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Dear All

Using Lie algebra package in Maple we can easily find nilradical for given abstract algebra, but how we can find all the ideal in lower central series by taking new basis as nilradical itself?

Please see following;

 

with(DifferentialGeometry); with(LieAlgebras)

DGsetup([x, y, t, u, v])

`frame name: Euc`

(1)
Euc > 

VectorFields := evalDG([D_v, D_v*x+D_y*t, 2*D_t*t-2*D_u*u-D_v*v+D_y*y, t*D_v, D_v*y+D_u, D_t, D_x, D_x*t+D_u, 2*D_v*x+D_x*y, -D_t*t+2*D_u*u+2*D_v*v+D_x*x, D_y])

[_DG([["vector", "Euc", []], [[[5], 1]]]), _DG([["vector", "Euc", []], [[[2], t], [[5], x]]]), _DG([["vector", "Euc", []], [[[2], y], [[3], 2*t], [[4], -2*u], [[5], -v]]]), _DG([["vector", "Euc", []], [[[5], t]]]), _DG([["vector", "Euc", []], [[[4], 1], [[5], y]]]), _DG([["vector", "Euc", []], [[[3], 1]]]), _DG([["vector", "Euc", []], [[[1], 1]]]), _DG([["vector", "Euc", []], [[[1], t], [[4], 1]]]), _DG([["vector", "Euc", []], [[[1], y], [[5], 2*x]]]), _DG([["vector", "Euc", []], [[[1], x], [[3], -t], [[4], 2*u], [[5], 2*v]]]), _DG([["vector", "Euc", []], [[[2], 1]]])]

(2)
Euc > 

L1 := LieAlgebraData(VectorFields)

_DG([["LieAlgebra", "L1", [11]], [[[1, 3, 1], -1], [[1, 10, 1], 2], [[2, 3, 2], -1], [[2, 5, 4], 1], [[2, 6, 11], -1], [[2, 7, 1], -1], [[2, 8, 4], -1], [[2, 9, 5], -1], [[2, 9, 8], 1], [[2, 10, 2], 1], [[3, 4, 4], 3], [[3, 5, 5], 2], [[3, 6, 6], -2], [[3, 8, 8], 2], [[3, 9, 9], 1], [[3, 11, 11], -1], [[4, 6, 1], -1], [[4, 10, 4], 3], [[5, 10, 5], 2], [[5, 11, 1], -1], [[6, 8, 7], 1], [[6, 10, 6], -1], [[7, 9, 1], 2], [[7, 10, 7], 1], [[8, 9, 4], 2], [[8, 10, 8], 2], [[9, 10, 9], 1], [[9, 11, 7], -1]]])

(3)
Euc > 

DGsetup(L1)

`Lie algebra: L1`

(4)
L1 > 

MultiplicationTable("LieTable"):

L1 > 

N := Nilradical(L1)

[_DG([["vector", "L1", []], [[[1], 1]]]), _DG([["vector", "L1", []], [[[2], 1]]]), _DG([["vector", "L1", []], [[[4], 1]]]), _DG([["vector", "L1", []], [[[5], 1]]]), _DG([["vector", "L1", []], [[[6], 1]]]), _DG([["vector", "L1", []], [[[7], 1]]]), _DG([["vector", "L1", []], [[[8], 1]]]), _DG([["vector", "L1", []], [[[9], 1]]]), _DG([["vector", "L1", []], [[[11], 1]]])]

(5)
L1 > 

Query(N, "Nilpotent")

true

(6)
L1 > 

Query(N, "Solvable")

true

(7)

Taking N as new basis , how we can find all ideals in lower central series of this solvable ideal N?

 

Download [944]_Structure_of_Lie_algebra.mw

Regards

In this question, I asked for a way to simplify an expression containing radicals. The discussion led us to that as default field for simplicfication is the Complex number system we should use assume or assuming command to simplify the radicals. However, the mothod suggested there seems to not work in this new case that I have. For details please see the attached file. The terms sqrt{u} and sqrt{u-1} should cancel in denominator.

