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:

2002-08-07, 2003-11-28 revised to 1.1

Version:

1 (Revision 1)

Review Date:

2003-08-07

Status:

experimental

Uses CD:

freealg1, arith1, list1, relation1

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="weylalgebra1" name="diffop"/>
      <OMA><OMS cd="freealg1" name="times"/>
        <OMV name="dq"/>
        <OMV name="q"/>
      </OMA>
      <OMA><OMS cd="list1" name="list"/>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="q"/>
        </OMA>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="dq"/>
        </OMA>
      </OMA>
    </OMA>
    <OMA><OMS cd="weylalgebra1" name="diffop"/>
      <OMA><OMS cd="arith1" name="plus"/>
        <OMA><OMS cd="arith1" name="times"/>
          <OMV name="q"/>
          <OMV name="dq"/>
        </OMA>
        <OMI> 1 </OMI>
      </OMA>
      <OMA><OMS cd="list1" name="list"/>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="q"/>
        </OMA>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="dq"/>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (diffop (times ( dq, q) , list (list ( q) , list ( dq) ) ) , diffop (plus (times ( q, dq) , 1 ) , list (list ( q) , list ( dq) ) ) )

$$\mathrm{diffop}(\mathrm{times}(\mathrm{dq},q),(\left(q\right),\left(\mathrm{dq}\right)))=\mathrm{diffop}(q\mathrm{dq}+1,(\left(q\right),\left(\mathrm{dq}\right)))$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="weylalgebra1" name="gr"/>
      <OMA><OMS cd="weylalgebra1" name="diffop"/>
        <OMA><OMS cd="arith1" name="plus"/>
          <OMA><OMS cd="arith1" name="times"/>
            <OMV name="q"/>
            <OMV name="dq"/>
          </OMA>
          <OMI> 1 </OMI>
        </OMA>
        <OMA><OMS cd="list1" name="list"/>
          <OMA><OMS cd="list1" name="list"/>
            <OMV name="q"/>
          </OMA>
          <OMA><OMS cd="list1" name="list"/>
            <OMV name="dq"/>
          </OMA>
        </OMA>
      </OMA>
      <OMA><OMS cd="list1" name="list"/>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="dq"/>
        </OMA>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="p"/>
        </OMA>
      </OMA>
    </OMA>
    <OMA><OMS cd="arith1" name="plus"/>
      <OMA><OMS cd="arith1" name="times"/>
        <OMV name="q"/>
        <OMV name="p"/>
      </OMA>
      <OMI> 1 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

eq (gr (diffop (plus (times ( q, dq) , 1 ) , list (list ( q) , list ( dq) ) ) , list (list ( dq) , list ( p) ) ) , plus (times ( q, p) , 1 ) )

$$\mathrm{gr}(\mathrm{diffop}(q\mathrm{dq}+1,(\left(q\right),\left(\mathrm{dq}\right))),(\left(\mathrm{dq}\right),\left(p\right)))=qp+1$$

Signatures:

sts

diff

Description:

Differentiation of a given function in one variable.

Example:

$\frac{d x^2}{dx} = 2 x$

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="weylalgebra1" name="diff"/>
      <OMA><OMS cd="arith1" name="power"/>
        <OMV name="x"/>
        <OMI> 2 </OMI>
      </OMA>
      <OMV name="x"/>
    </OMA>
    <OMA><OMS cd="arith1" name="times"/>
      <OMI> 2 </OMI>
      <OMV name="x"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (diff (power ( x, 2 ) , x) , times ( 2 , x) )

$$\mathrm{diff}(x^2,x)=2x$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="weylalgebra1" name="partialdiff"/>
      <OMA><OMS cd="arith1" name="times"/>
        <OMA><OMS cd="arith1" name="times"/>
          <OMV name="x"/>
          <OMV name="x"/>
        </OMA>
        <OMV name="y"/>
      </OMA>
      <OMA><OMS cd="list1" name="list"/>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="x"/>
          <OMI> 2 </OMI>
        </OMA>
      </OMA>
    </OMA>
    <OMA><OMS cd="arith1" name="times"/>
      <OMI> 2 </OMI>
      <OMV name="y"/>
    </OMA>
  </OMA>
</OMOBJ>

eq (partialdiff (times (times ( x, x) , y) , list (list ( x, 2 ) ) ) , times ( 2 , y) )

$$\mathrm{partialdiff}(xxy,\left((x,2)\right))=2y$$

Signatures:

sts

times

Description:

multiplication in D

Commented Mathematical property (CMP):

