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      <h1><a name="fund" id="fund"></a>2 MathML Fundamentals
      </h1>
      <!-- TOP NAVIGATION BAR -->
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           Overview: <a href="overview.html">Mathematical Markup Language (MathML) Version 2.0 (Second Edition)</a><br>
           Previous: 1 <a href="chapter1.html">Introduction</a><br>
           Next: 3 <a href="chapter3.html">Presentation Markup</a><br><br>2 <a href="chapter2.html">MathML Fundamentals</a><br>&nbsp;&nbsp;&nbsp;&nbsp;2.1 <a href="chapter2.html#fund.overview">MathML Overview</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.1.1 <a href="chapter2.html#id.2.1.1">Taxonomy of MathML Elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.1.2 <a href="chapter2.html#id.2.1.2">Presentation Markup</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.1.3 <a href="chapter2.html#id.2.1.3">Content Markup</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.1.4 <a href="chapter2.html#id.2.1.4">Mixing Presentation and Content</a><br>&nbsp;&nbsp;&nbsp;&nbsp;2.2 <a href="chapter2.html#fund.document">MathML in a Document</a><br>&nbsp;&nbsp;&nbsp;&nbsp;2.3 <a href="chapter2.html#fund.examples">Some MathML Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.3.1 <a href="chapter2.html#fund.pres">Presentation Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.3.2 <a href="chapter2.html#fund.cont">Content Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.3.3 <a href="chapter2.html#fund.mix">Mixed Markup Examples</a><br>&nbsp;&nbsp;&nbsp;&nbsp;2.4 <a href="chapter2.html#fund.syntax">MathML Syntax and Grammar</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.1 <a href="chapter2.html#id.2.4.1">MathML Syntax and Grammar</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.2 <a href="chapter2.html#fund.xmlsyntax">An XML Syntax Primer</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.3 <a href="chapter2.html#id.2.4.3">Children versus Arguments</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.4 <a href="chapter2.html#fund.attval">MathML Attribute Values</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.4.1 <a href="chapter2.html#id.2.4.4.1">Syntax notations used in the MathML specification</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.4.2 <a href="chapter2.html#fund.units">Attributes with units</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.4.3 <a href="chapter2.html#fund.cssatt">CSS-compatible attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.4.4 <a href="chapter2.html#id.2.4.4.4">Default values of attributes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.4.5 <a href="chapter2.html#id.2.4.4.5">Attribute values in the MathML DTD</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.5 <a href="chapter2.html#fund.globatt">Attributes Shared by all MathML Elements</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;2.4.6 <a href="chapter2.html#fund.collapse">Collapsing Whitespace in Input</a><br></div>
      <div class="div1">
         <div class="div2">
            
            <h2><a name="fund.overview" id="fund.overview"></a>2.1 MathML Overview
            </h2>
            <p>This chapter introduces the basic ideas of MathML.  The first section
               describes the overall design of MathML.  The second section presents a
               number of motivating examples, to give the reader something concrete to
               refer to while reading subsequent chapters of the MathML specification.
               The final section describes basic features of the MathML syntax and
               grammar, which apply to all MathML markup. In particular, <a href="chapter2.html#fund.syntax">Section&nbsp;2.4 MathML Syntax and Grammar</a> should be read <em>before</em> <a href="chapter3.html">Chapter&nbsp;3 Presentation Markup</a>, <a href="chapter4.html">Chapter&nbsp;4 Content Markup</a> and <a href="chapter5.html">Chapter&nbsp;5 Combining Presentation and Content Markup</a>.
            </p>
            <p>A fundamental challenge in defining a markup language for mathematics on
               the Web is reconciling the need to encode both the presentation of a
               mathematical notation and the content of the mathematical idea or object
               which it represents.
            </p>
            <p>The relationship between a mathematical notation and a mathematical idea
               is subtle and deep. On a formal level, the results of mathematical logic
               raise unsettling questions about the correspondence between systems of
               symbolic logic and the phenomena they model. At a more intuitive level,
               anyone who uses mathematical notation knows the difference that a good
               choice of notation can make; the symbolic structure of the notation
               suggests the logical structure.  For example, the Leibniz notation for
               derivatives "suggests" the chain rule of calculus through the
               symbolic cancellation of fractions: <img src="image/f2001.gif" alt="\frac{d f}{d x}\frac{d x}{d t}=\frac{d f}{d t}" align="middle">.
            </p>
            <p>Mathematicians and teachers intuitively understand this very well; part
               of their expertise lies in choosing notation that emphasizes key aspects of
               a problem while hiding or diminishing extraneous aspects.  It is
               commonplace in mathematics and science to write one thing when strictly
               technically something else is meant, because long experience shows this
               actually communicates the idea better at some higher level than rigorous
               detail.
            </p>
            <p>In many other settings, though, mathematical notation is used to encode
               the full, precise meaning of a mathematical object.  Mathematical notation
               is capable of prodigious rigor, and when used carefully, it can be
               virtually free of ambiguity. Moreover, it is precisely this lack of
               ambiguity which makes it possible to describe mathematical objects so that
               they can be used by software applications such as computer algebra systems
               and voice renderers.  In situations where such inter-application
               communication is of paramount importance, the nuances of visual
               presentation generally play a minimal role.
            </p>
            <p>MathML allows authors to encode both the notation which represents a
               mathematical object and the mathematical structure of the object
               itself. Moreover, authors can mix both kinds of encoding in order to
               specify both the presentation and content of a mathematical idea. The
               remainder of this section gives a basic overview of how MathML can be used
               in each of these ways.
            </p>
            <div class="div3">
               
               <h3><a name="id.2.1.1" id="id.2.1.1"></a>2.1.1 Taxonomy of MathML Elements
               </h3>
               <p>All MathML elements fall into one of three categories: presentation
                  elements, content elements and interface elements.  Each of these categories
                  is described in detail in <a href="chapter3.html">Chapter&nbsp;3 Presentation Markup</a>, <a href="chapter4.html">Chapter&nbsp;4 Content Markup</a>
                  and <a href="chapter7.html">Chapter&nbsp;7 The MathML Interface</a>, respectively.
               </p>
               <p>Presentation elements describe mathematical notation's visually oriented
                  two-dimensional structure.  Typical examples are the 
                  <a href="chapter3.html#presm.mrow"><code>mrow</code></a>
                   element, which is usually employed to indicate a
                  horizontal row of pieces of expressions, and the 
                  <a href="chapter3.html#presm.msup"><code>msup</code></a> element, which is used to mark up a base
                  expression and a superscript to it.  As a general rule, each presentation
                  element corresponds to a single kind of notational "schema" 
                  such as a row, a superscript, a subscript, an underscript and so on.
                  Any formula is made by putting together parts which ultimately can 
                  be analyzed down to the  simplest items such as digits, letters, or 
                  other symbol characters.
               </p>
               <p>Although the previous paragraph was concerned with the display
                  aspect of mathematical notation, and hence with presentation markup,
                  the same observation about decomposition applies equally well to
                  abstract mathematical objects, and hence to content markup.  For
                  example, in the context of content markup a superscript would
                  typically be denoted by an exponentiation operation that would require
                  two operands: a "base" and an
                  "exponent". This is no coincidence, since as a general
                  rule, mathematical notation's layout closely follows the logical
                  structure of the underlying mathematical objects.
               </p>
               <p>The recursive nature of mathematical objects and notation is strongly
                  reflected in MathML markup.  In use, most presentation or content elements
                  contain some number of other MathML elements corresponding to the
                  constituent pieces out of which the original object is recursively
                  built. The original schema is commonly called the <em>parent</em>
                  schema, and the constituent pieces are called <em>child</em>
                  schemata. More generally, MathML expressions can be regarded as trees,
                  where each node corresponds to a MathML element, the branches under a
                  "parent" node correspond to its "children", and
                  the leaves in the tree correspond to atomic notation or content units such
                  as numbers, characters, etc.
               </p>
               <p>Most leaf nodes in a MathML expression tree are either <em>canonically
                     empty elements</em> with no bodies, or <em>token elements</em>.
                  Canonically empty elements represent symbols directly in MathML, for
                  example, the content element 
                  <a href="chapter4.html#contm.plus"><code>&lt;plus/&gt;</code></a> does
                  this.  MathML token elements are the only MathML elements permitted to
                  contain MathML character data.  MathML character data consists of
                  Unicode characters with the infrequent addition of special
                  character constructions done with the 
                  <a href="chapter3.html#presm.mglyph"><code>mglyph</code></a>
                  element.  A third kind of leaf node
                  permitted in MathML is the 
                  <a href="chapter4.html#contm.annotation"><code>annotation</code></a> element,
                  which is used to hold data which is not in MathML format.
               </p>
               <p>The most important presentation token elements are 
                  <a href="chapter3.html#presm.mi"><code>mi</code></a>, 
                  <a href="chapter3.html#presm.mn"><code>mn</code></a> and 
                  <a href="chapter3.html#presm.mo"><code>mo</code></a> for representing identifiers, numbers and operators
                  respectively. Typically a renderer will employ slightly different
                  typesetting styles for each of these kinds of character data: numbers are
                  usually in upright font, identifiers in italics, and operators have extra
                  space around them.  In content markup, there are only three tokens, 
                  <a href="chapter4.html#contm.ci"><code>ci</code></a>, 
                  <a href="chapter4.html#contm.cn"><code>cn</code></a> and 
                  <a href="chapter4.html#contm.csymbol"><code>csymbol</code></a>, for identifiers, numbers and new symbols
                  introduced in the document itself, respectively.  In content markup,
                  separate elements are provided for commonly used functions and
                  operators. The 
                  <a href="chapter4.html#contm.apply"><code>apply</code></a> element is provided for
                  user-defined extensions to the base set.
               </p>
               <p>In terms of markup, most MathML elements are denoted by a
                  <em>start</em> tag and an <em>end</em> tag, which enclose the
                  markup for their contents. In the case of tokens, the content is
                  character data, and in most other cases, the content is the markup for
                  child elements. Elements in a third category, called canonically empty
                  elements, do not require any contents, and are denoted by a single tag of
                  the form <code>&lt;name/&gt;</code>. An example of this kind of
                  markup is <code>&lt;plus/&gt;</code> in content markup.
               </p>
               <p>Let us take the very simple example of (<var>a</var> +
                  <var>b</var>)<sup>2</sup>, and we can now see how the
                  principles discussed above play out in practice. One form of
                  presentation markup for this example is:
                  