 What Maple Does

restart

`ϕ` := (1+sqrt(5))*(1/2)

1/2+(1/2)*5^(1/2)

(1)

f := (1/2)*sqrt(-(u-1)*(u+1)*(u^2-u-1))*u*(4*u-3)/sqrt(u*(u-1))

(1/2)*(-(u-1)*(u+1)*(u^2-u-1))^(1/2)*u*(4*u-3)/(u*(u-1))^(1/2)

(2)

`assuming`([combine(f)], [1 < u and u < `&varphi;`])

(1/2)*u*(4*u-3)*((u+1)*(-u^2+u+1)/u)^(1/2)

(3)

`assuming`([simplify(f)], [1 < u and u < `&varphi;`])

(1/2)*(-u^2+u+1)^(1/2)*(u^2-1)^(1/2)*u^(1/2)*(4*u-3)/(u-1)^(1/2)

(4)

`assuming`([combine(f, radical)], [1 < u and u < `&varphi;`])

(1/2)*u*(4*u-3)*((u+1)*(-u^2+u+1)/u)^(1/2)

(5)

`assuming`([simplify(f, radical)], [1 < u and u < `&varphi;`])

(1/2)*((u-1)*(u+1)*(-u^2+u+1))^(1/2)*u*(4*u-3)/(u*(u-1))^(1/2)

(6)

``

Radical.mw

 Remark by Markiyan Hirnyk. The below content is added by the questionner on 08.02.2016 .

What Mathematica Does

 

I have the following expression

f=u/(sqrt(u*(u-1)))

and I want to simplify it. Eventhough that I tell Maple that u is real and greater than 1 but it does not simplify the expression. What is wrong? Please see the attached file.

Radical.mw

I want to cancel some expressions in numerator and denominator of a quotient. But Maple deos not cancel it!

Please see the attached file.


Simplifying_Radicals.mw

In the running of an example I faced to computation of radical ideal of the following ideal:

<-c*m*u+d*c*n+m*b*v+m*c*t>

 

I used from Radical command in PolynomialIdeals package. But I dno't now why it's computation is very hard and Time-consuming?

What I have to do? I think there is a bug, since this ideal is simple, apparently.

why the the software can't plot the function like x^(4/3)*sin(1/x) or x^(1/3)

it could only plot where x>0,but the value is does exist where x<0.

Thanks in advance for your help.

Which of Maple commands is better for computing the radical of a polynomial f (less time of computation and less complexity)?

For example if f=x^3+3x^2y+3xy^2+y^3 then rad(f)=x+y.

is there a built in function which calculates the radical of a number?

e.g. 54=2*3^3

rad(54)=2*3=6

Hello all. I have a probably simple problem, that is drinving me totally crazy. I have a expression having:

 (sqrt(M+m)*sqrt(M-m).(sqrt(M+m)*sqrt(M-m))

 

and it refuses to simplify. I told maple assume(m>0, M>0), etc. But it doesn't want to simplify. Anyone has any ideas?

 

Regards,

Jelmew

 

Edit: uploaded the mw file, so it can be seen what the problem is.10.1.mw10.1.mw

[I added Physics to the tags.--Carl Love as moderator]

The simplification of 1/sqrt(2) is always simplified or unsimplified as the case may be to sqrt(2)/2.  It is a matter of opinion which is simpler I suppose, but throughout mathematics teachings I've always learned cos(45) as 1/sqrt(2) as I'm sure the rest of you all have as well. Yes it is merely aesthetic, but a quirk to see it as sqrt(2)/2

Is the simplification process to get radicals in the numerator rather than the denominator?  I think yes if the answer...

Hi,

with great interest and surprise I read the post
"Converting Half-Angle Trig Formulas to Radicals".
Isnt it possible to evaluate cos(arccos(13/14)/3) also
to an exact expression in radicals ?
I simply do not succeed with my humble knowledge of
the Maple commands/internal workings...
Would be great if someone finds a solution ( of an 
unsolvable problem ??? ).

[Edit: Excess white space deleted.---Carl Love]

Hello,

I am experiencing wrong results using the PolynomialIdeals package in Maple 16 (see radical_error.mw). Creating the same ideal with different generators, the radicals computed differ which of course is wrong:

with(PolynomialIdeals);
J := <t*(a+A), A*(b+B+t), b*(a+A), B*a-A*(b+t), (variables = {A, B, a, b, t})>:
J2 := <t*(a+A), A*(b+B+t), b*(a+A), B*(a+A...

Maybe many questions on forum because method is not universal. What about this one:

JA1S := (2*sqrt(2*y+3)*y+3*sqrt(2*y+3)-3*sqrt(3))/((2*y+3)^(3/2)*y);

 

simplify(JA1S, power, radical, symbolic); - nothing

combine(JA1S, power, radical, symbolic); - nothing

 

Moreover, can i somehow also reduce by y?

GAL := 3*sqrt(6)/((2*x^2+3)*sqrt(4*x^2+6));

simplify(GAL) and combine(GAL,power)

have same result.

How to make them return

3*sqrt(3)/(2*x^2+3)^(3/2) ?

Let a:= (2x-1)/sqrt(1-4x2)  , b:=sqrt(1-2x)/sqrt(1+2x)

then c:=combine(a/b,radical,symbolic) is 1, it should be -1 !

(Multiply b top and bottom by sqrt(1-2x) to see this).

 

Am I misunderstanding combine?

 

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