From ab8cc85adde879fb963c94d15675783f2cf4b183 Mon Sep 17 00:00:00 2001 From: dos-reis Date: Tue, 14 Aug 2007 05:14:52 +0000 Subject: Initial population. --- src/hyper/pages/ug08.pht | 7664 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 7664 insertions(+) create mode 100644 src/hyper/pages/ug08.pht (limited to 'src/hyper/pages/ug08.pht') diff --git a/src/hyper/pages/ug08.pht b/src/hyper/pages/ug08.pht new file mode 100644 index 00000000..43637c24 --- /dev/null +++ b/src/hyper/pages/ug08.pht @@ -0,0 +1,7664 @@ +\begin{patch}{ugxProblemSeriesBernoulliPagePatch1} +\begin{paste}{ugxProblemSeriesBernoulliPageFull1}{ugxProblemSeriesBernoulliPageEmpty1} +\pastebutton{ugxProblemSeriesBernoulliPageFull1}{\hidepaste} +\tab{5}\spadcommand{reduce(+,[m**4 for m in 1..10])} +\indentrel{3}\begin{verbatim} + (1) 25333 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty1} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty1}{ugxProblemSeriesBernoulliPagePatch1} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty1}{\showpaste} +\tab{5}\spadcommand{reduce(+,[m**4 for m in 1..10])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch2} +\begin{paste}{ugxProblemSeriesBernoulliPageFull2}{ugxProblemSeriesBernoulliPageEmpty2} +\pastebutton{ugxProblemSeriesBernoulliPageFull2}{\hidepaste} +\tab{5}\spadcommand{sum4 := sum(m**4, m = 1..k)\bound{sum4 }} +\indentrel{3}\begin{verbatim} + 5 4 3 + 6k + 15k + 10k - k + (2) 컴컴컴컴컴컴컴컴컴컴� + 30 + Type: Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty2} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty2}{ugxProblemSeriesBernoulliPagePatch2} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty2}{\showpaste} +\tab{5}\spadcommand{sum4 := sum(m**4, m = 1..k)\bound{sum4 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch3} +\begin{paste}{ugxProblemSeriesBernoulliPageFull3}{ugxProblemSeriesBernoulliPageEmpty3} +\pastebutton{ugxProblemSeriesBernoulliPageFull3}{\hidepaste} +\tab{5}\spadcommand{eval(sum4, k = 10)\free{sum4 }} +\indentrel{3}\begin{verbatim} + (3) 25333 + Type: Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty3} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty3}{ugxProblemSeriesBernoulliPagePatch3} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty3}{\showpaste} +\tab{5}\spadcommand{eval(sum4, k = 10)\free{sum4 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch4} +\begin{paste}{ugxProblemSeriesBernoulliPageFull4}{ugxProblemSeriesBernoulliPageEmpty4} +\pastebutton{ugxProblemSeriesBernoulliPageFull4}{\hidepaste} +\tab{5}\spadcommand{f := t*exp(x*t) / (exp(t) - 1)\bound{f }} +\indentrel{3}\begin{verbatim} + t x + t %e + (4) 컴컴컴� + t + %e - 1 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty4} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty4}{ugxProblemSeriesBernoulliPagePatch4} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty4}{\showpaste} +\tab{5}\spadcommand{f := t*exp(x*t) / (exp(t) - 1)\bound{f }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch5} +\begin{paste}{ugxProblemSeriesBernoulliPageFull5}{ugxProblemSeriesBernoulliPageEmpty5} +\pastebutton{ugxProblemSeriesBernoulliPageFull5}{\hidepaste} +\tab{5}\spadcommand{)set streams calculate 5\bound{set }} +\indentrel{3}\begin{verbatim} +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty5} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty5}{ugxProblemSeriesBernoulliPagePatch5} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty5}{\showpaste} +\tab{5}\spadcommand{)set streams calculate 5\bound{set }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch6} +\begin{paste}{ugxProblemSeriesBernoulliPageFull6}{ugxProblemSeriesBernoulliPageEmpty6} +\pastebutton{ugxProblemSeriesBernoulliPageFull6}{\hidepaste} +\tab{5}\spadcommand{ff := taylor(f,t = 0)\free{f set }\bound{ff }} +\indentrel{3}\begin{verbatim} + (5) + 2 3 2 + 2x - 1 6x - 6x + 1 2 2x - 3x + x 3 + 1 + 컴컴컴 t + 컴컴컴컴컴컴 t + 컴컴컴컴컴컴� t + 2 12 12 + + + 4 3 2 5 4 3 + 30x - 60x + 30x - 1 4 6x - 15x + 10x - x 5 + 컴컴컴컴컴컴컴컴컴컴컴 t + 컴컴컴컴컴컴컴컴컴컴� t + 720 720 + + + 6 + O(t ) + Type: UnivariateTaylorSeries(Expression Integer,t,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty6} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty6}{ugxProblemSeriesBernoulliPagePatch6} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty6}{\showpaste} +\tab{5}\spadcommand{ff := taylor(f,t = 0)\free{f set }\bound{ff }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch7} +\begin{paste}{ugxProblemSeriesBernoulliPageFull7}{ugxProblemSeriesBernoulliPageEmpty7} +\pastebutton{ugxProblemSeriesBernoulliPageFull7}{\hidepaste} +\tab{5}\spadcommand{factorial(6) * coefficient(ff,6)\free{ff }} +\indentrel{3}\begin{verbatim} + 6 5 4 2 + 42x - 126x + 105x - 21x + 1 + (6) 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� + 42 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty7} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty7}{ugxProblemSeriesBernoulliPagePatch7} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty7}{\showpaste} +\tab{5}\spadcommand{factorial(6) * coefficient(ff,6)\free{ff }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch8} +\begin{paste}{ugxProblemSeriesBernoulliPageFull8}{ugxProblemSeriesBernoulliPageEmpty8} +\pastebutton{ugxProblemSeriesBernoulliPageFull8}{\hidepaste} +\tab{5}\spadcommand{g := eval(f, x = x + 1) - f\bound{g }\free{f }} +\indentrel{3}\begin{verbatim} + t x + t t x + t %e - t %e + (7) 컴컴컴컴컴컴컴컴컴컴� + t + %e - 1 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty8} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty8}{ugxProblemSeriesBernoulliPagePatch8} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty8}{\showpaste} +\tab{5}\spadcommand{g := eval(f, x = x + 1) - f\bound{g }\free{f }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch9} +\begin{paste}{ugxProblemSeriesBernoulliPageFull9}{ugxProblemSeriesBernoulliPageEmpty9} +\pastebutton{ugxProblemSeriesBernoulliPageFull9}{\hidepaste} +\tab{5}\spadcommand{normalize(g)\free{g }} +\indentrel{3}\begin{verbatim} + t x + (8) t %e + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty9} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty9}{ugxProblemSeriesBernoulliPagePatch9} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty9}{\showpaste} +\tab{5}\spadcommand{normalize(g)\free{g }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch10} +\begin{paste}{ugxProblemSeriesBernoulliPageFull10}{ugxProblemSeriesBernoulliPageEmpty10} +\pastebutton{ugxProblemSeriesBernoulliPageFull10}{\hidepaste} +\tab{5}\spadcommand{taylor(g,t = 0)\free{g }} +\indentrel{3}\begin{verbatim} + 2 3 4 + 2 x 3 x 4 x 5 6 + (9) t + x t + 컴 t + 컴 t + 컴 t + O(t ) + 2 6 24 + Type: UnivariateTaylorSeries(Expression Integer,t,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty10} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty10}{ugxProblemSeriesBernoulliPagePatch10} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty10}{\showpaste} +\tab{5}\spadcommand{taylor(g,t = 0)\free{g }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch11} +\begin{paste}{ugxProblemSeriesBernoulliPageFull11}{ugxProblemSeriesBernoulliPageEmpty11} +\pastebutton{ugxProblemSeriesBernoulliPageFull11}{\hidepaste} +\tab{5}\spadcommand{B5 := factorial(5) * coefficient(ff,5)\free{ff }\bound{B5 }} +\indentrel{3}\begin{verbatim} + 5 4 3 + 6x - 15x + 10x - x + (10) 컴컴컴컴컴컴컴컴컴컴� + 6 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty11} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty11}{ugxProblemSeriesBernoulliPagePatch11} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty11}{\showpaste} +\tab{5}\spadcommand{B5 := factorial(5) * coefficient(ff,5)\free{ff }\bound{B5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch12} +\begin{paste}{ugxProblemSeriesBernoulliPageFull12}{ugxProblemSeriesBernoulliPageEmpty12} +\pastebutton{ugxProblemSeriesBernoulliPageFull12}{\hidepaste} +\tab{5}\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1))\free{B5 }} +\indentrel{3}\begin{verbatim} + 5 4 3 + 6k + 15k + 10k - k + (11) 컴컴컴컴컴컴컴컴컴컴� + 30 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty12} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty12}{ugxProblemSeriesBernoulliPagePatch12} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty12}{\showpaste} +\tab{5}\spadcommand{1/5 * (eval(B5, x = k + 1) - eval(B5, x = 1))\free{B5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPagePatch13} +\begin{paste}{ugxProblemSeriesBernoulliPageFull13}{ugxProblemSeriesBernoulliPageEmpty13} +\pastebutton{ugxProblemSeriesBernoulliPageFull13}{\hidepaste} +\tab{5}\spadcommand{sum4\free{sum4 }} +\indentrel{3}\begin{verbatim} + 5 4 3 + 6k + 15k + 10k - k + (12) 컴컴컴컴컴컴컴컴컴컴� + 30 + Type: Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesBernoulliPageEmpty13} +\begin{paste}{ugxProblemSeriesBernoulliPageEmpty13}{ugxProblemSeriesBernoulliPagePatch13} +\pastebutton{ugxProblemSeriesBernoulliPageEmpty13}{\showpaste} +\tab{5}\spadcommand{sum4\free{sum4 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePagePatch1} +\begin{paste}{ugProblemLaplacePageFull1}{ugProblemLaplacePageEmpty1} +\pastebutton{ugProblemLaplacePageFull1}{\hidepaste} +\tab{5}\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)} +\indentrel{3}\begin{verbatim} + 3 + 4a + (1) 컴컴컴컴 + 4 4 + s + 4a + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePageEmpty1} +\begin{paste}{ugProblemLaplacePageEmpty1}{ugProblemLaplacePagePatch1} +\pastebutton{ugProblemLaplacePageEmpty1}{\showpaste} +\tab{5}\spadcommand{laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t), t, s)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePagePatch2} +\begin{paste}{ugProblemLaplacePageFull2}{ugProblemLaplacePageEmpty2} +\pastebutton{ugProblemLaplacePageFull2}{\hidepaste} +\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)} +\indentrel{3}\begin{verbatim} + (2) - log(s - a) + log(s - b) + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePageEmpty2} +\begin{paste}{ugProblemLaplacePageEmpty2}{ugProblemLaplacePagePatch2} +\pastebutton{ugProblemLaplacePageEmpty2}{\showpaste} +\tab{5}\spadcommand{laplace((exp(a*t) - exp(b*t))/t, t, s)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePagePatch3} +\begin{paste}{ugProblemLaplacePageFull3}{ugProblemLaplacePageEmpty3} +\pastebutton{ugProblemLaplacePageFull3}{\hidepaste} +\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)} +\indentrel{3}\begin{verbatim} + 2 2 + (3) log(s + a ) - 2log(s) + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePageEmpty3} +\begin{paste}{ugProblemLaplacePageEmpty3}{ugProblemLaplacePagePatch3} +\pastebutton{ugProblemLaplacePageEmpty3}{\showpaste} +\tab{5}\spadcommand{laplace(2/t * (1 - cos(a*t)), t, s)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePagePatch4} +\begin{paste}{ugProblemLaplacePageFull4}{ugProblemLaplacePageEmpty4} +\pastebutton{ugProblemLaplacePageFull4}{\hidepaste} +\tab{5}\spadcommand{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)} +\indentrel{3}\begin{verbatim} + 1 + (4) 컴컴컴컴컴컴컴컴컴컴컴컴 + 2 3 2 + b s + 2a b s + b + a b + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePageEmpty4} +\begin{paste}{ugProblemLaplacePageEmpty4}{ugProblemLaplacePagePatch4} +\pastebutton{ugProblemLaplacePageEmpty4}{\showpaste} +\tab{5}\spadcommand{laplace(exp(-a*t) * sin(b*t) / b**2, t, s)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePagePatch5} +\begin{paste}{ugProblemLaplacePageFull5}{ugProblemLaplacePageEmpty5} +\pastebutton{ugProblemLaplacePageFull5}{\hidepaste} +\tab{5}\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)} +\indentrel{3}\begin{verbatim} + 2 2 2 2 + log(s + b ) - log(s + a ) + (5) 컴컴컴컴컴컴컴컴컴컴컴컴컴� + 2 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePageEmpty5} +\begin{paste}{ugProblemLaplacePageEmpty5}{ugProblemLaplacePagePatch5} +\pastebutton{ugProblemLaplacePageEmpty5}{\showpaste} +\tab{5}\spadcommand{laplace((cos(a*t) - cos(b*t))/t, t, s)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePagePatch6} +\begin{paste}{ugProblemLaplacePageFull6}{ugProblemLaplacePageEmpty6} +\pastebutton{ugProblemLaplacePageFull6}{\hidepaste} +\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)} +\indentrel{3}\begin{verbatim} + b s + c - a + %e log(컴컴컴컴�) + c + (6) 컴컴컴컴컴컴컴컴� + s - a + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePageEmpty6} +\begin{paste}{ugProblemLaplacePageEmpty6}{ugProblemLaplacePagePatch6} +\pastebutton{ugProblemLaplacePageEmpty6}{\showpaste} +\tab{5}\spadcommand{laplace(exp(a*t+b)*Ei(c*t), t, s)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePagePatch7} +\begin{paste}{ugProblemLaplacePageFull7}{ugProblemLaplacePageEmpty7} +\pastebutton{ugProblemLaplacePageFull7}{\hidepaste} +\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)} +\indentrel{3}\begin{verbatim} + 2 2 + s + b d + a log(컴컴컴�) + 2c atan(�) + 2 s + b + (7) 컴컴컴컴컴컴컴컴컴컴컴컴컴� + 2s + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePageEmpty7} +\begin{paste}{ugProblemLaplacePageEmpty7}{ugProblemLaplacePagePatch7} +\pastebutton{ugProblemLaplacePageEmpty7}{\showpaste} +\tab{5}\spadcommand{laplace(a*Ci(b*t) + c*Si(d*t), t, s)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePagePatch8} +\begin{paste}{ugProblemLaplacePageFull8}{ugProblemLaplacePageEmpty8} +\pastebutton{ugProblemLaplacePageFull8}{\hidepaste} +\tab{5}\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)} +\indentrel{3}\begin{verbatim} + 2 + 4 2 2 4 t 3 + (s + 2a s + a )laplace(%e ,t,s) + 2a + (8) 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 + 4 2 2 4 + s + 2a s + a + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLaplacePageEmpty8} +\begin{paste}{ugProblemLaplacePageEmpty8}{ugProblemLaplacePagePatch8} +\pastebutton{ugProblemLaplacePageEmpty8}{\showpaste} +\tab{5}\spadcommand{laplace(sin(a*t) - a*t*cos(a*t) + exp(t**2), t, s)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch1} +\begin{paste}{ugxProblemSeriesFunctionsPageFull1}{ugxProblemSeriesFunctionsPageEmpty1} +\pastebutton{ugxProblemSeriesFunctionsPageFull1}{\hidepaste} +\tab{5}\spadcommand{x := series 'x\bound{x }} +\indentrel{3}\begin{verbatim} + (1) x + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty1} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty1}{ugxProblemSeriesFunctionsPagePatch1} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty1}{\showpaste} +\tab{5}\spadcommand{x := series 'x\bound{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch2} +\begin{paste}{ugxProblemSeriesFunctionsPageFull2}{ugxProblemSeriesFunctionsPageEmpty2} +\pastebutton{ugxProblemSeriesFunctionsPageFull2}{\hidepaste} +\tab{5}\spadcommand{rat := x**2 / (1 - 6*x + x**2)\free{x }\bound{rat }} +\indentrel{3}\begin{verbatim} + (2) + 2 3 4 5 6 7 8 + x + 6x + 35x + 204x + 1189x + 6930x + 40391x + + + 9 10 11 12 + 235416x + 1372105x + 7997214x + 46611179x + + + 13 + O(x ) + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty2} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty2}{ugxProblemSeriesFunctionsPagePatch2} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty2}{\showpaste} +\tab{5}\spadcommand{rat := x**2 / (1 - 6*x + x**2)\free{x }\bound{rat }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch3} +\begin{paste}{ugxProblemSeriesFunctionsPageFull3}{ugxProblemSeriesFunctionsPageEmpty3} +\pastebutton{ugxProblemSeriesFunctionsPageFull3}{\hidepaste} +\tab{5}\spadcommand{sin(rat)\free{rat }} +\indentrel{3}\begin{verbatim} + (3) + 2 3 4 5 7133 6 7 80711 8 + x + 6x + 35x + 204x + 컴컴 x + 6927x + 컴컴� x + 6 2 + + + 9 164285281 10 31888513 11 + 235068x + 컴컴컴컴� x + 컴컴컴컴 x + 120 4 + + + 371324777 12 13 + 컴컴컴컴� x + O(x ) + 8 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty3} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty3}{ugxProblemSeriesFunctionsPagePatch3} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty3}{\showpaste} +\tab{5}\spadcommand{sin(rat)\free{rat }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch4} +\begin{paste}{ugxProblemSeriesFunctionsPageFull4}{ugxProblemSeriesFunctionsPageEmpty4} +\pastebutton{ugxProblemSeriesFunctionsPageFull4}{\hidepaste} +\tab{5}\spadcommand{y : UTS(FRAC INT,y,0) := y\bound{y }} +\indentrel{3}\begin{verbatim} + (4) y + Type: UnivariateTaylorSeries(Fraction Integer,y,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty4} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty4}{ugxProblemSeriesFunctionsPagePatch4} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty4}{\showpaste} +\tab{5}\spadcommand{y : UTS(FRAC INT,y,0) := y\bound{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch5} +\begin{paste}{ugxProblemSeriesFunctionsPageFull5}{ugxProblemSeriesFunctionsPageEmpty5} +\pastebutton{ugxProblemSeriesFunctionsPageFull5}{\hidepaste} +\tab{5}\spadcommand{exp(y)\free{y }} +\indentrel{3}\begin{verbatim} + (5) + 1 2 1 3 1 4 1 5 1 6 + 1 + y + � y + � y + 컴 y + 컴� y + 컴� y + 2 6 24 120 720 + + + 1 7 1 8 1 9 1 10 11 + 컴컴 y + 컴컴� y + 컴컴컴 y + 컴컴컴� y + O(y ) + 5040 40320 362880 3628800 + Type: UnivariateTaylorSeries(Fraction Integer,y,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty5} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty5}{ugxProblemSeriesFunctionsPagePatch5} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty5}{\showpaste} +\tab{5}\spadcommand{exp(y)\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch6} +\begin{paste}{ugxProblemSeriesFunctionsPageFull6}{ugxProblemSeriesFunctionsPageEmpty6} +\pastebutton{ugxProblemSeriesFunctionsPageFull6}{\hidepaste} +\tab{5}\spadcommand{tan(y**2)\free{y }} +\indentrel{3}\begin{verbatim} + 2 1 6 2 10 11 + (6) y + � y + 컴 y + O(y ) + 3 15 + Type: UnivariateTaylorSeries(Fraction Integer,y,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty6} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty6}{ugxProblemSeriesFunctionsPagePatch6} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty6}{\showpaste} +\tab{5}\spadcommand{tan(y**2)\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch7} +\begin{paste}{ugxProblemSeriesFunctionsPageFull7}{ugxProblemSeriesFunctionsPageEmpty7} +\pastebutton{ugxProblemSeriesFunctionsPageFull7}{\hidepaste} +\tab{5}\spadcommand{cos(y + y**5)\free{y }} +\indentrel{3}\begin{verbatim} + (7) + 1 2 1 4 721 6 6721 8 1844641 10 + 1 - � y + 컴 y - 컴� y + 컴컴� y - 컴컴컴� y + 2 24 720 40320 3628800 + + + 11 + O(y ) + Type: UnivariateTaylorSeries(Fraction Integer,y,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty7} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty7}{ugxProblemSeriesFunctionsPagePatch7} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty7}{\showpaste} +\tab{5}\spadcommand{cos(y + y**5)\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch8} +\begin{paste}{ugxProblemSeriesFunctionsPageFull8}{ugxProblemSeriesFunctionsPageEmpty8} +\pastebutton{ugxProblemSeriesFunctionsPageFull8}{\hidepaste} +\tab{5}\spadcommand{log(1 + sin(y))\free{y }} +\indentrel{3}\begin{verbatim} + (8) + 1 2 1 3 1 4 1 5 1 6 61 7 + y - � y + � y - 컴 y + 컴 y - 컴 y + 컴컴 y + 2 6 12 24 45 5040 + + + 17 8 277 9 31 10 11 + - 컴컴 y + 컴컴� y - 컴컴� y + O(y ) + 2520 72576 14175 + Type: UnivariateTaylorSeries(Fraction Integer,y,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty8} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty8}{ugxProblemSeriesFunctionsPagePatch8} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty8}{\showpaste} +\tab{5}\spadcommand{log(1 + sin(y))\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch9} +\begin{paste}{ugxProblemSeriesFunctionsPageFull9}{ugxProblemSeriesFunctionsPageEmpty9} +\pastebutton{ugxProblemSeriesFunctionsPageFull9}{\hidepaste} +\tab{5}\spadcommand{z : UTS(EXPR INT,z,0) := z\bound{z }} +\indentrel{3}\begin{verbatim} + (9) z + Type: UnivariateTaylorSeries(Expression Integer,z,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty9} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty9}{ugxProblemSeriesFunctionsPagePatch9} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty9}{\showpaste} +\tab{5}\spadcommand{z : UTS(EXPR INT,z,0) := z\bound{z }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch10} +\begin{paste}{ugxProblemSeriesFunctionsPageFull10}{ugxProblemSeriesFunctionsPageEmpty10} +\pastebutton{ugxProblemSeriesFunctionsPageFull10}{\hidepaste} +\tab{5}\spadcommand{exp(2 + tan(z))\free{z }} +\indentrel{3}\begin{verbatim} + (10) + 2 2 2 2 + 2 2 %e 2 %e 3 3%e 4 37%e 5 + %e + %e z + 컴� z + 컴� z + 컴컴 z + 컴컴� z + 2 2 8 120 + + + 2 2 2 2 + 59%e 6 137%e 7 871%e 8 41641%e 9 + 컴컴� z + 컴컴컴 z + 컴컴컴 z + 컴컴컴컴 z + 240 720 5760 362880 + + + 2 + 325249%e 10 11 + 컴컴컴컴� z + O(z ) + 3628800 + Type: UnivariateTaylorSeries(Expression Integer,z,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty10} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty10}{ugxProblemSeriesFunctionsPagePatch10} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty10}{\showpaste} +\tab{5}\spadcommand{exp(2 + tan(z))\free{z }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch11} +\begin{paste}{ugxProblemSeriesFunctionsPageFull11}{ugxProblemSeriesFunctionsPageEmpty11} +\pastebutton{ugxProblemSeriesFunctionsPageFull11}{\hidepaste} +\tab{5}\spadcommand{w := taylor 'w\bound{w }} +\indentrel{3}\begin{verbatim} + (11) w + Type: UnivariateTaylorSeries(Expression Integer,w,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty11} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty11}{ugxProblemSeriesFunctionsPagePatch11} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty11}{\showpaste} +\tab{5}\spadcommand{w := taylor 'w\bound{w }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPagePatch12} +\begin{paste}{ugxProblemSeriesFunctionsPageFull12}{ugxProblemSeriesFunctionsPageEmpty12} +\pastebutton{ugxProblemSeriesFunctionsPageFull12}{\hidepaste} +\tab{5}\spadcommand{exp(2 + tan(w))\free{w }} +\indentrel{3}\begin{verbatim} + (12) + 2 2 2 2 + 2 2 %e 2 %e 3 3%e 4 37%e 5 + %e + %e w + 컴� w + 컴� w + 컴컴 w + 컴컴� w + 2 2 8 120 + + + 2 2 2 2 + 59%e 6 137%e 7 871%e 8 41641%e 9 + 컴컴� w + 컴컴컴 w + 컴컴컴 w + 컴컴컴컴 w + 240 720 5760 362880 + + + 2 + 325249%e 10 11 + 컴컴컴컴� w + O(w ) + 3628800 + Type: UnivariateTaylorSeries(Expression Integer,w,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFunctionsPageEmpty12} +\begin{paste}{ugxProblemSeriesFunctionsPageEmpty12}{ugxProblemSeriesFunctionsPagePatch12} +\pastebutton{ugxProblemSeriesFunctionsPageEmpty12}{\showpaste} +\tab{5}\spadcommand{exp(2 + tan(w))\free{w }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch1} +\begin{paste}{ugProblemGeneticPageFull1}{ugProblemGeneticPageEmpty1} +\pastebutton{ugProblemGeneticPageFull1}{\hidepaste} +\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 }} +\indentrel{3}\begin{verbatim} + (1) + � 1 1 9� � 1 1� � 1 1� + �1 � � 컴� �0 � 0 컴� �0 0 � 컴� + � 2 2 20� � 2 20� � 2 20� + � � � � � � + �1 1 � �1 9 1 � � 9 � + 냐 0 컴 0 � 냐 1 컴 � � �0 0 컴 0 � + �2 20 � �2 20 2 � � 20 � + [� �, � �, � �, + �1 1 � � 9 � �1 9 1 � + 냐 컴 0 0 � �0 컴 0 0 � 냐 컴 1 � � + �2 20 � � 20 � �2 20 2 � + � � � � � � + � 9 � � 1 1 � � 1 1 � + 냐� 0 0 0 � 냐� � 0 0 � 냐� 0 � 0 � + �20 � �20 2 � �20 2 � + � 9� + �0 0 0 컴� + � 20� + � � + � 1 1 � + �0 0 컴 � � + � 20 2 � + � �] + � 1 1 � + �0 컴 0 � � + � 20 2 � + � � + � 9 1 1 � + 냐� � � 1 � + �20 2 2 � + Type: List SquareMatrix(4,Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty1} +\begin{paste}{ugProblemGeneticPageEmpty1}{ugProblemGeneticPagePatch1} +\pastebutton{ugProblemGeneticPageEmpty1}{\showpaste} +\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 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch2} +\begin{paste}{ugProblemGeneticPageFull2}{ugProblemGeneticPageEmpty2} +\pastebutton{ugProblemGeneticPageFull2}{\hidepaste} +\tab{5}\spadcommand{gametes := ['AB,'Ab,'aB,'ab]\bound{gametes }} +\indentrel{3}\begin{verbatim} + (2) [AB,Ab,aB,ab] + Type: List OrderedVariableList [AB,Ab,aB,ab] +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty2} +\begin{paste}{ugProblemGeneticPageEmpty2}{ugProblemGeneticPagePatch2} +\pastebutton{ugProblemGeneticPageEmpty2}{\showpaste} +\tab{5}\spadcommand{gametes := ['AB,'Ab,'aB,'ab]\bound{gametes }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch3} +\begin{paste}{ugProblemGeneticPageFull3}{ugProblemGeneticPageEmpty3} +\pastebutton{ugProblemGeneticPageFull3}{\hidepaste} +\tab{5}\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A }\free{gametes segregationRates }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty3} +\begin{paste}{ugProblemGeneticPageEmpty3}{ugProblemGeneticPagePatch3} +\pastebutton{ugProblemGeneticPageEmpty3}{\showpaste} +\tab{5}\spadcommand{A := ALGSC(FRAC INT, 4, gametes, segregationRates);\bound{A }\free{gametes segregationRates }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch4} +\begin{paste}{ugProblemGeneticPageFull4}{ugProblemGeneticPageEmpty4} +\pastebutton{ugProblemGeneticPageFull4}{\hidepaste} +\tab{5}\spadcommand{a := basis()$A; a.1*a.4} +\indentrel{3}\begin{verbatim} + 9 1 1 9 + (4) 컴 ab + 컴 aB + 컴 Ab + 컴 AB + 20 20 20 20 +Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX]) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty4} +\begin{paste}{ugProblemGeneticPageEmpty4}{ugProblemGeneticPagePatch4} +\pastebutton{ugProblemGeneticPageEmpty4}{\showpaste} +\tab{5}\spadcommand{a := basis()$A; a.1*a.4} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch5} +\begin{paste}{ugProblemGeneticPageFull5}{ugProblemGeneticPageEmpty5} +\pastebutton{ugProblemGeneticPageFull5}{\hidepaste} +\tab{5}\spadcommand{commutative?()$A\free{A }} +\indentrel{3}\begin{verbatim} + algebra is commutative + (5) true + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty5} +\begin{paste}{ugProblemGeneticPageEmpty5}{ugProblemGeneticPagePatch5} +\pastebutton{ugProblemGeneticPageEmpty5}{\showpaste} +\tab{5}\spadcommand{commutative?()$A\free{A }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch6} +\begin{paste}{ugProblemGeneticPageFull6}{ugProblemGeneticPageEmpty6} +\pastebutton{ugProblemGeneticPageFull6}{\hidepaste} +\tab{5}\spadcommand{associative?()$A\free{A }} +\indentrel{3}\begin{verbatim} + algebra is not associative + (6) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty6} +\begin{paste}{ugProblemGeneticPageEmpty6}{ugProblemGeneticPagePatch6} +\pastebutton{ugProblemGeneticPageEmpty6}{\showpaste} +\tab{5}\spadcommand{associative?