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chapter4.html
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<!DOCTYPE HTML>
<html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Content Markup</title>
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
<style type="text/css">
/** Table of Contents taken from html 5.2 spec
stop flicker in toc **/
.toc a {
/* More spacing; use padding to make it part of the click target. */
padding-top: 0.1rem;
/* Larger, more consistently-sized click target */
display: block;
/* Reverse color scheme */
color: black;
border-color: #3980B5;
border-bottom-width: 3px !important;
margin-bottom: 0px !important;
}
.egmeta {
color:#5555AA;font-style:italic;font-family:serif;font-weight:bold;
}
div.graphic{margin-left:2em}
table.syntax {
font-size: 75%;
background-color: #DDDDDD;
border: thin solid;
}
table.syntax td {
border: solid thin;
}
table.syntax th {
text-align: left;
}
table.attributes td { padding-left:0.5em; padding-right:0.5em; border: solid thin; }
table.attributes td.attname { white-space:nowrap; vertical-align:top;}
table.attributes td.attdesc { background-color:#F0F0FF; padding-left:2em; padding-right:2em}
th.uname {font-size: 50%; text-align:left;}
code { font-family: monospace; }
div.constraint,
div.issue,
div.note,
div.notice { margin-left: 2em; }
li p { margin-top: 0.3em;
margin-bottom: 0.3em; }
div.exampleInner pre { margin-left: 1em;
margin-top: 0em; margin-bottom: 0em}
div.exampleOuter {border: 4px double gray;
margin: 0em; padding: 0em}
div.exampleInner { background-color: #d5dee3;
border-top-width: 4px;
border-top-style: double;
border-top-color: #d3d3d3;
border-bottom-width: 4px;
border-bottom-style: double;
border-bottom-color: #d3d3d3;
padding: 4px; margin: 0em }
div.exampleWrapper { margin: 4px }
div.exampleHeader { font-weight: bold;
margin: 4px}
a.mainindex {font-weight: bold;}
li.sitem {list-style-type: none;}
.error { color: red }
div.mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example h5 {
margin-top: -0.3em;
margin-bottom: -0.5em;}
var.meta {background-color:green}
var.transmeta {background-color:red}
pre.mathml {padding: 0.5em;
background-color: #FFFFDD;}
pre.mathml-fragment {padding: 0.5em;
background-color: #FFFFDD;}
pre.strict-mathml {padding: 0.5em;
background-color: #FFFFDD;}
span.uname {color:#999900;font-size:75%;font-family:sans-serif;}
.minitoc { border-style: solid;
border-color: #0050B2;
border-width: 1px ;
padding: 0.3em;}
.attention { border-style: solid;
border-width: 1px ;
color: #5D0091;
background: #F9F5DE;
border-color: red;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.attribute-Name { background: #F9F5C0; }
.method-Name { background: #C0C0F9; }
.IDL-definition { border-style: solid;
border-width: 1px ;
color: #001000;
background: #E0FFE0;
border-color: #206020;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.baseline {vertical-align: baseline}
#eqnoc1 {width: 10%}
#eqnoc2 {width: 80%; text-align: center; }
#eqnoc3 {width: 10%; text-align: right; }
div.div1 {margin-bottom: 1em;}
.h3style {
text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2;
font-size: 125%;
}
h4 { text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2; }
h5 { text-align: left;
font-family: sans-serif;
font-weight: bold;
color: #0050B2; }
th {background: #E0FFE0;}
p, blockquote, h4 { font-family: sans-serif; }
dt, dd, dl, ul, li { font-family: sans-serif; }
pre, code { font-family: monospace }
a.termref {
text-decoration: none;
color: black;
}
.mathml-render {
font-family: serif;
font-size: 130%;
border: solid 4px green;
padding-left: 1em;
padding-right: 1em;
}
</style>
<link rel="stylesheet" type="text/css" href="https://www.w3.org/StyleSheets/TR/2016/W3C-ED.css"><script src="//www.w3.org/scripts/TR/2016/fixup.js"> </script></head>
<body>
<h1><a id="contm"></a>4 Content Markup</h1>
<!-- TOP NAVIGATION BAR -->
<nav id="toc">
<ol class="toc">
<li><a href="Overview.html"><span class="secno"></span> Mathematical Markup Language (MathML) Version 4.0</a></li>
<li><a href="chapter3.html"><span class="secno">3 </span> Presentation Markup</a></li>
<li><a href="chapter5.html"><span class="secno">5 </span> Mixing Markup Languages for Mathematical Expressions</a></li>
<li><a href="chapter4.html"><span class="secno">4 </span> Content Markup</a>
<ol class="toc">
<li><a href="chapter4.html#contm.intro"><span class="secno">4.1 </span> Introduction</a>
<ol class="toc">
<li><a href="chapter4.html#id.4.1.1"><span class="secno">4.1.1 </span> The Intent of Content Markup</a>
</li>
<li><a href="chapter4.html#contm.rendering"><span class="secno">4.1.2 </span> The Structure and Scope of Content MathML Expressions</a>
</li>
<li><a href="chapter4.html#contm.strict"><span class="secno">4.1.3 </span> Strict Content MathML</a>
</li>
<li><a href="chapter4.html#contm.cds"><span class="secno">4.1.4 </span> Content Dictionaries</a>
</li>
<li><a href="chapter4.html#contm.concepts"><span class="secno">4.1.5 </span> Content MathML Concepts</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.core"><span class="secno">4.2 </span> Content MathML Elements Encoding Expression Structure</a>
