \|- 3 - 1 \|- 3 - 1 1126 (7) [x = ------------, x = ----------] 1127 2 2 1128 Type: List(Equation(Expression(Integer))) 1129\end{verbatim} 1130\indentrel{-3}\end{paste}\end{patch} 1131 1132\begin{patch}{ugxProblemOnePolPageEmpty7} 1133\begin{paste}{ugxProblemOnePolPageEmpty7}{ugxProblemOnePolPagePatch7} 1134\pastebutton{ugxProblemOnePolPageEmpty7}{\showpaste} 1135\tab{5}\spadcommand{radicalSolve(1/x^3 + 1/x^2 + 1/x = 0,x)} 1136\end{paste}\end{patch} 1137 1138\begin{patch}{ugProblemNumericPagePatch1} 1139\begin{paste}{ugProblemNumericPageFull1}{ugProblemNumericPageEmpty1} 1140\pastebutton{ugProblemNumericPageFull1}{\hidepaste} 1141\tab{5}\spadcommand{exp(3.1)} 1142\indentrel{3}\begin{verbatim} 1143 (1) 22.1979512814_41633405 1144 Type: Float 1145\end{verbatim} 1146\indentrel{-3}\end{paste}\end{patch} 1147 1148\begin{patch}{ugProblemNumericPageEmpty1} 1149\begin{paste}{ugProblemNumericPageEmpty1}{ugProblemNumericPagePatch1} 1150\pastebutton{ugProblemNumericPageEmpty1}{\showpaste} 1151\tab{5}\spadcommand{exp(3.1)} 1152\end{paste}\end{patch} 1153 1154\begin{patch}{ugProblemNumericPagePatch2} 1155\begin{paste}{ugProblemNumericPageFull2}{ugProblemNumericPageEmpty2} 1156\pastebutton{ugProblemNumericPageFull2}{\hidepaste} 1157\tab{5}\spadcommand{exp(3.1 + 4.5 * \%i)} 1158\indentrel{3}\begin{verbatim} 1159 (2) 1160 - 4.6792348860_969899118 - 21.6991659280_71731864 %i 1161 Type: Complex(Float) 1162\end{verbatim} 1163\indentrel{-3}\end{paste}\end{patch} 1164 1165\begin{patch}{ugProblemNumericPageEmpty2} 1166\begin{paste}{ugProblemNumericPageEmpty2}{ugProblemNumericPagePatch2} 1167\pastebutton{ugProblemNumericPageEmpty2}{\showpaste} 1168\tab{5}\spadcommand{exp(3.1 + 4.5 * \%i)} 1169\end{paste}\end{patch} 1170 1171\begin{patch}{ugProblemNumericPagePatch3} 1172\begin{paste}{ugProblemNumericPageFull3}{ugProblemNumericPageEmpty3} 1173\pastebutton{ugProblemNumericPageFull3}{\hidepaste} 1174\tab{5}\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)} 1175\indentrel{3}\begin{verbatim} 1176 (3) - 4.679234886096988 - 21.69916592807172 %i 1177 Type: Complex(DoubleFloat) 1178\end{verbatim} 1179\indentrel{-3}\end{paste}\end{patch} 1180 1181\begin{patch}{ugProblemNumericPageEmpty3} 1182\begin{paste}{ugProblemNumericPageEmpty3}{ugProblemNumericPagePatch3} 1183\pastebutton{ugProblemNumericPageEmpty3}{\showpaste} 1184\tab{5}\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)} 1185\end{paste}\end{patch} 1186 1187\begin{patch}{ugProblemNumericPagePatch4} 1188\begin{paste}{ugProblemNumericPageFull4}{ugProblemNumericPageEmpty4} 1189\pastebutton{ugProblemNumericPageFull4}{\hidepaste} 1190\tab{5}\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)} 1191\indentrel{3}\begin{verbatim} 1192 (4) - 4.679234886096988 - 21.69916592807172 %i 1193 Type: Complex(DoubleFloat) 1194\end{verbatim} 1195\indentrel{-3}\end{paste}\end{patch} 1196 1197\begin{patch}{ugProblemNumericPageEmpty4} 1198\begin{paste}{ugProblemNumericPageEmpty4}{ugProblemNumericPagePatch4} 1199\pastebutton{ugProblemNumericPageEmpty4}{\showpaste} 1200\tab{5}\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)} 1201\end{paste}\end{patch} 1202 1203\begin{patch}{ugProblemNumericPagePatch5} 1204\begin{paste}{ugProblemNumericPageFull5}{ugProblemNumericPageEmpty5} 1205\pastebutton{ugProblemNumericPageFull5}{\hidepaste} 1206\tab{5}\spadcommand{Gamma(0.5)^2} 1207\indentrel{3}\begin{verbatim} 1208 (5) 3.1415926535_897932385 1209 Type: Float 1210\end{verbatim} 1211\indentrel{-3}\end{paste}\end{patch} 1212 1213\begin{patch}{ugProblemNumericPageEmpty5} 1214\begin{paste}{ugProblemNumericPageEmpty5}{ugProblemNumericPagePatch5} 1215\pastebutton{ugProblemNumericPageEmpty5}{\showpaste} 1216\tab{5}\spadcommand{Gamma(0.5)^2} 1217\end{paste}\end{patch} 1218 1219\begin{patch}{ugProblemNumericPagePatch6} 1220\begin{paste}{ugProblemNumericPageFull6}{ugProblemNumericPageEmpty6} 1221\pastebutton{ugProblemNumericPageFull6}{\hidepaste} 1222\tab{5}\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)} 1223\indentrel{3}\begin{verbatim} 1224 (6) 2.4894824175473698 - 2.365846038146814 %i 1225 Type: Complex(DoubleFloat) 1226\end{verbatim} 1227\indentrel{-3}\end{paste}\end{patch} 1228 1229\begin{patch}{ugProblemNumericPageEmpty6} 1230\begin{paste}{ugProblemNumericPageEmpty6}{ugProblemNumericPagePatch6} 1231\pastebutton{ugProblemNumericPageEmpty6}{\showpaste} 1232\tab{5}\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)} 1233\end{paste}\end{patch} 1234 1235\begin{patch}{ugProblemNumericPagePatch7} 1236\begin{paste}{ugProblemNumericPageFull7}{ugProblemNumericPageEmpty7} 1237\pastebutton{ugProblemNumericPageFull7}{\hidepaste} 1238\tab{5}\spadcommand{[chebyshevT(i, z) for i in 0..5]} 1239\indentrel{3}\begin{verbatim} 1240 (7) 1241 2 3 4 2 1242 [1, z, 2 z - 1, 4 z - 3 z, 8 z - 8 z + 1, 1243 5 3 1244 16 z - 20 z + 5 z] 1245 Type: List(Polynomial(Integer)) 1246\end{verbatim} 1247\indentrel{-3}\end{paste}\end{patch} 1248 1249\begin{patch}{ugProblemNumericPageEmpty7} 1250\begin{paste}{ugProblemNumericPageEmpty7}{ugProblemNumericPagePatch7} 1251\pastebutton{ugProblemNumericPageEmpty7}{\showpaste} 1252\tab{5}\spadcommand{[chebyshevT(i, z) for i in 0..5]} 1253\end{paste}\end{patch} 1254 1255\begin{patch}{ugProblemNumericPagePatch8} 1256\begin{paste}{ugProblemNumericPageFull8}{ugProblemNumericPageEmpty8} 1257\pastebutton{ugProblemNumericPageFull8}{\hidepaste} 1258\tab{5}\spadcommand{chebyshevT(3, 5.0 + 6.0*\%i)} 1259\indentrel{3}\begin{verbatim} 1260 (8) - 1675.0 + 918.0 %i 1261 Type: Complex(Float) 1262\end{verbatim} 1263\indentrel{-3}\end{paste}\end{patch} 1264 1265\begin{patch}{ugProblemNumericPageEmpty8} 1266\begin{paste}{ugProblemNumericPageEmpty8}{ugProblemNumericPagePatch8} 1267\pastebutton{ugProblemNumericPageEmpty8}{\showpaste} 1268\tab{5}\spadcommand{chebyshevT(3, 5.0 + 6.0*\%i)} 1269\end{paste}\end{patch} 1270 1271\begin{patch}{ugProblemNumericPagePatch9} 1272\begin{paste}{ugProblemNumericPageFull9}{ugProblemNumericPageEmpty9} 1273\pastebutton{ugProblemNumericPageFull9}{\hidepaste} 1274\tab{5}\spadcommand{chebyshevT(3, 5.0::DoubleFloat)} 1275\indentrel{3}\begin{verbatim} 1276 (9) 485.0 1277 Type: DoubleFloat 1278\end{verbatim} 1279\indentrel{-3}\end{paste}\end{patch} 1280 1281\begin{patch}{ugProblemNumericPageEmpty9} 1282\begin{paste}{ugProblemNumericPageEmpty9}{ugProblemNumericPagePatch9} 1283\pastebutton{ugProblemNumericPageEmpty9}{\showpaste} 1284\tab{5}\spadcommand{chebyshevT(3, 5.0::DoubleFloat)} 1285\end{paste}\end{patch} 1286 1287\begin{patch}{ugProblemNumericPagePatch10} 1288\begin{paste}{ugProblemNumericPageFull10}{ugProblemNumericPageEmpty10} 1289\pastebutton{ugProblemNumericPageFull10}{\hidepaste} 1290\tab{5}\spadcommand{[chebyshevU(i, z) for i in 0..5]} 1291\indentrel{3}\begin{verbatim} 1292 (10) 1293 2 3 4 2 1294 [1, 2 z, 4 z - 1, 8 z - 4 z, 16 z - 12 z + 1, 1295 5 3 1296 32 z - 32 z + 6 z] 1297 Type: List(Polynomial(Integer)) 1298\end{verbatim} 1299\indentrel{-3}\end{paste}\end{patch} 1300 1301\begin{patch}{ugProblemNumericPageEmpty10} 1302\begin{paste}{ugProblemNumericPageEmpty10}{ugProblemNumericPagePatch10} 1303\pastebutton{ugProblemNumericPageEmpty10}{\showpaste} 1304\tab{5}\spadcommand{[chebyshevU(i, z) for i in 0..5]} 1305\end{paste}\end{patch} 1306 1307\begin{patch}{ugProblemNumericPagePatch11} 1308\begin{paste}{ugProblemNumericPageFull11}{ugProblemNumericPageEmpty11} 1309\pastebutton{ugProblemNumericPageFull11}{\hidepaste} 1310\tab{5}\spadcommand{chebyshevU(3, 0.2)} 1311\indentrel{3}\begin{verbatim} 1312 (11) - 0.736 1313 Type: Float 1314\end{verbatim} 1315\indentrel{-3}\end{paste}\end{patch} 1316 1317\begin{patch}{ugProblemNumericPageEmpty11} 1318\begin{paste}{ugProblemNumericPageEmpty11}{ugProblemNumericPagePatch11} 1319\pastebutton{ugProblemNumericPageEmpty11}{\showpaste} 1320\tab{5}\spadcommand{chebyshevU(3, 0.2)} 1321\end{paste}\end{patch} 1322 1323\begin{patch}{ugProblemNumericPagePatch12} 1324\begin{paste}{ugProblemNumericPageFull12}{ugProblemNumericPageEmpty12} 1325\pastebutton{ugProblemNumericPageFull12}{\hidepaste} 1326\tab{5}\spadcommand{[hermiteH(i, z) for i in 0..5]} 1327\indentrel{3}\begin{verbatim} 1328 (12) 1329 2 3 4 2 1330 [1, 2 z, 4 z - 2, 8 z - 12 z, 16 z - 48 z + 12, 1331 5 3 1332 32 z - 160 z + 120 z] 1333 Type: List(Polynomial(Integer)) 1334\end{verbatim} 1335\indentrel{-3}\end{paste}\end{patch} 1336 1337\begin{patch}{ugProblemNumericPageEmpty12} 1338\begin{paste}{ugProblemNumericPageEmpty12}{ugProblemNumericPagePatch12} 1339\pastebutton{ugProblemNumericPageEmpty12}{\showpaste} 1340\tab{5}\spadcommand{[hermiteH(i, z) for i in 0..5]} 1341\end{paste}\end{patch} 1342 1343\begin{patch}{ugProblemNumericPagePatch13} 1344\begin{paste}{ugProblemNumericPageFull13}{ugProblemNumericPageEmpty13} 1345\pastebutton{ugProblemNumericPageFull13}{\hidepaste} 1346\tab{5}\spadcommand{hermiteH(100, 1.0)} 1347\indentrel{3}\begin{verbatim} 1348 (13) - 0.1448706729_337934088 E 93 1349 Type: Float 1350\end{verbatim} 1351\indentrel{-3}\end{paste}\end{patch} 1352 1353\begin{patch}{ugProblemNumericPageEmpty13} 1354\begin{paste}{ugProblemNumericPageEmpty13}{ugProblemNumericPagePatch13} 1355\pastebutton{ugProblemNumericPageEmpty13}{\showpaste} 1356\tab{5}\spadcommand{hermiteH(100, 1.0)} 1357\end{paste}\end{patch} 1358 1359\begin{patch}{ugProblemNumericPagePatch14} 1360\begin{paste}{ugProblemNumericPageFull14}{ugProblemNumericPageEmpty14} 1361\pastebutton{ugProblemNumericPageFull14}{\hidepaste} 1362\tab{5}\spadcommand{[laguerreL(i, z) for i in 0..4]} 1363\indentrel{3}\begin{verbatim} 1364 (14) 1365 2 3 2 1366 [1, - z + 1, z - 4 z + 2, - z + 9 z - 18 z + 6, 1367 4 3 2 1368 z - 16 z + 72 z - 96 z + 24] 1369 Type: List(Polynomial(Integer)) 1370\end{verbatim} 1371\indentrel{-3}\end{paste}\end{patch} 1372 1373\begin{patch}{ugProblemNumericPageEmpty14} 1374\begin{paste}{ugProblemNumericPageEmpty14}{ugProblemNumericPagePatch14} 1375\pastebutton{ugProblemNumericPageEmpty14}{\showpaste} 1376\tab{5}\spadcommand{[laguerreL(i, z) for i in 0..4]} 1377\end{paste}\end{patch} 1378 1379\begin{patch}{ugProblemNumericPagePatch15} 1380\begin{paste}{ugProblemNumericPageFull15}{ugProblemNumericPageEmpty15} 1381\pastebutton{ugProblemNumericPageFull15}{\hidepaste} 1382\tab{5}\spadcommand{laguerreL(4, 1.2)} 1383\indentrel{3}\begin{verbatim} 1384 (15) - 13.0944 1385 Type: Float 1386\end{verbatim} 1387\indentrel{-3}\end{paste}\end{patch} 1388 1389\begin{patch}{ugProblemNumericPageEmpty15} 1390\begin{paste}{ugProblemNumericPageEmpty15}{ugProblemNumericPagePatch15} 1391\pastebutton{ugProblemNumericPageEmpty15}{\showpaste} 1392\tab{5}\spadcommand{laguerreL(4, 1.2)} 1393\end{paste}\end{patch} 1394 1395\begin{patch}{ugProblemNumericPagePatch16} 1396\begin{paste}{ugProblemNumericPageFull16}{ugProblemNumericPageEmpty16} 1397\pastebutton{ugProblemNumericPageFull16}{\hidepaste} 1398\tab{5}\spadcommand{[laguerreL(j, 3, z) for j in 0..4]} 1399\indentrel{3}\begin{verbatim} 1400 (16) 1401 3 2 2 1402 [- z + 9 z - 18 z + 6, - 3 z + 18 z - 18, 1403 - 6 z + 18, - 6, 0] 1404 Type: List(Polynomial(Integer)) 1405\end{verbatim} 1406\indentrel{-3}\end{paste}\end{patch} 1407 1408\begin{patch}{ugProblemNumericPageEmpty16} 1409\begin{paste}{ugProblemNumericPageEmpty16}{ugProblemNumericPagePatch16} 1410\pastebutton{ugProblemNumericPageEmpty16}{\showpaste} 1411\tab{5}\spadcommand{[laguerreL(j, 3, z) for j in 0..4]} 1412\end{paste}\end{patch} 1413 1414\begin{patch}{ugProblemNumericPagePatch17} 1415\begin{paste}{ugProblemNumericPageFull17}{ugProblemNumericPageEmpty17} 1416\pastebutton{ugProblemNumericPageFull17}{\hidepaste} 1417\tab{5}\spadcommand{laguerreL(1, 3, 2.1)} 1418\indentrel{3}\begin{verbatim} 1419 (17) 6.57 1420 Type: Float 1421\end{verbatim} 1422\indentrel{-3}\end{paste}\end{patch} 1423 1424\begin{patch}{ugProblemNumericPageEmpty17} 1425\begin{paste}{ugProblemNumericPageEmpty17}{ugProblemNumericPagePatch17} 1426\pastebutton{ugProblemNumericPageEmpty17}{\showpaste} 1427\tab{5}\spadcommand{laguerreL(1, 3, 2.1)} 1428\end{paste}\end{patch} 1429 1430\begin{patch}{ugProblemNumericPagePatch18} 1431\begin{paste}{ugProblemNumericPageFull18}{ugProblemNumericPageEmpty18} 1432\pastebutton{ugProblemNumericPageFull18}{\hidepaste} 1433\tab{5}\spadcommand{[legendreP(i,z) for i in 0..5]} 1434\indentrel{3}\begin{verbatim} 1435 (18) 1436 3 2 1 5 3 3 35 4 15 2 3 1437 [1, z, - z - -, - z - - z, -- z - -- z + -, 1438 2 2 2 2 8 4 8 1439 63 5 35 3 15 1440 -- z - -- z + -- z] 1441 8 4 8 1442 Type: List(Polynomial(Fraction(Integer))) 1443\end{verbatim} 1444\indentrel{-3}\end{paste}\end{patch} 1445 1446\begin{patch}{ugProblemNumericPageEmpty18} 1447\begin{paste}{ugProblemNumericPageEmpty18}{ugProblemNumericPagePatch18} 1448\pastebutton{ugProblemNumericPageEmpty18}{\showpaste} 1449\tab{5}\spadcommand{[legendreP(i,z) for i in 0..5]} 1450\end{paste}\end{patch} 1451 1452\begin{patch}{ugProblemNumericPagePatch19} 1453\begin{paste}{ugProblemNumericPageFull19}{ugProblemNumericPageEmpty19} 1454\pastebutton{ugProblemNumericPageFull19}{\hidepaste} 1455\tab{5}\spadcommand{legendreP(3, 3.0*\%i)} 1456\indentrel{3}\begin{verbatim} 1457 (19) - 72.0 %i 1458 Type: Complex(Float) 1459\end{verbatim} 1460\indentrel{-3}\end{paste}\end{patch} 1461 1462\begin{patch}{ugProblemNumericPageEmpty19} 1463\begin{paste}{ugProblemNumericPageEmpty19}{ugProblemNumericPagePatch19} 1464\pastebutton{ugProblemNumericPageEmpty19}{\showpaste} 1465\tab{5}\spadcommand{legendreP(3, 3.0*\%i)} 1466\end{paste}\end{patch} 1467 1468\begin{patch}{ugProblemNumericPagePatch20} 1469\begin{paste}{ugProblemNumericPageFull20}{ugProblemNumericPageEmpty20} 1470\pastebutton{ugProblemNumericPageFull20}{\hidepaste} 1471\tab{5}\spadcommand{bernoulliB(3, z)} 1472\indentrel{3}\begin{verbatim} 1473 3 3 2 1 1474 (20) z - - z + - z 1475 2 2 1476 Type: Polynomial(Fraction(Integer)) 1477\end{verbatim} 1478\indentrel{-3}\end{paste}\end{patch} 1479 1480\begin{patch}{ugProblemNumericPageEmpty20} 1481\begin{paste}{ugProblemNumericPageEmpty20}{ugProblemNumericPagePatch20} 1482\pastebutton{ugProblemNumericPageEmpty20}{\showpaste} 1483\tab{5}\spadcommand{bernoulliB(3, z)} 1484\end{paste}\end{patch} 1485 1486\begin{patch}{ugProblemNumericPagePatch21} 1487\begin{paste}{ugProblemNumericPageFull21}{ugProblemNumericPageEmpty21} 1488\pastebutton{ugProblemNumericPageFull21}{\hidepaste} 1489\tab{5}\spadcommand{bernoulliB(3, 0.7 + 0.4 * \%i)} 1490\indentrel{3}\begin{verbatim} 1491 (21) - 0.138 - 0.116 %i 1492 Type: Complex(Float) 1493\end{verbatim} 1494\indentrel{-3}\end{paste}\end{patch} 1495 1496\begin{patch}{ugProblemNumericPageEmpty21} 1497\begin{paste}{ugProblemNumericPageEmpty21}{ugProblemNumericPagePatch21} 1498\pastebutton{ugProblemNumericPageEmpty21}{\showpaste} 1499\tab{5}\spadcommand{bernoulliB(3, 0.7 + 0.4 * \%i)} 1500\end{paste}\end{patch} 1501 1502\begin{patch}{ugProblemNumericPagePatch22} 1503\begin{paste}{ugProblemNumericPageFull22}{ugProblemNumericPageEmpty22} 1504\pastebutton{ugProblemNumericPageFull22}{\hidepaste} 1505\tab{5}\spadcommand{eulerE(3, z)} 1506\indentrel{3}\begin{verbatim} 1507 3 3 2 1 1508 (22) z - - z + - 1509 2 4 1510 Type: Polynomial(Fraction(Integer)) 1511\end{verbatim} 1512\indentrel{-3}\end{paste}\end{patch} 1513 1514\begin{patch}{ugProblemNumericPageEmpty22} 1515\begin{paste}{ugProblemNumericPageEmpty22}{ugProblemNumericPagePatch22} 1516\pastebutton{ugProblemNumericPageEmpty22}{\showpaste} 1517\tab{5}\spadcommand{eulerE(3, z)} 1518\end{paste}\end{patch} 1519 1520\begin{patch}{ugProblemNumericPagePatch23} 1521\begin{paste}{ugProblemNumericPageFull23}{ugProblemNumericPageEmpty23} 1522\pastebutton{ugProblemNumericPageFull23}{\hidepaste} 1523\tab{5}\spadcommand{eulerE(3, 0.7 + 0.4 * \%i)} 1524\indentrel{3}\begin{verbatim} 1525 (23) - 0.238 - 0.316 %i 1526 Type: Complex(Float) 1527\end{verbatim} 1528\indentrel{-3}\end{paste}\end{patch} 1529 1530\begin{patch}{ugProblemNumericPageEmpty23} 1531\begin{paste}{ugProblemNumericPageEmpty23}{ugProblemNumericPagePatch23} 1532\pastebutton{ugProblemNumericPageEmpty23}{\showpaste} 1533\tab{5}\spadcommand{eulerE(3, 0.7 + 0.4 * \%i)} 1534\end{paste}\end{patch} 1535 1536\begin{patch}{ugProblemNumericPagePatch24} 1537\begin{paste}{ugProblemNumericPageFull24}{ugProblemNumericPageEmpty24} 1538\pastebutton{ugProblemNumericPageFull24}{\hidepaste} 1539\tab{5}\spadcommand{cyclotomic(3, z)} 1540\indentrel{3}\begin{verbatim} 1541 2 1542 (24) z + z + 1 1543 Type: Polynomial(Integer) 1544\end{verbatim} 1545\indentrel{-3}\end{paste}\end{patch} 1546 1547\begin{patch}{ugProblemNumericPageEmpty24} 1548\begin{paste}{ugProblemNumericPageEmpty24}{ugProblemNumericPagePatch24} 1549\pastebutton{ugProblemNumericPageEmpty24}{\showpaste} 1550\tab{5}\spadcommand{cyclotomic(3, z)} 1551\end{paste}\end{patch} 1552 1553\begin{patch}{ugProblemNumericPagePatch25} 1554\begin{paste}{ugProblemNumericPageFull25}{ugProblemNumericPageEmpty25} 1555\pastebutton{ugProblemNumericPageFull25}{\hidepaste} 1556\tab{5}\spadcommand{cyclotomic(3, (-1.0 + 0.0 * \%i)^(2/3))} 1557\indentrel{3}\begin{verbatim} 1558 (25) 0.0 1559 Type: Complex(Float) 1560\end{verbatim} 1561\indentrel{-3}\end{paste}\end{patch} 1562 1563\begin{patch}{ugProblemNumericPageEmpty25} 1564\begin{paste}{ugProblemNumericPageEmpty25}{ugProblemNumericPagePatch25} 1565\pastebutton{ugProblemNumericPageEmpty25}{\showpaste} 1566\tab{5}\spadcommand{cyclotomic(3, (-1.0 + 0.0 * \%i)^(2/3))} 1567\end{paste}\end{patch} 1568 1569\begin{patch}{ugProblemNumericPagePatch26} 1570\begin{paste}{ugProblemNumericPageFull26}{ugProblemNumericPageEmpty26} 1571\pastebutton{ugProblemNumericPageFull26}{\hidepaste} 1572\tab{5}\spadgraph{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +-> imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")} 1573\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugProblemNumericPage26.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugProblemNumericPage26}} 1574\end{paste}\end{patch} 1575 1576\begin{patch}{ugProblemNumericPageEmpty26} 1577\begin{paste}{ugProblemNumericPageEmpty26}{ugProblemNumericPagePatch26} 1578\pastebutton{ugProblemNumericPageEmpty26}{\showpaste} 1579\tab{5}\spadgraph{draw((x,y)+-> real exp complex(x,y), -2..2, -2*\%pi..2*\%pi, colorFunction == (x, y) +-> imag exp complex(x,y), title=="exp(x+\%i*y)", style=="smooth")} 1580\end{paste}\end{patch} 1581 1582\begin{patch}{ugProblemNumericPagePatch27} 1583\begin{paste}{ugProblemNumericPageFull27}{ugProblemNumericPageEmpty27} 1584\pastebutton{ugProblemNumericPageFull27}{\hidepaste} 1585\tab{5}\spadgraph{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp} 1586\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugProblemNumericPage27.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugProblemNumericPage27}} 1587\end{paste}\end{patch} 1588 1589\begin{patch}{ugProblemNumericPageEmpty27} 1590\begin{paste}{ugProblemNumericPageEmpty27}{ugProblemNumericPagePatch27} 1591\pastebutton{ugProblemNumericPageEmpty27}{\showpaste} 1592\tab{5}\spadgraph{vp := draw((x,y) +-> real atan complex(x,y), -\%pi..\%pi, -\%pi..\%pi, colorFunction==(x,y) +->argument atan complex(x,y), title=="atan(x+\%i*y)", style=="shade"); rotate(vp,-160,-45); vp} 1593\end{paste}\end{patch} 1594 1595\begin{patch}{ugProblemNumericPagePatch28} 1596\begin{paste}{ugProblemNumericPageFull28}{ugProblemNumericPageEmpty28} 1597\pastebutton{ugProblemNumericPageFull28}{\hidepaste} 1598\tab{5}\spadgraph{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)} 1599\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugProblemNumericPage28.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugProblemNumericPage28}} 1600\end{paste}\end{patch} 1601 1602\begin{patch}{ugProblemNumericPageEmpty28} 1603\begin{paste}{ugProblemNumericPageEmpty28}{ugProblemNumericPagePatch28} 1604\pastebutton{ugProblemNumericPageEmpty28}{\showpaste} 1605\tab{5}\spadgraph{draw((x,y) +-> max(min(real Gamma complex(x,y),4),-4), -\%pi..\%pi, -\%pi..\%pi, style=="shade", colorFunction == (x,y) +-> argument Gamma complex(x,y), title == "Gamma(x+\%i*y)", var1Steps == 50, var2Steps== 50)} 1606\end{paste}\end{patch} 1607 1608\begin{patch}{ugProblemNumericPagePatch29} 1609\begin{paste}{ugProblemNumericPageFull29}{ugProblemNumericPageEmpty29} 1610\pastebutton{ugProblemNumericPageFull29}{\hidepaste} 1611\tab{5}\spadgraph{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)} 1612\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugProblemNumericPage29.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugProblemNumericPage29}} 1613\end{paste}\end{patch} 1614 1615\begin{patch}{ugProblemNumericPageEmpty29} 1616\begin{paste}{ugProblemNumericPageEmpty29}{ugProblemNumericPagePatch29} 1617\pastebutton{ugProblemNumericPageEmpty29}{\showpaste} 1618\tab{5}\spadgraph{draw(Beta(x,y)/100, x=-1.6..1.7, y = -1.6..1.7, style=="shade", title=="Beta(x,y)", var1Steps==40, var2Steps==40)} 1619\end{paste}\end{patch} 1620 1621\begin{patch}{ugProblemNumericPagePatch30} 1622\begin{paste}{ugProblemNumericPageFull30}{ugProblemNumericPageEmpty30} 1623\pastebutton{ugProblemNumericPageFull30}{\hidepaste} 1624\tab{5}\spadgraph{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)} 1625\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugProblemNumericPage30.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugProblemNumericPage30}} 1626\end{paste}\end{patch} 1627 1628\begin{patch}{ugProblemNumericPageEmpty30} 1629\begin{paste}{ugProblemNumericPageEmpty30}{ugProblemNumericPagePatch30} 1630\pastebutton{ugProblemNumericPageEmpty30}{\showpaste} 1631\tab{5}\spadgraph{draw((alpha,x) +-> min(max(besselJ(alpha, x+8), -6), 6), -6..4, -6..6, title=="besselJ(alpha,x)", style=="shade", var1Steps==40, var2Steps==40)} 1632\end{paste}\end{patch} 1633 1634\begin{patch}{ugProblemNumericPagePatch31} 1635\begin{paste}{ugProblemNumericPageFull31}{ugProblemNumericPageEmpty31} 1636\pastebutton{ugProblemNumericPageFull31}{\hidepaste} 1637\tab{5}\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])} 1638\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugProblemNumericPage31.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugProblemNumericPage31}} 1639\end{paste}\end{patch} 1640 1641\begin{patch}{ugProblemNumericPageEmpty31} 1642\begin{paste}{ugProblemNumericPageEmpty31}{ugProblemNumericPagePatch31} 1643\pastebutton{ugProblemNumericPageEmpty31}{\showpaste} 1644\tab{5}\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])} 1645\end{paste}\end{patch} 1646 1647\begin{patch}{ugProblemNumericPagePatch32} 1648\begin{paste}{ugProblemNumericPageFull32}{ugProblemNumericPageEmpty32} 1649\pastebutton{ugProblemNumericPageFull32}{\hidepaste} 1650\tab{5}\spadgraph{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")} 1651\center{\unixcommand{\inputimage{\env{FRICAS}/share/viewports/ugProblemNumericPage32.VIEW/image}}{\env{FRICAS}/bin/viewAlone\space{1} \env{FRICAS}/share/viewports/ugProblemNumericPage32}} 1652\end{paste}\end{patch} 1653 1654\begin{patch}{ugProblemNumericPageEmpty32} 1655\begin{paste}{ugProblemNumericPageEmpty32}{ugProblemNumericPagePatch32} 1656\pastebutton{ugProblemNumericPageEmpty32}{\showpaste} 1657\tab{5}\spadgraph{draw((x,y) +-> real besselI(complex(x/20, y/20),5), -60..60, -60..60, colorFunction == (x,y)+-> argument besselI(complex(x/20,y/20),5), title=="besselI(x+i*y,5)", style=="shade")} 1658\end{paste}\end{patch} 1659 1660\begin{patch}{ugxProblemLinSysPagePatch1} 1661\begin{paste}{ugxProblemLinSysPageFull1}{ugxProblemLinSysPageEmpty1} 1662\pastebutton{ugxProblemLinSysPageFull1}{\hidepaste} 1663\tab{5}\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])} 1664\indentrel{3}\begin{verbatim} 1665 (1) [[x = - 1, y = 2, z = 7]] 1666Type: List(List(Equation(Fraction(Polynomial(Integer))))) 1667\end{verbatim} 1668\indentrel{-3}\end{paste}\end{patch} 1669 1670\begin{patch}{ugxProblemLinSysPageEmpty1} 1671\begin{paste}{ugxProblemLinSysPageEmpty1}{ugxProblemLinSysPagePatch1} 1672\pastebutton{ugxProblemLinSysPageEmpty1}{\showpaste} 1673\tab{5}\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])} 1674\end{paste}\end{patch} 1675 1676\begin{patch}{ugxProblemLinSysPagePatch2} 1677\begin{paste}{ugxProblemLinSysPageFull2}{ugxProblemLinSysPageEmpty2} 1678\pastebutton{ugxProblemLinSysPageFull2}{\hidepaste} 1679\tab{5}\spadcommand{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])} 1680\indentrel{3}\begin{verbatim} 1681 (2) [[x = %Z - 2, y = - 2 %Z + 2, z = %Z]] 1682Type: List(List(Equation(Fraction(Polynomial(Integer))))) 1683\end{verbatim} 1684\indentrel{-3}\end{paste}\end{patch} 1685 1686\begin{patch}{ugxProblemLinSysPageEmpty2} 1687\begin{paste}{ugxProblemLinSysPageEmpty2}{ugxProblemLinSysPagePatch2} 1688\pastebutton{ugxProblemLinSysPageEmpty2}{\showpaste} 1689\tab{5}\spadcommand{solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z])} 1690\end{paste}\end{patch} 1691 1692\begin{patch}{ugxProblemLinSysPagePatch3} 1693\begin{paste}{ugxProblemLinSysPageFull3}{ugxProblemLinSysPageEmpty3} 1694\pastebutton{ugxProblemLinSysPageFull3}{\hidepaste} 1695\tab{5}\spadcommand{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])} 1696\indentrel{3}\begin{verbatim} 1697 (3) [particular = [- 1, 2, 7], basis = []] 1698Type: Record(particular: Union(Vector(Fraction(Integer)),"failed"),basis: List(Vector(Fraction(Integer)))) 1699\end{verbatim} 1700\indentrel{-3}\end{paste}\end{patch} 1701 1702\begin{patch}{ugxProblemLinSysPageEmpty3} 1703\begin{paste}{ugxProblemLinSysPageEmpty3}{ugxProblemLinSysPagePatch3} 1704\pastebutton{ugxProblemLinSysPageEmpty3}{\showpaste} 1705\tab{5}\spadcommand{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])} 1706\end{paste}\end{patch} 1707 1708\begin{patch}{ugxProblemLinSysPagePatch4} 1709\begin{paste}{ugxProblemLinSysPageFull4}{ugxProblemLinSysPageEmpty4} 1710\pastebutton{ugxProblemLinSysPageFull4}{\hidepaste} 1711\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])} 1712\indentrel{3}\begin{verbatim} 1713 (4) 1714 [particular = [- 2, 2, 0], basis = [[1, - 2, 1]]] 1715Type: Record(particular: Union(Vector(Fraction(Integer)),"failed"),basis: List(Vector(Fraction(Integer)))) 1716\end{verbatim} 1717\indentrel{-3}\end{paste}\end{patch} 1718 1719\begin{patch}{ugxProblemLinSysPageEmpty4} 1720\begin{paste}{ugxProblemLinSysPageEmpty4}{ugxProblemLinSysPagePatch4} 1721\pastebutton{ugxProblemLinSysPageEmpty4}{\showpaste} 1722\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])} 1723\end{paste}\end{patch} 1724 1725\begin{patch}{ugxProblemLinSysPagePatch5} 1726\begin{paste}{ugxProblemLinSysPageFull5}{ugxProblemLinSysPageEmpty5} 1727\pastebutton{ugxProblemLinSysPageFull5}{\hidepaste} 1728\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])} 1729\indentrel{3}\begin{verbatim} 1730 (5) [particular = "failed", basis = [[1, - 2, 1]]] 1731Type: Record(particular: Union(Vector(Fraction(Integer)),"failed"),basis: List(Vector(Fraction(Integer)))) 1732\end{verbatim} 1733\indentrel{-3}\end{paste}\end{patch} 1734 1735\begin{patch}{ugxProblemLinSysPageEmpty5} 1736\begin{paste}{ugxProblemLinSysPageEmpty5}{ugxProblemLinSysPagePatch5} 1737\pastebutton{ugxProblemLinSysPageEmpty5}{\showpaste} 1738\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])} 1739\end{paste}\end{patch} 1740 1741\begin{patch}{ugxProblemLinSysPagePatch6} 1742\begin{paste}{ugxProblemLinSysPageFull6}{ugxProblemLinSysPageEmpty6} 1743\pastebutton{ugxProblemLinSysPageFull6}{\hidepaste} 1744\tab{5}\spadcommand{nullSpace([[1,2,3],[2,3,4],[3,4,5]])} 1745\indentrel{3}\begin{verbatim} 1746 (6) [[1, - 2, 1]] 1747 Type: List(Vector(Integer)) 1748\end{verbatim} 1749\indentrel{-3}\end{paste}\end{patch} 1750 1751\begin{patch}{ugxProblemLinSysPageEmpty6} 1752\begin{paste}{ugxProblemLinSysPageEmpty6}{ugxProblemLinSysPagePatch6} 1753\pastebutton{ugxProblemLinSysPageEmpty6}{\showpaste} 1754\tab{5}\spadcommand{nullSpace([[1,2,3],[2,3,4],[3,4,5]])} 1755\end{paste}\end{patch} 1756 1757\begin{patch}{ugxProblemPolSysPagePatch1} 1758\begin{paste}{ugxProblemPolSysPageFull1}{ugxProblemPolSysPageEmpty1} 1759\pastebutton{ugxProblemPolSysPageFull1}{\hidepaste} 1760\tab{5}\spadcommand{solve([3*x^3 + y + 1,y^2 -4],[x,y])} 1761\indentrel{3}\begin{verbatim} 1762 (1) 1763 2 1764 [[x = - 1, y = 2], [x - x + 1 = 0, y = 2], 1765 3 1766 [3 x - 1 = 0, y = - 2]] 1767Type: List(List(Equation(Fraction(Polynomial(Integer))))) 1768\end{verbatim} 1769\indentrel{-3}\end{paste}\end{patch} 1770 1771\begin{patch}{ugxProblemPolSysPageEmpty1} 1772\begin{paste}{ugxProblemPolSysPageEmpty1}{ugxProblemPolSysPagePatch1} 1773\pastebutton{ugxProblemPolSysPageEmpty1}{\showpaste} 1774\tab{5}\spadcommand{solve([3*x^3 + y + 1,y^2 -4],[x,y])} 1775\end{paste}\end{patch} 1776 1777\begin{patch}{ugxProblemPolSysPagePatch2} 1778\begin{paste}{ugxProblemPolSysPageFull2}{ugxProblemPolSysPageEmpty2} 1779\pastebutton{ugxProblemPolSysPageFull2}{\hidepaste} 1780\tab{5}\spadcommand{solve([x = y^2-19,y = z^2+x+3,z = 3*x],[x,y,z])} 1781\indentrel{3}\begin{verbatim} 1782 (2) 1783 [ 1784 2 1785 z 3 z + z + 9 1786 [x = -, y = ------------, 1787 3 3 1788 4 3 2 1789 9 z + 6 z + 55 z + 15 z - 90 = 0] 1790 ] 1791Type: List(List(Equation(Fraction(Polynomial(Integer))))) 1792\end{verbatim} 1793\indentrel{-3}\end{paste}\end{patch} 1794 1795\begin{patch}{ugxProblemPolSysPageEmpty2} 1796\begin{paste}{ugxProblemPolSysPageEmpty2}{ugxProblemPolSysPagePatch2} 1797\pastebutton{ugxProblemPolSysPageEmpty2}{\showpaste} 1798\tab{5}\spadcommand{solve([x = y^2-19,y = z^2+x+3,z = 3*x],[x,y,z])} 1799\end{paste}\end{patch} 1800 1801\begin{patch}{ugxProblemPolSysPagePatch3} 1802\begin{paste}{ugxProblemPolSysPageFull3}{ugxProblemPolSysPageEmpty3} 1803\pastebutton{ugxProblemPolSysPageFull3}{\hidepaste} 1804\tab{5}\spadcommand{radicalSolve([3*x^3 + y + 1,y^2 -4],[x,y])} 1805\indentrel{3}\begin{verbatim} 1806 (3) 1807 +---+ +---+ 1808 \|- 3 + 1 - \|- 3 + 1 1809 [[x = ----------, y = 2], [x = ------------, y = 2], 1810 2 2 1811 +---+ +-+ 1812 - \|- 1 \|3 - 1 1813 [x = ----------------, y = - 2], 1814 3+-+ 1815 2 \|3 1816 +---+ +-+ 1817 \|- 1 \|3 - 1 1 1818 [x = --------------, y = - 2], [x = ----, y = - 2], 1819 3+-+ 3+-+ 1820 2 \|3 \|3 1821 [x = - 1, y = 2]] 1822 Type: List(List(Equation(Expression(Integer)))) 1823\end{verbatim} 1824\indentrel{-3}\end{paste}\end{patch} 1825 1826\begin{patch}{ugxProblemPolSysPageEmpty3} 1827\begin{paste}{ugxProblemPolSysPageEmpty3}{ugxProblemPolSysPagePatch3} 1828\pastebutton{ugxProblemPolSysPageEmpty3}{\showpaste} 1829\tab{5}\spadcommand{radicalSolve([3*x^3 + y + 1,y^2 -4],[x,y])} 1830\end{paste}\end{patch} 1831 1832\begin{patch}{ugxProblemPolSysPagePatch4} 1833\begin{paste}{ugxProblemPolSysPageFull4}{ugxProblemPolSysPageEmpty4} 1834\pastebutton{ugxProblemPolSysPageFull4}{\hidepaste} 1835\tab{5}\spadcommand{solve([x^2*y - 1,x*y^2 - 2],.01)} 1836\indentrel{3}\begin{verbatim} 1837 (4) 1838 [ 1839 [y = 1.5874011516_5710449218_75, 1840 x = 0.7937005758_2855224609_375] 1841 ] 1842 Type: List(List(Equation(Polynomial(Float)))) 1843\end{verbatim} 1844\indentrel{-3}\end{paste}\end{patch} 1845 1846\begin{patch}{ugxProblemPolSysPageEmpty4} 1847\begin{paste}{ugxProblemPolSysPageEmpty4}{ugxProblemPolSysPagePatch4} 1848\pastebutton{ugxProblemPolSysPageEmpty4}{\showpaste} 1849\tab{5}\spadcommand{solve([x^2*y - 1,x*y^2 - 2],.01)} 1850\end{paste}\end{patch} 1851 1852\begin{patch}{ugxProblemPolSysPagePatch5} 1853\begin{paste}{ugxProblemPolSysPageFull5}{ugxProblemPolSysPageEmpty5} 1854\pastebutton{ugxProblemPolSysPageFull5}{\hidepaste} 1855\tab{5}\spadcommand{complexSolve([x^2*y - 1,x*y^2 - 2],1/1000)} 1856\indentrel{3}\begin{verbatim} 1857 (5) 1858 27925854633327 27925854633327 1859 [[y = --------------, x = --------------], 1860 17592186044416 35184372088832 1861 1862 [ 1863 y 1864 = 1865 - 1866 92799569462155921798031506794235389730378_ 1867 14343717 1868 / 1869 11692013098647223345629478661730264157247_ 1870 460343808 1871 + 1872 3023062441857 1873 - ------------- %i 1874 2199023255552 1875 , 1876 1877 x 1878 = 1879 - 1880 92799569462155921798031506794235389730378_ 1881 14343717 1882 / 1883 23384026197294446691258957323460528314494_ 1884 920687616 1885 + 1886 3023062441857 1887 - ------------- %i 1888 4398046511104 1889 ] 1890 , 1891 1892 [ 1893 y 1894 = 1895 - 1896 92799569462155921798031506794235389730378_ 1897 14343717 1898 / 1899 11692013098647223345629478661730264157247_ 1900 460343808 1901 + 1902 3023062441857 1903 ------------- %i 1904 2199023255552 1905 , 1906 1907 x 1908 = 1909 - 1910 92799569462155921798031506794235389730378_ 1911 14343717 1912 / 1913 23384026197294446691258957323460528314494_ 1914 920687616 1915 + 1916 3023062441857 1917 ------------- %i 1918 4398046511104 1919 ] 1920 ] 1921Type: List(List(Equation(Polynomial(Complex(Fraction(Integer)))))) 1922\end{verbatim} 1923\indentrel{-3}\end{paste}\end{patch} 1924 1925\begin{patch}{ugxProblemPolSysPageEmpty5} 1926\begin{paste}{ugxProblemPolSysPageEmpty5}{ugxProblemPolSysPagePatch5} 1927\pastebutton{ugxProblemPolSysPageEmpty5}{\showpaste} 1928\tab{5}\spadcommand{complexSolve([x^2*y - 1,x*y^2 - 2],1/1000)} 1929\end{paste}\end{patch} 1930 1931\begin{patch}{ugxProblemPolSysPagePatch6} 1932\begin{paste}{ugxProblemPolSysPageFull6}{ugxProblemPolSysPageEmpty6} 1933\pastebutton{ugxProblemPolSysPageFull6}{\hidepaste} 1934\tab{5}\spadcommand{solve([x^2/a = a,a = a*x],.001)} 1935\indentrel{3}\begin{verbatim} 1936 (6) [[x = 1.0, a = - 1.0], [x = 1.0, a = 1.0]] 1937 Type: List(List(Equation(Polynomial(Float)))) 1938\end{verbatim} 1939\indentrel{-3}\end{paste}\end{patch} 1940 1941\begin{patch}{ugxProblemPolSysPageEmpty6} 1942\begin{paste}{ugxProblemPolSysPageEmpty6}{ugxProblemPolSysPagePatch6} 1943\pastebutton{ugxProblemPolSysPageEmpty6}{\showpaste} 1944\tab{5}\spadcommand{solve([x^2/a = a,a = a*x],.001)} 1945\end{paste}\end{patch} 1946 1947\begin{patch}{ugxProblemPolSysPagePatch7} 1948\begin{paste}{ugxProblemPolSysPageFull7}{ugxProblemPolSysPageEmpty7} 1949\pastebutton{ugxProblemPolSysPageFull7}{\hidepaste} 1950\tab{5}\spadcommand{radicalSolve([x^2/a + a + y^3 - 1,a*y + a + 1],[x,y])} 1951\indentrel{3}\begin{verbatim} 1952 (7) 1953 +----------------------------+ 1954 | 4 3 2 1955 |- a + 2 a + 3 a + 3 a + 1 - a - 1 1956 [[x = - |---------------------------- , y = -------], 1957 | 2 a 1958 \| a 1959 +----------------------------+ 1960 | 4 3 2 1961 |- a + 2 a + 3 a + 3 a + 1 - a - 1 1962 [x = |---------------------------- , y = -------]] 1963 | 2 a 1964 \| a 1965 Type: List(List(Equation(Expression(Integer)))) 1966\end{verbatim} 1967\indentrel{-3}\end{paste}\end{patch} 1968 1969\begin{patch}{ugxProblemPolSysPageEmpty7} 1970\begin{paste}{ugxProblemPolSysPageEmpty7}{ugxProblemPolSysPagePatch7} 1971\pastebutton{ugxProblemPolSysPageEmpty7}{\showpaste} 1972\tab{5}\spadcommand{radicalSolve([x^2/a + a + y^3 - 1,a*y + a + 1],[x,y])} 1973\end{paste}\end{patch} 1974 1975\begin{patch}{ugProblemFactorFFPagePatch1} 1976\begin{paste}{ugProblemFactorFFPageFull1}{ugProblemFactorFFPageEmpty1} 1977\pastebutton{ugProblemFactorFFPageFull1}{\hidepaste} 1978\tab{5}\spadcommand{u : POLY(PF(19)) :=3*x^4+2*x^2+15*x+18\bound{u }} 1979\indentrel{3}\begin{verbatim} 1980 4 2 1981 (1) 3 x + 2 x + 15 x + 18 1982 Type: Polynomial(PrimeField(19)) 1983\end{verbatim} 1984\indentrel{-3}\end{paste}\end{patch} 1985 1986\begin{patch}{ugProblemFactorFFPageEmpty1} 1987\begin{paste}{ugProblemFactorFFPageEmpty1}{ugProblemFactorFFPagePatch1} 1988\pastebutton{ugProblemFactorFFPageEmpty1}{\showpaste} 1989\tab{5}\spadcommand{u : POLY(PF(19)) :=3*x^4+2*x^2+15*x+18\bound{u }} 1990\end{paste}\end{patch} 1991 1992\begin{patch}{ugProblemFactorFFPagePatch2} 1993\begin{paste}{ugProblemFactorFFPageFull2}{ugProblemFactorFFPageEmpty2} 1994\pastebutton{ugProblemFactorFFPageFull2}{\hidepaste} 1995\tab{5}\spadcommand{factor u\free{u }} 1996\indentrel{3}\begin{verbatim} 1997 3 2 1998 (2) 3 (x + x + 8 x + 13)(x + 18) 1999 Type: Factored(Polynomial(PrimeField(19))) 2000\end{verbatim} 2001\indentrel{-3}\end{paste}\end{patch} 2002 2003\begin{patch}{ugProblemFactorFFPageEmpty2} 2004\begin{paste}{ugProblemFactorFFPageEmpty2}{ugProblemFactorFFPagePatch2} 2005\pastebutton{ugProblemFactorFFPageEmpty2}{\showpaste} 2006\tab{5}\spadcommand{factor u\free{u }} 2007\end{paste}\end{patch} 2008 2009\begin{patch}{ugProblemFactorFFPagePatch3} 2010\begin{paste}{ugProblemFactorFFPageFull3}{ugProblemFactorFFPageEmpty3} 2011\pastebutton{ugProblemFactorFFPageFull3}{\hidepaste} 2012\tab{5}\spadcommand{factor(u :: POLY FFX(PF 19,3))\free{u }} 2013\indentrel{3}\begin{verbatim} 2014 (3) 2015 2 2016 3 (x + 18)(x + 5 %BD + 3 %BD + 13) 2017 * 2018 2 2 2019 (x + 16 %BD + 14 %BD + 13)(x + 17 %BD + 2 %BD + 13) 2020Type: Factored(Polynomial(FiniteFieldExtension(PrimeField(19),3))) 2021\end{verbatim} 2022\indentrel{-3}\end{paste}\end{patch} 2023 2024\begin{patch}{ugProblemFactorFFPageEmpty3} 2025\begin{paste}{ugProblemFactorFFPageEmpty3}{ugProblemFactorFFPagePatch3} 2026\pastebutton{ugProblemFactorFFPageEmpty3}{\showpaste} 2027\tab{5}\spadcommand{factor(u :: POLY FFX(PF 19,3))\free{u }} 2028\end{paste}\end{patch} 2029 2030\begin{patch}{ugxProblemSeriesArithmeticPagePatch1} 