               </p><pre>
 &lt;mrow&gt;
   &lt;msup&gt;
     &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;
     &lt;mn&gt;2&lt;/mn&gt;
   &lt;/msup&gt;
 &lt;/mrow&gt;
</pre><p>This example demonstrates a number of presentation elements.  The first
                  element, one that is used a great deal is 
                  <a href="chapter3.html#presm.mrow"><code>mrow</code></a>.
                  This element is used to denote a row of horizontally aligned material.  The
                  material contained between the <code>&lt;mrow&gt;</code> and <code>&lt;/mrow&gt;</code> tags is considered to be an argument to the <code>mrow</code> element.  Thus the whole expression here is
                  contained in an <code>mrow</code> element.  As previously noted,
                  almost all mathematical expressions decompose into subexpressions.  These
                  subexpressions can, in turn, also be contained in an <code>mrow</code> element.  For example, a + b is also contained in an
                  <code>mrow</code>.
               </p>
               <p>The 
                  <a href="chapter3.html#presm.mfenced"><code>mfenced</code></a> element is used to provide fences
                  (braces, brackets, and parentheses) around formula material.  It defaults
                  to using parentheses.
               </p>
               <p>Note the use of the 
                  <a href="chapter3.html#presm.mi"><code>mi</code></a> element for displaying
                  the variables a and b and the 
                  <a href="chapter3.html#presm.mo"><code>mo</code></a> element for
                  marking the + operator.
               </p>
               <p>The 
                  <a href="chapter3.html#presm.msup"><code>msup</code></a> element is for expressions involving
                  superscripts and takes two arguments, in order, the base expression (here,
                  (<var>a</var>+<var>b</var>)) and the exponent expression (here, 2).
               </p>
               <p>The content markup for the same example is:
                  
               </p><pre>
 &lt;mrow&gt;
   &lt;apply&gt;
     &lt;power/&gt;
     &lt;apply&gt;
       &lt;plus/&gt;
       &lt;ci&gt;a&lt;/ci&gt;
       &lt;ci&gt;b&lt;/ci&gt;
     &lt;/apply&gt;
     &lt;cn&gt;2&lt;/cn&gt;
   &lt;/apply&gt;
 &lt;/mrow&gt;
</pre><p>Here, the 
                  <a href="chapter4.html#contm.apply"><code>apply</code></a> content element means apply an
                  operation to an expression.  In this example, the <code>power</code> element (for exponentiation), which requires no
                  body, and the similar 
                  <a href="chapter4.html#contm.plus"><code>plus</code></a> element (for addition)
                  are both <em>applied</em>.  Observe that both operators take two
                  arguments, the order being particularly significant in the case of the
                  power operator.  But the order of the children is crucial in the use
                  of the  <code>apply</code> since the first child, the operator,
                  takes as argument list the remaining ones. 
               </p>
               <p>Note the use of the 
                  <a href="chapter4.html#contm.ci"><code>ci</code></a> element to mark up the
                  variables a and b, and the 
                  <a href="chapter4.html#contm.cn"><code>cn</code></a> element to mark up
                  the number 2.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="id.2.1.2" id="id.2.1.2"></a>2.1.2 Presentation Markup
               </h3>
               <p>MathML presentation markup consists of about 30 elements which accept
                  over 50 attributes.  Most of the elements correspond to <em>layout
                     schemata</em>, which contain other presentation elements. Each layout
                  schema corresponds to a two-dimensional notational device, such as a
                  superscript or subscript, fraction or table.  In addition, there are the
                  presentation token elements 
                  <a href="chapter3.html#presm.mi"><code>mi</code></a>, 
                  <a href="chapter3.html#presm.mo"><code>mo</code></a> and 
                  <a href="chapter3.html#presm.mn"><code>mn</code></a> introduced above, as
                  well as several other less commonly used token elements. The remaining few
                  presentation elements are empty elements, and are used mostly in connection
                  with alignment.
               </p>
               <p>The layout schemata fall into several classes. One group of
                  elements is concerned with scripts, and contains elements such as
                  <a href="chapter3.html#presm.msub"><code>msub</code></a>,
                  <a href="chapter3.html#presm.munder"><code>munder</code></a>,
                  and <a href="chapter3.html#presm.mmultiscripts"><code>mmultiscripts</code></a>.  Another group focuses on
                  more general layout and includes <a href="chapter3.html#presm.mrow"><code>mrow</code></a>, <a href="chapter3.html#presm.mstyle"><code>mstyle</code></a>, and <a href="chapter3.html#presm.mfrac"><code>mfrac</code></a>.  A third group deals with tables.
                  The <a href="chapter3.html#presm.maction"><code>maction</code></a> element is in a category by
                  itself, and allows coding of various kinds of actions on notation,
                  such as occur in an expression which toggles between two pieces of
                  notation.
               </p>
               <p>An important feature of many layout schemata is that the order of child
                  schemata is significant. For example, the first child of an 
                  <a href="chapter3.html#presm.mfrac"><code>mfrac</code></a> element is the numerator and the second child is
                  the denominator. Since the order of child schemata is not enforced at the
                  XML level by the MathML DTD, the information added by ordering is only
                  available to a MathML processor, as opposed to a generic XML processor.
                  When we want to emphasize that a MathML element such as 
                  <a href="chapter3.html#presm.mfrac"><code>mfrac</code></a> requires children in a specific order, we will
                  refer to them as <em>arguments</em>, and think of the 
                  <code>mfrac</code> element as a notational
                  "constructor".
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="id.2.1.3" id="id.2.1.3"></a>2.1.3 Content Markup
               </h3>
               <p>Content markup consists of about 120 elements accepting roughly a dozen
                  attributes. The majority of these elements are empty elements corresponding
                  to a wide variety of operators, relations and named functions. Examples of
                  this sort include 
                  <a href="chapter4.html#contm.partialdiff"><code>partialdiff</code></a>, 
                  <a href="chapter4.html#contm.leq"><code>leq</code></a> and 
                  <a href="chapter4.html#contm.trig"><code>tan</code></a>.  Others such as
                  <a href="chapter4.html#contm.matrix"><code>matrix</code></a> and 
                  <a href="chapter4.html#contm.set"><code>set</code></a> are used to
                  encode various mathematical data types, and a third, important category of
                  content elements such as 
                  <a href="chapter4.html#contm.apply"><code>apply</code></a> are used to apply
                  operations to expressions and also to make new mathematical objects from
                  others.
               </p>
               <p>The 
                  <code>apply</code> element is perhaps the single most
                  important content element. It is used to apply a function or operation to a
                  collection of arguments.  The positions of the child schemata are again
                  significant, with the first child denoting the function to be applied, and
                  the remaining children denoting the arguments of the function in
                  order. Note that the <code>apply</code> construct always uses
                  prefix notation, like the programming language LISP. In particular, even
                  binary operations such as subtraction are marked up by applying a prefix
                  subtraction operator to two arguments.  For example, <var>a</var> -
                  <var>b</var> would be marked up as
                  
               </p><pre>
&lt;mrow&gt;
&lt;apply&gt;
   &lt;minus/&gt;
   &lt;ci&gt;a&lt;/ci&gt;
   &lt;ci&gt;b&lt;/ci&gt;
&lt;/apply&gt;
&lt;/mrow&gt;
</pre><p>A number of functions and operations require one or more quantifiers to
                  be well-defined. For example, in addition to an integrand, a definite
                  integral must specify the limits of integration and the bound variable. For
                  this reason, there are several <em>qualifier</em> schemata such as 
                  <a href="chapter4.html#contm.bvar"><code>bvar</code></a> and 
                  <a href="chapter4.html#contm.lowlimit"><code>lowlimit</code></a>.  They are
                  used with operators such as 
                  <a href="chapter4.html#contm.diff"><code>diff</code></a> and 
                  <a href="chapter4.html#contm.int"><code>int</code></a>.
               </p>
               <p>The 
                  <a href="chapter4.html#contm.declare"><code>declare</code></a> construct is especially important
                  for content markup that might be evaluated by a computer algebra system.
                  The <code>declare</code> element provides a basic assignment
                  mechanism, where a variable can be declared to be of a certain type, with a
                  certain value.
               </p>
               <p>
                  In both the presentation and content markup examples, mathematical
                  expressions are recursively decomposed into nested, simpler MathML
                  elements specifying each stage of the decomposition.  The examples in
                  the following sections illustrate this with more complex expressions.
                  