()$A\free{A }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch7} +\begin{paste}{ugProblemGeneticPageFull7}{ugProblemGeneticPageEmpty7} +\pastebutton{ugProblemGeneticPageFull7}{\hidepaste} +\tab{5}\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]\bound{x }\free{gametes segregationRates }} +\indentrel{3}\begin{verbatim} + (7) 0.4ab + 0.1aB + 0.2Ab + 0.3AB +Type: AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX]) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty7} +\begin{paste}{ugProblemGeneticPageEmpty7}{ugProblemGeneticPagePatch7} +\pastebutton{ugProblemGeneticPageEmpty7}{\showpaste} +\tab{5}\spadcommand{x : ALGSC(DECIMAL, 4, gametes, segregationRates) := convert [3/10, 1/5, 1/10, 2/5]\bound{x }\free{gametes segregationRates }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch8} +\begin{paste}{ugProblemGeneticPageFull8}{ugProblemGeneticPageEmpty8} +\pastebutton{ugProblemGeneticPageFull8}{\hidepaste} +\tab{5}\spadcommand{plenaryPower(x,5)\free{x }} +\indentrel{3}\begin{verbatim} + (8) 0.36561ab + 0.13439aB + 0.23439Ab + 0.26561AB +Type: AlgebraGivenByStructuralConstants(DecimalExpansion,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX]) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty8} +\begin{paste}{ugProblemGeneticPageEmpty8}{ugProblemGeneticPagePatch8} +\pastebutton{ugProblemGeneticPageEmpty8}{\showpaste} +\tab{5}\spadcommand{plenaryPower(x,5)\free{x }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch9} +\begin{paste}{ugProblemGeneticPageFull9}{ugProblemGeneticPageEmpty9} +\pastebutton{ugProblemGeneticPageFull9}{\hidepaste} +\tab{5}\spadcommand{q := leftRankPolynomial()$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q }\free{gametes segregationRates }} +\indentrel{3}\begin{verbatim} + (9) + 3 29 29 29 29 2 + Y + (- 컴 %x4 - 컴 %x3 - 컴 %x2 - 컴 %x1)Y + 20 20 20 20 + + + 9 2 9 9 9 9 2 + 컴 %x4 + (컴 %x3 + 컴 %x2 + 컴 %x1)%x4 + 컴 %x3 + 20 10 10 10 20 + + + 9 9 9 2 9 + (컴 %x2 + 컴 %x1)%x3 + 컴 %x2 + 컴 %x1 %x2 + 10 10 20 10 + + + 9 2 + 컴 %x1 + 20 + * + Y +Type: UnivariatePolynomial(Y,Polynomial Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty9} +\begin{paste}{ugProblemGeneticPageEmpty9}{ugProblemGeneticPagePatch9} +\pastebutton{ugProblemGeneticPageEmpty9}{\showpaste} +\tab{5}\spadcommand{q := leftRankPolynomial()$GCNAALG(FRAC INT, 4, gametes, segregationRates) :: UP(Y, POLY FRAC INT)\bound{q }\free{gametes segregationRates }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch10} +\begin{paste}{ugProblemGeneticPageFull10}{ugProblemGeneticPageEmpty10} +\pastebutton{ugProblemGeneticPageFull10}{\hidepaste} +\tab{5}\spadcommand{factor(q :: POLY FRAC INT)\free{q }} +\indentrel{3}\begin{verbatim} + (10) + (Y - %x4 - %x3 - %x2 - %x1) + * + 9 9 9 9 + (Y - 컴 %x4 - 컴 %x3 - 컴 %x2 - 컴 %x1)Y + 20 20 20 20 + Type: Factored Polynomial Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty10} +\begin{paste}{ugProblemGeneticPageEmpty10}{ugProblemGeneticPagePatch10} +\pastebutton{ugProblemGeneticPageEmpty10}{\showpaste} +\tab{5}\spadcommand{factor(q :: POLY FRAC INT)\free{q }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch11} +\begin{paste}{ugProblemGeneticPageFull11}{ugProblemGeneticPageEmpty11} +\pastebutton{ugProblemGeneticPageFull11}{\hidepaste} +\tab{5}\spadcommand{cI := conditionsForIdempotents()$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI }\free{gametes segregationRates }} +\indentrel{3}\begin{verbatim} + (11) + 9 1 2 + [컴 %x1 %x4 + (컴 %x2 + %x1)%x3 + %x1 %x2 + %x1 - %x1, + 10 10 + 1 9 2 + (%x2 + 컴 %x1)%x4 + 컴 %x2 %x3 + %x2 + (%x1 - 1)%x2, + 10 10 + 1 2 9 + (%x3 + 컴 %x1)%x4 + %x3 + (컴 %x2 + %x1 - 1)%x3, + 10 10 + 2 9 1 + %x4 + (%x3 + %x2 + 컴 %x1 - 1)%x4 + 컴 %x2 %x3] + 10 10 + Type: List Polynomial Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty11} +\begin{paste}{ugProblemGeneticPageEmpty11}{ugProblemGeneticPagePatch11} +\pastebutton{ugProblemGeneticPageEmpty11}{\showpaste} +\tab{5}\spadcommand{cI := conditionsForIdempotents()$GCNAALG(FRAC INT, 4, gametes, segregationRates)\bound{cI }\free{gametes segregationRates }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch12} +\begin{paste}{ugProblemGeneticPageFull12}{ugProblemGeneticPageEmpty12} +\pastebutton{ugProblemGeneticPageFull12}{\hidepaste} +\tab{5}\spadcommand{gbs:= groebnerFactorize cI; gbs.1\free{cI }\bound{gbs }} +\indentrel{3}\begin{verbatim} + (12) + [%x4 + %x3 + %x2 + %x1 - 1, + 2 + (%x2 + %x1)%x3 + %x1 %x2 + %x1 - %x1] + Type: List Polynomial Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty12} +\begin{paste}{ugProblemGeneticPageEmpty12}{ugProblemGeneticPagePatch12} +\pastebutton{ugProblemGeneticPageEmpty12}{\showpaste} +\tab{5}\spadcommand{gbs:= groebnerFactorize cI; gbs.1\free{cI }\bound{gbs }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch13} +\begin{paste}{ugProblemGeneticPageFull13}{ugProblemGeneticPageEmpty13} +\pastebutton{ugProblemGeneticPageFull13}{\hidepaste} +\tab{5}\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol }\free{gbs }} +\indentrel{3}\begin{verbatim} + 2 2 1 1 + (13) [[%x4= �,%x3= �,%x2= 컴,%x1= 컴]] + 5 5 10 10 + Type: List List Equation Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty13} +\begin{paste}{ugProblemGeneticPageEmpty13}{ugProblemGeneticPagePatch13} +\pastebutton{ugProblemGeneticPageEmpty13}{\showpaste} +\tab{5}\spadcommand{sol := solve concat(gbs.1,[\%x1-1/10,\%x2-1/10])\bound{sol }\free{gbs }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch14} +\begin{paste}{ugProblemGeneticPageFull14}{ugProblemGeneticPageEmpty14} +\pastebutton{ugProblemGeneticPageFull14}{\hidepaste} +\tab{5}\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e }\free{A sol }} +\indentrel{3}\begin{verbatim} + 2 2 1 1 + (14) � ab + � aB + 컴 Ab + 컴 AB + 5 5 10 10 +Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX]) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty14} +\begin{paste}{ugProblemGeneticPageEmpty14}{ugProblemGeneticPagePatch14} +\pastebutton{ugProblemGeneticPageEmpty14}{\showpaste} +\tab{5}\spadcommand{e : A := represents reverse (map(rhs, sol.1) :: List FRAC INT)\bound{e }\free{A sol }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPagePatch15} +\begin{paste}{ugProblemGeneticPageFull15}{ugProblemGeneticPageEmpty15} +\pastebutton{ugProblemGeneticPageFull15}{\hidepaste} +\tab{5}\spadcommand{e*e-e\free{e }} +\indentrel{3}\begin{verbatim} + (15) 0 +Type: AlgebraGivenByStructuralConstants(Fraction Integer,4,[AB,Ab,aB,ab],[MATRIX,MATRIX,MATRIX,MATRIX]) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGeneticPageEmpty15} +\begin{paste}{ugProblemGeneticPageEmpty15}{ugProblemGeneticPagePatch15} +\pastebutton{ugProblemGeneticPageEmpty15}{\showpaste} +\tab{5}\spadcommand{e*e-e\free{e }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPagePatch1} +\begin{paste}{ugxProblemOnePolPageFull1}{ugxProblemOnePolPageEmpty1} +\pastebutton{ugxProblemOnePolPageFull1}{\hidepaste} +\tab{5}\spadcommand{solve(x**3 = 8,x)} +\indentrel{3}\begin{verbatim} + 2 + (1) [x= 2,x + 2x + 4= 0] + Type: List Equation Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPageEmpty1} +\begin{paste}{ugxProblemOnePolPageEmpty1}{ugxProblemOnePolPagePatch1} +\pastebutton{ugxProblemOnePolPageEmpty1}{\showpaste} +\tab{5}\spadcommand{solve(x**3 = 8,x)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPagePatch2} +\begin{paste}{ugxProblemOnePolPageFull2}{ugxProblemOnePolPageEmpty2} +\pastebutton{ugxProblemOnePolPageFull2}{\hidepaste} +\tab{5}\spadcommand{radicalSolve(x**3 = 8,x)} +\indentrel{3}\begin{verbatim} + 旼컴� 旼컴� + (2) [x= - \�- 3 - 1,x= \�- 3 - 1,x= 2] + Type: List Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPageEmpty2} +\begin{paste}{ugxProblemOnePolPageEmpty2}{ugxProblemOnePolPagePatch2} +\pastebutton{ugxProblemOnePolPageEmpty2}{\showpaste} +\tab{5}\spadcommand{radicalSolve(x**3 = 8,x)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPagePatch3} +\begin{paste}{ugxProblemOnePolPageFull3}{ugxProblemOnePolPageEmpty3} +\pastebutton{ugxProblemOnePolPageFull3}{\hidepaste} +\tab{5}\spadcommand{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)} +\indentrel{3}\begin{verbatim} + (3) [x= 3.6180114746 09375,x= 1.3819885253 90625] + Type: List Equation Polynomial Float +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPageEmpty3} +\begin{paste}{ugxProblemOnePolPageEmpty3}{ugxProblemOnePolPagePatch3} +\pastebutton{ugxProblemOnePolPageEmpty3}{\showpaste} +\tab{5}\spadcommand{solve(x**4 - 10*x**3 + 35*x**2 - 50*x + 25,.0001)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPagePatch4} +\begin{paste}{ugxProblemOnePolPageFull4}{ugxProblemOnePolPageEmpty4} +\pastebutton{ugxProblemOnePolPageFull4}{\hidepaste} +\tab{5}\spadcommand{solve(x**3-2,1/1000)} +\indentrel{3}\begin{verbatim} + 2581 + (4) [x= 컴컴] + 2048 + Type: List Equation Polynomial Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPageEmpty4} +\begin{paste}{ugxProblemOnePolPageEmpty4}{ugxProblemOnePolPagePatch4} +\pastebutton{ugxProblemOnePolPageEmpty4}{\showpaste} +\tab{5}\spadcommand{solve(x**3-2,1/1000)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPagePatch5} +\begin{paste}{ugxProblemOnePolPageFull5}{ugxProblemOnePolPageEmpty5} +\pastebutton{ugxProblemOnePolPageFull5}{\hidepaste} +\tab{5}\spadcommand{complexSolve(x**3-2,.0001)} +\indentrel{3}\begin{verbatim} + (5) + [x= 1.2599182128 90625, + x= - 0.6298943279 5395613131 - 1.0910949707 03125 %i, + x= - 0.6298943279 5395613131 + 1.0910949707 03125 %i] + Type: List Equation Polynomial Complex Float +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPageEmpty5} +\begin{paste}{ugxProblemOnePolPageEmpty5}{ugxProblemOnePolPagePatch5} +\pastebutton{ugxProblemOnePolPageEmpty5}{\showpaste} +\tab{5}\spadcommand{complexSolve(x**3-2,.0001)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPagePatch6} +\begin{paste}{ugxProblemOnePolPageFull6}{ugxProblemOnePolPageEmpty6} +\pastebutton{ugxProblemOnePolPageFull6}{\hidepaste} +\tab{5}\spadcommand{complexSolve(x**2-2*\%i+1,1/100)} +\indentrel{3}\begin{verbatim} + 13028925 325 13028925 325 + (6) [x= - 컴컴컴컴 - 컴� %i,x= 컴컴컴컴 + 컴� %i] + 16777216 256 16777216 256 +Type: List Equation Polynomial Complex Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPageEmpty6} +\begin{paste}{ugxProblemOnePolPageEmpty6}{ugxProblemOnePolPagePatch6} +\pastebutton{ugxProblemOnePolPageEmpty6}{\showpaste} +\tab{5}\spadcommand{complexSolve(x**2-2*\%i+1,1/100)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPagePatch7} +\begin{paste}{ugxProblemOnePolPageFull7}{ugxProblemOnePolPageEmpty7} +\pastebutton{ugxProblemOnePolPageFull7}{\hidepaste} +\tab{5}\spadcommand{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)} +\indentrel{3}\begin{verbatim} + 旼컴� 旼컴� + - \�- 3 - 1 \�- 3 - 1 + (7) [x= 컴컴컴컴컴컴,x= 컴컴컴컴컴] + 2 2 + Type: List Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemOnePolPageEmpty7} +\begin{paste}{ugxProblemOnePolPageEmpty7}{ugxProblemOnePolPagePatch7} +\pastebutton{ugxProblemOnePolPageEmpty7}{\showpaste} +\tab{5}\spadcommand{radicalSolve(1/x**3 + 1/x**2 + 1/x = 0,x)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch1} +\begin{paste}{ugProblemNumericPageFull1}{ugProblemNumericPageEmpty1} +\pastebutton{ugProblemNumericPageFull1}{\hidepaste} +\tab{5}\spadcommand{exp(3.1)} +\indentrel{3}\begin{verbatim} + (1) 22.1979512814 41633405 + Type: Float +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty1} +\begin{paste}{ugProblemNumericPageEmpty1}{ugProblemNumericPagePatch1} +\pastebutton{ugProblemNumericPageEmpty1}{\showpaste} +\tab{5}\spadcommand{exp(3.1)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch2} +\begin{paste}{ugProblemNumericPageFull2}{ugProblemNumericPageEmpty2} +\pastebutton{ugProblemNumericPageFull2}{\hidepaste} +\tab{5}\spadcommand{exp(3.1 + 4.5 * \%i)} +\indentrel{3}\begin{verbatim} + (2) + - 4.6792348860 969899118 - 21.6991659280 71731864 %i + Type: Complex Float +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty2} +\begin{paste}{ugProblemNumericPageEmpty2}{ugProblemNumericPagePatch2} +\pastebutton{ugProblemNumericPageEmpty2}{\showpaste} +\tab{5}\spadcommand{exp(3.1 + 4.5 * \%i)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch3} +\begin{paste}{ugProblemNumericPageFull3}{ugProblemNumericPageEmpty3} +\pastebutton{ugProblemNumericPageFull3}{\hidepaste} +\tab{5}\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)} +\indentrel{3}\begin{verbatim} + (3) - 4.6792348860969906 - 21.699165928071732%i + Type: Complex DoubleFloat +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty3} +\begin{paste}{ugProblemNumericPageEmpty3}{ugProblemNumericPagePatch3} +\pastebutton{ugProblemNumericPageEmpty3}{\showpaste} +\tab{5}\spadcommand{r: DFLOAT := 3.1; t: DFLOAT := 4.5; exp(r + t*\%i)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch4} +\begin{paste}{ugProblemNumericPageFull4}{ugProblemNumericPageEmpty4} +\pastebutton{ugProblemNumericPageFull4}{\hidepaste} +\tab{5}\spadcommand{exp(3.1::DFLOAT + 4.5::DFLOAT * \%i)} +\indentrel{3}\begin{verbatim} + (4) - 4.6792348860969906 - 21.699165928071732%i + Type: Complex 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+\pastebutton{ugProblemNumericPageFull6}{\hidepaste} +\tab{5}\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)} +\indentrel{3}\begin{verbatim} + (6) 2.489482417547372 - 2.3658460381468371%i + Type: Complex DoubleFloat +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty6} +\begin{paste}{ugProblemNumericPageEmpty6}{ugProblemNumericPagePatch6} +\pastebutton{ugProblemNumericPageEmpty6}{\showpaste} +\tab{5}\spadcommand{a := 2.1; b := 1.1; besselI(a + \%i*b, b*a + 1)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch7} +\begin{paste}{ugProblemNumericPageFull7}{ugProblemNumericPageEmpty7} +\pastebutton{ugProblemNumericPageFull7}{\hidepaste} +\tab{5}\spadcommand{[chebyshevT(i, z) for i in 0..5]} +\indentrel{3}\begin{verbatim} + (7) + 2 3 4 2 5 3 + [1,z,2z - 1,4z - 3z,8z - 8z + 1,16z - 20z + 5z] + Type: List Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + 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+\pastebutton{ugProblemNumericPageFull9}{\hidepaste} +\tab{5}\spadcommand{chebyshevT(3, 5.0::DoubleFloat)} +\indentrel{3}\begin{verbatim} + (9) 485.0 + Type: DoubleFloat +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty9} +\begin{paste}{ugProblemNumericPageEmpty9}{ugProblemNumericPagePatch9} +\pastebutton{ugProblemNumericPageEmpty9}{\showpaste} +\tab{5}\spadcommand{chebyshevT(3, 5.0::DoubleFloat)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch10} +\begin{paste}{ugProblemNumericPageFull10}{ugProblemNumericPageEmpty10} +\pastebutton{ugProblemNumericPageFull10}{\hidepaste} +\tab{5}\spadcommand{[chebyshevU(i, z) for i in 0..5]} +\indentrel{3}\begin{verbatim} + (10) + 2 3 4 2 + [1, 2z, 4z - 1, 8z - 4z, 16z - 12z + 1, + 5 3 + 32z - 32z + 6z] + Type: List Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty10} 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+\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage27.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage27}} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty27} +\begin{paste}{ugProblemNumericPageEmpty27}{ugProblemNumericPagePatch27} +\pastebutton{ugProblemNumericPageEmpty27}{\showpaste} +\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} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch28} +\begin{paste}{ugProblemNumericPageFull28}{ugProblemNumericPageEmpty28} +\pastebutton{ugProblemNumericPageFull28}{\hidepaste} +\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)} +\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage28.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage28}} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty28} +\begin{paste}{ugProblemNumericPageEmpty28}{ugProblemNumericPagePatch28} +\pastebutton{ugProblemNumericPageEmpty28}{\showpaste} +\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)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch29} +\begin{paste}{ugProblemNumericPageFull29}{ugProblemNumericPageEmpty29} +\pastebutton{ugProblemNumericPageFull29}{\hidepaste} +\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)} +\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage29.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage29}} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty29} +\begin{paste}{ugProblemNumericPageEmpty29}{ugProblemNumericPagePatch29} +\pastebutton{ugProblemNumericPageEmpty29}{\showpaste} +\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)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch30} +\begin{paste}{ugProblemNumericPageFull30}{ugProblemNumericPageEmpty30} +\pastebutton{ugProblemNumericPageFull30}{\hidepaste} +\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)} +\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage30}} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty30} +\begin{paste}{ugProblemNumericPageEmpty30}{ugProblemNumericPagePatch30} +\pastebutton{ugProblemNumericPageEmpty30}{\showpaste} +\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)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch31} +\begin{paste}{ugProblemNumericPageFull31}{ugProblemNumericPageEmpty31} +\pastebutton{ugProblemNumericPageFull31}{\hidepaste} +\tab{5}\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])} +\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage31.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage31}} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty31} +\begin{paste}{ugProblemNumericPageEmpty31}{ugProblemNumericPagePatch31} +\pastebutton{ugProblemNumericPageEmpty31}{\showpaste} +\tab{5}\spadgraph{draw(besselI(alpha, 5), alpha = -12..12, unit==[5,20])} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch32} +\begin{paste}{ugProblemNumericPageFull32}{ugProblemNumericPageEmpty32} +\pastebutton{ugProblemNumericPageFull32}{\hidepaste} +\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")} +\center{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage32.VIEW/image}}{viewAlone\space{1} \env{AXIOM}/doc/viewports/ugProblemNumericPage32}} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty32} +\begin{paste}{ugProblemNumericPageEmpty32}{ugProblemNumericPagePatch32} +\pastebutton{ugProblemNumericPageEmpty32}{\showpaste} +\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")} +\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPagePatch33} +\begin{paste}{ugProblemNumericPageFull33}{ugProblemNumericPageEmpty33} +\pastebutton{ugProblemNumericPageFull33}{\hidepaste} +\tab{5}\spadcommand{all} +\indentrel{3}\begin{verbatim} + (33) all + Type: Variable all +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemNumericPageEmpty33} +\begin{paste}{ugProblemNumericPageEmpty33}{ugProblemNumericPagePatch33} +\pastebutton{ugProblemNumericPageEmpty33}{\showpaste} +\tab{5}\spadcommand{all} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPagePatch1} +\begin{paste}{ugxProblemLinSysPageFull1}{ugxProblemLinSysPageEmpty1} +\pastebutton{ugxProblemLinSysPageFull1}{\hidepaste} +\tab{5}\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])} +\indentrel{3}\begin{verbatim} + (1) [[x= - 1,y= 2,z= 7]] + Type: List List Equation Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPageEmpty1} +\begin{paste}{ugxProblemLinSysPageEmpty1}{ugxProblemLinSysPagePatch1} +\pastebutton{ugxProblemLinSysPageEmpty1}{\showpaste} +\tab{5}\spadcommand{solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPagePatch2} +\begin{paste}{ugxProblemLinSysPageFull2}{ugxProblemLinSysPageEmpty2} +\pastebutton{ugxProblemLinSysPageFull2}{\hidepaste} +\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])} +\indentrel{3}\begin{verbatim} + (2) [[x= %Y - 2,y= - 2%Y + 2,z= %Y]] + Type: List List Equation Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPageEmpty2} +\begin{paste}{ugxProblemLinSysPageEmpty2}{ugxProblemLinSysPagePatch2} +\pastebutton{ugxProblemLinSysPageEmpty2}{\showpaste} +\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])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPagePatch3} +\begin{paste}{ugxProblemLinSysPageFull3}{ugxProblemLinSysPageEmpty3} +\pastebutton{ugxProblemLinSysPageFull3}{\hidepaste} +\tab{5}\spadcommand{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])} +\indentrel{3}\begin{verbatim} + (3) [particular= [- 1,2,7],basis= [[0,0,0]]] +Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPageEmpty3} +\begin{paste}{ugxProblemLinSysPageEmpty3}{ugxProblemLinSysPagePatch3} +\pastebutton{ugxProblemLinSysPageEmpty3}{\showpaste} +\tab{5}\spadcommand{solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPagePatch4} +\begin{paste}{ugxProblemLinSysPageFull4}{ugxProblemLinSysPageEmpty4} +\pastebutton{ugxProblemLinSysPageFull4}{\hidepaste} +\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])} +\indentrel{3}\begin{verbatim} + (4) [particular= [- 2,2,0],basis= [[1,- 2,1]]] +Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPageEmpty4} +\begin{paste}{ugxProblemLinSysPageEmpty4}{ugxProblemLinSysPagePatch4} +\pastebutton{ugxProblemLinSysPageEmpty4}{\showpaste} +\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPagePatch5} +\begin{paste}{ugxProblemLinSysPageFull5}{ugxProblemLinSysPageEmpty5} +\pastebutton{ugxProblemLinSysPageFull5}{\hidepaste} +\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])} +\indentrel{3}\begin{verbatim} + (5) [particular= "failed",basis= [[1,- 2,1]]] +Type: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPageEmpty5} +\begin{paste}{ugxProblemLinSysPageEmpty5}{ugxProblemLinSysPagePatch5} +\pastebutton{ugxProblemLinSysPageEmpty5}{\showpaste} +\tab{5}\spadcommand{solve([[1,2,3],[2,3,4],[3,4,5]],[2,3,2])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPagePatch6} +\begin{paste}{ugxProblemLinSysPageFull6}{ugxProblemLinSysPageEmpty6} +\pastebutton{ugxProblemLinSysPageFull6}{\hidepaste} +\tab{5}\spadcommand{nullSpace([[1,2,3],[2,3,4],[3,4,5]])} +\indentrel{3}\begin{verbatim} + (6) [[1,- 2,1]] + Type: List Vector Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLinSysPageEmpty6} +\begin{paste}{ugxProblemLinSysPageEmpty6}{ugxProblemLinSysPagePatch6} +\pastebutton{ugxProblemLinSysPageEmpty6}{\showpaste} +\tab{5}\spadcommand{nullSpace([[1,2,3],[2,3,4],[3,4,5]])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPagePatch1} +\begin{paste}{ugxProblemPolSysPageFull1}{ugxProblemPolSysPageEmpty1} +\pastebutton{ugxProblemPolSysPageFull1}{\hidepaste} +\tab{5}\spadcommand{solve([3*x**3 + y + 1,y**2 -4],[x,y])} +\indentrel{3}\begin{verbatim} + (1) + 2 + [[x= - 1,y= 2], [x - x + 1= 0,y= 2], + 3 + [3x - 1= 0,y= - 2]] + Type: List List Equation Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPageEmpty1} +\begin{paste}{ugxProblemPolSysPageEmpty1}{ugxProblemPolSysPagePatch1} +\pastebutton{ugxProblemPolSysPageEmpty1}{\showpaste} +\tab{5}\spadcommand{solve([3*x**3 + y + 1,y**2 -4],[x,y])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPagePatch2} +\begin{paste}{ugxProblemPolSysPageFull2}{ugxProblemPolSysPageEmpty2} +\pastebutton{ugxProblemPolSysPageFull2}{\hidepaste} +\tab{5}\spadcommand{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])} +\indentrel{3}\begin{verbatim} + (2) + 2 + z 3z + z + 9 4 3 2 + [[x= �,y= 컴컴컴컴컴�,9z + 6z + 55z + 15z - 90= 0]] + 3 3 + Type: List List Equation Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPageEmpty2} +\begin{paste}{ugxProblemPolSysPageEmpty2}{ugxProblemPolSysPagePatch2} +\pastebutton{ugxProblemPolSysPageEmpty2}{\showpaste} +\tab{5}\spadcommand{solve([x = y**2-19,y = z**2+x+3,z = 3*x],[x,y,z])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPagePatch3} +\begin{paste}{ugxProblemPolSysPageFull3}{ugxProblemPolSysPageEmpty3} +\pastebutton{ugxProblemPolSysPageFull3}{\hidepaste} +\tab{5}\spadcommand{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])} +\indentrel{3}\begin{verbatim} + (3) + 旼컴� 旼컴� + \�- 3 + 1 - \�- 3 + 1 + [[x= 컴컴컴컴컴,y= 2], [x= 컴컴컴컴컴컴,y= 2], + 2 2 + 旼컴� 旼� + - \�- 1 \�3 - 1 + [x= 컴컴컴컴컴컴컴컴,y= - 2], + 3旼� + 2\�3 + 旼컴� 旼� + \�- 1 \�3 - 1 1 + [x= 컴컴컴컴컴컴컴,y= - 2], [x= 컴컴,y= - 2], + 3旼� 3旼� + 2\�3 \�3 + [x= - 1,y= 2]] + Type: List List Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPageEmpty3} +\begin{paste}{ugxProblemPolSysPageEmpty3}{ugxProblemPolSysPagePatch3} +\pastebutton{ugxProblemPolSysPageEmpty3}{\showpaste} +\tab{5}\spadcommand{radicalSolve([3*x**3 + y + 1,y**2 -4],[x,y])} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPagePatch4} +\begin{paste}{ugxProblemPolSysPageFull4}{ugxProblemPolSysPageEmpty4} +\pastebutton{ugxProblemPolSysPageFull4}{\hidepaste} +\tab{5}\spadcommand{solve([x**2*y - 1,x*y**2 - 2],.01)} +\indentrel{3}\begin{verbatim} + (4) [[y= 1.5859375,x= 0.79296875]] + Type: List List Equation Polynomial Float +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPageEmpty4} +\begin{paste}{ugxProblemPolSysPageEmpty4}{ugxProblemPolSysPagePatch4} +\pastebutton{ugxProblemPolSysPageEmpty4}{\showpaste} +\tab{5}\spadcommand{solve([x**2*y - 1,x*y**2 - 2],.01)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPagePatch5} +\begin{paste}{ugxProblemPolSysPageFull5}{ugxProblemPolSysPageEmpty5} +\pastebutton{ugxProblemPolSysPageFull5}{\hidepaste} +\tab{5}\spadcommand{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)} +\indentrel{3}\begin{verbatim} + (5) + 1625 1625 + [[y= 컴컴,x= 컴컴], + 1024 2048 + + 435445573689 1407 + [y= - 컴컴컴컴컴컴 - 컴컴 %i, + 549755813888 1024 + 435445573689 1407 + x= - 컴컴컴컴컴컴� - 컴컴 %i] + 1099511627776 2048 + , + + 435445573689 1407 + [y= - 컴컴컴컴컴컴 + 컴컴 %i, + 549755813888 1024 + 435445573689 1407 + x= - 컴컴컴컴컴컴� + 컴컴 %i] + 1099511627776 2048 + ] +Type: List List Equation Polynomial Complex Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPageEmpty5} +\begin{paste}{ugxProblemPolSysPageEmpty5}{ugxProblemPolSysPagePatch5} +\pastebutton{ugxProblemPolSysPageEmpty5}{\showpaste} +\tab{5}\spadcommand{complexSolve([x**2*y - 1,x*y**2 - 2],1/1000)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPagePatch6} +\begin{paste}{ugxProblemPolSysPageFull6}{ugxProblemPolSysPageEmpty6} +\pastebutton{ugxProblemPolSysPageFull6}{\hidepaste} +\tab{5}\spadcommand{solve([x**2/a = a,a = a*x],.001)} +\indentrel{3}\begin{verbatim} + (6) [[x= 1.0,a= - 1.0],[x= 1.0,a= 1.0]] + Type: List List Equation Polynomial Float +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPageEmpty6} +\begin{paste}{ugxProblemPolSysPageEmpty6}{ugxProblemPolSysPagePatch6} +\pastebutton{ugxProblemPolSysPageEmpty6}{\showpaste} +\tab{5}\spadcommand{solve([x**2/a = a,a = a*x],.001)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPagePatch7} +\begin{paste}{ugxProblemPolSysPageFull7}{ugxProblemPolSysPageEmpty7} +\pastebutton{ugxProblemPolSysPageFull7}{\hidepaste} +\tab{5}\spadcommand{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])} +\indentrel{3}\begin{verbatim} + (7) + 旼컴컴컴컴컴컴컴컴컴컴컴컴� + � 4 3 2 + �- a + 2a + 3a + 3a + 1 - a - 1 + [[x= - 냐컴컴컴컴컴컴컴컴컴컴컴컴 ,y= 컴컴컴�], + � 2 a + \� a + 旼컴컴컴컴컴컴컴컴컴컴컴컴� + � 4 3 2 + �- a + 2a + 3a + 3a + 1 - a - 1 + [x= 냐컴컴컴컴컴컴컴컴컴컴컴컴 ,y= 컴컴컴�]] + � 2 a + \� a + Type: List List Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemPolSysPageEmpty7} +\begin{paste}{ugxProblemPolSysPageEmpty7}{ugxProblemPolSysPagePatch7} +\pastebutton{ugxProblemPolSysPageEmpty7}{\showpaste} +\tab{5}\spadcommand{radicalSolve([x**2/a + a + y**3 - 1,a*y + a + 1],[x,y])} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorFFPagePatch1} +\begin{paste}{ugProblemFactorFFPageFull1}{ugProblemFactorFFPageEmpty1} +\pastebutton{ugProblemFactorFFPageFull1}{\hidepaste} +\tab{5}\spadcommand{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18\bound{u }} +\indentrel{3}\begin{verbatim} + 4 2 + (1) 3x + 2x + 15x + 18 + Type: Polynomial PrimeField 19 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorFFPageEmpty1} +\begin{paste}{ugProblemFactorFFPageEmpty1}{ugProblemFactorFFPagePatch1} +\pastebutton{ugProblemFactorFFPageEmpty1}{\showpaste} +\tab{5}\spadcommand{u : POLY(PF(19)) :=3*x**4+2*x**2+15*x+18\bound{u }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorFFPagePatch2} +\begin{paste}{ugProblemFactorFFPageFull2}{ugProblemFactorFFPageEmpty2} +\pastebutton{ugProblemFactorFFPageFull2}{\hidepaste} +\tab{5}\spadcommand{factor u\free{u }} +\indentrel{3}\begin{verbatim} + 3 2 + (2) 3(x + 18)(x + x + 8x + 13) + Type: Factored Polynomial PrimeField 19 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorFFPageEmpty2} +\begin{paste}{ugProblemFactorFFPageEmpty2}{ugProblemFactorFFPagePatch2} +\pastebutton{ugProblemFactorFFPageEmpty2}{\showpaste} +\tab{5}\spadcommand{factor u\free{u }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorFFPagePatch3} +\begin{paste}{ugProblemFactorFFPageFull3}{ugProblemFactorFFPageEmpty3} +\pastebutton{ugProblemFactorFFPageFull3}{\hidepaste} +\tab{5}\spadcommand{factor(u :: POLY FFX(PF 19,3))\free{u }} +\indentrel{3}\begin{verbatim} + (3) + 2 + 3(x + 18)(x + 5%BC + 3%BC + 13) + * + 2 2 + (x + 16%BC + 14%BC + 13)(x + 17%BC + 2%BC + 13) +Type: Factored Polynomial FiniteFieldExtension(PrimeField 19,3) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorFFPageEmpty3} +\begin{paste}{ugProblemFactorFFPageEmpty3}{ugProblemFactorFFPagePatch3} +\pastebutton{ugProblemFactorFFPageEmpty3}{\showpaste} +\tab{5}\spadcommand{factor(u :: POLY FFX(PF 19,3))\free{u }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPagePatch1} +\begin{paste}{ugxProblemSeriesArithmeticPageFull1}{ugxProblemSeriesArithmeticPageEmpty1} +\pastebutton{ugxProblemSeriesArithmeticPageFull1}{\hidepaste} +\tab{5}\spadcommand{x := series x\bound{x }} +\indentrel{3}\begin{verbatim} + (1) x + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPageEmpty1} +\begin{paste}{ugxProblemSeriesArithmeticPageEmpty1}{ugxProblemSeriesArithmeticPagePatch1} +\pastebutton{ugxProblemSeriesArithmeticPageEmpty1}{\showpaste} +\tab{5}\spadcommand{x := series x\bound{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPagePatch2} +\begin{paste}{ugxProblemSeriesArithmeticPageFull2}{ugxProblemSeriesArithmeticPageEmpty2} +\pastebutton{ugxProblemSeriesArithmeticPageFull2}{\hidepaste} +\tab{5}\spadcommand{(3 + x) / (1 + 7*x)} +\indentrel{3}\begin{verbatim} + (2) + 2 3 4 5 6 + 3 - 20x + 140x - 980x + 6860x - 48020x + 336140x + + + 7 8 9 10 + - 2352980x + 16470860x - 115296020x + 807072140x + + + 11 + O(x ) + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPageEmpty2} +\begin{paste}{ugxProblemSeriesArithmeticPageEmpty2}{ugxProblemSeriesArithmeticPagePatch2} +\pastebutton{ugxProblemSeriesArithmeticPageEmpty2}{\showpaste} +\tab{5}\spadcommand{(3 + x) / (1 + 7*x)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPagePatch3} +\begin{paste}{ugxProblemSeriesArithmeticPageFull3}{ugxProblemSeriesArithmeticPageEmpty3} +\pastebutton{ugxProblemSeriesArithmeticPageFull3}{\hidepaste} +\tab{5}\spadcommand{base := 1 / (1 - x)\free{x }\bound{base }} +\indentrel{3}\begin{verbatim} + (3) + 2 3 4 5 6 7 8 9 10 + 1 + x + x + x + x + x + x + x + x + x + x + + + 11 + O(x ) + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPageEmpty3} +\begin{paste}{ugxProblemSeriesArithmeticPageEmpty3}{ugxProblemSeriesArithmeticPagePatch3} +\pastebutton{ugxProblemSeriesArithmeticPageEmpty3}{\showpaste} +\tab{5}\spadcommand{base := 1 / (1 - x)\free{x }\bound{base }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPagePatch4} +\begin{paste}{ugxProblemSeriesArithmeticPageFull4}{ugxProblemSeriesArithmeticPageEmpty4} +\pastebutton{ugxProblemSeriesArithmeticPageFull4}{\hidepaste} +\tab{5}\spadcommand{expon := x * base\free{x base }\bound{expon }} +\indentrel{3}\begin{verbatim} + (4) + 2 3 4 5 6 7 8 9 10 11 + x + x + x + x + x + x + x + x + x + x + x + + + 12 + O(x ) + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPageEmpty4} +\begin{paste}{ugxProblemSeriesArithmeticPageEmpty4}{ugxProblemSeriesArithmeticPagePatch4} +\pastebutton{ugxProblemSeriesArithmeticPageEmpty4}{\showpaste} +\tab{5}\spadcommand{expon := x * base\free{x base }\bound{expon }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPagePatch5} +\begin{paste}{ugxProblemSeriesArithmeticPageFull5}{ugxProblemSeriesArithmeticPageEmpty5} +\pastebutton{ugxProblemSeriesArithmeticPageFull5}{\hidepaste} +\tab{5}\spadcommand{base ** expon\free{base expon }} +\indentrel{3}\begin{verbatim} + (5) + 2 3 3 7 4 43 5 649 6 241 7 + 1 + x + � x + � x + 컴 x + 컴� x + 컴� x + 2 3 12 120 30 + + + 3706 8 85763 9 245339 10 11 + 컴컴 x + 컴컴� x + 컴컴컴 x + O(x ) + 315 5040 10080 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesArithmeticPageEmpty5} +\begin{paste}{ugxProblemSeriesArithmeticPageEmpty5}{ugxProblemSeriesArithmeticPagePatch5} +\pastebutton{ugxProblemSeriesArithmeticPageEmpty5}{\showpaste} +\tab{5}\spadcommand{base ** expon\free{base expon }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch1} +\begin{paste}{ugxProblemFiniteConversionPageFull1}{ugxProblemFiniteConversionPageEmpty1} +\pastebutton{ugxProblemFiniteConversionPageFull1}{\hidepaste} +\tab{5}\spadcommand{K := PrimeField 3\bound{K }} +\indentrel{3}\begin{verbatim} + (1) PrimeField 3 + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty1} +\begin{paste}{ugxProblemFiniteConversionPageEmpty1}{ugxProblemFiniteConversionPagePatch1} +\pastebutton{ugxProblemFiniteConversionPageEmpty1}{\showpaste} +\tab{5}\spadcommand{K := PrimeField 3\bound{K }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch2} +\begin{paste}{ugxProblemFiniteConversionPageFull2}{ugxProblemFiniteConversionPageEmpty2} +\pastebutton{ugxProblemFiniteConversionPageFull2}{\hidepaste} +\tab{5}\spadcommand{(m,n) := (4,8)\bound{m n }} +\indentrel{3}\begin{verbatim} + (2) 8 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty2} +\begin{paste}{ugxProblemFiniteConversionPageEmpty2}{ugxProblemFiniteConversionPagePatch2} +\pastebutton{ugxProblemFiniteConversionPageEmpty2}{\showpaste} +\tab{5}\spadcommand{(m,n) := (4,8)\bound{m n }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch3} +\begin{paste}{ugxProblemFiniteConversionPageFull3}{ugxProblemFiniteConversionPageEmpty3} +\pastebutton{ugxProblemFiniteConversionPageFull3}{\hidepaste} +\tab{5}\spadcommand{Km := FiniteFieldExtension(K,m)\bound{Km }\free{K m }} +\indentrel{3}\begin{verbatim} + (3) FiniteFieldExtension(PrimeField 3,4) + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty3} +\begin{paste}{ugxProblemFiniteConversionPageEmpty3}{ugxProblemFiniteConversionPagePatch3} +\pastebutton{ugxProblemFiniteConversionPageEmpty3}{\showpaste} +\tab{5}\spadcommand{Km := FiniteFieldExtension(K,m)\bound{Km }\free{K m }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch4} +\begin{paste}{ugxProblemFiniteConversionPageFull4}{ugxProblemFiniteConversionPageEmpty4} +\pastebutton{ugxProblemFiniteConversionPageFull4}{\hidepaste} +\tab{5}\spadcommand{Kn := FiniteFieldExtension(K,n)\bound{Kn }\free{K n }} +\indentrel{3}\begin{verbatim} + (4) FiniteFieldExtension(PrimeField 3,8) + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty4} +\begin{paste}{ugxProblemFiniteConversionPageEmpty4}{ugxProblemFiniteConversionPagePatch4} +\pastebutton{ugxProblemFiniteConversionPageEmpty4}{\showpaste} +\tab{5}\spadcommand{Kn := FiniteFieldExtension(K,n)\bound{Kn }\free{K n }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch5} +\begin{paste}{ugxProblemFiniteConversionPageFull5}{ugxProblemFiniteConversionPageEmpty5} +\pastebutton{ugxProblemFiniteConversionPageFull5}{\hidepaste} +\tab{5}\spadcommand{a1 := random()$Km\bound{a1 }\free{Km }} +\indentrel{3}\begin{verbatim} + 3 2 + (5) 2%BD + 2%BD + 2%BD + 1 + Type: FiniteFieldExtension(PrimeField 3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty5} +\begin{paste}{ugxProblemFiniteConversionPageEmpty5}{ugxProblemFiniteConversionPagePatch5} +\pastebutton{ugxProblemFiniteConversionPageEmpty5}{\showpaste} +\tab{5}\spadcommand{a1 := random()$Km\bound{a1 }\free{Km }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch6} +\begin{paste}{ugxProblemFiniteConversionPageFull6}{ugxProblemFiniteConversionPageEmpty6} +\pastebutton{ugxProblemFiniteConversionPageFull6}{\hidepaste} +\tab{5}\spadcommand{b1 := random()$Km\bound{b1 }\free{Km }} +\indentrel{3}\begin{verbatim} + 3 2 + (6) %BD + %BD + 2%BD + 2 + Type: FiniteFieldExtension(PrimeField 3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty6} +\begin{paste}{ugxProblemFiniteConversionPageEmpty6}{ugxProblemFiniteConversionPagePatch6} +\pastebutton{ugxProblemFiniteConversionPageEmpty6}{\showpaste} +\tab{5}\spadcommand{b1 := random()$Km\bound{b1 }\free{Km }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch7} +\begin{paste}{ugxProblemFiniteConversionPageFull7}{ugxProblemFiniteConversionPageEmpty7} +\pastebutton{ugxProblemFiniteConversionPageFull7}{\hidepaste} +\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a2 }\free{a1 Kn }} +\indentrel{3}\begin{verbatim} + 6 + (7) 2%BE + 1 + Type: FiniteFieldExtension(PrimeField 3,8) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty7} +\begin{paste}{ugxProblemFiniteConversionPageEmpty7}{ugxProblemFiniteConversionPagePatch7} +\pastebutton{ugxProblemFiniteConversionPageEmpty7}{\showpaste} +\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a2 }\free{a1 Kn }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch8} +\begin{paste}{ugxProblemFiniteConversionPageFull8}{ugxProblemFiniteConversionPageEmpty8} +\pastebutton{ugxProblemFiniteConversionPageFull8}{\hidepaste} +\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b2 }\free{b1 Kn }} +\indentrel{3}\begin{verbatim} + 6 4 2 + (8) 2%BE + 2%BE + %BE + 2 + Type: FiniteFieldExtension(PrimeField 3,8) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty8} +\begin{paste}{ugxProblemFiniteConversionPageEmpty8}{ugxProblemFiniteConversionPagePatch8} +\pastebutton{ugxProblemFiniteConversionPageEmpty8}{\showpaste} +\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b2 }\free{b1 Kn }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch9} +\begin{paste}{ugxProblemFiniteConversionPageFull9}{ugxProblemFiniteConversionPageEmpty9} +\pastebutton{ugxProblemFiniteConversionPageFull9}{\hidepaste} +\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a1 a2 b1 b2 Km Kn }} +\indentrel{3}\begin{verbatim} + (9) 0 + Type: FiniteFieldExtension(PrimeField 3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty9} +\begin{paste}{ugxProblemFiniteConversionPageEmpty9}{ugxProblemFiniteConversionPagePatch9} +\pastebutton{ugxProblemFiniteConversionPageEmpty9}{\showpaste} +\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a1 a2 b1 b2 Km Kn }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch10} +\begin{paste}{ugxProblemFiniteConversionPageFull10}{ugxProblemFiniteConversionPageEmpty10} +\pastebutton{ugxProblemFiniteConversionPageFull10}{\hidepaste} +\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a1 a2 b1 b2 Km Kn }} +\indentrel{3}\begin{verbatim} + (10) 0 + Type: FiniteFieldExtension(PrimeField 3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty10} +\begin{paste}{ugxProblemFiniteConversionPageEmpty10}{ugxProblemFiniteConversionPagePatch10} +\pastebutton{ugxProblemFiniteConversionPageEmpty10}{\showpaste} +\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a1 a2 b1 b2 Km Kn }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch11} +\begin{paste}{ugxProblemFiniteConversionPageFull11}{ugxProblemFiniteConversionPageEmpty11} +\pastebutton{ugxProblemFiniteConversionPageFull11}{\hidepaste} +\tab{5}\spadcommand{Km := FFCGX(K,m)\bound{Km2 }\free{K m }} +\indentrel{3}\begin{verbatim} + (11) FiniteFieldCyclicGroupExtension(PrimeField 3,4) + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty11} +\begin{paste}{ugxProblemFiniteConversionPageEmpty11}{ugxProblemFiniteConversionPagePatch11} +\pastebutton{ugxProblemFiniteConversionPageEmpty11}{\showpaste} +\tab{5}\spadcommand{Km := FFCGX(K,m)\bound{Km2 }\free{K m }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch12} +\begin{paste}{ugxProblemFiniteConversionPageFull12}{ugxProblemFiniteConversionPageEmpty12} +\pastebutton{ugxProblemFiniteConversionPageFull12}{\hidepaste} +\tab{5}\spadcommand{Kn := FFNBX(K,n)\bound{Kn2 }\free{K n }} +\indentrel{3}\begin{verbatim} + (12) FiniteFieldNormalBasisExtension(PrimeField 3,8) + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty12} +\begin{paste}{ugxProblemFiniteConversionPageEmpty12}{ugxProblemFiniteConversionPagePatch12} +\pastebutton{ugxProblemFiniteConversionPageEmpty12}{\showpaste} +\tab{5}\spadcommand{Kn := FFNBX(K,n)\bound{Kn2 }\free{K n }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch13} +\begin{paste}{ugxProblemFiniteConversionPageFull13}{ugxProblemFiniteConversionPageEmpty13} +\pastebutton{ugxProblemFiniteConversionPageFull13}{\hidepaste} +\tab{5}\spadcommand{(a1,b1) := (random()$Km,random()$Km)\bound{a12 b12 }\free{Km2 }} +\indentrel{3}\begin{verbatim} + 13 + (13) %BF + Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty13} +\begin{paste}{ugxProblemFiniteConversionPageEmpty13}{ugxProblemFiniteConversionPagePatch13} +\pastebutton{ugxProblemFiniteConversionPageEmpty13}{\showpaste} +\tab{5}\spadcommand{(a1,b1) := (random()$Km,random()$Km)\bound{a12 b12 }\free{Km2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch14} +\begin{paste}{ugxProblemFiniteConversionPageFull14}{ugxProblemFiniteConversionPageEmpty14} +\pastebutton{ugxProblemFiniteConversionPageFull14}{\hidepaste} +\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a22 }\free{a12 Kn2 }} +\indentrel{3}\begin{verbatim} + (14) + 6 5 4 2 + q q q q q + 2%BG + 2%BG + 2%BG + 2%BG + 2%BG + 2%BG + Type: FiniteFieldNormalBasisExtension(PrimeField 3,8) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty14} +\begin{paste}{ugxProblemFiniteConversionPageEmpty14}{ugxProblemFiniteConversionPagePatch14} +\pastebutton{ugxProblemFiniteConversionPageEmpty14}{\showpaste} +\tab{5}\spadcommand{a2 := a1 :: Kn\bound{a22 }\free{a12 Kn2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch15} +\begin{paste}{ugxProblemFiniteConversionPageFull15}{ugxProblemFiniteConversionPageEmpty15} +\pastebutton{ugxProblemFiniteConversionPageFull15}{\hidepaste} +\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b22 }\free{b12 Kn2 }} +\indentrel{3}\begin{verbatim} + (15) + 7 6 5 4 3 2 + q q q q q q + 2%BG + %BG + %BG + %BG + 2%BG + %BG + + + q + %BG + %BG + Type: FiniteFieldNormalBasisExtension(PrimeField 3,8) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty15} +\begin{paste}{ugxProblemFiniteConversionPageEmpty15}{ugxProblemFiniteConversionPagePatch15} +\pastebutton{ugxProblemFiniteConversionPageEmpty15}{\showpaste} +\tab{5}\spadcommand{b2 := b1 :: Kn\bound{b22 }\free{b12 Kn2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch16} +\begin{paste}{ugxProblemFiniteConversionPageFull16}{ugxProblemFiniteConversionPageEmpty16} +\pastebutton{ugxProblemFiniteConversionPageFull16}{\hidepaste} +\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a12 a22 b12 b22 Km2 }} +\indentrel{3}\begin{verbatim} + (16) 0 + Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty16} +\begin{paste}{ugxProblemFiniteConversionPageEmpty16}{ugxProblemFiniteConversionPagePatch16} +\pastebutton{ugxProblemFiniteConversionPageEmpty16}{\showpaste} +\tab{5}\spadcommand{a1+b1 - ((a2+b2) :: Km)\free{a12 a22 b12 b22 Km2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPagePatch17} +\begin{paste}{ugxProblemFiniteConversionPageFull17}{ugxProblemFiniteConversionPageEmpty17} +\pastebutton{ugxProblemFiniteConversionPageFull17}{\hidepaste} +\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a12 a22 b12 b22 Km2 }} +\indentrel{3}\begin{verbatim} + (17) 0 + Type: FiniteFieldCyclicGroupExtension(PrimeField 3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteConversionPageEmpty17} +\begin{paste}{ugxProblemFiniteConversionPageEmpty17}{ugxProblemFiniteConversionPagePatch17} +\pastebutton{ugxProblemFiniteConversionPageEmpty17}{\showpaste} +\tab{5}\spadcommand{a1*b1 - ((a2*b2) :: Km)\free{a12 a22 b12 b22 Km2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesSubstitutePagePatch1} +\begin{paste}{ugxProblemSeriesSubstitutePageFull1}{ugxProblemSeriesSubstitutePageEmpty1} +\pastebutton{ugxProblemSeriesSubstitutePageFull1}{\hidepaste} +\tab{5}\spadcommand{f := taylor(exp(x))\bound{f }} +\indentrel{3}\begin{verbatim} + (1) + 1 2 1 3 1 4 1 5 1 6 + 1 + x + � x + � x + 컴 x + 컴� x + 컴� x + 2 6 24 120 720 + + + 1 7 1 8 1 9 1 10 11 + 컴컴 x + 컴컴� x + 컴컴컴 x + 컴컴컴� x + O(x ) + 5040 40320 362880 3628800 + Type: UnivariateTaylorSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesSubstitutePageEmpty1} +\begin{paste}{ugxProblemSeriesSubstitutePageEmpty1}{ugxProblemSeriesSubstitutePagePatch1} +\pastebutton{ugxProblemSeriesSubstitutePageEmpty1}{\showpaste} +\tab{5}\spadcommand{f := taylor(exp(x))\bound{f }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesSubstitutePagePatch2} +\begin{paste}{ugxProblemSeriesSubstitutePageFull2}{ugxProblemSeriesSubstitutePageEmpty2} +\pastebutton{ugxProblemSeriesSubstitutePageFull2}{\hidepaste} +\tab{5}\spadcommand{eval(f,1.0)} +\indentrel{3}\begin{verbatim} + (2) + [1.0, 2.0, 2.5, 2.6666666666 666666667, + 2.7083333333 333333333, 2.7166666666 666666667, + 2.7180555555 555555556, 2.7182539682 53968254, + 2.7182787698 412698413, 2.7182815255 731922399, ...] + Type: Stream Expression Float +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesSubstitutePageEmpty2} +\begin{paste}{ugxProblemSeriesSubstitutePageEmpty2}{ugxProblemSeriesSubstitutePagePatch2} +\pastebutton{ugxProblemSeriesSubstitutePageEmpty2}{\showpaste} +\tab{5}\spadcommand{eval(f,1.0)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch1} +\begin{paste}{ugProblemFactorAlgPageFull1}{ugProblemFactorAlgPageEmpty1} +\pastebutton{ugProblemFactorAlgPageFull1}{\hidepaste} +\tab{5}\spadcommand{aa := rootOf(aa**2+aa+1)\bound{aa }} +\indentrel{3}\begin{verbatim} + (1) aa + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty1} +\begin{paste}{ugProblemFactorAlgPageEmpty1}{ugProblemFactorAlgPagePatch1} +\pastebutton{ugProblemFactorAlgPageEmpty1}{\showpaste} +\tab{5}\spadcommand{aa := rootOf(aa**2+aa+1)\bound{aa }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch2} +\begin{paste}{ugProblemFactorAlgPageFull2}{ugProblemFactorAlgPageEmpty2} +\pastebutton{ugProblemFactorAlgPageFull2}{\hidepaste} +\tab{5}\spadcommand{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2\free{aa }\bound{p }} +\indentrel{3}\begin{verbatim} + (2) + 5 3 4 + (- aa - 1)y + ((- aa - 1)x + aa x)y + + + 2 3 + ((- 2aa - 2)x + (- 2aa - 2)x)y + + + 5 4 3 2 2 + ((- 2aa - 2)x + (- 2aa - 2)x + 2aa x + 2aa x )y + + + 4 3 2 + ((- aa - 1)x + (- 2aa - 2)x + (- aa - 1)x )y + + + 7 6 5 4 3 + (- aa - 1)x + (- 2aa - 2)x - x + 2aa x + aa x + Type: Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty2} +\begin{paste}{ugProblemFactorAlgPageEmpty2}{ugProblemFactorAlgPagePatch2} +\pastebutton{ugProblemFactorAlgPageEmpty2}{\showpaste} +\tab{5}\spadcommand{p:=(x**3+aa**2*x+y)*(aa*x**2+aa*x+aa*y**2)**2\free{aa }\bound{p }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch3} +\begin{paste}{ugProblemFactorAlgPageFull3}{ugProblemFactorAlgPageEmpty3} +\pastebutton{ugProblemFactorAlgPageFull3}{\hidepaste} +\tab{5}\spadcommand{factor(p,[aa])\free{p aa }} +\indentrel{3}\begin{verbatim} + 3 2 2 2 + (3) (- aa - 1)(y + x + (- aa - 1)x)(y + x + x) + Type: Factored Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty3} +\begin{paste}{ugProblemFactorAlgPageEmpty3}{ugProblemFactorAlgPagePatch3} +\pastebutton{ugProblemFactorAlgPageEmpty3}{\showpaste} +\tab{5}\spadcommand{factor(p,[aa])\free{p aa }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch4} +\begin{paste}{ugProblemFactorAlgPageFull4}{ugProblemFactorAlgPageEmpty4} +\pastebutton{ugProblemFactorAlgPageFull4}{\hidepaste} +\tab{5}\spadcommand{factor(x**2+3)} +\indentrel{3}\begin{verbatim} + 2 + (4) x + 3 + Type: Factored Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty4} +\begin{paste}{ugProblemFactorAlgPageEmpty4}{ugProblemFactorAlgPagePatch4} +\pastebutton{ugProblemFactorAlgPageEmpty4}{\showpaste} +\tab{5}\spadcommand{factor(x**2+3)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch5} +\begin{paste}{ugProblemFactorAlgPageFull5}{ugProblemFactorAlgPageEmpty5} +\pastebutton{ugProblemFactorAlgPageFull5}{\hidepaste} +\tab{5}\spadcommand{factor(x**2+3,[aa])\free{aa }} +\indentrel{3}\begin{verbatim} + (5) (x - 2aa - 1)(x + 2aa + 1) + Type: Factored Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty5} +\begin{paste}{ugProblemFactorAlgPageEmpty5}{ugProblemFactorAlgPagePatch5} +\pastebutton{ugProblemFactorAlgPageEmpty5}{\showpaste} +\tab{5}\spadcommand{factor(x**2+3,[aa])\free{aa }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch6} +\begin{paste}{ugProblemFactorAlgPageFull6}{ugProblemFactorAlgPageEmpty6} +\pastebutton{ugProblemFactorAlgPageFull6}{\hidepaste} +\tab{5}\spadcommand{factor(x**6+108,[aa])\free{aa }} +\indentrel{3}\begin{verbatim} + 3 3 + (6) (x - 12aa - 6)(x + 12aa + 6) + Type: Factored Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty6} +\begin{paste}{ugProblemFactorAlgPageEmpty6}{ugProblemFactorAlgPagePatch6} +\pastebutton{ugProblemFactorAlgPageEmpty6}{\showpaste} +\tab{5}\spadcommand{factor(x**6+108,[aa])\free{aa }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch7} +\begin{paste}{ugProblemFactorAlgPageFull7}{ugProblemFactorAlgPageEmpty7} +\pastebutton{ugProblemFactorAlgPageFull7}{\hidepaste} +\tab{5}\spadcommand{bb:=rootOf(bb**3-2)\bound{bb }} +\indentrel{3}\begin{verbatim} + (7) bb + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty7} +\begin{paste}{ugProblemFactorAlgPageEmpty7}{ugProblemFactorAlgPagePatch7} +\pastebutton{ugProblemFactorAlgPageEmpty7}{\showpaste} +\tab{5}\spadcommand{bb:=rootOf(bb**3-2)\bound{bb }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch8} +\begin{paste}{ugProblemFactorAlgPageFull8}{ugProblemFactorAlgPageEmpty8} +\pastebutton{ugProblemFactorAlgPageFull8}{\hidepaste} +\tab{5}\spadcommand{factor(x**6+108,[bb])\free{bb }} +\indentrel{3}\begin{verbatim} + (8) + 2 2 2 2 2 2 + (x - 3bb x + 3bb )(x + 3bb )(x + 3bb x + 3bb ) + Type: Factored Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty8} +\begin{paste}{ugProblemFactorAlgPageEmpty8}{ugProblemFactorAlgPagePatch8} +\pastebutton{ugProblemFactorAlgPageEmpty8}{\showpaste} +\tab{5}\spadcommand{factor(x**6+108,[bb])\free{bb }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPagePatch9} +\begin{paste}{ugProblemFactorAlgPageFull9}{ugProblemFactorAlgPageEmpty9} +\pastebutton{ugProblemFactorAlgPageFull9}{\hidepaste} +\tab{5}\spadcommand{factor(x**6+108,[aa,bb])\free{aa bb }} +\indentrel{3}\begin{verbatim} + (9) + (x + (- 2aa - 1)bb)(x + (- aa - 2)bb) + * + (x + (- aa + 1)bb)(x + (aa - 1)bb)(x + (aa + 2)bb) + * + (x + (2aa + 1)bb) + Type: Factored Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorAlgPageEmpty9} +\begin{paste}{ugProblemFactorAlgPageEmpty9}{ugProblemFactorAlgPagePatch9} +\pastebutton{ugProblemFactorAlgPageEmpty9}{\showpaste} +\tab{5}\spadcommand{factor(x**6+108,[aa,bb])\free{aa bb }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch1} +\begin{paste}{ugxProblemSeriesConversionsPageFull1}{ugxProblemSeriesConversionsPageEmpty1} +\pastebutton{ugxProblemSeriesConversionsPageFull1}{\hidepaste} +\tab{5}\spadcommand{taylor(sin(x),x = 0)} +\indentrel{3}\begin{verbatim} + 1 3 1 5 1 7 1 9 11 + (1) x - � x + 컴� x - 컴컴 x + 컴컴컴 x + O(x ) + 6 120 5040 362880 + Type: UnivariateTaylorSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty1} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty1}{ugxProblemSeriesConversionsPagePatch1} +\pastebutton{ugxProblemSeriesConversionsPageEmpty1}{\showpaste} +\tab{5}\spadcommand{taylor(sin(x),x = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch2} +\begin{paste}{ugxProblemSeriesConversionsPageFull2}{ugxProblemSeriesConversionsPageEmpty2} +\pastebutton{ugxProblemSeriesConversionsPageFull2}{\hidepaste} +\tab{5}\spadcommand{taylor(sin(x),x = \%pi/6)} +\indentrel{3}\begin{verbatim} + (2) + 旼� 旼� + 1 \�3 %pi 1 %pi 2 \�3 %pi 3 + � + 컴컴 (x - 컴�) - � (x - 컴�) - 컴컴 (x - 컴�) + 2 2 6 4 6 12 6 + + + 旼� + 1 %pi 4 \�3 %pi 5 1 %pi 6 + 컴 (x - 컴�) + 컴컴 (x - 컴�) - 컴컴 (x - 컴�) + 48 6 240 6 1440 6 + + + 旼� + \�3 %pi 7 1 %pi 8 + - 컴컴� (x - 컴�) + 컴컴� (x - 컴�) + 10080 6 80640 6 + + + 旼� + \�3 %pi 9 1 %pi 10 + 컴컴컴 (x - 컴�) - 컴컴컴� (x - 컴�) + 725760 6 7257600 6 + + + %pi 11 + O((x - 컴�) ) + 6 +Type: UnivariateTaylorSeries(Expression Integer,x,pi/6) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty2} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty2}{ugxProblemSeriesConversionsPagePatch2} +\pastebutton{ugxProblemSeriesConversionsPageEmpty2}{\showpaste} +\tab{5}\spadcommand{taylor(sin(x),x = \%pi/6)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch3} +\begin{paste}{ugxProblemSeriesConversionsPageFull3}{ugxProblemSeriesConversionsPageEmpty3} +\pastebutton{ugxProblemSeriesConversionsPageFull3}{\hidepaste} +\tab{5}\spadcommand{taylor(tan(x*y),x = 0)} +\indentrel{3}\begin{verbatim} + (3) + 3 5 7 9 + y 3 2y 5 17y 7 62y 9 11 + y x + 컴 x + 컴� x + 컴컴 x + 컴컴 x + O(x ) + 3 15 315 2835 + Type: UnivariateTaylorSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty3} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty3}{ugxProblemSeriesConversionsPagePatch3} +\pastebutton{ugxProblemSeriesConversionsPageEmpty3}{\showpaste} +\tab{5}\spadcommand{taylor(tan(x*y),x = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch4} +\begin{paste}{ugxProblemSeriesConversionsPageFull4}{ugxProblemSeriesConversionsPageEmpty4} +\pastebutton{ugxProblemSeriesConversionsPageFull4}{\hidepaste} +\tab{5}\spadcommand{taylor(tan(x*y),y = 0)} +\indentrel{3}\begin{verbatim} + (4) + 3 5 7 9 + x 3 2x 5 17x 7 62x 9 11 + x y + 컴 y + 컴� y + 컴컴 y + 컴컴 y + O(y ) + 3 15 315 2835 + Type: UnivariateTaylorSeries(Expression Integer,y,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty4} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty4}{ugxProblemSeriesConversionsPagePatch4} +\pastebutton{ugxProblemSeriesConversionsPageEmpty4}{\showpaste} +\tab{5}\spadcommand{taylor(tan(x*y),y = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch5} +\begin{paste}{ugxProblemSeriesConversionsPageFull5}{ugxProblemSeriesConversionsPageEmpty5} +\pastebutton{ugxProblemSeriesConversionsPageFull5}{\hidepaste} +\tab{5}\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0)\bound{bern }} +\indentrel{3}\begin{verbatim} + (5) + 2 3 2 + 2x - 1 6x - 6x + 1 2 2x - 3x + x 3 + 1 + 컴컴컴 t + 컴컴컴컴컴컴 t + 컴컴컴컴컴컴� t + 2 12 12 + + + 4 3 2 5 4 3 + 30x - 60x + 30x - 1 4 6x - 15x + 10x - x 5 + 컴컴컴컴컴컴컴컴컴컴컴 t + 컴컴컴컴컴컴컴컴컴컴� t + 720 720 + + + 6 5 4 2 + 42x - 126x + 105x - 21x + 1 6 + 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� t + 30240 + + + 7 6 5 3 + 6x - 21x + 21x - 7x + x 7 + 컴컴컴컴컴컴컴컴컴컴컴컴컴� t + 30240 + + + 8 7 6 4 2 + 30x - 120x + 140x - 70x + 20x - 1 8 + 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 t + 1209600 + + + 9 8 7 5 3 + 10x - 45x + 60x - 42x + 20x - 3x 9 + 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� t + 3628800 + + + 10 9 8 6 4 2 + 66x - 330x + 495x - 462x + 330x - 99x + 5 10 + 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 t + 239500800 + + + 11 + O(t ) + Type: UnivariateTaylorSeries(Expression Integer,t,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty5} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty5}{ugxProblemSeriesConversionsPagePatch5} +\pastebutton{ugxProblemSeriesConversionsPageEmpty5}{\showpaste} +\tab{5}\spadcommand{bern := taylor(t*exp(x*t)/(exp(t) - 1),t = 0)\bound{bern }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch6} +\begin{paste}{ugxProblemSeriesConversionsPageFull6}{ugxProblemSeriesConversionsPageEmpty6} +\pastebutton{ugxProblemSeriesConversionsPageFull6}{\hidepaste} +\tab{5}\spadcommand{factorial(6) * coefficient(bern,6)\free{bern }} +\indentrel{3}\begin{verbatim} + 6 5 4 2 + 42x - 126x + 105x - 21x + 1 + (6) 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� + 42 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty6} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty6}{ugxProblemSeriesConversionsPagePatch6} +\pastebutton{ugxProblemSeriesConversionsPageEmpty6}{\showpaste} +\tab{5}\spadcommand{factorial(6) * coefficient(bern,6)\free{bern }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch7} +\begin{paste}{ugxProblemSeriesConversionsPageFull7}{ugxProblemSeriesConversionsPageEmpty7} +\pastebutton{ugxProblemSeriesConversionsPageFull7}{\hidepaste} +\tab{5}\spadcommand{bernoulliB(6,x)} +\indentrel{3}\begin{verbatim} + 6 5 5 4 1 2 1 + (7) x - 3x + � x - � x + 컴 + 2 2 42 + Type: Polynomial Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty7} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty7}{ugxProblemSeriesConversionsPagePatch7} +\pastebutton{ugxProblemSeriesConversionsPageEmpty7}{\showpaste} +\tab{5}\spadcommand{bernoulliB(6,x)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch8} +\begin{paste}{ugxProblemSeriesConversionsPageFull8}{ugxProblemSeriesConversionsPageEmpty8} +\pastebutton{ugxProblemSeriesConversionsPageFull8}{\hidepaste} +\tab{5}\spadcommand{laurent(x/log(x),x = 1)} +\indentrel{3}\begin{verbatim} + (8) + - 1 3 5 1 2 + (x - 1) + � + 컴 (x - 1) - 컴 (x - 1) + 2 12 