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      <h1><a name="presm" id="presm"></a>3 Presentation Markup
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         Overview: <a href="Overview-d.html">Mathematical Markup Language (MathML) Version 3.0</a><br>
         Previous: 2 <a href="chapter2-d.html">MathML Fundamentals</a><br>
         Next: 4 <a href="chapter4-d.html">Content Markup</a><br><br>3 <a href="chapter3-d.html">Presentation Markup</a><br>&nbsp;&nbsp;&nbsp;&nbsp;3.1 <a href="chapter3-d.html#presm.intro">Introduction</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.1 <a href="chapter3-d.html#id.3.1.1">What Presentation Elements Represent</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.2 <a href="chapter3-d.html#id.3.1.2">Terminology Used In This Chapter</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.2.1 <a href="chapter3-d.html#id.3.1.2.1">Types of presentation elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.2.2 <a href="chapter3-d.html#id.3.1.2.2">Terminology for other classes of elements and their relationships</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.3 <a href="chapter3-d.html#presm.reqarg">Required Arguments</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.3.1 <a href="chapter3-d.html#presm.inferredmrow">Inferred <code>&lt;mrow&gt;</code>s</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.3.2 <a href="chapter3-d.html#id.3.1.3.2">Table of argument requirements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.4 <a href="chapter3-d.html#id.3.1.4">Elements with Special Behaviors</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.5 <a href="chapter3-d.html#presm.bidi">Directionality</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.5.1 <a href="chapter3-d.html#presm.bidi.math">Overall Directionality of Mathematics Formulas</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.5.2 <a href="chapter3-d.html#presm.bidi.token">Bidirectional Layout in Token Elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.6 <a href="chapter3-d.html#presm.scriptlevel">Displaystyle and Scriptlevel</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.7 <a href="chapter3-d.html#presm.linebreaking">Linebreaking of Expressions</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.7.1 <a href="chapter3-d.html#id.3.1.7.1">Control of Linebreaks</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.7.2 <a href="chapter3-d.html#presm.lbalgorithm">Automatic Linebreaking Algorithm (Informative)</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.7.3 <a href="chapter3-d.html#presm.inline.lbalgorithm">Linebreaking Algorithm for Inline Expressions (Informative)</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.8 <a href="chapter3-d.html#presm.warnfinetuning">Warning about fine-tuning of presentation</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.8.1 <a href="chapter3-d.html#presm.warntweaking">Warning: non-portability of tweaking</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.8.2 <a href="chapter3-d.html#presm.warnspacing">Warning: spacing should not be used to convey meaning</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.9 <a href="chapter3-d.html#presm.summary">Summary of Presentation Elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.9.1 <a href="chapter3-d.html#id.3.1.9.1">Token Elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.9.2 <a href="chapter3-d.html#id.3.1.9.2">General Layout Schemata</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.9.3 <a href="chapter3-d.html#id.3.1.9.3">Script and Limit Schemata</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.9.4 <a href="chapter3-d.html#id.3.1.9.4">Tables and Matrices</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.9.5 <a href="chapter3-d.html#id.3.1.9.5">Elementary Math Layout</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.9.6 <a href="chapter3-d.html#id.3.1.9.6">Enlivening Expressions</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.1.10 <a href="chapter3-d.html#presm.presatt">Mathematics style attributes common to presentation elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;3.2 <a href="chapter3-d.html#presm.tokel">Token Elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.1 <a href="chapter3-d.html#presm.tokenchars">MathML characters in token elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.1.1 <a href="chapter3-d.html#presm.symbolchars">Alphanumeric symbol characters</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.1.2 <a href="chapter3-d.html#presm.mglyph"> <code>&lt;mglyph/&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.2 <a href="chapter3-d.html#presm.commatt">Mathematics style attributes common to token elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.2.1 <a href="chapter3-d.html#presm.deprecatt">Deprecated style attributes on token elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.3 <a href="chapter3-d.html#presm.mi">Identifier <code>&lt;mi&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.3.1 <a href="chapter3-d.html#id.3.2.3.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.3.2 <a href="chapter3-d.html#id.3.2.3.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.3.3 <a href="chapter3-d.html#id.3.2.3.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.4 <a href="chapter3-d.html#presm.mn">Number <code>&lt;mn&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.4.1 <a href="chapter3-d.html#id.3.2.4.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.4.2 <a href="chapter3-d.html#id.3.2.4.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.4.3 <a href="chapter3-d.html#id.3.2.4.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.4.4 <a href="chapter3-d.html#id.3.2.4.4">Numbers that should not be written 
            using <code>&lt;mn&gt;</code> alone</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5 <a href="chapter3-d.html#presm.mo">Operator, Fence, Separator or Accent
            <code>&lt;mo&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.1 <a href="chapter3-d.html#id.3.2.5.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.2 <a href="chapter3-d.html#id.3.2.5.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.3 <a href="chapter3-d.html#id.3.2.5.3">Examples with ordinary operators</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.4 <a href="chapter3-d.html#id.3.2.5.4">Examples with fences and separators</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.5 <a href="chapter3-d.html#presm.invisibleops">Invisible operators</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.6 <a href="chapter3-d.html#id.3.2.5.6">Names for other special operators</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.7 <a href="chapter3-d.html#id.3.2.5.7">Detailed rendering rules for <code>&lt;mo&gt;</code> elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.8 <a href="chapter3-d.html#presm.op.stretch">Stretching of operators, fences and accents</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.5.9 <a href="chapter3-d.html#presm.mo.linebreaks">Examples of Linebreaking</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.6 <a href="chapter3-d.html#presm.mtext">Text <code>&lt;mtext&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.6.1 <a href="chapter3-d.html#id.3.2.6.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.6.2 <a href="chapter3-d.html#id.3.2.6.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.6.3 <a href="chapter3-d.html#id.3.2.6.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.6.4 <a href="chapter3-d.html#presm.mixtextmath">Mixing text and mathematics</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.7 <a href="chapter3-d.html#presm.mspace">Space <code>&lt;mspace/&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.7.1 <a href="chapter3-d.html#id.3.2.7.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.7.2 <a href="chapter3-d.html#id.3.2.7.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.7.3 <a href="chapter3-d.html#id.3.2.7.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.7.4 <a href="chapter3-d.html#id.3.2.7.4">Definition of space-like elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.7.5 <a href="chapter3-d.html#id.3.2.7.5">Legal grouping of space-like elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.8 <a href="chapter3-d.html#presm.ms">String Literal <code>&lt;ms&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.8.1 <a href="chapter3-d.html#id.3.2.8.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.2.8.2 <a href="chapter3-d.html#id.3.2.8.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;3.3 <a href="chapter3-d.html#presm.genlayout">General Layout Schemata</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.1 <a href="chapter3-d.html#presm.mrow">Horizontally Group Sub-Expressions
            <code>&lt;mrow&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.1.1 <a href="chapter3-d.html#id.3.3.1.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.1.2 <a href="chapter3-d.html#id.3.3.1.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.1.3 <a href="chapter3-d.html#id.3.3.1.3">Proper grouping of sub-expressions using <code>&lt;mrow&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.1.4 <a href="chapter3-d.html#id.3.3.1.4">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.2 <a href="chapter3-d.html#presm.mfrac">Fractions <code>&lt;mfrac&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.2.1 <a href="chapter3-d.html#id.3.3.2.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.2.2 <a href="chapter3-d.html#id.3.3.2.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.2.3 <a href="chapter3-d.html#id.3.3.2.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.3 <a href="chapter3-d.html#presm.mroot">Radicals <code>&lt;msqrt&gt;</code>, <code>&lt;mroot&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.3.1 <a href="chapter3-d.html#id.3.3.3.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.3.2 <a href="chapter3-d.html#id.3.3.3.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.4 <a href="chapter3-d.html#presm.mstyle">Style Change <code>&lt;mstyle&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.4.1 <a href="chapter3-d.html#id.3.3.4.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.4.2 <a href="chapter3-d.html#presm.mstyle.attrs">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.4.3 <a href="chapter3-d.html#id.3.3.4.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.5 <a href="chapter3-d.html#presm.merror">Error Message <code>&lt;merror&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.5.1 <a href="chapter3-d.html#id.3.3.5.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.5.2 <a href="chapter3-d.html#id.3.3.5.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.5.3 <a href="chapter3-d.html#id.3.3.5.3">Example</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.6 <a href="chapter3-d.html#presm.mpadded">Adjust Space Around Content
            <code>&lt;mpadded&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.6.1 <a href="chapter3-d.html#id.3.3.6.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.6.2 <a href="chapter3-d.html#id.3.3.6.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.6.3 <a href="chapter3-d.html#id.3.3.6.3">Meanings of  attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.7 <a href="chapter3-d.html#presm.mphantom">Making Sub-Expressions Invisible <code>&lt;mphantom&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.7.1 <a href="chapter3-d.html#id.3.3.7.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.7.2 <a href="chapter3-d.html#id.3.3.7.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.7.3 <a href="chapter3-d.html#id.3.3.7.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.8 <a href="chapter3-d.html#presm.mfenced">Expression Inside Pair of Fences
            <code>&lt;mfenced&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.8.1 <a href="chapter3-d.html#id.3.3.8.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.8.2 <a href="chapter3-d.html#id.3.3.8.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.8.3 <a href="chapter3-d.html#id.3.3.8.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.9 <a href="chapter3-d.html#presm.menclose">Enclose Expression Inside Notation
            <code>&lt;menclose&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.9.1 <a href="chapter3-d.html#id.3.3.9.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.9.2 <a href="chapter3-d.html#id.3.3.9.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.3.9.3 <a href="chapter3-d.html#id.3.3.9.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;3.4 <a href="chapter3-d.html#presm.scrlim">Script and Limit Schemata</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.1 <a href="chapter3-d.html#presm.msub">Subscript <code>&lt;msub&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.1.1 <a href="chapter3-d.html#id.3.4.1.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.1.2 <a href="chapter3-d.html#id.3.4.1.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.2 <a href="chapter3-d.html#presm.msup">Superscript <code>&lt;msup&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.2.1 <a href="chapter3-d.html#id.3.4.2.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.2.2 <a href="chapter3-d.html#id.3.4.2.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.3 <a href="chapter3-d.html#presm.msubsup">Subscript-superscript Pair <code>&lt;msubsup&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.3.1 <a href="chapter3-d.html#id.3.4.3.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.3.2 <a href="chapter3-d.html#presm.subsupatt">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.3.3 <a href="chapter3-d.html#id.3.4.3.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.4 <a href="chapter3-d.html#presm.munder">Underscript <code>&lt;munder&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.4.1 <a href="chapter3-d.html#id.3.4.4.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.4.2 <a href="chapter3-d.html#id.3.4.4.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.4.3 <a href="chapter3-d.html#id.3.4.4.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.5 <a href="chapter3-d.html#presm.mover">Overscript <code>&lt;mover&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.5.1 <a href="chapter3-d.html#id.3.4.5.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.5.2 <a href="chapter3-d.html#id.3.4.5.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.5.3 <a href="chapter3-d.html#id.3.4.5.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.6 <a href="chapter3-d.html#presm.munderover">Underscript-overscript Pair
            <code>&lt;munderover&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.6.1 <a href="chapter3-d.html#id.3.4.6.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.6.2 <a href="chapter3-d.html#id.3.4.6.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.6.3 <a href="chapter3-d.html#id.3.4.6.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.7 <a href="chapter3-d.html#presm.mmultiscripts">Prescripts and Tensor Indices
            <code>&lt;mmultiscripts&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.7.1 <a href="chapter3-d.html#id.3.4.7.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.7.2 <a href="chapter3-d.html#id.3.4.7.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.4.7.3 <a href="chapter3-d.html#id.3.4.7.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;3.5 <a href="chapter3-d.html#presm.tabmat">Tabular Math</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.1 <a href="chapter3-d.html#presm.mtable">Table or Matrix
            <code>&lt;mtable&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.1.1 <a href="chapter3-d.html#id.3.5.1.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.1.2 <a href="chapter3-d.html#presm.mtable.attrs">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.1.3 <a href="chapter3-d.html#id.3.5.1.3">Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.2 <a href="chapter3-d.html#presm.mtr">Row in Table or Matrix <code>&lt;mtr&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.2.1 <a href="chapter3-d.html#id.3.5.2.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.2.2 <a href="chapter3-d.html#id.3.5.2.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.3 <a href="chapter3-d.html#presm.mlabeledtr">Labeled Row in Table or Matrix
            <code>&lt;mlabeledtr&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.3.1 <a href="chapter3-d.html#id.3.5.3.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.3.2 <a href="chapter3-d.html#id.3.5.3.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.3.3 <a href="chapter3-d.html#id.3.5.3.3">Equation Numbering</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.4 <a href="chapter3-d.html#presm.mtd">Entry in Table or Matrix <code>&lt;mtd&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.4.1 <a href="chapter3-d.html#id.3.5.4.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.4.2 <a href="chapter3-d.html#presm.mtdatts">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5 <a href="chapter3-d.html#presm.malign">Alignment Markers
            <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.1 <a href="chapter3-d.html#id.3.5.5.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.2 <a href="chapter3-d.html#id.3.5.5.2">Specifying alignment groups</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.3 <a href="chapter3-d.html#id.3.5.5.3">Table cells that are not divided into alignment groups</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.4 <a href="chapter3-d.html#id.3.5.5.4">Specifying alignment points using <code>&lt;malignmark/&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.5 <a href="chapter3-d.html#id.3.5.5.5"><code>&lt;malignmark/&gt;</code> Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.6 <a href="chapter3-d.html#id.3.5.5.6"><code>&lt;maligngroup/&gt;</code> Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.7 <a href="chapter3-d.html#id.3.5.5.7">Inheritance of groupalign values</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.8 <a href="chapter3-d.html#id.3.5.5.8">MathML representation of an alignment example</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.9 <a href="chapter3-d.html#id.3.5.5.9">Further details of alignment elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.5.5.10 <a href="chapter3-d.html#id.3.5.5.10">A simple alignment algorithm</a><br>&nbsp;&nbsp;&nbsp;&nbsp;3.6 <a href="chapter3-d.html#presm.elementary">Elementary Math</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.1 <a href="chapter3-d.html#presm.mstack">Stacks of Characters <code>&lt;mstack&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.1.1 <a href="chapter3-d.html#id.3.6.1.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.1.2 <a href="chapter3-d.html#id.3.6.1.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.2 <a href="chapter3-d.html#presm.mlongdiv">Long Division <code>&lt;mlongdiv&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.2.1 <a href="chapter3-d.html#id.3.6.2.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.2.2 <a href="chapter3-d.html#presm.mlongdiv.attrs">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.3 <a href="chapter3-d.html#presm.msgroup">Group Rows with Similiar Positions <code>&lt;msgroup&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.3.1 <a href="chapter3-d.html#id.3.6.3.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.3.2 <a href="chapter3-d.html#id.3.6.3.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.4 <a href="chapter3-d.html#presm.msrow">Rows in Elementary Math <code>&lt;msrow&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.4.1 <a href="chapter3-d.html#id.3.6.4.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.4.2 <a href="chapter3-d.html#id.3.6.4.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.5 <a href="chapter3-d.html#presm.mscarries">Carries, Borrows, and Crossouts <code>&lt;mscarries&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.5.1 <a href="chapter3-d.html#id.3.6.5.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.5.2 <a href="chapter3-d.html#id.3.6.5.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.6 <a href="chapter3-d.html#presm.mscarry">A Single Carry <code>&lt;mscarry&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.6.1 <a href="chapter3-d.html#id.3.6.6.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.6.2 <a href="chapter3-d.html#id.3.6.6.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.7 <a href="chapter3-d.html#presm.msline">Horizontal Line <code>&lt;msline/&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.7.1 <a href="chapter3-d.html#id.3.6.7.1">Description</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.7.2 <a href="chapter3-d.html#id.3.6.7.2">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.8 <a href="chapter3-d.html#presm.elemmath.examples">Elementary Math Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.8.1 <a href="chapter3-d.html#presm.addsub">Addition and Subtraction</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.8.2 <a href="chapter3-d.html#presm.mult">Multiplication</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.8.3 <a href="chapter3-d.html#presm.mlongdiv.ex">Long Division</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.6.8.4 <a href="chapter3-d.html#presm.repeatdec">Repeating decimal</a><br>&nbsp;&nbsp;&nbsp;&nbsp;3.7 <a href="chapter3-d.html#presm.enliven">Enlivening Expressions</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.7.1 <a href="chapter3-d.html#presm.maction">Bind Action to Sub-Expression
            <code>&lt;maction&gt;</code></a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3.7.1.1 <a href="chapter3-d.html#id.3.7.1.1">Attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;3.8 <a href="chapter3-d.html#presm.semantics">Semantics and Presentation</a><br></div>
      <div class="div1">
         <div class="div2">
            
            <h2><a name="presm.intro" id="presm.intro"></a>3.1 Introduction
            </h2>
            <p>This chapter specifies the "presentation" elements of
               MathML, which can be used to describe the layout structure of mathematical
               notation.
            </p>
            <div class="div3">
               
               <h3><a name="id.3.1.1" id="id.3.1.1"></a>3.1.1 What Presentation Elements Represent
               </h3>
               <p>Presentation elements correspond to the "constructors"
                  of traditional mathematical notation —  that is, to the basic
                  kinds of symbols and expression-building structures out of which any
                  particular piece of traditional mathematical notation is built.
                  Because of the importance of traditional visual notation, the
                  descriptions of the notational constructs the elements represent are
                  usually given here in visual terms.  However, the elements are
                  medium-independent in the sense that they have been designed to
                  contain enough information for good spoken renderings as well.  Some
                  attributes of these elements may make sense only for visual media, but
                  most attributes can be treated in an analogous way in audio as well
                  (for example, by a correspondence between time duration and horizontal
                  extent).
               </p>
               <p>MathML presentation elements only suggest (i.e. do not require)
                  specific ways of rendering in order to allow for medium-dependent
                  rendering and for individual preferences of style.  This specification
                  describes suggested visual rendering rules in some detail, but a
                  particular MathML renderer is free to use its own rules as long as its
                  renderings are intelligible.
               </p>
               <p>The presentation elements are meant to express the syntactic
                  structure of mathematical notation in much the same way as titles, sections,
                  and paragraphs capture the higher-level syntactic structure of a
                  textual document. Because of this, a single row of identifiers and operators
                  will often be represented by multiple nested <code>mrow</code> elements rather than
                  a single <code>mrow</code>.  For example, "<var>x</var> + <var>a</var> /
                  <var>b</var>" typically is represented as:
                  
               </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt; x &lt;/mi&gt;
  &lt;mo&gt; + &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mi&gt; a &lt;/mi&gt;
    &lt;mo&gt; / &lt;/mo&gt;
    &lt;mi&gt; b &lt;/mi&gt;
  &lt;/mrow&gt;
&lt;/mrow&gt;
</pre><p>Similarly, superscripts are attached to the full expression constituting 
                  their base rather than to the just preceding character.  This
                  structure permits better-quality rendering of mathematics, especially when
                  details of the rendering environment, such as display widths, are not
                  known ahead of time to the document author. It also greatly eases automatic
                  interpretation of the represented mathematical structures.
               </p>
               <p>Certain MathML characters are used
                  to name identifiers or operators that in traditional notation render the
                  same as other symbols or usually rendered invisibly.  For example, the entities
                  <code>&amp;DifferentialD;</code>, <code>&amp;ExponentialE;</code>, and 
                  <code>&amp;ImaginaryI;</code> denote notational symbols semantically distinct from visually 
                  identical letters used as simple variables.  Likewise, the entities <code>&amp;InvisibleTimes;</code>, 
                  <code>&amp;ApplyFunction;</code>, 
                  <code>&amp;InvisibleComma;</code> and the character U+2064 
                  (INVISIBLE PLUS) usually render invisibly but represent significant information.  
                  These entities have distinct spoken renderings, may influence visual linebreaking and spacing, and may 
                  effect the evaluation or meaning of particular expressions. Accordingly, authors should use these entities wherever they are
                  applicable.
                  For instance, the expression represented visually as
                  "<var>f</var>(<var>x</var>)" would usually be spoken in English as
                  "<var>f</var> of <var>x</var>" rather than just
                  "<var>f</var> <var>x</var>".  MathML conveys this meaning by using 
                  the <code>&amp;ApplyFunction;</code> operator after the
                  "<var>f</var>", which, in this case, can be aurally rendered as
                  "of".
               </p>
               <p>The complete list of MathML entities is described in <a href="appendixg-d.html#Entities">[Entities]</a>.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="id.3.1.2" id="id.3.1.2"></a>3.1.2 Terminology Used In This Chapter
               </h3>
               <p>It is strongly recommended that, before reading the present
                  chapter, one read <a href="chapter2-d.html#fund.syntax">Section&nbsp;2.1 MathML Syntax and Grammar</a> on MathML syntax and
                  grammar, which contains important information on MathML notations and
                  conventions.  In particular, in this chapter it is assumed that the
                  reader has an understanding of basic XML terminology described in
                  <a href="chapter2-d.html#fund.xmlsyntax">Section&nbsp;2.1.3 Children versus Arguments</a>, and the attribute value notations and
                  conventions described in <a href="chapter2-d.html#fund.attval">Section&nbsp;2.1.5 MathML Attribute Values</a>.
               </p>
               <p>The remainder of this section introduces MathML-specific
                  terminology and conventions used in this chapter.
               </p>
               <div class="div4">
                  
                  <h4><a name="id.3.1.2.1" id="id.3.1.2.1"></a>3.1.2.1 Types of presentation elements
                  </h4>
                  <p>The presentation elements are divided into two classes.
                     <a href="chapter3-d.html#presm.tokel"><em>Token elements</em></a>
                     represent individual symbols, names, numbers, labels, etc. 
                     <em>Layout schemata</em> build expressions out of parts and can have
                     only elements as content (except for whitespace, which they ignore).
                     These are subdivided into 
                     <a href="chapter3-d.html#presm.genlayout">General Layout</a>,
                     <a href="chapter3-d.html#presm.scrlim">Script and Limit</a>,
                     <a href="chapter3-d.html#presm.tabmat">Tabular Math</a> and
                     <a href="chapter3-d.html#presm.elementary">Elementary Math</a> schemata.
                     There
                     are also a few empty elements used only in conjunction with certain layout
                     schemata.
                  </p>
                  <p>All individual "symbols" in a mathematical expression should be
                     represented by MathML token elements. The primary MathML token element
                     types are identifiers (e.g. variables or function names), numbers, and
                     operators (including fences, such as parentheses, and separators, such
                     as commas). There are also token elements used to represent text or
                     whitespace that has more aesthetic than mathematical significance
                     and other elements representing "string literals" for compatibility with
                     computer algebra systems. Note that although a token element
                     represents a single meaningful "symbol" (name, number, label,
                     mathematical symbol, etc.), such symbols may be comprised of more than
                     one character.  For example <code>sin</code> and <code>24</code> are
                     represented by the single tokens <code>&lt;mi&gt;sin&lt;/mi&gt;</code>
                     and <code>&lt;mn&gt;24&lt;/mn&gt;</code> respectively.
                  </p>
                  <p>In traditional mathematical notation, expressions are recursively
                     constructed out of smaller expressions, and ultimately out of single
                     symbols, with the parts grouped and positioned using one of a small
                     set of notational structures, which can be thought of as "expression
                     constructors". In MathML, expressions are constructed in the same way,
                     with the layout schemata playing the role of the expression
                     constructors. The layout schemata specify the way in which
                     sub-expressions are built into larger expressions. The terminology
                     derives from the fact that each layout schema corresponds to a
                     different way of "laying out" its sub-expressions to form a larger
                     expression in traditional mathematical typesetting.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.1.2.2" id="id.3.1.2.2"></a>3.1.2.2 Terminology for other classes of elements and their relationships
                  </h4>
                  <p>The terminology used in this chapter for special classes of elements, and for
                     relationships between elements, is as follows: The <em>presentation elements</em> are
                     the MathML elements defined in this chapter. These elements are listed in <a href="chapter3-d.html#presm.summary">Section&nbsp;3.1.9 Summary of Presentation Elements</a>. The <em>content elements</em> are the MathML elements defined
                     in <a href="chapter4-d.html">Chapter&nbsp;4 Content Markup</a>.
                  </p>
                  <p>A MathML <em>expression</em> is a single instance of any of the
                     presentation elements with the exception of the empty elements <code>none</code> or <code>mprescripts</code>, or is
                     a single instance of any of the content elements which are allowed as
                     content of presentation elements (described in <a href="chapter5-d.html#mixing.cminpm">Section&nbsp;5.3.2 Content Markup in Presentation Markup</a>). A <em>sub-expression</em> of an expression
                     <var>E</var> is any MathML expression that is part of the content of
                     <var>E</var>, whether <em>directly</em> or <em>indirectly</em>,
                     i.e. whether it is a "child" of <var>E</var> or not.
                  </p>
                  <p>Since layout schemata attach special meaning to the number and/or
                     positions of their children, a child of a layout schema is also called
                     an <em>argument</em> of that element.  As a consequence of the
                     above definitions, the content of a layout schema consists exactly of
                     a sequence of zero or more elements that are its
                     arguments. 
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.reqarg" id="presm.reqarg"></a>3.1.3 Required Arguments
               </h3>
               <p>Many of the elements described herein require a specific number of
                  arguments (always 1, 2, or 3).  In the detailed descriptions of
                  element syntax given below, the number of required arguments is
                  implicitly indicated by giving names for the arguments at various
                  positions.  A few elements have additional requirements on the number
                  or type of arguments, which are described with the individual
                  element. For example, some elements accept sequences of zero or more
                  arguments — that is, they are allowed to occur with no arguments
                  at all.
               </p>
               <p>Note that MathML elements encoding rendered space <em>do</em>
                  count as arguments of the elements in which they appear. See <a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a> for a discussion of the proper use of such
                  space-like elements.
               </p>
               <div class="div4">
                  
                  <h4><a name="presm.inferredmrow" id="presm.inferredmrow"></a>3.1.3.1 Inferred <code>&lt;mrow&gt;</code>s
                  </h4>
                  <p>The elements listed in the following table as requiring 1*
                     argument (<code>msqrt</code>, <code>mstyle</code>, <code>merror</code>,
                     <code>mpadded</code>, <code>mphantom</code>, <code>menclose</code>,
                     <code>mtd</code>, <code>mscarry</code>,
                     and <code>math</code>) 
                     conceptually accept a single argument,
                     but actually accept any number of children.
                     If the number of children is 0 or is more than 1, they treat their contents
                     as a single <em>inferred</em> <code>mrow</code> formed from all their children,
                     and treat this <code>mrow</code> as the argument.
                     
                  </p>
                  <p>For example,
                     
                  </p><pre class="mathml-fragment">
&lt;mtd&gt;
&lt;/mtd&gt;
</pre><p>
                     is treated as if it were
                     
                  </p><pre class="mathml-fragment">
&lt;mtd&gt;
  &lt;mrow&gt;
  &lt;/mrow&gt;
&lt;/mtd&gt;
</pre><p>
                     and
                     
                  </p><pre class="mathml">
&lt;msqrt&gt;
  &lt;mo&gt; - &lt;/mo&gt;
  &lt;mn&gt; 1 &lt;/mn&gt;
&lt;/msqrt&gt;
</pre><p>
                     is treated as if it were
                     
                  </p><pre class="mathml">
&lt;msqrt&gt;
  &lt;mrow&gt;
    &lt;mo&gt; - &lt;/mo&gt;
    &lt;mn&gt; 1 &lt;/mn&gt;
  &lt;/mrow&gt;
&lt;/msqrt&gt;
</pre><p>This feature allows MathML data not to contain (and its authors to
                     leave out)  many <code>mrow</code> elements that would otherwise be
                     necessary.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.1.3.2" id="id.3.1.3.2"></a>3.1.3.2 Table of argument requirements
                  </h4>
                  <p>For convenience, here is a table of each element's argument count
                     requirements and the roles of individual arguments when these are
                     distinguished.  An argument count of 1* indicates an inferred <code>mrow</code> as described above. 
                     Although the <code>math</code> element is
                     not a presentation element, it is listed below for completeness.
                     
                  </p>
                  <table id="presm.table-reqarg" border="1">
                     <thead>
                        <tr>
                           <th>Element</th>
                           <th>Required argument count</th>
                           <th>Argument roles (when these differ by position)</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mrow"><code>mrow</code></a></td>
                           <td>0 or more</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mfrac"><code>mfrac</code></a></td>
                           <td>2</td>
                           <td><em>numerator</em> <em>denominator</em></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mroot"><code>msqrt</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mroot"><code>mroot</code></a></td>
                           <td>2</td>
                           <td><em>base</em> <em>index</em></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mstyle"><code>mstyle</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.merror"><code>merror</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mpadded"><code>mpadded</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mphantom"><code>mphantom</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mfenced"><code>mfenced</code></a></td>
                           <td>0 or more</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.menclose"><code>menclose</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msub"><code>msub</code></a></td>
                           <td>2</td>
                           <td><em>base</em> <em>subscript</em></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msup"><code>msup</code></a></td>
                           <td>2</td>
                           <td><em>base</em> <em>superscript</em></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msubsup"><code>msubsup</code></a></td>
                           <td>3</td>
                           <td><em>base</em> <em>subscript</em> <em>superscript</em></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.munder"><code>munder</code></a></td>
                           <td>2</td>
                           <td><em>base</em> <em>underscript</em></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mover"><code>mover</code></a></td>
                           <td>2</td>
                           <td><em>base</em> <em>overscript</em></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.munderover"><code>munderover</code></a></td>
                           <td>3</td>
                           <td><em>base</em> <em>underscript</em> <em>overscript</em></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mmultiscripts"><code>mmultiscripts</code></a></td>
                           <td>1 or more</td>
                           <td><em>base</em>
                              (<em>subscript</em> <em>superscript</em>)*
                              [<code>&lt;mprescripts/&gt;</code>
                              (<em>presubscript</em> <em>presuperscript</em>)*]
                           </td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mtable"><code>mtable</code></a></td>
                           <td>0 or more rows</td>
                           <td>0 or more <code>mtr</code> or <code>mlabeledtr</code> elements
                           </td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mlabeledtr"><code>mlabeledtr</code></a></td>
                           <td>1 or more</td>
                           <td>a label and 0 or more <code>mtd</code> elements
                           </td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mtr"><code>mtr</code></a></td>
                           <td>0 or more</td>
                           <td>0 or more <code>mtd</code> elements
                           </td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mtd"><code>mtd</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mstack"><code>mstack</code></a></td>
                           <td>0 or more</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mlongdiv"><code>mlongdiv</code></a></td>
                           <td>3 or more</td>
                           <td><em>divisor</em> <em>result</em>  <em>dividend</em> (<em>msrow</em> | <em>msgroup</em> | <em>mscarries</em> | <em>msline</em>)*
                           </td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msgroup"><code>msgroup</code></a></td>
                           <td>0 or more</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msrow"><code>msrow</code></a></td>
                           <td>0 or more</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mscarries"><code>mscarries</code></a></td>
                           <td>0 or more</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mscarry"><code>mscarry</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.maction"><code>maction</code></a></td>
                           <td>1 or more</td>
                           <td>depend on <code>actiontype</code> attribute
                           </td>
                        </tr>
                        <tr>
                           <td><a href="chapter2-d.html#interf.toplevel"><code>math</code></a></td>
                           <td>1*</td>
                           <td></td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="id.3.1.4" id="id.3.1.4"></a>3.1.4 Elements with Special Behaviors
               </h3>
               <p>Certain MathML presentation elements exhibit special behaviors in
                  certain contexts. Such special behaviors are discussed in the
                  detailed element descriptions below. However, for convenience, some
                  of the most important classes of special behavior are listed here.
               </p>
               <p>Certain elements are considered space-like; these are defined in
                  <a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a>. This definition affects some of the suggested rendering
                  rules for <code>mo</code> elements (<a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                     <code>&lt;mo&gt;</code></a>).
               </p>
               <p>Certain elements, e.g. <code>msup</code>, are able to
                  embellish operators that are their first argument. These elements are
                  listed in <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                     <code>&lt;mo&gt;</code></a>, which precisely defines an "embellished
                  operator" and explains how this affects the suggested rendering rules
                  for stretchy operators.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="presm.bidi" id="presm.bidi"></a>3.1.5 Directionality
               </h3>
               <p>
                  In the notations familiar to most readers,
                  both the overall layout and the textual symbols are arranged
                  from left to right (LTR).  Yet, as alluded to in the introduction,
                  mathematics written in Hebrew or in locales such
                  as  Morocco or Persia, the overall layout is used unchanged, but
                  the embedded symbols (often Hebrew or Arabic) are written right to left (RTL).
                  Moreover, in most of the Arabic speaking world, the notation
                  is arranged entirely RTL; thus a superscript is still raised,
                  but it follows the base on the left rather than the right.
               </p>
               <p>MathML 3.0 therefore recognizes two distinct directionalities:
                  the directionality of the text and symbols within token elements
                  and the overall directionality represented by Layout Schemata.
                  These two facets are discussed below.
               </p>
               <div class="div4">
                  
                  <h4><a name="presm.bidi.math" id="presm.bidi.math"></a>3.1.5.1 Overall Directionality of Mathematics Formulas
                  </h4>
                  <p>
                     The overall directionality for a formula, basically
                     the direction of the Layout Schemata, is specified by
                     the <code>dir</code> attribute on the containing <code>math</code> element
                     (see <a href="chapter2-d.html#interf.toplevel">Section&nbsp;2.2 The Top-Level 
                        <code>math</code> Element</a>).
                     The default is <code>ltr</code>.  When <code>dir='rtl'</code>
                     is used, the layout is simply the mirror image of the conventional
                     European layout.  That is, shifts up or down are unchanged,
                     but the progression in laying out is from right to left.
                  </p>
                  <p>For example, in a RTL layout, sub- and superscripts appear to the left of the base;
                     the surd for a root appears at the right, with the bar continuing over
                     the base to the left.
                     The layout details for elements whose behaviour depends on directionality
                     are given in the discussion of the element.  In those discussions, the
                     terms leading and trailing are used to specify a side of an object
                     when which side to use depends on the directionality; ie. leading
                     means left in LTR but right in RTL.
                     The terms left and right may otherwise be safely assumed to mean left and right.
                     
                  </p>
                  <p>
                     The overall directionality is usually set on the <code>math</code>, but
                     may also be switched for individual subformula by using the <code>dir</code>
                     attribute on <code>mrow</code> or <code>mstyle</code> elements.
                     When not specified, all elements inherit the directionality of their container.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.bidi.token" id="presm.bidi.token"></a>3.1.5.2 Bidirectional Layout in Token Elements
                  </h4>
                  <p>The text directionality comes into play for the MathML token elements
                     that can contain text (<code>mtext</code>, <code>mo</code>, <code>mi</code>, <code>mn</code>
                     and <code>ms</code>) and is determined by the Unicode properties of that text.
                     A token element containing exclusively LTR or RTL characters
                     is displayed straightforwardly in the given direction.
                     When a mixture of directions is involved used, such as RTL Arabic
                     and LTR numbers, the Unicode bidirectional algorithm <a href="appendixg-d.html#Bidi">[Bidi]</a>
                     is applied.  This algorithm specifies how runs of characters
                     with the same direction are processed and how the runs are (re)ordered.
                     The base, or initial, direction is given by the overall directionality
                     described above (<a href="chapter3-d.html#presm.bidi.math">Section&nbsp;3.1.5.1 Overall Directionality of Mathematics Formulas</a>) and affects
                     how weakly directional characters are treated and how runs are nested.
                     (The <code>dir</code> attribute is thus allowed on token elements to specify
                     the initial directionality that may be needed in rare cases.)
                     Any <code>mglyph</code> or <code>malignmark</code> elements appearing within
                     a token element are effectively <em>neutral</em> and have no effect
                     on ordering.
                  </p>
                  <p>The important thing to notice is that the bidirectional algorithm
                     is applied independently to the contents of each token element;
                     each token element is an independent run of characters.
                     This is in contrast to the application of bidirectionality to HTML, where
                     the algorithm applies to the entire sequence of characters within each
                     block level element.
                  </p>
                  <p>Other features of Unicode and scripts that should be respected
                     are ‘mirroring’ and ‘glyph shaping’. Some Unicode characters are marked as
                     being mirrored when presented in a RTL context; that is, the character
                     is drawn as if it were mirrored or replaced by a corresponding character.
                     Thus an opening parenthesis, ‘(’, in RTL will display as ‘)’.
                     Conversely, the solidus (/ U+002F) is <em>not</em> marked
                     as mirrored. Thus, an Arabic author that desires the slash to be reversed
                     in an inline division should explicitly use reverse solidus (\ U+005C)
                     or an alternative such as the mirroring DIVISION SLASH (U+2215).
                  </p>
                  <p>Additionally, calligraphic scripts such as Arabic blend, or connect
                     sequences of characters together, changing their appearance.
                     As this can have an significant impact on readability, as well as aesthetics,
                     it is important to apply such shaping if possible.  Glyph shaping,
                     like directionality, applies to each token element's contents individually.
                  </p>
                  <p>Please note that for the transfinite cardinals represented
                     by Hebrew characters, the code points U+2135-U+2138 (ALEF SYMBOL,
                     BET SYMBOL, GIMEL SYMBOL, DALET SYMBOL) should be used.
                     These are strong left-to-right.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.scriptlevel" id="presm.scriptlevel"></a>3.1.6 Displaystyle and Scriptlevel
               </h3>
               <p>So-called ‘displayed’ formulas, those appearing on a line by themselves,
                  typically make more generous use of vertical space than inline formulas,
                  which should blend into the adjacent text without intruding into
                  neighboring lines. For example, in a displayed summation, the limits
                  are placed above and below the summation symbol, while when it appears inline
                  the limits would appear in the sub and superscript position.
                  For similar reasons, sub- and superscripts,
                  nested fractions and other constructs typically display in a
                  smaller size than the main part of the formula.
                  MathML implicitly associates with every presentation node
                  a <code>displaystyle</code> and <code>scriptlevel</code> reflecting whether
                  a more expansive vertical layout applies and the level of scripting
                  in the current context.  
               </p>
               <p>These values are 
                  initialized by the <a href="chapter2-d.html#interf.toplevel"><code>math</code></a> element
                  according to the <code>display</code> attribute.
                  They are automatically adjusted by the 
                  various <a href="chapter3-d.html#presm.scrlim">script and limit schemata</a> elements,
                  and the elements
                  <a href="chapter3-d.html#presm.mfrac"><code>mfrac</code></a> and
                  <a href="chapter3-d.html#presm.mroot"><code>mroot</code></a>, 
                  which typically set <code>displaystyle</code> false and increment <code>scriptlevel</code>
                  for some or all of their arguments.
                  (See the description for each element for the specific rules used.)
                  They also may be set explicitly via the <code>displaystyle</code> and <code>scriptlevel</code> attributes
                  on the <a href="chapter3-d.html#presm.mstyle"><code>mstyle</code></a> element
                  or the <code>displaystyle</code> attribute of <a href="chapter3-d.html#presm.mtable"><code>mtable</code></a>.
                  In all other cases, they are inherited from the node's parent.
               </p>
               <p>The <code>displaystyle</code> affects the amount of vertical space used to lay out a formula: 
                  when true, the more spacious layout of displayed equations is used,
                  whereas when false a more compact layout of inline formula is used.
                  This primarily affects the interpretation
                  of the <code>largeop</code> and <code>movablelimits</code> attributes of
                  the <a href="chapter3-d.html#presm.mo"><code>mo</code></a> element.
                  However, more sophisticated renderers are free to use
                  this attribute to render more or less compactly.
                  
               </p>
               <p>The main effect of <code>scriptlevel</code> is to control the font size.
                  Typically, the higher the <code>scriptlevel</code>, the smaller the font size.
                  (Non-visual renderers can respond to the font size in an analogous way for their medium.)
                  Whenever the <code>scriptlevel</code> is changed, whether automatically or explicitly,
                  the current font size is multiplied by the value of
                  <code>scriptsizemultiplier</code> to the power of the <em>change</em> in <code>scriptlevel</code>.
                  However, changes to the font size due to <code>scriptlevel</code> changes should
                  never reduce the size below <code>scriptminsize</code> to prevent scripts
                  becoming unreadably small.
                  The default <code>scriptsizemultiplier</code> is approximately the square root of 1/2
                  whereas <code>scriptminsize</code> defaults to 8 points;
                  these values may be changed on <code>mstyle</code>; see  <a href="chapter3-d.html#presm.mstyle">Section&nbsp;3.3.4 Style Change <code>&lt;mstyle&gt;</code></a>.
                  Note that the <code>scriptlevel</code> attribute of <code>mstyle</code> allows arbitrary
                  values of <code>scriptlevel</code> to be obtained, including negative values which
                  result in increased font sizes.
                  
               </p>
               <p>The changes to the font size due to <code>scriptlevel</code> should be viewed
                  as being imposed from ‘outside’ the node.
                  This means that the effect of <code>scriptlevel</code> is applied 
                  before an explicit <code>mathsize</code> (see <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>)
                  on a token child of <code>mfrac</code>.
                  Thus, the <code>mathsize</code>  effectively overrides the effect of <code>scriptlevel</code>.
                  However, that change to <code>scriptlevel</code> changes the current font size,
                  which affects the meaning of an "em" length
                  (see <a href="chapter2-d.html#fund.units">Section&nbsp;2.1.5.2 Length Valued Attributes</a>)
                  and so the <code>scriptlevel</code> still may have an effect in such cases.
                  Note also that since <code>mathsize</code> is not constrained by <code>scriptminsize</code>,
                  such direct changes to font size can result in scripts smaller than <code>scriptminsize</code>.
                  
               </p>
               <p>Note that direct changes to current font size, whether by
                  CSS or by the <code>mathsize</code> attribute (See <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>),
                  have no effect on the value of <code>scriptlevel</code>.
                  
               </p>
               <p>T<sub>E</sub>X's \displaystyle, \textstyle, \scriptstyle, and \scriptscriptstyle 
                  correspond to  <code>displaystyle</code> and <code>scriptlevel</code>
                  as
                  "true" and "0",
                  "false" and "0",
                  "false" and "1",
                  and "false" and "2", respectively.
                  Thus,  <a href="chapter2-d.html#interf.toplevel"><code>math</code></a>'s
                  <code>display</code>="block" corresponds to \displaystyle,
                  while <code>display</code>="inline" corresponds to \textstyle.
                  
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="presm.linebreaking" id="presm.linebreaking"></a>3.1.7 Linebreaking of Expressions
               </h3>
               <div class="div4">
                  
                  <h4><a name="id.3.1.7.1" id="id.3.1.7.1"></a>3.1.7.1 Control of Linebreaks
                  </h4>
                  <p>MathML provides support for both automatic and manual (forced)
                     linebreaking of  expressions to break excessively long
                     expressions into several lines. 
                     All such linebreaks take place within <code>mrow</code>
                     (including inferred <code>mrow</code>; see <a href="chapter3-d.html#presm.inferredmrow">Section&nbsp;3.1.3.1 Inferred <code>&lt;mrow&gt;</code>s</a>)
                     or <code>mfenced</code>.
                     The breaks typically take place at <code>mo</code> elements
                     and also, for backwards compatibility, at <code>mspace</code>.
                     Renderers may also choose to place automatic linebreaks at other points such as between adjacent <code>mi</code> elements or even within a token element such as a very long <code>mn</code> element. MathML does not provide a means to specify such linebreaks, but if a render chooses to linebreak at such a point,
                     it should indent the following line according to the <a href="chapter3-d.html#presm.lbindent.attrs">indentation attributes</a> that are in effect at that point.
                     
                  </p>
                  <p>
                     Automatic linebreaking occurs when the containing <code>math</code> element
                     has <code>overflow</code>="linebreak"
                     and the display engine determines that there is not enough space available to
                     display the entire formula.  The available width must therefore be known
                     to the renderer. Like font properties, one is assumed to be inherited from the environment in
                     which the MathML element lives. If no width can be determined, an
                     infinite width should be assumed. Inside of a <code>mtable</code>,
                     each column has some width. This width may be specified as an attribute
                     or determined by the contents. This width should be used as the
                     line wrapping width for linebreaking, and each entry in an <code>mtable</code> is linewrapped as needed. 
                  </p>
                  <p>Forced linebreaks are specified by using
                     <code>linebreak</code>="newline"
                     on a <code>mo</code> or <code>mspace</code> element.
                     Both automatic and manual linebreaking can occur within the same formula.
                     
                  </p>
                  <p>Automatic linebreaking of subexpressions of <code>mfrac</code>, <code>msqrt</code>, <code>mroot</code>
                     and <code>menclose</code> and the various script elements is not required.
                     Renderers are free to ignore forced breaks within those elements if they choose.
                  </p>
                  <p>Attributes on <code>mo</code> and possibly on <code>mspace</code> elements control
                     linebreaking and indentation of the following line. The aspects of linebreaking
                     that can be controlled are:
                  </p>
                  <ul>
                     <li>
                        <p><em>Where</em> — attributes determine the desirability of
                           a linebreak at a specific operator or space, in particular whether a
                           break is required or inhibited.  These can only be set on
                           <code>mo</code> and <code>mspace</code> elements.
                           (See <a href="chapter3-d.html#presm.lbattrs">Section&nbsp;3.2.5.2.2 Linebreaking attributes</a>.)
                        </p>
                     </li>
                     <li>
                        <p><em>Operator Display/Position</em> — when a linebreak occurs,
                           determines whether the operator will appear
                           at the end of the line, at the beginning of the next line, or in both positions;
                           and how much vertical space should be added after the linebreak.
                           These attributes can be set on <code>mo</code> elements or inherited from
                           <code>mstyle</code> or  <code>math</code> elements.
                           (See <a href="chapter3-d.html#presm.lbattrs">Section&nbsp;3.2.5.2.2 Linebreaking attributes</a>.)
                        </p>
                     </li>
                     <li>
                        <p><em>Indentation</em> — determines the indentation of the
                           line following a linebreak, including indenting so that the next line aligns
                           with some point in a previous line.
                           These attributes can be set on <code>mo</code> elements or
                           inherited from <code>mstyle</code> or <code>math</code> elements.
                           (See <a href="chapter3-d.html#presm.lbindent.attrs">Section&nbsp;3.2.5.2.3 Indentation attributes</a>.)
                           