<ol class="toc">
<li><a href="chapter4.html#contm.cn"><span class="secno">4.2.1 </span> Numbers <code><cn></code></a>
<ol class="toc">
<li><a href="chapter4.html#contm.rendering.numbers"><span class="secno">4.2.1.1 </span> Rendering <code><cn></code>,<code><sep/></code>-Represented Numbers </a>
</li>
<li><a href="chapter4.html#contm.cn.strict"><span class="secno">4.2.1.2 </span> Strict uses of <code><cn></code></a>
</li>
<li><a href="chapter4.html#contm.cn.extended"><span class="secno">4.2.1.3 </span> Non-Strict uses of <code><cn></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.ci"><span class="secno">4.2.2 </span> Content Identifiers <code><ci></code></a>
<ol class="toc">
<li><a href="chapter4.html#contm.ci.strict"><span class="secno">4.2.2.1 </span> Strict uses of <code><ci></code></a>
</li>
<li><a href="chapter4.html#contm.ci.extended"><span class="secno">4.2.2.2 </span> Non-Strict uses of <code><ci></code></a>
</li>
<li><a href="chapter4.html#contm.rendering.ci"><span class="secno">4.2.2.3 </span> Rendering Content Identifiers</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.csymbol"><span class="secno">4.2.3 </span> Content Symbols <code><csymbol></code></a>
<ol class="toc">
<li><a href="chapter4.html#contm.csymbol.strict"><span class="secno">4.2.3.1 </span> Strict uses of <code><csymbol></code></a>
</li>
<li><a href="chapter4.html#contm.csymbol.extended"><span class="secno">4.2.3.2 </span> Non-Strict uses of <code><csymbol></code></a>
</li>
<li><a href="chapter4.html#contm.rendering.csymbol"><span class="secno">4.2.3.3 </span> Rendering Symbols</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.cs"><span class="secno">4.2.4 </span> String Literals <code><cs></code></a>
</li>
<li><a href="chapter4.html#contm.apply"><span class="secno">4.2.5 </span> Function Application <code><apply></code></a>
<ol class="toc">
<li><a href="chapter4.html#contm.applications.strict"><span class="secno">4.2.5.1 </span> Strict Content MathML</a>
</li>
<li><a href="chapter4.html#contm.rendering.applications"><span class="secno">4.2.5.2 </span> Rendering Applications</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.binding"><span class="secno">4.2.6 </span> Bindings and Bound Variables <code><bind></code>
and <code><bvar></code></a>
<ol class="toc">
<li><a href="chapter4.html#contm.bind"><span class="secno">4.2.6.1 </span> Bindings</a>
</li>
<li><a href="chapter4.html#contm.bvar"><span class="secno">4.2.6.2 </span> Bound Variables</a>
</li>
<li><a href="chapter4.html#contm.alpharenmaing"><span class="secno">4.2.6.3 </span> Renaming Bound Variables</a>
</li>
<li><a href="chapter4.html#contm.rendering.binders"><span class="secno">4.2.6.4 </span> Rendering Binding Constructions</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.sharing"><span class="secno">4.2.7 </span> Structure Sharing <code><share></code></a>
<ol class="toc">
<li><a href="chapter4.html#contm.share"><span class="secno">4.2.7.1 </span> The <code>share</code> element</a>
</li>
<li><a href="chapter4.html#contm.acyclicity"><span class="secno">4.2.7.2 </span> An Acyclicity Constraint</a>
</li>
<li><a href="chapter4.html#contm.share.binding"><span class="secno">4.2.7.3 </span> Structure Sharing and Binding</a>
</li>
<li><a href="chapter4.html#contm.rendering.share"><span class="secno">4.2.7.4 </span> Rendering Expressions with Structure Sharing</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.semantics"><span class="secno">4.2.8 </span> Attribution via <code>semantics</code></a>
</li>
<li><a href="chapter4.html#contm.cerror"><span class="secno">4.2.9 </span> Error Markup <code><cerror></code></a>
</li>
<li><a href="chapter4.html#contm.cbytes"><span class="secno">4.2.10 </span> Encoded Bytes <code><cbytes></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.structure.extended"><span class="secno">4.3 </span> Content MathML for Specific Structures</a>
<ol class="toc">
<li><a href="chapter4.html#contm.container"><span class="secno">4.3.1 </span> Container Markup</a>
<ol class="toc">
<li><a href="chapter4.html#contm.container.constructor"><span class="secno">4.3.1.1 </span> Container Markup for Constructor Symbols</a>
</li>
<li><a href="chapter4.html#contm.lambda.container"><span class="secno">4.3.1.2 </span> Container Markup for Binding Constructors</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.bind.apply"><span class="secno">4.3.2 </span> Bindings with <code><apply></code></a>
</li>
<li><a href="chapter4.html#contm.qualifiers"><span class="secno">4.3.3 </span> Qualifiers</a>
<ol class="toc">
<li><a href="chapter4.html#contm.domainofapplication.qualifier"><span class="secno">4.3.3.1 </span> Uses of
<code><domainofapplication></code>,
<code><interval></code>,
<code><condition></code>,
<code><lowlimit></code> and
<code><uplimit></code></a>
</li>
<li><a href="chapter4.html#contm.degree"><span class="secno">4.3.3.2 </span> Uses of <code><degree></code></a>
</li>
