OpenMath Content Dictionary: hypergeon0

Canonical URL:

http://www.math.kobe-u.ac.jp/OCD/hypergeon0.tfb

CD File:

hypergeon0.ocd

CD as XML Encoded OpenMath:

hypergeon0.omcd

Defines:

cartesian_product_n, kernel, minus_part, multi_power, plus_part, where

Date:

2002-11-11, 2003-11-30

Version:

1 (Revision 1)

Review Date:

2003-11-11

Status:

experimental

Uses CD:

arith1, linalg1, linalg4, linalg5, quant1, relation1, set1

This CD defines some supplementary symbols necessary for hypergeon1 and hypergeon2 (hypergeometric series of n variables). These symbols may be included in CD's linalg, logic, poly, and set.

plus_part

Description:

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

Signatures:

sts

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):

xml prefix mathml

<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="hypergeon0" name="plus_part"/>
        <OMV name="u"/>
      </OMA>
      <OMA><OMS cd="hypergeon0" name="minus_part"/>
        <OMV name="u"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

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

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

Signatures:

sts

kernel

Description:

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

Commented Mathematical property (CMP):

$\kernel(D,A) = \{ x \in D | A x = 0 \}$

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon0" 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>

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

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

Signatures:

sts

where

Description:

The word "where" is often used in mathematical expressions to set variables or to say side conditions. CDname logic1.implies can be used for these purposes, but "where" will be more intuitive and more friendly expression for authors.

Commented Mathematical property (CMP):

$x^n=x x x \where{n=3}$

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="hypergeon0" name="where"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="n"/>
      </OMBVAR>
      <OMA><OMS cd="relation1" name="eq"/>
        <OMA><OMS cd="arith1" name="power"/>
          <OMV name="x"/>
          <OMV name="n"/>
        </OMA>
        <OMA><OMS cd="arith1" name="times"/>
          <OMA><OMS cd="arith1" name="times"/>
            <OMV name="x"/>
            <OMV name="x"/>
          </OMA>
          <OMV name="x"/>
        </OMA>
      </OMA>
    </OMBIND>
    <OMA><OMS cd="relation1" name="eq"/>
      <OMV name="n"/>
      <OMI> 3 </OMI>
    </OMA>
  </OMA>
</OMOBJ>

where (lambda [ n ] . (eq (power ( x, n) , times (times ( x, x) , x) ) ) , eq ( n, 3 ) )

$$\mathrm{where}(\lambda \hspace{0.17em}n.x^n=xxx,n=3)$$

Signatures:

sts

multi_power

Description:

multi_power is for using the multi-index notation.

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="hypergeon0" 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>

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

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

Signatures:

sts

cartesian_product_n

Description:

the cartesian product of n copies of the first argument. Binary function.

Commented Mathematical property (CMP):

$ Z^m \times Z^n = Z^{m+n} $

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath">
  <OMA><OMS cd="relation1" name="eq"/>
    <OMA><OMS cd="set1" name="cartesian_product"/>
      <OMA><OMS cd="hypergeon0" name="cartesian_product_n"/>
        <OMS cd="setname1" name="Z"/>
        <OMV name="m"/>
      </OMA>
      <OMA><OMS cd="hypergeon0" name="cartesian_product_n"/>
        <OMS cd="setname1" name="Z"/>
        <OMV name="n"/>
      </OMA>
    </OMA>
    <OMA><OMS cd="hypergeon0" name="cartesian_product_n"/>
      <OMS cd="setname1" name="Z"/>
      <OMA><OMS cd="arith1" name="plus"/>
        <OMV name="m"/>
        <OMV name="n"/>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

eq (cartesian_product (cartesian_product_n (Z, m) , cartesian_product_n (Z, n) ) , cartesian_product_n (Z, plus ( m, n) ) )

$$\mathrm{cartesian\_product\_n}(\mathbb{Z},m)\times \mathrm{cartesian\_product\_n}(\mathbb{Z},n)=\mathrm{cartesian\_product\_n}(\mathbb{Z},m+n)$$

Signatures:

sts