                        </p>
                     </li>
                  </ul>
                  <p>
                     When a math element appears in an inline context, it may obey whatever paragraph flow rules
                     are employed by the document's text rendering engine.
                     Such rules are necessarily outside of the scope of this specification.
                     Alternatively, it may use the value of the <code>math</code> element's overflow attribute.
                     (See <a href="chapter2-d.html#interf.toplevel.atts">Section&nbsp;2.2.1 Attributes</a>.)
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.lbalgorithm" id="presm.lbalgorithm"></a>3.1.7.2 Automatic Linebreaking Algorithm (Informative)
                  </h4>
                  <p>One method of linebreaking that works reasonably well is sometimes referred
                     to as a "best-fit" algorithm. It works by computing a "penalty" for
                     each potential break point on a line. The break point with the smallest
                     penalty is chosen and the algorithm then works on the next line. Three
                     useful factors in a penalty calculation are:
                     
                  </p>
                  <ol type="1">
                     <li>
                        <p>How much of the line width (after subtracting of the indent) is unused? The more unused, the higher the penalty. </p>
                     </li>
                     <li>
                        <p>How deeply nested is the breakpoint in the expression tree? The expression
                           tree's depth is roughly similar to the nesting depth of <code>mrow</code>s.  The more deeply nested the break point, the higher the penalty. 
                        </p>
                     </li>
                     <li>
                        <p>Does a linebreak here make layout of the next line difficult?
                           If the next line is not the last line and if the indentingstyle uses
                           information about the linebreak point to determine how much to indent,
                           then the amount of room left for linebreaking on the next line must be considered;
                           i.e., linebreaks that leave very little room to draw the next line
                           result in a higher penalty. 
                        </p>
                     </li>
                     <li>
                        <p>Whether "linebreak" has been specified: "nobreak" effectively sets the penalty to infinity, "badbreak" increases the penalty,
                           "goodbreak" decreases the penalty, and "newline" effectively sets the penalty to 0. 
                        </p>
                     </li>
                  </ol>
                  <p>This algorithm takes time proportional to the number of token elements times the number of lines. </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.inline.lbalgorithm" id="presm.inline.lbalgorithm"></a>3.1.7.3 Linebreaking Algorithm for Inline Expressions (Informative)
                  </h4>
                  <p>
                     A common method for breaking inline expressions that are too long for the
                     space remaining on the current line is to pick an appropriate break point
                     for the expression and place the expression up to that point on the current line and place the remainder of the expression
                     on the following line.
                     This can be done by:
                     
                  </p>
                  <ol type="1">
                     <li>
                        <p>Querying the text processing engine for the minimum and maximum amount of space available on the current line. </p>
                     </li>
                     <li>
                        <p>Using a variation of the automatic linebreaking algorithm
                           given <a href="chapter3-d.html#presm.lbalgorithm">above</a>),
                           and/or using hints provided by <a href="chapter3-d.html#presm.lbattrs">linebreak attributes</a>
                           on <code>mo</code> or <code>mspace</code> elements, to choose a line break.
                           The goal is that the first part of the formula fits "comfortably" on the current line
                           while breaking at a point that results in keeping related
                           parts of an expression on the same line. 
                        </p>
                     </li>
                     <li>
                        <p>The remainder of the formula begins on the next line,
                           positioned both vertically and horizontally according
                           to the paragraph flow; MathML's <a href="chapter3-d.html#presm.lbindent.attrs">indentation attributes</a> are ignored in this algorithm. 
                        </p>
                     </li>
                     <li>
                        <p>If the remainder does not fit on a line, steps 1 - 3 are repeated for the second and subsequent lines.   Unlike the for the
                           first line, some part of the expression must be placed these lines so that the algorithm terminates.
                        </p>
                     </li>
                  </ol>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.warnfinetuning" id="presm.warnfinetuning"></a>3.1.8 Warning about fine-tuning of presentation
               </h3>
               <p>Some use-cases require precise control of the math layout and presentation.  
                  Several MathML elements and attributes expressly support such fine-tuning of the 
                  rendering.  However, MathML rendering agents exhibit wide variability in their 
                  presentation of the the same MathML expression due to difference in platforms, 
                  font availability, and requirements particular to the agent itself (see <a href="chapter3-d.html#presm.intro">Section&nbsp;3.1 Introduction</a>).  
                  The overuse of explicit rendering control may yield a ‘perfect’ layout on one platform, but give much
                  worse presentation on others.  The following sections clarify the kinds of problems that can occur.
               </p>
               <div class="div4">
                  
                  <h4><a name="presm.warntweaking" id="presm.warntweaking"></a>3.1.8.1 Warning: non-portability of "tweaking"
                  </h4>
                  <p>For particular expressions, authors may be tempted to use the 
                     <a href="chapter3-d.html#presm.mpadded"><code>mpadded</code></a>,
                     <a href="chapter3-d.html#presm.mspace"><code>mspace</code></a>,
                     <a href="chapter3-d.html#presm.mphantom"><code>mphantom</code></a>, and
                     <a href="chapter3-d.html#presm.mtext"><code>mtext</code></a> elements to improve 
                     ("tweak") the spacing generated by a specific renderer.
                  </p>
                  <p>Without explicit spacing rules, various MathML renders may use different spacing 
                     algorithms.  Consequently, different MathML renderers may position symbols in different 
                     locations relative to each other.  Say that renderer B, for example, provides improved 
                     spacing for a particular expression over renderer A.  Authors are strongly warned that 
                     "tweaking" the layout for renderer A may produce very poor results in renderer B, 
                     very likely worse than without any explicit adjustment at all.
                  </p>
                  <p>Even when a specific choice of renderer can be assumed, its spacing
                     rules may be improved in successive versions, so that the effect of
                     tweaking in a given MathML document may grow worse with time. Also,
                     when style sheet mechanisms are extended to MathML, even one version
                     of a renderer may use different spacing rules for users with different
                     style sheets.
                  </p>
                  <p>Therefore, it is suggested that MathML markup never use
                     <code>mpadded</code> or <code>mspace</code> elements
                     to tweak the rendering of specific expressions, unless the MathML is
                     generated solely to be viewed using one specific version of one MathML
                     renderer, using one specific style sheet (if style sheets are
                     available in that renderer).
                  </p>
                  <p>In cases where the temptation to improve spacing proves too strong,
                     careful use of <code>mpadded</code>,
                     <code>mphantom</code>, or the alignment elements (<a href="chapter3-d.html#presm.malign">Section&nbsp;3.5.5 Alignment Markers
                        <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></a>) may give more portable results than the
                     direct insertion of extra space using <code>mspace</code> or
                     <code>mtext</code>. Advice given to the implementers of MathML
                     renderers might be still more productive, in the long run.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.warnspacing" id="presm.warnspacing"></a>3.1.8.2 Warning: spacing should not be used to convey meaning
                  </h4>
                  <p>MathML elements that permit "negative spacing", namely
                     <code>mspace</code>, <code>mpadded</code>, and
                     <code>mtext</code>, could in theory be used to simulate new
                     notations or "overstruck" characters by the visual overlap of the
                     renderings of more than one MathML sub-expression.
                  </p>
                  <p>This practice is <em>strongly discouraged in all situations</em>,
                     for the following reasons:
                     
                  </p>
                  <ul>
                     <li>
                        <p>it will give different results in different MathML renderers
                           (so the warning about "tweaking" applies), especially
                           if attempts are made to render glyphs outside the bounding box of
                           the MathML expression;
                           
                        </p>
                     </li>
                     <li>
                        <p>it is likely to appear much worse than a more standard construct
                           supported by good renderers;
                        </p>
                     </li>
                     <li>
                        <p>such expressions are almost certain to be uninterpretable
                           by audio renderers, computer algebra systems,
                           text searches for standard symbols,
                           or other processors of MathML input.
                        </p>
                     </li>
                  </ul>
                  <p>More generally, any construct that uses spacing to convey
                     mathematical meaning, rather than simply as an aid to viewing
                     expression structure, is discouraged. That is, the constructs that
                     are discouraged are those that would be interpreted differently by a
                     human viewer of rendered MathML if all explicit spacing was
                     removed.
                  </p>
                  <p>Consider using the <a href="chapter3-d.html#presm.mglyph"><code>mglyph</code></a> element
                     for cases such as this.  If such spacing constructs are used in spite of this warning, they should
                     be enclosed in a <code>semantics</code> element that also
                     provides an additional MathML expression that can be interpreted in a
                     standard way. See <a href="chapter5-d.html#mixing.semantic.annotations">Section&nbsp;5.1 Annotation Framework</a> for further discussion.
                     
                  </p>
                  <p>The above warning also applies to most uses of rendering
                     attributes to alter the meaning conveyed by an expression, with the
                     exception of attributes on <code>mi</code> (such as <code>mathvariant</code>)
                     used to distinguish one variable from another.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.summary" id="presm.summary"></a>3.1.9 Summary of Presentation Elements
               </h3>
               <div class="div4">
                  
                  <h4><a name="id.3.1.9.1" id="id.3.1.9.1"></a>3.1.9.1 Token Elements
                  </h4>
                  <table border="1" class="attributes">
                     <tbody>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mi"><code>mi</code></a></td>
                           <td>identifier</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mn"><code>mn</code></a></td>
                           <td>number</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mo"><code>mo</code></a></td>
                           <td>operator, fence, or  separator</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mtext"><code>mtext</code></a></td>
                           <td>text</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mspace"><code>mspace</code></a></td>
                           <td>space</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.ms"><code>ms</code></a></td>
                           <td>string literal</td>
                        </tr>
                     </tbody>
                  </table>
                  <p>Additionally, the <a href="chapter3-d.html#presm.mglyph"><code>mglyph</code></a> element
                     may be used within Token elements to represent non-standard symbols as images.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.1.9.2" id="id.3.1.9.2"></a>3.1.9.2 General Layout Schemata
                  </h4>
                  <table border="1">
                     <tbody>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mrow"><code>mrow</code></a></td>
                           <td>group any number of sub-expressions horizontally</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mfrac"><code>mfrac</code></a></td>
                           <td>form a fraction from two sub-expressions</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mroot"><code>msqrt</code></a></td>
                           <td>form a square root (radical without an index)</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mroot"><code>mroot</code></a></td>
                           <td>form a radical with specified index</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mstyle"><code>mstyle</code></a></td>
                           <td>style change</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.merror"><code>merror</code></a></td>
                           <td>enclose a syntax error message from a preprocessor</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mpadded"><code>mpadded</code></a></td>
                           <td>adjust space around content</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mphantom"><code>mphantom</code></a></td>
                           <td>make content invisible but preserve its size</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mfenced"><code>mfenced</code></a></td>
                           <td>surround content with a pair of fences</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.menclose"><code>menclose</code></a></td>
                           <td>enclose content with a stretching symbol such as a long division sign.</td>
                        </tr>
                     </tbody>
                  </table>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.1.9.3" id="id.3.1.9.3"></a>3.1.9.3 Script and Limit Schemata
                  </h4>
                  <table border="1">
                     <tbody>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msub"><code>msub</code></a></td>
                           <td>attach a subscript to a base</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msup"><code>msup</code></a></td>
                           <td>attach a superscript to a base</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msubsup"><code>msubsup</code></a></td>
                           <td>attach a subscript-superscript pair to a base</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.munder"><code>munder</code></a></td>
                           <td>attach an underscript to a base</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mover"><code>mover</code></a></td>
                           <td>attach an overscript to a base</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.munderover"><code>munderover</code></a></td>
                           <td>attach an underscript-overscript pair to a base</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mmultiscripts"><code>mmultiscripts</code></a></td>
                           <td>attach prescripts and tensor indices to a base</td>
                        </tr>
                     </tbody>
                  </table>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.1.9.4" id="id.3.1.9.4"></a>3.1.9.4 Tables and Matrices
                  </h4>
                  <table border="1">
                     <tbody>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mtable"><code>mtable</code></a></td>
                           <td>table or matrix</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mlabeledtr"><code>mlabeledtr</code></a></td>
                           <td>row in a table or matrix with a label or equation number</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mtr"><code>mtr</code></a></td>
                           <td>row in a table or matrix</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mtd"><code>mtd</code></a></td>
                           <td>one entry in a table or matrix</td>
                        </tr>
                        <tr>
                           <td>
                              <a href="chapter3-d.html#presm.malign"><code>maligngroup</code></a> and
                              <a href="chapter3-d.html#presm.malign"><code>malignmark</code></a></td>
                           <td>alignment markers</td>
                        </tr>
                     </tbody>
                  </table>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.1.9.5" id="id.3.1.9.5"></a>3.1.9.5 Elementary Math Layout
                  </h4>
                  <table border="1">
                     <tbody>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mstack"><code>mstack</code></a></td>
                           <td>columns of aligned characters</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mlongdiv"><code>mlongdiv</code></a></td>
                           <td>similar to msgroup, with the addition of a divisor and result</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msgroup"><code>msgroup</code></a></td>
                           <td>a group of rows in an mstack that are shifted by similar amounts</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msrow"><code>msrow</code></a></td>
                           <td>a row in an mstack</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mscarries"><code>mscarries</code></a></td>
                           <td>row in an mstack that whose contents represent carries or borrows</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.mscarry"><code>mscarry</code></a></td>
                           <td>one entry in an mscarries</td>
                        </tr>
                        <tr>
                           <td><a href="chapter3-d.html#presm.msline"><code>msline</code></a></td>
                           <td>horizontal line inside of <code>mstack</code></td>
                        </tr>
                     </tbody>
                  </table>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.1.9.6" id="id.3.1.9.6"></a>3.1.9.6 Enlivening Expressions
                  </h4>
                  <table border="1">
                     <tbody>
                        <tr>
                           <td><a href="chapter3-d.html#presm.maction"><code>maction</code></a></td>
                           <td>bind actions to a sub-expression</td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.presatt" id="presm.presatt"></a>3.1.10 Mathematics style attributes common to presentation elements
               </h3>
               <p>In addition to the attributes listed in <a href="chapter2-d.html#fund.globatt">Section&nbsp;2.1.6 Attributes Shared by all MathML Elements</a>,
                  all MathML presentation elements accept the following two attributes:
                  
                  
               </p>
               <table border="1" class="attributes">
                  <thead>
                     <tr>
                        <th>Name</th>
                        <th>values</th>
                        <th>default</th>
                     </tr>
                  </thead>
                  <tbody>
                     <tr>
                        <td rowspan="2" class="attname">mathcolor</td>
                        <td><a href="chapter2-d.html#type.color"><em>color</em></a></td>
                        <td><em>inherited</em></td>
                     </tr>
                     <tr>
                        <td colspan="2" class="attdesc">
                           Specifies the foreground color to use when drawing the components of this element,
                           such as the content for token elements or any lines, surds, or other decorations.
                           It also establishes the default <code>mathcolor</code> used for child elements
                           when used on a layout element.
                           
                        </td>
                     </tr>
                     <tr>
                        <td rowspan="2" class="attname">mathbackground</td>
                        <td><a href="chapter2-d.html#type.color"><em>color</em></a> | "transparent"
                        </td>
                        <td>transparent</td>
                     </tr>
                     <tr>
                        <td colspan="2" class="attdesc">
                           Specifies the background color to be used to fill in the bounding box
                           of the element and its children. The default, "transparent", lets the
                           background color, if any, used in the current rendering context to show through.
                           
                        </td>
                     </tr>
                  </tbody>
               </table>
               <p>These style attributes are primarily intended for visual media.
                  They are not expected to affect the intended semantics of displayed
                  expressions, but are for use in highlighting or drawing attention
                  to the affected subexpressions.  For example, a red "x" is not assumed
                  to be semantically different than a black "x", in contrast to
                  variables with different <code>mathvariant</code> (See <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>).
               </p>
               <p>Since MathML expressions are often embedded in a textual data
                  format such as XHTML,  the MathML renderer should inherit the
                  foreground color used in the context in which the MathML appears.
                  Note, however, that MathML  doesn't specify the mechanism by which
                  style information is inherited from the rendering environment.
                  
               </p>
               <p>Note that the suggested MathML visual rendering rules do not define the
                  precise extent of the region whose background is affected by the
                  <code>mathbackground</code> attribute,
                  except that, when the content does not have
                  negative dimensions and its drawing region is not overlapped by other
                  drawing due to surrounding negative spacing, this region should lie
                  behind all the drawing done to render the content, but should not lie behind any of the
                  drawing done to render surrounding expressions. The effect of overlap
                  of drawing regions caused by negative spacing on the extent of the
                  region affected by the <code>mathbackground</code> attribute is not
                  defined by these rules.
               </p>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="presm.tokel" id="presm.tokel"></a>3.2 Token Elements
            </h2>
            <p>Token elements in presentation markup are broadly intended to
               represent the smallest units of mathematical notation which carry
               meaning.  Tokens are roughly analogous to words in text.  However,
               because of the precise, symbolic nature of mathematical notation, the
               various categories and properties of token elements figure prominently in
               MathML markup.  By contrast, in textual data, individual words rarely
               need to be marked up or styled specially.
            </p>
            <p>Frequently, tokens consist of a single character denoting a
               mathematical symbol.  Other cases, e.g. function names, involve
               multi-character tokens.  Further, because traditional mathematical
               notation makes wide use of symbols distinguished by their
               typographical properties (e.g. a Fraktur 'g' for a Lie algebra, or a
               bold 'x' for a vector), care must be taken to insure that styling
               mechanisms respect typographical properties which carry meaning.
               Consequently, characters, tokens, and typographical properties of
               symbols are closely related to one another in MathML.
               
            </p>
            <p>Token elements represent
               identifiers (<a href="chapter3-d.html#presm.mi"><code>mi</code></a>),
               numbers (<a href="chapter3-d.html#presm.mn"><code>mn</code></a>),
               operators (<a href="chapter3-d.html#presm.mo"><code>mo</code></a>),
               text (<a href="chapter3-d.html#presm.mtext"><code>mtext</code></a>),
               strings (<a href="chapter3-d.html#presm.ms"><code>ms</code></a>)
               and spacing (<a href="chapter3-d.html#presm.mspace"><code>mspace</code></a>).
               The <a href="chapter3-d.html#presm.mglyph"><code>mglyph</code></a> element
               may be used <em>within</em> token elements
               to represent non-standard symbols by images.
               Preceding detailed discussion of the individual elements,
               the next two subsections discuss the allowable content of
               token elements and the attributes common to them.
               
            </p>
            <div class="div3">
               
               <h3><a name="presm.tokenchars" id="presm.tokenchars"></a>3.2.1 MathML characters in token elements
               </h3>
               <p>Character data in MathML markup is only allowed to occur as part of
                  the content of token elements.  Whitespace between elements is ignored.
                  With the exception of the empty <a href="chapter3-d.html#presm.mspace"><code>mspace</code></a> element,
                  token elements can contain any sequence of zero or more Unicode characters,
                  or <a href="chapter3-d.html#presm.mglyph"><code>mglyph</code></a> or 
                  <code>malignmark</code> elements.
                  The <a href="chapter3-d.html#presm.mglyph"><code>mglyph</code></a> element is used
                  to represent non-standard characters or symbols by images;
                  the <code>malignmark</code> element establishes an alignment point for use within
                  table constructs, and is otherwise invisible (See <a href="chapter3-d.html#presm.malign">Section&nbsp;3.5.5 Alignment Markers
                     <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></a>).
               </p>
               <p>MathML characters can be either represented
                  directly as Unicode character data, or indirectly via numeric or
                  character entity references.  See <a href="chapter7-d.html">Chapter&nbsp;7 Characters, Entities and Fonts</a> for a
                  discussion of the advantages and disadvantages of numeric 
                  character references versus
                  entity references, and <a href="appendixg-d.html#Entities">[Entities]</a> for a full list of the entity names available.
                  Also, see <a href="chapter7-d.html#chars.anomalous">Section&nbsp;7.7 Anomalous Mathematical Characters</a> for a discussion of the
                  appropriate character content to choose for certain applications.
               </p>
               <p>Token elements (other than <code>mspace</code>) should
                  be rendered as their content, if any, (i.e. in the visual case, as a
                  closely-spaced horizontal row of standard glyphs for the characters
                  or images for the <code>mglyph</code>s in their content).
                  An <code>mspace</code> element is rendered as a blank space of a width determined by its attributes.
                  Rendering algorithms should also take into account the
                  mathematics style attributes as described below, and modify surrounding
                  spacing by rules or attributes specific to each type of token
                  element.  The directional characteristics of the content must
                  also be respected (see <a href="chapter3-d.html#presm.bidi.token">Section&nbsp;3.1.5.2 Bidirectional Layout in Token Elements</a>).
                  
               </p>
               <div class="div4">
                  
                  <h4><a name="presm.symbolchars" id="presm.symbolchars"></a>3.2.1.1 Alphanumeric symbol characters
                  </h4>
                  <p>A large class of mathematical symbols are single letter identifiers
                     typically used as variable names in formulas.  Different font variants
                     of a letter are treated as separate symbols.  For example, a Fraktur
                     'g' might denote a Lie algebra, while a Roman 'g' denotes the
                     corresponding Lie group.  These letter-like symbols are traditionally
                     typeset differently than the same characters appearing in text, using
                     different spacing and ligature conventions.  These characters must
                     also be treated specially by style mechanisms, since arbitrary style
                     transformations can change meaning in an expression.
                  </p>
                  <p>For these reasons, Unicode contains
                     more than nine hundred Math Alphanumeric Symbol characters
                     corresponding to letter-like symbols.  These characters are in the
                     Secondary Multilingual Plane (SMP).  See <a href="appendixg-d.html#Entities">[Entities]</a> for
                     more information.  As valid Unicode data, these characters are
                     permitted in MathML and, as tools and fonts for them become widely
                     available, we anticipate they will be the predominant way of denoting
                     letter-like symbols.
                  </p>
                  <p>MathML also provides an alternative encoding
                     for these characters using only Basic Multilingual Plane
                     (BMP) characters together with markup.  MathML defines a
                     correspondence between token elements with certain combinations of BMP
                     character data and the <code>mathvariant</code> attribute and tokens
                     containing SMP Math Alphanumeric Symbol characters.  Processing
                     applications that accept SMP characters are required to treat the
                     corresponding BMP and attribute combinations identically.  This is particularly important for applications that
                     support searching and/or equality testing.
                  </p>
                  <p>The <code>mathvariant</code> attribute is described in more detail in <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>,
                     and a complete technical description of the corresponding characters is given in <a href="chapter7-d.html#chars.BMP-SMP">Section&nbsp;7.5 Mathematical Alphanumeric Symbols</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.mglyph" id="presm.mglyph"></a>3.2.1.2 Using images to represent
                     symbols <code>&lt;mglyph/&gt;</code></h4>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.1.2.1" id="id.3.2.1.2.1"></a>3.2.1.2.1 Description
                     </h5>
                     <p>The <code>mglyph</code> element provides a mechanism
                        for displaying images to represent non-standard symbols.
                        It may be used within the content of the token elements
                        <code>mi</code>, <code>mn</code>, <code>mo</code>, <code>mtext</code> or <code>ms</code>
                        where existing Unicode characters are not adequate.
                     </p>
                     <p>Unicode defines a large number of characters used in mathematics
                        and, in most cases, glyphs representing these characters are widely
                        available in a variety of fonts.  Although these characters should
                        meet almost all users needs, MathML recognizes that mathematics is not
                        static and that new characters and symbols are added when convenient.  Characters
                        that become well accepted will likely be eventually incorporated by
                        the Unicode Consortium or other standards bodies, but that is often a
                        lengthy process.
                     </p>
                     <p>Note that the glyph's <code>src</code> attribute uniquely identifies the <code>mglyph</code>;
                        two <code>mglyph</code>s with the same values for <code>src</code> should
                        be considered identical by applications that must determine whether
                        two characters/glyphs are identical.
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.1.2.2" id="id.3.2.1.2.2"></a>3.2.1.2.2 Attributes
                     </h5>
                     <p>The <code>mglyph</code> element accepts the attributes listed in
                        <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>, but note that <code>mathcolor</code> has no effect.
                        The background color, <code>mathbackground</code>, should show through
                        if the specified image has transparency.
                        
                     </p>
                     <p>
                        <code>mglyph</code> also accepts the additional attributes listed here.
                     </p>
                     <table border="1" class="attributes">
                        <thead>
                           <tr>
                              <th>Name</th>
                              <th>values</th>
                              <th>default</th>
                           </tr>
                        </thead>
                        <tbody>
                           <tr>
                              <td rowspan="2" class="attname">src</td>
                              <td><a href="chapter2-d.html#type.uri"><em>URI</em></a></td>
                              <td><em>required</em></td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the location of the image resource;
                                 it may be a URI relative to the base-URI of the source of the MathML, if any.
                                 Examples of widely recognized image formats include GIF, JPEG and PNG; However,
                                 it may be advisable to omit the extension from the <code>src</code> URI, so
                                 that a user agent may use content-negotiation to choose the most appropriate format.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">width</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td><em>from image</em></td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the desired width of the glyph; see <code>height</code>.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">height</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td><em>from image</em></td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the desired height of the glyph.
                                 If only one of <code>width</code> and <code>height</code> are given,
                                 the image should be scaled to preserve the aspect ratio;
                                 if neither are given, the image should be displayed at its natural size.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">valign</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a> 
                              </td>
                              <td>0ex</td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the baseline alignment point of the image with respect to the current baseline.
                                 A positive value shifts the bottom of the image above the current baseline, while a negative value lowers it.  A value of
                                 0 (the default) means that the baseline of the image is at the bottom of the image.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">alt</td>
                              <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                              <td><em>required</em></td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Provides an alternate name for the glyph.  If the specified image can't be found or displayed,
                                 the renderer may use this name in a warning message or some unknown glyph notation.
                                 The name might also be used by an audio renderer or symbol processing
                                 system and should be chosen to be descriptive.
                                 
                              </td>
                           </tr>
                        </tbody>
                     </table>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.1.2.3" id="id.3.2.1.2.3"></a>3.2.1.2.3 Example
                     </h5>
                     <p>The following example illustrates how a researcher might use
                        the <code>mglyph</code> construct with a set of images to work
                        with braid group notation.
                        
                     </p><pre class="mathml-fragment">
&lt;mrow&gt;
  &lt;mi&gt;&lt;mglyph src="my-braid-23" alt="2 3 braid"/&gt;&lt;/mi&gt;
  &lt;mo&gt;+&lt;/mo&gt;
  &lt;mi&gt;&lt;mglyph src="my-braid-132" alt="1 3 2 braid"/&gt;&lt;/mi&gt;
  &lt;mo&gt;=&lt;/mo&gt;
  &lt;mi&gt;&lt;mglyph src="my-braid-13" alt="1 3 braid"/&gt;&lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>
                        This might render as:
                        
                     </p>
                     <blockquote>
                        <p><img src="image/f3006.gif" alt="\includegraphics{braids}"></p>
                     </blockquote>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.1.2.4" id="id.3.2.1.2.4"></a>3.2.1.2.4 Deprecated Attributes
                     </h5>
                     <p>Originally, <code>mglyph</code> was designed to provide access to non-standard
                        fonts. Since this functionality was seldom implemented, nor were downloadable
                        web fonts widely available, this use of <code>mglyph</code> has been deprecated.
                        For reference, the following attributes were previously defined:
                        
                     </p>
                     <table border="1" class="attributes">
                        <thead>
                           <tr>
                              <th>Name</th>
                              <th>values</th>
                           </tr>
                        </thead>
                        <tbody>
                           <tr>
                              <td rowspan="2" class="attname">fontfamily</td>
                              <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                           </tr>
                           <tr>
                              <td class="attdesc">
                                 the name of a font that may be available to a MathML renderer,
                                 or a CSS font specification; See  <a href="chapter6-d.html#world-int-style">Section&nbsp;6.5 Using CSS with MathML</a>
                                 and CSS <a href="appendixh-d.html#CSS21">[CSS21]</a> for more information.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">index</td>
                              <td><a href="chapter2-d.html#type.integer"><em>integer</em></a></td>
                           </tr>
                           <tr>
                              <td class="attdesc">
                                 Specified a position of the desired glyph within the font named
                                 by the <code>fontfamily</code> attribute (see <a href="chapter3-d.html#presm.deprecatt">Section&nbsp;3.2.2.1 Deprecated style attributes on token elements</a>).
                                 
                              </td>
                           </tr>
                        </tbody>
                     </table>
                     <p>
                        In MathML 1 and 2, both were required attributes; they are now optional
                        and should be ignored unless the <code>src</code> attribute is missing.
                        
                     </p>
                     <p>Additionally, in MathML 2, <code>mglyph</code> accepted the attributes described in  <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>
                        (<code>mathvariant</code> and <code>mathsize</code>, along with the attributes deprecated there);
                        to make clear that <code>mglyph</code> is <em>not</em> a token element, and since
                        these attributes have no effect in any case, these attributes have been deprecated.
                        
                     </p>
                  </div>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.commatt" id="presm.commatt"></a>3.2.2 Mathematics style attributes common to token elements
               </h3>
               <p>In addition to the attributes defined for all presentation elements
                  (<a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>), MathML includes two  <em>mathematics style</em> attributes
                  as well as a directionality attribute
                  valid on all presentation token elements,
                  as well as the <code>math</code> and <code>mstyle</code> elements;
                  <code>dir</code> is also valid on <code>mrow</code> elements.
                  The attributes are:
                  
                  
               </p>
               <table border="1" class="attributes">
                  <thead>
                     <tr>
                        <th>Name</th>
                        <th>values</th>
                        <th>default</th>
                     </tr>
                  </thead>
                  <tbody>
                     <tr>
                        <td rowspan="2" class="attname">mathvariant</td>
                        <td>
                           "normal" | "bold" | "italic" | "bold-italic" | "double-struck" |
                           "bold-fraktur" | "script" | "bold-script" |
                           "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" |
                           "sans-serif-bold-italic" | "monospace" |
                           "initial" | "tailed" | "looped" | "stretched"
                           
                        </td>
                        <td>normal (<em>except on</em> <code>&lt;mi&gt;</code>)
                        </td>
                     </tr>
                     <tr>
                        <td colspan="2" class="attdesc">
                           Specifies the logical class of the token. Note that this class
                           is more than styling, it typically conveys semantic intent; see the discussion below.
                           
                        </td>
                     </tr>
                     <tr>
                        <td rowspan="2" class="attname">mathsize</td>
                        <td>"small" | "normal" | "big" | <a href="chapter2-d.html#type.length"><em>length</em></a></td>
                        <td><em>inherited</em></td>
                     </tr>
                     <tr>
                        <td colspan="2" class="attdesc">
                           Specifies the size to display the token content.
                           The values "small" and "big" choose a size
                           smaller or larger than the current font size, but leave the exact proportions
                           unspecified; "normal" is allowed for completeness, but since
                           it is equivalent to "100%" or "1em", it has no effect.
                           
                        </td>
                     </tr>
                     <tr>
                        <td rowspan="2" class="attname">dir</td>
                        <td>"ltr" | "rtl"</td>
                        <td><em>inherited</em></td>
                     </tr>
                     <tr>
                        <td colspan="2" class="attdesc">
                           specifies the initial directionality for text within the token:
                           <code>ltr</code> (Left To Right) or <code>rtl</code> (Right To Left).
                           This attribute should only be needed in rare cases involving weak or neutral characters;
                           see <a href="chapter3-d.html#presm.bidi.math">Section&nbsp;3.1.5.1 Overall Directionality of Mathematics Formulas</a> for further discussion.
                           It has no effect on <code>mspace</code>.
                           
                        </td>
                     </tr>
                  </tbody>
               </table>
               <p>The <code>mathvariant</code> attribute defines logical classes of token
                  elements.  Each class provides a collection of typographically-related
                  symbolic tokens.  Each token has a specific meaning within a given
                  mathematical expression and, therefore, needs to be visually
                  distinguished and protected from inadvertent document-wide style
                  changes which might change its meaning.  Each token is identified
                  by the combination of the <code>mathvariant</code> attribute value
                  and the character data in the token element.
               </p>
               <p>When MathML rendering takes place in an environment where CSS is
                  available, the mathematics style attributes can be viewed as
                  predefined selectors for CSS style rules.
                  See <a href="chapter6-d.html#world-int-style">Section&nbsp;6.5 Using CSS with MathML</a> for discussion of the
                  interaction of MathML and CSS.
                  Also, see <a href="appendixh-d.html#MathMLforCSS">[MathMLforCSS]</a> for discussion of rendering MathML by CSS
                  and a sample CSS style sheet.
                  When CSS is not available, it is up to the internal style mechanism of the rendering application
                  to visually distinguish the different logical classes.
                  Most MathML renderers will probably want to rely on some degree to additional,
                  internal style processing algorithms.
                  In particular, the <code>mathvariant</code> attribute does not follow the CSS inheritance model;
                  the default value is "normal" (non-slanted)
                  for all tokens except for <code>mi</code> with single-character content.
                  See <a href="chapter3-d.html#presm.mi">Section&nbsp;3.2.3 Identifier <code>&lt;mi&gt;</code></a> for details.
               </p>
               <p>Renderers have complete freedom in
                  mapping mathematics style attributes to specific rendering properties.
                  However, in practice, the mathematics style attribute names and values
                  suggest obvious typographical properties, and renderers should attempt
                  to respect these natural interpretations as far as possible.  For
                  example, it is reasonable to render a token with the
                  <code>mathvariant</code> attribute set to "sans-serif" in
                  Helvetica or Arial.  However, rendering the token in a Times Roman
                  font could be seriously misleading and should be avoided. 
               </p>
               <p>In principle, any <code>mathvariant</code> value may be used with any
                  character data to define a specific symbolic token.  In practice,
                  only certain combinations of character data and <code>mathvariant</code>
                  values will be visually distinguished by a given renderer.  For example,
                  there is no clear-cut rendering for a "fraktur alpha" or a "bold italic
                  Kanji" character, and the <code>mathvariant</code> values "initial",
                  "tailed", "looped", and "stretched" are appropriate only for Arabic
                  characters.
               </p>
               <p>Certain combinations of character data and <code>mathvariant</code>
                  values are equivalent to assigned Unicode code points that encode
                  mathematical alphanumeric symbols.  These Unicode code points are
                  the ones in the <a href="http://www.unicode.org/charts/PDF/U1D400.pdf">SMP
                     Mathematical Alphanumeric Symbol</a> range U+1D400 to U+1D7FF,
                  listed in the Unicode standard, and the ones in the
                  <a href="http://www.unicode.org/charts/PDF/U2100.pdf">Letterlike
                     Symbols</a> range U+2100 to U+214F that represent "holes" in the
                  alphabets in the SMP, listed in <a href="chapter7-d.html#chars.BMP-SMP">Section&nbsp;7.5 Mathematical Alphanumeric Symbols</a>.
                  These characters are described in detail in section 2.2 of
                  <a href="http://www.unicode.org/reports/tr25/">UTR #25</a>.
                  The description of each such character in the Unicode standard
                  provides an unstyled character to which it would be equivalent
                  except for a font change that corresponds to a <code>mathvariant</code>
                  value.  A token element that uses the unstyled character in combination
                  with the corresponding <code>mathvariant</code> value is equivalent to a
                  token element that uses the mathematical alphanumeric symbol character
                  without the <code>mathvariant</code> attribute.  Note that the appearance
                  of a mathematical alphanumeric symbol character should not be altered
                  by surrounding <code>mathvariant</code> or other style declarations.
               </p>
               <p>Renderers should support those combinations of character data and
                  <code>mathvariant</code> values that correspond to Unicode characters,
                  and that they can visually distinguish using available font characters.
                  Renderers may ignore or support those combinations of character data
                  and <code>mathvariant</code> values that do not correspond to an assigned
                  Unicode code point, and authors should recognize that support for
                  mathematical symbols that do not correspond to assigned Unicode code
                  points may vary widely from one renderer to another.
               </p>
               <p>Since MathML expressions are often embedded in a textual data
                  format such as XHTML, the surrounding text and the MathML must share
                  rendering attributes such as font size, so that the renderings will be
                  compatible in style. For this reason, most attribute values affecting
                  text rendering are inherited from the rendering environment, as shown
                  in the "default" column in the table above. (In
                  cases where the surrounding text and the MathML are being rendered by
                  separate software, e.g. a browser and a plug-in, it is also important
                  for the rendering environment to provide the MathML renderer with
                  additional information, such as the baseline position of surrounding
                  text, which is not specified by any MathML attributes.)
                  Note, however, that MathML  doesn't specify the mechanism by which
                  style information is inherited from the rendering environment.
                  
                  
               </p>
               <p>If the requested <code>mathsize</code> of the current font is not available, the
                  renderer should approximate it in the manner likely to lead to the
                  most intelligible, highest quality rendering.
                  Note that many MathML elements automatically change the font size
                  in some of their children; see the discussion in <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.
               </p>
               <div class="div4">
                  
                  <h4><a name="presm.deprecatt" id="presm.deprecatt"></a>3.2.2.1 Deprecated style attributes on token elements
                  </h4>
                  <p>The MathML 1.01 style attributes listed below
                     are <a href="chapter2-d.html#interf.deprec">deprecated</a> in MathML 2 and 3.  
                     These attributes were aligned to CSS but, in rendering environments that support CSS,
                     it is preferable to use CSS directly to control the rendering properties
                     corresponding to these attributes, rather than the attributes themselves.
                     However as explained above, direct manipulation of these
                     rendering properties by whatever means should usually be avoided.
                     As a general rule, whenever there is a conflict between these
                     deprecated attributes and the corresponding attributes (<a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>),
                     the former attributes should be ignored.
                  </p>
                  <p>The deprecated attributes are:
                     
                     
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">fontfamily</td>
                           <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Should be the name of a font that may be available to a MathML renderer,
                              or a CSS font specification; See  <a href="chapter6-d.html#world-int-style">Section&nbsp;6.5 Using CSS with MathML</a>
                              and CSS <a href="appendixh-d.html#CSS21">[CSS21]</a> for more information.
                              Deprecated in favor of <code>mathvariant</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">fontweight</td>
                           <td>"normal" | "bold"</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specified the font weight for the token.
                              Deprecated in favor of <code>mathvariant</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">fontstyle</td>
                           <td>"normal" | "italic"</td>
                           <td>normal (<em>except on</em> <code>&lt;mi&gt;</code>)
                           </td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specified the font style to use for the token.
                              Deprecated in favor of <code>mathvariant</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">fontsize</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specified the size for the token.
                              Deprecated in favor of <code>mathsize</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">color</td>
                           <td><a href="chapter2-d.html#type.color"><em>color</em></a></td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specified the color for the token.
                              Deprecated in favor of <code>mathcolor</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">background</td>
                           <td><a href="chapter2-d.html#type.color"><em>color</em></a> | "transparent"
                           </td>
                           <td>transparent</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specified the background color to be used to fill in the bounding box
                              of the element and its children. Deprecated in favor of <code>mathbackground</code>.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mi" id="presm.mi"></a>3.2.3 Identifier <code>&lt;mi&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.2.3.1" id="id.3.2.3.1"></a>3.2.3.1 Description
                  </h4>
                  <p>An <code>mi</code> element represents a symbolic name or
                     arbitrary text that should be rendered as an identifier. Identifiers
                     can include variables, function names, and symbolic constants.
                     A typical graphical renderer would render an <code>mi</code> element
                     as its content (See <a href="chapter3-d.html#presm.tokenchars">Section&nbsp;3.2.1 MathML characters in token elements</a>),
                     with no extra spacing around it (except spacing associated with
                     neighboring elements).
                  </p>
                  <p>Not all "mathematical identifiers" are represented by
                     <code>mi</code> elements — for example, subscripted or primed
                     variables should be represented using <code>msub</code> or
                     <code>msup</code> respectively. Conversely, arbitrary text
                     playing the role of a "term" (such as an ellipsis in a summed series)
                     can be represented using an <code>mi</code> element, as shown
                     in an example in <a href="chapter3-d.html#presm.mixtextmath">Section&nbsp;3.2.6.4 Mixing text and mathematics</a>.
                  </p>
                  <p>It should be stressed that <code>mi</code> is a
                     presentation element, and as such, it only indicates that its content
                     should be rendered as an identifier. In the majority of cases, the
                     contents of an <code>mi</code> will actually represent a
                     mathematical identifier such as a variable or function name. However,
                     as the preceding paragraph indicates, the correspondence between
                     notations that should render as identifiers and notations that are
                     actually intended to represent mathematical identifiers is not
                     perfect. For an element whose semantics is guaranteed to be that of an
                     identifier, see the description of <code>ci</code> in 
                     <a href="chapter4-d.html">Chapter&nbsp;4 Content Markup</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.3.2" id="id.3.2.3.2"></a>3.2.3.2 Attributes
                  </h4>
                  <p><code>mi</code> elements accept the attributes listed in
                     <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>, but in one case with a different default value:
                     
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">mathvariant</td>
                           <td>"normal" | "bold" | "italic" | "bold-italic" | "double-struck" |
                              "bold-fraktur" | "script" | "bold-script" |
                              "fraktur" | "sans-serif" | "bold-sans-serif" | "sans-serif-italic" |
                              "sans-serif-bold-italic" | "monospace" |
                              "initial" | "tailed" | "looped" | "stretched"
                              
                           </td>
                           <td><em>(depends on content; described below)</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              
                              Specifies the logical class of the token.
                              The default is "normal" (non-slanted) unless the content
                              is a single character, in which case it would be "italic".
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>Note that the deprecated <code>fontstyle</code> attribute
                     defaults in the same way as <code>mathvariant</code>, depending on the content.
                  </p>
                  <p>Note that for purposes of determining equivalences of Math
                     Alphanumeric Symbol
                     characters (See <a href="chapter7-d.html#chars.BMP-SMP">Section&nbsp;7.5 Mathematical Alphanumeric Symbols</a> and <a href="chapter3-d.html#presm.symbolchars">Section&nbsp;3.2.1.1 Alphanumeric symbol characters</a>) the value of the <code>mathvariant</code> attribute should be resolved first,
                     including the special defaulting behavior described above.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.3.3" id="id.3.2.3.3"></a>3.2.3.3 Examples
                  </h4><pre class="mathml">
&lt;mi&gt; x &lt;/mi&gt;
&lt;mi&gt; D &lt;/mi&gt;
&lt;mi&gt; sin &lt;/mi&gt;
&lt;mi mathvariant='script'&gt; L &lt;/mi&gt;
&lt;mi&gt;&lt;/mi&gt;
</pre><p>An <code>mi</code> element with no content is allowed;
                     <code>&lt;mi&gt;&lt;/mi&gt;</code> might, for example, be used by an
                     "expression editor" to represent a location in a MathML expression
                     which requires a "term" (according to conventional syntax for
                     mathematics) but does not yet contain one.
                  </p>
                  <p>Identifiers include function names such as
                     "sin". Expressions such as "sin <var>x</var>"
                     should be written using the character U+2061 
                     (which also has the entity names <code>&amp;af;</code> and <code>&amp;ApplyFunction;</code>) as shown below;
                     see also the discussion of invisible operators in <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                        <code>&lt;mo&gt;</code></a>.
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt; sin &lt;/mi&gt;
  &lt;mo&gt; &amp;#x2061;<span style="color:#999900">&lt;!--FUNCTION APPLICATION--&gt;</span> &lt;/mo&gt;
  &lt;mi&gt; x &lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>Miscellaneous text that should be treated as a "term" can also be
                     represented by an <code>mi</code> element, as in:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mn&gt; 1 &lt;/mn&gt;
  &lt;mo&gt; + &lt;/mo&gt;
  &lt;mi&gt; ... &lt;/mi&gt;
  &lt;mo&gt; + &lt;/mo&gt;
  &lt;mi&gt; n &lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>When an <code>mi</code> is used in such exceptional
                     situations, explicitly setting the <code>mathvariant</code> attribute
                     may give better results than the default behavior of some
                     renderers.
                  </p>
                  <p>The names of symbolic constants should be represented as
                     <code>mi</code> elements:
                     
                  </p><pre class="mathml">
&lt;mi&gt; &amp;#x3C0;<span style="color:#999900">&lt;!--GREEK SMALL LETTER PI--&gt;</span> &lt;/mi&gt;
&lt;mi&gt; &amp;#x2148;<span style="color:#999900">&lt;!--DOUBLE-STRUCK ITALIC SMALL I--&gt;</span> &lt;/mi&gt;
&lt;mi&gt; &amp;#x2147;<span style="color:#999900">&lt;!--DOUBLE-STRUCK ITALIC SMALL E--&gt;</span> &lt;/mi&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mn" id="presm.mn"></a>3.2.4 Number <code>&lt;mn&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.2.4.1" id="id.3.2.4.1"></a>3.2.4.1 Description
                  </h4>
                  <p>An <code>mn</code> element represents a "numeric
                     literal" or other data that should be rendered as a numeric
                     literal. Generally speaking, a numeric literal is a sequence of digits,
                     perhaps including a decimal point, representing an unsigned integer or real
                     number.
                     A typical graphical renderer would render an <code>mn</code> element as
                     its content (See <a href="chapter3-d.html#presm.tokenchars">Section&nbsp;3.2.1 MathML characters in token elements</a>), with no extra spacing around them
                     (except spacing from neighboring elements such as <code>mo</code>).
                     <code>mn</code> elements are typically rendered in an unslanted font.
                     