2031\begin{paste}{ugxProblemSeriesArithmeticPageFull1}{ugxProblemSeriesArithmeticPageEmpty1} 2032\pastebutton{ugxProblemSeriesArithmeticPageFull1}{\hidepaste} 2033\tab{5}\spadcommand{x := series x\bound{x }} 2034\indentrel{3}\begin{verbatim} 2035 (1) x 2036 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 2037\end{verbatim} 2038\indentrel{-3}\end{paste}\end{patch} 2039 2040\begin{patch}{ugxProblemSeriesArithmeticPageEmpty1} 2041\begin{paste}{ugxProblemSeriesArithmeticPageEmpty1}{ugxProblemSeriesArithmeticPagePatch1} 2042\pastebutton{ugxProblemSeriesArithmeticPageEmpty1}{\showpaste} 2043\tab{5}\spadcommand{x := series x\bound{x }} 2044\end{paste}\end{patch} 2045 2046\begin{patch}{ugxProblemSeriesArithmeticPagePatch2} 2047\begin{paste}{ugxProblemSeriesArithmeticPageFull2}{ugxProblemSeriesArithmeticPageEmpty2} 2048\pastebutton{ugxProblemSeriesArithmeticPageFull2}{\hidepaste} 2049\tab{5}\spadcommand{(3 + x) / (1 + 7*x)} 2050\indentrel{3}\begin{verbatim} 2051 (2) 2052 2 3 4 5 2053 3 - 20 x + 140 x - 980 x + 6860 x - 48020 x 2054 + 2055 6 7 8 9 2056 336140 x - 2352980 x + 16470860 x - 115296020 x 2057 + 2058 10 11 2059 807072140 x + O(x ) 2060 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 2061\end{verbatim} 2062\indentrel{-3}\end{paste}\end{patch} 2063 2064\begin{patch}{ugxProblemSeriesArithmeticPageEmpty2} 2065\begin{paste}{ugxProblemSeriesArithmeticPageEmpty2}{ugxProblemSeriesArithmeticPagePatch2} 2066\pastebutton{ugxProblemSeriesArithmeticPageEmpty2}{\showpaste} 2067\tab{5}\spadcommand{(3 + x) / (1 + 7*x)} 2068\end{paste}\end{patch} 2069 2070\begin{patch}{ugxProblemSeriesArithmeticPagePatch3} 2071\begin{paste}{ugxProblemSeriesArithmeticPageFull3}{ugxProblemSeriesArithmeticPageEmpty3} 2072\pastebutton{ugxProblemSeriesArithmeticPageFull3}{\hidepaste} 2073\tab{5}\spadcommand{base := 1 / (1 - x)\free{x }\bound{base }} 2074\indentrel{3}\begin{verbatim} 2075 (3) 2076 2 3 4 5 6 7 8 9 10 2077 1 + x + x + x + x + x + x + x + x + x + x 2078 + 2079 11 2080 O(x ) 2081 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 2082\end{verbatim} 2083\indentrel{-3}\end{paste}\end{patch} 2084 2085\begin{patch}{ugxProblemSeriesArithmeticPageEmpty3} 2086\begin{paste}{ugxProblemSeriesArithmeticPageEmpty3}{ugxProblemSeriesArithmeticPagePatch3} 2087\pastebutton{ugxProblemSeriesArithmeticPageEmpty3}{\showpaste} 2088\tab{5}\spadcommand{base := 1 / (1 - x)\free{x }\bound{base }} 2089\end{paste}\end{patch} 2090 2091\begin{patch}{ugxProblemSeriesArithmeticPagePatch4} 2092\begin{paste}{ugxProblemSeriesArithmeticPageFull4}{ugxProblemSeriesArithmeticPageEmpty4} 2093\pastebutton{ugxProblemSeriesArithmeticPageFull4}{\hidepaste} 2094\tab{5}\spadcommand{expon := x * base\free{x base }\bound{expon }} 2095\indentrel{3}\begin{verbatim} 2096 (4) 2097 2 3 4 5 6 7 8 9 10 11 2098 x + x + x + x + x + x + x + x + x + x + x 2099 + 2100 12 2101 O(x ) 2102 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 2103\end{verbatim} 2104\indentrel{-3}\end{paste}\end{patch} 2105 2106\begin{patch}{ugxProblemSeriesArithmeticPageEmpty4} 2107\begin{paste}{ugxProblemSeriesArithmeticPageEmpty4}{ugxProblemSeriesArithmeticPagePatch4} 2108\pastebutton{ugxProblemSeriesArithmeticPageEmpty4}{\showpaste} 2109\tab{5}\spadcommand{expon := x * base\free{x base }\bound{expon }} 2110\end{paste}\end{patch} 2111 2112\begin{patch}{ugxProblemSeriesArithmeticPagePatch5} 2113\begin{paste}{ugxProblemSeriesArithmeticPageFull5}{ugxProblemSeriesArithmeticPageEmpty5} 2114\pastebutton{ugxProblemSeriesArithmeticPageFull5}{\hidepaste} 2115\tab{5}\spadcommand{base ^ expon\free{base expon }} 2116\indentrel{3}\begin{verbatim} 2117 (5) 2118 2 3 3 7 4 43 5 649 6 241 7 2119 1 + x + - x + - x + -- x + --- x + --- x 2120 2 3 12 120 30 2121 + 2122 3706 8 85763 9 245339 10 11 2123 ---- x + ----- x + ------ x + O(x ) 2124 315 5040 10080 2125 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 2126\end{verbatim} 2127\indentrel{-3}\end{paste}\end{patch} 2128 2129\begin{patch}{ugxProblemSeriesArithmeticPageEmpty5} 2130\begin{paste}{ugxProblemSeriesArithmeticPageEmpty5}{ugxProblemSeriesArithmeticPagePatch5} 2131\pastebutton{ugxProblemSeriesArithmeticPageEmpty5}{\showpaste} 2132\tab{5}\spadcommand{base ^ expon\free{base expon }} 2133\end{paste}\end{patch} 2134 2135\begin{patch}{ugxProblemFiniteConversionPagePatch1} 2136\begin{paste}{ugxProblemFiniteConversionPageFull1}{ugxProblemFiniteConversionPageEmpty1} 2137\pastebutton{ugxProblemFiniteConversionPageFull1}{\hidepaste} 2138\tab{5}\spadcommand{K := PrimeField 3\bound{K }} 2139\indentrel{3}\begin{verbatim} 2140 (1) PrimeField(3) 2141 Type: Type 2142\end{verbatim} 2143\indentrel{-3}\end{paste}\end{patch} 2144 2145\begin{patch}{ugxProblemFiniteConversionPageEmpty1} 2146\begin{paste}{ugxProblemFiniteConversionPageEmpty1}{ugxProblemFiniteConversionPagePatch1} 2147\pastebutton{ugxProblemFiniteConversionPageEmpty1}{\showpaste} 2148\tab{5}\spadcommand{K := PrimeField 3\bound{K }} 2149\end{paste}\end{patch} 2150 2151\begin{patch}{ugxProblemFiniteConversionPagePatch2} 2152\begin{paste}{ugxProblemFiniteConversionPageFull2}{ugxProblemFiniteConversionPageEmpty2} 2153\pastebutton{ugxProblemFiniteConversionPageFull2}{\hidepaste} 2154\tab{5}\spadcommand{(m,n) := (4,8)\bound{m n }} 2155\indentrel{3}\begin{verbatim} 2156 (2) 8 2157 Type: PositiveInteger 2158\end{verbatim} 2159\indentrel{-3}\end{paste}\end{patch} 2160 2161\begin{patch}{ugxProblemFiniteConversionPageEmpty2} 2162\begin{paste}{ugxProblemFiniteConversionPageEmpty2}{ugxProblemFiniteConversionPagePatch2} 2163\pastebutton{ugxProblemFiniteConversionPageEmpty2}{\showpaste} 2164\tab{5}\spadcommand{(m,n) := (4,8)\bound{m n }} 2165\end{paste}\end{patch} 2166 2167\begin{patch}{ugxProblemFiniteConversionPagePatch3} 2168\begin{paste}{ugxProblemFiniteConversionPageFull3}{ugxProblemFiniteConversionPageEmpty3} 2169\pastebutton{ugxProblemFiniteConversionPageFull3}{\hidepaste} 2170\tab{5}\spadcommand{Km := FiniteFieldExtension(K,m)\bound{Km }\free{K m }} 2171\indentrel{3}\begin{verbatim} 2172 (3) FiniteFieldExtension(PrimeField(3),4) 2173 Type: Type 2174\end{verbatim} 2175\indentrel{-3}\end{paste}\end{patch} 2176 2177\begin{patch}{ugxProblemFiniteConversionPageEmpty3} 2178\begin{paste}{ugxProblemFiniteConversionPageEmpty3}{ugxProblemFiniteConversionPagePatch3} 2179\pastebutton{ugxProblemFiniteConversionPageEmpty3}{\showpaste} 2180\tab{5}\spadcommand{Km := FiniteFieldExtension(K,m)\bound{Km }\free{K m }} 2181\end{paste}\end{patch} 2182 2183\begin{patch}{ugxProblemFiniteConversionPagePatch4} 2184\begin{paste}{ugxProblemFiniteConversionPageFull4}{ugxProblemFiniteConversionPageEmpty4} 2185\pastebutton{ugxProblemFiniteConversionPageFull4}{\hidepaste} 2186\tab{5}\spadcommand{Kn := FiniteFieldExtension(K,n)\bound{Kn }\free{K n }} 2187\indentrel{3}\begin{verbatim} 2188 (4) FiniteFieldExtension(PrimeField(3),8) 2189 Type: Type 2190\end{verbatim} 2191\indentrel{-3}\end{paste}\end{patch} 2192 2193\begin{patch}{ugxProblemFiniteConversionPageEmpty4} 2194\begin{paste}{ugxProblemFiniteConversionPageEmpty4}{ugxProblemFiniteConversionPagePatch4} 2195\pastebutton{ugxProblemFiniteConversionPageEmpty4}{\showpaste} 2196\tab{5}\spadcommand{Kn := FiniteFieldExtension(K,n)\bound{Kn }\free{K n }} 2197\end{paste}\end{patch} 2198 2199\begin{patch}{ugxProblemFiniteConversionPagePatch5} 2200\begin{paste}{ugxProblemFiniteConversionPageFull5}{ugxProblemFiniteConversionPageEmpty5} 2201\pastebutton{ugxProblemFiniteConversionPageFull5}{\hidepaste} 2202\tab{5}\spadcommand{a1 := random()$Km\bound{a1 }\free{Km }} 2203\indentrel{3}\begin{verbatim} 2204 3 2205 (5) %BE + %BE 2206 Type: FiniteFieldExtension(PrimeField(3),4) 2207\end{verbatim} 2208\indentrel{-3}\end{paste}\end{patch} 2209 2210\begin{patch}{ugxProblemFiniteConversionPageEmpty5} 2211\begin{paste}{ugxProblemFiniteConversionPageEmpty5}{ugxProblemFiniteConversionPagePatch5} 2212\pastebutton{ugxProblemFiniteConversionPageEmpty5}{\showpaste} 2213\tab{5}\spadcommand{a1 := random()$Km\bound{a1 }\free{Km }} 2214\end{paste}\end{patch} 2215 2216\begin{patch}{ugxProblemFiniteConversionPagePatch6} 2217\begin{paste}{ugxProblemFiniteConversionPageFull6}{ugxProblemFiniteConversionPageEmpty6} 2218\pastebutton{ugxProblemFiniteConversionPageFull6}{\hidepaste} 2219\tab{5}\spadcommand{b1 := random()$Km\bound{b1 }\free{Km }} 2220\indentrel{3}\begin{verbatim} 2221 3 2 2222 (6) %BE + 2 %BE 2223 Type: FiniteFieldExtension(PrimeField(3),4) 2224\end{verbatim} 2225\indentrel{-3}\end{paste}\end{patch} 2226 2227\begin{patch}{ugxProblemFiniteConversionPageEmpty6} 2228\begin{paste}{ugxProblemFiniteConversionPageEmpty6}{ugxProblemFiniteConversionPagePatch6} 2229\pastebutton{ugxProblemFiniteConversionPageEmpty6}{\showpaste} 2230\tab{5}\spadcommand{b1 := random()$Km\bound{b1 }\free{Km }} 2231\end{paste}\end{patch} 2232 2233\begin{patch}{ugxProblemFiniteConversionPagePatch7} 2234\begin{paste}{ugxProblemFiniteConversionPageFull7}{ugxProblemFiniteConversionPageEmpty7} 2235\pastebutton{ugxProblemFiniteConversionPageFull7}{\hidepaste} 2236\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a2 }\free{a1 Kn }} 2237\indentrel{3}\begin{verbatim} 2238 6 4 2 2239 (7) %BF + 2 %BF + 2 %BF 2240 Type: FiniteFieldExtension(PrimeField(3),8) 2241\end{verbatim} 2242\indentrel{-3}\end{paste}\end{patch} 2243 2244\begin{patch}{ugxProblemFiniteConversionPageEmpty7} 2245\begin{paste}{ugxProblemFiniteConversionPageEmpty7}{ugxProblemFiniteConversionPagePatch7} 2246\pastebutton{ugxProblemFiniteConversionPageEmpty7}{\showpaste} 2247\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a2 }\free{a1 Kn }} 2248\end{paste}\end{patch} 2249 2250\begin{patch}{ugxProblemFiniteConversionPagePatch8} 2251\begin{paste}{ugxProblemFiniteConversionPageFull8}{ugxProblemFiniteConversionPageEmpty8} 2252\pastebutton{ugxProblemFiniteConversionPageFull8}{\hidepaste} 2253\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b2 }\free{b1 Kn }} 2254\indentrel{3}\begin{verbatim} 2255 4 2256 (8) 2 %BF 2257 Type: FiniteFieldExtension(PrimeField(3),8) 2258\end{verbatim} 2259\indentrel{-3}\end{paste}\end{patch} 2260 2261\begin{patch}{ugxProblemFiniteConversionPageEmpty8} 2262\begin{paste}{ugxProblemFiniteConversionPageEmpty8}{ugxProblemFiniteConversionPagePatch8} 2263\pastebutton{ugxProblemFiniteConversionPageEmpty8}{\showpaste} 2264\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b2 }\free{b1 Kn }} 2265\end{paste}\end{patch} 2266 2267\begin{patch}{ugxProblemFiniteConversionPagePatch9} 2268\begin{paste}{ugxProblemFiniteConversionPageFull9}{ugxProblemFiniteConversionPageEmpty9} 2269\pastebutton{ugxProblemFiniteConversionPageFull9}{\hidepaste} 2270\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a1 a2 b1 b2 Km Kn }} 2271\indentrel{3}\begin{verbatim} 2272 (9) 0 2273 Type: FiniteFieldExtension(PrimeField(3),4) 2274\end{verbatim} 2275\indentrel{-3}\end{paste}\end{patch} 2276 2277\begin{patch}{ugxProblemFiniteConversionPageEmpty9} 2278\begin{paste}{ugxProblemFiniteConversionPageEmpty9}{ugxProblemFiniteConversionPagePatch9} 2279\pastebutton{ugxProblemFiniteConversionPageEmpty9}{\showpaste} 2280\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a1 a2 b1 b2 Km Kn }} 2281\end{paste}\end{patch} 2282 2283\begin{patch}{ugxProblemFiniteConversionPagePatch10} 2284\begin{paste}{ugxProblemFiniteConversionPageFull10}{ugxProblemFiniteConversionPageEmpty10} 2285\pastebutton{ugxProblemFiniteConversionPageFull10}{\hidepaste} 2286\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a1 a2 b1 b2 Km Kn }} 2287\indentrel{3}\begin{verbatim} 2288 (10) 0 2289 Type: FiniteFieldExtension(PrimeField(3),4) 2290\end{verbatim} 2291\indentrel{-3}\end{paste}\end{patch} 2292 2293\begin{patch}{ugxProblemFiniteConversionPageEmpty10} 2294\begin{paste}{ugxProblemFiniteConversionPageEmpty10}{ugxProblemFiniteConversionPagePatch10} 2295\pastebutton{ugxProblemFiniteConversionPageEmpty10}{\showpaste} 2296\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a1 a2 b1 b2 Km Kn }} 2297\end{paste}\end{patch} 2298 2299\begin{patch}{ugxProblemFiniteConversionPagePatch11} 2300\begin{paste}{ugxProblemFiniteConversionPageFull11}{ugxProblemFiniteConversionPageEmpty11} 2301\pastebutton{ugxProblemFiniteConversionPageFull11}{\hidepaste} 2302\tab{5}\spadcommand{Km := FFCGX(K,m)\bound{Km2 }\free{K m }} 2303\indentrel{3}\begin{verbatim} 2304 (11) 2305 FiniteFieldCyclicGroupExtension(PrimeField(3),4) 2306 Type: Type 2307\end{verbatim} 2308\indentrel{-3}\end{paste}\end{patch} 2309 2310\begin{patch}{ugxProblemFiniteConversionPageEmpty11} 2311\begin{paste}{ugxProblemFiniteConversionPageEmpty11}{ugxProblemFiniteConversionPagePatch11} 2312\pastebutton{ugxProblemFiniteConversionPageEmpty11}{\showpaste} 2313\tab{5}\spadcommand{Km := FFCGX(K,m)\bound{Km2 }\free{K m }} 2314\end{paste}\end{patch} 2315 2316\begin{patch}{ugxProblemFiniteConversionPagePatch12} 2317\begin{paste}{ugxProblemFiniteConversionPageFull12}{ugxProblemFiniteConversionPageEmpty12} 2318\pastebutton{ugxProblemFiniteConversionPageFull12}{\hidepaste} 2319\tab{5}\spadcommand{Kn := FFNBX(K,n)\bound{Kn2 }\free{K n }} 2320\indentrel{3}\begin{verbatim} 2321 (12) 2322 FiniteFieldNormalBasisExtension(PrimeField(3),8) 2323 Type: Type 2324\end{verbatim} 2325\indentrel{-3}\end{paste}\end{patch} 2326 2327\begin{patch}{ugxProblemFiniteConversionPageEmpty12} 2328\begin{paste}{ugxProblemFiniteConversionPageEmpty12}{ugxProblemFiniteConversionPagePatch12} 2329\pastebutton{ugxProblemFiniteConversionPageEmpty12}{\showpaste} 2330\tab{5}\spadcommand{Kn := FFNBX(K,n)\bound{Kn2 }\free{K n }} 2331\end{paste}\end{patch} 2332 2333\begin{patch}{ugxProblemFiniteConversionPagePatch13} 2334\begin{paste}{ugxProblemFiniteConversionPageFull13}{ugxProblemFiniteConversionPageEmpty13} 2335\pastebutton{ugxProblemFiniteConversionPageFull13}{\hidepaste} 2336\tab{5}\spadcommand{(a1,b1) := (random()$Km,random()$Km)\bound{a12 b12 }\free{Km2 }} 2337\indentrel{3}\begin{verbatim} 2338 70 2339 (13) %BG 2340 Type: FiniteFieldCyclicGroupExtension(PrimeField(3),4) 2341\end{verbatim} 2342\indentrel{-3}\end{paste}\end{patch} 2343 2344\begin{patch}{ugxProblemFiniteConversionPageEmpty13} 2345\begin{paste}{ugxProblemFiniteConversionPageEmpty13}{ugxProblemFiniteConversionPagePatch13} 2346\pastebutton{ugxProblemFiniteConversionPageEmpty13}{\showpaste} 2347\tab{5}\spadcommand{(a1,b1) := (random()$Km,random()$Km)\bound{a12 b12 }\free{Km2 }} 2348\end{paste}\end{patch} 2349 2350\begin{patch}{ugxProblemFiniteConversionPagePatch14} 2351\begin{paste}{ugxProblemFiniteConversionPageFull14}{ugxProblemFiniteConversionPageEmpty14} 2352\pastebutton{ugxProblemFiniteConversionPageFull14}{\hidepaste} 2353\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a22 }\free{a12 Kn2 }} 2354\indentrel{3}\begin{verbatim} 2355 (14) 2356 6 5 4 2 2357 q q q q q 2358 2 %BH + %BH + 2 %BH + 2 %BH + %BH + 2 %BH 2359 Type: FiniteFieldNormalBasisExtension(PrimeField(3),8) 2360\end{verbatim} 2361\indentrel{-3}\end{paste}\end{patch} 2362 2363\begin{patch}{ugxProblemFiniteConversionPageEmpty14} 2364\begin{paste}{ugxProblemFiniteConversionPageEmpty14}{ugxProblemFiniteConversionPagePatch14} 2365\pastebutton{ugxProblemFiniteConversionPageEmpty14}{\showpaste} 2366\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a22 }\free{a12 Kn2 }} 2367\end{paste}\end{patch} 2368 2369\begin{patch}{ugxProblemFiniteConversionPagePatch15} 2370\begin{paste}{ugxProblemFiniteConversionPageFull15}{ugxProblemFiniteConversionPageEmpty15} 2371\pastebutton{ugxProblemFiniteConversionPageFull15}{\hidepaste} 2372\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b22 }\free{b12 Kn2 }} 2373\indentrel{3}\begin{verbatim} 2374 6 4 2 2375 q q q 2376 (15) %BH + %BH + %BH + %BH 2377 Type: FiniteFieldNormalBasisExtension(PrimeField(3),8) 2378\end{verbatim} 2379\indentrel{-3}\end{paste}\end{patch} 2380 2381\begin{patch}{ugxProblemFiniteConversionPageEmpty15} 2382\begin{paste}{ugxProblemFiniteConversionPageEmpty15}{ugxProblemFiniteConversionPagePatch15} 2383\pastebutton{ugxProblemFiniteConversionPageEmpty15}{\showpaste} 2384\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b22 }\free{b12 Kn2 }} 2385\end{paste}\end{patch} 2386 2387\begin{patch}{ugxProblemFiniteConversionPagePatch16} 2388\begin{paste}{ugxProblemFiniteConversionPageFull16}{ugxProblemFiniteConversionPageEmpty16} 2389\pastebutton{ugxProblemFiniteConversionPageFull16}{\hidepaste} 2390\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a12 a22 b12 b22 Km2 }} 2391\indentrel{3}\begin{verbatim} 2392 (16) 0 2393 Type: FiniteFieldCyclicGroupExtension(PrimeField(3),4) 2394\end{verbatim} 2395\indentrel{-3}\end{paste}\end{patch} 2396 2397\begin{patch}{ugxProblemFiniteConversionPageEmpty16} 2398\begin{paste}{ugxProblemFiniteConversionPageEmpty16}{ugxProblemFiniteConversionPagePatch16} 2399\pastebutton{ugxProblemFiniteConversionPageEmpty16}{\showpaste} 2400\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a12 a22 b12 b22 Km2 }} 2401\end{paste}\end{patch} 2402 2403\begin{patch}{ugxProblemFiniteConversionPagePatch17} 2404\begin{paste}{ugxProblemFiniteConversionPageFull17}{ugxProblemFiniteConversionPageEmpty17} 2405\pastebutton{ugxProblemFiniteConversionPageFull17}{\hidepaste} 2406\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a12 a22 b12 b22 Km2 }} 2407\indentrel{3}\begin{verbatim} 2408 (17) 0 2409 Type: FiniteFieldCyclicGroupExtension(PrimeField(3),4) 2410\end{verbatim} 2411\indentrel{-3}\end{paste}\end{patch} 2412 2413\begin{patch}{ugxProblemFiniteConversionPageEmpty17} 2414\begin{paste}{ugxProblemFiniteConversionPageEmpty17}{ugxProblemFiniteConversionPagePatch17} 2415\pastebutton{ugxProblemFiniteConversionPageEmpty17}{\showpaste} 2416\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a12 a22 b12 b22 Km2 }} 2417\end{paste}\end{patch} 2418 2419\begin{patch}{ugxProblemSeriesSubstitutePagePatch1} 2420\begin{paste}{ugxProblemSeriesSubstitutePageFull1}{ugxProblemSeriesSubstitutePageEmpty1} 2421\pastebutton{ugxProblemSeriesSubstitutePageFull1}{\hidepaste} 2422\tab{5}\spadcommand{f := taylor(exp(x))\bound{f }} 2423\indentrel{3}\begin{verbatim} 2424 (1) 2425 1 2 1 3 1 4 1 5 1 6 2426 1 + x + - x + - x + -- x + --- x + --- x 2427 2 6 24 120 720 2428 + 2429 1 7 1 8 1 9 1 10 11 2430 ---- x + ----- x + ------ x + ------- x + O(x ) 2431 5040 40320 362880 3628800 2432 Type: UnivariateTaylorSeries(Expression(Integer),x,0) 2433\end{verbatim} 2434\indentrel{-3}\end{paste}\end{patch} 2435 2436\begin{patch}{ugxProblemSeriesSubstitutePageEmpty1} 2437\begin{paste}{ugxProblemSeriesSubstitutePageEmpty1}{ugxProblemSeriesSubstitutePagePatch1} 2438\pastebutton{ugxProblemSeriesSubstitutePageEmpty1}{\showpaste} 2439\tab{5}\spadcommand{f := taylor(exp(x))\bound{f }} 2440\end{paste}\end{patch} 2441 2442\begin{patch}{ugxProblemSeriesSubstitutePagePatch2} 2443\begin{paste}{ugxProblemSeriesSubstitutePageFull2}{ugxProblemSeriesSubstitutePageEmpty2} 2444\pastebutton{ugxProblemSeriesSubstitutePageFull2}{\hidepaste} 2445\tab{5}\spadcommand{eval(f,1.0)\free{f }} 2446\indentrel{3}\begin{verbatim} 2447 (2) 2448 [1.0, 2.0, 2.5, 2.6666666666_666666667, 2449 2.7083333333_333333333, 2.7166666666_666666667, 2450 2.7180555555_555555556, 2.7182539682_53968254, 2451 2.7182787698_412698413, 2.7182815255_731922399, ...] 2452 Type: Stream(Expression(Float)) 2453\end{verbatim} 2454\indentrel{-3}\end{paste}\end{patch} 2455 2456\begin{patch}{ugxProblemSeriesSubstitutePageEmpty2} 2457\begin{paste}{ugxProblemSeriesSubstitutePageEmpty2}{ugxProblemSeriesSubstitutePagePatch2} 2458\pastebutton{ugxProblemSeriesSubstitutePageEmpty2}{\showpaste} 2459\tab{5}\spadcommand{eval(f,1.0)\free{f }} 2460\end{paste}\end{patch} 2461 2462\begin{patch}{ugProblemFactorAlgPagePatch1} 2463\begin{paste}{ugProblemFactorAlgPageFull1}{ugProblemFactorAlgPageEmpty1} 2464\pastebutton{ugProblemFactorAlgPageFull1}{\hidepaste} 2465\tab{5}\spadcommand{aa := rootOf(aa^2+aa+1)\bound{aa }} 2466\indentrel{3}\begin{verbatim} 2467 (1) aa 2468 Type: AlgebraicNumber 2469\end{verbatim} 2470\indentrel{-3}\end{paste}\end{patch} 2471 2472\begin{patch}{ugProblemFactorAlgPageEmpty1} 2473\begin{paste}{ugProblemFactorAlgPageEmpty1}{ugProblemFactorAlgPagePatch1} 2474\pastebutton{ugProblemFactorAlgPageEmpty1}{\showpaste} 2475\tab{5}\spadcommand{aa := rootOf(aa^2+aa+1)\bound{aa }} 2476\end{paste}\end{patch} 2477 2478\begin{patch}{ugProblemFactorAlgPagePatch2} 2479\begin{paste}{ugProblemFactorAlgPageFull2}{ugProblemFactorAlgPageEmpty2} 2480\pastebutton{ugProblemFactorAlgPageFull2}{\hidepaste} 2481\tab{5}\spadcommand{p:=(x^3+aa^2*x+y)*(aa*x^2+aa*x+aa*y^2)^2\free{aa }\bound{p }} 2482\indentrel{3}\begin{verbatim} 2483 (2) 2484 5 3 4 2485 (- aa - 1)y + ((- aa - 1)x + aa x)y 2486 + 2487 2 3 2488 ((- 2 aa - 2)x + (- 2 aa - 2)x)y 2489 + 2490 5 4 3 2491 (- 2 aa - 2)x + (- 2 aa - 2)x + 2 aa x 2492 + 2493 2 2494 2 aa x 2495 * 2496 2 2497 y 2498 + 2499 4 3 2 2500 ((- aa - 1)x + (- 2 aa - 2)x + (- aa - 1)x )y 2501 + 2502 7 6 5 4 3 2503 (- aa - 1)x + (- 2 aa - 2)x - x + 2 aa x + aa x 2504 Type: Polynomial(AlgebraicNumber) 2505\end{verbatim} 2506\indentrel{-3}\end{paste}\end{patch} 2507 2508\begin{patch}{ugProblemFactorAlgPageEmpty2} 2509\begin{paste}{ugProblemFactorAlgPageEmpty2}{ugProblemFactorAlgPagePatch2} 2510\pastebutton{ugProblemFactorAlgPageEmpty2}{\showpaste} 2511\tab{5}\spadcommand{p:=(x^3+aa^2*x+y)*(aa*x^2+aa*x+aa*y^2)^2\free{aa }\bound{p }} 2512\end{paste}\end{patch} 2513 2514\begin{patch}{ugProblemFactorAlgPagePatch3} 2515\begin{paste}{ugProblemFactorAlgPageFull3}{ugProblemFactorAlgPageEmpty3} 2516\pastebutton{ugProblemFactorAlgPageFull3}{\hidepaste} 2517\tab{5}\spadcommand{factor(p,[aa])\free{p aa }} 2518\indentrel{3}\begin{verbatim} 2519 3 2 2 2 2520 (3) (- aa - 1)(y + x + (- aa - 1)x)(y + x + x) 2521 Type: Factored(Polynomial(AlgebraicNumber)) 2522\end{verbatim} 2523\indentrel{-3}\end{paste}\end{patch} 2524 2525\begin{patch}{ugProblemFactorAlgPageEmpty3} 2526\begin{paste}{ugProblemFactorAlgPageEmpty3}{ugProblemFactorAlgPagePatch3} 2527\pastebutton{ugProblemFactorAlgPageEmpty3}{\showpaste} 2528\tab{5}\spadcommand{factor(p,[aa])\free{p aa }} 2529\end{paste}\end{patch} 2530 2531\begin{patch}{ugProblemFactorAlgPagePatch4} 2532\begin{paste}{ugProblemFactorAlgPageFull4}{ugProblemFactorAlgPageEmpty4} 2533\pastebutton{ugProblemFactorAlgPageFull4}{\hidepaste} 2534\tab{5}\spadcommand{factor(x^2+3)} 2535\indentrel{3}\begin{verbatim} 2536 2 2537 (4) x + 3 2538 Type: Factored(Polynomial(Integer)) 2539\end{verbatim} 2540\indentrel{-3}\end{paste}\end{patch} 2541 2542\begin{patch}{ugProblemFactorAlgPageEmpty4} 2543\begin{paste}{ugProblemFactorAlgPageEmpty4}{ugProblemFactorAlgPagePatch4} 2544\pastebutton{ugProblemFactorAlgPageEmpty4}{\showpaste} 2545\tab{5}\spadcommand{factor(x^2+3)} 2546\end{paste}\end{patch} 2547 2548\begin{patch}{ugProblemFactorAlgPagePatch5} 2549\begin{paste}{ugProblemFactorAlgPageFull5}{ugProblemFactorAlgPageEmpty5} 2550\pastebutton{ugProblemFactorAlgPageFull5}{\hidepaste} 2551\tab{5}\spadcommand{factor(x^2+3,[aa])\free{aa }} 2552\indentrel{3}\begin{verbatim} 2553 (5) (x - 2 aa - 1)(x + 2 aa + 1) 2554 Type: Factored(Polynomial(AlgebraicNumber)) 2555\end{verbatim} 2556\indentrel{-3}\end{paste}\end{patch} 2557 2558\begin{patch}{ugProblemFactorAlgPageEmpty5} 2559\begin{paste}{ugProblemFactorAlgPageEmpty5}{ugProblemFactorAlgPagePatch5} 2560\pastebutton{ugProblemFactorAlgPageEmpty5}{\showpaste} 2561\tab{5}\spadcommand{factor(x^2+3,[aa])\free{aa }} 2562\end{paste}\end{patch} 2563 2564\begin{patch}{ugProblemFactorAlgPagePatch6} 2565\begin{paste}{ugProblemFactorAlgPageFull6}{ugProblemFactorAlgPageEmpty6} 2566\pastebutton{ugProblemFactorAlgPageFull6}{\hidepaste} 2567\tab{5}\spadcommand{factor(x^6+108,[aa])\free{aa }} 2568\indentrel{3}\begin{verbatim} 2569 3 3 2570 (6) (x - 12 aa - 6)(x + 12 aa + 6) 2571 Type: Factored(Polynomial(AlgebraicNumber)) 2572\end{verbatim} 2573\indentrel{-3}\end{paste}\end{patch} 2574 2575\begin{patch}{ugProblemFactorAlgPageEmpty6} 2576\begin{paste}{ugProblemFactorAlgPageEmpty6}{ugProblemFactorAlgPagePatch6} 2577\pastebutton{ugProblemFactorAlgPageEmpty6}{\showpaste} 2578\tab{5}\spadcommand{factor(x^6+108,[aa])\free{aa }} 2579\end{paste}\end{patch} 2580 2581\begin{patch}{ugProblemFactorAlgPagePatch7} 2582\begin{paste}{ugProblemFactorAlgPageFull7}{ugProblemFactorAlgPageEmpty7} 2583\pastebutton{ugProblemFactorAlgPageFull7}{\hidepaste} 2584\tab{5}\spadcommand{bb:=rootOf(bb^3-2)\bound{bb }} 2585\indentrel{3}\begin{verbatim} 2586 (7) bb 2587 Type: AlgebraicNumber 2588\end{verbatim} 2589\indentrel{-3}\end{paste}\end{patch} 2590 2591\begin{patch}{ugProblemFactorAlgPageEmpty7} 2592\begin{paste}{ugProblemFactorAlgPageEmpty7}{ugProblemFactorAlgPagePatch7} 2593\pastebutton{ugProblemFactorAlgPageEmpty7}{\showpaste} 2594\tab{5}\spadcommand{bb:=rootOf(bb^3-2)\bound{bb }} 2595\end{paste}\end{patch} 2596 2597\begin{patch}{ugProblemFactorAlgPagePatch8} 2598\begin{paste}{ugProblemFactorAlgPageFull8}{ugProblemFactorAlgPageEmpty8} 2599\pastebutton{ugProblemFactorAlgPageFull8}{\hidepaste} 2600\tab{5}\spadcommand{factor(x^6+108,[bb])\free{bb }} 2601\indentrel{3}\begin{verbatim} 2602 (8) 2603 2 2 2 2 2 2 2604 (x - 3 bb x + 3 bb )(x + 3 bb )(x + 3 bb x + 3 bb ) 2605 Type: Factored(Polynomial(AlgebraicNumber)) 2606\end{verbatim} 2607\indentrel{-3}\end{paste}\end{patch} 2608 2609\begin{patch}{ugProblemFactorAlgPageEmpty8} 2610\begin{paste}{ugProblemFactorAlgPageEmpty8}{ugProblemFactorAlgPagePatch8} 2611\pastebutton{ugProblemFactorAlgPageEmpty8}{\showpaste} 2612\tab{5}\spadcommand{factor(x^6+108,[bb])\free{bb }} 2613\end{paste}\end{patch} 2614 2615\begin{patch}{ugProblemFactorAlgPagePatch9} 2616\begin{paste}{ugProblemFactorAlgPageFull9}{ugProblemFactorAlgPageEmpty9} 2617\pastebutton{ugProblemFactorAlgPageFull9}{\hidepaste} 2618\tab{5}\spadcommand{factor(x^6+108,[aa,bb])\free{aa bb }} 2619\indentrel{3}\begin{verbatim} 2620 (9) 2621 (x + (- 2 aa - 1)bb)(x + (- aa - 2)bb) 2622 * 2623 (x + (- aa + 1)bb)(x + (aa - 1)bb)(x + (aa + 2)bb) 2624 * 2625 (x + (2 aa + 1)bb) 2626 Type: Factored(Polynomial(AlgebraicNumber)) 2627\end{verbatim} 2628\indentrel{-3}\end{paste}\end{patch} 2629 2630\begin{patch}{ugProblemFactorAlgPageEmpty9} 2631\begin{paste}{ugProblemFactorAlgPageEmpty9}{ugProblemFactorAlgPagePatch9} 2632\pastebutton{ugProblemFactorAlgPageEmpty9}{\showpaste} 2633\tab{5}\spadcommand{factor(x^6+108,[aa,bb])\free{aa bb }} 2634\end{paste}\end{patch} 2635 2636\begin{patch}{ugxProblemSeriesConversionsPagePatch1} 2637\begin{paste}{ugxProblemSeriesConversionsPageFull1}{ugxProblemSeriesConversionsPageEmpty1} 2638\pastebutton{ugxProblemSeriesConversionsPageFull1}{\hidepaste} 2639\tab{5}\spadcommand{taylor(sin(x),x = 0)} 2640\indentrel{3}\begin{verbatim} 2641 1 3 1 5 1 7 1 9 11 2642 (1) x - - x + --- x - ---- x + ------ x + O(x ) 2643 6 120 5040 362880 2644 Type: UnivariateTaylorSeries(Expression(Integer),x,0) 2645\end{verbatim} 2646\indentrel{-3}\end{paste}\end{patch} 2647 2648\begin{patch}{ugxProblemSeriesConversionsPageEmpty1} 2649\begin{paste}{ugxProblemSeriesConversionsPageEmpty1}{ugxProblemSeriesConversionsPagePatch1} 2650\pastebutton{ugxProblemSeriesConversionsPageEmpty1}{\showpaste} 2651\tab{5}\spadcommand{taylor(sin(x),x = 0)} 2652\end{paste}\end{patch} 2653 2654\begin{patch}{ugxProblemSeriesConversionsPagePatch2} 2655\begin{paste}{ugxProblemSeriesConversionsPageFull2}{ugxProblemSeriesConversionsPageEmpty2} 2656\pastebutton{ugxProblemSeriesConversionsPageFull2}{\hidepaste} 2657\tab{5}\spadcommand{taylor(sin(x),x = \%pi/6)} 2658\indentrel{3}\begin{verbatim} 2659 (2) 2660 +-+ +-+ 2661 1 \|3 %pi 1 %pi 2 \|3 %pi 3 2662 - + ---- (x - ---) - - (x - ---) - ---- (x - ---) 2663 2 2 6 4 6 12 6 2664 + 2665 +-+ 2666 1 %pi 4 \|3 %pi 5 1 %pi 6 2667 -- (x - ---) + ---- (x - ---) - ---- (x - ---) 2668 48 6 240 6 1440 6 2669 + 2670 +-+ 2671 \|3 %pi 7 1 %pi 8 2672 - ----- (x - ---) + ----- (x - ---) 2673 10080 6 80640 6 2674 + 2675 +-+ 2676 \|3 %pi 9 1 %pi 10 2677 ------ (x - ---) - ------- (x - ---) 2678 725760 6 7257600 6 2679 + 2680 %pi 11 2681 O((x - ---) ) 2682 6 2683Type: UnivariateTaylorSeries(Expression(Integer),x,%pi/6) 2684\end{verbatim} 2685\indentrel{-3}\end{paste}\end{patch} 2686 2687\begin{patch}{ugxProblemSeriesConversionsPageEmpty2} 2688\begin{paste}{ugxProblemSeriesConversionsPageEmpty2}{ugxProblemSeriesConversionsPagePatch2} 2689\pastebutton{ugxProblemSeriesConversionsPageEmpty2}{\showpaste} 2690\tab{5}\spadcommand{taylor(sin(x),x = \%pi/6)} 2691\end{paste}\end{patch} 2692 2693\begin{patch}{ugxProblemSeriesConversionsPagePatch3} 2694\begin{paste}{ugxProblemSeriesConversionsPageFull3}{ugxProblemSeriesConversionsPageEmpty3} 2695\pastebutton{ugxProblemSeriesConversionsPageFull3}{\hidepaste} 2696\tab{5}\spadcommand{taylor(tan(x*y),x = 0)} 2697\indentrel{3}\begin{verbatim} 2698 (3) 2699 3 5 7 9 2700 y 3 2 y 5 17 y 7 62 y 9 11 2701 y x + -- x + ---- x + ----- x + ----- x + O(x ) 2702 3 15 315 2835 2703 Type: UnivariateTaylorSeries(Expression(Integer),x,0) 2704\end{verbatim} 2705\indentrel{-3}\end{paste}\end{patch} 2706 2707\begin{patch}{ugxProblemSeriesConversionsPageEmpty3} 2708\begin{paste}{ugxProblemSeriesConversionsPageEmpty3}{ugxProblemSeriesConversionsPagePatch3} 2709\pastebutton{ugxProblemSeriesConversionsPageEmpty3}{\showpaste} 2710\tab{5}\spadcommand{taylor(tan(x*y),x = 0)} 2711\end{paste}\end{patch} 2712 2713\begin{patch}{ugxProblemSeriesConversionsPagePatch4} 2714\begin{paste}{ugxProblemSeriesConversionsPageFull4}{ugxProblemSeriesConversionsPageEmpty4} 2715\pastebutton{ugxProblemSeriesConversionsPageFull4}{\hidepaste} 2716\tab{5}\spadcommand{taylor(tan(x*y),y = 0)} 2717\indentrel{3}\begin{verbatim} 2718 (4) 2719 3 5 7 9 2720 x 3 2 x 5 17 x 7 62 x 9 11 2721 x y + -- y + ---- y + ----- y + ----- y + O(y ) 2722 3 15 315 2835 2723 Type: UnivariateTaylorSeries(Expression(Integer),y,0) 2724\end{verbatim} 2725\indentrel{-3}\end{paste}\end{patch} 2726 2727\begin{patch}{ugxProblemSeriesConversionsPageEmpty4} 2728\begin{paste}{ugxProblemSeriesConversionsPageEmpty4}{ugxProblemSeriesConversionsPagePatch4} 2729\pastebutton{ugxProblemSeriesConversionsPageEmpty4}{\showpaste} 2730\tab{5}\spadcommand{taylor(tan(x*y),y = 0)} 2731\end{paste}\end{patch} 2732 2733\begin{patch}{ugxProblemSeriesConversionsPagePatch5} 2734\begin{paste}{ugxProblemSeriesConversionsPageFull5}{ugxProblemSeriesConversionsPageEmpty5} 2735\pastebutton{ugxProblemSeriesConversionsPageFull5}{\hidepaste} 2736\tab{5}\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0)\bound{bern }} 2737\indentrel{3}\begin{verbatim} 2738 (5) 2739 2 2740 2 x - 1 6 x - 6 x + 1 2 2741 1 + ------- t + -------------- t 2742 2 12 2743 + 2744 3 2 4 3 2 2745 2 x - 3 x + x 3 30 x - 60 x + 30 x - 1 4 2746 --------------- t + ------------------------- t 2747 12 720 2748 + 2749 5 4 3 2750 6 x - 15 x + 10 x - x 5 2751 ------------------------ t 2752 720 2753 + 2754 6 5 4 2 2755 42 x - 126 x + 105 x - 21 x + 1 6 2756 ----------------------------------- t 2757 30240 2758 + 2759 7 6 5 3 2760 6 x - 21 x + 21 x - 7 x + x 7 2761 ------------------------------- t 2762 30240 2763 + 2764 8 7 6 4 2 2765 30 x - 120 x + 140 x - 70 x + 20 x - 1 8 2766 ------------------------------------------- t 2767 1209600 2768 + 2769 9 8 7 5 3 2770 10 x - 45 x + 60 x - 42 x + 20 x - 3 x 9 2771 ------------------------------------------- t 2772 3628800 2773 + 2774 10 9 8 6 4 2775 66 x - 330 x + 495 x - 462 x + 330 x 2776 + 2777 2 2778 - 99 x + 5 2779 / 2780 239500800 2781 * 2782 10 2783 t 2784 + 2785 11 2786 O(t ) 2787 Type: UnivariateTaylorSeries(Expression(Integer),t,0) 2788\end{verbatim} 2789\indentrel{-3}\end{paste}\end{patch} 2790 2791\begin{patch}{ugxProblemSeriesConversionsPageEmpty5} 2792\begin{paste}{ugxProblemSeriesConversionsPageEmpty5}{ugxProblemSeriesConversionsPagePatch5} 2793\pastebutton{ugxProblemSeriesConversionsPageEmpty5}{\showpaste} 2794\tab{5}\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0)\bound{bern }} 2795\end{paste}\end{patch} 2796 2797\begin{patch}{ugxProblemSeriesConversionsPagePatch6} 2798\begin{paste}{ugxProblemSeriesConversionsPageFull6}{ugxProblemSeriesConversionsPageEmpty6} 2799\pastebutton{ugxProblemSeriesConversionsPageFull6}{\hidepaste} 2800\tab{5}\spadcommand{factorial(6) * coefficient(bern,6)\free{bern }} 2801\indentrel{3}\begin{verbatim} 2802 6 5 4 2 2803 42 x - 126 x + 105 x - 21 x + 1 2804 (6) ----------------------------------- 2805 42 2806 Type: Expression(Integer) 2807\end{verbatim} 2808\indentrel{-3}\end{paste}\end{patch} 2809 2810\begin{patch}{ugxProblemSeriesConversionsPageEmpty6} 2811\begin{paste}{ugxProblemSeriesConversionsPageEmpty6}{ugxProblemSeriesConversionsPagePatch6} 2812\pastebutton{ugxProblemSeriesConversionsPageEmpty6}{\showpaste} 2813\tab{5}\spadcommand{factorial(6) * coefficient(bern,6)\free{bern }} 2814\end{paste}\end{patch} 2815 2816\begin{patch}{ugxProblemSeriesConversionsPagePatch7} 2817\begin{paste}{ugxProblemSeriesConversionsPageFull7}{ugxProblemSeriesConversionsPageEmpty7} 2818\pastebutton{ugxProblemSeriesConversionsPageFull7}{\hidepaste} 2819\tab{5}\spadcommand{bernoulliB(6,x)} 2820\indentrel{3}\begin{verbatim} 2821 6 5 5 4 1 2 1 2822 (7) x - 3 x + - x - - x + -- 2823 2 2 42 2824 Type: Polynomial(Fraction(Integer)) 2825\end{verbatim} 2826\indentrel{-3}\end{paste}\end{patch} 2827 2828\begin{patch}{ugxProblemSeriesConversionsPageEmpty7} 2829\begin{paste}{ugxProblemSeriesConversionsPageEmpty7}{ugxProblemSeriesConversionsPagePatch7} 2830\pastebutton{ugxProblemSeriesConversionsPageEmpty7}{\showpaste} 2831\tab{5}\spadcommand{bernoulliB(6,x)} 2832\end{paste}\end{patch} 2833 2834\begin{patch}{ugxProblemSeriesConversionsPagePatch8} 2835\begin{paste}{ugxProblemSeriesConversionsPageFull8}{ugxProblemSeriesConversionsPageEmpty8} 2836\pastebutton{ugxProblemSeriesConversionsPageFull8}{\hidepaste} 2837\tab{5}\spadcommand{laurent(x/log(x),x = 1)} 2838\indentrel{3}\begin{verbatim} 2839 (8) 2840 - 1 3 5 1 2 2841 (x - 1) + - + -- (x - 1) - -- (x - 1) 2842 2 12 24 2843 + 2844 11 3 11 4 271 5 2845 --- (x - 1) - ---- (x - 1) + ----- (x - 1) 2846 720 1440 60480 2847 + 2848 13 6 7297 7 425 8 2849 - ---- (x - 1) + ------- (x - 1) - ------ (x - 1) 2850 4480 3628800 290304 2851 + 2852 530113 9 10 2853 --------- (x - 1) + O((x - 1) ) 2854 479001600 2855 Type: UnivariateLaurentSeries(Expression(Integer),x,1) 2856\end{verbatim} 2857\indentrel{-3}\end{paste}\end{patch} 2858 2859\begin{patch}{ugxProblemSeriesConversionsPageEmpty8} 2860\begin{paste}{ugxProblemSeriesConversionsPageEmpty8}{ugxProblemSeriesConversionsPagePatch8} 2861\pastebutton{ugxProblemSeriesConversionsPageEmpty8}{\showpaste} 2862\tab{5}\spadcommand{laurent(x/log(x),x = 1)} 2863\end{paste}\end{patch} 2864 2865\begin{patch}{ugxProblemSeriesConversionsPagePatch9} 2866\begin{paste}{ugxProblemSeriesConversionsPageFull9}{ugxProblemSeriesConversionsPageEmpty9} 2867\pastebutton{ugxProblemSeriesConversionsPageFull9}{\hidepaste} 2868\tab{5}\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)} 2869\indentrel{3}\begin{verbatim} 2870 (9) 2871 1 3 7 2872 - - - - 2873 3 %pi 2 1 3 %pi 2 1 3 %pi 2 2874 (x - -----) + -- (x - -----) + --- (x - -----) 2875 2 12 2 160 2 2876 + 2877 3 %pi 5 2878 O((x - -----) ) 2879 2 2880Type: UnivariatePuiseuxSeries(Expression(Integer),x,(3*%pi)/2) 2881\end{verbatim} 2882\indentrel{-3}\end{paste}\end{patch} 2883 2884\begin{patch}{ugxProblemSeriesConversionsPageEmpty9} 2885\begin{paste}{ugxProblemSeriesConversionsPageEmpty9}{ugxProblemSeriesConversionsPagePatch9} 2886\pastebutton{ugxProblemSeriesConversionsPageEmpty9}{\showpaste} 2887\tab{5}\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)} 2888\end{paste}\end{patch} 2889 2890\begin{patch}{ugxProblemSeriesConversionsPagePatch10} 2891\begin{paste}{ugxProblemSeriesConversionsPageFull10}{ugxProblemSeriesConversionsPageEmpty10} 2892\pastebutton{ugxProblemSeriesConversionsPageFull10}{\hidepaste} 2893\tab{5}\spadcommand{series(x^x,x=0)} 2894\indentrel{3}\begin{verbatim} 2895 (10) 2896 2 3 4 2897 log(x) 2 log(x) 3 log(x) 4 2898 1 + log(x)x + ------- x + ------- x + ------- x 2899 2 6 24 2900 + 2901 5 6 7 8 2902 log(x) 5 log(x) 6 log(x) 7 log(x) 8 2903 ------- x + ------- x + ------- x + ------- x 2904 120 720 5040 40320 2905 + 2906 9 10 2907 log(x) 9 log(x) 10 11 2908 ------- x + -------- x + O(x ) 2909 362880 3628800 2910Type: GeneralUnivariatePowerSeries(Expression(Integer),x,0) 2911\end{verbatim} 2912\indentrel{-3}\end{paste}\end{patch} 2913 2914\begin{patch}{ugxProblemSeriesConversionsPageEmpty10} 2915\begin{paste}{ugxProblemSeriesConversionsPageEmpty10}{ugxProblemSeriesConversionsPagePatch10} 2916\pastebutton{ugxProblemSeriesConversionsPageEmpty10}{\showpaste} 2917\tab{5}\spadcommand{series(x^x,x=0)} 2918\end{paste}\end{patch} 2919 2920\begin{patch}{ugxProblemDEQSeriesPagePatch1} 2921\begin{paste}{ugxProblemDEQSeriesPageFull1}{ugxProblemDEQSeriesPageEmpty1} 2922\pastebutton{ugxProblemDEQSeriesPageFull1}{\hidepaste} 2923\tab{5}\spadcommand{)set streams calculate 7\bound{c7 }} 2924\indentrel{3}\begin{verbatim} 2925\end{verbatim} 2926\indentrel{-3}\end{paste}\end{patch} 2927 2928\begin{patch}{ugxProblemDEQSeriesPageEmpty1} 2929\begin{paste}{ugxProblemDEQSeriesPageEmpty1}{ugxProblemDEQSeriesPagePatch1} 