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mskalsi

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9 years, 190 days
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Cool !!...

mskalsi 290
June 26 2016
0 1

@Kitonum 

I was not aware of user lighting, it is wonderful

Typo...

mskalsi 290
June 20 2016
0 2

@John Fredsted 

I am sorry for typo, read this as

e1 -> -2e3, e2 -> e2, e3 -> e1

also check on link I have given you.

Explicit isomorphism...

mskalsi 290
June 20 2016
0 0

@John Fredsted 

The example of Lie algebras that I have given you are really isomorphic through transformations:

e1 -> - e3, e1-> e2 and e3 -> e2

see query on stack exchange regarding isomorphism among Lie algebra where isomorphism is preserving Lie bracket.

http://math.stackexchange.com/questions/1815341/how-many-three-and-four-dimensional-lie-algebras-are-there

Example...

mskalsi 290
June 18 2016
0 0

@John Fredsted 

For If I consider following case


> 

with(DifferentialGeometry):
> 

L1 := LieAlgebraData([[e1, e2] = e2, [e1, e3] = 2*e3], [e1, e2, e3], alg1);

_DG([["LieAlgebra", alg1, [3, table( [ ] )]], [[[1, 2, 2], 1], [[1, 3, 3], 2]]])

 (1)
> 

L2 := LieAlgebraData([[e1, e3] = e1, [e2, e3] = (1/2)*e2], [e1, e2, e3], alg2);

_DG([["LieAlgebra", alg2, [3, table( [ ] )]], [[[1, 3, 1], 1], [[2, 3, 2], 1/2]]])

 (2)
> 

DGsetup(L1);

`Lie algebra: alg1`

 (3)
alg1 > 

DGsetup(L2);

`Lie algebra: alg2`

 (4)
alg2 > 

A := Matrix(3, symbol = a)

A := Matrix(3, 3, {(1, 1) = a[1, 1], (1, 2) = a[1, 2], (1, 3) = a[1, 3], (2, 1) = a[2, 1], (2, 2) = a[2, 2], (2, 3) = a[2, 3], (3, 1) = a[3, 1], (3, 2) = a[3, 2], (3, 3) = a[3, 3]})

 (5)
alg2 > 

Query(alg1, alg2, A, indets(A), "Homomorphism")[3];

[{a[1, 1] = a[1, 1], a[1, 2] = a[1, 2], a[1, 3] = 0, a[2, 1] = a[2, 1], a[2, 2] = 0, a[2, 3] = 0, a[3, 1] = -1, a[3, 2] = 0, a[3, 3] = 0}, {a[1, 1] = a[1, 1], a[1, 2] = 0, a[1, 3] = a[1, 3], a[2, 1] = a[2, 1], a[2, 2] = a[2, 2], a[2, 3] = 0, a[3, 1] = -2, a[3, 2] = 0, a[3, 3] = 0}, {a[1, 1] = a[1, 1], a[1, 2] = 0, a[1, 3] = 0, a[2, 1] = a[2, 1], a[2, 2] = 0, a[2, 3] = 0, a[3, 1] = a[3, 1], a[3, 2] = 0, a[3, 3] = 0}, {a[1, 1] = a[1, 1], a[1, 2] = 0, a[1, 3] = 0, a[2, 1] = a[2, 1], a[2, 2] = 0, a[2, 3] = a[2, 3], a[3, 1] = -4, a[3, 2] = 0, a[3, 3] = 0}]

 (6)
alg2 > 

``


Download Homomorphism.mw

Atleats one should be non-singular ?...

mskalsi 290
June 18 2016
0 0

@John Fredsted 

I can see that all three matrices are singular, suppose if there were alteast one non-singular matrix then can we assume the existence of homomorphism ?

Exactly !!...

mskalsi 290
May 31 2016
0 0

@Thomas Richard This is exactly what I was wanting. Thanks

Wonderful....!!!!...

mskalsi 290
May 31 2016
0 0

@Carl Love This iis really amazing solution to my problem. I can manage without equation number, instead I would try to assign a name to equation. Are you aready using this feature anyway ?

Upload your Maple Code !!...

mskalsi 290
May 28 2016
0 0

You should try uploading Maple code while asking question, there is big green upward arrow, click and insert content. No need of Latex.