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="id.2.1.4" id="id.2.1.4"></a>2.1.4 Mixing Presentation and Content
               </h3>
               <p>Different kinds of markup will be found most appropriate for different
                  kinds of tasks.  Documents written before the World Wide Web became
                  important were most often intended only for visual communication of
                  information, so that legacy data is probably best translated into pure
                  presentation markup, since semantic information about what the author meant
                  can only be guessed at heuristically.  By contrast, some mathematical
                  applications and pedagogically-oriented authoring tools will likely choose
                  to be entirely content-based.  The majority of applications fall somewhere
                  in between these extremes.  For these applications, the most appropriate
                  markup is a mixture of both presentation and content markup.
               </p>
               <p>The rules for mixing presentation and content markup derive from the
                  general principle that mixed content should only be allowed in places where
                  it makes sense. For content markup embedded in presentation markup this
                  basically means that any content fragments should be semantically
                  meaningful, and should not require additional arguments or quantifiers to
                  be fully specified.  For presentation markup embedded in content markup,
                  this usually means that presentation markup must be contained in a content
                  token element, so that it will be treated as an indivisible notational unit
                  used as a variable or function name.
               </p>
               <p>Another option is to use a 
                  <a href="chapter4.html#contm.semantics"><code>semantics</code></a> element.
                  The <code>semantics</code> element is used to bind MathML
                  expressions to various kinds of annotations. One common use for the <code>semantics</code> element is to bind a piece of content markup
                  to some presentation markup as a semantic annotation. In this way, an
                  author can specify a non-standard notation to be used when displaying a
                  particular content expression.  Another use of the <code>semantics</code> element is to bind some other kind of
                  semantic specification, such as an OpenMath expression, to a MathML
                  expression.  In this way, the <code>semantics</code> element can
                  be used to extend the scope of MathML content markup.
               </p>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="fund.document" id="fund.document"></a>2.2 MathML in a Document
            </h2>
            <p>The discussion above has actually been of fragmentary formulas
               outside the context of any document.  To be more specific let us look
               at what corresponds to a programming language's "Hello
               World!" example.  We shall provide more complete code for an <a href="http://www.w3.org/TR/xhtml1/#well-formed">XHTML 1.0
                  document</a> containing the square of a sum of two variables
               mentioned above.  It would be
               
               
            </p><pre>

&lt;html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"&gt;
 
&lt;head&gt;
&lt;title&gt;MathML's Hello Square&lt;/title&gt;
&lt;/head&gt;

&lt;body&gt;

&lt;p&gt; This is a perfect square:&lt;/p&gt;

&lt;math xmlns="http://www.w3.org/1998/Math/MathML"&gt;
 &lt;mrow&gt;
   &lt;msup&gt;
     &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;
     &lt;mn&gt;2&lt;/mn&gt;
   &lt;/msup&gt;
 &lt;/mrow&gt; 
&lt;/math&gt;

&lt;/body&gt;
&lt;/html&gt;
</pre><p>Here we have the normal structure of an XHTML document.  It begins
               with the start tag <code>&lt;html&gt;</code> embellished with an
               XML namespace declaration and language assertions.  A <code>head</code> element contains a title as is customary.
               Within the <code>body</code> is 
                a simple paragraph followed by a <code>math</code> element
               which has the MathML namespace declared.  Inside the <code>math</code> element is MathML markup.
               
               
            </p>
            <p>For the next level of technical detail concerning such matters
               as <code>!DOCTYPE</code> statements and the like, see the 
               discussion in <a href="chapter7.html">Chapter&nbsp;7 The MathML Interface</a>.
               
            </p>
         </div>
         <div class="div2">
            
            <h2><a name="fund.examples" id="fund.examples"></a>2.3 Some MathML Examples
            </h2>
            <p>
               We continue below to display examples in the form of fragments of
               MathML markup such as would appear inside <code>math</code>
               elements in real documents.  For the sake of clearer exposition of
               principles, the examples in Chapters 3, 4, 5 and 6 follow this form of
               giving examples as MathML fragments.
               
            </p>
            <div class="div3">
               
               <h3><a name="fund.pres" id="fund.pres"></a>2.3.1 Presentation Examples
               </h3>
               <p>Notation: <var>x</var><sup>2</sup> + 4<var>x</var> + 4 = 0.
               </p>
               <p>Markup:
                  
               </p><pre>
&lt;mrow&gt;
  &lt;mrow&gt;
    &lt;msup&gt;
      &lt;mi&gt;x&lt;/mi&gt;
      &lt;mn&gt;2&lt;/mn&gt;
    &lt;/msup&gt;
    &lt;mo&gt;+&lt;/mo&gt;
    &lt;mrow&gt;
      &lt;mn&gt;4&lt;/mn&gt;
      &lt;mo&gt;&amp;InvisibleTimes;&lt;/mo&gt;
      &lt;mi&gt;x&lt;/mi&gt;
    &lt;/mrow&gt;
    &lt;mo&gt;+&lt;/mo&gt;
    &lt;mn&gt;4&lt;/mn&gt;
  &lt;/mrow&gt;
  &lt;mo&gt;=&lt;/mo&gt;
  &lt;mn&gt;0&lt;/mn&gt;
&lt;/mrow&gt;
</pre><p>
                  Note the use of nested <a href="chapter3.html#presm.mrow"><code>mrow</code></a> elements to denote terms, for
                  example, the left-hand side of the equation functioning as an operand
                  of "=".  Marking terms greatly facilitates spacing for
                  visual rendering, voice rendering, and line breaking.  The <code>&amp;InvisibleTimes;</code> MathML character entity is used here
                  to indicate to a renderer that there are special spacing rules between
                  the 4 and the x, and that the 4 and the x should not be broken onto
                  separate lines.
                  
                  
               </p>
               <p>Notation:
                  <img src="image/f1002.gif" alt="x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}" align="middle">.
               </p>
               <p>Markup:
                  
               </p><pre>
&lt;mrow&gt;
  &lt;mi&gt;x&lt;/mi&gt;
  &lt;mo&gt;=&lt;/mo&gt;
  &lt;mfrac&gt;
    &lt;mrow&gt;
      &lt;mrow&gt;
        &lt;mo&gt;-&lt;/mo&gt;
        &lt;mi&gt;b&lt;/mi&gt;
      &lt;/mrow&gt;
      &lt;mo&gt;&amp;PlusMinus;&lt;/mo&gt;
      &lt;msqrt&gt;
        &lt;mrow&gt;
          &lt;msup&gt;
            &lt;mi&gt;b&lt;/mi&gt;
            &lt;mn&gt;2&lt;/mn&gt;
          &lt;/msup&gt;
          &lt;mo&gt;-&lt;/mo&gt;
          &lt;mrow&gt;
            &lt;mn&gt;4&lt;/mn&gt;
            &lt;mo&gt;&amp;InvisibleTimes;&lt;/mo&gt;
            &lt;mi&gt;a&lt;/mi&gt;
            &lt;mo&gt;&amp;InvisibleTimes;&lt;/mo&gt;
            &lt;mi&gt;c&lt;/mi&gt;
          &lt;/mrow&gt;
        &lt;/mrow&gt;
      &lt;/msqrt&gt;
    &lt;/mrow&gt;
    &lt;mrow&gt;
      &lt;mn&gt;2&lt;/mn&gt;
      &lt;mo&gt;&amp;InvisibleTimes;&lt;/mo&gt;
      &lt;mi&gt;a&lt;/mi&gt;
    &lt;/mrow&gt;
  &lt;/mfrac&gt;
&lt;/mrow&gt;
</pre><p>
                  
                  The 
                  <a href="chapter3.html#presm.mfrac"><code>mfrac</code></a>  and 
                  <a href="chapter3.html#presm.mroot"><code>msqrt</code></a>  
                  elements are used for generating fractions
                  and square roots, respectively.
               </p>
               <p>Notice that the "plus or minus" sign is given by
                  the entity name <code>&amp;PlusMinus;</code>,
                  this is equivalent to using the character reference &amp;#00B1;.
                  MathML provides a very comprehensive list of character names for
                  mathematical symbols. In addition to the mathematical symbols needed for
                  screen and print rendering, MathML provides symbols to facilitate audio
                  rendering.  For audio rendering, it is important to be able to
                  automatically determine whether
                  
               </p><pre>
&lt;mrow&gt;
  &lt;mi&gt;z&lt;/mi&gt;
  &lt;mfenced&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;/mfenced&gt;
&lt;/mrow&gt;
</pre><p>
                  
                  should be read as '<var>z</var> times the quantity <var>x</var> plus
                  <var>y</var>' or '<var>z</var> of <var>x</var> plus <var>y</var>'.  The
                  characters <code>&amp;InvisibleTimes;</code> (U+2062) and <code>&amp;ApplyFunction;</code> (U+2061) provide a way for authors to
                  directly encode the distinction for audio renderers.  For instance, in
                  the first case <code>&amp;InvisibleTimes;</code> (U+2062) should
                  be inserted after the line containing the <var>z</var>.  MathML also
                  introduces entities like <code>&amp;dd;</code> (U+2146) 
                  representing a "differential d", which renders with slightly
                  different spacing in print and can be rendered as "d" or
                  "with respect to" in speech. Unless content tags, or some
                  other mechanism, are used to eliminate the ambiguity, authors should
                  always use these characters here referred to as entities, in order to
                  make their documents more accessible.
               </p>
               <p>Notation: 
                  <img src="image/f2008.gif" alt="A=\left[\begin{array}{cc} x &amp; y \\ z &amp; w \end{array}\right]" align="middle">.
               </p>
               <p>Markup:
                  
