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mz6687

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Determinant: The system hangs...

mz6687 30 Maple 2022
determinant
September 21 2023
1 4

When I try to solve the determinant the system hangs and Maple doesn't give any result. Here $\lambda_1,2$ is not a function of 'x' and 't'.

> 

restart
> 

with(LinearAlgebra)
> 

with(plots)
> 

with(Physics)
> 

``
> 

Setup(mathematicalnotation = true)

[mathematicalnotation = true]

 (1)
> 

assume(x::real); assume(t::real); assume(`α__1`::real); assume(`α__2`::real); assume(nu::real)
> 

alias(v = v(x, t))

v

 (2)
> 

``
> 

B1 := Matrix([[exp(I*v__11)/(`λ__1`-conjugate(`λ__1`)), 0, 0, 0, exp(I*v__12)/(`λ__2`-conjugate(`λ__1`)), 0, 0, 0], [0, exp(I*v__11)/(`λ__1`-conjugate(`λ__1`)), 0, 0, 0, exp(I*v__12)/(`λ__2`-conjugate(`λ__1`)), 0, 0], [0, 0, exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`)), 0, 0, 0, exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`)), 0], [0, 0, 0, exp(-I*v__11)/(`λ__1`-conjugate(`λ__1`)), 0, 0, 0, exp(-I*v__12)/(`λ__2`-conjugate(`λ__1`))], [exp(I*v__21)/(`λ__1`-conjugate(`λ__2`)), 0, 0, 0, exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), 0, 0, 0], [0, exp(I*v__21)/(`λ__1`-conjugate(`λ__2`)), 0, 0, 0, exp(I*v__22)/(`λ__2`-conjugate(`λ__2`)), 0, 0], [0, 0, exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), 0, 0, 0, exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`)), 0], [0, 0, 0, exp(-I*v__21)/(`λ__1`-conjugate(`λ__2`)), 0, 0, 0, exp(-I*v__22)/(`λ__2`-conjugate(`λ__2`))]]); H := Matrix([[H__11, H__12, H__13, H__14, H__15, H__16, H__17, H__18], [H__12, H__11, H__14, H__13, H__16, H__15, H__18, H__17], [H__13, H__14, H__33, H__34, H__17, H__18, H__55, H__38], [H__14, H__13, H__34, H__33, H__18, H__17, H__38, H__55], [H__15, H__16, H__17, H__18, H__11, H__12, H__13, H__14], [H__16, H__15, H__18, H__17, H__12, H__11, H__14, H__13], [H__17, H__18, H__55, H__38, H__13, H__14, H__33, H__34], [H__18, H__17, H__38, H__55, H__14, H__13, H__34, H__33]]); B := H.B1; idn8 := Matrix([[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 1]])
> 

Omeg := B+idn8
> 

``
> 

vvalue := {v__11 = (conjugate(`λ__1`)-`λ__1`)*x+(4*`α__1`*(conjugate(`λ__1`)^3-`λ__1`^3)+2*`α__2`*(conjugate(`λ__1`)^2-`λ__1`^2)-8*nu*(conjugate(`λ__1`)^4-`λ__1`^4))*t, v__12 = (conjugate(`λ__1`)-`λ__2`)*x+(4*`α__1`*(conjugate(`λ__1`)^3-`λ__2`^3)+2*`α__2`*(conjugate(`λ__1`)^2-`λ__2`^2)-8*nu*(conjugate(`λ__1`)^4-`λ__2`^4))*t, v__21 = (conjugate(`λ__2`)-`λ__1`)*x+(4*`α__1`*(conjugate(`λ__2`)^3-`λ__1`^3)+2*`α__2`*(conjugate(`λ__2`)^2-`λ__1`^2)-8*nu*(conjugate(`λ__2`)^4-`λ__1`^4))*t, v__22 = (conjugate(`λ__2`)-`λ__2`)*x+(4*`α__1`*(conjugate(`λ__2`)^3-`λ__2`^3)+2*`α__2`*(conjugate(`λ__2`)^2-`λ__2`^2)-8*nu*(conjugate(`λ__2`)^4-`λ__2`^4))*t}
> 

B2 := Determinant(B)
> 

Omegdet := Determinant(Omeg)
> 

NULL
NULL

Download determinant.mw

Why is Maple not calculating 'sqrt' ?...

mz6687 30 Maple 2022
sqrt
September 03 2023
1 0

Why is Maple not calculating 'sqrt' and continuously showing 'Evaluating'?


