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Minimize the number of tensor components according to its symmetries
(and relabel, redefine or count the number of independent tensor components)

 

 

The nice development described below is work in collaboration with Pascal Szriftgiser from Laboratoire PhLAM, Université Lille 1, France, used in the Mapleprimes post Magnetic traps in cold-atom physics

 

A new keyword in Define  and Setup : minimizetensorcomponents, allows for automatically minimizing the number of tensor components taking into account the tensor symmetries. For example, if a tensor with two indices in a 4D spacetime is defined as antisymmetric using Define with this new keyword, the number of different tensor components will be exactly 6, and the elements of the diagonal are automatically set equal to 0. After setting this keyword to true with Setup , all subsequent definitions of tensors automatically minimize the number of components while using this keyword with Define  makes this minimization only happen with the tensors being defined in the call to Define .

 

Related to this new functionality, 4 new Library routines were added: MinimizeTensorComponents, NumberOfIndependentTensorComponents, RelabelTensorComponents and RedefineTensorComponents

 

Example:

restart; with(Physics)

 

Define an antisymmetric tensor with two indices

Define(F[mu, nu], antisymmetric)

`Defined objects with tensor properties`

 

{Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}

(1.1)

Although the system knows that F[mu, nu] is antisymmetric, you need to use Simplify to apply the (anti)symmetry

F[mu, nu]+F[nu, mu]

F[mu, nu]+F[nu, mu]

(1.2)

 

Simplify(F[mu, nu]+F[nu, mu])

0

(1.3)

so by default the components of F[mu, nu] do not automatically reflect the (anti)symmetry; likewise

F[1, 2]+F[2, 1]

F[1, 2]+F[2, 1]

(1.4)

Simplify(F[1, 2]+F[2, 1])

0

(1.5)

and computing the array form of F[mu, nu]we do not see the elements of the diagonal equal to zero nor the lower-left triangle equal to the upper-right triangle but for a different sign:

TensorArray(F[mu, nu])

Matrix(%id = 18446744078270093062)

(1.6)

 

On the other hand, this new functionality, here called minimizetensorcomponents, makes the symmetries of the tensor be explicitly reflected in its components.

 

There are three ways to use it. First, one can minimize the number of tensor components of a tensor previously defined. For example

 

Library:-MinimizeTensorComponents(F)

Matrix(%id = 18446744078270064630)

(1.7)

After this, both (1.2) and (1.3) are automatically equal to 0 without having to use Simplify

F[mu, nu]+F[nu, mu]

0

(1.8)

0

0

(1.9)

And the output of TensorArray  in (1.6) becomes equal to (1.7).

 

NOTE: in addition, after using minimizetensorcomponents in the definition of a tensor, say F, all the keywords implemented for Physics tensors are available for F:

 

F[]

F[mu, nu] = Matrix(%id = 18446744078247910206)

(1.10)

F[trace]

0

(1.11)

F[nonzero]

F[mu, nu] = {(1, 2) = F[1, 2], (1, 3) = F[1, 3], (1, 4) = F[1, 4], (2, 1) = -F[1, 2], (2, 3) = F[2, 3], (2, 4) = F[2, 4], (3, 1) = -F[1, 3], (3, 2) = -F[2, 3], (3, 4) = F[3, 4], (4, 1) = -F[1, 4], (4, 2) = -F[2, 4], (4, 3) = -F[3, 4]}

(1.12)

"F[~1,mu,matrix]"

F[`~1`, mu] = Vector[row](%id = 18446744078247885990)

(1.13)

Alternatively, one can define a tensor, specifying that the symmetries should be taken into account to minimize the number of its components passing the keyword minimizetensorcomponents to Define .