               </p><pre>
&lt;mrow&gt;
  &lt;mi&gt;A&lt;/mi&gt;
  &lt;mo&gt;=&lt;/mo&gt;
  &lt;mfenced open="[" close="]"&gt;
    &lt;mtable&gt;
      &lt;mtr&gt;
         &lt;mtd&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mtd&gt;
         &lt;mtd&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;/mtd&gt;
      &lt;/mtr&gt;
      &lt;mtr&gt;
         &lt;mtd&gt;&lt;mi&gt;z&lt;/mi&gt;&lt;/mtd&gt;
         &lt;mtd&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;/mtd&gt;
      &lt;/mtr&gt;
    &lt;/mtable&gt;
  &lt;/mfenced&gt;
&lt;/mrow&gt;
</pre><p> The <code>mtable</code> element denotes that a
                  MathML table is being created.  The <code>mtr</code>
                  specifies a row of the table and the <code>mtd</code>
                  element holds the data for an element of a row.  Most elements have a
                  number of attributes that control the details of their screen and
                  print rendering. For example, there are several attributes for the <code>mfenced</code> element that controls what delimiters
                  should be used at the beginning and the end of the grouped expression
                  above. The attributes for operator elements given using <code>&lt;mo&gt;</code> are set to default values determined by a
                  dictionary. For the suggested MathML operator dictionary, see <a href="appendixf.html">Appendix&nbsp;F Operator Dictionary</a>.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="fund.cont" id="fund.cont"></a>2.3.2 Content Examples
               </h3>
               <p>Notation: <var>x</var><sup>2</sup> + 4<var>x</var> + 4 = 0.
               </p>
               <p>Markup:
                  
               </p><pre>
&lt;mrow&gt;
&lt;apply&gt;
  &lt;eq/&gt;
  &lt;apply&gt;
    &lt;plus/&gt;
    &lt;apply&gt;
      &lt;power/&gt;
      &lt;ci&gt;x&lt;/ci&gt;
      &lt;cn&gt;2&lt;/cn&gt;
    &lt;/apply&gt;
    &lt;apply&gt;
      &lt;times/&gt;
      &lt;cn&gt;4&lt;/cn&gt;
      &lt;ci&gt;x&lt;/ci&gt;
    &lt;/apply&gt;
    &lt;cn&gt;4&lt;/cn&gt;
  &lt;/apply&gt;
  &lt;cn&gt;0&lt;/cn&gt;
&lt;/apply&gt;
&lt;/mrow&gt;
</pre><p>
                  Note that the <code>apply</code> element is used for
                  relations, operators and functions.
               </p>
               <p>Notation:
                  <img src="image/f2007.gif" alt="x = \frac{\mathop{\pm}(-b,\sqrt{b^2 - 4ac})}{2a}" align="middle">.
               </p>
               <p>Markup:
                  
               </p><pre>
&lt;apply&gt;
  &lt;eq/&gt;
  &lt;ci&gt;x&lt;/ci&gt;
  &lt;apply&gt;
    &lt;divide/&gt;
    &lt;apply&gt;
      &lt;csymbol&gt;&amp;PlusMinus;&lt;/csymbol&gt;
      &lt;apply&gt;
        &lt;minus/&gt;
        &lt;ci&gt;b&lt;/ci&gt;
      &lt;/apply&gt;
      &lt;apply&gt;
        &lt;root/&gt;
        &lt;degree&gt;&lt;cn&gt;2&lt;/cn&gt;&lt;/degree&gt;
        &lt;apply&gt;
          &lt;minus/&gt;
          &lt;apply&gt;
            &lt;power/&gt;
            &lt;ci&gt;b&lt;/ci&gt;
            &lt;cn&gt;2&lt;/cn&gt;
          &lt;/apply&gt;
          &lt;apply&gt;
            &lt;times/&gt;
            &lt;cn&gt;4&lt;/cn&gt;
            &lt;ci&gt;a&lt;/ci&gt;
            &lt;ci&gt;c&lt;/ci&gt;
          &lt;/apply&gt;
        &lt;/apply&gt;
      &lt;/apply&gt;
    &lt;/apply&gt;
    &lt;apply&gt;
      &lt;times/&gt;
      &lt;cn&gt;2&lt;/cn&gt;
      &lt;ci&gt;a&lt;/ci&gt;
    &lt;/apply&gt;
  &lt;/apply&gt;
&lt;/apply&gt;
</pre><p>
                  MathML content markup does not directly contain an element for the
                  "plus or minus" operation. Therefore, we use the 
                  <code>csymbol</code> element to specify this operator.
                  Note that the default presentation is given here in prefix form,
                  although a renderer may recognize this operator and render it as
                  infix.
                  Alternatively the Mixed Markup style shown below may be used to
                  specify a presentation form for this expression as well as the Content
                  Markup.
               </p>
               <p>Notation:
                  <img src="image/f2011.gif" alt="A=\left(\begin{array}{cc} x &amp; y \\ z &amp; w \end{array}\right)" align="middle">.
               </p>
               <p>Markup:
                  
               </p><pre>
&lt;mrow&gt;
&lt;apply&gt;
  &lt;eq/&gt;
  &lt;ci&gt;A&lt;/ci&gt;
  &lt;matrix&gt;
    &lt;matrixrow&gt;
      &lt;ci&gt;x&lt;/ci&gt;
      &lt;ci&gt;y&lt;/ci&gt;
    &lt;/matrixrow&gt;
    &lt;matrixrow&gt;
      &lt;ci&gt;z&lt;/ci&gt;
      &lt;ci&gt;w&lt;/ci&gt;
    &lt;/matrixrow&gt;
  &lt;/matrix&gt;
&lt;/apply&gt;
&lt;/mrow&gt;
</pre><p>
                  
                  Here we have used the <code>matrix</code> element, and the
                  <code>matrixrow</code> element to wrap the entries in a row of
                  the matrix.  Note that, by default, the rendering of the content
                  element <code>matrix</code> includes enclosing parentheses,
                  so we need not directly encode them. This is quite different from the
                  presentation element <code>mtable</code> which may or may
                  not refer to a matrix, and hence requires explicit encoding of
                  parentheses if they are desired.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="fund.mix" id="fund.mix"></a>2.3.3 Mixed Markup Examples
               </h3>
               <p>Notation:
                  <img src="image/f2012.gif" alt="\displaystyle \int_1^t \frac{\diffd x}{x}" align="middle">.
               </p>
               <p>Markup:
                  
               </p><pre>
&lt;mrow&gt;
&lt;semantics&gt;
  &lt;mrow&gt;
    &lt;msubsup&gt;
      &lt;mo&gt;&amp;int;&lt;/mo&gt;
      &lt;mn&gt;1&lt;/mn&gt;
      &lt;mi&gt;t&lt;/mi&gt;
    &lt;/msubsup&gt;
    &lt;mfrac&gt;
      &lt;mrow&gt;
        &lt;mo&gt;&amp;dd;&lt;/mo&gt;
        &lt;mi&gt;x&lt;/mi&gt;
      &lt;/mrow&gt;
      &lt;mi&gt;x&lt;/mi&gt;
    &lt;/mfrac&gt;
  &lt;/mrow&gt;
  &lt;annotation-xml encoding="MathML-Content"&gt;
    &lt;apply&gt;
      &lt;int/&gt;
      &lt;bvar&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;/bvar&gt;
      &lt;lowlimit&gt;&lt;cn&gt;1&lt;/cn&gt;&lt;/lowlimit&gt;
      &lt;uplimit&gt;&lt;ci&gt;t&lt;/ci&gt;&lt;/uplimit&gt;
      &lt;apply&gt;
        &lt;divide/&gt;
        &lt;cn&gt;1&lt;/cn&gt;
        &lt;ci&gt;x&lt;/ci&gt;
      &lt;/apply&gt;
    &lt;/apply&gt;
  &lt;/annotation-xml&gt;
&lt;/semantics&gt;
&lt;/mrow&gt;
</pre><p> In this example, we use the <code>semantics</code>
                  element to provide a MathML content expression to serve as a
                  "semantic annotation" for a presentation expression.  In
                  the display markup, we have used the <code>msubsup</code>
                  element to attach a subscript and a superscript to an expression, in
                  this case the integral sign. We also used entities <code>&amp;int;</code> and <code>&amp;dd;</code> to specify the
                  integral and differential symbols.
                  
               </p>
               <p>
                  The <code>semantics</code> element has as its first child
                  the expression being annotated, and the subsequent children are the
                  annotations.  There is no restriction on the kind of annotation that
                  can be attached using the <code>semantics</code> element.
                  For example, one might give a T<sub>E</sub>X encoding, or computer algebra
                  input in an annotation. The type of annotation is specified by the <code>encoding</code> attribute and the <code>annotation</code> and <code>annotation-xml</code> elements.
               </p>
               <p>Another common use of the <code>semantics</code>
                  element arises  when one  wants to use  a content  coding, and
                  provide  a suggestion for  its presentation.  In such  a case,
                  applied to  the formula  above we would  have the  markup: 
                  
               </p><pre>
&lt;semantics&gt;
  &lt;apply&gt;
    &lt;int/&gt;
    &lt;bvar&gt;&lt;ci&gt;x&lt;/ci&gt;&lt;/bvar&gt;
    &lt;lowlimit&gt;&lt;cn&gt;1&lt;/cn&gt;&lt;/lowlimit&gt;
    &lt;uplimit&gt;&lt;ci&gt;t&lt;/ci&gt;&lt;/uplimit&gt;
    &lt;apply&gt;
      &lt;divide/&gt;
      &lt;cn&gt;1&lt;/cn&gt;
      &lt;ci&gt;x&lt;/ci&gt;
    &lt;/apply&gt;
  &lt;/apply&gt;
  &lt;annotation-xml encoding="MathML-Presentation"&gt;
    &lt;mrow&gt;
      &lt;msubsup&gt;
        &lt;mo&gt;&amp;int;&lt;/mo&gt;
        &lt;mn&gt;1&lt;/mn&gt;
        &lt;mi&gt;t&lt;/mi&gt;
      &lt;/msubsup&gt;
      &lt;mfrac&gt;
        &lt;mrow&gt;
          &lt;mo&gt;&amp;dd;&lt;/mo&gt;
          &lt;mi&gt;x&lt;/mi&gt;
        &lt;/mrow&gt;
        &lt;mi&gt;x&lt;/mi&gt;
      &lt;/mfrac&gt;
    &lt;/mrow&gt;
  &lt;/annotation-xml&gt;
&lt;/semantics&gt;
</pre><p>This kind of annotation is useful when something other than the default
                  rendering of the content encoding is desired. For example, by default, some
                  renderers might layout the integrand something like "(1/<var>x</var>)
                  d<var>x</var>".  Specifying that the integrand should by preference
                  render as "d<var>x</var>/<var>x</var>" instead can be accomplished
                  with the use of a MathML Presentation annotation as shown.  Be aware,
                  however, that renderers are not required to take into account information
                  contained in annotations, and what use is made of them, if any, will depend
                  on the renderer.
               </p>
            </div>
         </div>
         <div class="div2">
            
