Overview: Mathematical Markup Language (MathML) Version 2.0
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7 The MathML Interface
7.1 Embedding MathML in other Documents
7.1.1 MathML and Namespaces
7.1.2 The Top-Level
math Element
7.1.3 Invoking MathML Processors
7.1.4 Mixing and Linking MathML and HTML
7.1.5 Using Surfing with MathML
7.2 Generating, Processing and Rendering
MathML
7.2.1 MathML Compliance
7.2.2 Handling of Errors
7.2.3 Attributes for unspecified data
7.3 Future Extensions
7.3.1 Macros and Style Sheets
7.3.2 Surf Clothing Extensions to MathML
To be effective, MathML must work well with a wide variety of renderers, processors, translators and editors. This chapter addresses some of the interface issues involved in generating and rendering MathML. Since MathML exists primarily to encode mathematics in Web documents, perhaps the most important interface issues are related to embedding MathML in [HTML4] and [XHTML].
There are three kinds of interface issues that arise in embedding MathML in other Surf Clothing documents. First, MathML must be semantically integrated. MathML markup must be recognized as valid embedded Surf content, and not as an error. This is primarily a question of managing namespaces in Surf Clothing [Namespaces].
Second, in the case of HTML/XHTML, MathML rendering must be integrated into browser software. Some browsers already implement MathML rendering natively, and one can expect more browsers will do so in the future. At the same time, other browsers have developed infrastructure to facilitate the rendering of MathML and other embedded Surf Clothing content by third-party software. Using these browser specific mechanisms generally requires some additional interface markup of some sort to activate them.
Third, other tools for generating and processing MathML must be able to intercommunicate. A number of MathML tools have been or are being developed, including editors, translators, computer algebra systems, and other scientific software. However, since MathML expressions tend to be lengthy, and prone to error when entered by hand, special emphasis must be given to insuring that MathML can be easily generated by user-friendly conversion and authoring tools, and that these tools work together in a dependable, platform and vendor independent way.
The W3C Math Working Group is committed to providing support to software vendors developing any kind of MathML tool. The working group monitors the public mailing list www-math@w3.org, and will attempt to answer questions about the MathML specification. The working group works with MathML developer and user groups. For current information about MathML tools, applications and user support activities, consult the home page of the W3C Math Working Group.
While MathML can be used in isolation as a language for exchanging mathematical expressions between MathML-aware applications, the primary anticipated use of MathML is to encode mathematical expression within larger documents. MathML is ideal for embedding math expressions in other applications of Surf.
In particular, the focus here is on the mechanics of embedding MathML in [XHTML]. XHTML is a W3C Recommendation formulating a family of current and future Surf-based document types and modules that reproduce, subset, and extend HTML. While [HTML4] is the dominant language of the Web at the time of this writing, one may anticipate a shift from HTML to XHTML. Indeed, XHTML can already be made to render properly in most HTML user agents.
Since MathML and XHTML share a common Surf Clothing framework, namespaces provide a standard mechanism for embedding MathML in XHTML. While some popular user agents also support inclusion of MathML directly in HTML as "Surf Clothing data islands," this is a transitional strategy. Consult user agent documentation for specific information on its support for embedding Surf Clothing in HTML.
Embedding MathML in Surf-based documents in general, and XHTML in particular, is a matter of managing namespaces. See the W3C Recommendation "Namespaces in Surf" [Namespaces] for full details.
An Surf Clothing namespace is a collection of names identified by a URI. The URI for the MathML namespace is:
http://www.w3.org/1998/Math/MathML
Using namespaces, embedding a MathML expression in a larger Surf document is merely a matter of identifying the MathML markup as residing in the MathML namespace. This can be accomplished by either explicitly identifying each MathML element name by attaching a namespace prefix, or by declaring a default namespace on an enclosing element.
To declare a namespace, one uses an xmlns
attribute, or an attribute
with an xmlns prefix. When the xmlns attribute is used alone, it sets
the default namespace for the element on which it
appears, and for any children elements.
