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Dear All

I intend to convert Lie algebra into abstract Lie algebra with the aid of Maple packages DifferentialGeometry and Lie algebras. I want to do this for classification of Lie algebra for group inavriant solution, can anybody help me out.....?

Regards

Dear All

Is there anybody who is working on contruction of optimal Lie algebra using Maple packages like DifferentialGeometry and LieAlgebra, I tried to find commands for constructing algebra in these packages but could not find such commands. I am sure these are only package that might help me. Following is Lie algebra whose optimal system is required:

 

with(PDEtools, SymmetryCommutator, InfinitesimalGenerator):

S[1], S[2], S[3], S[4], S[5], S[6], S[7], S[8], S[9], S[10], S[11] := [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 1], [_xi[x] = 0, _xi[y] = t, _xi[t] = 0, _eta[u] = 0, _eta[v] = x], [_xi[x] = 0, _xi[y] = y, _xi[t] = 2*t, _eta[u] = -2*u, _eta[v] = -v], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = t], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 1, _eta[v] = y], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 1, _eta[u] = 0, _eta[v] = 0], [_xi[x] = 1, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 0], [_xi[x] = t, _xi[y] = 0, _xi[t] = 0, _eta[u] = 1, _eta[v] = 0], [_xi[x] = y, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 2*x], [_xi[x] = x, _xi[y] = 0, _xi[t] = -t, _eta[u] = 2*u, _eta[v] = 2*v], [_xi[x] = 0, _xi[y] = 1, _xi[t] = 0, _eta[u] = 0, _eta[v] = 0]

[_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 1], [_xi[x] = 0, _xi[y] = t, _xi[t] = 0, _eta[u] = 0, _eta[v] = x], [_xi[x] = 0, _xi[y] = y, _xi[t] = 2*t, _eta[u] = -2*u, _eta[v] = -v], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = t], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 0, _eta[u] = 1, _eta[v] = y], [_xi[x] = 0, _xi[y] = 0, _xi[t] = 1, _eta[u] = 0, _eta[v] = 0], [_xi[x] = 1, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 0], [_xi[x] = t, _xi[y] = 0, _xi[t] = 0, _eta[u] = 1, _eta[v] = 0], [_xi[x] = y, _xi[y] = 0, _xi[t] = 0, _eta[u] = 0, _eta[v] = 2*x], [_xi[x] = x, _xi[y] = 0, _xi[t] = -t, _eta[u] = 2*u, _eta[v] = 2*v], [_xi[x] = 0, _xi[y] = 1, _xi[t] = 0, _eta[u] = 0, _eta[v] = 0]

(1)

G[1] := InfinitesimalGenerator(S[1], [u(x, y, t), v(x, y, t)]); 1; G[2] := InfinitesimalGenerator(S[2], [u(x, y, t), v(x, y, t)]); 1; G[3] := InfinitesimalGenerator(S[3], [u(x, y, t), v(x, y, t)]); 1; G[4] := InfinitesimalGenerator(S[4], [u(x, y, t), v(x, y, t)]); 1; G[5] := InfinitesimalGenerator(S[5], [u(x, y, t), v(x, y, t)]); 1; G[6] := InfinitesimalGenerator(S[6], [u(x, y, t), v(x, y, t)]); 1; G[7] := InfinitesimalGenerator(S[7], [u(x, y, t), v(x, y, t)]); 1; G[8] := InfinitesimalGenerator(S[8], [u(x, y, t), v(x, y, t)]); 1; G[9] := InfinitesimalGenerator(S[9], [u(x, y, t), v(x, y, t)]); 1; G[10] := InfinitesimalGenerator(S[10], [u(x, y, t), v(x, y, t)]); 1; G[11] := InfinitesimalGenerator(S[11], [u(x, y, t), v(x, y, t)])

proc (f) options operator, arrow; diff(f, v) end proc

 

proc (f) options operator, arrow; t*(diff(f, y))+x*(diff(f, v)) end proc

 

proc (f) options operator, arrow; y*(diff(f, y))+2*t*(diff(f, t))-2*u*(diff(f, u))-v*(diff(f, v)) end proc

 

proc (f) options operator, arrow; t*(diff(f, v)) end proc

 

proc (f) options operator, arrow; diff(f, u)+y*(diff(f, v)) end proc

 

proc (f) options operator, arrow; diff(f, t) end proc

 

proc (f) options operator, arrow; diff(f, x) end proc

 

proc (f) options operator, arrow; t*(diff(f, x))+diff(f, u) end proc

 

proc (f) options operator, arrow; y*(diff(f, x))+2*x*(diff(f, v)) end proc

 

proc (f) options operator, arrow; x*(diff(f, x))-t*(diff(f, t))+2*u*(diff(f, u))+2*v*(diff(f, v)) end proc

 

proc (f) options operator, arrow; diff(f, y) end proc

(2)

``

 

Download Lie_Algebra_Classification.mwLie_Algebra_Classification.mw

   
 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(2)
 

``

Regards


I am trying to setup a Dual Quaternion Multiplication Table. I found the table on Wikki. I  need some help here.