$\partial_{q} q = \partial{q} q + 1 $

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="weylalgebra1" name="times"/>
      <OMV name="dq"/>
      <OMV name="q"/>
    </OMA>
    <OMA><OMS cd="arith1" name="plus"/>
      <OMA><OMS cd="weylalgebra1" name="times"/>
        <OMV name="q"/>
        <OMV name="dq"/>
      </OMA>
      <OMI> 1 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

eq (times ( dq, q) , plus (times ( q, dq) , 1 ) )

$$\mathrm{times}(\mathrm{dq},q)=\mathrm{times}(q,\mathrm{dq})+1$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="weylalgebra1" name="act"/>
      <OMA><OMS cd="weylalgebra1" name="diffop"/>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="x"/>
              <OMV name="m"/>
            </OMA>
            <OMA><OMS cd="arith1" name="power"/>
              <OMV name="dx"/>
              <OMV name="n"/>
            </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="dy"/>
            <OMV name="r"/>
          </OMA>
        </OMA>
        <OMA><OMS cd="list1" name="list"/>
          <OMA><OMS cd="list1" name="list"/>
            <OMV name="x"/>
            <OMV name="y"/>
          </OMA>
          <OMA><OMS cd="list1" name="list"/>
            <OMV name="dx"/>
            <OMV name="dy"/>
          </OMA>
        </OMA>
      </OMA>
      <OMV name="f"/>
    </OMA>
    <OMA><OMS cd="arith1" name="times"/>
      <OMA><OMS cd="arith1" name="power"/>
        <OMV name="x"/>
        <OMV name="m"/>
      </OMA>
      <OMA><OMS cd="weylalgebra1" name="partialdiff"/>
        <OMV name="f"/>
        <OMA><OMS cd="list1" name="list"/>
          <OMA><OMS cd="list1" name="list"/>
            <OMV name="x"/>
            <OMV name="n"/>
          </OMA>
          <OMA><OMS cd="list1" name="list"/>
            <OMV name="y"/>
            <OMV name="r"/>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (act (diffop (times (times (power ( x, m) , power ( dx, n) ) , power ( dy, r) ) , list (list ( x, y) , list ( dx, dy) ) ) , f) , times (power ( x, m) , partialdiff ( f, list (list ( x, n) , list ( y, r) ) ) ) )

$$\mathrm{act}(\mathrm{diffop}(x^m{\mathrm{dx}}^n{\mathrm{dy}}^r,((x,y),(\mathrm{dx},\mathrm{dy}))),f)=x^m\mathrm{partialdiff}(f,((x,n),(y,r)))$$

Signatures:

sts

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):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="weylalgebra" name="act_of_poly"/>
      <OMA><OMS cd="arith1" name="times"/>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="x"/>
            <OMV name="m"/>
          </OMA>
          <OMA><OMS cd="arith1" name="power"/>
            <OMV name="dx"/>
            <OMV name="n"/>
          </OMA>
        </OMA>
        <OMA><OMS cd="arith1" name="power"/>
          <OMV name="dy"/>
          <OMV name="r"/>
        </OMA>
      </OMA>
      <OMA><OMS cd="list1" name="list"/>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="x"/>
          <OMV name="y"/>
        </OMA>
        <OMA><OMS cd="list1" name="list"/>
          <OMV name="dx"/>
          <OMV name="dy"/>
        </OMA>
      </OMA>
      <OMV name="f"/>
    </OMA>
    <OMA><OMS cd="arith1" name="times"/>
      <OMA><OMS cd="arith1" name="power"/>
        <OMV name="x"/>
        <OMV name="m"/>
      </OMA>
      <OMA><OMS cd="weylalgebra" name="partialdiff"/>
        <OMV name="f"/>
        <OMA><OMS cd="list1" name="list"/>
          <OMA><OMS cd="list1" name="list"/>
            <OMV name="x"/>
            <OMV name="n"/>
          </OMA>
          <OMA><OMS cd="list1" name="list"/>
            <OMV name="y"/>
            <OMV name="r"/>
          </OMA>
        </OMA>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (act_of_poly (times (times (power ( x, m) , power ( dx, n) ) , power ( dy, r) ) , list (list ( x, y) , list ( dx, dy) ) , f) , times (power ( x, m) , partialdiff ( f, list (list ( x, n) , list ( y, r) ) ) ) )

$$\mathrm{act\_of\_poly}(x^m{\mathrm{dx}}^n{\mathrm{dy}}^r,((x,y),(\mathrm{dx},\mathrm{dy})),f)=x^m\mathrm{partialdiff}(f,((x,n),(y,r)))$$

Signatures:

sts