OpenMath Content Dictionary: poly1p

Canonical URL:

http://www.math.kobe-u.ac.jp/OCD/poly1p.tfb

CD File:

poly1p.ocd

CD as XML Encoded OpenMath:

poly1p.omcd

Defines:

index, indexed_variable, multi_power, sorted_set_of_indexed_variables, vector_of_indexed_variables

Date:

2002-08-08

Version:

1 (Revision 2)

Review Date:

2017-12-31

Status:

experimental

---

  Author: Nobuki Takayama

This CD defines symbols for concerning multi-index and indexed variables.

---

multi_power

Description:

multi_power is for using the multi-index notation.

Commented Mathematical property (CMP):

$\prod_{i=1}^n x_i ^ {e_i}$

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="poly1p" name="multi_power"/>
      <OMV name="x"/>
      <OMV name="e"/>
    </OMA>
    <OMA><OMS cd="arith1" name="product"/>
      <OMA><OMS cd="interval1" name="integer_interval"/>
        <OMI> 1 </OMI>
        <OMA><OMS cd="linalg4" name="size"/>
          <OMV name="x"/>
        </OMA>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="i"/>
        </OMBVAR>
        <OMA><OMS cd="arith1" name="power"/>
          <OMA><OMS cd="linalg1" name="vector_selector"/>
            <OMV name="i"/>
            <OMV name="x"/>
          </OMA>
          <OMA><OMS cd="linalg1" name="vector_selector"/>
            <OMV name="i"/>
            <OMV name="e"/>
          </OMA>
        </OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="poly1p">multi_power</csymbol><ci>x</ci><ci>e</ci></apply>
  <apply><csymbol cd="arith1">product</csymbol>
   <apply><csymbol cd="interval1">integer_interval</csymbol>
    <cn type="integer">1</cn>
    <apply><csymbol cd="linalg4">size</csymbol><ci>x</ci></apply>
   </apply>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>i</ci></bvar>
    <apply><csymbol cd="arith1">power</csymbol>
     <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>x</ci></apply>
     <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>i</ci><ci>e</ci></apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (multi_power ( x, e) , product (integer_interval ( 1 , size ( x) ) , lambda [ i ] . (power (vector_selector ( i, x) , vector_selector ( i, e) ) ) ) )

Popcorn

poly1p.multi_power($x, $e) = arith1.product(interval1.integer_interval(1, linalg4.size($x)), fns1.lambda[$i -> linalg1.vector_selector($i, $x) ^ linalg1.vector_selector($i, $e)])

Rendered Presentation MathML

$$\mathrm{multi\_power}(x,e)=\prod _{i=1}^{\mathrm{size}\left(x\right)}{x_i}^{e_i}$$

Signatures:

sts

---

[Next: index] [Last: sorted_set_of_indexed_variables] [Top]

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index

Description:

index returns the index of a given indexed variable.

Signatures:

sts

---

[Next: indexed_variable] [Previous: multi_power] [Top]

---

indexed_variable

Description:

indexed_variable(x,i) returns the variable x_i

Commented Mathematical property (CMP):

index(indexed_variable(x,i)) = i

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="poly1p" name="index"/>
      <OMA><OMS cd="poly1p" name="indexed_variable"/>
        <OMV name="x"/>
        <OMV name="i"/>
      </OMA>
    </OMA>
    <OMV name="i"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="poly1p">index</csymbol>
   <apply><csymbol cd="poly1p">indexed_variable</csymbol><ci>x</ci><ci>i</ci></apply>
  </apply>
  <ci>i</ci>
 </apply>
</math>

Prefix

eq (index (indexed_variable ( x, i) ) , i)

Popcorn

poly1p.index(poly1p.indexed_variable($x, $i)) = $i

Rendered Presentation MathML

$$\mathrm{index}\left(\mathrm{indexed\_variable}(x,i)\right)=i$$

Commented Mathematical property (CMP):

index(indexed_variable(x,[1,2])) = [1,2]