                  </p>
                  <p>The mathematical concept of a "number" can be quite 
                     subtle and involved, depending on the context. As a consequence, not all
                     mathematical numbers should be represented using <code>mn</code>; examples of mathematical numbers that should be
                     represented differently are shown below, and include 
                     complex numbers, ratios of numbers shown as fractions, and names of numeric
                     constants.
                  </p>
                  <p>Conversely, since <code>mn</code> is a presentation
                     element, there are a few situations where it may be desirable to include
                     arbitrary text in the content of an <code>mn</code> that
                     should merely render as a numeric literal, even though that content
                     may not be unambiguously interpretable as a number according to any
                     particular standard encoding of numbers as character sequences. As a
                     general rule, however, the <code>mn</code> element should be
                     reserved for situations where its content is actually intended to
                     represent a numeric quantity in some fashion.  For an element whose
                     semantics are guaranteed to be that of a particular kind of
                     mathematical number, see the description of <code>cn</code> in
                     <a href="chapter4-d.html">Chapter&nbsp;4 Content Markup</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.4.2" id="id.3.2.4.2"></a>3.2.4.2 Attributes
                  </h4>
                  <p><code>mn</code> elements accept the attributes listed in <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.4.3" id="id.3.2.4.3"></a>3.2.4.3 Examples
                  </h4><pre class="mathml">
&lt;mn&gt; 2 &lt;/mn&gt;
&lt;mn&gt; 0.123 &lt;/mn&gt;
&lt;mn&gt; 1,000,000 &lt;/mn&gt;
&lt;mn&gt; 2.1e10 &lt;/mn&gt;
&lt;mn&gt; 0xFFEF &lt;/mn&gt;
&lt;mn&gt; MCMLXIX &lt;/mn&gt;
&lt;mn&gt; twenty one &lt;/mn&gt;
</pre></div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.4.4" id="id.3.2.4.4"></a>3.2.4.4 Numbers that should <em>not</em> be written 
                     using <code>&lt;mn&gt;</code> alone
                  </h4>
                  <p>Many mathematical numbers should be represented using presentation
                     elements other than <code>mn</code> alone; this includes
                     complex numbers, ratios of numbers shown as fractions, and 
                     names of numeric constants. Examples of MathML representations of
                     such numbers include:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mn&gt; 2 &lt;/mn&gt;
  &lt;mo&gt; + &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mn&gt; 3 &lt;/mn&gt;
    &lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
    &lt;mi&gt; &amp;#x2148;<span style="color:#999900">&lt;!--DOUBLE-STRUCK ITALIC SMALL I--&gt;</span> &lt;/mi&gt;
  &lt;/mrow&gt;
&lt;/mrow&gt;
&lt;mfrac&gt; &lt;mn&gt; 1 &lt;/mn&gt; &lt;mn&gt; 2 &lt;/mn&gt; &lt;/mfrac&gt;
&lt;mi&gt; &amp;#x3C0;<span style="color:#999900">&lt;!--GREEK SMALL LETTER PI--&gt;</span> &lt;/mi&gt;
&lt;mi&gt; &amp;#x2147;<span style="color:#999900">&lt;!--DOUBLE-STRUCK ITALIC SMALL E--&gt;</span> &lt;/mi&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mo" id="presm.mo"></a>3.2.5 Operator, Fence, Separator or Accent
                  <code>&lt;mo&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.2.5.1" id="id.3.2.5.1"></a>3.2.5.1 Description
                  </h4>
                  <p>An <code>mo</code> element represents an operator or
                     anything that should be rendered as an operator. In general, the
                     notational conventions for mathematical operators are quite
                     complicated, and therefore MathML provides a relatively sophisticated
                     mechanism for specifying the rendering behavior of an
                     <code>mo</code> element.  As a consequence, in MathML the list
                     of things that should "render as an operator" includes a number of
                     notations that are not mathematical operators in the ordinary
                     sense. Besides ordinary operators with infix, prefix, or postfix
                     forms, these include fence characters such as braces, parentheses, and
                     "absolute value" bars; separators 
                     such as comma and semicolon; and
                     mathematical accents such as a bar or tilde over a symbol.
                     We will use the term "operator" in this chapter to refer to operators in this broad sense.
                  </p>
                  <p>Typical graphical renderers show all <code>mo</code>
                     elements as the content (See <a href="chapter3-d.html#presm.tokenchars">Section&nbsp;3.2.1 MathML characters in token elements</a>),
                     with additional spacing around the element determined by its attributes and
                     further described below. 
                     Renderers without access to complete fonts for the MathML character
                     set may choose  to render an <code>mo</code> element as
                     not precisely the characters in its content in some cases. For example,
                     <code>&lt;mo&gt; &amp;le; &lt;/mo&gt;</code> might be rendered as
                     <code>&lt;=</code> to a terminal.  However, as a general rule,
                     renderers should attempt to render the content of an
                     <code>mo</code> element as literally as possible.
                     That is,
                     <code>&lt;mo&gt; &amp;le; &lt;/mo&gt;</code> and
                     <code>&lt;mo&gt; &amp;lt;= &lt;/mo&gt;</code> should render differently.
                     The first one should render as a single character
                     representing a less-than-or-equal-to sign, and the second one as the
                     two-character sequence <code>&lt;=</code>.
                  </p>
                  <p>All operators, in the general sense used here, 
                     are subject to essentially the same rendering
                     attributes and rules. Subtle distinctions in the 
                     rendering of these classes of symbols,
                     when they exist, are supported using the Boolean attributes <code>fence</code>,
                     <code>separator</code> and <code>accent</code>, which can be used to distinguish these cases.
                  </p>
                  <p>A key feature of the <code>mo</code> element is that its
                     default attribute values are set on a case-by-case basis from an
                     "operator dictionary" as explained below. In particular, default
                     values for <code>fence</code>, <code>separator</code> and
                     <code>accent</code> can usually be found in the operator dictionary
                     and therefore need not be specified on each <code>mo</code>
                     element.
                  </p>
                  <p>Note that some mathematical operators are represented not by <code>mo</code> elements alone, but by <code>mo</code>
                     elements "embellished" with (for example) surrounding
                     superscripts; this is further described below. Conversely, as presentation
                     elements, <code>mo</code> elements can contain arbitrary text,
                     even when that text has no standard interpretation as an operator; for an
                     example, see the discussion "Mixing text and mathematics" in
                     <a href="chapter3-d.html#presm.mtext">Section&nbsp;3.2.6 Text <code>&lt;mtext&gt;</code></a>. See also <a href="chapter4-d.html">Chapter&nbsp;4 Content Markup</a> for
                     definitions of MathML content elements that are guaranteed to have the
                     semantics of specific mathematical operators.
                  </p>
                  <p>Note also that linebreaking, as discussed in
                     <a href="chapter3-d.html#presm.linebreaking">Section&nbsp;3.1.7 Linebreaking of Expressions</a>, usually takes place at operators
                     (either before or after, depending on local conventions).
                     Thus, <code>mo</code> accepts attributes to encode the desirability
                     of breaking at a particular operator, as well as attributes
                     describing the treatment of the operator and indentation in case
                     the a linebreak is made at that operator.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.5.2" id="id.3.2.5.2"></a>3.2.5.2 Attributes
                  </h4>
                  <p><code>mo</code> elements accept
                     the attributes listed in <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>
                     and the additional attributes listed here.
                     Since the display of operators is so critical in mathematics,
                     the <code>mo</code> element accepts a large number of attributes;
                     these are described in the next three subsections.
                     
                  </p>
                  <p>
                     Most attributes get their default values from an enclosing
                     <code>mstyle</code> element, <code>math</code> element, or from the
                     <a href="chapter3-d.html#presm.opdict">Section&nbsp;3.2.5.7.1 The operator dictionary</a>, as described later in this 
                     section. When a value that is listed as "inherited" is not explicitly given on an 
                     <code>mo</code>,  <code>mstyle</code> element, <code>math</code> element, or found in the operator
                     dictionary for a given <code>mo</code> element, the default value shown in
                     parentheses is used.
                     The attributes may also appear on any ancestor of the <code>math</code> element, if permitted by
                     the containing document, to provide defaults for all contained <code>math</code> elements.
                     In such cases, the attributes would be in the MathML namespace.
                  </p>
                  <div class="div5">
                     
                     <h5><a name="presm.mo.attrs" id="presm.mo.attrs"></a>3.2.5.2.1 Dictionary-based attributes
                     </h5>
                     <table border="1" id="presm.table-mo" class="attributes">
                        <thead>
                           <tr>
                              <th>Name</th>
                              <th>values</th>
                              <th>default</th>
                           </tr>
                        </thead>
                        <tbody>
                           <tr>
                              <td rowspan="2" class="attname">form</td>
                              <td>"prefix" | "infix" | "postfix"</td>
                              <td><em>set by position of operator in an</em> <code>mrow</code></td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the role of the operator in the enclosing expression.
                                 This role and the operator content affect the lookup of the operator in the operator dictionary
                                 which affects the spacing and other default properties;
                                 see <a href="chapter3-d.html#presm.formdefval">Section&nbsp;3.2.5.7.2 Default value of the form attribute</a>.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">fence</td>
                              <td>"true" | "false"</td>
                              <td><em>set by dictionary</em> (false)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies whether the operator represents a ‘fence’, such as a parenthesis.
                                 This attribute generally has no direct effect on the visual rendering,
                                 but may be useful in specific cases, such as non-visual renderers.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">separator</td>
                              <td>"true" | "false"</td>
                              <td><em>set by dictionary</em> (false)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies whether the operator represents a ‘separator’, or punctuation.
                                 This attribute generally has no direct effect on the visual rendering,
                                 but may be useful in specific cases, such as non-visual renderers.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">lspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td><em>set by dictionary</em> (thickmathspace)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the leading space appearing before the operator;
                                 see <a href="chapter3-d.html#presm.opspacing">Section&nbsp;3.2.5.7.5 Spacing around an operator</a>.
                                 (Note that before is on the right in a RTL context; see <a href="chapter3-d.html#presm.bidi">Section&nbsp;3.1.5 Directionality</a>).
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">rspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td><em>set by dictionary</em> (thickmathspace)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the trailing space appearing after the operator;
                                 see <a href="chapter3-d.html#presm.opspacing">Section&nbsp;3.2.5.7.5 Spacing around an operator</a>.
                                 (Note that after is on the left in a RTL context; see <a href="chapter3-d.html#presm.bidi">Section&nbsp;3.1.5 Directionality</a>).
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">stretchy</td>
                              <td>"true" | "false"</td>
                              <td><em>set by dictionary</em> (false)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies whether the operator should stretch to the size of adjacent material;
                                 see <a href="chapter3-d.html#presm.op.stretch">Section&nbsp;3.2.5.8 Stretching of operators, fences and accents</a>.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">symmetric</td>
                              <td>"true" | "false"</td>
                              <td><em>set by dictionary</em> (false)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies whether the operator should be kept symmetric around the math 
                                 <a href="appendixd-d.html#dt-axis">axis</a> when stretchy. glossary
                                 Note that the default is true, but this property only applies to vertically stretched symbols.
                                 See <a href="chapter3-d.html#presm.op.stretch">Section&nbsp;3.2.5.8 Stretching of operators, fences and accents</a>.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">maxsize</td>
                              <td> <a href="chapter2-d.html#type.length"><em>length</em></a> | "infinity"
                              </td>
                              <td><em>set by dictionary</em> (infinity)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the maximum size of the operator when stretchy;
                                 see <a href="chapter3-d.html#presm.op.stretch">Section&nbsp;3.2.5.8 Stretching of operators, fences and accents</a>.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">minsize</td>
                              <td> <a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td><em>set by dictionary</em> (1em)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the minimum size of the operator when stretchy;
                                 see <a href="chapter3-d.html#presm.op.stretch">Section&nbsp;3.2.5.8 Stretching of operators, fences and accents</a>.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">largeop</td>
                              <td>"true" | "false"</td>
                              <td><em>set by dictionary</em> (false)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies whether the operator is considered a ‘large’ operator,
                                 that is, whether it should be drawn larger than normal when
                                 <code>displaystyle</code>="true"
                                 (similar to using T<sub>E</sub>X's <b>\displaystyle</b>).
                                 Examples of large operators include <code>&amp;int;</code>
                                 and <code>&amp;prod;</code>.
                                 See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a> for more discussion.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">movablelimits</td>
                              <td>"true" | "false"</td>
                              <td><em>set by dictionary</em> (false)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies whether under- and overscripts attached to
                                 this operator ‘move’ to the more compact sub- and superscript positions
                                 when <code>displaystyle</code> is false.
                                 Examples of operators that typically have <code>movablelimits</code>="true"
                                 are <code>&amp;sum;</code>, <code>&amp;prod;</code>, and <b>lim</b>.
                                 See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a> for more discussion.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">accent</td>
                              <td>"true" | "false"</td>
                              <td><em>set by dictionary</em> (false)
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies whether this operator should be treated as an accent (diacritical mark)
                                 when used as an underscript or overscript;
                                 see <a href="chapter3-d.html#presm.munder"><code>munder</code></a>,
                                 <a href="chapter3-d.html#presm.mover"><code>mover</code></a>
                                 and <a href="chapter3-d.html#presm.munderover"><code>munderover</code></a>.
                                 
                              </td>
                           </tr>
                        </tbody>
                     </table>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="presm.lbattrs" id="presm.lbattrs"></a>3.2.5.2.2 Linebreaking attributes
                     </h5>
                     <p>The following attributes affect when a linebreak does or does not occur,
                        and the appearance of the linebreak when it does occur.
                     </p>
                     <table border="1" class="attributes">
                        <thead>
                           <tr>
                              <th> Name </th>
                              <th> values </th>
                              <th> default </th>
                           </tr>
                        </thead>
                        <tbody>
                           <tr>
                              <td rowspan="2" class="attname">linebreak</td>
                              <td>"auto" | "newline" | "nobreak" | "goodbreak" | "badbreak"</td>
                              <td>auto</td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the desirability of a linebreak occurring at this operator:
                                 the default "auto" indicates the renderer should use its default
                                 linebreaking algorithm to determine whether to break;
                                 "newline" is used to force a linebreak;
                                 For automatic linebreaking, "nobreak" forbids a break;
                                 "goodbreak" suggests a good position;
                                 "badbreak" suggests a poor position.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">lineleading </td>
                              <td> <a href="chapter2-d.html#type.length"><em>length</em></a> 
                              </td>
                              <td><em>inherited</em> (100%) 
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the amount of vertical space to use after a linebreak.
                                 For tall lines, it is often clearer to use more leading at linebreaks.
                                 Rendering agents are free to choose an appropriate default.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">linebreakstyle</td>
                              <td> "before" | "after" | "duplicate"
                                 | "infixlinebreakstyle"
                              </td>
                              <td><em>set by dictionary</em> (before) 
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies whether a linebreak occurs ‘before’ or ‘after’ the operator
                                 when a linebreaks occur on this operator; or whether the operator is duplicated.
                                 "before" causes the operator to appears at the beginning of the new line
                                 (but possibly indented);
                                 "after" causes it to appear at the end of the line before the break.
                                 "duplicate" places the operator at both positions.
                                 "infixlinebreakstyle" uses the value  that has been specified for
                                 infix operators; This value (one of "before",
                                 "after" or "duplicate") can be specified by
                                 the application or bound by <a href="chapter3-d.html#presm.mstyle"><code>mstyle</code></a>
                                 ("before" corresponds to the most common style of linebreaking).
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">linebreakmultchar</td>
                              <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                              <td><em>inherited</em> (&amp;InvisibleTimes;) 
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the character used to make an &amp;InvisibleTimes; operator visible at a linebreak.
                                 For example, <code>linebreakmultchar</code>="&amp;#xB7;"  would make the
                                 multiplication visible as a center dot.
                                 
                              </td>
                           </tr>
                        </tbody>
                     </table>
                     <p>
                        <code>linebreak</code> values on adjacent <code>mo</code> and <code>mspace</code>elements do
                        not interact; <code>linebreak</code>="nobreak" on a <code>mo</code> does
                        not, in itself, inhibit a break on a preceding or following (possibly nested) <code>mo</code> or <code>mspace</code> element and does not interact with the <code>linebreakstyle</code> attribute value of the preceding or following <code>mo</code> element. It does prevent breaks from occurring on either side of the <code>mo</code> element in all other situations.
                        
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="presm.lbindent.attrs" id="presm.lbindent.attrs"></a>3.2.5.2.3 Indentation attributes
                     </h5>
                     <p>The following attributes affect indentation of the lines making up a formula.
                        Primarily these attributes control the positioning of new lines following a linebreak,
                        whether automatic or manual. However, <code>indentalignfirst</code> and <code>indentshiftfirst</code>
                        also control the positioning of single line formula without any linebreaks.
                        When these attributes appear on <code>mo</code> or <code>mspace</code> they apply if a linebreak occurs
                        at that element.
                        When they appear on <code>mstyle</code> or <code>math</code> elements, they determine
                        defaults for the style to be used for any linebreaks occurring within.
                        Note that except for cases where heavily marked-up manual linebreaking is desired,
                        many of these attributes are most useful when bound on an
                        <code>mstyle</code> or <code>math</code> element.
                        
                     </p>
                     <p>Note that since the rendering context, such as available the width and current font,
                        is not always available to the author of the MathML, 
                        a render may ignore the values of these attributes if they result in a line in which
                        the remaining width is too small to usefully display the expression or if they result in a line in
                        which the remaining width exceeds the available linewrapping width. 
                     </p>
                     <table border="1" class="attributes">
                        <thead>
                           <tr>
                              <th> Name   </th>
                              <th> values </th>
                              <th> default </th>
                           </tr>
                        </thead>
                        <tbody>
                           <tr>
                              <td rowspan="2" class="attname">indentalign</td>
                              <td>"left" | "center" | "right" | "auto" | "id"</td>
                              <td><em>inherited</em> (auto)       
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the positioning of lines when linebreaking takes place within an <code>mrow</code>;
                                 see below for discussion of the attribute values.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">indentshift      </td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td><em>inherited</em> (0) 
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies an additional indentation offset relative to the position determined
                                 by <code>indentalign</code>.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">indenttarget</td>
                              <td><a href="chapter2-d.html#type.idref"><em>idref</em></a></td>
                              <td><em> inherited (none)</em></td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the <a href="chapter2-d.html#type.id">id</a> of another element
                                 whose horizontal position determines the position of indented lines
                                 when <code>indentalign</code>="id".
                                 Note that the identified element may be outside of the current
                                 <code>math</code> element, allowing for inter-expression alignment,
                                 or may be within invisible content such as <code>mphantom</code>;
                                 it must appear <em>before</em> being referenced, however.
                                 This may lead to an id being unavailable to a given renderer;
                                 in such cases, the <code>indentalign</code> should revert to "auto".
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">indentalignfirst  </td>
                              <td>"left" | "center" | "right" | "auto" | "id" | "indentalign"</td>
                              <td><em>inherited</em> (indentalign) 
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the indentation style to use for the first line of a formula;
                                 the value "indentalign" (the default) means
                                 to indent the same way as used for the general line.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">indentshiftfirst </td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a> | "indentshift"  
                              </td>
                              <td><em>inherited</em> (indentshift) 
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the offset to use for the first line of a formula;
                                 the value "indentshift" (the default) means
                                 to use the same offset as used for the general line.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">indentalignlast   </td>
                              <td>"left" | "center" | "right" | "auto" | "id" | "indentalign"</td>
                              <td><em>inherited</em> (indentalign) 
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the indentation style to use for the last line when a linebreak
                                 occurs within a given <code>mrow</code>;
                                 the value "indentalign" (the default) means 
                                 to indent the same way as used for the general line.
                                 When there are exactly two lines, the value of this attribute should
                                 be used for the second line in preference to <code>indentalign</code>.
                                 
                              </td>
                           </tr>
                           <tr>
                              <td rowspan="2" class="attname">indentshiftlast  </td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a> | "indentshift" 
                              </td>
                              <td><em>inherited</em> (indentshift) 
                              </td>
                           </tr>
                           <tr>
                              <td colspan="2" class="attdesc">
                                 Specifies the offset to use for the last line when a linebreak
                                 occurs within a given <code>mrow</code>;
                                 the value "indentshift" (the default) means
                                 to indent the same way as used for the general line.
                                 When there are exactly two lines, the value of this attribute should
                                 be used for the second line in preference to <code>indentshift</code>.
                                 
                              </td>
                           </tr>
                        </tbody>
                     </table>
                     <p>The legal values of indentalign are:
                        
                     </p>
                     <table id="presm.table-indentalign" border="1">
                        <thead>
                           <tr>
                              <th> Value   </th>
                              <th> Meaning</th>
                           </tr>
                        </thead>
                        <tbody>
                           <tr>
                              <td> left         </td>
                              <td> Align the left side of the next line to the left side of the line wrapping width </td>
                           </tr>
                           <tr>
                              <td> center       </td>
                              <td> Align the center of the next line to the center of the line wrapping width </td>
                           </tr>
                           <tr>
                              <td> right        </td>
                              <td> Align the right side of the next line to the right side of the line wrapping width </td>
                           </tr>
                           <tr>
                              <td> auto         </td>
                              <td>
                                 (default) indent using the renderer's default indenting style; this may
                                 be a fixed amount or one that varies with the depth of the element in
                                 the mrow nesting or some other similar method. 
                              </td>
                           </tr>
                           <tr>
                              <td> id           </td>
                              <td> Align the left side of the next line to the left side of the element
                                 referenced by the <a href="chapter2-d.html#type.idref"><em>idref</em></a>
                                 (given by <code>indenttarget</code>);
                                 if no such element exists, use "auto" as the <code>indentalign</code> value
                              </td>
                           </tr>
                        </tbody>
                     </table>
                  </div>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.5.3" id="id.3.2.5.3"></a>3.2.5.3 Examples with ordinary operators
                  </h4><pre class="mathml">
&lt;mo&gt; + &lt;/mo&gt;
&lt;mo&gt; &amp;lt; &lt;/mo&gt;
&lt;mo&gt; &amp;#x2264;<span style="color:#999900">&lt;!--LESS-THAN OR EQUAL TO--&gt;</span> &lt;/mo&gt;
&lt;mo&gt; &amp;lt;= &lt;/mo&gt;
&lt;mo&gt; ++ &lt;/mo&gt;
&lt;mo&gt; &amp;#x2211;<span style="color:#999900">&lt;!--N-ARY SUMMATION--&gt;</span> &lt;/mo&gt;
&lt;mo&gt; .NOT. &lt;/mo&gt;
&lt;mo&gt; and &lt;/mo&gt;
&lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
&lt;mo mathvariant='bold'&gt; + &lt;/mo&gt;
</pre></div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.5.4" id="id.3.2.5.4"></a>3.2.5.4 Examples with fences and separators
                  </h4>
                  <p>Note that the <code>mo</code> elements in these examples
                     don't need explicit <code>fence</code> or <code>separator</code> attributes,
                     since these can be found using the
                     operator dictionary as described below. Some of these examples could also
                     be encoded using the <code>mfenced</code> element described in
                     <a href="chapter3-d.html#presm.mfenced">Section&nbsp;3.3.8 Expression Inside Pair of Fences
                        <code>&lt;mfenced&gt;</code></a>.
                  </p>
                  <p>(<var>a</var>+<var>b</var>)
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mo&gt; ( &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mi&gt; a &lt;/mi&gt;
    &lt;mo&gt; + &lt;/mo&gt;
    &lt;mi&gt; b &lt;/mi&gt;
  &lt;/mrow&gt;
  &lt;mo&gt; ) &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>[0,1)
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mo&gt; [ &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mn&gt; 0 &lt;/mn&gt;
    &lt;mo&gt; , &lt;/mo&gt;
    &lt;mn&gt; 1 &lt;/mn&gt;
  &lt;/mrow&gt;
  &lt;mo&gt; ) &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p><var>f</var>(<var>x</var>,<var>y</var>)
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt; f &lt;/mi&gt;
  &lt;mo&gt; &amp;#x2061;<span style="color:#999900">&lt;!--FUNCTION APPLICATION--&gt;</span> &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mo&gt; ( &lt;/mo&gt;
    &lt;mrow&gt;
      &lt;mi&gt; x &lt;/mi&gt;
      &lt;mo&gt; , &lt;/mo&gt;
      &lt;mi&gt; y &lt;/mi&gt;
    &lt;/mrow&gt;
    &lt;mo&gt; ) &lt;/mo&gt;
  &lt;/mrow&gt;
&lt;/mrow&gt;
</pre></div>
               <div class="div4">
                  
                  <h4><a name="presm.invisibleops" id="presm.invisibleops"></a>3.2.5.5 Invisible operators
                  </h4>
                  <p>Certain operators that are "invisible" in traditional
                     mathematical notation should be represented using specific entity
                     references within <code>mo</code> elements, rather than simply
                     by nothing. The characters used for these "invisible
                     operators" are:
                     
                  </p>
                  <table border="1">
                     <thead>
                        <tr>
                           <th>Character</th>
                           <th>Entity name</th>
                           <th>Short name</th>
                           <th>Examples of use</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td>U+2061</td>
                           <td><code>&amp;ApplyFunction;</code></td>
                           <td><code>&amp;af;</code></td>
                           <td><var>f</var>(<var>x</var>) sin <var>x</var></td>
                        </tr>
                        <tr>
                           <td>U+2062</td>
                           <td><code>&amp;InvisibleTimes;</code></td>
                           <td><code>&amp;it;</code></td>
                           <td><var>x</var><var>y</var></td>
                        </tr>
                        <tr>
                           <td>U+2063</td>
                           <td><code>&amp;InvisibleComma;</code></td>
                           <td><code>&amp;ic;</code></td>
                           <td><var>m</var><sub>12</sub></td>
                        </tr>
                        <tr>
                           <td>U+2064</td>
                           <td></td>
                           <td></td>
                           <td>2¾</td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The MathML representations of the examples in the above table are:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt; f &lt;/mi&gt;
  &lt;mo&gt; &amp;#x2061;<span style="color:#999900">&lt;!--FUNCTION APPLICATION--&gt;</span> &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mo&gt; ( &lt;/mo&gt;
    &lt;mi&gt; x &lt;/mi&gt;
    &lt;mo&gt; ) &lt;/mo&gt;
  &lt;/mrow&gt;
&lt;/mrow&gt;

&lt;mrow&gt;
  &lt;mi&gt; sin &lt;/mi&gt;
  &lt;mo&gt; &amp;#x2061;<span style="color:#999900">&lt;!--FUNCTION APPLICATION--&gt;</span> &lt;/mo&gt;
  &lt;mi&gt; x &lt;/mi&gt;
&lt;/mrow&gt;

&lt;mrow&gt;
  &lt;mi&gt; x &lt;/mi&gt;
  &lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
  &lt;mi&gt; y &lt;/mi&gt;
&lt;/mrow&gt;

&lt;msub&gt;
  &lt;mi&gt; m &lt;/mi&gt;
  &lt;mrow&gt;
    &lt;mn&gt; 1 &lt;/mn&gt;
    &lt;mo&gt; &amp;#x2063;<span style="color:#999900">&lt;!--INVISIBLE SEPARATOR--&gt;</span> &lt;/mo&gt;
    &lt;mn&gt; 2 &lt;/mn&gt;
  &lt;/mrow&gt;
&lt;/msub&gt;

&lt;mrow&gt;
  &lt;mn&gt; 2 &lt;/mn&gt;
  &lt;mo&gt; &amp;#x2064;&lt;!-- INVISIBLE PLUS --&gt; &lt;/mo&gt;
  &lt;mfrac&gt;
    &lt;mn&gt; 3 &lt;/mn&gt;
    &lt;mn&gt; 4 &lt;/mn&gt;
  &lt;/mfrac&gt;
&lt;/mrow&gt;
</pre><p>The reasons for using specific <code>mo</code> elements for
                     invisible operators include:
                     
                  </p>
                  <ul>
                     <li>
                        <p>such operators should often have specific effects on visual
                           rendering (particularly spacing and linebreaking rules) that are not
                           the same as either the lack of any operator, or spacing represented by
                           <code>mspace</code> or <code>mtext</code>
                           elements;
                        </p>
                     </li>
                     <li>
                        <p>these operators should often have specific audio renderings
                           different than that of the lack of any operator;
                        </p>
                     </li>
                     <li>
                        <p>automatic semantic interpretation of MathML presentation elements
                           is made easier by the explicit specification of such operators.
                        </p>
                     </li>
                  </ul>
                  <p>For example, an audio renderer might render <var>f</var>(<var>x</var>)
                     (represented as in the above examples) by speaking "f of x", but use
                     the word "times" in its rendering of <var>x</var><var>y</var>.
                     Although its rendering must still be different depending on the structure
                     of neighboring elements (sometimes leaving out "of" or
                     "times" entirely), its task is made much easier by the use of
                     a different <code>mo</code> element for each invisible
                     operator.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.5.6" id="id.3.2.5.6"></a>3.2.5.6 Names for other special operators
                  </h4>
                  <p>MathML also includes <code>&amp;DifferentialD;</code> (U+2146) for use
                     in an <code>mo</code> element representing the differential
                     operator symbol usually denoted by "d".  The reasons for
                     explicitly using this special character are similar to those for using
                     the special characters for invisible operators described in the
                     preceding section.  
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.5.7" id="id.3.2.5.7"></a>3.2.5.7 Detailed rendering rules for <code>&lt;mo&gt;</code> elements
                  </h4>
                  <p>Typical visual rendering behaviors for <code>mo</code>
                     elements are more complex than for the other MathML token elements, so
                     the rules for rendering them are described in this separate
                     subsection.
                  </p>
                  <p>Note that, like all rendering rules in MathML, these rules are
                     suggestions rather than requirements.  Furthermore, no attempt is made
                     to specify the rendering completely; rather, enough information is
                     given to make the intended effect of the various rendering attributes
                     as clear as possible.
                  </p>
                  <div class="div5">
                     
                     <h5><a name="presm.opdict" id="presm.opdict"></a>3.2.5.7.1 The operator dictionary
                     </h5>
                     <p>Many mathematical symbols, such as an integral sign, a plus sign,
                        or a parenthesis, have a well-established, predictable, traditional
                        notational usage. Typically, this usage amounts to certain default
                        attribute values for <code>mo</code> elements with specific
                        contents and a specific <code>form</code> attribute. Since these
                        defaults vary from symbol to symbol, MathML anticipates that renderers
                        will have an "operator dictionary" of default attributes for
                        <code>mo</code> elements (see <a href="appendixc-d.html">Appendix&nbsp;C Operator Dictionary</a>) indexed by each
                        <code>mo</code> element's content and <code>form</code>
                        attribute.  If an <code>mo</code> element is not listed in the
                        dictionary, the default values shown in parentheses in the table of
                        attributes for <code>mo</code> should be used, since these
                        values are typically acceptable for a generic operator.
                     </p>
                     <p>Some operators are "overloaded", in the sense that they can occur
                        in more than one form (prefix, infix, or postfix), with possibly
                        different rendering properties for each form. For example, "+" can be
                        either a prefix or an infix operator. Typically, a visual renderer
                        would add space around both sides of an infix operator, while only in
                        front of a prefix operator. The <code>form</code> attribute allows
                        specification of which form to use, in case more than one form is
                        possible according to the operator dictionary and the default value
                        described below is not suitable.
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="presm.formdefval" id="presm.formdefval"></a>3.2.5.7.2 Default value of the <code>form</code> attribute
                     </h5>
                     <p>The <code>form</code> attribute does not usually have to be
                        specified explicitly, since there are effective heuristic rules for
                        inferring the value of the <code>form</code> attribute from the
                        context. If it is not specified, and there is more than one possible
                        form in the dictionary for an <code>mo</code> element with
                        given content, the renderer should choose which form to use as follows
                        (but see the exception for embellished operators, described later):
                        
                     </p>
                     <ul>
                        <li>
                           <p>If the operator is the first argument in an <code>mrow</code>
                              with more than one argument
                              (ignoring all space-like arguments (see <a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a>) in the
                              determination of both the length and the first argument), the prefix form
                              is used;
                           </p>
                        </li>
                        <li>
                           <p>if it is the last argument in an <code>mrow</code> with more than one argument (ignoring all space-like arguments), the postfix
                              form is used;
                           </p>
                        </li>
                        <li>
                           <p>if it is the only element in an implicit or explicit <code>mrow</code> and if it is in a script position of one of the elements listed in <a href="chapter3-d.html#presm.scrlim">Section&nbsp;3.4 Script and Limit Schemata</a>, the postfix form is used;
                           </p>
                        </li>
                        <li>
                           <p>in all other cases, including when the operator is not part of an
                              <code>mrow</code>, the infix form is used.
                           </p>
                        </li>
                     </ul>
                     <p>Note that the <code>mrow</code> discussed above may be <em>inferred</em>;
                        See <a href="chapter3-d.html#presm.inferredmrow">Section&nbsp;3.1.3.1 Inferred <code>&lt;mrow&gt;</code>s</a>.
                     </p>
                     <p>Opening fences should have  <code>form</code><code>="prefix"</code>,
                        and closing fences should have <code>form</code><code>="postfix"</code>;
                        separators are usually "infix", but not always,
                        depending on their surroundings. As with ordinary operators,
                        these values do not usually need to be specified explicitly.
                     </p>
                     <p>If the operator does not occur in the dictionary with the specified
                        form, the renderer should use one of the forms that is available
                        there, in the order of preference: infix, postfix, prefix; if no forms
                        are available for the given <code>mo</code> element content, the
                        renderer should use the defaults given in parentheses in the table of
                        attributes for <code>mo</code>.
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.5.7.3" id="id.3.2.5.7.3"></a>3.2.5.7.3 Exception for embellished operators
                     </h5>
                     <p>There is one exception to the above rules for choosing an <code>mo</code> element's default <code>form</code>
                        attribute. An <code>mo</code> element that is
                        "embellished" by one or more nested subscripts, superscripts,
                        surrounding text or whitespace, or style changes behaves differently. It is
                        the embellished operator as a whole (this is defined precisely, below)
                        whose position in an <code>mrow</code> is examined by the above
                        rules and whose surrounding spacing is affected by its form, not the <code>mo</code> element at its core; however, the attributes
                        influencing this surrounding spacing are taken from the <code>mo</code> element at the core (or from that element's
                        dictionary entry).
                     </p>
                     <p>For example, the "+<sub>4</sub>" in 
                        <var>a</var>+<sub>4</sub><var>b</var>
                        should be considered an infix operator as a whole, due to its position
                        in the middle of an <code>mrow</code>, but its rendering
                        attributes should be taken from the <code>mo</code> element
                        representing the "+", or when those are not specified explicitly,
                        from the operator dictionary entry for <code>&lt;mo form="infix"&gt; +
                           &lt;/mo&gt;</code>.
                        The precise definition of an "embellished operator" is: 
                        
                     </p>
                     <ul>
                        <li>
                           <p>an <code>mo</code> element;
                           </p>
                        </li>
                        <li>
                           <p>or one of the elements
                              <code>msub</code>,
                              <code>msup</code>,
                              <code>msubsup</code>,
                              <code>munder</code>,
                              <code>mover</code>,
                              <code>munderover</code>,
                              <code>mmultiscripts</code>,
                              <code>mfrac</code>, or
                              <code>semantics</code>
                              (<a href="chapter5-d.html#mixing.semantic.annotations">Section&nbsp;5.1 Annotation Framework</a>), whose first argument exists and is an embellished
                              operator;
                           </p>
                        </li>
                        <li>
                           <p>or one of the elements
                              <code>mstyle</code>,
                              <code>mphantom</code>, or
                              <code>mpadded</code>,
                              such that an <code>mrow</code> containing the same
                              arguments would be an embellished operator;
                           </p>
                        </li>
                        <li>
                           <p>or an <code>maction</code> element whose selected
                              sub-expression exists and is an embellished operator;
                           </p>
                        </li>
                        <li>
                           <p>or an <code>mrow</code> whose arguments consist (in any order)
                              of one embellished operator and zero or more space-like elements.
                           </p>
                        </li>
                     </ul>
                     <p>
                        Note that this definition permits nested embellishment only when
                        there are no intervening enclosing elements not in the above list.
                     </p>
                     <p>The above rules for choosing operator forms and defining
                        embellished operators are chosen so that in all ordinary cases it will
                        not be necessary for the author to specify a <code>form</code>
                        attribute.
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.5.7.4" id="id.3.2.5.7.4"></a>3.2.5.7.4 Rationale for definition of embellished operators
                     </h5>
                     <p>The following notes are included as a rationale for certain aspects
                        of the above definitions, but should not be important for most users
                        of MathML.
                     </p>
                     <p>An <code>mfrac</code> is included as an
                        "embellisher" because of the common notation for a
                        differential operator:
                        
                     </p><pre class="mathml">
&lt;mfrac&gt;
  &lt;mo&gt; &amp;#x2146;<span style="color:#999900">&lt;!--DOUBLE-STRUCK ITALIC SMALL D--&gt;</span> &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mo&gt; &amp;#x2146;<span style="color:#999900">&lt;!--DOUBLE-STRUCK ITALIC SMALL D--&gt;</span> &lt;/mo&gt;
    &lt;mi&gt; x &lt;/mi&gt;
  &lt;/mrow&gt;
&lt;/mfrac&gt;
</pre><p>Since the definition of embellished operator affects the use of the
                        attributes related to stretching, it is important that it includes
                        embellished fences as well as ordinary operators; thus it applies to
                        any <code>mo</code> element.
                     </p>
                     <p>Note that an <code>mrow</code> containing a single argument
                        is an embellished operator if and only if its argument is an embellished
                        operator. This is because an <code>mrow</code> with a single
                        argument must be equivalent in all respects to that argument alone (as
                        discussed in <a href="chapter3-d.html#presm.mrow">Section&nbsp;3.3.1 Horizontally Group Sub-Expressions
                           <code>&lt;mrow&gt;</code></a>). This means that an <code>mo</code> element that is the sole argument of an <code>mrow</code> will determine its default <code>form</code> attribute based on that <code>mrow</code>'s position in a surrounding, perhaps inferred, <code>mrow</code> (if there is one), rather than based on its own
                        position in the <code>mrow</code> in which it is the sole
                        argument.
                     </p>
                     <p>Note that the above definition defines every
                        <code>mo</code> element to be "embellished" — that is,
                        "embellished operator" can be considered (and implemented in
                        renderers) as a special class of MathML expressions, of which
                        <code>mo</code> is a specific case.
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="presm.opspacing" id="presm.opspacing"></a>3.2.5.7.5 Spacing around an operator
                     </h5>
                     <p>The amount of horizontal space added around an operator (or embellished operator),
                        when it occurs in an <code>mrow</code>, can be directly
                        specified by the <code>lspace</code> and <code>rspace</code>
                        attributes. Note that <code>lspace</code> and <code>rspace</code> should
                        be interpreted as leading and trailing space, in the case of RTL direction.
                        By convention, operators that tend to bind tightly to their
                        arguments have smaller values for spacing than operators that tend to bind
                        less tightly. This convention should be followed in the operator dictionary
                        included with a MathML renderer.
                     </p>
                     <p>Some renderers may choose to use no space around most operators
                        appearing within subscripts or superscripts, as is done in T<sub>E</sub>X.
                     </p>
                     <p>Non-graphical renderers should treat spacing attributes, and other
                        rendering attributes described here, in analogous ways for their
                        rendering medium.  For example, more space might translate into a
                        longer pause in an audio rendering.
                     </p>
                  </div>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.op.stretch" id="presm.op.stretch"></a>3.2.5.8 Stretching of operators, fences and accents
                  </h4>
                  <p>Four attributes govern whether and how an operator (perhaps embellished)
                     stretches so that it matches the size of other elements: <code>stretchy</code>, <code>symmetric</code>, <code>maxsize</code>, and <code>minsize</code>. If an
                     operator has the attribute <code>stretchy</code>="true", then it (that is, each character in its content)
                     obeys the stretching rules listed below, given the constraints imposed by
                     the fonts and font rendering system. In practice, typical renderers will
                     only be able to stretch a small set of characters, and quite possibly will
                     only be able to generate a discrete set of character sizes.
                  </p>
                  <p>There is no provision in MathML for specifying in which direction
                     (horizontal or vertical) to stretch a specific character or operator;
                     rather, when <code>stretchy</code>="true" it
                     should be stretched in each direction for which stretching is possible
                     and reasonable for that character.
                     It is up to the renderer to know in which directions it is reasonable to
                     stretch a character, if it can stretch the character.
                     Most characters can be stretched in at most one direction
                     by typical renderers, but some renderers may be able to stretch certain
                     characters, such as diagonal arrows, in both directions independently.
                  </p>
                  <p>The <code>minsize</code> and <code>maxsize</code>
                     attributes limit the amount of stretching (in either direction). These two
                     attributes are given as multipliers of the operator's normal size in the
                     direction or directions of stretching, or as absolute sizes using units.
                     For example, if a character has <code>maxsize</code>="3", then it
                     can grow to be no more than three times its normal (unstretched) size.
                  </p>
                  <p>The <code>symmetric</code> attribute governs whether the
                     height and 
                     depth above and below the <a class="termref" title="" href="appendixd-d.html#dt-axis">axis</a> of the
                     character are forced to be equal  
                     (by forcing both height and depth to become the maximum of the two).
                     An example of a situation where one might set
                     <code>symmetric</code>="false" 
                     arises with parentheses around a matrix not aligned on the <a class="termref" title="" href="appendixd-d.html#dt-axis">axis</a>, which
                     frequently occurs when multiplying non-square matrices. In this case, one
                     wants the parentheses to stretch to cover the matrix, whereas stretching
                     the parentheses symmetrically would cause them to protrude beyond one edge
                     of the matrix. The <code>symmetric</code> attribute only applies
                     to characters that stretch vertically (otherwise it is ignored).
                  </p>
                  <p>If a stretchy <code>mo</code> element is embellished (as defined
                     earlier in this section), the <code>mo</code> element at its core is
                     stretched to a size based on the context of the embellished operator
                     as a whole, i.e. to the same size as if the embellishments were not
                     present. For example, the parentheses in the following example (which
                     would typically be set to be stretchy by the operator dictionary) will be
                     stretched to the same size as each other, and the same size they would
                     have if they were not underlined and overlined, and furthermore will
                     cover the same vertical interval:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;munder&gt;
    &lt;mo&gt; ( &lt;/mo&gt;
    &lt;mo&gt; &amp;#x5F;<span style="color:#999900">&lt;!--LOW LINE--&gt;</span> &lt;/mo&gt;
  &lt;/munder&gt;
  &lt;mfrac&gt;
    &lt;mi&gt; a &lt;/mi&gt;
    &lt;mi&gt; b &lt;/mi&gt;
  &lt;/mfrac&gt;
  &lt;mover&gt;
    &lt;mo&gt; ) &lt;/mo&gt;
    &lt;mo&gt; &amp;#x203E;<span style="color:#999900">&lt;!--OVERLINE--&gt;</span> &lt;/mo&gt;
  &lt;/mover&gt;
&lt;/mrow&gt;
</pre><p>Note that this means that the stretching rules given below must
                     refer to the context of the embellished operator as a whole, not just
                     to the <code>mo</code> element itself.
                  </p>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.5.8.1" id="id.3.2.5.8.1"></a>3.2.5.8.1 Example of stretchy attributes
                     </h5>
                     <p>This shows one way to set the maximum size of a parenthesis so that
                        it does not grow, even though its default value is
                        <code>stretchy</code>="true".
                        
                     </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mo maxsize="1"&gt; ( &lt;/mo&gt;
  &lt;mfrac&gt;
    &lt;mi&gt; a &lt;/mi&gt; &lt;mi&gt; b &lt;/mi&gt;
  &lt;/mfrac&gt;
  &lt;mo maxsize="1"&gt; ) &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>The above should render as 
                        <img src="image/f3001.gif" alt="(\frac{a}{b})" align="middle"> 
                        as opposed to the default rendering
                        <img src="image/f3002.gif" alt="\left(\frac{a}{b}\right)" align="middle">.
                     </p>
                     <p>Note that each parenthesis is sized independently; if only one of
                        them had <code>maxsize</code>="1", they would render with different
                        sizes.
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.5.8.2" id="id.3.2.5.8.2"></a>3.2.5.8.2 Vertical Stretching Rules
                     </h5>
                     <p>The general rules governing stretchy operators are:</p>
                     <ul>
                        <li>
                           <p>If a stretchy operator is a direct sub-expression of an <code>mrow</code> element, or is the sole direct sub-expression of an
                              <code>mtd</code> element in some row of a table, then it should
                              stretch to cover the height and depth (above and below the axis) of the <em>non</em>-stretchy direct sub-expressions in the
                              <code>mrow</code> element or table row, unless stretching is
                              constrained by <code>minsize</code> or <code>maxsize</code> attributes.
                           </p>
                        </li>
                        <li>
                           <p>In the case of an embellished stretchy operator, the preceding
                              rule applies to the stretchy operator at its core.
                           </p>
                        </li>
                        <li>
                           <p>The preceding rules also apply in situations where the <code>mrow</code> element is inferred.
                           </p>
                        </li>
                        <li>
                           <p>The rules for symmetric stretching only apply if <code>symmetric</code>="true" and if the stretching occurs in an <code>mrow</code> or in an <code>mtr</code> whose <code>rowalign</code> value is either "baseline" or "axis".
                           </p>
                        </li>
                     </ul>
                     <p>The following algorithm specifies the height and depth of vertically stretched characters:
                        
                     </p>
                     <ol type="1">
                        <li>
                           <p>Let <code>maxheight</code> and <code>maxdepth</code> be the maximum height and depth of the <em>non</em>-stretchy
                              siblings within the same <code>mrow</code> or <code>mtr</code>. Let axis be the
                              height of the math axis above the baseline.
                           </p>
                           <p>Note that even if a <code>minsize</code> or <code>maxsize</code> value is set on a stretchy operator, it is <em>not</em> used in the initial calculation of the maximum height and depth of an <code>mrow</code>.
                           </p>
                        </li>
                        <li>
                           <p>
                              If <code>symmetric</code>="true", then the computed height
                              and depth of the stretchy operator are:
                              
                           </p><pre class="algorithm">
   height=max(maxheight-axis, maxdepth+axis) + axis
   depth =max(maxheight-axis, maxdepth+axis) - axis
</pre><p>
                              Otherwise the height and depth are:
                              
                           </p><pre class="algorithm">
   height= maxheight
   depth = maxdepth
</pre></li>
                        <li>
                           <p>
                              If the total size = height+depth is less than minsize
                              or greater than maxsize, increase or decrease both
                              height and depth proportionately so that the effective
                              size meets the constraint.
                              