<li><a href="chapter4.html#contm.otherqualifiers"><span class="secno">4.3.3.3 </span> Uses of <code><momentabout></code> and <code><logbase></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.opclasses"><span class="secno">4.3.4 </span> Operator Classes</a>
<ol class="toc">
<li><a href="chapter4.html#contm.nary"><span class="secno">4.3.4.1 </span> N-ary Operators (classes nary-arith, nary-functional, nary-logical,
nary-linalg, nary-set, nary-constructor)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.nary.schema"><span class="secno">4.3.4.1.1 </span> Schema Patterns</a>
</li>
<li><a href="chapter4.html#contm.nary.rewrite"><span class="secno">4.3.4.1.2 </span> Rewriting to Strict Content MathML</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.nary.setlist"><span class="secno">4.3.4.2 </span> N-ary Constructors for set and list (class nary-setlist-constructor)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.narysetlist.schema"><span class="secno">4.3.4.2.1 </span> Schema Patterns</a>
</li>
<li><a href="chapter4.html#contm.narysetlist.rewrite"><span class="secno">4.3.4.2.2 </span> Rewriting to Strict Content MathML</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.nary.reln"><span class="secno">4.3.4.3 </span> N-ary Relations (classes nary-reln, nary-set-reln)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.nary.reln.schema"><span class="secno">4.3.4.3.1 </span> Schema Patterns</a>
</li>
<li><a href="chapter4.html#contm.nary.reln.rewrite"><span class="secno">4.3.4.3.2 </span> Rewriting to Strict Content MathML</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.nary.unary"><span class="secno">4.3.4.4 </span> N-ary/Unary Operators (classes nary-minmax, nary-stats)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.nary.unary.schema"><span class="secno">4.3.4.4.1 </span> Schema Patterns</a>
</li>
<li><a href="chapter4.html#contm.nary.unary.rewrite"><span class="secno">4.3.4.4.2 </span> Rewriting to Strict Content MathML</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.binary"><span class="secno">4.3.4.5 </span> Binary Operators (classes binary-arith, binary-logical, binary-reln, binary-linalg,
binary-set)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.binary.schema"><span class="secno">4.3.4.5.1 </span> Schema Patterns</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.unary"><span class="secno">4.3.4.6 </span> Unary Operators (classes unary-arith, unary-linalg, unary-functional, unary-set,
unary-elementary, unary-veccalc)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.unary.schema"><span class="secno">4.3.4.6.1 </span> Schema Patterns</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.constant"><span class="secno">4.3.4.7 </span> Constants (classes constant-arith, constant-set)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.constant.schema"><span class="secno">4.3.4.7.1 </span> Schema Patterns</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.quantifier"><span class="secno">4.3.4.8 </span> Quantifiers (class quantifier)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.quantifier.schema"><span class="secno">4.3.4.8.1 </span> Schema Patterns</a>
</li>
<li><a href="chapter4.html#contm.quantifier.rewrite"><span class="secno">4.3.4.8.2 </span> Rewriting to Strict Content MathML</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.otherclass"><span class="secno">4.3.4.9 </span> Other Operators (classes lambda, interval, int, diff partialdiff, sum, product, limit)</a>
<ol class="toc">
<li><a href="chapter4.html#contm.otherclass.schema"><span class="secno">4.3.4.9.1 </span> Schema Patterns</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter4.html#id.4.3.5"><span class="secno">4.3.5 </span> Non-strict Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.opel"><span class="secno">4.4 </span> Content MathML for Specific Operators and Constants</a>
<ol class="toc">
<li><a href="chapter4.html#contm.basicfun"><span class="secno">4.4.1 </span> Functions and Inverses</a>
<ol class="toc">
<li><a href="chapter4.html#contm.interval"><span class="secno">4.4.1.1 </span> Interval <code><interval></code></a>
</li>
<li><a href="chapter4.html#contm.inverse"><span class="secno">4.4.1.2 </span> Inverse <code><inverse></code></a>
</li>
<li><a href="chapter4.html#contm.lambda"><span class="secno">4.4.1.3 </span> Lambda <code><lambda></code></a>
</li>
<li><a href="chapter4.html#contm.compose"><span class="secno">4.4.1.4 </span> Function composition <code><compose/></code></a>
</li>
<li><a href="chapter4.html#contm.ident"><span class="secno">4.4.1.5 </span> Identity function <code><ident/></code></a>
</li>
<li><a href="chapter4.html#contm.domain"><span class="secno">4.4.1.6 </span> Domain <code><domain/></code></a>
</li>
<li><a href="chapter4.html#contm.codomain"><span class="secno">4.4.1.7 </span> codomain <code><codomain/></code></a>
</li>
<li><a href="chapter4.html#contm.image"><span class="secno">4.4.1.8 </span> Image <code><image/></code></a>
</li>
<li><a href="chapter4.html#contm.piecewise"><span class="secno">4.4.1.9 </span> Piecewise declaration <code><piecewise></code>, <code><piece></code>, <code><otherwise></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#id.4.4.2"><span class="secno">4.4.2 </span> Arithmetic, Algebra and Logic</a>