Regards

I was also thinking so !...

mskalsi 290
May 28 2016
0 0

@Carl Love 

I took this code from Maple help and it worked as desired. This also work for first example too, but now constant become functions of xi and eta, see below:


> 

with(PDEtools):
> 

DepVars := [F(xi), G(eta), u(x, t)];

[F(xi), G(eta), u(x, t)]

 (1)
> 

alias(F = F(xi), G = G(eta), u = u(x, t));

F, G, u

 (2)
> 

declare(F, G(xi), u)

F(xi)*`will now be displayed as`*F

  

u(x, t)*`will now be displayed as`*u

  

G(xi)*`will now be displayed as`*G

 (3)
> 

dchange({t = (eta*k[1]-xi*k[2])/(k[1]*k[2]*(c[1]-c[2])), x = (eta*c[1]*k[1]-xi*c[2]*k[2])/(k[1]*k[2]*(c[1]-c[2]))}, Diff(u, t), [xi, eta], params = [c[1], c[2], k[1], k[2]])

-c[1]*k[1]*(diff(u(xi, eta), xi))-c[2]*k[2]*(diff(u(xi, eta), eta))

 (4)
> 

u := a[0]+(F*a[1]+G*a[2]+kappa[1])/(mu[0]+mu[1]*(diff(F, xi))+mu[2]*(diff(G, eta)))

a[0]+(a[1]*F+a[2]*G+kappa[1])/(mu[0]+mu[1]*(diff(F, xi))+mu[2]*(diff(G, eta)))

 (5)
> 

-c[1]*k[1]*(diff(u(xi, eta), xi))-c[2]*k[2]*(diff(u(xi, eta), eta))

-c[1]*k[1]*(diff(a[0](xi, eta), xi)+((diff(a[1](xi, eta), xi))*F(xi, eta)+a[1](xi, eta)*(diff(F(xi, eta), xi))+(diff(a[2](xi, eta), xi))*G(xi, eta)+a[2](xi, eta)*(diff(G(xi, eta), xi))+diff(kappa[1](xi, eta), xi))/(mu[0](xi, eta)+mu[1](xi, eta)*(diff(F, xi))(xi, eta)+mu[2](xi, eta)*(diff(G, eta))(xi, eta))-(a[1](xi, eta)*F(xi, eta)+a[2](xi, eta)*G(xi, eta)+kappa[1](xi, eta))*(diff(mu[0](xi, eta), xi)+(diff(mu[1](xi, eta), xi))*(diff(F, xi))(xi, eta)+mu[1](xi, eta)*(diff((diff(F, xi))(xi, eta), xi))+(diff(mu[2](xi, eta), xi))*(diff(G, eta))(xi, eta)+mu[2](xi, eta)*(diff((diff(G, eta))(xi, eta), xi)))/(mu[0](xi, eta)+mu[1](xi, eta)*(diff(F, xi))(xi, eta)+mu[2](xi, eta)*(diff(G, eta))(xi, eta))^2)-c[2]*k[2]*(diff(a[0](xi, eta), eta)+((diff(a[1](xi, eta), eta))*F(xi, eta)+a[1](xi, eta)*(diff(F(xi, eta), eta))+(diff(a[2](xi, eta), eta))*G(xi, eta)+a[2](xi, eta)*(diff(G(xi, eta), eta))+diff(kappa[1](xi, eta), eta))/(mu[0](xi, eta)+mu[1](xi, eta)*(diff(F, xi))(xi, eta)+mu[2](xi, eta)*(diff(G, eta))(xi, eta))-(a[1](xi, eta)*F(xi, eta)+a[2](xi, eta)*G(xi, eta)+kappa[1](xi, eta))*(diff(mu[0](xi, eta), eta)+(diff(mu[1](xi, eta), eta))*(diff(F, xi))(xi, eta)+mu[1](xi, eta)*(diff((diff(F, xi))(xi, eta), eta))+(diff(mu[2](xi, eta), eta))*(diff(G, eta))(xi, eta)+mu[2](xi, eta)*(diff((diff(G, eta))(xi, eta), eta)))/(mu[0](xi, eta)+mu[1](xi, eta)*(diff(F, xi))(xi, eta)+mu[2](xi, eta)*(diff(G, eta))(xi, eta))^2)

 (6)

