OpenMath Content Dictionary: weylalgebra1

Canonical URL:
http://www.math.kobe-u.ac.jp/OCD/weylalgebra1.tfb
CD File:
weylalgebra1.ocd
CD as XML Encoded OpenMath:
weylalgebra1.omcd
Defines:
act, act_of_poly, diff, diffop, gr, partialdiff, times
Date:
2003-11-28
Version:
1 (Revision 2)
Review Date:
2017-12-31
Status:
experimental

  Author: Yasushi Tamura and Nobuki Takayama

This CD defines elements of the ring of differential operators with coefficients in the polynomial ring.


diffop

Description:

constructor of a differential operator from a polynomial or from an element of the finitely generated free algebra. The inverse of gr.

Commented Mathematical property (CMP):
d/dq q = q d/dq + 1
Formal Mathematical property (FMP):
diffop ( times ( dq , q ) , ( ( q ) , ( dq ) ) ) = diffop ( q dq + 1 , ( ( q ) , ( dq ) ) )
Signatures:
sts


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gr

Description:

the symbol polynomial of a given differential operator. The inverse of diffop.

Commented Mathematical property (CMP):
$\gr( q \partial_{q} + 1) = q p + 1 $
Formal Mathematical property (FMP):
gr ( diffop ( q dq + 1 , ( ( q ) , ( dq ) ) ) , ( ( dq ) , ( p ) ) ) = q p + 1
Signatures:
sts


[Next: diff] [Previous: diffop] [Top]

diff

Description:

Differentiation of a given function in one variable.

Example:
$\frac{d x^2}{dx} = 2 x$
diff ( x 2 , x ) = 2 x
Signatures:
sts


[Next: partialdiff] [Previous: gr] [Top]

partialdiff

Description:

partial differentiation of a given function.

Commented Mathematical property (CMP):
$\frac{\partial^{2} x^{2} y}{\partial x^{2}} = 2 y $
Formal Mathematical property (FMP):
partialdiff ( x x y , ( ( x , 2 ) ) ) = 2 y
Signatures:
sts


[Next: times] [Previous: diff] [Top]

times

Description:

multiplication in D

Commented Mathematical property (CMP):
$\partial_{q} q = \partial{q} q + 1 $
Formal Mathematical property (FMP):
times ( dq , q ) = times ( q , dq ) + 1
Signatures:
sts


[Next: act] [Previous: partialdiff] [Top]

act

Description:

action of a differential operator to a function.

Commented Mathematical property (CMP):
$ x^{m} \partial_{x}^{n} \partial_{y}^{r} \cdot f = x^{m} \frac{partial^{n+r} f}{\partial x^{n} \partial y^{r}} $
Formal Mathematical property (FMP):
act ( diffop ( x m dx n dy r , ( ( x , y ) , ( dx , dy ) ) ) , f ) = x m partialdiff ( f , ( ( x , n ) , ( y , r ) ) )
Signatures:
sts


[Next: act_of_poly] [Previous: times] [Top]

act_of_poly

Description:

action of a polynomial as a differential operator to a function. act_of_poly is equivalent to the composition of act and diffop.

Commented Mathematical property (CMP):
$ x^{m} \partial_{x}^{n} \partial_{y}^{r} \cdot f = x^{m} \frac{partial^{n+r} f}{\partial x^{n} \partial y^{r}} $
Formal Mathematical property (FMP):
act_of_poly ( x m dx n dy r , ( ( x , y ) , ( dx , dy ) ) , f ) = x m partialdiff ( f , ( ( x , n ) , ( y , r ) ) )
Signatures:
sts


[First: diffop] [Previous: act] [Top]