            <h2><a name="fund.syntax" id="fund.syntax"></a>2.4 MathML Syntax and Grammar
            </h2>
            <div class="div3">
               
               <h3><a name="id.2.4.1" id="id.2.4.1"></a>2.4.1 MathML Syntax and Grammar
               </h3>
               <p>MathML is an application of <a href="appendixk.html#XML">[XML]</a>, or Extensible
                  Markup Language, and as such its syntax is governed by the rules of
                  XML syntax, and its grammar is in part specified by a DTD, or Document
                  Type Definition.  In other words, the details of using tags,
                  attributes, entity references and so on are defined in the XML
                  language specification, and the details about MathML element and
                  attribute names, which elements can be nested inside each other, and
                  so on are specified in the MathML DTD. This is in <a href="appendixa.html">Appendix&nbsp;A Parsing MathML</a>.
               </p>
               <p>The W3C in seeking to increase the flexibility of the use of XML
                  for the Web, and to encourage modularization of applications built
                  with XML, has found that the basic form of a DTD is not sufficiently
                  flexible. Therefore, a W3C Working Group was created to develop a
                  specification for XML Schemas <a href="appendixk.html#XMLSchemas">[XMLSchemas]</a>, which are
                  specification documents that will eventually supersede DTDs.  MathML
                  2.0 is consciously designed so that mathematics may take advantage of the
                  latest in the evolving Web technology.  Thus, there is to be a schema
                  for MathML.  For further information on a MathML schema see <a href="appendixa.html">Appendix&nbsp;A Parsing MathML</a> and the <a href="http://www.w3.org/Math/">MathML
                     Home Page</a>.
               </p>
               <p>However, MathML also specifies some syntax and grammar rules in addition
                  to the general rules it inherits as an XML application.  These rules allow
                  MathML to encode a great deal more information than would ordinarily be
                  possible with pure XML, without introducing many more elements, and using a
                  substantially more complex DTD or schema.  A grammar for content markup
                  expressions is given in <a href="appendixb.html">Appendix&nbsp;B Content Markup Validation Grammar</a>.  Of course, one drawback to
                  using MathML specific rules is that they are invisible to generic XML
                  processors and validators.
               </p>
               <p>There are basically two kinds of additional MathML grammar and syntax
                  rules.  One kind involves placing additional criteria on attribute values.
                  For example, it is not possible in pure XML to require that an attribute
                  value be a positive integer.  The second kind of rule specifies more
                  detailed restrictions on the child elements (for example on ordering) than
                  are given in the DTD or even a schema. For example, it is not possible in
                  XML to specify that the first child be interpreted one way, and the second
                  in another.
               </p>
               <p>The following sections discuss features both of XML syntax and grammar
                  in general, and of MathML in particular.  Throughout the remainder of the
                  MathML specification, we will usually take care to distinguish between
                  usage required by XML syntax and the MathML DTD (and schema) and usage
                  required by MathML specific rules. However, we will frequently allude to
                  "MathML errors" without identifying which part of the
                  specification is being violated.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="fund.xmlsyntax" id="fund.xmlsyntax"></a>2.4.2 An XML Syntax Primer
               </h3>
               <p>Since MathML is an application of XML, the MathML specification uses the
                  terminology of XML to describe it. Briefly, XML data is composed of Unicode
                  characters (which include ordinary ASCII characters), "entity
                  references" (informally called "entities") such as <code>&amp;lt;</code> which usually represent "extended
                  characters", and "elements" such as <code>&lt;mi
                     fontstyle="normal"&gt; x &lt;/mi&gt;</code>.
               </p>
               <p>An element quite often encloses other XML data called its
                  "content", or "body", between a "start
                  tag" (sometimes called a "begin tag") and an "end
                  tag", much as in HTML.  There are also "empty elements"
                  such as <code>&lt;plus/&gt;</code>, whose start tag ends with
                  <code>/&gt;</code> to indicate that the element has no content or end
                  tag. The start tag can contain named parameters called
                  "attributes", such as <code>fontstyle="normal"</code> in the
                  example above.  For further details on XML, consult the XML specification
                  <a href="appendixk.html#XML">[XML]</a>.
               </p>
               <p>As XML is case-sensitive, MathML element and attribute names are
                  case-sensitive.  For reasons of legibility, the MathML specification
                  defines them almost all in lowercase.
               </p>
               <p>In formal discussions of XML markup, a distinction is maintained
                  between an element, such as an <code>mrow</code> element,
                  and the tags <code>&lt;mrow&gt;</code> and <code>&lt;/mrow&gt;</code> marking it. What is between the <code>&lt;mrow&gt;</code> start tag and the <code>&lt;/mrow&gt;</code> end tag is the content, or body, of the <code>mrow</code> element.  An "empty element"
                  such as <code>none</code> is defined to have no body, and
                  so has a single tag of the form <code>&lt;none/&gt;</code>.
                  Usually, the distinction between elements and tags will not be so
                  finely drawn in this specification. For instance, we will sometimes
                  refer to the <code>&lt;mrow&gt;</code> and <code>&lt;none/&gt;</code> elements, really meaning the elements whose
                  tags these are, in order that references to elements are visually
                  distinguishable from references to attributes.  However, the words
                  "element" and "tag" themselves will be used
                  strictly in accordance with XML terminology.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="id.2.4.3" id="id.2.4.3"></a>2.4.3 Children versus Arguments
               </h3>
               <p>Many MathML elements require a specific number of child elements or
                  attach additional meanings to children in certain positions.  As noted
                  above, these kinds of requirements are MathML specific, and cannot be
                  given entirely using XML syntax and grammar. When the children of a
                  given MathML element are subject to these kinds of additional
                  conditions, we will often refer to them as <em>arguments</em>
                  instead of merely as children, in order to emphasize their MathML
                  specific usage. Note that, especially in <a href="chapter3.html">Chapter&nbsp;3 Presentation Markup</a>, the
                  term "argument" is usually used in this technical sense,
                  unless otherwise noted, and therefore refers to a child element.
               </p>
               <p>In the detailed discussions of element syntax given with each
                  element throughout the MathML specification, the number of required
                  arguments and their order is implicitly indicated by giving names for
                  the arguments at various positions. This information is also given for
                  presentation elements in the table of argument requirements in
                  <a href="chapter3.html#presm.reqarg">Section&nbsp;3.1.3 Required Arguments</a>, and for content elements in <a href="appendixb.html">Appendix&nbsp;B Content Markup Validation Grammar</a>.
               </p>
               <p>A  few elements have  other requirements  on the  number or
                  type of arguments. These additional requirements are described
                  together with the individual elements.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="fund.attval" id="fund.attval"></a>2.4.4 MathML Attribute Values
               </h3>
               <p>According  to the  XML  language specification,  attributes
                  given to elements must have one of the forms
                  
               </p><pre>
attribute-name = "value"
</pre><p>
                  or
                  
               </p><pre>
attribute-name = 'value'
</pre><p>
                  where whitespace around the '=' is optional.
               </p>
               <p>Attribute names are generally shown in a <code>monospaced</code> font within descriptive text in this
                  specification, just as the <code>monospaced</code> font is used
                  for examples.
               </p>
               <p>An attribute's value, which in general in MathML can be a string of
                  arbitrary characters, must be surrounded by a pair of either double quotes
                  (<code>"</code>) or single quotes (<code>'</code>). The kind of quotes
                  not used to surround the value may be included within it.
               </p>
               <p>MathML uses a more complicated syntax for attribute values than the
                  generic XML syntax required by the MathML DTD. These additional rules are
                  intended for use by MathML applications, and it is a MathML error to
                  violate them, though they cannot be enforced by XML processing. The MathML
                  syntax of each attribute value is specified in the table of attributes
                  provided with the description of each element, using a notation described
                  below.  When MathML applications process attribute values, whitespace
                  is ignored except to separate letter and digit sequences into
                  individual words or numbers.  Attribute values may contain any MathML
                  characters listed in <a href="chapter6.html#chars.mathmlchars">Section&nbsp;6.2 MathML Characters</a> permitted by the syntax
                  restrictions for an attribute.  Character data can be
                  included directly in attribute values, or by using entity references
                  as described in <a href="chapter6.html#chars.unicodechars">Section&nbsp;6.2.1 Unicode Character Data</a>.
                  
               </p>
               <p>In particular, the characters <code>"</code>, <code>'</code>,
                  <code>&amp;</code> and <code>&lt;</code> can be included in MathML
                  attribute values (when permitted by the attribute value syntax) using the
                  entity references <code>&amp;quot;</code>, <code>&amp;apos;</code>,
                  <code>&amp;amp;</code> and <code>&amp;lt;</code>, respectively.
                  