<ol class="toc">
<li><a href="chapter4.html#contm.quotient"><span class="secno">4.4.2.1 </span> Quotient <code><quotient/></code></a>
</li>
<li><a href="chapter4.html#contm.factorial"><span class="secno">4.4.2.2 </span> Factorial <code><factorial/></code></a>
</li>
<li><a href="chapter4.html#contm.divide"><span class="secno">4.4.2.3 </span> Division <code><divide/></code></a>
</li>
<li><a href="chapter4.html#contm.max"><span class="secno">4.4.2.4 </span> Maximum <code><max/></code></a>
</li>
<li><a href="chapter4.html#contm.min"><span class="secno">4.4.2.5 </span> Minimum <code><min/></code></a>
</li>
<li><a href="chapter4.html#contm.minus"><span class="secno">4.4.2.6 </span> Subtraction <code><minus/></code></a>
</li>
<li><a href="chapter4.html#contm.plus"><span class="secno">4.4.2.7 </span> Addition <code><plus/></code></a>
</li>
<li><a href="chapter4.html#contm.power"><span class="secno">4.4.2.8 </span> Exponentiation <code><power/></code></a>
</li>
<li><a href="chapter4.html#contm.rem"><span class="secno">4.4.2.9 </span> Remainder <code><rem/></code></a>
</li>
<li><a href="chapter4.html#contm.times"><span class="secno">4.4.2.10 </span> Multiplication <code><times/></code></a>
</li>
<li><a href="chapter4.html#contm.root"><span class="secno">4.4.2.11 </span> Root <code><root/></code></a>
</li>
<li><a href="chapter4.html#contm.gcd"><span class="secno">4.4.2.12 </span> Greatest common divisor <code><gcd/></code></a>
</li>
<li><a href="chapter4.html#contm.and"><span class="secno">4.4.2.13 </span> And <code><and/></code></a>
</li>
<li><a href="chapter4.html#contm.or"><span class="secno">4.4.2.14 </span> Or <code><or/></code></a>
</li>
<li><a href="chapter4.html#contm.xor"><span class="secno">4.4.2.15 </span> Exclusive Or <code><xor/></code></a>
</li>
<li><a href="chapter4.html#contm.not"><span class="secno">4.4.2.16 </span> Not <code><not/></code></a>
</li>
<li><a href="chapter4.html#contm.implies"><span class="secno">4.4.2.17 </span> Implies <code><implies/></code></a>
</li>
<li><a href="chapter4.html#contm.forall"><span class="secno">4.4.2.18 </span> Universal quantifier <code><forall/></code></a>
</li>
<li><a href="chapter4.html#contm.exists"><span class="secno">4.4.2.19 </span> Existential quantifier <code><exists/></code></a>
</li>
<li><a href="chapter4.html#contm.abs"><span class="secno">4.4.2.20 </span> Absolute Value <code><abs/></code></a>
</li>
<li><a href="chapter4.html#contm.conjugate"><span class="secno">4.4.2.21 </span> Complex conjugate <code><conjugate/></code></a>
</li>
<li><a href="chapter4.html#contm.arg"><span class="secno">4.4.2.22 </span> Argument <code><arg/></code></a>
</li>
<li><a href="chapter4.html#contm.real"><span class="secno">4.4.2.23 </span> Real part <code><real/></code></a>
</li>
<li><a href="chapter4.html#contm.imaginary"><span class="secno">4.4.2.24 </span> Imaginary part <code><imaginary/></code></a>
</li>
<li><a href="chapter4.html#contm.lcm"><span class="secno">4.4.2.25 </span> Lowest common multiple <code><lcm/></code></a>
</li>
<li><a href="chapter4.html#contm.floor"><span class="secno">4.4.2.26 </span> Floor <code><floor/></code></a>
</li>
<li><a href="chapter4.html#contm.ceiling"><span class="secno">4.4.2.27 </span> Ceiling <code><ceiling/></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#id.4.4.3"><span class="secno">4.4.3 </span> Relations</a>
<ol class="toc">
<li><a href="chapter4.html#contm.eq"><span class="secno">4.4.3.1 </span> Equals <code><eq/></code></a>
</li>
<li><a href="chapter4.html#contm.neq"><span class="secno">4.4.3.2 </span> Not Equals <code><neq/></code></a>
</li>
<li><a href="chapter4.html#contm.gt"><span class="secno">4.4.3.3 </span> Greater than <code><gt/></code></a>
</li>
<li><a href="chapter4.html#contm.lt"><span class="secno">4.4.3.4 </span> Less Than <code><lt/></code></a>
</li>
<li><a href="chapter4.html#contm.geq"><span class="secno">4.4.3.5 </span> Greater Than or Equal <code><geq/></code></a>
</li>
<li><a href="chapter4.html#contm.leq"><span class="secno">4.4.3.6 </span> Less Than or Equal <code><leq/></code></a>
</li>
<li><a href="chapter4.html#contm.equivalent"><span class="secno">4.4.3.7 </span> Equivalent <code><equivalent/></code></a>
</li>
<li><a href="chapter4.html#contm.approx"><span class="secno">4.4.3.8 </span> Approximately <code><approx/></code></a>
</li>
<li><a href="chapter4.html#contm.factorof"><span class="secno">4.4.3.9 </span> Factor Of <code><factorof/></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#id.4.4.4"><span class="secno">4.4.4 </span> Calculus and Vector Calculus</a>
<ol class="toc">
<li><a href="chapter4.html#contm.int"><span class="secno">4.4.4.1 </span> Integral <code><int/></code></a>
</li>
<li><a href="chapter4.html#contm.diff"><span class="secno">4.4.4.2 </span> Differentiation <code><diff/></code></a>
</li>
<li><a href="chapter4.html#contm.partialdiff"><span class="secno">4.4.4.3 </span> Partial Differentiation <code><partialdiff/></code></a>
</li>
<li><a href="chapter4.html#contm.divergence"><span class="secno">4.4.4.4 </span> Divergence <code><divergence/></code></a>