Have set

x1  =1   x2 = i   x3  =j   x4   =k   x5 =e   x6 = ei   x7 = ej   x8 =ek

 

restart

                                                                                                              #    x1   x2    x3   x4    x5   x6    x7   x8

with(DifferentialGeometry):

NULL

 

StructureEquations := [[x1, x1] = x1, [x1, x2] = x2, [x1, x3] = x3, [x1, x4] = x4, [x1, x5] = x1*x5, [x1, x6] = x6, [x1, x7] = x7, [x1, x8] = x8, [x2, x1] = x2, [x2, x2] = -1, [x2, x3] = x4, [x2, x4] = -x3, [x2, x5] = x6, [x2, x6] = -x5, [x2, x7] = x8, [x2, x8] = -x7, [x3, x1] = x3, [x3, x2] = -x4, [x3, x3] = -1, [x3, x4] = x2, [x3, x5] = x7, [x3, x6] = -x8, [x3, x7] = -x5, [x3, x8] = x6, [x4, x1] = x4, [x4, x2] = x3, [x4, x3] = -x2, [x4, x4] = -1, [x4, x5] = x8, [x4, x6] = x7, [x4, x7] = -x6, [x4, x8] = -x5, [x5, x1] = x5, [x5, x2] = x6, [x5, x3] = x7, [x5, x4] = x8, [x5, x5] = 0, [x6, x1] = x6, [x6, x2] = -x5, [x6, x3] = x8, [x6, x4] = -x7, [x7, x1] = x7, [x7, x2] = -x8, [x7, x3] = -x5, [x7, x4] = x6, [x8, x1] = x8, [x8, x2] = x7, [x8, x3] = -x6, [x8, x4] = -x5]

[[x1, x1] = x1, [x1, x2] = x2, [x1, x3] = x3, [x1, x4] = x4, [x1, x5] = x1*x5, [x1, x6] = x6, [x1, x7] = x7, [x1, x8] = x8, [x2, x1] = x2, [x2, x2] = -1, [x2, x3] = x4, [x2, x4] = -x3, [x2, x5] = x6, [x2, x6] = -x5, [x2, x7] = x8, [x2, x8] = -x7, [x3, x1] = x3, [x3, x2] = -x4, [x3, x3] = -1, [x3, x4] = x2, [x3, x5] = x7, [x3, x6] = -x8, [x3, x7] = -x5, [x3, x8] = x6, [x4, x1] = x4, [x4, x2] = x3, [x4, x3] = -x2, [x4, x4] = -1, [x4, x5] = x8, [x4, x6] = x7, [x4, x7] = -x6, [x4, x8] = -x5, [x5, x1] = x5, [x5, x2] = x6, [x5, x3] = x7, [x5, x4] = x8, [x5, x5] = 0, [x6, x1] = x6, [x6, x2] = -x5, [x6, x3] = x8, [x6, x4] = -x7, [x7, x1] = x7, [x7, x2] = -x8, [x7, x3] = -x5, [x7, x4] = x6, [x8, x1] = x8, [x8, x2] = x7, [x8, x3] = -x6, [x8, x4] = -x5]

(1)

``

(2)

DQ := LieAlgebraData(StructureEquations, [x1, x2, x3, x4, x5, x6, x7, x8])

_DG([["LieAlgebra", "L1", [8, table( [ ] )]], [[[1, 2, 2], 1], [[1, 3, 3], 1], [[1, 4, 4], 1], [[1, 5, 1], x5], [[1, 5, 5], x1], [[1, 6, 6], 1], [[1, 7, 7], 1], [[1, 8, 8], 1], [[1, 2, 2], -1], [[2, 3, 4], 1], [[2, 4, 3], -1], [[2, 5, 6], 1], [[2, 6, 5], -1], [[2, 7, 8], 1], [[2, 8, 7], -1], [[1, 3, 3], -1], [[2, 3, 4], 1], [[3, 4, 2], 1], [[3, 5, 7], 1], [[3, 6, 8], -1], [[3, 7, 5], -1], [[3, 8, 6], 1], [[1, 4, 4], -1], [[2, 4, 3], -1], [[3, 4, 2], 1], [[4, 5, 8], 1], [[4, 6, 7], 1], [[4, 7, 6], -1], [[4, 8, 5], -1], [[1, 5, 5], -1], [[2, 5, 6], -1], [[3, 5, 7], -1], [[4, 5, 8], -1], [[1, 6, 6], -1], [[2, 6, 5], 1], [[3, 6, 8], -1], [[4, 6, 7], 1], [[1, 7, 7], -1], [[2, 7, 8], 1], [[3, 7, 5], 1], [[4, 7, 6], -1], [[1, 8, 8], -1], [[2, 8, 7], -1], [[3, 8, 6], 1], [[4, 8, 5], 1]]])