               </p>
               <p>The MathML DTD provided in <a href="appendixa.html">Appendix&nbsp;A Parsing MathML</a> declares most
                  attribute value types as <b>CDATA</b> strings. This permits increased
                  interoperability with existing SGML and XML software and allows extension
                  to the lists of predefined values.  Similar sorts of considerations apply
                  with XML schemas.
               </p>
               <div class="div4">
                  
                  <h4><a name="id.2.4.4.1" id="id.2.4.4.1"></a>2.4.4.1 Syntax notations used in the MathML specification
                  </h4>
                  <p>To  describe  the  MathML-specific  syntax  of  permissible
                     attribute values, the  following conventions and notations are
                     used for most attributes in the present document.
                     
                  </p>
                  <table id="fund.table-attval">
                     <thead>
                        <tr>
                           <th rowspan="1" colspan="1">Notation</th>
                           <th rowspan="1" colspan="1">What it matches</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="1" colspan="1">number</td>
                           <td rowspan="1" colspan="1">decimal integer or rational number (a string of digits
                              with one decimal point), 
                              optionally starting with '-'
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">unsigned-number</td>
                           <td rowspan="1" colspan="1">decimal integer or real number, no sign</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">integer</td>
                           <td rowspan="1" colspan="1">decimal integer, optionally starting with '-'</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">positive-integer</td>
                           <td rowspan="1" colspan="1">decimal integer, unsigned, not 0</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">string</td>
                           <td rowspan="1" colspan="1">arbitrary string (always the entire attribute value)</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">character</td>
                           <td rowspan="1" colspan="1">single non-whitespace character, or MathML entity reference;
                              whitespace separation is optional
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">#rrggbb</td>
                           <td rowspan="1" colspan="1">RGB color value; the three pairs of
                              hexadecimal digits in the example #5599dd define proportions
                              of red, green and blue on a scale of x00 through xFF, which
                              gives a strong sky blue.
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">h-unit</td>
                           <td rowspan="1" colspan="1">unit of horizontal length (allowable units are listed below)</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">v-unit</td>
                           <td rowspan="1" colspan="1">unit of vertical length (allowable units are listed below)</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">css-fontfamily</td>
                           <td rowspan="1" colspan="1">explained in the CSS subsection below</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">css-color-name</td>
                           <td rowspan="1" colspan="1">explained in the CSS subsection below</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">other italicized words</td>
                           <td rowspan="1" colspan="1">explained in the text for each attribute</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">form +</td>
                           <td rowspan="1" colspan="1">one or more instances of 'form'</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">form *</td>
                           <td rowspan="1" colspan="1">zero or more instances of 'form'</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">f1 f2 ... fn</td>
                           <td rowspan="1" colspan="1">one instance of each form, in sequence, perhaps separated by whitespace</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">f1 | f2 | ... | fn</td>
                           <td rowspan="1" colspan="1">any one of the specified forms</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">[ form ]</td>
                           <td rowspan="1" colspan="1">an optional instance of 'form'</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">( form )</td>
                           <td rowspan="1" colspan="1">same as form</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">word in plain text</td>
                           <td rowspan="1" colspan="1">that word, literally present in the attribute value (unless it is 
                              obviously part of an explanatory phrase)
                           </td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">quoted symbol</td>
                           <td rowspan="1" colspan="1">that symbol, literally present in attribute value (e.g. "+" or '+')</td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The order of precedence of the syntax notation operators is,
                     from highest to lowest precedence:
                     
                  </p>
                  <ul>
                     <li class="sitem">form +    or    form *</li>
                     <li class="sitem">f1 f2 ... fn    (sequence of forms)</li>
                     <li class="sitem">f1 | f2 | ... | fn    (alternative forms)</li>
                  </ul>
                  <p>A <em>string</em> can contain arbitrary characters which are
                     specifiable within XML <b>CDATA</b> attribute values.  See <a href="chapter6.html">Chapter&nbsp;6 Characters, Entities and Fonts</a> for a discussion and complete listing of MathML
                     characters.  No syntax rule in MathML includes a <em>string</em>
                     as only part of an attribute value, only as the entire value.
                  </p>
                  <p>Adjacent keywords and numbers must be separated by whitespace in the
                     actual attribute values, except for unit identifiers (denoted by <code>h-unit</code> or <code>v-unit</code> syntax symbols)
                     following numbers.  Whitespace is not otherwise required, but is permitted
                     between any of the tokens listed above, except (for compatibility with
                     CSS) immediately before unit identifiers, between the '-' signs and digits
                     of negative numbers, or between <code>#</code> or "rrggbb" and "rgb".
                  </p>
                  <p>Numerical attribute values for dimensions that should depend upon the
                     current font can be given in font-related units, or in named absolute units
                     (described in a separate subsection below). Horizontal dimensions are
                     conventionally given in <code>em</code>'s, and vertical dimensions in
                     <code>ex</code>'s, by immediately following a number by one of the unit
                     identifiers "em" or "ex".  For
                     example, the horizontal spacing around an operator such as "+"
                     is conventionally given in "em"s, though other units
                     can be used. Using font-related units is usually preferable to using
                     absolute units, since it allows renderings to grow or shrink in proportion
                     to the current font size.
                  </p>
                  <p>For most numerical attributes, only those in a subset of the expressible
                     values are sensible; values outside this subset are not errors, unless
                     otherwise specified, but rather are rounded up or down (at the discretion
                     of the renderer) to the closest value within the allowed subset.  The set
                     of allowed values may depend on the renderer, and is not specified by
                     MathML.
                  </p>
                  <p>If a numerical value within an attribute value syntax description is
                     declared to allow a minus sign ('-'), e.g. <code>number</code> or
                     <code>integer</code>, it is not a syntax error when one is
                     provided in cases where a negative value is not sensible. Instead, the
                     value should be handled by the processing application as described in the
                     preceding paragraph.  An explicit plus sign ('+') is not allowed as part of
                     a numerical value except when it is specifically listed in the syntax (as a
                     quoted '+' or "+"), and its presence can change the meaning of the
                     attribute value (as documented with each attribute which permits it).
                  </p>
                  <p>The    symbols     <code>h-unit</code>,    <code>v-unit</code>, <code>css-fontfamily</code>,
                     and <code>css-color-name</code> are explained in the
                     following subsections.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="fund.units" id="fund.units"></a>2.4.4.2 Attributes with units
                  </h4>
                  <p>Some attributes accept horizontal or vertical lengths as numbers
                     followed by a "unit identifier" (often just called a
                     "unit"). The syntax symbols <code>h-unit</code> and
                     <code>v-unit</code> refer to a unit for horizontal or vertical
                     length, respectively. The possible units and the lengths they refer to are
                     shown in the table below; they are the same for horizontal and vertical
                     lengths, but the syntax symbols are distinguished in attribute syntaxes as
                     a reminder of the direction each is used in.
                  </p>
                  <p>The unit identifiers and meanings are taken from CSS.  However, the
                     syntax of numbers followed by unit identifiers in MathML is not identical
                     to the syntax of length values with units in CSS style sheets, since
                     numbers in CSS cannot end with decimal points, and are allowed to start
                     with '+' signs.
                  </p>
                  <p>The possible horizontal or vertical units in MathML are:
                     
                  </p>
                  <table>
                     <thead>
                        <tr>
                           <th rowspan="1" colspan="1">Unit identifier</th>
                           <th rowspan="1" colspan="1">Unit description</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="1" colspan="1">em</td>
                           <td rowspan="1" colspan="1">em (font-relative unit traditionally used for horizontal lengths)</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">ex</td>
                           <td rowspan="1" colspan="1">ex (font-relative unit traditionally used for vertical lengths)</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">px</td>
                           <td rowspan="1" colspan="1">pixels, or pixel size of the current display</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">in</td>
                           <td rowspan="1" colspan="1">inches (1 inch = 2.54 centimeters)</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">cm</td>
                           <td rowspan="1" colspan="1">centimeters</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">mm</td>
                           <td rowspan="1" colspan="1">millimeters</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">pt</td>
                           <td rowspan="1" colspan="1">points (1 point = 1/72 inch)</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">pc</td>
                           <td rowspan="1" colspan="1">picas (1 pica = 12 points)</td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">%</td>
                           <td rowspan="1" colspan="1">percentage of default value</td>
                        </tr>
                     </tbody>
                  </table>
                  <p>The typesetting units "em" and "ex" are defined in <a href="appendixh.html">Appendix&nbsp;H Glossary</a>, and
                     discussed further under "Additional notes" below.
                  </p>
                  <p><code>%</code> is a "relative unit"; when an attribute
                     value is given as "n%" (for any numerical value "n"), the value being specified is the default value for
                     the property being controlled multiplied by "n"
                     divided by 100. The default value (or the way in which it is obtained, when
                     it is not constant) is listed in the table of attributes for each element,
                     and its meaning is described in the subsequent documentation about that
                     attribute.  (The <code>mpadded</code> element has its own syntax
                     for <code>%</code> and does not allow it as a unit identifier.)
                  </p>
                  <p>For consistency with CSS, length units in MathML are rarely
                     optional. When they are, the unit symbol is enclosed in square brackets in
                     the attribute syntax, following the number to which it applies,
                     e.g. <code>number [ h-unit ]</code>. The meaning of specifying no unit is
                     given in the documentation for each attribute; in general it is that the
                     number given is a multiplier for the default value of the attribute. (In
                     such cases, specifying the number "nnn" without a unit
                     is equivalent to specifying the number "nnn" times 100
                     followed by <code>%</code>.  For example, <code>&lt;mo maxsize="2"&gt; (
                        &lt;/mo&gt;</code> is equivalent to <code>&lt;mo maxsize="200%"&gt; (
                        &lt;/mo&gt;</code>.)
                  </p>
                  <p>As a special exception (also consistent with CSS), a numerical value
                     equal to 0 need not be followed by a unit identifier even if the syntax
                     specified here requires one. In such cases, the unit identifier (or lack of
                     one) would not matter, since 0 times any unit is 0.
                  </p>
                  <p>For most attributes, the typical unit which would be used to describe
                     them in typesetting is chosen as the one used in that attribute's default
                     value in this specification; when a specific default value is not given,
                     the typical unit is usually mentioned in the syntax table or in the
                     documentation for that attribute. The most common units are <code>em</code>
                     or <code>ex</code>. However, any unit can be used, unless otherwise
                     specified for a specific attribute.
                  </p>
                  <div class="div5">
                     