</li>
<li><a href="chapter4.html#contm.grad"><span class="secno">4.4.4.5 </span> Gradient <code><grad/></code></a>
</li>
<li><a href="chapter4.html#contm.curl"><span class="secno">4.4.4.6 </span> Curl <code><curl/></code></a>
</li>
<li><a href="chapter4.html#contm.laplacian"><span class="secno">4.4.4.7 </span> Laplacian <code><laplacian/></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.sets"><span class="secno">4.4.5 </span> Theory of Sets</a>
<ol class="toc">
<li><a href="chapter4.html#contm.set"><span class="secno">4.4.5.1 </span> Set <code><set></code></a>
</li>
<li><a href="chapter4.html#contm.list"><span class="secno">4.4.5.2 </span> List <code><list></code></a>
</li>
<li><a href="chapter4.html#contm.union"><span class="secno">4.4.5.3 </span> Union <code><union/></code></a>
</li>
<li><a href="chapter4.html#contm.intersect"><span class="secno">4.4.5.4 </span> Intersect <code><intersect/></code></a>
</li>
<li><a href="chapter4.html#contm.in"><span class="secno">4.4.5.5 </span> Set inclusion <code><in/></code></a>
</li>
<li><a href="chapter4.html#contm.notin"><span class="secno">4.4.5.6 </span> Set exclusion <code><notin/></code></a>
</li>
<li><a href="chapter4.html#contm.subset"><span class="secno">4.4.5.7 </span> Subset <code><subset/></code></a>
</li>
<li><a href="chapter4.html#contm.prsubset"><span class="secno">4.4.5.8 </span> Proper Subset <code><prsubset/></code></a>
</li>
<li><a href="chapter4.html#contm.notsubset"><span class="secno">4.4.5.9 </span> Not Subset <code><notsubset/></code></a>
</li>
<li><a href="chapter4.html#contm.notprsubset"><span class="secno">4.4.5.10 </span> Not Proper Subset <code><notprsubset/></code></a>
</li>
<li><a href="chapter4.html#contm.setdiff"><span class="secno">4.4.5.11 </span> Set Difference <code><setdiff/></code></a>
</li>
<li><a href="chapter4.html#contm.card"><span class="secno">4.4.5.12 </span> Cardinality <code><card/></code></a>
</li>
<li><a href="chapter4.html#contm.cartesianproduct"><span class="secno">4.4.5.13 </span> Cartesian product <code><cartesianproduct/></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#id.4.4.6"><span class="secno">4.4.6 </span> Sequences and Series</a>
<ol class="toc">
<li><a href="chapter4.html#contm.sum"><span class="secno">4.4.6.1 </span> Sum <code><sum/></code></a>
</li>
<li><a href="chapter4.html#contm.product"><span class="secno">4.4.6.2 </span> Product <code><product/></code></a>
</li>
<li><a href="chapter4.html#contm.limit"><span class="secno">4.4.6.3 </span> Limits <code><limit/></code></a>
</li>
<li><a href="chapter4.html#contm.tendsto"><span class="secno">4.4.6.4 </span> Tends To <code><tendsto/></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.elemclass"><span class="secno">4.4.7 </span> Elementary classical functions</a>
<ol class="toc">
<li><a href="chapter4.html#contm.trig"><span class="secno">4.4.7.1 </span> Common trigonometric functions
<code><sin/></code>,
<code><cos/></code>,
<code><tan/></code>,
<code><sec/></code>,
<code><csc/></code>,
<code><cot/></code>
</a>
</li>
<li><a href="chapter4.html#contm.invtrig"><span class="secno">4.4.7.2 </span> Common inverses of trigonometric functions
<code><arcsin/></code>,
<code><arccos/></code>,
<code><arctan/></code>,
<code><arcsec/></code>,
<code><arccsc/></code>,
<code><arccot/></code></a>
</li>
<li><a href="chapter4.html#contm.hyperbolic"><span class="secno">4.4.7.3 </span> Common hyperbolic functions
<code><sinh/></code>,
<code><cosh/></code>,
<code><tanh/></code>,
<code><sech/></code>,
<code><csch/></code>,
<code><coth/></code></a>
</li>
<li><a href="chapter4.html#contm.invhyperbolic"><span class="secno">4.4.7.4 </span> Common inverses of hyperbolic functions
<code><arcsinh/></code>,
<code><arccosh/></code>,
<code><arctanh/></code>,
<code><arcsech/></code>,
<code><arccsch/></code>,
<code><arccoth/></code> </a>
</li>
<li><a href="chapter4.html#contm.exp"><span class="secno">4.4.7.5 </span> Exponential <code><exp/></code></a>
</li>
<li><a href="chapter4.html#contm.ln"><span class="secno">4.4.7.6 </span> Natural Logarithm <code><ln/></code></a>
</li>
<li><a href="chapter4.html#contm.log"><span class="secno">4.4.7.7 </span> Logarithm <code><log/></code>
, <code><logbase></code>
</a>
</li>
</ol>
</li>
<li><a href="chapter4.html#id.4.4.8"><span class="secno">4.4.8 </span> Statistics</a>
<ol class="toc">
<li><a href="chapter4.html#contm.mean"><span class="secno">4.4.8.1 </span> Mean <code><mean/></code></a>
</li>
<li><a href="chapter4.html#contm.sdev"><span class="secno">4.4.8.2 </span> Standard Deviation <code><sdev/></code></a>
</li>
<li><a href="chapter4.html#contm.variance"><span class="secno">4.4.8.3 </span> Variance <code><variance/></code></a>
</li>
<li><a href="chapter4.html#contm.median"><span class="secno">4.4.8.4 </span> Median <code><median/></code></a>
</li>
<li><a href="chapter4.html#contm.mode"><span class="secno">4.4.8.5 </span> Mode <code><mode/></code></a>
</li>