24 + + + 11 3 11 4 271 5 + 컴� (x - 1) - 컴컴 (x - 1) + 컴컴� (x - 1) + 720 1440 60480 + + + 13 6 7297 7 425 8 + - 컴컴 (x - 1) + 컴컴컴� (x - 1) - 컴컴컴 (x - 1) + 4480 3628800 290304 + + + 530113 9 10 + 컴컴컴컴� (x - 1) + O((x - 1) ) + 479001600 + Type: UnivariateLaurentSeries(Expression Integer,x,1) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty8} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty8}{ugxProblemSeriesConversionsPagePatch8} +\pastebutton{ugxProblemSeriesConversionsPageEmpty8}{\showpaste} +\tab{5}\spadcommand{laurent(x/log(x),x = 1)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch9} +\begin{paste}{ugxProblemSeriesConversionsPageFull9}{ugxProblemSeriesConversionsPageEmpty9} +\pastebutton{ugxProblemSeriesConversionsPageFull9}{\hidepaste} +\tab{5}\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)} +\indentrel{3}\begin{verbatim} + (9) + 1 3 7 + - � � � + 3%pi 2 1 3%pi 2 1 3%pi 2 + (x - 컴컴) + 컴 (x - 컴컴) + 컴� (x - 컴컴) + 2 12 2 160 2 + + + 3%pi 5 + O((x - 컴컴) ) + 2 +Type: UnivariatePuiseuxSeries(Expression Integer,x,(3*pi)/2) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty9} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty9}{ugxProblemSeriesConversionsPagePatch9} +\pastebutton{ugxProblemSeriesConversionsPageEmpty9}{\showpaste} +\tab{5}\spadcommand{puiseux(sqrt(sec(x)),x = 3 * \%pi/2)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPagePatch10} +\begin{paste}{ugxProblemSeriesConversionsPageFull10}{ugxProblemSeriesConversionsPageEmpty10} +\pastebutton{ugxProblemSeriesConversionsPageFull10}{\hidepaste} +\tab{5}\spadcommand{series(x**x,x=0)} +\indentrel{3}\begin{verbatim} + (10) + 2 3 4 + log(x) 2 log(x) 3 log(x) 4 + 1 + log(x)x + 컴컴컴� x + 컴컴컴� x + 컴컴컴� x + 2 6 24 + + + 5 6 7 8 + log(x) 5 log(x) 6 log(x) 7 log(x) 8 + 컴컴컴� x + 컴컴컴� x + 컴컴컴� x + 컴컴컴� x + 120 720 5040 40320 + + + 9 10 + log(x) 9 log(x) 10 11 + 컴컴컴� x + 컴컴컴컴 x + O(x ) + 362880 3628800 +Type: GeneralUnivariatePowerSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesConversionsPageEmpty10} +\begin{paste}{ugxProblemSeriesConversionsPageEmpty10}{ugxProblemSeriesConversionsPagePatch10} +\pastebutton{ugxProblemSeriesConversionsPageEmpty10}{\showpaste} +\tab{5}\spadcommand{series(x**x,x=0)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPagePatch1} +\begin{paste}{ugxProblemDEQSeriesPageFull1}{ugxProblemDEQSeriesPageEmpty1} +\pastebutton{ugxProblemDEQSeriesPageFull1}{\hidepaste} +\tab{5}\spadcommand{)set streams calculate 7\bound{c7 }} +\indentrel{3}\begin{verbatim} +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPageEmpty1} +\begin{paste}{ugxProblemDEQSeriesPageEmpty1}{ugxProblemDEQSeriesPagePatch1} +\pastebutton{ugxProblemDEQSeriesPageEmpty1}{\showpaste} +\tab{5}\spadcommand{)set streams calculate 7\bound{c7 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPagePatch2} +\begin{paste}{ugxProblemDEQSeriesPageFull2}{ugxProblemDEQSeriesPageEmpty2} +\pastebutton{ugxProblemDEQSeriesPageFull2}{\hidepaste} +\tab{5}\spadcommand{y := operator 'y\bound{y }} +\indentrel{3}\begin{verbatim} + (1) y + Type: BasicOperator +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPageEmpty2} +\begin{paste}{ugxProblemDEQSeriesPageEmpty2}{ugxProblemDEQSeriesPagePatch2} +\pastebutton{ugxProblemDEQSeriesPageEmpty2}{\showpaste} +\tab{5}\spadcommand{y := operator 'y\bound{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPagePatch3} +\begin{paste}{ugxProblemDEQSeriesPageFull3}{ugxProblemDEQSeriesPageEmpty3} +\pastebutton{ugxProblemDEQSeriesPageFull3}{\hidepaste} +\tab{5}\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq }\free{y }} +\indentrel{3}\begin{verbatim} + ,,, y(x) ,, + (2) y (x) - %e sin(y (x))= cos(x) + + Type: Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPageEmpty3} +\begin{paste}{ugxProblemDEQSeriesPageEmpty3}{ugxProblemDEQSeriesPagePatch3} +\pastebutton{ugxProblemDEQSeriesPageEmpty3}{\showpaste} +\tab{5}\spadcommand{eq := D(y(x), x, 3) - sin(D(y(x), x, 2))*exp(y(x)) = cos(x)\bound{eq }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPagePatch4} +\begin{paste}{ugxProblemDEQSeriesPageFull4}{ugxProblemDEQSeriesPageEmpty4} +\pastebutton{ugxProblemDEQSeriesPageFull4}{\hidepaste} +\tab{5}\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y }\free{eq }\free{c7 }} +\indentrel{3}\begin{verbatim} + (3) + 2 3 + 1 3 %e 4 %e - 1 5 %e - 2%e 6 + 1 + � x + 컴 x + 컴컴컴� x + 컴컴컴컴� x + 6 24 120 720 + + + 4 2 + %e - 8%e + 4%e + 1 7 8 + 컴컴컴컴컴컴컴컴컴컴 x + O(x ) + 5040 + Type: UnivariateTaylorSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPageEmpty4} +\begin{paste}{ugxProblemDEQSeriesPageEmpty4}{ugxProblemDEQSeriesPagePatch4} +\pastebutton{ugxProblemDEQSeriesPageEmpty4}{\showpaste} +\tab{5}\spadcommand{seriesSolve(eq, y, x = 0, [1, 0, 0])\free{y }\free{eq }\free{c7 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPagePatch5} +\begin{paste}{ugxProblemDEQSeriesPageFull5}{ugxProblemDEQSeriesPageEmpty5} +\pastebutton{ugxProblemDEQSeriesPageFull5}{\hidepaste} +\tab{5}\spadcommand{x := operator 'x\bound{x }} +\indentrel{3}\begin{verbatim} + (4) x + Type: BasicOperator +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPageEmpty5} +\begin{paste}{ugxProblemDEQSeriesPageEmpty5}{ugxProblemDEQSeriesPagePatch5} +\pastebutton{ugxProblemDEQSeriesPageEmpty5}{\showpaste} +\tab{5}\spadcommand{x := operator 'x\bound{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPagePatch6} +\begin{paste}{ugxProblemDEQSeriesPageFull6}{ugxProblemDEQSeriesPageEmpty6} +\pastebutton{ugxProblemDEQSeriesPageFull6}{\hidepaste} +\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }} +\indentrel{3}\begin{verbatim} + , 2 + (5) x (t)= x(t) + 1 + + Type: Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPageEmpty6} +\begin{paste}{ugxProblemDEQSeriesPageEmpty6}{ugxProblemDEQSeriesPagePatch6} +\pastebutton{ugxProblemDEQSeriesPageEmpty6}{\showpaste} +\tab{5}\spadcommand{eq1 := D(x(t), t) = 1 + x(t)**2\free{x }\free{y }\bound{eq1 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPagePatch7} +\begin{paste}{ugxProblemDEQSeriesPageFull7}{ugxProblemDEQSeriesPageEmpty7} +\pastebutton{ugxProblemDEQSeriesPageFull7}{\hidepaste} +\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }} +\indentrel{3}\begin{verbatim} + , + (6) y (t)= x(t)y(t) + + Type: Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPageEmpty7} +\begin{paste}{ugxProblemDEQSeriesPageEmpty7}{ugxProblemDEQSeriesPagePatch7} +\pastebutton{ugxProblemDEQSeriesPageEmpty7}{\showpaste} +\tab{5}\spadcommand{eq2 := D(y(t), t) = x(t) * y(t)\free{x }\free{y }\bound{eq2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPagePatch8} +\begin{paste}{ugxProblemDEQSeriesPageFull8}{ugxProblemDEQSeriesPageEmpty8} +\pastebutton{ugxProblemDEQSeriesPageFull8}{\hidepaste} +\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 }} +\indentrel{3}\begin{verbatim} + (7) + 1 3 2 5 17 7 8 + [t + � t + 컴 t + 컴� t + O(t ), + 3 15 315 + 1 2 5 4 61 6 8 + 1 + � t + 컴 t + 컴� t + O(t )] + 2 24 720 +Type: List UnivariateTaylorSeries(Expression Integer,t,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemDEQSeriesPageEmpty8} +\begin{paste}{ugxProblemDEQSeriesPageEmpty8}{ugxProblemDEQSeriesPagePatch8} +\pastebutton{ugxProblemDEQSeriesPageEmpty8}{\showpaste} +\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 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPagePatch1} +\begin{paste}{ugxProblemSeriesCoefficientsPageFull1}{ugxProblemSeriesCoefficientsPageEmpty1} +\pastebutton{ugxProblemSeriesCoefficientsPageFull1}{\hidepaste} +\tab{5}\spadcommand{x := series(x)\bound{x }} +\indentrel{3}\begin{verbatim} + (1) x + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty1} +\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty1}{ugxProblemSeriesCoefficientsPagePatch1} +\pastebutton{ugxProblemSeriesCoefficientsPageEmpty1}{\showpaste} +\tab{5}\spadcommand{x := series(x)\bound{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPagePatch2} +\begin{paste}{ugxProblemSeriesCoefficientsPageFull2}{ugxProblemSeriesCoefficientsPageEmpty2} +\pastebutton{ugxProblemSeriesCoefficientsPageFull2}{\hidepaste} +\tab{5}\spadcommand{y := exp(x) * sin(x)\free{x }\bound{y }} +\indentrel{3}\begin{verbatim} + (2) + 2 1 3 1 5 1 6 1 7 1 9 + x + x + � x - 컴 x - 컴 x - 컴� x + 컴컴� x + 3 30 90 630 22680 + + + 1 10 1 11 12 + 컴컴컴 x + 컴컴컴� x + O(x ) + 113400 1247400 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty2} +\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty2}{ugxProblemSeriesCoefficientsPagePatch2} +\pastebutton{ugxProblemSeriesCoefficientsPageEmpty2}{\showpaste} +\tab{5}\spadcommand{y := exp(x) * sin(x)\free{x }\bound{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPagePatch3} +\begin{paste}{ugxProblemSeriesCoefficientsPageFull3}{ugxProblemSeriesCoefficientsPageEmpty3} +\pastebutton{ugxProblemSeriesCoefficientsPageFull3}{\hidepaste} +\tab{5}\spadcommand{coefficient(y,6)\free{y }} +\indentrel{3}\begin{verbatim} + 1 + (3) - 컴 + 90 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty3} +\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty3}{ugxProblemSeriesCoefficientsPagePatch3} +\pastebutton{ugxProblemSeriesCoefficientsPageEmpty3}{\showpaste} +\tab{5}\spadcommand{coefficient(y,6)\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPagePatch4} +\begin{paste}{ugxProblemSeriesCoefficientsPageFull4}{ugxProblemSeriesCoefficientsPageEmpty4} +\pastebutton{ugxProblemSeriesCoefficientsPageFull4}{\hidepaste} +\tab{5}\spadcommand{coefficient(y,15)\free{y }\bound{y15 }} +\indentrel{3}\begin{verbatim} + 1 + (4) - 컴컴컴컴컴� + 10216206000 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty4} +\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty4}{ugxProblemSeriesCoefficientsPagePatch4} +\pastebutton{ugxProblemSeriesCoefficientsPageEmpty4}{\showpaste} +\tab{5}\spadcommand{coefficient(y,15)\free{y }\bound{y15 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPagePatch5} +\begin{paste}{ugxProblemSeriesCoefficientsPageFull5}{ugxProblemSeriesCoefficientsPageEmpty5} +\pastebutton{ugxProblemSeriesCoefficientsPageFull5}{\hidepaste} +\tab{5}\spadcommand{y\free{y15 }} +\indentrel{3}\begin{verbatim} + (5) + 2 1 3 1 5 1 6 1 7 1 9 + x + x + � x - 컴 x - 컴 x - 컴� x + 컴컴� x + 3 30 90 630 22680 + + + 1 10 1 11 1 13 + 컴컴컴 x + 컴컴컴� x - 컴컴컴컴 x + 113400 1247400 97297200 + + + 1 14 1 15 16 + - 컴컴컴컴� x - 컴컴컴컴컴� x + O(x ) + 681080400 10216206000 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCoefficientsPageEmpty5} +\begin{paste}{ugxProblemSeriesCoefficientsPageEmpty5}{ugxProblemSeriesCoefficientsPagePatch5} +\pastebutton{ugxProblemSeriesCoefficientsPageEmpty5}{\showpaste} +\tab{5}\spadcommand{y\free{y15 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch1} +\begin{paste}{ugxProblemLDEQClosedPageFull1}{ugxProblemLDEQClosedPageEmpty1} +\pastebutton{ugxProblemLDEQClosedPageFull1}{\hidepaste} +\tab{5}\spadcommand{y := operator 'y\bound{y }} +\indentrel{3}\begin{verbatim} + (1) y + Type: BasicOperator +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty1} +\begin{paste}{ugxProblemLDEQClosedPageEmpty1}{ugxProblemLDEQClosedPagePatch1} +\pastebutton{ugxProblemLDEQClosedPageEmpty1}{\showpaste} +\tab{5}\spadcommand{y := operator 'y\bound{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch2} +\begin{paste}{ugxProblemLDEQClosedPageFull2}{ugxProblemLDEQClosedPageEmpty2} +\pastebutton{ugxProblemLDEQClosedPageFull2}{\hidepaste} +\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }} +\indentrel{3}\begin{verbatim} + ,, , + (2) y (x) + y (x) + y(x)= 0 + + Type: Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty2} +\begin{paste}{ugxProblemLDEQClosedPageEmpty2}{ugxProblemLDEQClosedPagePatch2} +\pastebutton{ugxProblemLDEQClosedPageEmpty2}{\showpaste} +\tab{5}\spadcommand{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1 }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch3} +\begin{paste}{ugxProblemLDEQClosedPageFull3}{ugxProblemLDEQClosedPageEmpty3} +\pastebutton{ugxProblemLDEQClosedPageFull3}{\hidepaste} +\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }} +\indentrel{3}\begin{verbatim} + (3) + [particular= 0, + x x + 旼� - � - � 旼� + x\�3 2 2 x\�3 + basis= [cos(컴컴�)%e ,%e sin(컴컴�)]] + 2 2 +Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty3} +\begin{paste}{ugxProblemLDEQClosedPageEmpty3}{ugxProblemLDEQClosedPagePatch3} +\pastebutton{ugxProblemLDEQClosedPageEmpty3}{\showpaste} +\tab{5}\spadcommand{solve(deq, y, x)\free{e1 }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch4} +\begin{paste}{ugxProblemLDEQClosedPageFull4}{ugxProblemLDEQClosedPageEmpty4} +\pastebutton{ugxProblemLDEQClosedPageFull4}{\hidepaste} +\tab{5}\spadcommand{deq := D(y x, x, 2) + y x\bound{e2 }\free{y }} +\indentrel{3}\begin{verbatim} + ,, + (4) y (x) + y(x) + + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty4} +\begin{paste}{ugxProblemLDEQClosedPageEmpty4}{ugxProblemLDEQClosedPagePatch4} +\pastebutton{ugxProblemLDEQClosedPageEmpty4}{\showpaste} +\tab{5}\spadcommand{deq := D(y x, x, 2) + y x\bound{e2 }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch5} +\begin{paste}{ugxProblemLDEQClosedPageFull5}{ugxProblemLDEQClosedPageEmpty5} +\pastebutton{ugxProblemLDEQClosedPageFull5}{\hidepaste} +\tab{5}\spadcommand{solve(deq, y, x = 0, [1, 1])\free{e2 }\free{y }} +\indentrel{3}\begin{verbatim} + (5) sin(x) + cos(x) + Type: Union(Expression Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty5} +\begin{paste}{ugxProblemLDEQClosedPageEmpty5}{ugxProblemLDEQClosedPagePatch5} +\pastebutton{ugxProblemLDEQClosedPageEmpty5}{\showpaste} +\tab{5}\spadcommand{solve(deq, y, x = 0, [1, 1])\free{e2 }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch6} +\begin{paste}{ugxProblemLDEQClosedPageFull6}{ugxProblemLDEQClosedPageEmpty6} +\pastebutton{ugxProblemLDEQClosedPageFull6}{\hidepaste} +\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 }} +\indentrel{3}\begin{verbatim} + 3 ,,, 2 ,, , 4 + (6) x y (x) + x y (x) - 2xy (x) + 2y(x)= 2x + + Type: Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty6} +\begin{paste}{ugxProblemLDEQClosedPageEmpty6}{ugxProblemLDEQClosedPagePatch6} +\pastebutton{ugxProblemLDEQClosedPageEmpty6}{\showpaste} +\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 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch7} +\begin{paste}{ugxProblemLDEQClosedPageFull7}{ugxProblemLDEQClosedPageEmpty7} +\pastebutton{ugxProblemLDEQClosedPageFull7}{\hidepaste} +\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }} +\indentrel{3}\begin{verbatim} + (7) + 5 3 2 + x - 10x + 20x + 4 + [particular= 컴컴컴컴컴컴컴컴컴컴, + 15x + 3 2 3 3 2 + 2x - 3x + 1 x - 1 x - 3x - 1 + basis= [컴컴컴컴컴컴�,컴컴컴,컴컴컴컴컴컴]] + x x x +Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty7} +\begin{paste}{ugxProblemLDEQClosedPageEmpty7}{ugxProblemLDEQClosedPagePatch7} +\pastebutton{ugxProblemLDEQClosedPageEmpty7}{\showpaste} +\tab{5}\spadcommand{solve(deq, y, x)\free{e3 }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch8} +\begin{paste}{ugxProblemLDEQClosedPageFull8}{ugxProblemLDEQClosedPageEmpty8} +\pastebutton{ugxProblemLDEQClosedPageFull8}{\hidepaste} +\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 }} +\indentrel{3}\begin{verbatim} + (8) + 9 3 ,,, 8 ,, , + (x + x )y (x) + 18x y (x) - 90xy (x) + + + + 6 + (- 330x + 90)y(x) + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty8} +\begin{paste}{ugxProblemLDEQClosedPageEmpty8}{ugxProblemLDEQClosedPagePatch8} +\pastebutton{ugxProblemLDEQClosedPageEmpty8}{\showpaste} +\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 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch9} +\begin{paste}{ugxProblemLDEQClosedPageFull9}{ugxProblemLDEQClosedPageEmpty9} +\pastebutton{ugxProblemLDEQClosedPageFull9}{\hidepaste} +\tab{5}\spadcommand{solve(deq, y, x)\free{e4 }\free{y }} +\indentrel{3}\begin{verbatim} + (9) + [particular= 0, + 旼커 旼커 + - \�91 log(x) \�91 log(x) + x x %e x %e + basis= [컴컴컴,컴컴컴컴컴컴컴컴�,컴컴컴컴컴컴컴�]] + 6 6 6 + x + 1 x + 1 x + 1 +Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty9} +\begin{paste}{ugxProblemLDEQClosedPageEmpty9}{ugxProblemLDEQClosedPagePatch9} +\pastebutton{ugxProblemLDEQClosedPageEmpty9}{\showpaste} +\tab{5}\spadcommand{solve(deq, y, x)\free{e4 }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch10} +\begin{paste}{ugxProblemLDEQClosedPageFull10}{ugxProblemLDEQClosedPageEmpty10} +\pastebutton{ugxProblemLDEQClosedPageFull10}{\hidepaste} +\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 }} +\indentrel{3}\begin{verbatim} + 2 ,, , + (10) (x + 1)y (x) + 3xy (x) + y(x)= 0 + + Type: Equation Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty10} +\begin{paste}{ugxProblemLDEQClosedPageEmpty10}{ugxProblemLDEQClosedPagePatch10} +\pastebutton{ugxProblemLDEQClosedPageEmpty10}{\showpaste} +\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 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPagePatch11} +\begin{paste}{ugxProblemLDEQClosedPageFull11}{ugxProblemLDEQClosedPageEmpty11} +\pastebutton{ugxProblemLDEQClosedPageFull11}{\hidepaste} +\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }} +\indentrel{3}\begin{verbatim} + (11) + 旼컴컴커 + � 2 + 1 log(\퀈 + 1 - x) + [particular= 0,basis= [컴컴컴컴�,컴컴컴컴컴컴컴컴컴]] + 旼컴컴커 旼컴컴커 + � 2 � 2 + \퀈 + 1 \퀈 + 1 +Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemLDEQClosedPageEmpty11} +\begin{paste}{ugxProblemLDEQClosedPageEmpty11}{ugxProblemLDEQClosedPagePatch11} +\pastebutton{ugxProblemLDEQClosedPageEmpty11}{\showpaste} +\tab{5}\spadcommand{solve(deq, y, x)\free{e5 }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch1} +\begin{paste}{ugxProblemNLDEQClosedPageFull1}{ugxProblemNLDEQClosedPageEmpty1} +\pastebutton{ugxProblemNLDEQClosedPageFull1}{\hidepaste} +\tab{5}\spadcommand{m := -y\bound{m }} +\indentrel{3}\begin{verbatim} + (1) - y + Type: Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty1} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty1}{ugxProblemNLDEQClosedPagePatch1} +\pastebutton{ugxProblemNLDEQClosedPageEmpty1}{\showpaste} +\tab{5}\spadcommand{m := -y\bound{m }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch2} +\begin{paste}{ugxProblemNLDEQClosedPageFull2}{ugxProblemNLDEQClosedPageEmpty2} +\pastebutton{ugxProblemNLDEQClosedPageFull2}{\hidepaste} +\tab{5}\spadcommand{n := x + y * log y\bound{n }} +\indentrel{3}\begin{verbatim} + (2) y log(y) + x + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty2} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty2}{ugxProblemNLDEQClosedPagePatch2} +\pastebutton{ugxProblemNLDEQClosedPageEmpty2}{\showpaste} +\tab{5}\spadcommand{n := x + y * log y\bound{n }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch3} +\begin{paste}{ugxProblemNLDEQClosedPageFull3}{ugxProblemNLDEQClosedPageEmpty3} +\pastebutton{ugxProblemNLDEQClosedPageFull3}{\hidepaste} +\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m n }} +\indentrel{3}\begin{verbatim} + (3) - 2 + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty3} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty3}{ugxProblemNLDEQClosedPagePatch3} +\pastebutton{ugxProblemNLDEQClosedPageEmpty3}{\showpaste} +\tab{5}\spadcommand{D(m, y) - D(n, x)\free{m n }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch4} +\begin{paste}{ugxProblemNLDEQClosedPageFull4}{ugxProblemNLDEQClosedPageEmpty4} +\pastebutton{ugxProblemNLDEQClosedPageFull4}{\hidepaste} +\tab{5}\spadcommand{mu := operator 'mu\bound{mu }} +\indentrel{3}\begin{verbatim} + (4) mu + Type: BasicOperator +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty4} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty4}{ugxProblemNLDEQClosedPagePatch4} +\pastebutton{ugxProblemNLDEQClosedPageEmpty4}{\showpaste} +\tab{5}\spadcommand{mu := operator 'mu\bound{mu }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch5} +\begin{paste}{ugxProblemNLDEQClosedPageFull5}{ugxProblemNLDEQClosedPageEmpty5} +\pastebutton{ugxProblemNLDEQClosedPageFull5}{\hidepaste} +\tab{5}\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x)\bound{a }\free{m n mu }} +\indentrel{3}\begin{verbatim} + , + (5) (- y log(y) - x)mu (x) - 2mu(x) + + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty5} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty5}{ugxProblemNLDEQClosedPagePatch5} +\pastebutton{ugxProblemNLDEQClosedPageEmpty5}{\showpaste} +\tab{5}\spadcommand{a := D(mu(x) * m, y) - D(mu(x) * n, x)\bound{a }\free{m n mu }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch6} +\begin{paste}{ugxProblemNLDEQClosedPageFull6}{ugxProblemNLDEQClosedPageEmpty6} +\pastebutton{ugxProblemNLDEQClosedPageFull6}{\hidepaste} +\tab{5}\spadcommand{solve(a = 0, mu, x)\free{mu a }} +\indentrel{3}\begin{verbatim} + (6) + 1 + [particular= 0,basis= [컴컴컴컴컴컴컴컴컴컴컴컴컴컴]] + 2 2 2 + y log(y) + 2x y log(y) + x +Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty6} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty6}{ugxProblemNLDEQClosedPagePatch6} +\pastebutton{ugxProblemNLDEQClosedPageEmpty6}{\showpaste} +\tab{5}\spadcommand{solve(a = 0, mu, x)\free{mu a }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch7} +\begin{paste}{ugxProblemNLDEQClosedPageFull7}{ugxProblemNLDEQClosedPageEmpty7} +\pastebutton{ugxProblemNLDEQClosedPageFull7}{\hidepaste} +\tab{5}\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x)\bound{b }\free{mu m }} +\indentrel{3}\begin{verbatim} + , + (7) - ymu (y) - 2mu(y) + + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty7} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty7}{ugxProblemNLDEQClosedPagePatch7} +\pastebutton{ugxProblemNLDEQClosedPageEmpty7}{\showpaste} +\tab{5}\spadcommand{b := D(mu(y) * m, y) - D(mu(y) * n, x)\bound{b }\free{mu m }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch8} +\begin{paste}{ugxProblemNLDEQClosedPageFull8}{ugxProblemNLDEQClosedPageEmpty8} +\pastebutton{ugxProblemNLDEQClosedPageFull8}{\hidepaste} +\tab{5}\spadcommand{sb := solve(b = 0, mu, y)\free{mu b }\bound{sb }} +\indentrel{3}\begin{verbatim} + 1 + (8) [particular= 0,basis= [컴]] + 2 + y +Type: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty8} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty8}{ugxProblemNLDEQClosedPagePatch8} +\pastebutton{ugxProblemNLDEQClosedPageEmpty8}{\showpaste} +\tab{5}\spadcommand{sb := solve(b = 0, mu, y)\free{mu b }\bound{sb }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch9} +\begin{paste}{ugxProblemNLDEQClosedPageFull9}{ugxProblemNLDEQClosedPageEmpty9} +\pastebutton{ugxProblemNLDEQClosedPageFull9}{\hidepaste} +\tab{5}\spadcommand{intFactor := sb.basis.1\bound{intFactor }\free{sb }} +\indentrel{3}\begin{verbatim} + 1 + (9) 컴 + 2 + y + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty9} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty9}{ugxProblemNLDEQClosedPagePatch9} +\pastebutton{ugxProblemNLDEQClosedPageEmpty9}{\showpaste} +\tab{5}\spadcommand{intFactor := sb.basis.1\bound{intFactor }\free{sb }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch10} +\begin{paste}{ugxProblemNLDEQClosedPageFull10}{ugxProblemNLDEQClosedPageEmpty10} +\pastebutton{ugxProblemNLDEQClosedPageFull10}{\hidepaste} +\tab{5}\spadcommand{m := intFactor * m\bound{m1 }\free{m intFactor }} +\indentrel{3}\begin{verbatim} + 1 + (10) - � + y + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty10} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty10}{ugxProblemNLDEQClosedPagePatch10} +\pastebutton{ugxProblemNLDEQClosedPageEmpty10}{\showpaste} +\tab{5}\spadcommand{m := intFactor * m\bound{m1 }\free{m intFactor }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch11} +\begin{paste}{ugxProblemNLDEQClosedPageFull11}{ugxProblemNLDEQClosedPageEmpty11} +\pastebutton{ugxProblemNLDEQClosedPageFull11}{\hidepaste} +\tab{5}\spadcommand{n := intFactor * n\bound{n1 }\free{n intFactor }} +\indentrel{3}\begin{verbatim} + y log(y) + x + (11) 컴컴컴컴컴컴 + 2 + y + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPageEmpty11} +\begin{paste}{ugxProblemNLDEQClosedPageEmpty11}{ugxProblemNLDEQClosedPagePatch11} +\pastebutton{ugxProblemNLDEQClosedPageEmpty11}{\showpaste} +\tab{5}\spadcommand{n := intFactor * n\bound{n1 }\free{n intFactor }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemNLDEQClosedPagePatch12} +\begin{paste}{ugxProblemNLDEQClosedPageFull12}{ugxProblemNLDEQClosedPageEmpty12} 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+\begin{patch}{ugProblemIntegrationPageEmpty6} +\begin{paste}{ugProblemIntegrationPageEmpty6}{ugProblemIntegrationPagePatch6} +\pastebutton{ugProblemIntegrationPageEmpty6}{\showpaste} +\tab{5}\spadcommand{complexIntegrate(x**2 / (x**4 - a**2), x)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIntegrationPagePatch7} +\begin{paste}{ugProblemIntegrationPageFull7}{ugProblemIntegrationPageEmpty7} +\pastebutton{ugProblemIntegrationPageFull7}{\hidepaste} +\tab{5}\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)} +\indentrel{3}\begin{verbatim} + x 旼컴컴컴커 + 悶 log(\쿫 + %E a + 1) + (7) � 컴컴컴컴컴컴컴컴컴컴 d%E + 잎 %E + Type: Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIntegrationPageEmpty7} +\begin{paste}{ugProblemIntegrationPageEmpty7}{ugProblemIntegrationPagePatch7} +\pastebutton{ugProblemIntegrationPageEmpty7}{\showpaste} +\tab{5}\spadcommand{complexIntegrate(log(1 + sqrt(a * x + b)) / x, x)} +\end{paste}\end{patch} + 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+\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2)} +\indentrel{3}\begin{verbatim} + (9) potentialPole + Type: Union(pole: potentialPole,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIntegrationPageEmpty9} +\begin{paste}{ugProblemIntegrationPageEmpty9}{ugProblemIntegrationPagePatch9} +\pastebutton{ugProblemIntegrationPageEmpty9}{\showpaste} +\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIntegrationPagePatch10} +\begin{paste}{ugProblemIntegrationPageFull10}{ugProblemIntegrationPageEmpty10} +\pastebutton{ugProblemIntegrationPageFull10}{\hidepaste} +\tab{5}\spadcommand{integrate(1/(x**2-a), x = 1..2, "noPole")} +\indentrel{3}\begin{verbatim} + (10) + [ + 2 旼� 3 2 + (- 4a - 4a)\쿪 + a + 6a + a + - log(컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�) + 2 + a - 2a + 1 + + + 2 旼� 3 2 + (- 8a - 32a)\쿪 + a + 24a + 16a + log(컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�) + 2 + a - 8a + 16 + / + 旼� + 4\쿪 + , + 旼컴� 旼컴� + 2\�- a \�- a + - atan(컴컴컴�) + 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+\pastebutton{ugProblemFactorIntRatPageFull1}{\hidepaste} +\tab{5}\spadcommand{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12)\bound{v }} +\indentrel{3}\begin{verbatim} + (1) + 3 3 5 2 6 3 8 5 + - 2x y + (24x + 24)y + (- 4x - x )y + 48x + 12x + + + 3 + 48x + 12 + Type: Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorIntRatPageEmpty1} +\begin{paste}{ugProblemFactorIntRatPageEmpty1}{ugProblemFactorIntRatPagePatch1} +\pastebutton{ugProblemFactorIntRatPageEmpty1}{\showpaste} +\tab{5}\spadcommand{v := (4*x**3+2*y**2+1)*(12*x**5-x**3*y+12)\bound{v }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorIntRatPagePatch2} +\begin{paste}{ugProblemFactorIntRatPageFull2}{ugProblemFactorIntRatPageEmpty2} +\pastebutton{ugProblemFactorIntRatPageFull2}{\hidepaste} +\tab{5}\spadcommand{factor v\free{v }} +\indentrel{3}\begin{verbatim} + 3 5 2 3 + (2) - (x y - 12x - 12)(2y + 4x + 1) + Type: Factored Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorIntRatPageEmpty2} +\begin{paste}{ugProblemFactorIntRatPageEmpty2}{ugProblemFactorIntRatPagePatch2} +\pastebutton{ugProblemFactorIntRatPageEmpty2}{\showpaste} +\tab{5}\spadcommand{factor v\free{v }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorIntRatPagePatch3} +\begin{paste}{ugProblemFactorIntRatPageFull3}{ugProblemFactorIntRatPageEmpty3} +\pastebutton{ugProblemFactorIntRatPageFull3}{\hidepaste} +\tab{5}\spadcommand{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12)\bound{w }} +\indentrel{3}\begin{verbatim} + 8 7 6 35 5 95 3 2 + (3) 48x + 8x - 2x + 컴 x + 컴 x + 8x + 12 + 3 2 + Type: Polynomial Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorIntRatPageEmpty3} +\begin{paste}{ugProblemFactorIntRatPageEmpty3}{ugProblemFactorIntRatPagePatch3} +\pastebutton{ugProblemFactorIntRatPageEmpty3}{\showpaste} +\tab{5}\spadcommand{w := (4*x**3+(2/3)*x**2+1)*(12*x**5-(1/2)*x**3+12)\bound{w }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorIntRatPagePatch4} +\begin{paste}{ugProblemFactorIntRatPageFull4}{ugProblemFactorIntRatPageEmpty4} +\pastebutton{ugProblemFactorIntRatPageFull4}{\hidepaste} +\tab{5}\spadcommand{factor w\free{w }} +\indentrel{3}\begin{verbatim} + 3 1 2 1 5 1 3 + (4) 48(x + � x + �)(x - 컴 x + 1) + 6 4 24 + Type: Factored Polynomial Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemFactorIntRatPageEmpty4} +\begin{paste}{ugProblemFactorIntRatPageEmpty4}{ugProblemFactorIntRatPagePatch4} +\pastebutton{ugProblemFactorIntRatPageEmpty4}{\showpaste} +\tab{5}\spadcommand{factor w\free{w }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch1} +\begin{paste}{ugxProblemFinitePrimePageFull1}{ugxProblemFinitePrimePageEmpty1} +\pastebutton{ugxProblemFinitePrimePageFull1}{\hidepaste} +\tab{5}\spadcommand{(a,b) : IntegerMod 12\bound{abdec }} +\indentrel{3}\begin{verbatim} + Type: Void +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty1} +\begin{paste}{ugxProblemFinitePrimePageEmpty1}{ugxProblemFinitePrimePagePatch1} +\pastebutton{ugxProblemFinitePrimePageEmpty1}{\showpaste} +\tab{5}\spadcommand{(a,b) : IntegerMod 12\bound{abdec }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch2} +\begin{paste}{ugxProblemFinitePrimePageFull2}{ugxProblemFinitePrimePageEmpty2} +\pastebutton{ugxProblemFinitePrimePageFull2}{\hidepaste} +\tab{5}\spadcommand{(a, b) := (16, 7)\free{abdec }\bound{a b }} +\indentrel{3}\begin{verbatim} + (2) 7 + Type: IntegerMod 12 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty2} +\begin{paste}{ugxProblemFinitePrimePageEmpty2}{ugxProblemFinitePrimePagePatch2} +\pastebutton{ugxProblemFinitePrimePageEmpty2}{\showpaste} +\tab{5}\spadcommand{(a, b) := (16, 7)\free{abdec }\bound{a b }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch3} +\begin{paste}{ugxProblemFinitePrimePageFull3}{ugxProblemFinitePrimePageEmpty3} +\pastebutton{ugxProblemFinitePrimePageFull3}{\hidepaste} +\tab{5}\spadcommand{[a - b, a * b]\free{a b }} +\indentrel{3}\begin{verbatim} + (3) [9,4] + Type: List IntegerMod 12 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty3} +\begin{paste}{ugxProblemFinitePrimePageEmpty3}{ugxProblemFinitePrimePagePatch3} +\pastebutton{ugxProblemFinitePrimePageEmpty3}{\showpaste} +\tab{5}\spadcommand{[a - b, a * b]\free{a b }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch4} +\begin{paste}{ugxProblemFinitePrimePageFull4}{ugxProblemFinitePrimePageEmpty4} +\pastebutton{ugxProblemFinitePrimePageFull4}{\hidepaste} +\tab{5}\spadcommand{a / b\free{a b }} +\indentrel{3}\begin{verbatim} +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty4} +\begin{paste}{ugxProblemFinitePrimePageEmpty4}{ugxProblemFinitePrimePagePatch4} +\pastebutton{ugxProblemFinitePrimePageEmpty4}{\showpaste} +\tab{5}\spadcommand{a / b\free{a b }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch5} +\begin{paste}{ugxProblemFinitePrimePageFull5}{ugxProblemFinitePrimePageEmpty5} +\pastebutton{ugxProblemFinitePrimePageFull5}{\hidepaste} +\tab{5}\spadcommand{recip a\free{a }} +\indentrel{3}\begin{verbatim} + (4) "failed" + Type: Union("failed",...