OpenMath Content Dictionary: norm1

Canonical URL:

http://www.openmath.org/cd/norm1.ocd

CD File:

norm1.ocd

CD as XML Encoded OpenMath:

norm1.omcd

Defines:

Euclidean_norm, L_infinity_norm, L_norm

Date:

2003-09-16

Version:

1 (Revision 2)

Review Date:

2017-12-31

Status:

experimental

---

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     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
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  Author: James Davenport

This CD contains definitions of various norms.

---

L_norm

Description:

This symbol signifies the $L_p$ norm for any $p$ (the case of $L_\infty$ is handled specially).

Commented Mathematical property (CMP):

$L_p(v)=\left(\sum_{i=1}^{size v}|v_i|^p\right)^{1/p}$

Formal Mathematical property (FMP):

OpenMath XML (source)

     <OMOBJ xmlns="http://www.openmath.org/OpenMath">
       <OMA>
         <OMS name="eq" cd="relation1"/>
         <OMA>
           <OMS name="L_norm" cd="norm1"/>
	   <OMV name="p"/>
	   <OMV name="v"/>
         </OMA>
         <OMA>
           <OMS name="power" cd="arith1"/>
           <OMA>
             <OMS name="sum" cd="arith1"/>
             <OMA>
               <OMS name="integer_interval" cd="interval1"/>
	       <OMS name="one" cd="alg1"/>
               <OMA>
	         <OMS name="size" cd="set1"/>
	         <OMV name="v"/>
               </OMA>
             </OMA>
             <OMBIND>
               <OMS cd="fns1" name="lambda"/>
                 <OMBVAR>
                   <OMV name="i"/>
                 </OMBVAR>
                 <OMA>
                   <OMS cd="arith1" name="power"/>
                   <OMA>
		     <OMS name="abs" cd="arith1"/>
                     <OMA>
                       <OMS cd="linalg1" name="vector_selector"/>
                       <OMV name="v"/>
                       <OMV name="i"/>
                     </OMA>
                   </OMA>
	           <OMV name="p"/>
                 </OMA>
             </OMBIND>
           </OMA>
           <OMA>
             <OMS name="divide" cd="arith1"/>
	     <OMS name="one" cd="alg1"/>
	     <OMV name="p"/>
           </OMA>
         </OMA>
       </OMA>
     </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="norm1">L_norm</csymbol><ci>p</ci><ci>v</ci></apply>
  <apply><csymbol cd="arith1">power</csymbol>
   <apply><csymbol cd="arith1">sum</csymbol>
    <apply><csymbol cd="interval1">integer_interval</csymbol>
     <csymbol cd="alg1">one</csymbol>
     <apply><csymbol cd="set1">size</csymbol><ci>v</ci></apply>
    </apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>i</ci></bvar>
     <apply><csymbol cd="arith1">power</csymbol>
      <apply><csymbol cd="arith1">abs</csymbol>
       <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>v</ci><ci>i</ci></apply>
      </apply>
      <ci>p</ci>
     </apply>
    </bind>
   </apply>
   <apply><csymbol cd="arith1">divide</csymbol><csymbol cd="alg1">one</csymbol><ci>p</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (L_norm ( p, v) , power (sum (integer_interval (one, size ( v) ) , lambda [ i ] . (power (abs (vector_selector ( v, i) ) , p) ) ) , divide (one, p) ) )

Popcorn

norm1.L_norm($p, $v) = arith1.sum(interval1.integer_interval(alg1.one, set1.size($v)), fns1.lambda[$i -> arith1.abs(linalg1.vector_selector($v, $i)) ^ $p]) ^ (alg1.one / $p)

Rendered Presentation MathML

$$\mathrm{L\_norm}(p,v)={\sum _{i=1}^{\mathrm{size}\left(v\right)}}^{{\left|i_v\right|}^p}$$

Signatures:

sts

---

[Next: L_infinity_norm] [Last: Euclidean_norm] [Top]

---

L_infinity_norm

Description:

This symbol signifies the $L_\infty$ norm.