<li><a href="chapter4.html#contm.moment"><span class="secno">4.4.8.6 </span> Moment <code><moment/></code>, <code><momentabout></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#id.4.4.9"><span class="secno">4.4.9 </span> Linear Algebra</a>
<ol class="toc">
<li><a href="chapter4.html#contm.vector"><span class="secno">4.4.9.1 </span> Vector <code><vector></code></a>
</li>
<li><a href="chapter4.html#contm.matrix"><span class="secno">4.4.9.2 </span> Matrix <code><matrix></code></a>
</li>
<li><a href="chapter4.html#contm.matrixrow"><span class="secno">4.4.9.3 </span> Matrix row <code><matrixrow></code></a>
</li>
<li><a href="chapter4.html#contm.determinant"><span class="secno">4.4.9.4 </span> Determinant <code><determinant/></code></a>
</li>
<li><a href="chapter4.html#contm.transpose"><span class="secno">4.4.9.5 </span> Transpose <code><transpose/></code></a>
</li>
<li><a href="chapter4.html#contm.selector"><span class="secno">4.4.9.6 </span> Selector <code><selector/></code></a>
</li>
<li><a href="chapter4.html#contm.vectorproduct"><span class="secno">4.4.9.7 </span> Vector product <code><vectorproduct/></code></a>
</li>
<li><a href="chapter4.html#contm.scalarproduct"><span class="secno">4.4.9.8 </span> Scalar product <code><scalarproduct/></code></a>
</li>
<li><a href="chapter4.html#contm.outerproduct"><span class="secno">4.4.9.9 </span> Outer product <code><outerproduct/></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.constantsandsymbols"><span class="secno">4.4.10 </span> Constant and Symbol Elements</a>
<ol class="toc">
<li><a href="chapter4.html#contm.integers"><span class="secno">4.4.10.1 </span> integers <code><integers/></code></a>
</li>
<li><a href="chapter4.html#contm.reals"><span class="secno">4.4.10.2 </span> reals <code><reals/></code></a>
</li>
<li><a href="chapter4.html#contm.rationals"><span class="secno">4.4.10.3 </span> Rational Numbers <code><rationals/></code></a>
</li>
<li><a href="chapter4.html#contm.naturalnumbers"><span class="secno">4.4.10.4 </span> Natural Numbers <code><naturalnumbers/></code></a>
</li>
<li><a href="chapter4.html#contm.complexes"><span class="secno">4.4.10.5 </span> complexes <code><complexes/></code></a>
</li>
<li><a href="chapter4.html#contm.primes"><span class="secno">4.4.10.6 </span> primes <code><primes/></code></a>
</li>
<li><a href="chapter4.html#contm.exponentiale"><span class="secno">4.4.10.7 </span> Exponential e <code><exponentiale/></code></a>
</li>
<li><a href="chapter4.html#contm.imaginaryi"><span class="secno">4.4.10.8 </span> Imaginary i <code><imaginaryi/></code></a>
</li>
<li><a href="chapter4.html#contm.notanumber"><span class="secno">4.4.10.9 </span> Not A Number <code><notanumber/></code></a>
</li>
<li><a href="chapter4.html#contm.true"><span class="secno">4.4.10.10 </span> True <code><true/></code></a>
</li>
<li><a href="chapter4.html#contm.false"><span class="secno">4.4.10.11 </span> False <code><false/></code></a>
</li>
<li><a href="chapter4.html#contm.emptyset"><span class="secno">4.4.10.12 </span> Empty Set <code><emptyset/></code></a>
</li>
<li><a href="chapter4.html#contm.pi"><span class="secno">4.4.10.13 </span> pi <code><pi/></code></a>
</li>
<li><a href="chapter4.html#contm.eulergamma"><span class="secno">4.4.10.14 </span> Euler gamma <code><eulergamma/></code></a>
</li>
<li><a href="chapter4.html#contm.infinity"><span class="secno">4.4.10.15 </span> infinity <code><infinity/></code></a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.deprecated"><span class="secno">4.5 </span> Deprecated Content Elements</a>
<ol class="toc">
<li><a href="chapter4.html#contm.declare"><span class="secno">4.5.1 </span> Declare <code><declare></code></a>
</li>
<li><a href="chapter4.html#contm.reln"><span class="secno">4.5.2 </span> Relation <code><reln></code></a>
</li>
<li><a href="chapter4.html#contm.fn"><span class="secno">4.5.3 </span> Relation <code><fn></code></a>
</li>
</ol>
</li>
<li><a href="chapter4.html#contm.p2s"><span class="secno">4.6 </span> The Strict Content MathML Transformation</a>
</li>
</ol>
</li>
</ol>
</nav>
<div class="div1">
<div class="div2">
<h2><a id="contm.intro"></a>4.1 Introduction</h2>
<div class="div3">
<h3><a id="id.4.1.1"></a>4.1.1 The Intent of Content Markup</h3>
<p>The intent of Content Markup is to provide an explicit encoding of
the <em>underlying mathematical meaning</em> of an expression,
rather than any particular rendering for the expression. Mathematics
is distinguished both by its use of rigorous formal logic to define
and analyze mathematical concepts, and by the use of a (relatively)
formal notational system to represent and communicate those concepts.
However, mathematics and its presentation should not be viewed as one
and the same thing. Mathematical notation, though more rigorous than
natural language, is nonetheless at times ambiguous,
context-dependent, and varies from community to community. In some
cases, heuristics may adequately infer mathematical semantics from
mathematical notation. But in many others cases, it is preferable to
work directly with the underlying, formal, mathematical objects.
Content Markup provides a rigorous, extensible semantic framework and
a markup language for this purpose.