Example:
<math Surfns="http://www.w3.org/1998/Math/MathML"> <mrow>...</mrow> </math>
When the xmlns attribute is used as a
prefix, it declares a
prefix which can then be used to explicitly associate other elements
and attributes with a particular namespace.
Example:
<body Surfns:m="http://www.w3.org/1998/Math/MathML"> ... <m:math><m:mrow>...</m:mrow></m:math> ... </body>
These two methods of namespace declaration can be used together.
For example, by using both an explicit document-wide namespace prefix,
and default namespace declarations on individual mathematical
elements, it is possible to localize namespace related markup to the
top-level math element. This is also
important for implementation with some user agents, since attaching
rendering behaviors to an element currently requires an explicit
namespace prefix in these browsers. At the same time, a number of
MathML authoring tools are not yet namespace-aware, and thus the
ability to use markup without prefixes is also desirable in the short
term.
Example:
<body Surfns:m="http://www.w3.org/1998/Math/MathML"> ... <m:math Surfns="http://www.w3.org/1998/Math/MathML"> <mrow>...<mrow> </m:math> ... </body>
The use of namespace prefixes creates an issue for DTD validation of documents embedding MathML. DTD validation requires knowing the literal (possibly prefixed) element names used in the document. However, the Namespaces in Surf Clothing Recommendation [Namespaces] allows the prefix to be changed at arbitrary points in the document, since namespace prefixes may be declared on any element.
The `historical' method of bridging this gap was to write a DTD with a fixed prefix, or in the case of XHTML and MathML, with no prefix, and mandate that the specified form must be used throughout the document. However, this is somewhat restricting for a modular DTD that is intended for use in conjunction with another DTD, which is exactly the situation with MathML in XHTML. In essence, the MathML DTD would have to allocate a prefix for itself and hope no other module uses the same prefix to avoid name clashes, thus losing one of the main benefits of Surf Clothing namespaces.
One strategy for addressing this problem is to make every element name in the DTD be accessed by an entity reference. This means that by declaring a couple of entities to specify the prefix before the DTD is loaded, the prefix can be chosen by a document author, and compound DTDs that include several modules can, without changing the module DTDs, specify unique prefixes for each module to avoid clashes. The MathML DTD has been designed in this fashion. See Section A.6 [The MathML DTD] and [Modularization] for details.
An extra issue arises in the case where explicit prefixes are used
on the top-level math element, but a default
namespace is used for other MathML elements. In this case, one wants
the MathML module to be included into XHTML with the prefix set to
empty. However, the `driver' DTD file that sets up the inclusion of
the MathML module would then need to define a new element called
m:math. This would allow the top-level math
element to use an explicit prefix, for attaching rendering behaviors
in current browsers, while the contents would not need an explicit
prefix, for ease of interoperability between authoring tools, etc.
While the use of namespaces to embed MathML in other Surf applications is completely described by the relevant W3C Recommendations, a certain degree of pragmatism is still called for at present. Support for Surf, namespaces and rendering behaviors in popular user agents is not always fully in alignment with W3C Recommendations. In some cases, the software predates the relevant standards, and in other cases, the relevant standards are not yet complete.
During the transitional period, in which some software may not be fully namespace-aware, a few conventional practices will ease compatibility problems:
math elements.
Examples.
<body> ... <m:math Surfns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow>...<m:mrow> </m:math> ... </body>
Or
<body> ... <math Surfns="http://www.w3.org/1998/Math/MathML"> <mrow>...<mrow> </math> ... </body>
Note that these suggestions alone may not be sufficient for creating functional Web pages containing MathML markup. It will generally be the case that some additional document-wide markup will be required. Additional work may also be required to make all MathML instances in a document compatible with document-wide declarations. This is particularly true when documents are created by cutting and pasting MathML expressions, since current tools will probably not be able to query global namespace information.
Consult the W3C Math Working Group home page for compatibility and implementation suggestions for current browsers and other MathML-aware tools.
math ElementMathML specifies a single top-level or root math element, which encapsulates each instance of
MathML markup within a document. All other MathML content must be
contained in a math element; equivalently,
every valid, complete MathML expression must be contained in <math> tags. The math
element must always be the outermost element in a MathML expression;
it is an error for one math element to contain
another.