                           </p>
                        </li>
                     </ol>
                     <p>By default, most vertical arrows, along with most opening and closing fences are defined in the operator
                        dictionary to stretch by default.  
                     </p>
                     <p>In the case of a stretchy operator in a table cell (i.e. within an
                        <code>mtd</code> element), the above rules assume each cell of
                        the table row containing the stretchy operator covers exactly one row.
                        (Equivalently, the value of the <code>rowspan</code> attribute is
                        assumed to be 1 for all the table cells in the table row, including
                        the cell containing the operator.)  When this is not the case, the
                        operator should only be stretched vertically to cover those table
                        cells that are entirely within the set of table rows that the
                        operator's cell covers. Table cells that extend into rows not covered
                        by the stretchy operator's table cell should be ignored.  See 
                        <a href="chapter3-d.html#presm.mtdatts">Section&nbsp;3.5.4.2 Attributes</a> for details about the <code>rowspan</code> attribute. 
                        
                        
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="presm.horiz.stretch.rules" id="presm.horiz.stretch.rules"></a>3.2.5.8.3 Horizontal Stretching Rules
                     </h5>
                     <ul>
                        <li>
                           <p>If a stretchy operator, or an embellished stretchy operator,
                              is a direct sub-expression of an <code>munder</code>,
                              <code>mover</code>, or <code>munderover</code> element, 
                              or if it is the sole direct sub-expression of an <code>mtd</code> element in some
                              column of a table (see <code>mtable</code>), then it, or the <code>mo</code> element at its core, should stretch to cover
                              the width of the other direct sub-expressions in the given element (or
                              in the same table column), given the constraints mentioned above.
                           </p>
                        </li>
                        <li>
                           <p>In the case of an embellished stretchy operator, the preceding
                              rule applies to the stretchy operator at its core.
                           </p>
                        </li>
                     </ul>
                     <p>By default, most horizontal arrows and some accents stretch
                        horizontally.
                     </p>
                     <p>In the case of a stretchy operator in a table cell (i.e. within an
                        <code>mtd</code> element), the above rules assume each cell of
                        the table column containing the stretchy operator covers exactly one
                        column.  (Equivalently, the value of the <code>columnspan</code>
                        attribute is assumed to be 1 for all the table cells in the table row,
                        including the cell containing the operator.)  When this is not the
                        case, the operator should only be stretched horizontally to cover
                        those table cells that are entirely within the set of table columns
                        that the operator's cell covers. Table cells that extend into columns
                        not covered by the stretchy operator's table cell should be
                        ignored.  See <a href="chapter3-d.html#presm.mtdatts">Section&nbsp;3.5.4.2 Attributes</a> for details about the <code>rowspan</code> attribute. 
                     </p>
                     <p>The rules for horizontal stretching include <code>mtd</code>
                        elements to allow arrows to stretch for use in commutative diagrams
                        laid out using <code>mtable</code>. The rules for the horizontal
                        stretchiness include scripts to make examples such as the following
                        work:
                        
                     </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt; x &lt;/mi&gt;
  &lt;munder&gt;
    &lt;mo&gt; &amp;#x2192;<span style="color:#999900">&lt;!--RIGHTWARDS ARROW--&gt;</span> &lt;/mo&gt;
    &lt;mtext&gt; maps to &lt;/mtext&gt;
  &lt;/munder&gt;
  &lt;mi&gt; y &lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>This displays as 
                        <img src="image/f3003.gif" alt="x \widearrow{\mathrm{maps~to}} y" align="middle">.
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="id.3.2.5.8.4" id="id.3.2.5.8.4"></a>3.2.5.8.4 Rules Common to both Vertical and Horizontal Stretching
                     </h5>
                     <p>If a stretchy operator is not required to stretch (i.e. if it is
                        not in one of the locations mentioned above, or if there are no other
                        expressions whose size it should stretch to match), then it has the
                        standard (unstretched) size determined by the font and current
                        <code>mathsize</code>.
                     </p>
                     <p>If a stretchy operator is required to stretch, but all other expressions
                        in the containing element (as described above) are also stretchy,
                        all elements that can stretch should grow to the maximum of the normal
                        unstretched sizes of all elements in the containing object, if they can
                        grow that large. If the value of <code>minsize</code> or <code>maxsize</code> prevents 
                        that, then the specified (min or max) size is
                        used.
                     </p>
                     <p>For example, in an <code>mrow</code> containing nothing but
                        vertically stretchy operators, each of the operators should stretch to
                        the maximum of all of their normal unstretched sizes, provided no
                        other attributes are set that override this behavior. Of course,
                        limitations in fonts or font rendering may result in the final,
                        stretched sizes being only approximately the same.
                     </p>
                  </div>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.mo.linebreaks" id="presm.mo.linebreaks"></a>3.2.5.9 Examples of Linebreaking
                  </h4>
                  <p>The following example demonstrates forced linebreaks and forced alignment:</p><pre class="mathml">
 &lt;mrow&gt;
  &lt;mrow&gt;  &lt;mi&gt;f&lt;/mi&gt; &lt;mo&gt;&amp;#x2061;<span style="color:#999900">&lt;!--FUNCTION APPLICATION--&gt;</span>&lt;/mo&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;  &lt;/mrow&gt;

  &lt;mo id='eq1-equals'&gt;=&lt;/mo&gt;
  &lt;mrow&gt;
   &lt;msup&gt;
    &lt;mrow&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;mrow&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mn&gt;1&lt;/mn&gt; &lt;/mrow&gt; &lt;mo&gt;)&lt;/mo&gt; &lt;/mrow&gt;
    &lt;mn&gt;4&lt;/mn&gt;
   &lt;/msup&gt;
   &lt;mo linebreak='newline' linebreakstyle='before'
                     indentalign='id' indenttarget='eq1-equals'&gt;=&lt;/mo&gt;
   &lt;mrow&gt;
    &lt;msup&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;mn&gt;4&lt;/mn&gt; &lt;/msup&gt;
    &lt;mo id='eq1-plus'&gt;+&lt;/mo&gt;
    &lt;mrow&gt;  &lt;mn&gt;4&lt;/mn&gt; &lt;mo&gt;&amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span>&lt;/mo&gt; &lt;msup&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;mn&gt;3&lt;/mn&gt; &lt;/msup&gt;  &lt;/mrow&gt;
    &lt;mo&gt;+&lt;/mo&gt;
    &lt;mrow&gt;  &lt;mn&gt;6&lt;/mn&gt; &lt;mo&gt;&amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span>&lt;/mo&gt; &lt;msup&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;mn&gt;2&lt;/mn&gt; &lt;/msup&gt;  &lt;/mrow&gt;

    &lt;mo linebreak='newline' linebreakstyle='before'
                      indentalignlast='id' indenttarget='eq1-plus'&gt;+&lt;/mo&gt;
    &lt;mrow&gt;  &lt;mn&gt;4&lt;/mn&gt; &lt;mo&gt;&amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span>&lt;/mo&gt; &lt;mi&gt;x&lt;/mi&gt;  &lt;/mrow&gt;
    &lt;mo&gt;+&lt;/mo&gt;
    &lt;mn&gt;1&lt;/mn&gt;
   &lt;/mrow&gt;
  &lt;/mrow&gt;
 &lt;/mrow&gt;
</pre><p>This displays as </p>
                  <blockquote>
                     <p><img src="image/linebreak-example1.png" alt="example with equal and plus signs aligned"></p>
                  </blockquote>
                  <p>Note that because <code>indentalignlast</code> defaults to "indentalign",
                     in the above example <code>indentalign</code> could have been used in place of
                     <code>indentalignlast</code>.  Also, the specifying <code>linebreakstyle='before'</code>
                     is not needed because that is the default value.  
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mtext" id="presm.mtext"></a>3.2.6 Text <code>&lt;mtext&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.2.6.1" id="id.3.2.6.1"></a>3.2.6.1 Description
                  </h4>
                  <p>An <code>mtext</code> element is used to represent
                     arbitrary text that should be rendered as itself. In general, the
                     <code>mtext</code> element is intended to denote commentary
                     text.
                  </p>
                  <p>Note that some text with a clearly defined notational role might be
                     more appropriately marked up using <code>mi</code> or
                     <code>mo</code>; this is discussed further below.
                  </p>
                  <p>An <code>mtext</code> element can be used to contain
                     "renderable whitespace", i.e. invisible characters that are
                     intended to alter the positioning of surrounding elements. In non-graphical
                     media, such characters are intended to have an analogous effect, such as
                     introducing positive or negative time delays or affecting rhythm in an
                     audio renderer. This is not related to any whitespace in the source MathML
                     consisting of blanks, newlines, tabs, or carriage returns; whitespace
                     present directly in the source is trimmed and collapsed, as described in
                     <a href="chapter2-d.html#fund.collapse">Section&nbsp;2.1.7 Collapsing Whitespace in Input</a>. Whitespace that is intended to be rendered
                     as part of an element's content must be represented by entity references
                     or <code>mspace</code> elements
                     (unless it consists only of single blanks between non-whitespace
                     characters).
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.6.2" id="id.3.2.6.2"></a>3.2.6.2 Attributes
                  </h4>
                  <p><code>mtext</code> elements accept the attributes listed in
                     <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>.
                  </p>
                  <p>See also the warnings about the legal grouping of "space-like elements"
                     in <a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a>, and about the use of
                     such elements for "tweaking" in <a href="chapter3-d.html#presm.warnfinetuning">Section&nbsp;3.1.8 Warning about fine-tuning of presentation</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.6.3" id="id.3.2.6.3"></a>3.2.6.3 Examples
                  </h4><pre class="mathml">
&lt;mtext&gt; Theorem 1: &lt;/mtext&gt;
&lt;mtext&gt; &amp;#x2009;<span style="color:#999900">&lt;!--THIN SPACE--&gt;</span> &lt;/mtext&gt;
&lt;mtext&gt; &amp;#x205F;&amp;#x200A;<span style="color:#999900">&lt;!--space of width 5/18 em--&gt;</span>&amp;#x205F;&amp;#x200A;<span style="color:#999900">&lt;!--space of width 5/18 em--&gt;</span> &lt;/mtext&gt;
&lt;mtext&gt; /* a comment */ &lt;/mtext&gt;
</pre></div>
               <div class="div4">
                  
                  <h4><a name="presm.mixtextmath" id="presm.mixtextmath"></a>3.2.6.4 Mixing text and mathematics
                  </h4>
                  <p>In some cases, text embedded in mathematics could be more appropriately
                     represented using <code>mo</code> or <code>mi</code> elements.
                     For example, the expression 'there exists
                     <img src="image/f3004.gif" alt="\delta&gt;0" align="middle">
                     such that <var>f</var>(<var>x</var>) &lt;1' is equivalent to
                     <img src="image/f3005.gif" alt="\exists \delta&gt;0 \backepsilon f(x)<1" align="middle"> 
                     and could be represented as:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mo&gt; there exists &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mrow&gt;
      &lt;mi&gt; &amp;#x3B4;<span style="color:#999900">&lt;!--GREEK SMALL LETTER DELTA--&gt;</span> &lt;/mi&gt;
      &lt;mo&gt; &amp;gt; &lt;/mo&gt;
      &lt;mn&gt; 0 &lt;/mn&gt;
    &lt;/mrow&gt;
    &lt;mo&gt; such that &lt;/mo&gt;
    &lt;mrow&gt;
      &lt;mrow&gt;
        &lt;mi&gt; f &lt;/mi&gt;
        &lt;mo&gt; &amp;#x2061;<span style="color:#999900">&lt;!--FUNCTION APPLICATION--&gt;</span> &lt;/mo&gt;
        &lt;mrow&gt;
          &lt;mo&gt; ( &lt;/mo&gt;
          &lt;mi&gt; x &lt;/mi&gt;
          &lt;mo&gt; ) &lt;/mo&gt;
        &lt;/mrow&gt;
      &lt;/mrow&gt;
      &lt;mo&gt; &amp;lt; &lt;/mo&gt;
      &lt;mn&gt; 1 &lt;/mn&gt;
    &lt;/mrow&gt;
  &lt;/mrow&gt;
&lt;/mrow&gt;
</pre><p>An example involving an <code>mi</code> element is:
                     <var>x</var>+<var>x</var><sup>2</sup>+···+<var>x</var><sup><var>n</var></sup>.
                     In this example, ellipsis should be represented using an <code>mi</code> element, since it takes the place of a term in the
                     sum; (see <a href="chapter3-d.html#presm.mi">Section&nbsp;3.2.3 Identifier <code>&lt;mi&gt;</code></a>).
                  </p>
                  <p>On the other hand, expository text within MathML is best
                     represented with an <code>mtext</code> element. An example
                     of this is: 
                  </p>
                  <blockquote>
                     <p> Theorem 1: if <var>x</var> &gt; 1, then
                        <var>x</var><sup>2</sup> &gt; <var>x</var>.  
                     </p>
                  </blockquote>
                  <p> However, when MathML is
                     embedded in HTML, or another document markup language, the example is
                     probably best rendered with only the two inequalities represented as
                     MathML at all, letting the text be part of the surrounding HTML.
                  </p>
                  <p>Another factor to consider in deciding how to mark up text is the
                     effect on rendering. Text enclosed in an <code>mo</code>
                     element is unlikely to be found in a renderer's operator dictionary,
                     so it will be rendered with the format and spacing appropriate for an
                     "unrecognized operator", which may or may not be better than the
                     format and spacing for "text" obtained by using an
                     <code>mtext</code> element. An ellipsis entity in an
                     <code>mi</code> element is apt to be spaced more appropriately
                     for taking the place of a term within a series than if it appeared in
                     an <code>mtext</code> element.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mspace" id="presm.mspace"></a>3.2.7 Space <code>&lt;mspace/&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.2.7.1" id="id.3.2.7.1"></a>3.2.7.1 Description
                  </h4>
                  <p>An <code>mspace</code> empty element represents a blank
                     space of any desired size, as set by its attributes.  It can also be
                     used to make linebreaking suggestions to a visual renderer.
                     Note that the default values for attributes have been chosen so that
                     they typically will have no effect on rendering.  Thus, the <code>mspace</code> element is generally used with one
                     or more attribute values explicitly specified.
                     
                  </p>
                  <p>Note the warning about the legal grouping of "space-like
                     elements" given below, and the warning about the use of such
                     elements for "tweaking" in <a href="chapter3-d.html#presm.warnfinetuning">Section&nbsp;3.1.8 Warning about fine-tuning of presentation</a>.
                     See also the other elements that can render as
                     whitespace, namely <code>mtext</code>, <code>mphantom</code>, and
                     <code>maligngroup</code>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.7.2" id="id.3.2.7.2"></a>3.2.7.2 Attributes
                  </h4>
                  <p>In addition to the attributes listed below,
                     <code>mspace</code> elements accept the attributes described in <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>,
                     but note that <code>mathvariant</code> and <code>mathcolor</code> have no effect and that
                     <code>mathsize</code> only affects the interpretation of units in sizing
                     attributes (see <a href="chapter2-d.html#fund.units">Section&nbsp;2.1.5.2 Length Valued Attributes</a>).
                     <code>mspace</code> also accepts the indentation attributes described in <a href="chapter3-d.html#presm.lbindent.attrs">Section&nbsp;3.2.5.2.3 Indentation attributes</a>.
                     
                     
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">width</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>0em</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the desired width of the space.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">height</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>0ex</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the desired height (above the baseline) of the space.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">depth</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>0ex</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the desired depth (below the baseline) of the space.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">linebreak</td>
                           <td>"auto" | "newline" | "nobreak" | "goodbreak" | "badbreak"</td>
                           <td>auto</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the desirability of a linebreak at this space.
                              This attribute should be ignored if any dimensional attribute is set.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>Linebreaking was originally specified on <code>mspace</code> in MathML2,
                     but controlling linebreaking on <a href="chapter3-d.html#presm.mo"><code>mo</code></a>
                     is to be preferred  starting with MathML 3.  MathML 3 adds new linebreaking attributes only to <code>mo</code>, not <code>mspace</code>. However, because a linebreak can be specified on <code>mspace</code>, control over the indentation that follows that break can be specified using the attributes listed in <a href="chapter3-d.html#presm.lbindent.attrs">Section&nbsp;3.2.5.2.3 Indentation attributes</a>.
                     
                  </p>
                  <p>
                     The value "indentingnewline" was defined in MathML2 for <code>mspace</code>;
                     it is now deprecated.  Its meaning is the same as <code>newline</code>, which is compatible with its earlier use when no other linebreaking attributes are specified.
                     Note that <code>linebreak</code> values on adjacent <code>mo</code> and <code>mspace</code> elements do
                     not interact; a "nobreak" on an <code>mspace</code> will
                     not, in itself, inhibit a break on an adjacent <code>mo</code> element.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.7.3" id="id.3.2.7.3"></a>3.2.7.3 Examples
                  </h4><pre class="mathml-fragment">
&lt;mspace height="3ex" depth="2ex"/&gt;

&lt;mrow&gt;
  &lt;mi&gt;a&lt;/mi&gt;
  &lt;mo id="firstop"&gt;+&lt;/mo&gt;
  &lt;mi&gt;b&lt;/mi&gt;
  &lt;mspace linebreak="newline" indentalign="id" indenttarget="firstop"/&gt;
  &lt;mo&gt;+&lt;/mo&gt;
  &lt;mi&gt;c&lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>
                     In the last example, <code>mspace</code> will cause the line to end after the "b" and the following line to be indented so that the "+" that follows will align with
                     the "+" with <code>id</code>="firstop".
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.7.4" id="id.3.2.7.4"></a>3.2.7.4 Definition of space-like elements
                  </h4>
                  <p>A number of MathML presentation elements are "space-like" in the
                     sense that they typically render as whitespace, and do not affect the
                     mathematical meaning of the expressions in which they appear. As a
                     consequence, these elements often function in somewhat exceptional
                     ways in other MathML expressions. For example, space-like elements are
                     handled specially in the suggested rendering rules for
                     <code>mo</code> given in <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                        <code>&lt;mo&gt;</code></a>. 
                     The following MathML elements are defined to be "space-like":
                     
                  </p>
                  <ul>
                     <li>
                        <p>an <code>mtext</code>, <code>mspace</code>,
                           <code>maligngroup</code>, or <code>malignmark</code>
                           element;
                        </p>
                     </li>
                     <li>
                        <p>an <code>mstyle</code>, <code>mphantom</code>, or
                           <code>mpadded</code> element, all of whose direct sub-expressions
                           are space-like;
                        </p>
                     </li>
                     <li>
                        <p>an <code>maction</code> element whose selected
                           sub-expression exists and is space-like;
                        </p>
                     </li>
                     <li>
                        <p>an <code>mrow</code> all of whose direct
                           sub-expressions are space-like.
                        </p>
                     </li>
                  </ul>
                  <p>Note that an <code>mphantom</code> is <em>not</em>
                     automatically defined to be space-like, unless its content is
                     space-like. This is because operator spacing is affected by whether
                     adjacent elements are space-like. Since the
                     <code>mphantom</code> element is primarily intended as an aid
                     in aligning expressions, operators adjacent to an
                     <code>mphantom</code> should behave as if they were adjacent
                     to the <em>contents</em> of the <code>mphantom</code>,
                     rather than to an equivalently sized area of whitespace.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.7.5" id="id.3.2.7.5"></a>3.2.7.5 Legal grouping of space-like elements
                  </h4>
                  <p>Authors who insert space-like elements or
                     <code>mphantom</code> elements into an existing MathML
                     expression should note that such elements <em>are</em> counted as
                     arguments, in elements that require a specific number of arguments,
                     or that interpret different argument positions differently.
                  </p>
                  <p>Therefore, space-like elements inserted into such a MathML element
                     should be grouped with a neighboring argument of that element by
                     introducing an <code>mrow</code> for that purpose. For example,
                     to allow for vertical alignment on the right edge of the base of a
                     superscript, the expression
                     
                  </p><pre class="error">
&lt;msup&gt;
  &lt;mi&gt; x &lt;/mi&gt;
  &lt;malignmark edge="right"/&gt;
  &lt;mn&gt; 2 &lt;/mn&gt;
&lt;/msup&gt;
</pre><p>
                     is illegal, because <code>msup</code> must have exactly 2 arguments;
                     the correct expression would be:
                     
                  </p><pre class="mathml">
&lt;msup&gt;
  &lt;mrow&gt;
    &lt;mi&gt; x &lt;/mi&gt;
    &lt;malignmark edge="right"/&gt;
  &lt;/mrow&gt;
  &lt;mn&gt; 2 &lt;/mn&gt;
&lt;/msup&gt;
</pre><p>See also the warning about "tweaking" in
                     <a href="chapter3-d.html#presm.warnfinetuning">Section&nbsp;3.1.8 Warning about fine-tuning of presentation</a>.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.ms" id="presm.ms"></a>3.2.8 String Literal <code>&lt;ms&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.2.8.1" id="id.3.2.8.1"></a>3.2.8.1 Description
                  </h4>
                  <p>The <code>ms</code> element is used to represent
                     "string literals" in expressions meant to be interpreted by
                     computer algebra systems or other systems containing "programming
                     languages". By default, string literals are displayed surrounded by
                     double quotes, with no extra spacing added around the string.
                     As explained in <a href="chapter3-d.html#presm.mtext">Section&nbsp;3.2.6 Text <code>&lt;mtext&gt;</code></a>, ordinary text
                     embedded in a mathematical expression should be marked up with <code>mtext</code>,
                     or in some cases <code>mo</code> or <code>mi</code>, but never with <code>ms</code>.
                  </p>
                  <p>Note that the string literals encoded by <code>ms</code> are made up of characters, <code>mglyph</code>s and
                     <code>malignmark</code>s rather than "ASCII
                     strings".  For
                     example, <code>&lt;ms&gt;&amp;amp;&lt;/ms&gt;</code> represents a string
                     literal containing a single character, <code>&amp;</code>, and
                     <code>&lt;ms&gt;&amp;amp;amp;&lt;/ms&gt;</code> represents a string literal
                     containing 5 characters, the first one of which is
                     <code>&amp;</code>.
                  </p>
                  <p>The content of <code>ms</code> elements should be rendered with visible
                     "escaping" of certain characters in the content,
                     including at least the left and right quoting
                     characters, and preferably whitespace other than individual
                     space characters. The intent is for the viewer to see that the
                     expression is a string literal, and to see exactly which characters
                     form its content. For example, <code>&lt;ms&gt;double quote is
                        "&lt;/ms&gt;</code> might be rendered as "double quote is \"".
                  </p>
                  <p>Like all token elements, <code>ms</code> <em>does</em> trim and
                     collapse whitespace in its content according to the rules of 
                     <a href="chapter2-d.html#fund.collapse">Section&nbsp;2.1.7 Collapsing Whitespace in Input</a>, so whitespace intended to remain in
                     the content should be encoded as described in that section.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.2.8.2" id="id.3.2.8.2"></a>3.2.8.2 Attributes
                  </h4>
                  <p><code>ms</code> elements accept the attributes listed in
                     <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>, and additionally: 
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">lquote</td>
                           <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                           <td><code>&amp;quot;</code></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the opening quote to enclose the content.
                              (not necessarily ‘left quote’ in RTL context).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">rquote</td>
                           <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                           <td><code>&amp;quot;</code></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the closing quote to enclose the content.
                              (not necessarily ‘right quote’ in RTL context).
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="presm.genlayout" id="presm.genlayout"></a>3.3 General Layout Schemata
            </h2>
            <p>Besides tokens there are several families of MathML presentation
               elements. One family of elements deals with various
               "scripting" notations, such as subscript and
               superscript. Another family is concerned with matrices and tables. The
               remainder of the elements, discussed in this section, describe other basic
               notations such as fractions and radicals, or deal with general functions
               such as setting style properties and error handling.
            </p>
            <div class="div3">
               
               <h3><a name="presm.mrow" id="presm.mrow"></a>3.3.1 Horizontally Group Sub-Expressions
                  <code>&lt;mrow&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.1.1" id="id.3.3.1.1"></a>3.3.1.1 Description
                  </h4>
                  <p>An <code>mrow</code> element is used to group together any
                     number of sub-expressions, usually consisting of one or more <code>mo</code> elements acting as "operators" on one
                     or more other expressions that are their "operands".
                  </p>
                  <p>Several elements automatically treat their arguments as if they were
                     contained in an <code>mrow</code> element. See the discussion of
                     inferred <code>mrow</code>s in <a href="chapter3-d.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>. See also <code>mfenced</code> (<a href="chapter3-d.html#presm.mfenced">Section&nbsp;3.3.8 Expression Inside Pair of Fences
                        <code>&lt;mfenced&gt;</code></a>), which can effectively form an <code>mrow</code> containing its arguments separated by commas.
                  </p>
                  <p><code>mrow</code> elements are typically rendered visually
                     as a horizontal row of their arguments, left to right in the order in
                     which the arguments occur within a context with LTR directionality,
                     or right to left within a context with RTL directionality.  
                     The <code>dir</code> attribute can be used to specify 
                     the directionality for a specific <code>mrow</code>, otherwise it inherits the
                     directionality from the context. For aural agents, the arguments would be
                     rendered audibly as a sequence of renderings of
                     the arguments. The description in <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                        <code>&lt;mo&gt;</code></a> of suggested rendering
                     rules for <code>mo</code> elements assumes that all horizontal
                     spacing between operators and their operands is added by the rendering
                     of <code>mo</code> elements (or, more generally, embellished
                     operators), not by the rendering of the <code>mrow</code>s
                     they are contained in.
                  </p>
                  <p>MathML provides support for both automatic and manual
                     linebreaking of  expressions (that is, to break excessively long
                     expressions into several lines). All such linebreaks take place
                     within <code>mrow</code>s, whether they are explicitly marked up
                     in the document, or inferred (See <a href="chapter3-d.html#presm.inferredmrow">Section&nbsp;3.1.3.1 Inferred <code>&lt;mrow&gt;</code>s</a>),
                     although the control of linebreaking is effected through attributes
                     on other elements (See <a href="chapter3-d.html#presm.linebreaking">Section&nbsp;3.1.7 Linebreaking of Expressions</a>).
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.1.2" id="id.3.3.1.2"></a>3.3.1.2 Attributes
                  </h4>
                  <p><code>mrow</code> elements accept the attribute listed below in addition to
                     those listed in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">dir</td>
                           <td>"ltr" | "rtl"</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the overall directionality <code>ltr</code> (Left To Right) or
                              <code>rtl</code> (Right To Left) to use to layout the children of the row.
                              See <a href="chapter3-d.html#presm.bidi.math">Section&nbsp;3.1.5.1 Overall Directionality of Mathematics Formulas</a> for further discussion.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.1.3" id="id.3.3.1.3"></a>3.3.1.3 Proper grouping of sub-expressions using <code>&lt;mrow&gt;</code></h4>
                  <p>Sub-expressions should be grouped by the document author in the same way
                     as they are grouped in the mathematical interpretation of the expression;
                     that is, according to the underlying "syntax tree" of the
                     expression. Specifically, operators and their mathematical arguments should
                     occur in a single <code>mrow</code>; more than one operator
                     should occur directly in one <code>mrow</code> only when they
                     can be considered (in a syntactic sense) to act together on the interleaved
                     arguments, e.g. for a single parenthesized term and its parentheses, for
                     chains of relational operators, or for sequences of terms separated by
                     <code>+</code> and <code>-</code>. A precise rule is given below.
                  </p>
                  <p>Proper grouping has several purposes: it improves display by
                     possibly affecting spacing; it allows for more intelligent
                     linebreaking and indentation; and it simplifies possible semantic
                     interpretation of presentation elements by computer algebra systems,
                     and audio renderers.
                  </p>
                  <p>Although improper grouping will sometimes result in suboptimal
                     renderings, and will often make interpretation other than pure visual
                     rendering difficult or impossible, any grouping of expressions using
                     <code>mrow</code> is allowed in MathML syntax; that is,
                     renderers should not assume the rules for proper grouping will be
                     followed.
                  </p>
                  <div class="div5">
                     
                     <h5><a name="id.3.3.1.3.1" id="id.3.3.1.3.1"></a>3.3.1.3.1 <code>&lt;mrow&gt;</code> of one argument
                     </h5>
                     <p>MathML renderers are required to treat an <code>mrow</code>
                        element containing exactly one argument as equivalent in all ways to
                        the single argument occurring alone, provided there are no attributes
                        on the <code>mrow</code> element. If there are
                        attributes on the <code>mrow</code> element, no
                        requirement of equivalence is imposed. This equivalence condition is
                        intended to simplify the implementation of MathML-generating software
                        such as template-based authoring tools. It directly affects the
                        definitions of embellished operator and space-like element and the
                        rules for determining the default value of the <code>form</code>
                        attribute of an <code>mo</code> element;
                        see <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                           <code>&lt;mo&gt;</code></a> and <a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a>. See also the discussion of equivalence of MathML
                        expressions in <a href="chapter2-d.html#interf.genproc">Section&nbsp;2.3 Conformance</a>.
                     </p>
                  </div>
                  <div class="div5">
                     
                     <h5><a name="id.3.3.1.3.2" id="id.3.3.1.3.2"></a>3.3.1.3.2 Precise rule for proper grouping
                     </h5>
                     <p>A precise rule for when and how to nest sub-expressions using
                        <code>mrow</code> is especially desirable when generating
                        MathML automatically by conversion from other formats for displayed
                        mathematics, such as T<sub>E</sub>X, which don't always specify how sub-expressions
                        nest. When a precise rule for grouping is desired, the following rule
                        should be used:
                     </p>
                     <p>Two adjacent operators, possibly embellished, possibly separated by operands (i.e. anything
                        other than operators), should occur in the same
                        <code>mrow</code> only when the leading operator has an infix or
                        prefix form (perhaps inferred), the following operator has an infix or
                        postfix form, and the operators have the same priority in the 
                        operator dictionary  (<a href="appendixc-d.html">Appendix&nbsp;C Operator Dictionary</a>).
                        In all other cases, nested <code>mrow</code>s should be used.
                     </p>
                     <p>When forming a nested <code>mrow</code> (during generation
                        of MathML) that includes just one of two successive operators with
                        the forms mentioned above (which mean that either operator could in
                        principle act on the intervening operand or operands), it is necessary
                        to decide which operator acts on those operands directly (or would do
                        so, if they were present). Ideally, this should be determined from the
                        original expression; for example, in conversion from an
                        operator-precedence-based format, it would be the operator with the
                        higher precedence.
                     </p>
                     <p>Note that the above rule has no effect on whether any MathML
                        expression is valid, only on the recommended way of generating MathML
                        from other formats for displayed mathematics or directly from written
                        notation.
                     </p>
                     <p>(Some of the terminology used in stating the above rule in defined
                        in <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                           <code>&lt;mo&gt;</code></a>.)
                     </p>
                  </div>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.1.4" id="id.3.3.1.4"></a>3.3.1.4 Examples
                  </h4>
                  <p>As an example, 2<var>x</var>+<var>y</var>-<var>z</var>
                     should be written as:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mrow&gt;
    &lt;mn&gt; 2 &lt;/mn&gt;
    &lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
    &lt;mi&gt; x &lt;/mi&gt;
  &lt;/mrow&gt;
  &lt;mo&gt; + &lt;/mo&gt;
  &lt;mi&gt; y &lt;/mi&gt;
  &lt;mo&gt; - &lt;/mo&gt;
  &lt;mi&gt; z &lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>The proper encoding of (<var>x</var>, <var>y</var>) furnishes a less obvious
                     example of nesting <code>mrow</code>s:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mo&gt; ( &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mi&gt; x &lt;/mi&gt;
    &lt;mo&gt; , &lt;/mo&gt;
    &lt;mi&gt; y &lt;/mi&gt;
  &lt;/mrow&gt;
  &lt;mo&gt; ) &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>In this case, a nested <code>mrow</code> is required inside
                     the parentheses, since parentheses and commas, thought of as fence and
                     separator "operators", do not act together on their arguments.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mfrac" id="presm.mfrac"></a>3.3.2 Fractions <code>&lt;mfrac&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.2.1" id="id.3.3.2.1"></a>3.3.2.1 Description
                  </h4>
                  <p>The <code>mfrac</code> element is used for fractions. It can
                     also be used to mark up fraction-like objects such as binomial coefficients
                     and Legendre symbols. The syntax for <code>mfrac</code> is
                     
                  </p><pre>
&lt;mfrac&gt; <em>numerator</em> <em>denominator</em> &lt;/mfrac&gt;
</pre><p>The <code>mfrac</code> element sets <code>displaystyle</code> to "false", or if it
                     was already false increments <code>scriptlevel</code> by 1,
                     within <em>numerator</em> and <em>denominator</em>.
                     (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.) 
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.2.2" id="id.3.3.2.2"></a>3.3.2.2 Attributes
                  </h4>
                  <p><code>mfrac</code> elements accept the attributes listed below
                     in addition to those listed in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                     The fraction line, if any, should be drawn using the color specified by <code>mathcolor</code>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">linethickness</td>
                           <td> <a href="chapter2-d.html#type.length"><em>length</em></a> | "thin" | "medium" | "thick"
                           </td>
                           <td>medium</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the thickness of the horizontal "fraction bar", or "rule"
                              The default value is "medium",
                              "thin" is thinner, but visible, 
                              "thick" is thicker;
                              the exact thickness of these is left up to the rendering agent.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">numalign</td>
                           <td>"left" | "center" | "right"</td>
                           <td>center</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the alignment of the numerator over the fraction.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">denomalign</td>
                           <td>"left" | "center" | "right"</td>
                           <td>center</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the alignment of the denominator under the fraction.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">bevelled</td>
                           <td>"true" | "false"</td>
                           <td>false</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies whether the fraction should be displayed in a beveled style
                              (the numerator slightly raised, the denominator slightly lowered
                              and both separated by a slash), rather than "build up" vertically.
                              See below for an example.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>Thicker lines (e.g. <code>linethickness</code>="thick") might be used with nested fractions;
                     a value of "0" renders without the bar such as for  binomial coefficients.
                     These cases are shown below:
                     
                  </p>
                  <blockquote>
                     <p><img src="image/f3007.gif" alt="\binom{a}{b} \quad \genfrac{}{}{1pt}{}{\frac{a}{b}}{\frac{c}{d}}"></p>
                  </blockquote>
                  <p>An example illustrating the bevelled form is shown below:
                     
                  </p>
                  <blockquote>
                     <p><img src="image/f3008.gif" alt="\frac{1}{x^3 + \frac{x}{3}} = \raisebox{1ex}{$1$}\!        \left/ \!\raisebox{-1ex}{$x^3+\frac{x}{3}$} \right."></p>
                  </blockquote>
                  <p>
                     In a RTL directionality context,  the numerator leads (on the right),
                     the denominator follows (on the left) and the diagonal line slants upwards going from right to left (See <a href="chapter3-d.html#presm.bidi.math">Section&nbsp;3.1.5.1 Overall Directionality of Mathematics Formulas</a> for clarification).
                     Although this format is an established convention, it is not universally
                     followed; for situations where a forward slash is desired in a RTL context,
                     alternative markup, such as an <code>mo</code> within an <code>mrow</code> should be used.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.2.3" id="id.3.3.2.3"></a>3.3.2.3 Examples
                  </h4>
                  <p>The examples shown above can be represented in MathML as:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
   &lt;mo&gt; ( &lt;/mo&gt;
   &lt;mfrac linethickness="0"&gt;
      &lt;mi&gt; a &lt;/mi&gt;
      &lt;mi&gt; b &lt;/mi&gt;
   &lt;/mfrac&gt;
   &lt;mo&gt; ) &lt;/mo&gt;
&lt;/mrow&gt;
&lt;mfrac linethickness="2"&gt;
   &lt;mfrac&gt;
      &lt;mi&gt; a &lt;/mi&gt;
      &lt;mi&gt; b &lt;/mi&gt;
   &lt;/mfrac&gt;
   &lt;mfrac&gt;
      &lt;mi&gt; c &lt;/mi&gt;
      &lt;mi&gt; d &lt;/mi&gt;
   &lt;/mfrac&gt;
&lt;/mfrac&gt;
</pre><pre class="mathml">
&lt;mfrac&gt;
   &lt;mn&gt; 1 &lt;/mn&gt;
   &lt;mrow&gt;
      &lt;msup&gt;
         &lt;mi&gt; x &lt;/mi&gt;
         &lt;mn&gt; 3 &lt;/mn&gt;
       &lt;/msup&gt;
       &lt;mo&gt; + &lt;/mo&gt;
       &lt;mfrac&gt;
         &lt;mi&gt; x &lt;/mi&gt;
         &lt;mn&gt; 3 &lt;/mn&gt;
       &lt;/mfrac&gt;
   &lt;/mrow&gt;
&lt;/mfrac&gt;
&lt;mo&gt; = &lt;/mo&gt;
&lt;mfrac bevelled="true"&gt;
   &lt;mn&gt; 1 &lt;/mn&gt;
   &lt;mrow&gt;
      &lt;msup&gt;
         &lt;mi&gt; x &lt;/mi&gt;
         &lt;mn&gt; 3 &lt;/mn&gt;
       &lt;/msup&gt;
       &lt;mo&gt; + &lt;/mo&gt;
       &lt;mfrac&gt;
         &lt;mi&gt; x &lt;/mi&gt;
         &lt;mn&gt; 3 &lt;/mn&gt;
       &lt;/mfrac&gt;
   &lt;/mrow&gt;
&lt;/mfrac&gt;
</pre><p>A more generic example is:
                     
                  </p><pre class="mathml">
&lt;mfrac&gt;
   &lt;mrow&gt;
      &lt;mn&gt; 1 &lt;/mn&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;msqrt&gt;
         &lt;mn&gt; 5 &lt;/mn&gt;
      &lt;/msqrt&gt;
   &lt;/mrow&gt;
   &lt;mn&gt; 2 &lt;/mn&gt;
&lt;/mfrac&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mroot" id="presm.mroot"></a>3.3.3 Radicals <code>&lt;msqrt&gt;</code>, <code>&lt;mroot&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.3.1" id="id.3.3.3.1"></a>3.3.3.1 Description
                  </h4>
                  <p>These elements construct radicals. The <code>msqrt</code> element is
                     used for square roots, while the <code>mroot</code> element is used
                     to draw radicals with indices, e.g. a cube root. The syntax for these
                     elements is:
                     
                  </p><pre>
&lt;msqrt&gt; <em>base</em> &lt;/msqrt&gt;
&lt;mroot&gt; <em>base</em> <em>index</em> &lt;/mroot&gt;
</pre><p>The <code>mroot</code> element requires exactly 2 arguments.
                     However, <code>msqrt</code> accepts a single argument, possibly
                     being an inferred <code>mrow</code> of multiple children; see <a href="chapter3-d.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>.
                     The <code>mroot</code> element increments <code>scriptlevel</code> by 2,
                     and sets <code>displaystyle</code> to "false", within
                     <em>index</em>, but leaves both attributes unchanged within <em>base</em>. 
                     The <code>msqrt</code> element leaves both
                     attributes unchanged within its argument.
                     (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.)
                  </p>
                  <p>Note that in a RTL directionality, the surd begins
                     on the right, rather than the left, along with the index in the case
                     of <code>mroot</code>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.3.2" id="id.3.3.3.2"></a>3.3.3.2 Attributes
                  </h4>
                  <p><code>msqrt</code> and <code>mroot</code> elements accept the attributes listed in
                     <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.  The surd and overbar should be drawn using the
                     color specified by <code>mathcolor</code>.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mstyle" id="presm.mstyle"></a>3.3.4 Style Change <code>&lt;mstyle&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.4.1" id="id.3.3.4.1"></a>3.3.4.1 Description
                  </h4>
                  <p>The <code>mstyle</code> element is used to make style
                     changes that affect the rendering of its
                     contents.  
                     Firstly, as a presentation element, it accepts
                     the attributes described in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                     Additionally, it
                     can be given any attribute
                     accepted by any other presentation element, except for the
                     attributes described below.  
                     Finally,
                     the <code>mstyle</code> element can be given certain special
                     attributes listed in the next subsection.
                  </p>
                  <p>The <code>mstyle</code> element accepts a single argument,
                     possibly being an inferred <code>mrow</code> of multiple children;
                     see <a href="chapter3-d.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>.
                  </p>
                  <p>Loosely speaking, the effect of the <code>mstyle</code> element
                     is to change the default value of an attribute for the elements it
                     contains. Style changes work in one of several ways, depending on 
                     the way in which default values are specified for an attribute.
                     The cases are:
                     
                  </p>
                  <ul>
                     <li>
                        <p>Some attributes, such as <code>displaystyle</code> or 
                           <code>scriptlevel</code> (explained below), are inherited 
                           from the surrounding context when they are not explicitly set. Specifying
                           such an attribute on an <code>mstyle</code> element sets the
                           value that will be inherited by its child elements. Unless a child element
                           overrides this inherited value, it will pass it on to its children, and
                           they will pass it to their children, and so on. But if a child element does
                           override it, either by an explicit attribute setting or automatically (as
                           is common for <code>scriptlevel</code>), the new (overriding)
                           value will be passed on to that element's children, and then to their
                           children, etc, unless it is again overridden.
                        </p>
                     </li>
                     <li>
                        <p>Other attributes, such as <code>linethickness</code> on
                           <code>mfrac</code>, have default values that are not normally
                           inherited. That is, if the <code>linethickness</code> attribute
                           is not set on the  <code>mfrac</code> element,
                           it will normally use the default value of "1", even if it was
                           contained in a larger <code>mfrac</code> element that set this
                           attribute to a different value. For attributes like this, specifying a
                           value with an <code>mstyle</code> element has the effect of
                           changing the default value for all elements within its scope. The net
                           effect is that setting the attribute value with <code>mstyle</code> propagates the change to all the elements it
                           contains directly or indirectly, except for the individual elements on
                           which the value is overridden. Unlike in the case of inherited attributes,
                           elements that explicitly override this attribute have no effect on this
                           attribute's value in their children.
                        </p>
                     </li>
                     <li>
                        <p>Another group of attributes, such as <code>stretchy</code> and <code>form</code>, are
                           computed from operator dictionary information, position in the
                           enclosing <code>mrow</code>, and other similar data.  For
                           these attributes, a value specified by an enclosing <code>mstyle</code> overrides the value that would normally be
                           computed.
                        </p>
                     </li>
                  </ul>
                  <p>Note that attribute values inherited from an
                     <code>mstyle</code> in any manner affect a descendant element
                     in the <code>mstyle</code>'s content only if that attribute is
                     not given a value by the descendant element. On any element for
                     which the attribute is set explicitly, the value specified  overrides the inherited value. The only exception to this
                     rule is when the attribute value 
                     is documented as
                     specifying an incremental change to the value inherited from that
                     element's context or rendering environment.
                  </p>
                  <p>Note also that the difference between inherited and non-inherited
                     attributes set by <code>mstyle</code>, explained above, only
                     matters when the attribute is set on some element within the
                     <code>mstyle</code>'s contents that has descendants also
                     setting it. Thus it never matters for attributes, such as
                     <code>mathcolor</code>, which can only be set on token elements (or on
                     <code>mstyle</code> itself).
                  </p>
                  <p>MathML specifies that when
                     the attributes <code>height</code>, <code>depth</code> or <code>width</code>
                     are specified on an <code>mstyle</code> element, they apply only to
                     <code>mspace</code> elements, and not to the corresponding attributes of
                     <code>mglyph</code>, <code>mpadded</code>, or <code>mtable</code>.  Similarly, when
                     <code>rowalign</code>, <code>columnalign</code>, or <code>groupalign</code>
                     are specified on an <code>mstyle</code> element, they apply only to the
                     <code>mtable</code> element, and not the <code>mtr</code>, <code>mlabeledtr</code>,
                     <code>mtd</code>, and <code>maligngroup</code> elements.
                     When the <code>lspace</code> attribute is set with <code>mstyle</code>, it
                     applies only to the <code>mo</code> element and not to <code>mpadded</code>.
                     To be consistent, the <code>voffset</code> attribute of the
                     <code>mpadded</code> element can not be set on <code>mstyle</code>.
                     When the deprecated <code>fontfamily</code> attribute is specified on an
                     <code>mstyle</code> element, it does not apply to the <code>mglyph</code> element.
                     The deprecated <code>index</code> attribute cannot be set on <code>mstyle</code>.
                     When the <code>align</code> attribute is set with <code>mstyle</code>, it
                     applies only to the <code>munder</code>, <code>mover</code>, and <code>munderover</code>
                     elements, and not to the <code>mtable</code> and <code>mstack</code> elements.
                     The required attributes <code>src</code> and <code>alt</code> on <code>mglyph</code>,
                     and <code>actiontype</code> on <code>maction</code>, cannot be set on <code>mstyle</code>.
                     