2930\pastebutton{ugxProblemDEQSeriesPageEmpty1}{\showpaste} 2931\tab{5}\spadcommand{)set streams calculate 7\bound{c7 }} 2932\end{paste}\end{patch} 2933 2934\begin{patch}{ugxProblemDEQSeriesPagePatch2} 2935\begin{paste}{ugxProblemDEQSeriesPageFull2}{ugxProblemDEQSeriesPageEmpty2} 2936\pastebutton{ugxProblemDEQSeriesPageFull2}{\hidepaste} 2937\tab{5}\spadcommand{y := operator 'y\bound{y }} 2938\indentrel{3}\begin{verbatim} 2939 (1) y 2940 Type: BasicOperator 2941\end{verbatim} 2942\indentrel{-3}\end{paste}\end{patch} 2943 2944\begin{patch}{ugxProblemDEQSeriesPageEmpty2} 2945\begin{paste}{ugxProblemDEQSeriesPageEmpty2}{ugxProblemDEQSeriesPagePatch2} 2946\pastebutton{ugxProblemDEQSeriesPageEmpty2}{\showpaste} 2947\tab{5}\spadcommand{y := operator 'y\bound{y }} 2948\end{paste}\end{patch} 2949 2950\begin{patch}{ugxProblemDEQSeriesPagePatch3} 2951\begin{paste}{ugxProblemDEQSeriesPageFull3}{ugxProblemDEQSeriesPageEmpty3} 2952\pastebutton{ugxProblemDEQSeriesPageFull3}{\hidepaste} 2953\tab{5}\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq }\free{y }} 2954\indentrel{3}\begin{verbatim} 2955 ,,, y(x) ,, 2956 (2) y (x) - %e sin(y (x)) = cos(x) 2957 2958 Type: Equation(Expression(Integer)) 2959\end{verbatim} 2960\indentrel{-3}\end{paste}\end{patch} 2961 2962\begin{patch}{ugxProblemDEQSeriesPageEmpty3} 2963\begin{paste}{ugxProblemDEQSeriesPageEmpty3}{ugxProblemDEQSeriesPagePatch3} 2964\pastebutton{ugxProblemDEQSeriesPageEmpty3}{\showpaste} 2965\tab{5}\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq }\free{y }} 2966\end{paste}\end{patch} 2967 2968\begin{patch}{ugxProblemDEQSeriesPagePatch4} 2969\begin{paste}{ugxProblemDEQSeriesPageFull4}{ugxProblemDEQSeriesPageEmpty4} 2970\pastebutton{ugxProblemDEQSeriesPageFull4}{\hidepaste} 2971\tab{5}\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y }\free{eq }\free{c7 }} 2972\indentrel{3}\begin{verbatim} 2973 (3) 2974 2 3 2975 1 3 %e 4 %e - 1 5 %e - 2 %e 6 2976 1 + - x + -- x + ------- x + ---------- x 2977 6 24 120 720 2978 + 2979 4 2 2980 %e - 8 %e + 4 %e + 1 7 8 2981 ---------------------- x + O(x ) 2982 5040 2983 Type: UnivariateTaylorSeries(Expression(Integer),x,0) 2984\end{verbatim} 2985\indentrel{-3}\end{paste}\end{patch} 2986 2987\begin{patch}{ugxProblemDEQSeriesPageEmpty4} 2988\begin{paste}{ugxProblemDEQSeriesPageEmpty4}{ugxProblemDEQSeriesPagePatch4} 2989\pastebutton{ugxProblemDEQSeriesPageEmpty4}{\showpaste} 2990\tab{5}\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y }\free{eq }\free{c7 }} 2991\end{paste}\end{patch} 2992 2993\begin{patch}{ugxProblemDEQSeriesPagePatch5} 2994\begin{paste}{ugxProblemDEQSeriesPageFull5}{ugxProblemDEQSeriesPageEmpty5} 2995\pastebutton{ugxProblemDEQSeriesPageFull5}{\hidepaste} 2996\tab{5}\spadcommand{x := operator 'x\bound{x }} 2997\indentrel{3}\begin{verbatim} 2998 (4) x 2999 Type: BasicOperator 3000\end{verbatim} 3001\indentrel{-3}\end{paste}\end{patch} 3002 3003\begin{patch}{ugxProblemDEQSeriesPageEmpty5} 3004\begin{paste}{ugxProblemDEQSeriesPageEmpty5}{ugxProblemDEQSeriesPagePatch5} 3005\pastebutton{ugxProblemDEQSeriesPageEmpty5}{\showpaste} 3006\tab{5}\spadcommand{x := operator 'x\bound{x }} 3007\end{paste}\end{patch} 3008 3009\begin{patch}{ugxProblemDEQSeriesPagePatch6} 3010\begin{paste}{ugxProblemDEQSeriesPageFull6}{ugxProblemDEQSeriesPageEmpty6} 3011\pastebutton{ugxProblemDEQSeriesPageFull6}{\hidepaste} 3012\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)^2\free{x }\free{y }\bound{eq1 }} 3013\indentrel{3}\begin{verbatim} 3014 , 2 3015 (5) x (t) = x(t) + 1 3016 3017 Type: Equation(Expression(Integer)) 3018\end{verbatim} 3019\indentrel{-3}\end{paste}\end{patch} 3020 3021\begin{patch}{ugxProblemDEQSeriesPageEmpty6} 3022\begin{paste}{ugxProblemDEQSeriesPageEmpty6}{ugxProblemDEQSeriesPagePatch6} 3023\pastebutton{ugxProblemDEQSeriesPageEmpty6}{\showpaste} 3024\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)^2\free{x }\free{y }\bound{eq1 }} 3025\end{paste}\end{patch} 3026 3027\begin{patch}{ugxProblemDEQSeriesPagePatch7} 3028\begin{paste}{ugxProblemDEQSeriesPageFull7}{ugxProblemDEQSeriesPageEmpty7} 3029\pastebutton{ugxProblemDEQSeriesPageFull7}{\hidepaste} 3030\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }} 3031\indentrel{3}\begin{verbatim} 3032 , 3033 (6) y (t) = x(t)y(t) 3034 3035 Type: Equation(Expression(Integer)) 3036\end{verbatim} 3037\indentrel{-3}\end{paste}\end{patch} 3038 3039\begin{patch}{ugxProblemDEQSeriesPageEmpty7} 3040\begin{paste}{ugxProblemDEQSeriesPageEmpty7}{ugxProblemDEQSeriesPagePatch7} 3041\pastebutton{ugxProblemDEQSeriesPageEmpty7}{\showpaste} 3042\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }} 3043\end{paste}\end{patch} 3044 3045\begin{patch}{ugxProblemDEQSeriesPagePatch8} 3046\begin{paste}{ugxProblemDEQSeriesPageFull8}{ugxProblemDEQSeriesPageEmpty8} 3047\pastebutton{ugxProblemDEQSeriesPageFull8}{\hidepaste} 3048\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }\free{c7 }} 3049\indentrel{3}\begin{verbatim} 3050 (7) 3051 1 3 2 5 17 7 8 3052 [t + - t + -- t + --- t + O(t ), 3053 3 15 315 3054 1 2 5 4 61 6 8 3055 1 + - t + -- t + --- t + O(t )] 3056 2 24 720 3057Type: List(UnivariateTaylorSeries(Expression(Integer),t,0)) 3058\end{verbatim} 3059\indentrel{-3}\end{paste}\end{patch} 3060 3061\begin{patch}{ugxProblemDEQSeriesPageEmpty8} 3062\begin{paste}{ugxProblemDEQSeriesPageEmpty8}{ugxProblemDEQSeriesPagePatch8} 3063\pastebutton{ugxProblemDEQSeriesPageEmpty8}{\showpaste} 3064\tab{5}\spadcommand{seriesSolve([eq2, eq1], [x, y], t = 0, [y(0) = 1, x(0) = 0])\free{x }\free{y }\free{eq1 }\free{eq2 }\free{c7 }} 3065\end{paste}\end{patch} 3066 3067\begin{patch}{ugxProblemSeriesCoefficientsPagePatch1} 3068\begin{paste}{ugxProblemSeriesCoefficientsPageFull1}{ugxProblemSeriesCoefficientsPageEmpty1} 3069\pastebutton{ugxProblemSeriesCoefficientsPageFull1}{\hidepaste} 3070\tab{5}\spadcommand{x := series(x)\bound{x }} 3071\indentrel{3}\begin{verbatim} 3072 (1) x 3073 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 3074\end{verbatim} 3075\indentrel{-3}\end{paste}\end{patch} 3076 3077\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty1} 3078\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty1}{ugxProblemSeriesCoefficientsPagePatch1} 3079\pastebutton{ugxProblemSeriesCoefficientsPageEmpty1}{\showpaste} 3080\tab{5}\spadcommand{x := series(x)\bound{x }} 3081\end{paste}\end{patch} 3082 3083\begin{patch}{ugxProblemSeriesCoefficientsPagePatch2} 3084\begin{paste}{ugxProblemSeriesCoefficientsPageFull2}{ugxProblemSeriesCoefficientsPageEmpty2} 3085\pastebutton{ugxProblemSeriesCoefficientsPageFull2}{\hidepaste} 3086\tab{5}\spadcommand{y := exp(x) * sin(x)\free{x }\bound{y }} 3087\indentrel{3}\begin{verbatim} 3088 (2) 3089 2 1 3 1 5 1 6 1 7 1 9 3090 x + x + - x - -- x - -- x - --- x + ----- x 3091 3 30 90 630 22680 3092 + 3093 1 10 1 11 12 3094 ------ x + ------- x + O(x ) 3095 113400 1247400 3096 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 3097\end{verbatim} 3098\indentrel{-3}\end{paste}\end{patch} 3099 3100\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty2} 3101\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty2}{ugxProblemSeriesCoefficientsPagePatch2} 3102\pastebutton{ugxProblemSeriesCoefficientsPageEmpty2}{\showpaste} 3103\tab{5}\spadcommand{y := exp(x) * sin(x)\free{x }\bound{y }} 3104\end{paste}\end{patch} 3105 3106\begin{patch}{ugxProblemSeriesCoefficientsPagePatch3} 3107\begin{paste}{ugxProblemSeriesCoefficientsPageFull3}{ugxProblemSeriesCoefficientsPageEmpty3} 3108\pastebutton{ugxProblemSeriesCoefficientsPageFull3}{\hidepaste} 3109\tab{5}\spadcommand{coefficient(y,6)\free{y }} 3110\indentrel{3}\begin{verbatim} 3111 1 3112 (3) - -- 3113 90 3114 Type: Expression(Integer) 3115\end{verbatim} 3116\indentrel{-3}\end{paste}\end{patch} 3117 3118\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty3} 3119\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty3}{ugxProblemSeriesCoefficientsPagePatch3} 3120\pastebutton{ugxProblemSeriesCoefficientsPageEmpty3}{\showpaste} 3121\tab{5}\spadcommand{coefficient(y,6)\free{y }} 3122\end{paste}\end{patch} 3123 3124\begin{patch}{ugxProblemSeriesCoefficientsPagePatch4} 3125\begin{paste}{ugxProblemSeriesCoefficientsPageFull4}{ugxProblemSeriesCoefficientsPageEmpty4} 3126\pastebutton{ugxProblemSeriesCoefficientsPageFull4}{\hidepaste} 3127\tab{5}\spadcommand{coefficient(y,15)\free{y }\bound{y15 }} 3128\indentrel{3}\begin{verbatim} 3129 1 3130 (4) - ----------- 3131 10216206000 3132 Type: Expression(Integer) 3133\end{verbatim} 3134\indentrel{-3}\end{paste}\end{patch} 3135 3136\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty4} 3137\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty4}{ugxProblemSeriesCoefficientsPagePatch4} 3138\pastebutton{ugxProblemSeriesCoefficientsPageEmpty4}{\showpaste} 3139\tab{5}\spadcommand{coefficient(y,15)\free{y }\bound{y15 }} 3140\end{paste}\end{patch} 3141 3142\begin{patch}{ugxProblemSeriesCoefficientsPagePatch5} 3143\begin{paste}{ugxProblemSeriesCoefficientsPageFull5}{ugxProblemSeriesCoefficientsPageEmpty5} 3144\pastebutton{ugxProblemSeriesCoefficientsPageFull5}{\hidepaste} 3145\tab{5}\spadcommand{)set stream showall on} 3146\indentrel{3}\begin{verbatim} 3147\end{verbatim} 3148\indentrel{-3}\end{paste}\end{patch} 3149 3150\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty5} 3151\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty5}{ugxProblemSeriesCoefficientsPagePatch5} 3152\pastebutton{ugxProblemSeriesCoefficientsPageEmpty5}{\showpaste} 3153\tab{5}\spadcommand{)set stream showall on} 3154\end{paste}\end{patch} 3155 3156\begin{patch}{ugxProblemSeriesCoefficientsPagePatch6} 3157\begin{paste}{ugxProblemSeriesCoefficientsPageFull6}{ugxProblemSeriesCoefficientsPageEmpty6} 3158\pastebutton{ugxProblemSeriesCoefficientsPageFull6}{\hidepaste} 3159\tab{5}\spadcommand{y\free{y15 }} 3160\indentrel{3}\begin{verbatim} 3161 (5) 3162 2 1 3 1 5 1 6 1 7 1 9 3163 x + x + - x - -- x - -- x - --- x + ----- x 3164 3 30 90 630 22680 3165 + 3166 1 10 1 11 1 13 3167 ------ x + ------- x - -------- x 3168 113400 1247400 97297200 3169 + 3170 1 14 1 15 16 3171 - --------- x - ----------- x + O(x ) 3172 681080400 10216206000 3173 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 3174\end{verbatim} 3175\indentrel{-3}\end{paste}\end{patch} 3176 3177\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty6} 3178\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty6}{ugxProblemSeriesCoefficientsPagePatch6} 3179\pastebutton{ugxProblemSeriesCoefficientsPageEmpty6}{\showpaste} 3180\tab{5}\spadcommand{y\free{y15 }} 3181\end{paste}\end{patch} 3182 3183\begin{patch}{ugxProblemSeriesCoefficientsPagePatch7} 3184\begin{paste}{ugxProblemSeriesCoefficientsPageFull7}{ugxProblemSeriesCoefficientsPageEmpty7} 3185\pastebutton{ugxProblemSeriesCoefficientsPageFull7}{\hidepaste} 3186\tab{5}\spadcommand{)set stream showall off} 3187\indentrel{3}\begin{verbatim} 3188\end{verbatim} 3189\indentrel{-3}\end{paste}\end{patch} 3190 3191\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty7} 3192\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty7}{ugxProblemSeriesCoefficientsPagePatch7} 3193\pastebutton{ugxProblemSeriesCoefficientsPageEmpty7}{\showpaste} 3194\tab{5}\spadcommand{)set stream showall off} 3195\end{paste}\end{patch} 3196 3197\begin{patch}{ugxProblemLDEQClosedPagePatch1} 3198\begin{paste}{ugxProblemLDEQClosedPageFull1}{ugxProblemLDEQClosedPageEmpty1} 3199\pastebutton{ugxProblemLDEQClosedPageFull1}{\hidepaste} 3200\tab{5}\spadcommand{y := operator 'y\bound{y }} 3201\indentrel{3}\begin{verbatim} 3202 (1) y 3203 Type: BasicOperator 3204\end{verbatim} 3205\indentrel{-3}\end{paste}\end{patch} 3206 3207\begin{patch}{ugxProblemLDEQClosedPageEmpty1} 3208\begin{paste}{ugxProblemLDEQClosedPageEmpty1}{ugxProblemLDEQClosedPagePatch1} 3209\pastebutton{ugxProblemLDEQClosedPageEmpty1}{\showpaste} 3210\tab{5}\spadcommand{y := operator 'y\bound{y }} 3211\end{paste}\end{patch} 3212 3213\begin{patch}{ugxProblemLDEQClosedPagePatch2} 3214\begin{paste}{ugxProblemLDEQClosedPageFull2}{ugxProblemLDEQClosedPageEmpty2} 3215\pastebutton{ugxProblemLDEQClosedPageFull2}{\hidepaste} 3216\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }} 3217\indentrel{3}\begin{verbatim} 3218 ,, , 3219 (2) y (x) + y (x) + y(x) = 0 3220 3221 Type: Equation(Expression(Integer)) 3222\end{verbatim} 3223\indentrel{-3}\end{paste}\end{patch} 3224 3225\begin{patch}{ugxProblemLDEQClosedPageEmpty2} 3226\begin{paste}{ugxProblemLDEQClosedPageEmpty2}{ugxProblemLDEQClosedPagePatch2} 3227\pastebutton{ugxProblemLDEQClosedPageEmpty2}{\showpaste} 3228\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }} 3229\end{paste}\end{patch} 3230 3231\begin{patch}{ugxProblemLDEQClosedPagePatch3} 3232\begin{paste}{ugxProblemLDEQClosedPageFull3}{ugxProblemLDEQClosedPageEmpty3} 3233\pastebutton{ugxProblemLDEQClosedPageFull3}{\hidepaste} 3234\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }} 3235\indentrel{3}\begin{verbatim} 3236 (3) 3237 [particular = 0, 3238 x x 3239 +-+ - - - - +-+ 3240 x\|3 2 2 x\|3 3241 basis = [cos(-----)%e , %e sin(-----)]] 3242 2 2 3243Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3244\end{verbatim} 3245\indentrel{-3}\end{paste}\end{patch} 3246 3247\begin{patch}{ugxProblemLDEQClosedPageEmpty3} 3248\begin{paste}{ugxProblemLDEQClosedPageEmpty3}{ugxProblemLDEQClosedPagePatch3} 3249\pastebutton{ugxProblemLDEQClosedPageEmpty3}{\showpaste} 3250\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }} 3251\end{paste}\end{patch} 3252 3253\begin{patch}{ugxProblemLDEQClosedPagePatch4} 3254\begin{paste}{ugxProblemLDEQClosedPageFull4}{ugxProblemLDEQClosedPageEmpty4} 3255\pastebutton{ugxProblemLDEQClosedPageFull4}{\hidepaste} 3256\tab{5}\spadcommand{deq := D(y x, x, 2) + y x\bound{e2 }\free{y }} 3257\indentrel{3}\begin{verbatim} 3258 ,, 3259 (4) y (x) + y(x) 3260 3261 Type: Expression(Integer) 3262\end{verbatim} 3263\indentrel{-3}\end{paste}\end{patch} 3264 3265\begin{patch}{ugxProblemLDEQClosedPageEmpty4} 3266\begin{paste}{ugxProblemLDEQClosedPageEmpty4}{ugxProblemLDEQClosedPagePatch4} 3267\pastebutton{ugxProblemLDEQClosedPageEmpty4}{\showpaste} 3268\tab{5}\spadcommand{deq := D(y x, x, 2) + y x\bound{e2 }\free{y }} 3269\end{paste}\end{patch} 3270 3271\begin{patch}{ugxProblemLDEQClosedPagePatch5} 3272\begin{paste}{ugxProblemLDEQClosedPageFull5}{ugxProblemLDEQClosedPageEmpty5} 3273\pastebutton{ugxProblemLDEQClosedPageFull5}{\hidepaste} 3274\tab{5}\spadcommand{solve(deq, y, x = 0, [1, 1])\free{e2 }\free{y }} 3275\indentrel{3}\begin{verbatim} 3276 (5) sin(x) + cos(x) 3277 Type: Union(Expression(Integer),...) 3278\end{verbatim} 3279\indentrel{-3}\end{paste}\end{patch} 3280 3281\begin{patch}{ugxProblemLDEQClosedPageEmpty5} 3282\begin{paste}{ugxProblemLDEQClosedPageEmpty5}{ugxProblemLDEQClosedPagePatch5} 3283\pastebutton{ugxProblemLDEQClosedPageEmpty5}{\showpaste} 3284\tab{5}\spadcommand{solve(deq, y, x = 0, [1, 1])\free{e2 }\free{y }} 3285\end{paste}\end{patch} 3286 3287\begin{patch}{ugxProblemLDEQClosedPagePatch6} 3288\begin{paste}{ugxProblemLDEQClosedPageFull6}{ugxProblemLDEQClosedPageEmpty6} 3289\pastebutton{ugxProblemLDEQClosedPageFull6}{\hidepaste} 3290\tab{5}\spadcommand{deq := x^3 * D(y x, x, 3) + x^2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x^4\bound{e3 }\free{y }} 3291\indentrel{3}\begin{verbatim} 3292 3 ,,, 2 ,, , 4 3293 (6) x y (x) + x y (x) - 2 x y (x) + 2 y(x) = 2 x 3294 3295 Type: Equation(Expression(Integer)) 3296\end{verbatim} 3297\indentrel{-3}\end{paste}\end{patch} 3298 3299\begin{patch}{ugxProblemLDEQClosedPageEmpty6} 3300\begin{paste}{ugxProblemLDEQClosedPageEmpty6}{ugxProblemLDEQClosedPagePatch6} 3301\pastebutton{ugxProblemLDEQClosedPageEmpty6}{\showpaste} 3302\tab{5}\spadcommand{deq := x^3 * D(y x, x, 3) + x^2 * D(y x, x, 2) - 2 * x * D(y x, x) + 2 * y x = 2 * x^4\bound{e3 }\free{y }} 3303\end{paste}\end{patch} 3304 3305\begin{patch}{ugxProblemLDEQClosedPagePatch7} 3306\begin{paste}{ugxProblemLDEQClosedPageFull7}{ugxProblemLDEQClosedPageEmpty7} 3307\pastebutton{ugxProblemLDEQClosedPageFull7}{\hidepaste} 3308\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }} 3309\indentrel{3}\begin{verbatim} 3310 (7) 3311 5 3 2 3312 x - 10 x + 20 x + 4 3313 [particular = ----------------------, 3314 15 x 3315 3 2 3 3 2 3316 2 x - 3 x + 1 x - 1 x - 3 x - 1 3317 basis = [---------------, ------, -------------]] 3318 x x x 3319Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3320\end{verbatim} 3321\indentrel{-3}\end{paste}\end{patch} 3322 3323\begin{patch}{ugxProblemLDEQClosedPageEmpty7} 3324\begin{paste}{ugxProblemLDEQClosedPageEmpty7}{ugxProblemLDEQClosedPagePatch7} 3325\pastebutton{ugxProblemLDEQClosedPageEmpty7}{\showpaste} 3326\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }} 3327\end{paste}\end{patch} 3328 3329\begin{patch}{ugxProblemLDEQClosedPagePatch8} 3330\begin{paste}{ugxProblemLDEQClosedPageFull8}{ugxProblemLDEQClosedPageEmpty8} 3331\pastebutton{ugxProblemLDEQClosedPageFull8}{\hidepaste} 3332\tab{5}\spadcommand{deq := (x^9+x^3) * D(y x, x, 3) + 18 * x^8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x^6 - 3) * y x\bound{e4 }\free{y }} 3333\indentrel{3}\begin{verbatim} 3334 (8) 3335 9 3 ,,, 8 ,, , 3336 (x + x )y (x) + 18 x y (x) - 90 x y (x) 3337 3338 + 3339 6 3340 (- 330 x + 90)y(x) 3341 Type: Expression(Integer) 3342\end{verbatim} 3343\indentrel{-3}\end{paste}\end{patch} 3344 3345\begin{patch}{ugxProblemLDEQClosedPageEmpty8} 3346\begin{paste}{ugxProblemLDEQClosedPageEmpty8}{ugxProblemLDEQClosedPagePatch8} 3347\pastebutton{ugxProblemLDEQClosedPageEmpty8}{\showpaste} 3348\tab{5}\spadcommand{deq := (x^9+x^3) * D(y x, x, 3) + 18 * x^8 * D(y x, x, 2) - 90 * x * D(y x, x) - 30 * (11 * x^6 - 3) * y x\bound{e4 }\free{y }} 3349\end{paste}\end{patch} 3350 3351\begin{patch}{ugxProblemLDEQClosedPagePatch9} 3352\begin{paste}{ugxProblemLDEQClosedPageFull9}{ugxProblemLDEQClosedPageEmpty9} 3353\pastebutton{ugxProblemLDEQClosedPageFull9}{\hidepaste} 3354\tab{5}\spadcommand{solve(deq, y, x)\free{e4 }\free{y }} 3355\indentrel{3}\begin{verbatim} 3356 (9) 3357 [particular = 0, 3358 +--+ +--+ 3359 - \|91 log(x) \|91 log(x) 3360 x x %e x %e 3361 basis = [------, -----------------, ---------------]] 3362 6 6 6 3363 x + 1 x + 1 x + 1 3364Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3365\end{verbatim} 3366\indentrel{-3}\end{paste}\end{patch} 3367 3368\begin{patch}{ugxProblemLDEQClosedPageEmpty9} 3369\begin{paste}{ugxProblemLDEQClosedPageEmpty9}{ugxProblemLDEQClosedPagePatch9} 3370\pastebutton{ugxProblemLDEQClosedPageEmpty9}{\showpaste} 3371\tab{5}\spadcommand{solve(deq, y, x)\free{e4 }\free{y }} 3372\end{paste}\end{patch} 3373 3374\begin{patch}{ugxProblemLDEQClosedPagePatch10} 3375\begin{paste}{ugxProblemLDEQClosedPageFull10}{ugxProblemLDEQClosedPageEmpty10} 3376\pastebutton{ugxProblemLDEQClosedPageFull10}{\hidepaste} 3377\tab{5}\spadcommand{deq := (x^2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }} 3378\indentrel{3}\begin{verbatim} 3379 2 ,, , 3380 (10) (x + 1)y (x) + 3 x y (x) + y(x) = 0 3381 3382 Type: Equation(Expression(Integer)) 3383\end{verbatim} 3384\indentrel{-3}\end{paste}\end{patch} 3385 3386\begin{patch}{ugxProblemLDEQClosedPageEmpty10} 3387\begin{paste}{ugxProblemLDEQClosedPageEmpty10}{ugxProblemLDEQClosedPagePatch10} 3388\pastebutton{ugxProblemLDEQClosedPageEmpty10}{\showpaste} 3389\tab{5}\spadcommand{deq := (x^2 + 1) * D(y x, x, 2) + 3 * x * D(y x, x) + y x = 0\bound{e5 }\free{y }} 3390\end{paste}\end{patch} 3391 3392\begin{patch}{ugxProblemLDEQClosedPagePatch11} 3393\begin{paste}{ugxProblemLDEQClosedPageFull11}{ugxProblemLDEQClosedPageEmpty11} 3394\pastebutton{ugxProblemLDEQClosedPageFull11}{\hidepaste} 3395\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }} 3396\indentrel{3}\begin{verbatim} 3397 (11) 3398 [particular = 0, 3399 +------+ 3400 | 2 3401 1 log(\|x + 1 - x) 3402 basis = [---------, ------------------]] 3403 +------+ +------+ 3404 | 2 | 2 3405 \|x + 1 \|x + 1 3406Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3407\end{verbatim} 3408\indentrel{-3}\end{paste}\end{patch} 3409 3410\begin{patch}{ugxProblemLDEQClosedPageEmpty11} 3411\begin{paste}{ugxProblemLDEQClosedPageEmpty11}{ugxProblemLDEQClosedPagePatch11} 3412\pastebutton{ugxProblemLDEQClosedPageEmpty11}{\showpaste} 3413\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }} 3414\end{paste}\end{patch} 3415 3416\begin{patch}{ugxProblemNLDEQClosedPagePatch1} 3417\begin{paste}{ugxProblemNLDEQClosedPageFull1}{ugxProblemNLDEQClosedPageEmpty1} 3418\pastebutton{ugxProblemNLDEQClosedPageFull1}{\hidepaste} 3419\tab{5}\spadcommand{m := -y\bound{m }} 3420\indentrel{3}\begin{verbatim} 3421 (1) - y 3422 Type: Polynomial(Integer) 3423\end{verbatim} 3424\indentrel{-3}\end{paste}\end{patch} 3425 3426\begin{patch}{ugxProblemNLDEQClosedPageEmpty1} 3427\begin{paste}{ugxProblemNLDEQClosedPageEmpty1}{ugxProblemNLDEQClosedPagePatch1} 3428\pastebutton{ugxProblemNLDEQClosedPageEmpty1}{\showpaste} 3429\tab{5}\spadcommand{m := -y\bound{m }} 3430\end{paste}\end{patch} 3431 3432\begin{patch}{ugxProblemNLDEQClosedPagePatch2} 3433\begin{paste}{ugxProblemNLDEQClosedPageFull2}{ugxProblemNLDEQClosedPageEmpty2} 3434\pastebutton{ugxProblemNLDEQClosedPageFull2}{\hidepaste} 3435\tab{5}\spadcommand{n := x + y * log y\bound{n }} 3436\indentrel{3}\begin{verbatim} 3437 (2) y log(y) + x 3438 Type: Expression(Integer) 3439\end{verbatim} 3440\indentrel{-3}\end{paste}\end{patch} 3441 3442\begin{patch}{ugxProblemNLDEQClosedPageEmpty2} 3443\begin{paste}{ugxProblemNLDEQClosedPageEmpty2}{ugxProblemNLDEQClosedPagePatch2} 3444\pastebutton{ugxProblemNLDEQClosedPageEmpty2}{\showpaste} 3445\tab{5}\spadcommand{n := x + y * log y\bound{n }} 3446\end{paste}\end{patch} 3447 3448\begin{patch}{ugxProblemNLDEQClosedPagePatch3} 3449\begin{paste}{ugxProblemNLDEQClosedPageFull3}{ugxProblemNLDEQClosedPageEmpty3} 3450\pastebutton{ugxProblemNLDEQClosedPageFull3}{\hidepaste} 3451\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m n }} 3452\indentrel{3}\begin{verbatim} 3453 (3) - 2 3454 Type: Expression(Integer) 3455\end{verbatim} 3456\indentrel{-3}\end{paste}\end{patch} 3457 3458\begin{patch}{ugxProblemNLDEQClosedPageEmpty3} 3459\begin{paste}{ugxProblemNLDEQClosedPageEmpty3}{ugxProblemNLDEQClosedPagePatch3} 3460\pastebutton{ugxProblemNLDEQClosedPageEmpty3}{\showpaste} 3461\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m n }} 3462\end{paste}\end{patch} 3463 3464\begin{patch}{ugxProblemNLDEQClosedPagePatch4} 3465\begin{paste}{ugxProblemNLDEQClosedPageFull4}{ugxProblemNLDEQClosedPageEmpty4} 3466\pastebutton{ugxProblemNLDEQClosedPageFull4}{\hidepaste} 3467\tab{5}\spadcommand{mu := operator 'mu\bound{mu }} 3468\indentrel{3}\begin{verbatim} 3469 (4) mu 3470 Type: BasicOperator 3471\end{verbatim} 3472\indentrel{-3}\end{paste}\end{patch} 3473 3474\begin{patch}{ugxProblemNLDEQClosedPageEmpty4} 3475\begin{paste}{ugxProblemNLDEQClosedPageEmpty4}{ugxProblemNLDEQClosedPagePatch4} 3476\pastebutton{ugxProblemNLDEQClosedPageEmpty4}{\showpaste} 3477\tab{5}\spadcommand{mu := operator 'mu\bound{mu }} 3478\end{paste}\end{patch} 3479 3480\begin{patch}{ugxProblemNLDEQClosedPagePatch5} 3481\begin{paste}{ugxProblemNLDEQClosedPageFull5}{ugxProblemNLDEQClosedPageEmpty5} 3482\pastebutton{ugxProblemNLDEQClosedPageFull5}{\hidepaste} 3483\tab{5}\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x)\bound{a }\free{m n mu }} 3484\indentrel{3}\begin{verbatim} 3485 , 3486 (5) (- y log(y) - x)mu (x) - 2 mu(x) 3487 3488 Type: Expression(Integer) 3489\end{verbatim} 3490\indentrel{-3}\end{paste}\end{patch} 3491 3492\begin{patch}{ugxProblemNLDEQClosedPageEmpty5} 3493\begin{paste}{ugxProblemNLDEQClosedPageEmpty5}{ugxProblemNLDEQClosedPagePatch5} 3494\pastebutton{ugxProblemNLDEQClosedPageEmpty5}{\showpaste} 3495\tab{5}\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x)\bound{a }\free{m n mu }} 3496\end{paste}\end{patch} 3497 3498\begin{patch}{ugxProblemNLDEQClosedPagePatch6} 3499\begin{paste}{ugxProblemNLDEQClosedPageFull6}{ugxProblemNLDEQClosedPageEmpty6} 3500\pastebutton{ugxProblemNLDEQClosedPageFull6}{\hidepaste} 3501\tab{5}\spadcommand{solve(a = 0, mu, x)\free{mu a }} 3502\indentrel{3}\begin{verbatim} 3503 (6) 3504 [particular = 0, 3505 1 3506 basis = [-----------------------------]] 3507 2 2 2 3508 y log(y) + 2 x y log(y) + x 3509Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3510\end{verbatim} 3511\indentrel{-3}\end{paste}\end{patch} 3512 3513\begin{patch}{ugxProblemNLDEQClosedPageEmpty6} 3514\begin{paste}{ugxProblemNLDEQClosedPageEmpty6}{ugxProblemNLDEQClosedPagePatch6} 3515\pastebutton{ugxProblemNLDEQClosedPageEmpty6}{\showpaste} 3516\tab{5}\spadcommand{solve(a = 0, mu, x)\free{mu a }} 3517\end{paste}\end{patch} 3518 3519\begin{patch}{ugxProblemNLDEQClosedPagePatch7} 3520\begin{paste}{ugxProblemNLDEQClosedPageFull7}{ugxProblemNLDEQClosedPageEmpty7} 3521\pastebutton{ugxProblemNLDEQClosedPageFull7}{\hidepaste} 3522\tab{5}\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x)\bound{b }\free{mu m }} 3523\indentrel{3}\begin{verbatim} 3524 , 3525 (7) - y mu (y) - 2 mu(y) 3526 3527 Type: Expression(Integer) 3528\end{verbatim} 3529\indentrel{-3}\end{paste}\end{patch} 3530 3531\begin{patch}{ugxProblemNLDEQClosedPageEmpty7} 3532\begin{paste}{ugxProblemNLDEQClosedPageEmpty7}{ugxProblemNLDEQClosedPagePatch7} 3533\pastebutton{ugxProblemNLDEQClosedPageEmpty7}{\showpaste} 3534\tab{5}\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x)\bound{b }\free{mu m }} 3535\end{paste}\end{patch} 3536 3537\begin{patch}{ugxProblemNLDEQClosedPagePatch8} 3538\begin{paste}{ugxProblemNLDEQClosedPageFull8}{ugxProblemNLDEQClosedPageEmpty8} 3539\pastebutton{ugxProblemNLDEQClosedPageFull8}{\hidepaste} 3540\tab{5}\spadcommand{sb := solve(b = 0, mu, y)\free{mu b }\bound{sb }} 3541\indentrel{3}\begin{verbatim} 3542 1 3543 (8) [particular = 0, basis = [--]] 3544 2 3545 y 3546Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3547\end{verbatim} 3548\indentrel{-3}\end{paste}\end{patch} 3549 3550\begin{patch}{ugxProblemNLDEQClosedPageEmpty8} 3551\begin{paste}{ugxProblemNLDEQClosedPageEmpty8}{ugxProblemNLDEQClosedPagePatch8} 3552\pastebutton{ugxProblemNLDEQClosedPageEmpty8}{\showpaste} 3553\tab{5}\spadcommand{sb := solve(b = 0, mu, y)\free{mu b }\bound{sb }} 3554\end{paste}\end{patch} 3555 3556\begin{patch}{ugxProblemNLDEQClosedPagePatch9} 3557\begin{paste}{ugxProblemNLDEQClosedPageFull9}{ugxProblemNLDEQClosedPageEmpty9} 3558\pastebutton{ugxProblemNLDEQClosedPageFull9}{\hidepaste} 3559\tab{5}\spadcommand{intFactor := sb.basis.1\bound{intFactor }\free{sb }} 3560\indentrel{3}\begin{verbatim} 3561 1 3562 (9) -- 3563 2 3564 y 3565 Type: Expression(Integer) 3566\end{verbatim} 3567\indentrel{-3}\end{paste}\end{patch} 3568 3569\begin{patch}{ugxProblemNLDEQClosedPageEmpty9} 3570\begin{paste}{ugxProblemNLDEQClosedPageEmpty9}{ugxProblemNLDEQClosedPagePatch9} 3571\pastebutton{ugxProblemNLDEQClosedPageEmpty9}{\showpaste} 3572\tab{5}\spadcommand{intFactor := sb.basis.1\bound{intFactor }\free{sb }} 3573\end{paste}\end{patch} 3574 3575\begin{patch}{ugxProblemNLDEQClosedPagePatch10} 3576\begin{paste}{ugxProblemNLDEQClosedPageFull10}{ugxProblemNLDEQClosedPageEmpty10} 3577\pastebutton{ugxProblemNLDEQClosedPageFull10}{\hidepaste} 3578\tab{5}\spadcommand{m := intFactor * m\bound{m1 }\free{m intFactor }} 3579\indentrel{3}\begin{verbatim} 3580 1 3581 (10) - - 3582 y 3583 Type: Expression(Integer) 3584\end{verbatim} 3585\indentrel{-3}\end{paste}\end{patch} 3586 3587\begin{patch}{ugxProblemNLDEQClosedPageEmpty10} 3588\begin{paste}{ugxProblemNLDEQClosedPageEmpty10}{ugxProblemNLDEQClosedPagePatch10} 3589\pastebutton{ugxProblemNLDEQClosedPageEmpty10}{\showpaste} 3590\tab{5}\spadcommand{m := intFactor * m\bound{m1 }\free{m intFactor }} 3591\end{paste}\end{patch} 3592 3593\begin{patch}{ugxProblemNLDEQClosedPagePatch11} 3594\begin{paste}{ugxProblemNLDEQClosedPageFull11}{ugxProblemNLDEQClosedPageEmpty11} 3595\pastebutton{ugxProblemNLDEQClosedPageFull11}{\hidepaste} 3596\tab{5}\spadcommand{n := intFactor * n\bound{n1 }\free{n intFactor }} 3597\indentrel{3}\begin{verbatim} 3598 y log(y) + x 3599 (11) ------------ 3600 2 3601 y 3602 Type: Expression(Integer) 3603\end{verbatim} 3604\indentrel{-3}\end{paste}\end{patch} 3605 3606\begin{patch}{ugxProblemNLDEQClosedPageEmpty11} 3607\begin{paste}{ugxProblemNLDEQClosedPageEmpty11}{ugxProblemNLDEQClosedPagePatch11} 3608\pastebutton{ugxProblemNLDEQClosedPageEmpty11}{\showpaste} 3609\tab{5}\spadcommand{n := intFactor * n\bound{n1 }\free{n intFactor }} 3610\end{paste}\end{patch} 3611 3612\begin{patch}{ugxProblemNLDEQClosedPagePatch12} 3613\begin{paste}{ugxProblemNLDEQClosedPageFull12}{ugxProblemNLDEQClosedPageEmpty12} 3614\pastebutton{ugxProblemNLDEQClosedPageFull12}{\hidepaste} 3615\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m1 n1 }} 3616\indentrel{3}\begin{verbatim} 3617 (12) 0 3618 Type: Expression(Integer) 3619\end{verbatim} 3620\indentrel{-3}\end{paste}\end{patch} 3621 3622\begin{patch}{ugxProblemNLDEQClosedPageEmpty12} 3623\begin{paste}{ugxProblemNLDEQClosedPageEmpty12}{ugxProblemNLDEQClosedPagePatch12} 3624\pastebutton{ugxProblemNLDEQClosedPageEmpty12}{\showpaste} 3625\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m1 n1 }} 3626\end{paste}\end{patch} 3627 3628\begin{patch}{ugxProblemNLDEQClosedPagePatch13} 3629\begin{paste}{ugxProblemNLDEQClosedPageFull13}{ugxProblemNLDEQClosedPageEmpty13} 3630\pastebutton{ugxProblemNLDEQClosedPageFull13}{\hidepaste} 3631\tab{5}\spadcommand{h := operator 'h\bound{h }} 3632\indentrel{3}\begin{verbatim} 3633 (13) h 3634 Type: BasicOperator 3635\end{verbatim} 3636\indentrel{-3}\end{paste}\end{patch} 3637 3638\begin{patch}{ugxProblemNLDEQClosedPageEmpty13} 3639\begin{paste}{ugxProblemNLDEQClosedPageEmpty13}{ugxProblemNLDEQClosedPagePatch13} 3640\pastebutton{ugxProblemNLDEQClosedPageEmpty13}{\showpaste} 3641\tab{5}\spadcommand{h := operator 'h\bound{h }} 3642\end{paste}\end{patch} 3643 3644\begin{patch}{ugxProblemNLDEQClosedPagePatch14} 3645\begin{paste}{ugxProblemNLDEQClosedPageFull14}{ugxProblemNLDEQClosedPageEmpty14} 3646\pastebutton{ugxProblemNLDEQClosedPageFull14}{\hidepaste} 3647\tab{5}\spadcommand{sol := h y + integrate(m, x)\bound{sol }\free{h m1 }} 3648\indentrel{3}\begin{verbatim} 3649 y h(y) - x 3650 (14) ---------- 3651 y 3652 Type: Expression(Integer) 3653\end{verbatim} 3654\indentrel{-3}\end{paste}\end{patch} 3655 3656\begin{patch}{ugxProblemNLDEQClosedPageEmpty14} 3657\begin{paste}{ugxProblemNLDEQClosedPageEmpty14}{ugxProblemNLDEQClosedPagePatch14} 3658\pastebutton{ugxProblemNLDEQClosedPageEmpty14}{\showpaste} 3659\tab{5}\spadcommand{sol := h y + integrate(m, x)\bound{sol }\free{h m1 }} 3660\end{paste}\end{patch} 3661 3662\begin{patch}{ugxProblemNLDEQClosedPagePatch15} 3663\begin{paste}{ugxProblemNLDEQClosedPageFull15}{ugxProblemNLDEQClosedPageEmpty15} 3664\pastebutton{ugxProblemNLDEQClosedPageFull15}{\hidepaste} 3665\tab{5}\spadcommand{dsol := D(sol, y)\free{sol }\bound{dsol }} 3666\indentrel{3}\begin{verbatim} 3667 2 , 3668 y h (y) + x 3669 3670 (15) ----------- 3671 2 3672 y 3673 Type: Expression(Integer) 3674\end{verbatim} 3675\indentrel{-3}\end{paste}\end{patch} 3676 3677\begin{patch}{ugxProblemNLDEQClosedPageEmpty15} 3678\begin{paste}{ugxProblemNLDEQClosedPageEmpty15}{ugxProblemNLDEQClosedPagePatch15} 3679\pastebutton{ugxProblemNLDEQClosedPageEmpty15}{\showpaste} 3680\tab{5}\spadcommand{dsol := D(sol, y)\free{sol }\bound{dsol }} 3681\end{paste}\end{patch} 3682 3683\begin{patch}{ugxProblemNLDEQClosedPagePatch16} 3684\begin{paste}{ugxProblemNLDEQClosedPageFull16}{ugxProblemNLDEQClosedPageEmpty16} 3685\pastebutton{ugxProblemNLDEQClosedPageFull16}{\hidepaste} 3686\tab{5}\spadcommand{nsol := solve(dsol = n, h, y)\free{dsol n1 h }\bound{nsol }} 3687\indentrel{3}\begin{verbatim} 3688 2 3689 log(y) 3690 (16) [particular = -------, basis = [1]] 3691 2 3692Type: Union(Record(particular: Expression(Integer),basis: List(Expression(Integer))),...) 3693\end{verbatim} 3694\indentrel{-3}\end{paste}\end{patch} 3695 3696\begin{patch}{ugxProblemNLDEQClosedPageEmpty16} 3697\begin{paste}{ugxProblemNLDEQClosedPageEmpty16}{ugxProblemNLDEQClosedPagePatch16} 3698\pastebutton{ugxProblemNLDEQClosedPageEmpty16}{\showpaste} 3699\tab{5}\spadcommand{nsol := solve(dsol = n, h, y)\free{dsol n1 h }\bound{nsol }} 3700\end{paste}\end{patch} 3701 3702\begin{patch}{ugxProblemNLDEQClosedPagePatch17} 3703\begin{paste}{ugxProblemNLDEQClosedPageFull17}{ugxProblemNLDEQClosedPageEmpty17} 3704\pastebutton{ugxProblemNLDEQClosedPageFull17}{\hidepaste} 3705\tab{5}\spadcommand{eval(sol, h y = nsol.particular)\free{sol h nsol }} 3706\indentrel{3}\begin{verbatim} 3707 2 3708 y log(y) - 2 x 3709 (17) --------------- 3710 2 y 3711 Type: Expression(Integer) 3712\end{verbatim} 3713\indentrel{-3}\end{paste}\end{patch} 3714 3715\begin{patch}{ugxProblemNLDEQClosedPageEmpty17} 3716\begin{paste}{ugxProblemNLDEQClosedPageEmpty17}{ugxProblemNLDEQClosedPagePatch17} 3717\pastebutton{ugxProblemNLDEQClosedPageEmpty17}{\showpaste} 3718\tab{5}\spadcommand{eval(sol, h y = nsol.particular)\free{sol h nsol }} 3719\end{paste}\end{patch} 3720 3721\begin{patch}{ugxProblemNLDEQClosedPagePatch18} 3722\begin{paste}{ugxProblemNLDEQClosedPageFull18}{ugxProblemNLDEQClosedPageEmpty18} 3723\pastebutton{ugxProblemNLDEQClosedPageFull18}{\hidepaste} 3724\tab{5}\spadcommand{y := operator 'y\bound{y }} 3725\indentrel{3}\begin{verbatim} 3726 (18) y 3727 Type: BasicOperator 3728\end{verbatim} 3729\indentrel{-3}\end{paste}\end{patch} 3730 3731\begin{patch}{ugxProblemNLDEQClosedPageEmpty18} 3732\begin{paste}{ugxProblemNLDEQClosedPageEmpty18}{ugxProblemNLDEQClosedPagePatch18} 3733\pastebutton{ugxProblemNLDEQClosedPageEmpty18}{\showpaste} 3734\tab{5}\spadcommand{y := operator 'y\bound{y }} 3735\end{paste}\end{patch} 3736 3737\begin{patch}{ugxProblemNLDEQClosedPagePatch19} 3738\begin{paste}{ugxProblemNLDEQClosedPageFull19}{ugxProblemNLDEQClosedPageEmpty19} 3739\pastebutton{ugxProblemNLDEQClosedPageFull19}{\hidepaste} 3740\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }} 3741\indentrel{3}\begin{verbatim} 3742 , y(x) 3743 (19) y (x) = ----------------- 3744 y(x)log(y(x)) + x 3745 Type: Equation(Expression(Integer)) 3746\end{verbatim} 3747\indentrel{-3}\end{paste}\end{patch} 3748 3749\begin{patch}{ugxProblemNLDEQClosedPageEmpty19} 3750\begin{paste}{ugxProblemNLDEQClosedPageEmpty19}{ugxProblemNLDEQClosedPagePatch19} 3751\pastebutton{ugxProblemNLDEQClosedPageEmpty19}{\showpaste} 3752\tab{5}\spadcommand{deq := D(y x, x) = y(x) / (x + y(x) * log y x)\bound{deqi }\free{y }} 3753\end{paste}\end{patch} 3754 3755\begin{patch}{ugxProblemNLDEQClosedPagePatch20} 3756\begin{paste}{ugxProblemNLDEQClosedPageFull20}{ugxProblemNLDEQClosedPageEmpty20} 3757\pastebutton{ugxProblemNLDEQClosedPageFull20}{\hidepaste} 3758\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }} 3759\indentrel{3}\begin{verbatim} 3760 2 3761 y(x)log(y(x)) - 2 x 3762 (20) -------------------- 3763 2 y(x) 3764 Type: Union(Expression(Integer),...) 