(3)

DGsetup(DQ)

`Lie algebra: L1`

(4)

MultiplicationTable(DQ, "AlgebraTable")

Error, (in DifferentialGeometry:-LieAlgebras:-MultiplicationTable) invalid input: DifferentialGeometry:-ChangeFrame expects its 1st argument, frame_name, to be of type {name, string}, but received _DG([["LieAlgebra", "L1", [8, table( [ ] )]], [[[1, 2, 2], 1], [[1, 3, 3], 1], [[1, 4, 4], 1], [[1, 5, 1], x5], [[1, 5, 5], x1], [[1, 6, 6], 1], [[1, 7, 7], 1], [[1, 8, 8], 1], [[1, 2, 2], -1], [[2, 3, 4], 1], [[2, 4, 3], -1], [[2, 5, 6], 1], [[2, 6, 5], -1], [[2, 7, 8], 1], [[2, 8, 7], -1], [[1, 3, 3], -1], [[2, 3, 4], 1], [[3, 4, 2], 1], [[3, 5, 7], 1], [[3, 6, 8], -1], [[3, 7, 5], -1], [[3, 8, 6], 1], [[1, 4, 4], -1], [[2, 4, 3], -1], [[3, 4, 2]...

 

NULL

 

Download Dual_Quaternion_Defining_Algebra.mw

 

I'd like to have an algorithm for computing the ring of invariants of a reductive group (i.e. G acts on V, compute C[V]^G). I know that there are algorithms in existance for this, e.g. see http://www.math.lsa.umich.edu/~hderksen/Publications/invar1999.pdf. Does anyone know if this has been implemented in Maple, or where to find code in another language? Any ideas on what sort of data structure would be appropriate

how to calculate Ferrari resolvent of x^4-c1*x^3+c2*x^2-c3*x+c4

I have a multivariate polynomial in x and y. How can I firstly collect the terms with respect to the powers of x and y and then simplifying their coefficients.

restart

with(RealDomain):

with(LinearAlgebra):

A := Matrix(5, 5, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (1, 4) = 1, (1, 5) = 1, (2, 1) = 1, (2, 2) = 0, (2, 3) = c__1, (2, 4) = y, (2, 5) = c__2, (3, 1) = 1, (3, 2) = c__1, (3, 3) = 0, (3, 4) = c__3, (3, 5) = x, (4, 1) = 1, (4, 2) = y, (4, 3) = c__3, (4, 4) = 0, (4, 5) = c__4, (5, 1) = 1, (5, 2) = c__2, (5, 3) = x, (5, 4) = c__4, (5, 5) = 0})

Matrix(%id = 4510803138)

(1)

B := Determinant(A)

-2*c__1^2*c__4+2*c__1*c__2*c__3+2*c__1*c__2*c__4-2*c__1*c__2*x+2*c__1*c__3*c__4-2*c__1*c__3*y-2*c__1*c__4^2+2*c__1*c__4*x+2*c__1*c__4*y+2*c__1*x*y-2*c__2^2*c__3-2*c__2*c__3^2+2*c__2*c__3*c__4+2*c__2*c__3*x+2*c__2*c__3*y-2*c__2*c__4*y+2*c__2*x*y-2*c__3*c__4*x+2*c__3*x*y+2*c__4*x*y-2*x^2*y-2*x*y^2

(2)

``

 

Download A.mw

Hello,

i need help to translate a system which is given below to a for loop.

Other wise i am writing it with myself. 

instead of doing it like that 

with(LinearAlgebra);

sys := [galerkin_funcs[1], galerkin_funcs[2], galerkin_funcs[3], galerkin_funcs[4], galerkin_funcs[5], galerkin_funcs[6], galerkin_funcs[7], galerkin_funcs[8], galerkin_funcs[9], galerkin_funcs[10]];

var := [w[1], w[2], w[3], w[4], w[5], w[6]];

Kmat, Fmat := GenerateMatrix(sys, var);

i want to do it like that.