 

> 

restart
> 

with(LinearAlgebra); with(PDEtools); with(DifferentialGeometry)
> 

with(plots)
> 

with(Physics)
> 

q := (31.00000000*exp(-4.976*t+2.*x)*exp((-2.488+.8336000001*I)*t+(1.+.2*I)*x)-3.000000000*exp(-4.976*t+2.*x)*exp((2.488+.8336000001*I)*t+(-1.+.2*I)*x)+(94.0*I)*exp(I*(.8336*t+.2000*x))*exp(-4.976*t+2.*x))/((11.50000000*I)*exp(-2.488*t+x)+132.44*exp(-4.976*t+2.*x)-(104.5000000*I)*exp(-7.464*t+3.*x)-5.25*exp(-9.952*t+4.*x)+.25)

(31.00000000*exp(-4.976*t+2.*x)*exp((-2.488+.8336000001*I)*t+(1.+.2*I)*x)-3.000000000*exp(-4.976*t+2.*x)*exp((2.488+.8336000001*I)*t+(-1.+.2*I)*x)+(94.0*I)*exp(I*(.8336*t+.2000*x))*exp(-4.976*t+2.*x))/((11.50000000*I)*exp(-2.488*t+x)+132.44*exp(-4.976*t+2.*x)-(104.5000000*I)*exp(-7.464*t+3.*x)-5.25*exp(-9.952*t+4.*x)+.25)

 (1)
> 

assume(x::real); assume(t::real)
> 

q1 := simplify(subs({I = -I}, q))

(-(94.*I)*exp(-I*(.8336*t+.2000*x))+31.*exp((-2.488-.8336000001*I)*t+(1.-.2*I)*x)-3.*exp((2.488-.8336000001*I)*t+(-1.-.2*I)*x))*exp(-4.976*t+2.*x)/(-(11.50000000*I)*exp(-2.488*t+x)+132.44*exp(-4.976*t+2.*x)+(104.5000000*I)*exp(-7.464*t+3.*x)-5.25*exp(-9.952*t+4.*x)+.25)

 (2)
> 

q2 := simplify(sqrt(q*q1))
> 

NULL

NULL


 

Download q_sqrt.mw

Error, invalid if statement termination...

mz6687 30 Maple 2022
optimization syntax if-then
February 06 2023
2 5

Why there is an error? Is any part missing in the if statement?

> 

restart
> 

rel := w+(1/2)*sqrt(960*c^6*gamma^2*k^2-336*c^4*gamma^2*k^4+64*c^2*gamma^2*k^6-4*gamma^2*k^8+576*alpha*c^6*gamma*k-288*alpha*c^4*gamma*k^3-16*alpha*c^2*gamma*k^5+8*alpha*gamma*k^7-144*alpha^2*c^4*k^2-48*alpha^2*c^2*k^4-4*alpha^2*k^6+128*c^4*gamma*k^2-40*c^2*gamma*k^4+4*gamma*k^6+48*alpha*c^4*k-16*alpha*c^2*k^3-4*alpha*k^5+4*c^2*k^2-k^4)

w+(1/2)*(960*c^6*gamma^2*k^2-336*c^4*gamma^2*k^4+64*c^2*gamma^2*k^6-4*gamma^2*k^8+576*alpha*c^6*gamma*k-288*alpha*c^4*gamma*k^3-16*alpha*c^2*gamma*k^5+8*alpha*gamma*k^7-144*alpha^2*c^4*k^2-48*alpha^2*c^2*k^4-4*alpha^2*k^6+128*c^4*gamma*k^2-40*c^2*gamma*k^4+4*gamma*k^6+48*alpha*c^4*k-16*alpha*c^2*k^3-4*alpha*k^5+4*c^2*k^2-k^4)^(1/2)

 (1)
> 

crit_points := solve(diff(rel, k) = 0, k); rel_at_crit := subs(k = crit_points, rel)
> 

"#` determine the maxima and minima by checking the sign of the second derivative at the critical points` for i from 1 to nops(crit_points) do if (eval(diff(diff(rel, k), k), k = crit_points[i]) > 0) then print("Minimum at k = ", crit_points[i], " with value ", rel_at_crit[i]); else if (eval(diff(diff(rel, k), k), k = crit_points[i]) < 0) then print("Maximum at k = ", crit_points[i], " with value ", rel_at_crit[i]); else print("Saddle point at k = ", crit_points[i], " with value ", rel_at_crit[i]); end if; end do; "

Error, invalid if statement termination

"#` determine the maxima and minima by checking the sign of the second derivative at the critical points` for i from 1 to nops(crit_points) do if (eval(diff(diff(rel, k), k), k = crit_points[i]) > 0) then print("Minimum at k = ", crit_points[i], " with value ", rel_at_crit[i]); else if (eval(diff(diff(rel, k), k), k = crit_points[i]) < 0) then print("Maximum at k = ", crit_points[i], " with value ", rel_at_crit[i]); else print("Saddle point at k = ", crit_points[i], " with value ", rel_at_crit[i]); end if; end do; "

  

NULL

Download maxnmin.mw

How do I find maximum value of k?...

mz6687 30 Maple 2022
optimization solve differentiation
January 11 2023
1 3

How do I find the maximum value of k by putting dw/dk = 0? Also, how to find the range of k for which w is real? 