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty5} +\begin{paste}{ugxProblemFinitePrimePageEmpty5}{ugxProblemFinitePrimePagePatch5} +\pastebutton{ugxProblemFinitePrimePageEmpty5}{\showpaste} +\tab{5}\spadcommand{recip a\free{a }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch6} +\begin{paste}{ugxProblemFinitePrimePageFull6}{ugxProblemFinitePrimePageEmpty6} +\pastebutton{ugxProblemFinitePrimePageFull6}{\hidepaste} +\tab{5}\spadcommand{recip b\free{b }} +\indentrel{3}\begin{verbatim} + (5) 7 + Type: Union(IntegerMod 12,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty6} +\begin{paste}{ugxProblemFinitePrimePageEmpty6}{ugxProblemFinitePrimePagePatch6} +\pastebutton{ugxProblemFinitePrimePageEmpty6}{\showpaste} +\tab{5}\spadcommand{recip b\free{b }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch7} +\begin{paste}{ugxProblemFinitePrimePageFull7}{ugxProblemFinitePrimePageEmpty7} +\pastebutton{ugxProblemFinitePrimePageFull7}{\hidepaste} +\tab{5}\spadcommand{c : PrimeField 11 := 8\bound{c }} +\indentrel{3}\begin{verbatim} + (6) 8 + Type: PrimeField 11 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty7} +\begin{paste}{ugxProblemFinitePrimePageEmpty7}{ugxProblemFinitePrimePagePatch7} +\pastebutton{ugxProblemFinitePrimePageEmpty7}{\showpaste} +\tab{5}\spadcommand{c : PrimeField 11 := 8\bound{c }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch8} +\begin{paste}{ugxProblemFinitePrimePageFull8}{ugxProblemFinitePrimePageEmpty8} +\pastebutton{ugxProblemFinitePrimePageFull8}{\hidepaste} +\tab{5}\spadcommand{inv c\free{c }} +\indentrel{3}\begin{verbatim} + (7) 7 + Type: PrimeField 11 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty8} +\begin{paste}{ugxProblemFinitePrimePageEmpty8}{ugxProblemFinitePrimePagePatch8} +\pastebutton{ugxProblemFinitePrimePageEmpty8}{\showpaste} +\tab{5}\spadcommand{inv c\free{c }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch9} +\begin{paste}{ugxProblemFinitePrimePageFull9}{ugxProblemFinitePrimePageEmpty9} +\pastebutton{ugxProblemFinitePrimePageFull9}{\hidepaste} +\tab{5}\spadcommand{9/c\free{c }} +\indentrel{3}\begin{verbatim} + (8) 8 + Type: PrimeField 11 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty9} +\begin{paste}{ugxProblemFinitePrimePageEmpty9}{ugxProblemFinitePrimePagePatch9} +\pastebutton{ugxProblemFinitePrimePageEmpty9}{\showpaste} +\tab{5}\spadcommand{9/c\free{c }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch10} +\begin{paste}{ugxProblemFinitePrimePageFull10}{ugxProblemFinitePrimePageEmpty10} +\pastebutton{ugxProblemFinitePrimePageFull10}{\hidepaste} +\tab{5}\spadcommand{GF101 := PF 101\bound{GF101 }} +\indentrel{3}\begin{verbatim} + (9) PrimeField 101 + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty10} +\begin{paste}{ugxProblemFinitePrimePageEmpty10}{ugxProblemFinitePrimePagePatch10} +\pastebutton{ugxProblemFinitePrimePageEmpty10}{\showpaste} +\tab{5}\spadcommand{GF101 := PF 101\bound{GF101 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch11} +\begin{paste}{ugxProblemFinitePrimePageFull11}{ugxProblemFinitePrimePageEmpty11} +\pastebutton{ugxProblemFinitePrimePageFull11}{\hidepaste} +\tab{5}\spadcommand{x := random()$GF101\bound{x }\free{GF101 }} +\indentrel{3}\begin{verbatim} + (10) 78 + Type: PrimeField 101 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty11} +\begin{paste}{ugxProblemFinitePrimePageEmpty11}{ugxProblemFinitePrimePagePatch11} +\pastebutton{ugxProblemFinitePrimePageEmpty11}{\showpaste} +\tab{5}\spadcommand{x := random()$GF101\bound{x }\free{GF101 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch12} +\begin{paste}{ugxProblemFinitePrimePageFull12}{ugxProblemFinitePrimePageEmpty12} +\pastebutton{ugxProblemFinitePrimePageFull12}{\hidepaste} +\tab{5}\spadcommand{y : GF101 := 37\bound{y }\free{GF101 }} +\indentrel{3}\begin{verbatim} + (11) 37 + Type: PrimeField 101 +\end{verbatim} 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+\begin{paste}{ugxProblemFinitePrimePageFull14}{ugxProblemFinitePrimePageEmpty14} +\pastebutton{ugxProblemFinitePrimePageFull14}{\hidepaste} +\tab{5}\spadcommand{z * y - x\free{x y z }} +\indentrel{3}\begin{verbatim} + (13) 0 + Type: PrimeField 101 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty14} +\begin{paste}{ugxProblemFinitePrimePageEmpty14}{ugxProblemFinitePrimePagePatch14} +\pastebutton{ugxProblemFinitePrimePageEmpty14}{\showpaste} +\tab{5}\spadcommand{z * y - x\free{x y z }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch15} +\begin{paste}{ugxProblemFinitePrimePageFull15}{ugxProblemFinitePrimePageEmpty15} +\pastebutton{ugxProblemFinitePrimePageFull15}{\hidepaste} +\tab{5}\spadcommand{pe := primitiveElement()$GF101\bound{pe }\free{GF101 }} +\indentrel{3}\begin{verbatim} + (14) 2 + Type: PrimeField 101 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty15} +\begin{paste}{ugxProblemFinitePrimePageEmpty15}{ugxProblemFinitePrimePagePatch15} +\pastebutton{ugxProblemFinitePrimePageEmpty15}{\showpaste} +\tab{5}\spadcommand{pe := primitiveElement()$GF101\bound{pe }\free{GF101 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch16} +\begin{paste}{ugxProblemFinitePrimePageFull16}{ugxProblemFinitePrimePageEmpty16} +\pastebutton{ugxProblemFinitePrimePageFull16}{\hidepaste} +\tab{5}\spadcommand{[pe**i for i in 0..99]\free{pe }} +\indentrel{3}\begin{verbatim} + (15) + [1, 2, 4, 8, 16, 32, 64, 27, 54, 7, 14, 28, 56, 11, + 22, 44, 88, 75, 49, 98, 95, 89, 77, 53, 5, 10, 20, + 40, 80, 59, 17, 34, 68, 35, 70, 39, 78, 55, 9, 18, + 36, 72, 43, 86, 71, 41, 82, 63, 25, 50, 100, 99, 97, + 93, 85, 69, 37, 74, 47, 94, 87, 73, 45, 90, 79, 57, + 13, 26, 52, 3, 6, 12, 24, 48, 96, 91, 81, 61, 21, 42, + 84, 67, 33, 66, 31, 62, 23, 46, 92, 83, 65, 29, 58, + 15, 30, 60, 19, 38, 76, 51] + Type: List PrimeField 101 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty16} +\begin{paste}{ugxProblemFinitePrimePageEmpty16}{ugxProblemFinitePrimePagePatch16} +\pastebutton{ugxProblemFinitePrimePageEmpty16}{\showpaste} +\tab{5}\spadcommand{[pe**i for i in 0..99]\free{pe }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch17} +\begin{paste}{ugxProblemFinitePrimePageFull17}{ugxProblemFinitePrimePageEmpty17} +\pastebutton{ugxProblemFinitePrimePageFull17}{\hidepaste} +\tab{5}\spadcommand{ex := discreteLog(y)\bound{ex }\free{y }} +\indentrel{3}\begin{verbatim} + (16) 56 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty17} +\begin{paste}{ugxProblemFinitePrimePageEmpty17}{ugxProblemFinitePrimePagePatch17} +\pastebutton{ugxProblemFinitePrimePageEmpty17}{\showpaste} +\tab{5}\spadcommand{ex := discreteLog(y)\bound{ex }\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch18} +\begin{paste}{ugxProblemFinitePrimePageFull18}{ugxProblemFinitePrimePageEmpty18} +\pastebutton{ugxProblemFinitePrimePageFull18}{\hidepaste} +\tab{5}\spadcommand{pe ** ex\free{ex pe }} +\indentrel{3}\begin{verbatim} + (17) 37 + Type: PrimeField 101 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty18} +\begin{paste}{ugxProblemFinitePrimePageEmpty18}{ugxProblemFinitePrimePagePatch18} +\pastebutton{ugxProblemFinitePrimePageEmpty18}{\showpaste} +\tab{5}\spadcommand{pe ** ex\free{ex pe }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch19} +\begin{paste}{ugxProblemFinitePrimePageFull19}{ugxProblemFinitePrimePageEmpty19} +\pastebutton{ugxProblemFinitePrimePageFull19}{\hidepaste} +\tab{5}\spadcommand{order y\free{y }} +\indentrel{3}\begin{verbatim} + (18) 25 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty19} +\begin{paste}{ugxProblemFinitePrimePageEmpty19}{ugxProblemFinitePrimePagePatch19} +\pastebutton{ugxProblemFinitePrimePageEmpty19}{\showpaste} +\tab{5}\spadcommand{order y\free{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePagePatch20} +\begin{paste}{ugxProblemFinitePrimePageFull20}{ugxProblemFinitePrimePageEmpty20} +\pastebutton{ugxProblemFinitePrimePageFull20}{\hidepaste} +\tab{5}\spadcommand{order pe\free{pe }} +\indentrel{3}\begin{verbatim} + (19) 100 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFinitePrimePageEmpty20} +\begin{paste}{ugxProblemFinitePrimePageEmpty20}{ugxProblemFinitePrimePagePatch20} +\pastebutton{ugxProblemFinitePrimePageEmpty20}{\showpaste} +\tab{5}\spadcommand{order pe\free{pe }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch1} +\begin{paste}{ugxProblemFiniteCyclicPageFull1}{ugxProblemFiniteCyclicPageEmpty1} +\pastebutton{ugxProblemFiniteCyclicPageFull1}{\hidepaste} +\tab{5}\spadcommand{GF81 := FFCG(3,4);\bound{GF81 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty1} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty1}{ugxProblemFiniteCyclicPagePatch1} +\pastebutton{ugxProblemFiniteCyclicPageEmpty1}{\showpaste} +\tab{5}\spadcommand{GF81 := FFCG(3,4);\bound{GF81 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch2} +\begin{paste}{ugxProblemFiniteCyclicPageFull2}{ugxProblemFiniteCyclicPageEmpty2} +\pastebutton{ugxProblemFiniteCyclicPageFull2}{\hidepaste} +\tab{5}\spadcommand{a := primitiveElement()$GF81\bound{a }\free{GF81 }} +\indentrel{3}\begin{verbatim} + 1 + (2) %BF + Type: FiniteFieldCyclicGroup(3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty2} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty2}{ugxProblemFiniteCyclicPagePatch2} +\pastebutton{ugxProblemFiniteCyclicPageEmpty2}{\showpaste} +\tab{5}\spadcommand{a := primitiveElement()$GF81\bound{a }\free{GF81 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch3} +\begin{paste}{ugxProblemFiniteCyclicPageFull3}{ugxProblemFiniteCyclicPageEmpty3} +\pastebutton{ugxProblemFiniteCyclicPageFull3}{\hidepaste} +\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }} +\indentrel{3}\begin{verbatim} + 72 + (3) %BF + Type: FiniteFieldCyclicGroup(3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty3} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty3}{ugxProblemFiniteCyclicPagePatch3} +\pastebutton{ugxProblemFiniteCyclicPageEmpty3}{\showpaste} +\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch4} +\begin{paste}{ugxProblemFiniteCyclicPageFull4}{ugxProblemFiniteCyclicPageEmpty4} +\pastebutton{ugxProblemFiniteCyclicPageFull4}{\hidepaste} +\tab{5}\spadcommand{b\free{b }} +\indentrel{3}\begin{verbatim} + 72 + (4) %BF + Type: FiniteFieldCyclicGroup(3,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty4} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty4}{ugxProblemFiniteCyclicPagePatch4} +\pastebutton{ugxProblemFiniteCyclicPageEmpty4}{\showpaste} +\tab{5}\spadcommand{b\free{b }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch5} +\begin{paste}{ugxProblemFiniteCyclicPageFull5}{ugxProblemFiniteCyclicPageEmpty5} +\pastebutton{ugxProblemFiniteCyclicPageFull5}{\hidepaste} +\tab{5}\spadcommand{discreteLog b\free{b }} +\indentrel{3}\begin{verbatim} + (5) 72 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty5} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty5}{ugxProblemFiniteCyclicPagePatch5} +\pastebutton{ugxProblemFiniteCyclicPageEmpty5}{\showpaste} +\tab{5}\spadcommand{discreteLog b\free{b }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch6} +\begin{paste}{ugxProblemFiniteCyclicPageFull6}{ugxProblemFiniteCyclicPageEmpty6} +\pastebutton{ugxProblemFiniteCyclicPageFull6}{\hidepaste} +\tab{5}\spadcommand{GF9 := FF(3,2);\bound{GF9 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty6} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty6}{ugxProblemFiniteCyclicPagePatch6} +\pastebutton{ugxProblemFiniteCyclicPageEmpty6}{\showpaste} +\tab{5}\spadcommand{GF9 := FF(3,2);\bound{GF9 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch7} +\begin{paste}{ugxProblemFiniteCyclicPageFull7}{ugxProblemFiniteCyclicPageEmpty7} +\pastebutton{ugxProblemFiniteCyclicPageFull7}{\hidepaste} +\tab{5}\spadcommand{GF729 := FFCGX(GF9,3);\bound{GF729 }\free{GF9 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty7} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty7}{ugxProblemFiniteCyclicPagePatch7} +\pastebutton{ugxProblemFiniteCyclicPageEmpty7}{\showpaste} +\tab{5}\spadcommand{GF729 := FFCGX(GF9,3);\bound{GF729 }\free{GF9 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch8} +\begin{paste}{ugxProblemFiniteCyclicPageFull8}{ugxProblemFiniteCyclicPageEmpty8} +\pastebutton{ugxProblemFiniteCyclicPageFull8}{\hidepaste} +\tab{5}\spadcommand{r := (random()$GF729) ** 20\free{GF729 }\bound{r }} +\indentrel{3}\begin{verbatim} + 68 + (8) %CN +Type: FiniteFieldCyclicGroupExtension(FiniteField(3,2),3) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty8} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty8}{ugxProblemFiniteCyclicPagePatch8} +\pastebutton{ugxProblemFiniteCyclicPageEmpty8}{\showpaste} +\tab{5}\spadcommand{r := (random()$GF729) ** 20\free{GF729 }\bound{r }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch9} +\begin{paste}{ugxProblemFiniteCyclicPageFull9}{ugxProblemFiniteCyclicPageEmpty9} +\pastebutton{ugxProblemFiniteCyclicPageFull9}{\hidepaste} +\tab{5}\spadcommand{trace(r)\free{r }} +\indentrel{3}\begin{verbatim} + (9) 2 + Type: FiniteField(3,2) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty9} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty9}{ugxProblemFiniteCyclicPagePatch9} +\pastebutton{ugxProblemFiniteCyclicPageEmpty9}{\showpaste} +\tab{5}\spadcommand{trace(r)\free{r }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch10} +\begin{paste}{ugxProblemFiniteCyclicPageFull10}{ugxProblemFiniteCyclicPageEmpty10} +\pastebutton{ugxProblemFiniteCyclicPageFull10}{\hidepaste} +\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty10} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty10}{ugxProblemFiniteCyclicPagePatch10} +\pastebutton{ugxProblemFiniteCyclicPageEmpty10}{\showpaste} +\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch11} +\begin{paste}{ugxProblemFiniteCyclicPageFull11}{ugxProblemFiniteCyclicPageEmpty11} +\pastebutton{ugxProblemFiniteCyclicPageFull11}{\hidepaste} +\tab{5}\spadcommand{f := createPrimitivePoly(4)$FFPOLY(GF3)\bound{f }\free{GF3 }} +\indentrel{3}\begin{verbatim} + 4 + (11) ? + ? + 2 + Type: SparseUnivariatePolynomial PrimeField 3 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty11} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty11}{ugxProblemFiniteCyclicPagePatch11} +\pastebutton{ugxProblemFiniteCyclicPageEmpty11}{\showpaste} +\tab{5}\spadcommand{f := createPrimitivePoly(4)$FFPOLY(GF3)\bound{f }\free{GF3 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch12} +\begin{paste}{ugxProblemFiniteCyclicPageFull12}{ugxProblemFiniteCyclicPageEmpty12} +\pastebutton{ugxProblemFiniteCyclicPageFull12}{\hidepaste} +\tab{5}\spadcommand{GF81 := FFCGP(GF3,f);\bound{GF81x }\free{f GF3 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty12} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty12}{ugxProblemFiniteCyclicPagePatch12} +\pastebutton{ugxProblemFiniteCyclicPageEmpty12}{\showpaste} +\tab{5}\spadcommand{GF81 := FFCGP(GF3,f);\bound{GF81x }\free{f GF3 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPagePatch13} +\begin{paste}{ugxProblemFiniteCyclicPageFull13}{ugxProblemFiniteCyclicPageEmpty13} +\pastebutton{ugxProblemFiniteCyclicPageFull13}{\hidepaste} +\tab{5}\spadcommand{random()$GF81\free{GF81x }} +\indentrel{3}\begin{verbatim} + 8 + (13) %BF +Type: FiniteFieldCyclicGroupExtensionByPolynomial(PrimeField 3,?**4+?+2) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteCyclicPageEmpty13} +\begin{paste}{ugxProblemFiniteCyclicPageEmpty13}{ugxProblemFiniteCyclicPagePatch13} +\pastebutton{ugxProblemFiniteCyclicPageEmpty13}{\showpaste} +\tab{5}\spadcommand{random()$GF81\free{GF81x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePagePatch1} +\begin{paste}{ugxProblemSeriesCreatePageFull1}{ugxProblemSeriesCreatePageEmpty1} +\pastebutton{ugxProblemSeriesCreatePageFull1}{\hidepaste} +\tab{5}\spadcommand{x := series 'x\bound{x }} +\indentrel{3}\begin{verbatim} + (1) x + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePageEmpty1} +\begin{paste}{ugxProblemSeriesCreatePageEmpty1}{ugxProblemSeriesCreatePagePatch1} +\pastebutton{ugxProblemSeriesCreatePageEmpty1}{\showpaste} +\tab{5}\spadcommand{x := series 'x\bound{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePagePatch2} +\begin{paste}{ugxProblemSeriesCreatePageFull2}{ugxProblemSeriesCreatePageEmpty2} +\pastebutton{ugxProblemSeriesCreatePageFull2}{\hidepaste} +\tab{5}\spadcommand{1/(1 - x - x**2)\free{x }} +\indentrel{3}\begin{verbatim} + (2) + 2 3 4 5 6 7 8 + 1 + x + 2x + 3x + 5x + 8x + 13x + 21x + 34x + + + 9 10 11 + 55x + 89x + O(x ) + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePageEmpty2} +\begin{paste}{ugxProblemSeriesCreatePageEmpty2}{ugxProblemSeriesCreatePagePatch2} +\pastebutton{ugxProblemSeriesCreatePageEmpty2}{\showpaste} +\tab{5}\spadcommand{1/(1 - x - x**2)\free{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePagePatch3} +\begin{paste}{ugxProblemSeriesCreatePageFull3}{ugxProblemSeriesCreatePageEmpty3} +\pastebutton{ugxProblemSeriesCreatePageFull3}{\hidepaste} +\tab{5}\spadcommand{sin(x)\free{x }} +\indentrel{3}\begin{verbatim} + (3) + 1 3 1 5 1 7 1 9 + x - � x + 컴� x - 컴컴 x + 컴컴컴 x + 6 120 5040 362880 + + + 1 11 12 + - 컴컴컴컴 x + O(x ) + 39916800 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePageEmpty3} +\begin{paste}{ugxProblemSeriesCreatePageEmpty3}{ugxProblemSeriesCreatePagePatch3} +\pastebutton{ugxProblemSeriesCreatePageEmpty3}{\showpaste} +\tab{5}\spadcommand{sin(x)\free{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePagePatch4} +\begin{paste}{ugxProblemSeriesCreatePageFull4}{ugxProblemSeriesCreatePageEmpty4} +\pastebutton{ugxProblemSeriesCreatePageFull4}{\hidepaste} +\tab{5}\spadcommand{sin(1 + x)\free{x }} +\indentrel{3}\begin{verbatim} + (4) + sin(1) 2 cos(1) 3 sin(1) 4 + sin(1) + cos(1)x - 컴컴컴 x - 컴컴컴 x + 컴컴컴 x + 2 6 24 + + + cos(1) 5 sin(1) 6 cos(1) 7 sin(1) 8 + 컴컴컴 x - 컴컴컴 x - 컴컴컴 x + 컴컴컴 x + 120 720 5040 40320 + + + cos(1) 9 sin(1) 10 11 + 컴컴컴 x - 컴컴컴� x + O(x ) + 362880 3628800 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePageEmpty4} +\begin{paste}{ugxProblemSeriesCreatePageEmpty4}{ugxProblemSeriesCreatePagePatch4} +\pastebutton{ugxProblemSeriesCreatePageEmpty4}{\showpaste} +\tab{5}\spadcommand{sin(1 + x)\free{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePagePatch5} +\begin{paste}{ugxProblemSeriesCreatePageFull5}{ugxProblemSeriesCreatePageEmpty5} +\pastebutton{ugxProblemSeriesCreatePageFull5}{\hidepaste} +\tab{5}\spadcommand{sin(a * x)\free{x }} +\indentrel{3}\begin{verbatim} + (5) + 3 5 7 9 + a 3 a 5 a 7 a 9 + a x - 컴 x + 컴� x - 컴컴 x + 컴컴컴 x + 6 120 5040 362880 + + + 11 + a 11 12 + - 컴컴컴컴 x + O(x ) + 39916800 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePageEmpty5} +\begin{paste}{ugxProblemSeriesCreatePageEmpty5}{ugxProblemSeriesCreatePagePatch5} +\pastebutton{ugxProblemSeriesCreatePageEmpty5}{\showpaste} +\tab{5}\spadcommand{sin(a * x)\free{x }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePagePatch6} +\begin{paste}{ugxProblemSeriesCreatePageFull6}{ugxProblemSeriesCreatePageEmpty6} +\pastebutton{ugxProblemSeriesCreatePageFull6}{\hidepaste} +\tab{5}\spadcommand{series(1/log(y),y = 1)} +\indentrel{3}\begin{verbatim} + (6) + - 1 1 1 1 2 + (y - 1) + � - 컴 (y - 1) + 컴 (y - 1) + 2 12 24 + + + 19 3 3 4 863 5 + - 컴� (y - 1) + 컴� (y - 1) - 컴컴� (y - 1) + 720 160 60480 + + + 275 6 33953 7 8183 8 + 컴컴� (y - 1) - 컴컴컴� (y - 1) + 컴컴컴� (y - 1) + 24192 3628800 1036800 + + + 3250433 9 10 + - 컴컴컴컴� (y - 1) + O((y - 1) ) + 479001600 + Type: UnivariatePuiseuxSeries(Expression Integer,y,1) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePageEmpty6} +\begin{paste}{ugxProblemSeriesCreatePageEmpty6}{ugxProblemSeriesCreatePagePatch6} +\pastebutton{ugxProblemSeriesCreatePageEmpty6}{\showpaste} +\tab{5}\spadcommand{series(1/log(y),y = 1)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePagePatch7} +\begin{paste}{ugxProblemSeriesCreatePageFull7}{ugxProblemSeriesCreatePageEmpty7} +\pastebutton{ugxProblemSeriesCreatePageFull7}{\hidepaste} +\tab{5}\spadcommand{y : UTS(FLOAT,'z,0) := exp(z)\bound{y }} +\indentrel{3}\begin{verbatim} + (7) + 2 3 + 1.0 + z + 0.5 z + 0.1666666666 6666666667 z + + + 4 + 0.0416666666 6666666666 7 z + + + 5 + 0.0083333333 3333333333 34 z + + + 6 + 0.0013888888 8888888888 89 z + + + 7 + 0.0001984126 9841269841 27 z + + + 8 + 0.0000248015 8730158730 1587 z + + + 9 + 0.0000027557 3192239858 90653 z + + + 10 11 + 0.2755731922 3985890653 E -6 z + O(z ) + Type: UnivariateTaylorSeries(Float,z,0.0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePageEmpty7} +\begin{paste}{ugxProblemSeriesCreatePageEmpty7}{ugxProblemSeriesCreatePagePatch7} +\pastebutton{ugxProblemSeriesCreatePageEmpty7}{\showpaste} +\tab{5}\spadcommand{y : UTS(FLOAT,'z,0) := exp(z)\bound{y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePagePatch8} +\begin{paste}{ugxProblemSeriesCreatePageFull8}{ugxProblemSeriesCreatePageEmpty8} +\pastebutton{ugxProblemSeriesCreatePageFull8}{\hidepaste} +\tab{5}\spadcommand{series(1/factorial(n),n,w = 0)} +\indentrel{3}\begin{verbatim} + (8) + 1 2 1 3 1 4 1 5 1 6 + 1 + w + � w + � w + 컴 w + 컴� w + 컴� w + 2 6 24 120 720 + + + 1 7 1 8 1 9 1 10 11 + 컴컴 w + 컴컴� w + 컴컴컴 w + 컴컴컴� w + O(w ) + 5040 40320 362880 3628800 + Type: UnivariatePuiseuxSeries(Expression Integer,w,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesCreatePageEmpty8} +\begin{paste}{ugxProblemSeriesCreatePageEmpty8}{ugxProblemSeriesCreatePagePatch8} +\pastebutton{ugxProblemSeriesCreatePageEmpty8}{\showpaste} +\tab{5}\spadcommand{series(1/factorial(n),n,w = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch1} +\begin{paste}{ugxProblemFiniteNormalPageFull1}{ugxProblemFiniteNormalPageEmpty1} +\pastebutton{ugxProblemFiniteNormalPageFull1}{\hidepaste} +\tab{5}\spadcommand{K := FFNB(3,8)\bound{K }} +\indentrel{3}\begin{verbatim} + (1) FiniteFieldNormalBasis(3,8) + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty1} +\begin{paste}{ugxProblemFiniteNormalPageEmpty1}{ugxProblemFiniteNormalPagePatch1} +\pastebutton{ugxProblemFiniteNormalPageEmpty1}{\showpaste} +\tab{5}\spadcommand{K := FFNB(3,8)\bound{K }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch2} +\begin{paste}{ugxProblemFiniteNormalPageFull2}{ugxProblemFiniteNormalPageEmpty2} +\pastebutton{ugxProblemFiniteNormalPageFull2}{\hidepaste} +\tab{5}\spadcommand{a := normalElement()$K\bound{a }\free{K }} +\indentrel{3}\begin{verbatim} + (2) %CO + Type: FiniteFieldNormalBasis(3,8) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty2} +\begin{paste}{ugxProblemFiniteNormalPageEmpty2}{ugxProblemFiniteNormalPagePatch2} +\pastebutton{ugxProblemFiniteNormalPageEmpty2}{\showpaste} +\tab{5}\spadcommand{a := normalElement()$K\bound{a }\free{K }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch3} +\begin{paste}{ugxProblemFiniteNormalPageFull3}{ugxProblemFiniteNormalPageEmpty3} +\pastebutton{ugxProblemFiniteNormalPageFull3}{\hidepaste} +\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }} +\indentrel{3}\begin{verbatim} + 7 5 + q q q + (3) 2%CO + %CO + %CO + Type: FiniteFieldNormalBasis(3,8) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty3} +\begin{paste}{ugxProblemFiniteNormalPageEmpty3}{ugxProblemFiniteNormalPagePatch3} +\pastebutton{ugxProblemFiniteNormalPageEmpty3}{\showpaste} +\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch4} +\begin{paste}{ugxProblemFiniteNormalPageFull4}{ugxProblemFiniteNormalPageEmpty4} +\pastebutton{ugxProblemFiniteNormalPageFull4}{\hidepaste} +\tab{5}\spadcommand{GF9 := FFNB(3,2);\bound{GF9 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty4} +\begin{paste}{ugxProblemFiniteNormalPageEmpty4}{ugxProblemFiniteNormalPagePatch4} +\pastebutton{ugxProblemFiniteNormalPageEmpty4}{\showpaste} +\tab{5}\spadcommand{GF9 := FFNB(3,2);\bound{GF9 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch5} +\begin{paste}{ugxProblemFiniteNormalPageFull5}{ugxProblemFiniteNormalPageEmpty5} +\pastebutton{ugxProblemFiniteNormalPageFull5}{\hidepaste} +\tab{5}\spadcommand{GF729 := FFNBX(GF9,3);\bound{GF729 }\free{GF9 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty5} +\begin{paste}{ugxProblemFiniteNormalPageEmpty5}{ugxProblemFiniteNormalPagePatch5} +\pastebutton{ugxProblemFiniteNormalPageEmpty5}{\showpaste} +\tab{5}\spadcommand{GF729 := FFNBX(GF9,3);\bound{GF729 }\free{GF9 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch6} +\begin{paste}{ugxProblemFiniteNormalPageFull6}{ugxProblemFiniteNormalPageEmpty6} +\pastebutton{ugxProblemFiniteNormalPageFull6}{\hidepaste} +\tab{5}\spadcommand{r := random()$GF729\bound{r }\free{GF729 }} +\indentrel{3}\begin{verbatim} + 2 + q q + (6) 2%CP %CQ +Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty6} +\begin{paste}{ugxProblemFiniteNormalPageEmpty6}{ugxProblemFiniteNormalPagePatch6} +\pastebutton{ugxProblemFiniteNormalPageEmpty6}{\showpaste} +\tab{5}\spadcommand{r := random()$GF729\bound{r }\free{GF729 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch7} +\begin{paste}{ugxProblemFiniteNormalPageFull7}{ugxProblemFiniteNormalPageEmpty7} +\pastebutton{ugxProblemFiniteNormalPageFull7}{\hidepaste} +\tab{5}\spadcommand{r + r**3 + r**9 + r**27\free{r }} +\indentrel{3}\begin{verbatim} + 2 + q q q q + (7) (2%CP + 2%CP)%CQ + 2%CP %CQ + 2%CP %CQ +Type: FiniteFieldNormalBasisExtension(FiniteFieldNormalBasis(3,2),3) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty7} +\begin{paste}{ugxProblemFiniteNormalPageEmpty7}{ugxProblemFiniteNormalPagePatch7} +\pastebutton{ugxProblemFiniteNormalPageEmpty7}{\showpaste} +\tab{5}\spadcommand{r + r**3 + r**9 + r**27\free{r }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch8} +\begin{paste}{ugxProblemFiniteNormalPageFull8}{ugxProblemFiniteNormalPageEmpty8} +\pastebutton{ugxProblemFiniteNormalPageFull8}{\hidepaste} +\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty8} +\begin{paste}{ugxProblemFiniteNormalPageEmpty8}{ugxProblemFiniteNormalPagePatch8} +\pastebutton{ugxProblemFiniteNormalPageEmpty8}{\showpaste} +\tab{5}\spadcommand{GF3 := PrimeField 3;\bound{GF3 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch9} +\begin{paste}{ugxProblemFiniteNormalPageFull9}{ugxProblemFiniteNormalPageEmpty9} +\pastebutton{ugxProblemFiniteNormalPageFull9}{\hidepaste} +\tab{5}\spadcommand{f := createNormalPoly(4)$FFPOLY(GF3)\free{GF3 }\bound{f }} +\indentrel{3}\begin{verbatim} + 4 3 + (9) ? + 2? + 2 + Type: SparseUnivariatePolynomial PrimeField 3 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty9} +\begin{paste}{ugxProblemFiniteNormalPageEmpty9}{ugxProblemFiniteNormalPagePatch9} +\pastebutton{ugxProblemFiniteNormalPageEmpty9}{\showpaste} +\tab{5}\spadcommand{f := createNormalPoly(4)$FFPOLY(GF3)\free{GF3 }\bound{f }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch10} +\begin{paste}{ugxProblemFiniteNormalPageFull10}{ugxProblemFiniteNormalPageEmpty10} +\pastebutton{ugxProblemFiniteNormalPageFull10}{\hidepaste} +\tab{5}\spadcommand{GF81 := FFNBP(GF3,f);\bound{GF81 }\free{f GF3 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty10} +\begin{paste}{ugxProblemFiniteNormalPageEmpty10}{ugxProblemFiniteNormalPagePatch10} +\pastebutton{ugxProblemFiniteNormalPageEmpty10}{\showpaste} +\tab{5}\spadcommand{GF81 := FFNBP(GF3,f);\bound{GF81 }\free{f GF3 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch11} +\begin{paste}{ugxProblemFiniteNormalPageFull11}{ugxProblemFiniteNormalPageEmpty11} +\pastebutton{ugxProblemFiniteNormalPageFull11}{\hidepaste} +\tab{5}\spadcommand{r := random()$GF81\free{GF81 }\bound{r1 }} +\indentrel{3}\begin{verbatim} + 3 2 + q q q + (11) %CR + %CR + %CR +Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty11} +\begin{paste}{ugxProblemFiniteNormalPageEmpty11}{ugxProblemFiniteNormalPagePatch11} +\pastebutton{ugxProblemFiniteNormalPageEmpty11}{\showpaste} +\tab{5}\spadcommand{r := random()$GF81\free{GF81 }\bound{r1 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch12} +\begin{paste}{ugxProblemFiniteNormalPageFull12}{ugxProblemFiniteNormalPageEmpty12} +\pastebutton{ugxProblemFiniteNormalPageFull12}{\hidepaste} +\tab{5}\spadcommand{r * r**3 * r**9 * r**27\free{r1 }} +\indentrel{3}\begin{verbatim} + 3 2 + q q q + (12) 2%CR + 2%CR + 2%CR + 2%CR +Type: FiniteFieldNormalBasisExtensionByPolynomial(PrimeField 3,?