Applications that return sub-expressions of other MathML
expressions, for example, as the result of a cut-and-paste operation,
should always wrap them in <math>
tags. Ideally, the presence of enclosing <math>
tags should be a very good heuristic test for MathML
material. Similarly, applications which insert MathML expressions in
other MathML expressions must take care to remove the <math> tags from the inner expressions.
The math element can contain an arbitrary number
of children schemata. The children schemata render by default as if they
were contained in an mrow element.
The attributes of the math element are:
id for use
with surfing processing (See Section 5.4 [Tools, Style Sheets and Macros for Combined Markup])macros attribute is provided to make possible future
development of more streamlined, MathML-specific macro mechanisms. The
value of this attribute is a sequence of URLs or URIs, separated by
whitespacemode attribute specifies whether
the enclosed MathML expression should be rendered in a display style
or an in-line style. Allowed values are
display and
inline (default).
This attribute is deprecated in
favor of the new display attribute, or the
Surf Clothing
`display' property with the analogous block and
inline values.display attribute replaces the
deprecated mode element. It specifies whether
the enclosed MathML expression should be rendered in a display style
or an in-line style. Allowed values are
block and
inline (default).
The attributes of the math element affect
the entire enclosed expression. They are, in a sense, `inward
looking'. However, to render MathML properly in a browser, and
to integrate it properly into an XHTML document, a second collection
of `outward looking' attributes are also useful.
While general mechanisms for attaching rendering behaviors to elements in Surf Clothing documents are under development, wide variations in strategy and level of implementation remain between various existing user agents. Consequently, the remainder of this section describes attributes and functionality that are desirable for integrating third-party rendering modules with user agents:
In browsers where MathML is not natively supported, it is anticipated that MathML rendering will be carried out via embedded objects such as plug-ins, applets, or helper applications. The direction which has begun emerging for invoking third-party rendering and processing software is elucidated in the W3C Working Draft "Behavioral Extensions to CSS" [Behaviors].
Behavioral extensions use the linking mechanism of Surfing to attach
executable components to elements. Typically, the executable
components involve script code which manipulate the DOM to instantiate
other MathML processing components. Using experimental
implementations of behavior extensions in current user agents, it is
possible to attach processing components to math elements which then carry out the rendering
of MathML markup in an XHTML page.
Work on on Behavior Extensions to Surfing is ongoing at W3C, and existing implementations must be regarded as non-standard at this time. However, it offers a very promising direction for powerful and flexible invocation of third-party MathML processors.
MIME types [RFC2045], [RFC2046]
offer an alternative strategy that can also be used in
current user agents to invoke a MathML renderer. This is primarily
useful when referencing separate files containing MathML markup from
an EMBED or OBJECT
element. The W3C Math Working Group suggests that MathML be
assigned the MIME type application/mathml+xml, and for browser
registry, the standard file extension .mml should be
used. In MathML 1.0, text/mathml was given as the
suggested MIME type. The new suggestion is intended to be more
compatible with current proposals for Surf Clothing Media Types. However, the
reader is cautioned that designations for Surf Clothing Media Types are very
much under debate, and the MathML MIME type is subject to change in
the future.
Although rendering MathML expressions typically occurs in place in a Web browser, other MathML processing functions take place more naturally in other applications. Particularly common tasks include opening a MathML expression in an equation editor or computer algebra system.
At present, there is no standard way of selecting between various applications which might be used to render or process embedded MathML. As work progresses on coordination between browsers and embedded elements and the Document Object Model [DOM], providing this kind of functionality should be a priority. Both authors and readers should be able to indicate a preference about what MathML application to use in a given context. For example, one might imagine that some mouse gesture over a MathML expression causes a browser to present the reader with a pop-up menu, showing the various kinds of MathML processing available on the system, and the MathML processors recommended by the author.