This looks fine,  but constants a[0], a[1], a[2], mu[0], mu[1], mu[2]are now become functions of xiand eta


Download First_example_using_dchange.mw

No need of Value...

mskalsi 290
May 27 2016
0 0

@Carl Love 

I have tried dchange in following manner:

 

> 

with(PDEtools):
> 

DepVars := [F(xi), G(eta), u(x, t)]

[F(xi), G(eta), u(x, t)]

 (1)
> 

alias(F = F(xi), G = G(eta), u = u(x, t))

F, G, u

 (2)
> 

declare(F, G(xi), u)

F(xi)*`will now be displayed as`*F

  

u(x, t)*`will now be displayed as`*u

  

G(xi)*`will now be displayed as`*G

 (3)
> 

dchange({t = (eta*k[1]-xi*k[2])/(k[1]*k[2]*(c[1]-c[2])), x = (eta*c[1]*k[1]-xi*c[2]*k[2])/(k[1]*k[2]*(c[1]-c[2]))}, Diff(u, t)+zeta*u*(Diff(u, x))+rho*(Diff(u, x, x, x)), [xi, eta], params = [c[1], c[2], k[1], k[2]])

-c[1]*k[1]*(diff(u(xi, eta), xi))-c[2]*k[2]*(diff(u(xi, eta), eta))+zeta*u(xi, eta)*(k[1]*(diff(u(xi, eta), xi))+k[2]*(diff(u(xi, eta), eta)))+rho*(k[1]*(k[1]*(k[1]*(diff(diff(diff(u(xi, eta), xi), xi), xi))+k[2]*(diff(diff(diff(u(xi, eta), eta), xi), xi)))+k[2]*(k[1]*(diff(diff(diff(u(xi, eta), eta), xi), xi))+k[2]*(diff(diff(diff(u(xi, eta), eta), eta), xi))))+k[2]*(k[1]*(k[1]*(diff(diff(diff(u(xi, eta), eta), xi), xi))+k[2]*(diff(diff(diff(u(xi, eta), eta), eta), xi)))+k[2]*(k[1]*(diff(diff(diff(u(xi, eta), eta), eta), xi))+k[2]*(diff(diff(diff(u(xi, eta), eta), eta), eta)))))

 (4)
> 

u := a[0]+(F*a[1]+G*a[2]+kappa[1])/(mu[0]+mu[1]*(diff(F, xi))+mu[2]*(diff(G, eta)))+(F^2*a[3]+F*G*a[4]+G^2*a[5]+kappa[2])/(mu[0]+mu[1]*(diff(F, xi))+mu[2]*(diff(G, eta)))^2

a[0]+(a[1]*F+a[2]*G+kappa[1])/(mu[0]+mu[1]*(diff(F, xi))+mu[2]*(diff(G, eta)))+(a[3]*F^2+a[4]*F*G+a[5]*G^2+kappa[2])/(mu[0]+mu[1]*(diff(F, xi))+mu[2]*(diff(G, eta)))^2

 (5)
> 

``

Download Use_of_Dchange.mw

Please ignore equation (1)...

mskalsi 290
May 26 2016
0 0

The equation (1) is accidently left after deletion, please ignore it.

Regards

Anyway, it worked as desired...

mskalsi 290
May 12 2016
0 0

@Carl Love 

ohh....!!!!...

mskalsi 290
May 12 2016
0 0

@Carl Love 

But, that what I was expecting from operator to operate even on expression.

This worked as desired but not completel...

mskalsi 290
May 12 2016
0 0

@Carl Love 

Please see following:


> 

with(PDEtools):
> 

DepVars = [f(x, y, t, u, v)]; 1; alias(f = f(x, y, t, u, v)); 1; declare(f(x, y, t, u, v))

DepVars = [f(x, y, t, u, v)]

  

f

  

f(x, y, t, u, v)*`will now be displayed as`*f

 (1)
> 

V[1] := t*D[2]+x*D[5];

t*D[2]+x*D[5]

 (2)
> 

f := y+v

y+v

 (3)
> 

V[1](f)

t(y+v)*(D[2](y)+D[2](v))+x(y+v)*(D[5](y)+D[5](v))

 (4)

I was expecting this to be like t+x


Download operator2.mw

Yes,...

mskalsi 290
May 09 2016
0 0

@Carl Love 

The files are in E drive

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