Options Available When Declaring Ore Algebras - Maple Help

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Options Available When Declaring Ore Algebras

 

Description

Examples

Description

• 

The option characteristic=p, where p is 0 or any positive integer different from 1, is used to declare the characteristic of the algebra.

• 

The option alg_relations=s, where s is an equation of a list or set of polynomial equations, introduces algebraic relations between commutative parameters; a polynomial p is meant as the equation p=0.

• 

The option comm=s, where s is a name or a list or set of names, introduces commutative parameters; in case of a commutative algebra of polynomials, use Ore_algebra[poly_algebra] instead of Ore_algebra[skew_algebra].

• 

The option polynom=s, where s is a name or a list or set of names, introduces indeterminates that are to be viewed as polynomial indeterminates (that is, may not appear rationally).

• 

The option func=s, where s is a name or a list or set of names, introduces names of functions that are allowed to appear in the coefficients of the elements of the algebra.

• 

The option action=s, where s is a set or list of equations of the form

u

=

proc(f,n) ... end proc

  

overloads the default actions of the operators on Maple objects.  u is any of the indeterminates of the algebra that was declared in a commutation, and the right-hand side is a procedure that implements the action of the operator u on Maple objects.  More specifically, a call to this procedure with an expression f and a non-negative integer n as arguments returns the (u@@n)(f) (see the Examples section below).

Examples

Changing the Ground Field

  

Here is an example of operators over a finite field.

withOre_algebra:

Askew_algebradiff=Dx,x:

skew_productDx,x5,A

Dxx5+5x4

(1)

Askew_algebradiff=Dx,x,characteristic=5:

skew_productDx,x5,A

Dxx5

(2)
  

Here are Ore algebras on a polynomial ring and on a rational function field.  The types of coefficients allowed differ accordingly.  In particular, generic functions are allowed in the rational case only, and have to be explicitly declared.

Askew_algebradiff=Dx,x,polynom=x:

skew_productDx,x,A

Dxx+1

(3)
  

On the other hand, both following inputs are illegal:

skew_productDx,xx1,A

Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra

skew_productDx,ηx,A

Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra

Askew_algebradiff=Dx,x:

skew_productDx,x,A

Dxx+1

(4)

skew_productDx,xx1,A

1x12+Dxxx1

(5)
  

This is an error:

skew_productDx,ηx,A

Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra

Askew_algebradiff=Dx,x,func=η:

skew_productDx,x,A

Dxx+1

(6)

skew_productDx,xx1,A

1x12+Dxxx1

(7)

skew_productDx,ηx,A

Dxηx+ⅆⅆxηx

(8)
  

This is an error:

Askew_algebradiff=Dx,x,comm=i,alg_relations=i2+1:

skew_productiDx,iηx,A

Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra

  

This is not:

Askew_algebradiff=Dx,x,comm=i,alg_relations=i2+1,func=η:

skew_productiDx,iηx,A

Dxηx+ⅆⅆxηx

(9)

Action on Maple Objects

  

Each commutation type has its default action on Maple objects.  For instance, the diff commutation acts on functions f(x) and not on sequences u(n):

Askew_algebradiff=Dx,x:

applyoprx,fx,A

xfx

(10)

applyoprDx,fx,A

ⅆⅆxfx

(11)

applyoprx,un,A

xun

(12)

applyoprDx,un,A

0

(13)
  

By changing the action, you can view the previous Weyl algebra as acting on sequences u(n) rather than on functions f(x).

A:=skew_algebra(diff=[Dx,x],polynom=x,action={
    Dx=proc(u,order) local res; global n;
        res:=u; to order do res:=subs(n=n+1,n*res) end do; res
    end proc,
    x=proc(u,order) global n;
        subs(n=n-order,u)
    end proc}):

applyoprx,fx,A

fx

(14)

applyoprDx,fx,A

n+1fx

(15)

applyoprx,un,A

un1

(16)

applyoprDx,un,A

n+1un+1

(17)

See Also

Ore_algebra/skew_algebra

 


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