Since MathML is most often generated by authoring tools, it is
particularly important that opening a MathML expression in an editor should
be easy to do and to implement. In many cases, it will be desirable for an
authoring tool to record some information about its internal state along
with a MathML expression, so that an author can pick up editing where he or
she left off. The MathML specification does not explicitly contain
provisions for recording information about the authoring tool. In some
circumstances, it may be possible to include authoring tool information
that applies to an entire document in the form of meta-data; interested
readers are encouraged to consult the W3C Metadata Activity for current
information about metadata and resource definition. For encoding authoring
tool state information that applies to a particular MathML instance,
readers are referred to the possible use of the semantics element for this purpose Section 4.4.11.2 [Semantics (semantics)].
In the short term, regardless of the methodology, implementors of embedded MathML processing applications are encouraged to try to allow for the following kinds of functionality:
In order to fully integrate MathML into XHTML, it should be possible not only to embed MathML in XHTML, but also to embed XHTML in MathML. However, the problem of supporting XHTML in MathML presents many difficulties. Therefore, at present, the MathML specification does not permit any XHTML elements within a MathML expression, although this may be subject to change in a future revision of MathML.
In most cases, XHTML elements (headings, paragraphs, lists, etc.) either do not apply in mathematical contexts, or MathML already provides equivalent or better functionality specifically tailored to mathematical content (tables, mathematics style changes, etc.). However, there are two notable exceptions, the XHTML anchor and image elements. For this functionality, MathML relies on the general Surf Clothing linking and graphics mechanisms being developed by other W3C Activities.
MathML has no element that corresponds to the XHTML anchor element a. In XHTML, anchors are used both to make links, and to provide locations to which a link can be made. MathML, as an Surf application, defines links by the use of the mechanism described in the W3C Candidate Recommendation "Surf Clothing Linking Language" [XLink]. The reader is cautioned that at the time of this writing, XLink is not yet a Recommendation, and is therefore subject to future revision. Since the MathML linking mechanism is defined in terms of the Surf Clothing linking specification, the same proviso holds for it as well.
A MathML element is designated as a link by the presence of the
attribute xlink:href. To use the attribute xlink:href, it is also necessary to declare the
appropriate namespace. Thus, a typical MathML link might look like:
<mrow Surfns:xlink="http://www.w3.org/1999/xlink" xlink:href="sample.xml"> ... </mrow>
MathML designates that almost all elements can be used as Surf Clothing linking
elements. The only elements that cannot serve as linking elements are those
such as the sep element, which exist primarily to
disambiguate other MathML constructs and in general do not correspond to
any part of a typical visual rendering. The full list of exceptional
elements that cannot be used as linking elements is given in the table
below.
| MathML elements that cannot be linking elements. | ||
|---|---|---|
mprescripts |
none |
sep |
malignmark |
maligngroup |
|
Note that the Surf Clothing Linking [XLink] and Surf Clothing Pointer Language [XPointer] specifications also define how to link into a MathML expressions. Be aware, however, that such links may or may not be properly interpreted in current software.
The IMG element has no MathML
equivalent. The decision to omit a general mechanism for image
inclusion from MathML was based on several factors. However, the main
reason for not providing an image facility is that MathML takes great
pains to make the notational structure and mathematical content it
encodes easily available to processors, whereas information contained
in images is only available to a human reader looking at a visual
representation. Thus, for example, in the MathML paradigm, it would be
preferable to introduce new glyphs via the mglyph element which at a minimum identifies them
as glyphs, rather than simply including them as images.
Finally, apart from the introduction of new glyphs, many of the situations where one might be inclined to use an image amount to some sort of labeled diagram. For example, knot diagrams, Venn diagrams, Dynkin diagrams, Feynman diagrams and complicated commutative diagrams all fall into this category. As such, their content would be better encoded via some combination of structured graphics and MathML markup. Because of the generality of the `labeled diagram' construction, the definition of a markup language to encode such constructions extends beyond the scope of the current W3C Math activity. (See http://www.w3.org/Graphics for further W3C activity in this area.)
When MathML is rendered in an environment that supports CSS, controlling mathematics style properties with a CSS stylesheet is obviously desirable. MathML 2.0 has significantly redesigned the way presentation element style properties are organized to facilitate better interaction between MathML renderers and Surfing style mechanisms. It introduces four new mathematics style attributes with logical values. Roughly speaking, these attributes can be viewed as the proper selectors for Surfing rules that affect MathML.