                  </p>
                  <p>As a presentation element, <code>mstyle</code> directly accepts
                     the <code>mathcolor</code> and  <code>mathbackground</code> attributes.
                     Thus, the <code>mathbackground</code> specifies the color to fill the bounding
                     box of the <code>mstyle</code> element itself; it does <em>not</em>
                     specify the default background color.
                     This is an incompatible change from MathML 2, but we feel it is more useful
                     and intuitive.  Since the default for <code>mathcolor</code> is inherited,
                     this is no change in its behaviour.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.mstyle.attrs" id="presm.mstyle.attrs"></a>3.3.4.2 Attributes
                  </h4>
                  <p>As stated above, <code>mstyle</code> accepts all
                     attributes of all MathML presentation elements which do not have
                     required values.  That is, all attributes which have an explicit
                     default value or a default value which is inherited or computed are
                     accepted by the <code>mstyle</code> element.
                  </p>
                  <p><code>mstyle</code> elements accept the attributes listed in
                     <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <p>Additionally, <code>mstyle</code> can be given the following special
                     attributes that are implicitly inherited by every MathML element as
                     part of its rendering environment:
                     
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">scriptlevel</td>
                           <td>( "+" | "-" )? <a href="chapter2-d.html#type.unsigned-integer"><em>unsigned-integer</em></a></td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Changes the <code>scriptlevel</code> in effect for the children.
                              When the value is given without a sign, it sets <code>scriptlevel</code> to the specified value;
                              when a sign is given, it increments ("+") or decrements ("-") the current value.
                              (Note that large decrements can result in negative values of <code>scriptlevel</code>,
                              but these values are considered legal.)
                              See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">displaystyle</td>
                           <td>"true" | "false"</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Changes the <code>displaystyle</code> in effect for the children.
                              See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">scriptsizemultiplier</td>
                           <td><a href="chapter2-d.html#type.number"><em>number</em></a></td>
                           <td>0.71</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the multiplier to be used to adjust font size due
                              to changes in <code>scriptlevel</code>.
                              See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">scriptminsize</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>8pt</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the minimum font size allowed due to changes in <code>scriptlevel</code>.
                              Note that this does not limit the font size due to changes to <code>mathsize</code>.
                              See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">infixlinebreakstyle</td>
                           <td>"before" | "after" | "duplicate"</td>
                           <td>before</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the default linebreakstyle to use for infix operators;
                              see <a href="chapter3-d.html#presm.lbattrs">Section&nbsp;3.2.5.2.2 Linebreaking attributes</a>
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">decimalpoint</td>
                           <td><a href="chapter2-d.html#type.character"><em>character</em></a></td>
                           <td>.</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the character used to determine the alignment point within
                              <a href="chapter3-d.html#presm.mstack"><code>mstack</code></a>
                              and 
                              <a href="chapter3-d.html#presm.mtable"><code>mtable</code></a> columns
                              when the "decimalpoint" value is used to specify the alignment.
                              The default, ".", is the decimal separator used to separate the integral
                              and decimal fractional parts of floating point numbers in many countries.
                              (See <a href="chapter3-d.html#presm.elementary">Section&nbsp;3.6 Elementary Math</a> and <a href="chapter3-d.html#presm.malign">Section&nbsp;3.5.5 Alignment Markers
                                 <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></a>).
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>If <code>scriptlevel</code> is changed incrementally by an
                     <code>mstyle</code> element that also sets certain other
                     attributes, the overall effect of the changes may depend on the order
                     in which they are processed. In such cases, the attributes in the
                     following list should be processed in the following order, regardless
                     of the order in which they occur in the XML-format attribute list of
                     the <code>mstyle</code> start tag:
                     <code>scriptsizemultiplier</code>, <code>scriptminsize</code>,
                     <code>scriptlevel</code>, <code>mathsize</code>.
                  </p>
                  <div class="div5">
                     
                     <h5><a name="id.3.3.4.2.1" id="id.3.3.4.2.1"></a>3.3.4.2.1 Deprecated Attributes
                     </h5>
                     <p>MathML2 allowed the binding of <a href="chapter2-d.html#type.namedspace"><em>namedspaces</em></a>
                        to new values.
                        It appears that this capability was never implemented, and is now deprecated;
                        <a href="chapter2-d.html#type.namedspace"><em>namedspaces</em></a> are now considered constants.
                        For backwards compatibility, the following attributes are accepted
                        on the <code>mstyle</code> element, but are expected to have no effect.
                        
                     </p>
                     <table border="1" class="attributes">
                        <thead>
                           <tr>
                              <th>Name</th>
                              <th>values</th>
                              <th>default</th>
                           </tr>
                        </thead>
                        <tbody>
                           <tr>
                              <td class="attname">veryverythinmathspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td>0.0555556em</td>
                           </tr>
                           <tr>
                              <td class="attname">verythinmathspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td>0.111111em</td>
                           </tr>
                           <tr>
                              <td class="attname">thinmathspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td>0.166667em</td>
                           </tr>
                           <tr>
                              <td class="attname">mediummathspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td>0.222222em</td>
                           </tr>
                           <tr>
                              <td class="attname">thickmathspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td>0.277778em</td>
                           </tr>
                           <tr>
                              <td class="attname">verythickmathspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td>0.333333em</td>
                           </tr>
                           <tr>
                              <td class="attname">veryverythickmathspace</td>
                              <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                              <td>0.388889em</td>
                           </tr>
                        </tbody>
                     </table>
                  </div>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.4.3" id="id.3.3.4.3"></a>3.3.4.3 Examples
                  </h4>
                  <p>The example of limiting the stretchiness of a parenthesis shown in the
                     section on &lt;mo&gt;,
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
   &lt;mo maxsize="1"&gt; ( &lt;/mo&gt;
   &lt;mfrac&gt; &lt;mi&gt; a &lt;/mi&gt; &lt;mi&gt; b &lt;/mi&gt; &lt;/mfrac&gt;
   &lt;mo maxsize="1"&gt; ) &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>can be rewritten using <code>mstyle</code> as:
                     
                  </p><pre class="mathml">
&lt;mstyle maxsize="1"&gt;
   &lt;mrow&gt;
      &lt;mo&gt; ( &lt;/mo&gt;
      &lt;mfrac&gt; &lt;mi&gt; a &lt;/mi&gt; &lt;mi&gt; b &lt;/mi&gt; &lt;/mfrac&gt;
      &lt;mo&gt; ) &lt;/mo&gt;
   &lt;/mrow&gt;
&lt;/mstyle&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.merror" id="presm.merror"></a>3.3.5 Error Message <code>&lt;merror&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.5.1" id="id.3.3.5.1"></a>3.3.5.1 Description
                  </h4>
                  <p>The <code>merror</code> element displays its contents as an
                     "error message". This might be done, for example, by displaying the
                     contents in red, flashing the contents, or changing the background
                     color. The contents can be any expression or expression sequence.
                  </p>
                  <p><code>merror</code> accepts 
                     a single argument possibly being an inferred <code>mrow</code> of multiple children;
                     see <a href="chapter3-d.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>.
                  </p>
                  <p>The intent of this element is to provide a standard way for
                     programs that <em>generate</em> MathML from other input to report
                     syntax errors in their input. Since it is anticipated that
                     preprocessors that parse input syntaxes designed for easy hand entry
                     will be developed to generate MathML, it is important that they have
                     the ability to indicate that a syntax error occurred at a certain
                     point. See <a href="chapter2-d.html#interf.error">Section&nbsp;2.3.2 Handling of Errors</a>.
                  </p>
                  <p>The suggested use of <code>merror</code> for reporting
                     syntax errors is for a preprocessor to replace the erroneous part of
                     its input with an <code>merror</code> element containing a
                     description of the error, while processing the surrounding expressions
                     normally as far as possible. By this means, the error message will be
                     rendered where the erroneous input would have appeared, had it been
                     correct; this makes it easier for an author to determine from the
                     rendered output what portion of the input was in error.
                  </p>
                  <p>No specific error message format is suggested here, but as with
                     error messages from any program, the format should be designed to make
                     as clear as possible (to a human viewer of the rendered error message)
                     what was wrong with the input and how it can be fixed. If the
                     erroneous input contains correctly formatted subsections, it may be
                     useful for these to be preprocessed normally and included in the error
                     message (within the contents of the <code>merror</code>
                     element), taking advantage of the ability of
                     <code>merror</code> to contain arbitrary MathML expressions
                     rather than only text.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.5.2" id="id.3.3.5.2"></a>3.3.5.2 Attributes
                  </h4>
                  <p><code>merror</code> elements accept the attributes listed in
                     <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.5.3" id="id.3.3.5.3"></a>3.3.5.3 Example
                  </h4>
                  <p>If a MathML syntax-checking preprocessor
                     received the input
                     
                  </p><pre class="error">
&lt;mfraction&gt;
   &lt;mrow&gt; &lt;mn&gt; 1 &lt;/mn&gt; &lt;mo&gt; + &lt;/mo&gt; &lt;msqrt&gt; &lt;mn&gt; 5 &lt;/mn&gt; &lt;/msqrt&gt; &lt;/mrow&gt;
   &lt;mn&gt; 2 &lt;/mn&gt;
&lt;/mfraction&gt;
</pre><p>
                     which contains the non-MathML element <code>mfraction</code>
                     (presumably in place of the MathML element <code>mfrac</code>),
                     it might generate the error message
                     
                  </p><pre class="mathml">
&lt;merror&gt;
   &lt;mtext&gt; Unrecognized element: mfraction;
           arguments were:  &lt;/mtext&gt;
   &lt;mrow&gt; &lt;mn&gt; 1 &lt;/mn&gt; &lt;mo&gt; + &lt;/mo&gt; &lt;msqrt&gt; &lt;mn&gt; 5 &lt;/mn&gt; &lt;/msqrt&gt; &lt;/mrow&gt;
   &lt;mtext&gt;  and  &lt;/mtext&gt;
   &lt;mn&gt; 2 &lt;/mn&gt;
&lt;/merror&gt;
</pre><p>Note that the preprocessor's input is not, in this case, valid MathML,
                     but the error message it outputs is valid MathML.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mpadded" id="presm.mpadded"></a>3.3.6 Adjust Space Around Content
                  <code>&lt;mpadded&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.6.1" id="id.3.3.6.1"></a>3.3.6.1 Description
                  </h4>
                  <p>An <code>mpadded</code> element renders the same as its child content,
                     but with the size of the child's bounding box and the relative positioning
                     point of its content modified according to 
                     <code>mpadded</code>'s attributes.  It
                     does not rescale (stretch or shrink) its content.  The name of the
                     element reflects the typical use of <code>mpadded</code> to add padding,
                     or extra space, around its content.  However, <code>mpadded</code> can be
                     used to make more general adjustments of size and positioning, and some
                     combinations, e.g. negative padding, can cause the content of
                     <code>mpadded</code> to overlap the rendering of neighboring content.  See
                     <a href="chapter3-d.html#presm.warnfinetuning">Section&nbsp;3.1.8 Warning about fine-tuning of presentation</a> for warnings about several
                     potential pitfalls of this effect.
                  </p>
                  <p>The <code>mpadded</code> element accepts
                     a single argument which may be an inferred <code>mrow</code> of multiple children;
                     see <a href="chapter3-d.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>.
                  </p>
                  <p>It is suggested that audio renderers add (or shorten) time delays
                     based on the attributes representing horizontal space
                     (<code>width</code> and <code>lspace</code>).
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.6.2" id="id.3.3.6.2"></a>3.3.6.2 Attributes
                  </h4>
                  <p><code>mpadded</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">height</td>
                           <td>( "+" | "-" )?
                              <a href="chapter2-d.html#type.unsigned-number"><em>unsigned-number</em></a>
                              ( ("%"  <em>pseudo-unit</em>?)
                              | <em>pseudo-unit</em>
                              | <a href="chapter2-d.html#type.unit"><em>unit</em></a>
                              | <a href="chapter2-d.html#type.namedspace"><em>namedspace</em></a>
                              )
                           </td>
                           <td><em>same as content</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Sets or increments the height of the <code>mpadded</code> element.
                              See below for discussion.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">depth</td>
                           <td>( "+" | "-" )?
                              <a href="chapter2-d.html#type.unsigned-number"><em>unsigned-number</em></a>
                              (("%"  <em>pseudo-unit</em>?)
                              | <em>pseudo-unit</em>
                              | <a href="chapter2-d.html#type.unit"><em>unit</em></a>
                              | <a href="chapter2-d.html#type.namedspace"><em>namedspace</em></a>
                              )
                           </td>
                           <td><em>same as content</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Sets or increments the depth of the <code>mpadded</code> element.
                              See below for discussion.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">width</td>
                           <td>( "+" | "-" )?
                              <a href="chapter2-d.html#type.unsigned-number"><em>unsigned-number</em></a>
                              ( ("%" <em>pseudo-unit</em>?)
                              | <em>pseudo-unit</em> 
                              | <a href="chapter2-d.html#type.unit"><em>unit</em></a>
                              | <a href="chapter2-d.html#type.namedspace"><em>namedspace</em></a>
                              )
                           </td>
                           <td><em>same as content</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Sets or increments the width of the <code>mpadded</code> element.
                              See below for discussion.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">lspace</td>
                           <td>( "+" | "-" )?
                              <a href="chapter2-d.html#type.unsigned-number"><em>unsigned-number</em></a>
                              ( ("%" <em>pseudo-unit</em>?)
                              | <em>pseudo-unit</em>
                              | <a href="chapter2-d.html#type.unit"><em>unit</em></a>
                              | <a href="chapter2-d.html#type.namedspace"><em>namedspace</em></a>
                              )
                           </td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Sets the horizontal position of the child content.
                              See below for discussion.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">voffset</td>
                           <td>( "+" | "-" )?
                              <a href="chapter2-d.html#type.unsigned-number"><em>unsigned-number</em></a>
                              ( ("%" <em>pseudo-unit</em>?)
                              | <em>pseudo-unit</em>
                              | <a href="chapter2-d.html#type.unit"><em>unit</em></a>
                              | <a href="chapter2-d.html#type.namedspace"><em>namedspace</em></a>
                              )
                           </td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Sets the vertical position of the child content.
                              See below for discussion.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The <em>pseudo-unit</em> syntax symbol is described below.
                     Also, <code>height</code>, <code>depth</code> and 
                     <code>width</code> attributes are
                     referred to as size attributes, while <code>lspace</code> and <code>voffset</code> attributes 
                     are position attributes.
                  </p>
                  <p>These attributes specify the size of the bounding box of the <code>mpadded</code>
                     element relative to the size of the bounding box of its child content, and specify
                     the position of the child content of the <code>mpadded</code> element relative to the
                     natural positioning of the <code>mpadded</code> element.  The typographical
                     layout parameters determined by these attributes are described in the next subsection.
                     Depending on the form of the attribute value, a dimension may be set to a new value,
                     or specified relative to the child content's corresponding dimension.  Values may be given as
                     multiples or percentages of any of the  
                     dimensions of the normal rendering of the child content using so-called pseudo-units,
                     or they can be set directly using standard units <a href="chapter2-d.html#fund.units">Section&nbsp;2.1.5.2 Length Valued Attributes</a>.
                  </p>
                  <p>If the value of a size attribute begins with a <code>+</code> or <code>-</code> sign,
                     it specifies an <em>increment</em> or <em>decrement</em> to the corresponding
                     dimension by the following length value. Otherwise the corresponding
                     dimension is set directly to the following length value. 
                     Note that since a leading minus sign indicates a decrement, the size
                     attributes (<code>height</code>, <code>depth</code>, <code>width</code>) 
                     cannot be set directly to negative values.  In addition, specifying a
                     decrement that would produce a net negative value for these attributes
                     has the same effect as
                     setting the attribute to zero.  In other words, the effective
                     bounding box of an <code>mpadded</code> element always has non-negative
                     dimensions. However, negative values are allowed for the relative positioning
                     attributes <code>lspace</code> and <code>voffset</code>.
                  </p>
                  <p>Length values (excluding any sign) can be specified in several formats.
                     Each format begins with an <a href="chapter2-d.html#type.unsigned-number"><em>unsigned-number</em></a>,
                     which may be followed by
                     a <code>%</code> sign (effectively scaling the number)
                     and an optional <em>pseudo-unit</em>,
                     by a <em>pseudo-unit</em> alone,
                     or by a <a href="chapter2-d.html#type.unit"><em>unit</em></a> (excepting <code>%</code>).
                     The possible <em>pseudo-units</em> are the keywords <code>height</code>,
                     <code>depth</code>, and <code>width</code>.  They represent the length of the same-named dimension of the
                     <code>mpadded</code> element's child content.
                     
                  </p>
                  <p>For any of these length formats, the resulting length
                     is the product of the number (possibly including the <code>%</code>)
                     and the following <em>pseudo-unit</em>, 
                     <a href="chapter2-d.html#type.unit"><em>unit</em></a>,
                     <a href="chapter2-d.html#type.namedspace"><em>namedspace</em></a>
                     or the default value for the attribute if no such unit or space is given.
                  </p>
                  <p>Some examples of attribute formats using pseudo-units (explicit or
                     default) are as follows: <code>depth="100%height"</code> and
                     <code>depth="1.0height"</code> both set the depth of the
                     <code>mpadded</code> element to the height of its content.
                     <code>depth="105%"</code> sets the depth to 1.05 times the content's
                     depth, and either <code>depth="+100%"</code> or
                     <code>depth="200%"</code> sets the depth to twice the content's
                     depth.
                  </p>
                  <p>The rules given above imply that all of the following attribute
                     settings have the same effect, which is to leave the content's
                     dimensions unchanged:
                     
                  </p><pre>
&lt;mpadded width="+0em"&gt; ... &lt;/mpadded&gt;
&lt;mpadded width="+0%"&gt; ... &lt;/mpadded&gt;
&lt;mpadded width="-0em"&gt; ... &lt;/mpadded&gt;
&lt;mpadded width="-0height"&gt; ... &lt;/mpadded&gt;
&lt;mpadded width="100%"&gt; ... &lt;/mpadded&gt;
&lt;mpadded width="100%width"&gt; ... &lt;/mpadded&gt;
&lt;mpadded width="1width"&gt; ... &lt;/mpadded&gt;
&lt;mpadded width="1.0width"&gt; ... &lt;/mpadded&gt;
&lt;mpadded&gt; ... &lt;/mpadded&gt;
</pre><p>
                     Note that the examples in the Version 2 of the MathML specification showed
                     spaces within the attribute values, suggesting that this was the intended format.
                     Formally, spaces are not allowed within these values, but implementers may
                     wish to ignore such spaces to maximize backward compatibility. 
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.6.3" id="id.3.3.6.3"></a>3.3.6.3 Meanings of size and position attributes
                  </h4>
                  <p>See <a href="appendixd-d.html">Appendix&nbsp;D Glossary</a> for definitions of some of the typesetting 
                     terms used here.
                  </p>
                  <p>The content of an <code>mpadded</code> element defines a fragment of mathematical 
                     notation, such as a character, fraction, or expression, that can be regarded as 
                     a single typographical element with a natural positioning point relative to its 
                     natural bounding box.
                  </p>
                  <p>The size of the bounding box of an <code>mpadded</code> element is
                     defined as the size of the bounding box of its content, except as
                     modified by the <code>mpadded</code> element's 
                     <code>height</code>, <code>depth</code>, and
                     <code>width</code> attributes.  The natural positioning point of the
                     child content of the <code>mpadded</code> element is located to coincide
                     with the natural positioning point of the <code>mpadded</code> element,
                     except as modified by the <code>lspace</code> and <code>voffset</code>
                     attributes.  Thus, the size attributes of <code>mpadded</code> can be used
                     to expand or shrink the apparent bounding box of its content, and the
                     position attributes of <code>mpadded</code> can be used to move the
                     content relative to the bounding box (and hence also neighboring elements).
                     Note that MathML doesn't define the precise relationship between "ink",
                     bounding boxes and positioning points, which are implementation
                     specific.  Thus, absolute values for mpadded attributes may not be
                     portable between implementations.
                     
                  </p>
                  <p>The <code>height</code> attribute specifies the vertical extent of the 
                     bounding box of the <code>mpadded</code> element above its baseline.
                     Increasing the <code>height</code> increases the space between the baseline 
                     of the <code>mpadded</code> element and the content above it, and introduces 
                     padding above the rendering of the child content.  Decreasing the 
                     <code>height</code> reduces the space between the baseline of the 
                     <code>mpadded</code> element and the content above it, and removes 
                     space above the rendering of the child content.  Decreasing the 
                     <code>height</code> may cause content above the <code>mpadded</code> 
                     element to overlap the rendering of the child content, and should 
                     generally be avoided.
                  </p>
                  <p>The <code>depth</code> attribute specifies the vertical extent of the 
                     bounding box of the <code>mpadded</code> element below its baseline.
                     Increasing the <code>depth</code> increases the space between the baseline 
                     of the <code>mpadded</code> element and the content below it, and introduces 
                     padding below the rendering of the child content.  Decreasing the 
                     <code>depth</code> reduces the space between the baseline of the <code>mpadded</code> 
                     element and the content below it, and removes space below the rendering 
                     of the child content.  Decreasing the <code>depth</code> may cause content 
                     below the <code>mpadded</code> element to overlap the rendering of the child 
                     content, and should generally be avoided.
                  </p>
                  <p>The <code>width</code> attribute specifies the horizontal distance 
                     between the positioning point of the <code>mpadded</code> element and the 
                     positioning point of the following content.
                     Increasing the <code>width</code> increases the space between the 
                     positioning point of the <code>mpadded</code> element and the content 
                     that follows it, and introduces padding after the rendering of the 
                     child content.  Decreasing the <code>width</code> reduces the space 
                     between the positioning point of the <code>mpadded</code> element and 
                     the content that follows it, and removes space after the rendering 
                     of the child content.  Setting the <code>width</code> to zero causes 
                     following content to be positioned at the positioning point of the 
                     <code>mpadded</code> element.  Decreasing the <code>width</code> should 
                     generally be avoided, as it may cause overprinting of the following 
                     content.
                  </p>
                  <p>The <code>lspace</code> attribute ("leading" space; 
                     see <a href="chapter3-d.html#presm.bidi.math">Section&nbsp;3.1.5.1 Overall Directionality of Mathematics Formulas</a>) specifies the horizontal 
                     location of the positioning point of the child content with respect to 
                     the positioning point of the <code>mpadded</code> element.  By default they 
                     coincide, and therefore absolute values for lspace have the same effect 
                     as relative values.
                     Positive values for the <code>lspace</code> attribute increase the space 
                     between the preceding content and the child content, and introduce padding 
                     before the rendering of the child content.  Negative values for the 
                     <code>lspace</code> attributes reduce the space between the preceding 
                     content and the child content, and may cause overprinting of the 
                     preceding content, and should generally be avoided.  Note that the 
                     <code>lspace</code> attribute does not affect the <code>width</code> of 
                     the <code>mpadded</code> element, and so the <code>lspace</code> attribute 
                     will also affect the space between the child content and following 
                     content, and may cause overprinting of the following content, unless 
                     the <code>width</code> is adjusted accordingly.
                  </p>
                  <p>The <code>voffset</code> attribute specifies the vertical location 
                     of the positioning point of the child content with respect to the 
                     positioning point of the <code>mpadded</code> element.  Positive values 
                     for the <code>voffset</code> attribute raise the rendering of the child 
                     content above the baseline.  Negative values for the <code>voffset</code> 
                     attribute lower the rendering of the child content below the baseline.  
                     In either case, the <code>voffset</code> attribute may cause overprinting 
                     of neighboring content, which should generally be avoided.  Note that t
                     he <code>voffset</code> attribute does not affect the <code>height</code> 
                     or <code>depth</code> of the <code>mpadded</code> element, and so the <code>voffset</code> 
                     attribute will also affect the space between the child content and neighboring 
                     content, and may cause overprinting of the neighboring content, unless the 
                     <code>height</code> or <code>depth</code> is adjusted accordingly.
                  </p>
                  <p>MathML renderers should ensure that, except for the effects of the 
                     attributes, the relative spacing between the contents of the 
                     <code>mpadded</code> element and surrounding MathML elements would 
                     not be modified by replacing an <code>mpadded</code> element with an 
                     <code>mrow</code> element with the same content, even if linebreaking 
                     occurs within the <code>mpadded</code> element.  MathML does not define 
                     how non-default attribute values of an <code>mpadded</code> element interact 
                     with the linebreaking algorithm.
                  </p>
                  <p>The effects of the size and position attributes are illustrated
                     below. The following diagram illustrates the use of <code>lspace</code>
                     and <code>voffset</code> to shift the position of child content without
                     modifying the <code>mpadded</code> bounding box.
                  </p>
                  <blockquote>
                     <p><img src="image/mpadded-shift.png" alt="illustration of the use of mpadded to shift the position of child content without modifying the bounding box"></p>
                  </blockquote>
                  <p>The corresponding MathML is:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt;x&lt;/mi&gt;
  &lt;mpadded lspace="0.2em" voffset="0.3ex"&gt;
    &lt;mi&gt;y&lt;/mi&gt;
  &lt;/mpadded&gt;
  &lt;mi&gt;z&lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>The next diagram illustrates the use of 
                     <code>width</code>, <code>height</code> and <code>depth</code> 
                     to modifying the <code>mpadded</code> bounding box without changing the relative position 
                     of the child content. 
                  </p>
                  <blockquote>
                     <p><img src="image/mpadded-resize.png" alt="illustration of the use of mpadded to modifying its bounding box without shifting the relative location of its child content"></p>
                  </blockquote>
                  <p>The corresponding MathML is:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt;x&lt;/mi&gt;
  &lt;mpadded width="+90%width" height="+0.3ex" depth="+0.3ex"&gt;
    &lt;mi&gt;y&lt;/mi&gt;
  &lt;/mpadded&gt;
  &lt;mi&gt;z&lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>The final diagram illustrates the generic use of <code>mpadded</code> to modify both 
                     the bounding box and relative position of child content.
                  </p>
                  <blockquote>
                     <p><img src="image/mpadded-combined.png" alt="illustration of the use of mpadded to modify both the bounding box size and position of child content"></p>
                  </blockquote>
                  <p>The corresponding MathML is:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt;x&lt;/mi&gt;
  &lt;mpadded lspace="0.3em" width="+0.6em"&gt;
    &lt;mi&gt;y&lt;/mi&gt;
  &lt;/mpadded&gt;
  &lt;mi&gt;z&lt;/mi&gt;
&lt;/mrow&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mphantom" id="presm.mphantom"></a>3.3.7 Making Sub-Expressions Invisible <code>&lt;mphantom&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.7.1" id="id.3.3.7.1"></a>3.3.7.1 Description
                  </h4>
                  <p>The <code>mphantom</code> element renders invisibly, but
                     with the same size and other dimensions, including baseline position,
                     that its contents would have if they were rendered
                     normally. <code>mphantom</code> can be used to align parts of
                     an expression by invisibly duplicating sub-expressions.
                  </p>
                  <p>The <code>mphantom</code> element accepts
                     a single argument possibly being an inferred <code>mrow</code> of multiple children;
                     see <a href="chapter3-d.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>.
                  </p>
                  <p>Note that it is possible to wrap both an
                     <code>mphantom</code> and an <code>mpadded</code>
                     element around one MathML expression, as in
                     <code>&lt;mphantom&gt;&lt;mpadded attribute-settings&gt;
                        ... &lt;/mpadded&gt;&lt;/mphantom&gt;</code>, to change its size and make it
                     invisible at the same time.
                  </p>
                  <p>MathML renderers should ensure that the relative spacing between
                     the contents of an <code>mphantom</code> element and the
                     surrounding MathML elements is the same as it would be if the
                     <code>mphantom</code> element were replaced by an
                     <code>mrow</code> element with the same content. This holds
                     even if linebreaking occurs within the <code>mphantom</code>
                     element.
                  </p>
                  <p>For the above reason, <code>mphantom</code> is
                     <em>not</em> considered space-like (<a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a>) unless its
                     content is space-like, since the suggested rendering rules for
                     operators are affected by whether nearby elements are space-like.  Even
                     so, the warning about the legal grouping of space-like elements may
                     apply to uses of <code>mphantom</code>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.7.2" id="id.3.3.7.2"></a>3.3.7.2 Attributes
                  </h4>
                  <p><code>mphantom</code> elements accept the attributes listed in
                     <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a> (the <code>mathcolor</code> has no effect).
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.7.3" id="id.3.3.7.3"></a>3.3.7.3 Examples
                  </h4>
                  <p>There is one situation where the preceding rules for rendering an
                     <code>mphantom</code> may not give the desired effect. When an
                     <code>mphantom</code> is wrapped around a subsequence of the
                     arguments of an <code>mrow</code>, the default determination
                     of the <code>form</code> attribute for an <code>mo</code>
                     element within the subsequence can change. (See the default value of
                     the <code>form</code> attribute described in <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                        <code>&lt;mo&gt;</code></a>.) It may be
                     necessary to add an explicit <code>form</code> attribute to such an
                     <code>mo</code> in these cases. This is illustrated in the
                     following example.
                  </p>
                  <p>In this example, <code>mphantom</code> is used to ensure
                     alignment of corresponding parts of the numerator and denominator of a
                     fraction:
                     
                  </p><pre class="mathml">
&lt;mfrac&gt;
  &lt;mrow&gt;
    &lt;mi&gt; x &lt;/mi&gt;
    &lt;mo&gt; + &lt;/mo&gt;
    &lt;mi&gt; y &lt;/mi&gt;
    &lt;mo&gt; + &lt;/mo&gt;
    &lt;mi&gt; z &lt;/mi&gt;
  &lt;/mrow&gt;
  &lt;mrow&gt;
    &lt;mi&gt; x &lt;/mi&gt;
    &lt;mphantom&gt;
      &lt;mo form="infix"&gt; + &lt;/mo&gt;
      &lt;mi&gt; y &lt;/mi&gt;
    &lt;/mphantom&gt;
    &lt;mo&gt; + &lt;/mo&gt;
    &lt;mi&gt; z &lt;/mi&gt;
  &lt;/mrow&gt;
&lt;/mfrac&gt;
</pre><p>This would render as something like
                     
                  </p>
                  <blockquote>
                     <p><img src="image/mphantom.gif" alt="\frac{x+y+x}{x\phantom{{}+y}+z}"></p>
                  </blockquote>
                  <p>
                     rather than as
                     
                  </p>
                  <blockquote>
                     <p><img src="image/mphantom-bad.gif" alt="\frac{x+y+z}{x+z}"></p>
                  </blockquote>
                  <p>The explicit attribute setting <code>form</code><code>="infix"</code> on the
                     <code>mo</code> element inside the <code>mphantom</code> sets the
                     <code>form</code> attribute to what it would have been in the absence of the
                     surrounding <code>mphantom</code>. This is necessary since
                     otherwise, the <code>+</code> sign would be interpreted as a prefix
                     operator, which might have slightly different spacing.
                  </p>
                  <p>Alternatively, this problem could be avoided without any explicit
                     attribute settings, by wrapping each of the arguments
                     <code>&lt;mo&gt;+&lt;/mo&gt;</code> and <code>&lt;mi&gt;y&lt;/mi&gt;</code> in its
                     own <code>mphantom</code> element, i.e.
                     
                  </p><pre class="mathml">
&lt;mfrac&gt;
   &lt;mrow&gt;
      &lt;mi&gt; x &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; y &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; z &lt;/mi&gt;
   &lt;/mrow&gt;
   &lt;mrow&gt;
      &lt;mi&gt; x &lt;/mi&gt;
      &lt;mphantom&gt;
         &lt;mo&gt; + &lt;/mo&gt;
      &lt;/mphantom&gt;
      &lt;mphantom&gt;
         &lt;mi&gt; y &lt;/mi&gt;
      &lt;/mphantom&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; z &lt;/mi&gt;
   &lt;/mrow&gt;
&lt;/mfrac&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mfenced" id="presm.mfenced"></a>3.3.8 Expression Inside Pair of Fences
                  <code>&lt;mfenced&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.8.1" id="id.3.3.8.1"></a>3.3.8.1 Description
                  </h4>
                  <p>The <code>mfenced</code> element provides a convenient form
                     in which to express common constructs involving fences (i.e. braces,
                     brackets, and parentheses), possibly including separators (such as
                     comma) between the arguments.
                  </p>
                  <p>For example, <code>&lt;mfenced&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;/mfenced&gt;</code>
                     renders as "(<var>x</var>)" and is equivalent to
                     
                  </p><pre class="mathml">
&lt;mrow&gt; &lt;mo&gt; ( &lt;/mo&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;mo&gt; ) &lt;/mo&gt; &lt;/mrow&gt;
</pre><p>
                     and
                     <code>&lt;mfenced&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;mi&gt;y&lt;/mi&gt; &lt;/mfenced&gt;</code>
                     renders as "(<var>x</var>, <var>y</var>)"
                     and is equivalent to
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mo&gt; ( &lt;/mo&gt;
  &lt;mrow&gt; &lt;mi&gt;x&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mi&gt;y&lt;/mi&gt; &lt;/mrow&gt;
  &lt;mo&gt; ) &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>Individual fences or separators are represented using
                     <code>mo</code> elements, as described in <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                        <code>&lt;mo&gt;</code></a>. Thus, any <code>mfenced</code>
                     element is completely equivalent to an expanded form described below;
                     either form can be used in MathML, at the convenience of an author or
                     of a MathML-generating program. A MathML renderer is required to
                     render either of these forms in exactly the same way.
                  </p>
                  <p>In general, an <code>mfenced</code> element can contain
                     zero or more arguments, and will enclose them between fences in an
                     <code>mrow</code>; if there is more than one argument, it will
                     insert separators between adjacent arguments, using an additional
                     nested <code>mrow</code> around the arguments and separators
                     for proper grouping (<a href="chapter3-d.html#presm.mrow">Section&nbsp;3.3.1 Horizontally Group Sub-Expressions
                        <code>&lt;mrow&gt;</code></a>). The general expanded form is
                     shown below. The fences and separators will be parentheses and comma
                     by default, but can be changed using attributes, as shown in the
                     following table.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.8.2" id="id.3.3.8.2"></a>3.3.8.2 Attributes
                  </h4>
                  <p><code>mfenced</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                     The delimiters and separators should be drawn using the color specified by <code>mathcolor</code>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">open</td>
                           <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                           <td>(</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the opening delimiter.
                              Since it is used as the content of an <code>mo</code> element, any whitespace
                              will be trimmed and collapsed as described in <a href="chapter2-d.html#fund.collapse">Section&nbsp;2.1.7 Collapsing Whitespace in Input</a>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">close</td>
                           <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                           <td>)</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the closing delimiter.
                              Since it is used as the content of an <code>mo</code> element, any whitespace
                              will be trimmed and collapsed as described in <a href="chapter2-d.html#fund.collapse">Section&nbsp;2.1.7 Collapsing Whitespace in Input</a>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">separators</td>
                           <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                           <td>,</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies a sequence of zero or more separator characters, optionally separated by whitespace.
                              Each pair of arguments is displayed separated by the corresponding separator
                              (none appears after the last argument).
                              If there are too many separators, the excess are ignored;
                              if there are too few, the last separator is repeated.
                              Any whitespace within <code>separators</code> is ignored.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>A generic <code>mfenced</code> element, with all attributes
                     explicit, looks as follows:
                     
                  </p><pre>
&lt;mfenced open="opening-fence"
         close="closing-fence"
         separators="sep#1 sep#2 ... sep#(n-1)" &gt;
   arg#1
   ...
   arg#n
&lt;/mfenced&gt;
</pre><p>In an RTL directionality context, since the initial text
                     direction is RTL, characters in the <code>open</code> and <code>close</code>
                     attributes that have a mirroring counterpart will be rendered in that
                     mirrored form.  In particular, the default values will render correctly
                     as a parenthesized sequence in both LTR and RTL contexts.
                  </p>
                  <p>The general <code>mfenced</code> element shown above is
                     equivalent to the following expanded form:
                     
                  </p><pre>
&lt;mrow&gt;
   &lt;mo fence="true"&gt; opening-fence &lt;/mo&gt;
   &lt;mrow&gt;
      arg#1
      &lt;mo separator="true"&gt; sep#1 &lt;/mo&gt;
      ...
      &lt;mo separator="true"&gt; sep#(n-1) &lt;/mo&gt;
      arg#n
   &lt;/mrow&gt;
   &lt;mo fence="true"&gt; closing-fence &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>Each argument except the last is followed by a separator. The inner
                     <code>mrow</code> is added for proper grouping, as described in
                     <a href="chapter3-d.html#presm.mrow">Section&nbsp;3.3.1 Horizontally Group Sub-Expressions
                        <code>&lt;mrow&gt;</code></a>.
                  </p>
                  <p>When there is only one argument, the above form has no separators;
                     since <code>&lt;mrow&gt; arg#1 &lt;/mrow&gt;</code> is equivalent to
                     <code>arg#1</code> (as described in <a href="chapter3-d.html#presm.mrow">Section&nbsp;3.3.1 Horizontally Group Sub-Expressions
                        <code>&lt;mrow&gt;</code></a>), this case is also equivalent to:
                     
                  </p><pre>
&lt;mrow&gt;
   &lt;mo fence="true"&gt; opening-fence &lt;/mo&gt;
   arg#1
   &lt;mo fence="true"&gt; closing-fence &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>If there are too many separator characters, the extra ones are
                     ignored. If separator characters are given, but there are too few, the
                     last one is repeated as necessary. Thus, the default value of
                     <code>separators</code>="," is equivalent to
                     <code>separators</code>=",,", <code>separators</code>=",,,", etc. If
                     there are no separator characters provided but some are needed, for
                     example if <code>separators</code>=" " or "" and there is more than
                     one argument, then no separator elements are inserted at all — that
                     is, the elements <code>&lt;mo separator="true"&gt; sep#i
                        &lt;/mo&gt;</code> are left out entirely. Note that this is different
                     from inserting separators consisting of <code>mo</code>
                     elements with empty content.
                  </p>
                  <p>Finally, for the case with no arguments, i.e.
                     
                  </p><pre>
&lt;mfenced open="opening-fence"
         close="closing-fence"
         separators="anything" &gt;
&lt;/mfenced&gt;
</pre><p>
                     the equivalent expanded form is defined to include just
                     the fences within an <code>mrow</code>:
                     
                  </p><pre>
&lt;mrow&gt;
   &lt;mo fence="true"&gt; opening-fence &lt;/mo&gt;
   &lt;mo fence="true"&gt; closing-fence &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>Note that not all "fenced expressions" can be encoded by an
                     <code>mfenced</code> element. Such exceptional expressions
                     include those with an "embellished" separator or fence or one
                     enclosed in an <code>mstyle</code> element, a missing or extra
                     separator or fence, or a separator with multiple content
                     characters. In these cases, it is necessary to encode the expression
                     using an appropriately modified version of an expanded form. As
                     discussed above, it is always permissible to use the expanded form
                     directly, even when it is not necessary. In particular, authors cannot
                     be guaranteed that MathML preprocessors won't replace occurrences of
                     <code>mfenced</code> with equivalent expanded forms.
                  </p>
                  <p>Note that the equivalent expanded forms shown above include
                     attributes on the <code>mo</code> elements that identify them as fences or
                     separators. Since the most common choices of fences and separators
                     already occur in the operator dictionary with those attributes,
                     authors would not normally need to specify those attributes explicitly
                     when using the expanded form directly. Also, the rules for the default
                     <code>form</code> attribute (<a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                        <code>&lt;mo&gt;</code></a>) cause the
                     opening and closing fences to be effectively given the values
                     <code>form</code><code>="prefix"</code> and
                     <code>form</code><code>="postfix"</code> respectively, and the
                     separators to be given the value
                     <code>form</code><code>="infix"</code>.
                  </p>
                  <p>Note that it would be incorrect to use <code>mfenced</code>
                     with a separator of, for instance, "+", as an abbreviation for an
                     expression using "+" as an ordinary operator, e.g.
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt;x&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mi&gt;y&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mi&gt;z&lt;/mi&gt;
&lt;/mrow&gt;
</pre><p>
                     This is because the <code>+</code> signs would be treated as separators,
                     not infix operators. That is, it would render as if they were marked up as
                     <code>&lt;mo separator="true"&gt;+&lt;/mo&gt;</code>, which might therefore
                     render inappropriately.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.8.3" id="id.3.3.8.3"></a>3.3.8.3 Examples
                  </h4>
                  <p>(<var>a</var>+<var>b</var>)
                     
                  </p><pre class="mathml">
&lt;mfenced&gt;
  &lt;mrow&gt;
    &lt;mi&gt; a &lt;/mi&gt;
    &lt;mo&gt; + &lt;/mo&gt;
    &lt;mi&gt; b &lt;/mi&gt;
  &lt;/mrow&gt;
&lt;/mfenced&gt;
</pre><p>Note that the above <code>mrow</code> is necessary so that
                     the <code>mfenced</code> has just one argument. Without it, this
                     would render incorrectly as "(<var>a</var>, +,
                     <var>b</var>)".
                  </p>
                  <p>[0,1)
                     
                  </p><pre class="mathml">
&lt;mfenced open="["&gt;
  &lt;mn&gt; 0 &lt;/mn&gt;
  &lt;mn&gt; 1 &lt;/mn&gt;
&lt;/mfenced&gt;
</pre><p><var>f</var>(<var>x</var>,<var>y</var>)
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mi&gt; f &lt;/mi&gt;
  &lt;mo&gt; &amp;#x2061;<span style="color:#999900">&lt;!--FUNCTION APPLICATION--&gt;</span> &lt;/mo&gt;
  &lt;mfenced&gt;
    &lt;mi&gt; x &lt;/mi&gt;
    &lt;mi&gt; y &lt;/mi&gt;
  &lt;/mfenced&gt;
&lt;/mrow&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.menclose" id="presm.menclose"></a>3.3.9 Enclose Expression Inside Notation
                  <code>&lt;menclose&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.3.9.1" id="id.3.3.9.1"></a>3.3.9.1 Description
                  </h4>
                  <p>The <code>menclose</code> element renders its content
                     inside the enclosing notation specified by its <code>notation</code> attribute. 
                     <code>menclose</code> accepts 
                     a single argument possibly being an inferred <code>mrow</code> of multiple children;
                     see <a href="chapter3-d.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.9.2" id="id.3.3.9.2"></a>3.3.9.2 Attributes
                  </h4>
                  <p><code>menclose</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                     The notations should be drawn using the color specified by <code>mathcolor</code>.
                  </p>
                  <p>
                     The values allowed for <code>notation</code> are open-ended.
                     Conforming renderers may ignore any value they do not handle, although
                     renderers are encouraged to render as many of the values listed below as
                     possible.
                     
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">notation</td>
                           <td>("longdiv" | "actuarial" | "radical" | "box" | "roundedbox" | "circle" |
                              "left" | "right" | "top" | "bottom" |
                              "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike"
                              | "madruwb" | text) + 
                              
                           </td>
                           <td>longdiv</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies a space separated list of notations to be used to enclose the children.
                              See below for a description of each type of notation.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>Any number of values can be given for
                     <code>notation</code> separated by whitespace; all of those given and
                     understood by a MathML renderer should be rendered.
                     Each should be rendered as if the others were not present; they should not nest one inside of the other.  For example,
                     <code>notation</code>="circle box" should
                     result in circle and a box around the contents of <code>menclose</code>; the circle and box may overlap.  This is shown in the first example below.
                     Of the predefined notations, only "radical"
                     is affected by the directionality (see <a href="chapter3-d.html#presm.bidi.math">Section&nbsp;3.1.5.1 Overall Directionality of Mathematics Formulas</a>).
                     
                  </p>
                  <p>When <code>notation</code> has the value "longdiv",
                     the contents are drawn enclosed by a long division symbol.  A complete
                     example of long division is accomplished by also using <code>mtable</code>
                     and <code>malign</code>.  When <code>notation</code> is specified as
                     "actuarial", the contents are drawn enclosed by an
                     actuarial symbol. A similar result can be achieved
                     with the value "top right". The case of
                     <code>notation</code>="radical" is equivalent to the
                     <code>msqrt</code> schema.
                  </p>
                  <p> The values "box",
                     "roundedbox", and "circle" should
                     enclose the contents as indicated by the values.  The amount of
                     distance between the box, roundedbox, or circle, and the contents are
                     not specified by MathML, and is left to the renderer.  In practice,
                     paddings on each side of 0.4em in the horizontal direction and .5ex in
                     the vertical direction seem to work well.
                  </p>
                  <p>The values "left",
                     "right", "top" and
                     "bottom" should result in lines drawn on those sides of
                     the contents.  The values "updiagonalstrike",
                     "downdiagonalstrike", "verticalstrike"
                     and "horizontalstrike" should result in the indicated
                     strikeout lines being superimposed over the content of the
                     <code>menclose</code>, e.g. a strikeout that extends from the lower left 
                     corner to the upper right corner of the <code>menclose</code> element for
                     "updiagonalstrike", etc.
                     
                  </p>
                  <p>The value "madruwb" should generate an enclosure
                     representing an Arabic factorial (‘madruwb’ is the transliteration
                     of the Arabic مضروب for factorial).
                     This is shown in the third example below.
                  </p>
                  <p>The baseline of an <code>menclose</code> element is the baseline of its child (which might be an implied <code>mrow</code>).
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.3.9.3" id="id.3.3.9.3"></a>3.3.9.3 Examples
                  </h4>
                  <p>An example of using multiple attributes is
                     
                  </p><pre class="mathml">
&lt;menclose notation='circle box'&gt;
  &lt;mi&gt; x &lt;/mi&gt;&lt;mo&gt; + &lt;/mo&gt;&lt;mi&gt; y &lt;/mi&gt;
&lt;/menclose&gt;
</pre><p>
                     which renders with the box and circle overlapping roughly as
                     
                  </p>
                  <blockquote>
                     <p><img src="image/circlebox.png" alt="[Image of a circle and box around x plus y]"></p>
                  </blockquote>
                  <p>.
                     