3765\end{verbatim} 3766\indentrel{-3}\end{paste}\end{patch} 3767 3768\begin{patch}{ugxProblemNLDEQClosedPageEmpty20} 3769\begin{paste}{ugxProblemNLDEQClosedPageEmpty20}{ugxProblemNLDEQClosedPagePatch20} 3770\pastebutton{ugxProblemNLDEQClosedPageEmpty20}{\showpaste} 3771\tab{5}\spadcommand{solve(deq, y, x)\free{deqi y }} 3772\end{paste}\end{patch} 3773 3774\begin{patch}{ugxProblemSymRootAllPagePatch1} 3775\begin{paste}{ugxProblemSymRootAllPageFull1}{ugxProblemSymRootAllPageEmpty1} 3776\pastebutton{ugxProblemSymRootAllPageFull1}{\hidepaste} 3777\tab{5}\spadcommand{l := rootsOf(x^4 + x + 1,x)\bound{l }} 3778\indentrel{3}\begin{verbatim} 3779 (1) [%x0, %x1, %x2, - %x2 - %x1 - %x0] 3780 Type: List(Expression(Integer)) 3781\end{verbatim} 3782\indentrel{-3}\end{paste}\end{patch} 3783 3784\begin{patch}{ugxProblemSymRootAllPageEmpty1} 3785\begin{paste}{ugxProblemSymRootAllPageEmpty1}{ugxProblemSymRootAllPagePatch1} 3786\pastebutton{ugxProblemSymRootAllPageEmpty1}{\showpaste} 3787\tab{5}\spadcommand{l := rootsOf(x^4 + x + 1,x)\bound{l }} 3788\end{paste}\end{patch} 3789 3790\begin{patch}{ugxProblemSymRootAllPagePatch2} 3791\begin{paste}{ugxProblemSymRootAllPageFull2}{ugxProblemSymRootAllPageEmpty2} 3792\pastebutton{ugxProblemSymRootAllPageFull2}{\hidepaste} 3793\tab{5}\spadcommand{\%x0^5\free{l }} 3794\indentrel{3}\begin{verbatim} 3795 2 3796 (2) - %x0 - %x0 3797 Type: Expression(Integer) 3798\end{verbatim} 3799\indentrel{-3}\end{paste}\end{patch} 3800 3801\begin{patch}{ugxProblemSymRootAllPageEmpty2} 3802\begin{paste}{ugxProblemSymRootAllPageEmpty2}{ugxProblemSymRootAllPagePatch2} 3803\pastebutton{ugxProblemSymRootAllPageEmpty2}{\showpaste} 3804\tab{5}\spadcommand{\%x0^5\free{l }} 3805\end{paste}\end{patch} 3806 3807\begin{patch}{ugxProblemSymRootAllPagePatch3} 3808\begin{paste}{ugxProblemSymRootAllPageFull3}{ugxProblemSymRootAllPageEmpty3} 3809\pastebutton{ugxProblemSymRootAllPageFull3}{\hidepaste} 3810\tab{5}\spadcommand{definingPolynomial \%x0\free{l }} 3811\indentrel{3}\begin{verbatim} 3812 4 3813 (3) %x0 + %x0 + 1 3814 Type: Expression(Integer) 3815\end{verbatim} 3816\indentrel{-3}\end{paste}\end{patch} 3817 3818\begin{patch}{ugxProblemSymRootAllPageEmpty3} 3819\begin{paste}{ugxProblemSymRootAllPageEmpty3}{ugxProblemSymRootAllPagePatch3} 3820\pastebutton{ugxProblemSymRootAllPageEmpty3}{\showpaste} 3821\tab{5}\spadcommand{definingPolynomial \%x0\free{l }} 3822\end{paste}\end{patch} 3823 3824\begin{patch}{ugxProblemSymRootAllPagePatch4} 3825\begin{paste}{ugxProblemSymRootAllPageFull4}{ugxProblemSymRootAllPageEmpty4} 3826\pastebutton{ugxProblemSymRootAllPageFull4}{\hidepaste} 3827\tab{5}\spadcommand{definingPolynomial \%x1\free{l }} 3828\indentrel{3}\begin{verbatim} 3829 3 2 2 3 3830 (4) %x0 + %x1 %x0 + %x1 %x0 + %x1 + 1 3831 Type: Expression(Integer) 3832\end{verbatim} 3833\indentrel{-3}\end{paste}\end{patch} 3834 3835\begin{patch}{ugxProblemSymRootAllPageEmpty4} 3836\begin{paste}{ugxProblemSymRootAllPageEmpty4}{ugxProblemSymRootAllPagePatch4} 3837\pastebutton{ugxProblemSymRootAllPageEmpty4}{\showpaste} 3838\tab{5}\spadcommand{definingPolynomial \%x1\free{l }} 3839\end{paste}\end{patch} 3840 3841\begin{patch}{ugxProblemSymRootAllPagePatch5} 3842\begin{paste}{ugxProblemSymRootAllPageFull5}{ugxProblemSymRootAllPageEmpty5} 3843\pastebutton{ugxProblemSymRootAllPageFull5}{\hidepaste} 3844\tab{5}\spadcommand{definingPolynomial \%x2\free{l }} 3845\indentrel{3}\begin{verbatim} 3846 2 2 2 3847 (5) %x1 + (%x0 + %x2)%x1 + %x0 + %x2 %x0 + %x2 3848 Type: Expression(Integer) 3849\end{verbatim} 3850\indentrel{-3}\end{paste}\end{patch} 3851 3852\begin{patch}{ugxProblemSymRootAllPageEmpty5} 3853\begin{paste}{ugxProblemSymRootAllPageEmpty5}{ugxProblemSymRootAllPagePatch5} 3854\pastebutton{ugxProblemSymRootAllPageEmpty5}{\showpaste} 3855\tab{5}\spadcommand{definingPolynomial \%x2\free{l }} 3856\end{paste}\end{patch} 3857 3858\begin{patch}{ugxProblemSymRootAllPagePatch6} 3859\begin{paste}{ugxProblemSymRootAllPageFull6}{ugxProblemSymRootAllPageEmpty6} 3860\pastebutton{ugxProblemSymRootAllPageFull6}{\hidepaste} 3861\tab{5}\spadcommand{x3 := last l\free{l }\bound{x3 }} 3862\indentrel{3}\begin{verbatim} 3863 (6) - %x2 - %x1 - %x0 3864 Type: Expression(Integer) 3865\end{verbatim} 3866\indentrel{-3}\end{paste}\end{patch} 3867 3868\begin{patch}{ugxProblemSymRootAllPageEmpty6} 3869\begin{paste}{ugxProblemSymRootAllPageEmpty6}{ugxProblemSymRootAllPagePatch6} 3870\pastebutton{ugxProblemSymRootAllPageEmpty6}{\showpaste} 3871\tab{5}\spadcommand{x3 := last l\free{l }\bound{x3 }} 3872\end{paste}\end{patch} 3873 3874\begin{patch}{ugxProblemSymRootAllPagePatch7} 3875\begin{paste}{ugxProblemSymRootAllPageFull7}{ugxProblemSymRootAllPageEmpty7} 3876\pastebutton{ugxProblemSymRootAllPageFull7}{\hidepaste} 3877\tab{5}\spadcommand{\%x0 + \%x1 + \%x2 + x3\free{x3 }} 3878\indentrel{3}\begin{verbatim} 3879 (7) 0 3880 Type: Expression(Integer) 3881\end{verbatim} 3882\indentrel{-3}\end{paste}\end{patch} 3883 3884\begin{patch}{ugxProblemSymRootAllPageEmpty7} 3885\begin{paste}{ugxProblemSymRootAllPageEmpty7}{ugxProblemSymRootAllPagePatch7} 3886\pastebutton{ugxProblemSymRootAllPageEmpty7}{\showpaste} 3887\tab{5}\spadcommand{\%x0 + \%x1 + \%x2 + x3\free{x3 }} 3888\end{paste}\end{patch} 3889 3890\begin{patch}{ugxProblemSymRootAllPagePatch8} 3891\begin{paste}{ugxProblemSymRootAllPageFull8}{ugxProblemSymRootAllPageEmpty8} 3892\pastebutton{ugxProblemSymRootAllPageFull8}{\hidepaste} 3893\tab{5}\spadcommand{\%x0 * \%x1 * \%x2 * x3\free{x3 }} 3894\indentrel{3}\begin{verbatim} 3895 (8) 1 3896 Type: Expression(Integer) 3897\end{verbatim} 3898\indentrel{-3}\end{paste}\end{patch} 3899 3900\begin{patch}{ugxProblemSymRootAllPageEmpty8} 3901\begin{paste}{ugxProblemSymRootAllPageEmpty8}{ugxProblemSymRootAllPagePatch8} 3902\pastebutton{ugxProblemSymRootAllPageEmpty8}{\showpaste} 3903\tab{5}\spadcommand{\%x0 * \%x1 * \%x2 * x3\free{x3 }} 3904\end{paste}\end{patch} 3905 3906\begin{patch}{ugxProblemSymRootAllPagePatch9} 3907\begin{paste}{ugxProblemSymRootAllPageFull9}{ugxProblemSymRootAllPageEmpty9} 3908\pastebutton{ugxProblemSymRootAllPageFull9}{\hidepaste} 3909\tab{5}\spadcommand{rootsOf(x^4 + 1, x)} 3910\indentrel{3}\begin{verbatim} 3911 (9) [%x4, %x4 %x5, - %x4, - %x4 %x5] 3912 Type: List(Expression(Integer)) 3913\end{verbatim} 3914\indentrel{-3}\end{paste}\end{patch} 3915 3916\begin{patch}{ugxProblemSymRootAllPageEmpty9} 3917\begin{paste}{ugxProblemSymRootAllPageEmpty9}{ugxProblemSymRootAllPagePatch9} 3918\pastebutton{ugxProblemSymRootAllPageEmpty9}{\showpaste} 3919\tab{5}\spadcommand{rootsOf(x^4 + 1, x)} 3920\end{paste}\end{patch} 3921 3922\begin{patch}{ugxProblemSymRootAllPagePatch10} 3923\begin{paste}{ugxProblemSymRootAllPageFull10}{ugxProblemSymRootAllPageEmpty10} 3924\pastebutton{ugxProblemSymRootAllPageFull10}{\hidepaste} 3925\tab{5}\spadcommand{zerosOf(y^4 + y + 1, y)\bound{z }} 3926\indentrel{3}\begin{verbatim} 3927 (10) 3928 [%x0, %x1, 3929 +-----------------------------+ 3930 | 2 2 3931 \|- 3 %x1 - 2 %x0 %x1 - 3 %x0 - %x1 - %x0 3932 --------------------------------------------, 3933 2 3934 +-----------------------------+ 3935 | 2 2 3936 - \|- 3 %x1 - 2 %x0 %x1 - 3 %x0 - %x1 - %x0 3937 ----------------------------------------------] 3938 2 3939 Type: List(Expression(Integer)) 3940\end{verbatim} 3941\indentrel{-3}\end{paste}\end{patch} 3942 3943\begin{patch}{ugxProblemSymRootAllPageEmpty10} 3944\begin{paste}{ugxProblemSymRootAllPageEmpty10}{ugxProblemSymRootAllPagePatch10} 3945\pastebutton{ugxProblemSymRootAllPageEmpty10}{\showpaste} 3946\tab{5}\spadcommand{zerosOf(y^4 + y + 1, y)\bound{z }} 3947\end{paste}\end{patch} 3948 3949\begin{patch}{ugxProblemSymRootAllPagePatch11} 3950\begin{paste}{ugxProblemSymRootAllPageFull11}{ugxProblemSymRootAllPageEmpty11} 3951\pastebutton{ugxProblemSymRootAllPageFull11}{\hidepaste} 3952\tab{5}\spadcommand{definingPolynomial \%y0\free{z }} 3953\indentrel{3}\begin{verbatim} 3954 4 3955 (11) %x0 + %x0 + 1 3956 Type: Expression(Integer) 3957\end{verbatim} 3958\indentrel{-3}\end{paste}\end{patch} 3959 3960\begin{patch}{ugxProblemSymRootAllPageEmpty11} 3961\begin{paste}{ugxProblemSymRootAllPageEmpty11}{ugxProblemSymRootAllPagePatch11} 3962\pastebutton{ugxProblemSymRootAllPageEmpty11}{\showpaste} 3963\tab{5}\spadcommand{definingPolynomial \%y0\free{z }} 3964\end{paste}\end{patch} 3965 3966\begin{patch}{ugxProblemSymRootAllPagePatch12} 3967\begin{paste}{ugxProblemSymRootAllPageFull12}{ugxProblemSymRootAllPageEmpty12} 3968\pastebutton{ugxProblemSymRootAllPageFull12}{\hidepaste} 3969\tab{5}\spadcommand{definingPolynomial \%y1\free{z }} 3970\indentrel{3}\begin{verbatim} 3971 3 2 2 3 3972 (12) %x0 + %x1 %x0 + %x1 %x0 + %x1 + 1 3973 Type: Expression(Integer) 3974\end{verbatim} 3975\indentrel{-3}\end{paste}\end{patch} 3976 3977\begin{patch}{ugxProblemSymRootAllPageEmpty12} 3978\begin{paste}{ugxProblemSymRootAllPageEmpty12}{ugxProblemSymRootAllPagePatch12} 3979\pastebutton{ugxProblemSymRootAllPageEmpty12}{\showpaste} 3980\tab{5}\spadcommand{definingPolynomial \%y1\free{z }} 3981\end{paste}\end{patch} 3982 3983\begin{patch}{ugxProblemSymRootAllPagePatch13} 3984\begin{paste}{ugxProblemSymRootAllPageFull13}{ugxProblemSymRootAllPageEmpty13} 3985\pastebutton{ugxProblemSymRootAllPageFull13}{\hidepaste} 3986\tab{5}\spadcommand{zerosOf(x^4 + 1)} 3987\indentrel{3}\begin{verbatim} 3988 (13) 3989 +---+ +---+ +---+ +---+ 3990 \|- 1 + 1 \|- 1 - 1 - \|- 1 - 1 - \|- 1 + 1 3991 [----------, ----------, ------------, ------------] 3992 +-+ +-+ +-+ +-+ 3993 \|2 \|2 \|2 \|2 3994 Type: List(AlgebraicNumber) 3995\end{verbatim} 3996\indentrel{-3}\end{paste}\end{patch} 3997 3998\begin{patch}{ugxProblemSymRootAllPageEmpty13} 3999\begin{paste}{ugxProblemSymRootAllPageEmpty13}{ugxProblemSymRootAllPagePatch13} 4000\pastebutton{ugxProblemSymRootAllPageEmpty13}{\showpaste} 4001\tab{5}\spadcommand{zerosOf(x^4 + 1)} 4002\end{paste}\end{patch} 4003 4004\begin{patch}{ugxProblemSymRootOnePagePatch1} 4005\begin{paste}{ugxProblemSymRootOnePageFull1}{ugxProblemSymRootOnePageEmpty1} 4006\pastebutton{ugxProblemSymRootOnePageFull1}{\hidepaste} 4007\tab{5}\spadcommand{a := rootOf(a^4+1,a)\bound{a }} 4008\indentrel{3}\begin{verbatim} 4009 (1) a 4010 Type: Expression(Integer) 4011\end{verbatim} 4012\indentrel{-3}\end{paste}\end{patch} 4013 4014\begin{patch}{ugxProblemSymRootOnePageEmpty1} 4015\begin{paste}{ugxProblemSymRootOnePageEmpty1}{ugxProblemSymRootOnePagePatch1} 4016\pastebutton{ugxProblemSymRootOnePageEmpty1}{\showpaste} 4017\tab{5}\spadcommand{a := rootOf(a^4+1,a)\bound{a }} 4018\end{paste}\end{patch} 4019 4020\begin{patch}{ugxProblemSymRootOnePagePatch2} 4021\begin{paste}{ugxProblemSymRootOnePageFull2}{ugxProblemSymRootOnePageEmpty2} 4022\pastebutton{ugxProblemSymRootOnePageFull2}{\hidepaste} 4023\tab{5}\spadcommand{definingPolynomial a\free{a }} 4024\indentrel{3}\begin{verbatim} 4025 4 4026 (2) a + 1 4027 Type: Expression(Integer) 4028\end{verbatim} 4029\indentrel{-3}\end{paste}\end{patch} 4030 4031\begin{patch}{ugxProblemSymRootOnePageEmpty2} 4032\begin{paste}{ugxProblemSymRootOnePageEmpty2}{ugxProblemSymRootOnePagePatch2} 4033\pastebutton{ugxProblemSymRootOnePageEmpty2}{\showpaste} 4034\tab{5}\spadcommand{definingPolynomial a\free{a }} 4035\end{paste}\end{patch} 4036 4037\begin{patch}{ugxProblemSymRootOnePagePatch3} 4038\begin{paste}{ugxProblemSymRootOnePageFull3}{ugxProblemSymRootOnePageEmpty3} 4039\pastebutton{ugxProblemSymRootOnePageFull3}{\hidepaste} 4040\tab{5}\spadcommand{b := rootOf(b^2-a-1,b)\free{a }\bound{b }} 4041\indentrel{3}\begin{verbatim} 4042 (3) b 4043 Type: Expression(Integer) 4044\end{verbatim} 4045\indentrel{-3}\end{paste}\end{patch} 4046 4047\begin{patch}{ugxProblemSymRootOnePageEmpty3} 4048\begin{paste}{ugxProblemSymRootOnePageEmpty3}{ugxProblemSymRootOnePagePatch3} 4049\pastebutton{ugxProblemSymRootOnePageEmpty3}{\showpaste} 4050\tab{5}\spadcommand{b := rootOf(b^2-a-1,b)\free{a }\bound{b }} 4051\end{paste}\end{patch} 4052 4053\begin{patch}{ugxProblemSymRootOnePagePatch4} 4054\begin{paste}{ugxProblemSymRootOnePageFull4}{ugxProblemSymRootOnePageEmpty4} 4055\pastebutton{ugxProblemSymRootOnePageFull4}{\hidepaste} 4056\tab{5}\spadcommand{a + b\free{a b }\bound{c }} 4057\indentrel{3}\begin{verbatim} 4058 (4) b + a 4059 Type: Expression(Integer) 4060\end{verbatim} 4061\indentrel{-3}\end{paste}\end{patch} 4062 4063\begin{patch}{ugxProblemSymRootOnePageEmpty4} 4064\begin{paste}{ugxProblemSymRootOnePageEmpty4}{ugxProblemSymRootOnePagePatch4} 4065\pastebutton{ugxProblemSymRootOnePageEmpty4}{\showpaste} 4066\tab{5}\spadcommand{a + b\free{a b }\bound{c }} 4067\end{paste}\end{patch} 4068 4069\begin{patch}{ugxProblemSymRootOnePagePatch5} 4070\begin{paste}{ugxProblemSymRootOnePageFull5}{ugxProblemSymRootOnePageEmpty5} 4071\pastebutton{ugxProblemSymRootOnePageFull5}{\hidepaste} 4072\tab{5}\spadcommand{\% ^ 5\free{c }} 4073\indentrel{3}\begin{verbatim} 4074 (5) 4075 3 2 3 2 4076 (10 a + 11 a + 2 a - 4)b + 15 a + 10 a + 4 a - 10 4077 Type: Expression(Integer) 4078\end{verbatim} 4079\indentrel{-3}\end{paste}\end{patch} 4080 4081\begin{patch}{ugxProblemSymRootOnePageEmpty5} 4082\begin{paste}{ugxProblemSymRootOnePageEmpty5}{ugxProblemSymRootOnePagePatch5} 4083\pastebutton{ugxProblemSymRootOnePageEmpty5}{\showpaste} 4084\tab{5}\spadcommand{\% ^ 5\free{c }} 4085\end{paste}\end{patch} 4086 4087\begin{patch}{ugxProblemSymRootOnePagePatch6} 4088\begin{paste}{ugxProblemSymRootOnePageFull6}{ugxProblemSymRootOnePageEmpty6} 4089\pastebutton{ugxProblemSymRootOnePageFull6}{\hidepaste} 4090\tab{5}\spadcommand{rootOf(c^2+c+1,c)} 4091\indentrel{3}\begin{verbatim} 4092 (6) c 4093 Type: Expression(Integer) 4094\end{verbatim} 4095\indentrel{-3}\end{paste}\end{patch} 4096 4097\begin{patch}{ugxProblemSymRootOnePageEmpty6} 4098\begin{paste}{ugxProblemSymRootOnePageEmpty6}{ugxProblemSymRootOnePagePatch6} 4099\pastebutton{ugxProblemSymRootOnePageEmpty6}{\showpaste} 4100\tab{5}\spadcommand{rootOf(c^2+c+1,c)} 4101\end{paste}\end{patch} 4102 4103\begin{patch}{ugxProblemSymRootOnePagePatch7} 4104\begin{paste}{ugxProblemSymRootOnePageFull7}{ugxProblemSymRootOnePageEmpty7} 4105\pastebutton{ugxProblemSymRootOnePageFull7}{\hidepaste} 4106\tab{5}\spadcommand{zeroOf(d^2+d+1,d)} 4107\indentrel{3}\begin{verbatim} 4108 +---+ 4109 \|- 3 - 1 4110 (7) ---------- 4111 2 4112 Type: Expression(Integer) 4113\end{verbatim} 4114\indentrel{-3}\end{paste}\end{patch} 4115 4116\begin{patch}{ugxProblemSymRootOnePageEmpty7} 4117\begin{paste}{ugxProblemSymRootOnePageEmpty7}{ugxProblemSymRootOnePagePatch7} 4118\pastebutton{ugxProblemSymRootOnePageEmpty7}{\showpaste} 4119\tab{5}\spadcommand{zeroOf(d^2+d+1,d)} 4120\end{paste}\end{patch} 4121 4122\begin{patch}{ugxProblemSymRootOnePagePatch8} 4123\begin{paste}{ugxProblemSymRootOnePageFull8}{ugxProblemSymRootOnePageEmpty8} 4124\pastebutton{ugxProblemSymRootOnePageFull8}{\hidepaste} 4125\tab{5}\spadcommand{rootOf(e^5-2,e)} 4126\indentrel{3}\begin{verbatim} 4127 (8) e 4128 Type: Expression(Integer) 4129\end{verbatim} 4130\indentrel{-3}\end{paste}\end{patch} 4131 4132\begin{patch}{ugxProblemSymRootOnePageEmpty8} 4133\begin{paste}{ugxProblemSymRootOnePageEmpty8}{ugxProblemSymRootOnePagePatch8} 4134\pastebutton{ugxProblemSymRootOnePageEmpty8}{\showpaste} 4135\tab{5}\spadcommand{rootOf(e^5-2,e)} 4136\end{paste}\end{patch} 4137 4138\begin{patch}{ugxProblemSymRootOnePagePatch9} 4139\begin{paste}{ugxProblemSymRootOnePageFull9}{ugxProblemSymRootOnePageEmpty9} 4140\pastebutton{ugxProblemSymRootOnePageFull9}{\hidepaste} 4141\tab{5}\spadcommand{zeroOf(f^5-2,f)} 4142\indentrel{3}\begin{verbatim} 4143 5+-+ 4144 (9) \|2 4145 Type: Expression(Integer) 4146\end{verbatim} 4147\indentrel{-3}\end{paste}\end{patch} 4148 4149\begin{patch}{ugxProblemSymRootOnePageEmpty9} 4150\begin{paste}{ugxProblemSymRootOnePageEmpty9}{ugxProblemSymRootOnePagePatch9} 4151\pastebutton{ugxProblemSymRootOnePageEmpty9}{\showpaste} 4152\tab{5}\spadcommand{zeroOf(f^5-2,f)} 4153\end{paste}\end{patch} 4154 4155\begin{patch}{ugProblemIntegrationPagePatch1} 4156\begin{paste}{ugProblemIntegrationPageFull1}{ugProblemIntegrationPageEmpty1} 4157\pastebutton{ugProblemIntegrationPageFull1}{\hidepaste} 4158\tab{5}\spadcommand{integrate(cosh(a*x)*sinh(a*x), x)} 4159\indentrel{3}\begin{verbatim} 4160 2 2 4161 sinh(a x) + cosh(a x) 4162 (1) ----------------------- 4163 4 a 4164 Type: Union(Expression(Integer),...) 4165\end{verbatim} 4166\indentrel{-3}\end{paste}\end{patch} 4167 4168\begin{patch}{ugProblemIntegrationPageEmpty1} 4169\begin{paste}{ugProblemIntegrationPageEmpty1}{ugProblemIntegrationPagePatch1} 4170\pastebutton{ugProblemIntegrationPageEmpty1}{\showpaste} 4171\tab{5}\spadcommand{integrate(cosh(a*x)*sinh(a*x), x)} 4172\end{paste}\end{patch} 4173 4174\begin{patch}{ugProblemIntegrationPagePatch2} 4175\begin{paste}{ugProblemIntegrationPageFull2}{ugProblemIntegrationPageEmpty2} 4176\pastebutton{ugProblemIntegrationPageFull2}{\hidepaste} 4177\tab{5}\spadcommand{integrate(log(1 + sqrt(a * x + b)) / x, x)} 4178\indentrel{3}\begin{verbatim} 4179 x +--------+ 4180 ++ log(\|b + %A a + 1) 4181 (2) | -------------------- d%A 4182 ++ %A 4183 Type: Union(Expression(Integer),...) 4184\end{verbatim} 4185\indentrel{-3}\end{paste}\end{patch} 4186 4187\begin{patch}{ugProblemIntegrationPageEmpty2} 4188\begin{paste}{ugProblemIntegrationPageEmpty2}{ugProblemIntegrationPagePatch2} 4189\pastebutton{ugProblemIntegrationPageEmpty2}{\showpaste} 4190\tab{5}\spadcommand{integrate(log(1 + sqrt(a * x + b)) / x, x)} 4191\end{paste}\end{patch} 4192 4193\begin{patch}{ugProblemIntegrationPagePatch3} 4194\begin{paste}{ugProblemIntegrationPageFull3}{ugProblemIntegrationPageEmpty3} 4195\pastebutton{ugProblemIntegrationPageFull3}{\hidepaste} 4196\tab{5}\spadcommand{integrate(1/(x^2 - 2),x)} 4197\indentrel{3}\begin{verbatim} 4198 2 +-+ 4199 (x + 2)\|2 - 4 x 4200 log(------------------) 4201 2 4202 x - 2 4203 (3) ----------------------- 4204 +-+ 4205 2 \|2 4206 Type: Union(Expression(Integer),...) 4207\end{verbatim} 4208\indentrel{-3}\end{paste}\end{patch} 4209 4210\begin{patch}{ugProblemIntegrationPageEmpty3} 4211\begin{paste}{ugProblemIntegrationPageEmpty3}{ugProblemIntegrationPagePatch3} 4212\pastebutton{ugProblemIntegrationPageEmpty3}{\showpaste} 4213\tab{5}\spadcommand{integrate(1/(x^2 - 2),x)} 4214\end{paste}\end{patch} 4215 4216\begin{patch}{ugProblemIntegrationPagePatch4} 4217\begin{paste}{ugProblemIntegrationPageFull4}{ugProblemIntegrationPageEmpty4} 4218\pastebutton{ugProblemIntegrationPageFull4}{\hidepaste} 4219\tab{5}\spadcommand{integrate(1/(x^2 + 2),x)} 4220\indentrel{3}\begin{verbatim} 4221 +-+ 4222 x\|2 4223 atan(-----) 4224 2 4225 (4) ----------- 4226 +-+ 4227 \|2 4228 Type: Union(Expression(Integer),...) 4229\end{verbatim} 4230\indentrel{-3}\end{paste}\end{patch} 4231 4232\begin{patch}{ugProblemIntegrationPageEmpty4} 4233\begin{paste}{ugProblemIntegrationPageEmpty4}{ugProblemIntegrationPagePatch4} 4234\pastebutton{ugProblemIntegrationPageEmpty4}{\showpaste} 4235\tab{5}\spadcommand{integrate(1/(x^2 + 2),x)} 4236\end{paste}\end{patch} 4237 4238\begin{patch}{ugProblemIntegrationPagePatch5} 4239\begin{paste}{ugProblemIntegrationPageFull5}{ugProblemIntegrationPageEmpty5} 4240\pastebutton{ugProblemIntegrationPageFull5}{\hidepaste} 4241\tab{5}\spadcommand{integrate(x^2 / (x^4 - a^2), x)} 4242\indentrel{3}\begin{verbatim} 4243 (5) 4244 2 +-+ +-+ 4245 (x + a)\|a - 2 a x x\|a 4246 log(--------------------) + 2 atan(-----) 4247 2 a 4248 x - a 4249 [-----------------------------------------, 4250 +-+ 4251 4 \|a 4252 2 +---+ +---+ 4253 (x - a)\|- a + 2 a x x\|- a 4254 log(----------------------) - 2 atan(-------) 4255 2 a 4256 x + a 4257 ---------------------------------------------] 4258 +---+ 4259 4 \|- a 4260 Type: Union(List(Expression(Integer)),...) 4261\end{verbatim} 4262\indentrel{-3}\end{paste}\end{patch} 4263 4264\begin{patch}{ugProblemIntegrationPageEmpty5} 4265\begin{paste}{ugProblemIntegrationPageEmpty5}{ugProblemIntegrationPagePatch5} 4266\pastebutton{ugProblemIntegrationPageEmpty5}{\showpaste} 4267\tab{5}\spadcommand{integrate(x^2 / (x^4 - a^2), x)} 4268\end{paste}\end{patch} 4269 4270\begin{patch}{ugProblemIntegrationPagePatch6} 4271\begin{paste}{ugProblemIntegrationPageFull6}{ugProblemIntegrationPageEmpty6} 4272\pastebutton{ugProblemIntegrationPageFull6}{\hidepaste} 4273\tab{5}\spadcommand{complexIntegrate(x^2 / (x^4 - a^2), x)} 4274\indentrel{3}\begin{verbatim} 4275 (6) 4276 +---+ +---+ 4277 | 1 | 1 4278 - |--- log(2 a |--- + x) 4279 \|4 a \|4 a 4280 + 4281 +-----+ +-----+ 4282 | 1 | 1 4283 |- --- log(2 a |- --- + x) 4284 \| 4 a \| 4 a 4285 + 4286 +-----+ +-----+ 4287 | 1 | 1 4288 - |- --- log(- 2 a |- --- + x) 4289 \| 4 a \| 4 a 4290 + 4291 +---+ +---+ 4292 | 1 | 1 4293 |--- log(- 2 a |--- + x) 4294 \|4 a \|4 a 4295 / 4296 2 4297 Type: Expression(Integer) 4298\end{verbatim} 4299\indentrel{-3}\end{paste}\end{patch} 4300 4301\begin{patch}{ugProblemIntegrationPageEmpty6} 4302\begin{paste}{ugProblemIntegrationPageEmpty6}{ugProblemIntegrationPagePatch6} 4303\pastebutton{ugProblemIntegrationPageEmpty6}{\showpaste} 4304\tab{5}\spadcommand{complexIntegrate(x^2 / (x^4 - a^2), x)} 4305\end{paste}\end{patch} 4306 4307\begin{patch}{ugProblemIntegrationPagePatch7} 4308\begin{paste}{ugProblemIntegrationPageFull7}{ugProblemIntegrationPageEmpty7} 4309\pastebutton{ugProblemIntegrationPageFull7}{\hidepaste} 4310\tab{5}\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)} 4311\indentrel{3}\begin{verbatim} 4312 x +--------+ 4313 ++ log(\|b + %A a + 1) 4314 (7) | -------------------- d%A 4315 ++ %A 4316 Type: Expression(Integer) 4317\end{verbatim} 4318\indentrel{-3}\end{paste}\end{patch} 4319 4320\begin{patch}{ugProblemIntegrationPageEmpty7} 4321\begin{paste}{ugProblemIntegrationPageEmpty7}{ugProblemIntegrationPagePatch7} 4322\pastebutton{ugProblemIntegrationPageEmpty7}{\showpaste} 4323\tab{5}\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)} 4324\end{paste}\end{patch} 4325 4326\begin{patch}{ugProblemIntegrationPagePatch8} 4327\begin{paste}{ugProblemIntegrationPageFull8}{ugProblemIntegrationPageEmpty8} 4328\pastebutton{ugProblemIntegrationPageFull8}{\hidepaste} 4329\tab{5}\spadcommand{integrate((x^4 - 3*x^2 + 6)/(x^6-5*x^4+5*x^2+4), x = 1..2)} 4330\indentrel{3}\begin{verbatim} 4331 (8) 4332 1 4333 2 atan(8) + 2 atan(5) + 2 atan(2) + 2 atan(-) - %pi 4334 2 4335 --------------------------------------------------- 4336 2 4337Type: Union(f1: OrderedCompletion(Expression(Integer)),...) 4338\end{verbatim} 4339\indentrel{-3}\end{paste}\end{patch} 4340 4341\begin{patch}{ugProblemIntegrationPageEmpty8} 4342\begin{paste}{ugProblemIntegrationPageEmpty8}{ugProblemIntegrationPagePatch8} 4343\pastebutton{ugProblemIntegrationPageEmpty8}{\showpaste} 4344\tab{5}\spadcommand{integrate((x^4 - 3*x^2 + 6)/(x^6-5*x^4+5*x^2+4), x = 1..2)} 4345\end{paste}\end{patch} 4346 4347\begin{patch}{ugProblemIntegrationPagePatch9} 4348\begin{paste}{ugProblemIntegrationPageFull9}{ugProblemIntegrationPageEmpty9} 4349\pastebutton{ugProblemIntegrationPageFull9}{\hidepaste} 4350\tab{5}\spadcommand{integrate(1/(x^2-a), x = 1..2)} 4351\indentrel{3}\begin{verbatim} 4352 (9) "potentialPole" 4353 Type: Union(pole: potentialPole,...) 4354\end{verbatim} 4355\indentrel{-3}\end{paste}\end{patch} 4356 4357\begin{patch}{ugProblemIntegrationPageEmpty9} 4358\begin{paste}{ugProblemIntegrationPageEmpty9}{ugProblemIntegrationPagePatch9} 4359\pastebutton{ugProblemIntegrationPageEmpty9}{\showpaste} 4360\tab{5}\spadcommand{integrate(1/(x^2-a), x = 1..2)} 4361\end{paste}\end{patch} 4362 4363\begin{patch}{ugProblemIntegrationPagePatch10} 4364\begin{paste}{ugProblemIntegrationPageFull10}{ugProblemIntegrationPageEmpty10} 4365\pastebutton{ugProblemIntegrationPageFull10}{\hidepaste} 4366\tab{5}\spadcommand{integrate(1/(x^2-a), x = 1..2, "noPole")} 4367\indentrel{3}\begin{verbatim} 4368 (10) 4369 [ 4370 2 +-+ 3 2 4371 (- 4 a - 4 a)\|a + a + 6 a + a 4372 - log(----------------------------------) 4373 2 4374 a - 2 a + 1 4375 + 4376 2 +-+ 3 2 4377 (- 8 a - 32 a)\|a + a + 24 a + 16 a 4378 log(---------------------------------------) 4379 2 4380 a - 8 a + 16 4381 / 4382 +-+ 4383 4 \|a 4384 , 4385 +---+ +---+ 4386 2 \|- a \|- a 4387 - atan(--------) + atan(------) 4388 a a 4389 -------------------------------] 4390 +---+ 4391 \|- a 4392Type: Union(f2: List(OrderedCompletion(Expression(Integer))),...) 4393\end{verbatim} 4394\indentrel{-3}\end{paste}\end{patch} 4395 4396\begin{patch}{ugProblemIntegrationPageEmpty10} 4397\begin{paste}{ugProblemIntegrationPageEmpty10}{ugProblemIntegrationPagePatch10} 4398\pastebutton{ugProblemIntegrationPageEmpty10}{\showpaste} 4399\tab{5}\spadcommand{integrate(1/(x^2-a), x = 1..2, "noPole")} 4400\end{paste}\end{patch} 4401 4402\begin{patch}{ugMatrixManipulationPagePatch1} 4403\begin{paste}{ugMatrixManipulationPageFull1}{ugMatrixManipulationPageEmpty1} 4404\pastebutton{ugMatrixManipulationPageFull1}{\hidepaste} 4405\tab{5}\spadcommand{m := matrix([[11, 12, 13, 14], [21, 22,23, 24], [31, 32, 33, 34]])\bound{m }} 4406\indentrel{3}\begin{verbatim} 4407 +11 12 13 14+ 4408 | | 4409 (1) |21 22 23 24| 4410 | | 4411 +31 32 33 34+ 4412 Type: Matrix(Integer) 4413\end{verbatim} 4414\indentrel{-3}\end{paste}\end{patch} 4415 4416\begin{patch}{ugMatrixManipulationPageEmpty1} 4417\begin{paste}{ugMatrixManipulationPageEmpty1}{ugMatrixManipulationPagePatch1} 4418\pastebutton{ugMatrixManipulationPageEmpty1}{\showpaste} 4419\tab{5}\spadcommand{m := matrix([[11, 12, 13, 14], [21, 22,23, 24], [31, 32, 33, 34]])\bound{m }} 4420\end{paste}\end{patch} 4421 4422\begin{patch}{ugMatrixManipulationPagePatch2} 4423\begin{paste}{ugMatrixManipulationPageFull2}{ugMatrixManipulationPageEmpty2} 4424\pastebutton{ugMatrixManipulationPageFull2}{\hidepaste} 4425\tab{5}\spadcommand{m(1..2, 3..4)\free{m }} 4426\indentrel{3}\begin{verbatim} 4427 +13 14+ 4428 (2) | | 4429 +23 24+ 4430 Type: Matrix(Integer) 4431\end{verbatim} 4432\indentrel{-3}\end{paste}\end{patch} 4433 4434\begin{patch}{ugMatrixManipulationPageEmpty2} 4435\begin{paste}{ugMatrixManipulationPageEmpty2}{ugMatrixManipulationPagePatch2} 4436\pastebutton{ugMatrixManipulationPageEmpty2}{\showpaste} 4437\tab{5}\spadcommand{m(1..2, 3..4)\free{m }} 4438\end{paste}\end{patch} 4439 4440\begin{patch}{ugMatrixManipulationPagePatch3} 4441\begin{paste}{ugMatrixManipulationPageFull3}{ugMatrixManipulationPageEmpty3} 4442\pastebutton{ugMatrixManipulationPageFull3}{\hidepaste} 4443\tab{5}\spadcommand{m(1..2, 1..3 by 2)\free{m }} 4444\indentrel{3}\begin{verbatim} 4445 +11 13+ 4446 (3) | | 4447 +21 23+ 4448 Type: Matrix(Integer) 4449\end{verbatim} 4450\indentrel{-3}\end{paste}\end{patch} 4451 4452\begin{patch}{ugMatrixManipulationPageEmpty3} 4453\begin{paste}{ugMatrixManipulationPageEmpty3}{ugMatrixManipulationPagePatch3} 4454\pastebutton{ugMatrixManipulationPageEmpty3}{\showpaste} 4455\tab{5}\spadcommand{m(1..2, 1..3 by 2)\free{m }} 4456\end{paste}\end{patch} 4457 4458\begin{patch}{ugMatrixManipulationPagePatch4} 4459\begin{paste}{ugMatrixManipulationPageFull4}{ugMatrixManipulationPageEmpty4} 4460\pastebutton{ugMatrixManipulationPageFull4}{\hidepaste} 4461\tab{5}\spadcommand{m([1,3], [2,4])\free{m }} 4462\indentrel{3}\begin{verbatim} 4463 +12 14+ 4464 (4) | | 4465 +32 34+ 4466 Type: Matrix(Integer) 4467\end{verbatim} 4468\indentrel{-3}\end{paste}\end{patch} 4469 4470\begin{patch}{ugMatrixManipulationPageEmpty4} 4471\begin{paste}{ugMatrixManipulationPageEmpty4}{ugMatrixManipulationPagePatch4} 4472\pastebutton{ugMatrixManipulationPageEmpty4}{\showpaste} 4473\tab{5}\spadcommand{m([1,3], [2,4])\free{m }} 4474\end{paste}\end{patch} 4475 4476\begin{patch}{ugMatrixManipulationPagePatch5} 4477\begin{paste}{ugMatrixManipulationPageFull5}{ugMatrixManipulationPageEmpty5} 4478\pastebutton{ugMatrixManipulationPageFull5}{\hidepaste} 4479\tab{5}\spadcommand{m(1, 2)\free{m }} 4480\indentrel{3}\begin{verbatim} 4481 (5) 12 4482 Type: PositiveInteger 4483\end{verbatim} 4484\indentrel{-3}\end{paste}\end{patch} 4485 4486\begin{patch}{ugMatrixManipulationPageEmpty5} 4487\begin{paste}{ugMatrixManipulationPageEmpty5}{ugMatrixManipulationPagePatch5} 4488\pastebutton{ugMatrixManipulationPageEmpty5}{\showpaste} 4489\tab{5}\spadcommand{m(1, 2)\free{m }} 4490\end{paste}\end{patch} 4491 4492\begin{patch}{ugMatrixManipulationPagePatch6} 4493\begin{paste}{ugMatrixManipulationPageFull6}{ugMatrixManipulationPageEmpty6} 4494\pastebutton{ugMatrixManipulationPageFull6}{\hidepaste} 4495\tab{5}\spadcommand{m([1], [2])\free{m }} 4496\indentrel{3}\begin{verbatim} 4497 (6) [12] 4498 Type: Matrix(Integer) 4499\end{verbatim} 4500\indentrel{-3}\end{paste}\end{patch} 4501 4502\begin{patch}{ugMatrixManipulationPageEmpty6} 4503\begin{paste}{ugMatrixManipulationPageEmpty6}{ugMatrixManipulationPagePatch6} 4504\pastebutton{ugMatrixManipulationPageEmpty6}{\showpaste} 4505\tab{5}\spadcommand{m([1], [2])\free{m }} 4506\end{paste}\end{patch} 4507 4508\begin{patch}{ugMatrixManipulationPagePatch7} 4509\begin{paste}{ugMatrixManipulationPageFull7}{ugMatrixManipulationPageEmpty7} 4510\pastebutton{ugMatrixManipulationPageFull7}{\hidepaste} 4511\tab{5}\spadcommand{m(1, [2])\free{m }} 4512\indentrel{3}\begin{verbatim} 4513 (7) [12] 4514 Type: Matrix(Integer) 4515\end{verbatim} 4516\indentrel{-3}\end{paste}\end{patch} 4517 4518\begin{patch}{ugMatrixManipulationPageEmpty7} 4519\begin{paste}{ugMatrixManipulationPageEmpty7}{ugMatrixManipulationPagePatch7} 4520\pastebutton{ugMatrixManipulationPageEmpty7}{\showpaste} 4521\tab{5}\spadcommand{m(1, [2])\free{m }} 4522\end{paste}\end{patch} 4523 4524\begin{patch}{ugMatrixManipulationPagePatch8} 4525\begin{paste}{ugMatrixManipulationPageFull8}{ugMatrixManipulationPageEmpty8} 4526\pastebutton{ugMatrixManipulationPageFull8}{\hidepaste} 4527\tab{5}\spadcommand{m(1, 2..2)\free{m }} 4528\indentrel{3}\begin{verbatim} 4529 (8) [12] 4530 Type: Matrix(Integer) 4531\end{verbatim} 4532\indentrel{-3}\end{paste}\end{patch} 4533 4534\begin{patch}{ugMatrixManipulationPageEmpty8} 4535\begin{paste}{ugMatrixManipulationPageEmpty8}{ugMatrixManipulationPagePatch8} 4536\pastebutton{ugMatrixManipulationPageEmpty8}{\showpaste} 4537\tab{5}\spadcommand{m(1, 2..2)\free{m }} 4538\end{paste}\end{patch} 4539 4540\begin{patch}{ugMatrixManipulationPagePatch9} 4541\begin{paste}{ugMatrixManipulationPageFull9}{ugMatrixManipulationPageEmpty9} 4542\pastebutton{ugMatrixManipulationPageFull9}{\hidepaste} 4543\tab{5}\spadcommand{m(1, [2..2])\free{m }} 4544\indentrel{3}\begin{verbatim} 4545 (9) [12] 4546 Type: Matrix(Integer) 4547\end{verbatim} 4548\indentrel{-3}\end{paste}\end{patch} 4549 4550\begin{patch}{ugMatrixManipulationPageEmpty9} 4551\begin{paste}{ugMatrixManipulationPageEmpty9}{ugMatrixManipulationPagePatch9} 4552\pastebutton{ugMatrixManipulationPageEmpty9}{\showpaste} 4553\tab{5}\spadcommand{m(1, [2..2])\free{m }} 4554\end{paste}\end{patch} 4555 4556\begin{patch}{ugMatrixManipulationPagePatch10} 4557\begin{paste}{ugMatrixManipulationPageFull10}{ugMatrixManipulationPageEmpty10} 4558\pastebutton{ugMatrixManipulationPageFull10}{\hidepaste} 4559\tab{5}\spadcommand{m([1..2], [1, 3..4])\free{m }} 4560\indentrel{3}\begin{verbatim} 4561 +11 13 14+ 4562 (10) | | 4563 +21 23 24+ 4564 Type: Matrix(Integer) 4565\end{verbatim} 4566\indentrel{-3}\end{paste}\end{patch} 4567 4568\begin{patch}{ugMatrixManipulationPageEmpty10} 4569\begin{paste}{ugMatrixManipulationPageEmpty10}{ugMatrixManipulationPagePatch10} 4570\pastebutton{ugMatrixManipulationPageEmpty10}{\showpaste} 4571\tab{5}\spadcommand{m([1..2], [1, 3..4])\free{m }} 4572\end{paste}\end{patch} 4573 4574\begin{patch}{ugMatrixManipulationPagePatch11} 4575\begin{paste}{ugMatrixManipulationPageFull11}{ugMatrixManipulationPageEmpty11} 4576\pastebutton{ugMatrixManipulationPageFull11}{\hidepaste} 4577\tab{5}\spadcommand{m([1..2], [3..4, 3..4])\free{m }} 4578\indentrel{3}\begin{verbatim} 4579 +13 14 13 14+ 4580 (11) | | 4581 +23 24 23 24+ 4582 Type: Matrix(Integer) 4583\end{verbatim} 4584\indentrel{-3}\end{paste}\end{patch} 4585 4586\begin{patch}{ugMatrixManipulationPageEmpty11} 4587\begin{paste}{ugMatrixManipulationPageEmpty11}{ugMatrixManipulationPagePatch11} 4588\pastebutton{ugMatrixManipulationPageEmpty11}{\showpaste} 4589\tab{5}\spadcommand{m([1..2], [3..4, 3..4])\free{m }} 4590\end{paste}\end{patch} 4591 4592\begin{patch}{ugMatrixManipulationPagePatch12} 4593\begin{paste}{ugMatrixManipulationPageFull12}{ugMatrixManipulationPageEmpty12} 4594\pastebutton{ugMatrixManipulationPageFull12}{\hidepaste} 4595\tab{5}\spadcommand{m(2, [1,4])\free{m }} 4596\indentrel{3}\begin{verbatim} 4597 (12) [21 24] 4598 Type: Matrix(Integer) 4599\end{verbatim} 4600\indentrel{-3}\end{paste}\end{patch} 4601 4602\begin{patch}{ugMatrixManipulationPageEmpty12} 4603\begin{paste}{ugMatrixManipulationPageEmpty12}{ugMatrixManipulationPagePatch12} 4604\pastebutton{ugMatrixManipulationPageEmpty12}{\showpaste} 4605\tab{5}\spadcommand{m(2, [1,4])\free{m }} 4606\end{paste}\end{patch} 4607 4608\begin{patch}{ugMatrixManipulationPagePatch13} 4609\begin{paste}{ugMatrixManipulationPageFull13}{ugMatrixManipulationPageEmpty13} 4610\pastebutton{ugMatrixManipulationPageFull13}{\hidepaste} 4611\tab{5}\spadcommand{m([1,2,3], 2..3)\free{m }} 4612\indentrel{3}\begin{verbatim} 4613 +12 13+ 4614 | | 4615 (13) |22 23| 4616 | | 4617 +32 33+ 4618 Type: Matrix(Integer) 4619\end{verbatim} 4620\indentrel{-3}\end{paste}\end{patch} 4621 4622\begin{patch}{ugMatrixManipulationPageEmpty13} 4623\begin{paste}{ugMatrixManipulationPageEmpty13}{ugMatrixManipulationPagePatch13} 4624\pastebutton{ugMatrixManipulationPageEmpty13}{\showpaste} 4625\tab{5}\spadcommand{m([1,2,3], 2..3)\free{m }} 4626\end{paste}\end{patch} 4627 4628\begin{patch}{ugMatrixManipulationPagePatch14} 4629\begin{paste}{ugMatrixManipulationPageFull14}{ugMatrixManipulationPageEmpty14} 4630\pastebutton{ugMatrixManipulationPageFull14}{\hidepaste} 4631\tab{5}\spadcommand{m([1,2], [1..3, 4])\free{m }} 4632\indentrel{3}\begin{verbatim} 4633 +11 12 13 14+ 4634 (14) | | 4635 +21 22 23 24+ 4636 Type: Matrix(Integer) 4637\end{verbatim} 4638\indentrel{-3}\end{paste}\end{patch} 4639 4640\begin{patch}{ugMatrixManipulationPageEmpty14} 4641\begin{paste}{ugMatrixManipulationPageEmpty14}{ugMatrixManipulationPagePatch14} 4642\pastebutton{ugMatrixManipulationPageEmpty14}{\showpaste} 4643\tab{5}\spadcommand{m([1,2], [1..3, 4])\free{m }} 4644\end{paste}\end{patch} 4645 4646\begin{patch}{ugMatrixManipulationPagePatch15} 4647\begin{paste}{ugMatrixManipulationPageFull15}{ugMatrixManipulationPageEmpty15} 4648\pastebutton{ugMatrixManipulationPageFull15}{\hidepaste} 4649\tab{5}\spadcommand{m([1..2], [3..3]) := m([1,2], [2])\bound{m }} 4650\indentrel{3}\begin{verbatim} 4651 +12+ 4652 (15) | | 4653 +22+ 4654 Type: Matrix(Integer) 4655\end{verbatim} 4656\indentrel{-3}\end{paste}\end{patch} 4657 4658\begin{patch}{ugMatrixManipulationPageEmpty15} 4659\begin{paste}{ugMatrixManipulationPageEmpty15}{ugMatrixManipulationPagePatch15} 4660\pastebutton{ugMatrixManipulationPageEmpty15}{\showpaste} 4661\tab{5}\spadcommand{m([1..2], [3..3]) := m([1,2], [2])\bound{m }} 4662\end{paste}\end{patch} 4663 4664\begin{patch}{ugMatrixManipulationPagePatch16} 4665\begin{paste}{ugMatrixManipulationPageFull16}{ugMatrixManipulationPageEmpty16} 4666\pastebutton{ugMatrixManipulationPageFull16}{\hidepaste} 4667\tab{5}\spadcommand{m\free{m }} 4668\indentrel{3}\begin{verbatim} 4669 +11 12 12 14+ 4670 | | 4671 (16) |21 22 22 24| 4672 | | 4673 +31 32 33 34+ 4674 Type: Matrix(Integer) 4675\end{verbatim} 4676\indentrel{-3}\end{paste}\end{patch} 4677 4678\begin{patch}{ugMatrixManipulationPageEmpty16} 4679\begin{paste}{ugMatrixManipulationPageEmpty16}{ugMatrixManipulationPagePatch16} 4680\pastebutton{ugMatrixManipulationPageEmpty16}{\showpaste} 4681\tab{5}\spadcommand{m\free{m }} 4682\end{paste}\end{patch} 4683 4684\begin{patch}{ugMatrixManipulationPagePatch17} 4685\begin{paste}{ugMatrixManipulationPageFull17}{ugMatrixManipulationPageEmpty17} 4686\pastebutton{ugMatrixManipulationPageFull17}{\hidepaste} 4687\tab{5}\spadcommand{m([2], [2]) := matrix([[4]])\bound{m }} 4688\indentrel{3}\begin{verbatim} 4689 (17) [4] 4690 Type: Matrix(Integer) 4691\end{verbatim} 4692\indentrel{-3}\end{paste}\end{patch} 4693 4694\begin{patch}{ugMatrixManipulationPageEmpty17} 4695\begin{paste}{ugMatrixManipulationPageEmpty17}{ugMatrixManipulationPagePatch17} 4696\pastebutton{ugMatrixManipulationPageEmpty17}{\showpaste} 4697\tab{5}\spadcommand{m([2], [2]) := matrix([[4]])\bound{m }} 4698\end{paste}\end{patch} 4699 4700\begin{patch}{ugMatrixManipulationPagePatch18} 4701\begin{paste}{ugMatrixManipulationPageFull18}{ugMatrixManipulationPageEmpty18} 4702\pastebutton{ugMatrixManipulationPageFull18}{\hidepaste} 4703\tab{5}\spadcommand{m\free{m }} 4704\indentrel{3}\begin{verbatim} 4705 +11 12 12 14+ 4706 | | 4707 (18) |21 4 22 24| 4708 | | 4709 +31 32 33 34+ 4710 Type: Matrix(Integer) 4711\end{verbatim} 4712\indentrel{-3}\end{paste}\end{patch} 4713 4714\begin{patch}{ugMatrixManipulationPageEmpty18} 4715\begin{paste}{ugMatrixManipulationPageEmpty18}{ugMatrixManipulationPagePatch18} 4716\pastebutton{ugMatrixManipulationPageEmpty18}{\showpaste} 4717\tab{5}\spadcommand{m\free{m }} 4718\end{paste}\end{patch} 4719 4720\begin{patch}{ugMatrixManipulationPagePatch19} 4721\begin{paste}{ugMatrixManipulationPageFull19}{ugMatrixManipulationPageEmpty19} 4722\pastebutton{ugMatrixManipulationPageFull19}{\hidepaste} 4723\tab{5}\spadcommand{r := rowSlice(m)\bound{r }} 4724\indentrel{3}\begin{verbatim} 4725 (19) 1..3 4726 Type: Segment(Integer) 4727\end{verbatim} 4728\indentrel{-3}\end{paste}\end{patch} 4729 4730\begin{patch}{ugMatrixManipulationPageEmpty19} 4731\begin{paste}{ugMatrixManipulationPageEmpty19}{ugMatrixManipulationPagePatch19} 4732\pastebutton{ugMatrixManipulationPageEmpty19}{\showpaste} 4733\tab{5}\spadcommand{r := rowSlice(m)\bound{r }} 4734\end{paste}\end{patch} 4735 4736\begin{patch}{ugMatrixManipulationPagePatch20} 4737\begin{paste}{ugMatrixManipulationPageFull20}{ugMatrixManipulationPageEmpty20} 4738\pastebutton{ugMatrixManipulationPageFull20}{\hidepaste} 4739\tab{5}\spadcommand{m(r, 2)\free{m }} 4740\indentrel{3}\begin{verbatim} 4741 +12+ 4742 | | 4743 (20) |4 | 4744 | | 4745 +32+ 4746 Type: Matrix(Integer) 4747\end{verbatim} 4748\indentrel{-3}\end{paste}\end{patch} 4749 4750\begin{patch}{ugMatrixManipulationPageEmpty20} 4751\begin{paste}{ugMatrixManipulationPageEmpty20}{ugMatrixManipulationPagePatch20} 4752\pastebutton{ugMatrixManipulationPageEmpty20}{\showpaste} 4753\tab{5}\spadcommand{m(r, 2)\free{m }} 4754\end{paste}\end{patch} 4755 4756\begin{patch}{ugMatrixManipulationPagePatch21} 4757\begin{paste}{ugMatrixManipulationPageFull21}{ugMatrixManipulationPageEmpty21} 4758\pastebutton{ugMatrixManipulationPageFull21}{\hidepaste} 4759\tab{5}\spadcommand{m(r, 4)\free{m }} 4760\indentrel{3}\begin{verbatim} 4761 +14+ 4762 | | 4763 (21) |24| 4764 | | 4765 +34+ 4766 Type: Matrix(Integer) 4767\end{verbatim} 4768\indentrel{-3}\end{paste}\end{patch} 4769 4770\begin{patch}{ugMatrixManipulationPageEmpty21} 4771\begin{paste}{ugMatrixManipulationPageEmpty21}{ugMatrixManipulationPagePatch21} 4772\pastebutton{ugMatrixManipulationPageEmpty21}{\showpaste} 