**4+2*?**3+2) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty12} +\begin{paste}{ugxProblemFiniteNormalPageEmpty12}{ugxProblemFiniteNormalPagePatch12} +\pastebutton{ugxProblemFiniteNormalPageEmpty12}{\showpaste} +\tab{5}\spadcommand{r * r**3 * r**9 * r**27\free{r1 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPagePatch13} +\begin{paste}{ugxProblemFiniteNormalPageFull13}{ugxProblemFiniteNormalPageEmpty13} +\pastebutton{ugxProblemFiniteNormalPageFull13}{\hidepaste} +\tab{5}\spadcommand{norm r\free{r1 }} +\indentrel{3}\begin{verbatim} + (13) 2 + Type: PrimeField 3 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteNormalPageEmpty13} +\begin{paste}{ugxProblemFiniteNormalPageEmpty13}{ugxProblemFiniteNormalPagePatch13} +\pastebutton{ugxProblemFiniteNormalPageEmpty13}{\showpaste} +\tab{5}\spadcommand{norm r\free{r1 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPagePatch1} +\begin{paste}{ugxProblemSeriesFormulaPageFull1}{ugxProblemSeriesFormulaPageEmpty1} +\pastebutton{ugxProblemSeriesFormulaPageFull1}{\hidepaste} +\tab{5}\spadcommand{series(n +-> 1/factorial(n),x = 0)} +\indentrel{3}\begin{verbatim} + (1) + 1 2 1 3 1 4 1 5 1 6 + 1 + x + � x + � x + 컴 x + 컴� x + 컴� x + 2 6 24 120 720 + + + 1 7 1 8 1 9 1 10 11 + 컴컴 x + 컴컴� x + 컴컴컴 x + 컴컴컴� x + O(x ) + 5040 40320 362880 3628800 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPageEmpty1} +\begin{paste}{ugxProblemSeriesFormulaPageEmpty1}{ugxProblemSeriesFormulaPagePatch1} +\pastebutton{ugxProblemSeriesFormulaPageEmpty1}{\showpaste} +\tab{5}\spadcommand{series(n +-> 1/factorial(n),x = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPagePatch2} +\begin{paste}{ugxProblemSeriesFormulaPageFull2}{ugxProblemSeriesFormulaPageEmpty2} +\pastebutton{ugxProblemSeriesFormulaPageFull2}{\hidepaste} +\tab{5}\spadcommand{series(n +-> (-1)**(n-1)/n,x = 1,1..)} +\indentrel{3}\begin{verbatim} + (2) + 1 2 1 3 1 4 + (x - 1) - � (x - 1) + � (x - 1) - � (x - 1) + 2 3 4 + + + 1 5 1 6 1 7 1 8 + � (x - 1) - � (x - 1) + � (x - 1) - � (x - 1) + 5 6 7 8 + + + 1 9 1 10 1 11 12 + � (x - 1) - 컴 (x - 1) + 컴 (x - 1) + O((x - 1) ) + 9 10 11 + Type: UnivariatePuiseuxSeries(Expression Integer,x,1) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPageEmpty2} +\begin{paste}{ugxProblemSeriesFormulaPageEmpty2}{ugxProblemSeriesFormulaPagePatch2} +\pastebutton{ugxProblemSeriesFormulaPageEmpty2}{\showpaste} +\tab{5}\spadcommand{series(n +-> (-1)**(n-1)/n,x = 1,1..)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPagePatch3} +\begin{paste}{ugxProblemSeriesFormulaPageFull3}{ugxProblemSeriesFormulaPageEmpty3} +\pastebutton{ugxProblemSeriesFormulaPageFull3}{\hidepaste} +\tab{5}\spadcommand{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)} +\indentrel{3}\begin{verbatim} + (3) + 1 3 1 5 1 7 1 9 + x - � x + 컴� x - 컴컴 x + 컴컴컴 x + 6 120 5040 362880 + + + 1 11 12 + - 컴컴컴컴 x + O(x ) + 39916800 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPageEmpty3} +\begin{paste}{ugxProblemSeriesFormulaPageEmpty3}{ugxProblemSeriesFormulaPagePatch3} +\pastebutton{ugxProblemSeriesFormulaPageEmpty3}{\showpaste} +\tab{5}\spadcommand{series(n +-> (-1)**((n-1)/2)/factorial(n),x = 0,1..,2)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPagePatch4} +\begin{paste}{ugxProblemSeriesFormulaPageFull4}{ugxProblemSeriesFormulaPageEmpty4} +\pastebutton{ugxProblemSeriesFormulaPageFull4}{\hidepaste} +\tab{5}\spadcommand{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)} +\indentrel{3}\begin{verbatim} + (4) + 1 5 7 + � � � + 3 1 1 3 1 3 1 3 + x - � x + 컴� x - 컴컴 x + 컴컴컴 x + 6 120 5040 362880 + + + 11 + 컴 + 1 3 4 + - 컴컴컴컴 x + O(x ) + 39916800 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPageEmpty4} +\begin{paste}{ugxProblemSeriesFormulaPageEmpty4}{ugxProblemSeriesFormulaPagePatch4} +\pastebutton{ugxProblemSeriesFormulaPageEmpty4}{\showpaste} +\tab{5}\spadcommand{series(n +-> (-1)**((3*n-1)/2)/factorial(3*n),x = 0,1/3..,2/3)} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPagePatch5} +\begin{paste}{ugxProblemSeriesFormulaPageFull5}{ugxProblemSeriesFormulaPageEmpty5} +\pastebutton{ugxProblemSeriesFormulaPageFull5}{\hidepaste} +\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 }} +\indentrel{3}\begin{verbatim} + (5) + - 1 1 7 3 31 5 127 7 + x + � x + 컴� x + 컴컴� x + 컴컴컴 x + 6 360 15120 604800 + + + 73 9 10 + 컴컴컴� x + O(x ) + 3421440 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPageEmpty5} +\begin{paste}{ugxProblemSeriesFormulaPageEmpty5}{ugxProblemSeriesFormulaPagePatch5} +\pastebutton{ugxProblemSeriesFormulaPageEmpty5}{\showpaste} +\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 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPagePatch6} +\begin{paste}{ugxProblemSeriesFormulaPageFull6}{ugxProblemSeriesFormulaPageEmpty6} +\pastebutton{ugxProblemSeriesFormulaPageFull6}{\hidepaste} +\tab{5}\spadcommand{1/cscx\free{cscx }} +\indentrel{3}\begin{verbatim} + (6) + 1 3 1 5 1 7 1 9 + x - � x + 컴� x - 컴컴 x + 컴컴컴 x + 6 120 5040 362880 + + + 1 11 12 + - 컴컴컴컴 x + O(x ) + 39916800 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPageEmpty6} +\begin{paste}{ugxProblemSeriesFormulaPageEmpty6}{ugxProblemSeriesFormulaPagePatch6} +\pastebutton{ugxProblemSeriesFormulaPageEmpty6}{\showpaste} +\tab{5}\spadcommand{1/cscx\free{cscx }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPagePatch7} +\begin{paste}{ugxProblemSeriesFormulaPageFull7}{ugxProblemSeriesFormulaPageEmpty7} +\pastebutton{ugxProblemSeriesFormulaPageFull7}{\hidepaste} +\tab{5}\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2)\bound{asinx }} +\indentrel{3}\begin{verbatim} + (7) + 1 3 3 5 5 7 35 9 63 11 12 + x + � x + 컴 x + 컴� x + 컴컴 x + 컴컴 x + O(x ) + 6 40 112 1152 2816 + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPageEmpty7} +\begin{paste}{ugxProblemSeriesFormulaPageEmpty7}{ugxProblemSeriesFormulaPagePatch7} +\pastebutton{ugxProblemSeriesFormulaPageEmpty7}{\showpaste} +\tab{5}\spadcommand{asinx := series(n +-> binomial(n-1,(n-1)/2)/(n*2**(n-1)),x=0,1..,2)\bound{asinx }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPagePatch8} +\begin{paste}{ugxProblemSeriesFormulaPageFull8}{ugxProblemSeriesFormulaPageEmpty8} +\pastebutton{ugxProblemSeriesFormulaPageFull8}{\hidepaste} +\tab{5}\spadcommand{sin(asinx)\free{asinx }} +\indentrel{3}\begin{verbatim} + 12 + (8) x + O(x ) + Type: UnivariatePuiseuxSeries(Expression Integer,x,0) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemSeriesFormulaPageEmpty8} +\begin{paste}{ugxProblemSeriesFormulaPageEmpty8}{ugxProblemSeriesFormulaPagePatch8} +\pastebutton{ugxProblemSeriesFormulaPageEmpty8}{\showpaste} +\tab{5}\spadcommand{sin(asinx)\free{asinx }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch1} +\begin{paste}{ugxProblemFiniteModulusPageFull1}{ugxProblemFiniteModulusPageEmpty1} +\pastebutton{ugxProblemFiniteModulusPageFull1}{\hidepaste} +\tab{5}\spadcommand{GF4096 := FF(2,12);\bound{GF4096 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty1} +\begin{paste}{ugxProblemFiniteModulusPageEmpty1}{ugxProblemFiniteModulusPagePatch1} +\pastebutton{ugxProblemFiniteModulusPageEmpty1}{\showpaste} +\tab{5}\spadcommand{GF4096 := FF(2,12);\bound{GF4096 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch2} +\begin{paste}{ugxProblemFiniteModulusPageFull2}{ugxProblemFiniteModulusPageEmpty2} +\pastebutton{ugxProblemFiniteModulusPageFull2}{\hidepaste} +\tab{5}\spadcommand{a := index(2)$GF4096\bound{a }\free{GF4096 }} +\indentrel{3}\begin{verbatim} + (2) %CS + Type: FiniteField(2,12) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty2} +\begin{paste}{ugxProblemFiniteModulusPageEmpty2}{ugxProblemFiniteModulusPagePatch2} +\pastebutton{ugxProblemFiniteModulusPageEmpty2}{\showpaste} +\tab{5}\spadcommand{a := index(2)$GF4096\bound{a }\free{GF4096 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch3} +\begin{paste}{ugxProblemFiniteModulusPageFull3}{ugxProblemFiniteModulusPageEmpty3} +\pastebutton{ugxProblemFiniteModulusPageFull3}{\hidepaste} +\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }} +\indentrel{3}\begin{verbatim} + 5 3 + (3) %CS + %CS + %CS + 1 + Type: FiniteField(2,12) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty3} +\begin{paste}{ugxProblemFiniteModulusPageEmpty3}{ugxProblemFiniteModulusPagePatch3} +\pastebutton{ugxProblemFiniteModulusPageEmpty3}{\showpaste} +\tab{5}\spadcommand{b := a**12 - a**5 + a\bound{b }\free{a }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch4} +\begin{paste}{ugxProblemFiniteModulusPageFull4}{ugxProblemFiniteModulusPageEmpty4} +\pastebutton{ugxProblemFiniteModulusPageFull4}{\hidepaste} +\tab{5}\spadcommand{b ** 1000\free{b }} +\indentrel{3}\begin{verbatim} + 10 9 7 5 4 3 + (4) %CS + %CS + %CS + %CS + %CS + %CS + %CS + Type: FiniteField(2,12) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty4} +\begin{paste}{ugxProblemFiniteModulusPageEmpty4}{ugxProblemFiniteModulusPagePatch4} +\pastebutton{ugxProblemFiniteModulusPageEmpty4}{\showpaste} +\tab{5}\spadcommand{b ** 1000\free{b }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch5} +\begin{paste}{ugxProblemFiniteModulusPageFull5}{ugxProblemFiniteModulusPageEmpty5} +\pastebutton{ugxProblemFiniteModulusPageFull5}{\hidepaste} +\tab{5}\spadcommand{c := a/b\free{a b }\bound{c }} +\indentrel{3}\begin{verbatim} + 11 8 7 5 4 3 2 + (5) %CS + %CS + %CS + %CS + %CS + %CS + %CS + Type: FiniteField(2,12) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty5} +\begin{paste}{ugxProblemFiniteModulusPageEmpty5}{ugxProblemFiniteModulusPagePatch5} +\pastebutton{ugxProblemFiniteModulusPageEmpty5}{\showpaste} +\tab{5}\spadcommand{c := a/b\free{a b }\bound{c }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch6} +\begin{paste}{ugxProblemFiniteModulusPageFull6}{ugxProblemFiniteModulusPageEmpty6} +\pastebutton{ugxProblemFiniteModulusPageFull6}{\hidepaste} +\tab{5}\spadcommand{norm c\free{c }} +\indentrel{3}\begin{verbatim} + (6) 1 + Type: PrimeField 2 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty6} +\begin{paste}{ugxProblemFiniteModulusPageEmpty6}{ugxProblemFiniteModulusPagePatch6} +\pastebutton{ugxProblemFiniteModulusPageEmpty6}{\showpaste} +\tab{5}\spadcommand{norm c\free{c }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch7} +\begin{paste}{ugxProblemFiniteModulusPageFull7}{ugxProblemFiniteModulusPageEmpty7} +\pastebutton{ugxProblemFiniteModulusPageFull7}{\hidepaste} +\tab{5}\spadcommand{trace c\free{c }} +\indentrel{3}\begin{verbatim} + (7) 0 + Type: PrimeField 2 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty7} +\begin{paste}{ugxProblemFiniteModulusPageEmpty7}{ugxProblemFiniteModulusPagePatch7} +\pastebutton{ugxProblemFiniteModulusPageEmpty7}{\showpaste} +\tab{5}\spadcommand{trace c\free{c }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch8} +\begin{paste}{ugxProblemFiniteModulusPageFull8}{ugxProblemFiniteModulusPageEmpty8} +\pastebutton{ugxProblemFiniteModulusPageFull8}{\hidepaste} +\tab{5}\spadcommand{dL := discreteLog a\free{a }\bound{dL }} +\indentrel{3}\begin{verbatim} + (8) 1729 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty8} +\begin{paste}{ugxProblemFiniteModulusPageEmpty8}{ugxProblemFiniteModulusPagePatch8} +\pastebutton{ugxProblemFiniteModulusPageEmpty8}{\showpaste} +\tab{5}\spadcommand{dL := discreteLog a\free{a }\bound{dL }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch9} +\begin{paste}{ugxProblemFiniteModulusPageFull9}{ugxProblemFiniteModulusPageEmpty9} +\pastebutton{ugxProblemFiniteModulusPageFull9}{\hidepaste} +\tab{5}\spadcommand{g ** dL\free{dL g }} +\indentrel{3}\begin{verbatim} + 1729 + (9) g + Type: Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty9} +\begin{paste}{ugxProblemFiniteModulusPageEmpty9}{ugxProblemFiniteModulusPagePatch9} +\pastebutton{ugxProblemFiniteModulusPageEmpty9}{\showpaste} +\tab{5}\spadcommand{g ** dL\free{dL g }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch10} +\begin{paste}{ugxProblemFiniteModulusPageFull10}{ugxProblemFiniteModulusPageEmpty10} +\pastebutton{ugxProblemFiniteModulusPageFull10}{\hidepaste} +\tab{5}\spadcommand{GF16 := FF(2,4);\bound{GF16 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty10} +\begin{paste}{ugxProblemFiniteModulusPageEmpty10}{ugxProblemFiniteModulusPagePatch10} +\pastebutton{ugxProblemFiniteModulusPageEmpty10}{\showpaste} +\tab{5}\spadcommand{GF16 := FF(2,4);\bound{GF16 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch11} +\begin{paste}{ugxProblemFiniteModulusPageFull11}{ugxProblemFiniteModulusPageEmpty11} +\pastebutton{ugxProblemFiniteModulusPageFull11}{\hidepaste} +\tab{5}\spadcommand{GF4096 := FFX(GF16,3);\bound{GF4096x }\free{GF16 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty11} +\begin{paste}{ugxProblemFiniteModulusPageEmpty11}{ugxProblemFiniteModulusPagePatch11} +\pastebutton{ugxProblemFiniteModulusPageEmpty11}{\showpaste} +\tab{5}\spadcommand{GF4096 := FFX(GF16,3);\bound{GF4096x }\free{GF16 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch12} +\begin{paste}{ugxProblemFiniteModulusPageFull12}{ugxProblemFiniteModulusPageEmpty12} +\pastebutton{ugxProblemFiniteModulusPageFull12}{\hidepaste} +\tab{5}\spadcommand{r := (random()$GF4096) ** 20\free{GF4096x }\bound{r }} +\indentrel{3}\begin{verbatim} + 3 2 + (12) (%CT + %CT + 1)%CU + %CT %CU + Type: FiniteFieldExtension(FiniteField(2,4),3) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty12} +\begin{paste}{ugxProblemFiniteModulusPageEmpty12}{ugxProblemFiniteModulusPagePatch12} +\pastebutton{ugxProblemFiniteModulusPageEmpty12}{\showpaste} +\tab{5}\spadcommand{r := (random()$GF4096) ** 20\free{GF4096x }\bound{r }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch13} +\begin{paste}{ugxProblemFiniteModulusPageFull13}{ugxProblemFiniteModulusPageEmpty13} +\pastebutton{ugxProblemFiniteModulusPageFull13}{\hidepaste} +\tab{5}\spadcommand{norm(r)\free{r }} +\indentrel{3}\begin{verbatim} + 2 + (13) %CT + %CT + Type: FiniteField(2,4) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty13} +\begin{paste}{ugxProblemFiniteModulusPageEmpty13}{ugxProblemFiniteModulusPagePatch13} +\pastebutton{ugxProblemFiniteModulusPageEmpty13}{\showpaste} +\tab{5}\spadcommand{norm(r)\free{r }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch14} +\begin{paste}{ugxProblemFiniteModulusPageFull14}{ugxProblemFiniteModulusPageEmpty14} +\pastebutton{ugxProblemFiniteModulusPageFull14}{\hidepaste} +\tab{5}\spadcommand{GF4 := FF(2,2);\bound{GF4 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty14} +\begin{paste}{ugxProblemFiniteModulusPageEmpty14}{ugxProblemFiniteModulusPagePatch14} +\pastebutton{ugxProblemFiniteModulusPageEmpty14}{\showpaste} +\tab{5}\spadcommand{GF4 := FF(2,2);\bound{GF4 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch15} +\begin{paste}{ugxProblemFiniteModulusPageFull15}{ugxProblemFiniteModulusPageEmpty15} +\pastebutton{ugxProblemFiniteModulusPageFull15}{\hidepaste} +\tab{5}\spadcommand{f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)\free{GF4 }\bound{f }} +\indentrel{3}\begin{verbatim} + 6 5 4 2 + (15) ? + ? + (%CV + 1)? + (%CV + 1)? + ? + %CV +Type: Union(SparseUnivariatePolynomial FiniteField(2,2),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty15} +\begin{paste}{ugxProblemFiniteModulusPageEmpty15}{ugxProblemFiniteModulusPagePatch15} +\pastebutton{ugxProblemFiniteModulusPageEmpty15}{\showpaste} +\tab{5}\spadcommand{f := nextIrreduciblePoly(random(6)$FFPOLY(GF4))$FFPOLY(GF4)\free{GF4 }\bound{f }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch16} +\begin{paste}{ugxProblemFiniteModulusPageFull16}{ugxProblemFiniteModulusPageEmpty16} +\pastebutton{ugxProblemFiniteModulusPageFull16}{\hidepaste} +\tab{5}\spadcommand{GF4096 := FFP(GF4,f);\bound{GF4096y }\free{f GF4 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty16} +\begin{paste}{ugxProblemFiniteModulusPageEmpty16}{ugxProblemFiniteModulusPagePatch16} +\pastebutton{ugxProblemFiniteModulusPageEmpty16}{\showpaste} +\tab{5}\spadcommand{GF4096 := FFP(GF4,f);\bound{GF4096y }\free{f GF4 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPagePatch17} +\begin{paste}{ugxProblemFiniteModulusPageFull17}{ugxProblemFiniteModulusPageEmpty17} +\pastebutton{ugxProblemFiniteModulusPageFull17}{\hidepaste} +\tab{5}\spadcommand{discreteLog random()$GF4096\free{GF4096y }} +\indentrel{3}\begin{verbatim} + (17) 3370 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteModulusPageEmpty17} +\begin{paste}{ugxProblemFiniteModulusPageEmpty17}{ugxProblemFiniteModulusPagePatch17} +\pastebutton{ugxProblemFiniteModulusPageEmpty17}{\showpaste} +\tab{5}\spadcommand{discreteLog random()$GF4096\free{GF4096y }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch1} +\begin{paste}{ugProblemIdealPageFull1}{ugProblemIdealPageEmpty1} +\pastebutton{ugProblemIdealPageFull1}{\hidepaste} +\tab{5}\spadcommand{(n,m) : List DMP([x,y],FRAC INT)\bound{nm }} +\indentrel{3}\begin{verbatim} + Type: Void +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty1} +\begin{paste}{ugProblemIdealPageEmpty1}{ugProblemIdealPagePatch1} +\pastebutton{ugProblemIdealPageEmpty1}{\showpaste} +\tab{5}\spadcommand{(n,m) : List DMP([x,y],FRAC INT)\bound{nm }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch2} +\begin{paste}{ugProblemIdealPageFull2}{ugProblemIdealPageEmpty2} +\pastebutton{ugProblemIdealPageFull2}{\hidepaste} +\tab{5}\spadcommand{m := [x**2+y**2-1]\free{nm }\bound{m }} +\indentrel{3}\begin{verbatim} + 2 2 + (2) [x + y - 1] +Type: List DistributedMultivariatePolynomial([x,y],Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty2} +\begin{paste}{ugProblemIdealPageEmpty2}{ugProblemIdealPagePatch2} +\pastebutton{ugProblemIdealPageEmpty2}{\showpaste} +\tab{5}\spadcommand{m := [x**2+y**2-1]\free{nm }\bound{m }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch3} +\begin{paste}{ugProblemIdealPageFull3}{ugProblemIdealPageEmpty3} +\pastebutton{ugProblemIdealPageFull3}{\hidepaste} +\tab{5}\spadcommand{n := [x**2-y**2]\free{nm }\bound{n }} +\indentrel{3}\begin{verbatim} + 2 2 + (3) [x - y ] +Type: List DistributedMultivariatePolynomial([x,y],Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty3} +\begin{paste}{ugProblemIdealPageEmpty3}{ugProblemIdealPagePatch3} +\pastebutton{ugProblemIdealPageEmpty3}{\showpaste} +\tab{5}\spadcommand{n := [x**2-y**2]\free{nm }\bound{n }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch4} +\begin{paste}{ugProblemIdealPageFull4}{ugProblemIdealPageEmpty4} +\pastebutton{ugProblemIdealPageFull4}{\hidepaste} +\tab{5}\spadcommand{id := ideal m + ideal n\free{n m }\bound{id }} +\indentrel{3}\begin{verbatim} + 2 1 2 1 + (4) [x - �,y - �] + 2 2 +Type: PolynomialIdeals(Fraction Integer,DirectProduct(2,NonNegativeInteger),OrderedVariableList [x,y],DistributedMultivariatePolynomial([x,y],Fraction Integer)) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty4} +\begin{paste}{ugProblemIdealPageEmpty4}{ugProblemIdealPagePatch4} +\pastebutton{ugProblemIdealPageEmpty4}{\showpaste} +\tab{5}\spadcommand{id := ideal m + ideal n\free{n m }\bound{id }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch5} +\begin{paste}{ugProblemIdealPageFull5}{ugProblemIdealPageEmpty5} +\pastebutton{ugProblemIdealPageFull5}{\hidepaste} +\tab{5}\spadcommand{zeroDim? id\free{id }} +\indentrel{3}\begin{verbatim} + (5) true + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty5} +\begin{paste}{ugProblemIdealPageEmpty5}{ugProblemIdealPagePatch5} +\pastebutton{ugProblemIdealPageEmpty5}{\showpaste} +\tab{5}\spadcommand{zeroDim? id\free{id }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch6} +\begin{paste}{ugProblemIdealPageFull6}{ugProblemIdealPageEmpty6} +\pastebutton{ugProblemIdealPageFull6}{\hidepaste} +\tab{5}\spadcommand{zeroDim?(ideal m)\free{m }} +\indentrel{3}\begin{verbatim} + (6) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty6} +\begin{paste}{ugProblemIdealPageEmpty6}{ugProblemIdealPagePatch6} +\pastebutton{ugProblemIdealPageEmpty6}{\showpaste} +\tab{5}\spadcommand{zeroDim?