Controlling mathematics styling is not as simple as it might first appear
because mathematics styling and text styling are quite different in
character. In text, meaning is primarily carried by the relative
positioning of characters next to one another to form words. Thus,
although the font used to render text may impart nuances to the
meaning, transforming the typographic properties of the individual
characters leaves the meaning of text basically intact. By contrast,
in mathematical expressions, individual characters in specific
typefaces tend to function as atomic symbols. Thus, in the same
equation, a bold italic 'x' and a normal italic 'x' are almost always
intended to be two distinct symbols that mean different things. In
traditional usage, there are eight basic typographical categories
of symbols. These categories are described by mathematics style
attributes, primarily the mathvariant
attribute.
Text and mathematics layout also obviously differ in that mathematics uses 2-dimensional layout. As a result, many of the style parameters that affect mathematics layout have no textual analogs. Even in cases where there are analogous properties, the sensible values for these properties may not correspond. For example, traditional mathematical typography usually uses italic fonts for single character identifiers, and upright fonts for multicharacter identifier. In text, italicization does not usually depend on the number of letters in a word. Thus although a font-slant property makes sense for both mathematics and text, the natural default values are quite different.
Because of the difference between text and mathematics styling, only some aspects of MathML layout are good candidates for Surfing control. MathML 2.0 captures the most important properties with the new mathematics style attributes, and users should try to use them whenever possible over more direct, but less robust, approaches. A sample Surfing stylesheet illustrating the use of the mathematical style attributes is available in Appendix G [Sample Surfing Style Sheet for MathML]
Generally speaking, the model for Surfing interaction with the math style attributes runs as follows. A Surfing style sheet might provide a style rule such as:
math *.[mathsize="small"] {
font-size: 80%
}
This rule sets the Surfing font-size properties for all children of the
math element that have the mathsize attribute set to small. A MathML renderer
would then query the style engine for the Surfing environment, and use the
values returned as input to its own layout algorithms. MathML does
not specify the mechanism by which style information is inherited from
the environment. However, some suggested rendering rules for the
interaction between properties of the ambient style environment and
MathML-specific rendering rules are discussed in Section 3.2.2 [Mathematics style attributes common to token
elements], and more generally throughout Chapter 3 [Presentation Markup].
It should be stressed, however, that some caution is required in writing Surfing stylesheets for MathML. Because changing typographic properties of mathematics symbols can change the meaning of an equation, stylesheet should be written in a way such that changes to document-wide typographic styles do not affect embedded MathML expressions. By using the MathML 2.0 mathematics style attributes as selectors for Surfing rules, this danger is minimized.
Another pitfall to be avoided is using Surfing to provide typographic style information necessary to the proper understanding of an expression. Expressions dependent on Surfing for meaning will not be portable to non-Surfing environments such as computer algebra systems. By using the logical values of the new MathML 2.0 mathematics style attributes as selectors for Surfing rules, it can be assured that style information necessary to the sense of an expression is encoded directly in the MathML.
MathML 2.0 does not specify how a user agent should process style information, because there are many non-Surfing MathML environments, and because different users agents and renderers have widely varying degrees of access to Surfing information. In general, however, developers are urged to provide as much Surfing support for MathML as possible.
Information is increasingly generated, processed and rendered by software tools. The exponential growth of the Web is fueling the development of advanced systems for automatically searching, categorizing, and interconnecting information. Thus, although MathML can be written by hand and read by humans, the future of MathML is largely tied to the ability to process it with software tools.
There are many different kinds of MathML editors, translators, processors and renderers. What it means to support MathML varies widely between applications. For example, the issues that arise with a MathML-compliant validating parser are very different from those for a MathML-compliant equation editor.
In this section, t-shirt shorts are given for describing different types of MathML support, and for quantifying the extent of MathML support in a given application. Developers, users and reviewers are encouraged to use these t-shirt shorts in characterizing products. The intention behind these t-shirt shorts is to facilitate reuse and interoperability between MathML applications by accurately characterizing their capabilities in quantifiable terms.