4773\tab{5}\spadcommand{m(r, 4)\free{m }} 4774\end{paste}\end{patch} 4775 4776\begin{patch}{ugMatrixManipulationPagePatch22} 4777\begin{paste}{ugMatrixManipulationPageFull22}{ugMatrixManipulationPageEmpty22} 4778\pastebutton{ugMatrixManipulationPageFull22}{\hidepaste} 4779\tab{5}\spadcommand{c := colSlice(m)\bound{c }} 4780\indentrel{3}\begin{verbatim} 4781 (22) 1..4 4782 Type: Segment(Integer) 4783\end{verbatim} 4784\indentrel{-3}\end{paste}\end{patch} 4785 4786\begin{patch}{ugMatrixManipulationPageEmpty22} 4787\begin{paste}{ugMatrixManipulationPageEmpty22}{ugMatrixManipulationPagePatch22} 4788\pastebutton{ugMatrixManipulationPageEmpty22}{\showpaste} 4789\tab{5}\spadcommand{c := colSlice(m)\bound{c }} 4790\end{paste}\end{patch} 4791 4792\begin{patch}{ugMatrixManipulationPagePatch23} 4793\begin{paste}{ugMatrixManipulationPageFull23}{ugMatrixManipulationPageEmpty23} 4794\pastebutton{ugMatrixManipulationPageFull23}{\hidepaste} 4795\tab{5}\spadcommand{m := m :: Matrix(Fraction(Integer))} 4796\indentrel{3}\begin{verbatim} 4797 +11 12 12 14+ 4798 | | 4799 (23) |21 4 22 24| 4800 | | 4801 +31 32 33 34+ 4802 Type: Matrix(Fraction(Integer)) 4803\end{verbatim} 4804\indentrel{-3}\end{paste}\end{patch} 4805 4806\begin{patch}{ugMatrixManipulationPageEmpty23} 4807\begin{paste}{ugMatrixManipulationPageEmpty23}{ugMatrixManipulationPagePatch23} 4808\pastebutton{ugMatrixManipulationPageEmpty23}{\showpaste} 4809\tab{5}\spadcommand{m := m :: Matrix(Fraction(Integer))} 4810\end{paste}\end{patch} 4811 4812\begin{patch}{ugMatrixManipulationPagePatch24} 4813\begin{paste}{ugMatrixManipulationPageFull24}{ugMatrixManipulationPageEmpty24} 4814\pastebutton{ugMatrixManipulationPageFull24}{\hidepaste} 4815\tab{5}\spadcommand{m(2, c) := m(2, c) - m(2,1)/m(1,1) * m(1, c)\free{m }} 4816\indentrel{3}\begin{verbatim} 4817 + 208 10 30+ 4818 (24) |0 - --- - -- - --| 4819 + 11 11 11+ 4820 Type: Matrix(Fraction(Integer)) 4821\end{verbatim} 4822\indentrel{-3}\end{paste}\end{patch} 4823 4824\begin{patch}{ugMatrixManipulationPageEmpty24} 4825\begin{paste}{ugMatrixManipulationPageEmpty24}{ugMatrixManipulationPagePatch24} 4826\pastebutton{ugMatrixManipulationPageEmpty24}{\showpaste} 4827\tab{5}\spadcommand{m(2, c) := m(2, c) - m(2,1)/m(1,1) * m(1, c)\free{m }} 4828\end{paste}\end{patch} 4829 4830\begin{patch}{ugMatrixManipulationPagePatch25} 4831\begin{paste}{ugMatrixManipulationPageFull25}{ugMatrixManipulationPageEmpty25} 4832\pastebutton{ugMatrixManipulationPageFull25}{\hidepaste} 4833\tab{5}\spadcommand{m(3, c) := m(3, c) - m(3,1)/m(1,1) * m(1, c)\free{m }} 4834\indentrel{3}\begin{verbatim} 4835 + 20 9 60+ 4836 (25) |0 - -- - -- - --| 4837 + 11 11 11+ 4838 Type: Matrix(Fraction(Integer)) 4839\end{verbatim} 4840\indentrel{-3}\end{paste}\end{patch} 4841 4842\begin{patch}{ugMatrixManipulationPageEmpty25} 4843\begin{paste}{ugMatrixManipulationPageEmpty25}{ugMatrixManipulationPagePatch25} 4844\pastebutton{ugMatrixManipulationPageEmpty25}{\showpaste} 4845\tab{5}\spadcommand{m(3, c) := m(3, c) - m(3,1)/m(1,1) * m(1, c)\free{m }} 4846\end{paste}\end{patch} 4847 4848\begin{patch}{ugMatrixManipulationPagePatch26} 4849\begin{paste}{ugMatrixManipulationPageFull26}{ugMatrixManipulationPageEmpty26} 4850\pastebutton{ugMatrixManipulationPageFull26}{\hidepaste} 4851\tab{5}\spadcommand{m(3, c) := m(3, c) - m(3,2)/m(2,2) * m(2, c)\free{m }} 4852\indentrel{3}\begin{verbatim} 4853 + 19 135+ 4854 (26) |0 0 - -- - ---| 4855 + 26 26+ 4856 Type: Matrix(Fraction(Integer)) 4857\end{verbatim} 4858\indentrel{-3}\end{paste}\end{patch} 4859 4860\begin{patch}{ugMatrixManipulationPageEmpty26} 4861\begin{paste}{ugMatrixManipulationPageEmpty26}{ugMatrixManipulationPagePatch26} 4862\pastebutton{ugMatrixManipulationPageEmpty26}{\showpaste} 4863\tab{5}\spadcommand{m(3, c) := m(3, c) - m(3,2)/m(2,2) * m(2, c)\free{m }} 4864\end{paste}\end{patch} 4865 4866\begin{patch}{ugMatrixManipulationPagePatch27} 4867\begin{paste}{ugMatrixManipulationPageFull27}{ugMatrixManipulationPageEmpty27} 4868\pastebutton{ugMatrixManipulationPageFull27}{\hidepaste} 4869\tab{5}\spadcommand{m\free{m }} 4870\indentrel{3}\begin{verbatim} 4871 +11 12 12 14 + 4872 | | 4873 | 208 10 30 | 4874 |0 - --- - -- - -- | 4875 (27) | 11 11 11 | 4876 | | 4877 | 19 135| 4878 |0 0 - -- - ---| 4879 + 26 26+ 4880 Type: Matrix(Fraction(Integer)) 4881\end{verbatim} 4882\indentrel{-3}\end{paste}\end{patch} 4883 4884\begin{patch}{ugMatrixManipulationPageEmpty27} 4885\begin{paste}{ugMatrixManipulationPageEmpty27}{ugMatrixManipulationPagePatch27} 4886\pastebutton{ugMatrixManipulationPageEmpty27}{\showpaste} 4887\tab{5}\spadcommand{m\free{m }} 4888\end{paste}\end{patch} 4889 4890\begin{patch}{ugMatrixManipulationPagePatch28} 4891\begin{paste}{ugMatrixManipulationPageFull28}{ugMatrixManipulationPageEmpty28} 4892\pastebutton{ugMatrixManipulationPageFull28}{\hidepaste} 4893\tab{5}\spadcommand{r := rowSlice(m)\bound{r }} 4894\indentrel{3}\begin{verbatim} 4895 (28) 1..3 4896 Type: Segment(Integer) 4897\end{verbatim} 4898\indentrel{-3}\end{paste}\end{patch} 4899 4900\begin{patch}{ugMatrixManipulationPageEmpty28} 4901\begin{paste}{ugMatrixManipulationPageEmpty28}{ugMatrixManipulationPagePatch28} 4902\pastebutton{ugMatrixManipulationPageEmpty28}{\showpaste} 4903\tab{5}\spadcommand{r := rowSlice(m)\bound{r }} 4904\end{paste}\end{patch} 4905 4906\begin{patch}{ugMatrixManipulationPagePatch29} 4907\begin{paste}{ugMatrixManipulationPageFull29}{ugMatrixManipulationPageEmpty29} 4908\pastebutton{ugMatrixManipulationPageFull29}{\hidepaste} 4909\tab{5}\spadcommand{c := colSlice(m)\bound{c }} 4910\indentrel{3}\begin{verbatim} 4911 (29) 1..4 4912 Type: Segment(Integer) 4913\end{verbatim} 4914\indentrel{-3}\end{paste}\end{patch} 4915 4916\begin{patch}{ugMatrixManipulationPageEmpty29} 4917\begin{paste}{ugMatrixManipulationPageEmpty29}{ugMatrixManipulationPagePatch29} 4918\pastebutton{ugMatrixManipulationPageEmpty29}{\showpaste} 4919\tab{5}\spadcommand{c := colSlice(m)\bound{c }} 4920\end{paste}\end{patch} 4921 4922\begin{patch}{ugMatrixManipulationPagePatch30} 4923\begin{paste}{ugMatrixManipulationPageFull30}{ugMatrixManipulationPageEmpty30} 4924\pastebutton{ugMatrixManipulationPageFull30}{\hidepaste} 4925\tab{5}\spadcommand{m(r,c)\free{m }} 4926\indentrel{3}\begin{verbatim} 4927 +11 12 12 14 + 4928 | | 4929 | 208 10 30 | 4930 |0 - --- - -- - -- | 4931 (30) | 11 11 11 | 4932 | | 4933 | 19 135| 4934 |0 0 - -- - ---| 4935 + 26 26+ 4936 Type: Matrix(Fraction(Integer)) 4937\end{verbatim} 4938\indentrel{-3}\end{paste}\end{patch} 4939 4940\begin{patch}{ugMatrixManipulationPageEmpty30} 4941\begin{paste}{ugMatrixManipulationPageEmpty30}{ugMatrixManipulationPagePatch30} 4942\pastebutton{ugMatrixManipulationPageEmpty30}{\showpaste} 4943\tab{5}\spadcommand{m(r,c)\free{m }} 4944\end{paste}\end{patch} 4945 4946\begin{patch}{ugProblemFactorRatFunPagePatch1} 4947\begin{paste}{ugProblemFactorRatFunPageFull1}{ugProblemFactorRatFunPageEmpty1} 4948\pastebutton{ugProblemFactorRatFunPageFull1}{\hidepaste} 4949\tab{5}\spadcommand{factorFraction((x^2-4)/(y^2-4))} 4950\indentrel{3}\begin{verbatim} 4951 (x - 2)(x + 2) 4952 (1) -------------- 4953 (y - 2)(y + 2) 4954 Type: Fraction(Factored(Polynomial(Integer))) 4955\end{verbatim} 4956\indentrel{-3}\end{paste}\end{patch} 4957 4958\begin{patch}{ugProblemFactorRatFunPageEmpty1} 4959\begin{paste}{ugProblemFactorRatFunPageEmpty1}{ugProblemFactorRatFunPagePatch1} 4960\pastebutton{ugProblemFactorRatFunPageEmpty1}{\showpaste} 4961\tab{5}\spadcommand{factorFraction((x^2-4)/(y^2-4))} 4962\end{paste}\end{patch} 4963 4964\begin{patch}{ugProblemFactorRatFunPagePatch2} 4965\begin{paste}{ugProblemFactorRatFunPageFull2}{ugProblemFactorRatFunPageEmpty2} 4966\pastebutton{ugProblemFactorRatFunPageFull2}{\hidepaste} 4967\tab{5}\spadcommand{map(factor,(x^2-4)/(y^2-4))} 4968\indentrel{3}\begin{verbatim} 4969 (x - 2)(x + 2) 4970 (2) -------------- 4971 (y - 2)(y + 2) 4972 Type: Fraction(Factored(Polynomial(Integer))) 4973\end{verbatim} 4974\indentrel{-3}\end{paste}\end{patch} 4975 4976\begin{patch}{ugProblemFactorRatFunPageEmpty2} 4977\begin{paste}{ugProblemFactorRatFunPageEmpty2}{ugProblemFactorRatFunPagePatch2} 4978\pastebutton{ugProblemFactorRatFunPageEmpty2}{\showpaste} 4979\tab{5}\spadcommand{map(factor,(x^2-4)/(y^2-4))} 4980\end{paste}\end{patch} 4981 4982\begin{patch}{ugProblemFactorIntRatPagePatch1} 4983\begin{paste}{ugProblemFactorIntRatPageFull1}{ugProblemFactorIntRatPageEmpty1} 4984\pastebutton{ugProblemFactorIntRatPageFull1}{\hidepaste} 4985\tab{5}\spadcommand{v := (4*x^3+2*y^2+1)*(12*x^5-x^3*y+12)\bound{v }} 4986\indentrel{3}\begin{verbatim} 4987 (1) 4988 3 3 5 2 6 3 8 4989 - 2 x y + (24 x + 24)y + (- 4 x - x )y + 48 x 4990 + 4991 5 3 4992 12 x + 48 x + 12 4993 Type: Polynomial(Integer) 4994\end{verbatim} 4995\indentrel{-3}\end{paste}\end{patch} 4996 4997\begin{patch}{ugProblemFactorIntRatPageEmpty1} 4998\begin{paste}{ugProblemFactorIntRatPageEmpty1}{ugProblemFactorIntRatPagePatch1} 4999\pastebutton{ugProblemFactorIntRatPageEmpty1}{\showpaste} 5000\tab{5}\spadcommand{v := (4*x^3+2*y^2+1)*(12*x^5-x^3*y+12)\bound{v }} 5001\end{paste}\end{patch} 5002 5003\begin{patch}{ugProblemFactorIntRatPagePatch2} 5004\begin{paste}{ugProblemFactorIntRatPageFull2}{ugProblemFactorIntRatPageEmpty2} 5005\pastebutton{ugProblemFactorIntRatPageFull2}{\hidepaste} 5006\tab{5}\spadcommand{factor v\free{v }} 5007\indentrel{3}\begin{verbatim} 5008 3 5 2 3 5009 (2) - (x y - 12 x - 12)(2 y + 4 x + 1) 5010 Type: Factored(Polynomial(Integer)) 5011\end{verbatim} 5012\indentrel{-3}\end{paste}\end{patch} 5013 5014\begin{patch}{ugProblemFactorIntRatPageEmpty2} 5015\begin{paste}{ugProblemFactorIntRatPageEmpty2}{ugProblemFactorIntRatPagePatch2} 5016\pastebutton{ugProblemFactorIntRatPageEmpty2}{\showpaste} 5017\tab{5}\spadcommand{factor v\free{v }} 5018\end{paste}\end{patch} 5019 5020\begin{patch}{ugProblemFactorIntRatPagePatch3} 5021\begin{paste}{ugProblemFactorIntRatPageFull3}{ugProblemFactorIntRatPageEmpty3} 5022\pastebutton{ugProblemFactorIntRatPageFull3}{\hidepaste} 5023\tab{5}\spadcommand{w := (4*x^3+(2/3)*x^2+1)*(12*x^5-(1/2)*x^3+12)\bound{w }} 5024\indentrel{3}\begin{verbatim} 5025 8 7 6 35 5 95 3 2 5026 (3) 48 x + 8 x - 2 x + -- x + -- x + 8 x + 12 5027 3 2 5028 Type: Polynomial(Fraction(Integer)) 5029\end{verbatim} 5030\indentrel{-3}\end{paste}\end{patch} 5031 5032\begin{patch}{ugProblemFactorIntRatPageEmpty3} 5033\begin{paste}{ugProblemFactorIntRatPageEmpty3}{ugProblemFactorIntRatPagePatch3} 5034\pastebutton{ugProblemFactorIntRatPageEmpty3}{\showpaste} 5035\tab{5}\spadcommand{w := (4*x^3+(2/3)*x^2+1)*(12*x^5-(1/2)*x^3+12)\bound{w }} 5036\end{paste}\end{patch} 5037 5038\begin{patch}{ugProblemFactorIntRatPagePatch4} 5039\begin{paste}{ugProblemFactorIntRatPageFull4}{ugProblemFactorIntRatPageEmpty4} 5040\pastebutton{ugProblemFactorIntRatPageFull4}{\hidepaste} 5041\tab{5}\spadcommand{factor w\free{w }} 5042\indentrel{3}\begin{verbatim} 5043 3 1 2 1 5 1 3 5044 (4) 48 (x + - x + -)(x - -- x + 1) 5045 6 4 24 5046 Type: Factored(Polynomial(Fraction(Integer))) 5047\end{verbatim} 5048\indentrel{-3}\end{paste}\end{patch} 5049 5050\begin{patch}{ugProblemFactorIntRatPageEmpty4} 5051\begin{paste}{ugProblemFactorIntRatPageEmpty4}{ugProblemFactorIntRatPagePatch4} 5052\pastebutton{ugProblemFactorIntRatPageEmpty4}{\showpaste} 5053\tab{5}\spadcommand{factor w\free{w }} 5054\end{paste}\end{patch} 5055 5056\begin{patch}{ugxProblemFinitePrimePagePatch1} 5057\begin{paste}{ugxProblemFinitePrimePageFull1}{ugxProblemFinitePrimePageEmpty1} 5058\pastebutton{ugxProblemFinitePrimePageFull1}{\hidepaste} 5059\tab{5}\spadcommand{(a,b) : IntegerMod 12\bound{abdec }} 5060\indentrel{3}\begin{verbatim} 5061 Type: Void 5062\end{verbatim} 5063\indentrel{-3}\end{paste}\end{patch} 5064 5065\begin{patch}{ugxProblemFinitePrimePageEmpty1} 5066\begin{paste}{ugxProblemFinitePrimePageEmpty1}{ugxProblemFinitePrimePagePatch1} 5067\pastebutton{ugxProblemFinitePrimePageEmpty1}{\showpaste} 5068\tab{5}\spadcommand{(a,b) : IntegerMod 12\bound{abdec }} 5069\end{paste}\end{patch} 5070 5071\begin{patch}{ugxProblemFinitePrimePagePatch2} 5072\begin{paste}{ugxProblemFinitePrimePageFull2}{ugxProblemFinitePrimePageEmpty2} 5073\pastebutton{ugxProblemFinitePrimePageFull2}{\hidepaste} 5074\tab{5}\spadcommand{(a, b) := (16, 7)\free{abdec }\bound{a b }} 5075\indentrel{3}\begin{verbatim} 5076 (2) 7 5077 Type: IntegerMod(12) 5078\end{verbatim} 5079\indentrel{-3}\end{paste}\end{patch} 5080 5081\begin{patch}{ugxProblemFinitePrimePageEmpty2} 5082\begin{paste}{ugxProblemFinitePrimePageEmpty2}{ugxProblemFinitePrimePagePatch2} 5083\pastebutton{ugxProblemFinitePrimePageEmpty2}{\showpaste} 5084\tab{5}\spadcommand{(a, b) := (16, 7)\free{abdec }\bound{a b }} 5085\end{paste}\end{patch} 5086 5087\begin{patch}{ugxProblemFinitePrimePagePatch3} 5088\begin{paste}{ugxProblemFinitePrimePageFull3}{ugxProblemFinitePrimePageEmpty3} 5089\pastebutton{ugxProblemFinitePrimePageFull3}{\hidepaste} 5090\tab{5}\spadcommand{[a - b, a * b]\free{a b }} 5091\indentrel{3}\begin{verbatim} 5092 (3) [9, 4] 5093 Type: List(IntegerMod(12)) 5094\end{verbatim} 5095\indentrel{-3}\end{paste}\end{patch} 5096 5097\begin{patch}{ugxProblemFinitePrimePageEmpty3} 5098\begin{paste}{ugxProblemFinitePrimePageEmpty3}{ugxProblemFinitePrimePagePatch3} 5099\pastebutton{ugxProblemFinitePrimePageEmpty3}{\showpaste} 5100\tab{5}\spadcommand{[a - b, a * b]\free{a b }} 5101\end{paste}\end{patch} 5102 5103\begin{patch}{ugxProblemFinitePrimePagePatch4} 5104\begin{paste}{ugxProblemFinitePrimePageFull4}{ugxProblemFinitePrimePageEmpty4} 5105\pastebutton{ugxProblemFinitePrimePageFull4}{\hidepaste} 5106\tab{5}\spadcommand{a / b\free{a b }} 5107\indentrel{3}\begin{verbatim} 5108error 5109\end{verbatim} 5110\indentrel{-3}\end{paste}\end{patch} 5111 5112\begin{patch}{ugxProblemFinitePrimePageEmpty4} 5113\begin{paste}{ugxProblemFinitePrimePageEmpty4}{ugxProblemFinitePrimePagePatch4} 5114\pastebutton{ugxProblemFinitePrimePageEmpty4}{\showpaste} 5115\tab{5}\spadcommand{a / b\free{a b }} 5116\end{paste}\end{patch} 5117 5118\begin{patch}{ugxProblemFinitePrimePagePatch5} 5119\begin{paste}{ugxProblemFinitePrimePageFull5}{ugxProblemFinitePrimePageEmpty5} 5120\pastebutton{ugxProblemFinitePrimePageFull5}{\hidepaste} 5121\tab{5}\spadcommand{recip a\free{a }} 5122\indentrel{3}\begin{verbatim} 5123 (4) "failed" 5124 Type: Union("failed",...) 5125\end{verbatim} 5126\indentrel{-3}\end{paste}\end{patch} 5127 5128\begin{patch}{ugxProblemFinitePrimePageEmpty5} 5129\begin{paste}{ugxProblemFinitePrimePageEmpty5}{ugxProblemFinitePrimePagePatch5} 5130\pastebutton{ugxProblemFinitePrimePageEmpty5}{\showpaste} 5131\tab{5}\spadcommand{recip a\free{a }} 5132\end{paste}\end{patch} 5133 5134\begin{patch}{ugxProblemFinitePrimePagePatch6} 5135\begin{paste}{ugxProblemFinitePrimePageFull6}{ugxProblemFinitePrimePageEmpty6} 5136\pastebutton{ugxProblemFinitePrimePageFull6}{\hidepaste} 5137\tab{5}\spadcommand{recip b\free{b }} 5138\indentrel{3}\begin{verbatim} 5139 (5) 7 5140 Type: Union(IntegerMod(12),...) 5141\end{verbatim} 5142\indentrel{-3}\end{paste}\end{patch} 5143 5144\begin{patch}{ugxProblemFinitePrimePageEmpty6} 5145\begin{paste}{ugxProblemFinitePrimePageEmpty6}{ugxProblemFinitePrimePagePatch6} 5146\pastebutton{ugxProblemFinitePrimePageEmpty6}{\showpaste} 5147\tab{5}\spadcommand{recip b\free{b }} 5148\end{paste}\end{patch} 5149 5150\begin{patch}{ugxProblemFinitePrimePagePatch7} 5151\begin{paste}{ugxProblemFinitePrimePageFull7}{ugxProblemFinitePrimePageEmpty7} 5152\pastebutton{ugxProblemFinitePrimePageFull7}{\hidepaste} 5153\tab{5}\spadcommand{c : PrimeField 11 := 8\bound{c }} 5154\indentrel{3}\begin{verbatim} 5155 (6) 8 5156 Type: PrimeField(11) 5157\end{verbatim} 5158\indentrel{-3}\end{paste}\end{patch} 5159 5160\begin{patch}{ugxProblemFinitePrimePageEmpty7} 5161\begin{paste}{ugxProblemFinitePrimePageEmpty7}{ugxProblemFinitePrimePagePatch7} 5162\pastebutton{ugxProblemFinitePrimePageEmpty7}{\showpaste} 5163\tab{5}\spadcommand{c : PrimeField 11 := 8\bound{c }} 5164\end{paste}\end{patch} 5165 5166\begin{patch}{ugxProblemFinitePrimePagePatch8} 5167\begin{paste}{ugxProblemFinitePrimePageFull8}{ugxProblemFinitePrimePageEmpty8} 5168\pastebutton{ugxProblemFinitePrimePageFull8}{\hidepaste} 5169\tab{5}\spadcommand{inv c\free{c }} 5170\indentrel{3}\begin{verbatim} 5171 (7) 7 5172 Type: PrimeField(11) 5173\end{verbatim} 5174\indentrel{-3}\end{paste}\end{patch} 5175 5176\begin{patch}{ugxProblemFinitePrimePageEmpty8} 5177\begin{paste}{ugxProblemFinitePrimePageEmpty8}{ugxProblemFinitePrimePagePatch8} 5178\pastebutton{ugxProblemFinitePrimePageEmpty8}{\showpaste} 5179\tab{5}\spadcommand{inv c\free{c }} 5180\end{paste}\end{patch} 5181 5182\begin{patch}{ugxProblemFinitePrimePagePatch9} 5183\begin{paste}{ugxProblemFinitePrimePageFull9}{ugxProblemFinitePrimePageEmpty9} 5184\pastebutton{ugxProblemFinitePrimePageFull9}{\hidepaste} 5185\tab{5}\spadcommand{9/c\free{c }} 5186\indentrel{3}\begin{verbatim} 5187 (8) 8 5188 Type: PrimeField(11) 5189\end{verbatim} 5190\indentrel{-3}\end{paste}\end{patch} 5191 5192\begin{patch}{ugxProblemFinitePrimePageEmpty9} 5193\begin{paste}{ugxProblemFinitePrimePageEmpty9}{ugxProblemFinitePrimePagePatch9} 5194\pastebutton{ugxProblemFinitePrimePageEmpty9}{\showpaste} 5195\tab{5}\spadcommand{9/c\free{c }} 5196\end{paste}\end{patch} 5197 5198\begin{patch}{ugxProblemFinitePrimePagePatch10} 5199\begin{paste}{ugxProblemFinitePrimePageFull10}{ugxProblemFinitePrimePageEmpty10} 5200\pastebutton{ugxProblemFinitePrimePageFull10}{\hidepaste} 5201\tab{5}\spadcommand{GF101 := PF 101\bound{GF101 }} 5202\indentrel{3}\begin{verbatim} 5203 (9) PrimeField(101) 5204 Type: Type 5205\end{verbatim} 5206\indentrel{-3}\end{paste}\end{patch} 5207 5208\begin{patch}{ugxProblemFinitePrimePageEmpty10} 5209\begin{paste}{ugxProblemFinitePrimePageEmpty10}{ugxProblemFinitePrimePagePatch10} 5210\pastebutton{ugxProblemFinitePrimePageEmpty10}{\showpaste} 5211\tab{5}\spadcommand{GF101 := PF 101\bound{GF101 }} 5212\end{paste}\end{patch} 5213 5214\begin{patch}{ugxProblemFinitePrimePagePatch11} 5215\begin{paste}{ugxProblemFinitePrimePageFull11}{ugxProblemFinitePrimePageEmpty11} 5216\pastebutton{ugxProblemFinitePrimePageFull11}{\hidepaste} 5217\tab{5}\spadcommand{x := random()$GF101\bound{x }\free{GF101 }} 5218\indentrel{3}\begin{verbatim} 5219 (10) 96 5220 Type: PrimeField(101) 5221\end{verbatim} 5222\indentrel{-3}\end{paste}\end{patch} 5223 5224\begin{patch}{ugxProblemFinitePrimePageEmpty11} 5225\begin{paste}{ugxProblemFinitePrimePageEmpty11}{ugxProblemFinitePrimePagePatch11} 5226\pastebutton{ugxProblemFinitePrimePageEmpty11}{\showpaste} 5227\tab{5}\spadcommand{x := random()$GF101\bound{x }\free{GF101 }} 5228\end{paste}\end{patch} 5229 5230\begin{patch}{ugxProblemFinitePrimePagePatch12} 5231\begin{paste}{ugxProblemFinitePrimePageFull12}{ugxProblemFinitePrimePageEmpty12} 5232\pastebutton{ugxProblemFinitePrimePageFull12}{\hidepaste} 5233\tab{5}\spadcommand{y : GF101 := 37\bound{y }\free{GF101 }} 5234\indentrel{3}\begin{verbatim} 5235 (11) 37 5236 Type: PrimeField(101) 5237\end{verbatim} 5238\indentrel{-3}\end{paste}\end{patch} 5239 5240\begin{patch}{ugxProblemFinitePrimePageEmpty12} 5241\begin{paste}{ugxProblemFinitePrimePageEmpty12}{ugxProblemFinitePrimePagePatch12} 5242\pastebutton{ugxProblemFinitePrimePageEmpty12}{\showpaste} 5243\tab{5}\spadcommand{y : GF101 := 37\bound{y }\free{GF101 }} 5244\end{paste}\end{patch} 5245 5246\begin{patch}{ugxProblemFinitePrimePagePatch13} 5247\begin{paste}{ugxProblemFinitePrimePageFull13}{ugxProblemFinitePrimePageEmpty13} 5248\pastebutton{ugxProblemFinitePrimePageFull13}{\hidepaste} 5249\tab{5}\spadcommand{z := x/y\bound{z }\free{x y }} 5250\indentrel{3}\begin{verbatim} 5251 (12) 49 5252 Type: PrimeField(101) 5253\end{verbatim} 5254\indentrel{-3}\end{paste}\end{patch} 5255 5256\begin{patch}{ugxProblemFinitePrimePageEmpty13} 5257\begin{paste}{ugxProblemFinitePrimePageEmpty13}{ugxProblemFinitePrimePagePatch13} 5258\pastebutton{ugxProblemFinitePrimePageEmpty13}{\showpaste} 5259\tab{5}\spadcommand{z := x/y\bound{z }\free{x y }} 5260\end{paste}\end{patch} 5261 5262\begin{patch}{ugxProblemFinitePrimePagePatch14} 5263\begin{paste}{ugxProblemFinitePrimePageFull14}{ugxProblemFinitePrimePageEmpty14} 5264\pastebutton{ugxProblemFinitePrimePageFull14}{\hidepaste} 5265\tab{5}\spadcommand{z * y - x\free{x y z }} 5266\indentrel{3}\begin{verbatim} 5267 (13) 0 5268 Type: PrimeField(101) 5269\end{verbatim} 5270\indentrel{-3}\end{paste}\end{patch} 5271 5272\begin{patch}{ugxProblemFinitePrimePageEmpty14} 5273\begin{paste}{ugxProblemFinitePrimePageEmpty14}{ugxProblemFinitePrimePagePatch14} 5274\pastebutton{ugxProblemFinitePrimePageEmpty14}{\showpaste} 5275\tab{5}\spadcommand{z * y - x\free{x y z }} 5276\end{paste}\end{patch} 5277 5278\begin{patch}{ugxProblemFinitePrimePagePatch15} 5279\begin{paste}{ugxProblemFinitePrimePageFull15}{ugxProblemFinitePrimePageEmpty15} 5280\pastebutton{ugxProblemFinitePrimePageFull15}{\hidepaste} 5281\tab{5}\spadcommand{pe := primitiveElement()$GF101\bound{pe }\free{GF101 }} 5282\indentrel{3}\begin{verbatim} 5283 (14) 2 5284 Type: PrimeField(101) 5285\end{verbatim} 5286\indentrel{-3}\end{paste}\end{patch} 5287 5288\begin{patch}{ugxProblemFinitePrimePageEmpty15} 5289\begin{paste}{ugxProblemFinitePrimePageEmpty15}{ugxProblemFinitePrimePagePatch15} 5290\pastebutton{ugxProblemFinitePrimePageEmpty15}{\showpaste} 5291\tab{5}\spadcommand{pe := primitiveElement()$GF101\bound{pe }\free{GF101 }} 5292\end{paste}\end{patch} 5293 5294\begin{patch}{ugxProblemFinitePrimePagePatch16} 5295\begin{paste}{ugxProblemFinitePrimePageFull16}{ugxProblemFinitePrimePageEmpty16} 5296\pastebutton{ugxProblemFinitePrimePageFull16}{\hidepaste} 5297\tab{5}\spadcommand{[pe^i for i in 0..99]\free{pe }} 5298\indentrel{3}\begin{verbatim} 5299 (15) 5300 [1, 2, 4, 8, 16, 32, 64, 27, 54, 7, 14, 28, 56, 11, 5301 22, 44, 88, 75, 49, 98, 95, 89, 77, 53, 5, 10, 20, 5302 40, 80, 59, 17, 34, 68, 35, 70, 39, 78, 55, 9, 18, 5303 36, 72, 43, 86, 71, 41, 82, 63, 25, 50, 100, 99, 97, 5304 93, 85, 69, 37, 74, 47, 94, 87, 73, 45, 90, 79, 57, 5305 13, 26, 52, 3, 6, 12, 24, 48, 96, 91, 81, 61, 21, 42, 5306 84, 67, 33, 66, 31, 62, 23, 46, 92, 83, 65, 29, 58, 5307 15, 30, 60, 19, 38, 76, 51] 5308 Type: List(PrimeField(101)) 5309\end{verbatim} 5310\indentrel{-3}\end{paste}\end{patch} 5311 5312\begin{patch}{ugxProblemFinitePrimePageEmpty16} 5313\begin{paste}{ugxProblemFinitePrimePageEmpty16}{ugxProblemFinitePrimePagePatch16} 5314\pastebutton{ugxProblemFinitePrimePageEmpty16}{\showpaste} 5315\tab{5}\spadcommand{[pe^i for i in 0..99]\free{pe }} 5316\end{paste}\end{patch} 5317 5318\begin{patch}{ugxProblemFinitePrimePagePatch17} 5319\begin{paste}{ugxProblemFinitePrimePageFull17}{ugxProblemFinitePrimePageEmpty17} 5320\pastebutton{ugxProblemFinitePrimePageFull17}{\hidepaste} 5321\tab{5}\spadcommand{ex := discreteLog(y)\bound{ex }\free{y }} 5322\indentrel{3}\begin{verbatim} 5323 (16) 56 5324 Type: PositiveInteger 5325\end{verbatim} 5326\indentrel{-3}\end{paste}\end{patch} 5327 5328\begin{patch}{ugxProblemFinitePrimePageEmpty17} 5329\begin{paste}{ugxProblemFinitePrimePageEmpty17}{ugxProblemFinitePrimePagePatch17} 5330\pastebutton{ugxProblemFinitePrimePageEmpty17}{\showpaste} 5331\tab{5}\spadcommand{ex := discreteLog(y)\bound{ex }\free{y }} 5332\end{paste}\end{patch} 5333 5334\begin{patch}{ugxProblemFinitePrimePagePatch18} 5335\begin{paste}{ugxProblemFinitePrimePageFull18}{ugxProblemFinitePrimePageEmpty18} 5336\pastebutton{ugxProblemFinitePrimePageFull18}{\hidepaste} 5337\tab{5}\spadcommand{pe ^ ex\free{ex pe }} 5338\indentrel{3}\begin{verbatim} 5339 (17) 37 5340 Type: PrimeField(101) 5341\end{verbatim} 5342\indentrel{-3}\end{paste}\end{patch} 5343 5344\begin{patch}{ugxProblemFinitePrimePageEmpty18} 5345\begin{paste}{ugxProblemFinitePrimePageEmpty18}{ugxProblemFinitePrimePagePatch18} 5346\pastebutton{ugxProblemFinitePrimePageEmpty18}{\showpaste} 5347\tab{5}\spadcommand{pe ^ ex\free{ex pe }} 5348\end{paste}\end{patch} 5349 5350\begin{patch}{ugxProblemFinitePrimePagePatch19} 5351\begin{paste}{ugxProblemFinitePrimePageFull19}{ugxProblemFinitePrimePageEmpty19} 5352\pastebutton{ugxProblemFinitePrimePageFull19}{\hidepaste} 5353\tab{5}\spadcommand{order y\free{y }} 5354\indentrel{3}\begin{verbatim} 5355 (18) 25 5356 Type: PositiveInteger 5357\end{verbatim} 5358\indentrel{-3}\end{paste}\end{patch} 5359 5360\begin{patch}{ugxProblemFinitePrimePageEmpty19} 5361\begin{paste}{ugxProblemFinitePrimePageEmpty19}{ugxProblemFinitePrimePagePatch19} 5362\pastebutton{ugxProblemFinitePrimePageEmpty19}{\showpaste} 5363\tab{5}\spadcommand{order y\free{y }} 5364\end{paste}\end{patch} 5365 5366\begin{patch}{ugxProblemFinitePrimePagePatch20} 5367\begin{paste}{ugxProblemFinitePrimePageFull20}{ugxProblemFinitePrimePageEmpty20} 5368\pastebutton{ugxProblemFinitePrimePageFull20}{\hidepaste} 5369\tab{5}\spadcommand{order pe\free{pe }} 5370\indentrel{3}\begin{verbatim} 5371 (19) 100 5372 Type: PositiveInteger 5373\end{verbatim} 5374\indentrel{-3}\end{paste}\end{patch} 5375 5376\begin{patch}{ugxProblemFinitePrimePageEmpty20} 5377\begin{paste}{ugxProblemFinitePrimePageEmpty20}{ugxProblemFinitePrimePagePatch20} 5378\pastebutton{ugxProblemFinitePrimePageEmpty20}{\showpaste} 5379\tab{5}\spadcommand{order pe\free{pe }} 5380\end{paste}\end{patch} 5381 5382\begin{patch}{ugxProblemFiniteCyclicPagePatch1} 5383\begin{paste}{ugxProblemFiniteCyclicPageFull1}{ugxProblemFiniteCyclicPageEmpty1} 5384\pastebutton{ugxProblemFiniteCyclicPageFull1}{\hidepaste} 5385\tab{5}\spadcommand{GF81 := FFCG(3,4);\bound{GF81 }} 5386\indentrel{3}\begin{verbatim} 5387 Type: Type 5388\end{verbatim} 5389\indentrel{-3}\end{paste}\end{patch} 5390 5391\begin{patch}{ugxProblemFiniteCyclicPageEmpty1} 5392\begin{paste}{ugxProblemFiniteCyclicPageEmpty1}{ugxProblemFiniteCyclicPagePatch1} 5393\pastebutton{ugxProblemFiniteCyclicPageEmpty1}{\showpaste} 5394\tab{5}\spadcommand{GF81 := FFCG(3,4);\bound{GF81 }} 5395\end{paste}\end{patch} 5396 5397\begin{patch}{ugxProblemFiniteCyclicPagePatch2} 5398\begin{paste}{ugxProblemFiniteCyclicPageFull2}{ugxProblemFiniteCyclicPageEmpty2} 5399\pastebutton{ugxProblemFiniteCyclicPageFull2}{\hidepaste} 5400\tab{5}\spadcommand{a := primitiveElement()$GF81\bound{a }\free{GF81 }} 5401\indentrel{3}\begin{verbatim} 5402 1 5403 (2) %BG 5404 Type: FiniteFieldCyclicGroup(3,4) 5405\end{verbatim} 5406\indentrel{-3}\end{paste}\end{patch} 5407 5408\begin{patch}{ugxProblemFiniteCyclicPageEmpty2} 5409\begin{paste}{ugxProblemFiniteCyclicPageEmpty2}{ugxProblemFiniteCyclicPagePatch2} 5410\pastebutton{ugxProblemFiniteCyclicPageEmpty2}{\showpaste} 5411\tab{5}\spadcommand{a := primitiveElement()$GF81\bound{a }\free{GF81 }} 5412\end{paste}\end{patch} 5413 5414\begin{patch}{ugxProblemFiniteCyclicPagePatch3} 5415\begin{paste}{ugxProblemFiniteCyclicPageFull3}{ugxProblemFiniteCyclicPageEmpty3} 5416\pastebutton{ugxProblemFiniteCyclicPageFull3}{\hidepaste} 5417\tab{5}\spadcommand{b := a^12 - a^5 + a\bound{b }\free{a }} 5418\indentrel{3}\begin{verbatim} 5419 72 5420 (3) %BG 5421 Type: FiniteFieldCyclicGroup(3,4) 5422\end{verbatim} 5423\indentrel{-3}\end{paste}\end{patch} 5424 5425\begin{patch}{ugxProblemFiniteCyclicPageEmpty3} 5426\begin{paste}{ugxProblemFiniteCyclicPageEmpty3}{ugxProblemFiniteCyclicPagePatch3} 5427\pastebutton{ugxProblemFiniteCyclicPageEmpty3}{\showpaste} 5428\tab{5}\spadcommand{b := a^12 - a^5 + a\bound{b }\free{a }} 5429\end{paste}\end{patch} 5430 5431\begin{patch}{ugxProblemFiniteCyclicPagePatch4} 5432\begin{paste}{ugxProblemFiniteCyclicPageFull4}{ugxProblemFiniteCyclicPageEmpty4} 5433\pastebutton{ugxProblemFiniteCyclicPageFull4}{\hidepaste} 5434\tab{5}\spadcommand{b\free{b }} 5435\indentrel{3}\begin{verbatim} 5436 72 5437 (4) %BG 5438 Type: FiniteFieldCyclicGroup(3,4) 5439\end{verbatim} 5440\indentrel{-3}\end{paste}\end{patch} 5441 5442\begin{patch}{ugxProblemFiniteCyclicPageEmpty4} 5443\begin{paste}{ugxProblemFiniteCyclicPageEmpty4}{ugxProblemFiniteCyclicPagePatch4} 5444\pastebutton{ugxProblemFiniteCyclicPageEmpty4}{\showpaste} 5445\tab{5}\spadcommand{b\free{b }} 5446\end{paste}\end{patch} 5447 5448\begin{patch}{ugxProblemFiniteCyclicPagePatch5} 5449\begin{paste}{ugxProblemFiniteCyclicPageFull5}{ugxProblemFiniteCyclicPageEmpty5} 5450\pastebutton{ugxProblemFiniteCyclicPageFull5}{\hidepaste} 5451\tab{5}\spadcommand{discreteLog b\free{b }} 5452\indentrel{3}\begin{verbatim} 5453 (5) 72 5454 Type: PositiveInteger 5455\end{verbatim} 5456\indentrel{-3}\end{paste}\end{patch} 5457 5458\begin{patch}{ugxProblemFiniteCyclicPageEmpty5} 5459\begin{paste}{ugxProblemFiniteCyclicPageEmpty5}{ugxProblemFiniteCyclicPagePatch5} 5460\pastebutton{ugxProblemFiniteCyclicPageEmpty5}{\showpaste} 5461\tab{5}\spadcommand{discreteLog b\free{b }} 5462\end{paste}\end{patch} 5463 5464\begin{patch}{ugxProblemFiniteCyclicPagePatch6} 5465\begin{paste}{ugxProblemFiniteCyclicPageFull6}{ugxProblemFiniteCyclicPageEmpty6} 5466\pastebutton{ugxProblemFiniteCyclicPageFull6}{\hidepaste} 5467\tab{5}\spadcommand{GF9 := FF(3,2);\bound{GF9 }} 5468\indentrel{3}\begin{verbatim} 5469 Type: Type 5470\end{verbatim} 5471\indentrel{-3}\end{paste}\end{patch} 5472 5473\begin{patch}{ugxProblemFiniteCyclicPageEmpty6} 5474\begin{paste}{ugxProblemFiniteCyclicPageEmpty6}{ugxProblemFiniteCyclicPagePatch6} 5475\pastebutton{ugxProblemFiniteCyclicPageEmpty6}{\showpaste} 5476\tab{5}\spadcommand{GF9 := FF(3,2);\bound{GF9 }} 5477\end{paste}\end{patch} 5478 5479\begin{patch}{ugxProblemFiniteCyclicPagePatch7} 5480\begin{paste}{ugxProblemFiniteCyclicPageFull7}{ugxProblemFiniteCyclicPageEmpty7} 5481\pastebutton{ugxProblemFiniteCyclicPageFull7}{\hidepaste} 5482\tab{5}\spadcommand{GF729 := FFCGX(GF9,3);\bound{GF729 }\free{GF9 }} 5483\indentrel{3}\begin{verbatim} 5484 Type: Type 5485\end{verbatim} 5486\indentrel{-3}\end{paste}\end{patch} 5487 5488\begin{patch}{ugxProblemFiniteCyclicPageEmpty7} 5489\begin{paste}{ugxProblemFiniteCyclicPageEmpty7}{ugxProblemFiniteCyclicPagePatch7} 5490\pastebutton{ugxProblemFiniteCyclicPageEmpty7}{\showpaste} 5491\tab{5}\spadcommand{GF729 := FFCGX(GF9,3);\bound{GF729 }\free{GF9 }} 5492\end{paste}\end{patch} 5493 5494\begin{patch}{ugxProblemFiniteCyclicPagePatch8} 5495\begin{paste}{ugxProblemFiniteCyclicPageFull8}{ugxProblemFiniteCyclicPageEmpty8} 5496\pastebutton{ugxProblemFiniteCyclicPageFull8}{\hidepaste} 5497\tab{5}\spadcommand{r := (random()$GF729) ^ 20\free{GF729 }\bound{r }} 5498\indentrel{3}\begin{verbatim} 5499 (8) 0 5500Type: FiniteFieldCyclicGroupExtension(FiniteField(3,2),3) 5501\end{verbatim} 5502\indentrel{-3}\end{paste}\end{patch} 5503 5504\begin{patch}{ugxProblemFiniteCyclicPageEmpty8} 5505\begin{paste}{ugxProblemFiniteCyclicPageEmpty8}{ugxProblemFiniteCyclicPagePatch8} 5506\pastebutton{ugxProblemFiniteCyclicPageEmpty8}{\showpaste} 5507\tab{5}\spadcommand{r := (random()$GF729) ^ 20\free{GF729 }\bound{r }} 5508\end{paste}\end{patch} 5509 5510\begin{patch}{ugxProblemFiniteCyclicPagePatch9} 5511\begin{paste}{ugxProblemFiniteCyclicPageFull9}{ugxProblemFiniteCyclicPageEmpty9} 5512\pastebutton{ugxProblemFiniteCyclicPageFull9}{\hidepaste} 5513\tab{5}\spadcommand{trace(r)\free{r }} 5514\indentrel{3}\begin{verbatim} 5515 (9) 0 5516 Type: FiniteField(3,2) 5517\end{verbatim} 5518\indentrel{-3}\end{paste}\end{patch} 5519 5520\begin{patch}{ugxProblemFiniteCyclicPageEmpty9} 5521\begin{paste}{ugxProblemFiniteCyclicPageEmpty9}{ugxProblemFiniteCyclicPagePatch9} 5522\pastebutton{ugxProblemFiniteCyclicPageEmpty9}{\showpaste} 5523\tab{5}\spadcommand{trace(r)\free{r }} 5524\end{paste}\end{patch} 5525 5526\begin{patch}{ugxProblemFiniteCyclicPagePatch10} 5527\begin{paste}{ugxProblemFiniteCyclicPageFull10}{ugxProblemFiniteCyclicPageEmpty10} 5528\pastebutton{ugxProblemFiniteCyclicPageFull10}{\hidepaste} 5529\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }} 5530\indentrel{3}\begin{verbatim} 5531 Type: Type 5532\end{verbatim} 5533\indentrel{-3}\end{paste}\end{patch} 5534 5535\begin{patch}{ugxProblemFiniteCyclicPageEmpty10} 5536\begin{paste}{ugxProblemFiniteCyclicPageEmpty10}{ugxProblemFiniteCyclicPagePatch10} 5537\pastebutton{ugxProblemFiniteCyclicPageEmpty10}{\showpaste} 5538\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }} 5539\end{paste}\end{patch} 5540 5541\begin{patch}{ugxProblemFiniteCyclicPagePatch11} 5542\begin{paste}{ugxProblemFiniteCyclicPageFull11}{ugxProblemFiniteCyclicPageEmpty11} 5543\pastebutton{ugxProblemFiniteCyclicPageFull11}{\hidepaste} 5544\tab{5}\spadcommand{f := createPrimitivePoly(4)$FFPOLY(GF3)\bound{f }\free{GF3 }} 5545\indentrel{3}\begin{verbatim} 5546 4 5547 (11) ? + ? + 2 5548 Type: SparseUnivariatePolynomial(PrimeField(3)) 5549\end{verbatim} 5550\indentrel{-3}\end{paste}\end{patch} 5551 5552\begin{patch}{ugxProblemFiniteCyclicPageEmpty11} 5553\begin{paste}{ugxProblemFiniteCyclicPageEmpty11}{ugxProblemFiniteCyclicPagePatch11} 5554\pastebutton{ugxProblemFiniteCyclicPageEmpty11}{\showpaste} 5555\tab{5}\spadcommand{f := createPrimitivePoly(4)$FFPOLY(GF3)\bound{f }\free{GF3 }} 5556\end{paste}\end{patch} 5557 5558\begin{patch}{ugxProblemFiniteCyclicPagePatch12} 5559\begin{paste}{ugxProblemFiniteCyclicPageFull12}{ugxProblemFiniteCyclicPageEmpty12} 5560\pastebutton{ugxProblemFiniteCyclicPageFull12}{\hidepaste} 5561\tab{5}\spadcommand{GF81 := FFCGP(GF3,f);\bound{GF81x }\free{f GF3 }} 5562\indentrel{3}\begin{verbatim} 5563 Type: Type 5564\end{verbatim} 5565\indentrel{-3}\end{paste}\end{patch} 5566 5567\begin{patch}{ugxProblemFiniteCyclicPageEmpty12} 5568\begin{paste}{ugxProblemFiniteCyclicPageEmpty12}{ugxProblemFiniteCyclicPagePatch12} 5569\pastebutton{ugxProblemFiniteCyclicPageEmpty12}{\showpaste} 5570\tab{5}\spadcommand{GF81 := FFCGP(GF3,f);\bound{GF81x }\free{f GF3 }} 5571\end{paste}\end{patch} 5572 5573\begin{patch}{ugxProblemFiniteCyclicPagePatch13} 5574\begin{paste}{ugxProblemFiniteCyclicPageFull13}{ugxProblemFiniteCyclicPageEmpty13} 5575\pastebutton{ugxProblemFiniteCyclicPageFull13}{\hidepaste} 5576\tab{5}\spadcommand{random()$GF81\free{GF81x }} 5577\indentrel{3}\begin{verbatim} 5578 5 5579 (13) %BG 5580Type: FiniteFieldCyclicGroupExtensionByPolynomial(PrimeField(3),?^4+?