N:=10

for i to N+4 do

sys(1,i) := galerkin_funcs[i] 

end do

for i to N do

var(1,i) := w[i] 

end do

After that i will generate matrix with this comman Kmat, Fmat := GenerateMatrix(sys, var);

But this for loop i wrote is not doing the i want to do.

Thanks for your help.

 

 

1. Take a group for example. First we can set up a group G by

G:=<<a,b>|<a2=1,b3=1,(a.b)2=1>>. Actually,G=S3. So how to simplify a long production,

such as "a.b.b.a.b.b.a.b"?

2. How to define a finitely presented algebra over some field, such as the enveloping algebra

or quantum group of a Lie algebra? And moreover how to do the similar computation about 

simplifying a long production?

Here we see the projection of a vector onto another using different concepts ranging from linear algebra to vector calculus. Implemented components thus seen in three-dimensional space.

 

Proyecciones_Vectoriales.mw

(in spanish)

L.Araujo C.

Hello people in mapleprimes,

I have a question about why what is shown by maple by simplify(-8)^(1/3) is 1+ Complex(1)* (3)^(1/2)?

Solutions of x^3=-8 are -2, 1+Complex(1)*(3)^(1/2) and 1- Complex(1)*(3)^(1/2). And, as for the last one, 1- Complex(1)*(3)^(1/2), it is 

the conjugate of the second, so it might not need to be written, because of it being easily seen so.

Is it the same reason why just -2 is not shown as the result of simplify((-8)^(1/3))?

PS. I know the instruction to use surd in such a case.

the reason I asked this question is this:

I am reading Essential Maple, where

ln(z) = ln(rho*exp(Complex(1)*theta));

ln(rho*exp(Complex(1)*theta))=ln(rho)+Complex(1)*theta;

ln(rho)+Complex(1)*theta;=ln(rho)+Complex(1)*arctan(y,x);

and

z^a=exp(a*ln(z));

and

"Because of

exp(w*Pi*Complex(1)*k)=cos(2*Pi*k) + Complex(1)*sin(2*Pi*k);

and

cos(2*Pi*k) + Complex(1)*sin(2*Pi*k)=1;

we could equally well have chosen

ln_k z = ln(z) + 2*Pi*Complex(1)*k"

are written.

 Supposing these, there is a sentence that

"we choose k=0, and thus -Pi<=theta<=Pi to be the one (that for our canonical logarithm).

Every computer algebra language and numerical language follows this standard and takes the

complex logarithm to have its imaginary part in this range.

With this definition, (-8)^(1/3)=1 + Complex(1)*sqrt(3), and not -2. (the end of quotation)"

 

And, I can't understand the last sentence"With this definition", so I asked the above question.

 

I hope someone give an answer to the above question.

 

Thanks in advance.

 

taro

f := x^2*(y/x+sqrt(-7*y^2/x^2))/(y^2*(x/y+sqrt(-7*x^2/y^2)));
v := parametrization(f, x, y, t);

it can not parametrize.

i do not know which book teach group theory and algebraic curve

can we call this algebraic curve over finite field ?

 

how to represent a function as an algebraic curve equation for parametrization?

 

Here we have an application to understand how algebraic expressions, calculating degrees relative abosulutos polynomial operations and introduction to work.Here we have an application to understand how algebraic expressions, calculating degrees relative abosulutos polynomial operations and introduction to work.

 

Grados_de_Polinomios.mw

(in spanish)

Atte.

L.AraujoC.

 

 

How would one in Maple solve this, which is an inequality equation in some variables, which can be nonlinear, with constraints on range of each variable. I.e. I want to find conditions on the variables to make the inequality satisfied.

In Mathematica, I use the Reduce command

Clear[x, y];
eq = 1/2 - x + x^2 - y + y^2;
Reduce[{eq > 0, 0 < x < 1 && 0 < y < 1}, {x, y}, Reals]

How would one do the same in Maple? I tried solve, but can't give constraints.

restart;
eq:=1/2 -x+x^2-y+y^2:
solve(eq>0 , {x1, x2});

So I need to do the same as in the Mathematica command, but in Maple. I do not want numerical solution, but algebraic as shown above.

Using Maple 18.2 on windows.

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This first webinar in a two part series will cover true/false, multiple choice, numeric, mathematical formula, fill in the blank, sketch, and FBD questions. A second webinar that demonstrates more advanced question types will follow.

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In this webinar, Dr. Robert Lopez will apply the techniques of “Clickable Calculus” to standard calculations in Linear Algebra.

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Let (G, ·) be a group and X any set. Let F be the set of functions with domain

X and range G. Define a binary operation ∗ on F by (f ∗ g)(x) := f(x) · g(x). Is

prove that this is so.

Yes, (F, ∗) is a group.

prove it.

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