                  </p>
                  <p>An example of using <code>menclose</code> for actuarial
                     notation is
                     
                  </p><pre class="mathml">
&lt;msub&gt;
  &lt;mi&gt;a&lt;/mi&gt;
  &lt;mrow&gt;
    &lt;menclose notation='actuarial'&gt;
      &lt;mi&gt;n&lt;/mi&gt;
    &lt;/menclose&gt;
    &lt;mo&gt;&amp;#x2063;<span style="color:#999900">&lt;!--INVISIBLE SEPARATOR--&gt;</span>&lt;/mo&gt;
    &lt;mi&gt;i&lt;/mi&gt;
  &lt;/mrow&gt;
&lt;/msub&gt;
</pre><p>
                     which renders roughly as
                     
                  </p>
                  <blockquote>
                     <p><img src="image/actuarial.png" alt="[image of actuarial notation for a angle n at i]"></p>
                  </blockquote>
                  <p>An example of "madruwb" is:
                     
                  </p><pre class="mathml">
  &lt;menclose notation="madruwb"&gt;
    &lt;mn&gt;12&lt;/mn&gt;
  &lt;/menclose&gt;
</pre><p>
                     which renders roughly as
                     
                  </p>
                  <blockquote>
                     <p><img src="image/madruwb12.png" alt="[Image of 12 factorial in Arabic style]"></p>
                  </blockquote>
                  <p>.
                     
                  </p>
               </div>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="presm.scrlim" id="presm.scrlim"></a>3.4 Script and Limit Schemata
            </h2>
            <p>The elements described in this section position one or more scripts
               around a base. Attaching various kinds of scripts and embellishments to
               symbols is a very common notational device in mathematics. For purely
               visual layout, a single general-purpose element could suffice for
               positioning scripts and embellishments in any of the traditional script
               locations around a given base. However, in order to capture the abstract
               structure of common notation better, MathML provides several more
               specialized scripting elements.
            </p>
            <p>In addition to sub/superscript elements, MathML has overscript
               and underscript elements that place scripts above and below the base. These
               elements can be used to place limits on large operators, or for placing
               accents and lines above or below the base. The rules for rendering accents
               differ from those for overscripts and underscripts, and this difference can
               be controlled with the <code>accent</code> and <code>accentunder</code> attributes, as described in the appropriate
               sections below.
            </p>
            <p>Rendering of scripts is affected by the <code>scriptlevel</code> and <code>displaystyle</code>
               attributes, which are part of the environment inherited by the rendering
               process of every MathML expression, and are described in <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.
               These attributes cannot be given explicitly on a scripting element, but can be
               specified on the start tag of a surrounding <code>mstyle</code>
               element if desired.
            </p>
            <p>MathML also provides an element for attachment of tensor indices.
               Tensor indices are distinct from ordinary subscripts and superscripts in
               that they must align in vertical columns. Tensor indices can also occur in
               prescript positions.  Note that ordinary scripts follow the base (on the right
               in LTR context, but on the left in RTL context); prescripts precede the base
               (on the left (right) in LTR (RTL) context).
            </p>
            <p>Because presentation elements should be used to describe the abstract
               notational structure of expressions, it is important that the base
               expression in all "scripting" elements (i.e. the first
               argument expression) should be the entire expression that is being
               scripted, not just the trailing character. For example,
               (<var>x</var>+<var>y</var>)<sup>2</sup> should be written as:
               
            </p><pre class="mathml">
&lt;msup&gt;
  &lt;mrow&gt;
    &lt;mo&gt; ( &lt;/mo&gt;
    &lt;mrow&gt;
      &lt;mi&gt; x &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; y &lt;/mi&gt;
    &lt;/mrow&gt;
    &lt;mo&gt; ) &lt;/mo&gt;
  &lt;/mrow&gt;
  &lt;mn&gt; 2 &lt;/mn&gt;
&lt;/msup&gt;
</pre><div class="div3">
               
               <h3><a name="presm.msub" id="presm.msub"></a>3.4.1 Subscript <code>&lt;msub&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.4.1.1" id="id.3.4.1.1"></a>3.4.1.1 Description
                  </h4>
                  <p>The <code>msub</code> element attaches a subscript to a base using the syntax
                     
                  </p><pre>
&lt;msub&gt; <em>base</em> <em>subscript</em> &lt;/msub&gt;
</pre><p>
                     It increments <code>scriptlevel</code> by 1, and sets <code>displaystyle</code> to
                     "false", within <em>subscript</em>, but leaves both attributes
                     unchanged within <em>base</em>. (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.)
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.1.2" id="id.3.4.1.2"></a>3.4.1.2 Attributes
                  </h4>
                  <p><code>msub</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">subscriptshift</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td><em>automatic</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the minimum amount to shift the baseline of <em>subscript</em> down;
                              the default is for the rendering agent to use its own positioning rules.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.msup" id="presm.msup"></a>3.4.2 Superscript <code>&lt;msup&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.4.2.1" id="id.3.4.2.1"></a>3.4.2.1 Description
                  </h4>
                  <p>The <code>msup</code> element attaches a superscript to a base using the syntax
                     
                  </p><pre>
&lt;msup&gt; <em>base</em> <em>superscript</em> &lt;/msup&gt;
</pre><p>
                     It increments <code>scriptlevel</code> by 1, and sets <code>displaystyle</code> to "false", within
                     <em>superscript</em>, but leaves both attributes unchanged within
                     <em>base</em>. (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.) 
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.2.2" id="id.3.4.2.2"></a>3.4.2.2 Attributes
                  </h4>
                  <p><code>msup</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">superscriptshift</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td><em>automatic</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the minimum amount to shift the baseline of <em>superscript</em> up;
                              the default is for the rendering agent to use its own positioning rules.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.msubsup" id="presm.msubsup"></a>3.4.3 Subscript-superscript Pair <code>&lt;msubsup&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.4.3.1" id="id.3.4.3.1"></a>3.4.3.1 Description
                  </h4>
                  <p>The <code>msubsup</code> element is used to attach both a subscript and
                     superscript to a base expression.
                     
                  </p><pre>
&lt;msubsup&gt; <em>base</em> <em>subscript</em> <em>superscript</em> &lt;/msubsup&gt;
</pre><p>
                     It increments <code>scriptlevel</code> by 1, and sets <code>displaystyle</code> to
                     "false", within <em>subscript</em> and <em>superscript</em>,
                     but leaves both attributes unchanged within <em>base</em>.
                     (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.) 
                     
                  </p>
                  <p> Note that both scripts are positioned tight against the base as shown here
                     <img src="image/f3014.gif" alt="x_1^2" align="middle">
                     versus the staggered positioning of nested scripts as shown here
                     <img src="image/f3013.gif" alt="x_1{}^2" align="middle">;
                     the latter can be achieved by nesting an <code>msub</code> inside an <code>msup</code>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.subsupatt" id="presm.subsupatt"></a>3.4.3.2 Attributes
                  </h4>
                  <p><code>msubsup</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">subscriptshift</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td><em>automatic</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the minimum amount to shift the baseline of <em>subscript</em> down;
                              the default is for the rendering agent to use its own positioning rules.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">superscriptshift</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td><em>automatic</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the minimum amount to shift the baseline of <em>superscript</em> up;
                              the default is for the rendering agent to use its own positioning rules.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.3.3" id="id.3.4.3.3"></a>3.4.3.3 Examples
                  </h4>
                  <p>The <code>msubsup</code> is most commonly used for adding
                     sub/superscript pairs to identifiers as illustrated above. However,
                     another important use is placing limits on certain large operators
                     whose limits are traditionally displayed in the script positions even
                     when rendered in display style. The most common of these is the
                     integral. For example,
                  </p>
                  <blockquote>
                     <p><img src="image/f3015.gif" alt="\int\nolimits_0^1 \eulere^x \,\diffd x"></p>
                  </blockquote>
                  <p>
                     would be represented as
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;msubsup&gt;
    &lt;mo&gt; &amp;#x222B;<span style="color:#999900">&lt;!--INTEGRAL--&gt;</span> &lt;/mo&gt;
    &lt;mn&gt; 0 &lt;/mn&gt;
    &lt;mn&gt; 1 &lt;/mn&gt;
  &lt;/msubsup&gt;
  &lt;mrow&gt;
    &lt;msup&gt;
      &lt;mi&gt; &amp;#x2147;<span style="color:#999900">&lt;!--DOUBLE-STRUCK ITALIC SMALL E--&gt;</span> &lt;/mi&gt;
      &lt;mi&gt; x &lt;/mi&gt;
    &lt;/msup&gt;
    &lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
    &lt;mrow&gt;
      &lt;mo&gt; &amp;#x2146;<span style="color:#999900">&lt;!--DOUBLE-STRUCK ITALIC SMALL D--&gt;</span> &lt;/mo&gt;
      &lt;mi&gt; x &lt;/mi&gt;
    &lt;/mrow&gt;
  &lt;/mrow&gt;
&lt;/mrow&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.munder" id="presm.munder"></a>3.4.4 Underscript <code>&lt;munder&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.4.4.1" id="id.3.4.4.1"></a>3.4.4.1 Description
                  </h4>
                  <p>The <code>munder</code> element attaches an accent or limit placed under a base using the syntax
                     
                  </p><pre>
&lt;munder&gt; <em>base</em> <em>underscript</em> &lt;/munder&gt;
</pre><p>
                     It always sets <code>displaystyle</code> to "false" within the underscript,
                     but increments <code>scriptlevel</code> by 1 only when <code>accentunder</code> is "false".
                     Within <em>base</em>, it always leaves both attributes unchanged.
                     (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.) 
                     
                  </p>
                  <p>If <em>base</em> is an operator with <code>movablelimits</code>="true"
                     (or an embellished operator whose <code>mo</code> element core has <code>movablelimits</code>="true"),
                     and <code>displaystyle</code>="false",
                     then <em>underscript</em> is drawn in a subscript position.
                     In this case, the <code>accentunder</code> attribute is ignored.
                     This is often used for limits on symbols such as <code>&amp;sum;</code>.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.4.2" id="id.3.4.4.2"></a>3.4.4.2 Attributes
                  </h4>
                  <p><code>munder</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">accentunder</td>
                           <td>"true" | "false"</td>
                           <td><em>automatic</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies whether <em>underscript</em> is drawn as an "accent" or as a limit.
                              An accent is drawn the same size as the base (without incrementing <code>scriptlevel</code>)
                              and is drawn closer to the base.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">align</td>
                           <td>"left" | "right" | "center"</td>
                           <td>center</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies whether the script is aligned left, center, or right under/over the base.
                              As specfied in <a href="chapter3-d.html#presm.horiz.stretch.rules">Section&nbsp;3.2.5.8.3 Horizontal Stretching Rules</a>, the core of underscripts that are embellished operators should stretch to cover the base, but the alignment is based on
                              the entire underscript.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The default value of <code>accentunder</code> is false, unless
                     <em>underscript</em> is an <code>mo</code> element or an
                     embellished operator (see <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                        <code>&lt;mo&gt;</code></a>). If
                     <em>underscript</em> is an <code>mo</code> element, the
                     value of its <code>accent</code> attribute is used as the default
                     value of <code>accentunder</code>. If <em>underscript</em> is an
                     embellished operator, the <code>accent</code> attribute of the
                     <code>mo</code> element at its core is used as the default
                     value.  As with all attributes, an explicitly given value overrides
                     the default.
                  </p>
                  <p>Here is an example (accent versus underscript):
                     <img src="image/f3016.gif" alt="\underbrace{x+y+z}" align="middle"> versus
                     <img src="image/f3017.gif" alt="\underbrace{\strut x+y+z}" align="middle">.
                     The MathML representation for this example is shown below.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.4.3" id="id.3.4.4.3"></a>3.4.4.3 Examples
                  </h4>
                  <p>The MathML representation for the example shown above is:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;munder accentunder="true"&gt;
    &lt;mrow&gt;
      &lt;mi&gt; x &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; y &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; z &lt;/mi&gt;
    &lt;/mrow&gt;
    &lt;mo&gt; &amp;#x23DF;<span style="color:#999900">&lt;!--BOTTOM CURLY BRACKET--&gt;</span> &lt;/mo&gt;
  &lt;/munder&gt;
  &lt;mtext&gt;&amp;#xA0;<span style="color:#999900">&lt;!--NO-BREAK SPACE--&gt;</span>versus&amp;#xA0;<span style="color:#999900">&lt;!--NO-BREAK SPACE--&gt;</span>&lt;/mtext&gt;
  &lt;munder accentunder="false"&gt;
    &lt;mrow&gt;
      &lt;mi&gt; x &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; y &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; z &lt;/mi&gt;
    &lt;/mrow&gt;
    &lt;mo&gt; &amp;#x23DF;<span style="color:#999900">&lt;!--BOTTOM CURLY BRACKET--&gt;</span> &lt;/mo&gt;
  &lt;/munder&gt;
&lt;/mrow&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mover" id="presm.mover"></a>3.4.5 Overscript <code>&lt;mover&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.4.5.1" id="id.3.4.5.1"></a>3.4.5.1 Description
                  </h4>
                  <p>The <code>mover</code> element attaches an accent or limit placed over a base using the syntax
                     
                  </p><pre>
&lt;mover&gt; <em>base</em> <em>overscript</em> &lt;/mover&gt;
</pre><p>
                     It always sets <code>displaystyle</code> to "false" within overscript,
                     but increments <code>scriptlevel</code> by 1 only when <code>accent</code> is "false".
                     Within <em>base</em>, it always leaves both attributes unchanged.
                     (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.)
                  </p>
                  <p>If <em>base</em> is an operator with <code>movablelimits</code>="true"
                     (or an embellished operator whose <code>mo</code> element core has <code>movablelimits</code>="true"),
                     and <code>displaystyle</code>="false",
                     then <em>overscript</em> is drawn in a superscript position.
                     In this case, the <code>accent</code> attribute is ignored.
                     This is often used for limits on symbols such as <code>&amp;sum;</code>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.5.2" id="id.3.4.5.2"></a>3.4.5.2 Attributes
                  </h4>
                  <p><code>mover</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">accent</td>
                           <td>"true" | "false"</td>
                           <td><em>automatic</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies whether <em>overscript</em> is drawn as an "accent" or as a limit.
                              An accent is drawn the same size as the base (without incrementing <code>scriptlevel</code>)
                              and is drawn closer to the base.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">align</td>
                           <td>"left" | "right" | "center"</td>
                           <td>center</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies whether the script is aligned left, center, or right under/over the base.
                              As specfied in <a href="chapter3-d.html#presm.horiz.stretch.rules">Section&nbsp;3.2.5.8.3 Horizontal Stretching Rules</a>, the core of overscripts that are embellished operators should stretch to cover the base, but the alignment is based on the
                              entire overscript.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The difference between an accent versus limit is shown here:
                     <img src="image/f3018.gif" alt="\hat{x}" align="middle"> versus
                     <img src="image/f3019.gif" alt="\hat{\strut x}" align="middle">.
                     These differences also apply to "mathematical accents" such as
                     bars or braces over expressions:
                     <img src="image/f3020.gif" alt="\overbrace{x+y+z}" align="middle"> versus
                     <img src="image/f3021.gif" alt="\overbrace{\strut x+y+z}" align="middle">.
                     The MathML representation for each of these examples is shown below.
                  </p>
                  <p>The default value of <em>accent</em> is false, unless
                     <em>overscript</em> is an <code>mo</code> element or an
                     embellished operator (see <a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                        <code>&lt;mo&gt;</code></a>). If
                     <em>overscript</em> is an <code>mo</code> element, the value
                     of its <code>accent</code> attribute is used as the default value
                     of <code>accent</code> for <code>mover</code>.  If
                     <em>overscript</em> is an embellished operator, the <code>accent</code> attribute of the <code>mo</code>
                     element at its core is used as the default value.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.5.3" id="id.3.4.5.3"></a>3.4.5.3 Examples
                  </h4>
                  <p>The MathML representation for the examples shown above is:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mover accent="true"&gt;
    &lt;mi&gt; x &lt;/mi&gt;
    &lt;mo&gt; &amp;#x5E;<span style="color:#999900">&lt;!--CIRCUMFLEX ACCENT--&gt;</span> &lt;/mo&gt;
  &lt;/mover&gt;
  &lt;mtext&gt;&amp;#xA0;<span style="color:#999900">&lt;!--NO-BREAK SPACE--&gt;</span>versus&amp;#xA0;<span style="color:#999900">&lt;!--NO-BREAK SPACE--&gt;</span>&lt;/mtext&gt;
  &lt;mover accent="false"&gt;
    &lt;mi&gt; x &lt;/mi&gt;
    &lt;mo&gt; &amp;#x5E;<span style="color:#999900">&lt;!--CIRCUMFLEX ACCENT--&gt;</span> &lt;/mo&gt;
  &lt;/mover&gt;
&lt;/mrow&gt;
</pre><pre class="mathml">
&lt;mrow&gt;
  &lt;mover accent="true"&gt;
    &lt;mrow&gt;
      &lt;mi&gt; x &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; y &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; z &lt;/mi&gt;
    &lt;/mrow&gt;
    &lt;mo&gt; &amp;#x23DE;<span style="color:#999900">&lt;!--TOP CURLY BRACKET--&gt;</span> &lt;/mo&gt;
  &lt;/mover&gt;
  &lt;mtext&gt;&amp;#xA0;<span style="color:#999900">&lt;!--NO-BREAK SPACE--&gt;</span>versus&amp;#xA0;<span style="color:#999900">&lt;!--NO-BREAK SPACE--&gt;</span>&lt;/mtext&gt;
  &lt;mover accent="false"&gt;
    &lt;mrow&gt;
      &lt;mi&gt; x &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; y &lt;/mi&gt;
      &lt;mo&gt; + &lt;/mo&gt;
      &lt;mi&gt; z &lt;/mi&gt;
    &lt;/mrow&gt;
    &lt;mo&gt; &amp;#x23DE;<span style="color:#999900">&lt;!--TOP CURLY BRACKET--&gt;</span> &lt;/mo&gt;
  &lt;/mover&gt;
&lt;/mrow&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.munderover" id="presm.munderover"></a>3.4.6 Underscript-overscript Pair
                  <code>&lt;munderover&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.4.6.1" id="id.3.4.6.1"></a>3.4.6.1 Description
                  </h4>
                  <p>The <code>munderover</code> element attaches accents or limits placed both over and under a base using the syntax
                     
                  </p><pre>
&lt;munderover&gt; <em>base</em> <em>underscript</em> <em>overscript</em> &lt;/munderover&gt;
</pre><p>
                     It always sets <code>displaystyle</code> to "false"
                     within <em>underscript</em> and <em>overscript</em>,
                     but increments <code>scriptlevel</code> by 1 only when 
                     <code>accentunder</code> or <code>accent</code>, respectively, are "false".
                     Within <em>base</em>, it always leaves both attributes unchanged.
                     (see <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>).
                  </p>
                  <p>If <em>base</em> is an operator with <code>movablelimits</code>="true"
                     (or an embellished operator whose <code>mo</code> element core has <code>movablelimits</code>="true"),
                     and <code>displaystyle</code>="false",
                     then <em>underscript</em> and <em>overscript</em> are drawn in a subscript and superscript position,
                     respectively. In this case, the <code>accentunder</code> and <code>accent</code> attributes are ignored.
                     This is often used for limits on symbols such as <code>&amp;sum;</code>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.6.2" id="id.3.4.6.2"></a>3.4.6.2 Attributes
                  </h4>
                  <p><code>munderover</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">accent</td>
                           <td>"true" | "false"</td>
                           <td><em>automatic</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies whether <em>overscript</em> is drawn as an "accent" or as a limit.
                              An accent is drawn the same size as the base (without incrementing <code>scriptlevel</code>)
                              and is drawn closer to the base.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">accentunder</td>
                           <td>"true" | "false"</td>
                           <td><em>automatic</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies whether <em>underscript</em> is drawn as an "accent" or as a limit.
                              An accent is drawn the same size as the base (without incrementing <code>scriptlevel</code>)
                              and is drawn closer to the base.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">align</td>
                           <td>"left" | "right" | "center"</td>
                           <td>center</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies whether the scripts are aligned left, center, or right under/over the base.
                              As specfied in <a href="chapter3-d.html#presm.horiz.stretch.rules">Section&nbsp;3.2.5.8.3 Horizontal Stretching Rules</a>, the core of underscripts and overscripts that are embellished operators should stretch to cover the base, but the alignment
                              is based on the entire underscript or overscript.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The <code>munderover</code> element is used instead of separate
                     <code>munder</code> and <code>mover</code> elements so that the
                     underscript and overscript are vertically spaced equally in relation
                     to the base and so that they follow the slant of the base as in the
                     second expression shown below:
                  </p>
                  <p><img src="image/f3022.gif" alt="\int_0^{\!\!\!\infty}" align="middle">
                     versus
                     <img src="image/f3023.gif" alt="\int_0^{\infty}" align="middle">.
                     The MathML representation for this example is shown below.
                  </p>
                  <p>The difference in the vertical spacing is too small to be noticed on a
                     low resolution display at a normal font size, but is noticeable on a higher
                     resolution device such as a printer and when using large font sizes.  In
                     addition to the visual differences, attaching both the underscript and
                     overscript to the same base more accurately reflects the semantics of the
                     expression.
                  </p>
                  <p>The defaults for <code>accent</code> and <code>accentunder</code>
                     are computed in the same way as for
                     <a href="chapter3-d.html#presm.munder"><code>munder</code></a> and
                     <a href="chapter3-d.html#presm.mover"><code>mover</code></a>, respectively.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.6.3" id="id.3.4.6.3"></a>3.4.6.3 Examples
                  </h4>
                  <p>The MathML representation for the example shown above with the first
                     expression made using separate <code>munder</code> and
                     <code>mover</code> elements, and the second one using an
                     <code>munderover</code> element, is:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mover&gt;
    &lt;munder&gt;
      &lt;mo&gt; &amp;#x222B;<span style="color:#999900">&lt;!--INTEGRAL--&gt;</span> &lt;/mo&gt;
      &lt;mn&gt; 0 &lt;/mn&gt;
    &lt;/munder&gt;
    &lt;mi&gt; &amp;#x221E;<span style="color:#999900">&lt;!--INFINITY--&gt;</span> &lt;/mi&gt;
  &lt;/mover&gt;
  &lt;mtext&gt;&amp;#xA0;<span style="color:#999900">&lt;!--NO-BREAK SPACE--&gt;</span>versus&amp;#xA0;<span style="color:#999900">&lt;!--NO-BREAK SPACE--&gt;</span>&lt;/mtext&gt;
  &lt;munderover&gt;
    &lt;mo&gt; &amp;#x222B;<span style="color:#999900">&lt;!--INTEGRAL--&gt;</span> &lt;/mo&gt;
    &lt;mn&gt; 0 &lt;/mn&gt;
    &lt;mi&gt; &amp;#x221E;<span style="color:#999900">&lt;!--INFINITY--&gt;</span> &lt;/mi&gt;
  &lt;/munderover&gt;
&lt;/mrow&gt;
</pre></div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mmultiscripts" id="presm.mmultiscripts"></a>3.4.7 Prescripts and Tensor Indices
                  <code>&lt;mmultiscripts&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.4.7.1" id="id.3.4.7.1"></a>3.4.7.1 Description
                  </h4>
                  <p>Presubscripts and tensor notations are represented by a single
                     element, <code>mmultiscripts</code>, using the syntax:
                     
                  </p><pre>
&lt;mmultiscripts&gt;
    <em>base</em>
     (<em>subscript superscript</em>)*
     [ &lt;mprescripts/&gt; (<em>presubscript presuperscript</em>)* ]
&lt;/mmultiscripts&gt;
</pre><p>This element allows the representation of any number of vertically-aligned pairs of subscripts
                     and superscripts, attached to one base expression. It supports both
                     postscripts  and
                     prescripts.
                     Missing scripts can be represented by the empty element
                     <code>none</code>.
                  </p>
                  <p>The prescripts are optional, and when present are given
                     <em>after</em> the postscripts, because prescripts are relatively
                     rare compared to tensor notation.
                  </p>
                  <p>The argument sequence consists of the base followed by zero or more
                     pairs of vertically-aligned subscripts and superscripts (in that
                     order) that represent all of the postscripts. This list is optionally
                     followed by an empty element <code>mprescripts</code> and a
                     list of zero or more pairs of vertically-aligned presubscripts and
                     presuperscripts that represent all of the prescripts. The pair lists
                     for postscripts and prescripts are displayed in the same order as the
                     directional context (ie. left-to-right order in LTR context). If
                     no subscript or superscript should be rendered in a given position,
                     then the empty element <code>none</code> should be used in
                     that position.
                  </p>
                  <p>The base, subscripts, superscripts, the optional separator element
                     <code>mprescripts</code>, the presubscripts, and the
                     presuperscripts, are all direct sub-expressions of the
                     <code>mmultiscripts</code> element, i.e. they are all at the
                     same level of the expression tree. Whether a script argument is a
                     subscript or a superscript, or whether it is a presubscript or a
                     presuperscript is determined by whether it occurs in an even-numbered
                     or odd-numbered argument position, respectively, ignoring the empty
                     element <code>mprescripts</code> itself when determining the
                     position. The first argument, the base, is considered to be in
                     position 1. The total number of arguments must be odd, if
                     <code>mprescripts</code> is not given, or even, if it is.
                  </p>
                  <p>The empty element <code>mprescripts</code> is only allowed as direct sub-expression
                     of <code>mmultiscripts</code>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.7.2" id="id.3.4.7.2"></a>3.4.7.2 Attributes
                  </h4>
                  <p>Same as the attributes of <code>msubsup</code>.  See
                     <a href="chapter3-d.html#presm.subsupatt">Section&nbsp;3.4.3.2 Attributes</a>.
                  </p>
                  <p>The <code>mmultiscripts</code> element increments <code>scriptlevel</code> by 1, and sets <code>displaystyle</code> to "false", within
                     each of its arguments except <em>base</em>, but leaves both attributes
                     unchanged within <em>base</em>. (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.)
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.4.7.3" id="id.3.4.7.3"></a>3.4.7.3 Examples
                  </h4>
                  <p>Two examples of the use of <code>mmultiscripts</code> are:
                  </p>
                  <p><sub>0</sub><var>F</var><sub>1</sub>(;<var>a</var>;<var>z</var>).
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mmultiscripts&gt;
    &lt;mi&gt; F &lt;/mi&gt;
    &lt;mn&gt; 1 &lt;/mn&gt;
    &lt;none/&gt;
    &lt;mprescripts/&gt;
    &lt;mn&gt; 0 &lt;/mn&gt;
    &lt;none/&gt;
  &lt;/mmultiscripts&gt;
  &lt;mo&gt; &amp;#x2061;<span style="color:#999900">&lt;!--FUNCTION APPLICATION--&gt;</span> &lt;/mo&gt;
  &lt;mrow&gt;
    &lt;mo&gt; ( &lt;/mo&gt;
    &lt;mrow&gt; 
      &lt;mo&gt; ; &lt;/mo&gt;
      &lt;mi&gt; a &lt;/mi&gt;
      &lt;mo&gt; ; &lt;/mo&gt;
      &lt;mi&gt; z &lt;/mi&gt;
    &lt;/mrow&gt; 
    &lt;mo&gt; ) &lt;/mo&gt;
  &lt;/mrow&gt;
&lt;/mrow&gt;
</pre><p><img src="image/tensorindices.png" alt="R_i{}^j{}_k{}_l" align="middle">
                     (where <var>k</var> and <var>l</var> are different indices)
                     
                  </p><pre class="mathml">
&lt;mmultiscripts&gt;
  &lt;mi&gt; R &lt;/mi&gt;
  &lt;mi&gt; i &lt;/mi&gt;
  &lt;none/&gt;
  &lt;none/&gt;
  &lt;mi&gt; j &lt;/mi&gt;
  &lt;mi&gt; k &lt;/mi&gt;
  &lt;none/&gt;
  &lt;mi&gt; l &lt;/mi&gt;
  &lt;none/&gt;
&lt;/mmultiscripts&gt;
</pre><p>An additional example of <code>mmultiscripts</code> shows how the binomial
                     coefficient
                     
                  </p>
                  <blockquote>
                     <p><img src="image/binom5-12.png" alt="[binomial(5,12) in english style]"></p>
                  </blockquote>
                  <p>
                     can be displayed in Arabic style
                     
                  </p>
                  <blockquote>
                     <p><img src="image/arbinom5-12.png" alt="[binomial(5,12) in Arabic style]"></p>
                  </blockquote><pre class="mathml">
&lt;mstyle dir="rtl"&gt;
  &lt;mmultiscripts&gt;&lt;mo&gt;&amp;#x0644;&lt;!--ARABIC LETTER LAM--&gt;&lt;/mo&gt;
    &lt;mn&gt;12&lt;/mn&gt;&lt;none/&gt;
    &lt;mprescripts/&gt;
    &lt;none/&gt;&lt;mn&gt;5&lt;/mn&gt;
  &lt;/mmultiscripts&gt;
&lt;/mstyle&gt;
</pre></div>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="presm.tabmat" id="presm.tabmat"></a>3.5 Tabular Math
            </h2>
            <p>Matrices, arrays and other table-like mathematical notation are marked
               up using <code>mtable</code>,
               <code>mtr</code>, <code>mlabeledtr</code> and
               <code>mtd</code> elements. These elements are similar to the
               <code>table</code>, <code>tr</code> and <code>td</code> elements of HTML, except that they provide
               specialized attributes for the fine layout control
               necessary for commutative diagrams, block matrices and so on.
            </p>
            <p>While the two-dimensional layouts used for elementary math such as addition and multiplication
               are somewhat similar to tables, they differ in important ways.
               For layout and for accessibility reasons, the <code>mstack</code> and <code>mlongdiv</code> elements discussed
               in <a href="chapter3-d.html#presm.elementary">Section&nbsp;3.6 Elementary Math</a> should be used for elementary math notations.
            </p>
            <p>In addition to the table elements mentioned above, the <code>mlabeledtr</code> element is used for labeling rows
               of a table.  This is useful for numbered equations.
               The first child of <code>mlabeledtr</code> is the label.
               A label is somewhat special in that it is not considered an expression
               in the matrix and is not counted when determining the number of columns
               in that row.
            </p>
            <div class="div3">
               
               <h3><a name="presm.mtable" id="presm.mtable"></a>3.5.1 Table or Matrix
                  <code>&lt;mtable&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.5.1.1" id="id.3.5.1.1"></a>3.5.1.1 Description
                  </h4>
                  <p>A matrix or table is specified using the <code>mtable</code> element. Inside of the <code>mtable</code> element, only <code>mtr</code>
                     or <code>mlabeledtr</code> elements may appear.
                     (In MathML 1.x, the <code>mtable</code> was allowed to ‘infer’ <code>mtr</code> elements around its arguments,
                     and the <code>mtr</code> element could infer <code>mtd</code> elements.
                     This behaviour is <a href="chapter2-d.html#interf.deprec">deprecated</a>.)
                     
                  </p>
                  <p>Table rows that have fewer columns than other rows of the same
                     table (whether the other rows precede or follow them) are effectively
                     padded on the right (or left in RTL context) with empty <code>mtd</code> elements so
                     that the number of columns in each row equals the maximum number of
                     columns in any row of the table. Note that the use of
                     <code>mtd</code> elements with non-default values of the
                     <code>rowspan</code> or <code>columnspan</code>
                     attributes may affect
                     the number of <code>mtd</code> elements that should be given
                     in subsequent <code>mtr</code> elements to cover a given
                     number of columns.
                     Note also that the label in an <code>mlabeledtr</code> element
                     is not considered a column in the table.
                  </p>
                  <p>MathML does not specify a table layout algorithm.  In
                     particular, it is the responsibility of a MathML renderer to resolve
                     conflicts between the <code>width</code> attribute and other
                     constraints on the width of a table, such as explicit values for <code>columnwidth</code> attributes,
                     and minimum sizes for table cell contents.  For a discussion of table layout algorithms, see
                     <a href="http://www.w3.org/TR/CSS2/tables.html#width-layout">Cascading
                        Style Sheets, level 2</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.mtable.attrs" id="presm.mtable.attrs"></a>3.5.1.2 Attributes
                  </h4>
                  <p><code>mtable</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                     Any rules drawn as part of the <code>mtable</code> should be drawn using the color
                     specified by <code>mathcolor</code>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">align</td>
                           <td>("top" | "bottom" | "center" | "baseline" | "axis"), <em>rownumber</em>?
                           </td>
                           <td>axis</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the vertical alignment of the table with respect to its environment.
                              "axis" means to align the vertical center of the table on
                              the environment's <a class="termref" title="" href="appendixd-d.html#dt-axis">axis</a>.
                              (The <a class="termref" title="" href="appendixd-d.html#dt-axis">axis</a> of an equation is an alignment line used by typesetters.
                              It is the line on which a minus sign typically lies.)
                              "center" and "baseline" both mean to align the center of the table
                              on the environment's baseline.
                              "top" or "bottom" aligns the top or bottom of the table on the environment's baseline.
                              
                              If the <code>align</code> attribute value ends with a <em>rownumber</em>,
                              the specified row (counting from 1 for the top row), rather than the table as a whole, is aligned in the way described above
                              with the exceptions noted below.
                              If <em>rownumber</em> is negative, it counts rows from the bottom.
                              When the value of <em>rownumber</em> is out of range or not an integer, it is ignored.
                              If a row number is specified and the alignment value is "baseline" or "axis", 
                              the row's baseline or axis is used for alignment. Note this is only well defined when the <code>rowalign</code> 
                              value is "baseline" or "axis"; MathML does not specify how 
                              "baseline" or "axis" alignment should occur for other values of <code>rowalign</code>. 
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">rowalign</td>
                           <td>("top" | "bottom" | "center" | "baseline" | "axis") +</td>
                           <td>baseline</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the vertical alignment of the cells with respect to other cells within the same row:
                              "top" aligns the tops of each entry across the row;
                              "bottom" aligns the bottoms of the cells,
                              "center" centers the cells;
                              "baseline" aligns the baselines of the cells;
                              "axis" aligns the axis of each cells.
                              (See the note below about multiple values).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">columnalign</td>
                           <td>("left" | "center" | "right") +</td>
                           <td>center</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the horizontal alignment of the cells with respect to other cells within the same column:
                              "left" aligns the left side of the cells;
                              "center" centers each cells;
                              "right" aligns the right side of the cells.
                              (See the note below about multiple values).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">groupalign</td>
                           <td><a href="chapter3-d.html#type.group-align"><em>group-alignment-list-list</em></a></td>
                           <td>{left}</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              [this attribute is described with the alignment elements, <code>maligngroup</code> and <code>malignmark</code>,
                              in <a href="chapter3-d.html#presm.malign">Section&nbsp;3.5.5 Alignment Markers
                                 <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></a>.]
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">alignmentscope</td>
                           <td>("true" | "false") +</td>
                           <td>true</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              [this attribute is described with the alignment elements, <code>maligngroup</code> and <code>malignmark</code>,
                              in <a href="chapter3-d.html#presm.malign">Section&nbsp;3.5.5 Alignment Markers
                                 <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></a>.]
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">columnwidth</td>
                           <td>("auto" | <a href="chapter2-d.html#type.length"><em>length</em></a> | "fit") +
                           </td>
                           <td>auto</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies how wide a column should be:
                              "auto" means that the column should be as wide as needed;
                              an explicit length means that the column is exactly that wide and the contents of that column are made to fit
                              by linewrapping or clipping at the discretion of the renderer;
                              "fit" means that the page width
                              remaining after subtracting the "auto" or fixed width columns
                              is divided equally among the "fit" columns.
                              
                              If insufficient room remains to hold the
                              contents of the "fit" columns, renderers may
                              linewrap or clip the contents of the "fit" columns.
                              Note that when the <code>columnwidth</code> is specified as
                              a percentage, the value is relative to the width of the table, not
                              as a percentage of the default (which is "auto").  That
                              is, a renderer should try to adjust the width of the column so that it
                              covers the specified percentage of the entire table width.
                              
                              (See the note below about multiple values).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">width</td>
                           <td>"auto" | <a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>auto</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the desired width of the entire table and is intended for visual user agents.
                              When the value is a percentage value, the value is relative to the
                              horizontal space a MathML renderer has available for the <code>math</code> element.
                              When the value is "auto", the MathML
                              renderer should calculate the table width from its contents using
                              whatever layout algorithm it chooses.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">rowspacing</td>
                           <td>(<a href="chapter2-d.html#type.length"><em>length</em></a>) +
                           </td>
                           <td>1.0ex</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies how much space to add between rows.
                              (See the note below about multiple values).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">columnspacing</td>
                           <td>(<a href="chapter2-d.html#type.length"><em>length</em></a>) +
                           </td>
                           <td>0.8em</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies how much space to add between <span class="diff-chg">columns</span>.
                              (See the note below about multiple values).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">rowlines</td>
                           <td>("none" | "solid" | "dashed") +</td>
                           <td>none</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies whether and what kind of lines should be added between each row:
                              "none" means no lines;
                              "solid" means solid lines;
                              "dashed" means dashed lines (how the dashes are spaced is implementation dependent).
                              (See the note below about multiple values).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">columnlines</td>
                           <td>("none" | "solid" | "dashed") +</td>
                           <td>none</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies whether and what kind of lines should be added between each column:
                              "none" means no lines;
                              "solid" means solid lines;
                              "dashed" means dashed lines (how the dashes are spaced is implementation dependent).
                              (See the note below about multiple values).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">frame</td>
                           <td>"none" | "solid" | "dashed"</td>
                           <td>none</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies whether and what kind of lines should be drawn around the table.
                              "none" means no lines;
                              "solid" means solid lines;
                              "dashed" means dashed lines (how the dashes are spaced is implementation dependent).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">framespacing</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a>, <a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>0.4em 0.5ex</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the additional spacing added between the table and frame,
                              if <code>frame</code> is not "none".
                              The first value specifies the spacing on the right and left;
                              the second value specifies the spacing above and below.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">equalrows</td>
                           <td>"true" | "false"</td>
                           <td>false</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies whether to force all rows to have the same total height.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">equalcolumns</td>
                           <td>"true" | "false"</td>
                           <td>false</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies whether to force all columns to have the same total width.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">displaystyle</td>
                           <td>"true" | "false"</td>
                           <td><em>inherited (false)</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the value of <code>displaystyle</code> within each cell,
                              (<code>scriptlevel</code> is not changed);
                              see <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">side</td>
                           <td>"left" | "right" | "leftoverlap" | "rightoverlap"</td>
                           <td>right</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies on what side of the table labels from enclosed <code>mlabeledtr</code> (if any) should be  placed.
                              The variants "leftoverlap" and "rightoverlap"
                              are useful when the table fits with the allowed width
                              when the labels are omitted, but not when they are included:
                              in such cases, the labels will overlap the row placed above it
                              if the <code>rowalign</code> for that row is "top",
                              otherwise it is placed below it.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">minlabelspacing</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>0.8em</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the minimum space allowed between a label and the adjacent cell in the row.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>In the above specifications for attributes affecting rows
                     (respectively, columns, or the gaps between rows or columns),
                     the notation <code>(...)+</code> means that multiple values can be given for the attribute
                     as a space separated list (see <a href="chapter2-d.html#fund.attval">Section&nbsp;2.1.5 MathML Attribute Values</a>).
                     In this context, a single value specifies the value to be used for all rows (resp., columns or gaps).
                     A list of values are taken to apply to corresponding rows (resp., columns or gaps)
                     in order, that is starting from the top row for rows or first column (left or right,
                     depending on directionality) for columns.
                     If there are more rows (resp., columns or gaps) than supplied values, the last value is repeated as needed.
                     If there are too many values supplied, the excess are ignored.
                  </p>
                  <p>Note that none of the areas occupied by lines
                     <code>frame</code>, <code>rowlines</code> and <code>columnlines</code>,
                     nor the spacing  <code>framespacing</code>, <code>rowspacing</code> or <code>columnspacing</code>,
                     nor the label in <code>mlabeledtr</code> are counted as rows or columns.
                  </p>
                  <p>The <code>displaystyle</code> attribute is allowed on the <code>mtable</code>
                     element to set the inherited value of the attribute.  If the attribute is
                     not present, the <code>mtable</code> element sets <code>displaystyle</code> to
                     "false" within the table elements.
                     (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.) 
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.1.3" id="id.3.5.1.3"></a>3.5.1.3 Examples
                  </h4>
                  <p>A 3 by 3 identity matrix could be represented as follows:
                     
                  </p><pre class="mathml">
&lt;mrow&gt;
  &lt;mo&gt; ( &lt;/mo&gt;
  &lt;mtable&gt;
    &lt;mtr&gt;
      &lt;mtd&gt; &lt;mn&gt;1&lt;/mn&gt; &lt;/mtd&gt;
      &lt;mtd&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;/mtd&gt;
      &lt;mtd&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;/mtd&gt;
    &lt;/mtr&gt;
    &lt;mtr&gt;
      &lt;mtd&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;/mtd&gt;
      &lt;mtd&gt; &lt;mn&gt;1&lt;/mn&gt; &lt;/mtd&gt;
      &lt;mtd&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;/mtd&gt;
    &lt;/mtr&gt;
    &lt;mtr&gt;
      &lt;mtd&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;/mtd&gt;
      &lt;mtd&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;/mtd&gt;
      &lt;mtd&gt; &lt;mn&gt;1&lt;/mn&gt; &lt;/mtd&gt;
    &lt;/mtr&gt;
  &lt;/mtable&gt;
  &lt;mo&gt; ) &lt;/mo&gt;
&lt;/mrow&gt;
</pre><p>This might be rendered as:
                     
                  </p>
                  <blockquote>
                     <p><img src="image/f3025.gif" alt="\left(\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 0 &amp; 1 &amp; 0 \\ 0 &amp; 0 &amp; 1\end{array}\right)"></p>
                  </blockquote>
                  <p>
                     Note that the parentheses must be represented explicitly; they are not
                     part of the <code>mtable</code> element's rendering. This allows
                     use of other surrounding fences, such as brackets, or none at all.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mtr" id="presm.mtr"></a>3.5.2 Row in Table or Matrix <code>&lt;mtr&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.5.2.1" id="id.3.5.2.1"></a>3.5.2.1 Description
                  </h4>
                  <p>An <code>mtr</code> element represents one row in a table
                     or matrix. An <code>mtr</code> element is only allowed as a
                     direct sub-expression of an <code>mtable</code> element, and
                     specifies that its contents should form one row of the table. Each
                     argument of <code>mtr</code> is placed in a different column
                     of the table, starting at the leftmost column in a LTR context or rightmost
                     column in a RTL context.
                  </p>
                  <p>As described in <a href="chapter3-d.html#presm.mtable">Section&nbsp;3.5.1 Table or Matrix
                        <code>&lt;mtable&gt;</code></a>,
                     <code>mtr</code> elements are
                     effectively padded with <code>mtd</code> 
                     elements when they are shorter than other rows in a table.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.2.2" id="id.3.5.2.2"></a>3.5.2.2 Attributes
                  </h4>
                  <p><code>mtr</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">rowalign</td>
                           <td>"top" | "bottom" | "center" | "baseline" | "axis"</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              overrides, for this row, the vertical alignment of cells specified 
                              by the <a href="chapter3-d.html#presm.mtable.attrs"><code>rowalign</code></a> attribute on the <code>mtable</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">columnalign</td>
                           <td>("left" | "center" | "right") +</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              overrides, for this row, the horizontal alignment of cells specified 
                              by the <a href="chapter3-d.html#presm.mtable.attrs"><code>columnalign</code></a> attribute on the <code>mtable</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">groupalign</td>
                           <td><a href="chapter3-d.html#type.group-align"><em>group-alignment-list-list</em></a></td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              [this attribute is described with the alignment elements, <code>maligngroup</code> and <code>malignmark</code>,
                              in <a href="chapter3-d.html#presm.malign">Section&nbsp;3.5.5 Alignment Markers
                                 <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></a>.]
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mlabeledtr" id="presm.mlabeledtr"></a>3.5.3 Labeled Row in Table or Matrix
                  <code>&lt;mlabeledtr&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.5.3.1" id="id.3.5.3.1"></a>3.5.3.1 Description
                  </h4>
                  <p>An <code>mlabeledtr</code> element represents one row in
                     a table that has a label on either the left or right side, as
                     determined by the <code>side</code> attribute.  The label is
                     the first child of <code>mlabeledtr</code>, and should be enclosed in an <code>mtd</code>.
                     The rest of the children represent the contents of the row and are treated identically
                     to the children of <code>mtr</code>; consequently all of the children
                     must be <code>mtd</code> elements.
                     