+2) 5581\end{verbatim} 5582\indentrel{-3}\end{paste}\end{patch} 5583 5584\begin{patch}{ugxProblemFiniteCyclicPageEmpty13} 5585\begin{paste}{ugxProblemFiniteCyclicPageEmpty13}{ugxProblemFiniteCyclicPagePatch13} 5586\pastebutton{ugxProblemFiniteCyclicPageEmpty13}{\showpaste} 5587\tab{5}\spadcommand{random()$GF81\free{GF81x }} 5588\end{paste}\end{patch} 5589 5590\begin{patch}{ugxProblemSeriesCreatePagePatch1} 5591\begin{paste}{ugxProblemSeriesCreatePageFull1}{ugxProblemSeriesCreatePageEmpty1} 5592\pastebutton{ugxProblemSeriesCreatePageFull1}{\hidepaste} 5593\tab{5}\spadcommand{x := series 'x\bound{x }} 5594\indentrel{3}\begin{verbatim} 5595 (1) x 5596 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 5597\end{verbatim} 5598\indentrel{-3}\end{paste}\end{patch} 5599 5600\begin{patch}{ugxProblemSeriesCreatePageEmpty1} 5601\begin{paste}{ugxProblemSeriesCreatePageEmpty1}{ugxProblemSeriesCreatePagePatch1} 5602\pastebutton{ugxProblemSeriesCreatePageEmpty1}{\showpaste} 5603\tab{5}\spadcommand{x := series 'x\bound{x }} 5604\end{paste}\end{patch} 5605 5606\begin{patch}{ugxProblemSeriesCreatePagePatch2} 5607\begin{paste}{ugxProblemSeriesCreatePageFull2}{ugxProblemSeriesCreatePageEmpty2} 5608\pastebutton{ugxProblemSeriesCreatePageFull2}{\hidepaste} 5609\tab{5}\spadcommand{1/(1 - x - x^2)\free{x }} 5610\indentrel{3}\begin{verbatim} 5611 (2) 5612 2 3 4 5 6 7 5613 1 + x + 2 x + 3 x + 5 x + 8 x + 13 x + 21 x 5614 + 5615 8 9 10 11 5616 34 x + 55 x + 89 x + O(x ) 5617 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 5618\end{verbatim} 5619\indentrel{-3}\end{paste}\end{patch} 5620 5621\begin{patch}{ugxProblemSeriesCreatePageEmpty2} 5622\begin{paste}{ugxProblemSeriesCreatePageEmpty2}{ugxProblemSeriesCreatePagePatch2} 5623\pastebutton{ugxProblemSeriesCreatePageEmpty2}{\showpaste} 5624\tab{5}\spadcommand{1/(1 - x - x^2)\free{x }} 5625\end{paste}\end{patch} 5626 5627\begin{patch}{ugxProblemSeriesCreatePagePatch3} 5628\begin{paste}{ugxProblemSeriesCreatePageFull3}{ugxProblemSeriesCreatePageEmpty3} 5629\pastebutton{ugxProblemSeriesCreatePageFull3}{\hidepaste} 5630\tab{5}\spadcommand{sin(x)\free{x }} 5631\indentrel{3}\begin{verbatim} 5632 (3) 5633 1 3 1 5 1 7 1 9 5634 x - - x + --- x - ---- x + ------ x 5635 6 120 5040 362880 5636 + 5637 1 11 12 5638 - -------- x + O(x ) 5639 39916800 5640 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 5641\end{verbatim} 5642\indentrel{-3}\end{paste}\end{patch} 5643 5644\begin{patch}{ugxProblemSeriesCreatePageEmpty3} 5645\begin{paste}{ugxProblemSeriesCreatePageEmpty3}{ugxProblemSeriesCreatePagePatch3} 5646\pastebutton{ugxProblemSeriesCreatePageEmpty3}{\showpaste} 5647\tab{5}\spadcommand{sin(x)\free{x }} 5648\end{paste}\end{patch} 5649 5650\begin{patch}{ugxProblemSeriesCreatePagePatch4} 5651\begin{paste}{ugxProblemSeriesCreatePageFull4}{ugxProblemSeriesCreatePageEmpty4} 5652\pastebutton{ugxProblemSeriesCreatePageFull4}{\hidepaste} 5653\tab{5}\spadcommand{sin(1 + x)\free{x }} 5654\indentrel{3}\begin{verbatim} 5655 (4) 5656 sin(1) 2 cos(1) 3 sin(1) 4 5657 sin(1) + cos(1)x - ------ x - ------ x + ------ x 5658 2 6 24 5659 + 5660 cos(1) 5 sin(1) 6 cos(1) 7 sin(1) 8 5661 ------ x - ------ x - ------ x + ------ x 5662 120 720 5040 40320 5663 + 5664 cos(1) 9 sin(1) 10 11 5665 ------ x - ------- x + O(x ) 5666 362880 3628800 5667 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 5668\end{verbatim} 5669\indentrel{-3}\end{paste}\end{patch} 5670 5671\begin{patch}{ugxProblemSeriesCreatePageEmpty4} 5672\begin{paste}{ugxProblemSeriesCreatePageEmpty4}{ugxProblemSeriesCreatePagePatch4} 5673\pastebutton{ugxProblemSeriesCreatePageEmpty4}{\showpaste} 5674\tab{5}\spadcommand{sin(1 + x)\free{x }} 5675\end{paste}\end{patch} 5676 5677\begin{patch}{ugxProblemSeriesCreatePagePatch5} 5678\begin{paste}{ugxProblemSeriesCreatePageFull5}{ugxProblemSeriesCreatePageEmpty5} 5679\pastebutton{ugxProblemSeriesCreatePageFull5}{\hidepaste} 5680\tab{5}\spadcommand{sin(a * x)\free{x }} 5681\indentrel{3}\begin{verbatim} 5682 (5) 5683 3 5 7 9 5684 a 3 a 5 a 7 a 9 5685 a x - -- x + --- x - ---- x + ------ x 5686 6 120 5040 362880 5687 + 5688 11 5689 a 11 12 5690 - -------- x + O(x ) 5691 39916800 5692 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 5693\end{verbatim} 5694\indentrel{-3}\end{paste}\end{patch} 5695 5696\begin{patch}{ugxProblemSeriesCreatePageEmpty5} 5697\begin{paste}{ugxProblemSeriesCreatePageEmpty5}{ugxProblemSeriesCreatePagePatch5} 5698\pastebutton{ugxProblemSeriesCreatePageEmpty5}{\showpaste} 5699\tab{5}\spadcommand{sin(a * x)\free{x }} 5700\end{paste}\end{patch} 5701 5702\begin{patch}{ugxProblemSeriesCreatePagePatch6} 5703\begin{paste}{ugxProblemSeriesCreatePageFull6}{ugxProblemSeriesCreatePageEmpty6} 5704\pastebutton{ugxProblemSeriesCreatePageFull6}{\hidepaste} 5705\tab{5}\spadcommand{series(1/log(y),y = 1)} 5706\indentrel{3}\begin{verbatim} 5707 (6) 5708 - 1 1 1 1 2 5709 (y - 1) + - - -- (y - 1) + -- (y - 1) 5710 2 12 24 5711 + 5712 19 3 3 4 863 5 5713 - --- (y - 1) + --- (y - 1) - ----- (y - 1) 5714 720 160 60480 5715 + 5716 275 6 33953 7 8183 8 5717 ----- (y - 1) - ------- (y - 1) + ------- (y - 1) 5718 24192 3628800 1036800 5719 + 5720 3250433 9 10 5721 - --------- (y - 1) + O((y - 1) ) 5722 479001600 5723 Type: UnivariatePuiseuxSeries(Expression(Integer),y,1) 5724\end{verbatim} 5725\indentrel{-3}\end{paste}\end{patch} 5726 5727\begin{patch}{ugxProblemSeriesCreatePageEmpty6} 5728\begin{paste}{ugxProblemSeriesCreatePageEmpty6}{ugxProblemSeriesCreatePagePatch6} 5729\pastebutton{ugxProblemSeriesCreatePageEmpty6}{\showpaste} 5730\tab{5}\spadcommand{series(1/log(y),y = 1)} 5731\end{paste}\end{patch} 5732 5733\begin{patch}{ugxProblemSeriesCreatePagePatch7} 5734\begin{paste}{ugxProblemSeriesCreatePageFull7}{ugxProblemSeriesCreatePageEmpty7} 5735\pastebutton{ugxProblemSeriesCreatePageFull7}{\hidepaste} 5736\tab{5}\spadcommand{y : UTS(FLOAT,'z,0) := exp(z)\bound{y }} 5737\indentrel{3}\begin{verbatim} 5738 (7) 5739 2 3 5740 1.0 + z + 0.5 z + 0.1666666666_6666666667 z 5741 + 5742 4 5743 0.0416666666_6666666666_7 z 5744 + 5745 5 5746 0.0083333333_3333333333_34 z 5747 + 5748 6 5749 0.0013888888_8888888888_89 z 5750 + 5751 7 5752 0.0001984126_9841269841_27 z 5753 + 5754 8 5755 0.0000248015_8730158730_1587 z 5756 + 5757 9 5758 0.0000027557_3192239858_90653 z 5759 + 5760 10 11 5761 0.2755731922_3985890653 E -6 z + O(z ) 5762 Type: UnivariateTaylorSeries(Float,z,0.0) 5763\end{verbatim} 5764\indentrel{-3}\end{paste}\end{patch} 5765 5766\begin{patch}{ugxProblemSeriesCreatePageEmpty7} 5767\begin{paste}{ugxProblemSeriesCreatePageEmpty7}{ugxProblemSeriesCreatePagePatch7} 5768\pastebutton{ugxProblemSeriesCreatePageEmpty7}{\showpaste} 5769\tab{5}\spadcommand{y : UTS(FLOAT,'z,0) := exp(z)\bound{y }} 5770\end{paste}\end{patch} 5771 5772\begin{patch}{ugxProblemSeriesCreatePagePatch8} 5773\begin{paste}{ugxProblemSeriesCreatePageFull8}{ugxProblemSeriesCreatePageEmpty8} 5774\pastebutton{ugxProblemSeriesCreatePageFull8}{\hidepaste} 5775\tab{5}\spadcommand{series(1/factorial(n),n,w = 0)} 5776\indentrel{3}\begin{verbatim} 5777 (8) 5778 1 2 1 3 1 4 1 5 1 6 5779 1 + w + - w + - w + -- w + --- w + --- w 5780 2 6 24 120 720 5781 + 5782 1 7 1 8 1 9 1 10 11 5783 ---- w + ----- w + ------ w + ------- w + O(w ) 5784 5040 40320 362880 3628800 5785 Type: UnivariatePuiseuxSeries(Expression(Integer),w,0) 5786\end{verbatim} 5787\indentrel{-3}\end{paste}\end{patch} 5788 5789\begin{patch}{ugxProblemSeriesCreatePageEmpty8} 5790\begin{paste}{ugxProblemSeriesCreatePageEmpty8}{ugxProblemSeriesCreatePagePatch8} 5791\pastebutton{ugxProblemSeriesCreatePageEmpty8}{\showpaste} 5792\tab{5}\spadcommand{series(1/factorial(n),n,w = 0)} 5793\end{paste}\end{patch} 5794 5795\begin{patch}{ugxProblemFiniteNormalPagePatch1} 5796\begin{paste}{ugxProblemFiniteNormalPageFull1}{ugxProblemFiniteNormalPageEmpty1} 5797\pastebutton{ugxProblemFiniteNormalPageFull1}{\hidepaste} 5798\tab{5}\spadcommand{K := FFNB(3,8)\bound{K }} 5799\indentrel{3}\begin{verbatim} 5800 (1) FiniteFieldNormalBasis(3,8) 5801 Type: Type 5802\end{verbatim} 5803\indentrel{-3}\end{paste}\end{patch} 5804 5805\begin{patch}{ugxProblemFiniteNormalPageEmpty1} 5806\begin{paste}{ugxProblemFiniteNormalPageEmpty1}{ugxProblemFiniteNormalPagePatch1} 5807\pastebutton{ugxProblemFiniteNormalPageEmpty1}{\showpaste} 5808\tab{5}\spadcommand{K := FFNB(3,8)\bound{K }} 5809\end{paste}\end{patch} 5810 5811\begin{patch}{ugxProblemFiniteNormalPagePatch2} 5812\begin{paste}{ugxProblemFiniteNormalPageFull2}{ugxProblemFiniteNormalPageEmpty2} 5813\pastebutton{ugxProblemFiniteNormalPageFull2}{\hidepaste} 5814\tab{5}\spadcommand{a := normalElement()$K\bound{a }\free{K }} 5815\indentrel{3}\begin{verbatim} 5816 (2) %JL 5817 Type: FiniteFieldNormalBasis(3,8) 5818\end{verbatim} 5819\indentrel{-3}\end{paste}\end{patch} 5820 5821\begin{patch}{ugxProblemFiniteNormalPageEmpty2} 5822\begin{paste}{ugxProblemFiniteNormalPageEmpty2}{ugxProblemFiniteNormalPagePatch2} 5823\pastebutton{ugxProblemFiniteNormalPageEmpty2}{\showpaste} 5824\tab{5}\spadcommand{a := normalElement()$K\bound{a }\free{K }} 5825\end{paste}\end{patch} 5826 5827\begin{patch}{ugxProblemFiniteNormalPagePatch3} 5828\begin{paste}{ugxProblemFiniteNormalPageFull3}{ugxProblemFiniteNormalPageEmpty3} 5829\pastebutton{ugxProblemFiniteNormalPageFull3}{\hidepaste} 5830\tab{5}\spadcommand{b := a^12 - a^5 + a\bound{b }\free{a }} 5831\indentrel{3}\begin{verbatim} 5832 7 5 5833 q q q 5834 (3) 2 %JL + %JL + %JL 5835 Type: FiniteFieldNormalBasis(3,8) 5836\end{verbatim} 5837\indentrel{-3}\end{paste}\end{patch} 5838 5839\begin{patch}{ugxProblemFiniteNormalPageEmpty3} 5840\begin{paste}{ugxProblemFiniteNormalPageEmpty3}{ugxProblemFiniteNormalPagePatch3} 5841\pastebutton{ugxProblemFiniteNormalPageEmpty3}{\showpaste} 5842\tab{5}\spadcommand{b := a^12 - a^5 + a\bound{b }\free{a }} 5843\end{paste}\end{patch} 5844 5845\begin{patch}{ugxProblemFiniteNormalPagePatch4} 5846\begin{paste}{ugxProblemFiniteNormalPageFull4}{ugxProblemFiniteNormalPageEmpty4} 5847\pastebutton{ugxProblemFiniteNormalPageFull4}{\hidepaste} 5848\tab{5}\spadcommand{GF9 := FFNB(3,2);\bound{GF9 }} 5849\indentrel{3}\begin{verbatim} 5850 Type: Type 5851\end{verbatim} 5852\indentrel{-3}\end{paste}\end{patch} 5853 5854\begin{patch}{ugxProblemFiniteNormalPageEmpty4} 5855\begin{paste}{ugxProblemFiniteNormalPageEmpty4}{ugxProblemFiniteNormalPagePatch4} 5856\pastebutton{ugxProblemFiniteNormalPageEmpty4}{\showpaste} 5857\tab{5}\spadcommand{GF9 := FFNB(3,2);\bound{GF9 }} 5858\end{paste}\end{patch} 5859 5860\begin{patch}{ugxProblemFiniteNormalPagePatch5} 5861\begin{paste}{ugxProblemFiniteNormalPageFull5}{ugxProblemFiniteNormalPageEmpty5} 5862\pastebutton{ugxProblemFiniteNormalPageFull5}{\hidepaste} 5863\tab{5}\spadcommand{GF729 := FFNBX(GF9,3);\bound{GF729 }\free{GF9 }} 5864\indentrel{3}\begin{verbatim} 5865 Type: Type 5866\end{verbatim} 5867\indentrel{-3}\end{paste}\end{patch} 5868 5869\begin{patch}{ugxProblemFiniteNormalPageEmpty5} 5870\begin{paste}{ugxProblemFiniteNormalPageEmpty5}{ugxProblemFiniteNormalPagePatch5} 5871\pastebutton{ugxProblemFiniteNormalPageEmpty5}{\showpaste} 5872\tab{5}\spadcommand{GF729 := FFNBX(GF9,3);\bound{GF729 }\free{GF9 }} 5873\end{paste}\end{patch} 5874 5875\begin{patch}{ugxProblemFiniteNormalPagePatch6} 5876\begin{paste}{ugxProblemFiniteNormalPageFull6}{ugxProblemFiniteNormalPageEmpty6} 5877\pastebutton{ugxProblemFiniteNormalPageFull6}{\hidepaste} 5878\tab{5}\spadcommand{r := random()$GF729\bound{r }\free{GF729 }} 5879\indentrel{3}\begin{verbatim} 5880 (6) 5881 2 5882 q q q q q 5883 2 %JM %JN + (%JM + 2 %JM)%JN + (%JM + 2 %JM)%JN 5884Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3) 5885\end{verbatim} 5886\indentrel{-3}\end{paste}\end{patch} 5887 5888\begin{patch}{ugxProblemFiniteNormalPageEmpty6} 5889\begin{paste}{ugxProblemFiniteNormalPageEmpty6}{ugxProblemFiniteNormalPagePatch6} 5890\pastebutton{ugxProblemFiniteNormalPageEmpty6}{\showpaste} 5891\tab{5}\spadcommand{r := random()$GF729\bound{r }\free{GF729 }} 5892\end{paste}\end{patch} 5893 5894\begin{patch}{ugxProblemFiniteNormalPagePatch7} 5895\begin{paste}{ugxProblemFiniteNormalPageFull7}{ugxProblemFiniteNormalPageEmpty7} 5896\pastebutton{ugxProblemFiniteNormalPageFull7}{\hidepaste} 5897\tab{5}\spadcommand{r + r^3 + r^9 + r^27\free{r }} 5898\indentrel{3}\begin{verbatim} 5899 2 5900 q q q 5901 (7) (2 %JM + 2 %JM)%JN + %JN + %JN 5902Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3) 5903\end{verbatim} 5904\indentrel{-3}\end{paste}\end{patch} 5905 5906\begin{patch}{ugxProblemFiniteNormalPageEmpty7} 5907\begin{paste}{ugxProblemFiniteNormalPageEmpty7}{ugxProblemFiniteNormalPagePatch7} 5908\pastebutton{ugxProblemFiniteNormalPageEmpty7}{\showpaste} 5909\tab{5}\spadcommand{r + r^3 + r^9 + r^27\free{r }} 5910\end{paste}\end{patch} 5911 5912\begin{patch}{ugxProblemFiniteNormalPagePatch8} 5913\begin{paste}{ugxProblemFiniteNormalPageFull8}{ugxProblemFiniteNormalPageEmpty8} 5914\pastebutton{ugxProblemFiniteNormalPageFull8}{\hidepaste} 5915\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }} 5916\indentrel{3}\begin{verbatim} 5917 Type: Type 5918\end{verbatim} 5919\indentrel{-3}\end{paste}\end{patch} 5920 5921\begin{patch}{ugxProblemFiniteNormalPageEmpty8} 5922\begin{paste}{ugxProblemFiniteNormalPageEmpty8}{ugxProblemFiniteNormalPagePatch8} 5923\pastebutton{ugxProblemFiniteNormalPageEmpty8}{\showpaste} 5924\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }} 5925\end{paste}\end{patch} 5926 5927\begin{patch}{ugxProblemFiniteNormalPagePatch9} 5928\begin{paste}{ugxProblemFiniteNormalPageFull9}{ugxProblemFiniteNormalPageEmpty9} 5929\pastebutton{ugxProblemFiniteNormalPageFull9}{\hidepaste} 5930\tab{5}\spadcommand{f := createNormalPoly(4)$FFPOLY(GF3)\free{GF3 }\bound{f }} 5931\indentrel{3}\begin{verbatim} 5932 4 3 5933 (9) ? + 2 ? + 2 5934 Type: SparseUnivariatePolynomial(PrimeField(3)) 5935\end{verbatim} 5936\indentrel{-3}\end{paste}\end{patch} 5937 5938\begin{patch}{ugxProblemFiniteNormalPageEmpty9} 5939\begin{paste}{ugxProblemFiniteNormalPageEmpty9}{ugxProblemFiniteNormalPagePatch9} 5940\pastebutton{ugxProblemFiniteNormalPageEmpty9}{\showpaste} 5941\tab{5}\spadcommand{f := createNormalPoly(4)$FFPOLY(GF3)\free{GF3 }\bound{f }} 5942\end{paste}\end{patch} 5943 5944\begin{patch}{ugxProblemFiniteNormalPagePatch10} 5945\begin{paste}{ugxProblemFiniteNormalPageFull10}{ugxProblemFiniteNormalPageEmpty10} 5946\pastebutton{ugxProblemFiniteNormalPageFull10}{\hidepaste} 5947\tab{5}\spadcommand{GF81 := FFNBP(GF3,f);\bound{GF81 }\free{f GF3 }} 5948\indentrel{3}\begin{verbatim} 5949 Type: Type 5950\end{verbatim} 5951\indentrel{-3}\end{paste}\end{patch} 5952 5953\begin{patch}{ugxProblemFiniteNormalPageEmpty10} 5954\begin{paste}{ugxProblemFiniteNormalPageEmpty10}{ugxProblemFiniteNormalPagePatch10} 5955\pastebutton{ugxProblemFiniteNormalPageEmpty10}{\showpaste} 5956\tab{5}\spadcommand{GF81 := FFNBP(GF3,f);\bound{GF81 }\free{f GF3 }} 5957\end{paste}\end{patch} 5958 5959\begin{patch}{ugxProblemFiniteNormalPagePatch11} 5960\begin{paste}{ugxProblemFiniteNormalPageFull11}{ugxProblemFiniteNormalPageEmpty11} 5961\pastebutton{ugxProblemFiniteNormalPageFull11}{\hidepaste} 5962\tab{5}\spadcommand{r := random()$GF81\free{GF81 }\bound{r1 }} 5963\indentrel{3}\begin{verbatim} 5964 3 2 5965 q q 5966 (11) 2 %JO + %JO 5967Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField(3),?^4+2*?^3+2) 5968\end{verbatim} 5969\indentrel{-3}\end{paste}\end{patch} 5970 5971\begin{patch}{ugxProblemFiniteNormalPageEmpty11} 5972\begin{paste}{ugxProblemFiniteNormalPageEmpty11}{ugxProblemFiniteNormalPagePatch11} 5973\pastebutton{ugxProblemFiniteNormalPageEmpty11}{\showpaste} 5974\tab{5}\spadcommand{r := random()$GF81\free{GF81 }\bound{r1 }} 5975\end{paste}\end{patch} 5976 5977\begin{patch}{ugxProblemFiniteNormalPagePatch12} 5978\begin{paste}{ugxProblemFiniteNormalPageFull12}{ugxProblemFiniteNormalPageEmpty12} 5979\pastebutton{ugxProblemFiniteNormalPageFull12}{\hidepaste} 5980\tab{5}\spadcommand{r * r^3 * r^9 * r^27\free{r1 }} 5981\indentrel{3}\begin{verbatim} 5982 3 2 5983 q q q 5984 (12) %JO + %JO + %JO + %JO 5985Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField(3),?^4+2*?^3+2) 5986\end{verbatim} 5987\indentrel{-3}\end{paste}\end{patch} 5988 5989\begin{patch}{ugxProblemFiniteNormalPageEmpty12} 5990\begin{paste}{ugxProblemFiniteNormalPageEmpty12}{ugxProblemFiniteNormalPagePatch12} 5991\pastebutton{ugxProblemFiniteNormalPageEmpty12}{\showpaste} 5992\tab{5}\spadcommand{r * r^3 * r^9 * r^27\free{r1 }} 5993\end{paste}\end{patch} 5994 5995\begin{patch}{ugxProblemFiniteNormalPagePatch13} 5996\begin{paste}{ugxProblemFiniteNormalPageFull13}{ugxProblemFiniteNormalPageEmpty13} 5997\pastebutton{ugxProblemFiniteNormalPageFull13}{\hidepaste} 5998\tab{5}\spadcommand{norm r\free{r1 }} 5999\indentrel{3}\begin{verbatim} 6000 (13) 1 6001 Type: PrimeField(3) 6002\end{verbatim} 6003\indentrel{-3}\end{paste}\end{patch} 6004 6005\begin{patch}{ugxProblemFiniteNormalPageEmpty13} 6006\begin{paste}{ugxProblemFiniteNormalPageEmpty13}{ugxProblemFiniteNormalPagePatch13} 6007\pastebutton{ugxProblemFiniteNormalPageEmpty13}{\showpaste} 6008\tab{5}\spadcommand{norm r\free{r1 }} 6009\end{paste}\end{patch} 6010 6011\begin{patch}{ugxProblemSeriesFormulaPagePatch1} 6012\begin{paste}{ugxProblemSeriesFormulaPageFull1}{ugxProblemSeriesFormulaPageEmpty1} 6013\pastebutton{ugxProblemSeriesFormulaPageFull1}{\hidepaste} 6014\tab{5}\spadcommand{series(n +-> 1/factorial(n),x = 0)} 6015\indentrel{3}\begin{verbatim} 6016 (1) 6017 1 2 1 3 1 4 1 5 1 6 6018 1 + x + - x + - x + -- x + --- x + --- x 6019 2 6 24 120 720 6020 + 6021 1 7 1 8 1 9 1 10 11 6022 ---- x + ----- x + ------ x + ------- x + O(x ) 6023 5040 40320 362880 3628800 6024 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 6025\end{verbatim} 6026\indentrel{-3}\end{paste}\end{patch} 6027 6028\begin{patch}{ugxProblemSeriesFormulaPageEmpty1} 6029\begin{paste}{ugxProblemSeriesFormulaPageEmpty1}{ugxProblemSeriesFormulaPagePatch1} 6030\pastebutton{ugxProblemSeriesFormulaPageEmpty1}{\showpaste} 6031\tab{5}\spadcommand{series(n +-> 1/factorial(n),x = 0)} 6032\end{paste}\end{patch} 6033 6034\begin{patch}{ugxProblemSeriesFormulaPagePatch2} 6035\begin{paste}{ugxProblemSeriesFormulaPageFull2}{ugxProblemSeriesFormulaPageEmpty2} 6036\pastebutton{ugxProblemSeriesFormulaPageFull2}{\hidepaste} 6037\tab{5}\spadcommand{series(n +-> (-1)^(n-1)/n,x = 1,1..)} 6038\indentrel{3}\begin{verbatim} 6039 (2) 6040 1 2 1 3 1 4 6041 (x - 1) - - (x - 1) + - (x - 1) - - (x - 1) 6042 2 3 4 6043 + 6044 1 5 1 6 1 7 1 8 6045 - (x - 1) - - (x - 1) + - (x - 1) - - (x - 1) 6046 5 6 7 8 6047 + 6048 1 9 1 10 1 11 12 6049 - (x - 1) - -- (x - 1) + -- (x - 1) + O((x - 1) ) 6050 9 10 11 6051 Type: UnivariatePuiseuxSeries(Expression(Integer),x,1) 6052\end{verbatim} 6053\indentrel{-3}\end{paste}\end{patch} 6054 6055\begin{patch}{ugxProblemSeriesFormulaPageEmpty2} 6056\begin{paste}{ugxProblemSeriesFormulaPageEmpty2}{ugxProblemSeriesFormulaPagePatch2} 6057\pastebutton{ugxProblemSeriesFormulaPageEmpty2}{\showpaste} 6058\tab{5}\spadcommand{series(n +-> (-1)^(n-1)/n,x = 1,1..)} 6059\end{paste}\end{patch} 6060 6061\begin{patch}{ugxProblemSeriesFormulaPagePatch3} 6062\begin{paste}{ugxProblemSeriesFormulaPageFull3}{ugxProblemSeriesFormulaPageEmpty3} 6063\pastebutton{ugxProblemSeriesFormulaPageFull3}{\hidepaste} 6064\tab{5}\spadcommand{series(n +-> (-1)^((n-1)/2)/factorial(n),x = 0,1..,2)} 6065\indentrel{3}\begin{verbatim} 6066 (3) 6067 1 3 1 5 1 7 1 9 6068 x - - x + --- x - ---- x + ------ x 6069 6 120 5040 362880 6070 + 6071 1 11 12 6072 - -------- x + O(x ) 6073 39916800 6074 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 6075\end{verbatim} 6076\indentrel{-3}\end{paste}\end{patch} 6077 6078\begin{patch}{ugxProblemSeriesFormulaPageEmpty3} 6079\begin{paste}{ugxProblemSeriesFormulaPageEmpty3}{ugxProblemSeriesFormulaPagePatch3} 6080\pastebutton{ugxProblemSeriesFormulaPageEmpty3}{\showpaste} 6081\tab{5}\spadcommand{series(n +-> (-1)^((n-1)/2)/factorial(n),x = 0,1..,2)} 6082\end{paste}\end{patch} 6083 6084\begin{patch}{ugxProblemSeriesFormulaPagePatch4} 6085\begin{paste}{ugxProblemSeriesFormulaPageFull4}{ugxProblemSeriesFormulaPageEmpty4} 6086\pastebutton{ugxProblemSeriesFormulaPageFull4}{\hidepaste} 6087\tab{5}\spadcommand{series(n +-> (-1)^((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)} 6088\indentrel{3}\begin{verbatim} 6089 (4) 6090 1 5 7 6091 - - - 6092 3 1 1 3 1 3 1 3 6093 x - - x + --- x - ---- x + ------ x 6094 6 120 5040 362880 6095 + 6096 11 6097 -- 6098 1 3 4 6099 - -------- x + O(x ) 6100 39916800 6101 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 6102\end{verbatim} 6103\indentrel{-3}\end{paste}\end{patch} 6104 6105\begin{patch}{ugxProblemSeriesFormulaPageEmpty4} 6106\begin{paste}{ugxProblemSeriesFormulaPageEmpty4}{ugxProblemSeriesFormulaPagePatch4} 6107\pastebutton{ugxProblemSeriesFormulaPageEmpty4}{\showpaste} 6108\tab{5}\spadcommand{series(n +-> (-1)^((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)} 6109\end{paste}\end{patch} 6110 6111\begin{patch}{ugxProblemSeriesFormulaPagePatch5} 6112\begin{paste}{ugxProblemSeriesFormulaPageFull5}{ugxProblemSeriesFormulaPageEmpty5} 6113\pastebutton{ugxProblemSeriesFormulaPageFull5}{\hidepaste} 6114\tab{5}\spadcommand{cscx := series(n +-> (-1)^((n-1)/2) * 2 * (2^n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2)\bound{cscx }} 6115\indentrel{3}\begin{verbatim} 6116 (5) 6117 - 1 1 7 3 31 5 127 7 6118 x + - x + --- x + ----- x + ------ x 6119 6 360 15120 604800 6120 + 6121 73 9 10 6122 ------- x + O(x ) 6123 3421440 6124 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 6125\end{verbatim} 6126\indentrel{-3}\end{paste}\end{patch} 6127 6128\begin{patch}{ugxProblemSeriesFormulaPageEmpty5} 6129\begin{paste}{ugxProblemSeriesFormulaPageEmpty5}{ugxProblemSeriesFormulaPagePatch5} 6130\pastebutton{ugxProblemSeriesFormulaPageEmpty5}{\showpaste} 6131\tab{5}\spadcommand{cscx := series(n +-> (-1)^((n-1)/2) * 2 * (2^n-1) * bernoulli(numer(n+1)) / factorial(n+1), x=0, -1..,2)\bound{cscx }} 6132\end{paste}\end{patch} 6133 6134\begin{patch}{ugxProblemSeriesFormulaPagePatch6} 6135\begin{paste}{ugxProblemSeriesFormulaPageFull6}{ugxProblemSeriesFormulaPageEmpty6} 6136\pastebutton{ugxProblemSeriesFormulaPageFull6}{\hidepaste} 6137\tab{5}\spadcommand{1/cscx\free{cscx }} 6138\indentrel{3}\begin{verbatim} 6139 (6) 6140 1 3 1 5 1 7 1 9 6141 x - - x + --- x - ---- x + ------ x 6142 6 120 5040 362880 6143 + 6144 1 11 12 6145 - -------- x + O(x ) 6146 39916800 6147 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 6148\end{verbatim} 6149\indentrel{-3}\end{paste}\end{patch} 6150 6151\begin{patch}{ugxProblemSeriesFormulaPageEmpty6} 6152\begin{paste}{ugxProblemSeriesFormulaPageEmpty6}{ugxProblemSeriesFormulaPagePatch6} 6153\pastebutton{ugxProblemSeriesFormulaPageEmpty6}{\showpaste} 6154\tab{5}\spadcommand{1/cscx\free{cscx }} 6155\end{paste}\end{patch} 6156 6157\begin{patch}{ugxProblemSeriesFormulaPagePatch7} 6158\begin{paste}{ugxProblemSeriesFormulaPageFull7}{ugxProblemSeriesFormulaPageEmpty7} 6159\pastebutton{ugxProblemSeriesFormulaPageFull7}{\hidepaste} 6160\tab{5}\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2^(n-1)),x=0,1..,2)\bound{asinx }} 6161\indentrel{3}\begin{verbatim} 6162 (7) 6163 1 3 3 5 5 7 35 9 63 11 12 6164 x + - x + -- x + --- x + ---- x + ---- x + O(x ) 6165 6 40 112 1152 2816 6166 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 6167\end{verbatim} 6168\indentrel{-3}\end{paste}\end{patch} 6169 6170\begin{patch}{ugxProblemSeriesFormulaPageEmpty7} 6171\begin{paste}{ugxProblemSeriesFormulaPageEmpty7}{ugxProblemSeriesFormulaPagePatch7} 6172\pastebutton{ugxProblemSeriesFormulaPageEmpty7}{\showpaste} 6173\tab{5}\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2^(n-1)),x=0,1..,2)\bound{asinx }} 6174\end{paste}\end{patch} 6175 6176\begin{patch}{ugxProblemSeriesFormulaPagePatch8} 6177\begin{paste}{ugxProblemSeriesFormulaPageFull8}{ugxProblemSeriesFormulaPageEmpty8} 6178\pastebutton{ugxProblemSeriesFormulaPageFull8}{\hidepaste} 6179\tab{5}\spadcommand{sin(asinx)\free{asinx }} 6180\indentrel{3}\begin{verbatim} 6181 12 6182 (8) x + O(x ) 6183 Type: UnivariatePuiseuxSeries(Expression(Integer),x,0) 6184\end{verbatim} 6185\indentrel{-3}\end{paste}\end{patch} 6186 6187\begin{patch}{ugxProblemSeriesFormulaPageEmpty8} 6188\begin{paste}{ugxProblemSeriesFormulaPageEmpty8}{ugxProblemSeriesFormulaPagePatch8} 6189\pastebutton{ugxProblemSeriesFormulaPageEmpty8}{\showpaste} 6190\tab{5}\spadcommand{sin(asinx)\free{asinx }} 6191\end{paste}\end{patch} 6192 6193\begin{patch}{ugxProblemFiniteModulusPagePatch1} 6194\begin{paste}{ugxProblemFiniteModulusPageFull1}{ugxProblemFiniteModulusPageEmpty1} 6195\pastebutton{ugxProblemFiniteModulusPageFull1}{\hidepaste} 6196\tab{5}\spadcommand{GF4096 := FF(2,12);\bound{GF4096 }} 6197\indentrel{3}\begin{verbatim} 6198 Type: Type 6199\end{verbatim} 6200\indentrel{-3}\end{paste}\end{patch} 6201 6202\begin{patch}{ugxProblemFiniteModulusPageEmpty1} 6203\begin{paste}{ugxProblemFiniteModulusPageEmpty1}{ugxProblemFiniteModulusPagePatch1} 6204\pastebutton{ugxProblemFiniteModulusPageEmpty1}{\showpaste} 6205\tab{5}\spadcommand{GF4096 := FF(2,12);\bound{GF4096 }} 6206\end{paste}\end{patch} 6207 6208\begin{patch}{ugxProblemFiniteModulusPagePatch2} 6209\begin{paste}{ugxProblemFiniteModulusPageFull2}{ugxProblemFiniteModulusPageEmpty2} 6210\pastebutton{ugxProblemFiniteModulusPageFull2}{\hidepaste} 6211\tab{5}\spadcommand{a := index(2)$GF4096\bound{a }\free{GF4096 }} 6212\indentrel{3}\begin{verbatim} 6213 (2) %JP 6214 Type: FiniteField(2,12) 6215\end{verbatim} 6216\indentrel{-3}\end{paste}\end{patch} 6217 6218\begin{patch}{ugxProblemFiniteModulusPageEmpty2} 6219\begin{paste}{ugxProblemFiniteModulusPageEmpty2}{ugxProblemFiniteModulusPagePatch2} 6220\pastebutton{ugxProblemFiniteModulusPageEmpty2}{\showpaste} 6221\tab{5}\spadcommand{a := index(2)$GF4096\bound{a }\free{GF4096 }} 6222\end{paste}\end{patch} 6223 6224\begin{patch}{ugxProblemFiniteModulusPagePatch3} 6225\begin{paste}{ugxProblemFiniteModulusPageFull3}{ugxProblemFiniteModulusPageEmpty3} 6226\pastebutton{ugxProblemFiniteModulusPageFull3}{\hidepaste} 6227\tab{5}\spadcommand{b := a^12 - a^5 + a\bound{b }\free{a }} 6228\indentrel{3}\begin{verbatim} 6229 5 3 6230 (3) %JP + %JP + %JP + 1 6231 Type: FiniteField(2,12) 6232\end{verbatim} 6233\indentrel{-3}\end{paste}\end{patch} 6234 6235\begin{patch}{ugxProblemFiniteModulusPageEmpty3} 6236\begin{paste}{ugxProblemFiniteModulusPageEmpty3}{ugxProblemFiniteModulusPagePatch3} 6237\pastebutton{ugxProblemFiniteModulusPageEmpty3}{\showpaste} 6238\tab{5}\spadcommand{b := a^12 - a^5 + a\bound{b }\free{a }} 6239\end{paste}\end{patch} 6240 6241\begin{patch}{ugxProblemFiniteModulusPagePatch4} 6242\begin{paste}{ugxProblemFiniteModulusPageFull4}{ugxProblemFiniteModulusPageEmpty4} 6243\pastebutton{ugxProblemFiniteModulusPageFull4}{\hidepaste} 6244\tab{5}\spadcommand{b ^ 1000\free{b }} 6245\indentrel{3}\begin{verbatim} 6246 10 9 7 5 4 3 6247 (4) %JP + %JP + %JP + %JP + %JP + %JP + %JP 6248 Type: FiniteField(2,12) 6249\end{verbatim} 6250\indentrel{-3}\end{paste}\end{patch} 6251 6252\begin{patch}{ugxProblemFiniteModulusPageEmpty4} 6253\begin{paste}{ugxProblemFiniteModulusPageEmpty4}{ugxProblemFiniteModulusPagePatch4} 6254\pastebutton{ugxProblemFiniteModulusPageEmpty4}{\showpaste} 6255\tab{5}\spadcommand{b ^ 1000\free{b }} 6256\end{paste}\end{patch} 6257 6258\begin{patch}{ugxProblemFiniteModulusPagePatch5} 6259\begin{paste}{ugxProblemFiniteModulusPageFull5}{ugxProblemFiniteModulusPageEmpty5} 6260\pastebutton{ugxProblemFiniteModulusPageFull5}{\hidepaste} 6261\tab{5}\spadcommand{c := a/b\free{a b }\bound{c }} 6262\indentrel{3}\begin{verbatim} 6263 11 8 7 5 4 3 2 6264 (5) %JP + %JP + %JP + %JP + %JP + %JP + %JP 6265 Type: FiniteField(2,12) 6266\end{verbatim} 6267\indentrel{-3}\end{paste}\end{patch} 6268 6269\begin{patch}{ugxProblemFiniteModulusPageEmpty5} 6270\begin{paste}{ugxProblemFiniteModulusPageEmpty5}{ugxProblemFiniteModulusPagePatch5} 6271\pastebutton{ugxProblemFiniteModulusPageEmpty5}{\showpaste} 6272\tab{5}\spadcommand{c := a/b\free{a b }\bound{c }} 6273\end{paste}\end{patch} 6274 6275\begin{patch}{ugxProblemFiniteModulusPagePatch6} 6276\begin{paste}{ugxProblemFiniteModulusPageFull6}{ugxProblemFiniteModulusPageEmpty6} 6277\pastebutton{ugxProblemFiniteModulusPageFull6}{\hidepaste} 6278\tab{5}\spadcommand{norm c\free{c }} 6279\indentrel{3}\begin{verbatim} 6280 (6) 1 6281 Type: PrimeField(2) 6282\end{verbatim} 6283\indentrel{-3}\end{paste}\end{patch} 6284 6285\begin{patch}{ugxProblemFiniteModulusPageEmpty6} 6286\begin{paste}{ugxProblemFiniteModulusPageEmpty6}{ugxProblemFiniteModulusPagePatch6} 6287\pastebutton{ugxProblemFiniteModulusPageEmpty6}{\showpaste} 6288\tab{5}\spadcommand{norm c\free{c }} 6289\end{paste}\end{patch} 6290 6291\begin{patch}{ugxProblemFiniteModulusPagePatch7} 6292\begin{paste}{ugxProblemFiniteModulusPageFull7}{ugxProblemFiniteModulusPageEmpty7} 6293\pastebutton{ugxProblemFiniteModulusPageFull7}{\hidepaste} 6294\tab{5}\spadcommand{trace c\free{c }} 6295\indentrel{3}\begin{verbatim} 6296 (7) 0 6297 Type: PrimeField(2) 6298\end{verbatim} 6299\indentrel{-3}\end{paste}\end{patch} 6300 6301\begin{patch}{ugxProblemFiniteModulusPageEmpty7} 6302\begin{paste}{ugxProblemFiniteModulusPageEmpty7}{ugxProblemFiniteModulusPagePatch7} 6303\pastebutton{ugxProblemFiniteModulusPageEmpty7}{\showpaste} 6304\tab{5}\spadcommand{trace c\free{c }} 6305\end{paste}\end{patch} 6306 6307\begin{patch}{ugxProblemFiniteModulusPagePatch8} 6308\begin{paste}{ugxProblemFiniteModulusPageFull8}{ugxProblemFiniteModulusPageEmpty8} 6309\pastebutton{ugxProblemFiniteModulusPageFull8}{\hidepaste} 6310\tab{5}\spadcommand{dL := discreteLog a\free{a }\bound{dL }} 6311\indentrel{3}\begin{verbatim} 6312 (8) 1729 6313 Type: PositiveInteger 6314\end{verbatim} 6315\indentrel{-3}\end{paste}\end{patch} 6316 6317\begin{patch}{ugxProblemFiniteModulusPageEmpty8} 6318\begin{paste}{ugxProblemFiniteModulusPageEmpty8}{ugxProblemFiniteModulusPagePatch8} 6319\pastebutton{ugxProblemFiniteModulusPageEmpty8}{\showpaste} 6320\tab{5}\spadcommand{dL := discreteLog a\free{a }\bound{dL }} 6321\end{paste}\end{patch} 6322 6323\begin{patch}{ugxProblemFiniteModulusPagePatch9} 6324\begin{paste}{ugxProblemFiniteModulusPageFull9}{ugxProblemFiniteModulusPageEmpty9} 6325\pastebutton{ugxProblemFiniteModulusPageFull9}{\hidepaste} 6326\tab{5}\spadcommand{g ^ dL\free{dL g }} 6327\indentrel{3}\begin{verbatim} 6328 1729 6329 (9) g 6330 Type: Polynomial(Integer) 6331\end{verbatim} 6332\indentrel{-3}\end{paste}\end{patch} 6333 6334\begin{patch}{ugxProblemFiniteModulusPageEmpty9} 6335\begin{paste}{ugxProblemFiniteModulusPageEmpty9}{ugxProblemFiniteModulusPagePatch9} 6336\pastebutton{ugxProblemFiniteModulusPageEmpty9}{\showpaste} 6337\tab{5}\spadcommand{g ^ dL\free{dL g }} 6338\end{paste}\end{patch} 6339 6340\begin{patch}{ugxProblemFiniteModulusPagePatch10} 6341\begin{paste}{ugxProblemFiniteModulusPageFull10}{ugxProblemFiniteModulusPageEmpty10} 6342\pastebutton{ugxProblemFiniteModulusPageFull10}{\hidepaste} 6343\tab{5}\spadcommand{GF16 := FF(2,4);\bound{GF16 }} 6344\indentrel{3}\begin{verbatim} 6345 Type: Type 6346\end{verbatim} 6347\indentrel{-3}\end{paste}\end{patch} 6348 6349\begin{patch}{ugxProblemFiniteModulusPageEmpty10} 6350\begin{paste}{ugxProblemFiniteModulusPageEmpty10}{ugxProblemFiniteModulusPagePatch10} 6351\pastebutton{ugxProblemFiniteModulusPageEmpty10}{\showpaste} 6352\tab{5}\spadcommand{GF16 := FF(2,4);\bound{GF16 }} 6353\end{paste}\end{patch} 6354 6355\begin{patch}{ugxProblemFiniteModulusPagePatch11} 6356\begin{paste}{ugxProblemFiniteModulusPageFull11}{ugxProblemFiniteModulusPageEmpty11} 6357\pastebutton{ugxProblemFiniteModulusPageFull11}{\hidepaste} 6358\tab{5}\spadcommand{GF4096 := FFX(GF16,3);\bound{GF4096x }\free{GF16 }} 6359\indentrel{3}\begin{verbatim} 6360 Type: Type 6361\end{verbatim} 6362\indentrel{-3}\end{paste}\end{patch} 6363 6364\begin{patch}{ugxProblemFiniteModulusPageEmpty11} 6365\begin{paste}{ugxProblemFiniteModulusPageEmpty11}{ugxProblemFiniteModulusPagePatch11} 6366\pastebutton{ugxProblemFiniteModulusPageEmpty11}{\showpaste} 6367\tab{5}\spadcommand{GF4096 := FFX(GF16,3);\bound{GF4096x }\free{GF16 }} 6368\end{paste}\end{patch} 6369 6370\begin{patch}{ugxProblemFiniteModulusPagePatch12} 6371\begin{paste}{ugxProblemFiniteModulusPageFull12}{ugxProblemFiniteModulusPageEmpty12} 6372\pastebutton{ugxProblemFiniteModulusPageFull12}{\hidepaste} 6373\tab{5}\spadcommand{r := (random()$GF4096) ^ 20\free{GF4096x }\bound{r }} 6374\indentrel{3}\begin{verbatim} 6375 2 3 2 6376 (12) %JQ %JR + (%JQ + %JQ + 1)%JR + 1 6377 Type: FiniteFieldExtension(FiniteField(2,4),3) 6378\end{verbatim} 6379\indentrel{-3}\end{paste}\end{patch} 6380 6381\begin{patch}{ugxProblemFiniteModulusPageEmpty12} 6382\begin{paste}{ugxProblemFiniteModulusPageEmpty12}{ugxProblemFiniteModulusPagePatch12} 6383\pastebutton{ugxProblemFiniteModulusPageEmpty12}{\showpaste} 6384\tab{5}\spadcommand{r := (random()$GF4096) ^ 20\free{GF4096x }\bound{r }} 6385\end{paste}\end{patch} 6386 6387\begin{patch}{ugxProblemFiniteModulusPagePatch13} 6388\begin{paste}{ugxProblemFiniteModulusPageFull13}{ugxProblemFiniteModulusPageEmpty13} 6389\pastebutton{ugxProblemFiniteModulusPageFull13}{\hidepaste} 6390\tab{5}\spadcommand{norm(r)\free{r }} 6391\indentrel{3}\begin{verbatim} 6392 (13) 1 6393 Type: FiniteField(2,4) 6394\end{verbatim} 6395\indentrel{-3}\end{paste}\end{patch} 6396 6397\begin{patch}{ugxProblemFiniteModulusPageEmpty13} 6398\begin{paste}{ugxProblemFiniteModulusPageEmpty13}{ugxProblemFiniteModulusPagePatch13} 6399\pastebutton{ugxProblemFiniteModulusPageEmpty13}{\showpaste} 6400\tab{5}\spadcommand{norm(r)\free{r }} 6401\end{paste}\end{patch} 6402 6403\begin{patch}{ugxProblemFiniteModulusPagePatch14} 6404\begin{paste}{ugxProblemFiniteModulusPageFull14}{ugxProblemFiniteModulusPageEmpty14} 6405\pastebutton{ugxProblemFiniteModulusPageFull14}{\hidepaste} 6406\tab{5}\spadcommand{GF4 := FF(2,2);\bound{GF4 }} 6407\indentrel{3}\begin{verbatim} 6408 Type: Type 6409\end{verbatim} 6410\indentrel{-3}\end{paste}\end{patch} 6411 6412\begin{patch}{ugxProblemFiniteModulusPageEmpty14} 6413\begin{paste}{ugxProblemFiniteModulusPageEmpty14}{ugxProblemFiniteModulusPagePatch14} 6414\pastebutton{ugxProblemFiniteModulusPageEmpty14}{\showpaste} 6415\tab{5}\spadcommand{GF4 := FF(2,2);\bound{GF4 }} 6416\end{paste}\end{patch} 6417 6418\begin{patch}{ugxProblemFiniteModulusPagePatch15} 6419\begin{paste}{ugxProblemFiniteModulusPageFull15}{ugxProblemFiniteModulusPageEmpty15} 6420\pastebutton{ugxProblemFiniteModulusPageFull15}{\hidepaste} 6421\tab{5}\spadcommand{f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)\free{GF4 }\bound{f }} 6422\indentrel{3}\begin{verbatim} 6423 (15) 6424 6 5 4 3 2 6425 ? + (%JS + 1)? + (%JS + 1)? + ? + (%JS + 1)? 6426 + 6427 (%JS + 1)? + %JS 6428Type: Union(SparseUnivariatePolynomial(FiniteField(2,2)),...) 6429\end{verbatim} 6430\indentrel{-3}\end{paste}\end{patch} 6431 6432\begin{patch}{ugxProblemFiniteModulusPageEmpty15} 6433\begin{paste}{ugxProblemFiniteModulusPageEmpty15}{ugxProblemFiniteModulusPagePatch15} 6434\pastebutton{ugxProblemFiniteModulusPageEmpty15}{\showpaste} 6435\tab{5}\spadcommand{f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)\free{GF4 }\bound{f }} 6436\end{paste}\end{patch} 6437 6438\begin{patch}{ugxProblemFiniteModulusPagePatch16} 6439\begin{paste}{ugxProblemFiniteModulusPageFull16}{ugxProblemFiniteModulusPageEmpty16} 6440\pastebutton{ugxProblemFiniteModulusPageFull16}{\hidepaste} 6441\tab{5}\spadcommand{GF4096 := FFP(GF4,f);\bound{GF4096y }\free{f GF4 }} 6442\indentrel{3}\begin{verbatim} 6443 Type: Type 6444\end{verbatim} 6445\indentrel{-3}\end{paste}\end{patch} 6446 6447\begin{patch}{ugxProblemFiniteModulusPageEmpty16} 6448\begin{paste}{ugxProblemFiniteModulusPageEmpty16}{ugxProblemFiniteModulusPagePatch16} 6449\pastebutton{ugxProblemFiniteModulusPageEmpty16}{\showpaste} 6450\tab{5}\spadcommand{GF4096 := FFP(GF4,f);\bound{GF4096y }\free{f GF4 }} 6451\end{paste}\end{patch} 6452 6453\begin{patch}{ugxProblemFiniteModulusPagePatch17} 6454\begin{paste}{ugxProblemFiniteModulusPageFull17}{ugxProblemFiniteModulusPageEmpty17} 6455\pastebutton{ugxProblemFiniteModulusPageFull17}{\hidepaste} 6456\tab{5}\spadcommand{discreteLog random()$GF4096\free{GF4096y }} 6457\indentrel{3}\begin{verbatim} 6458 (17) 3855 6459 Type: PositiveInteger 6460\end{verbatim} 6461\indentrel{-3}\end{paste}\end{patch} 6462 6463\begin{patch}{ugxProblemFiniteModulusPageEmpty17} 6464\begin{paste}{ugxProblemFiniteModulusPageEmpty17}{ugxProblemFiniteModulusPagePatch17} 6465\pastebutton{ugxProblemFiniteModulusPageEmpty17}{\showpaste} 6466\tab{5}\spadcommand{discreteLog random()$GF4096\free{GF4096y }} 6467\end{paste}\end{patch} 6468 6469\begin{patch}{ugProblemIdealPagePatch1} 6470\begin{paste}{ugProblemIdealPageFull1}{ugProblemIdealPageEmpty1} 6471\pastebutton{ugProblemIdealPageFull1}{\hidepaste} 6472\tab{5}\spadcommand{(n,m) : List DMP([x,y],FRAC INT)\bound{nm }} 6473\indentrel{3}\begin{verbatim} 6474 Type: Void 6475\end{verbatim} 6476\indentrel{-3}\end{paste}\end{patch} 6477 6478\begin{patch}{ugProblemIdealPageEmpty1} 6479\begin{paste}{ugProblemIdealPageEmpty1}{ugProblemIdealPagePatch1} 6480\pastebutton{ugProblemIdealPageEmpty1}{\showpaste} 6481\tab{5}\spadcommand{(n,m) : List DMP([x,y],FRAC INT)\bound{nm }} 6482\end{paste}\end{patch} 6483 6484\begin{patch}{ugProblemIdealPagePatch2} 6485\begin{paste}{ugProblemIdealPageFull2}{ugProblemIdealPageEmpty2} 6486\pastebutton{ugProblemIdealPageFull2}{\hidepaste} 6487\tab{5}\spadcommand{m := [x^2+y^2-1]\free{nm }\bound{m }} 6488\indentrel{3}\begin{verbatim} 6489 2 2 6490 (2) [x + y - 1] 6491Type: List(DistributedMultivariatePolynomial([x,y],Fraction(Integer))) 6492\end{verbatim} 6493\indentrel{-3}\end{paste}\end{patch} 6494 6495\begin{patch}{ugProblemIdealPageEmpty2} 6496\begin{paste}{ugProblemIdealPageEmpty2}{ugProblemIdealPagePatch2} 6497\pastebutton{ugProblemIdealPageEmpty2}{\showpaste} 6498\tab{5}\spadcommand{m := [x^2+y^2-1]\free{nm }\bound{m }} 6499\end{paste}\end{patch} 6500 6501\begin{patch}{ugProblemIdealPagePatch3} 6502\begin{paste}{ugProblemIdealPageFull3}{ugProblemIdealPageEmpty3} 6503\pastebutton{ugProblemIdealPageFull3}{\hidepaste} 6504\tab{5}\spadcommand{n := [x^2-y^2]\free{nm }\bound{n }} 6505\indentrel{3}\begin{verbatim} 6506 2 2 6507 (3) [x - y ] 6508Type: List(DistributedMultivariatePolynomial([x,y],Fraction(Integer))) 6509\end{verbatim} 6510\indentrel{-3}\end{paste}\end{patch} 6511 6512\begin{patch}{ugProblemIdealPageEmpty3} 6513\begin{paste}{ugProblemIdealPageEmpty3}{ugProblemIdealPagePatch3} 6514\pastebutton{ugProblemIdealPageEmpty3}{\showpaste} 6515\tab{5}\spadcommand{n := [x^2-y^2]\free{nm }\bound{n }} 6516\end{paste}\end{patch} 6517 6518\begin{patch}{ugProblemIdealPagePatch4} 6519\begin{paste}{ugProblemIdealPageFull4}{ugProblemIdealPageEmpty4} 6520\pastebutton{ugProblemIdealPageFull4}{\hidepaste} 6521\tab{5}\spadcommand{id := ideal m + ideal n\free{n m }\bound{id }} 6522\indentrel{3}\begin{verbatim} 6523 2 1 2 1 6524 (4) [x - -, y - -] 6525 2 2 6526Type: PolynomialIdeal(Fraction(Integer),DirectProduct(2,NonNegativeInteger),OrderedVariableList([x,y]),DistributedMultivariatePolynomial([x,y],Fraction(Integer))) 6527\end{verbatim} 6528\indentrel{-3}\end{paste}\end{patch} 6529 6530\begin{patch}{ugProblemIdealPageEmpty4} 6531\begin{paste}{ugProblemIdealPageEmpty4}{ugProblemIdealPagePatch4} 6532\pastebutton{ugProblemIdealPageEmpty4}{\showpaste} 6533\tab{5}\spadcommand{id := ideal m + ideal n\free{n m }\bound{id }} 6534\end{paste}\end{patch} 6535 6536\begin{patch}{ugProblemIdealPagePatch5} 6537\begin{paste}{ugProblemIdealPageFull5}{ugProblemIdealPageEmpty5} 6538\pastebutton{ugProblemIdealPageFull5}{\hidepaste} 6539\tab{5}\spadcommand{zeroDim? id\free{id }} 6540\indentrel{3}\begin{verbatim} 6541 (5) true 6542 Type: Boolean 6543\end{verbatim} 6544\indentrel{-3}\end{paste}\end{patch} 6545 6546\begin{patch}{ugProblemIdealPageEmpty5} 6547\begin{paste}{ugProblemIdealPageEmpty5}{ugProblemIdealPagePatch5} 6548\pastebutton{ugProblemIdealPageEmpty5}{\showpaste} 6549\tab{5}\spadcommand{zeroDim? id\free{id }} 6550\end{paste}\end{patch} 6551 6552\begin{patch}{ugProblemIdealPagePatch6} 6553\begin{paste}{ugProblemIdealPageFull6}{ugProblemIdealPageEmpty6} 6554\pastebutton{ugProblemIdealPageFull6}{\hidepaste} 6555\tab{5}\spadcommand{zeroDim?