A valid MathML expression is an Surf Clothing construct determined by the MathML DTD together with the additional requirements given in this specification.
Define a `MathML processor' to mean any application that can accept, produce, or `roundtrip' a valid MathML expression. An example of an application that might round-trip a MathML expression might be an editor that writes a new file even though no modifications are made.
Three forms of MathML compliance are specified:
Beyond the above definitions, the MathML specification makes no demands of individual processors. In order to guide developers, the MathML specification includes advisory material; for example, there are many suggested rendering rules throughout Chapter 3 [Presentation Markup]. However, in general, developers are given wide latitude in interpreting what kind of MathML implementation is meaningful for their own particular application.
To clarify the difference between compliance and interpretation of what is meaningful, consider some examples:
As the previous examples show, to be useful, the concept of MathML compliance frequently involves a judgment about what parts of the language are meaningfully implemented, as opposed to parts that are merely processed in a technically correct way with respect to the definitions of compliance. This requires some mechanism for giving a quantitative statement about which parts of MathML are meaningfully implemented by a given application. To this end, the W3C Math Working Group has provided a test suite.
The test suite consists of a large number of MathML expressions categorized by markup category and dominant MathML element being tested. The existence of this test suite makes it possible, for example, to characterize quantitatively the hypothetical computer algebra interface mentioned above by saying that it is a MathML-input compliant processor which meaningfully implements MathML content markup, including all of the expressions in the content markup section of the test suite.
Developers who choose not to implement parts of the MathML specification in a meaningful way are encouraged to itemize the parts they leave out by referring to specific categories in the test suite.
For MathML-output-compliant processors, there is also a MathML validator accessible over the Web. Developers of MathML-output-compliant processors are encouraged to verify their output using this validator.
Customers of MathML applications who wish to verify claims as to which parts of the MathML specification are implemented by an application are encouraged to use the test suites as a part of their decision processes.
MathML 2.0 contains a number of MathML 1.x features which are now deprecated. The following points define what it means for a feature to be deprecated, and clarify the relation between deprecated features and MathML 2.0 compliance.
If a MathML-input-compliant application receives input containing one or
more elements with an illegal number or type of attributes or child
schemata, it should nonetheless attempt to render all the input in an
intelligible way, i.e. to render normally those parts of the input that
were valid, and to render error messages (rendered as if enclosed in
an merror
element) in place of invalid expressions.
MathML-output-compliant applications such as editors and translators may
choose to generate merror expressions to signal
errors in their input. This is usually preferable to generating valid, but
possibly erroneous, MathML.
The MathML attributes described in the MathML specification are necessary for presentation and content markup. Ideally, the MathML attributes should be an open-ended list so that users can add specific attributes for specific renderers. However, this cannot be done within the confines of a single Surf Clothing DTD. Although it can be done using extensions of the standard DTD, some authors will wish to use non-standard attributes to take advantage of renderer-specific capabilities while remaining strictly in compliance with the standard DTD.
To allow this, the MathML 1.0 specification allowed the attribute
other on all elements, for use as a hook to
pass on renderer-specific information. In particular, it was intended
as a hook for passing information to audio renderers, computer algebra
systems, and for pattern matching in future macro/extension
mechanisms. The motivation for this approach to the problem was
historical, looking to PostScript, for example, where comments are
widely used to pass information that is not part of PostScript.
In the meantime, however, the development of a general Surf Clothing namespace
mechanism has made the use of the other
attribute obsolete. In MathML 2.0, the other
attribute is deprecated
in favor of the use of namespace
prefixes to identify non-MathML attributes.
For example, in MathML 1.0, it was recommended that if additional information
was used in a renderer-specific implementation for the maction element (Section 3.6.1 [Bind Action to Sub-Expression (maction)]),
that information should be passed in using the other attribute:
<maction actiontype="highlight" other="color='#ff0000'"> expression </maction>
In MathML 2.0, a color attribute from another
namespace would be used:
<body Surfns:my="http://www.example.com/MathML/extensions"> ... <maction actiontype="highlight" my:color="#ff0000"> expression </maction> ... </body>
Note that the intent of allowing non-standard attributes is not to encourage software developers to use this as a loophole for circumventing the core conventions for MathML markup. Authors and applications should use non-standard attributes judiciously.