Commented Mathematical property (CMP):

$L_\infty(v)=\max_{i=1}^{size v}|v_i|$

Formal Mathematical property (FMP):

OpenMath XML (source)

     <OMOBJ xmlns="http://www.openmath.org/OpenMath">
       <OMA>
         <OMS name="eq" cd="relation1"/>
         <OMA>
           <OMS name="L_infinity_norm" cd="norm1"/>
	   <OMV name="v"/>
         </OMA>
         <OMA>
           <OMS cd="fns2" name="apply_to_list"/>
           <OMS cd="minmax1" name="max"/>
           <OMA>
             <OMS cd="list1" name="make_list"/>
             <OMI> 1 </OMI>
             <OMA>
	       <OMS name="size" cd="set1"/>
	       <OMV name="v"/>
             </OMA>
             <OMBIND>
               <OMS cd="fns1" name="lambda"/>
               <OMBVAR>
                 <OMV name="i"/>
               </OMBVAR>
               <OMA>
                 <OMS cd="arith1" name="abs"/>
                 <OMA>
                   <OMS cd="linalg1" name="vector_selector"/>
                   <OMV name="v"/>
                   <OMV name="i"/>
                 </OMA>
               </OMA>
             </OMBIND>
           </OMA>
         </OMA>
       </OMA>
     </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="norm1">L_infinity_norm</csymbol><ci>v</ci></apply>
  <apply><csymbol cd="fns2">apply_to_list</csymbol>
   <csymbol cd="minmax1">max</csymbol>
   <apply><csymbol cd="list1">make_list</csymbol>
    <cn type="integer">1</cn>
    <apply><csymbol cd="set1">size</csymbol><ci>v</ci></apply>
    <bind><csymbol cd="fns1">lambda</csymbol>
     <bvar><ci>i</ci></bvar>
     <apply><csymbol cd="arith1">abs</csymbol>
      <apply><csymbol cd="linalg1">vector_selector</csymbol><ci>v</ci><ci>i</ci></apply>
     </apply>
    </bind>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (L_infinity_norm ( v) , apply_to_list (max, make_list ( 1 , size ( v) , lambda [ i ] . (abs (vector_selector ( v, i) ) ) ) ) )

Popcorn

norm1.L_infinity_norm($v) = fns2.apply_to_list(minmax1.max, list1.make_list(1, set1.size($v), fns1.lambda[$i -> arith1.abs(linalg1.vector_selector($v, $i))]))

Rendered Presentation MathML

$$\mathrm{L\_infinity\_norm}\left(v\right)=\mathrm{apply\_to\_list}(\max ,\mathrm{make\_list}(1,\mathrm{size}\left(v\right),\lambda \hspace{0.17em}i.\left|i_v\right|))$$

Signatures:

sts

---

[Next: Euclidean_norm] [Previous: L_norm] [Top]

---

Euclidean_norm

Description:

This symbol signifies the Euclidean ($L_2$) norm.

Commented Mathematical property (CMP):

$L_2(v)=$ Euclidean_norm(v)

Formal Mathematical property (FMP):

OpenMath XML (source)

     <OMOBJ xmlns="http://www.openmath.org/OpenMath">
       <OMA>
         <OMS name="eq" cd="relation1"/>
         <OMA>
           <OMS name="Euclidean_norm" cd="norm1"/>
	   <OMV name="v"/>
         </OMA>
           <OMS name="L_norm" cd="norm1"/>
	   <OMI> 2 </OMI>
	   <OMV name="v"/>
       </OMA>
     </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="norm1">Euclidean_norm</csymbol><ci>v</ci></apply>
  <csymbol cd="norm1">L_norm</csymbol>
  <cn type="integer">2</cn>
  <ci>v</ci>
 </apply>
</math>

Prefix

eq (Euclidean_norm ( v) , L_norm, 2 , v)

Popcorn

norm1.Euclidean_norm($v) = norm1.L_norm = 2 = $v

Rendered Presentation MathML

$$\mathrm{Euclidean\_norm}\left(v\right)=\mathrm{L\_norm}=2=v$$

Signatures:

sts

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[First: L_norm] [Previous: L_infinity_norm] [Top]