OpenMath Content Dictionary: linalg1p

Canonical URL:

http://www.math.kobe-u.ac.jp/OCD/linalg1p.tfb

CD File:

linalg1p.ocd

CD as XML Encoded OpenMath:

linalg1p.omcd

Defines:

kernel, minus_part, ones, plus_part

Date:

2002-07-30

Version:

1 (Revision 1)

Review Date:

2017-12-31

Status:

experimental

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  Author: Nobuki Takayama

This CD defines symbols for linear algebra used for hypergeon1 (hypergeometric series of n variables).

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plus_part

Description:

The argument is a vector. It replaces negative elements in the vector to zero.

Signatures:

sts

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[Next: minus_part] [Last: kernel] [Top]

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minus_part

Description:

The argument is a vector. It replaces positive elements in the vector to zero and negative elements to their absolute values.

Commented Mathematical property (CMP):

$u = u_{+} - u_{-}$

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMV name="u"/>
    <OMA><OMS cd="arith1" name="sub"/>
      <OMA><OMS cd="linalg1p" name="plus_part"/>
        <OMV name="u"/>
      </OMA>
      <OMA><OMS cd="linalg1p" name="minus_part"/>
        <OMV name="u"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <ci>u</ci>
  <apply><csymbol cd="arith1">sub</csymbol>
   <apply><csymbol cd="linalg1p">plus_part</csymbol><ci>u</ci></apply>
   <apply><csymbol cd="linalg1p">minus_part</csymbol><ci>u</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq ( u, sub (plus_part ( u) , minus_part ( u) ) )

Popcorn

$u = arith1.sub(linalg1p.plus_part($u), linalg1p.minus_part($u))

Rendered Presentation MathML

$$u=\mathrm{sub}(\mathrm{plus\_part}\left(u\right),\mathrm{minus\_part}\left(u\right))$$

Signatures:

sts

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[Next: ones] [Previous: plus_part] [Top]

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ones

Description:

It returns a vector of a specifed size of which elements are one. 1-ary function.

Commented Mathematical property (CMP):

for all $i$, ${\vec 1}[i] = 1$

Signatures:

sts

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[Next: kernel] [Previous: minus_part] [Top]

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kernel

Description:

It returns the kernel of the map defined by a matrix in a specified domain.

Commented Mathematical property (CMP):

$\{ x \in D | A x = 0 \}$

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="linalg1p" name="kernel"/>
      <OMV name="d"/>
      <OMV name="a"/>
    </OMA>
    <OMA><OMS cd="set1" name="suchthat"/>
      <OMV name="d"/>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="x"/>
        </OMBVAR>
        <OMA><OMS cd="relation1" name="eq"/>
          <OMA><OMS cd="arith1" name="times"/>
            <OMV name="a"/>
            <OMV name="x"/>
          </OMA>
          <OMA><OMS cd="linalg5" name="zero"/>
            <OMA><OMS cd="linalg4" name="size"/>
              <OMV name="x"/>
            </OMA>
            <OMI> 1 </OMI>
          </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="linalg1p">kernel</csymbol><ci>d</ci><ci>a</ci></apply>
  <apply><csymbol cd="set1">suchthat</csymbol>
   <ci>d</ci>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <apply><csymbol cd="relation1">eq</csymbol>
     <apply><csymbol cd="arith1">times</csymbol><ci>a</ci><ci>x</ci></apply>
     <apply><csymbol cd="linalg5">zero</csymbol>
      <apply><csymbol cd="linalg4">size</csymbol><ci>x</ci></apply>
      <cn type="integer">1</cn>
     </apply>
    </apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (kernel ( d, a) , suchthat ( d, lambda [ x ] . (eq (times ( a, x) , zero (size ( x) , 1 ) ) ) ) )

Popcorn

linalg1p.kernel($d, $a) = set1.suchthat($d, fns1.lambda[$x -> $a * $x = linalg5.zero(linalg4.size($x), 1)])

Rendered Presentation MathML

$$\mathrm{kernel}(d,a)=\left\{x\in d\right|ax=\mathrm{zero}(\mathrm{size}\left(x\right),1)\}$$

Signatures:

sts

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[First: plus_part] [Previous: ones] [Top]