(ideal m)\free{m }} 6556\indentrel{3}\begin{verbatim} 6557 (6) false 6558 Type: Boolean 6559\end{verbatim} 6560\indentrel{-3}\end{paste}\end{patch} 6561 6562\begin{patch}{ugProblemIdealPageEmpty6} 6563\begin{paste}{ugProblemIdealPageEmpty6}{ugProblemIdealPagePatch6} 6564\pastebutton{ugProblemIdealPageEmpty6}{\showpaste} 6565\tab{5}\spadcommand{zeroDim?(ideal m)\free{m }} 6566\end{paste}\end{patch} 6567 6568\begin{patch}{ugProblemIdealPagePatch7} 6569\begin{paste}{ugProblemIdealPageFull7}{ugProblemIdealPageEmpty7} 6570\pastebutton{ugProblemIdealPageFull7}{\hidepaste} 6571\tab{5}\spadcommand{dimension ideal m\free{m }} 6572\indentrel{3}\begin{verbatim} 6573 (7) 1 6574 Type: PositiveInteger 6575\end{verbatim} 6576\indentrel{-3}\end{paste}\end{patch} 6577 6578\begin{patch}{ugProblemIdealPageEmpty7} 6579\begin{paste}{ugProblemIdealPageEmpty7}{ugProblemIdealPagePatch7} 6580\pastebutton{ugProblemIdealPageEmpty7}{\showpaste} 6581\tab{5}\spadcommand{dimension ideal m\free{m }} 6582\end{paste}\end{patch} 6583 6584\begin{patch}{ugProblemIdealPagePatch8} 6585\begin{paste}{ugProblemIdealPageFull8}{ugProblemIdealPageEmpty8} 6586\pastebutton{ugProblemIdealPageFull8}{\hidepaste} 6587\tab{5}\spadcommand{(f,g):DMP([x,y],FRAC INT)\bound{fg }} 6588\indentrel{3}\begin{verbatim} 6589 Type: Void 6590\end{verbatim} 6591\indentrel{-3}\end{paste}\end{patch} 6592 6593\begin{patch}{ugProblemIdealPageEmpty8} 6594\begin{paste}{ugProblemIdealPageEmpty8}{ugProblemIdealPagePatch8} 6595\pastebutton{ugProblemIdealPageEmpty8}{\showpaste} 6596\tab{5}\spadcommand{(f,g):DMP([x,y],FRAC INT)\bound{fg }} 6597\end{paste}\end{patch} 6598 6599\begin{patch}{ugProblemIdealPagePatch9} 6600\begin{paste}{ugProblemIdealPageFull9}{ugProblemIdealPageEmpty9} 6601\pastebutton{ugProblemIdealPageFull9}{\hidepaste} 6602\tab{5}\spadcommand{f := x^2-1\free{fg }\bound{f }} 6603\indentrel{3}\begin{verbatim} 6604 2 6605 (9) x - 1 6606Type: DistributedMultivariatePolynomial([x,y],Fraction(Integer)) 6607\end{verbatim} 6608\indentrel{-3}\end{paste}\end{patch} 6609 6610\begin{patch}{ugProblemIdealPageEmpty9} 6611\begin{paste}{ugProblemIdealPageEmpty9}{ugProblemIdealPagePatch9} 6612\pastebutton{ugProblemIdealPageEmpty9}{\showpaste} 6613\tab{5}\spadcommand{f := x^2-1\free{fg }\bound{f }} 6614\end{paste}\end{patch} 6615 6616\begin{patch}{ugProblemIdealPagePatch10} 6617\begin{paste}{ugProblemIdealPageFull10}{ugProblemIdealPageEmpty10} 6618\pastebutton{ugProblemIdealPageFull10}{\hidepaste} 6619\tab{5}\spadcommand{g := x*(x^2-1)\free{fg }\bound{g }} 6620\indentrel{3}\begin{verbatim} 6621 3 6622 (10) x - x 6623Type: DistributedMultivariatePolynomial([x,y],Fraction(Integer)) 6624\end{verbatim} 6625\indentrel{-3}\end{paste}\end{patch} 6626 6627\begin{patch}{ugProblemIdealPageEmpty10} 6628\begin{paste}{ugProblemIdealPageEmpty10}{ugProblemIdealPagePatch10} 6629\pastebutton{ugProblemIdealPageEmpty10}{\showpaste} 6630\tab{5}\spadcommand{g := x*(x^2-1)\free{fg }\bound{g }} 6631\end{paste}\end{patch} 6632 6633\begin{patch}{ugProblemIdealPagePatch11} 6634\begin{paste}{ugProblemIdealPageFull11}{ugProblemIdealPageEmpty11} 6635\pastebutton{ugProblemIdealPageFull11}{\hidepaste} 6636\tab{5}\spadcommand{relationsIdeal [f,g]\free{f g }} 6637\indentrel{3}\begin{verbatim} 6638 (11) 6639 2 3 2 2 3 6640 [- %JV + %JU + %JU ] | [%JU = x - 1, %JV = x - x] 6641Type: SuchThat(List(Polynomial(Fraction(Integer))),List(Equation(Polynomial(Fraction(Integer))))) 6642\end{verbatim} 6643\indentrel{-3}\end{paste}\end{patch} 6644 6645\begin{patch}{ugProblemIdealPageEmpty11} 6646\begin{paste}{ugProblemIdealPageEmpty11}{ugProblemIdealPagePatch11} 6647\pastebutton{ugProblemIdealPageEmpty11}{\showpaste} 6648\tab{5}\spadcommand{relationsIdeal [f,g]\free{f g }} 6649\end{paste}\end{patch} 6650 6651\begin{patch}{ugProblemIdealPagePatch12} 6652\begin{paste}{ugProblemIdealPageFull12}{ugProblemIdealPageEmpty12} 6653\pastebutton{ugProblemIdealPageFull12}{\hidepaste} 6654\tab{5}\spadcommand{l: List DMP([x,y,z],FRAC INT) := [x^2+2*y^2,x*z^2-y*z,z^2-4]\bound{l }} 6655\indentrel{3}\begin{verbatim} 6656 2 2 2 2 6657 (12) [x + 2 y , x z - y z, z - 4] 6658Type: List(DistributedMultivariatePolynomial([x,y,z],Fraction(Integer))) 6659\end{verbatim} 6660\indentrel{-3}\end{paste}\end{patch} 6661 6662\begin{patch}{ugProblemIdealPageEmpty12} 6663\begin{paste}{ugProblemIdealPageEmpty12}{ugProblemIdealPagePatch12} 6664\pastebutton{ugProblemIdealPageEmpty12}{\showpaste} 6665\tab{5}\spadcommand{l: List DMP([x,y,z],FRAC INT) := [x^2+2*y^2,x*z^2-y*z,z^2-4]\bound{l }} 6666\end{paste}\end{patch} 6667 6668\begin{patch}{ugProblemIdealPagePatch13} 6669\begin{paste}{ugProblemIdealPageFull13}{ugProblemIdealPageEmpty13} 6670\pastebutton{ugProblemIdealPageFull13}{\hidepaste} 6671\tab{5}\spadcommand{ld:=primaryDecomp(ideal l)$IdealDecompositionPackage([x,y,z])\free{l }\bound{ld }} 6672\indentrel{3}\begin{verbatim} 6673 1 2 1 2 6674 (13) [[x + - y, y , z + 2], [x - - y, y , z - 2]] 6675 2 2 6676Type: List(PolynomialIdeal(Fraction(Integer),DirectProduct(3,NonNegativeInteger),OrderedVariableList([x,y,z]),DistributedMultivariatePolynomial([x,y,z],Fraction(Integer)))) 6677\end{verbatim} 6678\indentrel{-3}\end{paste}\end{patch} 6679 6680\begin{patch}{ugProblemIdealPageEmpty13} 6681\begin{paste}{ugProblemIdealPageEmpty13}{ugProblemIdealPagePatch13} 6682\pastebutton{ugProblemIdealPageEmpty13}{\showpaste} 6683\tab{5}\spadcommand{ld:=primaryDecomp(ideal l)$IdealDecompositionPackage([x,y,z])\free{l }\bound{ld }} 6684\end{paste}\end{patch} 6685 6686\begin{patch}{ugProblemIdealPagePatch14} 6687\begin{paste}{ugProblemIdealPageFull14}{ugProblemIdealPageEmpty14} 6688\pastebutton{ugProblemIdealPageFull14}{\hidepaste} 6689\tab{5}\spadcommand{reduce(intersect,ld)\free{ld }} 6690\indentrel{3}\begin{verbatim} 6691 1 2 2 6692 (14) [x - - y z, y , z - 4] 6693 4 6694Type: PolynomialIdeal(Fraction(Integer),DirectProduct(3,NonNegativeInteger),OrderedVariableList([x,y,z]),DistributedMultivariatePolynomial([x,y,z],Fraction(Integer))) 6695\end{verbatim} 6696\indentrel{-3}\end{paste}\end{patch} 6697 6698\begin{patch}{ugProblemIdealPageEmpty14} 6699\begin{paste}{ugProblemIdealPageEmpty14}{ugProblemIdealPagePatch14} 6700\pastebutton{ugProblemIdealPageEmpty14}{\showpaste} 6701\tab{5}\spadcommand{reduce(intersect,ld)\free{ld }} 6702\end{paste}\end{patch} 6703 6704\begin{patch}{ugProblemIdealPagePatch15} 6705\begin{paste}{ugProblemIdealPageFull15}{ugProblemIdealPageEmpty15} 6706\pastebutton{ugProblemIdealPageFull15}{\hidepaste} 6707\tab{5}\spadcommand{reduce(intersect,[radical(ld.i)$IdealDecompositionPackage([x,y,z]) for i in 1..2])\free{ld }} 6708\indentrel{3}\begin{verbatim} 6709 2 6710 (15) [x, y, z - 4] 6711Type: PolynomialIdeal(Fraction(Integer),DirectProduct(3,NonNegativeInteger),OrderedVariableList([x,y,z]),DistributedMultivariatePolynomial([x,y,z],Fraction(Integer))) 6712\end{verbatim} 6713\indentrel{-3}\end{paste}\end{patch} 6714 6715\begin{patch}{ugProblemIdealPageEmpty15} 6716\begin{paste}{ugProblemIdealPageEmpty15}{ugProblemIdealPagePatch15} 6717\pastebutton{ugProblemIdealPageEmpty15}{\showpaste} 6718\tab{5}\spadcommand{reduce(intersect,[radical(ld.i)$IdealDecompositionPackage([x,y,z]) for i in 1..2])\free{ld }} 6719\end{paste}\end{patch} 6720 6721\begin{patch}{ugProblemIdealPagePatch16} 6722\begin{paste}{ugProblemIdealPageFull16}{ugProblemIdealPageEmpty16} 6723\pastebutton{ugProblemIdealPageFull16}{\hidepaste} 6724\tab{5}\spadcommand{radical(ideal l)$IdealDecompositionPackage([x,y,z])\free{l }} 6725\indentrel{3}\begin{verbatim} 6726 2 6727 (16) [x, y, z - 4] 6728Type: PolynomialIdeal(Fraction(Integer),DirectProduct(3,NonNegativeInteger),OrderedVariableList([x,y,z]),DistributedMultivariatePolynomial([x,y,z],Fraction(Integer))) 6729\end{verbatim} 6730\indentrel{-3}\end{paste}\end{patch} 6731 6732\begin{patch}{ugProblemIdealPageEmpty16} 6733\begin{paste}{ugProblemIdealPageEmpty16}{ugProblemIdealPagePatch16} 6734\pastebutton{ugProblemIdealPageEmpty16}{\showpaste} 6735\tab{5}\spadcommand{radical(ideal l)$IdealDecompositionPackage([x,y,z])\free{l }} 6736\end{paste}\end{patch} 6737 6738\begin{patch}{ugxProblemFiniteUtilityPagePatch1} 6739\begin{paste}{ugxProblemFiniteUtilityPageFull1}{ugxProblemFiniteUtilityPageEmpty1} 6740\pastebutton{ugxProblemFiniteUtilityPageFull1}{\hidepaste} 6741\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }} 6742\indentrel{3}\begin{verbatim} 6743 Type: Type 6744\end{verbatim} 6745\indentrel{-3}\end{paste}\end{patch} 6746 6747\begin{patch}{ugxProblemFiniteUtilityPageEmpty1} 6748\begin{paste}{ugxProblemFiniteUtilityPageEmpty1}{ugxProblemFiniteUtilityPagePatch1} 6749\pastebutton{ugxProblemFiniteUtilityPageEmpty1}{\showpaste} 6750\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }} 6751\end{paste}\end{patch} 6752 6753\begin{patch}{ugxProblemFiniteUtilityPagePatch2} 6754\begin{paste}{ugxProblemFiniteUtilityPageFull2}{ugxProblemFiniteUtilityPageEmpty2} 6755\pastebutton{ugxProblemFiniteUtilityPageFull2}{\hidepaste} 6756\tab{5}\spadcommand{f := createIrreduciblePoly(8)$FFPOLY(GF5)\bound{f }\free{GF5 }} 6757\indentrel{3}\begin{verbatim} 6758 8 4 6759 (2) ? + ? + 2 6760 Type: SparseUnivariatePolynomial(PrimeField(5)) 6761\end{verbatim} 6762\indentrel{-3}\end{paste}\end{patch} 6763 6764\begin{patch}{ugxProblemFiniteUtilityPageEmpty2} 6765\begin{paste}{ugxProblemFiniteUtilityPageEmpty2}{ugxProblemFiniteUtilityPagePatch2} 6766\pastebutton{ugxProblemFiniteUtilityPageEmpty2}{\showpaste} 6767\tab{5}\spadcommand{f := createIrreduciblePoly(8)$FFPOLY(GF5)\bound{f }\free{GF5 }} 6768\end{paste}\end{patch} 6769 6770\begin{patch}{ugxProblemFiniteUtilityPagePatch3} 6771\begin{paste}{ugxProblemFiniteUtilityPageFull3}{ugxProblemFiniteUtilityPageEmpty3} 6772\pastebutton{ugxProblemFiniteUtilityPageFull3}{\hidepaste} 6773\tab{5}\spadcommand{primitive?(f)$FFPOLY(GF5)\free{f }} 6774\indentrel{3}\begin{verbatim} 6775 (3) false 6776 Type: Boolean 6777\end{verbatim} 6778\indentrel{-3}\end{paste}\end{patch} 6779 6780\begin{patch}{ugxProblemFiniteUtilityPageEmpty3} 6781\begin{paste}{ugxProblemFiniteUtilityPageEmpty3}{ugxProblemFiniteUtilityPagePatch3} 6782\pastebutton{ugxProblemFiniteUtilityPageEmpty3}{\showpaste} 6783\tab{5}\spadcommand{primitive?(f)$FFPOLY(GF5)\free{f }} 6784\end{paste}\end{patch} 6785 6786\begin{patch}{ugxProblemFiniteUtilityPagePatch4} 6787\begin{paste}{ugxProblemFiniteUtilityPageFull4}{ugxProblemFiniteUtilityPageEmpty4} 6788\pastebutton{ugxProblemFiniteUtilityPageFull4}{\hidepaste} 6789\tab{5}\spadcommand{normal?(f)$FFPOLY(GF5)\free{f }} 6790\indentrel{3}\begin{verbatim} 6791 (4) false 6792 Type: Boolean 6793\end{verbatim} 6794\indentrel{-3}\end{paste}\end{patch} 6795 6796\begin{patch}{ugxProblemFiniteUtilityPageEmpty4} 6797\begin{paste}{ugxProblemFiniteUtilityPageEmpty4}{ugxProblemFiniteUtilityPagePatch4} 6798\pastebutton{ugxProblemFiniteUtilityPageEmpty4}{\showpaste} 6799\tab{5}\spadcommand{normal?(f)$FFPOLY(GF5)\free{f }} 6800\end{paste}\end{patch} 6801 6802\begin{patch}{ugxProblemFiniteUtilityPagePatch5} 6803\begin{paste}{ugxProblemFiniteUtilityPageFull5}{ugxProblemFiniteUtilityPageEmpty5} 6804\pastebutton{ugxProblemFiniteUtilityPageFull5}{\hidepaste} 6805\tab{5}\spadcommand{p := createPrimitivePoly(8)$FFPOLY(GF5)\bound{p }\free{GF5 }} 6806\indentrel{3}\begin{verbatim} 6807 8 3 2 6808 (5) ? + ? + ? + ? + 2 6809 Type: SparseUnivariatePolynomial(PrimeField(5)) 6810\end{verbatim} 6811\indentrel{-3}\end{paste}\end{patch} 6812 6813\begin{patch}{ugxProblemFiniteUtilityPageEmpty5} 6814\begin{paste}{ugxProblemFiniteUtilityPageEmpty5}{ugxProblemFiniteUtilityPagePatch5} 6815\pastebutton{ugxProblemFiniteUtilityPageEmpty5}{\showpaste} 6816\tab{5}\spadcommand{p := createPrimitivePoly(8)$FFPOLY(GF5)\bound{p }\free{GF5 }} 6817\end{paste}\end{patch} 6818 6819\begin{patch}{ugxProblemFiniteUtilityPagePatch6} 6820\begin{paste}{ugxProblemFiniteUtilityPageFull6}{ugxProblemFiniteUtilityPageEmpty6} 6821\pastebutton{ugxProblemFiniteUtilityPageFull6}{\hidepaste} 6822\tab{5}\spadcommand{primitive?(p)$FFPOLY(GF5)\free{p }} 6823\indentrel{3}\begin{verbatim} 6824 (6) true 6825 Type: Boolean 6826\end{verbatim} 6827\indentrel{-3}\end{paste}\end{patch} 6828 6829\begin{patch}{ugxProblemFiniteUtilityPageEmpty6} 6830\begin{paste}{ugxProblemFiniteUtilityPageEmpty6}{ugxProblemFiniteUtilityPagePatch6} 6831\pastebutton{ugxProblemFiniteUtilityPageEmpty6}{\showpaste} 6832\tab{5}\spadcommand{primitive?(p)$FFPOLY(GF5)\free{p }} 6833\end{paste}\end{patch} 6834 6835\begin{patch}{ugxProblemFiniteUtilityPagePatch7} 6836\begin{paste}{ugxProblemFiniteUtilityPageFull7}{ugxProblemFiniteUtilityPageEmpty7} 6837\pastebutton{ugxProblemFiniteUtilityPageFull7}{\hidepaste} 6838\tab{5}\spadcommand{normal?(p)$FFPOLY(GF5)\free{p }} 6839\indentrel{3}\begin{verbatim} 6840 (7) false 6841 Type: Boolean 6842\end{verbatim} 6843\indentrel{-3}\end{paste}\end{patch} 6844 6845\begin{patch}{ugxProblemFiniteUtilityPageEmpty7} 6846\begin{paste}{ugxProblemFiniteUtilityPageEmpty7}{ugxProblemFiniteUtilityPagePatch7} 6847\pastebutton{ugxProblemFiniteUtilityPageEmpty7}{\showpaste} 6848\tab{5}\spadcommand{normal?(p)$FFPOLY(GF5)\free{p }} 6849\end{paste}\end{patch} 6850 6851\begin{patch}{ugxProblemFiniteUtilityPagePatch8} 6852\begin{paste}{ugxProblemFiniteUtilityPageFull8}{ugxProblemFiniteUtilityPageEmpty8} 6853\pastebutton{ugxProblemFiniteUtilityPageFull8}{\hidepaste} 6854\tab{5}\spadcommand{n := createNormalPoly(8)$FFPOLY(GF5)\bound{n }\free{GF5 }} 6855\indentrel{3}\begin{verbatim} 6856 8 7 3 6857 (8) ? + 4 ? + ? + 1 6858 Type: SparseUnivariatePolynomial(PrimeField(5)) 6859\end{verbatim} 6860\indentrel{-3}\end{paste}\end{patch} 6861 6862\begin{patch}{ugxProblemFiniteUtilityPageEmpty8} 6863\begin{paste}{ugxProblemFiniteUtilityPageEmpty8}{ugxProblemFiniteUtilityPagePatch8} 6864\pastebutton{ugxProblemFiniteUtilityPageEmpty8}{\showpaste} 6865\tab{5}\spadcommand{n := createNormalPoly(8)$FFPOLY(GF5)\bound{n }\free{GF5 }} 6866\end{paste}\end{patch} 6867 6868\begin{patch}{ugxProblemFiniteUtilityPagePatch9} 6869\begin{paste}{ugxProblemFiniteUtilityPageFull9}{ugxProblemFiniteUtilityPageEmpty9} 6870\pastebutton{ugxProblemFiniteUtilityPageFull9}{\hidepaste} 6871\tab{5}\spadcommand{primitive?(n)$FFPOLY(GF5)\free{n }} 6872\indentrel{3}\begin{verbatim} 6873 (9) false 6874 Type: Boolean 6875\end{verbatim} 6876\indentrel{-3}\end{paste}\end{patch} 6877 6878\begin{patch}{ugxProblemFiniteUtilityPageEmpty9} 6879\begin{paste}{ugxProblemFiniteUtilityPageEmpty9}{ugxProblemFiniteUtilityPagePatch9} 6880\pastebutton{ugxProblemFiniteUtilityPageEmpty9}{\showpaste} 6881\tab{5}\spadcommand{primitive?(n)$FFPOLY(GF5)\free{n }} 6882\end{paste}\end{patch} 6883 6884\begin{patch}{ugxProblemFiniteUtilityPagePatch10} 6885\begin{paste}{ugxProblemFiniteUtilityPageFull10}{ugxProblemFiniteUtilityPageEmpty10} 6886\pastebutton{ugxProblemFiniteUtilityPageFull10}{\hidepaste} 6887\tab{5}\spadcommand{createPrimitiveNormalPoly(8)$FFPOLY(GF5)\free{GF5 }} 6888\indentrel{3}\begin{verbatim} 6889 8 7 5 6890 (10) ? + 4 ? + 2 ? + 2 6891 Type: SparseUnivariatePolynomial(PrimeField(5)) 6892\end{verbatim} 6893\indentrel{-3}\end{paste}\end{patch} 6894 6895\begin{patch}{ugxProblemFiniteUtilityPageEmpty10} 6896\begin{paste}{ugxProblemFiniteUtilityPageEmpty10}{ugxProblemFiniteUtilityPagePatch10} 6897\pastebutton{ugxProblemFiniteUtilityPageEmpty10}{\showpaste} 6898\tab{5}\spadcommand{createPrimitiveNormalPoly(8)$FFPOLY(GF5)\free{GF5 }} 6899\end{paste}\end{patch} 6900 6901\begin{patch}{ugxProblemFiniteUtilityPagePatch11} 6902\begin{paste}{ugxProblemFiniteUtilityPageFull11}{ugxProblemFiniteUtilityPageEmpty11} 6903\pastebutton{ugxProblemFiniteUtilityPageFull11}{\hidepaste} 6904\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }} 6905\indentrel{3}\begin{verbatim} 6906 Type: Type 6907\end{verbatim} 6908\indentrel{-3}\end{paste}\end{patch} 6909 6910\begin{patch}{ugxProblemFiniteUtilityPageEmpty11} 6911\begin{paste}{ugxProblemFiniteUtilityPageEmpty11}{ugxProblemFiniteUtilityPagePatch11} 6912\pastebutton{ugxProblemFiniteUtilityPageEmpty11}{\showpaste} 6913\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }} 6914\end{paste}\end{patch} 6915 6916\begin{patch}{ugxProblemFiniteUtilityPagePatch12} 6917\begin{paste}{ugxProblemFiniteUtilityPageFull12}{ugxProblemFiniteUtilityPageEmpty12} 6918\pastebutton{ugxProblemFiniteUtilityPageFull12}{\hidepaste} 6919\tab{5}\spadcommand{h := monomial(1,8)$SUP(GF5)\bound{h }\free{GF5 }} 6920\indentrel{3}\begin{verbatim} 6921 8 6922 (12) ? 6923 Type: SparseUnivariatePolynomial(PrimeField(5)) 6924\end{verbatim} 6925\indentrel{-3}\end{paste}\end{patch} 6926 6927\begin{patch}{ugxProblemFiniteUtilityPageEmpty12} 6928\begin{paste}{ugxProblemFiniteUtilityPageEmpty12}{ugxProblemFiniteUtilityPagePatch12} 6929\pastebutton{ugxProblemFiniteUtilityPageEmpty12}{\showpaste} 6930\tab{5}\spadcommand{h := monomial(1,8)$SUP(GF5)\bound{h }\free{GF5 }} 6931\end{paste}\end{patch} 6932 6933\begin{patch}{ugxProblemFiniteUtilityPagePatch13} 6934\begin{paste}{ugxProblemFiniteUtilityPageFull13}{ugxProblemFiniteUtilityPageEmpty13} 6935\pastebutton{ugxProblemFiniteUtilityPageFull13}{\hidepaste} 6936\tab{5}\spadcommand{nh := nextIrreduciblePoly(h)$FFPOLY(GF5)\bound{nh }\free{h }} 6937\indentrel{3}\begin{verbatim} 6938 8 6939 (13) ? + 2 6940Type: Union(SparseUnivariatePolynomial(PrimeField(5)),...) 6941\end{verbatim} 6942\indentrel{-3}\end{paste}\end{patch} 6943 6944\begin{patch}{ugxProblemFiniteUtilityPageEmpty13} 6945\begin{paste}{ugxProblemFiniteUtilityPageEmpty13}{ugxProblemFiniteUtilityPagePatch13} 6946\pastebutton{ugxProblemFiniteUtilityPageEmpty13}{\showpaste} 6947\tab{5}\spadcommand{nh := nextIrreduciblePoly(h)$FFPOLY(GF5)\bound{nh }\free{h }} 6948\end{paste}\end{patch} 6949 6950\begin{patch}{ugxProblemFiniteUtilityPagePatch14} 6951\begin{paste}{ugxProblemFiniteUtilityPageFull14}{ugxProblemFiniteUtilityPageEmpty14} 6952\pastebutton{ugxProblemFiniteUtilityPageFull14}{\hidepaste} 6953\tab{5}\spadcommand{createIrreduciblePoly(3)$FFPOLY(GF5)\free{GF5 }} 6954\indentrel{3}\begin{verbatim} 6955 3 6956 (14) ? + ? + 1 6957 Type: SparseUnivariatePolynomial(PrimeField(5)) 6958\end{verbatim} 6959\indentrel{-3}\end{paste}\end{patch} 6960 6961\begin{patch}{ugxProblemFiniteUtilityPageEmpty14} 6962\begin{paste}{ugxProblemFiniteUtilityPageEmpty14}{ugxProblemFiniteUtilityPagePatch14} 6963\pastebutton{ugxProblemFiniteUtilityPageEmpty14}{\showpaste} 6964\tab{5}\spadcommand{createIrreduciblePoly(3)$FFPOLY(GF5)\free{GF5 }} 6965\end{paste}\end{patch} 6966 6967\begin{patch}{ugxProblemFiniteUtilityPagePatch15} 6968\begin{paste}{ugxProblemFiniteUtilityPageFull15}{ugxProblemFiniteUtilityPageEmpty15} 6969\pastebutton{ugxProblemFiniteUtilityPageFull15}{\hidepaste} 6970\tab{5}\spadcommand{nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)\free{nh }} 6971\indentrel{3}\begin{verbatim} 6972 8 6973 (15) ? + 3 6974Type: Union(SparseUnivariatePolynomial(PrimeField(5)),...) 6975\end{verbatim} 6976\indentrel{-3}\end{paste}\end{patch} 6977 6978\begin{patch}{ugxProblemFiniteUtilityPageEmpty15} 6979\begin{paste}{ugxProblemFiniteUtilityPageEmpty15}{ugxProblemFiniteUtilityPagePatch15} 6980\pastebutton{ugxProblemFiniteUtilityPageEmpty15}{\showpaste} 6981\tab{5}\spadcommand{nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)\free{nh }} 6982\end{paste}\end{patch} 6983 6984\begin{patch}{ugxProblemFiniteUtilityPagePatch16} 6985\begin{paste}{ugxProblemFiniteUtilityPageFull16}{ugxProblemFiniteUtilityPageEmpty16} 6986\pastebutton{ugxProblemFiniteUtilityPageFull16}{\hidepaste} 6987\tab{5}\spadcommand{numberOfIrreduciblePoly(5)$FFPOLY(GF5)\free{GF5 }} 6988\indentrel{3}\begin{verbatim} 6989 (16) 624 6990 Type: PositiveInteger 6991\end{verbatim} 6992\indentrel{-3}\end{paste}\end{patch} 6993 6994\begin{patch}{ugxProblemFiniteUtilityPageEmpty16} 6995\begin{paste}{ugxProblemFiniteUtilityPageEmpty16}{ugxProblemFiniteUtilityPagePatch16} 6996\pastebutton{ugxProblemFiniteUtilityPageEmpty16}{\showpaste} 6997\tab{5}\spadcommand{numberOfIrreduciblePoly(5)$FFPOLY(GF5)\free{GF5 }} 6998\end{paste}\end{patch} 6999 7000\begin{patch}{ugxProblemFiniteUtilityPagePatch17} 7001\begin{paste}{ugxProblemFiniteUtilityPageFull17}{ugxProblemFiniteUtilityPageEmpty17} 7002\pastebutton{ugxProblemFiniteUtilityPageFull17}{\hidepaste} 7003\tab{5}\spadcommand{numberOfPrimitivePoly(3)$FFPOLY(GF5)\free{GF5 }} 7004\indentrel{3}\begin{verbatim} 7005 (17) 20 7006 Type: PositiveInteger 7007\end{verbatim} 7008\indentrel{-3}\end{paste}\end{patch} 7009 7010\begin{patch}{ugxProblemFiniteUtilityPageEmpty17} 7011\begin{paste}{ugxProblemFiniteUtilityPageEmpty17}{ugxProblemFiniteUtilityPagePatch17} 7012\pastebutton{ugxProblemFiniteUtilityPageEmpty17}{\showpaste} 7013\tab{5}\spadcommand{numberOfPrimitivePoly(3)$FFPOLY(GF5)\free{GF5 }} 7014\end{paste}\end{patch} 7015 7016\begin{patch}{ugxProblemFiniteUtilityPagePatch18} 7017\begin{paste}{ugxProblemFiniteUtilityPageFull18}{ugxProblemFiniteUtilityPageEmpty18} 7018\pastebutton{ugxProblemFiniteUtilityPageFull18}{\hidepaste} 7019\tab{5}\spadcommand{m := monomial(1,1)$SUP(GF5)\bound{m }\free{GF5 }} 7020\indentrel{3}\begin{verbatim} 7021 (18) ? 7022 Type: SparseUnivariatePolynomial(PrimeField(5)) 7023\end{verbatim} 7024\indentrel{-3}\end{paste}\end{patch} 7025 7026\begin{patch}{ugxProblemFiniteUtilityPageEmpty18} 7027\begin{paste}{ugxProblemFiniteUtilityPageEmpty18}{ugxProblemFiniteUtilityPagePatch18} 7028\pastebutton{ugxProblemFiniteUtilityPageEmpty18}{\showpaste} 7029\tab{5}\spadcommand{m := monomial(1,1)$SUP(GF5)\bound{m }\free{GF5 }} 7030\end{paste}\end{patch} 7031 7032\begin{patch}{ugxProblemFiniteUtilityPagePatch19} 7033\begin{paste}{ugxProblemFiniteUtilityPageFull19}{ugxProblemFiniteUtilityPageEmpty19} 7034\pastebutton{ugxProblemFiniteUtilityPageFull19}{\hidepaste} 7035\tab{5}\spadcommand{f := m^3 + 4*m^2 + m + 2\bound{fx }\free{m }} 7036\indentrel{3}\begin{verbatim} 7037 3 2 7038 (19) ? + 4 ? + ? + 2 7039 Type: SparseUnivariatePolynomial(PrimeField(5)) 7040\end{verbatim} 7041\indentrel{-3}\end{paste}\end{patch} 7042 7043\begin{patch}{ugxProblemFiniteUtilityPageEmpty19} 7044\begin{paste}{ugxProblemFiniteUtilityPageEmpty19}{ugxProblemFiniteUtilityPagePatch19} 7045\pastebutton{ugxProblemFiniteUtilityPageEmpty19}{\showpaste} 7046\tab{5}\spadcommand{f := m^3 + 4*m^2 + m + 2\bound{fx }\free{m }} 7047\end{paste}\end{patch} 7048 7049\begin{patch}{ugxProblemFiniteUtilityPagePatch20} 7050\begin{paste}{ugxProblemFiniteUtilityPageFull20}{ugxProblemFiniteUtilityPageEmpty20} 7051\pastebutton{ugxProblemFiniteUtilityPageFull20}{\hidepaste} 7052\tab{5}\spadcommand{f1 := nextPrimitivePoly(f)$FFPOLY(GF5)\free{fx }\bound{f1 }} 7053\indentrel{3}\begin{verbatim} 7054 3 2 7055 (20) ? + 4 ? + 4 ? + 2 7056Type: Union(SparseUnivariatePolynomial(PrimeField(5)),...) 7057\end{verbatim} 7058\indentrel{-3}\end{paste}\end{patch} 7059 7060\begin{patch}{ugxProblemFiniteUtilityPageEmpty20} 7061\begin{paste}{ugxProblemFiniteUtilityPageEmpty20}{ugxProblemFiniteUtilityPagePatch20} 7062\pastebutton{ugxProblemFiniteUtilityPageEmpty20}{\showpaste} 7063\tab{5}\spadcommand{f1 := nextPrimitivePoly(f)$FFPOLY(GF5)\free{fx }\bound{f1 }} 7064\end{paste}\end{patch} 7065 7066\begin{patch}{ugxProblemFiniteUtilityPagePatch21} 7067\begin{paste}{ugxProblemFiniteUtilityPageFull21}{ugxProblemFiniteUtilityPageEmpty21} 7068\pastebutton{ugxProblemFiniteUtilityPageFull21}{\hidepaste} 7069\tab{5}\spadcommand{nextPrimitivePoly(f1)$FFPOLY(GF5)\free{f1 }} 7070\indentrel{3}\begin{verbatim} 7071 3 7072 (21) ? + 3 ? + 3 7073Type: Union(SparseUnivariatePolynomial(PrimeField(5)),...) 7074\end{verbatim} 7075\indentrel{-3}\end{paste}\end{patch} 7076 7077\begin{patch}{ugxProblemFiniteUtilityPageEmpty21} 7078\begin{paste}{ugxProblemFiniteUtilityPageEmpty21}{ugxProblemFiniteUtilityPagePatch21} 7079\pastebutton{ugxProblemFiniteUtilityPageEmpty21}{\showpaste} 7080\tab{5}\spadcommand{nextPrimitivePoly(f1)$FFPOLY(GF5)\free{f1 }} 7081\end{paste}\end{patch} 7082 7083\begin{patch}{ugxProblemFiniteUtilityPagePatch22} 7084\begin{paste}{ugxProblemFiniteUtilityPageFull22}{ugxProblemFiniteUtilityPageEmpty22} 7085\pastebutton{ugxProblemFiniteUtilityPageFull22}{\hidepaste} 7086\tab{5}\spadcommand{f := m^3 + m^2 + 4*m + 1\bound{fy }\free{m }} 7087\indentrel{3}\begin{verbatim} 7088 3 2 7089 (22) ? + ? + 4 ? + 1 7090 Type: SparseUnivariatePolynomial(PrimeField(5)) 7091\end{verbatim} 7092\indentrel{-3}\end{paste}\end{patch} 7093 7094\begin{patch}{ugxProblemFiniteUtilityPageEmpty22} 7095\begin{paste}{ugxProblemFiniteUtilityPageEmpty22}{ugxProblemFiniteUtilityPagePatch22} 7096\pastebutton{ugxProblemFiniteUtilityPageEmpty22}{\showpaste} 7097\tab{5}\spadcommand{f := m^3 + m^2 + 4*m + 1\bound{fy }\free{m }} 7098\end{paste}\end{patch} 7099 7100\begin{patch}{ugxProblemFiniteUtilityPagePatch23} 7101\begin{paste}{ugxProblemFiniteUtilityPageFull23}{ugxProblemFiniteUtilityPageEmpty23} 7102\pastebutton{ugxProblemFiniteUtilityPageFull23}{\hidepaste} 7103\tab{5}\spadcommand{f1 := nextNormalPoly(f)$FFPOLY(GF5)\free{fy }\bound{f1y }} 7104\indentrel{3}\begin{verbatim} 7105 3 2 7106 (23) ? + ? + 4 ? + 3 7107Type: Union(SparseUnivariatePolynomial(PrimeField(5)),...) 7108\end{verbatim} 7109\indentrel{-3}\end{paste}\end{patch} 7110 7111\begin{patch}{ugxProblemFiniteUtilityPageEmpty23} 7112\begin{paste}{ugxProblemFiniteUtilityPageEmpty23}{ugxProblemFiniteUtilityPagePatch23} 7113\pastebutton{ugxProblemFiniteUtilityPageEmpty23}{\showpaste} 7114\tab{5}\spadcommand{f1 := nextNormalPoly(f)$FFPOLY(GF5)\free{fy }\bound{f1y }} 7115\end{paste}\end{patch} 7116 7117\begin{patch}{ugxProblemFiniteUtilityPagePatch24} 7118\begin{paste}{ugxProblemFiniteUtilityPageFull24}{ugxProblemFiniteUtilityPageEmpty24} 7119\pastebutton{ugxProblemFiniteUtilityPageFull24}{\hidepaste} 7120\tab{5}\spadcommand{nextNormalPoly(f1)$FFPOLY(GF5)\free{f1y }} 7121\indentrel{3}\begin{verbatim} 7122 3 2 7123 (24) ? + 2 ? + 1 7124Type: Union(SparseUnivariatePolynomial(PrimeField(5)),...) 7125\end{verbatim} 7126\indentrel{-3}\end{paste}\end{patch} 7127 7128\begin{patch}{ugxProblemFiniteUtilityPageEmpty24} 7129\begin{paste}{ugxProblemFiniteUtilityPageEmpty24}{ugxProblemFiniteUtilityPagePatch24} 7130\pastebutton{ugxProblemFiniteUtilityPageEmpty24}{\showpaste} 7131\tab{5}\spadcommand{nextNormalPoly(f1)$FFPOLY(GF5)\free{f1y }} 7132\end{paste}\end{patch} 7133 7134\begin{patch}{ugxProblemFiniteUtilityPagePatch25} 7135\begin{paste}{ugxProblemFiniteUtilityPageFull25}{ugxProblemFiniteUtilityPageEmpty25} 7136\pastebutton{ugxProblemFiniteUtilityPageFull25}{\hidepaste} 7137\tab{5}\spadcommand{GF16 := FFX(FFX(PF 2,2),2);\bound{GF16 }} 7138\indentrel{3}\begin{verbatim} 7139 Type: Type 7140\end{verbatim} 7141\indentrel{-3}\end{paste}\end{patch} 7142 7143\begin{patch}{ugxProblemFiniteUtilityPageEmpty25} 7144\begin{paste}{ugxProblemFiniteUtilityPageEmpty25}{ugxProblemFiniteUtilityPagePatch25} 7145\pastebutton{ugxProblemFiniteUtilityPageEmpty25}{\showpaste} 7146\tab{5}\spadcommand{GF16 := FFX(FFX(PF 2,2),2);\bound{GF16 }} 7147\end{paste}\end{patch} 7148 7149\begin{patch}{ugxProblemFiniteUtilityPagePatch26} 7150\begin{paste}{ugxProblemFiniteUtilityPageFull26}{ugxProblemFiniteUtilityPageEmpty26} 7151\pastebutton{ugxProblemFiniteUtilityPageFull26}{\hidepaste} 7152\tab{5}\spadcommand{createIrreduciblePoly(5)$FFPOLY(GF16)\free{GF16 }} 7153\indentrel{3}\begin{verbatim} 7154 5 7155 (26) ? + ? + %JS 7156Type: SparseUnivariatePolynomial(FiniteFieldExtension(FiniteFieldExtension(PrimeField(2),2),2)) 7157\end{verbatim} 7158\indentrel{-3}\end{paste}\end{patch} 7159 7160\begin{patch}{ugxProblemFiniteUtilityPageEmpty26} 7161\begin{paste}{ugxProblemFiniteUtilityPageEmpty26}{ugxProblemFiniteUtilityPagePatch26} 7162\pastebutton{ugxProblemFiniteUtilityPageEmpty26}{\showpaste} 7163\tab{5}\spadcommand{createIrreduciblePoly(5)$FFPOLY(GF16)\free{GF16 }} 7164\end{paste}\end{patch} 7165 7166\begin{patch}{ugxProblemFiniteUtilityPagePatch27} 7167\begin{paste}{ugxProblemFiniteUtilityPageFull27}{ugxProblemFiniteUtilityPageEmpty27} 7168\pastebutton{ugxProblemFiniteUtilityPageFull27}{\hidepaste} 7169\tab{5}\spadcommand{random(5)$FFPOLY(GF16)\free{GF16 }} 7170\indentrel{3}\begin{verbatim} 7171 (27) 7172 5 4 3 7173 ? + (%JS %JW + 1)? + ((%JS + 1)%JW + %JS + 1)? 7174 + 7175 2 7176 ? + ? + %JS %JW + 1 7177Type: SparseUnivariatePolynomial(FiniteFieldExtension(FiniteFieldExtension(PrimeField(2),2),2)) 7178\end{verbatim} 7179\indentrel{-3}\end{paste}\end{patch} 7180 7181\begin{patch}{ugxProblemFiniteUtilityPageEmpty27} 7182\begin{paste}{ugxProblemFiniteUtilityPageEmpty27}{ugxProblemFiniteUtilityPagePatch27} 7183\pastebutton{ugxProblemFiniteUtilityPageEmpty27}{\showpaste} 7184\tab{5}\spadcommand{random(5)$FFPOLY(GF16)\free{GF16 }} 7185\end{paste}\end{patch} 7186 7187\begin{patch}{ugxProblemFiniteUtilityPagePatch28} 7188\begin{paste}{ugxProblemFiniteUtilityPageFull28}{ugxProblemFiniteUtilityPageEmpty28} 7189\pastebutton{ugxProblemFiniteUtilityPageFull28}{\hidepaste} 7190\tab{5}\spadcommand{random(3,9)$FFPOLY(GF16)\free{GF16 }} 7191\indentrel{3}\begin{verbatim} 7192 (28) 7193 7 6 5 4 7194 ? + (%JW + %JS + 1)? + (%JW + %JS + 1)? + ? 7195 + 7196 3 2 7197 ((%JS + 1)%JW + %JS + 1)? + (%JW + %JS + 1)? 7198 + 7199 ((%JS + 1)%JW + %JS + 1)? + (%JS + 1)%JW + %JS 7200Type: SparseUnivariatePolynomial(FiniteFieldExtension(FiniteFieldExtension(PrimeField(2),2),2)) 7201\end{verbatim} 7202\indentrel{-3}\end{paste}\end{patch} 7203 7204\begin{patch}{ugxProblemFiniteUtilityPageEmpty28} 7205\begin{paste}{ugxProblemFiniteUtilityPageEmpty28}{ugxProblemFiniteUtilityPagePatch28} 7206\pastebutton{ugxProblemFiniteUtilityPageEmpty28}{\showpaste} 7207\tab{5}\spadcommand{random(3,9)$FFPOLY(GF16)\free{GF16 }} 7208\end{paste}\end{patch} 7209 7210\begin{patch}{ugxProblemFiniteUtilityPagePatch29} 7211\begin{paste}{ugxProblemFiniteUtilityPageFull29}{ugxProblemFiniteUtilityPageEmpty29} 7212\pastebutton{ugxProblemFiniteUtilityPageFull29}{\hidepaste} 7213\tab{5}\spadcommand{GF2 := PrimeField 2;\bound{GF2 }} 7214\indentrel{3}\begin{verbatim} 7215 Type: Type 7216\end{verbatim} 7217\indentrel{-3}\end{paste}\end{patch} 7218 7219\begin{patch}{ugxProblemFiniteUtilityPageEmpty29} 7220\begin{paste}{ugxProblemFiniteUtilityPageEmpty29}{ugxProblemFiniteUtilityPagePatch29} 7221\pastebutton{ugxProblemFiniteUtilityPageEmpty29}{\showpaste} 7222\tab{5}\spadcommand{GF2 := PrimeField 2;\bound{GF2 }} 7223\end{paste}\end{patch} 7224 7225\begin{patch}{ugxProblemFiniteUtilityPagePatch30} 7226\begin{paste}{ugxProblemFiniteUtilityPageFull30}{ugxProblemFiniteUtilityPageEmpty30} 7227\pastebutton{ugxProblemFiniteUtilityPageFull30}{\hidepaste} 7228\tab{5}\spadcommand{F := FFX(GF2,12)\bound{F }\free{GF2 }} 7229\indentrel{3}\begin{verbatim} 7230 (30) FiniteFieldExtension(PrimeField(2),12) 7231 Type: Type 7232\end{verbatim} 7233\indentrel{-3}\end{paste}\end{patch} 7234 7235\begin{patch}{ugxProblemFiniteUtilityPageEmpty30} 7236\begin{paste}{ugxProblemFiniteUtilityPageEmpty30}{ugxProblemFiniteUtilityPagePatch30} 7237\pastebutton{ugxProblemFiniteUtilityPageEmpty30}{\showpaste} 7238\tab{5}\spadcommand{F := FFX(GF2,12)\bound{F }\free{GF2 }} 7239\end{paste}\end{patch} 7240 7241\begin{patch}{ugxProblemFiniteUtilityPagePatch31} 7242\begin{paste}{ugxProblemFiniteUtilityPageFull31}{ugxProblemFiniteUtilityPageEmpty31} 7243\pastebutton{ugxProblemFiniteUtilityPageFull31}{\hidepaste} 7244\tab{5}\spadcommand{f := createIrreduciblePoly(6)$FFPOLY(GF2)\bound{fz }\free{GF2 }} 7245\indentrel{3}\begin{verbatim} 7246 6 7247 (31) ? + ? + 1 7248 Type: SparseUnivariatePolynomial(PrimeField(2)) 7249\end{verbatim} 7250\indentrel{-3}\end{paste}\end{patch} 7251 7252\begin{patch}{ugxProblemFiniteUtilityPageEmpty31} 7253\begin{paste}{ugxProblemFiniteUtilityPageEmpty31}{ugxProblemFiniteUtilityPagePatch31} 7254\pastebutton{ugxProblemFiniteUtilityPageEmpty31}{\showpaste} 7255\tab{5}\spadcommand{f := createIrreduciblePoly(6)$FFPOLY(GF2)\bound{fz }\free{GF2 }} 7256\end{paste}\end{patch} 7257 7258\begin{patch}{ugxProblemFiniteUtilityPagePatch32} 7259\begin{paste}{ugxProblemFiniteUtilityPageFull32}{ugxProblemFiniteUtilityPageEmpty32} 7260\pastebutton{ugxProblemFiniteUtilityPageFull32}{\hidepaste} 7261\tab{5}\spadcommand{root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)\free{F GF2 fz }\bound{root }} 7262\indentrel{3}\begin{verbatim} 7263 11 8 7 5 7264 (32) %JP + %JP + %JP + %JP + %JP + 1 7265 Type: FiniteFieldExtension(PrimeField(2),12) 7266\end{verbatim} 7267\indentrel{-3}\end{paste}\end{patch} 7268 7269\begin{patch}{ugxProblemFiniteUtilityPageEmpty32} 7270\begin{paste}{ugxProblemFiniteUtilityPageEmpty32}{ugxProblemFiniteUtilityPagePatch32} 7271\pastebutton{ugxProblemFiniteUtilityPageEmpty32}{\showpaste} 7272\tab{5}\spadcommand{root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)\free{F GF2 fz }\bound{root }} 7273\end{paste}\end{patch} 7274 7275\begin{patch}{ugProblemEigenPagePatch1} 7276\begin{paste}{ugProblemEigenPageFull1}{ugProblemEigenPageEmpty1} 7277\pastebutton{ugProblemEigenPageFull1}{\hidepaste} 7278\tab{5}\spadcommand{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]]\bound{m1 }} 7279\indentrel{3}\begin{verbatim} 7280 +1 2 1 + 7281 | | 7282 (1) |2 1 - 2| 7283 | | 7284 +1 - 2 4 + 7285 Type: Matrix(Integer) 7286\end{verbatim} 7287\indentrel{-3}\end{paste}\end{patch} 7288 7289\begin{patch}{ugProblemEigenPageEmpty1} 7290\begin{paste}{ugProblemEigenPageEmpty1}{ugProblemEigenPagePatch1} 7291\pastebutton{ugProblemEigenPageEmpty1}{\showpaste} 7292\tab{5}\spadcommand{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]]\bound{m1 }} 7293\end{paste}\end{patch} 7294 7295\begin{patch}{ugProblemEigenPagePatch2} 7296\begin{paste}{ugProblemEigenPageFull2}{ugProblemEigenPageEmpty2} 7297\pastebutton{ugProblemEigenPageFull2}{\hidepaste} 7298\tab{5}\spadcommand{leig := eigenvalues(m1)\free{m1 }\bound{leig }} 7299\indentrel{3}\begin{verbatim} 7300 2 7301 (2) [5, %JY | %JY - %JY - 5] 7302Type: List(Union(Fraction(Polynomial(Integer)),SuchThat(Symbol,Polynomial(Integer)))) 7303\end{verbatim} 7304\indentrel{-3}\end{paste}\end{patch} 7305 7306\begin{patch}{ugProblemEigenPageEmpty2} 7307\begin{paste}{ugProblemEigenPageEmpty2}{ugProblemEigenPagePatch2} 7308\pastebutton{ugProblemEigenPageEmpty2}{\showpaste} 7309\tab{5}\spadcommand{leig := eigenvalues(m1)\free{m1 }\bound{leig }} 7310\end{paste}\end{patch} 7311 7312\begin{patch}{ugProblemEigenPagePatch3} 7313\begin{paste}{ugProblemEigenPageFull3}{ugProblemEigenPageEmpty3} 7314\pastebutton{ugProblemEigenPageFull3}{\hidepaste} 7315\tab{5}\spadcommand{eigenvector(first(leig),m1)\free{m1 leig }} 7316\indentrel{3}\begin{verbatim} 7317 + 0 + 7318 | | 7319 | 1| 7320 (3) [|- -|] 7321 | 2| 7322 | | 7323 + 1 + 7324Type: List(Matrix(Fraction(Polynomial(Fraction(Integer))))) 7325\end{verbatim} 7326\indentrel{-3}\end{paste}\end{patch} 7327 7328\begin{patch}{ugProblemEigenPageEmpty3} 7329\begin{paste}{ugProblemEigenPageEmpty3}{ugProblemEigenPagePatch3} 7330\pastebutton{ugProblemEigenPageEmpty3}{\showpaste} 7331\tab{5}\spadcommand{eigenvector(first(leig),m1)\free{m1 leig }} 7332\end{paste}\end{patch} 7333 7334\begin{patch}{ugProblemEigenPagePatch4} 7335\begin{paste}{ugProblemEigenPageFull4}{ugProblemEigenPageEmpty4} 7336\pastebutton{ugProblemEigenPageFull4}{\hidepaste} 7337\tab{5}\spadcommand{eigenvectors(m1)\free{m1 }} 7338\indentrel{3}\begin{verbatim} 7339 (4) 7340 + 0 + 7341 | | 7342 | 1| 7343 [[eigval = 5, eigmult = 1, eigvec = [|- -|]], 7344 | 2| 7345 | | 7346 + 1 + 7347 7348 2 7349 [eigval = (%KA | %KA - %KA - 5), eigmult = 1, 7350 +%KA+ 7351 | | 7352 eigvec = [| 2 |]] 7353 | | 7354 + 1 + 7355 ] 7356Type: List(Record(eigval: Union(Fraction(Polynomial(Integer)),SuchThat(Symbol,Polynomial(Integer))),eigmult: NonNegativeInteger,eigvec: List(Matrix(Fraction(Polynomial(Integer)))))) 7357\end{verbatim} 7358\indentrel{-3}\end{paste}\end{patch} 7359 7360\begin{patch}{ugProblemEigenPageEmpty4} 7361\begin{paste}{ugProblemEigenPageEmpty4}{ugProblemEigenPagePatch4} 7362\pastebutton{ugProblemEigenPageEmpty4}{\showpaste} 7363\tab{5}\spadcommand{eigenvectors(m1)\free{m1 }} 7364\end{paste}\end{patch} 7365 7366\begin{patch}{ugProblemEigenPagePatch5} 7367\begin{paste}{ugProblemEigenPageFull5}{ugProblemEigenPageEmpty5} 7368\pastebutton{ugProblemEigenPageFull5}{\hidepaste} 7369\tab{5}\spadcommand{radicalEigenvectors(m1)\free{m1 }} 7370\indentrel{3}\begin{verbatim} 7371 (5) 7372 [ 7373 +--+ 7374 \|21 + 1 7375 [radval = ---------, radmult = 1, 7376 2 7377 + +--+ + 7378 |\|21 + 1| 7379 |---------| 7380 radvect = [| 2 |]] 7381 | | 7382 | 2 | 7383 | | 7384 + 1 + 7385 , 7386 7387 +--+ 7388 - \|21 + 1 7389 [radval = -----------, radmult = 1, 7390 2 7391 + +--+ + 7392 |- \|21 + 1| 7393 |-----------| 7394 radvect = [| 2 |]] 7395 | | 7396 | 2 | 7397 | | 7398 + 1 + 7399 , 7400 + 0 + 7401 | | 7402 | 1| 7403 [radval = 5, radmult = 1, radvect = [|- -|]]] 7404 | 2| 7405 | | 7406 + 1 + 7407Type: List(Record(radval: Expression(Integer),radmult: Integer,radvect: List(Matrix(Expression(Integer))))) 7408\end{verbatim} 7409\indentrel{-3}\end{paste}\end{patch} 7410 7411\begin{patch}{ugProblemEigenPageEmpty5} 7412\begin{paste}{ugProblemEigenPageEmpty5}{ugProblemEigenPagePatch5} 7413\pastebutton{ugProblemEigenPageEmpty5}{\showpaste} 7414\tab{5}\spadcommand{radicalEigenvectors(m1)\free{m1 }} 7415\end{paste}\end{patch} 7416 7417\begin{patch}{ugProblemEigenPagePatch6} 7418\begin{paste}{ugProblemEigenPageFull6}{ugProblemEigenPageEmpty6} 7419\pastebutton{ugProblemEigenPageFull6}{\hidepaste} 7420\tab{5}\spadcommand{realEigenvectors(m1,1/1000)\free{m1 }} 7421\indentrel{3}\begin{verbatim} 7422 (6) 7423 + 0 + 7424 | | 7425 | 1| 7426 [[outval = 5, outmult = 1, outvect = [|- -|]], 7427 | 2| 7428 | | 7429 + 1 + 7430 +5717+ 7431 |----| 7432 5717 |2048| 7433 [outval = ----, outmult = 1, outvect = [| |]], 7434 2048 | 2 | 7435 | | 7436 + 1 + 7437 + 3669+ 7438 |- ----| 7439 3669 | 2048| 7440 [outval = - ----, outmult = 1, outvect = [| |]]] 7441 2048 | 2 | 7442 | | 7443 + 1 + 7444Type: List(Record(outval: Fraction(Integer),outmult: Integer,outvect: List(Matrix(Fraction(Integer))))) 7445\end{verbatim} 7446\indentrel{-3}\end{paste}\end{patch} 7447 7448\begin{patch}{ugProblemEigenPageEmpty6} 7449\begin{paste}{ugProblemEigenPageEmpty6}{ugProblemEigenPagePatch6} 7450\pastebutton{ugProblemEigenPageEmpty6}{\showpaste} 7451\tab{5}\spadcommand{realEigenvectors(m1,1/1000)\free{m1 }} 7452\end{paste}\end{patch} 7453 7454\begin{patch}{ugProblemEigenPagePatch7} 7455\begin{paste}{ugProblemEigenPageFull7}{ugProblemEigenPageEmpty7} 7456\pastebutton{ugProblemEigenPageFull7}{\hidepaste} 7457\tab{5}\spadcommand{eigenMatrix(m1)\free{m1 }} 7458\indentrel{3}\begin{verbatim} 7459 + +--+ +--+ + 7460 |\|21 + 1 - \|21 + 1 | 7461 |--------- ----------- 0 | 7462 | 2 2 | 7463 (7) | | 7464 | 1| 7465 | 2 2 - -| 7466 | 2| 7467 | | 7468 + 1 1 1 + 7469 Type: Union(Matrix(Expression(Integer)),...) 