</p>
<p>The difficulties in inferring semantics from a presentation stem
from the fact that there are many to one mappings from presentation to
semantics and vice versa. For example the mathematical construct
"<var>H</var> multiplied by <var>e</var>" is often
encoded using an explicit operator as in
<var>H</var> × <var>e</var>. In different
presentational contexts, the multiplication operator might be
invisible "<var>H</var> <var>e</var>", or rendered
as the spoken word "times". Generally, many different
presentations are possible depending on the context and style
preferences of the author or reader. Thus, given
"<var>H</var> <var>e</var>" out of context it may be
impossible to decide if this is the name of a chemical or a
mathematical product of two variables <var>H</var> and
<var>e</var>. Mathematical presentation also varies across cultures
and geographical regions. For example, many notations for long
division are in use in different parts of the world today. Notations
may lose currency, for example the use of musical sharp and flat
symbols to denote maxima and minima <a href="appendixh.html#Chaundy1954">[Chaundy1954]</a>. A
notation in use in 1644 for the multiplication mentioned above was
<img src="image/f4001.gif" alt="\blacksquare" style="vertical-align:middle"><var>H</var><var>e</var> <a href="appendixh.html#Cajori1928">[Cajori1928]</a>.</p>
<p>By encoding the underlying mathematical structure explicitly,
without regard to how it is presented aurally or visually, it is
possible to interchange information more precisely between systems
that semantically process mathematical objects. In the trivial example
above, such a system could substitute values for the variables
<var>H</var> and <var>e</var> and evaluate the result. Important
application areas include computer algebra systems, automatic
reasoning system, industrial and scientific applications,
multi-lingual translation systems, mathematical search, and
interactive textbooks.</p>
<p>The organization of this chapter is as follows. In <a href="#contm.core">Section 4.2 Content MathML Elements Encoding Expression Structure</a>, a core collection of elements comprising Strict
Content Markup are described. Strict Content Markup is sufficient to
encode general expression trees in a semantically rigorous way. It is
in one-to-one correspondence with OpenMath element set. OpenMath is a
standard for representing formal mathematical objects and semantics
through the use of extensible Content Dictionaries. Strict Content
Markup defines a mechanism for associating precise mathematical
semantics with expression trees by referencing OpenMath Content
Dictionaries. The next two sections introduce markup that is more
convenient than Strict markup for some purposes, somewhat less formal
and verbose. In <a href="#contm.structure.extended">Section 4.3 Content MathML for Specific Structures</a>, markup is
introduced for representing a small number of mathematical idioms,
such as limits on integrals, sums and product. These constructs may
all be rewritten as Strict Content Markup expressions, and rules for
doing so are given. In <a href="#contm.opel">Section 4.4 Content MathML for Specific Operators and Constants</a>, elements are
introduced for many common function, operators and constants. This
section contains many examples, including equivalent Strict Content
expressions. In <a href="#contm.deprecated">Section 4.5 Deprecated Content Elements</a>, elements from
MathML 1 and 2 whose use is now discouraged are listed.
Finally, <a href="#contm.p2s">Section 4.6 The Strict Content MathML Transformation</a> summarizes the algorithm for
translating arbitrary Content Markup into Strict Content Markup. It
collects together in sequence all the rewrite rules introduced
throughout the rest of the chapter.</p>
</div>
<div class="div3">
<h3><a id="contm.rendering"></a>4.1.2 The Structure and Scope of Content MathML Expressions</h3>
<p>Content MathML represents mathematical objects as <em>expression trees</em>. The
notion of constructing a general expression tree is e.g. that of applying an operator
to
sub-objects. For example, the sum "<var>x</var>+<var>y</var>" can be
thought of as an application of the addition operator to two arguments <var>x</var> and
<var>y</var>. And the expression "cos(π)" as the application of the
cosine function to the number π.</p>
<p>As a general rule, the terminal nodes in the tree represent basic mathematical
objects such as numbers, variables, arithmetic operations and so on. The internal
nodes
in the tree represent function application or other mathematical constructions that
build up a compound objects. Function application provides the most important example;
an internal node might represent the application of a function to several arguments,
which are themselves represented by the nodes underneath the internal node.</p>
<p>The semantics of general mathematical expressions is not a matter of consensus. It
would be an enormous job to systematically codify most of mathematics – a task
that can never be complete. Instead, MathML makes explicit a relatively small number
of
commonplace mathematical constructs, chosen carefully to be sufficient in a large
number
of applications. In addition, it provides a mechanism for referring
to mathematical concepts outside of the base collection, allowing
them to be represented, as well.</p>
<p>The base set of content elements is chosen to be adequate for simple coding of most
of the formulas used from kindergarten to the end of high school in the United States,
and probably beyond through the first two years of college, that is up to A-Level
or
Baccalaureate level in Europe.</p>
<p>While the primary role of the MathML content element set is to directly encode the
mathematical structure of expressions independent of the notation used to present
the
objects, rendering issues cannot be ignored. There are different approaches for
rendering Content MathML formulae, ranging from native implementations of the
MathML elements to declarative notation definitions, to XSLT style
sheets. Because rendering requirements for Content MathML vary
widely, MathML 3 does not provide a normative specification for
rendering. Instead, typical renderings are suggested by way of examples.</p>
</div>
<div class="div3">
<h3><a id="contm.strict"></a>4.1.3 Strict Content MathML</h3>
<p>In MathML 3, a subset, or profile, of Content MathML is defined: <em>Strict Content
MathML</em>. This uses a minimal, but sufficient, set of elements to represent the meaning of
a
mathematical expression in a uniform structure, while the full Content MathML grammar
is
backward compatible with MathML 2.0, and generally tries to strike a more pragmatic
balance between verbosity and formality.</p>
<p>Content MathML provides a large number of predefined functions
encoded as empty elements (e.g. <code>sin</code>, <code>log</code>, etc.)
and a variety of constructs for forming compound objects
(e.g. <code>set</code>, <code>interval</code>, etc.). By contrast, Strict
Content MathML uses a single element (<code>csymbol</code>) with an
attribute pointing to an external definition in extensible content
dictionaries to represent all functions, and uses only
<code>apply</code> and <code>bind</code> for building up compound
objects. The token elements such as <code>ci</code> and <code>cn</code> are
also considered part of Strict Content MathML, but with a more
restricted set of attributes and with content restricted to
text.</p>
<p>Strict Content MathML is designed to be compatible with OpenMath (in
fact it is an XML encoding of OpenMath Objects in the sense of <a href="appendixg.html#OpenMath2004">[OpenMath2004]</a>).
OpenMath is a standard for representing formal mathematical
objects and semantics through the use of extensible Content Dictionaries. The table
below
gives an element-by-element correspondence between the OpenMath XML encoding of OpenMath
objects and Strict Content MathML.