(ideal m)\free{m }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch7} +\begin{paste}{ugProblemIdealPageFull7}{ugProblemIdealPageEmpty7} +\pastebutton{ugProblemIdealPageFull7}{\hidepaste} +\tab{5}\spadcommand{dimension ideal m\free{m }} +\indentrel{3}\begin{verbatim} + (7) 1 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty7} +\begin{paste}{ugProblemIdealPageEmpty7}{ugProblemIdealPagePatch7} +\pastebutton{ugProblemIdealPageEmpty7}{\showpaste} +\tab{5}\spadcommand{dimension ideal m\free{m }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch8} +\begin{paste}{ugProblemIdealPageFull8}{ugProblemIdealPageEmpty8} +\pastebutton{ugProblemIdealPageFull8}{\hidepaste} +\tab{5}\spadcommand{(f,g):DMP([x,y],FRAC INT)\bound{fg }} +\indentrel{3}\begin{verbatim} + Type: Void +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty8} +\begin{paste}{ugProblemIdealPageEmpty8}{ugProblemIdealPagePatch8} +\pastebutton{ugProblemIdealPageEmpty8}{\showpaste} +\tab{5}\spadcommand{(f,g):DMP([x,y],FRAC INT)\bound{fg }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch9} +\begin{paste}{ugProblemIdealPageFull9}{ugProblemIdealPageEmpty9} +\pastebutton{ugProblemIdealPageFull9}{\hidepaste} +\tab{5}\spadcommand{f := x**2-1\free{fg }\bound{f }} +\indentrel{3}\begin{verbatim} + 2 + (9) x - 1 +Type: DistributedMultivariatePolynomial([x,y],Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty9} +\begin{paste}{ugProblemIdealPageEmpty9}{ugProblemIdealPagePatch9} +\pastebutton{ugProblemIdealPageEmpty9}{\showpaste} +\tab{5}\spadcommand{f := x**2-1\free{fg }\bound{f }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch10} +\begin{paste}{ugProblemIdealPageFull10}{ugProblemIdealPageEmpty10} +\pastebutton{ugProblemIdealPageFull10}{\hidepaste} +\tab{5}\spadcommand{g := x*(x**2-1)\free{fg }\bound{g }} +\indentrel{3}\begin{verbatim} + 3 + (10) x - x +Type: DistributedMultivariatePolynomial([x,y],Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty10} +\begin{paste}{ugProblemIdealPageEmpty10}{ugProblemIdealPagePatch10} +\pastebutton{ugProblemIdealPageEmpty10}{\showpaste} +\tab{5}\spadcommand{g := x*(x**2-1)\free{fg }\bound{g }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch11} +\begin{paste}{ugProblemIdealPageFull11}{ugProblemIdealPageEmpty11} +\pastebutton{ugProblemIdealPageFull11}{\hidepaste} +\tab{5}\spadcommand{relationsIdeal [f,g]\free{f g }} +\indentrel{3}\begin{verbatim} + (11) + 2 3 2 2 3 + [- %CY + %CX + %CX ] | [%CX= x - 1,%CY= x - x] +Type: SuchThat(List Polynomial Fraction Integer,List Equation Polynomial Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty11} +\begin{paste}{ugProblemIdealPageEmpty11}{ugProblemIdealPagePatch11} +\pastebutton{ugProblemIdealPageEmpty11}{\showpaste} +\tab{5}\spadcommand{relationsIdeal [f,g]\free{f g }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch12} +\begin{paste}{ugProblemIdealPageFull12}{ugProblemIdealPageEmpty12} +\pastebutton{ugProblemIdealPageFull12}{\hidepaste} +\tab{5}\spadcommand{l: List DMP([x,y,z],FRAC INT)\bound{ll }} +\indentrel{3}\begin{verbatim} + Type: Void +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty12} +\begin{paste}{ugProblemIdealPageEmpty12}{ugProblemIdealPagePatch12} +\pastebutton{ugProblemIdealPageEmpty12}{\showpaste} +\tab{5}\spadcommand{l: List DMP([x,y,z],FRAC INT)\bound{ll }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch13} +\begin{paste}{ugProblemIdealPageFull13}{ugProblemIdealPageEmpty13} +\pastebutton{ugProblemIdealPageFull13}{\hidepaste} +\tab{5}\spadcommand{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4]\free{ll }\bound{l }} +\indentrel{3}\begin{verbatim} + 2 2 2 2 + (13) [x + 2y ,x z - y z,z - 4] +Type: List DistributedMultivariatePolynomial([x,y,z],Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty13} +\begin{paste}{ugProblemIdealPageEmpty13}{ugProblemIdealPagePatch13} +\pastebutton{ugProblemIdealPageEmpty13}{\showpaste} +\tab{5}\spadcommand{l:=[x**2+2*y**2,x*z**2-y*z,z**2-4]\free{ll }\bound{l }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch14} +\begin{paste}{ugProblemIdealPageFull14}{ugProblemIdealPageEmpty14} +\pastebutton{ugProblemIdealPageFull14}{\hidepaste} +\tab{5}\spadcommand{ld:=primaryDecomp ideal l\free{l }\bound{ld }} +\indentrel{3}\begin{verbatim} + 1 2 1 2 + (14) [[x + � y,y ,z + 2],[x - � y,y ,z - 2]] + 2 2 +Type: List PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer)) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty14} +\begin{paste}{ugProblemIdealPageEmpty14}{ugProblemIdealPagePatch14} +\pastebutton{ugProblemIdealPageEmpty14}{\showpaste} +\tab{5}\spadcommand{ld:=primaryDecomp ideal l\free{l }\bound{ld }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch15} +\begin{paste}{ugProblemIdealPageFull15}{ugProblemIdealPageEmpty15} +\pastebutton{ugProblemIdealPageFull15}{\hidepaste} +\tab{5}\spadcommand{reduce(intersect,ld)\free{ld }} +\indentrel{3}\begin{verbatim} + 1 2 2 + (15) [x - � y z,y ,z - 4] + 4 +Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer)) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty15} +\begin{paste}{ugProblemIdealPageEmpty15}{ugProblemIdealPagePatch15} +\pastebutton{ugProblemIdealPageEmpty15}{\showpaste} +\tab{5}\spadcommand{reduce(intersect,ld)\free{ld }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch16} +\begin{paste}{ugProblemIdealPageFull16}{ugProblemIdealPageEmpty16} +\pastebutton{ugProblemIdealPageFull16}{\hidepaste} +\tab{5}\spadcommand{reduce(intersect,[radical ld.i for i in 1..2])\free{ld }} +\indentrel{3}\begin{verbatim} + 2 + (16) [x,y,z - 4] +Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer)) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty16} +\begin{paste}{ugProblemIdealPageEmpty16}{ugProblemIdealPagePatch16} +\pastebutton{ugProblemIdealPageEmpty16}{\showpaste} +\tab{5}\spadcommand{reduce(intersect,[radical ld.i for i in 1..2])\free{ld }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPagePatch17} +\begin{paste}{ugProblemIdealPageFull17}{ugProblemIdealPageEmpty17} +\pastebutton{ugProblemIdealPageFull17}{\hidepaste} +\tab{5}\spadcommand{radical ideal l\free{l }} +\indentrel{3}\begin{verbatim} + 2 + (17) [x,y,z - 4] +Type: PolynomialIdeals(Fraction Integer,DirectProduct(3,NonNegativeInteger),OrderedVariableList [x,y,z],DistributedMultivariatePolynomial([x,y,z],Fraction Integer)) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemIdealPageEmpty17} +\begin{paste}{ugProblemIdealPageEmpty17}{ugProblemIdealPagePatch17} +\pastebutton{ugProblemIdealPageEmpty17}{\showpaste} +\tab{5}\spadcommand{radical ideal l\free{l }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch1} +\begin{paste}{ugxProblemFiniteUtilityPageFull1}{ugxProblemFiniteUtilityPageEmpty1} +\pastebutton{ugxProblemFiniteUtilityPageFull1}{\hidepaste} +\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty1} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty1}{ugxProblemFiniteUtilityPagePatch1} +\pastebutton{ugxProblemFiniteUtilityPageEmpty1}{\showpaste} +\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch2} +\begin{paste}{ugxProblemFiniteUtilityPageFull2}{ugxProblemFiniteUtilityPageEmpty2} +\pastebutton{ugxProblemFiniteUtilityPageFull2}{\hidepaste} +\tab{5}\spadcommand{f := createIrreduciblePoly(8)$FFPOLY(GF5)\bound{f }\free{GF5 }} +\indentrel{3}\begin{verbatim} + 8 4 + (2) ? + ? + 2 + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty2} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty2}{ugxProblemFiniteUtilityPagePatch2} +\pastebutton{ugxProblemFiniteUtilityPageEmpty2}{\showpaste} +\tab{5}\spadcommand{f := createIrreduciblePoly(8)$FFPOLY(GF5)\bound{f }\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch3} +\begin{paste}{ugxProblemFiniteUtilityPageFull3}{ugxProblemFiniteUtilityPageEmpty3} +\pastebutton{ugxProblemFiniteUtilityPageFull3}{\hidepaste} +\tab{5}\spadcommand{primitive?(f)$FFPOLY(GF5)\free{f }} +\indentrel{3}\begin{verbatim} + (3) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty3} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty3}{ugxProblemFiniteUtilityPagePatch3} +\pastebutton{ugxProblemFiniteUtilityPageEmpty3}{\showpaste} +\tab{5}\spadcommand{primitive?(f)$FFPOLY(GF5)\free{f }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch4} +\begin{paste}{ugxProblemFiniteUtilityPageFull4}{ugxProblemFiniteUtilityPageEmpty4} +\pastebutton{ugxProblemFiniteUtilityPageFull4}{\hidepaste} +\tab{5}\spadcommand{normal?(f)$FFPOLY(GF5)\free{f }} +\indentrel{3}\begin{verbatim} + (4) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty4} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty4}{ugxProblemFiniteUtilityPagePatch4} +\pastebutton{ugxProblemFiniteUtilityPageEmpty4}{\showpaste} +\tab{5}\spadcommand{normal?(f)$FFPOLY(GF5)\free{f }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch5} +\begin{paste}{ugxProblemFiniteUtilityPageFull5}{ugxProblemFiniteUtilityPageEmpty5} +\pastebutton{ugxProblemFiniteUtilityPageFull5}{\hidepaste} +\tab{5}\spadcommand{p := createPrimitivePoly(8)$FFPOLY(GF5)\bound{p }\free{GF5 }} +\indentrel{3}\begin{verbatim} + 8 3 2 + (5) ? + ? + ? + ? + 2 + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty5} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty5}{ugxProblemFiniteUtilityPagePatch5} +\pastebutton{ugxProblemFiniteUtilityPageEmpty5}{\showpaste} +\tab{5}\spadcommand{p := createPrimitivePoly(8)$FFPOLY(GF5)\bound{p }\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch6} +\begin{paste}{ugxProblemFiniteUtilityPageFull6}{ugxProblemFiniteUtilityPageEmpty6} +\pastebutton{ugxProblemFiniteUtilityPageFull6}{\hidepaste} +\tab{5}\spadcommand{primitive?(p)$FFPOLY(GF5)\free{p }} +\indentrel{3}\begin{verbatim} + (6) true + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty6} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty6}{ugxProblemFiniteUtilityPagePatch6} +\pastebutton{ugxProblemFiniteUtilityPageEmpty6}{\showpaste} +\tab{5}\spadcommand{primitive?(p)$FFPOLY(GF5)\free{p }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch7} +\begin{paste}{ugxProblemFiniteUtilityPageFull7}{ugxProblemFiniteUtilityPageEmpty7} +\pastebutton{ugxProblemFiniteUtilityPageFull7}{\hidepaste} +\tab{5}\spadcommand{normal?(p)$FFPOLY(GF5)\free{p }} +\indentrel{3}\begin{verbatim} + (7) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty7} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty7}{ugxProblemFiniteUtilityPagePatch7} +\pastebutton{ugxProblemFiniteUtilityPageEmpty7}{\showpaste} +\tab{5}\spadcommand{normal?(p)$FFPOLY(GF5)\free{p }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch8} +\begin{paste}{ugxProblemFiniteUtilityPageFull8}{ugxProblemFiniteUtilityPageEmpty8} +\pastebutton{ugxProblemFiniteUtilityPageFull8}{\hidepaste} +\tab{5}\spadcommand{n := createNormalPoly(8)$FFPOLY(GF5)\bound{n }\free{GF5 }} +\indentrel{3}\begin{verbatim} + 8 7 3 + (8) ? + 4? + ? + 1 + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty8} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty8}{ugxProblemFiniteUtilityPagePatch8} +\pastebutton{ugxProblemFiniteUtilityPageEmpty8}{\showpaste} +\tab{5}\spadcommand{n := createNormalPoly(8)$FFPOLY(GF5)\bound{n }\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch9} +\begin{paste}{ugxProblemFiniteUtilityPageFull9}{ugxProblemFiniteUtilityPageEmpty9} +\pastebutton{ugxProblemFiniteUtilityPageFull9}{\hidepaste} +\tab{5}\spadcommand{primitive?(n)$FFPOLY(GF5)\free{n }} +\indentrel{3}\begin{verbatim} + (9) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty9} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty9}{ugxProblemFiniteUtilityPagePatch9} +\pastebutton{ugxProblemFiniteUtilityPageEmpty9}{\showpaste} +\tab{5}\spadcommand{primitive?(n)$FFPOLY(GF5)\free{n }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch10} +\begin{paste}{ugxProblemFiniteUtilityPageFull10}{ugxProblemFiniteUtilityPageEmpty10} +\pastebutton{ugxProblemFiniteUtilityPageFull10}{\hidepaste} +\tab{5}\spadcommand{createPrimitiveNormalPoly(8)$FFPOLY(GF5)\free{GF5 }} +\indentrel{3}\begin{verbatim} + 8 7 5 + (10) ? + 4? + 2? + 2 + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty10} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty10}{ugxProblemFiniteUtilityPagePatch10} +\pastebutton{ugxProblemFiniteUtilityPageEmpty10}{\showpaste} +\tab{5}\spadcommand{createPrimitiveNormalPoly(8)$FFPOLY(GF5)\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch11} +\begin{paste}{ugxProblemFiniteUtilityPageFull11}{ugxProblemFiniteUtilityPageEmpty11} +\pastebutton{ugxProblemFiniteUtilityPageFull11}{\hidepaste} +\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty11} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty11}{ugxProblemFiniteUtilityPagePatch11} +\pastebutton{ugxProblemFiniteUtilityPageEmpty11}{\showpaste} +\tab{5}\spadcommand{GF5 := PF 5;\bound{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch12} +\begin{paste}{ugxProblemFiniteUtilityPageFull12}{ugxProblemFiniteUtilityPageEmpty12} +\pastebutton{ugxProblemFiniteUtilityPageFull12}{\hidepaste} +\tab{5}\spadcommand{h := monomial(1,8)$SUP(GF5)\bound{h }\free{GF5 }} +\indentrel{3}\begin{verbatim} + 8 + (12) ? + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty12} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty12}{ugxProblemFiniteUtilityPagePatch12} +\pastebutton{ugxProblemFiniteUtilityPageEmpty12}{\showpaste} +\tab{5}\spadcommand{h := monomial(1,8)$SUP(GF5)\bound{h }\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch13} +\begin{paste}{ugxProblemFiniteUtilityPageFull13}{ugxProblemFiniteUtilityPageEmpty13} +\pastebutton{ugxProblemFiniteUtilityPageFull13}{\hidepaste} +\tab{5}\spadcommand{nh := nextIrreduciblePoly(h)$FFPOLY(GF5)\bound{nh }\free{h }} +\indentrel{3}\begin{verbatim} + 8 + (13) ? + 2 +Type: Union(SparseUnivariatePolynomial PrimeField 5,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty13} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty13}{ugxProblemFiniteUtilityPagePatch13} +\pastebutton{ugxProblemFiniteUtilityPageEmpty13}{\showpaste} +\tab{5}\spadcommand{nh := nextIrreduciblePoly(h)$FFPOLY(GF5)\bound{nh }\free{h }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch14} +\begin{paste}{ugxProblemFiniteUtilityPageFull14}{ugxProblemFiniteUtilityPageEmpty14} +\pastebutton{ugxProblemFiniteUtilityPageFull14}{\hidepaste} +\tab{5}\spadcommand{createIrreduciblePoly(3)$FFPOLY(GF5)\free{GF5 }} +\indentrel{3}\begin{verbatim} + 3 + (14) ? + ? + 1 + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty14} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty14}{ugxProblemFiniteUtilityPagePatch14} +\pastebutton{ugxProblemFiniteUtilityPageEmpty14}{\showpaste} +\tab{5}\spadcommand{createIrreduciblePoly(3)$FFPOLY(GF5)\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch15} +\begin{paste}{ugxProblemFiniteUtilityPageFull15}{ugxProblemFiniteUtilityPageEmpty15} +\pastebutton{ugxProblemFiniteUtilityPageFull15}{\hidepaste} +\tab{5}\spadcommand{nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)\free{nh }} +\indentrel{3}\begin{verbatim} + 8 + (15) ? + 3 +Type: Union(SparseUnivariatePolynomial PrimeField 5,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty15} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty15}{ugxProblemFiniteUtilityPagePatch15} +\pastebutton{ugxProblemFiniteUtilityPageEmpty15}{\showpaste} +\tab{5}\spadcommand{nh := nextIrreduciblePoly(nh)$FFPOLY(GF5)\free{nh }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch16} +\begin{paste}{ugxProblemFiniteUtilityPageFull16}{ugxProblemFiniteUtilityPageEmpty16} +\pastebutton{ugxProblemFiniteUtilityPageFull16}{\hidepaste} +\tab{5}\spadcommand{numberOfIrreduciblePoly(5)$FFPOLY(GF5)\free{GF5 }} +\indentrel{3}\begin{verbatim} + (16) 624 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty16} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty16}{ugxProblemFiniteUtilityPagePatch16} +\pastebutton{ugxProblemFiniteUtilityPageEmpty16}{\showpaste} +\tab{5}\spadcommand{numberOfIrreduciblePoly(5)$FFPOLY(GF5)\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch17} +\begin{paste}{ugxProblemFiniteUtilityPageFull17}{ugxProblemFiniteUtilityPageEmpty17} +\pastebutton{ugxProblemFiniteUtilityPageFull17}{\hidepaste} +\tab{5}\spadcommand{numberOfPrimitivePoly(3)$FFPOLY(GF5)\free{GF5 }} +\indentrel{3}\begin{verbatim} + (17) 20 + Type: PositiveInteger +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty17} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty17}{ugxProblemFiniteUtilityPagePatch17} +\pastebutton{ugxProblemFiniteUtilityPageEmpty17}{\showpaste} +\tab{5}\spadcommand{numberOfPrimitivePoly(3)$FFPOLY(GF5)\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch18} +\begin{paste}{ugxProblemFiniteUtilityPageFull18}{ugxProblemFiniteUtilityPageEmpty18} +\pastebutton{ugxProblemFiniteUtilityPageFull18}{\hidepaste} +\tab{5}\spadcommand{m := monomial(1,1)$SUP(GF5)\bound{m }\free{GF5 }} +\indentrel{3}\begin{verbatim} + (18) ? + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty18} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty18}{ugxProblemFiniteUtilityPagePatch18} +\pastebutton{ugxProblemFiniteUtilityPageEmpty18}{\showpaste} +\tab{5}\spadcommand{m := monomial(1,1)$SUP(GF5)\bound{m }\free{GF5 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch19} +\begin{paste}{ugxProblemFiniteUtilityPageFull19}{ugxProblemFiniteUtilityPageEmpty19} +\pastebutton{ugxProblemFiniteUtilityPageFull19}{\hidepaste} +\tab{5}\spadcommand{f := m**3 + 4*m**2 + m + 2\bound{fx }\free{m }} +\indentrel{3}\begin{verbatim} + 3 2 + (19) ? + 4? + ? + 2 + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty19} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty19}{ugxProblemFiniteUtilityPagePatch19} +\pastebutton{ugxProblemFiniteUtilityPageEmpty19}{\showpaste} +\tab{5}\spadcommand{f := m**3 + 4*m**2 + m + 2\bound{fx }\free{m }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch20} +\begin{paste}{ugxProblemFiniteUtilityPageFull20}{ugxProblemFiniteUtilityPageEmpty20} +\pastebutton{ugxProblemFiniteUtilityPageFull20}{\hidepaste} +\tab{5}\spadcommand{f1 := nextPrimitivePoly(f)$FFPOLY(GF5)\free{fx }\bound{f1 }} +\indentrel{3}\begin{verbatim} + 3 2 + (20) ? + 4? + 4? + 2 +Type: Union(SparseUnivariatePolynomial PrimeField 5,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty20} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty20}{ugxProblemFiniteUtilityPagePatch20} +\pastebutton{ugxProblemFiniteUtilityPageEmpty20}{\showpaste} +\tab{5}\spadcommand{f1 := nextPrimitivePoly(f)$FFPOLY(GF5)\free{fx }\bound{f1 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch21} +\begin{paste}{ugxProblemFiniteUtilityPageFull21}{ugxProblemFiniteUtilityPageEmpty21} +\pastebutton{ugxProblemFiniteUtilityPageFull21}{\hidepaste} +\tab{5}\spadcommand{nextPrimitivePoly(f1)$FFPOLY(GF5)\free{f1 }} +\indentrel{3}\begin{verbatim} + 3 2 + (21) ? + 2? + 3 +Type: Union(SparseUnivariatePolynomial PrimeField 5,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty21} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty21}{ugxProblemFiniteUtilityPagePatch21} +\pastebutton{ugxProblemFiniteUtilityPageEmpty21}{\showpaste} +\tab{5}\spadcommand{nextPrimitivePoly(f1)$FFPOLY(GF5)\free{f1 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch22} +\begin{paste}{ugxProblemFiniteUtilityPageFull22}{ugxProblemFiniteUtilityPageEmpty22} +\pastebutton{ugxProblemFiniteUtilityPageFull22}{\hidepaste} +\tab{5}\spadcommand{f := m**3 + m**2 + 4*m + 1\bound{fy }\free{m }} +\indentrel{3}\begin{verbatim} + 3 2 + (22) ? + ? + 4? + 1 + Type: SparseUnivariatePolynomial PrimeField 5 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty22} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty22}{ugxProblemFiniteUtilityPagePatch22} +\pastebutton{ugxProblemFiniteUtilityPageEmpty22}{\showpaste} +\tab{5}\spadcommand{f := m**3 + m**2 + 4*m + 1\bound{fy }\free{m }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch23} +\begin{paste}{ugxProblemFiniteUtilityPageFull23}{ugxProblemFiniteUtilityPageEmpty23} +\pastebutton{ugxProblemFiniteUtilityPageFull23}{\hidepaste} +\tab{5}\spadcommand{f1 := nextNormalPoly(f)$FFPOLY(GF5)\free{fy }\bound{f1y }} +\indentrel{3}\begin{verbatim} + 3 2 + (23) ? + ? + 4? + 3 +Type: Union(SparseUnivariatePolynomial PrimeField 5,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty23} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty23}{ugxProblemFiniteUtilityPagePatch23} +\pastebutton{ugxProblemFiniteUtilityPageEmpty23}{\showpaste} +\tab{5}\spadcommand{f1 := nextNormalPoly(f)$FFPOLY(GF5)\free{fy }\bound{f1y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch24} +\begin{paste}{ugxProblemFiniteUtilityPageFull24}{ugxProblemFiniteUtilityPageEmpty24} +\pastebutton{ugxProblemFiniteUtilityPageFull24}{\hidepaste} +\tab{5}\spadcommand{nextNormalPoly(f1)$FFPOLY(GF5)\free{f1y }} +\indentrel{3}\begin{verbatim} + 3 2 + (24) ? + 2? + 1 +Type: Union(SparseUnivariatePolynomial PrimeField 5,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty24} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty24}{ugxProblemFiniteUtilityPagePatch24} +\pastebutton{ugxProblemFiniteUtilityPageEmpty24}{\showpaste} +\tab{5}\spadcommand{nextNormalPoly(f1)$FFPOLY(GF5)\free{f1y }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch25} +\begin{paste}{ugxProblemFiniteUtilityPageFull25}{ugxProblemFiniteUtilityPageEmpty25} +\pastebutton{ugxProblemFiniteUtilityPageFull25}{\hidepaste} +\tab{5}\spadcommand{GF16 := FFX(FFX(PF 2,2),2);\bound{GF16 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty25} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty25}{ugxProblemFiniteUtilityPagePatch25} +\pastebutton{ugxProblemFiniteUtilityPageEmpty25}{\showpaste} +\tab{5}\spadcommand{GF16 := FFX(FFX(PF 2,2),2);\bound{GF16 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch26} +\begin{paste}{ugxProblemFiniteUtilityPageFull26}{ugxProblemFiniteUtilityPageEmpty26} +\pastebutton{ugxProblemFiniteUtilityPageFull26}{\hidepaste} +\tab{5}\spadcommand{createIrreduciblePoly(5)$FFPOLY(GF16)\free{GF16 }} +\indentrel{3}\begin{verbatim} + 5 + (26) ? + %CZ +Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty26} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty26}{ugxProblemFiniteUtilityPagePatch26} +\pastebutton{ugxProblemFiniteUtilityPageEmpty26}{\showpaste} +\tab{5}\spadcommand{createIrreduciblePoly(5)$FFPOLY(GF16)\free{GF16 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch27} +\begin{paste}{ugxProblemFiniteUtilityPageFull27}{ugxProblemFiniteUtilityPageEmpty27} +\pastebutton{ugxProblemFiniteUtilityPageFull27}{\hidepaste} +\tab{5}\spadcommand{random(5)$FFPOLY(GF16)\free{GF16 }} +\indentrel{3}\begin{verbatim} + (27) + 5 4 3 + ? + (%CV + 1)? + ((%CV + 1)%CZ + %CV + 1)? + + + 2 + (%CV + 1)? + ((%CV + 1)%CZ + %CV + 1)? + + + (%CV + 1)%CZ + 1 +Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty27} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty27}{ugxProblemFiniteUtilityPagePatch27} +\pastebutton{ugxProblemFiniteUtilityPageEmpty27}{\showpaste} +\tab{5}\spadcommand{random(5)$FFPOLY(GF16)\free{GF16 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch28} +\begin{paste}{ugxProblemFiniteUtilityPageFull28}{ugxProblemFiniteUtilityPageEmpty28} +\pastebutton{ugxProblemFiniteUtilityPageFull28}{\hidepaste} +\tab{5}\spadcommand{random(3,9)$FFPOLY(GF16)\free{GF16 }} +\indentrel{3}\begin{verbatim} + (28) + 4 3 2 + ? + (%CZ + %CV + 1)? + (%CV %CZ + %CV + 1)? + + + (%CZ + %CV)? + (%CV + 1)%CZ +Type: SparseUnivariatePolynomial FiniteFieldExtension(FiniteFieldExtension(PrimeField 2,2),2) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty28} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty28}{ugxProblemFiniteUtilityPagePatch28} +\pastebutton{ugxProblemFiniteUtilityPageEmpty28}{\showpaste} +\tab{5}\spadcommand{random(3,9)$FFPOLY(GF16)\free{GF16 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch29} +\begin{paste}{ugxProblemFiniteUtilityPageFull29}{ugxProblemFiniteUtilityPageEmpty29} +\pastebutton{ugxProblemFiniteUtilityPageFull29}{\hidepaste} +\tab{5}\spadcommand{GF2 := PrimeField 2;\bound{GF2 }} +\indentrel{3}\begin{verbatim} + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty29} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty29}{ugxProblemFiniteUtilityPagePatch29} +\pastebutton{ugxProblemFiniteUtilityPageEmpty29}{\showpaste} +\tab{5}\spadcommand{GF2 := PrimeField 2;\bound{GF2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch30} +\begin{paste}{ugxProblemFiniteUtilityPageFull30}{ugxProblemFiniteUtilityPageEmpty30} +\pastebutton{ugxProblemFiniteUtilityPageFull30}{\hidepaste} +\tab{5}\spadcommand{F := FFX(GF2,12)\bound{F }\free{GF2 }} +\indentrel{3}\begin{verbatim} + (30) FiniteFieldExtension(PrimeField 2,12) + Type: Domain +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty30} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty30}{ugxProblemFiniteUtilityPagePatch30} +\pastebutton{ugxProblemFiniteUtilityPageEmpty30}{\showpaste} +\tab{5}\spadcommand{F := FFX(GF2,12)\bound{F }\free{GF2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch31} +\begin{paste}{ugxProblemFiniteUtilityPageFull31}{ugxProblemFiniteUtilityPageEmpty31} +\pastebutton{ugxProblemFiniteUtilityPageFull31}{\hidepaste} +\tab{5}\spadcommand{f := createIrreduciblePoly(6)$FFPOLY(GF2)\bound{fz }\free{GF2 }} +\indentrel{3}\begin{verbatim} + 6 + (31) ? + ? + 1 + Type: SparseUnivariatePolynomial PrimeField 2 +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty31} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty31}{ugxProblemFiniteUtilityPagePatch31} +\pastebutton{ugxProblemFiniteUtilityPageEmpty31}{\showpaste} +\tab{5}\spadcommand{f := createIrreduciblePoly(6)$FFPOLY(GF2)\bound{fz }\free{GF2 }} +\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPagePatch32} +\begin{paste}{ugxProblemFiniteUtilityPageFull32}{ugxProblemFiniteUtilityPageEmpty32} +\pastebutton{ugxProblemFiniteUtilityPageFull32}{\hidepaste} +\tab{5}\spadcommand{root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)\free{F GF2 fz }\bound{root }} +\indentrel{3}\begin{verbatim} + 11 8 7 5 + (32) %CS + %CS + %CS + %CS + %CS + 1 + Type: FiniteFieldExtension(PrimeField 2,12) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugxProblemFiniteUtilityPageEmpty32} +\begin{paste}{ugxProblemFiniteUtilityPageEmpty32}{ugxProblemFiniteUtilityPagePatch32} +\pastebutton{ugxProblemFiniteUtilityPageEmpty32}{\showpaste} +\tab{5}\spadcommand{root := rootOfIrreduciblePoly(f)$FFPOLY2(F,GF2)\free{F GF2 fz }\bound{root }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch1} +\begin{paste}{ugProblemEigenPageFull1}{ugProblemEigenPageEmpty1} +\pastebutton{ugProblemEigenPageFull1}{\hidepaste} +\tab{5}\spadcommand{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]]\bound{m1 }} +\indentrel{3}\begin{verbatim} + �1 2 1 � + � � + (1) �2 1 - 2� + � � + �1 - 2 4 � + Type: Matrix Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty1} +\begin{paste}{ugProblemEigenPageEmpty1}{ugProblemEigenPagePatch1} +\pastebutton{ugProblemEigenPageEmpty1}{\showpaste} +\tab{5}\spadcommand{m1 := matrix [[1,2,1],[2,1,-2],[1,-2,4]]\bound{m1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch2} +\begin{paste}{ugProblemEigenPageFull2}{ugProblemEigenPageEmpty2} +\pastebutton{ugProblemEigenPageFull2}{\hidepaste} +\tab{5}\spadcommand{leig := eigenvalues(m1)\free{m1 }\bound{leig }} +\indentrel{3}\begin{verbatim} + 2 + (2) [5,%DA | %DA - %DA - 5] +Type: List Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty2} +\begin{paste}{ugProblemEigenPageEmpty2}{ugProblemEigenPagePatch2} +\pastebutton{ugProblemEigenPageEmpty2}{\showpaste} +\tab{5}\spadcommand{leig := eigenvalues(m1)\free{m1 }\bound{leig }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch3} +\begin{paste}{ugProblemEigenPageFull3}{ugProblemEigenPageEmpty3} +\pastebutton{ugProblemEigenPageFull3}{\hidepaste} +\tab{5}\spadcommand{eigenvector(first(leig),m1)\free{m1 leig }} +\indentrel{3}\begin{verbatim} + � 0 � + � � + � 1� + (3) [�- 캐] + � 2� + � � + � 1 � + Type: List Matrix Fraction Polynomial Fraction Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty3} +\begin{paste}{ugProblemEigenPageEmpty3}{ugProblemEigenPagePatch3} +\pastebutton{ugProblemEigenPageEmpty3}{\showpaste} +\tab{5}\spadcommand{eigenvector(first(leig),m1)\free{m1 leig }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch4} +\begin{paste}{ugProblemEigenPageFull4}{ugProblemEigenPageEmpty4} +\pastebutton{ugProblemEigenPageFull4}{\hidepaste} +\tab{5}\spadcommand{eigenvectors(m1)\free{m1 }} +\indentrel{3}\begin{verbatim} + (4) + � 0 � + � � + � 1� + [[eigval= 5,eigmult= 1,eigvec= [�- 캐]], + � 2� + � � + � 1 � + + 2 + [eigval= (%DB | %DB - %DB - 5), eigmult= 1, + �%DB� + � � + eigvec= [� 2 �]] + � � + � 1 � + ] +Type: List Record(eigval: Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)),eigmult: NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty4} +\begin{paste}{ugProblemEigenPageEmpty4}{ugProblemEigenPagePatch4} +\pastebutton{ugProblemEigenPageEmpty4}{\showpaste} +\tab{5}\spadcommand{eigenvectors(m1)\free{m1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch5} +\begin{paste}{ugProblemEigenPageFull5}{ugProblemEigenPageEmpty5} +\pastebutton{ugProblemEigenPageFull5}{\hidepaste} +\tab{5}\spadcommand{radicalEigenvectors(m1)\free{m1 }} +\indentrel{3}\begin{verbatim} + (5) + � 旼커 � + 旼커 �\�21 + 1� + \�21 + 1 냐컴컴컴컴� + [[radval= 컴컴컴컴�,radmult= 1,radvect= [� 2 �]], + 2 � � + � 2 � + � � + � 1 � + + 旼커 + - \�21 + 1 + [radval= 컴컴컴컴컴�, radmult= 1, + 2 + � 旼커 � + �- \�21 + 1� + 냐컴컴컴컴컴� + radvect= [� 2 �]] + � � + � 2 � + � � + � 1 � + , + � 0 � + � � + � 1� + [radval= 5,radmult= 1,radvect= [�- 캐]]] + � 2� + � � + � 1 � +Type: List Record(radval: Expression Integer,radmult: Integer,radvect: List Matrix Expression Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty5} +\begin{paste}{ugProblemEigenPageEmpty5}{ugProblemEigenPagePatch5} +\pastebutton{ugProblemEigenPageEmpty5}{\showpaste} +\tab{5}\spadcommand{radicalEigenvectors(m1)\free{m1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch6} +\begin{paste}{ugProblemEigenPageFull6}{ugProblemEigenPageEmpty6} +\pastebutton{ugProblemEigenPageFull6}{\hidepaste} +\tab{5}\spadcommand{realEigenvectors(m1,1/1000)\free{m1 }} +\indentrel{3}\begin{verbatim} + (6) + � 0 � + � � + � 1� + [[outval= 5,outmult= 1,outvect= [�- 캐]], + � 2� + � � + � 1 � + �5717� + 냐컴캐 + 5717 �2048� + [outval= 컴컴,outmult= 1,outvect= [� �]], + 2048 � 2 � + � � + � 1 � + � 3669� + �- 컴컴� + 3669 � 2048� + [outval= - 컴컴,outmult= 1,outvect= [� �]]] + 2048 � 2 � + � � + � 1 � +Type: List Record(outval: Fraction Integer,outmult: Integer,outvect: List Matrix Fraction Integer) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty6} +\begin{paste}{ugProblemEigenPageEmpty6}{ugProblemEigenPagePatch6} +\pastebutton{ugProblemEigenPageEmpty6}{\showpaste} +\tab{5}\spadcommand{realEigenvectors(m1,1/1000)\free{m1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch7} +\begin{paste}{ugProblemEigenPageFull7}{ugProblemEigenPageEmpty7} +\pastebutton{ugProblemEigenPageFull7}{\hidepaste} +\tab{5}\spadcommand{eigenMatrix(m1)\free{m1 }} +\indentrel{3}\begin{verbatim} + � 旼커 旼커 � + �\�21 + 1 - \�21 + 1 � + 냐컴컴컴컴 컴컴컴컴컴� 0 � + � 2 2 � + (7) � � + � 1� + � 2 2 - 캐 + � 2� + � � + � 1 1 1 � + Type: Union(Matrix Expression Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty7} +\begin{paste}{ugProblemEigenPageEmpty7}{ugProblemEigenPagePatch7} +\pastebutton{ugProblemEigenPageEmpty7}{\showpaste} +\tab{5}\spadcommand{eigenMatrix(m1)\free{m1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch8} +\begin{paste}{ugProblemEigenPageFull8}{ugProblemEigenPageEmpty8} +\pastebutton{ugProblemEigenPageFull8}{\hidepaste} +\tab{5}\spadcommand{m2 := matrix [[-5,-2],[18,7]]\bound{m2 }} +\indentrel{3}\begin{verbatim} + �- 5 - 2� + (8) � � + �18 7 � + Type: Matrix Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty8} +\begin{paste}{ugProblemEigenPageEmpty8}{ugProblemEigenPagePatch8} +\pastebutton{ugProblemEigenPageEmpty8}{\showpaste} +\tab{5}\spadcommand{m2 := matrix [[-5,-2],[18,7]]\bound{m2 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch9} +\begin{paste}{ugProblemEigenPageFull9}{ugProblemEigenPageEmpty9} +\pastebutton{ugProblemEigenPageFull9}{\hidepaste} +\tab{5}\spadcommand{eigenMatrix(m2)\free{m2 }} +\indentrel{3}\begin{verbatim} + (9) "failed" + Type: Union("failed",...