7470\end{verbatim} 7471\indentrel{-3}\end{paste}\end{patch} 7472 7473\begin{patch}{ugProblemEigenPageEmpty7} 7474\begin{paste}{ugProblemEigenPageEmpty7}{ugProblemEigenPagePatch7} 7475\pastebutton{ugProblemEigenPageEmpty7}{\showpaste} 7476\tab{5}\spadcommand{eigenMatrix(m1)\free{m1 }} 7477\end{paste}\end{patch} 7478 7479\begin{patch}{ugProblemEigenPagePatch8} 7480\begin{paste}{ugProblemEigenPageFull8}{ugProblemEigenPageEmpty8} 7481\pastebutton{ugProblemEigenPageFull8}{\hidepaste} 7482\tab{5}\spadcommand{m2 := matrix [[-5,-2],[18,7]]\bound{m2 }} 7483\indentrel{3}\begin{verbatim} 7484 +- 5 - 2+ 7485 (8) | | 7486 +18 7 + 7487 Type: Matrix(Integer) 7488\end{verbatim} 7489\indentrel{-3}\end{paste}\end{patch} 7490 7491\begin{patch}{ugProblemEigenPageEmpty8} 7492\begin{paste}{ugProblemEigenPageEmpty8}{ugProblemEigenPagePatch8} 7493\pastebutton{ugProblemEigenPageEmpty8}{\showpaste} 7494\tab{5}\spadcommand{m2 := matrix [[-5,-2],[18,7]]\bound{m2 }} 7495\end{paste}\end{patch} 7496 7497\begin{patch}{ugProblemEigenPagePatch9} 7498\begin{paste}{ugProblemEigenPageFull9}{ugProblemEigenPageEmpty9} 7499\pastebutton{ugProblemEigenPageFull9}{\hidepaste} 7500\tab{5}\spadcommand{eigenMatrix(m2)\free{m2 }} 7501\indentrel{3}\begin{verbatim} 7502 (9) "failed" 7503 Type: Union("failed",...) 7504\end{verbatim} 7505\indentrel{-3}\end{paste}\end{patch} 7506 7507\begin{patch}{ugProblemEigenPageEmpty9} 7508\begin{paste}{ugProblemEigenPageEmpty9}{ugProblemEigenPagePatch9} 7509\pastebutton{ugProblemEigenPageEmpty9}{\showpaste} 7510\tab{5}\spadcommand{eigenMatrix(m2)\free{m2 }} 7511\end{paste}\end{patch} 7512 7513\begin{patch}{ugProblemEigenPagePatch10} 7514\begin{paste}{ugProblemEigenPageFull10}{ugProblemEigenPageEmpty10} 7515\pastebutton{ugProblemEigenPageFull10}{\hidepaste} 7516\tab{5}\spadcommand{m3 := matrix [[1,2],[2,1]]\bound{m3 }} 7517\indentrel{3}\begin{verbatim} 7518 +1 2+ 7519 (10) | | 7520 +2 1+ 7521 Type: Matrix(Integer) 7522\end{verbatim} 7523\indentrel{-3}\end{paste}\end{patch} 7524 7525\begin{patch}{ugProblemEigenPageEmpty10} 7526\begin{paste}{ugProblemEigenPageEmpty10}{ugProblemEigenPagePatch10} 7527\pastebutton{ugProblemEigenPageEmpty10}{\showpaste} 7528\tab{5}\spadcommand{m3 := matrix [[1,2],[2,1]]\bound{m3 }} 7529\end{paste}\end{patch} 7530 7531\begin{patch}{ugProblemEigenPagePatch11} 7532\begin{paste}{ugProblemEigenPageFull11}{ugProblemEigenPageEmpty11} 7533\pastebutton{ugProblemEigenPageFull11}{\hidepaste} 7534\tab{5}\spadcommand{orthonormalBasis(m3)\free{m3 }} 7535\indentrel{3}\begin{verbatim} 7536 + 1 + + 1 + 7537 |- ----| |----| 7538 | +-+| | +-+| 7539 | \|2 | |\|2 | 7540 (11) [| |, | |] 7541 | 1 | | 1 | 7542 | ---- | |----| 7543 | +-+ | | +-+| 7544 + \|2 + +\|2 + 7545 Type: List(Matrix(Expression(Integer))) 7546\end{verbatim} 7547\indentrel{-3}\end{paste}\end{patch} 7548 7549\begin{patch}{ugProblemEigenPageEmpty11} 7550\begin{paste}{ugProblemEigenPageEmpty11}{ugProblemEigenPagePatch11} 7551\pastebutton{ugProblemEigenPageEmpty11}{\showpaste} 7552\tab{5}\spadcommand{orthonormalBasis(m3)\free{m3 }} 7553\end{paste}\end{patch} 7554 7555\begin{patch}{ugProblemGaloisPagePatch1} 7556\begin{paste}{ugProblemGaloisPageFull1}{ugProblemGaloisPageEmpty1} 7557\pastebutton{ugProblemGaloisPageFull1}{\hidepaste} 7558\tab{5}\spadcommand{p := x^5 - 5*x + 12\bound{p }} 7559\indentrel{3}\begin{verbatim} 7560 5 7561 (1) x - 5 x + 12 7562 Type: Polynomial(Integer) 7563\end{verbatim} 7564\indentrel{-3}\end{paste}\end{patch} 7565 7566\begin{patch}{ugProblemGaloisPageEmpty1} 7567\begin{paste}{ugProblemGaloisPageEmpty1}{ugProblemGaloisPagePatch1} 7568\pastebutton{ugProblemGaloisPageEmpty1}{\showpaste} 7569\tab{5}\spadcommand{p := x^5 - 5*x + 12\bound{p }} 7570\end{paste}\end{patch} 7571 7572\begin{patch}{ugProblemGaloisPagePatch2} 7573\begin{paste}{ugProblemGaloisPageFull2}{ugProblemGaloisPageEmpty2} 7574\pastebutton{ugProblemGaloisPageFull2}{\hidepaste} 7575\tab{5}\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y)\free{p }\bound{q }} 7576\indentrel{3}\begin{verbatim} 7577 (2) 7578 25 21 17 15 13 7579 x - 50 x - 2375 x + 90000 x - 5000 x 7580 + 7581 11 9 7 5 7582 2700000 x + 250000 x + 18000000 x + 64000000 x 7583 Type: Polynomial(Integer) 7584\end{verbatim} 7585\indentrel{-3}\end{paste}\end{patch} 7586 7587\begin{patch}{ugProblemGaloisPageEmpty2} 7588\begin{paste}{ugProblemGaloisPageEmpty2}{ugProblemGaloisPagePatch2} 7589\pastebutton{ugProblemGaloisPageEmpty2}{\showpaste} 7590\tab{5}\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y)\free{p }\bound{q }} 7591\end{paste}\end{patch} 7592 7593\begin{patch}{ugProblemGaloisPagePatch3} 7594\begin{paste}{ugProblemGaloisPageFull3}{ugProblemGaloisPageEmpty3} 7595\pastebutton{ugProblemGaloisPageFull3}{\hidepaste} 7596\tab{5}\spadcommand{q1 := exquo(q, x^5)\free{q }\bound{q1 }} 7597\indentrel{3}\begin{verbatim} 7598 (3) 7599 20 16 12 10 8 7600 x - 50 x - 2375 x + 90000 x - 5000 x 7601 + 7602 6 4 2 7603 2700000 x + 250000 x + 18000000 x + 64000000 7604 Type: Union(Polynomial(Integer),...) 7605\end{verbatim} 7606\indentrel{-3}\end{paste}\end{patch} 7607 7608\begin{patch}{ugProblemGaloisPageEmpty3} 7609\begin{paste}{ugProblemGaloisPageEmpty3}{ugProblemGaloisPagePatch3} 7610\pastebutton{ugProblemGaloisPageEmpty3}{\showpaste} 7611\tab{5}\spadcommand{q1 := exquo(q, x^5)\free{q }\bound{q1 }} 7612\end{paste}\end{patch} 7613 7614\begin{patch}{ugProblemGaloisPagePatch4} 7615\begin{paste}{ugProblemGaloisPageFull4}{ugProblemGaloisPageEmpty4} 7616\pastebutton{ugProblemGaloisPageFull4}{\hidepaste} 7617\tab{5}\spadcommand{factoredQ := factor q1\free{q1 }\bound{factoredQ }} 7618\indentrel{3}\begin{verbatim} 7619 (4) 7620 10 8 6 4 2 7621 (x - 10 x - 75 x + 1500 x - 5500 x + 16000) 7622 * 7623 10 8 6 4 2 7624 (x + 10 x + 125 x + 500 x + 2500 x + 4000) 7625 Type: Factored(Polynomial(Integer)) 7626\end{verbatim} 7627\indentrel{-3}\end{paste}\end{patch} 7628 7629\begin{patch}{ugProblemGaloisPageEmpty4} 7630\begin{paste}{ugProblemGaloisPageEmpty4}{ugProblemGaloisPagePatch4} 7631\pastebutton{ugProblemGaloisPageEmpty4}{\showpaste} 7632\tab{5}\spadcommand{factoredQ := factor q1\free{q1 }\bound{factoredQ }} 7633\end{paste}\end{patch} 7634 7635\begin{patch}{ugProblemGaloisPagePatch5} 7636\begin{paste}{ugProblemGaloisPageFull5}{ugProblemGaloisPageEmpty5} 7637\pastebutton{ugProblemGaloisPageFull5}{\hidepaste} 7638\tab{5}\spadcommand{r := factorList(factoredQ).1.factor\free{factoredQ }\bound{r }} 7639\indentrel{3}\begin{verbatim} 7640 10 8 6 4 2 7641 (5) x - 10 x - 75 x + 1500 x - 5500 x + 16000 7642 Type: Polynomial(Integer) 7643\end{verbatim} 7644\indentrel{-3}\end{paste}\end{patch} 7645 7646\begin{patch}{ugProblemGaloisPageEmpty5} 7647\begin{paste}{ugProblemGaloisPageEmpty5}{ugProblemGaloisPagePatch5} 7648\pastebutton{ugProblemGaloisPageEmpty5}{\showpaste} 7649\tab{5}\spadcommand{r := factorList(factoredQ).1.factor\free{factoredQ }\bound{r }} 7650\end{paste}\end{patch} 7651 7652\begin{patch}{ugProblemGaloisPagePatch6} 7653\begin{paste}{ugProblemGaloisPageFull6}{ugProblemGaloisPageEmpty6} 7654\pastebutton{ugProblemGaloisPageFull6}{\hidepaste} 7655\tab{5}\spadcommand{beta:AN := rootOf(eval(r,x,b))\free{r }\bound{beta }} 7656\indentrel{3}\begin{verbatim} 7657 (6) b 7658 Type: AlgebraicNumber 7659\end{verbatim} 7660\indentrel{-3}\end{paste}\end{patch} 7661 7662\begin{patch}{ugProblemGaloisPageEmpty6} 7663\begin{paste}{ugProblemGaloisPageEmpty6}{ugProblemGaloisPagePatch6} 7664\pastebutton{ugProblemGaloisPageEmpty6}{\showpaste} 7665\tab{5}\spadcommand{beta:AN := rootOf(eval(r,x,b))\free{r }\bound{beta }} 7666\end{paste}\end{patch} 7667 7668\begin{patch}{ugProblemGaloisPagePatch7} 7669\begin{paste}{ugProblemGaloisPageFull7}{ugProblemGaloisPageEmpty7} 7670\pastebutton{ugProblemGaloisPageFull7}{\hidepaste} 7671\tab{5}\spadcommand{p := p::UP(x,INT)::UP(x,AN)\free{p }\bound{declareP }} 7672\indentrel{3}\begin{verbatim} 7673 5 7674 (7) x - 5 x + 12 7675 Type: UnivariatePolynomial(x,AlgebraicNumber) 7676\end{verbatim} 7677\indentrel{-3}\end{paste}\end{patch} 7678 7679\begin{patch}{ugProblemGaloisPageEmpty7} 7680\begin{paste}{ugProblemGaloisPageEmpty7}{ugProblemGaloisPagePatch7} 7681\pastebutton{ugProblemGaloisPageEmpty7}{\showpaste} 7682\tab{5}\spadcommand{p := p::UP(x,INT)::UP(x,AN)\free{p }\bound{declareP }} 7683\end{paste}\end{patch} 7684 7685\begin{patch}{ugProblemGaloisPagePatch8} 7686\begin{paste}{ugProblemGaloisPageFull8}{ugProblemGaloisPageEmpty8} 7687\pastebutton{ugProblemGaloisPageFull8}{\hidepaste} 7688\tab{5}\spadcommand{algFactors := factor(p,[beta])\free{declareP beta }\bound{algFactors }} 7689\indentrel{3}\begin{verbatim} 7690 (8) 7691 x 7692 + 7693 9 8 7 6 5 7694 - 85 b - 116 b + 780 b + 2640 b + 14895 b 7695 + 7696 4 3 2 7697 - 8820 b - 127050 b - 327000 b - 405200 b 7698 + 7699 2062400 7700 / 7701 1339200 7702 * 7703 8 6 4 2 7704 - 17 b + 156 b + 2979 b - 25410 b - 14080 7705 (x + ---------------------------------------------) 7706 66960 7707 * 7708 x 7709 + 7710 8 6 4 2 7711 143 b - 2100 b - 10485 b + 290550 b 7712 + 7713 - 334800 b - 960800 7714 / 7715 669600 7716 * 7717 x 7718 + 7719 8 6 4 2 7720 143 b - 2100 b - 10485 b + 290550 b 7721 + 7722 334800 b - 960800 7723 / 7724 669600 7725 * 7726 x 7727 + 7728 9 8 7 6 5 7729 85 b - 116 b - 780 b + 2640 b - 14895 b 7730 + 7731 4 3 2 7732 - 8820 b + 127050 b - 327000 b + 405200 b 7733 + 7734 2062400 7735 / 7736 1339200 7737Type: Factored(UnivariatePolynomial(x,AlgebraicNumber)) 7738\end{verbatim} 7739\indentrel{-3}\end{paste}\end{patch} 7740 7741\begin{patch}{ugProblemGaloisPageEmpty8} 7742\begin{paste}{ugProblemGaloisPageEmpty8}{ugProblemGaloisPagePatch8} 7743\pastebutton{ugProblemGaloisPageEmpty8}{\showpaste} 7744\tab{5}\spadcommand{algFactors := factor(p,[beta])\free{declareP beta }\bound{algFactors }} 7745\end{paste}\end{patch} 7746 7747\begin{patch}{ugProblemGaloisPagePatch9} 7748\begin{paste}{ugProblemGaloisPageFull9}{ugProblemGaloisPageEmpty9} 7749\pastebutton{ugProblemGaloisPageFull9}{\hidepaste} 7750\tab{5}\spadcommand{factor(p)\free{declareP }} 7751\indentrel{3}\begin{verbatim} 7752 5 7753 (9) x - 5 x + 12 7754Type: Factored(UnivariatePolynomial(x,AlgebraicNumber)) 7755\end{verbatim} 7756\indentrel{-3}\end{paste}\end{patch} 7757 7758\begin{patch}{ugProblemGaloisPageEmpty9} 7759\begin{paste}{ugProblemGaloisPageEmpty9}{ugProblemGaloisPagePatch9} 7760\pastebutton{ugProblemGaloisPageEmpty9}{\showpaste} 7761\tab{5}\spadcommand{factor(p)\free{declareP }} 7762\end{paste}\end{patch} 7763 7764\begin{patch}{ugProblemGaloisPagePatch10} 7765\begin{paste}{ugProblemGaloisPageFull10}{ugProblemGaloisPageEmpty10} 7766\pastebutton{ugProblemGaloisPageFull10}{\hidepaste} 7767\tab{5}\spadcommand{factor1 := factorList(algFactors).1.factor\free{algFactors }\bound{factor1 }} 7768\indentrel{3}\begin{verbatim} 7769 (10) 7770 x 7771 + 7772 9 8 7 6 5 7773 - 85 b - 116 b + 780 b + 2640 b + 14895 b 7774 + 7775 4 3 2 7776 - 8820 b - 127050 b - 327000 b - 405200 b 7777 + 7778 2062400 7779 / 7780 1339200 7781 Type: UnivariatePolynomial(x,AlgebraicNumber) 7782\end{verbatim} 7783\indentrel{-3}\end{paste}\end{patch} 7784 7785\begin{patch}{ugProblemGaloisPageEmpty10} 7786\begin{paste}{ugProblemGaloisPageEmpty10}{ugProblemGaloisPagePatch10} 7787\pastebutton{ugProblemGaloisPageEmpty10}{\showpaste} 7788\tab{5}\spadcommand{factor1 := factorList(algFactors).1.factor\free{algFactors }\bound{factor1 }} 7789\end{paste}\end{patch} 7790 7791\begin{patch}{ugProblemGaloisPagePatch11} 7792\begin{paste}{ugProblemGaloisPageFull11}{ugProblemGaloisPageEmpty11} 7793\pastebutton{ugProblemGaloisPageFull11}{\hidepaste} 7794\tab{5}\spadcommand{root1 := -coefficient(factor1,0)\free{factor1 }\bound{root1 }} 7795\indentrel{3}\begin{verbatim} 7796 (11) 7797 9 8 7 6 5 7798 85 b + 116 b - 780 b - 2640 b - 14895 b 7799 + 7800 4 3 2 7801 8820 b + 127050 b + 327000 b + 405200 b - 2062400 7802 / 7803 1339200 7804 Type: AlgebraicNumber 7805\end{verbatim} 7806\indentrel{-3}\end{paste}\end{patch} 7807 7808\begin{patch}{ugProblemGaloisPageEmpty11} 7809\begin{paste}{ugProblemGaloisPageEmpty11}{ugProblemGaloisPagePatch11} 7810\pastebutton{ugProblemGaloisPageEmpty11}{\showpaste} 7811\tab{5}\spadcommand{root1 := -coefficient(factor1,0)\free{factor1 }\bound{root1 }} 7812\end{paste}\end{patch} 7813 7814\begin{patch}{ugProblemGaloisPagePatch12} 7815\begin{paste}{ugProblemGaloisPageFull12}{ugProblemGaloisPageEmpty12} 7816\pastebutton{ugProblemGaloisPageFull12}{\hidepaste} 7817\tab{5}\spadcommand{roots := [-coefficient(j.factor, 0) for j in factorList(algFactors)]\free{algFactors }\bound{roots }} 7818\indentrel{3}\begin{verbatim} 7819 (12) 7820 [ 7821 9 8 7 6 5 7822 85 b + 116 b - 780 b - 2640 b - 14895 b 7823 + 7824 4 3 2 7825 8820 b + 127050 b + 327000 b + 405200 b 7826 + 7827 - 2062400 7828 / 7829 1339200 7830 , 7831 8 6 4 2 7832 17 b - 156 b - 2979 b + 25410 b + 14080 7833 -------------------------------------------, 7834 66960 7835 7836 8 6 4 2 7837 - 143 b + 2100 b + 10485 b - 290550 b 7838 + 7839 334800 b + 960800 7840 / 7841 669600 7842 , 7843 7844 8 6 4 2 7845 - 143 b + 2100 b + 10485 b - 290550 b 7846 + 7847 - 334800 b + 960800 7848 / 7849 669600 7850 , 7851 7852 9 8 7 6 5 7853 - 85 b + 116 b + 780 b - 2640 b + 14895 b 7854 + 7855 4 3 2 7856 8820 b - 127050 b + 327000 b - 405200 b 7857 + 7858 - 2062400 7859 / 7860 1339200 7861 ] 7862 Type: List(AlgebraicNumber) 7863\end{verbatim} 7864\indentrel{-3}\end{paste}\end{patch} 7865 7866\begin{patch}{ugProblemGaloisPageEmpty12} 7867\begin{paste}{ugProblemGaloisPageEmpty12}{ugProblemGaloisPagePatch12} 7868\pastebutton{ugProblemGaloisPageEmpty12}{\showpaste} 7869\tab{5}\spadcommand{roots := [-coefficient(j.factor, 0) for j in factorList(algFactors)]\free{algFactors }\bound{roots }} 7870\end{paste}\end{patch} 7871 7872\begin{patch}{ugProblemGaloisPagePatch13} 7873\begin{paste}{ugProblemGaloisPageFull13}{ugProblemGaloisPageEmpty13} 7874\pastebutton{ugProblemGaloisPageFull13}{\hidepaste} 7875\tab{5}\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5)\free{roots }\bound{ais }} 7876\indentrel{3}\begin{verbatim} 7877 (13) 7878 9 8 7 6 5 7879 - 85 b + 116 b + 780 b - 2640 b + 14895 b 7880 + 7881 4 3 2 7882 8820 b - 127050 b + 327000 b - 405200 b - 2062400 7883 / 7884 1339200 7885 Type: AlgebraicNumber 7886\end{verbatim} 7887\indentrel{-3}\end{paste}\end{patch} 7888 7889\begin{patch}{ugProblemGaloisPageEmpty13} 7890\begin{paste}{ugProblemGaloisPageEmpty13}{ugProblemGaloisPagePatch13} 7891\pastebutton{ugProblemGaloisPageEmpty13}{\showpaste} 7892\tab{5}\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5)\free{roots }\bound{ais }} 7893\end{paste}\end{patch} 7894 7895\begin{patch}{ugProblemGaloisPagePatch14} 7896\begin{paste}{ugProblemGaloisPageFull14}{ugProblemGaloisPageEmpty14} 7897\pastebutton{ugProblemGaloisPageFull14}{\hidepaste} 7898\tab{5}\spadcommand{eval(r,x,a1 - a2)\free{ais }} 7899\indentrel{3}\begin{verbatim} 7900 (14) 0 7901 Type: Polynomial(AlgebraicNumber) 7902\end{verbatim} 7903\indentrel{-3}\end{paste}\end{patch} 7904 7905\begin{patch}{ugProblemGaloisPageEmpty14} 7906\begin{paste}{ugProblemGaloisPageEmpty14}{ugProblemGaloisPagePatch14} 7907\pastebutton{ugProblemGaloisPageEmpty14}{\showpaste} 7908\tab{5}\spadcommand{eval(r,x,a1 - a2)\free{ais }} 7909\end{paste}\end{patch} 7910 7911\begin{patch}{ugProblemGaloisPagePatch15} 7912\begin{paste}{ugProblemGaloisPageFull15}{ugProblemGaloisPageEmpty15} 7913\pastebutton{ugProblemGaloisPageFull15}{\hidepaste} 7914\tab{5}\spadcommand{eval(r,x,a1 - a3)\free{ais }} 7915\indentrel{3}\begin{verbatim} 7916 (15) 7917 9 8 7 6 7918 47905 b + 66920 b - 536100 b - 980400 b 7919 + 7920 5 4 3 7921 - 3345075 b - 5787000 b + 75572250 b 7922 + 7923 2 7924 161688000 b - 184600000 b - 710912000 7925 / 7926 4464 7927 Type: Polynomial(AlgebraicNumber) 7928\end{verbatim} 7929\indentrel{-3}\end{paste}\end{patch} 7930 7931\begin{patch}{ugProblemGaloisPageEmpty15} 7932\begin{paste}{ugProblemGaloisPageEmpty15}{ugProblemGaloisPagePatch15} 7933\pastebutton{ugProblemGaloisPageEmpty15}{\showpaste} 7934\tab{5}\spadcommand{eval(r,x,a1 - a3)\free{ais }} 7935\end{paste}\end{patch} 7936 7937\begin{patch}{ugProblemGaloisPagePatch16} 7938\begin{paste}{ugProblemGaloisPageFull16}{ugProblemGaloisPageEmpty16} 7939\pastebutton{ugProblemGaloisPageFull16}{\hidepaste} 7940\tab{5}\spadcommand{eval(r,x,a1 - a4)\free{ais }} 7941\indentrel{3}\begin{verbatim} 7942 (16) 0 7943 Type: Polynomial(AlgebraicNumber) 7944\end{verbatim} 7945\indentrel{-3}\end{paste}\end{patch} 7946 7947\begin{patch}{ugProblemGaloisPageEmpty16} 7948\begin{paste}{ugProblemGaloisPageEmpty16}{ugProblemGaloisPagePatch16} 7949\pastebutton{ugProblemGaloisPageEmpty16}{\showpaste} 7950\tab{5}\spadcommand{eval(r,x,a1 - a4)\free{ais }} 7951\end{paste}\end{patch} 7952 7953\begin{patch}{ugProblemGaloisPagePatch17} 7954\begin{paste}{ugProblemGaloisPageFull17}{ugProblemGaloisPageEmpty17} 7955\pastebutton{ugProblemGaloisPageFull17}{\hidepaste} 7956\tab{5}\spadcommand{eval(r,x,a1 - a5)\free{ais }} 7957\indentrel{3}\begin{verbatim} 7958 (17) 7959 8 6 4 2 7960 405 b + 3450 b - 19875 b - 198000 b - 588000 7961 ------------------------------------------------ 7962 31 7963 Type: Polynomial(AlgebraicNumber) 7964\end{verbatim} 7965\indentrel{-3}\end{paste}\end{patch} 7966 7967\begin{patch}{ugProblemGaloisPageEmpty17} 7968\begin{paste}{ugProblemGaloisPageEmpty17}{ugProblemGaloisPagePatch17} 7969\pastebutton{ugProblemGaloisPageEmpty17}{\showpaste} 7970\tab{5}\spadcommand{eval(r,x,a1 - a5)\free{ais }} 7971\end{paste}\end{patch} 7972 7973\begin{patch}{ugProblemGaloisPagePatch18} 7974\begin{paste}{ugProblemGaloisPageFull18}{ugProblemGaloisPageEmpty18} 7975\pastebutton{ugProblemGaloisPageFull18}{\hidepaste} 7976\tab{5}\spadcommand{bb := a1 - a4\free{ais }\bound{bb }} 7977\indentrel{3}\begin{verbatim} 7978 (18) 7979 9 8 7 6 5 7980 85 b + 402 b - 780 b - 6840 b - 14895 b 7981 + 7982 4 3 2 7983 - 12150 b + 127050 b + 908100 b + 1074800 b 7984 + 7985 - 3984000 7986 / 7987 1339200 7988 Type: AlgebraicNumber 7989\end{verbatim} 7990\indentrel{-3}\end{paste}\end{patch} 7991 7992\begin{patch}{ugProblemGaloisPageEmpty18} 7993\begin{paste}{ugProblemGaloisPageEmpty18}{ugProblemGaloisPagePatch18} 7994\pastebutton{ugProblemGaloisPageEmpty18}{\showpaste} 7995\tab{5}\spadcommand{bb := a1 - a4\free{ais }\bound{bb }} 7996\end{paste}\end{patch} 7997 7998\begin{patch}{ugProblemGaloisPagePatch19} 7999\begin{paste}{ugProblemGaloisPageFull19}{ugProblemGaloisPageEmpty19} 8000\pastebutton{ugProblemGaloisPageFull19}{\hidepaste} 8001\tab{5}\spadcommand{aa1 := subst(a1,beta = bb)\free{beta bb ais }\bound{aa1 }} 8002\indentrel{3}\begin{verbatim} 8003 (19) 8004 8 6 4 2 8005 - 143 b + 2100 b + 10485 b - 290550 b 8006 + 8007 334800 b + 960800 8008 / 8009 669600 8010 Type: AlgebraicNumber 8011\end{verbatim} 8012\indentrel{-3}\end{paste}\end{patch} 8013 8014\begin{patch}{ugProblemGaloisPageEmpty19} 8015\begin{paste}{ugProblemGaloisPageEmpty19}{ugProblemGaloisPagePatch19} 8016\pastebutton{ugProblemGaloisPageEmpty19}{\showpaste} 8017\tab{5}\spadcommand{aa1 := subst(a1,beta = bb)\free{beta bb ais }\bound{aa1 }} 8018\end{paste}\end{patch} 8019 8020\begin{patch}{ugProblemGaloisPagePatch20} 8021\begin{paste}{ugProblemGaloisPageFull20}{ugProblemGaloisPageEmpty20} 8022\pastebutton{ugProblemGaloisPageFull20}{\hidepaste} 8023\tab{5}\spadcommand{aa2 := subst(a2,beta = bb)\free{beta bb ais }\bound{aa2 }} 8024\indentrel{3}\begin{verbatim} 8025 (20) 8026 9 8 7 6 5 8027 - 85 b + 116 b + 780 b - 2640 b + 14895 b 8028 + 8029 4 3 2 8030 8820 b - 127050 b + 327000 b - 405200 b - 2062400 8031 / 8032 1339200 8033 Type: AlgebraicNumber 8034\end{verbatim} 8035\indentrel{-3}\end{paste}\end{patch} 8036 8037\begin{patch}{ugProblemGaloisPageEmpty20} 8038\begin{paste}{ugProblemGaloisPageEmpty20}{ugProblemGaloisPagePatch20} 8039\pastebutton{ugProblemGaloisPageEmpty20}{\showpaste} 8040\tab{5}\spadcommand{aa2 := subst(a2,beta = bb)\free{beta bb ais }\bound{aa2 }} 8041\end{paste}\end{patch} 8042 8043\begin{patch}{ugProblemGaloisPagePatch21} 8044\begin{paste}{ugProblemGaloisPageFull21}{ugProblemGaloisPageEmpty21} 8045\pastebutton{ugProblemGaloisPageFull21}{\hidepaste} 8046\tab{5}\spadcommand{aa3 := subst(a3,beta = bb)\free{beta bb ais }\bound{aa3 }} 8047\indentrel{3}\begin{verbatim} 8048 (21) 8049 9 8 7 6 5 8050 85 b + 116 b - 780 b - 2640 b - 14895 b 8051 + 8052 4 3 2 8053 8820 b + 127050 b + 327000 b + 405200 b - 2062400 8054 / 8055 1339200 8056 Type: AlgebraicNumber 8057\end{verbatim} 8058\indentrel{-3}\end{paste}\end{patch} 8059 8060\begin{patch}{ugProblemGaloisPageEmpty21} 8061\begin{paste}{ugProblemGaloisPageEmpty21}{ugProblemGaloisPagePatch21} 8062\pastebutton{ugProblemGaloisPageEmpty21}{\showpaste} 8063\tab{5}\spadcommand{aa3 := subst(a3,beta = bb)\free{beta bb ais }\bound{aa3 }} 8064\end{paste}\end{patch} 8065 8066\begin{patch}{ugProblemGaloisPagePatch22} 8067\begin{paste}{ugProblemGaloisPageFull22}{ugProblemGaloisPageEmpty22} 8068\pastebutton{ugProblemGaloisPageFull22}{\hidepaste} 8069\tab{5}\spadcommand{aa4 := subst(a4,beta = bb)\free{beta bb ais }\bound{aa4 }} 8070\indentrel{3}\begin{verbatim} 8071 (22) 8072 8 6 4 2 8073 - 143 b + 2100 b + 10485 b - 290550 b 8074 + 8075 - 334800 b + 960800 8076 / 8077 669600 8078 Type: AlgebraicNumber 8079\end{verbatim} 8080\indentrel{-3}\end{paste}\end{patch} 8081 8082\begin{patch}{ugProblemGaloisPageEmpty22} 8083\begin{paste}{ugProblemGaloisPageEmpty22}{ugProblemGaloisPagePatch22} 8084\pastebutton{ugProblemGaloisPageEmpty22}{\showpaste} 8085\tab{5}\spadcommand{aa4 := subst(a4,beta = bb)\free{beta bb ais }\bound{aa4 }} 8086\end{paste}\end{patch} 8087 8088\begin{patch}{ugProblemGaloisPagePatch23} 8089\begin{paste}{ugProblemGaloisPageFull23}{ugProblemGaloisPageEmpty23} 8090\pastebutton{ugProblemGaloisPageFull23}{\hidepaste} 8091\tab{5}\spadcommand{aa5 := subst(a5,beta = bb)\free{beta bb ais }\bound{aa5 }} 8092\indentrel{3}\begin{verbatim} 8093 8 6 4 2 8094 17 b - 156 b - 2979 b + 25410 b + 14080 8095 (23) ------------------------------------------- 8096 66960 8097 Type: AlgebraicNumber 8098\end{verbatim} 8099\indentrel{-3}\end{paste}\end{patch} 8100 8101\begin{patch}{ugProblemGaloisPageEmpty23} 8102\begin{paste}{ugProblemGaloisPageEmpty23}{ugProblemGaloisPagePatch23} 8103\pastebutton{ugProblemGaloisPageEmpty23}{\showpaste} 8104\tab{5}\spadcommand{aa5 := subst(a5,beta = bb)\free{beta bb ais }\bound{aa5 }} 8105\end{paste}\end{patch} 8106 8107\begin{patch}{ugProblemGaloisPagePatch24} 8108\begin{paste}{ugProblemGaloisPageFull24}{ugProblemGaloisPageEmpty24} 8109\pastebutton{ugProblemGaloisPageFull24}{\hidepaste} 8110\tab{5}\spadcommand{(aa1 = a1) :: Boolean\free{aa1 }} 8111\indentrel{3}\begin{verbatim} 8112 (24) false 8113 Type: Boolean 8114\end{verbatim} 8115\indentrel{-3}\end{paste}\end{patch} 8116 8117\begin{patch}{ugProblemGaloisPageEmpty24} 8118\begin{paste}{ugProblemGaloisPageEmpty24}{ugProblemGaloisPagePatch24} 8119\pastebutton{ugProblemGaloisPageEmpty24}{\showpaste} 8120\tab{5}\spadcommand{(aa1 = a1) :: Boolean\free{aa1 }} 8121\end{paste}\end{patch} 8122 8123\begin{patch}{ugProblemGaloisPagePatch25} 8124\begin{paste}{ugProblemGaloisPageFull25}{ugProblemGaloisPageEmpty25} 8125\pastebutton{ugProblemGaloisPageFull25}{\hidepaste} 8126\tab{5}\spadcommand{(aa1 = a2) :: Boolean\free{aa1 }} 8127\indentrel{3}\begin{verbatim} 8128 (25) false 8129 Type: Boolean 8130\end{verbatim} 8131\indentrel{-3}\end{paste}\end{patch} 8132 8133\begin{patch}{ugProblemGaloisPageEmpty25} 8134\begin{paste}{ugProblemGaloisPageEmpty25}{ugProblemGaloisPagePatch25} 8135\pastebutton{ugProblemGaloisPageEmpty25}{\showpaste} 8136\tab{5}\spadcommand{(aa1 = a2) :: Boolean\free{aa1 }} 8137\end{paste}\end{patch} 8138 8139\begin{patch}{ugProblemGaloisPagePatch26} 8140\begin{paste}{ugProblemGaloisPageFull26}{ugProblemGaloisPageEmpty26} 8141\pastebutton{ugProblemGaloisPageFull26}{\hidepaste} 8142\tab{5}\spadcommand{(aa1 = a3) :: Boolean\free{aa1 }} 8143\indentrel{3}\begin{verbatim} 8144 (26) true 8145 Type: Boolean 8146\end{verbatim} 8147\indentrel{-3}\end{paste}\end{patch} 8148 8149\begin{patch}{ugProblemGaloisPageEmpty26} 8150\begin{paste}{ugProblemGaloisPageEmpty26}{ugProblemGaloisPagePatch26} 8151\pastebutton{ugProblemGaloisPageEmpty26}{\showpaste} 8152\tab{5}\spadcommand{(aa1 = a3) :: Boolean\free{aa1 }} 8153\end{paste}\end{patch} 8154 8155\begin{patch}{ugProblemGaloisPagePatch27} 8156\begin{paste}{ugProblemGaloisPageFull27}{ugProblemGaloisPageEmpty27} 8157\pastebutton{ugProblemGaloisPageFull27}{\hidepaste} 8158\tab{5}\spadcommand{(aa1 = a4) :: Boolean\free{aa1 }} 8159\indentrel{3}\begin{verbatim} 8160 (27) false 8161 Type: Boolean 8162\end{verbatim} 8163\indentrel{-3}\end{paste}\end{patch} 8164 8165\begin{patch}{ugProblemGaloisPageEmpty27} 8166\begin{paste}{ugProblemGaloisPageEmpty27}{ugProblemGaloisPagePatch27} 8167\pastebutton{ugProblemGaloisPageEmpty27}{\showpaste} 8168\tab{5}\spadcommand{(aa1 = a4) :: Boolean\free{aa1 }} 8169\end{paste}\end{patch} 8170 8171\begin{patch}{ugProblemGaloisPagePatch28} 8172\begin{paste}{ugProblemGaloisPageFull28}{ugProblemGaloisPageEmpty28} 8173\pastebutton{ugProblemGaloisPageFull28}{\hidepaste} 8174\tab{5}\spadcommand{(aa1 = a5) :: Boolean\free{aa1 }} 8175\indentrel{3}\begin{verbatim} 8176 (28) false 8177 Type: Boolean 8178\end{verbatim} 8179\indentrel{-3}\end{paste}\end{patch} 8180 8181\begin{patch}{ugProblemGaloisPageEmpty28} 8182\begin{paste}{ugProblemGaloisPageEmpty28}{ugProblemGaloisPagePatch28} 8183\pastebutton{ugProblemGaloisPageEmpty28}{\showpaste} 8184\tab{5}\spadcommand{(aa1 = a5) :: Boolean\free{aa1 }} 8185\end{paste}\end{patch} 8186 8187\begin{patch}{ugProblemLimitsPagePatch1} 8188\begin{paste}{ugProblemLimitsPageFull1}{ugProblemLimitsPageEmpty1} 8189\pastebutton{ugProblemLimitsPageFull1}{\hidepaste} 8190\tab{5}\spadcommand{limit((x^2 - 3*x + 2)/(x^2 - 1),x = 1)} 8191\indentrel{3}\begin{verbatim} 8192 1 8193 (1) - - 8194 2 8195Type: Union(OrderedCompletion(Fraction(Polynomial(Integer))),...) 8196\end{verbatim} 8197\indentrel{-3}\end{paste}\end{patch} 8198 8199\begin{patch}{ugProblemLimitsPageEmpty1} 8200\begin{paste}{ugProblemLimitsPageEmpty1}{ugProblemLimitsPagePatch1} 8201\pastebutton{ugProblemLimitsPageEmpty1}{\showpaste} 8202\tab{5}\spadcommand{limit((x^2 - 3*x + 2)/(x^2 - 1),x = 1)} 8203\end{paste}\end{patch} 8204 8205\begin{patch}{ugProblemLimitsPagePatch2} 8206\begin{paste}{ugProblemLimitsPageFull2}{ugProblemLimitsPageEmpty2} 8207\pastebutton{ugProblemLimitsPageFull2}{\hidepaste} 8208\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")} 8209\indentrel{3}\begin{verbatim} 8210 (2) 0 8211Type: Union(OrderedCompletion(Expression(Integer)),...) 8212\end{verbatim} 8213\indentrel{-3}\end{paste}\end{patch} 8214 8215\begin{patch}{ugProblemLimitsPageEmpty2} 8216\begin{paste}{ugProblemLimitsPageEmpty2}{ugProblemLimitsPagePatch2} 8217\pastebutton{ugProblemLimitsPageEmpty2}{\showpaste} 8218\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")} 8219\end{paste}\end{patch} 8220 8221\begin{patch}{ugProblemLimitsPagePatch3} 8222\begin{paste}{ugProblemLimitsPageFull3}{ugProblemLimitsPageEmpty3} 8223\pastebutton{ugProblemLimitsPageFull3}{\hidepaste} 8224\tab{5}\spadcommand{limit(sin(1/x)*exp(1/x), x=0)} 8225\indentrel{3}\begin{verbatim} 8226 (3) [leftHandLimit = 0, rightHandLimit = "failed"] 8227Type: Union(Record(leftHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed"),rightHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed")),...) 8228\end{verbatim} 8229\indentrel{-3}\end{paste}\end{patch} 8230 8231\begin{patch}{ugProblemLimitsPageEmpty3} 8232\begin{paste}{ugProblemLimitsPageEmpty3}{ugProblemLimitsPagePatch3} 8233\pastebutton{ugProblemLimitsPageEmpty3}{\showpaste} 8234\tab{5}\spadcommand{limit(sin(1/x)*exp(1/x), x=0)} 8235\end{paste}\end{patch} 8236 8237\begin{patch}{ugProblemLimitsPagePatch4} 8238\begin{paste}{ugProblemLimitsPageFull4}{ugProblemLimitsPageEmpty4} 8239\pastebutton{ugProblemLimitsPageFull4}{\hidepaste} 8240\tab{5}\spadcommand{limit(sqrt(y^2)/y,y = 0)} 8241\indentrel{3}\begin{verbatim} 8242 (4) [leftHandLimit = - 1, rightHandLimit = 1] 8243Type: Union(Record(leftHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed"),rightHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed")),...) 8244\end{verbatim} 8245\indentrel{-3}\end{paste}\end{patch} 8246 8247\begin{patch}{ugProblemLimitsPageEmpty4} 8248\begin{paste}{ugProblemLimitsPageEmpty4}{ugProblemLimitsPagePatch4} 8249\pastebutton{ugProblemLimitsPageEmpty4}{\showpaste} 8250\tab{5}\spadcommand{limit(sqrt(y^2)/y,y = 0)} 8251\end{paste}\end{patch} 8252 8253\begin{patch}{ugProblemLimitsPagePatch5} 8254\begin{paste}{ugProblemLimitsPageFull5}{ugProblemLimitsPageEmpty5} 8255\pastebutton{ugProblemLimitsPageFull5}{\hidepaste} 8256\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)} 8257\indentrel{3}\begin{verbatim} 8258 1 1 8259 (5) [leftHandLimit = - ----, rightHandLimit = ----] 8260 +-+ +-+ 8261 \|2 \|2 8262Type: Union(Record(leftHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed"),rightHandLimit: Union(OrderedCompletion(Expression(Integer)),"failed")),...) 8263\end{verbatim} 8264\indentrel{-3}\end{paste}\end{patch} 8265 8266\begin{patch}{ugProblemLimitsPageEmpty5} 8267\begin{paste}{ugProblemLimitsPageEmpty5}{ugProblemLimitsPagePatch5} 8268\pastebutton{ugProblemLimitsPageEmpty5}{\showpaste} 8269\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)} 8270\end{paste}\end{patch} 8271 8272\begin{patch}{ugProblemLimitsPagePatch6} 8273\begin{paste}{ugProblemLimitsPageFull6}{ugProblemLimitsPageEmpty6} 8274\pastebutton{ugProblemLimitsPageFull6}{\hidepaste} 8275\tab{5}\spadcommand{limit(sqrt(3*x^2 + 1)/(5*x),x = \%plusInfinity)} 8276\indentrel{3}\begin{verbatim} 8277 +-+ 8278 \|3 8279 (6) ---- 8280 5 8281Type: Union(OrderedCompletion(Expression(Integer)),...) 8282\end{verbatim} 8283\indentrel{-3}\end{paste}\end{patch} 8284 8285\begin{patch}{ugProblemLimitsPageEmpty6} 8286\begin{paste}{ugProblemLimitsPageEmpty6}{ugProblemLimitsPagePatch6} 8287\pastebutton{ugProblemLimitsPageEmpty6}{\showpaste} 8288\tab{5}\spadcommand{limit(sqrt(3*x^2 + 1)/(5*x),x = \%plusInfinity)} 8289\end{paste}\end{patch} 8290 8291\begin{patch}{ugProblemLimitsPagePatch7} 8292\begin{paste}{ugProblemLimitsPageFull7}{ugProblemLimitsPageEmpty7} 8293\pastebutton{ugProblemLimitsPageFull7}{\hidepaste} 8294\tab{5}\spadcommand{limit(sqrt(3*x^2 + 1)/(5*x),x = \%minusInfinity)} 8295\indentrel{3}\begin{verbatim} 8296 +-+ 8297 \|3 8298 (7) - ---- 8299 5 8300Type: Union(OrderedCompletion(Expression(Integer)),...) 8301\end{verbatim} 8302\indentrel{-3}\end{paste}\end{patch} 8303 8304\begin{patch}{ugProblemLimitsPageEmpty7} 8305\begin{paste}{ugProblemLimitsPageEmpty7}{ugProblemLimitsPagePatch7} 8306\pastebutton{ugProblemLimitsPageEmpty7}{\showpaste} 8307\tab{5}\spadcommand{limit(sqrt(3*x^2 + 1)/(5*x),x = \%minusInfinity)} 8308\end{paste}\end{patch} 8309 8310\begin{patch}{ugProblemLimitsPagePatch8} 8311\begin{paste}{ugProblemLimitsPageFull8}{ugProblemLimitsPageEmpty8} 8312\pastebutton{ugProblemLimitsPageFull8}{\hidepaste} 8313\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)} 8314\indentrel{3}\begin{verbatim} 8315 a 8316 (8) - 8317 b 8318Type: Union(OrderedCompletion(Expression(Integer)),...) 8319\end{verbatim} 8320\indentrel{-3}\end{paste}\end{patch} 8321 8322\begin{patch}{ugProblemLimitsPageEmpty8} 8323\begin{paste}{ugProblemLimitsPageEmpty8}{ugProblemLimitsPagePatch8} 8324\pastebutton{ugProblemLimitsPageEmpty8}{\showpaste} 8325\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)} 8326\end{paste}\end{patch} 8327 8328\begin{patch}{ugProblemLimitsPagePatch9} 8329\begin{paste}{ugProblemLimitsPageFull9}{ugProblemLimitsPageEmpty9} 8330\pastebutton{ugProblemLimitsPageFull9}{\hidepaste} 8331\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)} 8332\indentrel{3}\begin{verbatim} 8333 (9) 0 8334Type: Union(OrderedCompletion(Expression(Integer)),...) 8335\end{verbatim} 8336\indentrel{-3}\end{paste}\end{patch} 8337 8338\begin{patch}{ugProblemLimitsPageEmpty9} 8339\begin{paste}{ugProblemLimitsPageEmpty9}{ugProblemLimitsPagePatch9} 8340\pastebutton{ugProblemLimitsPageEmpty9}{\showpaste} 8341\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)} 8342\end{paste}\end{patch} 8343 8344\begin{patch}{ugProblemLimitsPagePatch10} 8345\begin{paste}{ugProblemLimitsPageFull10}{ugProblemLimitsPageEmpty10} 8346\pastebutton{ugProblemLimitsPageFull10}{\hidepaste} 8347\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)} 8348\indentrel{3}\begin{verbatim} 8349 (10) "failed" 8350 Type: Union("failed",...) 8351\end{verbatim} 8352\indentrel{-3}\end{paste}\end{patch} 8353 8354\begin{patch}{ugProblemLimitsPageEmpty10} 8355\begin{paste}{ugProblemLimitsPageEmpty10}{ugProblemLimitsPagePatch10} 8356\pastebutton{ugProblemLimitsPageEmpty10}{\showpaste} 8357\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)} 8358\end{paste}\end{patch} 8359 8360\begin{patch}{ugProblemLimitsPagePatch11} 8361\begin{paste}{ugProblemLimitsPageFull11}{ugProblemLimitsPageEmpty11} 8362\pastebutton{ugProblemLimitsPageFull11}{\hidepaste} 8363\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)} 8364\indentrel{3}\begin{verbatim} 8365 (11) - 1 8366Type: OnePointCompletion(Fraction(Polynomial(Integer))) 8367\end{verbatim} 8368\indentrel{-3}\end{paste}\end{patch} 8369 8370\begin{patch}{ugProblemLimitsPageEmpty11} 8371\begin{paste}{ugProblemLimitsPageEmpty11}{ugProblemLimitsPagePatch11} 8372\pastebutton{ugProblemLimitsPageEmpty11}{\showpaste} 8373\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)} 8374\end{paste}\end{patch} 8375 8376\begin{patch}{ugProblemLimitsPagePatch12} 8377\begin{paste}{ugProblemLimitsPageFull12}{ugProblemLimitsPageEmpty12} 8378\pastebutton{ugProblemLimitsPageFull12}{\hidepaste} 8379\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)} 8380\indentrel{3}\begin{verbatim} 8381 (12) 0 8382Type: Union(OrderedCompletion(Expression(Integer)),...) 8383\end{verbatim} 8384\indentrel{-3}\end{paste}\end{patch} 8385 8386\begin{patch}{ugProblemLimitsPageEmpty12} 8387\begin{paste}{ugProblemLimitsPageEmpty12}{ugProblemLimitsPagePatch12} 8388\pastebutton{ugProblemLimitsPageEmpty12}{\showpaste} 8389\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)} 8390\end{paste}\end{patch} 8391 8392\begin{patch}{ugProblemLimitsPagePatch13} 8393\begin{paste}{ugProblemLimitsPageFull13}{ugProblemLimitsPageEmpty13} 8394\pastebutton{ugProblemLimitsPageFull13}{\hidepaste} 8395\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)} 8396\indentrel{3}\begin{verbatim} 8397 (13) "failed" 8398 Type: Union("failed",...) 8399\end{verbatim} 8400\indentrel{-3}\end{paste}\end{patch} 8401 8402\begin{patch}{ugProblemLimitsPageEmpty13} 8403\begin{paste}{ugProblemLimitsPageEmpty13}{ugProblemLimitsPagePatch13} 8404\pastebutton{ugProblemLimitsPageEmpty13}{\showpaste} 8405\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)} 8406\end{paste}\end{patch} 8407 8408