[Om3] Definite Integrals (was Re: [Fwd: MathML CDs])

[Michael Kohlhase]

Dear James, dear all,

to continue the discussion from the last e-mail, ...

I have seen that James introduced symbols for the MathML qualifiers 
upperbound and lowerbound. I do not think that this is a good way of 
doing things. I would propose instead to have two definite integration 
symbols, defint and defintbounds (or something like this. The first 
takes a set as an argument and the second two real numbers, they are 
used thus:

<OMA>
  <OMS name="defint" cd="newint"/>
  <OMS cd="sets" name="reals/>
  <OMBIND>
    <OMS cd="fn1" name="lambda"/>
    <OMBVAR><OMV name="x"/></OMBVAR>
    <OMA><OMS cd="specfun1" name="sin"/><OMV name="x"/></OMA>
 </OMBIND>
</OMA>

for $\int_\RR \sin(x) dx $ in traditional OM representation


<OMA>
  <OMS name="defintbounds" cd="newint"/>
  <OMI>0</OMI>
  <OMI>1</OMI>
  <OMBIND>
    <OMS cd="fn1" name="lambda"/>
    <OMBVAR><OMV name="x"/></OMBVAR>
    <OMA><OMS cd="specfun1" name="sin"/><OMV name="x"/></OMA>
 </OMBIND>
</OMA>
 
for $\int_0^1\sin(x) dx$ in traditional OM representation

If we follow the multi-role proposal in my earlier emails, then we would 
also have

<OMBIND>
  <OMA>
    <OMS name="defint" cd="newint"/>
    <OMS cd="sets" name="reals/>
  </OMA>
  <OMBVAR><OMV name="x"/></OMBVAR>
  <OMA><OMS cd="specfun1" name="sin"/><OMV name="x"/></OMA>
</OMBIND>

for $\int_\RR \sin(x) dx $ in traditional MathML representation

<OMBIND>
  <OMA>
    <OMS name="defintbounds" cd="newint"/>
    <OMI>0</OMI>
    <OMI>1</OMI>
  </OMA>
  <OMBVAR><OMV name="x"/></OMBVAR>
  <OMA><OMS cd="specfun1" name="sin"/><OMV name="x"/></OMA>
</OMBIND>


for $\int_0^1\sin(x) dx$ in traditional MathML representation

This uses the fact that we have introduced complex binding operators in 
OM2, and is (I think) a very natural conceptualization of the integrals 
involved.

In summary, I think that we need three integration symbols:

int (for indefinite integration)
defint (for definite integration over a set)
defintbounds (for definite integration with bounds)

If we do not adopt the multiple-role proposal, we need three more for 
the binding occurrances.

To see what we can do with this proposal, try to figure out the 
following multiple integral.

<!-- multiple integral example -->
<OMOBJ version="3.0">
  <OMBIND>
    <OMA><OMS cd="newint" name="int"/><OMV name="S1"/><OMV 
name="S2"/><OMV name="S3"/></OMA>
    <OMBVAR><OMV name="x1"/><OMV name="x2"/><OMV name="x3"/></OMBVAR>
    <OMC>
      <OMA>
    <OMS cd="logic1" name="and"/>
    <OMA><OMS cd="relation1" name="gt"/><OMV name="x1"/><OMI>1</OMI></OMA>
    <OMA><OMS cd="relation1" name="gt"/><OMV name="x2"/><OMI>1</OMI></OMA>
    <OMA><OMS cd="relation1" name="gt"/><OMV name="x3"/><OMI>1</OMI></OMA>
      </OMA>
    </OMC>
    <OMA>
      <OMS cd="arith1" name="plus"/>
      <OMV name="x1"/>
      <OMA><OMS cd="arith1" name="times"/>
    <OMI>2</OMI>
    <OMA><OMS cd="arith1" name="power"/><OMV name="x2"/><OMI>2</OMI></OMA>
      </OMA>
      <OMA><OMS cd="arith1" name="times"/>
    <OMI>3</OMI>
    <OMA><OMS cd="arith1" name="power"/><OMV name="x3"/><OMI>3</OMI></OMA>
      </OMA>
    </OMA>
  </OMBIND>
</OMOBJ>

that is a beauty isn't it?

Michael

-- 

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 Professor of Computer Science     Campus Ring 12, 
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 m.kohlhase at jacobs-university.de http://kwarc.info/kohlhase 
 skype: m.kohlhase   * International University Bremen until Feb. 2007
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