</p>
<table id="contm.om.correspondence" class="data">
<thead>
<tr>
<th>Strict Content MathML</th>
<th>OpenMath</th>
</tr>
</thead>
<tbody>
<tr>
<td><a href="#contm.cn"><code>cn</code></a></td>
<td><code>OMI</code>, <code>OMF</code></td>
</tr>
<tr>
<td><a href="#contm.csymbol"><code>csymbol</code></a></td>
<td><code>OMS</code></td>
</tr>
<tr>
<td><a href="#contm.ci"><code>ci</code></a></td>
<td><code>OMV</code></td>
</tr>
<tr>
<td><a href="#contm.cs"><code>cs</code></a></td>
<td><code>OMSTR</code></td>
</tr>
<tr>
<td><a href="#contm.apply"><code>apply</code></a></td>
<td><code>OMA</code></td>
</tr>
<tr>
<td><a href="#contm.binding"><code>bind</code></a></td>
<td><code>OMBIND</code></td>
</tr>
<tr>
<td><a href="#contm.binding"><code>bvar</code></a></td>
<td><code>OMBVAR</code></td>
</tr>
<tr>
<td><a href="#contm.sharing"><code>share</code></a></td>
<td><code>OMR</code></td>
</tr>
<tr>
<td><a href="#contm.semantics"><code>semantics</code></a></td>
<td><code>OMATTR</code></td>
</tr>
<tr>
<td><a href="#contm.semantics"><code>annotation</code></a>,
<a href="#contm.semantics"><code>annotation-xml</code></a></td>
<td><code>OMATP</code>, <code>OMFOREIGN</code></td>
</tr>
<tr>
<td><a href="#contm.cerror"><code>cerror</code></a></td>
<td><code>OME</code></td>
</tr>
<tr>
<td><a href="#contm.cbytes"><code>cbytes</code></a></td>
<td><code>OMB</code></td>
</tr>
</tbody>
</table>
<p>In MathML 3, formal semantics Content MathML expressions are
given by specifying equivalent Strict Content MathML expressions.
Since Strict Content MathML expressions all have carefully-defined
semantics given in terms of OpenMath Content Dictionaries, all
Content MathML expressions inherit well-defined semantics in this
way. To make the correspondence exact, an algorithm is
given in terms of transformation rules that are applied to
rewrite non-Strict MathML constructs into a strict equivalents. The
individual rules are introduced in context throughout the chapter.
In <a href="#contm.p2s">Section 4.6 The Strict Content MathML Transformation</a>, the algorithm as a whole is
described.</p>
<p>As most transformation rules relate to
classes of MathML elements that have similar argument structure,
they are introduced in <a href="#contm.opclasses">Section 4.3.4 Operator Classes</a> where these
classes are defined. Some special case rules for specific elements
are given in Section <a href="#contm.opel">Section 4.4 Content MathML for Specific Operators and Constants</a>. Transformations in
<a href="#contm.core">Section 4.2 Content MathML Elements Encoding Expression Structure</a> concern non-Strict usages of the core
Content MathML elements, those in <a href="#contm.structure.extended">Section 4.3 Content MathML for Specific Structures</a> concern the rewriting of some
additional structures not directly supported in Strict Content MathML.</p>
<p>The full algorithm described in<a href="#contm.p2s">Section 4.6 The Strict Content MathML Transformation</a> is
complete in the sense that it gives every Content MathML expression a specific
meaning in terms of a Strict Content MathML expression. This means
it has to give specific strict interpretations to some expressions
whose meaning was insufficiently specified in MathML2. The intention
of this algorithm is to be faithful to mathematical intuitions.
However edge cases may remain where the normative interpretation of
the algorithm may break earlier intuitions.</p>
<p>A conformant MathML processor need
not implement this transformation. The existence of these
transformation rules does not imply that a system must treat
equivalent expressions identically. In particular systems may give
different presentation renderings for expressions that the
transformation rules imply are mathematically equivalent.</p>
</div>
<div class="div3">
<h3><a id="contm.cds"></a>4.1.4 Content Dictionaries</h3>
<p>Due to the nature of mathematics, any method for formalizing
the meaning of the mathematical expressions must be
extensible. The key to extensibility is the ability to define
new functions and other symbols to expand the terrain of
mathematical discourse. To do this, two things are required: a
mechanism for representing symbols not already defined by
Content MathML, and a means of associating a specific
mathematical meaning with them in an unambiguous way. In MathML
3, the <code>csymbol</code> element provides the means to represent
new symbols, while <em>Content Dictionaries</em> are the way
in which mathematical semantics are described. The association
is accomplished via attributes of the <code>csymbol</code> element
that point at a definition in a CD. The syntax and usage of
these attributes are described in detail in <a href="#contm.csymbol">Section 4.2.3 Content Symbols <code><csymbol></code></a>.</p>
<p>Content Dictionaries are structured documents for the
definition of mathematical concepts; see the OpenMath standard,
<a href="appendixg.html#OpenMath2004">[OpenMath2004]</a>.
To maximize modularity and reuse, a
Content Dictionary typically contains a relatively small
collection of definitions for closely related concepts. The
OpenMath Society maintains a large set of public Content Dictionaries
including the MathML CD group that including contains definitions
for all pre-defined symbols in MathML.
There is a process for contributing privately
developed CDs to the OpenMath Society repository to facilitate
discovery and reuse. MathML 3 does not require CDs be publicly
available, though in most situations the goals of semantic
markup will be best served by referencing public CDs available
to all user agents.</p>
<p>In the text below, descriptions of semantics for predefined
MathML symbols refer to the Content Dictionaries developed by
the OpenMath Society in conjunction with the W3C Math Working
Group. It is important to note, however, that this information
is informative, and not normative. In general, the precise
mathematical semantics of predefined symbols are not not fully
specified by the MathML 3 Recommendation, and the only normative
statements about symbol semantics are those present in the text
of this chapter. The semantic definitions provided by the
OpenMath Content CDs are intended to be sufficient for
most applications, and are generally compatible with the
semantics specified for analogous constructs in the MathML 2.0
Recommendation. However, in contexts where highly precise
semantics are required (e.g. communication between computer
algebra systems, within formal systems such as theorem provers,
etc.) it is the responsibility of the relevant community of