                  </p>
                  <p>
                     An <code>mlabeledtr</code> element is only allowed as a
                     direct sub-expression of an <code>mtable</code> element.
                     Each argument of <code>mlabeledtr</code> except for the first
                     argument (the label) is placed in a different column
                     of the table, starting at the leftmost column.
                  </p>
                  <p>Note that the label element is not considered to be a cell in the
                     table row.  In particular, the label element is not taken into
                     consideration in the table layout for purposes of width and alignment
                     calculations.  For example, in the case of an <code>mlabeledtr</code> with a label and a single centered <code>mtd</code> child, the child is first centered in the
                     enclosing <code>mtable</code>, and then the label is
                     placed.  Specifically, the child is <em>not</em> centered in the
                     space that remains in the table after placing the label.
                  </p>
                  <p>While MathML does not specify an algorithm for placing labels,
                     implementers of visual renderers may find the following formatting
                     model useful.  To place a label, an implementor might think in terms
                     of creating a larger table, with an extra column on both ends.  The
                     <code>columnwidth</code> attributes of both these border
                     columns would be set to "fit" so that they expand
                     to fill whatever space remains after the inner columns have been laid
                     out.  Finally, depending on the values of <code>side</code>
                     and <code>minlabelspacing</code>, the label is placed
                     in whatever border column is appropriate, possibly shifted down if
                     necessary, and aligned according to <code>columnalignment</code>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.3.2" id="id.3.5.3.2"></a>3.5.3.2 Attributes
                  </h4>
                  <p>
                     The attributes for <code>mlabeledtr</code> are the same
                     as for <code>mtr</code>.  Unlike the attributes for the
                     <code>mtable</code> element, attributes of
                     <code>mlabeledtr</code> that apply to column elements
                     also apply to the label.  For example, in a one column table,
                     
                  </p><pre class="mathml-fragment">

&lt;mlabeledtr rowalign='top'&gt;

</pre><p>
                     means that the label and other entries in the row are vertically aligned
                     along their top.  To force a particular alignment on the label,
                     the appropriate attribute would normally be set on the
                     <code>mtd</code> element that surrounds the label content.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.3.3" id="id.3.5.3.3"></a>3.5.3.3 Equation Numbering
                  </h4>
                  <p>One of the important uses of <code>mlabeledtr</code> is
                     for numbered equations.  In a <code>mlabeledtr</code>, the
                     label represents the equation number and the elements in the row are
                     the equation being numbered.  The <code>side</code> and <code>minlabelspacing</code> attributes of <code>mtable</code> determine the placement of the equation
                     number.
                  </p>
                  <p>In larger documents with many numbered equations, automatic
                     numbering becomes important. While automatic equation numbering and
                     automatically resolving references to equation numbers is outside the
                     scope of MathML, these problems can be addressed by the use of style
                     sheets or other means. The mlabeledtr construction provides support
                     for both of these functions in a way that is intended to facilitate
                     XSLT processing.  The <code>mlabeledtr</code> element can be
                     used to indicate the presence of a numbered equation, and the first
                     child can be changed to the current equation number, along with
                     incrementing the global equation number.  For cross references, an
                     <code>id</code> on either the mlabeledtr element or on the first element
                     itself could be used as a target of any link.
                  </p><pre class="mathml">
&lt;mtable&gt;
  &lt;mlabeledtr id='e-is-m-c-square'&gt;
    &lt;mtd&gt;
      &lt;mtext&gt; (2.1) &lt;/mtext&gt;
    &lt;/mtd&gt;
    &lt;mtd&gt;
     &lt;mrow&gt;
       &lt;mi&gt;E&lt;/mi&gt;
       &lt;mo&gt;=&lt;/mo&gt;
       &lt;mrow&gt;
        &lt;mi&gt;m&lt;/mi&gt;
        &lt;mo&gt;&amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span>&lt;/mo&gt;
        &lt;msup&gt;
         &lt;mi&gt;c&lt;/mi&gt;
         &lt;mn&gt;2&lt;/mn&gt;
        &lt;/msup&gt;
       &lt;/mrow&gt;
     &lt;/mrow&gt;
    &lt;/mtd&gt;
  &lt;/mlabeledtr&gt;
&lt;/mtable&gt;
</pre><p>This should be rendered as:
                     
                  </p>
                  <table width="100%">
                     <tbody>
                        <tr>
                           <td id="eqnoc1">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
                           <td id="eqnoc2"><var>E</var> = <var>m</var><var>c</var><sup>2</sup></td>
                           <td id="eqnoc3">(2.1)</td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mtd" id="presm.mtd"></a>3.5.4 Entry in Table or Matrix <code>&lt;mtd&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.5.4.1" id="id.3.5.4.1"></a>3.5.4.1 Description
                  </h4>
                  <p>An <code>mtd</code> element represents one entry, or cell, in a
                     table or matrix. An <code>mtd</code> element is only
                     allowed as a direct sub-expression of an <code>mtr</code>
                     or an <code>mlabeledtr</code> element.
                  </p>
                  <p>The <code>mtd</code> element accepts 
                     a single argument possibly being an inferred <code>mrow</code> of multiple children;
                     see <a href="chapter3-d.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.mtdatts" id="presm.mtdatts"></a>3.5.4.2 Attributes
                  </h4>
                  <p><code>mtd</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">rowspan</td>
                           <td><a href="chapter2-d.html#type.positive-integer"><em>positive-integer</em></a></td>
                           <td>1</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              causes the cell to be treated as if it occupied the number of rows specified.
                              The corresponding <code>mtd</code> in the following "rowspan"-1 rows must be omitted.
                              The interpretation corresponds with the similar attributes for HTML 4.01 tables.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">columnspan</td>
                           <td><a href="chapter2-d.html#type.positive-integer"><em>positive-integer</em></a></td>
                           <td>1</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              causes the cell to be treated as if it occupied the number of columns specified.
                              The following "rowspan"-1 <code>mtd</code>s must be omitted.
                              The interpretation corresponds with the similar attributes for HTML 4.01 tables.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">rowalign</td>
                           <td>"top" | "bottom" | "center" | "baseline" | "axis"</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the vertical alignment of this cell, overriding any value
                              specified on the containing <code>mrow</code> and <code>mtable</code>.
                              See the <a href="chapter3-d.html#presm.mtable.attrs"><code>rowalign</code></a> attribute of <code>mtable</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">columnalign</td>
                           <td>"left" | "center" | "right"</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the horizontal alignment of this cell, overriding any value
                              specified on the containing <code>mrow</code> and <code>mtable</code>.
                              See the <a href="chapter3-d.html#presm.mtable.attrs"><code>columnalign</code></a> attribute of <code>mtable</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">groupalign</td>
                           <td><a href="chapter3-d.html#type.group-align"><em>group-alignment-list</em></a></td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              [this attribute is described with the alignment elements, <code>maligngroup</code> and <code>malignmark</code>,
                              in <a href="chapter3-d.html#presm.malign">Section&nbsp;3.5.5 Alignment Markers
                                 <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></a>.]
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The <code>rowspan</code> and <code>columnspan</code> attributes
                     can be used around an <code>mtd</code> element that represents
                     the label in a <code>mlabeledtr</code> element.
                     Also, the label of a <code>mlabeledtr</code> element is not
                     considered to be part of a previous <code>rowspan</code> and
                     <code>columnspan</code>.
                     
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.malign" id="presm.malign"></a>3.5.5 Alignment Markers
                  <code>&lt;maligngroup/&gt;</code>, <code>&lt;malignmark/&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.1" id="id.3.5.5.1"></a>3.5.5.1 Description
                  </h4>
                  <p>Alignment markers are space-like elements (see <a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a>) that can be used 
                     to vertically align specified points within a column of MathML
                     expressions by the automatic insertion of the necessary amount of
                     horizontal space between specified sub-expressions.
                  </p>
                  <p>The discussion that follows will use the example of a set of
                     simultaneous equations that should be rendered with vertical
                     alignment of the coefficients and variables of each term, by
                     inserting spacing somewhat like that shown here:
                     
                  </p><pre>
    8.44x + 55  y =  0
    3.1 x -  0.7y = -1.1
</pre><p>
                     If the example expressions shown above were arranged in a column
                     but not aligned, they would appear as:
                     
                  </p><pre>
    8.44x + 55y = 0
    3.1x - 0.7y = -1.1
</pre><p>
                     For audio renderers, it is suggested that the alignment elements
                     produce the analogous behavior of altering the rhythm of pronunciation
                     so that it is the same for several sub-expressions in a column, by the
                     insertion of the appropriate time delays in place of the extra
                     horizontal spacing described here.
                  </p>
                  <p>The expressions whose parts are to be aligned (each equation, in the
                     example above) must be given as the table elements (i.e. as the <code>mtd</code> elements) of one column of an
                     <code>mtable</code>. To avoid confusion, the term "table
                     cell" rather than "table element" will be used in the
                     remainder of this section.
                  </p>
                  <p>All interactions between alignment elements are limited to the
                     <code>mtable</code> column they arise in. That is, every
                     column of a table specified by an <code>mtable</code> element
                     acts as an "alignment scope" that contains within it all alignment
                     effects arising from its contents. It also excludes any interaction
                     between its own alignment elements and the alignment elements inside
                     any nested alignment scopes it might contain.
                  </p>
                  <p>The reason <code>mtable</code> columns are used as
                     alignment scopes is that they are the only general way in MathML to
                     arrange expressions into vertical columns. Future versions of MathML
                     may provide an <code>malignscope</code> element that allows
                     an alignment scope to be created around any MathML element, but even
                     then, table columns would still sometimes need to act as alignment
                     scopes, and since they are not elements themselves, but rather are
                     made from corresponding parts of the content of several
                     <code>mtr</code> elements, they could not individually be the
                     content of an alignment scope element.
                  </p>
                  <p>An <code>mtable</code> element can be given the attribute
                     <code>alignmentscope</code>="false" to cause 
                     its columns not to act as alignment scopes. This is discussed further at
                     the end of this section. Otherwise, the discussion in this section assumes
                     that this attribute has its default value of "true".
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.2" id="id.3.5.5.2"></a>3.5.5.2 Specifying alignment groups
                  </h4>
                  <p>To cause alignment, it is necessary to specify, within each
                     expression to be aligned, the points to be aligned with corresponding
                     points in other expressions, and the beginning of each <em>alignment
                        group</em> of sub-expressions that can be horizontally shifted as a
                     unit to effect the alignment. Each alignment group must contain one
                     alignment point. It is also necessary to specify which expressions in
                     the column have no alignment groups at all, but are affected only by
                     the ordinary column alignment for that column of the table, i.e. by
                     the <code>columnalign</code> attribute, described elsewhere.
                  </p>
                  <p>The alignment groups start at the locations of invisible
                     <code>maligngroup</code> elements, which are rendered with
                     zero width when they occur outside of an alignment scope, but within
                     an alignment scope are rendered with just enough horizontal space to
                     cause the desired alignment of the alignment group that follows
                     them. A simple algorithm by which a MathML application can achieve this is given
                     later. In the example above, each equation would have one
                     <code>maligngroup</code> element before each coefficient,
                     variable, and operator on the left-hand side, one before the
                     <code>=</code> sign, and one before the constant on the right-hand
                     side.
                  </p>
                  <p>In general, a table cell containing <var>n</var>
                     <code>maligngroup</code> elements contains <var>n</var>
                     alignment groups, with the <var>i</var>th group consisting of the
                     elements entirely after the <var>i</var>th
                     <code>maligngroup</code> element and before the
                     (<var>i</var>+1)-th; no element within the table cell's content
                     should occur entirely before its first
                     <code>maligngroup</code> element.
                  </p>
                  <p>Note that the division into alignment groups does <em>not</em>
                     necessarily fit the nested expression structure of the MathML
                     expression containing the groups — that is, it is permissible for one
                     alignment group to consist of the end of one
                     <code>mrow</code>, all of another one, and the beginning of a
                     third one, for example. This can be seen in the MathML markup for the
                     present example, given at the end of this section.
                  </p>
                  <p>The nested expression structure formed by <code>mrow</code>s
                     and other layout schemata should reflect the mathematical structure of the
                     expression, not the alignment-group structure, to make possible optimal
                     renderings and better automatic interpretations; see the discussion of
                     proper grouping in section <a href="chapter3-d.html#presm.mrow">Section&nbsp;3.3.1 Horizontally Group Sub-Expressions
                        <code>&lt;mrow&gt;</code></a>. Insertion of
                     alignment elements (or other space-like elements) should not alter the
                     correspondence between the structure of a MathML expression and the
                     structure of the mathematical expression it represents.
                  </p>
                  <p>Although alignment groups need not
                     coincide with the nested expression structure of layout schemata,
                     there are nonetheless restrictions on where an <code>maligngroup</code>
                     element is allowed within a table cell. The <code>maligngroup</code>
                     element may only be contained within elements (directly or indirectly) of the following types
                     (which are themselves contained in the table cell):
                     
                  </p>
                  <ul>
                     <li>
                        <p>an <code>mrow</code> element, including an inferred
                           <code>mrow</code> such as the one formed by a multi-child
                           <code>mtd</code> element, but excluding <code>mrow</code> which
                           contains a change of direction using the <code>dir</code> attribute;
                        </p>
                     </li>
                     <li>
                        <p>an <code>mstyle</code> element
                           , but excluding those which change direction
                           using the <code>dir</code> attribute;
                        </p>
                     </li>
                     <li>
                        <p>an <code>mphantom</code> element;
                        </p>
                     </li>
                     <li>
                        <p>an <code>mfenced</code> element;
                        </p>
                     </li>
                     <li>
                        <p>an <code>maction</code> element, though only its
                           selected sub-expression is checked;
                        </p>
                     </li>
                     <li>
                        <p>a <code>semantics</code> element.
                        </p>
                     </li>
                  </ul>
                  <p>These restrictions are intended to ensure that alignment can be
                     unambiguously specified, while avoiding complexities involving things
                     like overscripts, radical signs and fraction bars. They also ensure
                     that a simple algorithm suffices to accomplish the desired
                     alignment.
                  </p>
                  <p>Note that some positions for an <code>maligngroup</code>
                     element, although legal, are not useful, such as  an
                     <code>maligngroup</code> element that is an argument of an
                     <code>mfenced</code> element. Similarly, when inserting an <code>maligngroup</code>
                     element in an element whose arguments have positional significance, it
                     is necessary to introduce a new <code>mrow</code> element
                     containing just the <code>maligngroup</code> element and the child element
                     it precedes in order to preserve the proper expression structure.  For
                     example, to insert an <code>maligngroup</code> before the denominator
                     child of an <code>mfrac</code> element, it is necessary to enclose the
                     <code>maligngroup</code> and the denominator in an <code>mrow</code> to
                     avoid introducing an illegal third child in the
                     <code>mfrac</code>. In general, this will be necessary except
                     when the <code>maligngroup</code> element is inserted directly into an
                     <code>mrow</code> or into an element that can form an inferred
                     <code>mrow</code> from its contents.  See the warning about the legal
                     grouping of "space-like elements" in <a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a> for an analogous example 
                     involving <code>malignmark</code>.
                  </p>
                  <p>For the table cells that are divided into alignment groups, every
                     element in their content must be part of exactly one alignment group,
                     except for the elements from the above list that contain
                     <code>maligngroup</code> elements inside them and the
                     <code>maligngroup</code> elements themselves. This means
                     that, within any table cell containing alignment groups, the first
                     complete element must be an <code>maligngroup</code> element,
                     though this may be preceded by the start tags of other elements.
                  </p>
                  <p>This requirement removes a potential confusion about how to align
                     elements before the first <code>maligngroup</code> element,
                     and makes it easy to identify table cells that are left out of their
                     column's alignment process entirely.
                  </p>
                  <p>Note that it is not required that the table cells in a column that
                     are divided into alignment groups each contain the same number of
                     groups. If they don't, zero-width alignment groups are effectively
                     added on the right side (or left side, in a RTL context) of each table cell that has fewer groups than
                     other table cells in the same column.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.3" id="id.3.5.5.3"></a>3.5.5.3 Table cells that are not divided into alignment groups
                  </h4>
                  <p>Expressions in a column that are to have no alignment groups
                     should contain no <code>maligngroup</code>
                     elements. Expressions with no alignment groups are aligned using only
                     the <code>columnalign</code> attribute that applies to the table
                     column as a whole, and are not affected by the <code>groupalign</code>
                     attribute described below. If such an expression is wider than the
                     column width needed for the table cells containing alignment groups,
                     all the table cells containing alignment groups will be shifted as a
                     unit within the column as described by the <code>columnalign</code>
                     attribute for that column. For example, a column heading with no
                     internal alignment could be added to the column of two equations given
                     above by preceding them with another table row containing an
                     <code>mtext</code> element for the heading, and using the
                     default <code>columnalign</code>="center" for the table, to
                     produce:
                     
                  </p><pre>
equations with aligned variables
      8.44x + 55  y =  0
      3.1 x -  0.7y = -1.1
</pre><p>
                     or, with a shorter heading,
                     
                  </p><pre>
   some equations
8.44x + 55  y =  0
3.1 x -  0.7y = -1.1
</pre></div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.4" id="id.3.5.5.4"></a>3.5.5.4 Specifying alignment points using <code>&lt;malignmark/&gt;</code></h4>
                  <p>Each alignment group's alignment point can either be specified by
                     an <code>malignmark</code> element anywhere within the
                     alignment group (except within another alignment scope wholly
                     contained inside it), or it is determined automatically from the
                     <code>groupalign</code> attribute. The <code>groupalign</code>
                     attribute can be specified on the group's preceding
                     <code>maligngroup</code> element or on its surrounding
                     <code>mtd</code>, <code>mtr</code>, or
                     <code>mtable</code> elements. In typical cases, using the
                     <code>groupalign</code> attribute is sufficient to describe the
                     desired alignment points, so no <code>malignmark</code>
                     elements need to be provided.
                  </p>
                  <p>The <code>malignmark</code> element indicates that the
                     alignment point should occur on the right edge of the preceding
                     element, or the left edge of the following element or character,
                     depending on the <code>edge</code> attribute of
                     <code>malignmark</code>. Note that it may be necessary to
                     introduce an <code>mrow</code> to group an
                     <code>malignmark</code> element with a neighboring element,
                     in order not to alter the argument count of the containing
                     element. (See the warning about the legal grouping of "space-like
                     elements" in <a href="chapter3-d.html#presm.mspace">Section&nbsp;3.2.7 Space <code>&lt;mspace/&gt;</code></a>).
                  </p>
                  <p>When an <code>malignmark</code> element is provided within an
                     alignment group, it can occur in an arbitrarily deeply nested element
                     within the group, as long as it is not within a nested alignment scope. It
                     is not subject to the same restrictions on location as <code>maligngroup</code> elements. However, its immediate
                     surroundings need to be such that the element to its immediate right or
                     left (depending on its <code>edge</code> attribute) can be
                     unambiguously identified. If no such element is present, renderers should
                     behave as if a zero-width element had been inserted there.
                  </p>
                  <p>For the purposes of alignment, an element X is considered to be to the
                     immediate left of an element Y, and Y to the immediate right of X, whenever
                     X and Y are successive arguments of one (possibly inferred) <code>mrow</code> element,
                     with X coming before Y (in a LTR context; with X coming after Y in a RTL context). In the case of
                     <code>mfenced</code> elements, MathML applications should evaluate this
                     relation as if the <code>mfenced</code> element had been
                     replaced by the equivalent expanded form involving <code>mrow</code>. Similarly, an <code>maction</code>
                     element should be treated as if it were replaced by its currently selected
                     sub-expression. In all other cases, no relation of "to the immediate
                     left or right" is defined for two elements X and Y. However, in the
                     case of content elements interspersed in presentation markup, MathML applications
                     should attempt to evaluate this relation in a sensible way. For example, if
                     a renderer maintains an internal presentation structure for rendering
                     content elements, the relation could be evaluated with respect to
                     that. (See <a href="chapter4-d.html">Chapter&nbsp;4 Content Markup</a> and <a href="chapter5-d.html">Chapter&nbsp;5 Mixing Markup Languages for Mathematical Expressions</a> for further
                     details about mixing presentation and content markup.)
                  </p>
                  <p>
                     <code>malignmark</code> elements are allowed to occur within
                     the content of token elements, such as <code>mn</code>,
                     <code>mi</code>, or <code>mtext</code>. When this
                     occurs, the character immediately before or after the
                     <code>malignmark</code> element will carry the alignment
                     point; in all other cases, the element to its immediate left or right
                     will carry the alignment point. The rationale for this is that it is
                     sometimes desirable to align on the edges of specific characters
                     within multi-character token elements.
                  </p>
                  <p>If there is more than one <code>malignmark</code> element
                     in an alignment group, all but the first one will be ignored. MathML
                     applications may wish to provide a mode in which they will warn about
                     this situation, but it is not an error, and should trigger no warnings
                     by default. The rationale for this is that it would
                     be inconvenient to have to remove all 
                     unnecessary <code>malignmark</code> elements from
                     automatically generated data, in certain cases, such as when they are
                     used to specify alignment on "decimal points" other than the '.' 
                     character.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.5" id="id.3.5.5.5"></a>3.5.5.5 <code>&lt;malignmark/&gt;</code> Attributes
                  </h4>
                  <p><code>malignmark</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>
                     (however, neither <code>mathcolor</code> nor <code>mathbackground</code> have any effect).
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">edge</td>
                           <td>"left" | "right"</td>
                           <td>left</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              see the discussion below.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The
                     <code>edge</code> attribute specifies whether the alignment point will be
                     found on the left or right edge of some element or character. The
                     precise location meant by "left edge" or "right edge" is discussed
                     below. If <code>edge</code>="right", the alignment point is the right
                     edge of the element or character to the immediate left of the
                     <code>malignmark</code> element. If <code>edge</code>="left",
                     the alignment point is the left edge of the element or character to
                     the immediate right of the <code>malignmark</code>
                     element. Note that the attribute refers to the choice of edge rather
                     than to the direction in which to look for the element whose edge will
                     be used.
                  </p>
                  <p>For <code>malignmark</code> elements that occur within
                     the content of MathML token elements, the preceding or following
                     character in the token element's content is used; if there is no such
                     character, a zero-width character is effectively inserted for the
                     purpose of carrying the alignment point on its edge. For all other
                     <code>malignmark</code> elements, the preceding or following
                     element is used; if there is no such element, a zero-width element is
                     effectively inserted to carry the alignment point.
                  </p>
                  <p>The precise definition of the "left edge" or "right edge" of a
                     character or glyph (e.g. whether it should coincide with an edge of
                     the character's bounding box) is not specified by MathML, but is at
                     the discretion of the renderer; the renderer is allowed to let the
                     edge position depend on the character's context as well as on the
                     character itself.
                  </p>
                  <p>For proper alignment of columns of numbers (using <code>groupalign</code> values of "left", "right", or "decimalpoint"), it is
                     likely to be desirable for the effective width (i.e. the distance between
                     the left and right edges) of decimal digits to be constant, even if their
                     bounding box widths are not constant (e.g. if "1" is narrower
                     than other digits). For other characters, such as letters and operators, it
                     may be desirable for the aligned edges to coincide with the bounding
                     box.
                  </p>
                  <p>The "left edge" of a MathML element or alignment group
                     refers to the left edge of the leftmost glyph drawn to render the element
                     or group, except that explicit space represented by <code>mspace</code> or <code>mtext</code> elements
                     should also count as "glyphs" in this context, as should
                     glyphs that would be drawn if not for <code>mphantom</code>
                     elements around them. The "right edge" of an element or
                     alignment group is defined similarly.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.6" id="id.3.5.5.6"></a>3.5.5.6 <code>&lt;maligngroup/&gt;</code> Attributes
                  </h4>
                  <p><code>maligngroup</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>
                     (however, neither <code>mathcolor</code> nor <code>mathbackground</code> have any effect).
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">groupalign</td>
                           <td>"left" | "center" | "right" | "decimalpoint"</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              see the discussion below.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p><code>maligngroup</code> has one attribute,
                     <code>groupalign</code>, which is used to determine the position of
                     its group's alignment point when no <code>malignmark</code>
                     element is present. The following discussion assumes that no
                     <code>malignmark</code> element is found within a group.
                  </p>
                  <p>In the example given at the beginning of this section, there is one
                     column of 2 table cells, with 7 alignment groups in each table cell;
                     thus there are 7 columns of alignment groups, with 2 groups, one above
                     the other, in each column. These columns of alignment groups should be
                     given the 7 <code>groupalign</code> values "decimalpoint left left
                     decimalpoint left left decimalpoint", in that order. How to specify
                     this list of values for a table cell or table column as a whole, using
                     attributes on elements surrounding the
                     <code>maligngroup</code> element is described later.
                  </p>
                  <p>If <code>groupalign</code> is "left",
                     "right", or "center", the alignment point is
                     defined to be at the group's left edge, at its right edge, or halfway
                     between these edges, respectively. The meanings of "left edge"
                     and "right edge" are as discussed above in relation to <code>malignmark</code>.
                  </p>
                  <p>If <code>groupalign</code> is "decimalpoint", 
                     the alignment point is the right edge of the character immediately before the
                     left-most 'decimal point', i.e. matching the character specified by
                     the <code>decimalpoint</code> attribute of <code>mstyle</code> (default ".", U+002E)
                     in the first <code>mn</code> element found along
                     the alignment group's baseline. More precisely, the alignment group is
                     scanned recursively, depth-first, for the first <code>mn</code>
                     element, descending into all arguments of each element of the types
                     <code>mrow</code> (including inferred
                     <code>mrow</code>s), <code>mstyle</code>,
                     <code>mpadded</code>, <code>mphantom</code>, <code>menclose</code>,
                     <code>mfenced</code>, or <code>msqrt</code>,
                     descending into only the first argument of each "scripting" element
                     (<code>msub</code>, <code>msup</code>,
                     <code>msubsup</code>, <code>munder</code>,
                     <code>mover</code>, <code>munderover</code>,
                     <code>mmultiscripts</code>) or of each
                     <code>mroot</code> or <code>semantics</code> element,
                     descending into only the selected sub-expression of each
                     <code>maction</code> element, and skipping the content of all
                     other elements. The first <code>mn</code> so found always
                     contains the alignment point, which is the right edge of the last
                     character before the first decimal point in the content of the
                     <code>mn</code> element. If there is no decimal point in the
                     <code>mn</code> element, the alignment point is the right edge
                     of the last character in the content. If the decimal point is the
                     first character of the <code>mn</code> element's content, the
                     right edge of a zero-width character inserted before the decimal point
                     is used. If no <code>mn</code> element is found, the right
                     edge of the entire alignment group is used (as for
                     <code>groupalign</code>="right").
                  </p>
                  <p>In order to permit alignment on decimal points in
                     <code>cn</code> elements, a MathML application can convert a
                     content expression into a presentation expression that renders the
                     same way before searching for decimal points as described above.
                  </p>
                  <p>Characters other than "." can be used as
                     "decimal points" for alignment by using <code>mstyle</code>;
                     more arbitrary alignment points can chosen by embedding <code>malignmark</code> elements
                     within the <code>mn</code> token's content itself.
                  </p>
                  <p>For any of the <code>groupalign</code> values, if an explicit
                     <code>malignmark</code> element is present anywhere within
                     the group, the position it specifies (described earlier) overrides the
                     automatic determination of alignment point from the
                     <code>groupalign</code> value.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.7" id="id.3.5.5.7"></a>3.5.5.7 Inheritance of <code>groupalign</code> values
                  </h4>
                  <p>It is not usually necessary to put a <code>groupalign</code>
                     attribute on every <code>maligngroup</code> element. Since
                     this attribute is usually the same for every group in a column of
                     alignment groups to be aligned, it can be inherited from an attribute
                     on the <code>mtable</code> that was used to set up the
                     alignment scope as a whole, or from the <code>mtr</code> or
                     <code>mtd</code> elements surrounding the alignment group. It
                     is inherited via an "inheritance path" that proceeds from
                     <code>mtable</code> through successively contained
                     <code>mtr</code>, <code>mtd</code>, and
                     <code>maligngroup</code> elements. There is exactly one
                     element of each of these kinds in this path from an
                     <code>mtable</code> to any alignment group inside it.  In
                     general, the value of <code>groupalign</code> will be 
                     inherited by any given alignment group from the innermost element
                     that surrounds the alignment group and provides an explicit
                     setting for this attribute.  For example, if an
                     <code>mtable</code> element specifies values for <code>groupalign</code> and
                     a <code>maligngroup</code> element within the table also specifies an
                     explicit <code>groupalign</code> value, then then the value from the
                     <code>maligngroup</code> takes priority.
                  </p>
                  <p id="type.group-align">Note, however, that each <code>mtd</code> element needs, in
                     general, a list of <code>groupalign</code> values, one for each
                     <code>maligngroup</code> element inside it (from left to right, in an LTR context,
                     or from right to left in an RTL context), rather than just
                     a single value. Furthermore, an <code>mtr</code> or
                     <code>mtable</code> element needs, in general, a list of lists
                     of <code>groupalign</code> values, since it spans multiple
                     <code>mtable</code> columns, each potentially acting as an
                     alignment scope. Such lists of <em>group-alignment</em> values are specified
                     using the following syntax rules:
                     
                     
                  </p><pre>
group-alignment             = "left" | "right" | "center" | "decimalpoint"
group-alignment-list        = <em>group-alignment</em> +
group-alignment-list-list   = ( "{" <em>group-alignment-list</em> "}" ) +
</pre><p>As described in <a href="chapter2-d.html#fund.attval">Section&nbsp;2.1.5 MathML Attribute Values</a>, <code>|</code> separates
                     alternatives; <code>+</code> represents optional repetition (i.e. 1 or
                     more copies of what precedes it), with extra values ignored and the
                     last value repeated if necessary to cover additional table columns or
                     alignment group columns; <code>'{'</code> and <code>'}'</code>
                     represent literal braces; and <code>(</code> and <code>)</code> are
                     used for grouping, but do not literally appear in the attribute
                     value.
                  </p>
                  <p>The permissible values of the <code>groupalign</code> attribute of the
                     elements that have this attribute are specified using the above
                     syntax definitions as follows:
                     
                  </p>
                  <table border="1">
                     <thead>
                        <tr>
                           <th>Element type</th>
                           <th>groupalign attribute syntax</th>
                           <th>default value</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td><code>mtable</code></td>
                           <td><a href="chapter3-d.html#type.group-align"><em>group-alignment-list-list</em></a></td>
                           <td>{left}</td>
                        </tr>
                        <tr>
                           <td><code>mtr</code></td>
                           <td><a href="chapter3-d.html#type.group-align"><em>group-alignment-list-list</em></a></td>
                           <td><em>inherited from <code>mtable</code> attribute</em></td>
                        </tr>
                        <tr>
                           <td><code>mlabeledtr</code></td>
                           <td><a href="chapter3-d.html#type.group-align"><em>group-alignment-list-list</em></a></td>
                           <td><em>inherited from <code>mtable</code> attribute</em></td>
                        </tr>
                        <tr>
                           <td><code>mtd</code></td>
                           <td><a href="chapter3-d.html#type.group-align"><em>group-alignment-list</em></a></td>
                           <td><em>inherited from within <code>mtr</code> attribute</em></td>
                        </tr>
                        <tr>
                           <td><code>maligngroup</code></td>
                           <td><a href="chapter3-d.html#type.group-align"><em>group-alignment</em></a></td>
                           <td><em>inherited from within <code>mtd</code> attribute</em></td>
                        </tr>
                     </tbody>
                  </table>
                  <p>In the example near the beginning of this section, the group
                     alignment values could be specified on every <code>mtd</code>
                     element using <code>groupalign</code> = "decimalpoint left left
                     decimalpoint left left decimalpoint", or on every
                     <code>mtr</code> element using <code>groupalign</code> =
                     "{decimalpoint left left decimalpoint left left decimalpoint}", or
                     (most conveniently) on the <code>mtable</code> as a whole
                     using <code>groupalign</code> = "{decimalpoint left left decimalpoint
                     left left decimalpoint}", which provides a single braced list of
                     <a href="chapter3-d.html#type.group-align"><em>group-alignment</em></a> values for the single column of expressions to be
                     aligned.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.8" id="id.3.5.5.8"></a>3.5.5.8 MathML representation of an alignment example
                  </h4>
                  <p>The above rules are sufficient to explain the MathML representation
                     of the example given near the start of this section. 
                     To repeat the example, the desired rendering is:
                     
                  </p><pre>
    8.44x + 55  y =  0
    3.1 x -  0.7y = -1.1
</pre><p>One way to represent that in MathML is:
                     
                  </p><pre class="mathml">
&lt;mtable groupalign="{decimalpoint left left decimalpoint left left decimalpoint}"&gt;
  &lt;mtr&gt;
    &lt;mtd&gt;
      &lt;mrow&gt;
        &lt;mrow&gt;
          &lt;mrow&gt;
            &lt;maligngroup/&gt;
            &lt;mn&gt; 8.44 &lt;/mn&gt;
            &lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
            &lt;maligngroup/&gt;
            &lt;mi&gt; x &lt;/mi&gt;
          &lt;/mrow&gt;
          &lt;maligngroup/&gt;
          &lt;mo&gt; + &lt;/mo&gt;
          &lt;mrow&gt;
            &lt;maligngroup/&gt;
            &lt;mn&gt; 55 &lt;/mn&gt;
            &lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
            &lt;maligngroup/&gt;
            &lt;mi&gt; y &lt;/mi&gt;
          &lt;/mrow&gt;
        &lt;/mrow&gt;
      &lt;maligngroup/&gt;
      &lt;mo&gt; = &lt;/mo&gt;
      &lt;maligngroup/&gt;
      &lt;mn&gt; 0 &lt;/mn&gt;
    &lt;/mrow&gt;
    &lt;/mtd&gt;
  &lt;/mtr&gt;
  &lt;mtr&gt;
    &lt;mtd&gt;
      &lt;mrow&gt;
        &lt;mrow&gt;
          &lt;mrow&gt;
            &lt;maligngroup/&gt;
            &lt;mn&gt; 3.1 &lt;/mn&gt;
            &lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
            &lt;maligngroup/&gt;
            &lt;mi&gt; x &lt;/mi&gt;
          &lt;/mrow&gt;
          &lt;maligngroup/&gt;
          &lt;mo&gt; - &lt;/mo&gt;
          &lt;mrow&gt;
            &lt;maligngroup/&gt;
            &lt;mn&gt; 0.7 &lt;/mn&gt;
            &lt;mo&gt; &amp;#x2062;<span style="color:#999900">&lt;!--INVISIBLE TIMES--&gt;</span> &lt;/mo&gt;
            &lt;maligngroup/&gt;
            &lt;mi&gt; y &lt;/mi&gt;
          &lt;/mrow&gt;
        &lt;/mrow&gt;
        &lt;maligngroup/&gt;
        &lt;mo&gt; = &lt;/mo&gt;
        &lt;maligngroup/&gt;
        &lt;mrow&gt;
          &lt;mo&gt; - &lt;/mo&gt;
          &lt;mn&gt; 1.1 &lt;/mn&gt;
        &lt;/mrow&gt;
      &lt;/mrow&gt;
    &lt;/mtd&gt;
  &lt;/mtr&gt;
&lt;/mtable&gt;
</pre></div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.9" id="id.3.5.5.9"></a>3.5.5.9 Further details of alignment elements
                  </h4>
                  <p>The alignment elements <code>maligngroup</code> and
                     <code>malignmark</code> can occur outside of alignment
                     scopes, where they are ignored. The rationale behind this is that in
                     situations in which MathML is generated, or copied from another
                     document, without knowing whether it will be placed inside an
                     alignment scope, it would be inconvenient for this to be an error.
                  </p>
                  <p>An <code>mtable</code> element can be given the attribute <code>alignmentscope</code>="false" to cause its
                     columns not to act as alignment scopes. In general, this attribute has the
                     syntax <code>("true" | "false") +</code>; if its value is a list of Boolean
                     values, each Boolean value applies to one column, with the last value
                     repeated if necessary to cover additional columns, or with extra values
                     ignored. Columns that are not alignment scopes are part of the alignment
                     scope surrounding the <code>mtable</code> element, if there is
                     one. Use of <code>alignmentscope</code>="false" allows nested tables to contain <code>malignmark</code> elements for aligning the inner table in the
                     surrounding alignment scope.
                  </p>
                  <p>As discussed above, processing of alignment for content elements is
                     not well-defined, since MathML does not specify how content elements
                     should be rendered. However, many MathML applications are likely to find it
                     convenient to internally convert content elements to presentation
                     elements that render the same way. Thus, as a general rule, even if a
                     renderer does not perform such conversions internally, it is
                     recommended that the alignment elements should be processed as if it
                     did perform them.
                  </p>
                  <p>A particularly important case for renderers to handle gracefully is the
                     interaction of alignment elements with the <code>matrix</code>
                     content element, since this element may or may not be internally converted
                     to an expression containing an <code>mtable</code> element for
                     rendering. To partially resolve this ambiguity, it is suggested, but not
                     required, that if the <code>matrix</code> element is converted
                     to an expression involving an <code>mtable</code> element, that
                     the <code>mtable</code> element be given the attribute <code>alignmentscope</code>="false", which will
                     make the interaction of the <code>matrix</code> element with the
                     alignment elements no different than that of a generic presentation element
                     (in particular, it will allow it to contain <code>malignmark</code> elements that operate within the alignment
                     scopes created by the columns of an <code>mtable</code> that
                     contains the <code>matrix</code> element in one of its table
                     cells).
                  </p>
                  <p>The effect of alignment elements within table cells that have
                     non-default values of the <code>columnspan</code> or <code>rowspan</code> attributes is not specified, except that such
                     use of alignment elements is not an error. Future versions of MathML may
                     specify the behavior of alignment elements in such table cells.
                  </p>
                  <p>The effect of possible linebreaking of an <code>mtable</code>
                     element on the alignment elements is not specified.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.5.5.10" id="id.3.5.5.10"></a>3.5.5.10 A simple alignment algorithm
                  </h4>
                  <p>A simple algorithm by which a MathML renderer can perform the
                     alignment specified in this section is given here. Since the alignment
                     specification is deterministic (except for the definition of the left
                     and right edges of a character), any correct MathML alignment
                     algorithm will have the same behavior as this one. Each
                     <code>mtable</code> column (alignment scope) can be treated
                     independently; the algorithm given here applies to one
                     <code>mtable</code> column, and takes into account the
                     alignment elements, the <code>groupalign</code> attribute described in
                     this section, and the <code>columnalign</code> attribute described
                     under <code>mtable</code> (<a href="chapter3-d.html#presm.mtable">Section&nbsp;3.5.1 Table or Matrix
                        <code>&lt;mtable&gt;</code></a>).
                  </p>
                  <p>First, a rendering is computed for the contents of each table cell
                     in the column, using zero width for all
                     <code>maligngroup</code> and <code>malignmark</code>
                     elements. The final rendering will be identical except for horizontal
                     shifts applied to each alignment group and/or table cell. The
                     positions of alignment points specified by any
                     <code>malignmark</code> elements are noted, and the remaining
                     alignment points are determined using <code>groupalign</code>
                     values.
                  </p>
                  <p>For each alignment group, the horizontal positions of the left
                     edge, alignment point, and right edge are noted, allowing the width of
                     the group on each side of the alignment point (left and right) to be
                     determined. The sum of these two "side-widths", i.e. the sum of the
                     widths to the left and right of the alignment point, will equal the
                     width of the alignment group.
                  </p>
                  <p>Second, each column of alignment groups is
                     scanned. The <var>i</var>th scan covers the <var>i</var>th
                     alignment group in each table cell containing any alignment
                     groups. Table cells with no alignment groups, or with fewer than
                     <var>i</var> alignment groups, are ignored. Each scan computes two
                     maximums over the alignment groups scanned: the maximum width to the
                     left of the alignment point, and the maximum width to the right of the
                     alignment point, of any alignment group scanned.
                  </p>
                  <p>The sum of all the maximum widths computed (two for each column of
                     alignment groups) gives one total width, which will be the width of
                     each table cell containing alignment groups. Call the maximum number
                     of alignment groups in one cell <var>n</var>; each such cell
                     is divided into 2<var>n</var> horizontally adjacent sections, called
                     L(<var>i</var>) and R(<var>i</var>) for <var>i</var> from 1 to
                     <var>n</var>, using the 2<var>n</var> maximum side-widths computed
                     above; for each <var>i</var>, the width of all sections called
                     L(<var>i</var>) is the maximum width of any cell's <var>i</var>th
                     alignment group to the left of its alignment point, and the width of
                     all sections called R(<var>i</var>) is the maximum width of any
                     cell's <var>i</var>th alignment group to the right of its alignment
                     point.
                  </p>
                  <p>Each alignment group is then shifted horizontally as a block
                     to a unique position that places: in the section called L(<var>i</var>) that part
                     of the <var>i</var>th group to the left of its alignment point;
                     in the section called R(<var>i</var>) that part of the <var>i</var>th group
                     to the right of its alignment point. This results in the
                     alignment point of each <var>i</var>th group being on the boundary
                     between adjacent sections L(<var>i</var>) and R(<var>i</var>), so
                     that all alignment points of <var>i</var>th groups have the same
                     horizontal position.
                  </p>
                  <p>The widths of the table cells that contain no alignment groups
                     were computed as part of the initial rendering, and may be different
                     for each cell, and different from the single width used for cells
                     containing alignment groups. The maximum of all the cell widths (for
                     both kinds of cells) gives the width of the table column as a
                     whole.
                  </p>
                  <p>The position of each cell in the column is determined by the
                     applicable part of the value of the <code>columnalign</code> attribute
                     of the innermost surrounding <code>mtable</code>,
                     <code>mtr</code>, or <code>mtd</code> element that
                     has an explicit value for it, as described in the sections on those
                     elements. This may mean that the cells containing alignment groups
                     will be shifted within their column, in addition to their alignment
                     groups having been shifted within the cells as described above, but
                     since each such cell has the same width, it will be shifted the same
                     amount within the column, thus maintaining the vertical alignment of
                     the alignment points of the corresponding alignment groups in each
                     cell.
                  </p>
               </div>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="presm.elementary" id="presm.elementary"></a>3.6 Elementary Math
            </h2>
            <p>Mathematics used in the lower grades such as two-dimensional addition, multiplication,
               and long division tends to be tabular in nature. 
               However, the specific notations used varies among countries
               much more than for higher level math.
               Furthermore, elementary math often presents examples in some intermediate state
               and MathML must be able to capture these intermediate or intentionally missing
               partial forms. Indeed, these constructs represent memory aids or
               procedural guides, as much as they represent ‘mathematics’.
            </p>
            <p>
               The elements used for basic alignments in elementary math are:
               
            </p>
            <dl>
               <dt class="label"><code>mstack</code></dt>
               <dd>align rows of digits and operators</dd>
               <dt class="label"><code>msgroup</code></dt>
               <dd>groups rows with similar alignment</dd>
               <dt class="label"><code>msrow</code></dt>
               <dd>groups digits and operators into a row</dd>
               <dt class="label"><code>msline</code></dt>
               <dd>draws lines between rows of the stack</dd>
               <dt class="label"><code>mscarries</code></dt>
               <dd>annotates the following row with optional borrows/carries and/or crossouts</dd>
               <dt class="label"><code>mscarry</code></dt>
               <dd>a borrow/carry and/or crossout for a single digit</dd>
               <dt class="label"><code>mlongdiv</code></dt>
               <dd>specifies a divisor and a quotient for long division, along with a stack of the intermediate computations</dd>
            </dl>
            <p>
               
               <code>mstack</code> and <code>mlongdiv</code> are the parent elements for all elementary
               math layout.
               Any children of <code>mstack</code>, <code>mlongdiv</code>, and <code>msgroup</code>,
               besides <code>msrow</code>, <code>msgroup</code>, <code>mscarries</code> and <code>msline</code>,
               are treated as if implicitly surrounded by an <code>msrow</code>
               (See <a href="chapter3-d.html#presm.msrow">Section&nbsp;3.6.4 Rows in Elementary Math <code>&lt;msrow&gt;</code></a> for more details about rows).
               
            </p>
            <p>Since the primary use of these stacking constructs is to 
               stack rows of numbers aligned on their digits,
               and since numbers are always formatted left-to-right,
               the columns of an mstack are always processed left-to-right;
               the overall directionality in effect (ie. the <code>dir</code> attribute)
               does not affect to the ordering of display of columns or carries in rows
               and, in particular, does not affect the ordering of any operators within a row
               (See <a href="chapter3-d.html#presm.bidi">Section&nbsp;3.1.5 Directionality</a>).
               
            </p>
            <p>
               These elements are described in this section followed by examples of their use.
               In addition to two-dimensional addition, subtraction, multiplication, and long division,
               these elements can be used to represent several notations used for repeating decimals.
            </p>
            <p>A very simple example of two-dimensional addition is shown below:</p>
            <blockquote>
               <p><img src="image/em-plus.png" alt="\begin{array}{r}   424 \\   +33 \\   \hline \end{array}"></p>
            </blockquote>
            <p>The MathML for this is:</p><pre class="mathml">
&lt;mstack&gt;
  &lt;mn&gt;424&lt;/mn&gt;
  &lt;msrow&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mn&gt;33&lt;/mn&gt; &lt;/msrow&gt;
  &lt;msline/&gt;
&lt;/mstack&gt;
</pre><p> Many more examples are given in <a href="chapter3-d.html#presm.elemmath.examples">Section&nbsp;3.6.8 Elementary Math Examples</a>.
            </p>
            <div class="div3">
               
               <h3><a name="presm.mstack" id="presm.mstack"></a>3.6.1 Stacks of Characters <code>&lt;mstack&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.6.1.1" id="id.3.6.1.1"></a>3.6.1.1 Description
                  </h4>
                  <p><code>mstack</code> is used to lay out rows of numbers that are aligned on each digit.
                     This is common in many elementary math notations such as 2D addition, subtraction, and multiplication.
                  </p>
                  <p>The children of an <code>mstack</code> represent rows, or groups of them,
                     to be stacked each below the previous row; there can be any number of rows.
                     An <code>msrow</code> represents a row;
                     an <code>msgroup</code> groups a set of rows together
                     so that their horizontal alignment can be adjusted together;
                     an <code>mscarries</code> represents a set of carries to be
                     applied to the following row;
                     an <code>msline</code> represents a line separating rows.
                     Any other element is treated as if implicitly surrounded by <code>msrow</code>.
                     