                     <h5><a name="id.2.4.4.2.1" id="id.2.4.4.2.1"></a>2.4.4.2.1 Additional notes about units
                     </h5>
                     <p>Note that some attributes, e.g.  <code>framespacing</code> 
                        on a <code>&lt;mtable&gt;</code>,
                        can contain more than one numerical value, each followed by its own
                        unit.
                     </p>
                     <p>It is conventional to use the font-relative unit <code>ex</code> mainly
                        for vertical lengths, and <code>em</code> mainly for horizontal lengths,
                        but this is not required. These units are relative to the font and font size
                        which would be used for rendering the element in whose attribute value they
                        are specified, which means they should be interpreted <em>after</em>
                        attributes such as <code>fontfamily</code> and <code>fontsize</code> are processed, if those occur on the same
                        element, since changing the current font or font size can change the length
                        of one of these units.
                     </p>
                     <p>The definition of the length of each unit, but not the MathML syntax for
                        length values, is as specified in CSS, except that if a font provides
                        specific values for <code>em</code> and <code>ex</code> which differ from
                        the values defined by CSS (the font size and "x"-height
                        respectively), those values should be used.
                     </p>
                  </div>
               </div>
               <div class="div4">
                  
                  <h4><a name="fund.cssatt" id="fund.cssatt"></a>2.4.4.3 CSS-compatible attributes
                  </h4>
                  <p>Several MathML attributes, listed below, correspond closely to text
                     rendering properties defined originally in <a href="appendixk.html#CSS1">[CSS1]</a>.
                     In MathML 1.01, the names and values of these attributes were aligned
                     with the CSS Recommendation where possible.  This was done so that
                     renderers in CSS environments could query the environment for the
                     corresponding property when determining the default values for the
                     attributes. 
                  </p>
                  <p>Allowing style properties to be set both via MathML attributes and
                     CSS style sheets has drawbacks.  At a minimum, its confusing, and at
                     worst, it leads to the meaning of equations being inadvertently
                     changed by document-wide CSS changes.  For these reasons, these
                     attributes have been <a href="chapter7.html#interf.deprec">deprecated</a>.
                     In their place, MathML 2.0 introduces four new mathematical style
                     attributes.  These attributes use logical values to better capture the
                     abstract categories of letter-like symbols used in math, and afford a
                     much cleaner separation between MathML and CSS.  See <a href="chapter3.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token
                        elements</a> for more details.
                  </p>
                  <p>For reference, a table showing the correspondence of the deprecated
                     MathML 1.01 style attribute with the CSS counterparts is given below:
                  </p>
                  <table>
                     <thead>
                        <tr>
                           <th rowspan="1" colspan="1">MathML attribute</th>
                           <th rowspan="1" colspan="1">CSS property</th>
                           <th rowspan="1" colspan="1">syntax symbol</th>
                           <th rowspan="1" colspan="1">MathML elements</th>
                           <th rowspan="1" colspan="1">refer to</th>
                        </tr>
                     </thead>
                     <tbody>
                        <tr>
                           <td rowspan="1" colspan="1">fontsize</td>
                           <td rowspan="1" colspan="1">font-size</td>
                           <td rowspan="1" colspan="1">-</td>
                           <td rowspan="1" colspan="1">presentation tokens; <code>mstyle</code></td>
                           <td rowspan="1" colspan="1"><a href="chapter3.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token
                                 elements</a></td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">fontweight</td>
                           <td rowspan="1" colspan="1">font-weight</td>
                           <td rowspan="1" colspan="1">-</td>
                           <td rowspan="1" colspan="1">presentation tokens; <code>mstyle</code></td>
                           <td rowspan="1" colspan="1"><a href="chapter3.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token
                                 elements</a></td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">fontstyle</td>
                           <td rowspan="1" colspan="1">font-style</td>
                           <td rowspan="1" colspan="1">-</td>
                           <td rowspan="1" colspan="1">presentation tokens; <code>mstyle</code></td>
                           <td rowspan="1" colspan="1"><a href="chapter3.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token
                                 elements</a></td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">fontfamily</td>
                           <td rowspan="1" colspan="1">font-family</td>
                           <td rowspan="1" colspan="1">css-fontfamily</td>
                           <td rowspan="1" colspan="1">presentation tokens; <code>mstyle</code></td>
                           <td rowspan="1" colspan="1"><a href="chapter3.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token
                                 elements</a></td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">color</td>
                           <td rowspan="1" colspan="1">color</td>
                           <td rowspan="1" colspan="1">css-color-name</td>
                           <td rowspan="1" colspan="1">presentation tokens; <code>mstyle</code></td>
                           <td rowspan="1" colspan="1"><a href="chapter3.html#presm.mstyle">Section&nbsp;3.3.4 Style Change (mstyle)</a></td>
                        </tr>
                        <tr>
                           <td rowspan="1" colspan="1">background</td>
                           <td rowspan="1" colspan="1">background</td>
                           <td rowspan="1" colspan="1">css-color-name</td>
                           <td rowspan="1" colspan="1"><code>mstyle</code></td>
                           <td rowspan="1" colspan="1"><a href="chapter3.html#presm.mstyle">Section&nbsp;3.3.4 Style Change (mstyle)</a></td>
                        </tr>
                     </tbody>
                  </table>
                  <p>See also <a href="chapter2.html#fund.globatt">Section&nbsp;2.4.5 Attributes Shared by all MathML Elements</a> below for a discussion of the <code>class</code>, <code>style</code> and <code>id</code> attributes for use with style sheets.
                  </p>
                  <div class="div5">
                     
                     <h5><a name="id.2.4.4.3.1" id="id.2.4.4.3.1"></a>2.4.4.3.1 Order of processing attributes versus style sheets
                     </h5>
                     <p>CSS or analogous style sheets can specify changes to rendering
                        properties of selected MathML elements.  Since rendering properties
                        can also be changed by attributes on an element, or be changed automatically
                        by the renderer, it is necessary to specify the order in which changes
                        from various sources occur. An example of automatic adjustment is what
                        happens for <code>fontsize</code>, as explained in the
                        discussion on <code>scriptlevel</code> in <a href="chapter3.html#presm.mstyle">Section&nbsp;3.3.4 Style Change (mstyle)</a>.  In the case of "absolute" changes,
                        i.e., setting a new property value independent of the old value (as
                        opposed to "relative" changes, such as increments or
                        multiplications by a factor), the absolute change performed last will
                        be the only absolute change which is effective, so the sources of
                        changes which should have the highest priority must be processed
                        last.
                     </p>
                     <p>In the  case of  CSS, the order  of processing  of changes
                        from  various  sources   which  affect  one  MathML  element's
                        rendering properties should be as follows:
                     </p>
                     <blockquote>
                        <p>
                           (first changes; lowest priority)
                           
                        </p>
                     </blockquote>
                     <ul>
                        <li>
                           <p>Automatic changes to properties or attributes based on the type of the
                              parent element, and this element's position in the parent, as for the
                              changes to <code>fontsize</code> in relation to <code>scriptlevel</code> mentioned above; such changes will usually
                              be implemented by the parent element itself before it passes a set of
                              rendering properties to this element
                           </p>
                        </li>
                        <li>
                           <p>From a style sheet from the reader: styles which are <em>not</em>
                              declared "important"
                           </p>
                        </li>
                        <li>
                           <p>Explicit attribute settings on this MathML element</p>
                        </li>
                        <li>
                           <p>From a style sheet from the author: styles which are <em>not</em>
                              declared "important"
                           </p>
                        </li>
                        <li>
                           <p>From a style sheet from the author: styles which <em>are</em>
                              declared "important"
                           </p>
                        </li>
                        <li>
                           <p>From a style sheet from the reader: styles which <em>are</em>
                              declared "important"
                           </p>
                        </li>
                     </ul>
                     <blockquote>
                        <p>
                           (last changes; highest priority)
                           