 

Example:

 

Define a tensor with the symmetries of the Riemann  tensor, that is, a tensor of 4 indices that is symmetric with respect to interchanging the positions of the 1st and 2nd pair of indices and antisymmetric with respect to interchanging the position of its 1st and 2nd indices, or 3rd and 4th indices, and define it minimizing the number of tensor components

 

Define(R[alpha, beta, mu, nu], symmetric = {[[1, 2], [3, 4]]}, antisymmetric = {[1, 2], [3, 4]}, minimizetensorcomponents)

`Defined objects with tensor properties`

 

{Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], R[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}

(1.14)

We now have

R[1, 2, 3, 4]+R[2, 1, 3, 4]

0

(1.15)

R[alpha, beta, mu, nu]-R[mu, nu, alpha, beta]

0

(1.16)
• 

One can always retrieve the symmetry properties in the abstract notation used by the Define command using the new Library:-GetTensorSymmetryProperties, its output is ordered, first the symmetric then the antisymmetric properties

 

Library:-GetTensorSymmetryProperties(R)

{[[1, 2], [3, 4]]}, {[1, 2], [3, 4]}

(1.17)
• 

After making the symmetries explicit (and also before that), it is frequently useful to know the number of independent components of a given tensor. For this purpose you can use the new Library:-NumberOfIndependentTensorComponents

 

Library:-NumberOfIndependentTensorComponents(R)

21

(1.18)

and besides taking into account the symmetries, in the case of the Riemann  tensor, after taking into account the first Bianchi identity this number of components is further reduced to 20.

 

A third way of using the new minimizetensorcomponents functionality is using Setup , so that, automatically, every subsequent definition of tensors with symmetries is performed minimizing the number of its components using the indicated symmetries

 

Example:

Setup(minimizetensorcomponents = true)

[minimizetensorcomponents = true]

(1.19)

So from hereafter you can define tensors taking into account their symmetries explicitly and without having to include the keyword minimizetensorcomponents at each definition

 

Define(C[alpha, beta], antisymmetric)

`Defined objects with tensor properties`

 

{C[mu, nu], Physics:-Dgamma[mu], F[mu, nu], Physics:-Psigma[mu], R[mu, nu, alpha, beta], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}

(1.20)

 

C[]

C[mu, nu] = Matrix(%id = 18446744078408747598)

(1.21)
• 

Two new related functionalities are provided via Library:-RelabelTensorComponents and Library:-RedefineTensorComponent, the first one to have the number of tensor components directly reflected in the names of the components, the second one to redefine only one of these components

Library:-RelabelTensorComponents(C)

Matrix(%id = 18446744078408729774)

(1.22)

 

Suppose now we want to make one of these components equal to 1, say C__2

Library:-RedefineTensorComponent(C[1, 2] = 1)

C[mu, nu] = Matrix(%id = 18446744078270104390)

(1.23)

This nice development is work in collaboration with Pascal Szriftgiser from Laboratoire PhLAM, UMR CNRS 8523, Université Lille 1, F-59655, France.

``

 

Download MinimizeTensorComponents.mw

 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

 

Hello, 

I study mainly subjects that fall under umbrella of number theory, but i have specified a little further in the worksheet. This is really a request for assistance, because in as much as i have met so many brilliant people online via social media etc,  I would always love to meet more, and especially ones who are more experienced in this field. 

 

Basically i am too cheap and old to think about going to a good university, so I am trying to get free advice from the people who have probably completed doctorates in the relevant field. Got to be honest I say.

 

Anyway my contact email is at the top of the attached worksheet.

 

First thing that stood out to me about the distributions produced in this worksheet is how sparse the number of points is for N=17 relative to all the other values of N.

EXAMPLE_FOR_MAPLE3.mw

 

Edit: Another example worksheet added.

MAPLE_EXAMPLE_13.mw

Dear all , I' would like to join a group to produce quantum information tools in Maple

 

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As the title says, a lesson on functions:  eg the -> operator, f(2), eval, evalf etc 

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Lenin Araujo

Ambassador Of Maple

 

 

On the example of a manipulator with three degrees of freedom.
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