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty9} +\begin{paste}{ugProblemEigenPageEmpty9}{ugProblemEigenPagePatch9} +\pastebutton{ugProblemEigenPageEmpty9}{\showpaste} +\tab{5}\spadcommand{eigenMatrix(m2)\free{m2 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch10} +\begin{paste}{ugProblemEigenPageFull10}{ugProblemEigenPageEmpty10} +\pastebutton{ugProblemEigenPageFull10}{\hidepaste} +\tab{5}\spadcommand{m3 := matrix [[1,2],[2,1]]\bound{m3 }} +\indentrel{3}\begin{verbatim} + �1 2� + (10) � � + �2 1� + Type: Matrix Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty10} +\begin{paste}{ugProblemEigenPageEmpty10}{ugProblemEigenPagePatch10} +\pastebutton{ugProblemEigenPageEmpty10}{\showpaste} +\tab{5}\spadcommand{m3 := matrix [[1,2],[2,1]]\bound{m3 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPagePatch11} +\begin{paste}{ugProblemEigenPageFull11}{ugProblemEigenPageEmpty11} +\pastebutton{ugProblemEigenPageFull11}{\hidepaste} +\tab{5}\spadcommand{orthonormalBasis(m3)\free{m3 }} +\indentrel{3}\begin{verbatim} + � 1 � � 1 � + �- 컴컴� 냐컴캐 + � 旼엿 � 旼엿 + � \�2 � �\�2 � + (11) [� �,� �] + � 1 � � 1 � + � 컴컴 � 냐컴캐 + � 旼� � � 旼엿 + � \�2 � �\�2 � + Type: List Matrix Expression Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemEigenPageEmpty11} +\begin{paste}{ugProblemEigenPageEmpty11}{ugProblemEigenPagePatch11} +\pastebutton{ugProblemEigenPageEmpty11}{\showpaste} +\tab{5}\spadcommand{orthonormalBasis(m3)\free{m3 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch1} +\begin{paste}{ugProblemGaloisPageFull1}{ugProblemGaloisPageEmpty1} +\pastebutton{ugProblemGaloisPageFull1}{\hidepaste} +\tab{5}\spadcommand{p := x**5 - 5*x + 12\bound{p }} +\indentrel{3}\begin{verbatim} + 5 + (1) x - 5x + 12 + Type: Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty1} +\begin{paste}{ugProblemGaloisPageEmpty1}{ugProblemGaloisPagePatch1} +\pastebutton{ugProblemGaloisPageEmpty1}{\showpaste} +\tab{5}\spadcommand{p := x**5 - 5*x + 12\bound{p }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch2} +\begin{paste}{ugProblemGaloisPageFull2}{ugProblemGaloisPageEmpty2} +\pastebutton{ugProblemGaloisPageFull2}{\hidepaste} +\tab{5}\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y)\free{p }\bound{q }} +\indentrel{3}\begin{verbatim} + (2) + 25 21 17 15 13 + x - 50x - 2375x + 90000x - 5000x + + + 11 9 7 5 + 2700000x + 250000x + 18000000x + 64000000x + Type: Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty2} +\begin{paste}{ugProblemGaloisPageEmpty2}{ugProblemGaloisPagePatch2} +\pastebutton{ugProblemGaloisPageEmpty2}{\showpaste} +\tab{5}\spadcommand{q := resultant(eval(p,x,y),-eval(p,x,y-x),y)\free{p }\bound{q }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch3} +\begin{paste}{ugProblemGaloisPageFull3}{ugProblemGaloisPageEmpty3} +\pastebutton{ugProblemGaloisPageFull3}{\hidepaste} +\tab{5}\spadcommand{q1 := exquo(q, x**5)\free{q }\bound{q1 }} +\indentrel{3}\begin{verbatim} + (3) + 20 16 12 10 8 6 + x - 50x - 2375x + 90000x - 5000x + 2700000x + + + 4 2 + 250000x + 18000000x + 64000000 + Type: Union(Polynomial Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty3} +\begin{paste}{ugProblemGaloisPageEmpty3}{ugProblemGaloisPagePatch3} +\pastebutton{ugProblemGaloisPageEmpty3}{\showpaste} +\tab{5}\spadcommand{q1 := exquo(q, x**5)\free{q }\bound{q1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch4} +\begin{paste}{ugProblemGaloisPageFull4}{ugProblemGaloisPageEmpty4} +\pastebutton{ugProblemGaloisPageFull4}{\hidepaste} +\tab{5}\spadcommand{factoredQ := factor q1\free{q1 }\bound{factoredQ }} +\indentrel{3}\begin{verbatim} + (4) + 10 8 6 4 2 + (x - 10x - 75x + 1500x - 5500x + 16000) + * + 10 8 6 4 2 + (x + 10x + 125x + 500x + 2500x + 4000) + Type: Factored Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty4} +\begin{paste}{ugProblemGaloisPageEmpty4}{ugProblemGaloisPagePatch4} +\pastebutton{ugProblemGaloisPageEmpty4}{\showpaste} +\tab{5}\spadcommand{factoredQ := factor q1\free{q1 }\bound{factoredQ }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch5} +\begin{paste}{ugProblemGaloisPageFull5}{ugProblemGaloisPageEmpty5} +\pastebutton{ugProblemGaloisPageFull5}{\hidepaste} +\tab{5}\spadcommand{r := nthFactor(factoredQ,1)\free{factoredQ }\bound{r }} +\indentrel{3}\begin{verbatim} + 10 8 6 4 2 + (5) x - 10x - 75x + 1500x - 5500x + 16000 + Type: Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty5} +\begin{paste}{ugProblemGaloisPageEmpty5}{ugProblemGaloisPagePatch5} +\pastebutton{ugProblemGaloisPageEmpty5}{\showpaste} +\tab{5}\spadcommand{r := nthFactor(factoredQ,1)\free{factoredQ }\bound{r }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch6} +\begin{paste}{ugProblemGaloisPageFull6}{ugProblemGaloisPageEmpty6} +\pastebutton{ugProblemGaloisPageFull6}{\hidepaste} +\tab{5}\spadcommand{beta:AN := rootOf(eval(r,x,b))\free{r }\bound{beta }} +\indentrel{3}\begin{verbatim} + (6) b + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty6} +\begin{paste}{ugProblemGaloisPageEmpty6}{ugProblemGaloisPagePatch6} +\pastebutton{ugProblemGaloisPageEmpty6}{\showpaste} +\tab{5}\spadcommand{beta:AN := rootOf(eval(r,x,b))\free{r }\bound{beta }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch7} +\begin{paste}{ugProblemGaloisPageFull7}{ugProblemGaloisPageEmpty7} +\pastebutton{ugProblemGaloisPageFull7}{\hidepaste} +\tab{5}\spadcommand{p := p::UP(x,INT)::UP(x,AN)\free{p }\bound{declareP }} +\indentrel{3}\begin{verbatim} + 5 + (7) x - 5x + 12 + Type: UnivariatePolynomial(x,AlgebraicNumber) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty7} +\begin{paste}{ugProblemGaloisPageEmpty7}{ugProblemGaloisPagePatch7} +\pastebutton{ugProblemGaloisPageEmpty7}{\showpaste} +\tab{5}\spadcommand{p := p::UP(x,INT)::UP(x,AN)\free{p }\bound{declareP }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch8} +\begin{paste}{ugProblemGaloisPageFull8}{ugProblemGaloisPageEmpty8} +\pastebutton{ugProblemGaloisPageFull8}{\hidepaste} +\tab{5}\spadcommand{algFactors := factor(p,[beta])\free{declareP beta }\bound{algFactors }} +\indentrel{3}\begin{verbatim} + (8) + x + + + 9 8 7 6 5 + - 85b - 116b + 780b + 2640b + 14895b + + + 4 3 2 + - 8820b - 127050b - 327000b - 405200b + + + 2062400 + / + 1339200 + * + 8 6 4 2 + - 17b + 156b + 2979b - 25410b - 14080 + (x + 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�) + 66960 + * + x + + + 8 6 4 2 + 143b - 2100b - 10485b + 290550b - 334800b + + + - 960800 + / + 669600 + * + x + + + 8 6 4 2 + 143b - 2100b - 10485b + 290550b + 334800b + + + - 960800 + / + 669600 + * + x + + + 9 8 7 6 5 + 85b - 116b - 780b + 2640b - 14895b + + + 4 3 2 + - 8820b + 127050b - 327000b + 405200b + + + 2062400 + / + 1339200 + Type: Factored UnivariatePolynomial(x,AlgebraicNumber) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty8} +\begin{paste}{ugProblemGaloisPageEmpty8}{ugProblemGaloisPagePatch8} +\pastebutton{ugProblemGaloisPageEmpty8}{\showpaste} +\tab{5}\spadcommand{algFactors := factor(p,[beta])\free{declareP beta }\bound{algFactors }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch9} +\begin{paste}{ugProblemGaloisPageFull9}{ugProblemGaloisPageEmpty9} +\pastebutton{ugProblemGaloisPageFull9}{\hidepaste} +\tab{5}\spadcommand{factor(p)\free{declareP }} +\indentrel{3}\begin{verbatim} + 5 + (9) x - 5x + 12 + Type: Factored UnivariatePolynomial(x,AlgebraicNumber) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty9} +\begin{paste}{ugProblemGaloisPageEmpty9}{ugProblemGaloisPagePatch9} +\pastebutton{ugProblemGaloisPageEmpty9}{\showpaste} +\tab{5}\spadcommand{factor(p)\free{declareP }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch10} +\begin{paste}{ugProblemGaloisPageFull10}{ugProblemGaloisPageEmpty10} +\pastebutton{ugProblemGaloisPageFull10}{\hidepaste} +\tab{5}\spadcommand{factor1 := nthFactor(algFactors,1)\free{algFactors }\bound{factor1 }} +\indentrel{3}\begin{verbatim} + (10) + x + + + 9 8 7 6 5 + - 85b - 116b + 780b + 2640b + 14895b + + + 4 3 2 + - 8820b - 127050b - 327000b - 405200b + 2062400 + / + 1339200 + Type: UnivariatePolynomial(x,AlgebraicNumber) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty10} +\begin{paste}{ugProblemGaloisPageEmpty10}{ugProblemGaloisPagePatch10} +\pastebutton{ugProblemGaloisPageEmpty10}{\showpaste} +\tab{5}\spadcommand{factor1 := nthFactor(algFactors,1)\free{algFactors }\bound{factor1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch11} +\begin{paste}{ugProblemGaloisPageFull11}{ugProblemGaloisPageEmpty11} +\pastebutton{ugProblemGaloisPageFull11}{\hidepaste} +\tab{5}\spadcommand{root1 := -coefficient(factor1,0)\free{factor1 }\bound{root1 }} +\indentrel{3}\begin{verbatim} + (11) + 9 8 7 6 5 4 + 85b + 116b - 780b - 2640b - 14895b + 8820b + + + 3 2 + 127050b + 327000b + 405200b - 2062400 + / + 1339200 + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty11} +\begin{paste}{ugProblemGaloisPageEmpty11}{ugProblemGaloisPagePatch11} +\pastebutton{ugProblemGaloisPageEmpty11}{\showpaste} +\tab{5}\spadcommand{root1 := -coefficient(factor1,0)\free{factor1 }\bound{root1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch12} +\begin{paste}{ugProblemGaloisPageFull12}{ugProblemGaloisPageEmpty12} +\pastebutton{ugProblemGaloisPageFull12}{\hidepaste} +\tab{5}\spadcommand{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5]\free{algFactors }\bound{roots }} +\indentrel{3}\begin{verbatim} + (12) + [ + 9 8 7 6 5 4 + 85b + 116b - 780b - 2640b - 14895b + 8820b + + + 3 2 + 127050b + 327000b + 405200b - 2062400 + / + 1339200 + , + 8 6 4 2 + 17b - 156b - 2979b + 25410b + 14080 + 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�, + 66960 + + 8 6 4 2 + - 143b + 2100b + 10485b - 290550b + 334800b + + + 960800 + / + 669600 + , + + 8 6 4 2 + - 143b + 2100b + 10485b - 290550b - 334800b + + + 960800 + / + 669600 + , + + 9 8 7 6 5 + - 85b + 116b + 780b - 2640b + 14895b + + + 4 3 2 + 8820b - 127050b + 327000b - 405200b - 2062400 + / + 1339200 + ] + Type: List AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty12} +\begin{paste}{ugProblemGaloisPageEmpty12}{ugProblemGaloisPagePatch12} +\pastebutton{ugProblemGaloisPageEmpty12}{\showpaste} +\tab{5}\spadcommand{roots := [-coefficient(nthFactor(algFactors,i),0) for i in 1..5]\free{algFactors }\bound{roots }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch13} +\begin{paste}{ugProblemGaloisPageFull13}{ugProblemGaloisPageEmpty13} +\pastebutton{ugProblemGaloisPageFull13}{\hidepaste} +\tab{5}\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5)\free{roots }\bound{ais }} +\indentrel{3}\begin{verbatim} + (13) + 9 8 7 6 5 4 + - 85b + 116b + 780b - 2640b + 14895b + 8820b + + + 3 2 + - 127050b + 327000b - 405200b - 2062400 + / + 1339200 + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty13} +\begin{paste}{ugProblemGaloisPageEmpty13}{ugProblemGaloisPagePatch13} +\pastebutton{ugProblemGaloisPageEmpty13}{\showpaste} +\tab{5}\spadcommand{(a1,a2,a3,a4,a5) := (roots.1,roots.2,roots.3,roots.4,roots.5)\free{roots }\bound{ais }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch14} +\begin{paste}{ugProblemGaloisPageFull14}{ugProblemGaloisPageEmpty14} +\pastebutton{ugProblemGaloisPageFull14}{\hidepaste} +\tab{5}\spadcommand{eval(r,x,a1 - a2)\free{ais }} +\indentrel{3}\begin{verbatim} + (14) 0 + Type: Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty14} +\begin{paste}{ugProblemGaloisPageEmpty14}{ugProblemGaloisPagePatch14} +\pastebutton{ugProblemGaloisPageEmpty14}{\showpaste} +\tab{5}\spadcommand{eval(r,x,a1 - a2)\free{ais }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch15} +\begin{paste}{ugProblemGaloisPageFull15}{ugProblemGaloisPageEmpty15} +\pastebutton{ugProblemGaloisPageFull15}{\hidepaste} +\tab{5}\spadcommand{eval(r,x,a1 - a3)\free{ais }} +\indentrel{3}\begin{verbatim} + (15) + 9 8 7 6 5 + 47905b + 66920b - 536100b - 980400b - 3345075b + + + 4 3 2 + - 5787000b + 75572250b + 161688000b - 184600000b + + + - 710912000 + / + 4464 + Type: Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty15} +\begin{paste}{ugProblemGaloisPageEmpty15}{ugProblemGaloisPagePatch15} +\pastebutton{ugProblemGaloisPageEmpty15}{\showpaste} +\tab{5}\spadcommand{eval(r,x,a1 - a3)\free{ais }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch16} +\begin{paste}{ugProblemGaloisPageFull16}{ugProblemGaloisPageEmpty16} +\pastebutton{ugProblemGaloisPageFull16}{\hidepaste} +\tab{5}\spadcommand{eval(r,x,a1 - a4)\free{ais }} +\indentrel{3}\begin{verbatim} + (16) 0 + Type: Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty16} +\begin{paste}{ugProblemGaloisPageEmpty16}{ugProblemGaloisPagePatch16} +\pastebutton{ugProblemGaloisPageEmpty16}{\showpaste} +\tab{5}\spadcommand{eval(r,x,a1 - a4)\free{ais }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch17} +\begin{paste}{ugProblemGaloisPageFull17}{ugProblemGaloisPageEmpty17} +\pastebutton{ugProblemGaloisPageFull17}{\hidepaste} +\tab{5}\spadcommand{eval(r,x,a1 - a5)\free{ais }} +\indentrel{3}\begin{verbatim} + 8 6 4 2 + 405b + 3450b - 19875b - 198000b - 588000 + (17) 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 + 31 + Type: Polynomial AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty17} +\begin{paste}{ugProblemGaloisPageEmpty17}{ugProblemGaloisPagePatch17} +\pastebutton{ugProblemGaloisPageEmpty17}{\showpaste} +\tab{5}\spadcommand{eval(r,x,a1 - a5)\free{ais }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch18} +\begin{paste}{ugProblemGaloisPageFull18}{ugProblemGaloisPageEmpty18} +\pastebutton{ugProblemGaloisPageFull18}{\hidepaste} +\tab{5}\spadcommand{bb := a1 - a4\free{ais }\bound{bb }} +\indentrel{3}\begin{verbatim} + (18) + 9 8 7 6 5 4 + 85b + 402b - 780b - 6840b - 14895b - 12150b + + + 3 2 + 127050b + 908100b + 1074800b - 3984000 + / + 1339200 + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty18} +\begin{paste}{ugProblemGaloisPageEmpty18}{ugProblemGaloisPagePatch18} +\pastebutton{ugProblemGaloisPageEmpty18}{\showpaste} +\tab{5}\spadcommand{bb := a1 - a4\free{ais }\bound{bb }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch19} +\begin{paste}{ugProblemGaloisPageFull19}{ugProblemGaloisPageEmpty19} +\pastebutton{ugProblemGaloisPageFull19}{\hidepaste} +\tab{5}\spadcommand{aa1 := subst(a1,beta = bb)\free{beta bb ais }\bound{aa1 }} +\indentrel{3}\begin{verbatim} + (19) + 8 6 4 2 + - 143b + 2100b + 10485b - 290550b + 334800b + + + 960800 + / + 669600 + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty19} +\begin{paste}{ugProblemGaloisPageEmpty19}{ugProblemGaloisPagePatch19} +\pastebutton{ugProblemGaloisPageEmpty19}{\showpaste} +\tab{5}\spadcommand{aa1 := subst(a1,beta = bb)\free{beta bb ais }\bound{aa1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch20} +\begin{paste}{ugProblemGaloisPageFull20}{ugProblemGaloisPageEmpty20} +\pastebutton{ugProblemGaloisPageFull20}{\hidepaste} +\tab{5}\spadcommand{aa2 := subst(a2,beta = bb)\free{beta bb ais }\bound{aa2 }} +\indentrel{3}\begin{verbatim} + (20) + 9 8 7 6 5 4 + - 85b + 116b + 780b - 2640b + 14895b + 8820b + + + 3 2 + - 127050b + 327000b - 405200b - 2062400 + / + 1339200 + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty20} +\begin{paste}{ugProblemGaloisPageEmpty20}{ugProblemGaloisPagePatch20} +\pastebutton{ugProblemGaloisPageEmpty20}{\showpaste} +\tab{5}\spadcommand{aa2 := subst(a2,beta = bb)\free{beta bb ais }\bound{aa2 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch21} +\begin{paste}{ugProblemGaloisPageFull21}{ugProblemGaloisPageEmpty21} +\pastebutton{ugProblemGaloisPageFull21}{\hidepaste} +\tab{5}\spadcommand{aa3 := subst(a3,beta = bb)\free{beta bb ais }\bound{aa3 }} +\indentrel{3}\begin{verbatim} + (21) + 9 8 7 6 5 4 + 85b + 116b - 780b - 2640b - 14895b + 8820b + + + 3 2 + 127050b + 327000b + 405200b - 2062400 + / + 1339200 + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty21} +\begin{paste}{ugProblemGaloisPageEmpty21}{ugProblemGaloisPagePatch21} +\pastebutton{ugProblemGaloisPageEmpty21}{\showpaste} +\tab{5}\spadcommand{aa3 := subst(a3,beta = bb)\free{beta bb ais }\bound{aa3 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch22} +\begin{paste}{ugProblemGaloisPageFull22}{ugProblemGaloisPageEmpty22} +\pastebutton{ugProblemGaloisPageFull22}{\hidepaste} +\tab{5}\spadcommand{aa4 := subst(a4,beta = bb)\free{beta bb ais }\bound{aa4 }} +\indentrel{3}\begin{verbatim} + (22) + 8 6 4 2 + - 143b + 2100b + 10485b - 290550b - 334800b + + + 960800 + / + 669600 + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty22} +\begin{paste}{ugProblemGaloisPageEmpty22}{ugProblemGaloisPagePatch22} +\pastebutton{ugProblemGaloisPageEmpty22}{\showpaste} +\tab{5}\spadcommand{aa4 := subst(a4,beta = bb)\free{beta bb ais }\bound{aa4 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch23} +\begin{paste}{ugProblemGaloisPageFull23}{ugProblemGaloisPageEmpty23} +\pastebutton{ugProblemGaloisPageFull23}{\hidepaste} +\tab{5}\spadcommand{aa5 := subst(a5,beta = bb)\free{beta bb ais }\bound{aa5 }} +\indentrel{3}\begin{verbatim} + 8 6 4 2 + 17b - 156b - 2979b + 25410b + 14080 + (23) 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� + 66960 + Type: AlgebraicNumber +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty23} +\begin{paste}{ugProblemGaloisPageEmpty23}{ugProblemGaloisPagePatch23} +\pastebutton{ugProblemGaloisPageEmpty23}{\showpaste} +\tab{5}\spadcommand{aa5 := subst(a5,beta = bb)\free{beta bb ais }\bound{aa5 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch24} +\begin{paste}{ugProblemGaloisPageFull24}{ugProblemGaloisPageEmpty24} +\pastebutton{ugProblemGaloisPageFull24}{\hidepaste} +\tab{5}\spadcommand{(aa1 = a1) :: Boolean\free{aa1 }} +\indentrel{3}\begin{verbatim} + (24) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty24} +\begin{paste}{ugProblemGaloisPageEmpty24}{ugProblemGaloisPagePatch24} +\pastebutton{ugProblemGaloisPageEmpty24}{\showpaste} +\tab{5}\spadcommand{(aa1 = a1) :: Boolean\free{aa1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch25} +\begin{paste}{ugProblemGaloisPageFull25}{ugProblemGaloisPageEmpty25} +\pastebutton{ugProblemGaloisPageFull25}{\hidepaste} +\tab{5}\spadcommand{(aa1 = a2) :: Boolean\free{aa1 }} +\indentrel{3}\begin{verbatim} + (25) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty25} +\begin{paste}{ugProblemGaloisPageEmpty25}{ugProblemGaloisPagePatch25} +\pastebutton{ugProblemGaloisPageEmpty25}{\showpaste} +\tab{5}\spadcommand{(aa1 = a2) :: Boolean\free{aa1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch26} +\begin{paste}{ugProblemGaloisPageFull26}{ugProblemGaloisPageEmpty26} +\pastebutton{ugProblemGaloisPageFull26}{\hidepaste} +\tab{5}\spadcommand{(aa1 = a3) :: Boolean\free{aa1 }} +\indentrel{3}\begin{verbatim} + (26) true + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty26} +\begin{paste}{ugProblemGaloisPageEmpty26}{ugProblemGaloisPagePatch26} +\pastebutton{ugProblemGaloisPageEmpty26}{\showpaste} +\tab{5}\spadcommand{(aa1 = a3) :: Boolean\free{aa1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch27} +\begin{paste}{ugProblemGaloisPageFull27}{ugProblemGaloisPageEmpty27} +\pastebutton{ugProblemGaloisPageFull27}{\hidepaste} +\tab{5}\spadcommand{(aa1 = a4) :: Boolean\free{aa1 }} +\indentrel{3}\begin{verbatim} + (27) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty27} +\begin{paste}{ugProblemGaloisPageEmpty27}{ugProblemGaloisPagePatch27} +\pastebutton{ugProblemGaloisPageEmpty27}{\showpaste} +\tab{5}\spadcommand{(aa1 = a4) :: Boolean\free{aa1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPagePatch28} +\begin{paste}{ugProblemGaloisPageFull28}{ugProblemGaloisPageEmpty28} +\pastebutton{ugProblemGaloisPageFull28}{\hidepaste} +\tab{5}\spadcommand{(aa1 = a5) :: Boolean\free{aa1 }} +\indentrel{3}\begin{verbatim} + (28) false + Type: Boolean +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemGaloisPageEmpty28} +\begin{paste}{ugProblemGaloisPageEmpty28}{ugProblemGaloisPagePatch28} +\pastebutton{ugProblemGaloisPageEmpty28}{\showpaste} +\tab{5}\spadcommand{(aa1 = a5) :: Boolean\free{aa1 }} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch1} +\begin{paste}{ugProblemLimitsPageFull1}{ugProblemLimitsPageEmpty1} +\pastebutton{ugProblemLimitsPageFull1}{\hidepaste} +\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)} +\indentrel{3}\begin{verbatim} + 1 + (1) - � + 2 +Type: Union(OrderedCompletion Fraction Polynomial Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty1} +\begin{paste}{ugProblemLimitsPageEmpty1}{ugProblemLimitsPagePatch1} +\pastebutton{ugProblemLimitsPageEmpty1}{\showpaste} +\tab{5}\spadcommand{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch2} +\begin{paste}{ugProblemLimitsPageFull2}{ugProblemLimitsPageEmpty2} +\pastebutton{ugProblemLimitsPageFull2}{\hidepaste} +\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")} +\indentrel{3}\begin{verbatim} + (2) 0 + Type: Union(OrderedCompletion Expression Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty2} +\begin{paste}{ugProblemLimitsPageEmpty2}{ugProblemLimitsPagePatch2} +\pastebutton{ugProblemLimitsPageEmpty2}{\showpaste} +\tab{5}\spadcommand{limit(x * log(x),x = 0,"right")} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch3} +\begin{paste}{ugProblemLimitsPageFull3}{ugProblemLimitsPageEmpty3} +\pastebutton{ugProblemLimitsPageFull3}{\hidepaste} +\tab{5}\spadcommand{limit(x * log(x),x = 0)} +\indentrel{3}\begin{verbatim} + (3) [leftHandLimit= "failed",rightHandLimit= 0] +Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty3} +\begin{paste}{ugProblemLimitsPageEmpty3}{ugProblemLimitsPagePatch3} +\pastebutton{ugProblemLimitsPageEmpty3}{\showpaste} +\tab{5}\spadcommand{limit(x * log(x),x = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch4} +\begin{paste}{ugProblemLimitsPageFull4}{ugProblemLimitsPageEmpty4} +\pastebutton{ugProblemLimitsPageFull4}{\hidepaste} +\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)} +\indentrel{3}\begin{verbatim} + (4) [leftHandLimit= - 1,rightHandLimit= 1] +Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty4} +\begin{paste}{ugProblemLimitsPageEmpty4}{ugProblemLimitsPagePatch4} +\pastebutton{ugProblemLimitsPageEmpty4}{\showpaste} +\tab{5}\spadcommand{limit(sqrt(y**2)/y,y = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch5} +\begin{paste}{ugProblemLimitsPageFull5}{ugProblemLimitsPageEmpty5} +\pastebutton{ugProblemLimitsPageFull5}{\hidepaste} +\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)} +\indentrel{3}\begin{verbatim} + 1 1 + (5) [leftHandLimit= - 컴컴,rightHandLimit= 컴컴] + 旼� 旼� + \�2 \�2 +Type: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty5} +\begin{paste}{ugProblemLimitsPageEmpty5}{ugProblemLimitsPagePatch5} +\pastebutton{ugProblemLimitsPageEmpty5}{\showpaste} +\tab{5}\spadcommand{limit(sqrt(1 - cos(t))/t,t = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch6} +\begin{paste}{ugProblemLimitsPageFull6}{ugProblemLimitsPageEmpty6} +\pastebutton{ugProblemLimitsPageFull6}{\hidepaste} +\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)} +\indentrel{3}\begin{verbatim} + 旼� + \�3 + (6) 컴컴 + 5 + Type: Union(OrderedCompletion Expression Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty6} +\begin{paste}{ugProblemLimitsPageEmpty6}{ugProblemLimitsPagePatch6} +\pastebutton{ugProblemLimitsPageEmpty6}{\showpaste} +\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch7} +\begin{paste}{ugProblemLimitsPageFull7}{ugProblemLimitsPageEmpty7} +\pastebutton{ugProblemLimitsPageFull7}{\hidepaste} +\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)} +\indentrel{3}\begin{verbatim} + 旼� + \�3 + (7) - 컴컴 + 5 + Type: Union(OrderedCompletion Expression Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty7} +\begin{paste}{ugProblemLimitsPageEmpty7}{ugProblemLimitsPagePatch7} +\pastebutton{ugProblemLimitsPageEmpty7}{\showpaste} +\tab{5}\spadcommand{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch8} +\begin{paste}{ugProblemLimitsPageFull8}{ugProblemLimitsPageEmpty8} +\pastebutton{ugProblemLimitsPageFull8}{\hidepaste} +\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)} +\indentrel{3}\begin{verbatim} + a + (8) � + b + Type: Union(OrderedCompletion Expression Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty8} +\begin{paste}{ugProblemLimitsPageEmpty8}{ugProblemLimitsPagePatch8} +\pastebutton{ugProblemLimitsPageEmpty8}{\showpaste} +\tab{5}\spadcommand{limit(sinh(a*x)/tan(b*x),x = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch9} +\begin{paste}{ugProblemLimitsPageFull9}{ugProblemLimitsPageEmpty9} +\pastebutton{ugProblemLimitsPageFull9}{\hidepaste} +\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)} +\indentrel{3}\begin{verbatim} + (9) 0 + Type: Union(OrderedCompletion Expression Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty9} +\begin{paste}{ugProblemLimitsPageEmpty9}{ugProblemLimitsPagePatch9} +\pastebutton{ugProblemLimitsPageEmpty9}{\showpaste} +\tab{5}\spadcommand{limit(z * sin(1/z),z = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch10} +\begin{paste}{ugProblemLimitsPageFull10}{ugProblemLimitsPageEmpty10} +\pastebutton{ugProblemLimitsPageFull10}{\hidepaste} +\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)} +\indentrel{3}\begin{verbatim} + (10) "failed" + Type: Union("failed",...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty10} +\begin{paste}{ugProblemLimitsPageEmpty10}{ugProblemLimitsPagePatch10} +\pastebutton{ugProblemLimitsPageEmpty10}{\showpaste} +\tab{5}\spadcommand{complexLimit(z * sin(1/z),z = 0)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch11} +\begin{paste}{ugProblemLimitsPageFull11}{ugProblemLimitsPageEmpty11} +\pastebutton{ugProblemLimitsPageFull11}{\hidepaste} +\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)} +\indentrel{3}\begin{verbatim} + (11) - 1 + Type: OnePointCompletion Fraction Polynomial Integer +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty11} +\begin{paste}{ugProblemLimitsPageEmpty11}{ugProblemLimitsPagePatch11} +\pastebutton{ugProblemLimitsPageEmpty11}{\showpaste} +\tab{5}\spadcommand{complexLimit((2 + z)/(1 - z),z = \%infinity)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch12} +\begin{paste}{ugProblemLimitsPageFull12}{ugProblemLimitsPageEmpty12} +\pastebutton{ugProblemLimitsPageFull12}{\hidepaste} +\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)} +\indentrel{3}\begin{verbatim} + (12) 0 + Type: Union(OrderedCompletion Expression Integer,...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty12} +\begin{paste}{ugProblemLimitsPageEmpty12}{ugProblemLimitsPagePatch12} +\pastebutton{ugProblemLimitsPageEmpty12}{\showpaste} +\tab{5}\spadcommand{limit(sin(x)/x,x = \%plusInfinity)} +\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPagePatch13} +\begin{paste}{ugProblemLimitsPageFull13}{ugProblemLimitsPageEmpty13} +\pastebutton{ugProblemLimitsPageFull13}{\hidepaste} +\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)} +\indentrel{3}\begin{verbatim} + (13) "failed" + Type: Union("failed",...) +\end{verbatim} +\indentrel{-3}\end{paste}\end{patch} + +\begin{patch}{ugProblemLimitsPageEmpty13} +\begin{paste}{ugProblemLimitsPageEmpty13}{ugProblemLimitsPagePatch13} +\pastebutton{ugProblemLimitsPageEmpty13}{\showpaste} +\tab{5}\spadcommand{complexLimit(sin(x)/x,x = \%infinity)} +\end{paste}\end{patch} + -- cgit v1.2.3