> 

restart
> 

w = -(1/2)*sqrt(960*c^6*gamma^2*k^2-336*c^4*gamma^2*k^4+64*c^2*gamma^2*k^6-4*gamma^2*k^8+576*alpha*c^6*gamma*k-288*alpha*c^4*gamma*k^3-16*alpha*c^2*gamma*k^5+8*alpha*gamma*k^7-144*alpha^2*c^4*k^2-48*alpha^2*c^2*k^4-4*alpha^2*k^6+128*c^4*gamma*k^2-40*c^2*gamma*k^4+4*gamma*k^6+48*alpha*c^4*k-16*alpha*c^2*k^3-4*alpha*k^5+4*c^2*k^2-k^4)

w = -(1/2)*(960*c^6*gamma^2*k^2-336*c^4*gamma^2*k^4+64*c^2*gamma^2*k^6-4*gamma^2*k^8+576*alpha*c^6*gamma*k-288*alpha*c^4*gamma*k^3-16*alpha*c^2*gamma*k^5+8*alpha*gamma*k^7-144*alpha^2*c^4*k^2-48*alpha^2*c^2*k^4-4*alpha^2*k^6+128*c^4*gamma*k^2-40*c^2*gamma*k^4+4*gamma*k^6+48*alpha*c^4*k-16*alpha*c^2*k^3-4*alpha*k^5+4*c^2*k^2-k^4)^(1/2)

 (1)
> 

diff(w = -(1/2)*(960*c^6*gamma^2*k^2-336*c^4*gamma^2*k^4+64*c^2*gamma^2*k^6-4*gamma^2*k^8+576*alpha*c^6*gamma*k-288*alpha*c^4*gamma*k^3-16*alpha*c^2*gamma*k^5+8*alpha*gamma*k^7-144*alpha^2*c^4*k^2-48*alpha^2*c^2*k^4-4*alpha^2*k^6+128*c^4*gamma*k^2-40*c^2*gamma*k^4+4*gamma*k^6+48*alpha*c^4*k-16*alpha*c^2*k^3-4*alpha*k^5+4*c^2*k^2-k^4)^(1/2), k)

0 = -(1/4)*(1920*c^6*gamma^2*k-1344*c^4*gamma^2*k^3+384*c^2*gamma^2*k^5-32*gamma^2*k^7+576*alpha*c^6*gamma-864*alpha*c^4*gamma*k^2-80*alpha*c^2*gamma*k^4+56*alpha*gamma*k^6-288*alpha^2*c^4*k-192*alpha^2*c^2*k^3-24*alpha^2*k^5+256*c^4*gamma*k-160*c^2*gamma*k^3+24*gamma*k^5+48*alpha*c^4-48*alpha*c^2*k^2-20*alpha*k^4+8*c^2*k-4*k^3)/(960*c^6*gamma^2*k^2-336*c^4*gamma^2*k^4+64*c^2*gamma^2*k^6-4*gamma^2*k^8+576*alpha*c^6*gamma*k-288*alpha*c^4*gamma*k^3-16*alpha*c^2*gamma*k^5+8*alpha*gamma*k^7-144*alpha^2*c^4*k^2-48*alpha^2*c^2*k^4-4*alpha^2*k^6+128*c^4*gamma*k^2-40*c^2*gamma*k^4+4*gamma*k^6+48*alpha*c^4*k-16*alpha*c^2*k^3-4*alpha*k^5+4*c^2*k^2-k^4)^(1/2)

 (2)
> 

NULL

NULL

Download drelation.mw

possible transformations...

mz6687 30 Maple 2022
pde differential-equations invariant
December 14 2022
0 0

How can we unveil all the possible transformations that reduce the number of independent variables of a given pde? I tried it by using "InvarientSolutions" which gives eight possible transformations. But under those transformations, I got only two different odes. Is there any other method/command from which we can get other possible odes of a given pde?

 

pde_ode.mw

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