                  </p>
                  <p>Each row contains ‘digits’ that are placed into columns.
                     (see <a href="chapter3-d.html#presm.msrow">Section&nbsp;3.6.4 Rows in Elementary Math <code>&lt;msrow&gt;</code></a> for further details).
                     The <code>stackalign</code> attribute together with
                     the <code>position</code> and <code>shift</code> attributes of <code>msgroup</code>,
                     <code>mscarries</code>, and <code>msrow</code> determine
                     to which column a character belongs.
                  </p>
                  <p>The width of a column is the maximum of the widths of each ‘digit’ in that
                     column — carries do <em>not</em> participate in the
                     width calculation; they are treated as having zero width.
                     If an element is too wide to fit into a column, it overflows into the adjacent
                     column(s) as determined by the <code>charalign</code> attribute.
                     If there is no character in a column, its width is taken to be the width of a 0 
                     in the current language (in many fonts, all digits have the same width).
                     
                  </p>
                  <p>The method for laying out an mstack is:
                     
                  </p>
                  <ol type="1">
                     <li>
                        <p>The ‘digits’ in a row are determined.</p>
                     </li>
                     <li>
                        <p>All of the digits in a row are initially aligned according to the <code>stackalign</code> value.
                        </p>
                     </li>
                     <li>
                        <p>Each row is positioned relative to that alignment based on the <code>position</code> attribute (if any)
                           that controls that row.
                        </p>
                     </li>
                     <li>
                        <p>The maximum width of the digits in a column are determined and
                           shorter and wider entries in that column are aligned according to
                           the <code>charalign</code> attribute.
                        </p>
                     </li>
                     <li>
                        <p>The width and height of the mstack element are computed based on the
                           rows and columns.
                           Any overflow from a column is <em>not</em> used as part of that computation.
                           
                        </p>
                     </li>
                     <li>
                        <p>The baseline of the mstack element is determined by the <code>align</code> attribute.
                        </p>
                     </li>
                  </ol>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.6.1.2" id="id.3.6.1.2"></a>3.6.1.2 Attributes
                  </h4>
                  <p><code>mstack</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">align</td>
                           <td>("top" | "bottom" | "center" | "baseline" | "axis"), <em>rownumber</em>?
                           </td>
                           <td>baseline</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the vertical alignment of the <code>mstack</code> with respect to its environment.
                              The legal values and their meanings are the same as that for <code>mtable</code>'s
                              <a href="chapter3-d.html#presm.mtable.attrs"><code>align</code></a> attribute.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">stackalign</td>
                           <td>"left" | "center" | "right" | "decimalpoint"</td>
                           <td>decimalpoint</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies which column is used to horizontally align the rows.
                              For "left", rows are aligned flush on the left;
                              similarly for "right", rows are flush on the right;
                              for "center", the middle column (or to the right of the middle, for an even number of columns)
                              is used for alignment.
                              Rows with non-zero <code>position</code>, or affected by a <code>shift</code>,
                              are treated as if the
                              requisite number of empty columns were added on the appropriate side;
                              see <a href="chapter3-d.html#presm.msgroup">Section&nbsp;3.6.3 Group Rows with Similiar Positions <code>&lt;msgroup&gt;</code></a> and <a href="chapter3-d.html#presm.msrow">Section&nbsp;3.6.4 Rows in Elementary Math <code>&lt;msrow&gt;</code></a>.
                              For "decimalpoint", the column used is the left-most column in each
                              row that contains the decimalpoint character specified
                              using the <code>decimalpoint</code> attribute of <code>mstyle</code> (default ".").
                              If there is no decimalpoint character in the row, an implied decimal is assumed on the right of the first number in the row;
                              See <a href="chapter3-d.html#presm.mstyle">"decimalpoint"</a> for a discussion
                              of "decimalpoint".
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">charalign</td>
                           <td>"left" | "center" | "right" </td>
                           <td>right</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the horizontal alignment of digits within a column.
                              If the content is larger than the column width, then it overflows the opposite side from the alignment.
                              For example, for "right", the content will overflow on the left side; for center,
                              it overflows on both sides.
                              This excess does not participate in the column width calculation, nor does it participate
                              in the overall width of the <code>mstack</code>.
                              In these cases, authors should take care to avoid collisions between column overflows.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">charspacing</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a> | "loose" | "medium" | "tight"
                           </td>
                           <td>medium</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the amount of space to put between each column.
                              Larger spacing might be useful if carries are not placed above or are particularly wide.
                              The keywords "loose", "medium", and "tight" automatically adjust spacing to when carries or other entries in a column are
                              wide. The three values allow authors to some flexibility in choosing what the layout looks like without having to figure out
                              what values works well.  In all cases, the spacing between columns is a fixed amount and does not vary between different columns.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mlongdiv" id="presm.mlongdiv"></a>3.6.2 Long Division <code>&lt;mlongdiv&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.6.2.1" id="id.3.6.2.1"></a>3.6.2.1 Description
                  </h4>
                  <p>Long division notation varies quite a bit around the world, although the heart of the notation is often similar.
                     <code>mlongdiv</code> is similar to <code>mstack</code> and used to layout long division.
                     The first two children of <code>mlongdiv</code> are the divisor and the result of the division, in that order.
                     The remaining children are treated as if they were children of <code>mstack</code>.
                     The placement of these and the lines and separators used to display long division are controlled
                     by the <code>longdivstyle</code> attribute. 
                  </p>
                  <p>The result or divisor may be an elementary math element or may be <code>none</code>.  In particular, if <code>msgroup</code> is used, the elements in that group may or may not form their own mstack or be part of the dividend's <code>mstack</code>, depending upon the value of the <code>longdivstyle</code> attribute.
                     For example, in the US style for division, the result is treated as part of the dividend's <code>mstack</code>, but divisor is not.
                     MathML does not specify when the result and divisor form their own <code>mstack</code>, nor does it specify what should happen if <code>msline</code> or other elementary math elements are used for the result or divisor and they do not participate in the dividend's <code>mstack</code> layout.
                  </p>
                  <p>In the remainder of this section on elementary math, anything that is said about <code>mstack</code> applies
                     to <code>mlongdiv</code> unless stated otherwise.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="presm.mlongdiv.attrs" id="presm.mlongdiv.attrs"></a>3.6.2.2 Attributes
                  </h4>
                  <p><code>mlongdiv</code> elements accept all of the attributes that <code>mstack</code> elements
                     accept (including those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>), along with the attribute listed below.
                  </p>
                  <p>The values allowed for <code>longdivstyle</code> are open-ended. Conforming renderers may ignore any value they do not handle, although renderers are encouraged to render
                     as many of the values listed below as possible.
                     Any rules drawn as part of division layout should be drawn using the color specified by <code>mathcolor</code>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">longdivstyle</td>
                           <td> "lefttop" | "stackedrightright" | "mediumstackedrightright" | "shortstackedrightright" | "righttop" | "left/\right" | "left)(right"
                              | ":right=right" | "stackedleftleft" | "stackedleftlinetop" 
                           </td>
                           <td>lefttop</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              		  Controls the style of the long division layout.  The names are meant as a rough mnemonic that describes the position of
                              the divisor and result in relation to the dividend. 		  
                              	   
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>See <a href="chapter3-d.html#presm.mlongdiv.ex">Section&nbsp;3.6.8.3 Long Division</a> for examples of how these notations are drawn.  The values listed above are used for long division notations in different
                     countries around the world:
                  </p>
                  <dl>
                     <dt class="label">"lefttop"</dt>
                     <dd>a notation that is commonly used in the United States, Great Britain, and elsewhere</dd>
                     <dt class="label">"stackedrightright"</dt>
                     <dd>a notation that is commonly used in France and elsewhere</dd>
                     <dt class="label">"mediumrightright"</dt>
                     <dd>a notation that is commonly used in Russia and elsewhere</dd>
                     <dt class="label">"shortstackedrightright"</dt>
                     <dd>a notation that is commonly used in Brazil and elsewhere</dd>
                     <dt class="label">"righttop"</dt>
                     <dd>a notation that is commonly used in China, Sweden, and elsewhere</dd>
                     <dt class="label">"left/\right"</dt>
                     <dd>a notation that is commonly used in Netherlands</dd>
                     <dt class="label">"left)(right"</dt>
                     <dd>a notation that is commonly used in India</dd>
                     <dt class="label">":right=right "</dt>
                     <dd>a notation that is commonly used in Germany</dd>
                     <dt class="label">"stackedleftleft "</dt>
                     <dd>a notation that is commonly used in Arabic countries</dd>
                     <dt class="label">"stackedleftlinetop"</dt>
                     <dd>a notation that is commonly used in Arabic countries</dd>
                  </dl>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.msgroup" id="presm.msgroup"></a>3.6.3 Group Rows with Similiar Positions <code>&lt;msgroup&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.6.3.1" id="id.3.6.3.1"></a>3.6.3.1 Description
                  </h4>
                  <p><code>msgroup</code> is used to group rows inside of the <code>mstack</code> and <code>mlongdiv</code> elements
                     that have a similar position relative to the alignment of stack.
                     If not explicitly given, the children representing the stack in <code>mstack</code> and <code>mlongdiv</code>
                     are treated as if they are implicitly surrounded by an <code>msgroup</code> element.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.6.3.2" id="id.3.6.3.2"></a>3.6.3.2 Attributes
                  </h4>
                  <p><code>msgroup</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">position</td>
                           <td><a href="chapter2-d.html#type.integer"><em>integer</em></a></td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the horizontal position of the rows within this group relative
                              the position determined by the containing <code>msgroup</code> (according to its
                              <code>position</code> and <code>shift</code> attributes).
                              The resulting position value is relative to the column specified by <code>stackalign</code> of the containing <code>mstack</code> or <code>mlongdiv</code>.
                              Positive values move each row towards the tens digit,
                              like multiplying by a power of 10,
                              effectively padding with empty columns on the right;
                              negative values move towards the ones digit,
                              effectively padding on the left.
                              The decimal point is counted as a column and should be taken into account for negative values.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">shift</td>
                           <td><a href="chapter2-d.html#type.integer"><em>integer</em></a></td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies an incremental shift of position for successive children (rows or groups)
                              within this group. The value is interpreted as with position, but specifies the
                              position of each child (except the first) with respect to the previous child in the group.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.msrow" id="presm.msrow"></a>3.6.4 Rows in Elementary Math <code>&lt;msrow&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.6.4.1" id="id.3.6.4.1"></a>3.6.4.1 Description
                  </h4>
                  <p>An <code>msrow</code> represents a row in an <code>mstack</code>.
                     In most cases it is implied by the context, but is useful
                     explicitly for putting multiple elements in a single row,
                     such as when placing an operator "+" or "-" alongside a number
                     within an addition or subtraction.
                  </p>
                  <p>If an <code>mn</code> element is a child of <code>msrow</code>
                     (whether implicit or not), then the number is split into its digits
                     and the digits are placed into successive columns.
                     Any other element, with the exception of <code>mstyle</code> is treated effectively
                     as a single digit occupying the next column.
                     An <code>mstyle</code> is treated as if its children were 
                     directly the children of the <code>msrow</code>, but with their style affected
                     by the attributes of the <code>mstyle</code>.
                     The empty element <code>none</code> may be used to create an empty column.
                     
                  </p>
                  <p>Note that a row is considered primarily as if it were a number,
                     which are always displayed left-to-right,
                     and so the directionality used to display the columns is always left-to-right;
                     textual bidirectionality within token elements (other than <code>mn</code>) still applies,
                     as does the overall directionality <em>within</em> any children of the <code>msrow</code>
                     (which end up treated as single digits);
                     see <a href="chapter3-d.html#presm.bidi">Section&nbsp;3.1.5 Directionality</a>.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.6.4.2" id="id.3.6.4.2"></a>3.6.4.2 Attributes
                  </h4>
                  <p><code>msrow</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">position</td>
                           <td><a href="chapter2-d.html#type.integer"><em>integer</em></a></td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the horizontal position of the rows within this group relative
                              the position determined by the containing <code>msgroup</code> (according to its
                              <code>position</code> and <code>shift</code> attributes).
                              The resulting position value is relative to the column specified by <code>stackalign</code> of the containing <code>mstack</code> or <code>mlongdiv</code>.
                              Positive values move each row towards the tens digit,
                              like multiplying by a power of 10,
                              effectively padding with empty columns on the right;
                              negative values move towards the ones digit,
                              effectively padding on the left.
                              The decimal point is counted as a column and should be taken into account for negative values.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mscarries" id="presm.mscarries"></a>3.6.5 Carries, Borrows, and Crossouts <code>&lt;mscarries&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.6.5.1" id="id.3.6.5.1"></a>3.6.5.1 Description
                  </h4>
                  <p>The <code>mscarries</code> element is used for various annotations such as carries, borrows, and crossouts that occur in elementary math.
                     The children are associated with elements in the <em>following</em> row of the <code>mstack</code>.
                     It is an error for <code>mscarries</code> to be the last element of an <code>mstack</code> or <code>mlongdiv</code> element. Each child of the <code>mscarries</code> applies to the same column in the following row.
                     As these annotations are used to adorn what are treated as
                     numbers, the attachment of carries to columns proceeds from left-to-right;
                     The overall directionality does not apply to the ordering of the carries,
                     although it may apply to the contents of each carry;
                     see <a href="chapter3-d.html#presm.bidi">Section&nbsp;3.1.5 Directionality</a>.
                     
                  </p>
                  <p>
                     Each child of <code>mscarries</code> other than <code>mscarry</code> or <code>none</code> is
                     treated as if implicitly surrounded by <code>mscarry</code>;
                     the element <code>none</code> is used when no carry for a particular column is needed.
                     The <code>mscarries</code> element sets <code>displaystyle</code> to "false", and increments <code>scriptlevel</code> by 1, so the children are 
                     typically displayed in a smaller font. (See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a>.)
                     It also changes the default value of <code>scriptsizemultiplier</code>.
                     The effect is that the inherited value of
                     <code>scriptsizemultiplier</code> should still override the default value,
                     but the default value, inside <code>mscarries</code>, should be "0.6".
                     <code>scriptsizemultiplier</code> can be set on the <code>mscarries</code> element,
                     and the value should override the inherited value as usual.
                     
                  </p>
                  <p>
                     If two rows of carries are adjacent to each other, the first row of carries annotates the second (following) row as if the
                     second row had <code>location</code>="n". This means that the second row, even if it does not draw, visually uses some (undefined by this specification) amount
                     of space when displayed.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.6.5.2" id="id.3.6.5.2"></a>3.6.5.2 Attributes
                  </h4>
                  <p><code>mscarries</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">position</td>
                           <td><a href="chapter2-d.html#type.integer"><em>integer</em></a></td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the horizontal position of the rows within this group relative
                              the position determined by the containing <code>msgroup</code> (according to its
                              <code>position</code> and <code>shift</code> attributes).
                              The resulting position value is relative to the column specified by <code>stackalign</code> of the containing <code>mstack</code> or <code>mlongdiv</code>.
                              The interpretation of the value is the same as <code>position</code> for <code>msgroup</code> or <code>msrow</code>,
                              but it alters the association of each carry with the column below.
                              For example, <code>position</code>=1 would cause the rightmost carry to be associated with
                              the second digit column from the right.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">location</td>
                           <td>"w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw" </td>
                           <td>n</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the location of the carry or borrow relative to the character below it in the associated column.
                              Compass directions are used for the values; the default is to place the carry above the character.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">crossout</td>
                           <td>("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")*  </td>
                           <td>none</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies how the column content below each carry is "crossed out";
                              one or more values may be given and all values are drawn.
                              If "none" is given with other values, it is ignored.
                              See <a href="chapter3-d.html#presm.elemmath.examples">Section&nbsp;3.6.8 Elementary Math Examples</a> for examples of the different values.
                              The crossout is only applied for columns which have a corresponding
                              <code>mscarry</code>.
                              The crossouts should be drawn using the color specified by <code>mathcolor</code>.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">scriptsizemultiplier</td>
                           <td> <a href="chapter2-d.html#type.number"><em>number</em></a></td>
                           <td><em>inherited (0.6)</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the factor to change the font size by.
                              See <a href="chapter3-d.html#presm.scriptlevel">Section&nbsp;3.1.6 Displaystyle and Scriptlevel</a> for a description of how this works with the <code>scriptsize</code> attribute.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.mscarry" id="presm.mscarry"></a>3.6.6 A Single Carry <code>&lt;mscarry&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.6.6.1" id="id.3.6.6.1"></a>3.6.6.1 Description
                  </h4>
                  <p><code>mscarry</code> is used inside of <code>mscarries</code> to
                     represent the carry for an individual column.
                     A carry is treated as if its width were zero; it does not participate in
                     the calculation of the width of its corresponding column;
                     as such, it may extend beyond the column boundaries.
                     Although it is usually implied, the element may be used explicitly to override the
                     <code>location</code> and/or <code>crossout</code> attributes of 
                     the containing <code>mscarries</code>.
                     It may also be useful with <code>none</code> as its content in order
                     to display no actual carry, but still enable a <code>crossout</code>
                     due to the enclosing <code>mscarries</code> to be drawn for the given column.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.6.6.2" id="id.3.6.6.2"></a>3.6.6.2 Attributes
                  </h4>
                  <p>The <code>mscarry</code> element accepts the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">location</td>
                           <td>"w" | "nw" | "n" | "ne" | "e" | "se" | "s" | "sw"</td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the location of the carry or borrow relative to the character in the corresponding column in the row below it.
                              Compass directions are used for the values.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">crossout</td>
                           <td>("none" | "updiagonalstrike" | "downdiagonalstrike" | "verticalstrike" | "horizontalstrike")*  </td>
                           <td><em>inherited</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies how the column content associated with the carry is "crossed out";
                              one or more values may be given and all values are drawn.
                              If "none" is given with other values, it is essentially ignored.
                              The crossout should be drawn using the color specified by <code>mathcolor</code>.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.msline" id="presm.msline"></a>3.6.7 Horizontal Line <code>&lt;msline/&gt;</code></h3>
               <div class="div4">
                  
                  <h4><a name="id.3.6.7.1" id="id.3.6.7.1"></a>3.6.7.1 Description
                  </h4>
                  <p><code>msline</code> draws a horizontal line inside of a <code>mstack</code> element.
                     The position, length, and thickness of the line are specified as attributes.
                     If the length is specified, the line is positioned and drawn as if it were a number with the given number of digits.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.3.6.7.2" id="id.3.6.7.2"></a>3.6.7.2 Attributes
                  </h4>
                  <p><code>msline</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                     The line should be drawn using the color specified by <code>mathcolor</code>.
                     
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">position</td>
                           <td><a href="chapter2-d.html#type.integer"><em>integer</em></a></td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              specifies the horizontal position of the rows within this group relative
                              the position determined by the containing <code>msgroup</code> (according to its
                              <code>position</code> and <code>shift</code> attributes).
                              The resulting position value is relative to the column specified by <code>stackalign</code> of the containing <code>mstack</code> or <code>mlongdiv</code>.
                              Positive values moves towards the tens digit (like multiplying by a power of 10);
                              negative values moves towards the ones digit.
                              The decimal point is counted as a column and should be taken into account for negative values.
                              Note that since the default line length spans the entire <code>mstack</code>,
                              the position has no effect unless the <code>length</code> is specified as non-zero.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">length</td>
                           <td><a href="chapter2-d.html#type.unsigned-integer"><em>unsigned-integer</em></a></td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies the the number of columns that should be spanned by the line.
                              A value of '0' (the default) means that <em>all</em> columns in
                              the row are spanned (in which case <code>position</code> and <code>stackalign</code> have no effect).
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">leftoverhang</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies an extra amount that the line should overhang on the left of the leftmost column spanned by the line.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">rightoverhang</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a></td>
                           <td>0</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies an extra amount that the line should overhang on the right of the rightmost column spanned by the line.
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">mslinethickness</td>
                           <td><a href="chapter2-d.html#type.length"><em>length</em></a> | "thin" | "medium" | "thick"
                           </td>
                           <td>medium</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies how thick the line should be drawn.
                              The line should have height=0, and depth=mslinethickness so that the top
                              of the <code>msline</code> is on the baseline of the surrounding context (if any).
                              (See <a href="chapter3-d.html#presm.mfrac">Section&nbsp;3.3.2 Fractions <code>&lt;mfrac&gt;</code></a> for discussion of the thickness keywords
                              "medium", "thin" and "thick".)
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="presm.elemmath.examples" id="presm.elemmath.examples"></a>3.6.8 Elementary Math Examples
               </h3>
               <div class="div4">
                  
                  <h4><a name="presm.addsub" id="presm.addsub"></a>3.6.8.1 Addition and Subtraction
                  </h4>
                  <p>Two-dimensional addition, subtraction, and multiplication typically
                     involve numbers, carrries/borrows, lines, and the sign of the
                     operation.
                  </p>
                  <p>Notice that the <code>msline</code> spans all of the columns and that <code>none</code> is used to make the "+" appear to the left of all of the operands.
                  </p>
                  <blockquote>
                     <p><img src="image/em-plus-shift.png" alt="\begin{array}{@{}r@{}}    424 \\   +\phantom0 33 \\   \hline \end{array}"></p>
                  </blockquote>
                  <p>The MathML for this is:</p><pre class="mathml">
&lt;mstack&gt;
  &lt;mn&gt;424&lt;/mn&gt;
  &lt;msrow&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;none/&gt; &lt;mn&gt;33&lt;/mn&gt; &lt;/msrow&gt;
  &lt;msline/&gt;
&lt;/mstack&gt;
</pre><p>Here is an example with the operator on the right. Placing the
                     operator on the right is standard in the Netherlands and some other countries.
                     Notice that although there are a total of four columns in the example,
                     because the default alignment in on the implied decimal point to the right of the numbers,
                     it is not necessary to pad any row.
                  </p>
                  <blockquote>
                     <p><img src="image/em-plus-right.png" alt="\begin{array}{l}   123 \\   456+ \\   \hline   579 \end{array}"></p>
                  </blockquote><pre class="mathml">
&lt;mstack&gt;
  &lt;mn&gt;123&lt;/mn&gt; 
  &lt;msrow&gt; &lt;mn&gt;456&lt;/mn&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;/msrow&gt;
  &lt;msline/&gt;
  &lt;mn&gt;579&lt;/mn&gt;
&lt;/mstack&gt;
</pre><p>Because the default alignment is placed to the right of number, the
                     numbers align properly and none of the rows need to be shifted.
                  </p>
                  <p>The following two examples illustrate the use of <code>mscarries</code>,
                     <code>mscarry</code> and using <code>none</code> to fill in a column.
                     The examples illustrate two different ways of displaying a borrow.
                  </p>
                  <table>
                     <tbody>
                        <tr>
                           <td>
                              <blockquote>
                                 <p><img src="image/em-sub-borrow-above.png" alt=""></p>
                              </blockquote>
                           </td>
                           <td>
                              <blockquote>
                                 <p><img src="image/em-sub-borrow-aboveleft.png" alt=""></p>
                              </blockquote>
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The MathML for the first example is:</p><pre class="mathml">
&lt;mstack&gt;
  &lt;mscarries crossout='updiagonalstrike'&gt;
    &lt;mn&gt;2&lt;/mn&gt;  &lt;mn&gt;12&lt;/mn&gt;  &lt;mscarry crossout='none'&gt; &lt;none/&gt; &lt;/mscarry&gt;
  &lt;/mscarries&gt;
  &lt;mn&gt;2,327&lt;/mn&gt;
  &lt;msrow&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mn&gt; 1,156&lt;/mn&gt; &lt;/msrow&gt;
  &lt;msline/&gt;
  &lt;mn&gt;1,171&lt;/mn&gt;
&lt;/mstack&gt;
</pre><p>The MathML for the second example uses <code>mscarry</code> because a crossout should only happen on a single column:
                  </p><pre class="mathml">
&lt;mstack&gt;
  &lt;mscarries location='nw'&gt;
    &lt;none/&gt;
    &lt;mscarry crossout='updiagonalstrike' location='n'&gt; &lt;mn&gt;2&lt;/mn&gt; &lt;/mscarry&gt;
    &lt;mn&gt;1&lt;/mn&gt;
    &lt;none/&gt;
  &lt;/mscarries&gt;
  &lt;mn&gt;2,327&lt;/mn&gt;
  &lt;msrow&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mn&gt; 1,156&lt;/mn&gt; &lt;/msrow&gt;
  &lt;msline/&gt;
  &lt;mn&gt;1,171&lt;/mn&gt;
&lt;/mstack&gt;
</pre><p>Here is an example of subtraction where there is a borrow with
                     multiple digits in a single column and a cross out. The borrowed
                     amount is underlined (the example is from a <a href="http://www.fritext.se/matte/grunder/posi2.html">Swedish
                        source</a>):
                  </p>
                  <blockquote>
                     <p><img src="image/em-sub-borrow-swedish.png" alt="\begin{array}{r} \underbar{\scriptsize 10}\!\\ 5\llap{$/$}2\\ {}-{}7\\ \hline 45 \end{array}"></p>
                  </blockquote>
                  <p>There are two things to notice.
                     The first is that <code>menclose</code> is used in the carry and that <code>none</code> is used for
                     the empty element so that <code>mscarry</code> can be used to create a crossout.
                  </p><pre class="mathml">
&lt;mstack&gt;
  &lt;mscarries&gt;
    &lt;mscarry crossout='updiagonalstrike'&gt;&lt;none/&gt;&lt;/mscarry&gt;
    &lt;menclose notation='bottom'&gt; &lt;mn&gt;10&lt;/mn&gt; &lt;/menclose&gt;
  &lt;/mscarries&gt;
  &lt;mn&gt;52&lt;/mn&gt;
  &lt;msrow&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mn&gt; 7&lt;/mn&gt; &lt;/msrow&gt;
  &lt;msline/&gt;
  &lt;mn&gt;45&lt;/mn&gt;
&lt;/mstack&gt;
</pre></div>
               <div class="div4">
                  
                  <h4><a name="presm.mult" id="presm.mult"></a>3.6.8.2 Multiplication
                  </h4>
                  <p>Below is a simple multiplication example that illustrates the use of <code>msgroup</code> and
                     the <code>shift</code> attribute. The first <code>msgroup</code> does nothing.
                     The second <code>msgroup</code> could also be removed, but <code>msrow</code> would be needed for its second and third children.
                     They would set the <code>position</code> or <code>shift</code> attributes, or would add <code>none</code> elements.
                     
                  </p>
                  <blockquote>
                     <p><img src="image/em-mult-simple.png" alt=""></p>
                  </blockquote><pre class="mathml">
&lt;mstack&gt;
  &lt;msgroup&gt;
    &lt;mn&gt;123&lt;/mn&gt;
    &lt;msrow&gt;&lt;mo&gt;&amp;#xD7;&lt;!--MULTIPLICATION SIGN--&gt;&lt;/mo&gt;&lt;mn&gt;321&lt;/mn&gt;&lt;/msrow&gt;
  &lt;/msgroup&gt;
  &lt;msline/&gt;
  &lt;msgroup shift="1"&gt;
    &lt;mn&gt;123&lt;/mn&gt;
    &lt;mn&gt;246&lt;/mn&gt;
    &lt;mn&gt;369&lt;/mn&gt;
  &lt;/msgroup&gt;
  &lt;msline/&gt;
&lt;/mstack&gt;
</pre><p>This example has multiple rows of carries. It also (somewhat
                     artificially) includes commas (",") as digit separators. The encoding includes
                     these separators in the spacing attribute value, along non-ASCII
                     values. 
                  </p>
                  <blockquote>
                     <p><img src="image/em-mult-carries.png" alt="\begin{array}{r}  {}_1 {\hspace{0.05em}}_1\phantom{0} \\  {}_1 {\hspace{0.05em}}_1\phantom{0} \\   1,234 \\   \times 4,321 \\   \hline  {}_1 \phantom{,} {\hspace{0.05em \,}}_1 {\hspace{0.05em}}_1  {\hspace{0.05em}}_1 \phantom{,} {\hspace{0.05em \,}}_1 \phantom{00} \\   1,234 \\   24,68\phantom{0} \\   370,2\phantom{00} \\   4,936\phantom{,000} \\   \hline   5,332,114 \end{array}"></p>
                  </blockquote><pre class="mathml">
&lt;mstack&gt;
  &lt;mscarries&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;none/&gt;&lt;/mscarries&gt;
  &lt;mscarries&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;none/&gt;&lt;/mscarries&gt;
  &lt;mn&gt;1,234&lt;/mn&gt;
  &lt;msrow&gt;&lt;mo&gt;&amp;#xD7;&lt;!--MULTIPLICATION SIGN--&gt;&lt;/mo&gt;&lt;mn&gt;4,321&lt;/mn&gt;&lt;/msrow&gt;
  &lt;msline/&gt;

  &lt;mscarries position='2'&gt;
    &lt;mn&gt;1&lt;/mn&gt;
    &lt;none/&gt;
    &lt;mn&gt;1&lt;/mn&gt;
    &lt;mn&gt;1&lt;/mn&gt;
    &lt;mn&gt;1&lt;/mn&gt;
    &lt;none/&gt;
    &lt;mn&gt;1&lt;/mn&gt;
  &lt;/mscarries&gt;
  &lt;msgroup shift="1"&gt;
    &lt;mn&gt;1,234&lt;/mn&gt;
    &lt;mn&gt;24,68&lt;/mn&gt;
    &lt;mn&gt;370,2&lt;/mn&gt;
    &lt;msrow position="1"&gt; &lt;mn&gt;4,936&lt;/mn&gt; &lt;/msrow&gt;
  &lt;/msgroup&gt;
  &lt;msline/&gt;
  
  &lt;mn&gt;5,332,114&lt;/mn&gt;
&lt;/mstack&gt;
</pre></div>
               <div class="div4">
                  
                  <h4><a name="presm.mlongdiv.ex" id="presm.mlongdiv.ex"></a>3.6.8.3 Long Division
                  </h4>
                  <p>
                     The notation used for long division varies considerably among
                     countries.  Most notations share the common characteristics of
                     aligning intermediate results and drawing lines for the operands to be
                     subtracted. Minus signs are sometimes shown for the intermediate calculations, and sometimes they are not. The line that is
                     drawn varies in length depending upon the notation. 
                     The most apparently difference among the notations is that the position of the divisor varies, as does the location of the
                     quotient, remainder, and intermediate terms.
                     
                  </p>
                  <p>
                     The layout used is controlled by the <code>longdivstyle</code> attribute.  Below are examples for the values listed in <a href="chapter3-d.html#presm.mlongdiv.attrs">Section&nbsp;3.6.2.2 Attributes</a>
                     
                  </p>
                  <table>
                     <thead>
                        <tr>
                           <th>"lefttop"</th>
                           <th>"stackedrightright"</th>
                           <th>"mediumstackedrightright"</th>
                           <th>"shortstackedrightright"</th>
                           <th>"righttop"</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td>
                              		
                              <blockquote>
                                 <p><img src="image/em-longdiv-us.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                           <td>
                              		
                              <blockquote>
                                 <p><img src="image/em-longdiv-french.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                           <td>
                              
                              <blockquote>
                                 <p><img src="image/em-longdiv-medium-stacked-rightright.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                           <td>
                              
                              <blockquote>
                                 <p><img src="image/em-longdiv-short-stacked-rightright.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                           <td>
                              
                              <blockquote>
                                 <p><img src="image/em-longdiv-righttop.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <table>
                     <thead>
                        <tr>
                           <th>"left/\right"</th>
                           <th>"left)(right"</th>
                           <th>":right=right"</th>
                           <th>"stackedleftleft"</th>
                           <th>"stackedleftlinetop"</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td>
                              		
                              <blockquote>
                                 <p><img src="image/em-longdiv-leftright.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                           <td>
                              		
                              <blockquote>
                                 <p><img src="image/em-longdiv-indian.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                           <td>
                              
                              <blockquote>
                                 <p><img src="image/em-longdiv-colonrightright.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                           <td>
                              
                              <blockquote>
                                 <p><img src="image/em-longdiv-mirrored-french.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                           <td>
                              
                              <blockquote>
                                 <p><img src="image/em-longdiv-arabic-lefttop.png" alt=""></p>
                              </blockquote>
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>
                     The MathML for the first example is shown below.  It illustrates the use of nested <code>msgroup</code>s and how the <code>position</code> is calculated in those usages.
                     
                  </p><pre class="mathml">
&lt;mlongdiv longdivstyle="lefttop"&gt;
  &lt;mn&gt; 3 &lt;/mn&gt;
  &lt;mn&gt; 435.3&lt;/mn&gt;

  &lt;mn&gt; 1306&lt;/mn&gt;

  &lt;msgroup position="2" shift="-1"&gt;
    &lt;msgroup&gt;
      &lt;mn&gt; 12&lt;/mn&gt;
      &lt;msline length="2"/&gt;
    &lt;/msgroup&gt;
    &lt;msgroup&gt;
      &lt;mn&gt; 10&lt;/mn&gt;
      &lt;mn&gt; 9&lt;/mn&gt;
      &lt;msline length="2"/&gt;
    &lt;/msgroup&gt;
    &lt;msgroup&gt;
      &lt;mn&gt; 16&lt;/mn&gt;
      &lt;mn&gt; 15&lt;/mn&gt;
      &lt;msline length="2"/&gt;
      &lt;mn&gt; 1.0&lt;/mn&gt;           &lt;!-- aligns on '.', not the right edge ('0') --&gt;
    &lt;/msgroup&gt;
    &lt;msgroup position='-1'&gt;   &lt;!-- extra shift to move to the right of the "." --&gt;
       &lt;mn&gt; 9&lt;/mn&gt;
      &lt;msline length="3"/&gt;
      &lt;mn&gt; 1&lt;/mn&gt;
    &lt;/msgroup&gt;
  &lt;/msgroup&gt;
&lt;/mlongdiv&gt;
</pre><p>
                     With the exception of the last example, the encodings for the other examples are the same except that the values for <code>longdivstyle</code> differ and that a "," is used instead of a "." for the decimal point.
                     For the last example, the only difference from the other examples besides a different value for <code>longdivstyle</code> is that Arabic numerals have been used in place of Latin numerals, 
                     as shown  below.
                  </p><pre class="mathml">
&lt;mstyle decimalpoint="&amp;#x066B;"&gt;
&lt;mlongdiv longdivstyle="stackedleftlinetop"&gt;
  &lt;mn&gt; &amp;#x0663; &lt;/mn&gt;
  &lt;mn&gt; &amp;#x0664;&amp;#x0663;&amp;#x0665;&amp;#x066B;&amp;#x0663;&lt;/mn&gt;

  &lt;mn&gt; &amp;#x0661;&amp;#x0663;&amp;#x0660;&amp;#x0666;&lt;/mn&gt;
  &lt;msgroup position="2" shift="-1"&gt;
    &lt;msgroup&gt;
      &lt;mn&gt; &amp;#x0661;&amp;#x0662;&lt;/mn&gt;
      &lt;msline length="2"/&gt;
    &lt;/msgroup&gt;
    &lt;msgroup&gt;
      &lt;mn&gt; &amp;#x0661;&amp;#x0660;&lt;/mn&gt;
      &lt;mn&gt; &amp;#x0669;&lt;/mn&gt;
      &lt;msline length="2"/&gt;
    &lt;/msgroup&gt;
    &lt;msgroup&gt;
      &lt;mn&gt; &amp;#x0661;&amp;#x0666;&lt;/mn&gt;
      &lt;mn&gt; &amp;#x0661;&amp;#x0665;&lt;/mn&gt;
      &lt;msline length="2"/&gt;
        &lt;mn&gt; &amp;#x0661;&amp;#x066B;&amp;#x0660;&lt;/mn&gt;
   &lt;/msgroup&gt;
   &lt;msgroup position='-1'&gt;
       &lt;mn&gt; &amp;#x0669;&lt;/mn&gt;
      &lt;msline length="3"/&gt;
      &lt;mn&gt; &amp;#x0661;&lt;/mn&gt;
    &lt;/msgroup&gt;
  &lt;/msgroup&gt;
&lt;/mlongdiv&gt;
&lt;/mstyle&gt;</pre></div>
               <div class="div4">
                  
                  <h4><a name="presm.repeatdec" id="presm.repeatdec"></a>3.6.8.4 Repeating decimal
                  </h4>
                  <p>
                     Decimal numbers that have digits that repeat infinitely such as 1/3
                     (.3333...) are represented using several notations. One common notation
                     is to put a horizontal line over the digits that repeat (in Portugal an underline is used).
                     Another notation involves putting dots over the digits that repeat. These notations are shown below:
                     
                  </p>
                  <blockquote>
                     <p><img src="image/repeat1.png" alt="0.33333 \overline{3}"></p>
                  </blockquote>
                  <blockquote>
                     <p><img src="image/repeat2.png" alt="0.\overline{142857}"></p>
                  </blockquote>
                  <blockquote>
                     <p><img src="image/repeat3.png" alt="0.\underline{142857}"></p>
                  </blockquote>
                  <blockquote>
                     <p><img src="image/repeat4.png" alt="0.\dot{1}4285\dot{7}"></p>
                  </blockquote>
                  <p>The MathML for these involves using <code>mstack</code>, <code>msrow</code>, and <code>msline</code>
                     in a straightforward manner.  The MathML for the preceding examples above is given below.
                  </p><pre class="mathml-fragment">
&lt;mstack stackalign="right"&gt;
  &lt;msline length="1"/&gt;
  &lt;mn&gt; 0.3333 &lt;/mn&gt;
&lt;/mstack&gt;
</pre><pre class="mathml-fragment">
&lt;mstack stackalign="right"&gt;
  &lt;msline length="6"/&gt;
  &lt;mn&gt; 0.142857 &lt;/mn&gt;
&lt;/mstack&gt;
</pre><pre class="mathml-fragment">
&lt;mstack stackalign="right"&gt;
  &lt;mn&gt; 0.142857 &lt;/mn&gt;
  &lt;msline length="6"/&gt;
&lt;/mstack&gt;
</pre><pre class="mathml-fragment">
&lt;mstack stackalign="right"&gt;
  &lt;msrow&gt; &lt;mo&gt;.&lt;/mo&gt; &lt;none/&gt;&lt;none/&gt;&lt;none/&gt;&lt;none/&gt; &lt;mo&gt;.&lt;/mo&gt; &lt;/msrow&gt;
  &lt;mn&gt; 0.142857 &lt;/mn&gt;
&lt;/mstack&gt;
</pre></div>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="presm.enliven" id="presm.enliven"></a>3.7 Enlivening Expressions
            </h2>
            <div class="div3">
               
               <h3><a name="presm.maction" id="presm.maction"></a>3.7.1 Bind Action to Sub-Expression
                  <code>&lt;maction&gt;</code></h3>
               <p>To provide a mechanism for binding actions to expressions, MathML
                  provides the <code>maction</code> element. This element accepts any
                  number of sub-expressions as arguments and the type of action that should happen
                  is controlled by the <code>actiontype</code> attribute.  Only three actions are predefined by MathML, but the list of possible actions is open.  Additional predefined
                  actions may be added in future versions of MathML.
               </p>
               <p>Linking to other elements, either locally within the <code>math</code> element or to some URL, is not handled by <code>maction</code>.  Instead, it is handled by adding a link directly on a MathML element as specified in <a href="chapter6-d.html#interf.link">Section&nbsp;6.4.3 Linking</a>.
               </p>
               <div class="div4">
                  
                  <h4><a name="id.3.7.1.1" id="id.3.7.1.1"></a>3.7.1.1 Attributes
                  </h4>
                  <p><code>maction</code> elements accept the attributes listed
                     below in addition to those specified in <a href="chapter3-d.html#presm.presatt">Section&nbsp;3.1.10 Mathematics style attributes common to presentation elements</a>.
                  </p>
                  <p>By default, MathML applications that do not recognize the specified
                     <code>actiontype</code> should render the selected sub-expression as
                     defined below. If no selected sub-expression exists, it is a MathML
                     error; the appropriate rendering in that case is as described in
                     <a href="chapter2-d.html#interf.error">Section&nbsp;2.3.2 Handling of Errors</a>.
                  </p>
                  <table border="1" class="attributes">
                     <thead>
                        <tr>
                           <th>Name</th>
                           <th>values</th>
                           <th>default</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="2" class="attname">actiontype</td>
                           <td><a href="chapter2-d.html#type.string"><em>string</em></a></td>
                           <td><em>required</em></td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies what should happen for this element. The values allowed are open-ended. Conforming renderers may ignore any value
                              they do not handle, although renderers are encouraged to render the values listed below. 
                              
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="2" class="attname">selection</td>
                           <td><a href="chapter2-d.html#type.positive-integer"><em>positive-integer</em></a></td>
                           <td>1</td>
                        </tr>
                        <tr>
                           <td colspan="2" class="attdesc">
                              Specifies which child should be used for viewing. Its value should be between 1 and the number of
                              children of the element. The specified child is referred to as the "selected sub-expression" of the
                              <code>maction</code> element. If the value specified is out of range, it is an error. When the
                              <code>selection</code> attribute is not specified (including for
                              action types for which it makes no sense), its default value is 1, so
                              the selected sub-expression will be the first sub-expression.
                              
                           </td>
                        </tr>
                     </tbody>
                  </table>
                  <p>If a MathML application responds to a user command to copy a MathML sub-expression to
                     the environment's "clipboard" (see <a href="chapter6-d.html#world-int-transfers">Section&nbsp;6.3 Transferring MathML</a>), any <code>maction</code> elements present in what is copied should
                     be given <code>selection</code> values that correspond to their selection
                     state in the MathML rendering at the time of the copy command.
                  </p>
                  <p>When a MathML application receives a mouse event that may be
                     processed by two or more nested maction elements, the innermost
                     maction element of each action type should respond to the event.
                  </p>
                  <p>The meanings of the various <code>actiontype</code> values is given below.
                     Note that not all renderers support all of the <code>actiontype</code> values, and  that the allowed values are open-ended.
                  </p>
                  <dl>
                     <dt class="label">&lt;maction actiontype="toggle" selection="positive-integer" &gt; (first expression) (second expression)... &lt;/maction&gt;</dt>
                     <dd>The renderer alternately display the
                        selected subexpression, cycling through them when there is a click on the selected subexpression.  Each click increments the
                        <code>selection</code> value, wrapping back to 1 when it reaches the last child.
                        Typical uses would be for exercises in education, ellipses in long
                        computer algebra output, or to illustrate alternate notations. Note
                        that the expressions may be of significantly different size, so that
                        size negotiation with the browser may be desirable. If size
                        negotiation is not available, scrolling, elision, panning, or some
                        other method may be necessary to allow full viewing.
                     </dd>
                     <dt class="label">&lt;maction actiontype="statusline"&gt; (expression) (message) &lt;/maction&gt;</dt>
                     <dd>The renderer displays the first child.
                        When a reader clicks on the expression or
                        moves the pointer over it, the renderer sends a rendering of the
                        message to the browser statusline. Because most browsers in the
                        foreseeable future are likely to be limited to displaying text on their
                        statusline, the second child should be an 
                        <code>mtext</code> element in most circumstances. 
                        For non-<code>mtext</code> messages, renderers might provide a
                        natural language translation of the markup, but this is not
                        required.
                     </dd>
                     <dt class="label">&lt;maction actiontype="tooltip"&gt; (expression) (message) &lt;/maction&gt;</dt>
                     <dd>The renderer displays the first child.
                        When the pointer pauses over the expression for a long
                        enough delay time, the renderer displays a rendering of the message in
                        a pop-up "tooltip" box near the expression. Many systems may limit
                        the popup to be text, so the second child should be an 
                        <code>mtext</code> element in most circumstances.
                        For non-<code>mtext</code> messages,
                        renderers may provide a natural language translation of the markup if
                        full MathML rendering is not practical, but this is not
                        required.
                     </dd>
                     <dt class="label">&lt;maction actiontype="input"&gt; (expression) &lt;/maction&gt;</dt>
                     <dd>The renderer displays the expression. For renderers that allow editing, when focus is passed to this element, the <code>maction</code> is replaced by what is entered, pasted, etc. MathML does not restrict what is allowed as input, nor does it require an editor
                        to allow arbitrary input.  Some renderers/editors may restrict the input to simple (linear) text.
                     </dd>
                  </dl>
                  <p>The <code>actiontype</code> values are open-ended. If another value is given and it requires additional attributes, the attributes must be in a different
                     namespace. This is shown below:
                     
                  </p>
                  <dl>
                     <dt class="label">&lt;maction actiontype="highlight" my:color="red" my:background="yellow"&gt; expression &lt;/maction&gt;</dt>
                     <dd>In the example,
                        non-standard attributes from another namespace are being used to pass
                        additional information to renderers that support them,
                        without violating the MathML Schema (see <a href="chapter2-d.html#interf.unspecified">Section&nbsp;2.3.3 Attributes for unspecified data</a>).
                        The <code>my:color</code> attributes
                        might change the color of the characters in the presentation, while the
                        <code>my:background</code> attribute might change the color of the background
                        behind the characters.
                     </dd>
                  </dl>
               </div>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="presm.semantics" id="presm.semantics"></a>3.8 Semantics and Presentation
            </h2>
            <p>MathML uses the <code>semantics</code> element to allow specifying semantic annotations to
               presentation MathML elements; these can be content MathML or other notations.  As such,
               <code>semantics</code> should be considered part of both presentation MathML and content
               MathML. All MathML processors should process the <code>semantics</code> element, even if they
               only process one of those subsets.
            </p>
            <p>In semantic annotations a presentation MathML expression is typically the first child
               of the <code>semantics</code> element.  However, it can also be given inside of an
               <code>annotation-xml</code> element inside the <code>semantics</code> element. If it is part of an
               <code>annotation-xml</code> element, then
               <code>encoding</code>="application/mathml-presentation+xml" or 
               <code>encoding</code>="MathML-Presentation" may be used and presentation
               MathML processors should use this value for the presentation.
            </p>
            <p>See <a href="chapter5-d.html#mixing.semantic.annotations">Section&nbsp;5.1 Annotation Framework</a> for more details about the
               <code>semantics</code> and <code>annotation-xml</code> elements.
            </p>
         </div>
      </div>
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         Overview: <a href="Overview-d.html">Mathematical Markup Language (MathML) Version 3.0</a><br>
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