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="poly1p" name="index"/>
      <OMA><OMS cd="poly1p" name="indexed_variable"/>
        <OMV name="x"/>
        <OMA><OMS cd="linalg2" name="vector"/>
          <OMI> 1 </OMI>
          <OMI> 2 </OMI>
        </OMA>
      </OMA>
    </OMA>
    <OMA><OMS cd="linalg2" name="vector"/>
      <OMI> 1 </OMI>
      <OMI> 2 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="poly1p">index</csymbol>
   <apply><csymbol cd="poly1p">indexed_variable</csymbol>
    <ci>x</ci>
    <apply><csymbol cd="linalg2">vector</csymbol>
     <cn type="integer">1</cn>
     <cn type="integer">2</cn>
    </apply>
   </apply>
  </apply>
  <apply><csymbol cd="linalg2">vector</csymbol>
   <cn type="integer">1</cn>
   <cn type="integer">2</cn>
  </apply>
 </apply>
</math>

Prefix

eq (index (indexed_variable ( x, vector ( 1 , 2 ) ) ) , vector ( 1 , 2 ) )

Popcorn

poly1p.index(poly1p.indexed_variable($x, linalg2.vector(1, 2))) = linalg2.vector(1, 2)

Rendered Presentation MathML

$$\mathrm{index}\left(\mathrm{indexed\_variable}(x,(1,2))\right)=(1,2)$$

Signatures:

sts

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[Next: vector_of_indexed_variables] [Previous: index] [Top]

---

vector_of_indexed_variables

Description:

vector_of_indexed_variables(x,n) returns the vector of variables (x_1, ..., x_n). vector_of_indexed_variables(x,[m,n]) returns the vector of variables (x_{1,1}, ..., x_{m,n}). Any vector of numbers can be given as an argument.

Commented Mathematical property (CMP):

i-th element of vector_of_indexed_variable(x,n) is indexed_variable(x,i)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="linalg1" name="vector_selector"/>
      <OMV name="i"/>
      <OMA><OMS cd="poly1p" name="vector_of_indexed_variables"/>
        <OMV name="x"/>
        <OMV name="n"/>
      </OMA>
    </OMA>
    <OMA><OMS cd="poly1p" name="indexed_variable"/>
      <OMV name="x"/>
      <OMV name="i"/>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalg1">vector_selector</csymbol>
   <ci>i</ci>
   <apply><csymbol cd="poly1p">vector_of_indexed_variables</csymbol><ci>x</ci><ci>n</ci></apply>
  </apply>
  <apply><csymbol cd="poly1p">indexed_variable</csymbol><ci>x</ci><ci>i</ci></apply>
 </apply>
</math>

Prefix

eq (vector_selector ( i, vector_of_indexed_variables ( x, n) ) , indexed_variable ( x, i) )

Popcorn

linalg1.vector_selector($i, poly1p.vector_of_indexed_variables($x, $n)) = poly1p.indexed_variable($x, $i)

Rendered Presentation MathML

$${\mathrm{vector\_of\_indexed\_variables}(x,n)}_i=\mathrm{indexed\_variable}(x,i)$$

Signatures:

sts

---

[Next: sorted_set_of_indexed_variables] [Previous: indexed_variable] [Top]

---

sorted_set_of_indexed_variables

Description:

sorted_set_of_indexed_variables(x,s) returns the vector of variables indexed by the sorted set s.

Commented Mathematical property (CMP):

m-element of sorted_set_of_indexed_variables(x,s) is indexed_variable(x,m)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="set1p" name="index_set_selector"/>
      <OMV name="m"/>
      <OMA><OMS cd="poly1p" name="sorted_set_of_indexed_variables"/>
        <OMV name="x"/>
        <OMV name="s"/>
      </OMA>
    </OMA>
    <OMA><OMS cd="poly1p" name="indexed_variable"/>
      <OMV name="x"/>
      <OMV name="m"/>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="set1p">index_set_selector</csymbol>
   <ci>m</ci>
   <apply><csymbol cd="poly1p">sorted_set_of_indexed_variables</csymbol><ci>x</ci><ci>s</ci></apply>
  </apply>
  <apply><csymbol cd="poly1p">indexed_variable</csymbol><ci>x</ci><ci>m</ci></apply>
 </apply>
</math>

Prefix

eq (index_set_selector ( m, sorted_set_of_indexed_variables ( x, s) ) , indexed_variable ( x, m) )

Popcorn

set1p.index_set_selector($m, poly1p.sorted_set_of_indexed_variables($x, $s)) = poly1p.indexed_variable($x, $m)

Rendered Presentation MathML

$$\mathrm{index\_set\_selector}(m,\mathrm{sorted\_set\_of\_indexed\_variables}(x,s))=\mathrm{indexed\_variable}(x,m)$$

Signatures:

sts

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[First: multi_power] [Previous: vector_of_indexed_variables] [Top]