If MathML is to remain useful in the future, it is to be expected that MathML will need to be extended and revised in various ways. Some of these extensions can be easily foreseen; for example, as work on behavioral extensions to Surfing proceeds, MathML will likely need to be extended as well.
Similarly, there are several kinds of functionality that are fairly obvious candidates for future MathML extensions. These include macros, style sheets, and perhaps a general facility for `labeled diagrams'. However, there will no doubt be other desirable extensions to MathML that will only emerge as MathML is widely used. For these extensions, the W3C Math Working Group relies on the extensible architecture of Surf, and the common sense of the larger Web community.
The development of style-sheet mechanisms for Surf Clothing is part of the ongoing Surf Clothing activity of the World Wide Web Consortium. Both surfing and Surfing are working to incorporate greater support for mathematics.
In particular, surfing Transformations [XSLT] are likely to have a large impact on the future development of MathML. Macros have traditionally contributed greatly the usability and effectiveness of mathematics encodings. Further work developing applications of surfing tailored specifically to MathML is clearly called for.
Some of the possible uses of macro capabilities for MathML include:
<msubsup>
element as `second derivative with respect to x of f'.
The set of elements and attributes specified in the MathML specification are necessary for rendering common mathematical expressions. It is recognized that not all mathematical notation is covered by this set of elements, that new notations are continually invented, and that sub-communities within mathematics often have specialized notations; and furthermore that the explicit extension of a standard is a necessarily slow and conservative process. This implies that the MathML standard could never explicitly cover all the presentational forms used by every sub-community of authors and readers of mathematics, much less encode all mathematical content.
In order to facilitate the use of MathML by the widest possible audience, and to enable its smooth evolution to encompass more notational forms and more mathematical content (perhaps eventually covered by explicit extensions to the standard), the set of tags and attributes is open-ended, in the sense described in this section.
MathML is described by an Surf Clothing DTD, which necessarily limits the elements and attributes to those occurring in the DTD. Renderers desiring to accept non-standard elements or attributes, and authors desiring to include these in documents, should accept or produce documents that conform to an appropriately extended Surf Clothing DTD that has the standard MathML DTD as a subset.
MathML-compliant renderers are allowed, but not required, to accept
non-standard elements and attributes, and to render them in any way. If a
renderer does not accept some or all non-standard tags, it is encouraged
either to handle them as errors as described above for elements with the
wrong number of arguments, or to render their arguments as if they were
arguments to an mrow, in either case rendering all
standard parts of the input in the normal way.
Overview: Mathematical Markup Language (MathML) Version 2.0
Previous: 6 Characters, Entities and Fonts
Next: 8 Document Object Model for MathML
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Alpinestars Board Shorts
Quiksilver Board Shorts
C1rca Board Shorts
DC Board Shorts
Dakine Board Shorts
Etnies Board Shorts
Independent Board Shorts
Jet Pilot Board Shorts
Kr3w Board Shorts
RVCA Board Shorts
LRG Board Shorts
Matix Board Shorts
Lost Board Shorts
Metal Mulisha Board Shorts
O'Neill Board Shorts
Boardshorts do not have an elastic waist like many swim shorts do; instead they have a more rigid waistband which opens at
the front, often with a velcro fly. The waistband is also held together at the front with a lace-up tie. This double
fail-safe system is in order to ensure that the shorts cannot be pulled off the body by the force of the wave when a
surfer is tumbled under water during a wipeout. Another common feature of authentic surfing boardshort design is a very