                        </p>
                     </blockquote>
                     <p>Note that the order of the changes derived from CSS style sheets is
                        specified by CSS itself (this is the order specified by CSS2).
                        The following rationale is related only to the
                        issue of where in this pre-existing order the changes caused by explicit
                        MathML attribute settings should be inserted.
                     </p>
                     <p>Rationale: MathML rendering attributes are analogous to HTML rendering
                        attributes such as <code>align</code>, which the CSS section on
                        cascading order specifies should be processed with the same priority.
                        Furthermore, this choice of priority permits readers, by declaring certain
                        CSS styles as "important", to decide which of their style
                        preferences should override explicit attribute settings in MathML.  Since
                        MathML expressions, whether composed of "presentation" or
                        "content" elements, are primarily intended to convey meaning,
                        with their "graphic design" (if any) intended mainly to aid in
                        that purpose but not to be essential in it, it is likely that readers will
                        often want their own style preferences to have priority; the main exception
                        will be when a rendering attribute is intended to alter the meaning
                        conveyed by an expression, which is generally discouraged in the
                        presentation attributes of MathML.
                     </p>
                  </div>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.2.4.4.4" id="id.2.4.4.4"></a>2.4.4.4 Default values of attributes
                  </h4>
                  <p>Default values for MathML attributes are in general given along with the
                     detailed descriptions of specific elements in the text.  Default values
                     shown in plain text in the tables of attributes for an element are literal
                     (unless they are obviously explanatory phrases), but when italicized are
                     descriptions of how default values can be computed.
                  </p>
                  <p>Default values described as <em>inherited</em> are taken from the
                     rendering environment, as described under <code>mstyle</code>,
                     or in some cases (described individually) from the values of other
                     attributes of surrounding elements, or from certain parts of those
                     values. The value used will always be one which could have been specified
                     explicitly, had it been known; it will never depend on the content or
                     attributes of the same element, only on its environment. (What it means
                     when used may, however, depend on those attributes or the content.)
                  </p>
                  <p>Default values described as <em>automatic</em> should be computed by
                     a MathML renderer in a way which will produce a high-quality rendering; how
                     to do this is not usually specified by the MathML specification. The value
                     computed will always be one which could have been specified explicitly, had
                     it been known, but it will usually depend on the element content and
                     possibly on the rendering environment.
                  </p>
                  <p>Other italicized descriptions of default values which appear in the
                     tables of attributes are explained for each attribute individually.
                  </p>
                  <p>The single or double quotes which are required around attribute values
                     in an XML start tag are not shown in the tables of attribute value syntax
                     for each element, but are shown around example attribute values in the
                     text.
                  </p>
                  <p>Note that, in general, there is no value which can be given explicitly
                     for a MathML attribute which will simulate the effect of not specifying the
                     attribute at all for attributes which are <em>inherited</em> or
                     <em>automatic</em>.  Giving the words "inherited" or
                     "automatic" explicitly will not work, and is not generally
                     allowed.  Furthermore, even for presentation attributes for which a
                     specific default value is documented here, the <code>mstyle</code> element (<a href="chapter3.html#presm.mstyle">Section&nbsp;3.3.4 Style Change (mstyle)</a>) can be
                     used to change this for the elements it contains. Therefore, the MathML DTD
                     declares most presentation attribute default values as <b>#IMPLIED</b>,
                     which prevents XML preprocessors from adding them with any specific default
                     value. This point of view is carried through to the MathML schema.
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.2.4.4.5" id="id.2.4.4.5"></a>2.4.4.5 Attribute values in the MathML DTD
                  </h4>
                  <p>In an XML DTD, allowed attribute values can be declared as general
                     strings, or they can be constrained in various ways, either by enumerating
                     the possible values, or by declaring them to be certain special data
                     types. The choice of an XML attribute type affects the extent to which
                     validity checks can be performed using a DTD.
                  </p>
                  <p>The MathML DTD specifies formal XML attribute types for all MathML
                     attributes, including enumerations of legitimate values in some cases.  In
                     general, however, the MathML DTD is relatively permissive, frequently
                     declaring attribute values as strings; this is done to provide for
                     interoperability with SGML parsers while allowing multiple attributes on
                     one MathML element to accept the same values (such as "true" and "false"), and also to
                     allow extension to the lists of predefined values.
                  </p>
                  <p>At the same time, even though an attribute value may be declared as a
                     string in the DTD, only certain values are legitimate in MathML, as
                     described above and in the rest of this specification.  For example, many
                     attributes expect numerical values.  In the sections which follow, the
                     allowed attribute values are described for each element. To determine when
                     these constraints are actually enforced in the MathML DTD, consult <a href="appendixa.html">Appendix&nbsp;A Parsing MathML</a>.  However, lack of enforcement of a requirement in the DTD
                     does <em>not</em> imply that the requirement is not part of the MathML
                     language itself, or that it will not be enforced by a particular MathML
                     renderer. (See <a href="chapter7.html#interf.error">Section&nbsp;7.2.2 Handling of Errors</a> for a description of how
                     MathML renderers should respond to MathML errors.)
                  </p>
                  <p>Furthermore, the MathML DTD is provided for convenience; although it is
                     intended to be fully compatible with the text of the specification, the
                     text should be taken as definitive if there is a contradiction.  (Any
                     contradictions which may exist between various chapters of the text should
                     be resolved by favoring <a href="chapter6.html">Chapter&nbsp;6 Characters, Entities and Fonts</a> first, then <a href="chapter3.html">Chapter&nbsp;3 Presentation Markup</a>, <a href="chapter4.html">Chapter&nbsp;4 Content Markup</a>, then <a href="chapter2.html#fund.syntax">Section&nbsp;2.4 MathML Syntax and Grammar</a>,
                     and then other parts of the text.)  For the MathML schema the situation
                     will be the same: the published Recommendation text takes precedence.
                     Though this is what is intended to happen, there is a practical difficulty.
                     If the system processing the MathML uses a validating parser, whether it be
                     based on a DTD or on a schema, the process will probably simply stop when
                     it hits something held to be incorrect syntax, whether or not further
                     MathML processing in full harmony with the specification would have
                     processed the piece correctly.
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="fund.globatt" id="fund.globatt"></a>2.4.5 Attributes Shared by all MathML Elements
               </h3>
               <p>In order to facilitate use with style sheet mechanisms such as
                  <a href="appendixk.html#XSLT">[XSLT]</a> and <a href="appendixk.html#CSS2">[CSS2]</a>
                  all MathML elements accept <code>class</code>, <code>style</code>, and <code>id</code> attributes in addition to the attributes described
                  specifically for each element.  MathML renderers not supporting CSS may
                  ignore these attributes. MathML specifies these attribute values as general
                  strings, even if style sheet mechanisms have more restrictive syntaxes for
                  them. That is, any value for them is valid in MathML.
               </p>
               <p>In order to facilitate compatibility with linking mechanisms, all
                  MathML elements accept the <code>xlink:href</code>
                  attribute.
               </p>
               <p>All MathML elements also accept the <code>xref</code>
                  attribute for use in parallel markup (<a href="chapter5.html#mixing.parallel">Section&nbsp;5.3 Parallel Markup</a>).  The <code>id</code> is also used
                  in this context.
               </p>
               <p>Every MathML element, because of a legacy from MathML 1.0, also
                  accepts the <a href="chapter7.html#interf.deprec">deprecated</a> attribute
                  <code>other</code> (<a href="chapter7.html#interf.unspecified">Section&nbsp;7.2.3 Attributes for unspecified data</a>)
                  which was conceived for passing non-standard attributes without
                  violating the MathML DTD. MathML renderers are only required to
                  process this attribute if they respond to any attributes which are not
                  standard in MathML.  However, the use of <code>other</code>
                  is strongly discouraged when there are already other ways within MathML
                  of passing specific information.
               </p>
               <p>See also <a href="chapter3.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token
                     elements</a> for a list of MathML attributes
                  which can be used on most presentation token elements.
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="fund.collapse" id="fund.collapse"></a>2.4.6 Collapsing Whitespace in Input
               </h3>
               <p>MathML ignores whitespace occurring outside token elements.
                  Non-whitespace characters are not allowed there. Whitespace occurring
                  within the content of token elements is "trimmed" from the
                  ends, i.e.,  all whitespace at the beginning and end of the content is
                  removed.  Whitespace internal to content of MathML elements is
                  "collapsed" canonically, i.e.,  each sequence of 1 or more
                  whitespace characters is replaced with one space character (sometimes
                  called a blank character).
               </p>
               <p>In MathML, as in XML, "whitespace" means simple spaces,
                  tabs, newlines, or carriage returns, i.e.,  characters with hexadecimal
                  Unicode codes <b>U+0020</b>, <b>U+0009</b>, <b>U+000A</b>, or
                  <b>U+000D</b>, respectively.
               </p>
               <p>For example, <code>&lt;mo&gt; ( &lt;/mo&gt;</code> is equivalent to
                  <code>&lt;mo&gt;(&lt;/mo&gt;</code>, and
                  
               </p><pre>
&lt;mtext&gt;
  Theorem
  1:
&lt;/mtext&gt;
</pre><p>
                  is equivalent to 
                  <code>&lt;mtext&gt;Theorem 1:&lt;/mtext&gt;</code>.
               </p>
               <p>Authors wishing to encode whitespace characters at the start or end of
                  the content of a token, or in sequences other than a single space, without
                  having them ignored, must use <code>&amp;nbsp;</code> or other
                  "whitespace" non-marking entities as described in <a href="chapter6.html#chars.nonmark">Section&nbsp;6.2.4 Non-Marking Characters</a>. For example, compare
                  
               </p><pre>
&lt;mtext&gt;
 Theorem
  1:
&lt;/mtext&gt;
</pre><p>
                  with
                  
               </p><pre>
&lt;mtext&gt;
&amp;nbsp;Theorem &amp;nbsp;1: 
&lt;/mtext&gt; </pre><p>When the first example is rendered, there is no whitespace before
                  "Theorem", one space between "Theorem" and
                  "1:", and no whitespace after "1:". In the
                  second example, a single space is rendered before
                  "Theorem", two spaces are rendered before
                  "1:", and there is no whitespace after the
                  "1:".
               </p>
               <p>Note that the <code>xml:space</code> attribute does not apply
                  in this situation since XML processors pass whitespace in tokens to a
                  MathML processor; it is the MathML processing rules which specify that
                  whitespace is trimmed and collapsed.
               </p>
               <p>For whitespace occurring outside the content of the token elements <code>mi</code>, <code>mn</code>, <code>mo</code>, <code>ms</code>, <code>mtext</code>, <code>ci</code>, <code>cn</code> and <code>annotation</code>, an <code>mspace</code> element should be used, as opposed to an <code>mtext</code> element containing only "whitespace"
                  entities.
               </p>
            </div>
         </div>
      </div>
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