http://en.wikipedia.org/wiki/Twisted_cubic

1.

How to find S^3 : S^2*T : S*T^2 : T^3

2.

How to find X*Z-Y^2, Y*W-Z^2, X*W - Y*Z

Real application running a very long time in elimination step

z1 := Diff(x1(t),t) = x2(t);

z2 := Diff(x2(t),t) = x3(t);

z3 := Diff(x3(t),t) = x4(t);

z4 := Diff(x4(t),t) = -1*(1.0000*x4(t)-.5109*x3(t)-0.595e-1*x2(t)+3.1086*10^(-15)*(Diff(u(t), t, t, t))-(Diff(u(t), t, t))+.5109*(Diff(u(t), t))+0.595e-1*u(t));

with(Involutive):

with(OreModules):

Alg := DefineOreAlgebra(diff=[D,t], polynom=[t], comm=[a1,a2,a3,u]):

DD := evalm([[D, -1, 0, 0, 0],[0, D, -1, 0, 0],[0, 0, D, -1, 0],[0, -0.0595, -0.5109, D+1, (3.1086*10^(-15))*D^3-D^2+0.5109*D+0.0595]]);

DDtilde := Involution(DD,Alg);

ApplyMatrix(DDtilde, [lambda1(t),lambda2(t),lambda3(t),lambda4(t)],Alg)=evalm([[miu1(t)],[miu2(t)],[miu3(t)],[miu4(t)],[miu(t)]]);

E := Elimination(DDtilde, [lambda1,lambda2,lambda3,lambda4],[miu1,miu2,miu3,miu4,miu],Alg);

Diff(x1(t),t) - x2(t) = 0;

Diff(x2(t),t) - x3(t) = 0;

Diff(x3(t),t) +3*x3(t) + 4*x2(t) + x1(t) = 2*Diff(u(t),t$2) + 5*Diff(u(t),t) + 7*u(t);

multiply above system by test functions -lambda1(t),-lambda2(t),-lambda3(t) and integrate by part

in order to find adjoint operator in the form

x1 -> Diff(lambda1(t),t) - lambda3(t) = miu1(t)

x2 -> Diff(lambda2(t),t) + lambda1(t) - 4*lambda3(t) = miu2(t)

Question 1

a := miu1 = Diff(lambda1(t),t) + lambda3(t);

b := miu2 = Diff(lambda2(t),t) + lambda1(t) - 4*lambda3(t);

c := miu3 = Diff(lambda3(t),t) + lambda2(t) - 3*lambda3(t);

d := miu = 2*Diff(lambda3(t),t$2) - 5*Diff(lambda3(t),t) + 7*lambda3(t);

how to eliminate all derivatives of lambda in miu

result in

miu - 2*Diff(miu3,t) - miu3 + 2*miu2 = 2*lambda1 - lambda2 + 2*lambda3;

sys := [1.8*a+.4000000000*b+103.9*c-.5000000000*e-102.6*g = 0, .5000000000*b+102.6*c+.3*d+.3500000000*e-102.1*g = 0, -3.3*a-1.050000000*b+102.4*c+.7*d+.2500000000*e-105.1*g = 0, -2.5*a+1.450000000*b+102.6*c+3.3*d+1.050000000*e-102.4*g = 0, -1.6*a+1.100000000*b+105.8*c+.3*d+1.750000000*e-105.5*g = 0, -.7*a-.2500000000*b+105.1*c-.2*d+.2000000000*e-105.7*g = 0, -.3*a-.3500000000*b+102.1*c+2.5*d-1.450000000*e-102.6*g = 0, .2*a-.2000000000*b+105.7*c+1.6*d-1.100000000*e-105.8*g = 0...