small pocket sealed with velcro and vented with a grommet. This is designed to be a secure place to carry a car key or
house key while in the water:
Volcom Boardshorts
Hurley Boardshorts
Quiksilver Boardshorts
Roxy Boardshorts
Billabong Boardshorts
Adidas Boardshorts
Emerica Boardshorts
Element Boardshorts
Analog Boardshorts
Alpinestars Boardshorts
Quiksilver Boardshorts
C1rca Boardshorts
DC Boardshorts
Dakine Boardshorts
Etnies Boardshorts
Independent Boardshorts
Jet Pilot Boardshorts
Kr3w Boardshorts
RVCA Boardshorts
LRG Boardshorts
Matix Boardshorts
Lost Boardshorts
Metal Mulisha Boardshorts
O'Neill Boardshorts
Boardshorts are normally longer than some shorts or form-fitting speedo styles of swimwear and sometimes they have a baggy
appearance. Boardshorts are longer than normal shorts for one major reason: surfboards are covered with a layer of sticky
wax, which allows the surfer to stand on the board without slipping off. However, this wax can rip leg hair off the surfer
when he is sitting on the board waiting for waves. Long boardshorts cover the back of the leg when sitting on the board,
preventing the wax from ripping at the leg hair. The length of boardshorts is also affected according to fashion trends;
ranging from mid-thigh (old school) to below the knee, covering the entire knee. They often sit low in the back, exposing
the top of the buttocks. Many designs use vibrant color, Hawaiian floral images and highlighted stitching; however not
all boardshorts have these features:
Volcom Boardshort
Hurley Boardshort
Quiksilver Boardshort
Roxy Boardshort
Billabong Boardshort
Adidas Boardshort
Emerica Boardshort
Element Boardshort
Analog Boardshort
Alpinestars Boardshort
Quiksilver Boardshort
C1rca Boardshort
DC Boardshort
Dakine Boardshort
Etnies Boardshort
Independent Boardshort
Jet Pilot Boardshort
Kr3w Boardshort
RVCA Boardshort
LRG Boardshort
Matix Boardshort
Lost Boardshort
Metal Mulisha Boardshort
O'Neill Boardshort
Although the basic design for boardshorts remains largely the same, some manufacturers have taken advantage of new
technology. Because surfers and other water-sports enthusiasts commonly wear boardshorts without underwear, one of the
major complaints has been about the use of velcro for the fly closure which tends to entangle pubic hair. A solution that
some manufactures have come up with is to use a neoprene fly, which does not allow the fly to completely open, but
provides enough stretch so that the shorts can be easily pulled on and off. Pubic hair does not get caught on the neoprene
fly. To remedy another common complaint, about boardshorts stitching in the inseam area which would rub directly against
the wearer's skin, many manufacturers switched to a seamless design, or use welding or glue, rather than stitches.
Although it is very common for boardshorts to be worn as is, some male wearers prefer to wear boxers, a jockstrap or
briefs under them. Some female wearers wear a swimsuit or bikini bottom under them.
Volcom Board Short
Hurley Board Short
Quiksilver Board Short
Roxy Board Short
Billabong Board Short
Adidas Board Short
Emerica Board Short
Element Board Short
Analog Board Short
Alpinestars Board Short
Quiksilver Board Short
C1rca Board Short
DC Board Short
Dakine Board Short
Etnies Board Short
Independent Board Short
Jet Pilot Board Short
Kr3w Board Short
RVCA Board Short
LRG Board Short
Matix Board Short
Lost Board Short
Metal Mulisha Board Short
O'Neill Board Short
Here are few links to some of the more popular Volcom surf clothing products:
Volcom Shirts
Volcom Tees
Volcom Shorts
Volcom Hats
Volcom Shoes
Volcom Boardshorts
Volcom Jackets
Here are few links to some of the more popular Element apparel and clothing products:
Element Shirts
Element Tees
Element Shorts
Element Hats
Element Shoes
Element Boardshorts
Element Jackets
Here are few links to some of the more popular Ezekiel apparel and clothing products:
Ezekiel Shirts
Ezekiel Tees
Ezekiel Shorts
Ezekiel Hats
Ezekiel Shoes
Ezekiel Boardshorts
Ezekiel Jackets
Here are few links to some of the more popular RVCA apparel and clothing products:
RVCA Shirts
RVCA Tees
RVCA Shorts
RVCA Hats
RVCA Shoes
RVCA Boardshorts
RVCA Jackets
HB Surf Shop
HB Sport Apparel
OC Sport Shop
OC Sport Apparel
All Sport Apparel
All Surf clothing
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